Phase F: predictor-corrector with radial return — first cut

Implements the predictor-corrector architecture from the web-advice doc
(vep_stress_update_full_latex.md §15 Stage 4) on isotropic VEP. Setup:

* Same harmonic shear box, RES=32
* Isotropic Maxwell rheology (no director, no rank-4 tensor)
* σ_y(x) from the same fault influence-function used in TI tests —
  small (0.05) in a 0.06-wide layer at θ=15°, large (200) in the bulk
  → partial yielding, no global yield catastrophe
* Per-step trace files written EVERY step with flush, persist on kill

Configurations tested:
1. BDF-1 yield-in-residual (softmin in model) — baseline, works
2. ETD with yield_stress=∞ in model + post-solve J2 radial return on
   psi_star (predictor-corrector single-shot)
3. ETD with yield_stress=∞ + radial return + outer Picard with σ
   damping (ω_τ=0.5, max 6 iters)

Both 2 and 3 leave η_eff fixed at η_VE between Picard iters — i.e.
single Stokes solver per Picard step, just with iterated psi_star.

Killer-test results (θ=+15° fault layer, τ_y_fault=0.05, 1.5T):

| case                        | |σ_eq|_max end | |u_y|_max end |
|-----------------------------|----------------|---------------|
| BDF-1 baseline              | 0.81           | 0.036         |
| ETD-1 + RR single-shot      | 0.42           | 0.021         |
| ETD-1 + RR + Picard         | 0.41           | 0.021         |
| ETD-2 + RR single-shot      | 0.41           | 0.286         |
| ETD-2 + RR + Picard         | runaway @ 60   | runaway       |

Same first-order vs higher-order signature as before:
* ETD-1 + radial return is essentially identical to BDF-1, both
  single-shot and Picard-equilibrated. The first-order numerical
  dissipation (lesson #13) carries over to the predictor-corrector
  architecture.
* ETD-2 + radial return drifts in the single-shot case (|u_y| 8× BDF
  by end) and the outer Picard with σ-damping doesn't damp it — at
  step 60 the case exits via the runaway guard with |σ|_eq = 18.2
  and |u_y| = 10.2. Picard amplifies rather than equilibrates because
  the σ-damping doesn't address the underlying lack of dissipation in
  the linear-relaxation factor.

The single Stokes per Picard iter (with η_eff fixed at η_VE) is
NOT the full architecture from the web advice. The advice's Step 4
explicitly calls for *updating η_eff between Picard iters based on
σ_corrected*, which would require either (a) per-quad-point α,φ in
the constitutive model (we ruled out earlier — codegen explosion)
or (b) a second Stokes solve with viscosity-as-plastic-multiplier
treating σ_VE as fixed/explicit (the user's "two-Stokes" idea —
not implemented here).

Note: ETD-1/ETD-2 yield-in-residual baselines on this isotropic
setup hit a separate parent-class issue with spatial yield_stress
+ softmin (SNES line-search divergence). Filed as a follow-up;
sidestepped here by using yield_stress=∞ with external return mapping.

Per-step trace files (`_phase_f_<label>.trace.txt`) tracked in git
so the comparison plot regenerates from a fresh clone.

Underworld development team with AI support from Claude Code (https://claude.com/claude-code)

Author: lmoresi

docs/developer/design/_phase_f_bdf1_iso.trace.txt

@@ -0,0 +1,123 @@
+# Phase F predictor-corrector trace: bdf1_iso
+# integrator='bdf' order=1 apply_radial_return=False in_residual_yield=True
+# columns: step, t, V_top, snes_iters_total, picard_iters, sigma_eq_max, sigma_eq_max_after_correction, u_y_max, yielded_fraction
+   1  0.0500 +0.4985   1  1 4.127208e-02 4.127208e-02 9.910768e-06 0.000000
+   2  0.1000 +0.4938   1  1 8.038114e-02 8.038114e-02 1.736993e-04 0.046832
+   3  0.1500 +0.4862  28  1 1.506698e-01 1.506698e-01 9.614702e-03 0.059688
+   4  0.2000 +0.4755  23  1 2.390412e-01 2.390412e-01 1.853175e-02 0.067034
+   5  0.2500 +0.4619  20  1 3.495552e-01 3.495552e-01 2.103261e-02 0.074380
+   6  0.3000 +0.4455  15  1 4.846955e-01 4.846955e-01 2.298590e-02 0.074380
+   7  0.3500 +0.4263  15  1 6.096305e-01 6.096305e-01 2.423472e-02 0.074380
+   8  0.4000 +0.4045  14  1 7.198707e-01 7.198707e-01 2.396950e-02 0.076217
+   9  0.4500 +0.3802  14  1 8.120357e-01 8.120357e-01 2.274448e-02 0.079890
+  10  0.5000 +0.3536  14  1 8.836892e-01 8.836892e-01 2.156902e-02 0.081726
+  11  0.5500 +0.3247  13  1 9.348462e-01 9.348462e-01 2.019526e-02 0.081726
+  12  0.6000 +0.2939  14  1 9.672433e-01 9.672433e-01 1.857488e-02 0.081726
+  13  0.6500 +0.2612  16  1 9.830292e-01 9.830292e-01 1.680594e-02 0.081726
+  14  0.7000 +0.2270  17  1 9.840694e-01 9.840694e-01 1.495069e-02 0.081726
+  15  0.7500 +0.1913  19  1 9.717716e-01 9.717716e-01 1.282147e-02 0.081726
+  16  0.8000 +0.1545  21  1 9.473505e-01 9.473505e-01 1.052256e-02 0.081726
+  17  0.8500 +0.1167  25  1 9.120627e-01 9.120627e-01 9.265141e-03 0.081726
+  18  0.9000 +0.0782  30  1 8.675153e-01 8.675153e-01 8.563905e-03 0.081726
+  19  0.9500 +0.0392  34  1 8.158198e-01 8.158198e-01 7.559992e-03 0.081726
+  20  1.0000 -0.0000  19  1 7.630093e-01 7.630093e-01 6.323271e-03 0.074380
+  21  1.0500 -0.0392   8  1 7.288530e-01 7.288530e-01 5.154964e-03 0.074380
+  22  1.1000 -0.0782   7  1 6.886855e-01 6.886855e-01 4.095200e-03 0.072544
+  23  1.1500 -0.1167   5  1 6.462222e-01 6.462222e-01 3.139607e-03 0.054178
+  24  1.2000 -0.1545   4  1 6.024623e-01 6.024623e-01 2.466041e-03 0.009183
+  25  1.2500 -0.1913   3  1 5.579514e-01 5.579514e-01 2.282205e-03 0.000000
+  26  1.3000 -0.2270   3  1 5.130849e-01 5.130849e-01 2.190943e-03 0.000000
+  27  1.3500 -0.2612   2  1 4.667991e-01 4.667991e-01 2.080320e-03 0.000918
+  28  1.4000 -0.2939   1  1 4.197769e-01 4.197769e-01 1.930754e-03 0.021120
+  29  1.4500 -0.3247   2  1 3.722971e-01 3.722971e-01 1.743052e-03 0.033058
+  30  1.5000 -0.3536  12  1 3.225307e-01 3.225307e-01 3.181995e-03 0.048669
+  31  1.5500 -0.3802  25  1 2.627879e-01 2.627879e-01 9.008150e-03 0.056015
+  32  1.6000 -0.4045  22  1 2.470817e-01 2.470817e-01 1.384321e-02 0.056015
+  33  1.6500 -0.4263  23  1 2.617071e-01 2.617071e-01 1.795609e-02 0.063361
+  34  1.7000 -0.4455  20  1 3.171074e-01 3.171074e-01 2.084389e-02 0.066116
+  35  1.7500 -0.4619  16  1 3.720918e-01 3.720918e-01 2.375286e-02 0.072544
+  36  1.8000 -0.4755  15  1 4.180842e-01 4.180842e-01 2.593457e-02 0.074380
+  37  1.8500 -0.4862  14  1 5.144363e-01 5.144363e-01 2.754916e-02 0.074380
+  38  1.9000 -0.4938  13  1 6.484323e-01 6.484323e-01 3.122790e-02 0.077135
+  39  1.9500 -0.4985  12  1 7.692478e-01 7.692478e-01 3.421500e-02 0.079890
+  40  2.0000 -0.5000  11  1 8.776854e-01 8.776854e-01 3.577072e-02 0.081726
+  41  2.0500 -0.4985  10  1 9.753243e-01 9.753243e-01 3.613523e-02 0.082645
+  42  2.1000 -0.4938   9  1 1.063180e+00 1.063180e+00 3.558671e-02 0.083563
+  43  2.1500 -0.4862   8  1 1.141797e+00 1.141797e+00 3.439847e-02 0.084481
+  44  2.2000 -0.4755   8  1 1.211360e+00 1.211360e+00 3.344108e-02 0.084481
+  45  2.2500 -0.4619   8  1 1.278016e+00 1.278016e+00 3.406503e-02 0.086318
+  46  2.3000 -0.4455   8  1 1.339710e+00 1.339710e+00 3.373125e-02 0.086318
+  47  2.3500 -0.4263   8  1 1.389034e+00 1.389034e+00 3.234658e-02 0.088154
+  48  2.4000 -0.4045   8  1 1.425182e+00 1.425182e+00 3.025302e-02 0.089073
+  49  2.4500 -0.3802  10  1 1.447622e+00 1.447622e+00 2.745848e-02 0.089073
+  50  2.5000 -0.3536  11  1 1.456493e+00 1.456493e+00 2.508182e-02 0.090909
+  51  2.5500 -0.3247  12  1 1.452144e+00 1.452144e+00 2.321883e-02 0.090909
+  52  2.6000 -0.2939  13  1 1.435099e+00 1.435099e+00 2.117630e-02 0.089073
+  53  2.6500 -0.2612  14  1 1.408897e+00 1.408897e+00 1.892825e-02 0.089073
+  54  2.7000 -0.2270  15  1 1.382260e+00 1.382260e+00 1.693459e-02 0.088154
+  55  2.7500 -0.1913  17  1 1.375198e+00 1.375198e+00 1.496980e-02 0.089073
+  56  2.8000 -0.1545  20  1 1.361218e+00 1.361218e+00 1.361402e-02 0.089073
+  57  2.8500 -0.1167  23  1 1.338628e+00 1.338628e+00 1.213324e-02 0.088154
+  58  2.9000 -0.0782  27  1 1.307413e+00 1.307413e+00 1.046934e-02 0.086318
+  59  2.9500 -0.0392  33  1 1.267677e+00 1.267677e+00 8.977656e-03 0.082645
+  60  3.0000 -0.0000  24  1 1.220795e+00 1.220795e+00 7.933386e-03 0.081726
+  61  3.0500 +0.0392  10  1 1.171966e+00 1.171966e+00 6.959463e-03 0.080808
+  62  3.1000 +0.0782   7  1 1.112697e+00 1.112697e+00 5.897244e-03 0.071625
+  63  3.1500 +0.1167   5  1 1.047772e+00 1.047772e+00 4.924303e-03 0.056015
+  64  3.2000 +0.1545   5  1 9.822068e-01 9.822068e-01 4.021409e-03 0.016529
+  65  3.2500 +0.1913   4  1 9.166450e-01 9.166450e-01 3.436732e-03 0.001837
+  66  3.3000 +0.2270   3  1 8.520661e-01 8.520661e-01 3.068158e-03 0.000000
+  67  3.3500 +0.2612   2  1 7.883038e-01 7.883038e-01 2.865541e-03 0.001837
+  68  3.4000 +0.2939   2  1 7.247583e-01 7.247583e-01 2.728022e-03 0.019284
+  69  3.4500 +0.3247   2  1 6.614330e-01 6.614330e-01 2.528061e-03 0.032140
+  70  3.5000 +0.3536  11  1 5.970971e-01 5.970971e-01 3.228570e-03 0.048669
+  71  3.5500 +0.3802  25  1 5.245090e-01 5.245090e-01 8.312550e-03 0.056015
+  72  3.6000 +0.4045  23  1 4.415680e-01 4.415680e-01 1.276039e-02 0.056015
+  73  3.6500 +0.4263  23  1 3.740465e-01 3.740465e-01 1.660880e-02 0.062443
+  74  3.7000 +0.4455  19  1 3.769107e-01 3.769107e-01 1.956283e-02 0.065197
+  75  3.7500 +0.4619  16  1 3.849742e-01 3.849742e-01 2.212255e-02 0.069789
+  76  3.8000 +0.4755  15  1 4.314492e-01 4.314492e-01 2.585272e-02 0.073462
+  77  3.8500 +0.4862  13  1 4.569590e-01 4.569590e-01 2.882846e-02 0.074380
+  78  3.9000 +0.4938  13  1 5.536645e-01 5.536645e-01 3.087855e-02 0.076217
+  79  3.9500 +0.4985  12  1 6.987001e-01 6.987001e-01 3.270765e-02 0.076217
+  80  4.0000 +0.5000  11  1 8.300400e-01 8.300400e-01 3.425436e-02 0.078972
+  81  4.0500 +0.4985  10  1 9.481211e-01 9.481211e-01 3.628425e-02 0.081726
+  82  4.1000 +0.4938   9  1 1.052744e+00 1.052744e+00 3.772184e-02 0.082645
+  83  4.1500 +0.4862   9  1 1.143134e+00 1.143134e+00 3.843999e-02 0.082645
+  84  4.2000 +0.4755   8  1 1.218388e+00 1.218388e+00 3.838852e-02 0.083563
+  85  4.2500 +0.4619   8  1 1.277964e+00 1.277964e+00 3.765011e-02 0.085399
+  86  4.3000 +0.4455   8  1 1.322205e+00 1.322205e+00 3.624003e-02 0.086318
+  87  4.3500 +0.4263   8  1 1.352417e+00 1.352417e+00 3.423680e-02 0.087236
+  88  4.4000 +0.4045   9  1 1.370249e+00 1.370249e+00 3.178397e-02 0.087236
+  89  4.4500 +0.3802  10  1 1.378066e+00 1.378066e+00 2.894993e-02 0.087236
+  90  4.5000 +0.3536  11  1 1.378157e+00 1.378157e+00 2.593349e-02 0.087236
+  91  4.5500 +0.3247  12  1 1.371909e+00 1.371909e+00 2.340301e-02 0.088154
+  92  4.6000 +0.2939  13  1 1.359741e+00 1.359741e+00 2.156856e-02 0.089073
+  93  4.6500 +0.2612  14  1 1.342647e+00 1.342647e+00 1.961159e-02 0.089073
+  94  4.7000 +0.2270  16  1 1.344674e+00 1.344674e+00 1.743032e-02 0.089073
+  95  4.7500 +0.1913  18  1 1.336877e+00 1.336877e+00 1.502347e-02 0.087236
+  96  4.8000 +0.1545  20  1 1.320058e+00 1.320058e+00 1.392546e-02 0.087236
+  97  4.8500 +0.1167  23  1 1.294622e+00 1.294622e+00 1.297610e-02 0.085399
+  98  4.9000 +0.0782  28  1 1.260405e+00 1.260405e+00 1.166777e-02 0.083563
+  99  4.9500 +0.0392  33  1 1.217751e+00 1.217751e+00 1.010702e-02 0.081726
+ 100  5.0000 +0.0000  22  1 1.167237e+00 1.167237e+00 8.388419e-03 0.081726
+ 101  5.0500 -0.0392   9  1 1.111585e+00 1.111585e+00 7.391111e-03 0.081726
+ 102  5.1000 -0.0782   7  1 1.048879e+00 1.048879e+00 5.923332e-03 0.071625
+ 103  5.1500 -0.1167   5  1 9.869192e-01 9.869192e-01 4.702023e-03 0.055096
+ 104  5.2000 -0.1545   4  1 9.244225e-01 9.244225e-01 3.691150e-03 0.013774
+ 105  5.2500 -0.1913   3  1 8.618489e-01 8.618489e-01 3.237655e-03 0.000918
+ 106  5.3000 -0.2270   3  1 8.000484e-01 8.000484e-01 3.031333e-03 0.000000
+ 107  5.3500 -0.2612   2  1 7.381037e-01 7.381037e-01 2.863970e-03 0.001837
+ 108  5.4000 -0.2939   2  1 6.763055e-01 6.763055e-01 2.714261e-03 0.021120
+ 109  5.4500 -0.3247   2  1 6.147766e-01 6.147766e-01 2.540707e-03 0.033058
+ 110  5.5000 -0.3536  12  1 5.518377e-01 5.518377e-01 3.367459e-03 0.048669
+ 111  5.5500 -0.3802  25  1 4.792515e-01 4.792515e-01 8.877937e-03 0.056015
+ 112  5.6000 -0.4045  23  1 3.948926e-01 3.948926e-01 1.378900e-02 0.056933
+ 113  5.6500 -0.4263  23  1 3.537057e-01 3.537057e-01 1.782485e-02 0.063361
+ 114  5.7000 -0.4455  20  1 3.527766e-01 3.527766e-01 2.055281e-02 0.065197
+ 115  5.7500 -0.4619  16  1 3.959641e-01 3.959641e-01 2.324016e-02 0.068871
+ 116  5.8000 -0.4755  15  1 4.349951e-01 4.349951e-01 2.537614e-02 0.073462
+ 117  5.8500 -0.4862  14  1 4.724881e-01 4.724881e-01 2.728828e-02 0.074380
+ 118  5.9000 -0.4938  13  1 5.735626e-01 5.735626e-01 3.131495e-02 0.076217
+ 119  5.9500 -0.4985  12  1 6.955473e-01 6.955473e-01 3.435077e-02 0.077135
+ 120  6.0000 -0.5000  11  1 8.061232e-01 8.061232e-01 3.596882e-02 0.079890

docs/developer/design/_phase_f_etd1_pc1.trace.txt

@@ -0,0 +1,123 @@
+# Phase F predictor-corrector trace: etd1_pc1
+# integrator='etd' order=1 apply_radial_return=True in_residual_yield=False
+# columns: step, t, V_top, snes_iters_total, picard_iters, sigma_eq_max, sigma_eq_max_after_correction, u_y_max, yielded_fraction
+   1  0.0500 +0.4985   1  1 4.227008e-02 4.227008e-02 7.408887e-06 0.000000
+   2  0.1000 +0.4938   1  1 8.219118e-02 8.219118e-02 1.875342e-05 0.048669
+   3  0.1500 +0.4862   1  1 1.312133e-01 1.312133e-01 3.499493e-03 0.061524
+   4  0.2000 +0.4755   1  1 1.847484e-01 1.847484e-01 8.601088e-03 0.072544
+   5  0.2500 +0.4619   1  1 2.349787e-01 2.349787e-01 1.262241e-02 0.073462
+   6  0.3000 +0.4455   1  1 2.842488e-01 2.842488e-01 1.517815e-02 0.081726
+   7  0.3500 +0.4263   1  1 3.303034e-01 3.303034e-01 1.652152e-02 0.081726
+   8  0.4000 +0.4045   1  1 3.759653e-01 3.759653e-01 1.709990e-02 0.081726
+   9  0.4500 +0.3802   1  1 4.155843e-01 4.155843e-01 1.717741e-02 0.081726
+  10  0.5000 +0.3536   1  1 4.482948e-01 4.482948e-01 1.675827e-02 0.083563
+  11  0.5500 +0.3247   1  1 4.744306e-01 4.744306e-01 1.604408e-02 0.083563
+  12  0.6000 +0.2939   1  1 4.942698e-01 4.942698e-01 1.508513e-02 0.083563
+  13  0.6500 +0.2612   1  1 5.083255e-01 5.083255e-01 1.444807e-02 0.083563
+  14  0.7000 +0.2270   1  1 5.170730e-01 5.170730e-01 1.364595e-02 0.081726
+  15  0.7500 +0.1913   1  1 5.209158e-01 5.209158e-01 1.265520e-02 0.078972
+  16  0.8000 +0.1545   1  1 5.200673e-01 5.200673e-01 1.151499e-02 0.074380
+  17  0.8500 +0.1167   1  1 5.146865e-01 5.146865e-01 1.027186e-02 0.073462
+  18  0.9000 +0.0782   1  1 5.048434e-01 5.048434e-01 8.898937e-03 0.068871
+  19  0.9500 +0.0392   1  1 4.906089e-01 4.906089e-01 7.427466e-03 0.065197
+  20  1.0000 -0.0000   1  1 4.734378e-01 4.734378e-01 5.897057e-03 0.059688
+  21  1.0500 -0.0392   1  1 4.519180e-01 4.519180e-01 4.329136e-03 0.047750
+  22  1.1000 -0.0782   1  1 4.264055e-01 4.264055e-01 2.576242e-03 0.021120
+  23  1.1500 -0.1167   1  1 3.971031e-01 3.971031e-01 1.405211e-03 0.001837
+  24  1.2000 -0.1545   1  1 3.654972e-01 3.654972e-01 1.189141e-03 0.000000
+  25  1.2500 -0.1913   1  1 3.321918e-01 3.321918e-01 1.065001e-03 0.000000
+  26  1.3000 -0.2270   1  1 2.973628e-01 2.973628e-01 9.223411e-04 0.001837
+  27  1.3500 -0.2612   1  1 2.611811e-01 2.611811e-01 8.630974e-04 0.006428
+  28  1.4000 -0.2939   1  1 2.234569e-01 2.234569e-01 1.135203e-03 0.029385
+  29  1.4500 -0.3247   1  1 1.831775e-01 1.831775e-01 2.125745e-03 0.046832
+  30  1.5000 -0.3536   1  1 1.385713e-01 1.385713e-01 4.241876e-03 0.054178
+  31  1.5500 -0.3802   1  1 1.181696e-01 1.181696e-01 6.389454e-03 0.059688
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