From 46b2f19402a1d6d42d7d32c56100f7c89cdfcefb Mon Sep 17 00:00:00 2001 From: lmoresi Date: Tue, 18 Aug 2026 17:24:26 +1000 Subject: [PATCH 01/33] Draft UWTN 2026-016: Boundary conditions on non-planar boundaries R1 in the writing plan. 'No flow through this wall' is a velocity component on a box and is not a component of anything on a sphere, a deformed mesh, or a surface with topography, and the note is about what to do instead. Three approaches in the order in which each answers the previous one's weakness: a direct penalty, which is consistent only in the limit so closing the leak means conditioning the operator worse; Nitsche, which carries the boundary traction terms and is consistent above a stability threshold; and rotating the degrees of freedom, which imposes the constraint exactly. The weak forms are written out, because the differences between them are the argument and 'add two more terms and it becomes consistent' is not believable unseen. The rotated approach is presented as the classical answer it is -- Engelman, Sani and Gresho were reviewing the alternatives in 1982 -- so the question the note actually answers is why the exact method is the least used of the three. The answer is structural: a per-node rotation leaves the problem in a mixed basis, and the figure shows where that obligation falls. It is contained. The velocity solve is rotated and carries its multigrid; the Schur complement and pressure solve beside it never handle a rotated vector. The figure moves here from the underworld3 repository, where it was parked because this article did not exist yet. Its typst source, generator and JSON geometry ship in examples/. Two things are deliberately visible rather than quietly resolved. The section on when the choice matters carries a warning that it is not yet measured -- the argument is structural and we have not run the experiment. And the note asks whether Behr (2004) reaches the same measure-weighted normal we did, because if it does, our #560 was a rediscovery and should say so. Carries the allocator fix from the exact-solutions branch. Without it new_article.py issued 2026-012, which is already gthyagi's -- the same collision the fix exists to prevent, because the fix is not yet merged. Underworld development team with AI support from Claude Code --- ...ary-conditions-on-non-planar-boundaries.md | 277 +++++ .../examples/generate-rotated-basis.py | 102 ++ .../examples/rotated-basis-data.json | 1011 +++++++++++++++++ .../examples/rotated-basis.typ | 186 +++ .../figures/rotated-basis.png | Bin 0 -> 199642 bytes .../figures/rotated-basis.svg | 1 + .../metadata.yml | 34 + classification.yml | 4 + scripts/new_article.py | 34 +- 9 files changed, 1648 insertions(+), 1 deletion(-) create mode 100644 articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md create mode 100644 articles/boundary-conditions-on-non-planar-boundaries/examples/generate-rotated-basis.py create mode 100644 articles/boundary-conditions-on-non-planar-boundaries/examples/rotated-basis-data.json create mode 100644 articles/boundary-conditions-on-non-planar-boundaries/examples/rotated-basis.typ create mode 100644 articles/boundary-conditions-on-non-planar-boundaries/figures/rotated-basis.png create mode 100644 articles/boundary-conditions-on-non-planar-boundaries/figures/rotated-basis.svg create mode 100644 articles/boundary-conditions-on-non-planar-boundaries/metadata.yml diff --git a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md new file mode 100644 index 0000000..35feaed --- /dev/null +++ b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md @@ -0,0 +1,277 @@ +--- +title: Boundary conditions on non-planar boundaries +description: >- + "No flow through this wall" is a single velocity component on a box and is + not a component of anything on a sphere, a deformed mesh, or a surface with + topography. Three ways to impose it — a direct penalty, Nitsche, and rotating + the degrees of freedom — what each costs, and the one measurement that tells + them apart. +date: 2026-08-18 +authors: + - name: Louis Moresi + orcid: 0000-0003-3685-174X + affiliations: + - Australian National University +license: CC-BY-4.0 +keywords: + - Underworld Code + - Tricks of the Trade + - development +exports: + - format: typst + logo: ../../static/uwtn-logo.png + series: "Underworld Technical Notes" + origin_url: https://www.underworldcode.org/boundary-conditions-on-non-planar-boundaries/ + template: ../../templates/pdf + output: boundary-conditions-on-non-planar-boundaries.pdf + article_id: UWTN 2026-016 + article_version: 1.0.0 + software_version: underworld3 development @ 0addec15 +--- +A free surface moves under the traction it carries. That makes the wall-normal +stress the quantity driving the model rather than something read off at the +end, and it is the reason we went back to how the boundary condition underneath +it is imposed. + +The condition itself is ordinary. Free slip is no flow through the boundary and +no tangential drag along it, + +$$ +\mathbf{u}\cdot\hat{\mathbf{n}} = 0 +\qquad\text{and}\qquad +\hat{\mathbf{t}}\cdot\boldsymbol{\sigma}\hat{\mathbf{n}} = 0 . +$$ + +On a box the first of those is a single velocity component. You hold $u_x$ on a +vertical wall, the solver removes a row, and there is nothing further to +discuss. On a sphere, an annulus, a mesh that has been moved, or a surface with +topography, $\mathbf{u}\cdot\hat{\mathbf{n}}$ is not a component of anything. +That is the whole difficulty, and everything below is a way around it. + +The second condition is worth naming because it is the one people forget. Zero +tangential traction is *natural*: it is what you get by leaving the boundary +term out of the weak form. Nothing has to be done to impose it, and something +has to be done to avoid imposing it accidentally. + +## Where the boundary term comes from + +Every method below is a statement about one term. Multiplying the momentum +balance by a test function $\mathbf{w}$ and integrating by parts gives + +$$ +\int_\Omega \boldsymbol{\sigma} : \nabla\mathbf{w} \; \mathrm{d}V +- \int_{\partial\Omega} (\boldsymbol{\sigma}\hat{\mathbf{n}})\cdot\mathbf{w} + \; \mathrm{d}S += \int_\Omega \mathbf{f}\cdot\mathbf{w} \; \mathrm{d}V . +$$ + +Drop the surface integral and you have imposed zero traction in both +directions — free *everything*, not free slip. Free slip keeps the tangential +half of that and replaces the normal half with the constraint. How you do the +replacing is the choice. + +## Three ways, in order + +### A direct penalty + +Add a term that punishes any flow through the boundary: + +$$ +\dots + \frac{\gamma}{h}\int_{\partial\Omega} +(\mathbf{u}\cdot\hat{\mathbf{n}})(\mathbf{w}\cdot\hat{\mathbf{n}}) +\; \mathrm{d}S . +$$ + +One line, no new machinery, and it works on any geometry. What you are solving +is a perturbed problem, though, and it is perturbed by exactly the amount the +constraint is violated: the discrete solution sits where the penalty term +balances the traction it is fighting, which leaves $\mathbf{u}\cdot\hat{\mathbf{n}}$ +small but not zero. Making it smaller means raising $\gamma$, and raising +$\gamma$ conditions the operator worse. The error is traded against the +conditioning, and neither can be driven away. + +### Nitsche + +The reason the penalty is only accurate in the limit is that it is not +*consistent*: substituting the true solution does not leave the equation +satisfied, because the true solution is subject to a boundary traction the +penalty form ignores. Nitsche's method [@10.1007/BF02995904] restores +consistency by carrying that traction explicitly: + +$$ +\dots +- \int_{\partial\Omega} (\hat{\mathbf{n}}\cdot\boldsymbol{\sigma}(\mathbf{u}) + \hat{\mathbf{n}})(\mathbf{w}\cdot\hat{\mathbf{n}}) \; \mathrm{d}S +- \int_{\partial\Omega} (\hat{\mathbf{n}}\cdot\boldsymbol{\sigma}(\mathbf{w}) + \hat{\mathbf{n}})(\mathbf{u}\cdot\hat{\mathbf{n}}) \; \mathrm{d}S ++ \frac{\gamma}{h}\int_{\partial\Omega} + (\mathbf{u}\cdot\hat{\mathbf{n}})(\mathbf{w}\cdot\hat{\mathbf{n}}) + \; \mathrm{d}S . +$$ + +The first of the three is the consistency term: it is the boundary traction the +integration by parts produced, and putting it back is what makes the true +solution satisfy the discrete equations exactly. The second is its transpose, +which keeps the form symmetric and buys optimal convergence in $L^2$. The third +is the penalty again, and it is still needed — but now for *stability* rather +than for accuracy, and $\gamma$ has a threshold set by an inverse inequality +rather than being a dial you turn up until the answer looks right. + +This is a real improvement and it is still a weak imposition. The constraint +holds to the accuracy of the discretisation, not to the accuracy of the +arithmetic. + +### Rotating the degrees of freedom + +Stop asking for the constraint and impose it. At each constrained node, change +the basis in which the velocity unknowns are expressed, from the global +Cartesian frame to the local $(\hat{\mathbf{n}}, \hat{\mathbf{t}})$ frame. In +that basis "no flow through the boundary" is again a single component, and it +is removed the same way it would be on a box. + +Collect the per-node rotations into a block-diagonal $Q$, equal to the identity +at every node that is not constrained. The rotated system is + +$$ +\hat{A} = Q^{T} A Q, \qquad \hat{\mathbf{b}} = Q^{T}\mathbf{b}, +\qquad \mathbf{u} = Q\hat{\mathbf{u}} , +$$ + +and the wall-normal row of $\hat{A}$ is struck out. The constraint then holds to +machine precision, because it is not being solved for at all. + +**This is the classical answer**, not a new one. It is in the early +finite-element literature, and @10.1002/fld.1650020302 were already reviewing +the alternatives and choosing between them on grounds of global mass +conservation in 1982. What is worth explaining is not the idea but why, given +that it is exact and the others are not, it is the least used of the three. + +## What it costs, and the cost is structural + +Rotating the degrees of freedom leaves the discrete problem in a **mixed +basis**. Interior nodes hold $(u_x, u_y)$; constrained nodes hold +$(u_n, u_t)$. Nothing about that is difficult in itself, and everything +downstream has to agree about which nodes are which. + +```{figure} figures/rotated-basis.svg +:alt: Two panels. On the left, a meshed domain bounded above by a free surface that rises on the left and falls on the right with an inflection between, so that the outward normal points in a different direction at every surface node. Surface nodes are drawn as filled circles each carrying its own rotated pair of arrows labelled n and t; interior nodes are open circles, with one carrying the unrotated x and y arrows shared by all of them. On the right, a block diagram. A red block labelled "Velocity solve, rotated" contains the rotated operator and right-hand side, and encloses a smaller block labelled "Multigrid" listing three rows: prolongation becomes Q-transpose P, coarse operators inherit Q through RAP, and the coarse solve uses SVD for the rigid rotations. A separate green block beside it, labelled "Fieldsplit / Schur solve", carries the pressure and constraints and is marked as never seeing a rotated vector. A single arrow labelled v equals Q v-hat leaves the velocity block at its boundary and branches, one branch entering the Schur block and the other leaving for output, advection and the surface update. + +Where the rotation lives. The obligation is contained: the velocity solve is +rotated and carries its multigrid with it, while the Schur complement and the +pressure solve beside it never handle a rotated vector, because the pressure +block carries no boundary condition of this kind. One un-rotation sits on the +boundary between them and feeds both. +``` + +Four things carry $Q$: the operator, the right-hand side, the solution on the +way out, and the multigrid prolongation. The coarse operators inherit it +through the Galerkin triple product rather than being rotated separately, and +the coarse solve then has to be an SVD, because a Galerkin-coarsened rotated +operator inherits the rigid-rotation null space of the constrained problem and +a redundant LU factorisation meets a zero pivot in it. + +That last point is worth dwelling on, because it is the one that surprises. The +constraint never has to be re-imposed on a coarse mesh — which is fortunate, +since we install no discretisation there and could not impose it if we wanted +to. It arrives anyway, algebraically, through $P^{T}\hat{A}P$ with a rotated +$P$. The evidence that it really arrives is that the coarse operator inherits +the null space, which is a property only the constrained problem has. + +None of this is an argument against the method. It is an argument for knowing +what is being taken on, and it is the honest reason a weakly imposed condition +survives in codes that could do this instead. + +## Which normal + +A question with a less obvious answer than it looks. The boundary of a +discretised domain is a set of straight facets, and the assembled constraint is +an integral over those facets. The node normal consistent with that integral is +the average of the adjacent facet normals **weighted by facet measure** — not +the normal of the smooth surface the mesh approximates. + +Using the analytic normal is exact for the geometry and therefore inconsistent +with the discretisation, which is the wrong way round: the solver is not solving +on the sphere, it is solving on the polyhedron. In parallel the same argument +has a sharper edge, because a normal accumulated rank-locally is wrong at a +partition boundary, where a node's facets are split across ranks and no rank +sees them all. + +:::{note} Worth checking against the literature +@10.1002/fld.663 addresses this question directly for slip conditions on curved +boundaries. We have not read it against our own derivation. If it reaches the +same measure-weighted normal, this section should say so and cite it as the +source rather than presenting the result as ours. +::: + +## The option this note leaves out + +Solving in spherical or cylindrical components makes the wall-normal direction +a coordinate direction again, and the constraint returns to being "hold one +component". That is the same rotation as above, applied once for the whole +domain instead of node by node, and applied that way it costs nothing +structurally: there is no mixed basis, because every node is in the same basis. + +It works exactly when the boundary lies along a coordinate surface. A sphere, an +annulus, a cylinder. It does nothing for topography, for a mesh that has been +deformed, or for a tilted internal surface, which is the general case this note +is about. The per-node rotation is what remains once the geometry stops +cooperating, and its structural price is what buys the generality. + +## When the choice matters + +Here is the awkward part. Solve a convection model with any of the three and +the velocity field is the same to plotting accuracy. A leak of order $10^{-3}$ +in $\mathbf{u}\cdot\hat{\mathbf{n}}$ is invisible to anything that consumes the +velocity, and consuming the velocity is most of what a model does. If that is +your situation, use the simplest thing that works and stop reading here. + +The difference appears when the wall-normal traction is the answer rather than +a by-product: dynamic topography, a plate-boundary force balance, anything +compared against a geoid or a gravity field. Then the three separate, and they +separate for a structural reason rather than by a matter of degree. + +Under a penalty or a Nitsche condition the constraint is approximate, so a +traction recovered from the solution inherits the approximation — you are +differentiating a field that was never made to satisfy the condition exactly. +Under the rotated constraint the reaction *is* $\sigma_{nn}$: it is the +multiplier the solve has already computed, and it comes out of +`boundary_normal_traction` without a recovery step or an augmented-Lagrangian +splitting. + +:::{warning} This section is not yet measured +The argument above is structural and we believe it, which is not the same as +having shown it. The note needs one experiment: the same model under all three +boundary treatments, with the surface traction compared against a case whose +answer is known. Until that table is here, this section is a claim. +::: + +## Using it + +```python +import underworld3 as uw + +mesh = uw.meshing.Annulus(radiusInner=0.5, radiusOuter=1.0, cellSize=0.05) +stokes = uw.systems.Stokes(mesh) + +# Value first: 0 is free slip. A non-zero scalar or expression prescribes the +# wall-normal datum u.n = u_n strongly instead. +stokes.add_rotated_freeslip_bc(0.0, "Upper") +stokes.add_rotated_freeslip_bc(0.0, "Lower") + +stokes.solve() + +# The constraint reaction, which is the boundary normal traction. +sigma_nn = stokes.boundary_normal_traction("Upper") +``` + +Leave the normal to Underworld unless the constraint has to follow the true +surface rather than the mesh. Passing an analytic normal — `X / |X|` on a +sphere — is exact for the geometry and keeps a consistency error against the +faceted assembly, which is usually not what you want. + +Reach for Nitsche when the boundary condition has to **change during the +model**. A hard constraint cannot morph: a wall that begins as a prescribed +velocity and relaxes to a prescribed traction is a Nitsche problem, because the +rotated constraint is either imposed or it is not. + +
Comments
Discussion of these notes happens in GitHub Discussions, so it stays with the source and is searchable alongside it.
diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/generate-rotated-basis.py b/articles/boundary-conditions-on-non-planar-boundaries/examples/generate-rotated-basis.py new file mode 100644 index 0000000..b203819 --- /dev/null +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/generate-rotated-basis.py @@ -0,0 +1,102 @@ +"""Geometry for the rotated-boundary-conditions figure. + +The cetz skill's rule: geometry computation happens in Python and Typst just +draws. Doing the triangulation here rather than in the figure is what stops +nodes being left out of the mesh -- an earlier version connected nodes by a +distance threshold and silently missed several. + +Emits the schema in the skill's `underworld-bridge.md`: + + {"vertices": [[x, y], ...], + "triangles": [[i, j, k], ...], + "surface": [i, ...], indices of the constrained nodes + "frames": [{"p": [x, y], "n": [nx, ny], "t": [tx, ty]}, ...], + "curve": [[x, y], ...]} the surface, finely sampled + +Run: python3 generate-rotated-basis.py +""" +import json +import pathlib + +import numpy as np +from scipy.spatial import Delaunay + +OUT = pathlib.Path(__file__).with_name("rotated-basis-data.json") + +X0, X1 = -3.5, 3.5 +BASE = -2.9 +NX = 9 # columns of nodes +NY = 4 # rows, surface included + + +def surface_y(x): + """A free surface: rises on the left, falls on the right, with an + inflection between, so the normal swings through a wide range and the + curvature changes sign. No global rotation straightens this out, which is + the reason the figure exists.""" + x = np.asarray(x, dtype=float) + return (0.95 * np.exp(-(((x + 1.75) / 1.30) ** 2)) + - 0.80 * np.exp(-(((x - 1.70) / 1.15) ** 2)) + + 0.75) + + +def surface_slope(x, eps=1.0e-4): + return (surface_y(x + eps) - surface_y(x - eps)) / (2 * eps) + + +# Nodes on a grid warped to sit under the surface. Columns are staggered on +# alternate rows so the Delaunay triangulation comes out as triangles rather +# than as near-degenerate right angles on a perfect lattice. +xs = np.linspace(X0, X1, NX) +pts = [] +surface_idx = [] +for row in range(NY): + frac = row / (NY - 1) # 0 at the base, 1 at the surface + offset = 0.0 if row % 2 == 0 else 0.5 * (xs[1] - xs[0]) + cols = xs + offset + if row == NY - 1: + cols = xs # surface row unstaggered + for x in cols: + if x < X0 - 1e-9 or x > X1 + 1e-9: + continue + top = float(surface_y(x)) + y = BASE + (top - BASE) * frac + if row == NY - 1: + surface_idx.append(len(pts)) + pts.append([float(x), float(y)]) + +pts = np.array(pts) +tri = Delaunay(pts) + +# Drop the slivers Delaunay leaves along a non-convex top edge: any triangle +# whose centroid sits above the surface is outside the domain. +keep = [] +for simplex in tri.simplices: + c = pts[simplex].mean(axis=0) + if c[1] <= float(surface_y(c[0])) + 1.0e-9: + keep.append([int(i) for i in simplex]) + +frames = [] +for i in surface_idx: + x = pts[i][0] + m = float(surface_slope(x)) + n = np.array([-m, 1.0]) + n /= np.linalg.norm(n) + t = np.array([1.0, m]) + t /= np.linalg.norm(t) + frames.append({"p": [float(pts[i][0]), float(pts[i][1])], + "n": [float(n[0]), float(n[1])], + "t": [float(t[0]), float(t[1])]}) + +curve_x = np.linspace(X0, X1, 121) +data = { + "vertices": [[float(a), float(b)] for a, b in pts], + "triangles": keep, + "surface": [int(i) for i in surface_idx], + "frames": frames, + "curve": [[float(a), float(b)] for a, b in zip(curve_x, surface_y(curve_x))], +} +OUT.write_text(json.dumps(data, indent=1)) +print("wrote %s: %d vertices, %d triangles, %d surface nodes" + % (OUT.name, len(data["vertices"]), len(data["triangles"]), + len(data["surface"]))) diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/rotated-basis-data.json b/articles/boundary-conditions-on-non-planar-boundaries/examples/rotated-basis-data.json new file mode 100644 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lives in the velocity +// block and its multigrid substructure, and the Schur/pressure machinery +// wrapping it never handles a rotated vector. +// +// House style follows the cetz-figures skill in the underworld3 repository +// (cetz 0.3.4, hex colours, helper-function pattern). +// +// Build, from this directory: +// +// python3 generate-rotated-basis.py # writes the JSON geometry +// typst compile --format svg rotated-basis.typ ../figures/rotated-basis.svg +// typst compile --ppi 200 rotated-basis.typ ../figures/rotated-basis.png + +#import "@preview/cetz:0.3.4" + +#set page(width: auto, height: auto, margin: 16pt) +#set text(size: 10pt) + +#cetz.canvas({ + import cetz.draw: * + + // Colours + let cart-fg = rgb("#4a7bf7") + let cart-bg = rgb("#dce8fc") + let rot-fg = rgb("#e57373") + let rot-bg = rgb("#fce4ec") + let plain-bg = rgb("#f2f2f0") + let mg-bg = rgb("#fdf1f4") + let schur-fg = rgb("#49a87c") // the un-rotated half + let schur-bg = rgb("#e8f4e8") + let ink = rgb("#1a1a1a") + let muted = rgb("#7a7a7a") + let hair = rgb("#d0d0d0") + + // ====================================================================== + // LEFT: a free surface, meshed. Geometry (nodes, Delaunay triangles, + // surface frames) comes from generate-rotated-basis.py via JSON -- the + // skill's rule, and the reason every node is now in the triangulation. + // An earlier version joined nodes by a distance threshold and left some out. + // ====================================================================== + let data = json("rotated-basis-data.json") + let vtx = data.vertices + let at(i) = (vtx.at(i).at(0), vtx.at(i).at(1)) + let is-surface(i) = data.surface.contains(i) + + // triangles first, then the surface, then the nodes: painter's algorithm + for tri in data.triangles { + line(at(tri.at(0)), at(tri.at(1)), at(tri.at(2)), close: true, + stroke: (paint: hair, thickness: 0.55pt)) + } + + line(..data.curve.map(q => (q.at(0), q.at(1))), + stroke: (paint: ink, thickness: 1.2pt)) + + // A rotated frame at every constrained node: n the outward surface normal, + // t the tangent. Both follow the node, which is the whole point. + for (k, f) in data.frames.enumerate() { + let p = (f.p.at(0), f.p.at(1)) + let n = f.n + let tg = f.t + let ln = 0.72 + let lt = 0.46 + line(p, (p.at(0) + ln * n.at(0), p.at(1) + ln * n.at(1)), + mark: (end: ">", scale: 0.4), stroke: (paint: rot-fg, thickness: 1.1pt)) + line(p, (p.at(0) + lt * tg.at(0), p.at(1) + lt * tg.at(1)), + mark: (end: ">", scale: 0.4), stroke: (paint: rot-fg, thickness: 1.1pt)) + if k == 4 { + content((p.at(0) + 1.20 * ln * n.at(0), p.at(1) + 1.20 * ln * n.at(1)), + text(fill: rot-fg, size: 9pt, $n$)) + content((p.at(0) + 1.55 * lt * tg.at(0) + 0.24 * n.at(0), + p.at(1) + 1.55 * lt * tg.at(1) + 0.24 * n.at(1)), + text(fill: rot-fg, size: 9pt, $t$)) + } + } + + // the Cartesian frame at one interior node -- identical at every other one, + // which is what makes the surface the odd one out + let ip = at(12) // an interior node with room around it + line(ip, (ip.at(0) + 0.66, ip.at(1)), mark: (end: ">", scale: 0.4), + stroke: (paint: cart-fg, thickness: 1.1pt)) + line(ip, (ip.at(0), ip.at(1) + 0.66), mark: (end: ">", scale: 0.4), + stroke: (paint: cart-fg, thickness: 1.1pt)) + content((ip.at(0) + 0.88, ip.at(1)), text(fill: cart-fg, size: 9pt, $x$)) + content((ip.at(0), ip.at(1) + 0.88), text(fill: cart-fg, size: 9pt, $y$)) + + for i in range(vtx.len()) { + if is-surface(i) { + circle(at(i), radius: 0.13, fill: rot-fg, + stroke: (paint: rot-fg, thickness: 1pt)) + } else { + circle(at(i), radius: 0.11, fill: cart-bg, + stroke: (paint: cart-fg, thickness: 1pt)) + } + } + + content((0, 3.05), text(weight: "bold", size: 10pt, fill: ink, + "A free surface has no preferred direction")) + + circle((-3.3, -3.55), radius: 0.13, fill: rot-fg, stroke: (paint: rot-fg)) + content((-3.05, -3.55), anchor: "west", + text(size: 9pt, fill: ink, [surface node --- solve for $(v_n, v_t)$, hold $v_n$])) + circle((-3.3, -4.1), radius: 0.11, fill: cart-bg, + stroke: (paint: cart-fg, thickness: 1pt)) + content((-3.05, -4.1), anchor: "west", + text(size: 9pt, fill: ink, [interior node --- solve for $(v_x, v_y)$])) + + line((4.35, -4.9), (4.35, 3.3), stroke: (paint: hair, thickness: 0.8pt)) + + // ====================================================================== + // RIGHT: where the rotation lives. Two blocks ABUTTING, not nested: the + // velocity solve is rotated, the Schur/pressure solve is not, and the + // single un-rotation on the boundary between them feeds both the Schur + // solve and everything outside. + // ====================================================================== + let panel(tl, br, title, subtitle, bg, edge) = { + import cetz.draw: * + rect(tl, br, fill: bg, stroke: (paint: edge, thickness: 1pt), radius: 5pt) + content((tl.at(0) + 0.30, tl.at(1) - 0.36), anchor: "west", + text(weight: "bold", size: 9.5pt, fill: edge, title)) + if subtitle != none { + // cetz content() lays out at natural width and spills over the rect. + // Box it to the panel's own width so the text wraps inside the border. + // The canvas default is 1cm per unit, so the arithmetic is direct. + let tw = (br.at(0) - tl.at(0) - 0.60) * 1cm + content((tl.at(0) + 0.30, tl.at(1) - 0.90), anchor: "north-west", + box(width: tw, text(size: 8.5pt, fill: muted, subtitle))) + } + } + + content((10.3, 3.05), text(weight: "bold", size: 10pt, fill: ink, + "Where the rotation lives")) + + // -- the rotated half --------------------------------------------------- + panel((4.9, 1.65), (11.75, -3.70), [Velocity solve --- rotated], + [$hat(A) = Q^T A Q$, #h(0.7em) $hat(b) = Q^T b$, #h(0.7em) + $v_n$ held strongly at surface nodes], rot-bg, rot-fg) + + panel((5.35, -0.15), (11.50, -3.25), "Multigrid", + [the transfers are the only further obligation], mg-bg, rot-fg) + + let mgrow(y, lhs, rhs) = { + import cetz.draw: * + content((5.65, y), anchor: "west", text(size: 9pt, fill: ink, lhs)) + content((8.05, y), anchor: "west", text(size: 9pt, fill: rot-fg, rhs)) + } + mgrow(-1.85, [prolongation], [$P -> Q^T P$]) + mgrow(-2.40, [coarse operators], [inherit $Q$ via $R A P$]) + mgrow(-2.93, [coarse solve], [SVD (rigid rotations)]) + + // -- the un-rotated half, abutting -------------------------------------- + panel((13.35, 1.65), (17.85, -1.35), [Fieldsplit / Schur solve], + [pressure and constraints. Isotropic, and carrying no boundary + condition of this kind, so it never sees a rotated vector.], + schur-bg, schur-fg) + + // -- one un-rotation on the boundary, feeding both ---------------------- + // The junction sits in the gap BETWEEN the two blocks, low enough to clear + // the velocity block's own subtitle -- one un-rotation, two consumers. + let jx = 12.55 + let jy = -0.40 + line((11.75, jy), (jx, jy), stroke: (paint: rot-fg, thickness: 1pt)) + circle((jx, jy), radius: 0.075, fill: rot-fg, stroke: (paint: rot-fg)) + content((jx, jy + 0.45), text(size: 9pt, fill: rot-fg, $v = Q hat(v)$)) + + // branch 1: into the Schur solve, which needs it un-rotated + line((jx, jy), (13.29, jy), mark: (end: ">", scale: 0.4), + stroke: (paint: rot-fg, thickness: 1pt)) + + // branch 2: out to everything else + line((jx, jy), (jx, -2.70), stroke: (paint: rot-fg, thickness: 1pt)) + line((jx, -2.70), (13.20, -2.70), mark: (end: ">", scale: 0.4), + stroke: (paint: rot-fg, thickness: 1pt)) + 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diff --git a/articles/boundary-conditions-on-non-planar-boundaries/metadata.yml b/articles/boundary-conditions-on-non-planar-boundaries/metadata.yml new file mode 100644 index 0000000..2778f0f --- /dev/null +++ b/articles/boundary-conditions-on-non-planar-boundaries/metadata.yml @@ -0,0 +1,34 @@ +# Validated in CI against schemas/article-metadata.schema.json. +# `pixi run validate` checks this and the cross-file invariants a schema cannot +# express -- that the article file is named .md, that canonical_path +# matches the slug, and that no legacy DOI is ever paired with a new registrant. +id: UWTN 2026-016 +slug: boundary-conditions-on-non-planar-boundaries +title: Boundary conditions on non-planar boundaries +article_type: technical-note +status: review +authors: + - name: Louis Moresi + orcid: 0000-0003-3685-174X + affiliation: Australian National University +publication_date: null +version: 1.0.0 +# The deposit writes archive_doi and repository_record_id when the note is +# published; leave them out until then. `doi` and `doi_registrant` were here +# once and are not fields the schema knows -- every note made from this +# template failed `pixi run validate` on all three of them. +license: CC-BY-4.0 +canonical_path: /boundary-conditions-on-non-planar-boundaries/ +legacy_paths: [] +# Facets, from vocabulary.yml. Both keys must be present even when empty: a +# note with no subject is normal -- many are purely about method. +subjects: + - surface-processes +methods: + - solvers + - finite-elements + - meshing +ghost_tags: + - Underworld Code +figures: 1 +source: native diff --git a/classification.yml b/classification.yml index 65201be..4925dd5 100644 --- a/classification.yml +++ b/classification.yml @@ -322,3 +322,7 @@ setting-up-full-multigrid: article_type: technical-note subjects: [mantle-convection] methods: [solvers, meshing, parallel-hpc] +boundary-conditions-on-non-planar-boundaries: + article_type: technical-note + subjects: [surface-processes] + methods: [solvers, finite-elements, meshing] diff --git a/scripts/new_article.py b/scripts/new_article.py index a010c81..c60324f 100644 --- a/scripts/new_article.py +++ b/scripts/new_article.py @@ -17,6 +17,7 @@ import datetime import pathlib import re +import subprocess import sys ROOT = pathlib.Path(__file__).resolve().parent.parent @@ -44,6 +45,37 @@ def load_authors(): return registry +def ids_on_branches(year): + """Article numbers claimed on a branch, merged or not. + + The rest of the allocator reads the working tree, which cannot see a number + claimed by a note still in review on its own branch. Two notes drafted in + parallel were therefore both offered the same number, and UWTN 2026-012 was + claimed twice before anyone noticed. + + Every local and remote-tracking ref is searched, so this covers open pull + requests as far as they have been fetched. A note on a branch that has never + been pushed to a remote this checkout tracks is still invisible, which is why + `pixi run validate` checks for duplicates as well. + """ + def git(*args): + return subprocess.run(("git",) + args, cwd=str(ROOT), + capture_output=True, text=True) + + refs = git("for-each-ref", "--format=%(refname)", "refs/heads", "refs/remotes") + if refs.returncode != 0: + return set() # no git, or not a checkout: the tree is all we have + + used = set() + for ref in refs.stdout.split(): + found = git("grep", "-h", "-E", r"^id: +UWTN +[0-9]{4}-[0-9]{3}", + ref, "--", "articles/*/metadata.yml") + for match in re.finditer(r"UWTN\s+(\d{4})-(\d{3})", found.stdout): + if match.group(1) == str(year): + used.add(int(match.group(2))) + return used + + def next_article_id(year): """Allocate an ID that no existing article uses. @@ -52,7 +84,7 @@ def next_article_id(year): number already present in that year keeps a new note clear of anything the backfill will produce, and an ID that has been published never moves. """ - used = set() + used = ids_on_branches(year) for meta in ARTICLES.glob("*/metadata.yml"): match = re.search(r"^id:\s*(UWTN\s+(\d{4})-(\d{3}))\s*$", meta.read_text(encoding="utf-8"), re.M) From 74680278967512210322eb3df588b83f8e0a2c94 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Tue, 18 Aug 2026 22:24:55 +1000 Subject: [PATCH 02/33] Measure the leak, so the central claim stops being a claim The note asserted that a weakly imposed constraint holds to the accuracy of the discretisation and a strong one to the accuracy of the arithmetic. Annulus, no slip inside, the treatment under test outside, degree-four radial forcing; largest normal velocity on the outer boundary against the true radial direction, over the flow speed: cell Nitsche rotated 0.150 4.6e-03 7.3e-11 0.100 2.2e-03 6.5e-11 0.075 1.7e-03 1.0e-10 0.050 5.8e-04 8.4e-11 Refinement is what separates them and one resolution would not have. Nitsche falls as about h^1.9 -- discretisation-limited, which is what consistency buys and all it buys. The rotated constraint does not move with the mesh at all. Three things the measurement cost, all worth recording: The first control was worthless. Two free-slip circles leave the rigid rotation unconstrained, and that nullspace is purely TANGENTIAL, so a radial leak metric read 2e-14 on a solve that had diverged with |u| = 2.7e5. The inner boundary is now no-slip and the control -- outer boundary left natural -- reads 0.98. A vector MeshVariable's .array is (N, 1, dim). It broadcasts silently against an (N, dim) array of normals and returns projections of 1e-16 for a velocity of 1e-2, without raising. Every number in the first run was that artefact. Underworld exposes no separate direct-penalty condition and building one from a natural BC would not converge, so that row has no measured counterpart. Said in the note rather than quietly dropped, alongside the other limit: this measures the constraint, not the traction recovered from it. Underworld development team with AI support from Claude Code --- ...ary-conditions-on-non-planar-boundaries.md | 45 +- .../uw_annulus_ro1.0_ri0.5_csize0.05.msh | 5034 +++++++++++++++++ .../uw_annulus_ro1.0_ri0.5_csize0.05.msh.h5 | Bin 0 -> 303224 bytes .../uw_annulus_ro1.0_ri0.5_csize0.075.msh | 2342 ++++++++ .../uw_annulus_ro1.0_ri0.5_csize0.075.msh.h5 | Bin 0 -> 159772 bytes .../uw_annulus_ro1.0_ri0.5_csize0.1.msh | 1434 +++++ .../uw_annulus_ro1.0_ri0.5_csize0.1.msh.h5 | Bin 0 -> 111992 bytes .../uw_annulus_ro1.0_ri0.5_csize0.15.msh | 730 +++ .../uw_annulus_ro1.0_ri0.5_csize0.15.msh.h5 | Bin 0 -> 75576 bytes .../examples/leak.py | 137 + 10 files changed, 9716 insertions(+), 6 deletions(-) create mode 100644 articles/boundary-conditions-on-non-planar-boundaries/examples/.meshes/uw_annulus_ro1.0_ri0.5_csize0.05.msh create mode 100644 articles/boundary-conditions-on-non-planar-boundaries/examples/.meshes/uw_annulus_ro1.0_ri0.5_csize0.05.msh.h5 create mode 100644 articles/boundary-conditions-on-non-planar-boundaries/examples/.meshes/uw_annulus_ro1.0_ri0.5_csize0.075.msh create mode 100644 articles/boundary-conditions-on-non-planar-boundaries/examples/.meshes/uw_annulus_ro1.0_ri0.5_csize0.075.msh.h5 create mode 100644 articles/boundary-conditions-on-non-planar-boundaries/examples/.meshes/uw_annulus_ro1.0_ri0.5_csize0.1.msh create mode 100644 articles/boundary-conditions-on-non-planar-boundaries/examples/.meshes/uw_annulus_ro1.0_ri0.5_csize0.1.msh.h5 create mode 100644 articles/boundary-conditions-on-non-planar-boundaries/examples/.meshes/uw_annulus_ro1.0_ri0.5_csize0.15.msh create mode 100644 articles/boundary-conditions-on-non-planar-boundaries/examples/.meshes/uw_annulus_ro1.0_ri0.5_csize0.15.msh.h5 create mode 100644 articles/boundary-conditions-on-non-planar-boundaries/examples/leak.py diff --git a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md index 35feaed..9a20ca6 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md +++ b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md @@ -119,7 +119,9 @@ rather than being a dial you turn up until the answer looks right. This is a real improvement and it is still a weak imposition. The constraint holds to the accuracy of the discretisation, not to the accuracy of the -arithmetic. +arithmetic — measured below, it leaks a few parts in a thousand on a mesh of +the resolution people actually run, and the leak falls with the mesh rather +than with the machine. ### Rotating the degrees of freedom @@ -238,11 +240,42 @@ multiplier the solve has already computed, and it comes out of `boundary_normal_traction` without a recovery step or an augmented-Lagrangian splitting. -:::{warning} This section is not yet measured -The argument above is structural and we believe it, which is not the same as -having shown it. The note needs one experiment: the same model under all three -boundary treatments, with the surface traction compared against a case whose -answer is known. Until that table is here, this section is a claim. +### The leak, measured + +An annulus, no slip on the inner radius, the treatment under test on the outer, +driven by a degree-four radial density anomaly. The number is the largest +normal velocity on the outer boundary, taken against the true radial direction +and divided by the flow speed, so it reads as the fraction of the flow that +goes through a boundary nothing should pass through. + +| cell size | Nitsche | rotated | +|---|---|---| +| 0.150 | 4.6 × 10⁻³ | 7.3 × 10⁻¹¹ | +| 0.100 | 2.2 × 10⁻³ | 6.5 × 10⁻¹¹ | +| 0.075 | 1.7 × 10⁻³ | 1.0 × 10⁻¹⁰ | +| 0.050 | 5.8 × 10⁻⁴ | 8.4 × 10⁻¹¹ | + +The refinement is what separates them, and a single resolution would not have. +Nitsche's leak falls as about $h^{1.9}$ — it is limited by the discretisation, +which is what "consistent" buys and all that it buys. The rotated constraint +does not move: it is at the solver's floor at every resolution, because the +mesh has nothing to do with it. + +The control matters here more than the result. With the outer boundary left +free, the same measurement reads **0.98** — nearly all the boundary flow is +normal — so the metric can see a leak when there is one. + +:::{note} What is measured, and what is still inferred +Two honest limits. Underworld exposes no separate direct-penalty condition, so +the first row of the table above has no measured counterpart here; the argument +for it is the standard one and it is not ours to demonstrate. + +And this measures the *constraint*, not the traction. That a traction recovered +from a constraint satisfied to $10^{-3}$ inherits that error, while the rotated +reaction is exact because it is the multiplier itself, follows from how each is +computed — but the surface-stress comparison against a known answer has not +been run, and until it is, the last two paragraphs of this section are +reasoning rather than measurement. ::: ## Using it diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/.meshes/uw_annulus_ro1.0_ri0.5_csize0.05.msh b/articles/boundary-conditions-on-non-planar-boundaries/examples/.meshes/uw_annulus_ro1.0_ri0.5_csize0.05.msh new file mode 100644 index 0000000..4a06cfb --- /dev/null +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/.meshes/uw_annulus_ro1.0_ri0.5_csize0.05.msh @@ -0,0 +1,5034 @@ +$MeshFormat +4.1 0 8 +$EndMeshFormat +$PhysicalNames +3 +1 1 "Lower" +1 2 "Upper" +2 666666 "Elements" +$EndPhysicalNames +$Entities +5 4 1 0 +1 0 0 0 0 +2 0.5 0 0 0 +3 -0.5 0 0 0 +4 1 0 0 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z%b%aW6hVXk`t`_1iGH!;pm&nIj^ka`{^;>*Q4@$0YsmInJbn$jHZj9$C=L6Mc>Ed^ zDZ##JDWE5sJeH!S?({lmpT{O{)y;sL0XG9~2HXs|8E`Y;X5b&3fq(V&!@ufr Date: Wed, 19 Aug 2026 10:47:08 +1000 Subject: [PATCH 03/33] Measure the direct penalty too: it was my mistake, not a missing feature The previous commit said Underworld exposes no direct-penalty condition. It does, it is documented in docs/advanced/curved-boundary-conditions.md, and it is in a good many working scripts: G = mesh.Gamma stokes.add_natural_bc(1.0e4 * G.dot(v.sym) * G, "Upper") What I had written used the nodal boundary_normal and a NEGATIVE coefficient. A negative one is anti-damping, so the linear solve failed on the first iteration, and I read that as the method being unavailable rather than as my sign being wrong. With all three measured, the resolution table gains its first column, and the two weak methods look alike in it -- both leak parts in a thousand and both improve with the mesh. What separates them is their own parameter: penalty 1e2 2.6e-01 | 1e3 2.3e-02 | 1e4 8.5e-04 | 1e5 9.6e-04 | 1e6 diverged gamma 1 diverged | 10 1.7e-03 | 100 2.7e-04 | 1000 3.0e-05 The penalty improves in proportion to how hard it pushes until the conditioning catches up: no gain from 1e4 to 1e5, failure at 1e6. Nitsche fails below its stability threshold and improves steadily above it, with no wall in this range, which is why gamma=10 can be a documented default instead of a number to fit per problem. Two runs in the first parameter sweep had failed the line search and were about to be tabulated as measurements. The sweep now records the converged reason and prints 'diverged' instead of a number. Underworld development team with AI support from Claude Code --- ...ary-conditions-on-non-planar-boundaries.md | 99 ++++++++++++------ .../uw_annulus_ro1.0_ri0.5_csize0.05.msh.h5 | Bin 303224 -> 303224 bytes .../uw_annulus_ro1.0_ri0.5_csize0.075.msh.h5 | Bin 159772 -> 159772 bytes .../uw_annulus_ro1.0_ri0.5_csize0.1.msh.h5 | Bin 111992 -> 111992 bytes .../uw_annulus_ro1.0_ri0.5_csize0.15.msh.h5 | Bin 75576 -> 75576 bytes .../examples/leak.py | 66 +++++++++--- 6 files changed, 118 insertions(+), 47 deletions(-) diff --git a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md index 9a20ca6..672e4ff 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md +++ b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md @@ -82,13 +82,23 @@ $$ \; \mathrm{d}S . $$ -One line, no new machinery, and it works on any geometry. What you are solving -is a perturbed problem, though, and it is perturbed by exactly the amount the -constraint is violated: the discrete solution sits where the penalty term -balances the traction it is fighting, which leaves $\mathbf{u}\cdot\hat{\mathbf{n}}$ -small but not zero. Making it smaller means raising $\gamma$, and raising -$\gamma$ conditions the operator worse. The error is traded against the -conditioning, and neither can be driven away. +One line, no new machinery, and it works on any geometry. It is still in a +good many working scripts, and deservedly. What you are solving is a perturbed +problem, though, and it is perturbed by exactly the amount the constraint is +violated: the discrete solution sits where the penalty term balances the +traction it is fighting, which leaves $\mathbf{u}\cdot\hat{\mathbf{n}}$ small +but not zero. Making it smaller means pushing harder, and pushing harder +conditions the operator worse. The error is traded against the conditioning, +and measured below, that trade runs out: past $10^4$ the leak stops improving, +and by $10^6$ the solve fails. + +Underworld spells it as a boundary traction opposing normal flow, using the +facet normal at the quadrature points: + +```python +G = mesh.Gamma +stokes.add_natural_bc(1.0e4 * G.dot(v.sym) * G, "Upper") +``` ### Nitsche @@ -245,36 +255,57 @@ splitting. An annulus, no slip on the inner radius, the treatment under test on the outer, driven by a degree-four radial density anomaly. The number is the largest normal velocity on the outer boundary, taken against the true radial direction -and divided by the flow speed, so it reads as the fraction of the flow that -goes through a boundary nothing should pass through. +and divided by the flow speed: the fraction of the flow going through a +boundary nothing should pass through. Penalty at $10^4$, Nitsche at +$\gamma = 10$. -| cell size | Nitsche | rotated | -|---|---|---| -| 0.150 | 4.6 × 10⁻³ | 7.3 × 10⁻¹¹ | -| 0.100 | 2.2 × 10⁻³ | 6.5 × 10⁻¹¹ | -| 0.075 | 1.7 × 10⁻³ | 1.0 × 10⁻¹⁰ | -| 0.050 | 5.8 × 10⁻⁴ | 8.4 × 10⁻¹¹ | +| cell size | direct penalty | Nitsche | rotated | +|---|---|---|---| +| 0.150 | 4.5 × 10⁻³ | 4.6 × 10⁻³ | 7.3 × 10⁻¹¹ | +| 0.100 | 2.5 × 10⁻³ | 2.2 × 10⁻³ | 6.5 × 10⁻¹¹ | +| 0.075 | 8.5 × 10⁻⁴ | 1.7 × 10⁻³ | 1.0 × 10⁻¹⁰ | +| 0.050 | 9.5 × 10⁻⁴ | 5.8 × 10⁻⁴ | 8.4 × 10⁻¹¹ | The refinement is what separates them, and a single resolution would not have. -Nitsche's leak falls as about $h^{1.9}$ — it is limited by the discretisation, -which is what "consistent" buys and all that it buys. The rotated constraint -does not move: it is at the solver's floor at every resolution, because the -mesh has nothing to do with it. - -The control matters here more than the result. With the outer boundary left -free, the same measurement reads **0.98** — nearly all the boundary flow is -normal — so the metric can see a leak when there is one. - -:::{note} What is measured, and what is still inferred -Two honest limits. Underworld exposes no separate direct-penalty condition, so -the first row of the table above has no measured counterpart here; the argument -for it is the standard one and it is not ours to demonstrate. - -And this measures the *constraint*, not the traction. That a traction recovered -from a constraint satisfied to $10^{-3}$ inherits that error, while the rotated -reaction is exact because it is the multiplier itself, follows from how each is -computed — but the surface-stress comparison against a known answer has not -been run, and until it is, the last two paragraphs of this section are +Both weak forms leak parts in a thousand and improve as the mesh improves. The +rotated constraint does not move: it sits at the solver's floor at every +resolution, because the mesh has nothing to do with it. + +The control matters more than the result. With the outer boundary left free the +same measurement reads **0.98** — nearly all the boundary flow is normal — so +the metric can see a leak when there is one. + +### What each parameter buys + +The two weak methods look alike in that table. They are not alike, and their +own parameters are what tells them apart. + +| penalty | leak | | $\gamma$ (Nitsche) | leak | +|---|---|---|---|---| +| 10² | 2.6 × 10⁻¹ | | 1 | diverged | +| 10³ | 2.3 × 10⁻² | | 10 | 1.7 × 10⁻³ | +| 10⁴ | 8.5 × 10⁻⁴ | | 100 | 2.7 × 10⁻⁴ | +| 10⁵ | 9.6 × 10⁻⁴ | | 1000 | 3.0 × 10⁻⁵ | +| 10⁶ | diverged | | | | + +The direct penalty improves in proportion to how hard it pushes, until it +stops: between $10^4$ and $10^5$ the leak does not fall, and at $10^6$ the +solve fails. That is the conditioning catching up, and it is the reason a +penalty has to be tuned — the useful range is bounded at both ends, and where +the ceiling sits depends on the problem. + +Nitsche fails at $\gamma = 1$ and improves steadily above it. The failure is the +stability threshold: below it the form is not coercive and there is nothing to +tune your way out of. Above it the method keeps improving without the +conditioning wall, which is what makes $\gamma = 10$ a documented default rather +than a number to fit each time. + +:::{note} What is measured here, and what is not +This measures the *constraint*. The claim that follows — that a traction +recovered from a constraint good to $10^{-3}$ inherits that error, while the +rotated reaction is exact because it is the multiplier itself — follows from +how each is computed, and the surface-stress comparison against a known answer +has not been run. Until it is, the two paragraphs above this section are reasoning rather than measurement. ::: diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/.meshes/uw_annulus_ro1.0_ri0.5_csize0.05.msh.h5 b/articles/boundary-conditions-on-non-planar-boundaries/examples/.meshes/uw_annulus_ro1.0_ri0.5_csize0.05.msh.h5 index 43d6ffcff5f266d16009fedbafc602a5bc99c83b..21bdce5c5bb78ec58ee7054afe40d4c2d936e194 100644 GIT binary patch delta 156 zcmexyKUgEU8?}n)9d7zH*NlL zeMu#dvwdD5W8O0mgU^Bq!T?I=)q+Ic&1T*v57GdXnzY?w53`OlocH5D^Be;>Ps5N! b#Sx^%uZTsCAH>LOWQh|5G0sh5S;Y$g=XQ2vKSaLCL6}&PCjs5V)F#%J5fv%{5I#sU8?}n)9d7zH*NlL zeMu#dvwdD5W8O0mgU^Bq!T?I=)q+Ic&1T*v57GdXnzY?w53`OlocH5D^Be;>Ps5N! b#Sx^%uZTsCAH>LOWQh|5G0sh5S;Y$gmgGD} diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/.meshes/uw_annulus_ro1.0_ri0.5_csize0.075.msh.h5 b/articles/boundary-conditions-on-non-planar-boundaries/examples/.meshes/uw_annulus_ro1.0_ri0.5_csize0.075.msh.h5 index b4c5d9b34069311ab1e4db236f32174bc0d24607..4a9988623400d2cba0a0a47dd3af43643fa4b97e 100644 GIT binary patch delta 160 zcmbR9fOF0R&J8Tg>=u7ovKSaLCSPREoxb5Dqr~P3%x4UkEdFecGudYYqU8$hY{C3> zcDX=$`#fnz2;*J{W0(U_WU^f0r0pk8GAMGBc^k!bBq!nPw=%I4!PB WcX(l(FTqST{1DFM?W`$G;yeIX4m=zH diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/.meshes/uw_annulus_ro1.0_ri0.5_csize0.1.msh.h5 b/articles/boundary-conditions-on-non-planar-boundaries/examples/.meshes/uw_annulus_ro1.0_ri0.5_csize0.1.msh.h5 index 2ef557ba52f84f5e609ad3e03b2528eaab9868d3..fbf86010956ea277f5ea034ad7548996492097c1 100644 GIT binary patch delta 163 zcmezIitWcMwhb)I>?VI&vKSaLCSPREom@~Mxp^YKb!T`?rDH%KL2MLEFeXb zd>uAhR2*>u3I5x|<_e@I^Z9SuzJrO;M;pSKw0(viqlh4ovwfXCBbzi#)FYblizrCc du8^@z3NG5i*ux9sWXxi`!w2KsSjo768vr0WK)e6| delta 163 zcmezIitWcMwhb)I?A?AXSquyrlP@yoPA;gB+&qzawh~jf-)23vdm12`&;OYQ3rNu< zUx&>W6-Qh^g8%lgxdQ3QeEyrZ?_gr|(S~p)ZJ(jXC?W{tY+q;3$R-UF^@wKtA_@|< dD`YH_f{XSr_VB_u8M7Gg@WD7YRx&Q&1^~hmJW&7u diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/.meshes/uw_annulus_ro1.0_ri0.5_csize0.15.msh.h5 b/articles/boundary-conditions-on-non-planar-boundaries/examples/.meshes/uw_annulus_ro1.0_ri0.5_csize0.15.msh.h5 index d677ad3878abc360bec25f6d4af4b161b704b894..d94d770ec05d764a909ef4bd5ddfe7a5fc29c954 100644 GIT binary patch delta 157 zcmdmSj%CL=mJKY-?D~INvKSaLCSPRCo&3R1V)F#%IpR$Ee>Uq$-;)H!2Zo)9J6SuwyfW-E_HiL_G a{I}tS%5RV0Wz+#_0va}HyMzp*0yh8zK0k*5 delta 157 zcmdmSj%CL=mJKY->>YkBSquyrlP@ynPX6F0v3UaX9C4-&zs-8m_as3ypJ|#T3rNu< zV~5Q<{9M#Pg8xd|K=foj(@mQ#=A4m+uqSPvu_J^JB>(Q1n=nl5#H}v^AhCU~&ER4k a|801o^4lYL8FfIKfQC)lE+NCHzzqPuMLRYC diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/leak.py b/articles/boundary-conditions-on-non-planar-boundaries/examples/leak.py index 74ff5d4..e26b229 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/examples/leak.py +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/leak.py @@ -38,7 +38,7 @@ GAMMA = 10.0 -def build(mode, cell=CELL): +def build(mode, cell=CELL, penalty=1.0e4, gamma=GAMMA): mesh = uw.meshing.Annulus(radiusInner=0.5, radiusOuter=1.0, cellSize=cell) x, y = mesh.X r = sympy.sqrt(x**2 + y**2) @@ -68,21 +68,29 @@ def build(mode, cell=CELL): if mode == "rotated": stokes.add_rotated_freeslip_bc(0.0, boundary) elif mode == "nitsche": - stokes.add_nitsche_bc(0.0, boundary, gamma=GAMMA, theta=1) + stokes.add_nitsche_bc(0.0, boundary, gamma=gamma, theta=1) elif mode == "penalty": - # A DIRECT penalty: a boundary traction opposing any normal flow, - # and nothing else. No consistency term, which is exactly what - # makes it consistent only in the limit. - # both boundary_normal and v.sym are (1, dim) row matrices, so the - # normal component is a dot product and the traction is a row again - n = mesh.boundary_normal(boundary) - un = v.sym.dot(n) - stokes.add_natural_bc(-(GAMMA / cell) * un * n, boundary) + # A DIRECT penalty: a boundary traction opposing normal flow, and + # nothing else -- no consistency term, which is exactly what leaves + # it consistent only in the limit. The documented form, from + # docs/advanced/curved-boundary-conditions.md, uses the + # quadrature-point facet normal mesh.Gamma and a POSITIVE + # coefficient. A negative one is anti-damping and the linear solve + # fails immediately, which is how this was got wrong the first time. + G = mesh.Gamma + stokes.add_natural_bc(penalty * G.dot(v.sym) * G, boundary) else: raise ValueError(mode) return mesh, stokes, v +def converged(stokes): + """A diverged solve still leaves numbers in the array, and they look like + measurements. Two runs in the first parameter sweep here had failed the + line search and were about to be tabulated.""" + return stokes.snes.getConvergedReason() > 0 + + def leaks(mesh, v): """max |u.n| on the outer boundary, against the facet normal and against the true radial direction, both normalised by the flow speed.""" @@ -103,7 +111,8 @@ def leaks(mesh, v): return leak_true, on_outer.sum(), speed -def sweep(cells=(0.15, 0.10, 0.075, 0.05), modes=("nitsche", "rotated")): +def sweep(cells=(0.15, 0.10, 0.075, 0.05), + modes=("penalty", "nitsche", "rotated")): """Does the leak fall with the mesh, or is it already at round-off? A weakly imposed constraint is satisfied to the accuracy of the @@ -119,14 +128,45 @@ def sweep(cells=(0.15, 0.10, 0.075, 0.05), modes=("nitsche", "rotated")): for mode in modes: mesh, stokes, v = build(mode, cell=cell) stokes.solve() - lk, _, _ = leaks(mesh, v) - row.append("%.2e" % lk) + row.append(("%.2e" % leaks(mesh, v)[0]) if converged(stokes) + else "diverged") print("| %.3f | %s |" % (cell, " | ".join(row))) +def parameter_sweep(): + """The distinction between a direct penalty and Nitsche, measured. + + A penalty holds the constraint only in proportion to how hard it pushes, so + its leak should track 1/penalty and it has to be tuned. Nitsche's penalty + term is there for STABILITY, not accuracy -- the consistency terms carry + the accuracy -- so its leak should barely move with gamma above the + stability threshold. That is the practical difference between them and it + is why gamma is documented as mesh- and problem-independent. + """ + print("\ndirect penalty: leak against penalty magnitude") + print("\n| penalty | leak/|u| |") + print("|---|---|") + for pen in (1.0e2, 1.0e3, 1.0e4, 1.0e5, 1.0e6): + mesh, stokes, v = build("penalty", penalty=pen) + stokes.solve() + cell = ("%.2e" % leaks(mesh, v)[0]) if converged(stokes) else "diverged" + print("| %.0e | %s |" % (pen, cell)) + + print("\nNitsche: leak against gamma") + print("\n| gamma | leak/|u| |") + print("|---|---|") + for g in (1.0, 10.0, 100.0, 1000.0): + mesh, stokes, v = build("nitsche", gamma=g) + stokes.solve() + cell = ("%.2e" % leaks(mesh, v)[0]) if converged(stokes) else "diverged" + print("| %g | %s |" % (g, cell)) + + if __name__ == "__main__": if sys.argv[1:2] == ["sweep"]: sweep() + elif sys.argv[1:2] == ["params"]: + parameter_sweep() else: modes = sys.argv[1:] or ["free", "nitsche", "rotated"] print("%-9s %10s %10s %8s" % ("mode", "leak/|u|", "|u|max", "nodes")) From e795432ff3dd140ca54d90ce7ac6672f4b7a0d59 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Wed, 19 Aug 2026 12:09:04 +1000 Subject: [PATCH 04/33] Nitsche has a conditioning wall too: it diverges from gamma = 1e4 The note said Nitsche 'keeps improving without the conditioning wall'. That was written from a sweep that stopped at gamma = 1000, which is one value short of where it fails. gamma 1e3 3.0e-05 converged 1e4 -- DIVERGED_LINE_SEARCH 1e5 -- DIVERGED_LINE_SEARCH It does not stop converging in the metric, it stops converging: reason -6, and |u| collapses toward zero, so the leak figures above the threshold are from solves that never finished. So both weak methods are bounded above by conditioning and neither escapes tuning. What differs is the floor each reaches before failing -- the penalty bottoms out near 1e-3, Nitsche at 3e-5 -- which is what consistency actually buys: a lower floor, not an unbounded parameter. Nitsche is bounded below as well, at gamma = 1, where the form stops being coercive. Its usable range has a threshold at each end, and gamma = 10 sits in the middle of that window on any mesh because gamma is dimensionless and the term it scales already carries mu/h. The penalty coefficient carries no such scaling, which is why its working value is a property of the problem. The shipped sweep now runs into the failing range rather than stopping just below it. Underworld development team with AI support from Claude Code --- ...ary-conditions-on-non-planar-boundaries.md | 37 +++++++++++------- .../uw_annulus_ro1.0_ri0.5_csize0.075.msh.h5 | Bin 159772 -> 159772 bytes .../examples/leak.py | 15 +++---- 3 files changed, 31 insertions(+), 21 deletions(-) diff --git a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md index 672e4ff..32e0e87 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md +++ b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md @@ -90,7 +90,8 @@ traction it is fighting, which leaves $\mathbf{u}\cdot\hat{\mathbf{n}}$ small but not zero. Making it smaller means pushing harder, and pushing harder conditions the operator worse. The error is traded against the conditioning, and measured below, that trade runs out: past $10^4$ the leak stops improving, -and by $10^6$ the solve fails. +and by $10^6$ the solve fails. Nitsche moves the floor down rather than +removing it — it fails too, at its own threshold. Underworld spells it as a boundary traction opposing normal flow, using the facet normal at the quadrature points: @@ -286,19 +287,27 @@ own parameters are what tells them apart. | 10³ | 2.3 × 10⁻² | | 10 | 1.7 × 10⁻³ | | 10⁴ | 8.5 × 10⁻⁴ | | 100 | 2.7 × 10⁻⁴ | | 10⁵ | 9.6 × 10⁻⁴ | | 1000 | 3.0 × 10⁻⁵ | -| 10⁶ | diverged | | | | - -The direct penalty improves in proportion to how hard it pushes, until it -stops: between $10^4$ and $10^5$ the leak does not fall, and at $10^6$ the -solve fails. That is the conditioning catching up, and it is the reason a -penalty has to be tuned — the useful range is bounded at both ends, and where -the ceiling sits depends on the problem. - -Nitsche fails at $\gamma = 1$ and improves steadily above it. The failure is the -stability threshold: below it the form is not coercive and there is nothing to -tune your way out of. Above it the method keeps improving without the -conditioning wall, which is what makes $\gamma = 10$ a documented default rather -than a number to fit each time. +| 10⁶ | diverged | | 10⁴ and above | diverged | + +**Both have a wall, and neither escapes tuning.** The direct penalty improves in +proportion to how hard it pushes until the conditioning catches up: no gain +from $10^4$ to $10^5$, and failure at $10^6$. Nitsche improves further and then +fails outright — at $\gamma = 10^4$ the line search stops converging, and the +figures above that are from solves that did not finish. + +What differs is how far down each gets before it fails. The penalty bottoms out +near $10^{-3}$; Nitsche reaches $3 \times 10^{-5}$, better by more than an order +of magnitude. That is what consistency buys — a smaller floor, not an +unbounded parameter. + +Nitsche also fails *below* $\gamma = 1$, and for the opposite reason: the form +is no longer coercive, and no amount of tuning recovers it. So its usable range +is bounded at both ends, with the stability threshold underneath and the +conditioning above. The virtue of $\gamma = 10$ is that it sits in the middle of +that window on any mesh, because $\gamma$ is dimensionless and the term it +scales already carries $\mu / h$. The penalty coefficient carries no such +scaling, which is why the value that works is a property of the problem rather +than a default. :::{note} What is measured here, and what is not This measures the *constraint*. The claim that follows — that a traction diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/.meshes/uw_annulus_ro1.0_ri0.5_csize0.075.msh.h5 b/articles/boundary-conditions-on-non-planar-boundaries/examples/.meshes/uw_annulus_ro1.0_ri0.5_csize0.075.msh.h5 index 4a9988623400d2cba0a0a47dd3af43643fa4b97e..eb680921b3bf3a93a17a3b8a61254bb7acda573d 100644 GIT binary patch delta 172 zcmbR9fOF0R&J8Tg>}UB~vltjMCSPREoxb5Dqr~P3%x4Ukfuft^P4?P=8S;g;wh)nZ zwz(k2_W9C`P}cnp#!v^4*kt*_iQ7+{WLzKu=l!|IxIhJ>o|#Ei7A_v4$TUM4&TDaH Xy2A_SeF=u7ovKSaLCSPREoxb5Dqr~P3%x4Ukfuft^P4?P=8S;g;wh)nZ zwz(k2_W9C`P}cnp#!v^4*kt*_iQ7+{WLzKu=l!|IxIhJ>o|#Ei7A_v4$TUM4&TDaH Xy2A_SeF Date: Wed, 19 Aug 2026 12:42:09 +1000 Subject: [PATCH 05/33] Add the multiplier constraint as the fourth method, and measure it The note had three approaches and there are four. add_constraint_bc carries a scalar multiplier field h coupled into the saddle point, so the constraint is a ROW of the system rather than a term added to one -- and at convergence h on the boundary is the normal traction, which makes it the second of the four that returns the stress rather than recovering it. Measured alongside the others, same annulus, same forcing: cell penalty nitsche multiplier rotated 0.150 4.5e-03 4.6e-03 8.3e-04 7.3e-11 0.100 2.5e-03 2.2e-03 1.6e-04 6.5e-11 0.075 8.5e-04 1.7e-03 5.6e-05 1.0e-10 0.050 9.5e-04 5.8e-04 1.2e-05 8.4e-11 It is an approximation, as expected, but a much better behaved one: it starts an order of magnitude below the weak forms and falls near h^3.9 against their h^1.9, because the only thing approximated is the screening of the interior multiplier degrees of freedom rather than the enforcement itself. The note says where that screening enters, and that the augmented-Lagrangian term conditions the [p,h] block without biasing the multiplier, so accuracy does not depend on it the way a penalty's does. Also started the surface-traction comparison, which is the test that matters, and it is NOT in the note. Two exact routes to sigma_nn -- the multiplier and the rotated reaction -- agree to about 1% at one resolution, but under refinement the vertex agreement converges while the midpoint agreement gets steadily worse, which is backwards: the solver documents midpoints as superconvergent on a curved boundary. The fault is most likely this script selecting the multiplier's boundary trace by a radius band that narrows with the mesh; it needs to come from the boundary label instead. The symptom and the suspicion are recorded in examples/traction.py rather than the numbers being quoted anywhere. Underworld development team with AI support from Claude Code --- ...ary-conditions-on-non-planar-boundaries.md | 64 +++++-- .../uw_annulus_ro1.0_ri0.5_csize0.05.msh.h5 | Bin 303224 -> 303224 bytes .../uw_annulus_ro1.0_ri0.5_csize0.075.msh.h5 | Bin 159772 -> 159772 bytes .../uw_annulus_ro1.0_ri0.5_csize0.1.msh.h5 | Bin 111992 -> 111992 bytes .../uw_annulus_ro1.0_ri0.5_csize0.15.msh.h5 | Bin 75576 -> 75576 bytes .../examples/leak.py | 17 +- .../examples/traction.py | 159 ++++++++++++++++++ 7 files changed, 226 insertions(+), 14 deletions(-) create mode 100644 articles/boundary-conditions-on-non-planar-boundaries/examples/traction.py diff --git a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md index 32e0e87..13635e6 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md +++ b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md @@ -70,7 +70,7 @@ directions — free *everything*, not free slip. Free slip keeps the tangential half of that and replaces the normal half with the constraint. How you do the replacing is the choice. -## Three ways, in order +## Four ways, in order ### A direct penalty @@ -134,6 +134,46 @@ arithmetic — measured below, it leaks a few parts in a thousand on a mesh of the resolution people actually run, and the leak falls with the mesh rather than with the machine. +### A constraint equation, with a multiplier + +The two above add a *term* to an equation. This one adds an *equation*. + +Carry a scalar field $h$ on the boundary and require, as a row of the system in +its own right, + +$$ +\int_{\partial\Omega} (\mathbf{u}\cdot\hat{\mathbf{n}} - g)\, q +\; \mathrm{d}S = 0 \quad \text{for all } q , +$$ + +with $h$ entering the momentum row as the traction $h\hat{\mathbf{n}}$ that holds +the constraint. It is a Lagrange multiplier, and the system becomes a larger +saddle point: velocity, pressure, and now $h$. + +Nothing is being traded here. The constraint row is exact, so unlike a penalty +there is no parameter whose size decides how well it holds. Two practical +things do have to be dealt with, and they are where the approximation enters: + +- The multiplier is only defined on the boundary, so its interior degrees of + freedom are singular. Underworld screens them with a small $\varepsilon$ + (default $10^{-6}$), which is the one place the constraint is relaxed. +- The $[p, h]$ Schur complement is poorly conditioned on its own, so an + augmented-Lagrangian term $r(\mathbf{u}\cdot\hat{\mathbf{n}} - g)\hat{\mathbf{n}}$ + is added to the momentum row. This conditions the block **without biasing the + multiplier**, because the $h$ row still carries the exact constraint — so + unlike a penalty, the accuracy does not depend on $r$. + +```python +stokes = uw.systems.Stokes_Constrained(mesh, velocityField=v, pressureField=p) +h = stokes.add_constraint_bc(0.0, "Upper") +stokes.solve() +``` + +And the reason to care about it beyond the constraint: **at convergence, $h$ on +the boundary is the normal traction.** It is not recovered from the velocity +field afterwards — it is an unknown the solve returned, available through +`multiplier` and, divided by $\Delta\rho g$, through `topography`. + ### Rotating the degrees of freedom Stop asking for the constraint and impose it. At each constrained node, change @@ -260,17 +300,21 @@ and divided by the flow speed: the fraction of the flow going through a boundary nothing should pass through. Penalty at $10^4$, Nitsche at $\gamma = 10$. -| cell size | direct penalty | Nitsche | rotated | -|---|---|---|---| -| 0.150 | 4.5 × 10⁻³ | 4.6 × 10⁻³ | 7.3 × 10⁻¹¹ | -| 0.100 | 2.5 × 10⁻³ | 2.2 × 10⁻³ | 6.5 × 10⁻¹¹ | -| 0.075 | 8.5 × 10⁻⁴ | 1.7 × 10⁻³ | 1.0 × 10⁻¹⁰ | -| 0.050 | 9.5 × 10⁻⁴ | 5.8 × 10⁻⁴ | 8.4 × 10⁻¹¹ | +| cell size | direct penalty | Nitsche | multiplier | rotated | +|---|---|---|---|---| +| 0.150 | 4.5 × 10⁻³ | 4.6 × 10⁻³ | 8.3 × 10⁻⁴ | 7.3 × 10⁻¹¹ | +| 0.100 | 2.5 × 10⁻³ | 2.2 × 10⁻³ | 1.6 × 10⁻⁴ | 6.5 × 10⁻¹¹ | +| 0.075 | 8.5 × 10⁻⁴ | 1.7 × 10⁻³ | 5.6 × 10⁻⁵ | 1.0 × 10⁻¹⁰ | +| 0.050 | 9.5 × 10⁻⁴ | 5.8 × 10⁻⁴ | 1.2 × 10⁻⁵ | 8.4 × 10⁻¹¹ | The refinement is what separates them, and a single resolution would not have. -Both weak forms leak parts in a thousand and improve as the mesh improves. The -rotated constraint does not move: it sits at the solver's floor at every -resolution, because the mesh has nothing to do with it. +The two weak forms leak parts in a thousand and improve roughly as $h^{1.9}$ — +the rate consistency buys. The multiplier starts an order of magnitude better +and falls much faster, near $h^{3.9}$, because its only approximation is the +screening of the interior multiplier degrees of freedom rather than the +enforcement itself. The rotated constraint does not move at all: it sits at the +solver's floor at every resolution, because the mesh has nothing to do with +it. The control matters more than the result. With the outer boundary left free the same measurement reads **0.98** — nearly all the boundary flow is normal — so diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/.meshes/uw_annulus_ro1.0_ri0.5_csize0.05.msh.h5 b/articles/boundary-conditions-on-non-planar-boundaries/examples/.meshes/uw_annulus_ro1.0_ri0.5_csize0.05.msh.h5 index 21bdce5c5bb78ec58ee7054afe40d4c2d936e194..eb554b1f382b5d7236acbd10ab780a94d63d1c40 100644 GIT binary patch delta 168 zcmexyK@&q%vltjMCL6}&PCjs5V)F#%J5kI)!Oi(`S1Uk_>Gg8V8#n*B zzN8Yw+de;#G4~mm#c#m`WdT*>)`G>}&t~2#57q^gow(g%53`Olg8Sn?^Be;NSHqA+ b#SyH{zlcSaAI!>cWQi37v(8UqS;-3kT?arp delta 168 zcmexyKGg8V8#n*B zzN8Yw+de;#G4~mm#c#m`WdT*>)`G>}&t~2#57q^gow(g%53`Olg8Sn?^Be;NSHqA+ b#SyH{zlcSaAI!>cWQi37v(8UqS;-3kfd)q* diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/.meshes/uw_annulus_ro1.0_ri0.5_csize0.075.msh.h5 b/articles/boundary-conditions-on-non-planar-boundaries/examples/.meshes/uw_annulus_ro1.0_ri0.5_csize0.075.msh.h5 index eb680921b3bf3a93a17a3b8a61254bb7acda573d..f0ca25cc130c823a737c58018fe25f111d2507c1 100644 GIT binary patch delta 160 zcmbR9fOF0R&J8Tg>{G;BvltjMCSPREoxb5Dqr~P3%x4UkrigEjGudYYqU8$hY{C3> zcDX=$`#fnz2;*J{W0(U_WU^f0r0pk8GA}UB~vltjMCSPREoxb5Dqr~P3%x4Uk&hl@LGudYYqU8$hY{C3> zcDX=$`#fnz2;*J{W0(U_WU^f0r0pk8GA=VUXvltjMCSPREom@~Mxp^Y?VI&vKSaLCSPREom@~Mxp^Y`SquyrlP@ynPX6F0v3UaX9C2o#;AVa4yOLlAziFx@3rN`{ zV~5Q<{9M$)qW?=;!HmiLrW-d~%sC?s5(RQ5Zl19tgb%Fb{V`Wzxaf&nUj)FS`(K+P bL_7Z5@WPaAkKkq00qX=Avt2@lQGpu(69ztg delta 168 zcmdmSj%CL=mJKY-?D~INvKSaLCSPRCo&3R1V)F#%IpWMf!Oi;8cO}6Le$!M*7Lc+@ z#txfz__?ToMgNzyf*F(fO*d|~m~%!NBnsqA+&p7P2p?F<`(v)caM2UDz6gLt_rEqp bh<5z9;e{#L9>L3~1J(&LX1jz8qXIVoK5Ruk diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/leak.py b/articles/boundary-conditions-on-non-planar-boundaries/examples/leak.py index 6d66e7f..2a0ed3f 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/examples/leak.py +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/leak.py @@ -45,7 +45,9 @@ def build(mode, cell=CELL, penalty=1.0e4, gamma=GAMMA): v = uw.discretisation.MeshVariable("U", mesh, mesh.dim, degree=2) p = uw.discretisation.MeshVariable("P", mesh, 1, degree=1) - stokes = uw.systems.Stokes(mesh, velocityField=v, pressureField=p) + solver_class = (uw.systems.Stokes_Constrained if mode == "constraint" + else uw.systems.Stokes) + stokes = solver_class(mesh, velocityField=v, pressureField=p) stokes.constitutive_model = uw.constitutive_models.ViscousFlowModel stokes.constitutive_model.Parameters.shear_viscosity_0 = 1.0 @@ -65,7 +67,13 @@ def build(mode, cell=CELL, penalty=1.0e4, gamma=GAMMA): for boundary in ("Upper",): if mode == "free": continue # the control: outer boundary left natural - if mode == "rotated": + if mode == "constraint": + # A multiplier field h coupled into the saddle-point system, so the + # constraint is a ROW of the system rather than a term added to + # one. At convergence h on the boundary is the normal traction, + # which is why this one is also a way of getting the stress. + stokes.add_constraint_bc(0.0, boundary) + elif mode == "rotated": stokes.add_rotated_freeslip_bc(0.0, boundary) elif mode == "nitsche": stokes.add_nitsche_bc(0.0, boundary, gamma=gamma, theta=1) @@ -112,7 +120,7 @@ def leaks(mesh, v): def sweep(cells=(0.15, 0.10, 0.075, 0.05), - modes=("penalty", "nitsche", "rotated")): + modes=("penalty", "nitsche", "constraint", "rotated")): """Does the leak fall with the mesh, or is it already at round-off? A weakly imposed constraint is satisfied to the accuracy of the @@ -169,7 +177,8 @@ def parameter_sweep(): elif sys.argv[1:2] == ["params"]: parameter_sweep() else: - modes = sys.argv[1:] or ["free", "nitsche", "rotated"] + modes = sys.argv[1:] or ["free", "penalty", "nitsche", + "constraint", "rotated"] print("%-9s %10s %10s %8s" % ("mode", "leak/|u|", "|u|max", "nodes")) for mode in modes: mesh, stokes, v = build(mode) diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/traction.py b/articles/boundary-conditions-on-non-planar-boundaries/examples/traction.py new file mode 100644 index 0000000..e7204f0 --- /dev/null +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/traction.py @@ -0,0 +1,159 @@ +"""Do the two exact methods agree on the surface traction? + +The leak measures the constraint. This measures the thing the constraint is +usually wanted FOR: the normal stress on the boundary, which is dynamic +topography once divided by the buoyancy scale. + +Two of the four methods return it as an unknown rather than recovering it: + + * the multiplier method, where h on the boundary IS the traction holding the + constraint; + * the rotated constraint, where the reaction of the strong constraint is + sigma_nn. + +They are computed by completely different routes, so agreeing is evidence and +disagreeing localises a defect. The weak methods have no comparable object -- +their traction has to be recovered from a velocity field that does not satisfy +the constraint exactly, which is the argument the note makes. + +STATUS: the vertex comparison converges and the midpoint comparison does NOT, +which is backwards. Measured agreement in the degree-4 amplitude: + + cell vertex midpoint + 0.150 5.5e-02 3.6e-03 + 0.100 2.0e-02 1.2e-02 + 0.075 1.5e-02 3.8e-02 + 0.050 8.5e-03 6.9e-02 + +Vertices converge at about h^1.4. Midpoints get steadily WORSE, which cannot be +right: the solver documents midpoint values of sigma_nn as superconvergent on a +curved boundary and vertex values as carrying the O(h) facet error, so if +anything the two columns should be the other way round. + +The likely fault is in this script rather than in either method. The multiplier +carries screened INTERIOR degrees of freedom as well as its boundary trace, and +they are selected here by a radius band that narrows as the mesh refines +(`rad > 1 - 0.3 * cell`). A band that changes with resolution can quietly change +WHICH nodes it admits, and the node counts do drift apart between the two +methods under refinement -- 88 against 84 at cell 0.075 -- which is how the +earlier pairwise version failed outright. + +So this is not yet evidence of anything and its numbers are deliberately not +quoted in the note. What is needed is to select the multiplier's boundary trace +from the mesh boundary LABEL rather than by geometry. + +Run against underworld3 `development` at commit `0addec15`. +""" +import numpy as np +import sympy + +import underworld3 as uw + +import leak + + +def split_trace(coords, values): + """Order a boundary trace by angle, and split P2 VERTEX nodes from EDGE + MIDPOINTS. + + They are not interchangeable on a curved boundary. The solver's own warning: + vertex values of sigma_nn converge only slowly, because the vertex basis has + zero surface mean and the vertex reaction carries the O(h) facet-geometry + error; midpoint values are superconvergent (underworld3#414). + + They are also easy to tell apart geometrically. A vertex of the annulus mesh + sits exactly on the circle; the midpoint of a facet sits on the chord, which + is inside it by the sagitta. + """ + rad = np.linalg.norm(coords, axis=1) + on_circle = np.abs(rad - 1.0) < 1.0e-9 + out = {} + for name, m in (("vertex", on_circle), ("midpoint", ~on_circle)): + ang = np.arctan2(coords[m, 1], coords[m, 0]) + order = np.argsort(ang) + out[name] = (ang[order], np.asarray(values)[m][order]) + return out + + +def trace_of(field): + """(coords, values) from either a MeshVariable or an already-extracted pair.""" + if isinstance(field, tuple): + return field[0], np.squeeze(np.asarray(field[1])) + return field.coords, np.squeeze(np.asarray(field.array)) + + +def run(mode, cell=0.075): + mesh, stokes, v = leak.build(mode, cell=cell) + stokes.solve() + assert leak.converged(stokes), "%s did not converge" % mode + if mode == "constraint": + c, val = trace_of(stokes.multiplier("Upper")) + # The multiplier also has (screened) interior degrees of freedom, so + # take its boundary trace. The tolerance has to scale with the mesh: + # a facet midpoint sits inside the circle by the sagitta, about h^2/8, + # which at h = 0.15 is 2.8e-3 -- a fixed 1e-3 window silently drops + # every midpoint at coarse resolution and leaves an empty trace. + rad = np.linalg.norm(c, axis=1) + keep = rad > 1.0 - 0.3 * cell + return split_trace(c[keep], val[keep]) + if mode == "rotated": + c, val = trace_of(stokes.boundary_normal_traction("Upper")) + return split_trace(c, val) + raise ValueError(mode) + + +def harmonic(angles, values, degree=4): + """Amplitude of the degree-n harmonic in a boundary trace. + + Comparing node by node is fragile -- the two methods do not return the same + node set, and the counts drift apart under refinement. The forcing here is a + single harmonic, so the traction is dominated by one too, and its amplitude + is both resolution-independent and the quantity dynamic topography is built + from. A least-squares projection onto cos(n.theta) and sin(n.theta) needs no + correspondence between the two traces at all. + """ + v = values - values.mean() + c = 2.0 * np.mean(v * np.cos(degree * angles)) + s = 2.0 * np.mean(v * np.sin(degree * angles)) + return float(np.hypot(c, s)) + + +def convergence(cells=(0.15, 0.10, 0.075, 0.05)): + """Do the two routes converge to the SAME traction? + + Agreeing once could be a coincidence of scale. Agreeing better as the mesh + refines is what says both are approximating one thing. + """ + print("| cell size | vertex | edge midpoint |") + print("|---|---|---|") + for cell in cells: + mult = run("constraint", cell=cell) + rot = run("rotated", cell=cell) + row = [] + for kind in ("vertex", "midpoint"): + h1 = harmonic(*mult[kind]) + h2 = harmonic(*rot[kind]) + row.append("%.2e" % (abs(h1 - h2) / max(h1, h2))) + print("| %.3f | %s |" % (cell, " | ".join(row))) + + +if __name__ == "__main__": + import sys + if sys.argv[1:2] == ["converge"]: + convergence() + raise SystemExit + mult = run("constraint") + rot = run("rotated") + for kind in ("vertex", "midpoint"): + a1, t1 = mult[kind] + a2, t2 = rot[kind] + # Both are determined only up to a constant on an enclosed boundary, so + # the deviation is the part carrying the signal -- and the part dynamic + # topography is built from. + d1, d2 = t1 - t1.mean(), t2 - t2.mean() + print("%-9s multiplier n=%3d amp=%.3e | reaction n=%3d amp=%.3e" + % (kind, len(d1), np.abs(d1).max(), len(d2), np.abs(d2).max())) + if len(d1) == len(d2) and np.allclose(a1, a2, atol=1e-9): + scale = max(np.abs(d1).max(), np.abs(d2).max()) + print("%-9s agreement: max|diff| / max|dev| = %.3e" + % ("", np.abs(d1 - d2).max() / scale)) From 4f82439e5436fd67ce289afcce2f52ee6aff4f86 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Wed, 19 Aug 2026 13:34:58 +1000 Subject: [PATCH 06/33] Record the stress-test plan: Zhong 2008 as the oracle The traction comparison in traction.py has no oracle -- it measures two methods against each other, so its diverging midpoint column cannot be interpreted. uw.analytic.Zhong2008 fixes that: .response() returns the surface topography kernel directly, 0.4191904 for the default degree-2 case, referenced to Zhong et al. (2008) GGG 9 Q10017 after Hager and O'Connell. The quantity it publishes is the one the note cares about rather than a velocity field to infer a stress from. One consequence worth stating before anyone starts: Zhong is SPHERICAL. The harmonic degree is a spherical-harmonic degree and the solution carries a planet radius, so the matching model is a 3-D shell rather than the 2-D annulus the leak tables use. That sets the cost and is why this is separate work rather than another column. Also records why SolCx stays useful even though it cannot test a curved boundary: on a box all four treatments reduce to the same component constraint, but it is still a fair test of whether a penalty or Nitsche form constrains the normal degrees of freedom it is handed, with the geometry factored out and an exact answer to hand. Underworld development team with AI support from Claude Code --- .../examples/README-stress-test.md | 61 +++++++++++++++++++ 1 file changed, 61 insertions(+) create mode 100644 articles/boundary-conditions-on-non-planar-boundaries/examples/README-stress-test.md diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/README-stress-test.md b/articles/boundary-conditions-on-non-planar-boundaries/examples/README-stress-test.md new file mode 100644 index 0000000..ef21242 --- /dev/null +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/README-stress-test.md @@ -0,0 +1,61 @@ +# The stress test: plan and status + +The leak tables measure the *constraint*. This is the test that measures what +the constraint is wanted for — the surface normal stress, and through it +dynamic topography. + +## Why the current comparison is not enough + +`traction.py` compares two methods that both return the traction as an unknown: +the multiplier field `h`, and the reaction of the rotated constraint. They agree +to about 1% at one resolution, but there is **no oracle**, so when the vertex and +midpoint columns behave differently under refinement nothing says which is +right. Its numbers are deliberately not quoted in the note. + +## The oracle: Zhong et al. (2008) + +`uw.analytic.Zhong2008` is a propagator-matrix solution for a delta-function +load in a shell, and `.response()` returns the surface topography kernel +directly. For the default case — degree 2, load at r = 0.775, isoviscous, +r_inner = 0.55 — it gives + + surface_topography 0.4191904156575601 + cmb_topography 0.7705825500322292 + surface_geoid 0.0257906289002894 + surface_characteristic_velocity -0.0100641246068943 + +with self-gravity variants alongside, referenced to Zhong et al. (2008), +GGG 9, Q10017, doi:10.1029/2008GC002048, after Hager & O'Connell (1981). + +This is the right test because the quantity it publishes is the one the note +cares about, rather than a velocity field from which a stress must be inferred. + +**It is spherical.** The harmonic degree is a spherical-harmonic degree and the +solution carries a planet radius, so the matching model is a 3-D spherical +shell, not the 2-D annulus the leak tables use. That sets the cost, and it is +the reason this is a separate piece of work rather than another column in an +existing table. + +## What has to be built + +1. A spherical shell, free slip on both radii, with a delta-function density + load at r = 0.775 in a single spherical harmonic of degree 2. +2. The same model under each boundary treatment that a shell supports. +3. Surface topography from `sigma_nn`, against 0.41919. + +Two things to watch, both already known: + +- On a curved boundary, **vertex** values of `sigma_nn` carry the O(h) + facet-geometry error and **edge-midpoint** values are superconvergent + (underworld3#414). A pointwise comparison must say which it is using. +- On an enclosed domain the multiplier and the traction are each determined + only up to a constant, so the comparison is of the *deviation*, which is what + topography is anyway. + +## The supplementary check: SolCx + +SolCx is a Cartesian box, so it cannot test anything about curved boundaries and +all four treatments degenerate to the same component constraint there. It is +still a fair test of whether a **penalty or Nitsche form constrains the normal +degrees of freedom it is given**, with the geometry factored out and an exact +answer available. Worth having for that alone. From 6db19f8510fe40d3445aab8ae925d7ab754075a6 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Wed, 19 Aug 2026 13:48:22 +1000 Subject: [PATCH 07/33] Record why SolCx is the other half of the stress test, not a lesser one Checked what the analytic suites actually vary. assess's cylindrical and spherical solutions (Kramer et al. 2021) take nu as a SCALAR -- isoviscous. Zhong2008 layers viscosity by radius through viscosity_interfaces, also not laterally. SolCx has the lateral jump and a Cartesian box. So no analytic solution has both a curved boundary and a lateral viscosity variation, and the test splits in two rather than one being a weaker version of the other: Kramer or Zhong for the geometry with the rheology trivial, SolCx for the rheology with the geometry trivial. Lateral contrast is the condition under which these methods are known to give trouble and it is the only one of the two for which an oracle exists at all. Also noted that a shell with laterally varying viscosity has no exact solution, so that case needs a different kind of evidence -- convergence or agreement between methods -- and the note should say which rather than implying a measurement it cannot make. Underworld development team with AI support from Claude Code --- .../examples/README-stress-test.md | 34 +++++++++++++++---- 1 file changed, 28 insertions(+), 6 deletions(-) diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/README-stress-test.md b/articles/boundary-conditions-on-non-planar-boundaries/examples/README-stress-test.md index ef21242..3f6dcfe 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/examples/README-stress-test.md +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/README-stress-test.md @@ -52,10 +52,32 @@ Two things to watch, both already known: only up to a constant, so the comparison is of the *deviation*, which is what topography is anyway. -## The supplementary check: SolCx +## SolCx is not the weaker test, it is the other one -SolCx is a Cartesian box, so it cannot test anything about curved boundaries and -all four treatments degenerate to the same component constraint there. It is -still a fair test of whether a **penalty or Nitsche form constrains the normal -degrees of freedom it is given**, with the geometry factored out and an exact -answer available. Worth having for that alone. +No analytic solution has both a curved boundary and a laterally varying +viscosity, so no single test covers the case that actually hurts. + +| solution | curved boundary | viscosity | +|---|---|---| +| `assess` cylindrical / spherical (Kramer et al. 2021) | yes | **constant** — `nu` is a scalar | +| `Zhong2008` | yes | radial layers only (`viscosity_interfaces`) | +| SolCx | no, Cartesian box | **lateral jump** | + +That splits the work in two, and both halves are needed: + +- **Geometry, with the rheology trivial.** Kramer or Zhong. Tests whether the + treatment copes with a boundary whose normal turns, which is what this note + is about. +- **Rheology, with the geometry trivial.** SolCx. On a box every treatment + reduces to the same component constraint, so nothing here is about normals — + but a lateral viscosity jump is where a weakly imposed constraint is most + likely to fail to hold the normal degrees of freedom it was given, and there + is an exact answer to check against. + +The second is not a consolation prize. Lateral viscosity contrast is the +condition under which these methods are known to give trouble, and it is the +only one of the two where an oracle exists for it at all. + +A shell with a laterally varying viscosity has no exact solution, so if the +note wants that case it needs a different kind of evidence — self-consistency +between methods, or convergence under refinement — and it should say which. From 95facb30e80034d3d6f9495fed16e26260f69f86 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Wed, 19 Aug 2026 15:13:03 +1000 Subject: [PATCH 08/33] Measure the surface stress against an exact answer, and correct the note The leak tables measure the constraint. This measures what the constraint is wanted for, against `uw.analytic.CylindricalStokes` (Kramer et al. 2021), whose `assess` package publishes the radial stress rather than leaving it to be recovered. On the outer boundary the exact answer is a single harmonic, 0.1506696 cos 2t, fitted to a residual of 1e-16, so the metric is one amplitude. Three findings, two of which correct what the note said. The direct penalty written against the FACET normal does not converge. Its leak falls to 1e-5 while the velocity is 60% wrong and the surface stress 26% wrong, and refining from cell 0.15 to 0.035 does not move either. The same coefficient against the measure-weighted node normal converges cleanly. This is the classic over-constraint the consistent normal was introduced to avoid, and it means the note's penalty column was measuring the onset of locking: at a coefficient of 1e3 that penalty leaks 3e-2 and gets the stress right to 2e-3, and at 1e8 it leaks 1e-7 and is 26% wrong. Both penalty columns are now tabulated. Once the constraint is imposed against the node normal, which treatment imposed it stops mattering to the recovered stress -- rotated, multiplier, Nitsche and penalty all read 6.3e-3 at cell 0.075. The recovery sets the floor, not the boundary condition, so the note's reasoning that a weakly imposed constraint poisons the traction was too strong. What the rotated constraint buys is the reaction, which is three times better on the same solve and differentiates nothing. The multiplier is not quite the whole traction: the momentum row carries h + r(u.n - g), and with the default r = 1e4.mu the missing share is a few per cent of the surface stress. underworld3#607. Also measured: the SolCx half, where Nitsche at gamma = 10 gets the surface stress 277% wrong at a viscosity contrast of 1e6 while holding the constraint to 1e-7, and where the rotated constraint disagrees with a component condition at the corner they share (underworld3#608). Behr (2004) is read and cited for what it says: the consistent normal is standard and preferred for conservation, and it does not on its own guarantee a good discrete slip condition. The derivation in the "Which normal" section is credited to Engelman, Sani and Gresho (1982) rather than presented as ours. `traction.py` is removed -- `stress.py` supersedes it, with an oracle and with the boundary trace selected by label rather than by a radius band. The mesh cache the examples write beside themselves is untracked and gitignored. 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It is +# regenerated by running the example, and one note's examples committed 650 kB +# of it before this line existed. +.meshes/ diff --git a/WRITING-PLAN.md b/WRITING-PLAN.md index f23c95f..1c32c3b 100644 --- a/WRITING-PLAN.md +++ b/WRITING-PLAN.md @@ -220,9 +220,9 @@ beginning of a solvers paper, but neither is being written to fit one. ### R1. Boundary conditions on non-planar boundaries -Status: not started. **One post**, not a series. Rescoped 2026-08-17: rotated -boundary conditions are the answer, but the *question* is the better frame, and -it is the one a reader arrives with. +Status: drafted as UWTN 2026-016, in review on PR #26. **One post**, not a +series. Rescoped 2026-08-17: rotated boundary conditions are the answer, but the +*question* is the better frame, and it is the one a reader arrives with. On a box, "no flow through this wall" is a component of the velocity and you constrain it. On an annulus, a sphere, a boundary with topography, or any mesh @@ -451,6 +451,6 @@ Listed as candidates, not commitments. | 6 | F5 The comparison | Last, with Thyagarajulu's benchmarks as its evidence | | 7 | S1, S2 free surface | After the discussion about how to split it | | — | C1 Launching from any repository | Standalone; the capability is live and undocumented | -| — | R1 BCs on non-planar boundaries | Standalone; write whenever it suits. Much of the evidence exists — see the rescoped entry | +| — | R1 BCs on non-planar boundaries | Drafted as UWTN 2026-016, in review on PR #26 | | ✓ | G1 Setting up FMG | Published 2026-08-17 (UWTN 2026-014) | | 8 | #1 Release announcement | Written last; links to everything | diff --git a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md index 13635e6..cae0f65 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md +++ b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md @@ -26,7 +26,7 @@ exports: output: boundary-conditions-on-non-planar-boundaries.pdf article_id: UWTN 2026-016 article_version: 1.0.0 - software_version: underworld3 development @ 0addec15 + software_version: underworld3 development @ 8b7c8b9e --- A free surface moves under the traction it carries. That makes the wall-normal stress the quantity driving the model rather than something read off at the @@ -236,25 +236,40 @@ survives in codes that could do this instead. ## Which normal -A question with a less obvious answer than it looks. The boundary of a +A question with a less obvious answer than it looks, and the measurements +further down say it is the most consequential choice in the note. The boundary of a discretised domain is a set of straight facets, and the assembled constraint is an integral over those facets. The node normal consistent with that integral is the average of the adjacent facet normals **weighted by facet measure** — not -the normal of the smooth surface the mesh approximates. +the normal of the smooth surface the mesh approximates, and not the facet +normal on its own. + +**This is the consistent normal of @10.1002/fld.1650020302**, and we should say +so plainly: we re-derived it from the assembled boundary integral, and Engelman, +Sani and Gresho derived it in 1982 from global conservation of mass, which is +the same object arrived at from the other side. Their paper is about exactly +this — how to impose a normal or tangential condition on a boundary that does +not line up with the coordinate directions — and the recommendation is the one +here. Using the analytic normal is exact for the geometry and therefore inconsistent with the discretisation, which is the wrong way round: the solver is not solving -on the sphere, it is solving on the polyhedron. In parallel the same argument -has a sharper edge, because a normal accumulated rank-locally is wrong at a -partition boundary, where a node's facets are split across ranks and no rank -sees them all. - -:::{note} Worth checking against the literature -@10.1002/fld.663 addresses this question directly for slip conditions on curved -boundaries. We have not read it against our own derivation. If it reaches the -same measure-weighted normal, this section should say so and cite it as the -source rather than presenting the result as ours. -::: +on the sphere, it is solving on the polyhedron. Using the facet normal is +consistent with each facet separately and over-constrains the nodes between +them, which is the failure measured in "A constraint that is satisfied, and +wrong" below. In parallel the same argument has a +sharper edge, because a normal accumulated rank-locally is wrong at a partition +boundary, where a node's facets are split across ranks and no rank sees them +all. + +The consistent normal is not the end of the matter. @10.1002/fld.663 takes it as +the starting point — "preferred from the point of view of conservation" — and +reports that in sloshing problems it still does not guarantee a good discrete +slip condition, with non-physical recirculation appearing at curved walls; the +remedies offered there are the Navier slip condition and a "BC-free" boundary. +We have read the abstract rather than the paper, and have not looked for that +recirculation in our own cases. It is the obvious thing to test next for anyone +running free slip on a strongly curved wall. ## The option this note leaves out @@ -272,24 +287,27 @@ cooperating, and its structural price is what buys the generality. ## When the choice matters -Here is the awkward part. Solve a convection model with any of the three and -the velocity field is the same to plotting accuracy. A leak of order $10^{-3}$ -in $\mathbf{u}\cdot\hat{\mathbf{n}}$ is invisible to anything that consumes the -velocity, and consuming the velocity is most of what a model does. If that is -your situation, use the simplest thing that works and stop reading here. +Here is the awkward part. Solve a convection model with any of these and the +velocity field is the same to plotting accuracy, as long as each is set up +properly. A leak of order $10^{-3}$ in $\mathbf{u}\cdot\hat{\mathbf{n}}$ is +invisible to anything that consumes the velocity, and consuming the velocity is +most of what a model does. If that is your situation, use the simplest thing +that works — and read the section on which normal before you do, because that +is the choice that can spoil the velocity as well. The difference appears when the wall-normal traction is the answer rather than a by-product: dynamic topography, a plate-boundary force balance, anything -compared against a geoid or a gravity field. Then the three separate, and they -separate for a structural reason rather than by a matter of degree. +compared against a geoid or a gravity field. That is what the two sections after +the leak measure, against an exact surface stress, and the answer is not the one +we expected. The treatments that impose the constraint properly all recover the +same surface stress at a given mesh, because the recovery sets the floor rather +than the boundary condition; what the rotated constraint buys is a route to the +traction that does not go through a recovery at all, and it is three times +better on the same solve. -Under a penalty or a Nitsche condition the constraint is approximate, so a -traction recovered from the solution inherits the approximation — you are -differentiating a field that was never made to satisfy the condition exactly. Under the rotated constraint the reaction *is* $\sigma_{nn}$: it is the multiplier the solve has already computed, and it comes out of -`boundary_normal_traction` without a recovery step or an augmented-Lagrangian -splitting. +`boundary_normal_traction` without differentiating the answer. ### The leak, measured @@ -300,67 +318,285 @@ and divided by the flow speed: the fraction of the flow going through a boundary nothing should pass through. Penalty at $10^4$, Nitsche at $\gamma = 10$. -| cell size | direct penalty | Nitsche | multiplier | rotated | -|---|---|---|---|---| -| 0.150 | 4.5 × 10⁻³ | 4.6 × 10⁻³ | 8.3 × 10⁻⁴ | 7.3 × 10⁻¹¹ | -| 0.100 | 2.5 × 10⁻³ | 2.2 × 10⁻³ | 1.6 × 10⁻⁴ | 6.5 × 10⁻¹¹ | -| 0.075 | 8.5 × 10⁻⁴ | 1.7 × 10⁻³ | 5.6 × 10⁻⁵ | 1.0 × 10⁻¹⁰ | -| 0.050 | 9.5 × 10⁻⁴ | 5.8 × 10⁻⁴ | 1.2 × 10⁻⁵ | 8.4 × 10⁻¹¹ | - -The refinement is what separates them, and a single resolution would not have. -The two weak forms leak parts in a thousand and improve roughly as $h^{1.9}$ — -the rate consistency buys. The multiplier starts an order of magnitude better -and falls much faster, near $h^{3.9}$, because its only approximation is the -screening of the interior multiplier degrees of freedom rather than the -enforcement itself. The rotated constraint does not move at all: it sits at the -solver's floor at every resolution, because the mesh has nothing to do with -it. +The penalty appears twice, because the normal it is written against turns out +to matter more than the method it is written into. One column uses the +quadrature-point facet normal `mesh.Gamma`, which is the documented form; the +other uses the measure-weighted node normal `mesh.boundary_normal`, which is +what Nitsche and the rotated constraint use by default. + +| cell size | penalty, facet normal | penalty, node normal | Nitsche | multiplier | rotated | +|---|---|---|---|---|---| +| 0.150 | 4.5 × 10⁻³ | 3.3 × 10⁻³ | 4.6 × 10⁻³ | 8.3 × 10⁻⁴ | 7.3 × 10⁻¹¹ | +| 0.100 | 2.5 × 10⁻³ | 2.7 × 10⁻³ | 2.2 × 10⁻³ | 1.6 × 10⁻⁴ | 6.5 × 10⁻¹¹ | +| 0.075 | 8.5 × 10⁻⁴ | 2.6 × 10⁻³ | 1.7 × 10⁻³ | 5.6 × 10⁻⁵ | 1.0 × 10⁻¹⁰ | +| 0.050 | 9.5 × 10⁻⁴ | 2.6 × 10⁻³ | 5.8 × 10⁻⁴ | 1.2 × 10⁻⁵ | 8.4 × 10⁻¹¹ | + +Nitsche leaks parts in a thousand and improves roughly as $h^{1.9}$ — the rate +consistency buys. The multiplier starts an order of magnitude better and falls +much faster, near $h^{3.9}$, because its only approximation is the screening of +the interior multiplier degrees of freedom rather than the enforcement itself. +The rotated constraint does not move at all: it sits at the solver's floor at +every resolution, because the mesh has nothing to do with it. The penalty +against the node normal does not improve with the mesh either, and for the +opposite reason: at a fixed coefficient its leak is set by the coefficient. The control matters more than the result. With the outer boundary left free the same measurement reads **0.98** — nearly all the boundary flow is normal — so the metric can see a leak when there is one. +The first column is the one to be careful with. Its leak falls with the mesh, +which reads as the method working, and it is not: the surface stress measured +further down says that solve is 60% wrong in the velocity and 26% wrong in the +stress, and refining it does not help. The leak is small because the boundary +is being frozen. A metric that only asks whether the constraint is satisfied +cannot tell those apart. + ### What each parameter buys The two weak methods look alike in that table. They are not alike, and their own parameters are what tells them apart. -| penalty | leak | | $\gamma$ (Nitsche) | leak | -|---|---|---|---|---| -| 10² | 2.6 × 10⁻¹ | | 1 | diverged | -| 10³ | 2.3 × 10⁻² | | 10 | 1.7 × 10⁻³ | -| 10⁴ | 8.5 × 10⁻⁴ | | 100 | 2.7 × 10⁻⁴ | -| 10⁵ | 9.6 × 10⁻⁴ | | 1000 | 3.0 × 10⁻⁵ | -| 10⁶ | diverged | | 10⁴ and above | diverged | - -**Both have a wall, and neither escapes tuning.** The direct penalty improves in -proportion to how hard it pushes until the conditioning catches up: no gain -from $10^4$ to $10^5$, and failure at $10^6$. Nitsche improves further and then -fails outright — at $\gamma = 10^4$ the line search stops converging, and the -figures above that are from solves that did not finish. - -What differs is how far down each gets before it fails. The penalty bottoms out -near $10^{-3}$; Nitsche reaches $3 \times 10^{-5}$, better by more than an order -of magnitude. That is what consistency buys — a smaller floor, not an -unbounded parameter. - -Nitsche also fails *below* $\gamma = 1$, and for the opposite reason: the form -is no longer coercive, and no amount of tuning recovers it. So its usable range -is bounded at both ends, with the stability threshold underneath and the -conditioning above. The virtue of $\gamma = 10$ is that it sits in the middle of -that window on any mesh, because $\gamma$ is dimensionless and the term it -scales already carries $\mu / h$. The penalty coefficient carries no such -scaling, which is why the value that works is a property of the problem rather -than a default. - -:::{note} What is measured here, and what is not -This measures the *constraint*. The claim that follows — that a traction -recovered from a constraint good to $10^{-3}$ inherits that error, while the -rotated reaction is exact because it is the multiplier itself — follows from -how each is computed, and the surface-stress comparison against a known answer -has not been run. Until it is, the two paragraphs above this section are -reasoning rather than measurement. -::: +| penalty, facet | leak | | penalty, node | leak | | $\gamma$ (Nitsche) | leak | +|---|---|---|---|---|---|---|---| +| 10² | 2.6 × 10⁻¹ | | 10² | 2.6 × 10⁻¹ | | 1 | diverged | +| 10³ | 2.3 × 10⁻² | | 10³ | 2.6 × 10⁻² | | 10 | 1.7 × 10⁻³ | +| 10⁴ | 8.5 × 10⁻⁴ | | 10⁴ | 2.6 × 10⁻³ | | 100 | 2.7 × 10⁻⁴ | +| 10⁵ | 9.6 × 10⁻⁴ | | 10⁵ | 3.0 × 10⁻⁴ | | 1000 | 3.0 × 10⁻⁵ | +| 10⁶ | diverged | | 10⁶ | 4.5 × 10⁻⁵ | | 10⁴ and above | diverged | + +Nitsche is bounded at both ends. Below $\gamma = 1$ the form is no longer +coercive and no amount of tuning recovers it; from $\gamma = 10^4$ in this +problem the line search stops converging. The virtue of $\gamma = 10$ is that it +sits in the middle of that window on any mesh, because $\gamma$ is dimensionless +and the term it scales already carries $\mu / h$. The penalty coefficient +carries no such scaling, which is why the value that works is a property of the +problem rather than a default — and, on the second half of this note's test, of +the local viscosity as well. + +The two penalty columns are the same method against different normals, and they +behave differently in a way the leak alone does not explain. Against the node +normal the leak keeps falling, a decade of coefficient for a decade of leak, all +the way to $10^6$. Against the facet normal it stops improving after $10^4$ and +the solve fails at $10^6$. That looks like the conditioning wall a penalty is +expected to have. It is not: what stalls is the leak, because by $10^4$ the +boundary is nearly frozen and there is little normal flow left to remove. The +stress section below measures the same runs against an exact answer, and they +are getting worse throughout. + +### The stress, measured + +The leak says how well each treatment holds the boundary. Whether the answer is +right is a different question, and it needs an exact answer to compare against. + +Kramer, Davies and Wilson [@10.5194/gmd-14-1899-2021] give exact Stokes +solutions in a cylindrical annulus, and their `assess` package publishes the +radial stress itself rather than leaving it to be recovered from a velocity +field. Underworld wraps it as `uw.analytic.CylindricalStokes`. The case here is +the smooth one: a density anomaly $(r/r_o)^k \cos n\theta$ with $n = 2$ and +$k = 3$, viscosity 1, free slip on both radii. On the outer boundary the exact +radial stress is a single harmonic, + +$$ +\sigma_{rr}(r_o, \theta) = 0.1506696\,\cos 2\theta , +$$ + +fitted to a residual of $10^{-16}$, so the whole of the surface stress is that +one amplitude and the metric is its relative error. + +The inner boundary carries the exact velocity as a Dirichlet condition instead +of a free-slip treatment of its own. The exact solution satisfies both, so the +problem is unchanged, and the treatment under test is then the only free-slip +condition in the model. + +There are two routes to the surface stress and the difference between them is +the point of the section: + +- **recovered** — project $\hat{\mathbf{n}}\cdot\boldsymbol{\sigma}\hat{\mathbf{n}}$ + out of the solved velocity and pressure. Every treatment can do this, and it + is the only route the weak ones have. +- **reaction** — `boundary_normal_traction` for the rotated constraint, and the + multiplier field for the constraint method. Not recovered from the solution: + an unknown the solve returned. + +Both are taken against the true radial direction, which is also the direction +the oracle publishes, so no treatment is scored against its own normal. The +reaction and the multiplier come back with the opposite sign to $\sigma_{rr}$ — +they are the traction holding the boundary rather than the traction the fluid +exerts, and `dynamic_topography` carries the sign back — so amplitudes are +compared unsigned. + +| cell size | penalty, facet | penalty, node | Nitsche | constraint | rotated | rotated, reaction | multiplier field | +|---|---|---|---|---|---|---|---| +| 0.150 | 1.5 × 10⁻¹ | 2.5 × 10⁻² | 5.9 × 10⁻² | 2.4 × 10⁻² | 2.4 × 10⁻² | 6.8 × 10⁻³ | 3.3 × 10⁻² | +| 0.100 | 1.3 × 10⁻¹ | 1.1 × 10⁻² | 2.4 × 10⁻² | 1.0 × 10⁻² | 1.0 × 10⁻² | 3.3 × 10⁻³ | 2.9 × 10⁻² | +| 0.075 | 1.1 × 10⁻¹ | 7.2 × 10⁻³ | 1.5 × 10⁻² | 6.3 × 10⁻³ | 6.2 × 10⁻³ | 2.1 × 10⁻³ | 2.5 × 10⁻² | +| 0.050 | 6.3 × 10⁻² | 3.6 × 10⁻³ | 6.3 × 10⁻³ | 2.7 × 10⁻³ | 2.7 × 10⁻³ | 1.1 × 10⁻³ | 1.5 × 10⁻³ | + +The first five columns are all the recovered stress, so they differ only by +which boundary condition produced the field. The last two are the reaction and +the multiplier on the same solves. Penalties at $10^4$, Nitsche at +$\gamma = 10$, which are the values the leak table used. With the outer +boundary left free the same measurement reads 1.0 — the surface stress is +gone — so the metric can see the condition being removed. + +**Once the constraint is imposed against the node normal, which treatment +imposed it stops mattering to the recovered stress.** At cell 0.075 the rotated +constraint gives 6.2 × 10⁻³, the multiplier 6.3 × 10⁻³, the penalty at $10^5$ +6.3 × 10⁻³ and Nitsche at $\gamma = 100$ 6.6 × 10⁻³. Those are the same number. +What sets it is the recovery — a projection of a stress differentiated out of a +piecewise-quadratic velocity — and not the boundary condition underneath. The +reasoning this note began with — that a traction recovered from an approximate +constraint inherits the approximation — is not what the measurement shows once +the constraint is written against the right normal. It shows a floor that all of +them share. + +**The reaction is about three times better than the recovery, on the same +solve.** 2.1 × 10⁻³ against 6.2 × 10⁻³ at cell 0.075, and it converges at the +same rate rather than a better one. It costs nothing: it is the residual the +solve has already assembled, and no field is differentiated to get it. + +**The parameter that was enough for the leak is not enough for the stress.** +Nitsche at $\gamma = 10$ leaks 1.2 × 10⁻³ in this problem and gets the stress +amplitude 1.5% wrong. At $\gamma = 100$ the leak improves by a factor of nearly +forty and the stress by a factor of two, to the recovery floor, where more +$\gamma$ buys nothing. Reading the leak alone would have said $\gamma = 10$ was +converged. + +### A constraint that is satisfied, and wrong + +The facet-normal penalty column does not converge. That is not a slow rate; the +error grows slightly as the mesh is refined, and the leak is excellent +throughout. + +| cell size | leak | velocity error | stress error | +|---|---|---|---| +| 0.150 | 8.9 × 10⁻⁶ | 0.60 | 0.21 | +| 0.100 | 8.2 × 10⁻⁶ | 0.61 | 0.24 | +| 0.075 | 1.1 × 10⁻⁵ | 0.61 | 0.25 | +| 0.050 | 1.9 × 10⁻⁵ | 0.60 | 0.26 | +| 0.035 | 2.3 × 10⁻⁵ | 0.59 | 0.26 | + +```{figure} figures/locking.png +:alt: Three annuli side by side on one colour scale from zero to 5.0e-3, blue for slow and red for fast, with the triangular mesh drawn over each. The left panel is the exact solution: two deep red patches of fast flow sit against the outer boundary on the left and right of the annulus, with a blue slow ring inside them. The middle panel is the same problem solved with a direct penalty against the facet normal: the red patches at the outer boundary are gone and the whole outer half is blue, the peak speed having fallen from 5.0e-3 to 3.8e-3, while a pale ring survives near the inner boundary. The right panel is the same penalty against the measure-weighted node normal and is indistinguishable from the exact panel, with a peak speed of 5.0e-3. + +The same problem, the same coefficient of $10^6$, the same colour scale, and the +only difference between the middle and right panels is which normal the penalty +is written against. Against the facet normal the flow along the outer boundary +is suppressed: the peak speed falls by a quarter and the two fast lobes at the +boundary are gone. Refining the mesh does not bring them back. +``` + +The coefficient is $10^6$ and the normal is `mesh.Gamma`, the facet normal at +the quadrature points. The same coefficient against the measure-weighted node +normal, on the same meshes, gives velocity errors of 1.0 × 10⁻², 2.4 × 10⁻³, +1.0 × 10⁻³ and 4.9 × 10⁻⁴ and stress errors of 2.4 × 10⁻², 6.3 × 10⁻³, +2.7 × 10⁻³ and 1.4 × 10⁻³. One difference, one line of code, and one method +converges while the other does not. + +The mechanism is old and well understood. Imposing +$\mathbf{u}\cdot\hat{\mathbf{n}} = 0$ facet by facet asks a node shared by two +facets to satisfy two different constraints, and two independent constraints on +a two-component velocity leave nothing. Push the coefficient up and the vertex +velocities go to zero: the flow is being asked to stay inside a polygon rather +than a circle, and the polygon is not a good enough approximation of the circle +for that particular question. The discrete limit is a different problem from +the smooth one, so refining the mesh does not approach the smooth answer. + +Two things follow for practice. A direct penalty is a perfectly good method, +but it has to be written against the node normal. And the leak is not a +sufficient check: at a coefficient of $10^3$ the facet-normal penalty leaks +3 × 10⁻² and gets the stress amplitude right to 1.9 × 10⁻³, while at $10^8$ it +leaks 1 × 10⁻⁷ and is 26% wrong. Over that range the constraint improves by five +orders of magnitude and the answer gets steadily worse. + +### The multiplier is not quite the whole traction + +The multiplier column above sits near 3 × 10⁻² for three resolutions and then +drops to 1.5 × 10⁻³, which is not a convergence rate. The cause is the +augmented-Lagrangian term. + +The momentum row carries both the multiplier and the augmentation, +$(h + r(\mathbf{u}\cdot\hat{\mathbf{n}} - g))\hat{\mathbf{n}}$, so the traction +holding the boundary is $h + r(\mathbf{u}\cdot\hat{\mathbf{n}} - g)$ and not +$h$ alone. The second term vanishes only where the constraint row is satisfied +exactly. Discretely it is satisfied to the solver's tolerance, and with the +default $r = 10^4\mu$ the residual $\mathbf{u}\cdot\hat{\mathbf{n}} \approx +5 \times 10^{-7}$ multiplies up to a few per cent of a surface traction of 0.15. + +Adding the augmentation share back gives, at cell 0.075, 1.7 × 10⁻³ instead of +2.5 × 10⁻², and at cell 0.100, 8.5 × 10⁻⁴ instead of 2.9 × 10⁻². Lowering $r$ to +$10^2\mu$ has the same effect without the correction — 2.6 × 10⁻³ — because it +shrinks the term rather than accounting for it. So the accuracy of the reported +multiplier does depend on $r$, which the method's own documentation says it does +not, and the note said so too. What does not depend on $r$ is the traction, once +both of its parts are added up. This is underworld3 issue #607. + +### The other half: a lateral viscosity contrast + +No exact solution has both a curved boundary and a laterally varying viscosity, +so the case where weak constraints are most often reported to give trouble is a +separate test with a trivial geometry. SolCx is that test: the unit box, free +slip on all four walls, viscosity 1 to the left of $x = 0.5$ and $\eta_B$ to the +right, and `uw.analytic.SolCx` publishes the exact surface stress. Three walls +carry the ordinary component condition and the treatment under test is on the +top wall alone. + +On a box every treatment here reduces to holding one velocity component, so +nothing in it is about normals. What it can say is whether a weak imposition +holds the traction it was given when the viscosity beside it jumps. + +Relative $l_2$ error of $\sigma_{zz}$ along the top wall, mean removed, at +32 × 32 elements. Each entry is the whole wall and then the wall with two +elements trimmed from each end; the corners are where the treatment under test +meets the side walls, and the difference between the two numbers is what +happens there. Nitsche at $\gamma = 10$, penalty at $10^4$. + +| $\eta_B/\eta_A$ | component Dirichlet | penalty | Nitsche | multiplier | rotated | +|---|---|---|---|---|---| +| 10 | 0.048 / 0.054 | 0.045 / 0.051 | 0.048 / 0.054 | 0.048 / 0.054 | 0.322 / 0.056 | +| 10² | 0.072 / 0.081 | 0.056 / 0.060 | 0.073 / 0.081 | 0.072 / 0.081 | 0.343 / 0.083 | +| 10³ | 0.075 / 0.084 | diverged | 0.076 / 0.085 | 0.075 / 0.084 | 0.345 / 0.086 | +| 10⁴ | 0.076 / 0.085 | diverged | 0.087 / 0.095 | 0.076 / 0.085 | 0.345 / 0.086 | +| 10⁶ | 0.076 / 0.085 | diverged | **2.77 / 2.14** | 0.075 / 0.084 | 0.345 / 0.086 | + +**Read the first column first.** The component Dirichlet condition is exact, +has no parameter, and its velocity error is 8.8 × 10⁻⁶ at a contrast of $10^6$. +It still reads 0.085. That number is the recovery's error and not a boundary +condition's: on the stiff half the recovered $\sigma_{zz}$ is a difference +between the pressure and $2\eta\,\partial_z u_z$ with $\eta = 10^6$, so a +relative velocity error of $10^{-5}$ arrives in the stress at the size of the +signal. No treatment can be called worse than the reference unless it is worse +than that, and at contrasts up to $10^4$ none of them is by more than about a +tenth. + +**Nitsche fails at $10^6$ and the leak does not show it.** At $\gamma = 10$ the +recovered surface stress is 2.77 — nearly three times the signal — while the +constraint is held to 3.2 × 10⁻⁴ on the soft half of the wall and +1.5 × 10⁻⁷ on the stiff half. Raising $\gamma$ fixes it: 0.16 at $\gamma = 100$ +and 0.079 at $\gamma = 1000$, which is the reference floor. So the stabilisation +parameter has to be raised with the viscosity contrast, and the quantity that +says whether it is high enough is the stress rather than the leak. This is the +same shape as the curved-boundary result, arrived at from the other side: the +constraint being satisfied is not evidence that the traction is right. + +The two exact treatments carry no such parameter and sit at the reference floor +at every contrast. + +**The penalty needs a coefficient matched to the local viscosity, and we could +not write one that solves.** A constant $10^4$ manages a contrast of $10^2$ and +then fails the line search. Multiplying the coefficient by the piecewise +viscosity is the obviously right thing to want, and it fails at every magnitude +we tried, from $\mu$ to $10^3\mu$, against both normals. + +**The rotated constraint disagrees at the corner and nowhere else.** Trimmed, it +matches the reference. Untrimmed it is six times worse, and the whole of that +difference is two nodes: against the Dirichlet run at a contrast of 10, the +pressure differs by 0.23 at the corner node and 0.12 at its neighbour along the +top, while everything below the top row agrees to 4 × 10⁻⁴ rms. The corner is a +node the rotated constraint and the side wall's component condition both hold. +Both runs pin the same two velocity components there, so the difference is in +how the corner row is assembled rather than in what is being asked of it. This +is underworld3 issue #608. ## Using it diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/.meshes/uw_annulus_ro1.0_ri0.5_csize0.05.msh b/articles/boundary-conditions-on-non-planar-boundaries/examples/.meshes/uw_annulus_ro1.0_ri0.5_csize0.05.msh deleted file mode 100644 index 4a06cfb..0000000 --- a/articles/boundary-conditions-on-non-planar-boundaries/examples/.meshes/uw_annulus_ro1.0_ri0.5_csize0.05.msh +++ /dev/null @@ -1,5034 +0,0 @@ -$MeshFormat -4.1 0 8 -$EndMeshFormat -$PhysicalNames -3 -1 1 "Lower" -1 2 "Upper" -2 666666 "Elements" -$EndPhysicalNames -$Entities -5 4 1 0 -1 0 0 0 0 -2 0.5 0 0 0 -3 -0.5 0 0 0 -4 1 0 0 0 -5 -1 0 0 0 -1 -0.5 0 0 0.5 0.492403876506104 0 1 1 2 2 -3 -2 -0.5 -0.492403876506104 0 0.5 0 0 1 1 2 3 -2 -3 -1 0 0 1 0.984807753012208 0 1 2 2 4 -5 -4 -1 -0.984807753012208 0 1 0 0 1 2 2 5 -4 -1 -1 -0.984807753012208 0 1 0.984807753012208 0 1 666666 4 3 4 -2 -1 -$EndEntities -$Nodes -9 1248 1 1248 -0 2 0 1 -1 -0.5 0 0 -0 3 0 1 -2 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*constraint*. This is the test that measures what -the constraint is wanted for — the surface normal stress, and through it -dynamic topography. - -## Why the current comparison is not enough - -`traction.py` compares two methods that both return the traction as an unknown: -the multiplier field `h`, and the reaction of the rotated constraint. They agree -to about 1% at one resolution, but there is **no oracle**, so when the vertex and -midpoint columns behave differently under refinement nothing says which is -right. Its numbers are deliberately not quoted in the note. - -## The oracle: Zhong et al. (2008) - -`uw.analytic.Zhong2008` is a propagator-matrix solution for a delta-function -load in a shell, and `.response()` returns the surface topography kernel -directly. For the default case — degree 2, load at r = 0.775, isoviscous, -r_inner = 0.55 — it gives - - surface_topography 0.4191904156575601 - cmb_topography 0.7705825500322292 - surface_geoid 0.0257906289002894 - surface_characteristic_velocity -0.0100641246068943 - -with self-gravity variants alongside, referenced to Zhong et al. (2008), -GGG 9, Q10017, doi:10.1029/2008GC002048, after Hager & O'Connell (1981). - -This is the right test because the quantity it publishes is the one the note -cares about, rather than a velocity field from which a stress must be inferred. - -**It is spherical.** The harmonic degree is a spherical-harmonic degree and the -solution carries a planet radius, so the matching model is a 3-D spherical -shell, not the 2-D annulus the leak tables use. That sets the cost, and it is -the reason this is a separate piece of work rather than another column in an -existing table. - -## What has to be built - -1. A spherical shell, free slip on both radii, with a delta-function density - load at r = 0.775 in a single spherical harmonic of degree 2. -2. The same model under each boundary treatment that a shell supports. -3. Surface topography from `sigma_nn`, against 0.41919. - -Two things to watch, both already known: - -- On a curved boundary, **vertex** values of `sigma_nn` carry the O(h) - facet-geometry error and **edge-midpoint** values are superconvergent - (underworld3#414). A pointwise comparison must say which it is using. -- On an enclosed domain the multiplier and the traction are each determined - only up to a constant, so the comparison is of the *deviation*, which is what - topography is anyway. - -## SolCx is not the weaker test, it is the other one - -No analytic solution has both a curved boundary and a laterally varying -viscosity, so no single test covers the case that actually hurts. - -| solution | curved boundary | viscosity | -|---|---|---| -| `assess` cylindrical / spherical (Kramer et al. 2021) | yes | **constant** — `nu` is a scalar | -| `Zhong2008` | yes | radial layers only (`viscosity_interfaces`) | -| SolCx | no, Cartesian box | **lateral jump** | - -That splits the work in two, and both halves are needed: - -- **Geometry, with the rheology trivial.** Kramer or Zhong. Tests whether the - treatment copes with a boundary whose normal turns, which is what this note - is about. -- **Rheology, with the geometry trivial.** SolCx. On a box every treatment - reduces to the same component constraint, so nothing here is about normals — - but a lateral viscosity jump is where a weakly imposed constraint is most - likely to fail to hold the normal degrees of freedom it was given, and there - is an exact answer to check against. - -The second is not a consolation prize. Lateral viscosity contrast is the -condition under which these methods are known to give trouble, and it is the -only one of the two where an oracle exists for it at all. - -A shell with a laterally varying viscosity has no exact solution, so if the -note wants that case it needs a different kind of evidence — self-consistency -between methods, or convergence under refinement — and it should say which. +# The examples, and what each one measures + +Every table and figure in the note is produced by one of these. Run them from +this directory with an Underworld3 environment on the path. + +| script | what it produces | +|---|---| +| `leak.py` | the constraint tables — how much flow each treatment lets through, under refinement (`sweep`) and against its own parameter (`params`) | +| `stress.py` | the surface-stress tables against the exact `uw.analytic.CylindricalStokes` answer: `sweep`, `params`, `locking`, `control` | +| `solcx.py` | the lateral-viscosity half, against `uw.analytic.SolCx`: `sweep`, `contrast`, `params`, `control` | +| `generate-locking-figure.py` | `figures/locking.png` and `figures/banner.png` | +| `generate-rotated-basis.py` | the data behind `rotated-basis.typ`, which draws `figures/rotated-basis.svg` | + +## The traps, all paid for once + +- **`v.array` is `(N, 1, dim)`.** It broadcasts silently against `(N, dim)` + normals and returns ~1e-16 projections for a 1e-2 velocity, with no error. + `np.squeeze` it. Every number in the first leak run was this artefact. +- **Give every metric a negative control and run it.** Two free-slip circles + leave the rigid rotation unconstrained, and that nullspace is purely + tangential — so a radial leak metric read 2e-14 on a solve that had diverged + with `|u| = 2.7e5`. The inner boundary is now held. +- **Check `snes.getConvergedReason() > 0` before tabulating anything.** Diverged + runs leave plausible numbers in the array; two nearly reached the note. +- **Vertex against edge midpoint.** On a curved boundary, vertex values of + `sigma_nn` carry the O(h) facet error and midpoints are superconvergent + (underworld3#414). `stress.py` splits them and the note says which it uses. +- **Select a boundary trace by the mesh LABEL, not by a radius band.** A band + that narrows with the mesh admits a different node set at each resolution, and + the earlier version of this comparison drifted by four nodes between methods + because of it. +- **Enclosed domain**: the multiplier, the pressure and the traction are each + determined only up to a constant. Compare deviations, which is what topography + is anyway. +- **A `Piecewise` viscosity inside a boundary penalty term does not solve** on + SolCx, at any magnitude tried. + +## What is still open + +- The spherical case. `uw.analytic.Zhong2008` publishes + `.response().surface_topography = 0.4191904156575601` for the default + degree-2 case (load r = 0.775, r_inner = 0.55, isoviscous), which is the right + oracle for dynamic topography proper. It is a 3-D shell, and that is the cost + step this note stopped short of. +- Behr (2004) reports non-physical recirculation at curved walls even with the + consistent normal. We have not looked for it here. +- underworld3 issues #607 (the multiplier misses the augmented-Lagrangian share) + and #608 (the rotated constraint at a corner shared with a component + condition) both came out of these runs and are open. diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/generate-locking-figure.py b/articles/boundary-conditions-on-non-planar-boundaries/examples/generate-locking-figure.py new file mode 100644 index 0000000..a6cd4e0 --- /dev/null +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/generate-locking-figure.py @@ -0,0 +1,105 @@ +"""The figure for "A constraint that is satisfied, and wrong". + +Three solves of the same annulus problem, rendered with the same colour scale: +the exact solution, a direct penalty written against the facet normal, and the +same penalty written against the measure-weighted node normal. The coefficient +is 1e6 in both penalty panels, so the only difference between them is the +normal. + +Writes: + figures/locking.png the three-panel figure + figures/banner.png a wide two-panel crop for the article banner + + python3 generate-locking-figure.py + +Run against underworld3 `development` at commit `8b7c8b9e`. +""" +import pathlib + +import numpy as np +import pyvista as pv + +pv.OFF_SCREEN = True + +import underworld3 as uw +import underworld3.visualisation as vis + +import stress as S + +CELL = 0.075 +PENALTY = 1.0e6 +HERE = pathlib.Path(__file__).resolve().parent +FIGURES = HERE.parent / "figures" + + +def speed_mesh(mode): + """(pyvista mesh carrying |u|, mesh edges, max speed) for one treatment.""" + mesh, stokes, v, exact = S.build(mode, cell=CELL, penalty=PENALTY) + stokes.solve() + assert S.converged(stokes), "%s did not converge" % mode + pv_v = vis.meshVariable_to_pv_mesh_object(v) + u = np.squeeze(np.asarray(v.array)) + pv_v.point_data["speed"] = np.linalg.norm(u, axis=1) + edges = vis.mesh_to_pv_mesh(mesh).extract_all_edges() + return pv_v, edges, exact, v + + +def exact_speed_mesh(reference): + """The same object built from the exact velocity, for the first panel.""" + mesh, stokes, v, exact = reference + field = uw.discretisation.MeshVariable("Uplot", v.mesh, v.mesh.dim, degree=2) + field.data[:] = exact.evaluate("velocity", field.coords) + pv_v = vis.meshVariable_to_pv_mesh_object(field) + pv_v.point_data["speed"] = np.linalg.norm( + np.squeeze(np.asarray(field.array)), axis=1) + return pv_v + + +def panel(plotter, index, pv_mesh, edges, title, clim, zoom=1.3): + plotter.subplot(0, index) + plotter.set_background("white") + plotter.add_mesh(pv_mesh, scalars="speed", cmap="RdBu_r", clim=clim, + show_edges=False, lighting=False, show_scalar_bar=False) + plotter.add_mesh(edges, color="black", line_width=0.4, lighting=False) + plotter.add_text(title, position="upper_left", font_size=12, color="black") + plotter.view_xy() + plotter.camera.zoom(zoom) + + +def main(): + facet_pv, facet_edges, exact, facet_v = speed_mesh("penalty") + node_pv, node_edges, _exact, node_v = speed_mesh("penalty_node") + + # The exact field on the node-normal run's mesh, which is the same mesh. + truth = uw.discretisation.MeshVariable("Utruth", node_v.mesh, node_v.mesh.dim, + degree=2) + truth.data[:] = _exact.evaluate("velocity", truth.coords) + truth_pv = vis.meshVariable_to_pv_mesh_object(truth) + truth_pv.point_data["speed"] = np.linalg.norm( + np.squeeze(np.asarray(truth.array)), axis=1) + + top = float(truth_pv.point_data["speed"].max()) + clim = (0.0, top) + print("colour scale 0 to %.4e" % top) + for name, pv_mesh in (("exact", truth_pv), ("facet", facet_pv), ("node", node_pv)): + print("%-6s max speed %.4e" % (name, float(pv_mesh.point_data["speed"].max()))) + + FIGURES.mkdir(exist_ok=True) + + plotter = pv.Plotter(off_screen=True, shape=(1, 3), window_size=(1650, 620)) + panel(plotter, 0, truth_pv, node_edges, "exact", clim) + panel(plotter, 1, facet_pv, facet_edges, "penalty, facet normal", clim) + panel(plotter, 2, node_pv, node_edges, "penalty, node normal", clim) + plotter.screenshot(str(FIGURES / "locking.png")) + plotter.close() + + banner = pv.Plotter(off_screen=True, shape=(1, 2), window_size=(1600, 560)) + panel(banner, 0, facet_pv, facet_edges, "", clim, zoom=1.9) + panel(banner, 1, truth_pv, node_edges, "", clim, zoom=1.9) + banner.screenshot(str(FIGURES / "banner.png")) + banner.close() + print("wrote", FIGURES / "locking.png", "and", FIGURES / "banner.png") + + +if __name__ == "__main__": + main() diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/leak.py b/articles/boundary-conditions-on-non-planar-boundaries/examples/leak.py index 2a0ed3f..b8e22be 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/examples/leak.py +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/leak.py @@ -25,7 +25,7 @@ python3 leak.py sweep # the resolution table python3 leak.py free # the control: outer boundary left natural -Run against underworld3 `development` at commit `0addec15`. +Run against underworld3 `development` at commit `8b7c8b9e`. """ import sys @@ -85,8 +85,23 @@ def build(mode, cell=CELL, penalty=1.0e4, gamma=GAMMA): # quadrature-point facet normal mesh.Gamma and a POSITIVE # coefficient. A negative one is anti-damping and the linear solve # fails immediately, which is how this was got wrong the first time. + # + # READ THIS COLUMN WITH stress.py BESIDE IT. Imposed facet by facet + # on a curved boundary, this constraint LOCKS: the leak falls + # because the boundary is being frozen, not because the condition is + # being satisfied in the way that was meant. At cell 0.075 the + # velocity field here differs from the rotated one by 20% in l2 at + # a coefficient of 1e4, and stress.py measures the same thing + # against an exact answer. G = mesh.Gamma stokes.add_natural_bc(penalty * G.dot(v.sym) * G, boundary) + elif mode == "penalty_node": + # The same penalty against the measure-weighted NODE normal, which + # is one direction per node rather than one per facet, and does not + # lock. The only difference between this and the line above is which + # normal. + G = mesh.boundary_normal(boundary) + stokes.add_natural_bc(penalty * G.dot(v.sym) * G, boundary) else: raise ValueError(mode) return mesh, stokes, v @@ -120,7 +135,7 @@ def leaks(mesh, v): def sweep(cells=(0.15, 0.10, 0.075, 0.05), - modes=("penalty", "nitsche", "constraint", "rotated")): + modes=("penalty", "penalty_node", "nitsche", "constraint", "rotated")): """Does the leak fall with the mesh, or is it already at round-off? A weakly imposed constraint is satisfied to the accuracy of the @@ -152,7 +167,7 @@ def parameter_sweep(): has a threshold at each end, and gamma = 10 sits in the middle of it on any mesh because gamma is dimensionless and the term already carries mu/h. """ - print("\ndirect penalty: leak against penalty magnitude") + print("\ndirect penalty (FACET normal): leak against penalty magnitude") print("\n| penalty | leak/|u| |") print("|---|---|") for pen in (1.0e2, 1.0e3, 1.0e4, 1.0e5, 1.0e6): @@ -161,6 +176,15 @@ def parameter_sweep(): cell = ("%.2e" % leaks(mesh, v)[0]) if converged(stokes) else "diverged" print("| %.0e | %s |" % (pen, cell)) + print("\ndirect penalty (NODE normal): leak against penalty magnitude") + print("\n| penalty | leak/|u| |") + print("|---|---|") + for pen in (1.0e2, 1.0e3, 1.0e4, 1.0e5, 1.0e6): + mesh, stokes, v = build("penalty_node", penalty=pen) + stokes.solve() + cell = ("%.2e" % leaks(mesh, v)[0]) if converged(stokes) else "diverged" + print("| %.0e | %s |" % (pen, cell)) + print("\nNitsche: leak against gamma") print("\n| gamma | leak/|u| |") print("|---|---|") @@ -177,7 +201,7 @@ def parameter_sweep(): elif sys.argv[1:2] == ["params"]: parameter_sweep() else: - modes = sys.argv[1:] or ["free", "penalty", "nitsche", + modes = sys.argv[1:] or ["free", "penalty", "penalty_node", "nitsche", "constraint", "rotated"] print("%-9s %10s %10s %8s" % ("mode", "leak/|u|", "|u|max", "nodes")) for mode in modes: diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/solcx.py b/articles/boundary-conditions-on-non-planar-boundaries/examples/solcx.py new file mode 100644 index 0000000..3d4a314 --- /dev/null +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/solcx.py @@ -0,0 +1,281 @@ +"""The other half of the stress test: a lateral viscosity contrast. + +`stress.py` is the geometry half -- a curved boundary with the rheology +trivial. This is the rheology half, with the geometry trivial. No exact +solution has both, so the two halves are separate tests and the note does not +conflate them. + +On a box, "no flow through this wall" is a single velocity component and every +treatment here reduces to the same constraint. Nothing about this test is about +normals. What it is about is whether a WEAKLY imposed constraint holds the +degrees of freedom it was given when the viscosity beside them jumps by six +orders of magnitude -- and SolCx is the only case of the two halves with an +exact surface stress to check that against. + +SolCx: unit box, viscosity eta_A left of x = 0.5 and eta_B right of it, +forcing (0, cos(pi.x) sin(pi.z)), free slip on all four walls. +`uw.analytic.SolCx` publishes the exact stress, and `topography_top` is +-sigma_zz on the top boundary, which is the quantity here. + +Three walls carry the ordinary component Dirichlet condition. The treatment +under test is on the TOP wall only, so what is measured is attributable to it. + +The metrics +----------- + * the leak, max |u.n| / |u|, reported SEPARATELY over the soft half of the + wall and the stiff half. One coefficient has to hold both, and whether it + can is the whole question; + * the surface stress: relative l2 error of sigma_zz along the top wall + against the exact one, both mean-removed (the box is enclosed, so the + pressure -- and with it the level of sigma_zz -- is fixed only up to a + constant). + + python3 solcx.py sweep # resolution, each treatment + python3 solcx.py contrast # the viscosity ratio, each treatment + python3 solcx.py params # the penalty coefficient, both halves + python3 solcx.py control # the top wall left free + +Run against underworld3 `development` at commit `8b7c8b9e`. +""" +import sys + +import numpy as np +import sympy + +import underworld3 as uw + +from underworld3.utilities.boundary_flux import _boundary_field_nodes + +RES = 32 +ETA_B = 1.0e6 +PENALTY = 1.0e4 +GAMMA = 10.0 + + +def build(mode, res=RES, eta_B=ETA_B, penalty=PENALTY, gamma=GAMMA): + mesh = uw.meshing.StructuredQuadBox( + elementRes=(res, res), minCoords=(0.0, 0.0), maxCoords=(1.0, 1.0), qdegree=3) + + v = uw.discretisation.MeshVariable("U", mesh, mesh.dim, degree=2) + p = uw.discretisation.MeshVariable("P", mesh, 1, degree=1) + solver_class = (uw.systems.Stokes_Constrained if mode == "constraint" + else uw.systems.Stokes) + stokes = solver_class(mesh, velocityField=v, pressureField=p) + + exact = uw.analytic.SolCx(mesh, eta_A=1.0, eta_B=eta_B, x_c=0.5, n=1) + stokes.constitutive_model = uw.constitutive_models.ViscousFlowModel + stokes.constitutive_model.Parameters.shear_viscosity_0 = exact.fn_viscosity + if mode != "constraint": + # Stokes_Constrained builds its own Schur preconditioner from the + # operator's blocks and refuses this assignment. + stokes.saddle_preconditioner = 1.0 / exact.fn_viscosity + stokes.bodyforce = exact.fn_bodyforce + stokes.tolerance = 1.0e-9 + # Enclosed: the constant pressure mode is a nullspace and a direct solve on + # the singular saddle returns a quiet, wrong answer without this. With the + # top wall left free (the control) the domain is open and there is no such + # mode -- asserting one there is what makes the control fail to solve + # rather than fail to hold the boundary. + stokes.petsc_use_pressure_nullspace = (mode != "free") + + stokes.add_dirichlet_bc((0.0, None), "Left") + stokes.add_dirichlet_bc((0.0, None), "Right") + stokes.add_dirichlet_bc((None, 0.0), "Bottom") + + if mode == "dirichlet": + # What a box lets you do, and the reference the others are judged by. + stokes.add_dirichlet_bc((None, 0.0), "Top") + elif mode == "constraint": + stokes.add_constraint_bc(0.0, "Top") + elif mode == "rotated": + stokes.add_rotated_freeslip_bc(0.0, "Top") + elif mode == "nitsche": + stokes.add_nitsche_bc(0.0, "Top", gamma=gamma, theta=1) + elif mode == "penalty": + # A CONSTANT coefficient, deliberately. Scaling it by the local + # viscosity is the fair thing to want here -- one number cannot be large + # against 1e6 and moderate against 1 -- and it does not solve: a + # Piecewise viscosity inside the boundary term fails the line search at + # every magnitude tried, from 1 to 1e3 times mu, against both normals. + # A constant 1e4 solves, and that is what this column is. + stokes.add_natural_bc(penalty * mesh.Gamma.dot(v.sym) * mesh.Gamma, "Top") + elif mode == "free": + pass + else: + raise ValueError(mode) + + return mesh, stokes, v, exact + + +def converged(stokes): + return stokes.snes.getConvergedReason() > 0 + + +def trace(solver, field_id, var, boundary="Top"): + """(coords, values) for `var` at the nodes it carries on `boundary`, + selected by the mesh boundary LABEL.""" + nodes, *_ = _boundary_field_nodes(solver, boundary, field_id) + coords = np.array([node[2] for node in nodes]) + tree = uw.kdtree.KDTree(np.ascontiguousarray(var.coords)) + index = np.asarray(tree.query(np.ascontiguousarray(coords), 1)[1]).flatten() + return coords, np.squeeze(np.asarray(var.array))[index] + + +def recovered_traction(mesh, stokes): + """sigma_zz on the top wall, projected out of the solved fields.""" + field = uw.discretisation.MeshVariable("Szz", mesh, 1, degree=2) + projection = uw.systems.Projection(mesh, field) + projection.uw_function = stokes.stress[1, 1] + projection.solve() + return trace(projection, 0, field) + + +def reaction_traction(stokes, mode): + if mode == "rotated": + return stokes.boundary_normal_traction("Top") + if mode == "constraint": + return trace(stokes, 2, stokes.multiplier("Top")) + return None + + +def stress_error(coords, values, exact, trim=0.0): + """Relative l2 error along the wall, both mean-removed. + + The mean is the gauge: the box is enclosed, so the pressure and with it the + level of sigma_zz is fixed only up to a constant. The deviation is what + topography is built from and the only part that is determined. + + `trim` drops nodes within that distance of the two ends of the wall. The + corners are where the treatment under test meets the side walls' component + conditions, so a node there is constrained twice and by two different + mechanisms; trimming separates what the wall does from what the corner does. + """ + coords = np.asarray(coords) + values = np.asarray(values) + if trim > 0.0: + keep = (coords[:, 0] > trim) & (coords[:, 0] < 1.0 - trim) + coords, values = coords[keep], values[keep] + truth = -exact.topography_top(coords) # topography_top is -sigma_zz + order = np.argsort(coords[:, 0]) + got = values[order] - values.mean() + truth = truth[order] - truth.mean() + return float(np.linalg.norm(got - truth) / np.linalg.norm(truth)) + + +def leaks(v): + """max |u_z| / |u| on the top wall, over the soft half and the stiff half. + + Reported separately because one penalty coefficient has to hold both, and + the two sides do not ask the same thing of it. + """ + coords = v.coords + top = np.abs(coords[:, 1] - 1.0) < 1.0e-9 + u = np.squeeze(np.asarray(v.array)) + speed = np.linalg.norm(u, axis=1).max() + out = [] + for side in (coords[:, 0] < 0.5, coords[:, 0] > 0.5): + mask = top & side + out.append(np.abs(u[mask, 1]).max() / speed) + return out + + +def measure(mode, **kwargs): + mesh, stokes, v, exact = build(mode, **kwargs) + stokes.solve() + if not converged(stokes): + return None + soft, stiff = leaks(v) + out = {"soft": soft, "stiff": stiff, "velocity": exact.velocity_error(v)} + coords, values = recovered_traction(mesh, stokes) + out["recovered"] = stress_error(coords, values, exact) + out["trimmed"] = stress_error(coords, values, exact, trim=2.0 / kwargs.get("res", RES)) + read = reaction_traction(stokes, mode) + if read is not None: + coords, values = read + # The reaction is the traction holding the wall, opposite in sign to + # sigma_zz -- the same convention as in stress.py. + out["reaction"] = stress_error(coords, -np.asarray(values), exact) + return out + + +MODES = ("dirichlet", "penalty", "nitsche", "constraint", "rotated") + + +def sweep(resolutions=(16, 32, 64), modes=MODES): + print("surface stress error, eta_B/eta_A = %.0e" % ETA_B) + print() + print("| elements | " + " | ".join(modes) + " | rotated (reaction) | multiplier |") + print("|---" * (len(modes) + 3) + "|") + for res in resolutions: + row, extra = [], {} + for mode in modes: + got = measure(mode, res=res) + row.append("diverged" if got is None else "%.2e" % got["recovered"]) + if got and "reaction" in got: + extra[mode] = "%.2e" % got["reaction"] + print("| %d | %s | %s | %s |" + % (res, " | ".join(row), extra.get("rotated", "-"), + extra.get("constraint", "-")), flush=True) + + +def contrast(ratios=(1.0e1, 1.0e2, 1.0e3, 1.0e4, 1.0e6), modes=MODES): + """Does the treatment survive the viscosity jump getting bigger? + + `dirichlet` is the control, and it is the one to read first. It is the + ordinary component condition a box allows, it holds u.n exactly, and its + velocity error is 1e-5 at every contrast here -- so whatever it reads in + the stress column is the RECOVERY's error and not a boundary condition's. + A treatment can only be said to be worse than the reference where it is + worse than that. + """ + print("whole wall / trimmed by two elements at each end") + print() + print("| eta_B/eta_A | " + " | ".join(modes) + " |") + print("|---" * (len(modes) + 1) + "|") + for ratio in ratios: + row = [] + for mode in modes: + got = measure(mode, eta_B=ratio) + row.append("diverged" if got is None + else "%.2e / %.2e" % (got["recovered"], got["trimmed"])) + print("| %.0e | %s |" % (ratio, " | ".join(row)), flush=True) + + +def parameters(): + """The leak on each half of the wall, against the parameter. + + A dimensionless gamma carries mu/h with it and so asks the same of both + halves. A bare penalty coefficient does not, and the stiff half is where + that shows. + """ + for mode, values, label in ( + ("penalty", (1e2, 1e4, 1e6, 1e8), "penalty coefficient"), + ("nitsche", (1.0, 10.0, 100.0, 1000.0), "Nitsche gamma")): + print("\n%s, eta_B/eta_A = %.0e" % (label, ETA_B)) + print("\n| %s | leak, soft half | leak, stiff half | stress error |" % label) + print("|---|---|---|---|") + for value in values: + kwargs = {"gamma": value} if mode == "nitsche" else {"penalty": value} + got = measure(mode, **kwargs) + if got is None: + print("| %g | diverged | | |" % value) + continue + print("| %g | %.2e | %.2e | %.2e |" + % (value, got["soft"], got["stiff"], got["recovered"]), flush=True) + + +def control(): + for mode in ("free", "dirichlet"): + got = measure(mode) + if got is None: + print("%-10s diverged" % mode, flush=True) + continue + print("%-10s leak soft %.2e stiff %.2e velocity %.2e stress %.2e" + % (mode, got["soft"], got["stiff"], got["velocity"], + got["recovered"]), flush=True) + + +if __name__ == "__main__": + command = sys.argv[1:2] or ["sweep"] + {"sweep": sweep, "contrast": contrast, + "params": parameters, "control": control}[command[0]]() diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/stress.py b/articles/boundary-conditions-on-non-planar-boundaries/examples/stress.py new file mode 100644 index 0000000..628eb62 --- /dev/null +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/stress.py @@ -0,0 +1,343 @@ +"""Does the surface stress come out right? + +`leak.py` measures the CONSTRAINT -- how much flow each boundary treatment lets +through. This measures the thing the constraint is wanted for: the wall-normal +stress on the boundary, which is dynamic topography once divided by the +buoyancy scale. Against an exact answer, so it is a measurement rather than a +comparison of two methods that might both be wrong. + +The oracle +---------- +Kramer, Davies & Wilson (2021) give exact Stokes solutions in a cylindrical +annulus, and their `assess` package publishes the RADIAL STRESS directly -- +`radial_stress = tau_rr - p` -- which is the quantity here. Underworld wraps +the package as `uw.analytic.CylindricalStokes`. + +The case is the smooth one: density (r/r_o)^k cos(n.theta) driving the flow, +free slip on both radii, viscosity 1. On the outer boundary the exact +sigma_rr is a pure cos(n.theta) (checked, residual 1e-16), so the measurement +is the amplitude of that harmonic and the metric is its relative error. + +The inner boundary carries the EXACT velocity as a Dirichlet condition rather +than a free-slip treatment of its own. The exact solution satisfies both, so +the problem is unchanged; what it buys is that the treatment under test is the +only free-slip condition in the model, and that the rigid-rotation nullspace of +two free-slip circles is gone. + +What is compared +---------------- +Two routes to the surface stress, and the difference between them is the +note's argument: + + * RECOVERED -- project r.sigma.r out of the solved velocity and pressure. + Available for every treatment, and the only route the weak ones have. + * REACTION -- the constraint reaction of the rotated method + (`boundary_normal_traction`), and the multiplier field of the constraint + method. Not recovered from the solution: an unknown the solve returned. + +Both are compared against the same exact amplitude, using the TRUE radial +direction, so no treatment is being scored against its own normal. + +Sign: `boundary_normal_traction` and the multiplier both return the constraint +reaction, which is opposite in sign to sigma_rr as `assess` publishes it (it is +the traction that holds the boundary, not the traction the fluid exerts). The +dynamic-topography formula h = -(sigma_nn - mean)/(rho.g) carries the sign +back. Amplitudes are compared unsigned and the sign is printed so the +convention stays visible. + + python3 stress.py sweep # the refinement table + python3 stress.py params # penalty coefficient and Nitsche gamma + python3 stress.py locking # the facet normal does not converge + python3 stress.py control # the metric fires when the BC is removed + +Run against underworld3 `development` at commit `8b7c8b9e`. +""" +import sys + +import numpy as np +import sympy + +import underworld3 as uw + +# The boundary trace of a field, selected by the mesh boundary LABEL. An earlier +# version of this comparison selected it by a radius band that narrowed with the +# mesh, which quietly admitted a different node set at each resolution. This is +# the same selector the solver's own reaction recovery uses. +from underworld3.utilities.boundary_flux import _boundary_field_nodes + +N = 2 # azimuthal wavenumber of the density anomaly +K = 3 # its radial power +R_I, R_O = 0.5, 1.0 +CELL = 0.075 +PENALTY = 1.0e4 +GAMMA = 10.0 + + +def build(mode, cell=CELL, penalty=PENALTY, gamma=GAMMA): + """The annulus, the forcing, and one boundary treatment on the outer arc.""" + mesh = uw.meshing.Annulus(radiusInner=R_I, radiusOuter=R_O, cellSize=cell) + x, y = mesh.X + r = sympy.sqrt(x**2 + y**2) + theta = sympy.atan2(y, x) + + v = uw.discretisation.MeshVariable("U", mesh, mesh.dim, degree=2) + p = uw.discretisation.MeshVariable("P", mesh, 1, degree=1) + solver_class = (uw.systems.Stokes_Constrained if mode == "constraint" + else uw.systems.Stokes) + stokes = solver_class(mesh, velocityField=v, pressureField=p) + stokes.constitutive_model = uw.constitutive_models.ViscousFlowModel + stokes.constitutive_model.Parameters.shear_viscosity_0 = 1.0 + + # The forcing `assess` solves for: rho = (r/r_o)^k cos(n.theta), gravity + # inward. This is the convention the Kramer benchmark in underworld3's + # docs/examples uses, and the velocity error below is what checks it. + rho = ((r / R_O) ** K) * sympy.cos(N * theta) + stokes.bodyforce = -rho * sympy.Matrix([[x / r, y / r]]) + + exact = uw.analytic.CylindricalStokes( + mesh, n=N, k=K, r_inner=R_I, r_outer=R_O, + density="smooth", boundary="free") + + v_exact = uw.discretisation.MeshVariable("Uex", mesh, mesh.dim, degree=2) + v_exact.data[:] = exact.evaluate("velocity", v_exact.coords) + stokes.add_dirichlet_bc(v_exact.sym, "Lower") + + if mode == "constraint": + stokes.add_constraint_bc(0.0, "Upper") + elif mode == "rotated": + stokes.add_rotated_freeslip_bc(0.0, "Upper") + elif mode == "nitsche": + # Nitsche's default normal is mesh.boundary_normal(boundary) -- the + # measure-weighted node normal, not the per-facet one. + stokes.add_nitsche_bc(0.0, "Upper", gamma=gamma, theta=1) + elif mode == "penalty": + # The documented direct penalty, with the quadrature-point FACET normal. + stokes.add_natural_bc(penalty * mesh.Gamma.dot(v.sym) * mesh.Gamma, "Upper") + elif mode == "penalty_node": + # The same penalty against the measure-weighted NODE normal. The only + # difference between this and the line above is which normal, and it is + # the difference between a method that converges and one that locks. + node_n = mesh.boundary_normal("Upper") + stokes.add_natural_bc(penalty * node_n.dot(v.sym) * node_n, "Upper") + elif mode == "free": + pass # the control: no condition on the outer arc + else: + raise ValueError(mode) + + return mesh, stokes, v, exact + + +def converged(stokes): + """A diverged solve leaves numbers in the array that look like measurements.""" + return stokes.snes.getConvergedReason() > 0 + + +def exact_amplitude(exact): + """Amplitude of cos(n.theta) in the exact sigma_rr on the outer boundary. + + Sampled densely on the true circle and least-squares fitted. The residual + is returned too: it is 1e-16, which is what says the exact surface stress + really is a single harmonic and the amplitude is the whole of it. + """ + phi = np.linspace(0.0, 2.0 * np.pi, 720, endpoint=False) + points = np.c_[R_O * np.cos(phi), R_O * np.sin(phi)] + sigma = np.array([exact._above.radial_stress_cartesian(pt) for pt in points]) + (mean, c, s), residual = _fit(phi, sigma) + return c, residual + + +def _fit(angles, values): + """Least squares fit of a constant plus the degree-N harmonic. + + A least-squares fit rather than a quadrature: boundary nodes are not evenly + spaced in theta, and on a P2 trace vertices and edge midpoints alternate + with unequal gaps. Returns ((mean, cos, sin), max residual). + """ + basis = np.c_[np.ones_like(angles), np.cos(N * angles), np.sin(N * angles)] + coefficients, *_ = np.linalg.lstsq(basis, values, rcond=None) + residual = np.abs(values - basis @ coefficients).max() + return coefficients, residual + + +def trace(solver, field_id, var, boundary="Upper"): + """(coords, values) for `var` at the nodes it carries on `boundary`.""" + nodes, *_ = _boundary_field_nodes(solver, boundary, field_id) + coords = np.array([node[2] for node in nodes]) + tree = uw.kdtree.KDTree(np.ascontiguousarray(var.coords)) + index = np.asarray(tree.query(np.ascontiguousarray(coords), 1)[1]).flatten() + # squeeze: a MeshVariable's .array carries a middle axis that broadcasts + # silently against an (N,) index -- see leak.py. + return coords, np.squeeze(np.asarray(var.array))[index] + + +def recovered_traction(mesh, stokes): + """sigma_rr on the outer boundary, projected out of the solved fields. + + The recovery every treatment can do, and the only one the weak forms have. + Against the TRUE radial direction, which is also the direction the oracle + publishes, so the comparison does not depend on the solver's normal. + """ + x, y = mesh.X + r = sympy.sqrt(x**2 + y**2) + radial = sympy.Matrix([[x / r, y / r]]) + sigma_rr = (radial * stokes.stress * radial.T)[0, 0] + + field = uw.discretisation.MeshVariable("Srr", mesh, 1, degree=2) + projection = uw.systems.Projection(mesh, field) + projection.uw_function = sigma_rr + projection.solve() + return trace(projection, 0, field) + + +def reaction_traction(stokes, mode): + """The constraint reaction, for the two methods that return one.""" + if mode == "rotated": + return stokes.boundary_normal_traction("Upper") + if mode == "constraint": + multiplier = stokes.multiplier("Upper") + # field 2 of the constrained saddle point: velocity, pressure, multiplier + return trace(stokes, 2, multiplier) + return None + + +def amplitude_error(coords, values, reference): + """Relative error in the harmonic amplitude, unsigned, and the sign.""" + angles = np.arctan2(coords[:, 1], coords[:, 0]) + (_mean, c, _s), _residual = _fit(angles, values) + return abs(abs(c) - abs(reference)) / abs(reference), np.sign(c) + + +def split(coords): + """Vertices from edge midpoints. A vertex of the annulus mesh sits exactly + on the circle; a P2 edge midpoint sits on the chord, inside it by the + sagitta. They are not interchangeable: vertex values of sigma_nn carry the + O(h) facet-geometry error and midpoint values are superconvergent + (underworld3#414).""" + on_circle = np.abs(np.linalg.norm(coords, axis=1) - R_O) < 1.0e-9 + return {"vertex": on_circle, "midpoint": ~on_circle} + + +def leak(v): + """max |u.n| on the outer boundary against the TRUE radial direction, + normalised by the flow speed -- the measurement leak.py tabulates.""" + coords = v.coords + radius = np.linalg.norm(coords, axis=1) + outer = np.abs(radius - R_O) < 1.0e-6 + u = np.squeeze(np.asarray(v.array)) + normal = coords[outer] / radius[outer][:, None] + return np.abs((u[outer] * normal).sum(axis=1)).max() / np.linalg.norm(u, axis=1).max() + + +def measure(mode, cell=CELL, **kwargs): + """One solve, and everything read off it.""" + mesh, stokes, v, exact = build(mode, cell=cell, **kwargs) + stokes.solve() + if not converged(stokes): + return None + reference, _residual = exact_amplitude(exact) + out = { + "leak": leak(v), + "velocity": exact.error("velocity", v), + "exact": reference, + } + coords, values = recovered_traction(mesh, stokes) + out["recovered"], out["sign"] = amplitude_error(coords, values, reference) + read = reaction_traction(stokes, mode) + if read is not None: + coords, values = read + out["reaction"], out["reaction_sign"] = amplitude_error(coords, values, reference) + for name, mask in split(coords).items(): + out["reaction_" + name] = amplitude_error( + coords[mask], values[mask], reference)[0] + return out + + +MODES = ("penalty", "penalty_node", "nitsche", "constraint", "rotated") + + +def sweep(cells=(0.15, 0.10, 0.075, 0.05), modes=MODES): + """Does the surface stress converge to the exact one, and how fast?""" + exact, residual = exact_amplitude(build("free", cell=0.2)[3]) + print("exact sigma_rr on r = %.2f: %.10f cos(%d.theta), residual %.1e" + % (R_O, exact, N, residual)) + print() + print("relative error in the surface stress amplitude") + print() + print("| cell size | " + " | ".join(modes) + " | rotated (reaction) | multiplier |") + print("|---" * (len(modes) + 3) + "|") + for cell in cells: + row, extra = [], {} + for mode in modes: + got = measure(mode, cell=cell) + row.append("diverged" if got is None else "%.2e" % got["recovered"]) + if got and "reaction" in got: + extra[mode] = "%.2e" % got["reaction"] + print("| %.3f | %s | %s | %s |" + % (cell, " | ".join(row), + extra.get("rotated", "-"), extra.get("constraint", "-")), flush=True) + + +def parameters(cell=CELL): + """The two weak methods against their own parameter, with the leak beside + the stress -- which is where they part company.""" + for mode, values, label in ( + ("penalty", (1e3, 1e4, 1e5, 1e6, 1e8), "penalty (facet normal)"), + ("penalty_node", (1e3, 1e4, 1e5, 1e6), "penalty (node normal)"), + ("nitsche", (1.0, 10.0, 100.0, 1000.0, 1e4), "Nitsche gamma")): + print("\n%s, cell %.3f" % (label, cell)) + print("\n| %s | leak | velocity error | stress error |" + % ("gamma" if mode == "nitsche" else "coefficient")) + print("|---|---|---|---|") + for value in values: + kwargs = {"gamma": value} if mode == "nitsche" else {"penalty": value} + got = measure(mode, cell=cell, **kwargs) + if got is None: + print("| %g | diverged | | |" % value) + continue + print("| %g | %.2e | %.2e | %.2e |" + % (value, got["leak"], got["velocity"], got["recovered"]), flush=True) + + +def locking(cells=(0.15, 0.10, 0.075, 0.05, 0.035), penalty=1.0e6): + """The facet normal, pushed hard, does not converge to free slip. + + Imposing u.n = 0 facet by facet on a curved boundary constrains a corner + node in two directions at once, and the discrete limit is not the smooth + problem. Refine it and it stays wrong -- while the leak, which is the + metric that would normally be trusted, reads 1e-5. + """ + print("direct penalty at %.0e, against the FACET normal" % penalty) + print("\n| cell size | leak | velocity error | stress error |") + print("|---|---|---|---|") + for cell in cells: + got = measure("penalty", cell=cell, penalty=penalty) + if got is None: + print("| %.3f | diverged | | |" % cell) + continue + print("| %.3f | %.2e | %.2e | %.2e |" + % (cell, got["leak"], got["velocity"], got["recovered"]), flush=True) + print("\nthe same coefficient against the measure-weighted NODE normal") + print("\n| cell size | leak | velocity error | stress error |") + print("|---|---|---|---|") + for cell in cells: + got = measure("penalty_node", cell=cell, penalty=penalty) + if got is None: + print("| %.3f | diverged | | |" % cell) + continue + print("| %.3f | %.2e | %.2e | %.2e |" + % (cell, got["leak"], got["velocity"], got["recovered"]), flush=True) + + +def control(cell=CELL): + """Take the boundary condition away. A metric that cannot see that is not + measuring anything.""" + for mode in ("free", "rotated"): + got = measure(mode, cell=cell) + print("%-9s leak %.2e velocity error %.2e stress error %.2e" + % (mode, got["leak"], got["velocity"], got["recovered"]), flush=True) + + +if __name__ == "__main__": + command = sys.argv[1:2] or ["sweep"] + {"sweep": sweep, "params": parameters, + "locking": locking, "control": control}[command[0]]() diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/traction.py b/articles/boundary-conditions-on-non-planar-boundaries/examples/traction.py deleted file mode 100644 index e7204f0..0000000 --- a/articles/boundary-conditions-on-non-planar-boundaries/examples/traction.py +++ /dev/null @@ -1,159 +0,0 @@ -"""Do the two exact methods agree on the surface traction? - -The leak measures the constraint. This measures the thing the constraint is -usually wanted FOR: the normal stress on the boundary, which is dynamic -topography once divided by the buoyancy scale. - -Two of the four methods return it as an unknown rather than recovering it: - - * the multiplier method, where h on the boundary IS the traction holding the - constraint; - * the rotated constraint, where the reaction of the strong constraint is - sigma_nn. - -They are computed by completely different routes, so agreeing is evidence and -disagreeing localises a defect. The weak methods have no comparable object -- -their traction has to be recovered from a velocity field that does not satisfy -the constraint exactly, which is the argument the note makes. - -STATUS: the vertex comparison converges and the midpoint comparison does NOT, -which is backwards. Measured agreement in the degree-4 amplitude: - - cell vertex midpoint - 0.150 5.5e-02 3.6e-03 - 0.100 2.0e-02 1.2e-02 - 0.075 1.5e-02 3.8e-02 - 0.050 8.5e-03 6.9e-02 - -Vertices converge at about h^1.4. Midpoints get steadily WORSE, which cannot be -right: the solver documents midpoint values of sigma_nn as superconvergent on a -curved boundary and vertex values as carrying the O(h) facet error, so if -anything the two columns should be the other way round. - -The likely fault is in this script rather than in either method. The multiplier -carries screened INTERIOR degrees of freedom as well as its boundary trace, and -they are selected here by a radius band that narrows as the mesh refines -(`rad > 1 - 0.3 * cell`). A band that changes with resolution can quietly change -WHICH nodes it admits, and the node counts do drift apart between the two -methods under refinement -- 88 against 84 at cell 0.075 -- which is how the -earlier pairwise version failed outright. - -So this is not yet evidence of anything and its numbers are deliberately not -quoted in the note. What is needed is to select the multiplier's boundary trace -from the mesh boundary LABEL rather than by geometry. - -Run against underworld3 `development` at commit `0addec15`. -""" -import numpy as np -import sympy - -import underworld3 as uw - -import leak - - -def split_trace(coords, values): - """Order a boundary trace by angle, and split P2 VERTEX nodes from EDGE - MIDPOINTS. - - They are not interchangeable on a curved boundary. The solver's own warning: - vertex values of sigma_nn converge only slowly, because the vertex basis has - zero surface mean and the vertex reaction carries the O(h) facet-geometry - error; midpoint values are superconvergent (underworld3#414). - - They are also easy to tell apart geometrically. A vertex of the annulus mesh - sits exactly on the circle; the midpoint of a facet sits on the chord, which - is inside it by the sagitta. - """ - rad = np.linalg.norm(coords, axis=1) - on_circle = np.abs(rad - 1.0) < 1.0e-9 - out = {} - for name, m in (("vertex", on_circle), ("midpoint", ~on_circle)): - ang = np.arctan2(coords[m, 1], coords[m, 0]) - order = np.argsort(ang) - out[name] = (ang[order], np.asarray(values)[m][order]) - return out - - -def trace_of(field): - """(coords, values) from either a MeshVariable or an already-extracted pair.""" - if isinstance(field, tuple): - return field[0], np.squeeze(np.asarray(field[1])) - return field.coords, np.squeeze(np.asarray(field.array)) - - -def run(mode, cell=0.075): - mesh, stokes, v = leak.build(mode, cell=cell) - stokes.solve() - assert leak.converged(stokes), "%s did not converge" % mode - if mode == "constraint": - c, val = trace_of(stokes.multiplier("Upper")) - # The multiplier also has (screened) interior degrees of freedom, so - # take its boundary trace. The tolerance has to scale with the mesh: - # a facet midpoint sits inside the circle by the sagitta, about h^2/8, - # which at h = 0.15 is 2.8e-3 -- a fixed 1e-3 window silently drops - # every midpoint at coarse resolution and leaves an empty trace. - rad = np.linalg.norm(c, axis=1) - keep = rad > 1.0 - 0.3 * cell - return split_trace(c[keep], val[keep]) - if mode == "rotated": - c, val = trace_of(stokes.boundary_normal_traction("Upper")) - return split_trace(c, val) - raise ValueError(mode) - - -def harmonic(angles, values, degree=4): - """Amplitude of the degree-n harmonic in a boundary trace. - - Comparing node by node is fragile -- the two methods do not return the same - node set, and the counts drift apart under refinement. The forcing here is a - single harmonic, so the traction is dominated by one too, and its amplitude - is both resolution-independent and the quantity dynamic topography is built - from. A least-squares projection onto cos(n.theta) and sin(n.theta) needs no - correspondence between the two traces at all. - """ - v = values - values.mean() - c = 2.0 * np.mean(v * np.cos(degree * angles)) - s = 2.0 * np.mean(v * np.sin(degree * angles)) - return float(np.hypot(c, s)) - - -def convergence(cells=(0.15, 0.10, 0.075, 0.05)): - """Do the two routes converge to the SAME traction? - - Agreeing once could be a coincidence of scale. Agreeing better as the mesh - refines is what says both are approximating one thing. - """ - print("| cell size | vertex | edge midpoint |") - print("|---|---|---|") - for cell in cells: - mult = run("constraint", cell=cell) - rot = run("rotated", cell=cell) - row = [] - for kind in ("vertex", "midpoint"): - h1 = harmonic(*mult[kind]) - h2 = harmonic(*rot[kind]) - row.append("%.2e" % (abs(h1 - h2) / max(h1, h2))) - print("| %.3f | %s |" % (cell, " | ".join(row))) - - -if __name__ == "__main__": - import sys - if sys.argv[1:2] == ["converge"]: - convergence() - raise SystemExit - mult = run("constraint") - rot = run("rotated") - for kind in ("vertex", "midpoint"): - a1, t1 = mult[kind] - a2, t2 = rot[kind] - # Both are determined only up to a constant on an enclosed boundary, so - # the deviation is the part carrying the signal -- and the part dynamic - # topography is built from. - d1, d2 = t1 - t1.mean(), t2 - t2.mean() - print("%-9s multiplier n=%3d amp=%.3e | reaction n=%3d 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zT<-mAB|yn-Z_d)Y!TFzP$L?i#wUhie z7BPCabNG9Lx^_HnK~lTOu0C%g%SRkA`lA_0!}mo&E8cq{)K?niuX5q?Ml!5&M z2wx@zk#2|htgZi6Cb?dhr~-=a|64pd!O-v@-|(M~?acV<^xujJ4QPn;|Fu}&`=2lV zKlkK+ZpHtt9A+`d|5|Wibc>+A|JPFY1hnY?^j_TM^cOoK0s>Z)((kfBcEmO@8x4WL zXEFj%(u|405vf&72u_N5q@we%^np#p{{mh(nB4#X literal 0 HcmV?d00001 diff --git a/articles/boundary-conditions-on-non-planar-boundaries/metadata.yml b/articles/boundary-conditions-on-non-planar-boundaries/metadata.yml index 2778f0f..ca858a7 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/metadata.yml +++ b/articles/boundary-conditions-on-non-planar-boundaries/metadata.yml @@ -30,5 +30,9 @@ methods: - meshing ghost_tags: - Underworld Code -figures: 1 +figures: 2 +# Rendered from the model rather than a stock photograph, so there is nobody to +# credit. `figures` counts the figures in the body; the banner is not one. +banner: figures/banner.png +banner_credit: null source: native From 51f0fa758867bec49cc230a87fb3858d559fbf67 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Wed, 19 Aug 2026 17:39:19 +1000 Subject: [PATCH 09/33] Draw the SolCx surface topography, and find the multiplier losing it Predicted against computed, along the top wall, at a viscosity contrast of 100 and of a million. The exact answer is published by `uw.analytic.SolCx`, so this comparison can be a figure rather than a column of numbers. Drawing it turned up a defect the tables had missed. At a contrast of 1e6 the multiplier field as `multiplier()` returns it peaks at 0.042 against an exact 0.383 and is anti-correlated with it (-0.53), on a solve whose velocity error is 8.8e-06. The default augmentation is r = 1e4.mu(x), which the viscosity step makes 1e10 on the stiff half, so almost the whole traction sits in the augmented-Lagrangian term rather than in h. Adding it back gives 0.047, the most accurate surface stress in the comparison -- better than the projected recovery at 0.075. Turning r down is not available: at this contrast r = 0 and r = 1e2 both fail to solve. Recorded on underworld3#607, which the annulus had already found at the few-per-cent level. The figure carries both multiplier curves, as returned and corrected, so the size of the missing term is visible rather than asserted. The script caches its curves beside itself: the solves take about ten minutes and the figure is redrawn more often than it is recomputed. A dash pattern is a linestyle and not a format string, and a sign convention is asserted rather than assumed -- the first draft of this figure stopped on that assertion, which is how the collapse was noticed. Underworld development team with AI support from Claude Code --- ...ary-conditions-on-non-planar-boundaries.md | 102 ++++---- .../examples/generate-topography-figure.py | 226 ++++++++++++++++++ .../examples/topography-data.json | 1 + .../figures/topography.png | Bin 0 -> 153534 bytes .../metadata.yml | 2 +- 5 files changed, 288 insertions(+), 43 deletions(-) create mode 100644 articles/boundary-conditions-on-non-planar-boundaries/examples/generate-topography-figure.py create mode 100644 articles/boundary-conditions-on-non-planar-boundaries/examples/topography-data.json create mode 100644 articles/boundary-conditions-on-non-planar-boundaries/figures/topography.png diff --git a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md index cae0f65..0e74d10 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md +++ b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md @@ -537,19 +537,27 @@ No exact solution has both a curved boundary and a laterally varying viscosity, so the case where weak constraints are most often reported to give trouble is a separate test with a trivial geometry. SolCx is that test: the unit box, free slip on all four walls, viscosity 1 to the left of $x = 0.5$ and $\eta_B$ to the -right, and `uw.analytic.SolCx` publishes the exact surface stress. Three walls -carry the ordinary component condition and the treatment under test is on the -top wall alone. - -On a box every treatment here reduces to holding one velocity component, so -nothing in it is about normals. What it can say is whether a weak imposition -holds the traction it was given when the viscosity beside it jumps. +right. `uw.analytic.SolCx` publishes the exact dynamic topography on the top +wall, so this one can be drawn rather than tabulated. Three walls carry the +ordinary component condition and the treatment under test is on the top wall +alone. + +On a box every treatment reduces to holding one velocity component, so nothing +here is about normals. What it can say is whether a weak imposition holds the +traction it was given when the viscosity beside it jumps. + +```{figure} figures/topography.png +:alt: Two line plots of surface topography along the top wall from x=0 to x=1, mean removed, at viscosity contrasts of 100 and a million. In both, the exact answer is a thick grey curve falling from +0.29 at the left, flattening near +0.21, jumping down sharply at the viscosity step at x=0.5 and continuing down to -0.38 at the right. At a contrast of 100 every treatment lies on the grey curve, apart from the rotated constraint which spikes to -0.50 and +0.27 in the last two nodes at x=1. At a contrast of a million the picture separates: the component Dirichlet, the rotated reaction and the corrected multiplier still lie on the exact curve; Nitsche at gamma=10 leaves the panel entirely and peaks at 4.09 against a signal of 0.38; and the multiplier as the API returns it is a nearly flat line near zero, reaching only 0.04, having lost almost all of the traction to the augmented-Lagrangian term. Adding that term back, drawn as a dotted blue line, puts it on the exact curve again. + +Predicted against computed, for each treatment. Every curve is mean-removed, +because the box is enclosed and the level of the topography is a gauge. At a +contrast of 100 the treatments agree with each other and with the exact answer; +at a million, three of the five have something wrong with them, and each is +wrong in a different way. +``` -Relative $l_2$ error of $\sigma_{zz}$ along the top wall, mean removed, at -32 × 32 elements. Each entry is the whole wall and then the wall with two -elements trimmed from each end; the corners are where the treatment under test -meets the side walls, and the difference between the two numbers is what -happens there. Nitsche at $\gamma = 10$, penalty at $10^4$. +Relative $l_2$ error along the wall, at 32 × 32 elements, over the whole wall +and then with two elements trimmed from each end: | $\eta_B/\eta_A$ | component Dirichlet | penalty | Nitsche | multiplier | rotated | |---|---|---|---|---|---| @@ -557,46 +565,56 @@ happens there. Nitsche at $\gamma = 10$, penalty at $10^4$. | 10² | 0.072 / 0.081 | 0.056 / 0.060 | 0.073 / 0.081 | 0.072 / 0.081 | 0.343 / 0.083 | | 10³ | 0.075 / 0.084 | diverged | 0.076 / 0.085 | 0.075 / 0.084 | 0.345 / 0.086 | | 10⁴ | 0.076 / 0.085 | diverged | 0.087 / 0.095 | 0.076 / 0.085 | 0.345 / 0.086 | -| 10⁶ | 0.076 / 0.085 | diverged | **2.77 / 2.14** | 0.075 / 0.084 | 0.345 / 0.086 | +| 10⁶ | 0.076 / 0.085 | diverged | **2.77 / 2.14** | **1.04** | 0.345 / 0.086 | -**Read the first column first.** The component Dirichlet condition is exact, -has no parameter, and its velocity error is 8.8 × 10⁻⁶ at a contrast of $10^6$. -It still reads 0.085. That number is the recovery's error and not a boundary +**Read the first column first.** The component Dirichlet condition is exact, has +no parameter, and its velocity error is 8.8 × 10⁻⁶ at a contrast of $10^6$. It +still reads 0.085. That number is the recovery's error and not a boundary condition's: on the stiff half the recovered $\sigma_{zz}$ is a difference between the pressure and $2\eta\,\partial_z u_z$ with $\eta = 10^6$, so a relative velocity error of $10^{-5}$ arrives in the stress at the size of the -signal. No treatment can be called worse than the reference unless it is worse -than that, and at contrasts up to $10^4$ none of them is by more than about a -tenth. +signal. Nothing here can be called worse than the reference unless it is worse +than that, and up to a contrast of $10^4$ nothing is by more than about a tenth. **Nitsche fails at $10^6$ and the leak does not show it.** At $\gamma = 10$ the -recovered surface stress is 2.77 — nearly three times the signal — while the -constraint is held to 3.2 × 10⁻⁴ on the soft half of the wall and -1.5 × 10⁻⁷ on the stiff half. Raising $\gamma$ fixes it: 0.16 at $\gamma = 100$ -and 0.079 at $\gamma = 1000$, which is the reference floor. So the stabilisation -parameter has to be raised with the viscosity contrast, and the quantity that -says whether it is high enough is the stress rather than the leak. This is the -same shape as the curved-boundary result, arrived at from the other side: the -constraint being satisfied is not evidence that the traction is right. - -The two exact treatments carry no such parameter and sit at the reference floor -at every contrast. +recovered topography peaks at 4.09 against a signal of 0.38, while the +constraint is held to 3.2 × 10⁻⁴ on the soft half of the wall and 1.5 × 10⁻⁷ on +the stiff half. Raising $\gamma$ fixes it — 0.16 at $\gamma = 100$, 0.079 at +$\gamma = 1000$, which is the reference floor — so the stabilisation parameter +has to be raised with the viscosity contrast, and the quantity that says whether +it is high enough is the stress rather than the leak. That is the same shape as +the curved-boundary result arrived at from the other side. + +**The multiplier loses the traction to the augmentation.** At a contrast of +$10^6$ the multiplier field as `multiplier()` returns it peaks at 0.042 against +an exact 0.383, and it is anti-correlated with the right answer (−0.53) — the +flat blue line in the figure. The solve is not at fault: the same run's velocity +error is 8.8 × 10⁻⁶ and its recovered stress is as good as the reference. The +default augmentation is $r = 10^4\mu(x)$, which the viscosity step makes $10^{10}$ +on the stiff half, and the momentum row carries +$h + r(\mathbf{u}\cdot\hat{\mathbf{n}} - g)$. Almost all of the traction is in +the second term. Adding it back gives 0.047 — better than every other route in +this test, including the recovery — and it is the dotted curve lying on the +exact one. Turning $r$ down instead is not available: at this contrast both +$r = 0$ and $r = 10^2$ fail to solve. This is the same defect as in the annulus, +underworld3#607, and the contrast is what makes it severe. + +**The rotated constraint disagrees at the corner and nowhere else.** Trimmed, it +matches the reference at every contrast; untrimmed it is six times worse, and +the whole of that is two nodes — the spike at $x = 1$ in both panels of the +figure. Against the Dirichlet run at a contrast of 10, the pressure differs by +0.23 at the corner node and 0.12 at its neighbour, while everything below the +top row agrees to 4 × 10⁻⁴ rms. The corner is a node the rotated constraint and +the side wall's component condition both hold, and both runs pin the same two +velocity components there, so the difference is in how the corner row is +assembled rather than in what is asked of it. This is underworld3#608. **The penalty needs a coefficient matched to the local viscosity, and we could not write one that solves.** A constant $10^4$ manages a contrast of $10^2$ and -then fails the line search. Multiplying the coefficient by the piecewise -viscosity is the obviously right thing to want, and it fails at every magnitude -we tried, from $\mu$ to $10^3\mu$, against both normals. +then fails the line search. Multiplying by the piecewise viscosity is the +obviously right thing to want, and it fails at every magnitude we tried, from +$\mu$ to $10^3\mu$, against both normals. -**The rotated constraint disagrees at the corner and nowhere else.** Trimmed, it -matches the reference. Untrimmed it is six times worse, and the whole of that -difference is two nodes: against the Dirichlet run at a contrast of 10, the -pressure differs by 0.23 at the corner node and 0.12 at its neighbour along the -top, while everything below the top row agrees to 4 × 10⁻⁴ rms. The corner is a -node the rotated constraint and the side wall's component condition both hold. -Both runs pin the same two velocity components there, so the difference is in -how the corner row is assembled rather than in what is being asked of it. This -is underworld3 issue #608. ## Using it diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/generate-topography-figure.py b/articles/boundary-conditions-on-non-planar-boundaries/examples/generate-topography-figure.py new file mode 100644 index 0000000..f5bab72 --- /dev/null +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/generate-topography-figure.py @@ -0,0 +1,226 @@ +"""Figure: the surface topography each treatment predicts, against the exact one. + +SolCx publishes the exact dynamic topography on the top wall +(`uw.analytic.SolCx.topography_top`, which is -sigma_zz), so the comparison can +be drawn rather than tabulated. Two panels, the same five curves in each, at a +viscosity contrast of 100 and of a million. + +The topography is mean-removed in every case: the box is enclosed, so the +pressure and with it the level of sigma_zz is fixed only up to a constant, and +the deviation is the part that is determined and the part topography is built +from. + +Two routes are drawn, and they are not the same measurement: + + * the recovered traction, projected out of the solved velocity and pressure, + which is the only route the weak treatments have; + * the constraint reaction -- `boundary_normal_traction` for the rotated + constraint and the multiplier field for the constraint method -- which the + solve returns as an unknown. Its sign convention is the traction holding the + wall, which is the topography's sign directly. + +Colour carries the treatment and the exact answer is a thick grey line behind +everything, so no curve is identified by colour alone against the reference. + + python3 generate-topography-figure.py + +Writes figures/topography.png. Run against underworld3 `development` at commit +`8b7c8b9e`. +""" +import json +import pathlib +import sys + +import numpy as np + +import matplotlib +matplotlib.use("Agg") +import matplotlib.pyplot as plt + +import solcx as C + +HERE = pathlib.Path(__file__).resolve().parent +OUT = HERE.parent / "figures" / "topography.png" +# The solves take about ten minutes and the figure is redrawn far more often +# than it is recomputed, so the curves are cached beside the script the way +# `rotated-basis-data.json` is. Delete it, or pass --recompute, to re-solve. +CACHE = HERE / "topography-data.json" + +RES = 32 +CONTRASTS = (1.0e2, 1.0e6) +MODES = ("dirichlet", "penalty", "nitsche", "constraint", "rotated") + +# Validated categorical palette, as in the multigrid note's figure. +COLOUR = { + "dirichlet": "#0b0b0b", + "penalty": "#eb6834", + "nitsche": "#c13ec1", + "constraint": "#2a78d6", + "constraint+r": "#2a78d6", + "rotated": "#1e9e6a", +} +LABEL = { + "dirichlet": "component Dirichlet", + "penalty": "penalty, $10^4$", + "nitsche": r"Nitsche, $\gamma = 10$", + "constraint": "multiplier, as returned", + "constraint+r": r"multiplier $+\;r(\mathbf{u}\cdot\hat{\mathbf{n}})$", + "rotated": "rotated (reaction)", +} +STYLE = {"dirichlet": (0, (4, 2)), "penalty": "-", "nitsche": "-", + "constraint": "-", "constraint+r": (0, (1, 1.6)), "rotated": "-"} +INK = "#0b0b0b" +INK_MUTED = "#52514e" +GRID = "#e4e3df" +EXACT = "#b9b7b2" + + +def profile(mode, eta_B): + """{name: (x, topography)} along the top wall, mean removed, and the exact one. + + The reaction where there is one, the recovered traction otherwise: that is + what a user of each treatment would actually have. The constraint method + returns two curves -- the multiplier as the API returns it, and the + multiplier plus the augmented-Lagrangian share r(u.n - g), which is the + other half of the traction the momentum row carries (underworld3#607). + """ + mesh, stokes, v, exact = C.build(mode, res=RES, eta_B=eta_B) + stokes.solve() + if not C.converged(stokes): + print("%-10s eta_B %.0e diverged" % (mode, eta_B), flush=True) + return None + read = C.reaction_traction(stokes, mode) + if read is None: + coords, values = C.recovered_traction(mesh, stokes) + curves = {mode: -np.asarray(values)} # h = -sigma_zz + else: + coords, values = read + curves = {mode: np.asarray(values)} # the reaction already carries it + if mode == "constraint": + # r is the default augmentation: augmentation_base (1e4) times the local + # viscosity, so 1e4 on the soft half and 1e4.eta_B on the stiff one. On a + # flat top wall u.n is u_y. + import underworld3 as uw + tree = uw.kdtree.KDTree(np.ascontiguousarray(v.coords)) + index = np.asarray(tree.query(np.ascontiguousarray(coords), 1)[1]).flatten() + u_n = np.squeeze(np.asarray(v.array))[index][:, 1] + r = 1.0e4 * np.where(coords[:, 0] < 0.5, 1.0, eta_B) + curves["constraint+r"] = curves[mode] + r * u_n + + order = np.argsort(coords[:, 0]) + x = coords[order, 0] + truth = exact.topography_top(coords)[order] + truth = truth - truth.mean() + out = {} + for name, values in curves.items(): + got = values[order] - values.mean() + # The sign convention is checked rather than assumed, and a curve that + # comes back anti-correlated is drawn AND named rather than flipped. + correlation = float(np.dot(got, truth) + / (np.linalg.norm(got) * np.linalg.norm(truth) + 1e-300)) + print("%-14s eta_B %.0e max|h| %.4f corr %+.3f l2 %.3f" + % (name, eta_B, np.abs(got).max(), correlation, + np.linalg.norm(got - truth) / np.linalg.norm(truth)), flush=True) + out[name] = (x, got) + return out, (x, truth) + + +def panel(ax, results, truth, title, ylim): + x, exact = truth + ax.plot(x, exact, color=EXACT, linewidth=5.0, solid_capstyle="round", + zorder=1, label="exact") + for mode, (xs, got) in results.items(): + ax.plot(xs, got, linestyle=STYLE[mode], color=COLOUR[mode], + linewidth=1.6, zorder=3, label=LABEL[mode]) + ax.axvline(0.5, color=GRID, linewidth=1.0, zorder=0) + ax.text(0.505, ylim[1] * 0.80, "viscosity step", fontsize=8.5, + color=INK_MUTED, ha="left", va="top") + ax.set_xlabel("$x$ along the top wall", fontsize=9.5, color=INK_MUTED) + ax.set_title(title, fontsize=10.5, color=INK, pad=10) + ax.set_xlim(0.0, 1.0) + ax.set_ylim(*ylim) + ax.grid(True, which="major", color=GRID, linewidth=0.8, zorder=0) + ax.set_axisbelow(True) + for side in ("top", "right"): + ax.spines[side].set_visible(False) + for side in ("left", "bottom"): + ax.spines[side].set_color(GRID) + ax.tick_params(colors=INK_MUTED, labelsize=9) + + +def compute(): + data, truth = {}, {} + for eta_B in CONTRASTS: + got = {} + for mode in MODES: + out = profile(mode, eta_B) + if out is None: + continue + curves, truth[eta_B] = out + got.update(curves) + data[eta_B] = got + CACHE.write_text(json.dumps( + {"contrasts": list(CONTRASTS), + "truth": {str(k): [v[0].tolist(), v[1].tolist()] for k, v in truth.items()}, + "data": {str(k): {m: [c[0].tolist(), c[1].tolist()] for m, c in v.items()} + for k, v in data.items()}})) + return data, truth + + +def cached(): + raw = json.loads(CACHE.read_text()) + truth = {float(k): (np.array(v[0]), np.array(v[1])) + for k, v in raw["truth"].items()} + data = {float(k): {m: (np.array(c[0]), np.array(c[1])) for m, c in v.items()} + for k, v in raw["data"].items()} + return data, truth + + +def main(): + if CACHE.exists() and "--recompute" not in sys.argv: + data, truth = cached() + print("drawn from", CACHE.name, "-- pass --recompute to re-solve") + else: + data, truth = compute() + + span = max(np.abs(truth[e][1]).max() for e in CONTRASTS) + ylim = (-1.6 * span, 1.6 * span) + + fig, axes = plt.subplots(1, 2, figsize=(9.6, 4.3), sharey=True) + fig.patch.set_facecolor("white") + for ax, eta_B in zip(axes, CONTRASTS): + panel(ax, data[eta_B], truth[eta_B], + r"$\eta_B/\eta_A = 10^{%d}$" % int(round(np.log10(eta_B))), ylim) + # Name what left the panel rather than leaving a curve to run off it. + for mode, (xs, got) in data[eta_B].items(): + if np.abs(got).max() > ylim[1]: + ax.annotate("%s leaves the panel:\npeaks at %.2f" + % (LABEL[mode], np.abs(got).max()), + xy=(0.03, ylim[0] * 0.72), fontsize=8.5, + color=COLOUR[mode], linespacing=1.35) + missing = [m for m in MODES if m not in data[eta_B]] + if missing: + ax.annotate("does not solve: %s" % ", ".join(LABEL[m] for m in missing), + xy=(0.03, ylim[1] * 0.86), fontsize=8.5, color=INK_MUTED) + axes[0].set_ylabel("surface topography, mean removed", + fontsize=9.5, color=INK_MUTED) + # The legend goes below the panels: at 1e6 every corner of both axes has a + # curve in it, and a legend inside covered the multiplier's collapse. + # Both panels, deduplicated: the penalty only converges in the left one and + # would otherwise be an unlabelled curve. + handles, labels = [], [] + for ax in axes: + for handle, label in zip(*ax.get_legend_handles_labels()): + if label not in labels: + handles.append(handle) + labels.append(label) + fig.legend(handles, labels, frameon=False, fontsize=9, labelcolor=INK_MUTED, + loc="lower center", ncol=3, handlelength=2.6, + bbox_to_anchor=(0.5, -0.01)) + fig.tight_layout(rect=(0, 0.15, 1, 1)) + fig.savefig(OUT, dpi=200, facecolor="white") + print("wrote", OUT) + + +if __name__ == "__main__": + main() diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/topography-data.json b/articles/boundary-conditions-on-non-planar-boundaries/examples/topography-data.json new file mode 100644 index 0000000..409aae1 --- /dev/null +++ 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zC=pv5B&E+m&yztg&QgZk{gV-sCd39;P-K-qJ*q9c_Y zE)VX?zX!`>WUuwD9W2XD$h0M5*L#yjb~O*;n;CZ^;9>8u63HbCN1-lU_Pd|LpK_yE`B Date: Wed, 19 Aug 2026 18:56:35 +1000 Subject: [PATCH 10/33] The multiplier IS the consistent boundary flux, and the interior DOFs are masked Two corrections and one identity, all from Louis reading the draft against the free-surface work. The note said the multiplier's interior degrees of freedom are screened and that the screening is where the approximation enters. They are not screened, they are CONSTRAINED OUT of the global system in the PetscSection before the solve sees them (_constrain_interior_multipliers_in_section), which is documented as lossless: the interior rows are the screening block alone, so the interior multiplier is determined by the boundary trace and pinning it moves a converged solve by ~1e-8 in velocity. They neither cost anything nor limit the accuracy. The identity: the momentum row's boundary load is M_Gamma (h + r(u.n - g)), and at convergence that balances the volume residual restricted to the boundary, which is the nodal load the consistent boundary flux back-calculation reads (Zhong, Gurnis & Hulbert 1993). So the corrected multiplier is the CBF traction de-smeared by the boundary mass -- the same computation, arrived at by carrying the traction as an unknown rather than reading it out of the residual. Measured across two solves they agree to 3.2% and 4.9%, inside each route's own error. That is why the free-surface work was right on both counts: it rejected the multiplier AS RETURNED, which is missing the augmentation share, and kept the rotated lid with the CBF reaction, which is the same quantity with nothing missing. The corner spike is re-framed. It is the documented behaviour of the reaction recovery at a node where a rotated wall meets an essential one -- the value reported there is the essential constraint's reaction, not the wall's traction -- and the practice, which the free-surface SolCx work already followed, is to leave the corner out of the rotation. Control: the same CBF read on a lid held by the component condition alone peaks at 0.381 against an exact 0.379 with no spike, where the rotated run peaks at 0.497. underworld3#608 is re-scoped to match. New: underworld3#614, boundary_flux() returns ~1e12 on Stokes_Constrained while reading the exact topography to 8% on an ordinary solve. It is the cross-check that would verify the identity within a single solve, and it is unavailable. The Zhong citation is pinned in a references.bib: the DOI contains parentheses and MyST's inline DOI form truncates at the first one. An at-sign in a bibtex COMMENT fails the whole file. Underworld development team with AI support from Claude Code --- ...ary-conditions-on-non-planar-boundaries.md | 102 +++++++++++++----- .../references.bib | 27 +++++ 2 files changed, 104 insertions(+), 25 deletions(-) create mode 100644 articles/boundary-conditions-on-non-planar-boundaries/references.bib diff --git a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md index 0e74d10..dcac089 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md +++ b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md @@ -150,18 +150,20 @@ with $h$ entering the momentum row as the traction $h\hat{\mathbf{n}}$ that hold the constraint. It is a Lagrange multiplier, and the system becomes a larger saddle point: velocity, pressure, and now $h$. -Nothing is being traded here. The constraint row is exact, so unlike a penalty -there is no parameter whose size decides how well it holds. Two practical -things do have to be dealt with, and they are where the approximation enters: - -- The multiplier is only defined on the boundary, so its interior degrees of - freedom are singular. Underworld screens them with a small $\varepsilon$ - (default $10^{-6}$), which is the one place the constraint is relaxed. +The constraint row is exact, so unlike a penalty there is no parameter whose +size decides how well it holds. Two practical things do have to be dealt with. + +- $h$ is carried as a full-domain field but only its boundary trace means + anything, so the interior degrees of freedom are constrained out of the global + system in the section before the solve sees it. They are not solved for and + they do not enlarge the $[p, h]$ block. The interior rows are the screening + block alone, so the interior multiplier is determined by the boundary trace and + carries no independent signal. - The $[p, h]$ Schur complement is poorly conditioned on its own, so an augmented-Lagrangian term $r(\mathbf{u}\cdot\hat{\mathbf{n}} - g)\hat{\mathbf{n}}$ - is added to the momentum row. This conditions the block **without biasing the - multiplier**, because the $h$ row still carries the exact constraint — so - unlike a penalty, the accuracy does not depend on $r$. + is added to the momentum row. It does not change what the constraint enforces, + because the $h$ row still carries the exact constraint. It does change what $h$ + *is*, which is the subject of a section below. ```python stokes = uw.systems.Stokes_Constrained(mesh, velocityField=v, pressureField=p) @@ -169,10 +171,12 @@ h = stokes.add_constraint_bc(0.0, "Upper") stokes.solve() ``` -And the reason to care about it beyond the constraint: **at convergence, $h$ on -the boundary is the normal traction.** It is not recovered from the velocity -field afterwards — it is an unknown the solve returned, available through -`multiplier` and, divided by $\Delta\rho g$, through `topography`. +And the reason to care about it beyond the constraint: **at convergence the +momentum row's boundary term is the normal traction.** It is not recovered from +the velocity field afterwards — it is an unknown the solve returned, available +through `multiplier` and, divided by $\Delta\rho g$, through `topography`. With +an augmentation in place that term is $h + r(\mathbf{u}\cdot\hat{\mathbf{n}} - g)$ +rather than $h$, and the difference between the two is measured below. ### Rotating the degrees of freedom @@ -333,8 +337,8 @@ what Nitsche and the rotated constraint use by default. Nitsche leaks parts in a thousand and improves roughly as $h^{1.9}$ — the rate consistency buys. The multiplier starts an order of magnitude better and falls -much faster, near $h^{3.9}$, because its only approximation is the screening of -the interior multiplier degrees of freedom rather than the enforcement itself. +much faster, near $h^{3.9}$, because the constraint row is an equation rather +than a term whose weight has to be chosen. The rotated constraint does not move at all: it sits at the solver's floor at every resolution, because the mesh has nothing to do with it. The penalty against the node normal does not improve with the mesh either, and for the @@ -531,6 +535,43 @@ multiplier does depend on $r$, which the method's own documentation says it does not, and the note said so too. What does not depend on $r$ is the traction, once both of its parts are added up. This is underworld3 issue #607. +### The multiplier and the consistent boundary flux are the same object + +The correction above is not a patch. Write the momentum row's boundary term out +and the identity is immediate: the assembled load is +$M_\Gamma\,(h + r(\mathbf{u}\cdot\hat{\mathbf{n}} - g))$, and at convergence it +balances the volume residual restricted to the boundary, which is precisely the +nodal load the consistent boundary flux back-calculation reads +[@Zhong_1993]. So + +$$ +h + r(\mathbf{u}\cdot\hat{\mathbf{n}} - g) + = -M_\Gamma^{-1} \left. (A\mathbf{u} - \mathbf{b}) \right|_\Gamma , +$$ + +the CBF traction de-smeared with the boundary mass. The multiplier is not a +second, independent estimate of the surface stress: it is the same computation +the rotated constraint's reaction performs, arrived at by carrying the traction +as an unknown instead of reading it out of the residual afterwards. Dropping the +$r$ term is dropping part of the load, which is why it fails exactly where $r$ is +large. + +Measured across the two solves — they are different discrete problems, so this +is agreement rather than an identity check — the corrected multiplier and the +rotated reaction differ by 3.2% at a contrast of 100 and 4.9% at $10^6$, both +inside each route's own error against the exact answer (5 to 9%). We could not +compare them within a single solve: `boundary_flux`, which is the CBF primitive, +returns values of order $10^{12}$ on `Stokes_Constrained` while reading the exact +topography to 8% on an ordinary solve. That is underworld3#614. + +This also settles a question the free-surface work left open. That work used +SolCx the same way, to choose among topography recoveries, and landed on a +rotated free-slip lid with the CBF reaction and a continuous pressure — corr +0.999, relative $l_2$ 0.04 — while rejecting the multiplier. Both conclusions +were right about what was in front of them: the multiplier *as returned* is +missing the augmentation share, and the CBF reaction is the same quantity with +nothing missing. + ### The other half: a lateral viscosity contrast No exact solution has both a curved boundary and a laterally varying viscosity, @@ -599,15 +640,26 @@ exact one. Turning $r$ down instead is not available: at this contrast both $r = 0$ and $r = 10^2$ fail to solve. This is the same defect as in the annulus, underworld3#607, and the contrast is what makes it severe. -**The rotated constraint disagrees at the corner and nowhere else.** Trimmed, it -matches the reference at every contrast; untrimmed it is six times worse, and -the whole of that is two nodes — the spike at $x = 1$ in both panels of the -figure. Against the Dirichlet run at a contrast of 10, the pressure differs by -0.23 at the corner node and 0.12 at its neighbour, while everything below the -top row agrees to 4 × 10⁻⁴ rms. The corner is a node the rotated constraint and -the side wall's component condition both hold, and both runs pin the same two -velocity components there, so the difference is in how the corner row is -assembled rather than in what is asked of it. This is underworld3#608. +**The rotated constraint's spike at $x = 1$ is the corner, and the corner is a +known one.** Trimmed by two elements at each end it matches the reference at +every contrast; untrimmed it is six times worse, and the whole of that is the +two end nodes. A node where a rotated wall meets a wall held by an ordinary +component condition is constrained by both. Its velocity degrees of freedom are +constrained out of the global vector, the rotation drops it, and the reaction +recovery still reports a value there — the reaction of the *essential* +constraint, which is not the wall's traction. Underworld's own implementation +notes say so. The same recovery on a lid held by the component condition alone +has no spike at all: the consistent boundary flux there reads the exact +topography to 8% with a peak of 0.381 against an exact 0.379, where the rotated +run peaks at 0.497. + +So this is a thing to know rather than a thing to fix in the solve: leave the +corner out of the rotation, or drop it from the recovered trace. It is also why +the free-surface work that used SolCx to choose a topography recovery never saw +it. The remaining difference is small and real — at a contrast of 10 the +pressure differs from the Dirichlet run by 0.23 at the corner node and 0.12 at +its neighbour, while everything below the top row agrees to 4 × 10⁻⁴ rms — +which is underworld3#608. **The penalty needs a coefficient matched to the local viscosity, and we could not write one that solves.** A constant $10^4$ manages a contrast of $10^2$ and diff --git a/articles/boundary-conditions-on-non-planar-boundaries/references.bib b/articles/boundary-conditions-on-non-planar-boundaries/references.bib new file mode 100644 index 0000000..261b510 --- /dev/null +++ b/articles/boundary-conditions-on-non-planar-boundaries/references.bib @@ -0,0 +1,27 @@ +% Pinned here rather than fetched at build time, following the convention in the +% MMPDE note: a build that depends on doi.org answering can publish a note with a +% broken citation, and a deposited PDF cannot be repaired afterwards. +% +% This one has a second reason. The DOI contains parentheses, +% 10.1016/0031-9201(93)90078-N, and MyST's inline DOI citation form stops at the +% first "(" -- the build reported: Could not find DOI "10.1016/0031-9201". A +% bibtex key sidesteps that. +% +% Do not write an at-sign in these comments. The bibtex reader scans for one +% wherever it appears and then expects an entry: an at-sign inside a comment +% here failed the whole file with "expected lbrace, got label". + +@article{Zhong_1993, + title = {Accurate determination of surface normal stress in viscous flow from a consistent boundary flux method}, + volume = {78}, + ISSN = {0031-9201}, + url = {http://dx.doi.org/10.1016/0031-9201(93)90078-N}, + DOI = {10.1016/0031-9201(93)90078-n}, + number = {1-2}, + journal = {Physics of the Earth and Planetary Interiors}, + publisher = {Elsevier BV}, + author = {Zhong, Shijie and Gurnis, Michael and Hulbert, Gregory}, + year = {1993}, + month = jun, + pages = {1--8} +} From 9d1ca02cefdfd2cabeac4d8d939d5ca8f67e6999 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Wed, 19 Aug 2026 21:59:25 +1000 Subject: [PATCH 11/33] Re-measure against the fixed solver, add the cost, and withdraw a contaminated table The three defects this note turned up are fixed in underworld3 PR #617, so every table here is re-measured against a solver that behaves. The SolCx half is the one that changes. The three exact treatments -- component Dirichlet, multiplier, rotated -- now agree to three figures at every viscosity contrast from 10 to 1e6, whole wall and trimmed alike. Before, the rotated column read 0.322 at a contrast of 10 against the reference's 0.048, all of it two corner nodes, and the multiplier was an order of magnitude out at 1e6. Getting those three to agree was the point. A bare penalty coefficient now has numbers instead of "diverged": 0.045 at a contrast of 10, and 0.992 at 1e6, which is to say none of the signal. Scaling it by the local viscosity still does not solve. WITHDRAWN: the earlier SolCx numbers, including the Nitsche column. They were taken while concurrent runs shared one mesh-cache file -- StructuredQuadBox keys its cache on the box corners and NOT on the element resolution, so runs at different resolutions silently swap meshes (underworld3#618, filed). A marginal solve then flips between converged and diverged for reasons that look like the method, which is exactly how it fooled me. Everything in that section was re-measured sequentially on a cleared cache. Nitsche has no column now: our configuration of it on this box converges at a contrast of 1e6 and fails the line search at 10, at both resolutions and at gamma = 10, 100 and 1000, and we are not confident enough in that setup to publish numbers from it. New: what each treatment COSTS. The solve is the same for all four within the spread; the recovery is 15 to 25 times cheaper for the reaction and the multiplier than for a projection, because they read the state the solve already returned. And a "which one to use" section, which the accuracy and the cost now point at together. The multiplier section is rewritten around the fix rather than the defect, with the clean SolCx numbers: h alone reads 0.042 against an exact 0.381 and is anti-correlated with it; h + r(u.n - g) reads 0.382. Underworld development team with AI support from Claude Code --- ...ary-conditions-on-non-planar-boundaries.md | 287 +++++++++++------- .../examples/generate-locking-figure.py | 3 +- .../examples/leak.py | 3 +- .../examples/solcx.py | 14 +- .../examples/stress.py | 38 ++- .../examples/timing.py | 99 ++++++ .../examples/topography-data.json | 2 +- .../figures/topography.png | Bin 153534 -> 140358 bytes 8 files changed, 323 insertions(+), 123 deletions(-) create mode 100644 articles/boundary-conditions-on-non-planar-boundaries/examples/timing.py diff --git a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md index dcac089..f1de09d 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md +++ b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md @@ -237,7 +237,6 @@ the null space, which is a property only the constrained problem has. None of this is an argument against the method. It is an argument for knowing what is being taken on, and it is the honest reason a weakly imposed condition survives in codes that could do this instead. - ## Which normal A question with a less obvious answer than it looks, and the measurements @@ -431,10 +430,10 @@ compared unsigned. | cell size | penalty, facet | penalty, node | Nitsche | constraint | rotated | rotated, reaction | multiplier field | |---|---|---|---|---|---|---|---| -| 0.150 | 1.5 × 10⁻¹ | 2.5 × 10⁻² | 5.9 × 10⁻² | 2.4 × 10⁻² | 2.4 × 10⁻² | 6.8 × 10⁻³ | 3.3 × 10⁻² | -| 0.100 | 1.3 × 10⁻¹ | 1.1 × 10⁻² | 2.4 × 10⁻² | 1.0 × 10⁻² | 1.0 × 10⁻² | 3.3 × 10⁻³ | 2.9 × 10⁻² | -| 0.075 | 1.1 × 10⁻¹ | 7.2 × 10⁻³ | 1.5 × 10⁻² | 6.3 × 10⁻³ | 6.2 × 10⁻³ | 2.1 × 10⁻³ | 2.5 × 10⁻² | -| 0.050 | 6.3 × 10⁻² | 3.6 × 10⁻³ | 6.3 × 10⁻³ | 2.7 × 10⁻³ | 2.7 × 10⁻³ | 1.1 × 10⁻³ | 1.5 × 10⁻³ | +| 0.150 | 1.5 × 10⁻¹ | 2.5 × 10⁻² | 5.9 × 10⁻² | 2.4 × 10⁻² | 2.4 × 10⁻² | 6.8 × 10⁻³ | 8.6 × 10⁻³ | +| 0.100 | 1.3 × 10⁻¹ | 1.1 × 10⁻² | 2.4 × 10⁻² | 1.0 × 10⁻² | 1.0 × 10⁻² | 3.3 × 10⁻³ | 8.5 × 10⁻⁴ | +| 0.075 | 1.1 × 10⁻¹ | 7.2 × 10⁻³ | 1.5 × 10⁻² | 6.3 × 10⁻³ | 6.2 × 10⁻³ | 2.1 × 10⁻³ | 1.7 × 10⁻³ | +| 0.050 | 6.3 × 10⁻² | 3.6 × 10⁻³ | 6.3 × 10⁻³ | 2.7 × 10⁻³ | 2.7 × 10⁻³ | 1.1 × 10⁻³ | 1.4 × 10⁻³ | The first five columns are all the recovered stress, so they differ only by which boundary condition produced the field. The last two are the reaction and @@ -456,8 +455,15 @@ them share. **The reaction is about three times better than the recovery, on the same solve.** 2.1 × 10⁻³ against 6.2 × 10⁻³ at cell 0.075, and it converges at the -same rate rather than a better one. It costs nothing: it is the residual the -solve has already assembled, and no field is differentiated to get it. +same rate rather than a better one. It costs nothing — it is the residual the +solve has already assembled, and no field is differentiated to get it — and the +timings below put a number on "nothing". + +The multiplier's traction is the same story from the other side, and lands in the +same place: 8.5 × 10⁻⁴ to 1.7 × 10⁻³ over the three finer meshes. Its column does +not fall smoothly with $h$, because part of what it reports is the augmentation +times the constraint residual, and how far a particular solve drove that residual +is not a function of the mesh. **The parameter that was enough for the leak is not enough for the stress.** Nitsche at $\gamma = 10$ leaks 1.2 × 10⁻³ in this problem and gets the stress @@ -513,27 +519,42 @@ sufficient check: at a coefficient of $10^3$ the facet-normal penalty leaks leaks 1 × 10⁻⁷ and is 26% wrong. Over that range the constraint improves by five orders of magnitude and the answer gets steadily worse. -### The multiplier is not quite the whole traction +### The multiplier was not the whole traction -The multiplier column above sits near 3 × 10⁻² for three resolutions and then -drops to 1.5 × 10⁻³, which is not a convergence rate. The cause is the -augmented-Lagrangian term. +The multiplier column of the table above used to sit near 3 × 10⁻² for three +resolutions and then drop, which is not a convergence rate. The cause was the +augmented-Lagrangian term, and it was a defect rather than a property of the +method. The momentum row carries both the multiplier and the augmentation, $(h + r(\mathbf{u}\cdot\hat{\mathbf{n}} - g))\hat{\mathbf{n}}$, so the traction -holding the boundary is $h + r(\mathbf{u}\cdot\hat{\mathbf{n}} - g)$ and not -$h$ alone. The second term vanishes only where the constraint row is satisfied -exactly. Discretely it is satisfied to the solver's tolerance, and with the -default $r = 10^4\mu$ the residual $\mathbf{u}\cdot\hat{\mathbf{n}} \approx -5 \times 10^{-7}$ multiplies up to a few per cent of a surface traction of 0.15. - -Adding the augmentation share back gives, at cell 0.075, 1.7 × 10⁻³ instead of -2.5 × 10⁻², and at cell 0.100, 8.5 × 10⁻⁴ instead of 2.9 × 10⁻². Lowering $r$ to -$10^2\mu$ has the same effect without the correction — 2.6 × 10⁻³ — because it -shrinks the term rather than accounting for it. So the accuracy of the reported -multiplier does depend on $r$, which the method's own documentation says it does -not, and the note said so too. What does not depend on $r$ is the traction, once -both of its parts are added up. This is underworld3 issue #607. +holding the boundary is $h + r(\mathbf{u}\cdot\hat{\mathbf{n}} - g)$ and not $h$ +alone. The second term vanishes only where the constraint row is satisfied +exactly. Discretely it is satisfied to the solver's tolerance, and $r$ multiplies +that residual straight back into the traction. Underworld returned $h$. + +On the annulus, with a uniform viscosity and the default $r = 10^4\mu$, the +omitted share is a few per cent. On SolCx it is nearly everything, because the +default $r$ is viscosity-weighted and the step takes it to $10^{10}$ on the stiff +half. At a contrast of $10^6$, against an exact surface topography whose +deviation peaks at 0.381: + +| read | peak | correlation with the exact | relative $l_2$ | +|---|---|---|---| +| the multiplier $h$ | 0.042 | **−0.52** | 0.96 | +| $h + r(\mathbf{u}\cdot\hat{\mathbf{n}} - g)$ | 0.382 | +0.999 | 0.084 | + +An order of magnitude low and pointing the wrong way, on a solve whose velocity +error is 8.8 × 10⁻⁶. Underworld3 issue #607: `traction()` now returns the sum and +`topography()` is built on it, so the tables here are the corrected read and a +caller who asks for topography gets it. + +Worth saying how it survived. The method's own validation scored a *correlation* +of 0.9999 between the multiplier and the recovered normal stress. A correlation +is scale-free, so it cannot see a systematic amplitude deficit — which is exactly +what a missing share of the load is. The replacement guard scores a relative +$l_2$ against an exact answer and carries the bare multiplier as its negative +control. ### The multiplier and the consistent boundary flux are the same object @@ -559,10 +580,13 @@ large. Measured across the two solves — they are different discrete problems, so this is agreement rather than an identity check — the corrected multiplier and the rotated reaction differ by 3.2% at a contrast of 100 and 4.9% at $10^6$, both -inside each route's own error against the exact answer (5 to 9%). We could not -compare them within a single solve: `boundary_flux`, which is the CBF primitive, -returns values of order $10^{12}$ on `Stokes_Constrained` while reading the exact -topography to 8% on an ordinary solve. That is underworld3#614. +inside each route's own error against the exact answer (5 to 9%). They cannot be compared within a single solve, and the reason is the identity +itself: on a multiplier-constrained boundary the constraint enters the row it +constrains, so the assembled residual there is balanced at convergence and the +CBF back-calculation reads zero (measured: rms 4 × 10⁻¹³ against a traction of +0.37). There is no reaction left to read because the multiplier is holding it. +`boundary_flux` now says so instead of returning that zero quietly +(underworld3#614). This also settles a question the free-surface work left open. That work used SolCx the same way, to choose among topography recoveries, and landed on a @@ -579,93 +603,132 @@ so the case where weak constraints are most often reported to give trouble is a separate test with a trivial geometry. SolCx is that test: the unit box, free slip on all four walls, viscosity 1 to the left of $x = 0.5$ and $\eta_B$ to the right. `uw.analytic.SolCx` publishes the exact dynamic topography on the top -wall, so this one can be drawn rather than tabulated. Three walls carry the -ordinary component condition and the treatment under test is on the top wall -alone. +wall. Three walls carry the ordinary component condition and the treatment under +test is on the top wall alone. On a box every treatment reduces to holding one velocity component, so nothing -here is about normals. What it can say is whether a weak imposition holds the -traction it was given when the viscosity beside it jumps. - -```{figure} figures/topography.png -:alt: Two line plots of surface topography along the top wall from x=0 to x=1, mean removed, at viscosity contrasts of 100 and a million. In both, the exact answer is a thick grey curve falling from +0.29 at the left, flattening near +0.21, jumping down sharply at the viscosity step at x=0.5 and continuing down to -0.38 at the right. At a contrast of 100 every treatment lies on the grey curve, apart from the rotated constraint which spikes to -0.50 and +0.27 in the last two nodes at x=1. At a contrast of a million the picture separates: the component Dirichlet, the rotated reaction and the corrected multiplier still lie on the exact curve; Nitsche at gamma=10 leaves the panel entirely and peaks at 4.09 against a signal of 0.38; and the multiplier as the API returns it is a nearly flat line near zero, reaching only 0.04, having lost almost all of the traction to the augmented-Lagrangian term. Adding that term back, drawn as a dotted blue line, puts it on the exact curve again. - -Predicted against computed, for each treatment. Every curve is mean-removed, -because the box is enclosed and the level of the topography is a gauge. At a -contrast of 100 the treatments agree with each other and with the exact answer; -at a million, three of the five have something wrong with them, and each is -wrong in a different way. -``` - -Relative $l_2$ error along the wall, at 32 × 32 elements, over the whole wall -and then with two elements trimmed from each end: - -| $\eta_B/\eta_A$ | component Dirichlet | penalty | Nitsche | multiplier | rotated | +here is about normals. What it can say is whether a treatment holds the traction +it was given when the viscosity beside it jumps. + +Relative $l_2$ error of the surface topography along the top wall, mean removed, +at 32 × 32 elements. Each entry is the whole wall and then the wall with two +elements trimmed from each end. + +| $\eta_B/\eta_A$ | component Dirichlet | penalty, $10^4$ | multiplier | rotated | +|---|---|---|---|---| +| 10 | 0.048 / 0.054 | 0.045 / 0.051 | 0.048 / 0.054 | 0.048 / 0.054 | +| 10² | 0.072 / 0.081 | 0.056 / 0.060 | 0.072 / 0.081 | 0.072 / 0.081 | +| 10³ | 0.075 / 0.084 | 0.234 / 0.230 | 0.075 / 0.084 | 0.075 / 0.084 | +| 10⁴ | 0.076 / 0.085 | 0.698 / 0.697 | 0.076 / 0.085 | 0.076 / 0.085 | +| 10⁶ | 0.076 / 0.085 | 0.992 / 1.000 | 0.075 / 0.084 | 0.076 / 0.085 | + +**The three exact treatments agree to three figures at every contrast**, whole +wall and trimmed alike. That is the result to take from this half, and it took +two fixes to get: the multiplier had to report the whole traction rather than +$h$, and the rotated constraint had to hold the corner where it meets the side +walls. Before them the rotated column read 0.322 at a contrast of 10 — six times +the reference, all of it two nodes — and the multiplier column was an order of +magnitude out at $10^6$. + +**Read the first column as the floor.** The component Dirichlet condition is +exact and has no parameter, and its velocity error is 8.8 × 10⁻⁶ at a contrast of +$10^6$. It still reads 0.085. That number is the recovery's error, not a boundary +condition's: on the stiff half the recovered $\sigma_{zz}$ is a difference between +the pressure and $2\eta\,\partial_z u_z$ with $\eta = 10^6$, so a relative velocity +error of $10^{-5}$ arrives in the stress at the size of the signal. Nothing here +is worse than the reference except the penalty. + +**A bare penalty coefficient cannot serve both halves.** At $10^4$ it is the best +column in the table at low contrast — the constraint is weak enough not to fight +the recovery — and by $10^6$ it is useless: 0.992, which is to say the recovered +topography carries none of the signal. Scaling the coefficient by the local +viscosity is the obviously right thing to want, and it does not solve here at any +magnitude we tried, from $\mu$ to $10^3\mu$ (a `Piecewise` viscosity inside the +boundary term fails the line search). This is the same lesson as $\gamma$'s window +on the annulus, in a place where the window closes entirely. + +**Nitsche is missing from this table**, and honestly so. Our configuration of it +on this box converges at a contrast of $10^6$ and fails the line search at $10$ — +the opposite way round from every expectation — at 16 × 16 and 32 × 32 alike and +at $\gamma = 10$, $100$ and $1000$. Where it does converge, $\gamma$ has to rise +with the contrast exactly as the annulus said: at $10^6$ and 16 × 16 the surface +stress error is 25 at $\gamma = 10$, 1.2 at $100$ and 0.17 at $1000$, while the +constraint is held to $10^{-3}$ or better throughout. We are not confident enough +in that configuration to put a column of numbers behind it. + +:::{note} A measurement we withdrew +An earlier draft of this section carried a Nitsche column and different numbers +throughout. They were taken while several runs shared one mesh-cache file: +`StructuredQuadBox` keys its cache on the box corners and not on the element +resolution, so concurrent runs at different resolutions silently swap meshes +(underworld3#618). A marginal solve then flips between converged and diverged for +reasons that look like the method. Every number in this section was re-measured +sequentially, on a cleared cache, with nothing else running. +::: + + +### What each one costs + +Accuracy is half the choice. Seconds on the annulus, median of three timed +repeats after an untimed warm-up, run sequentially — the solve, and then the +recovery of the surface traction by whatever route that treatment has. + +| cell size | velocity nodes | penalty | Nitsche | multiplier | rotated | |---|---|---|---|---|---| -| 10 | 0.048 / 0.054 | 0.045 / 0.051 | 0.048 / 0.054 | 0.048 / 0.054 | 0.322 / 0.056 | -| 10² | 0.072 / 0.081 | 0.056 / 0.060 | 0.073 / 0.081 | 0.072 / 0.081 | 0.343 / 0.083 | -| 10³ | 0.075 / 0.084 | diverged | 0.076 / 0.085 | 0.075 / 0.084 | 0.345 / 0.086 | -| 10⁴ | 0.076 / 0.085 | diverged | 0.087 / 0.095 | 0.076 / 0.085 | 0.345 / 0.086 | -| 10⁶ | 0.076 / 0.085 | diverged | **2.77 / 2.14** | **1.04** | 0.345 / 0.086 | - -**Read the first column first.** The component Dirichlet condition is exact, has -no parameter, and its velocity error is 8.8 × 10⁻⁶ at a contrast of $10^6$. It -still reads 0.085. That number is the recovery's error and not a boundary -condition's: on the stiff half the recovered $\sigma_{zz}$ is a difference -between the pressure and $2\eta\,\partial_z u_z$ with $\eta = 10^6$, so a -relative velocity error of $10^{-5}$ arrives in the stress at the size of the -signal. Nothing here can be called worse than the reference unless it is worse -than that, and up to a contrast of $10^4$ nothing is by more than about a tenth. - -**Nitsche fails at $10^6$ and the leak does not show it.** At $\gamma = 10$ the -recovered topography peaks at 4.09 against a signal of 0.38, while the -constraint is held to 3.2 × 10⁻⁴ on the soft half of the wall and 1.5 × 10⁻⁷ on -the stiff half. Raising $\gamma$ fixes it — 0.16 at $\gamma = 100$, 0.079 at -$\gamma = 1000$, which is the reference floor — so the stabilisation parameter -has to be raised with the viscosity contrast, and the quantity that says whether -it is high enough is the stress rather than the leak. That is the same shape as -the curved-boundary result arrived at from the other side. - -**The multiplier loses the traction to the augmentation.** At a contrast of -$10^6$ the multiplier field as `multiplier()` returns it peaks at 0.042 against -an exact 0.383, and it is anti-correlated with the right answer (−0.53) — the -flat blue line in the figure. The solve is not at fault: the same run's velocity -error is 8.8 × 10⁻⁶ and its recovered stress is as good as the reference. The -default augmentation is $r = 10^4\mu(x)$, which the viscosity step makes $10^{10}$ -on the stiff half, and the momentum row carries -$h + r(\mathbf{u}\cdot\hat{\mathbf{n}} - g)$. Almost all of the traction is in -the second term. Adding it back gives 0.047 — better than every other route in -this test, including the recovery — and it is the dotted curve lying on the -exact one. Turning $r$ down instead is not available: at this contrast both -$r = 0$ and $r = 10^2$ fail to solve. This is the same defect as in the annulus, -underworld3#607, and the contrast is what makes it severe. - -**The rotated constraint's spike at $x = 1$ is the corner, and the corner is a -known one.** Trimmed by two elements at each end it matches the reference at -every contrast; untrimmed it is six times worse, and the whole of that is the -two end nodes. A node where a rotated wall meets a wall held by an ordinary -component condition is constrained by both. Its velocity degrees of freedom are -constrained out of the global vector, the rotation drops it, and the reaction -recovery still reports a value there — the reaction of the *essential* -constraint, which is not the wall's traction. Underworld's own implementation -notes say so. The same recovery on a lid held by the component condition alone -has no spike at all: the consistent boundary flux there reads the exact -topography to 8% with a peak of 0.381 against an exact 0.379, where the rotated -run peaks at 0.497. - -So this is a thing to know rather than a thing to fix in the solve: leave the -corner out of the rotation, or drop it from the recovered trace. It is also why -the free-surface work that used SolCx to choose a topography recovery never saw -it. The remaining difference is small and real — at a contrast of 10 the -pressure differs from the Dirichlet run by 0.23 at the corner node and 0.12 at -its neighbour, while everything below the top row agrees to 4 × 10⁻⁴ rms — -which is underworld3#608. - -**The penalty needs a coefficient matched to the local viscosity, and we could -not write one that solves.** A constant $10^4$ manages a contrast of $10^2$ and -then fails the line search. Multiplying by the piecewise viscosity is the -obviously right thing to want, and it fails at every magnitude we tried, from -$\mu$ to $10^3\mu$, against both normals. +| 0.075 | 2 174 | 0.01 / 0.153 | 0.01 / 0.037 | 0.02 / 0.001 | 0.01 / 0.003 | +| 0.050 | 4 802 | 0.03 / 0.058 | 0.03 / 0.060 | 0.04 / 0.002 | 0.03 / 0.004 | +| 0.035 | 9 406 | 0.06 / 0.095 | 0.06 / 0.099 | 0.08 / 0.004 | 0.05 / 0.006 | + +**The solve costs the same whichever you choose.** Within the spread, all four +are the same number at every size — 0.06 s at 9 400 nodes. The multiplier is the +one that carries a visible premium, about 30%, which is the extra field and the +larger saddle point; its interior degrees of freedom are constrained out of the +global system, so what is left is the boundary trace. + +**The recovery is where they differ, by a factor of 15 to 25.** Projecting +$\hat{\mathbf{n}}\cdot\boldsymbol{\sigma}\hat{\mathbf{n}}$ is a second solve, and +it costs more than the Stokes solve did at the coarser sizes. The reaction and +the multiplier are arithmetic on the boundary trace: 4 to 6 ms where the +projection takes 95 to 99. + +That ratio is the practical argument, and it points the same way the accuracy +did. The route that does not differentiate the answer is both the more accurate +one and the cheaper one, and on a time-dependent model it is paid at every step. + +:::{note} What these timings are not +Two-dimensional problems of ten thousand nodes, on one core, with a direct +solver. They separate a recovery solve from boundary arithmetic, which is a +structural difference and will survive scaling. They say nothing about how the +four compare on a large parallel spherical shell, where the rotated velocity +block's multigrid and the multiplier's larger Schur complement are the terms that +matter and neither is exercised here. +::: + +### Which one to use + +For a model that consumes the velocity and nothing else, all four are the same to +plotting accuracy — provided the constraint is written against the node normal. +That proviso is the only one that can spoil the velocity, and it costs one line. + +When the wall-normal traction is the answer: + +- **Rotated free slip is the default.** It holds the constraint to machine + precision rather than to the discretisation, its reaction is the most accurate + surface stress measured here, and that reaction is nearly free. The price is + structural — a mixed basis that the multigrid has to carry — and it is paid + once, inside the solver, rather than by the person setting up the model. +- **The multiplier is its equal on accuracy** and returns the same object by a + different route; take it when you want the traction as an unknown of the + system, or when the constrained problem's conditioning suits you better. It + costs about 30% more to solve. +- **Nitsche is the one to reach for when the boundary condition must change + during the model** — a wall that begins as a prescribed velocity and relaxes to + a prescribed traction is a Nitsche problem, because a hard constraint cannot + morph. Budget for tuning $\gamma$ against the stress and not against the leak, + and expect the window to move with the viscosity contrast. +- **A direct penalty is fine for a velocity-only model** and needs the node + normal, a coefficient chosen per problem, and a check on something physical + before the answer is believed. ## Using it diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/generate-locking-figure.py b/articles/boundary-conditions-on-non-planar-boundaries/examples/generate-locking-figure.py index a6cd4e0..e2f51b3 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/examples/generate-locking-figure.py +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/generate-locking-figure.py @@ -12,7 +12,8 @@ python3 generate-locking-figure.py -Run against underworld3 `development` at commit `8b7c8b9e`. +Run against underworld3 `bugfix/multiplier-traction` (PR #617); the +constrained solver's `traction()` is the fix this note prompted. """ import pathlib diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/leak.py b/articles/boundary-conditions-on-non-planar-boundaries/examples/leak.py index b8e22be..df43248 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/examples/leak.py +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/leak.py @@ -25,7 +25,8 @@ python3 leak.py sweep # the resolution table python3 leak.py free # the control: outer boundary left natural -Run against underworld3 `development` at commit `8b7c8b9e`. +Run against underworld3 `bugfix/multiplier-traction` (PR #617); the +constrained solver's `traction()` is the fix this note prompted. """ import sys diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/solcx.py b/articles/boundary-conditions-on-non-planar-boundaries/examples/solcx.py index 3d4a314..89b2660 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/examples/solcx.py +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/solcx.py @@ -35,7 +35,8 @@ python3 solcx.py params # the penalty coefficient, both halves python3 solcx.py control # the top wall left free -Run against underworld3 `development` at commit `8b7c8b9e`. +Run against underworld3 `bugfix/multiplier-traction` (PR #617); the +constrained solver's `traction()` is the fix this note prompted. """ import sys @@ -131,10 +132,19 @@ def recovered_traction(mesh, stokes): def reaction_traction(stokes, mode): + """The traction the solve returned, for the two methods that return one.""" if mode == "rotated": return stokes.boundary_normal_traction("Top") if mode == "constraint": - return trace(stokes, 2, stokes.multiplier("Top")) + # `traction()` is h + r(u.n - g), the WHOLE boundary term. Reading the + # multiplier alone here was wrong by an order of magnitude and a sign at + # a 1e6 viscosity contrast, because the default r is viscosity-weighted + # (underworld3#607, fixed in #617). The box is flat, so evaluating the + # expression at the trace nodes is safe -- on a convex curved boundary it + # would extrapolate (#605), which is why stress.py reads arrays instead. + coords, _h = trace(stokes, 2, stokes.multiplier("Top")) + return coords, np.asarray( + uw.function.evaluate(stokes.traction("Top"), coords)).reshape(-1) return None diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/stress.py b/articles/boundary-conditions-on-non-planar-boundaries/examples/stress.py index 628eb62..d873c1c 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/examples/stress.py +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/stress.py @@ -50,7 +50,8 @@ python3 stress.py locking # the facet normal does not converge python3 stress.py control # the metric fires when the BC is removed -Run against underworld3 `development` at commit `8b7c8b9e`. +Run against underworld3 `bugfix/multiplier-traction` (PR #617); the +constrained solver's `traction()` is the fix this note prompted. """ import sys @@ -189,14 +190,39 @@ def recovered_traction(mesh, stokes): return trace(projection, 0, field) -def reaction_traction(stokes, mode): +# The augmented-Lagrangian parameter this problem gets by default: the base +# (1e4) times the local viscosity, which is 1 here. +AUGMENTATION = 1.0e4 + + +def multiplier_traction(stokes, v, boundary="Upper"): + """The WHOLE traction a multiplier constraint holds the boundary with. + + The momentum row carries `h + r(u.n - g)`, so `h` alone is short by `r` times + the discrete constraint residual. `stokes.traction(boundary)` is that sum as + an expression; this reads it off the node arrays instead, because evaluating + an expression at points sitting exactly on a convex curved boundary + extrapolates from the containing cell (underworld3#605) and that error would + land on top of the measurement. + + `u.n` is taken against the true radial direction rather than the constraint's + node normal. The two differ at O(h^2) and they multiply a term that is itself + a correction. + """ + coords, h = trace(stokes, 2, stokes.multiplier(boundary)) + tree = uw.kdtree.KDTree(np.ascontiguousarray(v.coords)) + index = np.asarray(tree.query(np.ascontiguousarray(coords), 1)[1]).flatten() + u = np.squeeze(np.asarray(v.array))[index] + normal = coords / np.linalg.norm(coords, axis=1)[:, None] + return coords, np.asarray(h) + AUGMENTATION * (u * normal).sum(axis=1) + + +def reaction_traction(stokes, mode, v=None): """The constraint reaction, for the two methods that return one.""" if mode == "rotated": return stokes.boundary_normal_traction("Upper") if mode == "constraint": - multiplier = stokes.multiplier("Upper") - # field 2 of the constrained saddle point: velocity, pressure, multiplier - return trace(stokes, 2, multiplier) + return multiplier_traction(stokes, v) return None @@ -242,7 +268,7 @@ def measure(mode, cell=CELL, **kwargs): } coords, values = recovered_traction(mesh, stokes) out["recovered"], out["sign"] = amplitude_error(coords, values, reference) - read = reaction_traction(stokes, mode) + read = reaction_traction(stokes, mode, v=v) if read is not None: coords, values = read out["reaction"], out["reaction_sign"] = amplitude_error(coords, values, reference) diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/timing.py b/articles/boundary-conditions-on-non-planar-boundaries/examples/timing.py new file mode 100644 index 0000000..b3c7076 --- /dev/null +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/timing.py @@ -0,0 +1,99 @@ +"""What each treatment costs. + +The note compares four ways of imposing free slip on accuracy. This measures the +other half of the choice: what each one costs to solve, and what its surface +traction costs to recover once solved. + +Two numbers per treatment, because they are paid at different times: + + * the SOLVE. The rotated constraint changes the operator; the multiplier adds a + field and enlarges the saddle point; the weak forms add a boundary term to a + system that is otherwise the plain Stokes one. + * the RECOVERY of the surface traction. The weak forms have to project + `n.sigma.n` out of the solution, which is a second (scalar, symmetric) solve. + The rotated constraint and the multiplier read theirs off the state the solve + already returned, which is arithmetic on the boundary trace and no solve at + all. + +Method: each configuration is built and solved once UNTIMED (JIT compilation, +PETSc setup and the first-touch allocations are not what is being measured), then +timed `repeats` times. Runs are sequential by construction -- concurrent PETSc +solves contend for memory bandwidth and inflate each other by tens of per cent. + + python3 timing.py # the table + python3 timing.py 0.05 # at one cell size + +Run against underworld3 `bugfix/multiplier-traction`. +""" +import sys +import time + +import numpy as np +import sympy + +import underworld3 as uw + +import stress as S + +REPEATS = 3 +CELLS = (0.075, 0.05, 0.035) +MODES = ("penalty_node", "nitsche", "constraint", "rotated") + + +def _time(call, repeats=REPEATS): + """Median of `repeats` timings, and the spread, in seconds. + + One UNTIMED call first. The first recovery in a process compiles its + projection: timed cold it read 1.145 s where the same call reads 0.038 s + warm, which is a measurement of the JIT and not of the method. + """ + call() + got = [] + for _ in range(repeats): + start = time.perf_counter() + call() + got.append(time.perf_counter() - start) + return float(np.median(got)), float(np.max(got) - np.min(got)) + + +def solve_cost(mode, cell): + """Seconds to solve, and the nonlinear/linear iteration counts.""" + mesh, stokes, v, exact = S.build(mode, cell=cell) + stokes.solve() # untimed: JIT, setup, first touch + assert S.converged(stokes) + iterations = stokes.snes.getIterationNumber() + median, spread = _time(stokes.solve) + return mesh, stokes, v, exact, median, iterations + + +def recovery_cost(mesh, stokes, mode): + """Seconds to get sigma_nn on the boundary, by the route that treatment has.""" + if mode == "rotated": + return _time(lambda: stokes.boundary_normal_traction("Upper")) + if mode == "constraint": + # The whole traction, h + r(u.n - g). Reading it is an expression build + # plus a boundary-trace evaluation -- no solve. + return _time(lambda: S.multiplier_traction(stokes, stokes.u)) + # The weak forms: project n.sigma.n and read its trace. A scalar solve. + return _time(lambda: S.recovered_traction(mesh, stokes)) + + +def table(cells=CELLS, modes=MODES): + print("seconds, median of %d timed repeats after one untimed warm-up" % REPEATS) + print() + print("| cell size | nodes | " + " | ".join( + "%s: solve / recover" % m for m in modes) + " |") + print("|---" * (len(modes) + 2) + "|") + for cell in cells: + row, nodes = [], None + for mode in modes: + mesh, stokes, v, exact, solve, its = solve_cost(mode, cell) + nodes = len(v.coords) if nodes is None else nodes + recover, _s = recovery_cost(mesh, stokes, mode) + row.append("%.2f / %.3f" % (solve, recover)) + print("| %.3f | %d | %s |" % (cell, nodes, " | ".join(row)), flush=True) + + +if __name__ == "__main__": + cells = (float(sys.argv[1]),) if sys.argv[1:] else CELLS + table(cells=cells) diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/topography-data.json b/articles/boundary-conditions-on-non-planar-boundaries/examples/topography-data.json 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zC=pv5B&E+m&yztg&QgZk{gV-sCd39;P-K-qJ*q9c_Y zE)VX?zX!`>WUuwD9W2XD$h0M5*L#yjb~O*;n;CZ^;9>8u63HbCN1-lU_Pd|LpK_yE`B Date: Wed, 19 Aug 2026 22:04:30 +1000 Subject: [PATCH 12/33] Pin every citation in a references.bib The preview build failed on CI with "Citation data from doi.org was not available or malformed" for the Behr and Kramer DOIs, while resolving fine locally. That is the failure the MMPDE note's references.bib was written to prevent: a build that depends on doi.org answering will eventually publish a note with a broken citation, and a deposited PDF cannot be repaired afterwards. All five are now pinned and cited by key. Underworld development team with AI support from Claude Code --- ...ndary-conditions-on-non-planar-boundaries.md | 8 ++++---- .../references.bib | 17 ++++++++++++++++- 2 files changed, 20 insertions(+), 5 deletions(-) diff --git a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md index f1de09d..2d9106f 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md +++ b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md @@ -106,7 +106,7 @@ stokes.add_natural_bc(1.0e4 * G.dot(v.sym) * G, "Upper") The reason the penalty is only accurate in the limit is that it is not *consistent*: substituting the true solution does not leave the equation satisfied, because the true solution is subject to a boundary traction the -penalty form ignores. Nitsche's method [@10.1007/BF02995904] restores +penalty form ignores. Nitsche's method [@Nitsche_1971] restores consistency by carrying that traction explicitly: $$ @@ -198,7 +198,7 @@ and the wall-normal row of $\hat{A}$ is struck out. The constraint then holds to machine precision, because it is not being solved for at all. **This is the classical answer**, not a new one. It is in the early -finite-element literature, and @10.1002/fld.1650020302 were already reviewing +finite-element literature, and @Engelman_1982 were already reviewing the alternatives and choosing between them on grounds of global mass conservation in 1982. What is worth explaining is not the idea but why, given that it is exact and the others are not, it is the least used of the three. @@ -265,7 +265,7 @@ sharper edge, because a normal accumulated rank-locally is wrong at a partition boundary, where a node's facets are split across ranks and no rank sees them all. -The consistent normal is not the end of the matter. @10.1002/fld.663 takes it as +The consistent normal is not the end of the matter. @Behr_2004 takes it as the starting point — "preferred from the point of view of conservation" — and reports that in sloshing problems it still does not guarantee a good discrete slip condition, with non-physical recirculation appearing at curved walls; the @@ -391,7 +391,7 @@ are getting worse throughout. The leak says how well each treatment holds the boundary. Whether the answer is right is a different question, and it needs an exact answer to compare against. -Kramer, Davies and Wilson [@10.5194/gmd-14-1899-2021] give exact Stokes +Kramer, Davies and Wilson [@Kramer_2021] give exact Stokes solutions in a cylindrical annulus, and their `assess` package publishes the radial stress itself rather than leaving it to be recovered from a velocity field. Underworld wraps it as `uw.analytic.CylindricalStokes`. The case here is diff --git a/articles/boundary-conditions-on-non-planar-boundaries/references.bib b/articles/boundary-conditions-on-non-planar-boundaries/references.bib index 261b510..3af1725 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/references.bib +++ b/articles/boundary-conditions-on-non-planar-boundaries/references.bib @@ -2,7 +2,13 @@ % MMPDE note: a build that depends on doi.org answering can publish a note with a % broken citation, and a deposited PDF cannot be repaired afterwards. % -% This one has a second reason. The DOI contains parentheses, +% Every citation in this note is pinned, not only the awkward one. The preview +% build failed on CI with "Citation data from doi.org was not available or +% malformed" for two of them while resolving fine locally: a build that depends +% on doi.org answering will eventually publish a note with a broken citation, and +% a deposited PDF cannot be repaired afterwards. +% +% The Zhong entry has a second reason. The DOI contains parentheses, % 10.1016/0031-9201(93)90078-N, and MyST's inline DOI citation form stops at the % first "(" -- the build reported: Could not find DOI "10.1016/0031-9201". A % bibtex key sidesteps that. @@ -25,3 +31,12 @@ @article{Zhong_1993 month = jun, pages = {1--8} } + + +@article{Nitsche_1971, title={Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind}, volume={36}, ISSN={1865-8784}, url={http://dx.doi.org/10.1007/BF02995904}, DOI={10.1007/bf02995904}, number={1}, journal={Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg}, publisher={Springer Science and Business Media LLC}, author={Nitsche, J.}, year={1971}, month=July, pages={9–15} } + +@article{Engelman_1982, title={The implementation of normal and/or tangential boundary conditions in finite element codes for incompressible fluid flow}, volume={2}, ISSN={1097-0363}, url={http://dx.doi.org/10.1002/fld.1650020302}, DOI={10.1002/fld.1650020302}, number={3}, journal={International Journal for Numerical Methods in Fluids}, publisher={Wiley}, author={Engelman, M. S. and Sani, R. L. and Gresho, P. M.}, year={1982}, month=July, pages={225–238} } + +@article{Behr_2004, title={On the application of slip boundary condition on curved boundaries}, volume={45}, ISSN={1097-0363}, url={http://dx.doi.org/10.1002/fld.663}, DOI={10.1002/fld.663}, number={1}, journal={International Journal for Numerical Methods in Fluids}, publisher={Wiley}, author={Behr, Marek}, year={2004}, month=Apr, pages={43–51} } + +@article{Kramer_2021, title={Analytical solutions for mantle flow in cylindrical and spherical shells}, volume={14}, ISSN={1991-9603}, url={http://dx.doi.org/10.5194/gmd-14-1899-2021}, DOI={10.5194/gmd-14-1899-2021}, number={4}, journal={Geoscientific Model Development}, publisher={Copernicus GmbH}, author={Kramer, Stephan C. and Davies, D. Rhodri and Wilson, Cian R.}, year={2021}, month=Apr, pages={1899–1919} } From c490843fc3dce2c70c17ff97400664969a171201 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Wed, 19 Aug 2026 22:36:34 +1000 Subject: [PATCH 13/33] Present the method, not the route we took to it Louis's note: the interesting thing here is two penalty approaches against two ways of imposing the constraint exactly -- by construction, or by a Lagrange multiplier -- and that pairing is now stated where the four are introduced. Removed: the section that told the multiplier's missing augmentation share as a defect story, the withdrawn-measurement note, the "how it survived validation" paragraph, and the issue numbers. What survives is the method. The traction a multiplier constraint holds the boundary with is h + r(u.n - g), which is stated where the method is described, and the two exact routes are shown to be the same computation, which is the point worth keeping. The timings are re-measured at sizes where they mean something: 71 000 velocity nodes rather than 2 000, where the numbers were hundredths of a second and inside the run-to-run spread. At that size the multiplier's solve costs 50% more than the others, and the recovery separates by a factor of 25 to 40 -- the projection costs MORE than the Stokes solve (0.63 s against 0.45 s) while the two exact routes read their traction off the state the solve already returned (27 ms and 16 ms). The smaller meshes are kept only to show the scaling, and the table says so. Underworld development team with AI support from Claude Code --- ...ary-conditions-on-non-planar-boundaries.md | 142 +++++++----------- .../examples/timing.py | 6 +- 2 files changed, 61 insertions(+), 87 deletions(-) diff --git a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md index 2d9106f..f271078 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md +++ b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md @@ -72,6 +72,16 @@ replacing is the choice. ## Four ways, in order +They fall into two pairs, and the pairing is the useful way to hold them. Two +impose the constraint **weakly**, by adding a term to the momentum equation and +letting the solution satisfy the condition to within the accuracy of that term: a +direct penalty, and Nitsche's method, which differ in whether the term is +consistent. Two impose it **exactly**: by construction, changing the basis so +that the constraint is a component that can be struck out, or by a Lagrange +multiplier, adding an equation that enforces it. The weak pair have a parameter +to choose and a floor they cannot go below. The exact pair have neither, and both +return the boundary traction as part of the answer. + ### A direct penalty Add a term that punishes any flow through the boundary: @@ -174,9 +184,15 @@ stokes.solve() And the reason to care about it beyond the constraint: **at convergence the momentum row's boundary term is the normal traction.** It is not recovered from the velocity field afterwards — it is an unknown the solve returned, available -through `multiplier` and, divided by $\Delta\rho g$, through `topography`. With -an augmentation in place that term is $h + r(\mathbf{u}\cdot\hat{\mathbf{n}} - g)$ -rather than $h$, and the difference between the two is measured below. +through `traction` and, divided by $\Delta\rho g$, through `topography`. + +That term is the whole boundary load, $h + r(\mathbf{u}\cdot\hat{\mathbf{n}} - g)$, +and not the multiplier alone. The second part vanishes only where the constraint +row is satisfied exactly; discretely it is satisfied to the solver's tolerance and +$r$ multiplies that residual back into the traction. With a viscosity-weighted +$r$ and a lateral viscosity contrast it is most of the answer, so the two parts +are not separable in practice — which is the first hint that this traction and +the rotated constraint's reaction are the same object. ### Rotating the degrees of freedom @@ -519,43 +535,6 @@ sufficient check: at a coefficient of $10^3$ the facet-normal penalty leaks leaks 1 × 10⁻⁷ and is 26% wrong. Over that range the constraint improves by five orders of magnitude and the answer gets steadily worse. -### The multiplier was not the whole traction - -The multiplier column of the table above used to sit near 3 × 10⁻² for three -resolutions and then drop, which is not a convergence rate. The cause was the -augmented-Lagrangian term, and it was a defect rather than a property of the -method. - -The momentum row carries both the multiplier and the augmentation, -$(h + r(\mathbf{u}\cdot\hat{\mathbf{n}} - g))\hat{\mathbf{n}}$, so the traction -holding the boundary is $h + r(\mathbf{u}\cdot\hat{\mathbf{n}} - g)$ and not $h$ -alone. The second term vanishes only where the constraint row is satisfied -exactly. Discretely it is satisfied to the solver's tolerance, and $r$ multiplies -that residual straight back into the traction. Underworld returned $h$. - -On the annulus, with a uniform viscosity and the default $r = 10^4\mu$, the -omitted share is a few per cent. On SolCx it is nearly everything, because the -default $r$ is viscosity-weighted and the step takes it to $10^{10}$ on the stiff -half. At a contrast of $10^6$, against an exact surface topography whose -deviation peaks at 0.381: - -| read | peak | correlation with the exact | relative $l_2$ | -|---|---|---|---| -| the multiplier $h$ | 0.042 | **−0.52** | 0.96 | -| $h + r(\mathbf{u}\cdot\hat{\mathbf{n}} - g)$ | 0.382 | +0.999 | 0.084 | - -An order of magnitude low and pointing the wrong way, on a solve whose velocity -error is 8.8 × 10⁻⁶. Underworld3 issue #607: `traction()` now returns the sum and -`topography()` is built on it, so the tables here are the corrected read and a -caller who asks for topography gets it. - -Worth saying how it survived. The method's own validation scored a *correlation* -of 0.9999 between the multiplier and the recovered normal stress. A correlation -is scale-free, so it cannot see a systematic amplitude deficit — which is exactly -what a missing share of the load is. The replacement guard scores a relative -$l_2$ against an exact answer and carries the bare multiplier as its negative -control. - ### The multiplier and the consistent boundary flux are the same object The correction above is not a patch. Write the momentum row's boundary term out @@ -580,13 +559,12 @@ large. Measured across the two solves — they are different discrete problems, so this is agreement rather than an identity check — the corrected multiplier and the rotated reaction differ by 3.2% at a contrast of 100 and 4.9% at $10^6$, both -inside each route's own error against the exact answer (5 to 9%). They cannot be compared within a single solve, and the reason is the identity -itself: on a multiplier-constrained boundary the constraint enters the row it -constrains, so the assembled residual there is balanced at convergence and the -CBF back-calculation reads zero (measured: rms 4 × 10⁻¹³ against a traction of -0.37). There is no reaction left to read because the multiplier is holding it. -`boundary_flux` now says so instead of returning that zero quietly -(underworld3#614). +inside each route's own error against the exact answer (5 to 9%). The identity also says why the two cannot be read off a single solve. On a +multiplier-constrained boundary the constraint enters the row it constrains, so +the assembled residual there is balanced at convergence and the back-calculation +reads zero — measured, rms 4 × 10⁻¹³ against a traction of 0.37. There is no +reaction left in the residual because the multiplier is holding it. The two +routes are alternatives, not a cross-check available at the same time. This also settles a question the free-surface work left open. That work used SolCx the same way, to choose among topography recoveries, and landed on a @@ -656,52 +634,46 @@ stress error is 25 at $\gamma = 10$, 1.2 at $100$ and 0.17 at $1000$, while the constraint is held to $10^{-3}$ or better throughout. We are not confident enough in that configuration to put a column of numbers behind it. -:::{note} A measurement we withdrew -An earlier draft of this section carried a Nitsche column and different numbers -throughout. They were taken while several runs shared one mesh-cache file: -`StructuredQuadBox` keys its cache on the box corners and not on the element -resolution, so concurrent runs at different resolutions silently swap meshes -(underworld3#618). A marginal solve then flips between converged and diverged for -reasons that look like the method. Every number in this section was re-measured -sequentially, on a cleared cache, with nothing else running. -::: ### What each one costs -Accuracy is half the choice. Seconds on the annulus, median of three timed -repeats after an untimed warm-up, run sequentially — the solve, and then the -recovery of the surface traction by whatever route that treatment has. +Seconds on the annulus, uniform viscosity, one core, direct solver: the solve, +and then the recovery of the surface traction by whatever route that treatment +has. Median of three timed repeats after an untimed warm-up, run sequentially. +The smaller meshes are here for the scaling; at ten thousand nodes the four +treatments are separated by less than the run-to-run spread and only the largest +row is worth reading. | cell size | velocity nodes | penalty | Nitsche | multiplier | rotated | |---|---|---|---|---|---| -| 0.075 | 2 174 | 0.01 / 0.153 | 0.01 / 0.037 | 0.02 / 0.001 | 0.01 / 0.003 | -| 0.050 | 4 802 | 0.03 / 0.058 | 0.03 / 0.060 | 0.04 / 0.002 | 0.03 / 0.004 | -| 0.035 | 9 406 | 0.06 / 0.095 | 0.06 / 0.099 | 0.08 / 0.004 | 0.05 / 0.006 | - -**The solve costs the same whichever you choose.** Within the spread, all four -are the same number at every size — 0.06 s at 9 400 nodes. The multiplier is the -one that carries a visible premium, about 30%, which is the extra field and the -larger saddle point; its interior degrees of freedom are constrained out of the -global system, so what is left is the boundary trace. - -**The recovery is where they differ, by a factor of 15 to 25.** Projecting -$\hat{\mathbf{n}}\cdot\boldsymbol{\sigma}\hat{\mathbf{n}}$ is a second solve, and -it costs more than the Stokes solve did at the coarser sizes. The reaction and -the multiplier are arithmetic on the boundary trace: 4 to 6 ms where the -projection takes 95 to 99. - -That ratio is the practical argument, and it points the same way the accuracy -did. The route that does not differentiate the answer is both the more accurate -one and the cheaper one, and on a time-dependent model it is paid at every step. +| 0.050 | 4 802 | 0.03 / 0.073 | 0.03 / 0.060 | 0.04 / 0.002 | 0.03 / 0.004 | +| 0.030 | 12 852 | 0.08 / 0.128 | 0.08 / 0.131 | 0.12 / 0.005 | 0.07 / 0.007 | +| 0.020 | 28 338 | 0.18 / 0.257 | 0.18 / 0.268 | 0.26 / 0.011 | 0.17 / 0.010 | +| 0.013 | 71 424 | 0.45 / 0.626 | 0.47 / 0.652 | 0.67 / 0.027 | 0.43 / 0.016 | + +Two things separate, and both are structural rather than incidental. + +**The multiplier's solve costs about 50% more**, consistently across the sweep — +0.67 s against 0.43 s at 71 000 nodes. That is the extra field and the larger +saddle point. The other three are the same solve to within the spread; rotating +the degrees of freedom costs nothing measurable here, because the rotation is a +sparse orthogonal transform applied to a boundary's worth of rows. + +**The recovery differs by a factor of 25 to 40, and it is the larger number.** +Projecting $\hat{\mathbf{n}}\cdot\boldsymbol{\sigma}\hat{\mathbf{n}}$ out of the +solution is a second solve, and at every size in this table it costs *more than +the Stokes solve did* — 0.63 s against 0.45 s at the largest. The two exact +routes read their traction off the state the solve already returned: 27 ms for +the multiplier, 16 ms for the rotated reaction. On a time-dependent model with a +free surface that difference is paid at every step. :::{note} What these timings are not -Two-dimensional problems of ten thousand nodes, on one core, with a direct -solver. They separate a recovery solve from boundary arithmetic, which is a -structural difference and will survive scaling. They say nothing about how the -four compare on a large parallel spherical shell, where the rotated velocity -block's multigrid and the multiplier's larger Schur complement are the terms that -matter and neither is exercised here. +Two-dimensional, one core, direct solver. What they measure is the difference +between a recovery *solve* and boundary arithmetic, which is structural and will +survive scaling. They say nothing about a large parallel spherical shell, where +the rotated velocity block's multigrid and the multiplier's larger Schur +complement are the terms that matter and neither is exercised here. ::: ### Which one to use @@ -720,7 +692,7 @@ When the wall-normal traction is the answer: - **The multiplier is its equal on accuracy** and returns the same object by a different route; take it when you want the traction as an unknown of the system, or when the constrained problem's conditioning suits you better. It - costs about 30% more to solve. + costs about 50% more to solve. - **Nitsche is the one to reach for when the boundary condition must change during the model** — a wall that begins as a prescribed velocity and relaxes to a prescribed traction is a Nitsche problem, because a hard constraint cannot diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/timing.py b/articles/boundary-conditions-on-non-planar-boundaries/examples/timing.py index b3c7076..5bf4c2b 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/examples/timing.py +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/timing.py @@ -36,7 +36,9 @@ import stress as S REPEATS = 3 -CELLS = (0.075, 0.05, 0.035) +# Big enough that a solve is seconds rather than hundredths: at 10k nodes the +# four treatments were separated by less than the run-to-run spread. +CELLS = (0.05, 0.03, 0.02, 0.0125) MODES = ("penalty_node", "nitsche", "constraint", "rotated") @@ -81,7 +83,7 @@ def recovery_cost(mesh, stokes, mode): def table(cells=CELLS, modes=MODES): print("seconds, median of %d timed repeats after one untimed warm-up" % REPEATS) print() - print("| cell size | nodes | " + " | ".join( + print("| cell size | velocity nodes | " + " | ".join( "%s: solve / recover" % m for m in modes) + " |") print("|---" * (len(modes) + 2) + "|") for cell in cells: From fd5110a7abcb3b36fee40efa3c4c4fbaacaa620f Mon Sep 17 00:00:00 2001 From: lmoresi Date: Thu, 20 Aug 2026 10:49:22 +1000 Subject: [PATCH 14/33] No solve is needed for either exact traction, and the example no longer implies one Louis asked why determining topography from the multiplier requires a solve. It does not, and neither does the rotated reaction: the multiplier's traction is h + r(u.n - g) evaluated on the boundary trace, and the rotated constraint's is its nodal reaction divided by the LUMPED boundary mass, which is a division. In 2-D neither route assembles anything. The 27 ms in the previous table was the example's bookkeeping, not the method. Profiled at 71 424 velocity nodes with 1 008 trace nodes: 2.7 ms to walk the DM for the boundary node list, 5.3 ms for the traction arithmetic, and 18 ms building a KD-tree over the WHOLE mesh -- twice, once for the multiplier and once for the velocity -- to line the trace up with the field arrays. The multiplier is carried at the velocity degree, so one lookup serves both, and it depends on the mesh rather than the solution: a time-stepping consumer builds it once. With that fixed the recovery reads 9 ms against the rotated reaction's 16 ms. So the table now says the thing that is actually structural: the weak forms have to RECOVER their traction with a projection solve, which costs more than the Stokes solve itself (0.63 s against 0.45 s), while the two exact routes read theirs off the state the solve already returned in 9 to 16 ms. Underworld development team with AI support from Claude Code --- ...ary-conditions-on-non-planar-boundaries.md | 57 +++++++++---------- .../examples/stress.py | 26 +++++++-- .../examples/timing.py | 2 +- 3 files changed, 51 insertions(+), 34 deletions(-) diff --git a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md index f271078..5bb7860 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md +++ b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md @@ -639,39 +639,38 @@ in that configuration to put a column of numbers behind it. ### What each one costs Seconds on the annulus, uniform viscosity, one core, direct solver: the solve, -and then the recovery of the surface traction by whatever route that treatment -has. Median of three timed repeats after an untimed warm-up, run sequentially. -The smaller meshes are here for the scaling; at ten thousand nodes the four -treatments are separated by less than the run-to-run spread and only the largest -row is worth reading. +and then the surface traction by whatever route that treatment has. Median of +three timed repeats after an untimed warm-up, run sequentially. Two sizes, +because below about ten thousand nodes the four are separated by less than the +run-to-run spread and there is nothing to read. -| cell size | velocity nodes | penalty | Nitsche | multiplier | rotated | -|---|---|---|---|---|---| -| 0.050 | 4 802 | 0.03 / 0.073 | 0.03 / 0.060 | 0.04 / 0.002 | 0.03 / 0.004 | -| 0.030 | 12 852 | 0.08 / 0.128 | 0.08 / 0.131 | 0.12 / 0.005 | 0.07 / 0.007 | -| 0.020 | 28 338 | 0.18 / 0.257 | 0.18 / 0.268 | 0.26 / 0.011 | 0.17 / 0.010 | -| 0.013 | 71 424 | 0.45 / 0.626 | 0.47 / 0.652 | 0.67 / 0.027 | 0.43 / 0.016 | - -Two things separate, and both are structural rather than incidental. - -**The multiplier's solve costs about 50% more**, consistently across the sweep — -0.67 s against 0.43 s at 71 000 nodes. That is the extra field and the larger -saddle point. The other three are the same solve to within the spread; rotating -the degrees of freedom costs nothing measurable here, because the rotation is a -sparse orthogonal transform applied to a boundary's worth of rows. - -**The recovery differs by a factor of 25 to 40, and it is the larger number.** -Projecting $\hat{\mathbf{n}}\cdot\boldsymbol{\sigma}\hat{\mathbf{n}}$ out of the -solution is a second solve, and at every size in this table it costs *more than -the Stokes solve did* — 0.63 s against 0.45 s at the largest. The two exact -routes read their traction off the state the solve already returned: 27 ms for -the multiplier, 16 ms for the rotated reaction. On a time-dependent model with a -free surface that difference is paid at every step. +| velocity nodes | penalty | Nitsche | multiplier | rotated | +|---|---|---|---|---| +| 28 338 | 0.17 / 0.273 | 0.18 / 0.267 | 0.26 / 0.004 | 0.16 / 0.010 | +| 71 424 | 0.45 / 0.631 | 0.47 / 0.657 | 0.67 / 0.009 | 0.43 / 0.016 | + +**The multiplier's solve costs about 50% more**, consistently — 0.67 s against +0.43 s at 71 000 nodes. That is the extra field and the larger saddle point. +Rotating the degrees of freedom costs nothing measurable against the weak forms: +the rotation is a sparse orthogonal transform on a boundary's worth of rows. + +**Neither exact route solves for its traction.** The multiplier's is +$h + r(\mathbf{u}\cdot\hat{\mathbf{n}} - g)$ evaluated on the boundary trace; the +rotated constraint's is its nodal reaction divided by the lumped boundary mass. +Both are arithmetic over the ~1 000 nodes of the trace, which is why they cost +milliseconds and why the two are within a factor of two of each other. + +**The weak forms have to recover theirs, and that is a second solve.** Projecting +$\hat{\mathbf{n}}\cdot\boldsymbol{\sigma}\hat{\mathbf{n}}$ out of the solution +costs *more than the Stokes solve did* — 0.63 s against 0.45 s — so asking a +penalty or Nitsche model for its surface stress roughly doubles the cost of the +timestep. Against 9 to 16 ms for the exact routes, that is a factor of forty to +seventy, and on a free-surface model it is paid at every step. :::{note} What these timings are not Two-dimensional, one core, direct solver. What they measure is the difference -between a recovery *solve* and boundary arithmetic, which is structural and will -survive scaling. They say nothing about a large parallel spherical shell, where +between a recovery *solve* and boundary arithmetic, which is structural and +survives scaling. They say nothing about a large parallel spherical shell, where the rotated velocity block's multigrid and the multiplier's larger Schur complement are the terms that matter and neither is exercised here. ::: diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/stress.py b/articles/boundary-conditions-on-non-planar-boundaries/examples/stress.py index d873c1c..33a4072 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/examples/stress.py +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/stress.py @@ -209,12 +209,30 @@ def multiplier_traction(stokes, v, boundary="Upper"): node normal. The two differ at O(h^2) and they multiply a term that is itself a correction. """ - coords, h = trace(stokes, 2, stokes.multiplier(boundary)) - tree = uw.kdtree.KDTree(np.ascontiguousarray(v.coords)) - index = np.asarray(tree.query(np.ascontiguousarray(coords), 1)[1]).flatten() + nodes, *_ = _boundary_field_nodes(stokes, boundary, 2) + coords = np.array([node[2] for node in nodes]) + # The multiplier is carried at the VELOCITY degree, so h and u share a node + # set and one lookup serves both. Building it is the only part of this that + # is not arithmetic, and it depends on the mesh rather than on the solution: + # a time-stepping consumer builds it once and reuses it every step. + index = _trace_index(v, coords) + h = np.squeeze(np.asarray(stokes.multiplier(boundary).array))[index] u = np.squeeze(np.asarray(v.array))[index] normal = coords / np.linalg.norm(coords, axis=1)[:, None] - return coords, np.asarray(h) + AUGMENTATION * (u * normal).sum(axis=1) + return coords, h + AUGMENTATION * (u * normal).sum(axis=1) + + +_TRACE_INDEX = {} + + +def _trace_index(var, coords): + """Row of `var`'s arrays for each trace coordinate, cached per (mesh, trace).""" + key = (id(var), coords.shape[0], float(coords.sum())) + if key not in _TRACE_INDEX: + tree = uw.kdtree.KDTree(np.ascontiguousarray(var.coords)) + _TRACE_INDEX[key] = np.asarray( + tree.query(np.ascontiguousarray(coords), 1)[1]).flatten() + return _TRACE_INDEX[key] def reaction_traction(stokes, mode, v=None): diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/timing.py b/articles/boundary-conditions-on-non-planar-boundaries/examples/timing.py index 5bf4c2b..e068184 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/examples/timing.py +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/timing.py @@ -38,7 +38,7 @@ REPEATS = 3 # Big enough that a solve is seconds rather than hundredths: at 10k nodes the # four treatments were separated by less than the run-to-run spread. -CELLS = (0.05, 0.03, 0.02, 0.0125) +CELLS = (0.02, 0.0125) MODES = ("penalty_node", "nitsche", "constraint", "rotated") From 74a08192648079f0cb49603a37cc259d3b27e827 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Thu, 20 Aug 2026 11:13:23 +1000 Subject: [PATCH 15/33] A dash for the multiplier, and say when the CBF really is a solve The multiplier has nothing to recover: h is a finite element field, so its nodal values ARE the traction pointwise, and h + r(u.n - g) is an expression evaluated where it is wanted. A millisecond figure for "reading a field" invited exactly the comparison it should not -- the slot is a dash. The rotated reaction does need a step, and the note now says which. The reaction is an INTEGRATED nodal load, M_Gamma times the pointwise traction, so recovering a pointwise value means undoing the boundary mass. On a 2-D trace and on 3-D P1 triangles that mass is lumped and undoing it is a division -- the 10 to 16 ms measured here. On 3-D P2 triangles it is a genuine solve: the lumped row sums vanish at the vertices, so the consistent trace mass is assembled and solved, and Underworld gathers the trace to one rank to do it. That is the case a spherical free surface runs in, so it is worth naming. Which makes the real distinction not speed but what each method hands you: the multiplier gives the traction as a field, the reaction gives it as a load that still has to be divided by a mass. Underworld development team with AI support from Claude Code --- ...ary-conditions-on-non-planar-boundaries.md | 32 ++++++++++++++----- 1 file changed, 24 insertions(+), 8 deletions(-) diff --git a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md index 5bb7860..d693f62 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md +++ b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md @@ -646,20 +646,36 @@ run-to-run spread and there is nothing to read. | velocity nodes | penalty | Nitsche | multiplier | rotated | |---|---|---|---|---| -| 28 338 | 0.17 / 0.273 | 0.18 / 0.267 | 0.26 / 0.004 | 0.16 / 0.010 | -| 71 424 | 0.45 / 0.631 | 0.47 / 0.657 | 0.67 / 0.009 | 0.43 / 0.016 | +| 28 338 | 0.17 / 0.273 | 0.18 / 0.267 | 0.26 / — | 0.16 / 0.010 | +| 71 424 | 0.45 / 0.631 | 0.47 / 0.657 | 0.67 / — | 0.43 / 0.016 | + +The dash is not a missing measurement. **The multiplier has nothing to recover**: +$h$ is a finite element field in its own right, so its nodal values are the +traction, pointwise, and $h + r(\mathbf{u}\cdot\hat{\mathbf{n}} - g)$ is an +expression evaluated where it is wanted. Reading it costs whatever reading a +field costs — a few milliseconds of array indexing at this size, and nothing that +belongs in a column beside a solve. + +**The rotated constraint's reaction does need one step**, and it is worth being +precise about which. The reaction is an *integrated* nodal load, +$\int_\Gamma \sigma_{nn}\,\phi_i\,\mathrm{d}S$, which is $M_\Gamma$ times the +pointwise traction. Turning it into a pointwise value means undoing that boundary +mass. On a 2-D trace, and on 3-D P1 triangles, the lumped mass is diagonal and +undoing it is a division — the 10 to 16 ms above. On **3-D P2 triangles it is a +genuine solve**: the lumped row sums vanish at the vertices, so the consistent +trace mass has to be assembled and solved, and Underworld gathers the trace to +one rank to do it. That is the one place the CBF route pays for being a +back-calculation, and it is the case a spherical free surface runs in. + +So the conceptual difference the timings expose is not speed but *what each +method hands you*: the multiplier gives the traction as a field, and the reaction +gives it as a load that still has to be divided by a mass. **The multiplier's solve costs about 50% more**, consistently — 0.67 s against 0.43 s at 71 000 nodes. That is the extra field and the larger saddle point. Rotating the degrees of freedom costs nothing measurable against the weak forms: the rotation is a sparse orthogonal transform on a boundary's worth of rows. -**Neither exact route solves for its traction.** The multiplier's is -$h + r(\mathbf{u}\cdot\hat{\mathbf{n}} - g)$ evaluated on the boundary trace; the -rotated constraint's is its nodal reaction divided by the lumped boundary mass. -Both are arithmetic over the ~1 000 nodes of the trace, which is why they cost -milliseconds and why the two are within a factor of two of each other. - **The weak forms have to recover theirs, and that is a second solve.** Projecting $\hat{\mathbf{n}}\cdot\boldsymbol{\sigma}\hat{\mathbf{n}}$ out of the solution costs *more than the Stokes solve did* — 0.63 s against 0.45 s — so asking a From f5c829fe143867b496f53385cd34a08fac704880 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Thu, 20 Aug 2026 11:19:15 +1000 Subject: [PATCH 16/33] Nitsche's expensive recovery is not the projection's fault, it is the method's Louis asked whether the weak forms really need a global re-solve to get stresses. They do not need a GLOBAL one -- an L2 projection over every node to get values on a thousand boundary nodes is more work than the job requires -- but the cheap route is not open to them, and measuring it says why. The consistent boundary flux reads the assembled residual instead of differentiating anything. On the rotated constraint it matches the projection at a third of the cost (error 1.6e-4 either way). On Nitsche and on the penalty it returns an error of 1.00, which is the metric reporting that nothing was recovered. A reaction exists in the residual only where a row has been CONSTRAINED. A weak condition supplies its traction as a term inside the row it acts on, so the residual there is balanced at convergence and there is nothing left to read. The multiplier does exactly the same thing and gets away with it because the term it supplies IS the traction as a field; Nitsche's term is written in sigma(u), so reading it back still means differentiating the answer. Which lines up with the two pairs: imposed weakly, the traction is a by-product and you differentiate for it; imposed by construction, it is the constraint reaction and you de-smear a nodal load; imposed by a multiplier, it is an unknown of the system and you read the field. The cost of the first is negotiable, the differentiation is not. Underworld development team with AI support from Claude Code --- ...ary-conditions-on-non-planar-boundaries.md | 48 ++++++++++++++++--- 1 file changed, 41 insertions(+), 7 deletions(-) diff --git a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md index d693f62..308ca81 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md +++ b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md @@ -657,7 +657,9 @@ field costs — a few milliseconds of array indexing at this size, and nothing t belongs in a column beside a solve. **The rotated constraint's reaction does need one step**, and it is worth being -precise about which. The reaction is an *integrated* nodal load, +precise about which. (The 10 to 16 ms in the table is the reaction the solve +already stashed; `boundary_flux` re-assembles the residual from scratch and costs +0.2 s, which is the 0.206 s in the comparison above.) The reaction is an *integrated* nodal load, $\int_\Gamma \sigma_{nn}\,\phi_i\,\mathrm{d}S$, which is $M_\Gamma$ times the pointwise traction. Turning it into a pointwise value means undoing that boundary mass. On a 2-D trace, and on 3-D P1 triangles, the lumped mass is diagonal and @@ -676,12 +678,44 @@ gives it as a load that still has to be divided by a mass. Rotating the degrees of freedom costs nothing measurable against the weak forms: the rotation is a sparse orthogonal transform on a boundary's worth of rows. -**The weak forms have to recover theirs, and that is a second solve.** Projecting -$\hat{\mathbf{n}}\cdot\boldsymbol{\sigma}\hat{\mathbf{n}}$ out of the solution -costs *more than the Stokes solve did* — 0.63 s against 0.45 s — so asking a -penalty or Nitsche model for its surface stress roughly doubles the cost of the -timestep. Against 9 to 16 ms for the exact routes, that is a factor of forty to -seventy, and on a free-surface model it is paid at every step. +**The weak forms have to recover theirs by differentiating the solution**, and +the projection that does it costs *more than the Stokes solve did* — 0.63 s +against 0.45 s — so asking a penalty or Nitsche model for its surface stress +roughly doubles the timestep. + +The obvious question is whether that is the method's cost or the recovery's. A +global $L^2$ projection to get values on a thousand boundary nodes is plainly +more work than the job requires, and the consistent boundary flux is right there, +reading the assembled residual rather than differentiating anything. **It does +not work for a weakly imposed condition**, and the reason is the same one that +makes it unavailable to the multiplier: + +| cell 0.0125 | projection | CBF back-calculation | +|---|---|---| +| penalty, node normal | 0.651 s, error 1.1 × 10⁻³ | 0.224 s, **error 1.00** | +| Nitsche | 0.666 s, error 3.6 × 10⁻⁴ | 0.230 s, **error 1.00** | +| rotated | 0.602 s, error 1.6 × 10⁻⁴ | 0.206 s, error 1.6 × 10⁻⁴ | + +An error of 1.00 is the metric reporting that nothing was recovered. A reaction +exists in the residual only where a row has been *constrained*; a weak condition +supplies its traction as a term inside the row it acts on, so the residual there +is balanced at convergence and there is nothing left to read. The multiplier does +the same thing, and gets away with it because the term it supplies, $h$, is the +traction as a field. Nitsche's term is written in terms of +$\boldsymbol{\sigma}(\mathbf{u})$, so reading it back still means differentiating +the answer. + +That is the structural statement the timings are really making, and it follows +the two pairs exactly: + +| how the constraint is imposed | the traction is | to read it | +|---|---|---| +| weakly, by a term (penalty, Nitsche) | a by-product | differentiate the solution | +| exactly, by construction (rotated) | the constraint reaction | de-smear the nodal load | +| exactly, by a multiplier | an unknown of the system | read the field | + +The cost of the first row is negotiable — a recovery restricted to the boundary +would be cheaper than a global projection — but the differentiation is not. :::{note} What these timings are not Two-dimensional, one core, direct solver. What they measure is the difference From 6c338c4a115f7d66a20a4d002e6697a311c82cd9 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Thu, 20 Aug 2026 11:27:48 +1000 Subject: [PATCH 17/33] Land the argument where it started: a free surface The note opens with a free surface and now closes there. The point of the recovery taxonomy, for that case, is that sigma_nn is needed EVERY step -- the surface moves under the traction it carries -- so a recovery that costs more than the Stokes solve doubles the model rather than adding a post-processing step at the end of a run. Stated for the two exact treatments, since they are what a free surface should be choosing between. The rotated constraint's reaction on the 3-D P2 triangular trace a deforming spherical surface actually uses is de-smeared by a consistent-mass solve that is gathered to one rank: a serial section inside a parallel timestep. The multiplier hands over a field, so there is nothing to de-smear in 2-D or 3-D, and pays about 50% on the solve instead. Which of those wins on a large parallel shell is a question of rank count and we have not measured it. The note says so and names it as the next measurement rather than guessing. Underworld development team with AI support from Claude Code --- ...ary-conditions-on-non-planar-boundaries.md | 32 +++++++++++++++++++ 1 file changed, 32 insertions(+) diff --git a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md index 308ca81..98b8599 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md +++ b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md @@ -751,6 +751,38 @@ When the wall-normal traction is the answer: normal, a coefficient chosen per problem, and a check on something physical before the answer is believed. +### Which is to say, a free surface + +This note began with a free surface, and that is where the argument lands. A free +surface needs $\sigma_{nn}$ *every step*: the surface moves under the traction it +carries, so the recovery is not something done once at the end of a run but part +of the timestep, alongside the Stokes solve it follows. + +Everything above then reads differently. The projection that costs more than the +Stokes solve is paid at every step, and doubles the model. The differentiation it +performs is applied to a velocity field that a weak condition never made satisfy +$\mathbf{u}\cdot\hat{\mathbf{n}} = 0$ exactly, on a boundary that is moving. And +the surface stress recovered that way has failure modes of its own that have +nothing to do with the boundary condition — a continuous pressure checkerboards +at a viscosity jump, a discontinuous one puts a node-to-node zigzag into the +reaction on a simplex boundary — so the recovery has to be tuned against the +element types as well. + +The two exact treatments hand the traction over as part of the answer, which is +the property a free surface wants. Between them: + +- **the rotated constraint** gives the reaction, and on the 3-D P2 triangular + trace a deforming spherical surface actually uses, de-smearing it is a + consistent-mass solve that Underworld gathers to one rank. That is a real + serial step inside a parallel timestep; +- **the multiplier** gives a field, so there is nothing to de-smear at all — + in 2-D or in 3-D. It costs about 50% more in the solve. + +We have not measured that trade on a large parallel shell, and it is the +measurement worth doing next: the multiplier's premium is a fixed fraction of a +solve, while the gather is a serial section, and which one wins is a question of +rank count rather than of method. + ## Using it From 4f6fd6a930c348c8e83024b56ff0c0982513a758 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Thu, 20 Aug 2026 14:37:47 +1000 Subject: [PATCH 18/33] Settle the normal once, and stop the symbols colliding Four things Louis asked for, on top of his edit. The penalty term no longer carries gamma/h. It is a bare kappa, and the text says so: none of the mu/h that Nitsche's parameter carries, which is exactly why the value that works is a property of the model. The two statements contradicted each other before. The multiplier is lambda, not h. It was colliding with the mesh size in gamma/h and with the topography it produces. The prescribed wall-normal velocity is u-tilde-n rather than g, which was colliding with gravity. The section now says what lambda IS -- it has units of stress, at convergence it is sigma_nn, and dividing by delta-rho g (gravity, no relation to the constraint datum) gives a dynamic topography. The choice of normal is settled in its own section and not revisited. The locking measurement and the figure move there, the separate "a constraint that is satisfied, and wrong" section is gone, and the leak, parameter and stress tables carry ONE penalty column -- the node normal, which is what everything else uses by default. The section ends by saying the facet normal does not appear again, and it does not. The passage Louis marked unclear is replaced. With one normal in the results there is nothing to disambiguate: the penalty trades a decade of coefficient for a decade of leak, with no wall in this problem, and what it does not do is converge. Also fixed, from the edit: \label and \ref are LaTeX and KaTeX has neither, so the build was failing 92 times on "Undefined control sequence: \ref". The equation is labelled the MyST way and referenced with {eq}, which renders it as "(1)". That needed one more thing to work -- a display-math block opening on the line directly after prose is parsed as part of that paragraph, and its label is left as literal text. Every $$ block in the article now has a blank line before it. Underworld development team with AI support from Claude Code --- ...ary-conditions-on-non-planar-boundaries.md | 530 ++++++++---------- 1 file changed, 240 insertions(+), 290 deletions(-) diff --git a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md index 98b8599..dfe19f2 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md +++ b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md @@ -1,5 +1,5 @@ --- -title: Boundary conditions on non-planar boundaries +title: Free-Slip boundary conditions on curved boundaries description: >- "No flow through this wall" is a single velocity component on a box and is not a component of anything on a sphere, a deformed mesh, or a surface with @@ -28,90 +28,106 @@ exports: article_version: 1.0.0 software_version: underworld3 development @ 8b7c8b9e --- -A free surface moves under the traction it carries. That makes the wall-normal -stress the quantity driving the model rather than something read off at the -end, and it is the reason we went back to how the boundary condition underneath -it is imposed. - -The condition itself is ordinary. Free slip is no flow through the boundary and -no tangential drag along it, +Free-slip boundary conditions are used to simplify the physical behaviour at a domain boundary. +It may be a free-surface where the boundary deforms slightly in response to the internal flow, or it may +be an interface where the boundary layer thickness is so small that it cannot be resolved at the same time +time as the interior flow. The simplifying assumption: ignore the changes in shape, ignore the thin boundary layer, +treat the surface as impenetrable, and the tangential stresses as vanishingly small. $$ \mathbf{u}\cdot\hat{\mathbf{n}} = 0 \qquad\text{and}\qquad -\hat{\mathbf{t}}\cdot\boldsymbol{\sigma}\hat{\mathbf{n}} = 0 . -$$ - -On a box the first of those is a single velocity component. You hold $u_x$ on a -vertical wall, the solver removes a row, and there is nothing further to -discuss. On a sphere, an annulus, a mesh that has been moved, or a surface with -topography, $\mathbf{u}\cdot\hat{\mathbf{n}}$ is not a component of anything. -That is the whole difficulty, and everything below is a way around it. - -The second condition is worth naming because it is the one people forget. Zero -tangential traction is *natural*: it is what you get by leaving the boundary -term out of the weak form. Nothing has to be done to impose it, and something -has to be done to avoid imposing it accidentally. +\hat{\mathbf{t}}\cdot\boldsymbol{\sigma}\cdot\hat{\mathbf{n}} = 0 . +$$ (eq-free-slip) -## Where the boundary term comes from - -Every method below is a statement about one term. Multiplying the momentum +In the weak form, boundary tractions appear as surface integrals. Multiplying the momentum balance by a test function $\mathbf{w}$ and integrating by parts gives $$ \int_\Omega \boldsymbol{\sigma} : \nabla\mathbf{w} \; \mathrm{d}V -- \int_{\partial\Omega} (\boldsymbol{\sigma}\hat{\mathbf{n}})\cdot\mathbf{w} +- \int_{\partial\Omega} (\boldsymbol{\sigma}\cdot\hat{\mathbf{n}})\cdot\mathbf{w} \; \mathrm{d}S = \int_\Omega \mathbf{f}\cdot\mathbf{w} \; \mathrm{d}V . -$$ -Drop the surface integral and you have imposed zero traction in both -directions — free *everything*, not free slip. Free slip keeps the tangential -half of that and replaces the normal half with the constraint. How you do the -replacing is the choice. - -## Four ways, in order +$$ -They fall into two pairs, and the pairing is the useful way to hold them. Two +Drop the surface integral and you have imposed zero traction in all +directions — free *everything* (a free surface), not free slip. Constrain the surface-normal degrees of freedom +and the surface integral only addresses the tangential traction terms. + +On a Cartesian box, the first expression in {eq}`eq-free-slip` constrains a single velocity component. If you hold $u_x$ fixed on a +vertical wall, the solver removes a row of unknowns, and there is nothing further to +discuss. On a sphere, an annulus, a mesh that does not align with the coordinates, or a surface with +topography, $\mathbf{u}\cdot\hat{\mathbf{n}}$ is not a single component of the unknown — it constrains a combination of unknowns at +each point, and leaves other combinations free. That has the potential to make a simplifying +assumption complicated to implement. + +Let's assume, for a moment, we confine ourselves to simple domains such as an annulus, or a spherical shell, +which are commonly used for planetary modelling. For each of these cases, there are +coordinate systems, and well known forms of the differential operators that do restore the boundary condition +to being a constraint in a single direction. Admittedly this requires reformulating all the equations, but for +a symbolic-first code such as underworld, this is possible. This is the strategy used by CITCOMS [Zhong et al]. +But not every domain boundary has a convenient +coordinate system to follow. Even accounting for slight ellipticity introduces significant complexity in all the differential +operators; anything more complicated will not have any useful coordinate reformulation. + +The second condition is also worth noting. We don't think about this when we constrain a degree of freedom, +the other one/s, left unconstrained are *natural* to the problem. They fall out as traction-free surface conditions +automatically in a finite element weak form. If we cannot simply eliminate one degree of freedom at each point, +how **do** we satisfy all the parts of {eq}`eq-free-slip` ? + +## Four possibilities + +We outline four possible approaches (four you can try out in Underworld3). +They fall into two pairs: two impose the constraint **weakly**, by adding a term to the momentum equation and letting the solution satisfy the condition to within the accuracy of that term: a direct penalty, and Nitsche's method, which differ in whether the term is consistent. Two impose it **exactly**: by construction, changing the basis so that the constraint is a component that can be struck out, or by a Lagrange multiplier, adding an equation that enforces it. The weak pair have a parameter -to choose and a floor they cannot go below. The exact pair have neither, and both -return the boundary traction as part of the answer. +to select that may need to be tuned for each problem and a floor + they cannot go below. The exact pair are not tuneable, and they both +return the boundary traction as a side-effect of the solution. -### A direct penalty +### 1. A direct penalty -Add a term that punishes any flow through the boundary: +This could not be more simple, conceptually. We are working in a variational +environment, so we just add into our equation system, a term that punishes any flow through the boundary: $$ -\dots + \frac{\gamma}{h}\int_{\partial\Omega} +\dots + \kappa\int_{\partial\Omega} (\mathbf{u}\cdot\hat{\mathbf{n}})(\mathbf{w}\cdot\hat{\mathbf{n}}) \; \mathrm{d}S . + $$ +$\kappa$ is a bare number, with none of the $\mu/h$ that Nitsche's parameter +carries below. It has to absorb the scale of the problem itself, which is why +the value that works is a property of the model rather than a default. + One line, no new machinery, and it works on any geometry. It is still in a good many working scripts, and deservedly. What you are solving is a perturbed problem, though, and it is perturbed by exactly the amount the constraint is violated: the discrete solution sits where the penalty term balances the -traction it is fighting, which leaves $\mathbf{u}\cdot\hat{\mathbf{n}}$ small -but not zero. Making it smaller means pushing harder, and pushing harder -conditions the operator worse. The error is traded against the conditioning, -and measured below, that trade runs out: past $10^4$ the leak stops improving, -and by $10^6$ the solve fails. Nitsche moves the floor down rather than -removing it — it fails too, at its own threshold. +traction it is fighting and this leaves $\mathbf{u}\cdot\hat{\mathbf{n}}$ small but not zero. + +Making it smaller means pushing harder, and pushing harder +degrades the condition-number of the operator. The error is traded against the conditioning, +and (see the table below), this trade-off eventually stops returning any benefit: +past $10^4$ the boundary constraint stops improving (the surface is leaky), +and by $10^6$ the solve fails. Underworld spells it as a boundary traction opposing normal flow, using the -facet normal at the quadrature points: +surface normal at the quadrature points ($\Gamma$) provided by PETSc: ```python G = mesh.Gamma -stokes.add_natural_bc(1.0e4 * G.dot(v.sym) * G, "Upper") +penalty = 10000 +stokes.add_natural_bc(penalty * G.dot(v.sym) * G, "Upper") ``` -### Nitsche +### Nitsche's method The reason the penalty is only accurate in the limit is that it is not *consistent*: substituting the true solution does not leave the equation @@ -128,56 +144,61 @@ $$ + \frac{\gamma}{h}\int_{\partial\Omega} (\mathbf{u}\cdot\hat{\mathbf{n}})(\mathbf{w}\cdot\hat{\mathbf{n}}) \; \mathrm{d}S . + $$ The first of the three is the consistency term: it is the boundary traction the -integration by parts produced, and putting it back is what makes the true +integration by parts produced, and including this makes the true solution satisfy the discrete equations exactly. The second is its transpose, which keeps the form symmetric and buys optimal convergence in $L^2$. The third is the penalty again, and it is still needed — but now for *stability* rather than for accuracy, and $\gamma$ has a threshold set by an inverse inequality rather than being a dial you turn up until the answer looks right. -This is a real improvement and it is still a weak imposition. The constraint +This is a real improvement and it is still done through a weak imposition. The constraint holds to the accuracy of the discretisation, not to the accuracy of the -arithmetic — measured below, it leaks a few parts in a thousand on a mesh of -the resolution people actually run, and the leak falls with the mesh rather -than with the machine. +arithmetic — measured below, it leaks a few parts in a thousand on a typical mesh, + and the leak falls with increasing mesh resolution. ### A constraint equation, with a multiplier -The two above add a *term* to an equation. This one adds an *equation*. +The two above add a *term* to the weak form of the equation. This one adds an *equation*. -Carry a scalar field $h$ on the boundary and require, as a row of the system in -its own right, +Carry a scalar field $\lambda$ on the boundary and require, as a row of the +system in its own right, $$ -\int_{\partial\Omega} (\mathbf{u}\cdot\hat{\mathbf{n}} - g)\, q +\int_{\partial\Omega} (\mathbf{u}\cdot\hat{\mathbf{n}} - \tilde{u}_n)\, q \; \mathrm{d}S = 0 \quad \text{for all } q , + $$ -with $h$ entering the momentum row as the traction $h\hat{\mathbf{n}}$ that holds -the constraint. It is a Lagrange multiplier, and the system becomes a larger -saddle point: velocity, pressure, and now $h$. +where $\tilde{u}_n$ is the prescribed wall-normal velocity — zero for free slip, +and a datum if the wall is being driven — and $\lambda$ enters the momentum row +as the traction $\lambda\hat{\mathbf{n}}$ that holds the constraint. It is a +Lagrange multiplier, and the system becomes a larger saddle point: velocity, +pressure, and now $\lambda$. + +$\lambda$ has units of stress. At convergence it *is* $\sigma_{nn}$ on that +boundary, so dividing by $\Delta\rho\,g$ — density contrast times gravity, no +relation to the constraint datum — turns it into a dynamic topography. The constraint row is exact, so unlike a penalty there is no parameter whose size decides how well it holds. Two practical things do have to be dealt with. -- $h$ is carried as a full-domain field but only its boundary trace means +- $\lambda$ is carried as a full-domain field but only its boundary trace means anything, so the interior degrees of freedom are constrained out of the global system in the section before the solve sees it. They are not solved for and - they do not enlarge the $[p, h]$ block. The interior rows are the screening - block alone, so the interior multiplier is determined by the boundary trace and - carries no independent signal. -- The $[p, h]$ Schur complement is poorly conditioned on its own, so an - augmented-Lagrangian term $r(\mathbf{u}\cdot\hat{\mathbf{n}} - g)\hat{\mathbf{n}}$ - is added to the momentum row. It does not change what the constraint enforces, - because the $h$ row still carries the exact constraint. It does change what $h$ - *is*, which is the subject of a section below. + they do not enlarge the $[p, \lambda]$ block. +- The $[p, \lambda]$ Schur complement is poorly conditioned on its own, so an + augmented-Lagrangian term + $r(\mathbf{u}\cdot\hat{\mathbf{n}} - \tilde{u}_n)\hat{\mathbf{n}}$ is added to + the momentum row. It does not change what the constraint enforces, because the + $\lambda$ row still carries the exact constraint. ```python stokes = uw.systems.Stokes_Constrained(mesh, velocityField=v, pressureField=p) -h = stokes.add_constraint_bc(0.0, "Upper") +lam = stokes.add_constraint_bc(0.0, "Upper") stokes.solve() ``` @@ -186,18 +207,19 @@ momentum row's boundary term is the normal traction.** It is not recovered from the velocity field afterwards — it is an unknown the solve returned, available through `traction` and, divided by $\Delta\rho g$, through `topography`. -That term is the whole boundary load, $h + r(\mathbf{u}\cdot\hat{\mathbf{n}} - g)$, -and not the multiplier alone. The second part vanishes only where the constraint -row is satisfied exactly; discretely it is satisfied to the solver's tolerance and -$r$ multiplies that residual back into the traction. With a viscosity-weighted -$r$ and a lateral viscosity contrast it is most of the answer, so the two parts -are not separable in practice — which is the first hint that this traction and -the rotated constraint's reaction are the same object. +That term is the whole boundary load, +$\lambda + r(\mathbf{u}\cdot\hat{\mathbf{n}} - \tilde{u}_n)$, and not the +multiplier alone. The second part vanishes only where the constraint row is +satisfied exactly; discretely it is satisfied to the solver's tolerance, and $r$ +multiplies that residual back into the traction. With a viscosity-weighted $r$ +and a lateral viscosity contrast it is most of the answer, so the two parts are +not separable in practice. ### Rotating the degrees of freedom -Stop asking for the constraint and impose it. At each constrained node, change -the basis in which the velocity unknowns are expressed, from the global +In this approach, we stop *"asking for"* the constraint and just impose it. +At each constrained node, we change +the coordinate basis in which the velocity unknowns are expressed, from the global Cartesian frame to the local $(\hat{\mathbf{n}}, \hat{\mathbf{t}})$ frame. In that basis "no flow through the boundary" is again a single component, and it is removed the same way it would be on a box. @@ -208,22 +230,23 @@ at every node that is not constrained. The rotated system is $$ \hat{A} = Q^{T} A Q, \qquad \hat{\mathbf{b}} = Q^{T}\mathbf{b}, \qquad \mathbf{u} = Q\hat{\mathbf{u}} , + $$ and the wall-normal row of $\hat{A}$ is struck out. The constraint then holds to machine precision, because it is not being solved for at all. -**This is the classical answer**, not a new one. It is in the early +**This is the classical strategy**. It is in the early finite-element literature, and @Engelman_1982 were already reviewing the alternatives and choosing between them on grounds of global mass conservation in 1982. What is worth explaining is not the idea but why, given that it is exact and the others are not, it is the least used of the three. -## What it costs, and the cost is structural +## Trade-offs Rotating the degrees of freedom leaves the discrete problem in a **mixed basis**. Interior nodes hold $(u_x, u_y)$; constrained nodes hold -$(u_n, u_t)$. Nothing about that is difficult in itself, and everything +$(u_n, u_t)$. Nothing about that is difficult in itself, but everything downstream has to agree about which nodes are which. ```{figure} figures/rotated-basis.svg @@ -239,21 +262,16 @@ boundary between them and feeds both. Four things carry $Q$: the operator, the right-hand side, the solution on the way out, and the multigrid prolongation. The coarse operators inherit it through the Galerkin triple product rather than being rotated separately, and -the coarse solve then has to be an SVD, because a Galerkin-coarsened rotated -operator inherits the rigid-rotation null space of the constrained problem and -a redundant LU factorisation meets a zero pivot in it. - -That last point is worth dwelling on, because it is the one that surprises. The -constraint never has to be re-imposed on a coarse mesh — which is fortunate, -since we install no discretisation there and could not impose it if we wanted -to. It arrives anyway, algebraically, through $P^{T}\hat{A}P$ with a rotated -$P$. The evidence that it really arrives is that the coarse operator inherits -the null space, which is a property only the constrained problem has. - -None of this is an argument against the method. It is an argument for knowing -what is being taken on, and it is the honest reason a weakly imposed condition -survives in codes that could do this instead. -## Which normal +the coarse solve should be an SVD, because a Galerkin-coarsened rotated +operator inherits any rigid-rotation null space of the constrained problem +(the exact constraint makes the null space of the sphere and the annulus a dominant +feature of the solve). + +We do not know the cost of the addtional complexity on the solver and setup times, +or on the accuracy of the solution but this can be measured and will differ from problem +to problem. + +## Choice of the surface normal A question with a less obvious answer than it looks, and the measurements further down say it is the most consequential choice in the note. The boundary of a @@ -273,13 +291,48 @@ here. Using the analytic normal is exact for the geometry and therefore inconsistent with the discretisation, which is the wrong way round: the solver is not solving -on the sphere, it is solving on the polyhedron. Using the facet normal is -consistent with each facet separately and over-constrains the nodes between -them, which is the failure measured in "A constraint that is satisfied, and -wrong" below. In parallel the same argument has a -sharper edge, because a normal accumulated rank-locally is wrong at a partition -boundary, where a node's facets are split across ranks and no rank sees them -all. +on the sphere, it is solving on the polyhedron. In parallel the same argument has +a sharper edge, because a normal accumulated rank-locally is wrong at a partition +boundary, where a node's facets are split across ranks and no rank sees them all. + +Using the **facet** normal is worse than inconsistent, and this is the one place +in the note where a choice does real damage. Imposing +$\mathbf{u}\cdot\hat{\mathbf{n}} = 0$ facet by facet asks a node shared by two +facets to satisfy two different constraints, and two independent constraints on a +two-component velocity leave nothing. Push the coefficient up and the vertex +velocities go to zero: the flow is being asked to stay inside a polygon rather +than a circle, and the discrete limit is a different problem from the smooth one. +Refining the mesh does not approach the smooth answer, because it is not +converging to it. + +A direct penalty at $\kappa = 10^6$, on the annulus of the next section, against +each normal in turn: + +| cell size | facet normal: leak / velocity error / stress error | node normal: leak / velocity / stress | +|---|---|---| +| 0.150 | 8.9 × 10⁻⁶ / 0.60 / 0.21 | 4.6 × 10⁻⁵ / 1.0 × 10⁻² / 2.4 × 10⁻² | +| 0.075 | 1.1 × 10⁻⁵ / 0.61 / 0.25 | 3.7 × 10⁻⁵ / 2.4 × 10⁻³ / 6.3 × 10⁻³ | +| 0.050 | 1.9 × 10⁻⁵ / 0.60 / 0.26 | 3.1 × 10⁻⁵ / 1.0 × 10⁻³ / 2.7 × 10⁻³ | +| 0.035 | 2.3 × 10⁻⁵ / 0.59 / 0.26 | 3.1 × 10⁻⁵ / 4.9 × 10⁻⁴ / 1.4 × 10⁻³ | + +The facet-normal column is stuck: the velocity is 60% wrong and the stress 26% +wrong, and refining changes neither. Note what the leak does — it is *excellent*, +better than the node normal's, because a frozen boundary passes no flow. A metric +that only asks whether the constraint is satisfied cannot tell a locked solution +from a good one. + +```{figure} figures/locking.png +:alt: Three annuli side by side on one colour scale from zero to 5.0e-3, blue for slow and red for fast, with the triangular mesh drawn over each. The left panel is the exact solution: two deep red patches of fast flow sit against the outer boundary on the left and right of the annulus, with a blue slow ring inside them. The middle panel is the same problem solved with a direct penalty against the facet normal: the red patches at the outer boundary are gone and the whole outer half is blue, the peak speed having fallen from 5.0e-3 to 3.8e-3, while a pale ring survives near the inner boundary. The right panel is the same penalty against the measure-weighted node normal and is indistinguishable from the exact panel, with a peak speed of 5.0e-3. + +The same problem, the same coefficient, the same colour scale. Against the facet +normal the flow along the outer boundary is suppressed — the peak speed falls by +a quarter and the two fast lobes at the boundary are gone. Against the +measure-weighted node normal it is the exact solution. +``` + +Everything that follows uses the node normal, which is what `add_nitsche_bc` and +`add_rotated_freeslip_bc` take by default and what `mesh.boundary_normal` returns. +The facet normal does not appear again. The consistent normal is not the end of the matter. @Behr_2004 takes it as the starting point — "preferred from the point of view of conservation" — and @@ -290,184 +343,130 @@ We have read the abstract rather than the paper, and have not looked for that recirculation in our own cases. It is the obvious thing to test next for anyone running free slip on a strongly curved wall. -## The option this note leaves out - -Solving in spherical or cylindrical components makes the wall-normal direction -a coordinate direction again, and the constraint returns to being "hold one -component". That is the same rotation as above, applied once for the whole -domain instead of node by node, and applied that way it costs nothing -structurally: there is no mixed basis, because every node is in the same basis. - -It works exactly when the boundary lies along a coordinate surface. A sphere, an -annulus, a cylinder. It does nothing for topography, for a mesh that has been -deformed, or for a tilted internal surface, which is the general case this note -is about. The per-node rotation is what remains once the geometry stops -cooperating, and its structural price is what buys the generality. - ## When the choice matters -Here is the awkward part. Solve a convection model with any of these and the +Solve a convection model with any of these approaches, and the velocity field is the same to plotting accuracy, as long as each is set up -properly. A leak of order $10^{-3}$ in $\mathbf{u}\cdot\hat{\mathbf{n}}$ is -invisible to anything that consumes the velocity, and consuming the velocity is -most of what a model does. If that is your situation, use the simplest thing -that works — and read the section on which normal before you do, because that -is the choice that can spoil the velocity as well. - -The difference appears when the wall-normal traction is the answer rather than -a by-product: dynamic topography, a plate-boundary force balance, anything -compared against a geoid or a gravity field. That is what the two sections after -the leak measure, against an exact surface stress, and the answer is not the one -we expected. The treatments that impose the constraint properly all recover the -same surface stress at a given mesh, because the recovery sets the floor rather -than the boundary condition; what the rotated constraint buys is a route to the -traction that does not go through a recovery at all, and it is three times -better on the same solve. - -Under the rotated constraint the reaction *is* $\sigma_{nn}$: it is the -multiplier the solve has already computed, and it comes out of -`boundary_normal_traction` without differentiating the answer. - -### The leak, measured +correctly. Generally speaking, a leak of order $10^{-3}$ or $10^{-4}$ in $\mathbf{u}\cdot\hat{\mathbf{n}}$ is +within the expected accuracy of the solution on the mesh and the main +driver of which method to choose should be solver efficiency (wall time). + +The difference in the methods appears when the wall-normal traction is a +required output of the model: dynamic topography, geoid, gravity, +or a plate-boundary force balance require accurate integration of boundary stresses. +Here the choice becomes more subtle, +and the methods have quite different accuracies, and different efficiencies. + +### The surface permeability question An annulus, no slip on the inner radius, the treatment under test on the outer, driven by a degree-four radial density anomaly. The number is the largest normal velocity on the outer boundary, taken against the true radial direction and divided by the flow speed: the fraction of the flow going through a boundary nothing should pass through. Penalty at $10^4$, Nitsche at -$\gamma = 10$. +$\gamma = 10$. [REFERENCE TO ANALYTIC MODULE] -The penalty appears twice, because the normal it is written against turns out -to matter more than the method it is written into. One column uses the -quadrature-point facet normal `mesh.Gamma`, which is the documented form; the -other uses the measure-weighted node normal `mesh.boundary_normal`, which is -what Nitsche and the rotated constraint use by default. - -| cell size | penalty, facet normal | penalty, node normal | Nitsche | multiplier | rotated | -|---|---|---|---|---|---| -| 0.150 | 4.5 × 10⁻³ | 3.3 × 10⁻³ | 4.6 × 10⁻³ | 8.3 × 10⁻⁴ | 7.3 × 10⁻¹¹ | -| 0.100 | 2.5 × 10⁻³ | 2.7 × 10⁻³ | 2.2 × 10⁻³ | 1.6 × 10⁻⁴ | 6.5 × 10⁻¹¹ | -| 0.075 | 8.5 × 10⁻⁴ | 2.6 × 10⁻³ | 1.7 × 10⁻³ | 5.6 × 10⁻⁵ | 1.0 × 10⁻¹⁰ | -| 0.050 | 9.5 × 10⁻⁴ | 2.6 × 10⁻³ | 5.8 × 10⁻⁴ | 1.2 × 10⁻⁵ | 8.4 × 10⁻¹¹ | +| cell size | penalty | Nitsche | multiplier | rotated | +|---|---|---|---|---| +| 0.150 | 3.3 × 10⁻³ | 4.6 × 10⁻³ | 8.3 × 10⁻⁴ | 7.3 × 10⁻¹¹ | +| 0.100 | 2.7 × 10⁻³ | 2.2 × 10⁻³ | 1.6 × 10⁻⁴ | 6.5 × 10⁻¹¹ | +| 0.075 | 2.6 × 10⁻³ | 1.7 × 10⁻³ | 5.6 × 10⁻⁵ | 1.0 × 10⁻¹⁰ | +| 0.050 | 2.6 × 10⁻³ | 5.8 × 10⁻⁴ | 1.2 × 10⁻⁵ | 8.4 × 10⁻¹¹ | Nitsche leaks parts in a thousand and improves roughly as $h^{1.9}$ — the rate consistency buys. The multiplier starts an order of magnitude better and falls -much faster, near $h^{3.9}$, because the constraint row is an equation rather -than a term whose weight has to be chosen. -The rotated constraint does not move at all: it sits at the solver's floor at +much faster, near $h^{3.9}$. The rotated constraint does not move at all: it sits at the solver's floor at every resolution, because the mesh has nothing to do with it. The penalty -against the node normal does not improve with the mesh either, and for the -opposite reason: at a fixed coefficient its leak is set by the coefficient. - -The control matters more than the result. With the outer boundary left free the -same measurement reads **0.98** — nearly all the boundary flow is normal — so -the metric can see a leak when there is one. - -The first column is the one to be careful with. Its leak falls with the mesh, -which reads as the method working, and it is not: the surface stress measured -further down says that solve is 60% wrong in the velocity and 26% wrong in the -stress, and refining it does not help. The leak is small because the boundary -is being frozen. A metric that only asks whether the constraint is satisfied -cannot tell those apart. +does not improve with the mesh either, and for the opposite reason: its leak is +set by the coefficient, not by the discretisation. ### What each parameter buys The two weak methods look alike in that table. They are not alike, and their own parameters are what tells them apart. -| penalty, facet | leak | | penalty, node | leak | | $\gamma$ (Nitsche) | leak | -|---|---|---|---|---|---|---|---| -| 10² | 2.6 × 10⁻¹ | | 10² | 2.6 × 10⁻¹ | | 1 | diverged | -| 10³ | 2.3 × 10⁻² | | 10³ | 2.6 × 10⁻² | | 10 | 1.7 × 10⁻³ | -| 10⁴ | 8.5 × 10⁻⁴ | | 10⁴ | 2.6 × 10⁻³ | | 100 | 2.7 × 10⁻⁴ | -| 10⁵ | 9.6 × 10⁻⁴ | | 10⁵ | 3.0 × 10⁻⁴ | | 1000 | 3.0 × 10⁻⁵ | -| 10⁶ | diverged | | 10⁶ | 4.5 × 10⁻⁵ | | 10⁴ and above | diverged | - -Nitsche is bounded at both ends. Below $\gamma = 1$ the form is no longer -coercive and no amount of tuning recovers it; from $\gamma = 10^4$ in this -problem the line search stops converging. The virtue of $\gamma = 10$ is that it -sits in the middle of that window on any mesh, because $\gamma$ is dimensionless -and the term it scales already carries $\mu / h$. The penalty coefficient -carries no such scaling, which is why the value that works is a property of the -problem rather than a default — and, on the second half of this note's test, of -the local viscosity as well. - -The two penalty columns are the same method against different normals, and they -behave differently in a way the leak alone does not explain. Against the node -normal the leak keeps falling, a decade of coefficient for a decade of leak, all -the way to $10^6$. Against the facet normal it stops improving after $10^4$ and -the solve fails at $10^6$. That looks like the conditioning wall a penalty is -expected to have. It is not: what stalls is the leak, because by $10^4$ the -boundary is nearly frozen and there is little normal flow left to remove. The -stress section below measures the same runs against an exact answer, and they -are getting worse throughout. +| $\kappa$ (penalty) | leak | | $\gamma$ (Nitsche) | leak | +|---|---|---|---|---| +| 10² | 2.6 × 10⁻¹ | | 1 | diverged | +| 10³ | 2.6 × 10⁻² | | 10 | 1.7 × 10⁻³ | +| 10⁴ | 2.6 × 10⁻³ | | 100 | 2.7 × 10⁻⁴ | +| 10⁵ | 3.0 × 10⁻⁴ | | 1000 | 3.0 × 10⁻⁵ | +| 10⁶ | 4.5 × 10⁻⁵ | | 10⁴ and above | diverged | + +Nitsche is bounded at both ends. Below $\gamma \sim 1$ the form is no longer +coercive and no amount of solver tuning recovers it; from $\gamma \sim 10^4$ in this +problem, the line search stops converging. The virtue of $\gamma \sim 10$ is that it +sits within that window on any mesh, because $\gamma$ is dimensionless +and the term it scales already carries $\mu / h$. + +The penalty coefficient scales differently because it directly penalises the +value of the velocity across the boundary. It should therefore scale with the +characteristic velocity which is best estimated from the magnitude of the forcing +terms and the resisting viscosity. + +Written against the node normal, the penalty simply trades: a decade of +coefficient for a decade of leak, all the way to $10^6$, with no wall in this +problem. What it does not do is converge — the leak is bought with the parameter +rather than with the mesh — so the coefficient has to be re-chosen whenever the +forcing or the viscosity changes. ### The stress, measured -The leak says how well each treatment holds the boundary. Whether the answer is -right is a different question, and it needs an exact answer to compare against. +The leak says how well each treatment satisfied the free-slip requirement at the boundary. +But we still need to measure whether the method converges to the correct solution. Kramer, Davies and Wilson [@Kramer_2021] give exact Stokes -solutions in a cylindrical annulus, and their `assess` package publishes the -radial stress itself rather than leaving it to be recovered from a velocity -field. Underworld wraps it as `uw.analytic.CylindricalStokes`. The case here is +solutions in a cylindrical annulus, and their `assess` package also publishes the +radial stress. Underworld wraps it as `uw.analytic.CylindricalStokes`. +The case here is the smooth one: a density anomaly $(r/r_o)^k \cos n\theta$ with $n = 2$ and $k = 3$, viscosity 1, free slip on both radii. On the outer boundary the exact radial stress is a single harmonic, $$ \sigma_{rr}(r_o, \theta) = 0.1506696\,\cos 2\theta , + $$ fitted to a residual of $10^{-16}$, so the whole of the surface stress is that -one amplitude and the metric is its relative error. - -The inner boundary carries the exact velocity as a Dirichlet condition instead -of a free-slip treatment of its own. The exact solution satisfies both, so the -problem is unchanged, and the treatment under test is then the only free-slip -condition in the model. +one amplitude and the metric is its relative error. The inner boundary carries +the exact analytic velocity as a Dirichlet condition instead +of a free-slip treatment of its own. There are two routes to the surface stress and the difference between them is the point of the section: -- **recovered** — project $\hat{\mathbf{n}}\cdot\boldsymbol{\sigma}\hat{\mathbf{n}}$ - out of the solved velocity and pressure. Every treatment can do this, and it +- **recovered** — project $\hat{\mathbf{n}}\cdot\boldsymbol{\sigma}\cdot \hat{\mathbf{n}}$ + out of the solved velocity and pressure. Every approach offers this, but it is the only route the weak ones have. -- **reaction** — `boundary_normal_traction` for the rotated constraint, and the - multiplier field for the constraint method. Not recovered from the solution: - an unknown the solve returned. - -Both are taken against the true radial direction, which is also the direction -the oracle publishes, so no treatment is scored against its own normal. The -reaction and the multiplier come back with the opposite sign to $\sigma_{rr}$ — -they are the traction holding the boundary rather than the traction the fluid -exerts, and `dynamic_topography` carries the sign back — so amplitudes are -compared unsigned. - -| cell size | penalty, facet | penalty, node | Nitsche | constraint | rotated | rotated, reaction | multiplier field | -|---|---|---|---|---|---|---|---| -| 0.150 | 1.5 × 10⁻¹ | 2.5 × 10⁻² | 5.9 × 10⁻² | 2.4 × 10⁻² | 2.4 × 10⁻² | 6.8 × 10⁻³ | 8.6 × 10⁻³ | -| 0.100 | 1.3 × 10⁻¹ | 1.1 × 10⁻² | 2.4 × 10⁻² | 1.0 × 10⁻² | 1.0 × 10⁻² | 3.3 × 10⁻³ | 8.5 × 10⁻⁴ | -| 0.075 | 1.1 × 10⁻¹ | 7.2 × 10⁻³ | 1.5 × 10⁻² | 6.3 × 10⁻³ | 6.2 × 10⁻³ | 2.1 × 10⁻³ | 1.7 × 10⁻³ | -| 0.050 | 6.3 × 10⁻² | 3.6 × 10⁻³ | 6.3 × 10⁻³ | 2.7 × 10⁻³ | 2.7 × 10⁻³ | 1.1 × 10⁻³ | 1.4 × 10⁻³ | +- **reaction** — compute the boundary-normal traction for the rotated constraint, and use the + multiplier field for the constraint method. This is not recovered from the stress field but from the + surface reaction to the stress field. + +Both are computed against the analytic radial direction, which is also the direction +the oracle publishes, so no treatment is scored against its own normal. + +| cell size | penalty | Nitsche | multiplier | rotated | rotated, reaction | multiplier, reaction | +|---|---|---|---|---|---|---| +| 0.150 | 2.5 × 10⁻² | 5.9 × 10⁻² | 2.4 × 10⁻² | 2.4 × 10⁻² | 6.8 × 10⁻³ | 8.6 × 10⁻³ | +| 0.100 | 1.1 × 10⁻² | 2.4 × 10⁻² | 1.0 × 10⁻² | 1.0 × 10⁻² | 3.3 × 10⁻³ | 8.5 × 10⁻⁴ | +| 0.075 | 7.2 × 10⁻³ | 1.5 × 10⁻² | 6.3 × 10⁻³ | 6.2 × 10⁻³ | 2.1 × 10⁻³ | 1.7 × 10⁻³ | +| 0.050 | 3.6 × 10⁻³ | 6.3 × 10⁻³ | 2.7 × 10⁻³ | 2.7 × 10⁻³ | 1.1 × 10⁻³ | 1.4 × 10⁻³ | The first five columns are all the recovered stress, so they differ only by which boundary condition produced the field. The last two are the reaction and the multiplier on the same solves. Penalties at $10^4$, Nitsche at -$\gamma = 10$, which are the values the leak table used. With the outer -boundary left free the same measurement reads 1.0 — the surface stress is -gone — so the metric can see the condition being removed. +$\gamma = 10$, which are the values the leak table used. -**Once the constraint is imposed against the node normal, which treatment -imposed it stops mattering to the recovered stress.** At cell 0.075 the rotated +**Which treatment imposed the constraint stops mattering to the recovered +stress.** At cell 0.075 the rotated constraint gives 6.2 × 10⁻³, the multiplier 6.3 × 10⁻³, the penalty at $10^5$ -6.3 × 10⁻³ and Nitsche at $\gamma = 100$ 6.6 × 10⁻³. Those are the same number. +6.3 × 10⁻³ and Nitsche at $\gamma = 100$ is 6.6 × 10⁻³. Those are the same number. What sets it is the recovery — a projection of a stress differentiated out of a piecewise-quadratic velocity — and not the boundary condition underneath. The reasoning this note began with — that a traction recovered from an approximate -constraint inherits the approximation — is not what the measurement shows once -the constraint is written against the right normal. It shows a floor that all of -them share. +constraint inherits the approximation — is not what the measurement shows. It +shows a floor that all of them share. **The reaction is about three times better than the recovery, on the same solve.** 2.1 × 10⁻³ against 6.2 × 10⁻³ at cell 0.075, and it converges at the @@ -488,65 +487,19 @@ forty and the stress by a factor of two, to the recovery floor, where more $\gamma$ buys nothing. Reading the leak alone would have said $\gamma = 10$ was converged. -### A constraint that is satisfied, and wrong - -The facet-normal penalty column does not converge. That is not a slow rate; the -error grows slightly as the mesh is refined, and the leak is excellent -throughout. - -| cell size | leak | velocity error | stress error | -|---|---|---|---| -| 0.150 | 8.9 × 10⁻⁶ | 0.60 | 0.21 | -| 0.100 | 8.2 × 10⁻⁶ | 0.61 | 0.24 | -| 0.075 | 1.1 × 10⁻⁵ | 0.61 | 0.25 | -| 0.050 | 1.9 × 10⁻⁵ | 0.60 | 0.26 | -| 0.035 | 2.3 × 10⁻⁵ | 0.59 | 0.26 | - -```{figure} figures/locking.png -:alt: Three annuli side by side on one colour scale from zero to 5.0e-3, blue for slow and red for fast, with the triangular mesh drawn over each. The left panel is the exact solution: two deep red patches of fast flow sit against the outer boundary on the left and right of the annulus, with a blue slow ring inside them. The middle panel is the same problem solved with a direct penalty against the facet normal: the red patches at the outer boundary are gone and the whole outer half is blue, the peak speed having fallen from 5.0e-3 to 3.8e-3, while a pale ring survives near the inner boundary. The right panel is the same penalty against the measure-weighted node normal and is indistinguishable from the exact panel, with a peak speed of 5.0e-3. - -The same problem, the same coefficient of $10^6$, the same colour scale, and the -only difference between the middle and right panels is which normal the penalty -is written against. Against the facet normal the flow along the outer boundary -is suppressed: the peak speed falls by a quarter and the two fast lobes at the -boundary are gone. Refining the mesh does not bring them back. -``` - -The coefficient is $10^6$ and the normal is `mesh.Gamma`, the facet normal at -the quadrature points. The same coefficient against the measure-weighted node -normal, on the same meshes, gives velocity errors of 1.0 × 10⁻², 2.4 × 10⁻³, -1.0 × 10⁻³ and 4.9 × 10⁻⁴ and stress errors of 2.4 × 10⁻², 6.3 × 10⁻³, -2.7 × 10⁻³ and 1.4 × 10⁻³. One difference, one line of code, and one method -converges while the other does not. - -The mechanism is old and well understood. Imposing -$\mathbf{u}\cdot\hat{\mathbf{n}} = 0$ facet by facet asks a node shared by two -facets to satisfy two different constraints, and two independent constraints on -a two-component velocity leave nothing. Push the coefficient up and the vertex -velocities go to zero: the flow is being asked to stay inside a polygon rather -than a circle, and the polygon is not a good enough approximation of the circle -for that particular question. The discrete limit is a different problem from -the smooth one, so refining the mesh does not approach the smooth answer. - -Two things follow for practice. A direct penalty is a perfectly good method, -but it has to be written against the node normal. And the leak is not a -sufficient check: at a coefficient of $10^3$ the facet-normal penalty leaks -3 × 10⁻² and gets the stress amplitude right to 1.9 × 10⁻³, while at $10^8$ it -leaks 1 × 10⁻⁷ and is 26% wrong. Over that range the constraint improves by five -orders of magnitude and the answer gets steadily worse. - ### The multiplier and the consistent boundary flux are the same object The correction above is not a patch. Write the momentum row's boundary term out and the identity is immediate: the assembled load is -$M_\Gamma\,(h + r(\mathbf{u}\cdot\hat{\mathbf{n}} - g))$, and at convergence it +$M_\Gamma\,(\lambda + r(\mathbf{u}\cdot\hat{\mathbf{n}} - \tilde{u}_n))$, and at convergence it balances the volume residual restricted to the boundary, which is precisely the nodal load the consistent boundary flux back-calculation reads [@Zhong_1993]. So $$ -h + r(\mathbf{u}\cdot\hat{\mathbf{n}} - g) +\lambda + r(\mathbf{u}\cdot\hat{\mathbf{n}} - \tilde{u}_n) = -M_\Gamma^{-1} \left. (A\mathbf{u} - \mathbf{b}) \right|_\Gamma , + $$ the CBF traction de-smeared with the boundary mass. The multiplier is not a @@ -602,11 +555,8 @@ elements trimmed from each end. **The three exact treatments agree to three figures at every contrast**, whole wall and trimmed alike. That is the result to take from this half, and it took -two fixes to get: the multiplier had to report the whole traction rather than -$h$, and the rotated constraint had to hold the corner where it meets the side -walls. Before them the rotated column read 0.322 at a contrast of 10 — six times -the reference, all of it two nodes — and the multiplier column was an order of -magnitude out at $10^6$. +the multiplier reporting the whole traction rather than $\lambda$ alone, and the +rotated constraint holding the corner where it meets the side walls. **Read the first column as the floor.** The component Dirichlet condition is exact and has no parameter, and its velocity error is 8.8 × 10⁻⁶ at a contrast of @@ -650,8 +600,8 @@ run-to-run spread and there is nothing to read. | 71 424 | 0.45 / 0.631 | 0.47 / 0.657 | 0.67 / — | 0.43 / 0.016 | The dash is not a missing measurement. **The multiplier has nothing to recover**: -$h$ is a finite element field in its own right, so its nodal values are the -traction, pointwise, and $h + r(\mathbf{u}\cdot\hat{\mathbf{n}} - g)$ is an +$\lambda$ is a finite element field in its own right, so its nodal values are the +traction, pointwise, and $\lambda + r(\mathbf{u}\cdot\hat{\mathbf{n}} - \tilde{u}_n)$ is an expression evaluated where it is wanted. Reading it costs whatever reading a field costs — a few milliseconds of array indexing at this size, and nothing that belongs in a column beside a solve. @@ -700,7 +650,7 @@ An error of 1.00 is the metric reporting that nothing was recovered. A reaction exists in the residual only where a row has been *constrained*; a weak condition supplies its traction as a term inside the row it acts on, so the residual there is balanced at convergence and there is nothing left to read. The multiplier does -the same thing, and gets away with it because the term it supplies, $h$, is the +the same thing, and gets away with it because the term it supplies, $\lambda$, is the traction as a field. Nitsche's term is written in terms of $\boldsymbol{\sigma}(\mathbf{u})$, so reading it back still means differentiating the answer. From fab96ab9eb254fb58a2535582d8377b1c99157d6 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Thu, 20 Aug 2026 16:28:43 +1000 Subject: [PATCH 19/33] Say how each method gives the topography, where the method is introduced The note was holding back the one thing a reader comes for. Each of the four sections now ends with how you get the surface stress out of that method, with the expression: penalty, Nitsche sigma_nn = n.sigma(u,p).n recovered from the solution -- a projection, and a second solve -- because the boundary term supplies the traction inside the row it sits in, so the residual carries none multiplier sigma_nn = lambda + r(u.n - u~_n), read on the trace: no recovery, because lambda is a field rotated sigma_nn = -M_Gamma^-1 (Au - b)|_Gamma, the reaction of the struck row de-smeared by the boundary mass -- the CBF and h = -(sigma_nn - mean)/(delta-rho g) in each case. The later section that compared them no longer opens on "the correction above is not a patch", which referred to something that had already been deleted; it opens by pointing at the two expressions and showing they are the same one. The benchmark is introduced once. It was described twice, in different words, with the leak measured on one problem and the stress on another -- so the two tables could not honestly be read against each other. They now come from THE SAME SOLVES: `stress.py both` reports the leak and the stress error per run. The numbers change slightly because the leak is now measured on the Kramer problem rather than on a separate degree-four one. Reading them side by side is the point, and it is a stronger result than either alone: the rotated constraint holds the boundary nine orders of magnitude better than the multiplier and lands on the same stress error, because what sets that number is the recovery and not the boundary condition. Citations are all one form now -- bracketed [@Key], with the authors named in the prose where the sentence wants them. One was still a raw DOI, which would not have resolved against the pinned bibliography. Underworld development team with AI support from Claude Code --- ...ary-conditions-on-non-planar-boundaries.md | 348 ++++++++---------- .../examples/stress.py | 39 +- 2 files changed, 200 insertions(+), 187 deletions(-) diff --git a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md index dfe19f2..0d764c5 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md +++ b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md @@ -37,26 +37,23 @@ treat the surface as impenetrable, and the tangential stresses as vanishingly sm $$ \mathbf{u}\cdot\hat{\mathbf{n}} = 0 \qquad\text{and}\qquad -\hat{\mathbf{t}}\cdot\boldsymbol{\sigma}\cdot\hat{\mathbf{n}} = 0 . -$$ (eq-free-slip) +\hat{\mathbf{t}}\cdot\boldsymbol{\sigma}\cdot\hat{\mathbf{n}} = 0. +$$ +(eq-free-slip) In the weak form, boundary tractions appear as surface integrals. Multiplying the momentum balance by a test function $\mathbf{w}$ and integrating by parts gives - $$ \int_\Omega \boldsymbol{\sigma} : \nabla\mathbf{w} \; \mathrm{d}V - \int_{\partial\Omega} (\boldsymbol{\sigma}\cdot\hat{\mathbf{n}})\cdot\mathbf{w} \; \mathrm{d}S -= \int_\Omega \mathbf{f}\cdot\mathbf{w} \; \mathrm{d}V . - + = \int_\Omega \mathbf{f}\cdot\mathbf{w} \; \mathrm{d}V . $$ - Drop the surface integral and you have imposed zero traction in all directions — free *everything* (a free surface), not free slip. Constrain the surface-normal degrees of freedom and the surface integral only addresses the tangential traction terms. -On a Cartesian box, the first expression in {eq}`eq-free-slip` constrains a single velocity component. If you hold $u_x$ fixed on a -vertical wall, the solver removes a row of unknowns, and there is nothing further to +On a Cartesian box, the first expression in {eq}`eq-free-slip` constrains a single velocity component. If you hold $u_x$ fixed on a vertical wall, the solver removes a row of unknowns, and there is nothing further to discuss. On a sphere, an annulus, a mesh that does not align with the coordinates, or a surface with topography, $\mathbf{u}\cdot\hat{\mathbf{n}}$ is not a single component of the unknown — it constrains a combination of unknowns at each point, and leaves other combinations free. That has the potential to make a simplifying @@ -66,19 +63,23 @@ Let's assume, for a moment, we confine ourselves to simple domains such as an an which are commonly used for planetary modelling. For each of these cases, there are coordinate systems, and well known forms of the differential operators that do restore the boundary condition to being a constraint in a single direction. Admittedly this requires reformulating all the equations, but for -a symbolic-first code such as underworld, this is possible. This is the strategy used by CITCOMS [Zhong et al]. -But not every domain boundary has a convenient -coordinate system to follow. Even accounting for slight ellipticity introduces significant complexity in all the differential -operators; anything more complicated will not have any useful coordinate reformulation. - -The second condition is also worth noting. We don't think about this when we constrain a degree of freedom, -the other one/s, left unconstrained are *natural* to the problem. They fall out as traction-free surface conditions -automatically in a finite element weak form. If we cannot simply eliminate one degree of freedom at each point, +a symbolic-first code such as underworld, this is possible. This is the strategy used by CITCOMS [Zhong et al, 2008](https://doi.org/10.1029/2008GC002048). But not every domain boundary has a convenient +coordinate system to follow. Even accounting for slight ellipticity introduces + significant complexity in all the differential +operators; anything more complicated will not have a useful +coordinate reformulation. + +The second condition in {eq}`eq-free-slip` is also worth noting. +We don't generally think about this when we constrain a degree of freedom, +the other one/s, left unconstrained are *natural* to the problem. +They fall out as traction-free surface conditions +automatically in a finite element weak form. +If we cannot simply eliminate one degree of freedom at each point, how **do** we satisfy all the parts of {eq}`eq-free-slip` ? ## Four possibilities -We outline four possible approaches (four you can try out in Underworld3). +We outline four possible approaches (all of which you can try out in Underworld3). They fall into two pairs: two impose the constraint **weakly**, by adding a term to the momentum equation and letting the solution satisfy the condition to within the accuracy of that term: a @@ -94,31 +95,25 @@ return the boundary traction as a side-effect of the solution. This could not be more simple, conceptually. We are working in a variational environment, so we just add into our equation system, a term that punishes any flow through the boundary: - $$ \dots + \kappa\int_{\partial\Omega} (\mathbf{u}\cdot\hat{\mathbf{n}})(\mathbf{w}\cdot\hat{\mathbf{n}}) \; \mathrm{d}S . - $$ - -$\kappa$ is a bare number, with none of the $\mu/h$ that Nitsche's parameter -carries below. It has to absorb the scale of the problem itself, which is why +$\kappa$ is a single scalar. It has to absorb the scale of the problem itself, which is why the value that works is a property of the model rather than a default. -One line, no new machinery, and it works on any geometry. It is still in a -good many working scripts, and deservedly. What you are solving is a perturbed -problem, though, and it is perturbed by exactly the amount the constraint is -violated: the discrete solution sits where the penalty term balances the -traction it is fighting and this leaves $\mathbf{u}\cdot\hat{\mathbf{n}}$ small but not zero. +One line, no new machinery, and it works on any geometry. +What we are solving is a mildly perturbed problem and it is perturbed by +exactly the amount the constraint cannot be satisfied: the discrete solution +sits where the penalty term balances the boundary +traction and this leaves $\mathbf{u}\cdot\hat{\mathbf{n}}$ small but not zero. -Making it smaller means pushing harder, and pushing harder +Making the residual $\mathbf{u}\cdot\hat{\mathbf{n}}$ smaller means pushing harder, and pushing harder degrades the condition-number of the operator. The error is traded against the conditioning, -and (see the table below), this trade-off eventually stops returning any benefit: -past $10^4$ the boundary constraint stops improving (the surface is leaky), -and by $10^6$ the solve fails. +and (discussed below), this trade-off eventually stops returning any benefit. -Underworld spells it as a boundary traction opposing normal flow, using the +Underworld codes this term it as a boundary traction opposing normal flow, using the surface normal at the quadrature points ($\Gamma$) provided by PETSc: ```python @@ -130,49 +125,49 @@ stokes.add_natural_bc(penalty * G.dot(v.sym) * G, "Upper") ### Nitsche's method The reason the penalty is only accurate in the limit is that it is not -*consistent*: substituting the true solution does not leave the equation -satisfied, because the true solution is subject to a boundary traction the +*consistent*: substituting the true solution does not fully satisfy the equation, +because the true solution is subject to a separate boundary traction the penalty form ignores. Nitsche's method [@Nitsche_1971] restores consistency by carrying that traction explicitly: - $$ \dots - \int_{\partial\Omega} (\hat{\mathbf{n}}\cdot\boldsymbol{\sigma}(\mathbf{u}) - \hat{\mathbf{n}})(\mathbf{w}\cdot\hat{\mathbf{n}}) \; \mathrm{d}S + \cdot \hat{\mathbf{n}})(\mathbf{w}\cdot\hat{\mathbf{n}}) \; \mathrm{d}S - \int_{\partial\Omega} (\hat{\mathbf{n}}\cdot\boldsymbol{\sigma}(\mathbf{w}) - \hat{\mathbf{n}})(\mathbf{u}\cdot\hat{\mathbf{n}}) \; \mathrm{d}S + \cdot \hat{\mathbf{n}})(\mathbf{u}\cdot\hat{\mathbf{n}}) \; \mathrm{d}S + \frac{\gamma}{h}\int_{\partial\Omega} (\mathbf{u}\cdot\hat{\mathbf{n}})(\mathbf{w}\cdot\hat{\mathbf{n}}) \; \mathrm{d}S . - $$ - The first of the three is the consistency term: it is the boundary traction the integration by parts produced, and including this makes the true solution satisfy the discrete equations exactly. The second is its transpose, which keeps the form symmetric and buys optimal convergence in $L^2$. The third is the penalty again, and it is still needed — but now for *stability* rather than for accuracy, and $\gamma$ has a threshold set by an inverse inequality -rather than being a dial you turn up until the answer looks right. +rather than being an unspecified free parameter. This is a real improvement and it is still done through a weak imposition. The constraint holds to the accuracy of the discretisation, not to the accuracy of the arithmetic — measured below, it leaks a few parts in a thousand on a typical mesh, and the leak falls with increasing mesh resolution. +**The topography** comes the same way as the penalty's, and for the same reason: +the consistency term supplies the traction inside the momentum row, so there is +no reaction left in the residual to read. Recover $\sigma_{nn}$ from the solved +fields and divide by $\Delta\rho\,g$. + ### A constraint equation, with a multiplier -The two above add a *term* to the weak form of the equation. This one adds an *equation*. +The two strategies above add a *term* to the weak form of the equation. +This approach adds an *equation*. Carry a scalar field $\lambda$ on the boundary and require, as a row of the system in its own right, - $$ \int_{\partial\Omega} (\mathbf{u}\cdot\hat{\mathbf{n}} - \tilde{u}_n)\, q \; \mathrm{d}S = 0 \quad \text{for all } q , - $$ - where $\tilde{u}_n$ is the prescribed wall-normal velocity — zero for free slip, and a datum if the wall is being driven — and $\lambda$ enters the momentum row as the traction $\lambda\hat{\mathbf{n}}$ that holds the constraint. It is a @@ -180,8 +175,8 @@ Lagrange multiplier, and the system becomes a larger saddle point: velocity, pressure, and now $\lambda$. $\lambda$ has units of stress. At convergence it *is* $\sigma_{nn}$ on that -boundary, so dividing by $\Delta\rho\,g$ — density contrast times gravity, no -relation to the constraint datum — turns it into a dynamic topography. +boundary, so dividing by $\Delta\rho\,g$ (density contrast $\times$ gravity) +is the dynamic topography. The constraint row is exact, so unlike a penalty there is no parameter whose size decides how well it holds. Two practical things do have to be dealt with. @@ -189,12 +184,11 @@ size decides how well it holds. Two practical things do have to be dealt with. - $\lambda$ is carried as a full-domain field but only its boundary trace means anything, so the interior degrees of freedom are constrained out of the global system in the section before the solve sees it. They are not solved for and - they do not enlarge the $[p, \lambda]$ block. + are not stored in the $[p, \lambda]$ block. - The $[p, \lambda]$ Schur complement is poorly conditioned on its own, so an augmented-Lagrangian term - $r(\mathbf{u}\cdot\hat{\mathbf{n}} - \tilde{u}_n)\hat{\mathbf{n}}$ is added to - the momentum row. It does not change what the constraint enforces, because the - $\lambda$ row still carries the exact constraint. + $r(\mathbf{u}\cdot\hat{\mathbf{n}} - \tilde{u}_n)\,\hat{\mathbf{n}}$ is added to the momentum row. It does not change what the constraint enforces, + because the $\lambda$ row still carries the exact constraint. ```python stokes = uw.systems.Stokes_Constrained(mesh, velocityField=v, pressureField=p) @@ -202,18 +196,18 @@ lam = stokes.add_constraint_bc(0.0, "Upper") stokes.solve() ``` -And the reason to care about it beyond the constraint: **at convergence the -momentum row's boundary term is the normal traction.** It is not recovered from -the velocity field afterwards — it is an unknown the solve returned, available +**Note**: **at convergence, the +momentum row's boundary term is the normal traction.** It does not need to be recovered from +the velocity field after the fact. It is an unknown the solve returns, available (on the solver) through `traction` and, divided by $\Delta\rho g$, through `topography`. That term is the whole boundary load, -$\lambda + r(\mathbf{u}\cdot\hat{\mathbf{n}} - \tilde{u}_n)$, and not the +$\lambda + r(\mathbf{u}\cdot\hat{\mathbf{n}} - \tilde{u}_n)$, not the multiplier alone. The second part vanishes only where the constraint row is satisfied exactly; discretely it is satisfied to the solver's tolerance, and $r$ multiplies that residual back into the traction. With a viscosity-weighted $r$ -and a lateral viscosity contrast it is most of the answer, so the two parts are -not separable in practice. +and a lateral viscosity contrast it can be the largest term of the result, +so the two parts are not separable in practice. ### Rotating the degrees of freedom @@ -226,18 +220,31 @@ is removed the same way it would be on a box. Collect the per-node rotations into a block-diagonal $Q$, equal to the identity at every node that is not constrained. The rotated system is - $$ \hat{A} = Q^{T} A Q, \qquad \hat{\mathbf{b}} = Q^{T}\mathbf{b}, \qquad \mathbf{u} = Q\hat{\mathbf{u}} , - $$ - and the wall-normal row of $\hat{A}$ is struck out. The constraint then holds to machine precision, because it is not being solved for at all. +**The topography** is the reaction of that struck row — the force the constraint +had to supply — de-smeared by the boundary mass to turn an integrated nodal load +into a pointwise stress, + +$$ +\sigma_{nn} = -M_\Gamma^{-1}\left.(A\mathbf{u} - \mathbf{b})\right|_\Gamma, +\qquad +h = -\frac{\sigma_{nn} - \overline{\sigma_{nn}}}{\Delta\rho\,g} , +$$ + +which is the consistent boundary flux of Zhong, Gurnis and Hulbert +[@Zhong_1993]. `boundary_normal_traction()` and `dynamic_topography()` return +these. Nothing is differentiated and nothing is solved: in two dimensions +$M_\Gamma$ is lumped and the de-smear is a division. + **This is the classical strategy**. It is in the early -finite-element literature, and @Engelman_1982 were already reviewing +finite-element literature, and Engelman, Sani and Gresho [@Engelman_1982] +were already reviewing the alternatives and choosing between them on grounds of global mass conservation in 1982. What is worth explaining is not the idea but why, given that it is exact and the others are not, it is the least used of the three. @@ -259,7 +266,7 @@ block carries no boundary condition of this kind. One un-rotation sits on the boundary between them and feeds both. ``` -Four things carry $Q$: the operator, the right-hand side, the solution on the +Four objects carry $Q$: the operator, the right-hand side, the solution on the way out, and the multigrid prolongation. The coarse operators inherit it through the Galerkin triple product rather than being rotated separately, and the coarse solve should be an SVD, because a Galerkin-coarsened rotated @@ -281,25 +288,22 @@ the average of the adjacent facet normals **weighted by facet measure** — not the normal of the smooth surface the mesh approximates, and not the facet normal on its own. -**This is the consistent normal of @10.1002/fld.1650020302**, and we should say -so plainly: we re-derived it from the assembled boundary integral, and Engelman, -Sani and Gresho derived it in 1982 from global conservation of mass, which is -the same object arrived at from the other side. Their paper is about exactly +**This is the consistent normal of Engelman, Sani and Gresho** +[@Engelman_1982]. They +derived this result in 1982 from global conservation of mass. Their paper is about exactly this — how to impose a normal or tangential condition on a boundary that does -not line up with the coordinate directions — and the recommendation is the one -here. +not line up with the coordinate directions — and their recommendation is the one we use +here. -Using the analytic normal is exact for the geometry and therefore inconsistent -with the discretisation, which is the wrong way round: the solver is not solving -on the sphere, it is solving on the polyhedron. In parallel the same argument has -a sharper edge, because a normal accumulated rank-locally is wrong at a partition -boundary, where a node's facets are split across ranks and no rank sees them all. +The analytic normal is exact for the geometry and therefore inconsistent +with the discretisation: the solver is not solving +on the sphere (or annulus), it is solving on the polyhedral approximation to the sphere. Using the **facet** normal is worse than inconsistent, and this is the one place in the note where a choice does real damage. Imposing $\mathbf{u}\cdot\hat{\mathbf{n}} = 0$ facet by facet asks a node shared by two facets to satisfy two different constraints, and two independent constraints on a -two-component velocity leave nothing. Push the coefficient up and the vertex +two-component velocity provide no freedom. Push the coefficient up and the vertex velocities go to zero: the flow is being asked to stay inside a polygon rather than a circle, and the discrete limit is a different problem from the smooth one. Refining the mesh does not approach the smooth answer, because it is not @@ -308,7 +312,7 @@ converging to it. A direct penalty at $\kappa = 10^6$, on the annulus of the next section, against each normal in turn: -| cell size | facet normal: leak / velocity error / stress error | node normal: leak / velocity / stress | +| cell size | facet normal: leak / velocity error / stress error | node normal: leak / velocity error / stress error | |---|---|---| | 0.150 | 8.9 × 10⁻⁶ / 0.60 / 0.21 | 4.6 × 10⁻⁵ / 1.0 × 10⁻² / 2.4 × 10⁻² | | 0.075 | 1.1 × 10⁻⁵ / 0.61 / 0.25 | 3.7 × 10⁻⁵ / 2.4 × 10⁻³ / 6.3 × 10⁻³ | @@ -334,7 +338,7 @@ Everything that follows uses the node normal, which is what `add_nitsche_bc` and `add_rotated_freeslip_bc` take by default and what `mesh.boundary_normal` returns. The facet normal does not appear again. -The consistent normal is not the end of the matter. @Behr_2004 takes it as +The consistent normal is not the end of the matter. Behr [@Behr_2004] takes it as the starting point — "preferred from the point of view of conservation" — and reports that in sloshing problems it still does not guarantee a good discrete slip condition, with non-physical recirculation appearing at curved walls; the @@ -357,28 +361,81 @@ or a plate-boundary force balance require accurate integration of boundary stres Here the choice becomes more subtle, and the methods have quite different accuracies, and different efficiencies. -### The surface permeability question +### The benchmark + +Kramer, Davies and Wilson [@Kramer_2021] give exact Stokes solutions in a +cylindrical annulus, and their `assess` package publishes the radial stress as +well as the velocity, which is what makes it an oracle for this question rather +than only for the flow. Underworld wraps it as `uw.analytic.CylindricalStokes`. + +The case used throughout is the smooth one: a density anomaly +$(r/r_o)^k \cos n\theta$ with $n = 2$ and $k = 3$, viscosity 1, free slip on both +radii. On the outer boundary the exact radial stress is a single harmonic, + +$$ +\sigma_{rr}(r_o, \theta) = 0.1506696\,\cos 2\theta , +$$ + +fitted to a residual of $10^{-16}$, so the whole of the surface stress is that one +amplitude and the error in it is one number. The treatment under test is on the +outer radius; the inner carries the exact analytic velocity as a Dirichlet +condition, so it is the only free-slip condition in the model. + +Two things are measured on every solve, and they answer different questions: -An annulus, no slip on the inner radius, the treatment under test on the outer, -driven by a degree-four radial density anomaly. The number is the largest -normal velocity on the outer boundary, taken against the true radial direction -and divided by the flow speed: the fraction of the flow going through a -boundary nothing should pass through. Penalty at $10^4$, Nitsche at -$\gamma = 10$. [REFERENCE TO ANALYTIC MODULE] +- **the leak** — the largest $\mathbf{u}\cdot\hat{\mathbf{n}}$ on the outer + boundary, against the true radial direction, divided by the flow speed. The + fraction of the flow going through a boundary nothing should pass through. +- **the stress error** — the relative error in that harmonic amplitude, recovered + from the solved fields by projection, which is the route every treatment has. + +Penalty at $\kappa = 10^4$, Nitsche at $\gamma = 10$. | cell size | penalty | Nitsche | multiplier | rotated | |---|---|---|---|---| -| 0.150 | 3.3 × 10⁻³ | 4.6 × 10⁻³ | 8.3 × 10⁻⁴ | 7.3 × 10⁻¹¹ | -| 0.100 | 2.7 × 10⁻³ | 2.2 × 10⁻³ | 1.6 × 10⁻⁴ | 6.5 × 10⁻¹¹ | -| 0.075 | 2.6 × 10⁻³ | 1.7 × 10⁻³ | 5.6 × 10⁻⁵ | 1.0 × 10⁻¹⁰ | -| 0.050 | 2.6 × 10⁻³ | 5.8 × 10⁻⁴ | 1.2 × 10⁻⁵ | 8.4 × 10⁻¹¹ | - -Nitsche leaks parts in a thousand and improves roughly as $h^{1.9}$ — the rate -consistency buys. The multiplier starts an order of magnitude better and falls -much faster, near $h^{3.9}$. The rotated constraint does not move at all: it sits at the solver's floor at -every resolution, because the mesh has nothing to do with it. The penalty -does not improve with the mesh either, and for the opposite reason: its leak is -set by the coefficient, not by the discretisation. +| 0.150 | 3.1 × 10⁻³ / 2.5 × 10⁻² | 1.0 × 10⁻² / 5.8 × 10⁻² | 2.3 × 10⁻⁴ / 2.4 × 10⁻² | 5.3 × 10⁻¹¹ / 2.4 × 10⁻² | +| 0.100 | 3.0 × 10⁻³ / 1.1 × 10⁻² | 2.4 × 10⁻³ / 2.4 × 10⁻² | 1.0 × 10⁻⁴ / 1.0 × 10⁻² | 5.8 × 10⁻¹¹ / 1.0 × 10⁻² | +| 0.075 | 3.0 × 10⁻³ / 7.2 × 10⁻³ | 1.2 × 10⁻³ / 1.5 × 10⁻² | 8.6 × 10⁻⁵ / 6.3 × 10⁻³ | 1.2 × 10⁻¹⁰ / 6.2 × 10⁻³ | +| 0.050 | 3.0 × 10⁻³ / 3.6 × 10⁻³ | 2.7 × 10⁻⁴ / 6.3 × 10⁻³ | 6.7 × 10⁻⁵ / 2.7 × 10⁻³ | 1.2 × 10⁻¹⁰ / 2.7 × 10⁻³ | + +Read the leak first. Nitsche leaks parts in a thousand and improves with the mesh +— the rate consistency buys. The multiplier is an order of magnitude better and +improves faster. The rotated constraint does not move: it sits at the solver's +floor at every resolution, because the mesh has nothing to do with it. The +penalty does not improve either, and for the opposite reason — its leak is set by +the coefficient rather than by the discretisation. + +Now read the stress beside it, and the ranking is not the same. **Every treatment +that imposes the constraint properly lands on the same stress error at a given +mesh**: 6.3 × 10⁻³ for the multiplier and 6.2 × 10⁻³ for the rotated constraint at +cell 0.075, where their leaks differ by nine orders of magnitude. What sets that +number is the recovery — a projection of a stress differentiated out of a +piecewise-quadratic velocity — and not the boundary condition underneath it. A +constraint held to $10^{-10}$ buys nothing over one held to $10^{-4}$ if the +answer is then recovered the same way. + +Nitsche is the exception, at twice the error of the others on the coarser meshes. +Its $\gamma = 10$ is enough for the leak and not for the stress: at +$\gamma = 100$ the leak improves by a factor of nearly forty and the stress by a +factor of two, onto the same floor as everything else, after which more $\gamma$ +buys nothing. Reading the leak alone would have called $\gamma = 10$ converged. + +### The traction the solve already has + +The two exact treatments do not have to recover anything, and the difference +shows up against the same exact answer: + +| cell size | rotated, reaction | multiplier, traction | either, recovered by projection | +|---|---|---|---| +| 0.150 | 6.8 × 10⁻³ | 8.6 × 10⁻³ | 2.4 × 10⁻² | +| 0.100 | 3.3 × 10⁻³ | 8.5 × 10⁻⁴ | 1.0 × 10⁻² | +| 0.075 | 2.1 × 10⁻³ | 1.7 × 10⁻³ | 6.3 × 10⁻³ | +| 0.050 | 1.1 × 10⁻³ | 1.4 × 10⁻³ | 2.7 × 10⁻³ | + +Three to five times better than the projection on the same solve, at every +resolution, using the expressions given with each method above. Neither column +falls smoothly with $h$: part of what they report is the constraint residual, and +how far a particular solve drove that is not a function of the mesh. ### What each parameter buys @@ -394,7 +451,7 @@ own parameters are what tells them apart. | 10⁶ | 4.5 × 10⁻⁵ | | 10⁴ and above | diverged | Nitsche is bounded at both ends. Below $\gamma \sim 1$ the form is no longer -coercive and no amount of solver tuning recovers it; from $\gamma \sim 10^4$ in this +coercive and no amount of solver tuning recovers a solution; from $\gamma \sim 10^4$ in this problem, the line search stops converging. The virtue of $\gamma \sim 10$ is that it sits within that window on any mesh, because $\gamma$ is dimensionless and the term it scales already carries $\mu / h$. @@ -406,108 +463,29 @@ terms and the resisting viscosity. Written against the node normal, the penalty simply trades: a decade of coefficient for a decade of leak, all the way to $10^6$, with no wall in this -problem. What it does not do is converge — the leak is bought with the parameter -rather than with the mesh — so the coefficient has to be re-chosen whenever the +problem. What it does not do is converge with resolution — the leak is bought with the parameter +rather than with the mesh resolution — so the coefficient has to be re-chosen whenever the forcing or the viscosity changes. -### The stress, measured - -The leak says how well each treatment satisfied the free-slip requirement at the boundary. -But we still need to measure whether the method converges to the correct solution. - -Kramer, Davies and Wilson [@Kramer_2021] give exact Stokes -solutions in a cylindrical annulus, and their `assess` package also publishes the -radial stress. Underworld wraps it as `uw.analytic.CylindricalStokes`. -The case here is -the smooth one: a density anomaly $(r/r_o)^k \cos n\theta$ with $n = 2$ and -$k = 3$, viscosity 1, free slip on both radii. On the outer boundary the exact -radial stress is a single harmonic, - -$$ -\sigma_{rr}(r_o, \theta) = 0.1506696\,\cos 2\theta , - -$$ +### The multiplier and the consistent boundary flux are the same -fitted to a residual of $10^{-16}$, so the whole of the surface stress is that -one amplitude and the metric is its relative error. The inner boundary carries -the exact analytic velocity as a Dirichlet condition instead -of a free-slip treatment of its own. - -There are two routes to the surface stress and the difference between them is -the point of the section: - -- **recovered** — project $\hat{\mathbf{n}}\cdot\boldsymbol{\sigma}\cdot \hat{\mathbf{n}}$ - out of the solved velocity and pressure. Every approach offers this, but it - is the only route the weak ones have. -- **reaction** — compute the boundary-normal traction for the rotated constraint, and use the - multiplier field for the constraint method. This is not recovered from the stress field but from the - surface reaction to the stress field. - -Both are computed against the analytic radial direction, which is also the direction -the oracle publishes, so no treatment is scored against its own normal. - -| cell size | penalty | Nitsche | multiplier | rotated | rotated, reaction | multiplier, reaction | -|---|---|---|---|---|---|---| -| 0.150 | 2.5 × 10⁻² | 5.9 × 10⁻² | 2.4 × 10⁻² | 2.4 × 10⁻² | 6.8 × 10⁻³ | 8.6 × 10⁻³ | -| 0.100 | 1.1 × 10⁻² | 2.4 × 10⁻² | 1.0 × 10⁻² | 1.0 × 10⁻² | 3.3 × 10⁻³ | 8.5 × 10⁻⁴ | -| 0.075 | 7.2 × 10⁻³ | 1.5 × 10⁻² | 6.3 × 10⁻³ | 6.2 × 10⁻³ | 2.1 × 10⁻³ | 1.7 × 10⁻³ | -| 0.050 | 3.6 × 10⁻³ | 6.3 × 10⁻³ | 2.7 × 10⁻³ | 2.7 × 10⁻³ | 1.1 × 10⁻³ | 1.4 × 10⁻³ | - -The first five columns are all the recovered stress, so they differ only by -which boundary condition produced the field. The last two are the reaction and -the multiplier on the same solves. Penalties at $10^4$, Nitsche at -$\gamma = 10$, which are the values the leak table used. - -**Which treatment imposed the constraint stops mattering to the recovered -stress.** At cell 0.075 the rotated -constraint gives 6.2 × 10⁻³, the multiplier 6.3 × 10⁻³, the penalty at $10^5$ -6.3 × 10⁻³ and Nitsche at $\gamma = 100$ is 6.6 × 10⁻³. Those are the same number. -What sets it is the recovery — a projection of a stress differentiated out of a -piecewise-quadratic velocity — and not the boundary condition underneath. The -reasoning this note began with — that a traction recovered from an approximate -constraint inherits the approximation — is not what the measurement shows. It -shows a floor that all of them share. - -**The reaction is about three times better than the recovery, on the same -solve.** 2.1 × 10⁻³ against 6.2 × 10⁻³ at cell 0.075, and it converges at the -same rate rather than a better one. It costs nothing — it is the residual the -solve has already assembled, and no field is differentiated to get it — and the -timings below put a number on "nothing". - -The multiplier's traction is the same story from the other side, and lands in the -same place: 8.5 × 10⁻⁴ to 1.7 × 10⁻³ over the three finer meshes. Its column does -not fall smoothly with $h$, because part of what it reports is the augmentation -times the constraint residual, and how far a particular solve drove that residual -is not a function of the mesh. - -**The parameter that was enough for the leak is not enough for the stress.** -Nitsche at $\gamma = 10$ leaks 1.2 × 10⁻³ in this problem and gets the stress -amplitude 1.5% wrong. At $\gamma = 100$ the leak improves by a factor of nearly -forty and the stress by a factor of two, to the recovery floor, where more -$\gamma$ buys nothing. Reading the leak alone would have said $\gamma = 10$ was -converged. - -### The multiplier and the consistent boundary flux are the same object - -The correction above is not a patch. Write the momentum row's boundary term out -and the identity is immediate: the assembled load is +The two expressions given above for the topography are the same expression. Write +the momentum row's boundary term out and the identity is immediate: the assembled +load is $M_\Gamma\,(\lambda + r(\mathbf{u}\cdot\hat{\mathbf{n}} - \tilde{u}_n))$, and at convergence it balances the volume residual restricted to the boundary, which is precisely the nodal load the consistent boundary flux back-calculation reads [@Zhong_1993]. So - $$ \lambda + r(\mathbf{u}\cdot\hat{\mathbf{n}} - \tilde{u}_n) = -M_\Gamma^{-1} \left. (A\mathbf{u} - \mathbf{b}) \right|_\Gamma , - $$ -the CBF traction de-smeared with the boundary mass. The multiplier is not a -second, independent estimate of the surface stress: it is the same computation -the rotated constraint's reaction performs, arrived at by carrying the traction -as an unknown instead of reading it out of the residual afterwards. Dropping the -$r$ term is dropping part of the load, which is why it fails exactly where $r$ is -large. +which is the rotated constraint's reaction, de-smeared with the same boundary +mass. The multiplier is not a second, independent estimate of the surface stress: +it is the same computation, arrived at by carrying the traction as an unknown +instead of reading it out of the residual afterwards. Dropping the $r$ term drops +part of the load, which is why it fails where $r$ is large. Measured across the two solves — they are different discrete problems, so this is agreement rather than an identity check — the corrected multiplier and the @@ -575,7 +553,7 @@ magnitude we tried, from $\mu$ to $10^3\mu$ (a `Piecewise` viscosity inside the boundary term fails the line search). This is the same lesson as $\gamma$'s window on the annulus, in a place where the window closes entirely. -**Nitsche is missing from this table**, and honestly so. Our configuration of it +**Nitsche is missing from this table**. Our configuration of it on this box converges at a contrast of $10^6$ and fails the line search at $10$ — the opposite way round from every expectation — at 16 × 16 and 32 × 32 alike and at $\gamma = 10$, $100$ and $1000$. Where it does converge, $\gamma$ has to rise diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/stress.py b/articles/boundary-conditions-on-non-planar-boundaries/examples/stress.py index 33a4072..977dc8a 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/examples/stress.py +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/stress.py @@ -296,7 +296,42 @@ def measure(mode, cell=CELL, **kwargs): return out -MODES = ("penalty", "penalty_node", "nitsche", "constraint", "rotated") +MODES = ("penalty_node", "nitsche", "constraint", "rotated") + + +def both(cells=(0.15, 0.10, 0.075, 0.05), modes=MODES): + """The constraint and the stress, from the SAME solves. + + The two questions -- does the boundary hold, and is the answer right -- are + one experiment. Reporting them from separate runs invited the reader to + compare numbers that came from different problems. + """ + exact, _residual = exact_amplitude(build("free", cell=0.2)[3]) + print("exact sigma_rr on r = %.2f: %.10f cos(%d.theta)" % (R_O, exact, N)) + print() + print("leak / stress error, from one solve each") + print() + print("| cell size | " + " | ".join(modes) + " |") + print("|---" * (len(modes) + 1) + "|") + for cell in cells: + row = [] + for mode in modes: + got = measure(mode, cell=cell) + row.append("diverged" if got is None + else "%.1e / %.1e" % (got["leak"], got["recovered"])) + print("| %.3f | %s |" % (cell, " | ".join(row)), flush=True) + print() + print("the reaction routes, same solves") + print() + print("| cell size | rotated reaction | multiplier traction |") + print("|---|---|---|") + for cell in cells: + entries = [] + for mode in ("rotated", "constraint"): + got = measure(mode, cell=cell) + entries.append("-" if not got or "reaction" not in got + else "%.1e" % got["reaction"]) + print("| %.3f | %s |" % (cell, " | ".join(entries)), flush=True) def sweep(cells=(0.15, 0.10, 0.075, 0.05), modes=MODES): @@ -383,5 +418,5 @@ def control(cell=CELL): if __name__ == "__main__": command = sys.argv[1:2] or ["sweep"] - {"sweep": sweep, "params": parameters, + {"sweep": sweep, "both": both, "params": parameters, "locking": locking, "control": control}[command[0]]() From ac3316ef30ae654c2108ed6aa1d64d2e7276498c Mon Sep 17 00:00:00 2001 From: lmoresi Date: Thu, 20 Aug 2026 19:02:40 +1000 Subject: [PATCH 20/33] Put the equation label back on the closing $$ The preview was failing while build and test passed, which is the signature of a PDF-only fault: the HTML tolerates an unresolved reference and Typst does not. $$ ... $$ (eq-free-slip) <- a paragraph of literal text MyST reads the label only when it is on the same line as the closing $$. On its own line the equation is never labelled, so the three {eq}`eq-free-slip` references have no target: HTML renders them as bare spans and carries on, while `typst compile` stops with "label does not exist in the document" and the archival PDF is never produced. Verified by building the PDFs with UWTN_PREVIEW=1, which is what CI does -- the note is at status: review, so a production build skips it and the fault does not show. 45 PDFs, this one 766 kB, no label error. Underworld development team with AI support from Claude Code --- .../boundary-conditions-on-non-planar-boundaries.md | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md index 0d764c5..87d034b 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md +++ b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md @@ -38,8 +38,7 @@ $$ \mathbf{u}\cdot\hat{\mathbf{n}} = 0 \qquad\text{and}\qquad \hat{\mathbf{t}}\cdot\boldsymbol{\sigma}\cdot\hat{\mathbf{n}} = 0. -$$ -(eq-free-slip) +$$ (eq-free-slip) In the weak form, boundary tractions appear as surface integrals. Multiplying the momentum balance by a test function $\mathbf{w}$ and integrating by parts gives From a9e81c5c5ebc5d8061dd41dc35d085960b00f0d4 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Thu, 20 Aug 2026 21:08:41 +1000 Subject: [PATCH 21/33] Louis's edit: tighten the argument and cut what the note cannot support Louis's revision, brought in from his working copy. The substance of it: the classical-strategy paragraph moves to where the rotated method is actually described rather than trailing it, the normals section states what a discrete normal IS before saying which one to use, the CITCOMS reference is given its place in the coordinate-rotation discussion, and the forward references to free-surface work that is not written yet are gone -- along with the passage motivated by the traction requirement of our own free-surface algorithm, which a reader cannot check and which belongs in its own note. Two mechanical things carried with it. The equation label needs a space: `$$ (eq-free-slip)`, not `$$(eq-free-slip)`. Same failure mode as the label on its own line -- the equation goes unlabelled, `{eq}` resolves to nothing, the HTML shrugs and `typst compile` stops. Removing the Behr caveat left `Behr_2004` in references.bib with nothing citing it, so the entry goes too. If that caveat should come back -- the consistent normal not guaranteeing a good discrete slip condition, with recirculation at curved walls -- the entry comes back with it. Verified before pushing: 137 tests, a full UWTN_PREVIEW build with no MyST errors, the archival PDF at 758 kB, and the equation rendering as a numbered, linked (1). Underworld development team with AI support from Claude Code --- ...ary-conditions-on-non-planar-boundaries.md | 212 +++++------------- .../references.bib | 3 - 2 files changed, 61 insertions(+), 154 deletions(-) diff --git a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md index 87d034b..0896d25 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md +++ b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md @@ -62,7 +62,8 @@ Let's assume, for a moment, we confine ourselves to simple domains such as an an which are commonly used for planetary modelling. For each of these cases, there are coordinate systems, and well known forms of the differential operators that do restore the boundary condition to being a constraint in a single direction. Admittedly this requires reformulating all the equations, but for -a symbolic-first code such as underworld, this is possible. This is the strategy used by CITCOMS [Zhong et al, 2008](https://doi.org/10.1029/2008GC002048). But not every domain boundary has a convenient +a symbolic-first code such as underworld, this is quite straightforward. +This is the strategy used by CITCOMS [Zhong et al, 2008](https://doi.org/10.1029/2008GC002048). But not every domain boundary has a convenient coordinate system to follow. Even accounting for slight ellipticity introduces significant complexity in all the differential operators; anything more complicated will not have a useful @@ -226,28 +227,26 @@ $$ and the wall-normal row of $\hat{A}$ is struck out. The constraint then holds to machine precision, because it is not being solved for at all. +**This is the classical strategy**. It is in the early +finite-element literature, and Engelman, Sani and Gresho [@Engelman_1982] +were already reviewing +the alternatives and choosing between them on grounds of global mass +conservation in 1982. What is worth explaining is not the idea but why, given +that it is exact and the others are not, it is the least used of the three. + **The topography** is the reaction of that struck row — the force the constraint had to supply — de-smeared by the boundary mass to turn an integrated nodal load into a pointwise stress, - $$ \sigma_{nn} = -M_\Gamma^{-1}\left.(A\mathbf{u} - \mathbf{b})\right|_\Gamma, \qquad h = -\frac{\sigma_{nn} - \overline{\sigma_{nn}}}{\Delta\rho\,g} , $$ - which is the consistent boundary flux of Zhong, Gurnis and Hulbert -[@Zhong_1993]. `boundary_normal_traction()` and `dynamic_topography()` return +[@Zhong_1993]. In Underworld3, the solver's `boundary_normal_traction()` and `dynamic_topography()` return these. Nothing is differentiated and nothing is solved: in two dimensions $M_\Gamma$ is lumped and the de-smear is a division. -**This is the classical strategy**. It is in the early -finite-element literature, and Engelman, Sani and Gresho [@Engelman_1982] -were already reviewing -the alternatives and choosing between them on grounds of global mass -conservation in 1982. What is worth explaining is not the idea but why, given -that it is exact and the others are not, it is the least used of the three. - ## Trade-offs Rotating the degrees of freedom leaves the discrete problem in a **mixed @@ -279,53 +278,35 @@ to problem. ## Choice of the surface normal -A question with a less obvious answer than it looks, and the measurements -further down say it is the most consequential choice in the note. The boundary of a +In a discrete representation of a curved surface, the normal can be defined in various +ways. The boundary of a discretised domain is a set of straight facets, and the assembled constraint is -an integral over those facets. The node normal consistent with that integral is +an integral over those facets. +The node normal consistent with that integral is the average of the adjacent facet normals **weighted by facet measure** — not the normal of the smooth surface the mesh approximates, and not the facet -normal on its own. - -**This is the consistent normal of Engelman, Sani and Gresho** -[@Engelman_1982]. They -derived this result in 1982 from global conservation of mass. Their paper is about exactly -this — how to impose a normal or tangential condition on a boundary that does -not line up with the coordinate directions — and their recommendation is the one we use -here. +normal on its own.**This is the consistent normal of Engelman, Sani and Gresho** +[@Engelman_1982]. They derived this result in 1982 from global conservation of mass. The analytic normal is exact for the geometry and therefore inconsistent with the discretisation: the solver is not solving on the sphere (or annulus), it is solving on the polyhedral approximation to the sphere. Using the **facet** normal is worse than inconsistent, and this is the one place -in the note where a choice does real damage. Imposing +where the wrong choice does real damage. In 2D, Imposing $\mathbf{u}\cdot\hat{\mathbf{n}} = 0$ facet by facet asks a node shared by two facets to satisfy two different constraints, and two independent constraints on a -two-component velocity provide no freedom. Push the coefficient up and the vertex +two-component velocity provide no freedom. Push the penalty higher, and the vertex velocities go to zero: the flow is being asked to stay inside a polygon rather than a circle, and the discrete limit is a different problem from the smooth one. Refining the mesh does not approach the smooth answer, because it is not converging to it. -A direct penalty at $\kappa = 10^6$, on the annulus of the next section, against -each normal in turn: - -| cell size | facet normal: leak / velocity error / stress error | node normal: leak / velocity error / stress error | -|---|---|---| -| 0.150 | 8.9 × 10⁻⁶ / 0.60 / 0.21 | 4.6 × 10⁻⁵ / 1.0 × 10⁻² / 2.4 × 10⁻² | -| 0.075 | 1.1 × 10⁻⁵ / 0.61 / 0.25 | 3.7 × 10⁻⁵ / 2.4 × 10⁻³ / 6.3 × 10⁻³ | -| 0.050 | 1.9 × 10⁻⁵ / 0.60 / 0.26 | 3.1 × 10⁻⁵ / 1.0 × 10⁻³ / 2.7 × 10⁻³ | -| 0.035 | 2.3 × 10⁻⁵ / 0.59 / 0.26 | 3.1 × 10⁻⁵ / 4.9 × 10⁻⁴ / 1.4 × 10⁻³ | - -The facet-normal column is stuck: the velocity is 60% wrong and the stress 26% -wrong, and refining changes neither. Note what the leak does — it is *excellent*, -better than the node normal's, because a frozen boundary passes no flow. A metric -that only asks whether the constraint is satisfied cannot tell a locked solution -from a good one. +On an annulus with a free slip boundary, the direct-penalty approach locks at high penalty values ($\sim 10^6$) if facet normals are used in the +constraint equation. In the figure below, the node-normal approach does solve and reproduces the analytic solution (described in detail in the next section) ```{figure} figures/locking.png -:alt: Three annuli side by side on one colour scale from zero to 5.0e-3, blue for slow and red for fast, with the triangular mesh drawn over each. The left panel is the exact solution: two deep red patches of fast flow sit against the outer boundary on the left and right of the annulus, with a blue slow ring inside them. The middle panel is the same problem solved with a direct penalty against the facet normal: the red patches at the outer boundary are gone and the whole outer half is blue, the peak speed having fallen from 5.0e-3 to 3.8e-3, while a pale ring survives near the inner boundary. The right panel is the same penalty against the measure-weighted node normal and is indistinguishable from the exact panel, with a peak speed of 5.0e-3. +:alt: Three annulus solutions side by side on one colour scale from zero to 5.0e-3, blue for slow speeds and red for fast, with the triangular mesh drawn over each. The left panel is the exact solution: two deep red patches of fast flow sit against the outer boundary on the left and right of the annulus, with a blue slow ring inside them. The middle panel is the same problem solved with a direct penalty against the facet normal: the red patches at the outer boundary are gone and the whole outer half is blue, the peak speed having fallen from 5.0e-3 to 3.8e-3, while a pale ring survives near the inner boundary. The right panel is the same penalty against the measure-weighted node normal and is indistinguishable from the exact panel, with a peak speed of 5.0e-3. The same problem, the same coefficient, the same colour scale. Against the facet normal the flow along the outer boundary is suppressed — the peak speed falls by @@ -337,20 +318,11 @@ Everything that follows uses the node normal, which is what `add_nitsche_bc` and `add_rotated_freeslip_bc` take by default and what `mesh.boundary_normal` returns. The facet normal does not appear again. -The consistent normal is not the end of the matter. Behr [@Behr_2004] takes it as -the starting point — "preferred from the point of view of conservation" — and -reports that in sloshing problems it still does not guarantee a good discrete -slip condition, with non-physical recirculation appearing at curved walls; the -remedies offered there are the Navier slip condition and a "BC-free" boundary. -We have read the abstract rather than the paper, and have not looked for that -recirculation in our own cases. It is the obvious thing to test next for anyone -running free slip on a strongly curved wall. - -## When the choice matters +## When the choice of constraint matters Solve a convection model with any of these approaches, and the -velocity field is the same to plotting accuracy, as long as each is set up -correctly. Generally speaking, a leak of order $10^{-3}$ or $10^{-4}$ in $\mathbf{u}\cdot\hat{\mathbf{n}}$ is +velocity field is the same to plotting accuracy. +Generally speaking, a leak of order $10^{-3}$ or $10^{-4}$ in $\mathbf{u}\cdot\hat{\mathbf{n}}$ is within the expected accuracy of the solution on the mesh and the main driver of which method to choose should be solver efficiency (wall time). @@ -370,23 +342,21 @@ than only for the flow. Underworld wraps it as `uw.analytic.CylindricalStokes`. The case used throughout is the smooth one: a density anomaly $(r/r_o)^k \cos n\theta$ with $n = 2$ and $k = 3$, viscosity 1, free slip on both radii. On the outer boundary the exact radial stress is a single harmonic, - $$ \sigma_{rr}(r_o, \theta) = 0.1506696\,\cos 2\theta , $$ - fitted to a residual of $10^{-16}$, so the whole of the surface stress is that one amplitude and the error in it is one number. The treatment under test is on the outer radius; the inner carries the exact analytic velocity as a Dirichlet condition, so it is the only free-slip condition in the model. -Two things are measured on every solve, and they answer different questions: +Two things are measured on every solve: -- **the leak** — the largest $\mathbf{u}\cdot\hat{\mathbf{n}}$ on the outer +- **the surface permeability** — the largest $\mathbf{u}\cdot\hat{\mathbf{n}}$ on the outer boundary, against the true radial direction, divided by the flow speed. The - fraction of the flow going through a boundary nothing should pass through. -- **the stress error** — the relative error in that harmonic amplitude, recovered - from the solved fields by projection, which is the route every treatment has. + fraction of the flow going through an impermeable boundary. +- **the boundary stress error** — the relative error in that harmonic amplitude, recovered + from the solved fields by projection, which is the route every method has available. Penalty at $\kappa = 10^4$, Nitsche at $\gamma = 10$. @@ -397,12 +367,12 @@ Penalty at $\kappa = 10^4$, Nitsche at $\gamma = 10$. | 0.075 | 3.0 × 10⁻³ / 7.2 × 10⁻³ | 1.2 × 10⁻³ / 1.5 × 10⁻² | 8.6 × 10⁻⁵ / 6.3 × 10⁻³ | 1.2 × 10⁻¹⁰ / 6.2 × 10⁻³ | | 0.050 | 3.0 × 10⁻³ / 3.6 × 10⁻³ | 2.7 × 10⁻⁴ / 6.3 × 10⁻³ | 6.7 × 10⁻⁵ / 2.7 × 10⁻³ | 1.2 × 10⁻¹⁰ / 2.7 × 10⁻³ | -Read the leak first. Nitsche leaks parts in a thousand and improves with the mesh +Reading the leak first, Nitsche leaks parts in a thousand and improves with the mesh — the rate consistency buys. The multiplier is an order of magnitude better and improves faster. The rotated constraint does not move: it sits at the solver's floor at every resolution, because the mesh has nothing to do with it. The penalty does not improve either, and for the opposite reason — its leak is set by -the coefficient rather than by the discretisation. +the penalty coefficient rather than by the discretisation. Now read the stress beside it, and the ranking is not the same. **Every treatment that imposes the constraint properly lands on the same stress error at a given @@ -417,7 +387,7 @@ Nitsche is the exception, at twice the error of the others on the coarser meshes Its $\gamma = 10$ is enough for the leak and not for the stress: at $\gamma = 100$ the leak improves by a factor of nearly forty and the stress by a factor of two, onto the same floor as everything else, after which more $\gamma$ -buys nothing. Reading the leak alone would have called $\gamma = 10$ converged. +buys nothing. ### The traction the solve already has @@ -438,8 +408,7 @@ how far a particular solve drove that is not a function of the mesh. ### What each parameter buys -The two weak methods look alike in that table. They are not alike, and their -own parameters are what tells them apart. +The two weak methods look alike in the comparison above, but this is for a fixed, tuned penalty parameter. | $\kappa$ (penalty) | leak | | $\gamma$ (Nitsche) | leak | |---|---|---|---|---| @@ -468,10 +437,10 @@ forcing or the viscosity changes. ### The multiplier and the consistent boundary flux are the same -The two expressions given above for the topography are the same expression. Write -the momentum row's boundary term out and the identity is immediate: the assembled -load is -$M_\Gamma\,(\lambda + r(\mathbf{u}\cdot\hat{\mathbf{n}} - \tilde{u}_n))$, and at convergence it +The two expressions given above for computing topography from the boundary reaction are exactly equivalent. +Write the momentum row's boundary term out and the identity is immediate: the assembled +load is $M_\Gamma\,(\lambda + r(\mathbf{u}\cdot\hat{\mathbf{n}} - \tilde{u}_n))$, +and at convergence it balances the volume residual restricted to the boundary, which is precisely the nodal load the consistent boundary flux back-calculation reads [@Zhong_1993]. So @@ -483,26 +452,7 @@ $$ which is the rotated constraint's reaction, de-smeared with the same boundary mass. The multiplier is not a second, independent estimate of the surface stress: it is the same computation, arrived at by carrying the traction as an unknown -instead of reading it out of the residual afterwards. Dropping the $r$ term drops -part of the load, which is why it fails where $r$ is large. - -Measured across the two solves — they are different discrete problems, so this -is agreement rather than an identity check — the corrected multiplier and the -rotated reaction differ by 3.2% at a contrast of 100 and 4.9% at $10^6$, both -inside each route's own error against the exact answer (5 to 9%). The identity also says why the two cannot be read off a single solve. On a -multiplier-constrained boundary the constraint enters the row it constrains, so -the assembled residual there is balanced at convergence and the back-calculation -reads zero — measured, rms 4 × 10⁻¹³ against a traction of 0.37. There is no -reaction left in the residual because the multiplier is holding it. The two -routes are alternatives, not a cross-check available at the same time. - -This also settles a question the free-surface work left open. That work used -SolCx the same way, to choose among topography recoveries, and landed on a -rotated free-slip lid with the CBF reaction and a continuous pressure — corr -0.999, relative $l_2$ 0.04 — while rejecting the multiplier. Both conclusions -were right about what was in front of them: the multiplier *as returned* is -missing the augmentation share, and the CBF reaction is the same quantity with -nothing missing. +instead of reading it out of the residual afterwards. ### The other half: a lateral viscosity contrast @@ -540,17 +490,14 @@ exact and has no parameter, and its velocity error is 8.8 × 10⁻⁶ at a contr $10^6$. It still reads 0.085. That number is the recovery's error, not a boundary condition's: on the stiff half the recovered $\sigma_{zz}$ is a difference between the pressure and $2\eta\,\partial_z u_z$ with $\eta = 10^6$, so a relative velocity -error of $10^{-5}$ arrives in the stress at the size of the signal. Nothing here -is worse than the reference except the penalty. +error of $10^{-5}$ appears in the stress. **A bare penalty coefficient cannot serve both halves.** At $10^4$ it is the best column in the table at low contrast — the constraint is weak enough not to fight -the recovery — and by $10^6$ it is useless: 0.992, which is to say the recovered +the recovery — and by $10^6$ it is meaningless: 0.992, which is to say the recovered topography carries none of the signal. Scaling the coefficient by the local -viscosity is the obviously right thing to want, and it does not solve here at any -magnitude we tried, from $\mu$ to $10^3\mu$ (a `Piecewise` viscosity inside the -boundary term fails the line search). This is the same lesson as $\gamma$'s window -on the annulus, in a place where the window closes entirely. +viscosity is the obviously right thing to want, but the solver does not converge here at any +magnitude we tried, from $\eta$ to $10^3\eta$. **Nitsche is missing from this table**. Our configuration of it on this box converges at a contrast of $10^6$ and fails the line search at $10$ — @@ -561,8 +508,6 @@ stress error is 25 at $\gamma = 10$, 1.2 at $100$ and 0.17 at $1000$, while the constraint is held to $10^{-3}$ or better throughout. We are not confident enough in that configuration to put a column of numbers behind it. - - ### What each one costs Seconds on the annulus, uniform viscosity, one core, direct solver: the solve, @@ -576,44 +521,37 @@ run-to-run spread and there is nothing to read. | 28 338 | 0.17 / 0.273 | 0.18 / 0.267 | 0.26 / — | 0.16 / 0.010 | | 71 424 | 0.45 / 0.631 | 0.47 / 0.657 | 0.67 / — | 0.43 / 0.016 | -The dash is not a missing measurement. **The multiplier has nothing to recover**: +The dash indicates that the multiplier does not require any additional *solver* — $\lambda$ is a finite element field in its own right, so its nodal values are the traction, pointwise, and $\lambda + r(\mathbf{u}\cdot\hat{\mathbf{n}} - \tilde{u}_n)$ is an -expression evaluated where it is wanted. Reading it costs whatever reading a -field costs — a few milliseconds of array indexing at this size, and nothing that -belongs in a column beside a solve. +expression evaluated where it is wanted. -**The rotated constraint's reaction does need one step**, and it is worth being -precise about which. (The 10 to 16 ms in the table is the reaction the solve +**The rotated constraint's reaction does need one step**. (The 10 to 16 ms in the table is the reaction the solve already stashed; `boundary_flux` re-assembles the residual from scratch and costs 0.2 s, which is the 0.206 s in the comparison above.) The reaction is an *integrated* nodal load, $\int_\Gamma \sigma_{nn}\,\phi_i\,\mathrm{d}S$, which is $M_\Gamma$ times the pointwise traction. Turning it into a pointwise value means undoing that boundary mass. On a 2-D trace, and on 3-D P1 triangles, the lumped mass is diagonal and undoing it is a division — the 10 to 16 ms above. On **3-D P2 triangles it is a -genuine solve**: the lumped row sums vanish at the vertices, so the consistent -trace mass has to be assembled and solved, and Underworld gathers the trace to -one rank to do it. That is the one place the CBF route pays for being a +true solve**: the lumped row sums vanish at the vertices, so the consistent +trace mass has to be assembled and solved. That is the one place the CBF +route pays for being a back-calculation, and it is the case a spherical free surface runs in. -So the conceptual difference the timings expose is not speed but *what each -method hands you*: the multiplier gives the traction as a field, and the reaction -gives it as a load that still has to be divided by a mass. - **The multiplier's solve costs about 50% more**, consistently — 0.67 s against 0.43 s at 71 000 nodes. That is the extra field and the larger saddle point. Rotating the degrees of freedom costs nothing measurable against the weak forms: the rotation is a sparse orthogonal transform on a boundary's worth of rows. -**The weak forms have to recover theirs by differentiating the solution**, and +**The weak constraints have to recover surface stress by differentiating the solution**, and the projection that does it costs *more than the Stokes solve did* — 0.63 s against 0.45 s — so asking a penalty or Nitsche model for its surface stress roughly doubles the timestep. The obvious question is whether that is the method's cost or the recovery's. A global $L^2$ projection to get values on a thousand boundary nodes is plainly -more work than the job requires, and the consistent boundary flux is right there, -reading the assembled residual rather than differentiating anything. **It does +more work than the job requires, and the consistent boundary flux is an available alternative, +reading the assembled residual rather than differentiating anything. However, we find that **it does not work for a weakly imposed condition**, and the reason is the same one that makes it unavailable to the multiplier: @@ -629,7 +567,7 @@ supplies its traction as a term inside the row it acts on, so the residual there is balanced at convergence and there is nothing left to read. The multiplier does the same thing, and gets away with it because the term it supplies, $\lambda$, is the traction as a field. Nitsche's term is written in terms of -$\boldsymbol{\sigma}(\mathbf{u})$, so reading it back still means differentiating +$\boldsymbol{\sigma}(\mathbf{u})$, so reading it back still requires differentiating the answer. That is the structural statement the timings are really making, and it follows @@ -656,15 +594,15 @@ complement are the terms that matter and neither is exercised here. For a model that consumes the velocity and nothing else, all four are the same to plotting accuracy — provided the constraint is written against the node normal. -That proviso is the only one that can spoil the velocity, and it costs one line. +That proviso is the only one that can spoil the velocity, and it costs one line. -When the wall-normal traction is the answer: +When the wall-normal traction is needed: - **Rotated free slip is the default.** It holds the constraint to machine precision rather than to the discretisation, its reaction is the most accurate surface stress measured here, and that reaction is nearly free. The price is structural — a mixed basis that the multigrid has to carry — and it is paid - once, inside the solver, rather than by the person setting up the model. + once, inside the solver, rather than by the person setting up the model. - **The multiplier is its equal on accuracy** and returns the same object by a different route; take it when you want the traction as an unknown of the system, or when the constrained problem's conditioning suits you better. It @@ -673,42 +611,14 @@ When the wall-normal traction is the answer: during the model** — a wall that begins as a prescribed velocity and relaxes to a prescribed traction is a Nitsche problem, because a hard constraint cannot morph. Budget for tuning $\gamma$ against the stress and not against the leak, - and expect the window to move with the viscosity contrast. + and expect the window to move with the viscosity contrast. If the viscosity + contrast is large with jumps or strong gradients along the boundary, be very careful + if you choose Nitsche. - **A direct penalty is fine for a velocity-only model** and needs the node normal, a coefficient chosen per problem, and a check on something physical - before the answer is believed. - -### Which is to say, a free surface - -This note began with a free surface, and that is where the argument lands. A free -surface needs $\sigma_{nn}$ *every step*: the surface moves under the traction it -carries, so the recovery is not something done once at the end of a run but part -of the timestep, alongside the Stokes solve it follows. - -Everything above then reads differently. The projection that costs more than the -Stokes solve is paid at every step, and doubles the model. The differentiation it -performs is applied to a velocity field that a weak condition never made satisfy -$\mathbf{u}\cdot\hat{\mathbf{n}} = 0$ exactly, on a boundary that is moving. And -the surface stress recovered that way has failure modes of its own that have -nothing to do with the boundary condition — a continuous pressure checkerboards -at a viscosity jump, a discontinuous one puts a node-to-node zigzag into the -reaction on a simplex boundary — so the recovery has to be tuned against the -element types as well. - -The two exact treatments hand the traction over as part of the answer, which is -the property a free surface wants. Between them: - -- **the rotated constraint** gives the reaction, and on the 3-D P2 triangular - trace a deforming spherical surface actually uses, de-smearing it is a - consistent-mass solve that Underworld gathers to one rank. That is a real - serial step inside a parallel timestep; -- **the multiplier** gives a field, so there is nothing to de-smear at all — - in 2-D or in 3-D. It costs about 50% more in the solve. - -We have not measured that trade on a large parallel shell, and it is the -measurement worth doing next: the multiplier's premium is a fixed fraction of a -solve, while the gather is a serial section, and which one wins is a question of -rank count rather than of method. + before the answer is believed. It has the advantage that this is pure, direct penalty + on the weak form and can be used for many things beyond simply boundary conditions. + Good for a first pass on a very general idea. ## Using it diff --git a/articles/boundary-conditions-on-non-planar-boundaries/references.bib b/articles/boundary-conditions-on-non-planar-boundaries/references.bib index 3af1725..d8c2b29 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/references.bib +++ b/articles/boundary-conditions-on-non-planar-boundaries/references.bib @@ -32,11 +32,8 @@ @article{Zhong_1993 pages = {1--8} } - @article{Nitsche_1971, title={Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind}, volume={36}, ISSN={1865-8784}, url={http://dx.doi.org/10.1007/BF02995904}, DOI={10.1007/bf02995904}, number={1}, journal={Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg}, publisher={Springer Science and Business Media LLC}, author={Nitsche, J.}, year={1971}, month=July, pages={9–15} } @article{Engelman_1982, title={The implementation of normal and/or tangential boundary conditions in finite element codes for incompressible fluid flow}, volume={2}, ISSN={1097-0363}, url={http://dx.doi.org/10.1002/fld.1650020302}, DOI={10.1002/fld.1650020302}, number={3}, journal={International Journal for Numerical Methods in Fluids}, publisher={Wiley}, author={Engelman, M. S. and Sani, R. L. and Gresho, P. M.}, year={1982}, month=July, pages={225–238} } -@article{Behr_2004, title={On the application of slip boundary condition on curved boundaries}, volume={45}, ISSN={1097-0363}, url={http://dx.doi.org/10.1002/fld.663}, DOI={10.1002/fld.663}, number={1}, journal={International Journal for Numerical Methods in Fluids}, publisher={Wiley}, author={Behr, Marek}, year={2004}, month=Apr, pages={43–51} } - @article{Kramer_2021, title={Analytical solutions for mantle flow in cylindrical and spherical shells}, volume={14}, ISSN={1991-9603}, url={http://dx.doi.org/10.5194/gmd-14-1899-2021}, DOI={10.5194/gmd-14-1899-2021}, number={4}, journal={Geoscientific Model Development}, publisher={Copernicus GmbH}, author={Kramer, Stephan C. and Davies, D. Rhodri and Wilson, Cian R.}, year={2021}, month=Apr, pages={1899–1919} } From fd70ec63cbd0edce980de6748be693e095e8aadb Mon Sep 17 00:00:00 2001 From: Louis Moresi Date: Thu, 20 Aug 2026 21:33:37 +1000 Subject: [PATCH 22/33] Update boundary conditions explanation and penalty term Clarified the weak form of boundary conditions and introduced a direct penalty term for flow through boundaries. --- .../boundary-conditions-on-non-planar-boundaries.md | 4 ++++ 1 file changed, 4 insertions(+) diff --git a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md index 0896d25..34dddf7 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md +++ b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md @@ -42,12 +42,14 @@ $$ (eq-free-slip) In the weak form, boundary tractions appear as surface integrals. Multiplying the momentum balance by a test function $\mathbf{w}$ and integrating by parts gives + $$ \int_\Omega \boldsymbol{\sigma} : \nabla\mathbf{w} \; \mathrm{d}V - \int_{\partial\Omega} (\boldsymbol{\sigma}\cdot\hat{\mathbf{n}})\cdot\mathbf{w} \; \mathrm{d}S = \int_\Omega \mathbf{f}\cdot\mathbf{w} \; \mathrm{d}V . $$ + Drop the surface integral and you have imposed zero traction in all directions — free *everything* (a free surface), not free slip. Constrain the surface-normal degrees of freedom and the surface integral only addresses the tangential traction terms. @@ -95,11 +97,13 @@ return the boundary traction as a side-effect of the solution. This could not be more simple, conceptually. We are working in a variational environment, so we just add into our equation system, a term that punishes any flow through the boundary: + $$ \dots + \kappa\int_{\partial\Omega} (\mathbf{u}\cdot\hat{\mathbf{n}})(\mathbf{w}\cdot\hat{\mathbf{n}}) \; \mathrm{d}S . $$ + $\kappa$ is a single scalar. It has to absorb the scale of the problem itself, which is why the value that works is a property of the model rather than a default. From 53f4d87fa055873e48dc5340903029e49e09922f Mon Sep 17 00:00:00 2001 From: Louis Moresi Date: Thu, 20 Aug 2026 21:47:39 +1000 Subject: [PATCH 23/33] Update boundary conditions section in documentation Mostly just adding newlines around $$ so they render correctly. I hope I got them all ! --- .../boundary-conditions-on-non-planar-boundaries.md | 11 +++++++++++ 1 file changed, 11 insertions(+) diff --git a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md index 34dddf7..eb490f8 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md +++ b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md @@ -133,6 +133,7 @@ The reason the penalty is only accurate in the limit is that it is not because the true solution is subject to a separate boundary traction the penalty form ignores. Nitsche's method [@Nitsche_1971] restores consistency by carrying that traction explicitly: + $$ \dots - \int_{\partial\Omega} (\hat{\mathbf{n}}\cdot\boldsymbol{\sigma}(\mathbf{u}) @@ -143,6 +144,7 @@ $$ (\mathbf{u}\cdot\hat{\mathbf{n}})(\mathbf{w}\cdot\hat{\mathbf{n}}) \; \mathrm{d}S . $$ + The first of the three is the consistency term: it is the boundary traction the integration by parts produced, and including this makes the true solution satisfy the discrete equations exactly. The second is its transpose, @@ -168,10 +170,12 @@ This approach adds an *equation*. Carry a scalar field $\lambda$ on the boundary and require, as a row of the system in its own right, + $$ \int_{\partial\Omega} (\mathbf{u}\cdot\hat{\mathbf{n}} - \tilde{u}_n)\, q \; \mathrm{d}S = 0 \quad \text{for all } q , $$ + where $\tilde{u}_n$ is the prescribed wall-normal velocity — zero for free slip, and a datum if the wall is being driven — and $\lambda$ enters the momentum row as the traction $\lambda\hat{\mathbf{n}}$ that holds the constraint. It is a @@ -224,10 +228,12 @@ is removed the same way it would be on a box. Collect the per-node rotations into a block-diagonal $Q$, equal to the identity at every node that is not constrained. The rotated system is + $$ \hat{A} = Q^{T} A Q, \qquad \hat{\mathbf{b}} = Q^{T}\mathbf{b}, \qquad \mathbf{u} = Q\hat{\mathbf{u}} , $$ + and the wall-normal row of $\hat{A}$ is struck out. The constraint then holds to machine precision, because it is not being solved for at all. @@ -241,11 +247,13 @@ that it is exact and the others are not, it is the least used of the three. **The topography** is the reaction of that struck row — the force the constraint had to supply — de-smeared by the boundary mass to turn an integrated nodal load into a pointwise stress, + $$ \sigma_{nn} = -M_\Gamma^{-1}\left.(A\mathbf{u} - \mathbf{b})\right|_\Gamma, \qquad h = -\frac{\sigma_{nn} - \overline{\sigma_{nn}}}{\Delta\rho\,g} , $$ + which is the consistent boundary flux of Zhong, Gurnis and Hulbert [@Zhong_1993]. In Underworld3, the solver's `boundary_normal_traction()` and `dynamic_topography()` return these. Nothing is differentiated and nothing is solved: in two dimensions @@ -346,9 +354,11 @@ than only for the flow. Underworld wraps it as `uw.analytic.CylindricalStokes`. The case used throughout is the smooth one: a density anomaly $(r/r_o)^k \cos n\theta$ with $n = 2$ and $k = 3$, viscosity 1, free slip on both radii. On the outer boundary the exact radial stress is a single harmonic, + $$ \sigma_{rr}(r_o, \theta) = 0.1506696\,\cos 2\theta , $$ + fitted to a residual of $10^{-16}$, so the whole of the surface stress is that one amplitude and the error in it is one number. The treatment under test is on the outer radius; the inner carries the exact analytic velocity as a Dirichlet @@ -448,6 +458,7 @@ and at convergence it balances the volume residual restricted to the boundary, which is precisely the nodal load the consistent boundary flux back-calculation reads [@Zhong_1993]. So + $$ \lambda + r(\mathbf{u}\cdot\hat{\mathbf{n}} - \tilde{u}_n) = -M_\Gamma^{-1} \left. (A\mathbf{u} - \mathbf{b}) \right|_\Gamma , From 6759ebf74e69ba2fd2c3d4b8eefde2f0f981f954 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Thu, 20 Aug 2026 22:07:44 +1000 Subject: [PATCH 24/33] Stop counting the augmentation twice, and label the surface as deformed The topography figure drew the multiplier's traction at twice its augmentation share, so at a 1e6 contrast it left the panel -- exactly the "out by 2" it looked like -- while the curve labelled "as returned" was the correct one. The cause was two copies of one expression. `solcx.reaction_traction` was changed to return `traction()` (lambda + r(u.n - u~_n), the whole boundary load, from the solver) when underworld3#617 landed, and this script went on adding r(u.n) to whatever that returned, which had been right while it returned the bare multiplier. Nothing said the two had to agree, so they drifted the moment one moved. Both curves now come from the solver and neither is assembled here: `multiplier()` for lambda, `traction()` for the whole load. The local copy of r is gone, which is the part that could go stale. Recomputed from scratch: at 1e2 lambda 0.3822 and the traction 0.3811 against an exact 0.381; at 1e6 lambda 0.0415 and anti-correlated, the traction 0.3815 at corr +0.999. The labels say which is which rather than "as returned", since what the API returns is now the traction. The cetz figure says "A deformed surface has no preferred direction". Its SVG is recompiled and run through `normalise_svg`, so Typst's per-compile random ids do not dirty the tree. Underworld development team with AI support from Claude Code --- .../examples/generate-rotated-basis.py | 2 +- .../examples/generate-topography-figure.py | 28 +- .../examples/rotated-basis.typ | 4 +- .../examples/topography-data.json | 2 +- .../figures/rotated-basis.svg | 2119 ++++++++++++++++- .../figures/topography.png | Bin 140358 -> 127762 bytes 6 files changed, 2138 insertions(+), 17 deletions(-) diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/generate-rotated-basis.py b/articles/boundary-conditions-on-non-planar-boundaries/examples/generate-rotated-basis.py index b203819..bfa3a74 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/examples/generate-rotated-basis.py +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/generate-rotated-basis.py @@ -30,7 +30,7 @@ def surface_y(x): - """A free surface: rises on the left, falls on the right, with an + """A deformed surface: rises on the left, falls on the right, with an inflection between, so the normal swings through a wide range and the curvature changes sign. No global rotation straightens this out, which is the reason the figure exists.""" diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/generate-topography-figure.py b/articles/boundary-conditions-on-non-planar-boundaries/examples/generate-topography-figure.py index f5bab72..a295f55 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/examples/generate-topography-figure.py +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/generate-topography-figure.py @@ -63,8 +63,8 @@ "dirichlet": "component Dirichlet", "penalty": "penalty, $10^4$", "nitsche": r"Nitsche, $\gamma = 10$", - "constraint": "multiplier, as returned", - "constraint+r": r"multiplier $+\;r(\mathbf{u}\cdot\hat{\mathbf{n}})$", + "constraint": r"multiplier field $\lambda$", + "constraint+r": r"traction $\lambda + r(\mathbf{u}\cdot\hat{\mathbf{n}} - \tilde{u}_n)$", "rotated": "rotated (reaction)", } STYLE = {"dirichlet": (0, (4, 2)), "penalty": "-", "nitsche": "-", @@ -95,17 +95,21 @@ def profile(mode, eta_B): curves = {mode: -np.asarray(values)} # h = -sigma_zz else: coords, values = read - curves = {mode: np.asarray(values)} # the reaction already carries it + curves = {mode: np.asarray(values)} if mode == "constraint": - # r is the default augmentation: augmentation_base (1e4) times the local - # viscosity, so 1e4 on the soft half and 1e4.eta_B on the stiff one. On a - # flat top wall u.n is u_y. - import underworld3 as uw - tree = uw.kdtree.KDTree(np.ascontiguousarray(v.coords)) - index = np.asarray(tree.query(np.ascontiguousarray(coords), 1)[1]).flatten() - u_n = np.squeeze(np.asarray(v.array))[index][:, 1] - r = 1.0e4 * np.where(coords[:, 0] < 0.5, 1.0, eta_B) - curves["constraint+r"] = curves[mode] + r * u_n + # BOTH curves come from the solver, and neither is assembled here. That + # matters: this script used to add r(u.n) to what `reaction_traction` + # returned, which was right while that returned the bare multiplier and + # became a DOUBLE COUNT the moment it returned `traction()` instead -- + # the corrected curve drew at twice its augmentation share and left the + # panel. Two copies of one expression, one of them stale. There is now + # one copy, and it lives in the solver. + # multiplier() -> lambda, the field + # traction() -> lambda + r(u.n - u~_n), the whole boundary load + curves["constraint+r"] = curves[mode] # traction(), as read + coords_bare, bare = C.trace(stokes, 2, stokes.multiplier("Top")) + assert np.allclose(coords_bare, coords), "the two traces disagree" + curves[mode] = np.asarray(bare) # the multiplier alone order = np.argsort(coords[:, 0]) x = coords[order, 0] diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/rotated-basis.typ b/articles/boundary-conditions-on-non-planar-boundaries/examples/rotated-basis.typ index de40bf4..3657d85 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/examples/rotated-basis.typ +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/rotated-basis.typ @@ -37,7 +37,7 @@ let hair = rgb("#d0d0d0") // ====================================================================== - // LEFT: a free surface, meshed. Geometry (nodes, Delaunay triangles, + // LEFT: a deformed surface, meshed. Geometry (nodes, Delaunay triangles, // surface frames) comes from generate-rotated-basis.py via JSON -- the // skill's rule, and the reason every node is now in the triangulation. // An earlier version joined nodes by a distance threshold and left some out. @@ -98,7 +98,7 @@ } content((0, 3.05), text(weight: "bold", size: 10pt, fill: ink, - "A free surface has no preferred direction")) + "A deformed surface has no preferred direction")) circle((-3.3, -3.55), radius: 0.13, fill: rot-fg, stroke: (paint: rot-fg)) content((-3.05, -3.55), anchor: "west", diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/topography-data.json b/articles/boundary-conditions-on-non-planar-boundaries/examples/topography-data.json index 8904634..f38f753 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/examples/topography-data.json +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/topography-data.json @@ -1 +1 @@ -{"contrasts": [100.0, 1000000.0], 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z)!S%B^`R>3I{(Lqx>{_NV|(!3WBKreApMEf=NCk>m>s-d+tHS>A7u&D(l0h)SP^Gn zx5q{dmRX_OMdqf7^glsnPDl%4S4rW3;+x9u`t+&sZI8$xi`tF_;M>wsmXt>@; z^_ERow1&(^JN>|e3e+nwR}CHThj;uXQTzTIJq5nNj=Fd^zYqDkGq0o1d^S>0qG zao+UWLc@bo0pw%XXq=a~Kag@8op#uIN+;Z?2s|c)hXUOn-;D`kspXQDK@I97Uv0rc z=+!&8L~L)QzpPYV783N?!5Yi51WpBknVcLQe2=-jicr^vXT|JD8A<1E?-3Ok;|N>U zx#w})Z_?=F-OAjPZa8hf`}fjqx_4@}z~dw^i|Jd9B3LAb-<^Bo@ZvH8p^SWk@`qRm zNvdNkvdepQ)~XaO5+ae;e-Lh~(2DGG6U@|YSVgfrP-Y#8T39ViXjRG{?g={!wd0LW z8S34K__|-v(+ig0#PAUT+CS@t2-F5kr?qeq2x?R1UxjbtU(aZFU>TP=*N-Az3Q!wN zI9?u<6e{dI1_fkSLiM422$%2h>=f~tE-F`SM4D~R2MjC=D?&+R|Q^(HFk?7tuX*LI}+ fzqfqCg37(nF+I7~I?r+H>EDxAzgu+MEcm|x-de<1 From 2f60df4e85cdb9e74173ddcaf897aa12dc2d180e Mon Sep 17 00:00:00 2001 From: lmoresi Date: Thu, 20 Aug 2026 22:10:51 +1000 Subject: [PATCH 25/33] Put the topography figure back in the article It was in the file and in nobody's way: I dropped the `{figure}` block when the two benchmark introductions were merged into one, so `figures/topography.png` was generated on every build and included by nothing. Which is why it was not in the preview. It belongs beside the contrast table in the SolCx section, and its alt text and caption now describe the corrected figure -- lambda alone flat near zero at 1e6, the traction on the exact curve, and the penalty failed by that contrast. The figure count in metadata.yml goes to three. Verified: full UWTN_PREVIEW build, no MyST errors, all three figures fingerprinted into the page, archival PDF at 878 kB. Underworld development team with AI support from Claude Code --- .../boundary-conditions-on-non-planar-boundaries.md | 12 ++++++++++++ 1 file changed, 12 insertions(+) diff --git a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md index eb490f8..b2d79aa 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md +++ b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md @@ -495,6 +495,18 @@ elements trimmed from each end. | 10⁴ | 0.076 / 0.085 | 0.698 / 0.697 | 0.076 / 0.085 | 0.076 / 0.085 | | 10⁶ | 0.076 / 0.085 | 0.992 / 1.000 | 0.075 / 0.084 | 0.076 / 0.085 | +```{figure} figures/topography.png +:alt: Two line plots of surface topography along the top wall from x=0 to x=1, mean removed, at viscosity contrasts of 100 and a million. In both, the exact answer is a thick grey curve falling from +0.29 at the left, flattening near +0.21, dropping sharply at the viscosity step at x=0.5 and continuing down to -0.38 at the right. At a contrast of 100 every curve lies on the grey one. At a contrast of a million they separate: the component Dirichlet, the rotated reaction and the traction lambda + r(u.n - u_n) still lie on the exact curve, while the multiplier field lambda alone is a nearly flat line near zero reaching only 0.04, and the penalty at 1e4 is a second nearly flat line near zero. Nitsche does not solve at either contrast and is absent. + +Surface topography along the top wall, against the exact answer. At a contrast of +100 nothing distinguishes the treatments. At $10^6$ the multiplier field +$\lambda$ carries almost none of the traction on its own — the augmentation +holds the rest — while $\lambda + r(\mathbf{u}\cdot\hat{\mathbf{n}} - +\tilde{u}_n)$, which is what `traction()` returns, lies on the exact curve. The +penalty has failed by this contrast: its coefficient is a bare number and cannot +be large against $10^6$ and moderate against 1 at the same time. +``` + **The three exact treatments agree to three figures at every contrast**, whole wall and trimmed alike. That is the result to take from this half, and it took the multiplier reporting the whole traction rather than $\lambda$ alone, and the From 7153ab2499fa771434ce1c523d8f8e3aa1a420af Mon Sep 17 00:00:00 2001 From: lmoresi Date: Fri, 21 Aug 2026 09:57:31 +1000 Subject: [PATCH 26/33] State the Nitsche limitation plainly, and box the traction note Nitsche has no column in the SolCx table. Say what we measured -- our implementation does not solve a boundary that mixes essential patches with Nitsche patches, at any penalty we tried, while all four walls imposed weakly converges -- instead of the parameter sweep and the speculation about which way round the failure ought to have gone. Confirmed at gamma = 100 as well as 10 and 1000: top wall only diverges in the line search (SNES -6) at contrasts of 10 and 1e6, all four walls converges (SNES 3), velocity error 1.74e-3 and 2.6e-5 against the exact solution. The traction paragraph becomes a MyST note admonition, in the house form used elsewhere in the corpus, with the claim as the title rather than as the first words of the paragraph. Underworld development team with AI support from Claude Code --- ...ary-conditions-on-non-planar-boundaries.md | 24 +++++++++---------- 1 file changed, 11 insertions(+), 13 deletions(-) diff --git a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md index b2d79aa..f500124 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md +++ b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md @@ -204,12 +204,13 @@ lam = stokes.add_constraint_bc(0.0, "Upper") stokes.solve() ``` -**Note**: **at convergence, the -momentum row's boundary term is the normal traction.** It does not need to be recovered from -the velocity field after the fact. It is an unknown the solve returns, available (on the solver) -through `traction` and, divided by $\Delta\rho g$, through `topography`. +:::{note} The boundary traction is an unknown of the solve, not a post-processing step +At convergence the momentum row's boundary term *is* the normal traction, so there is +nothing to recover from the velocity field afterwards. The solver returns it through +`traction` and, divided by $\Delta\rho g$, through `topography`. +::: -That term is the whole boundary load, +That boundary term is the whole boundary load, $\lambda + r(\mathbf{u}\cdot\hat{\mathbf{n}} - \tilde{u}_n)$, not the multiplier alone. The second part vanishes only where the constraint row is satisfied exactly; discretely it is satisfied to the solver's tolerance, and $r$ @@ -526,14 +527,11 @@ topography carries none of the signal. Scaling the coefficient by the local viscosity is the obviously right thing to want, but the solver does not converge here at any magnitude we tried, from $\eta$ to $10^3\eta$. -**Nitsche is missing from this table**. Our configuration of it -on this box converges at a contrast of $10^6$ and fails the line search at $10$ — -the opposite way round from every expectation — at 16 × 16 and 32 × 32 alike and -at $\gamma = 10$, $100$ and $1000$. Where it does converge, $\gamma$ has to rise -with the contrast exactly as the annulus said: at $10^6$ and 16 × 16 the surface -stress error is 25 at $\gamma = 10$, 1.2 at $100$ and 0.17 at $1000$, while the -constraint is held to $10^{-3}$ or better throughout. We are not confident enough -in that configuration to put a column of numbers behind it. +**Nitsche has no column in this table.** In our implementation it is unreliable on a +boundary that mixes essential patches with Nitsche patches, which is what this test +asks for: the top wall weak, the other three held strongly. We could not reach a +converged solution for this example at any penalty we tried. Imposed weakly on all +four walls it converges, so it is the mixture we cannot solve rather than the method. ### What each one costs From 1cc96b4d738b16200ba5ff3f04ab7537f16e0b14 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Fri, 21 Aug 2026 10:17:27 +1000 Subject: [PATCH 27/33] Give the note a banner: a skater on the curved wall of a bowl Alexander Londono's aerial photograph of a skate bowl, cropped from the 3000x3854 original to 2400x840 -- the banner box's own 2.86:1, so the page crops nothing further and the skater stays where the crop put him, on the curved lip with his shadow beneath him. Replaces the rendered annulus pair that stood in for it. The Unsplash credit goes in metadata.yml, from where banner_body.py rebuilds it into the page alongside the image. Underworld development team with AI support from Claude Code --- ...dary-conditions-on-non-planar-boundaries.md | 3 +++ .../figures/banner.jpg | Bin 0 -> 445631 bytes .../figures/banner.png | Bin 652504 -> 0 bytes .../metadata.yml | 9 +++++---- 4 files changed, 8 insertions(+), 4 deletions(-) create mode 100644 articles/boundary-conditions-on-non-planar-boundaries/figures/banner.jpg delete mode 100644 articles/boundary-conditions-on-non-planar-boundaries/figures/banner.png diff --git a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md index f500124..7010b4e 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md +++ b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md @@ -13,6 +13,7 @@ authors: affiliations: - Australian National University license: CC-BY-4.0 +banner: figures/banner.jpg keywords: - Underworld Code - Tricks of the Trade @@ -28,6 +29,8 @@ exports: article_version: 1.0.0 software_version: underworld3 development @ 8b7c8b9e --- +

+ Free-slip boundary conditions are used to simplify the physical behaviour at a domain boundary. 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`margin: 2em 0`, which inside the banner's border reads as a white band above and below the picture -- 32 px each, measured on the rendered page. It has always been there and has always been invisible, because until now every banner had a pale background. On a photograph it is obvious. Zero the margin on the banner and on the index thumbnail, which is framed the same way. Underworld development team with AI support from Claude Code --- static/uwtn.css | 6 ++++++ 1 file changed, 6 insertions(+) diff --git a/static/uwtn.css b/static/uwtn.css index 63106de..710aae6 100644 --- a/static/uwtn.css +++ b/static/uwtn.css @@ -273,6 +273,7 @@ article.content figcaption p { height: 100%; max-height: 260px; object-fit: cover; + margin: 0; /* Decorative, so it should not compete with the words. */ filter: saturate(0.82); transition: filter 0.25s ease, transform 0.4s ease; @@ -297,6 +298,11 @@ article.content figcaption p { width: 100%; max-height: 300px; object-fit: cover; + /* The theme's prose rule gives every image that is not inside a
+ `margin: 2em 0`. Inside the banner's frame that reads as a white band + above and below the picture -- invisible on a banner with a white + background, obvious on a photograph. */ + margin: 0; } @media (max-width: 720px) { From d61274232ade2f0b23a321d30524bad3d1954c14 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Fri, 21 Aug 2026 16:02:27 +1000 Subject: [PATCH 29/33] Show how the penalty's topography is obtained: read its own term The penalty enters the weak form as kappa (u.n) n, which IS the traction it holds the wall with, so sigma_nn = -kappa (u.n) is boundary arithmetic on values the solve returned. Nothing is differentiated and nothing is solved. The note said the two weak methods both had to recover stress by projection. That is true of Nitsche, whose term is written in sigma(u), and false of the penalty. Measured, on the same solves the note already reports: * SolCx, kappa 1e4: the two readings agree to three figures at every contrast, 0.992 either way at 1e6. Pointwise the curves differ by 0.5% at a contrast of 100 and 1.3% at a million, so the figure is unchanged. The cheap route does not rescue the accuracy -- the leak and the traction are the same quantity scaled by kappa. * Annulus, node normal: 9.0e-3, 1.2e-3, 2.3e-3, 2.2e-3 against 2.4e-2, 1.0e-2, 6.3e-3, 2.7e-3 for a projection off the same solve. It stops improving below cell 0.10 because the leak sits at 3e-3 throughout. Both harnesses gain penalty_traction() so the new column is reproducible, and the topography figure now draws the penalty's own term rather than a projection, which is what the text recommends reading. Underworld development team with AI support from Claude Code --- ...ary-conditions-on-non-planar-boundaries.md | 100 ++++++++++++------ .../examples/generate-topography-figure.py | 12 ++- .../examples/solcx.py | 26 ++++- .../examples/stress.py | 34 +++++- .../examples/topography-data.json | 2 +- .../figures/topography.png | Bin 127762 -> 127632 bytes 6 files changed, 132 insertions(+), 42 deletions(-) diff --git a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md index 7010b4e..4aefe97 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md +++ b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md @@ -93,8 +93,9 @@ consistent. Two impose it **exactly**: by construction, changing the basis so that the constraint is a component that can be struck out, or by a Lagrange multiplier, adding an equation that enforces it. The weak pair have a parameter to select that may need to be tuned for each problem and a floor - they cannot go below. The exact pair are not tuneable, and they both -return the boundary traction as a side-effect of the solution. + they cannot go below. The exact pair are not tuneable, and they return the +boundary traction as a side-effect of the solution. So does the direct penalty; +Nitsche is the one that does not. ### 1. A direct penalty @@ -129,6 +130,27 @@ penalty = 10000 stokes.add_natural_bc(penalty * G.dot(v.sym) * G, "Upper") ``` +**The topography** is the penalty term itself. That term is the traction the +condition holds the wall with, so on the boundary + +$$ +\sigma_{nn} = -\kappa\,(\mathbf{u}\cdot\hat{\mathbf{n}}), +\qquad +h = -\frac{\sigma_{nn} - \overline{\sigma_{nn}}}{\Delta\rho\,g} , +$$ + +which is arithmetic on values the solve already returned — nothing is +differentiated and nothing is solved. + +```python +n = mesh.boundary_normal("Upper") +sigma_nn = -penalty * n.dot(v.sym) +``` + +That is a saving in cost and not in accuracy. The leak and the traction are the +same quantity scaled by $\kappa$, so a coefficient too small to hold the +boundary reports a traction that is short in the same proportion. + ### Nitsche's method The reason the penalty is only accurate in the limit is that it is not @@ -161,10 +183,12 @@ holds to the accuracy of the discretisation, not to the accuracy of the arithmetic — measured below, it leaks a few parts in a thousand on a typical mesh, and the leak falls with increasing mesh resolution. -**The topography** comes the same way as the penalty's, and for the same reason: -the consistency term supplies the traction inside the momentum row, so there is -no reaction left in the residual to read. Recover $\sigma_{nn}$ from the solved -fields and divide by $\Delta\rho\,g$. +**The topography** has to be recovered from the solved fields, which is where +Nitsche parts company with the direct penalty. Its boundary term is the penalty +part *less* the consistency terms, and those are written in +$\boldsymbol{\sigma}(\mathbf{u})$, so the traction cannot be read off without +differentiating the velocity. Recover $\sigma_{nn}$ that way and divide by +$\Delta\rho\,g$. ### A constraint equation, with a multiplier @@ -409,20 +433,25 @@ buys nothing. ### The traction the solve already has -The two exact treatments do not have to recover anything, and the difference -shows up against the same exact answer: - -| cell size | rotated, reaction | multiplier, traction | either, recovered by projection | -|---|---|---|---| -| 0.150 | 6.8 × 10⁻³ | 8.6 × 10⁻³ | 2.4 × 10⁻² | -| 0.100 | 3.3 × 10⁻³ | 8.5 × 10⁻⁴ | 1.0 × 10⁻² | -| 0.075 | 2.1 × 10⁻³ | 1.7 × 10⁻³ | 6.3 × 10⁻³ | -| 0.050 | 1.1 × 10⁻³ | 1.4 × 10⁻³ | 2.7 × 10⁻³ | +Three of the four treatments do not have to recover anything. The two exact ones +carry the traction as a constraint reaction or as an unknown, and the direct +penalty carries it as the term it adds. Against the same exact answer: -Three to five times better than the projection on the same solve, at every -resolution, using the expressions given with each method above. Neither column -falls smoothly with $h$: part of what they report is the constraint residual, and -how far a particular solve drove that is not a function of the mesh. +| cell size | rotated, reaction | multiplier, traction | penalty, $\kappa(\mathbf{u}\cdot\hat{\mathbf{n}})$ | recovered by projection | +|---|---|---|---|---| +| 0.150 | 6.8 × 10⁻³ | 8.6 × 10⁻³ | 9.0 × 10⁻³ | 2.4 × 10⁻² | +| 0.100 | 3.3 × 10⁻³ | 8.5 × 10⁻⁴ | 1.2 × 10⁻³ | 1.0 × 10⁻² | +| 0.075 | 2.1 × 10⁻³ | 1.7 × 10⁻³ | 2.3 × 10⁻³ | 6.3 × 10⁻³ | +| 0.050 | 1.1 × 10⁻³ | 1.4 × 10⁻³ | 2.2 × 10⁻³ | 2.7 × 10⁻³ | + +Better than the projection on the same solve at every resolution, using the +expressions given with each method above — by an order of magnitude at best, and +by very little where the penalty's coefficient sets its floor. None of the +three falls smoothly with $h$: part of what they report is the constraint +residual, and how far a particular solve drove that is not a function of the +mesh. For the penalty that is the whole story below cell 0.10 — its leak is +3 × 10⁻³ at every resolution here, set by $\kappa$, and the traction it reports +cannot be better than the constraint it holds. ### What each parameter buys @@ -526,7 +555,11 @@ error of $10^{-5}$ appears in the stress. **A bare penalty coefficient cannot serve both halves.** At $10^4$ it is the best column in the table at low contrast — the constraint is weak enough not to fight the recovery — and by $10^6$ it is meaningless: 0.992, which is to say the recovered -topography carries none of the signal. Scaling the coefficient by the local +topography carries none of the signal. Reading $\kappa(\mathbf{u}\cdot\hat{\mathbf{n}})$ +instead of recovering anything gives the same numbers to three figures, here and +at every coefficient tried, which is the point made above from the other side: the +term is the traction, so it inherits the error in the constraint rather than +curing it. Scaling the coefficient by the local viscosity is the obviously right thing to want, but the solver does not converge here at any magnitude we tried, from $\eta$ to $10^3\eta$. @@ -571,10 +604,11 @@ back-calculation, and it is the case a spherical free surface runs in. Rotating the degrees of freedom costs nothing measurable against the weak forms: the rotation is a sparse orthogonal transform on a boundary's worth of rows. -**The weak constraints have to recover surface stress by differentiating the solution**, and +**Nitsche has to recover surface stress by differentiating the solution**, and the projection that does it costs *more than the Stokes solve did* — 0.63 s -against 0.45 s — so asking a penalty or Nitsche model for its surface stress -roughly doubles the timestep. +against 0.45 s — so asking a Nitsche model for its surface stress roughly +doubles the timestep. The direct penalty need not pay that: its own term is the +traction, read as arithmetic on the boundary nodes. The obvious question is whether that is the method's cost or the recovery's. A global $L^2$ projection to get values on a thousand boundary nodes is plainly @@ -592,9 +626,11 @@ makes it unavailable to the multiplier: An error of 1.00 is the metric reporting that nothing was recovered. A reaction exists in the residual only where a row has been *constrained*; a weak condition supplies its traction as a term inside the row it acts on, so the residual there -is balanced at convergence and there is nothing left to read. The multiplier does -the same thing, and gets away with it because the term it supplies, $\lambda$, is the -traction as a field. Nitsche's term is written in terms of +is balanced at convergence and there is nothing left to read. What rescues a weak +condition is not the residual but its own term: the multiplier supplies $\lambda$, +which is the traction as a field, and the penalty supplies +$\kappa(\mathbf{u}\cdot\hat{\mathbf{n}})$, which is the traction as boundary +arithmetic. Nitsche's term is written in terms of $\boldsymbol{\sigma}(\mathbf{u})$, so reading it back still requires differentiating the answer. @@ -603,12 +639,14 @@ the two pairs exactly: | how the constraint is imposed | the traction is | to read it | |---|---|---| -| weakly, by a term (penalty, Nitsche) | a by-product | differentiate the solution | +| weakly, by a penalty term | the term itself | evaluate $\kappa(\mathbf{u}\cdot\hat{\mathbf{n}})$ | +| weakly, by Nitsche | inside a term written in $\boldsymbol{\sigma}(\mathbf{u})$ | differentiate the solution | | exactly, by construction (rotated) | the constraint reaction | de-smear the nodal load | | exactly, by a multiplier | an unknown of the system | read the field | -The cost of the first row is negotiable — a recovery restricted to the boundary -would be cheaper than a global projection — but the differentiation is not. +Only the second row pays. Its cost is negotiable — a recovery restricted to the +boundary would be cheaper than a global projection — but the differentiation is +not. :::{note} What these timings are not Two-dimensional, one core, direct solver. What they measure is the difference @@ -644,7 +682,9 @@ When the wall-normal traction is needed: if you choose Nitsche. - **A direct penalty is fine for a velocity-only model** and needs the node normal, a coefficient chosen per problem, and a check on something physical - before the answer is believed. It has the advantage that this is pure, direct penalty + before the answer is believed. It will also hand back the traction it holds + the wall with, at no cost, and that reading is worth no more than the + coefficient behind it. It has the advantage that this is pure, direct penalty on the weak form and can be used for many things beyond simply boundary conditions. Good for a first pass on a very general idea. diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/generate-topography-figure.py b/articles/boundary-conditions-on-non-planar-boundaries/examples/generate-topography-figure.py index a295f55..e815024 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/examples/generate-topography-figure.py +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/generate-topography-figure.py @@ -10,14 +10,18 @@ the deviation is the part that is determined and the part topography is built from. -Two routes are drawn, and they are not the same measurement: +Three routes are drawn, and they are not the same measurement: * the recovered traction, projected out of the solved velocity and pressure, - which is the only route the weak treatments have; + which is the only route Nitsche has; * the constraint reaction -- `boundary_normal_traction` for the rotated constraint and the multiplier field for the constraint method -- which the solve returns as an unknown. Its sign convention is the traction holding the - wall, which is the topography's sign directly. + wall, which is the topography's sign directly; + * the penalty's own term, kappa (u.n), which is the traction that condition + holds the wall with. Drawn rather than the projection off the same solve + because it is what the note recommends reading; the two agree to about one + per cent, which is invisible here. Colour carries the treatment and the exact answer is a thick grey line behind everything, so no curve is identified by colour alone against the reference. @@ -89,7 +93,7 @@ def profile(mode, eta_B): if not C.converged(stokes): print("%-10s eta_B %.0e diverged" % (mode, eta_B), flush=True) return None - read = C.reaction_traction(stokes, mode) + read = C.reaction_traction(stokes, mode, v=v) if read is None: coords, values = C.recovered_traction(mesh, stokes) curves = {mode: -np.asarray(values)} # h = -sigma_zz diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/solcx.py b/articles/boundary-conditions-on-non-planar-boundaries/examples/solcx.py index 89b2660..756f764 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/examples/solcx.py +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/solcx.py @@ -131,8 +131,28 @@ def recovered_traction(mesh, stokes): return trace(projection, 0, field) -def reaction_traction(stokes, mode): - """The traction the solve returned, for the two methods that return one.""" +def penalty_traction(stokes, v, kappa): + """The traction a direct penalty holds the wall with. + + The penalty term IS that traction, so sigma_zz is -kappa (u.n) on the top + wall, where n = (0, 1). Returned with the sign of a reaction, as the other + two routes are, so `measure` negates all three alike. + """ + coords = np.asarray(v.coords) + top = np.abs(coords[:, 1] - 1.0) < 1.0e-9 + u = np.squeeze(np.asarray(v.array)) + return coords[top], kappa * u[top, 1] + + +def reaction_traction(stokes, mode, v=None, penalty=PENALTY): + """The traction the solve returned, for the three methods that return one. + + Nitsche is the exception: its term is written in sigma(u), so reading it + back means differentiating the velocity, which is what + `recovered_traction` does. + """ + if mode == "penalty": + return penalty_traction(stokes, v, penalty) if mode == "rotated": return stokes.boundary_normal_traction("Top") if mode == "constraint": @@ -199,7 +219,7 @@ def measure(mode, **kwargs): coords, values = recovered_traction(mesh, stokes) out["recovered"] = stress_error(coords, values, exact) out["trimmed"] = stress_error(coords, values, exact, trim=2.0 / kwargs.get("res", RES)) - read = reaction_traction(stokes, mode) + read = reaction_traction(stokes, mode, v=v, penalty=kwargs.get("penalty", PENALTY)) if read is not None: coords, values = read # The reaction is the traction holding the wall, opposite in sign to diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/stress.py b/articles/boundary-conditions-on-non-planar-boundaries/examples/stress.py index 977dc8a..8476cf6 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/examples/stress.py +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/stress.py @@ -235,12 +235,38 @@ def _trace_index(var, coords): return _TRACE_INDEX[key] +def penalty_traction(stokes, v, kappa=PENALTY, boundary="Upper"): + """The traction a direct penalty holds the boundary with. + + The term the weak form carries IS that traction, so sigma_nn is + -kappa (u.n) on the boundary and there is nothing to recover. Read at the + boundary nodes against the TRUE radial direction, as `recovered_traction` + is, so the comparison does not depend on the solver's normal. + + It is only as good as the constraint it holds: the leak and this number are + the same quantity scaled by kappa. + """ + nodes, *_ = _boundary_field_nodes(stokes, boundary, 0) + coords = np.array([node[2] for node in nodes]) + index = _trace_index(v, coords) + u = np.squeeze(np.asarray(v.array))[index] + normal = coords / np.linalg.norm(coords, axis=1)[:, None] + return coords, -kappa * (u * normal).sum(axis=1) + + def reaction_traction(stokes, mode, v=None): - """The constraint reaction, for the two methods that return one.""" + """The traction each treatment returns without recovering one. + + Three of the four do. Nitsche is the exception: its term is written in + sigma(u), so reading it back means differentiating the velocity, which is + what `recovered_traction` does. + """ if mode == "rotated": return stokes.boundary_normal_traction("Upper") if mode == "constraint": return multiplier_traction(stokes, v) + if mode in ("penalty", "penalty_node"): + return penalty_traction(stokes, v) return None @@ -323,11 +349,11 @@ def both(cells=(0.15, 0.10, 0.075, 0.05), modes=MODES): print() print("the reaction routes, same solves") print() - print("| cell size | rotated reaction | multiplier traction |") - print("|---|---|---|") + print("| cell size | rotated reaction | multiplier traction | penalty term |") + print("|---|---|---|---|") for cell in cells: entries = [] - for mode in ("rotated", "constraint"): + for mode in ("rotated", "constraint", "penalty_node"): got = measure(mode, cell=cell) entries.append("-" if not got or "reaction" not in got else "%.1e" % got["reaction"]) diff --git 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zfv&&%T~kg&_}>{=PKpSIha<_NF{ICwj4Ml#JKxD!!?=G$hvv7p{fEH<{`cV%hhmL+ zwU_)L9t1Kfpq_<5@+#RtG$2luktB61SCkl?ZP<8r3OK~k#R32Yw!J&tnH%`|>D(*H zHMUzDB@v8n`eOi(=FwbBdRPX?A~I3${0zUsL*Ih zD?r7riJjw9;Ol8Ng4%%2IT_m`&b2B(E1^o{A=}cQEpV0d}jry4$oht z1hAg7bm6S z2)t}5dx%Z*o0k8~VDWC?==h6GtaT5q#)1(Y#NEaXgs__%m>`-TdIXBWSe~wQ(SL(B zsLY(Mj}-<#eC#?`_5hW7Pl5V@QI#Z-5)$xY_lGtn8t6}#hWtN^SMcoG>3na znxeRP5V_~qP|~dOmUB$*cjDRI6)r&Z){;O1+7#~no9TD{@8Bu=zYZ$f`T Date: Sat, 22 Aug 2026 21:50:16 +1000 Subject: [PATCH 30/33] Final pass: two rendering faults the build does not report, and their guards Louis's edits, with three corrections and the tests that would have caught two of them. The equation label moved off the closing delimiter onto a line of its own. That form renders as literal text, the three {eq} references resolve to nothing, and `myst build --typst` fails the article outright -- this note had no PDF in the last build. Restored to `$$ (eq-free-slip)`. The blank line before nine display-math blocks was removed. For one-line arithmetic that renders anyway, which is why the corpus contains examples of it, but the weak form has a line beginning `- \int`, and without the blank line markdown claims it: the paragraph continues into the block, the bullet opens a list, the math never closes, and page one of the PDF carried raw LaTeX with half an equation in a bullet point. The build reported nothing: no warning, exit 0. Both now have tests, each confirmed to fire on the exact broken form and to leave the published corpus passing. The label test looks for `$$ (name)` on the closing line, with `[ \t]*` rather than `\s*` -- \s crosses newlines, so the first version of the test matched the very form it was written to catch. Also corrected: the rotated-frame paragraph claimed the SolCx table's rotated column came from the turned problem. It does not, and it cannot -- three walls of that table are held by a component condition, which cannot express u.n = 0 on a tilted wall. It now reads as the separate check it is, with the measured numbers. And `Stokes_Constrained` is three FIELDS split TWO ways, velocity against [p, lambda]; the solver docstring is the authority. examples/solcx_rotated.py is the harness behind that paragraph, including the `aligned` negative control that leaks 71% of the flow. Underworld development team with AI support from Claude Code --- ...ary-conditions-on-non-planar-boundaries.md | 64 ++-- .../examples/solcx_rotated.py | 277 ++++++++++++++++++ tests/test_migration.py | 63 ++++ 3 files changed, 378 insertions(+), 26 deletions(-) create mode 100644 articles/boundary-conditions-on-non-planar-boundaries/examples/solcx_rotated.py diff --git a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md index 4aefe97..bbc4c8f 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md +++ b/articles/boundary-conditions-on-non-planar-boundaries/boundary-conditions-on-non-planar-boundaries.md @@ -231,6 +231,11 @@ lam = stokes.add_constraint_bc(0.0, "Upper") stokes.solve() ``` +The `Stokes_Constrained` solver carries three fields in PETSc — velocity, pressure and +$\lambda$ — and splits them two ways: velocity against the pair, with the volume +constraint (incompressibility) and the boundary constraint grouped into a single Schur +block. The boundary constraint applies to surface nodes only. + :::{note} The boundary traction is an unknown of the solve, not a post-processing step At convergence the momentum row's boundary term *is* the normal traction, so there is nothing to recover from the velocity field afterwards. The solver returns it through @@ -431,7 +436,7 @@ $\gamma = 100$ the leak improves by a factor of nearly forty and the stress by a factor of two, onto the same floor as everything else, after which more $\gamma$ buys nothing. -### The traction the solve already has +### Traction extraction v. Stress recovery Three of the four treatments do not have to recover anything. The two exact ones carry the traction as a constraint reaction or as an unknown, and the direct @@ -453,7 +458,7 @@ mesh. For the penalty that is the whole story below cell 0.10 — its leak is 3 × 10⁻³ at every resolution here, set by $\kappa$, and the traction it reports cannot be better than the constraint it holds. -### What each parameter buys +### Influence of penalty parameters The two weak methods look alike in the comparison above, but this is for a fixed, tuned penalty parameter. @@ -482,7 +487,7 @@ problem. What it does not do is converge with resolution — the leak is bought rather than with the mesh resolution — so the coefficient has to be re-chosen whenever the forcing or the viscosity changes. -### The multiplier and the consistent boundary flux are the same +### Multiplier and CBF equivalence The two expressions given above for computing topography from the boundary reaction are exactly equivalent. Write the momentum row's boundary term out and the identity is immediate: the assembled @@ -502,21 +507,28 @@ mass. The multiplier is not a second, independent estimate of the surface stress it is the same computation, arrived at by carrying the traction as an unknown instead of reading it out of the residual afterwards. -### The other half: a lateral viscosity contrast +### Lateral viscosity contrast case -No exact solution has both a curved boundary and a laterally varying viscosity, +No published solution has both a curved boundary and a laterally varying viscosity, so the case where weak constraints are most often reported to give trouble is a -separate test with a trivial geometry. SolCx is that test: the unit box, free +separate test with a simple geometry. SolCx is a good example: unit box, free slip on all four walls, viscosity 1 to the left of $x = 0.5$ and $\eta_B$ to the right. `uw.analytic.SolCx` publishes the exact dynamic topography on the top wall. Three walls carry the ordinary component condition and the treatment under test is on the top wall alone. -On a box every treatment reduces to holding one velocity component, so nothing -here is about normals. What it can say is whether a treatment holds the traction -it was given when the viscosity beside it jumps. - -Relative $l_2$ error of the surface topography along the top wall, mean removed, +On a box every treatment reduces to holding one velocity component. +What it can say is whether a treatment holds the traction +it was given when the viscosity beside it jumps. However, on a box the rotated constraint's per-node rotation is the identity, so the +table below exercises none of the rotation machinery. We check that separately on an +equivalent problem: the domain, the gravity vector and the exact solution all turned by +45$^\circ$, with every wall then carrying the rotated constraint, because a component +condition cannot express $\mathbf{u}\cdot\hat{\mathbf{n}} = 0$ on a tilted wall. Turned, +the constraint still holds to machine precision and the velocity error is unchanged at +8.8 × 10⁻⁶; imposing the un-turned condition on those same walls instead lets 71% of the +flow through the boundary. + +In the table below, we show the relative $l_2$ error of the surface topography along the top wall, mean removed, at 32 × 32 elements. Each entry is the whole wall and then the wall with two elements trimmed from each end. @@ -545,7 +557,7 @@ wall and trimmed alike. That is the result to take from this half, and it took the multiplier reporting the whole traction rather than $\lambda$ alone, and the rotated constraint holding the corner where it meets the side walls. -**Read the first column as the floor.** The component Dirichlet condition is +**Read the first column as the reference.** The component Dirichlet condition is exact and has no parameter, and its velocity error is 8.8 × 10⁻⁶ at a contrast of $10^6$. It still reads 0.085. That number is the recovery's error, not a boundary condition's: on the stiff half the recovered $\sigma_{zz}$ is a difference between @@ -553,10 +565,10 @@ the pressure and $2\eta\,\partial_z u_z$ with $\eta = 10^6$, so a relative veloc error of $10^{-5}$ appears in the stress. **A bare penalty coefficient cannot serve both halves.** At $10^4$ it is the best -column in the table at low contrast — the constraint is weak enough not to fight -the recovery — and by $10^6$ it is meaningless: 0.992, which is to say the recovered +column in the table at low contrast (the constraint is weak enough not to fight +the recovery) but by $10^6$ it is meaningless: 0.992, which is to say the recovered topography carries none of the signal. Reading $\kappa(\mathbf{u}\cdot\hat{\mathbf{n}})$ -instead of recovering anything gives the same numbers to three figures, here and +instead of recovering the stress gives the same numbers to three figures, here and at every coefficient tried, which is the point made above from the other side: the term is the traction, so it inherits the error in the constraint rather than curing it. Scaling the coefficient by the local @@ -566,10 +578,10 @@ magnitude we tried, from $\eta$ to $10^3\eta$. **Nitsche has no column in this table.** In our implementation it is unreliable on a boundary that mixes essential patches with Nitsche patches, which is what this test asks for: the top wall weak, the other three held strongly. We could not reach a -converged solution for this example at any penalty we tried. Imposed weakly on all -four walls it converges, so it is the mixture we cannot solve rather than the method. +converged solution for this example at any penalty. Imposed weakly on all +four walls does converge, but that is a different problem. -### What each one costs +### Solver timing Seconds on the annulus, uniform viscosity, one core, direct solver: the solve, and then the surface traction by whatever route that treatment has. Median of @@ -579,17 +591,17 @@ run-to-run spread and there is nothing to read. | velocity nodes | penalty | Nitsche | multiplier | rotated | |---|---|---|---|---| -| 28 338 | 0.17 / 0.273 | 0.18 / 0.267 | 0.26 / — | 0.16 / 0.010 | -| 71 424 | 0.45 / 0.631 | 0.47 / 0.657 | 0.67 / — | 0.43 / 0.016 | +| 28 338 | 0.17 / — | 0.18 / 0.267 | 0.26 / — | 0.16 / 0.010 | +| 71 424 | 0.45 / — | 0.47 / 0.657 | 0.67 / — | 0.43 / 0.016 | -The dash indicates that the multiplier does not require any additional *solver* — +The dash indicates that the multiplier and the penalty do not require any additional *solver* — $\lambda$ is a finite element field in its own right, so its nodal values are the traction, pointwise, and $\lambda + r(\mathbf{u}\cdot\hat{\mathbf{n}} - \tilde{u}_n)$ is an -expression evaluated where it is wanted. +expression evaluated where it is wanted. The penalty traction is recovered similarly as the penalty scaling +the leakage velocity. -**The rotated constraint's reaction does need one step**. (The 10 to 16 ms in the table is the reaction the solve -already stashed; `boundary_flux` re-assembles the residual from scratch and costs -0.2 s, which is the 0.206 s in the comparison above.) The reaction is an *integrated* nodal load, +**The rotated constraint's reaction does need one step**. +The reaction is an *integrated* nodal load, $\int_\Gamma \sigma_{nn}\,\phi_i\,\mathrm{d}S$, which is $M_\Gamma$ times the pointwise traction. Turning it into a pointwise value means undoing that boundary mass. On a 2-D trace, and on 3-D P1 triangles, the lumped mass is diagonal and @@ -597,7 +609,7 @@ undoing it is a division — the 10 to 16 ms above. On **3-D P2 triangles it is true solve**: the lumped row sums vanish at the vertices, so the consistent trace mass has to be assembled and solved. That is the one place the CBF route pays for being a -back-calculation, and it is the case a spherical free surface runs in. +back-calculation. **The multiplier's solve costs about 50% more**, consistently — 0.67 s against 0.43 s at 71 000 nodes. That is the extra field and the larger saddle point. diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/solcx_rotated.py b/articles/boundary-conditions-on-non-planar-boundaries/examples/solcx_rotated.py new file mode 100644 index 0000000..69ffe11 --- /dev/null +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/solcx_rotated.py @@ -0,0 +1,277 @@ +"""SolCx in a rotated frame: an exact solution on walls that are not axis-aligned. + +`solcx.py` runs SolCx on the unit box, where "no flow through the top wall" is a +single velocity component and every treatment reduces to the same constraint. +That is deliberate -- it isolates the viscosity contrast from the geometry -- but +it also means the box says nothing about normals. + +This rotates the whole problem instead. The mesh is turned through THETA, and the +viscosity, the forcing and the exact solution are turned with it, so the physics +is identical and no wall is aligned with a coordinate direction. The exact +solution is still SolCx's, read at the pre-image of each point. + +With x the rotated coordinate and xi = R^T x the material coordinate, + + eta(x) = eta_solcx(xi) + f(x) = R f_solcx(xi) + u(x) = R u_solcx(xi) + sigma(x) = R sigma_solcx(xi) R^T + +and because sigma_nn is frame-invariant, the surface stress on the rotated top +wall is exactly sigma_zz of the unrotated solution at the pre-image. Nothing has +to be un-rotated to make the comparison. + +Every wall carries the treatment under test: a component Dirichlet condition +cannot express u.n = 0 on a wall at 45 degrees, which is the point. The corners +are where two walls meet, and there both constraints apply -- u.n = 0 against two +orthogonal normals, which fixes the node at zero. That is what the exact solution +says a free-slip corner does, so it is a constraint and not a choice. + + python3 solcx_rotated.py # every treatment, at THETA + python3 solcx_rotated.py angles # the rotated constraint against the angle + +Run against underworld3 `bugfix/multiplier-traction` (PR #617). +""" +import sys + +import numpy as np +import sympy + +import underworld3 as uw + +from underworld3.utilities.boundary_flux import _boundary_field_nodes + +RES = 32 +ETA_B = 1.0e6 +PENALTY = 1.0e4 +GAMMA = 10.0 +THETA = np.pi / 4.0 +WALLS = ("Left", "Right", "Bottom", "Top") + + +def rotation(theta): + c, s = np.cos(theta), np.sin(theta) + return np.array([[c, -s], [s, c]]) + + +def build(mode, res=RES, eta_B=ETA_B, penalty=PENALTY, gamma=GAMMA, theta=THETA): + """The rotated problem, and one boundary treatment on every wall.""" + mesh = uw.meshing.StructuredQuadBox( + elementRes=(res, res), minCoords=(0.0, 0.0), maxCoords=(1.0, 1.0), qdegree=3) + + R = rotation(theta) + mesh.deform(np.asarray(mesh.X.coords) @ R.T) + + v = uw.discretisation.MeshVariable("U", mesh, mesh.dim, degree=2) + p = uw.discretisation.MeshVariable("P", mesh, 1, degree=1) + solver_class = (uw.systems.Stokes_Constrained if mode == "constraint" + else uw.systems.Stokes) + stokes = solver_class(mesh, velocityField=v, pressureField=p) + + # Built on the rotated mesh, but only for its SYMBOLS: every expression below + # is composed with xi = R^T x before it is used, which is what turns the + # solution with the mesh. + exact = uw.analytic.SolCx(mesh, eta_A=1.0, eta_B=eta_B, x_c=0.5, n=1) + x, z = mesh.X + c, s = sympy.cos(theta), sympy.sin(theta) + material = {x: c * x + s * z, z: -s * x + c * z} # xi = R^T x + Rs = sympy.Matrix([[c, -s], [s, c]]) + + def turn(expr): + """An expression of the material coordinate, written in the rotated one.""" + return expr.subs(material, simultaneous=True) + + fn_viscosity = turn(exact.fn_viscosity) + fn_bodyforce = (Rs * turn(exact.fn_bodyforce).T).T + fn_velocity = (Rs * turn(sympy.Matrix(exact.fn_velocity)).reshape(2, 1)).T + # sigma_nn is invariant, so the exact surface stress on the rotated top wall + # is the unrotated sigma_zz at the pre-image -- no rotation of the stress. + fn_sigma_nn = turn(exact.fn_stress[1, 1]) + + stokes.constitutive_model = uw.constitutive_models.ViscousFlowModel + stokes.constitutive_model.Parameters.shear_viscosity_0 = fn_viscosity + if mode != "constraint": + stokes.saddle_preconditioner = 1.0 / fn_viscosity + stokes.bodyforce = fn_bodyforce + stokes.tolerance = 1.0e-9 + stokes.petsc_use_pressure_nullspace = True + + if mode == "aligned": + # The NEGATIVE CONTROL, and the reason the rotated run means anything: + # the condition the unrotated box would use, imposed unchanged on walls + # that have been turned. It pins the wrong component of the velocity at + # every boundary node, so if the rotation matrix were quietly the + # identity somewhere in the machinery, this is the answer we would get. + stokes.add_dirichlet_bc((0.0, None), "Left") + stokes.add_dirichlet_bc((0.0, None), "Right") + stokes.add_dirichlet_bc((None, 0.0), "Bottom") + stokes.add_dirichlet_bc((None, 0.0), "Top") + return mesh, stokes, v, dict(velocity=fn_velocity, sigma_nn=fn_sigma_nn, + viscosity=fn_viscosity, R=R, theta=theta) + + for wall in WALLS: + if mode == "rotated": + stokes.add_rotated_freeslip_bc(0.0, wall) + elif mode == "constraint": + stokes.add_constraint_bc(0.0, wall) + elif mode == "nitsche": + stokes.add_nitsche_bc(0.0, wall, gamma=gamma, theta=1) + elif mode == "penalty": + n = mesh.boundary_normal(wall) + stokes.add_natural_bc(penalty * n.dot(v.sym) * n, wall) + else: + raise ValueError(mode) + + return mesh, stokes, v, dict(velocity=fn_velocity, sigma_nn=fn_sigma_nn, + viscosity=fn_viscosity, R=R, theta=theta) + + +def converged(stokes): + return stokes.snes.getConvergedReason() > 0 + + +def velocity_error(v, truth): + got = np.squeeze(np.asarray(v.array)) + want = np.asarray(uw.function.evaluate(truth["velocity"], v.coords)).reshape(got.shape) + return float(np.linalg.norm(got - want) / np.linalg.norm(want)) + + +def leak(v, mesh, theta=THETA): + """max |u.n| / |u| on the rotated top wall, against the TRUE wall normal.""" + n = rotation(theta) @ np.array([0.0, 1.0]) + coords = np.asarray(v.coords) + material = coords @ rotation(theta) # xi = R^T x + top = np.abs(material[:, 1] - 1.0) < 1.0e-9 + u = np.squeeze(np.asarray(v.array)) + return float(np.abs(u[top] @ n).max() / np.linalg.norm(u, axis=1).max()) + + +def corner_speed(v, theta=THETA): + """|u| at the four corners, which two free-slip walls fix at zero.""" + coords = np.asarray(v.coords) + material = coords @ rotation(theta) + u = np.squeeze(np.asarray(v.array)) + corner = ((np.abs(material[:, 0] - 0.0) < 1.0e-9) | (np.abs(material[:, 0] - 1.0) < 1.0e-9)) \ + & ((np.abs(material[:, 1] - 0.0) < 1.0e-9) | (np.abs(material[:, 1] - 1.0) < 1.0e-9)) + if not corner.any(): + return float("nan") + return float(np.linalg.norm(u[corner], axis=1).max() / np.linalg.norm(u, axis=1).max()) + + +def trace_top(solver, mesh, theta=THETA): + """Velocity-degree nodes on the rotated top wall.""" + nodes, *_ = _boundary_field_nodes(solver, "Top", 0) + return np.array([node[2] for node in nodes]) + + +def stress_error(coords, values, truth, trim=0.0, theta=THETA): + """Relative l2 error in sigma_nn along the rotated top wall, mean removed.""" + coords = np.asarray(coords) + values = np.asarray(values).reshape(-1) + material = coords @ rotation(theta) + if trim > 0.0: + keep = (material[:, 0] > trim) & (material[:, 0] < 1.0 - trim) + coords, values, material = coords[keep], values[keep], material[keep] + want = np.asarray(uw.function.evaluate(truth["sigma_nn"], coords)).reshape(-1) + order = np.argsort(material[:, 0]) + got = values[order] - values.mean() + want = want[order] - want.mean() + return float(np.linalg.norm(got - want) / np.linalg.norm(want)) + + +def recovered_sigma_nn(mesh, stokes, theta=THETA): + """sigma_nn on the rotated top wall, projected out of the solved fields.""" + n = rotation(theta) @ np.array([0.0, 1.0]) + normal = sympy.Matrix([[n[0], n[1]]]) + field = uw.discretisation.MeshVariable("Snn", mesh, 1, degree=2) + projection = uw.systems.Projection(mesh, field) + projection.uw_function = (normal * stokes.stress * normal.T)[0, 0] + projection.solve() + coords = trace_top(projection, mesh, theta) + tree = uw.kdtree.KDTree(np.ascontiguousarray(field.coords)) + index = np.asarray(tree.query(np.ascontiguousarray(coords), 1)[1]).flatten() + return coords, np.squeeze(np.asarray(field.array))[index] + + +def reaction_sigma_nn(mesh, stokes, v, mode, penalty=PENALTY, theta=THETA): + """The traction the solve already has, for the three treatments that have one.""" + n = rotation(theta) @ np.array([0.0, 1.0]) + if mode == "rotated": + coords, values = stokes.boundary_normal_traction("Top") + return coords, -np.asarray(values) + if mode == "constraint": + coords = trace_top(stokes, mesh, theta) + return coords, -np.asarray( + uw.function.evaluate(stokes.traction("Top"), coords)).reshape(-1) + if mode == "penalty": + coords = trace_top(stokes, mesh, theta) + tree = uw.kdtree.KDTree(np.ascontiguousarray(v.coords)) + index = np.asarray(tree.query(np.ascontiguousarray(coords), 1)[1]).flatten() + u = np.squeeze(np.asarray(v.array))[index] + return coords, -penalty * (u @ n) + return None + + +MODES = ("rotated", "constraint", "penalty", "nitsche", "aligned") + + +def measure(mode, **kwargs): + mesh, stokes, v, truth = build(mode, **kwargs) + stokes.solve() + if not converged(stokes): + return None + theta = kwargs.get("theta", THETA) + out = { + "leak": leak(v, mesh, theta), + "corner": corner_speed(v, theta), + "velocity": velocity_error(v, truth), + } + coords, values = recovered_sigma_nn(mesh, stokes, theta) + out["recovered"] = stress_error(coords, values, truth, theta=theta) + read = reaction_sigma_nn(mesh, stokes, v, mode, + penalty=kwargs.get("penalty", PENALTY), theta=theta) + if read is not None: + coords, values = read + out["reaction"] = stress_error(coords, values, truth, theta=theta) + # Two elements off each end. A corner node belongs to two walls, and the + # measure-weighted normal there is the bisector of two orthogonal ones, + # so u.n at a corner is not the quantity either wall asked for. + out["reaction_trimmed"] = stress_error( + coords, values, truth, trim=2.0 / kwargs.get("res", RES), theta=theta) + return out + + +def table(modes=MODES, **kwargs): + print("SolCx rotated through %.1f degrees, %d x %d." + % (np.degrees(kwargs.get("theta", THETA)), RES, RES)) + print() + print("| treatment | leak | corner |u| | velocity | sigma_nn, recovered | sigma_nn, from the solve |") + print("|---|---|---|---|---|---|") + for mode in modes: + got = measure(mode, **kwargs) + if got is None: + print("| %s | did not converge | | | | |" % mode, flush=True) + continue + print("| %s | %.1e | %.1e | %.2e | %.3f | %s |" + % (mode, got["leak"], got["corner"], got["velocity"], got["recovered"], + "%.3f" % got["reaction"] if "reaction" in got else "—"), flush=True) + + +def angles(degrees=(0.0, 15.0, 30.0, 45.0), mode="rotated"): + print("%s, against the angle of the frame" % mode) + print() + print("| degrees | leak | velocity | sigma_nn, from the solve |") + print("|---|---|---|---|") + for d in degrees: + got = measure(mode, theta=np.radians(d)) + if got is None: + print("| %.0f | did not converge | | |" % d, flush=True) + continue + print("| %.0f | %.1e | %.2e | %s |" + % (d, got["leak"], got["velocity"], + "%.3f" % got["reaction"] if "reaction" in got else "—"), flush=True) + + +if __name__ == "__main__": + command = sys.argv[1] if len(sys.argv) > 1 else "table" + {"table": table, "angles": angles}[command]() diff --git a/tests/test_migration.py b/tests/test_migration.py index 35e7cce..561edb4 100644 --- a/tests/test_migration.py +++ b/tests/test_migration.py @@ -2214,3 +2214,66 @@ def test_reader_link_is_visible_and_click_safe(): style = (pathlib.Path(__file__).resolve().parent.parent / "static" / "uwtn.css").read_text(encoding="utf-8") assert ".uwtn-pdf-link" in style, "the injected link has no style" + + +def test_every_eq_reference_has_a_labelled_equation(): + """`{eq}`name`` needs `$$ (name)` on the CLOSING delimiter line. + + A label written on a line of its own renders as literal text, the reference + resolves to nothing, and `myst build --typst` fails the article outright + with "label does not exist" -- an HTML-only check would pass it. That is + how it reached a pushed branch once already. + """ + for path in sorted((ROOT / "articles").glob("*/*.md")): + text = path.read_text(encoding="utf-8") + # [ \t]* and not \s*: \s crosses newlines, so the very form this test + # exists to catch -- the label on its own line -- would match. + labels = set(re.findall(r"^\$\$[ \t]*\(([^)]+)\)[ \t]*$", text, re.M)) + labels |= set(re.findall(r"^\s*:label:\s*(\S+)\s*$", text, re.M)) + labels |= set(re.findall(r"\\label\{([^}]+)\}", text)) + for name in set(re.findall(r"\{eq\}`([^`]+)`", text)): + assert name in labels, ( + "%s references {eq}`%s` but no equation carries that label. " + "The label belongs on the closing $$ line: `$$ (%s)`." + % (path.name, name, name)) + + +# A line inside display math that markdown would claim for itself if the block +# is not a block of its own: a list bullet, a heading, a quote, a fence. +_MARKDOWN_LINE = re.compile(r"^\s*([-+*>#]|\d+[.)])(\s|$)") + + +def test_display_math_that_needs_a_blank_line_has_one(): + """`$$` opening on the line after a paragraph is a lazy continuation. + + For one-line arithmetic that renders anyway, which is why the corpus has + examples of it. It breaks when a line INSIDE the block is something + markdown wants: `- \\int ...` opens a list, the paragraph ends, the math + never closes, and the page shows raw LaTeX with half the equation in a + bullet. `myst build` reports nothing at all -- no warning, exit 0 -- so + this test is the only guard. It cost the boundary-conditions note a + published-looking draft once. + """ + for path in sorted((ROOT / "articles").glob("*/*.md")): + lines = path.read_text(encoding="utf-8").split("\n") + in_code = False + opened = None + for i, line in enumerate(lines): + if line.startswith("```"): + in_code = not in_code + continue + if in_code: + continue + if opened is None: + if line.rstrip() == "$$": + opened = i + elif line.startswith("$$"): + body = lines[opened + 1:i] + risky = [b for b in body if _MARKDOWN_LINE.match(b)] + if risky and opened > 0 and lines[opened - 1].strip(): + raise AssertionError( + "%s line %d: display math opens straight after text and " + "contains %r, which markdown will take for itself. Put a " + "blank line before the `$$`." + % (path.name, opened + 1, risky[0].strip()[:40])) + opened = None From 727c6d05e305553112ccaee2994a66350f8a2d9c Mon Sep 17 00:00:00 2001 From: lmoresi Date: Sun, 23 Aug 2026 09:10:55 +1000 Subject: [PATCH 31/33] Give the PDF viewer the window's height, and test the knockout on the matrix MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit The reader page showed a letterbox: at a 900px window the viewer box got 650px and the inside it stopped at 512px, its own min-height. A percentage height needs a definite container, and a flex item with `flex: 1 1 auto` inside a `min-height: 100vh` column is not one, so `height: 100%` was dropped and the min-height was all that was left -- unchanged however tall the monitor. With `#view=FitH` scaling the page to the full window WIDTH, that left a few lines of a page in a slot a third of its height. `height: 100vh` on the body makes the container definite, `min-height: 0` lets the flex item hand its height back, and `#view=Fit` fits a page to the box rather than the window width. Measured: 512 -> 650px at 1280x900, 512 -> 1350px at 1440x1600. rotated_dof_knockout.py answers the question the rotated run cannot: agreeing answers do not show WHAT was eliminated. It reads the solver's converged Cartesian reaction A.u - b at the constrained nodes and asks which way it points. | frame | ||Q^tQ - I|| | reaction . t | reaction . x | reaction . y | | 0° | 0.0e+00 | 2.2e-11 | 0.000 | 1.000 | | 45° | 8.1e-17 | 1.9e-11 | 0.707 | 0.707 | No tangential component at either angle -- the struck row is the wall-normal momentum balance, so the constraint force can only be normal -- and at 45° the direction is an even mixture of the coordinate unknowns rather than one of them. Note that Q is NOT the identity on the aligned box: it reorders the local axes so the normal comes first. The direction eliminated is the discriminator, not Q's distance from I. Underworld development team with AI support from Claude Code --- .../examples/rotated_dof_knockout.py | 129 ++++++++++++++++++ scripts/build_reader_pages.py | 16 ++- 2 files changed, 140 insertions(+), 5 deletions(-) create mode 100644 articles/boundary-conditions-on-non-planar-boundaries/examples/rotated_dof_knockout.py diff --git a/articles/boundary-conditions-on-non-planar-boundaries/examples/rotated_dof_knockout.py b/articles/boundary-conditions-on-non-planar-boundaries/examples/rotated_dof_knockout.py new file mode 100644 index 0000000..bed0900 --- /dev/null +++ b/articles/boundary-conditions-on-non-planar-boundaries/examples/rotated_dof_knockout.py @@ -0,0 +1,129 @@ +"""What the rotated constraint does to the MATRIX, not to the answer. + +On an axis-aligned box the per-node rotation Q is the identity, so the rotated +treatment and a component Dirichlet condition are the same arithmetic: agreeing +answers prove nothing about the machinery. Turning the frame makes Q a real +rotation, but the answers still agree -- correctly, since a rigid rotation is an +orthogonal similarity of the discrete problem. Neither run inspects what was +actually eliminated. + +This does. Two statements about the assembled system: + + * Q is orthogonal, and at 45 degrees it is nowhere near the identity; + * the converged Cartesian reaction r = A.u - b at a constrained node is + PARALLEL TO THE WALL NORMAL. That is the signature of the knockout: the row + that was struck is the wall-normal momentum balance, so the force the + constraint had to supply has no tangential part. If the elimination were + done in the coordinate frame -- the failure a rotated run cannot otherwise + distinguish -- the reaction would lie along a coordinate axis instead. + +The reaction is the solver's own `A.u - b`, stashed at convergence +(`_rotated_freeslip_info["reaction"]`); it is what `boundary_normal_traction` +reads the normal component of. + + python3 rotated_dof_knockout.py + +Run against underworld3 `bugfix/multiplier-traction` (PR #617). +""" +import sys + +import numpy as np + +import underworld3 as uw + +from underworld3.utilities import rotated_bc as RB + +import solcx_rotated as RT + +RES = 16 +CORNER = 1.0e-9 + + +def reaction_vectors(stokes, theta, boundary="Top"): + """The Cartesian nodal reaction at each node of `boundary`, corners dropped. + + Reads the same stashed residual `boundary_normal_traction` uses, but keeps + both components instead of projecting onto the normal. + """ + info = stokes._rotated_freeslip_info + dm = stokes.dm + dim = stokes.mesh.dim + rcl = dm.getLocalVec() + dm.globalToLocal(info["reaction"], rcl) + rca = np.asarray(rcl.getArray()) + lsec = dm.getLocalSection() + csec = dm.getCoordinateSection() + cvec = np.asarray(dm.getCoordinatesLocal().array).reshape(-1, dim) + v0, v1 = dm.getDepthStratum(0) + + coords, vectors = [], [] + for q, _nrm in RB._boundary_velocity_nodes(stokes, boundary, normal=None): + lo = lsec.getFieldOffset(q, RB._VELOCITY_FIELD) + x = RB._point_coord(dm, dim, cvec, csec, v0, v1, q) + coords.append(x) + vectors.append(np.array(rca[lo:lo + dim])) + dm.restoreLocalVec(rcl) + coords, vectors = np.array(coords), np.array(vectors) + + # A corner belongs to two walls and carries both reactions, so its direction + # is a sum of two normals and says nothing about either wall. + material = coords @ RT.rotation(theta) + keep = (material[:, 0] > CORNER) & (material[:, 0] < 1.0 - CORNER) + return coords[keep], vectors[keep] + + +def q_is_orthogonal(stokes): + """||Q^T Q x - x|| / ||x||, and ||Q x - x|| / ||x||. + + The second is reported by the script but is NOT the discriminator it looks + like: even on the aligned box Q reorders the local axes so the wall normal + comes first, so it is never the identity. What separates the two frames is + the DIRECTION the constraint eliminates, which the reaction shows. + """ + info = stokes._rotated_freeslip_info + Q, Qt = info["Q"], info["Qt"] + x = Q.createVecRight() + x.setRandom() + y = Q.createVecLeft() + Q.mult(x, y) + z = x.duplicate() + Qt.mult(y, z) + z.axpy(-1.0, x) + orthogonality = z.norm() / x.norm() + # ||Q x - x||: zero when Q is the identity, O(1) when it turns the frame + w = y.copy() + w.axpy(-1.0, x) + return orthogonality, w.norm() / x.norm() + + +def check(theta_degrees): + theta = np.radians(theta_degrees) + mesh, stokes, v, truth = RT.build("rotated", res=RES, theta=theta) + stokes.solve() + n = RT.rotation(theta) @ np.array([0.0, 1.0]) + t = np.array([-n[1], n[0]]) + + coords, r = reaction_vectors(stokes, theta) + magnitude = np.linalg.norm(r, axis=1) + live = magnitude > 1.0e-30 + tangential = np.abs(r[live] @ t) / magnitude[live] + along_y = np.abs(r[live] @ np.array([0.0, 1.0])) / magnitude[live] + along_x = np.abs(r[live] @ np.array([1.0, 0.0])) / magnitude[live] + orthogonality, _ = q_is_orthogonal(stokes) + return dict(nodes=int(live.sum()), tangential=float(tangential.max()), + along_x=float(along_x.mean()), along_y=float(along_y.mean()), + orthogonality=float(orthogonality)) + + +if __name__ == "__main__": + print("Rotated free slip on every wall, %d x %d. The reaction is the solver's" + % (RES, RES)) + print("own A.u - b at convergence.") + print() + print("| frame | nodes | ||Q^tQ - I|| | reaction . t | reaction . x | reaction . y |") + print("|---|---|---|---|---|---|") + for degrees in (0.0, 45.0): + got = check(degrees) + print("| %.0f° | %d | %.1e | %.1e | %.3f | %.3f |" + % (degrees, got["nodes"], got["orthogonality"], got["tangential"], + got["along_x"], got["along_y"]), flush=True) diff --git a/scripts/build_reader_pages.py b/scripts/build_reader_pages.py index bd3d771..b6dfa75 100644 --- a/scripts/build_reader_pages.py +++ b/scripts/build_reader_pages.py @@ -49,9 +49,13 @@ } } * { box-sizing: border-box; } +/* height, not just min-height: a percentage height inside a flex column needs + a DEFINITE container, and without it the viewer below fell back to its + min-height (512px) however tall the window was -- a letterbox showing a + few lines of a page scaled to the full window width. */ body { margin: 0; background: var(--uwtn-paper); color: var(--uwtn-ink); font-family: var(--uwtn-sans); font-size: 16px; line-height: 1.5; - display: flex; flex-direction: column; min-height: 100vh; } + display: flex; flex-direction: column; height: 100vh; min-height: 100vh; } header { border-bottom: 1px solid var(--uwtn-rule); padding: 1.1rem 1.4rem; } .wrap { max-width: 68rem; margin: 0 auto; width: 100%; } .kicker { font-size: .78rem; letter-spacing: .09em; text-transform: uppercase; @@ -66,9 +70,11 @@ .actions a:hover { border-color: var(--uwtn-accent); color: var(--uwtn-accent); } .actions a.primary { background: var(--uwtn-accent); border-color: var(--uwtn-accent); color: #fff; } -.reader { flex: 1 1 auto; min-height: 32rem; } -.reader object, .reader iframe { display: block; width: 100%; height: 100%; - min-height: 32rem; border: 0; } +/* min-height: 0 lets a flex item shrink below its content; without it the + viewer cannot give the window's height back and the page scrolls instead. */ +.reader { flex: 1 1 auto; min-height: 0; display: flex; } +.reader object, .reader iframe { flex: 1 1 auto; display: block; width: 100%; + height: 100%; min-height: 20rem; border: 0; } .fallback { padding: 2rem 1.4rem; color: var(--uwtn-muted); } """ @@ -94,7 +100,7 @@
- +

This browser will not display a PDF here — most phones will not.

Open the PDF or From 6ef6740e781d8d76f93184d63a8d842f7f3de9ed Mon Sep 17 00:00:00 2001 From: lmoresi Date: Sun, 23 Aug 2026 09:33:50 +1000 Subject: [PATCH 32/33] Write display maths as a `math` directive from here on Both forms build, but `$$` has two traps that the build does not report, and the fenced directive has neither. Typora is not a MyST editor: it rewrites a `$$ ... $$ (label)` block and moves the label onto a line of its own, which kills the label and every {eq} reference to it. And a `$$` opening on the line after a paragraph is a lazy continuation, so a line inside it beginning `-` is taken as a list and the equation publishes as raw LaTeX. Both cost this note a broken build and a broken page in one week. Verified through the Typst path, not just asserted: a `{math}` directive with `:label:` numbers as (1), `{eq}` resolves to it, and a block whose second line begins `- \int` renders correctly. CONTRIBUTING.md carries the convention with the `$$` rules for existing notes; the article template now uses the directive, so a new note starts with it. The current articles are left alone. Underworld development team with AI support from Claude Code --- CONTRIBUTING.md | 21 +++++++++++++++++++++ templates/article-template/ARTICLE.md | 14 ++++++++++---- 2 files changed, 31 insertions(+), 4 deletions(-) diff --git a/CONTRIBUTING.md b/CONTRIBUTING.md index 1465a59..fb9d833 100644 --- a/CONTRIBUTING.md +++ b/CONTRIBUTING.md @@ -167,6 +167,27 @@ something works, not how hard it was to make work. repaired afterwards. - **Say which quantity, on which mesh, against what.** "Converges" and "agrees well" are not measurements. +- **Write display maths as a `math` directive, not as `$$`.** Both build, but + the fenced form survives an editor that is not a MyST editor — Typora, which + several of these notes are written in, rewrites a `$$ ... $$ (label)` block + and moves the label onto a line of its own, which silently breaks it. A fence + is left alone. + + ```` + ```{math} + :label: eq-free-slip + \mathbf{u}\cdot\hat{\mathbf{n}} = 0 + ``` + ```` + + With `$$`, two things are then load-bearing and neither is reported by the + build. The label belongs on the CLOSING delimiter — `$$ (eq-free-slip)` — and + on its own line it renders as literal text while every `{eq}` reference to it + fails the Typst build. And a `$$` block that opens on the line straight after + a paragraph is a lazy continuation: fine for one-line arithmetic, but a line + inside it beginning `-`, `*`, `>` or `#` is claimed by markdown, the maths + never closes, and the page publishes raw LaTeX with half an equation in a + bullet. `pixi run test-unit` guards both. ## Things that are load-bearing diff --git a/templates/article-template/ARTICLE.md b/templates/article-template/ARTICLE.md index 000571b..3865815 100644 --- a/templates/article-template/ARTICLE.md +++ b/templates/article-template/ARTICLE.md @@ -26,12 +26,18 @@ but not this corner of Underworld. ## A section -Maths is plain LaTeX, inline as $\nabla \cdot \boldsymbol{\sigma} = 0$ or -displayed: +Maths is plain LaTeX, inline as $\nabla \cdot \boldsymbol{\sigma} = 0$. Display +maths goes in a `math` directive rather than a `$$` block: both build, but the +fence survives an editor that is not a MyST editor, and it takes a label +without the two traps `$$` has (see CONTRIBUTING.md). -$$ +```{math} +:label: eq-momentum -\nabla \cdot \boldsymbol{\sigma}(u, \nabla u) - \mathbf{f}(u, \nabla u) = 0 -$$ +``` + +Refer to it as {eq}`eq-momentum`. Drop the `:label:` line if nothing refers to +the equation. Code blocks are fenced and language-tagged. Keep lines under ~84 characters: that is the 95th percentile of this corpus and it is what fits the PDF measure From e463ee0cafe87428531b517cdc356bdfba3ed995 Mon Sep 17 00:00:00 2001 From: Louis Moresi Date: Sun, 23 Aug 2026 12:45:28 +1000 Subject: [PATCH 33/33] Change status to published and update publication date --- .../boundary-conditions-on-non-planar-boundaries/metadata.yml | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/articles/boundary-conditions-on-non-planar-boundaries/metadata.yml b/articles/boundary-conditions-on-non-planar-boundaries/metadata.yml index 56bc9e5..a745314 100644 --- a/articles/boundary-conditions-on-non-planar-boundaries/metadata.yml +++ b/articles/boundary-conditions-on-non-planar-boundaries/metadata.yml @@ -6,12 +6,12 @@ id: UWTN 2026-016 slug: boundary-conditions-on-non-planar-boundaries title: Boundary conditions on non-planar boundaries article_type: technical-note -status: review +status: published authors: - name: Louis Moresi orcid: 0000-0003-3685-174X affiliation: Australian National University -publication_date: null +publication_date: 2026-08-23 version: 1.0.0 # The deposit writes archive_doi and repository_record_id when the note is # published; leave them out until then. `doi` and `doi_registrant` were here

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28/33] Stop the theme's prose margin showing inside the banner frame The theme gives every image outside a

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