Couple URIS valve to tissue mechanics

[aabrown100-git]

Use Case

This feature would improve the physiological fidelity of cardiac FSI simulations that use unfitted resistive immersed surfaces (URIS) to model the cardiac valves.

Problem

In cardiac FSI simulations that use URIS valves, the force exerted by the blood on the valves is not transferred to the surrounding tissue. This results in a net force by the blood on the tissue, which significantly alters tissue deformation. This is unphysiological, since in reality, closed valves are mechanically supported by the surrounding cardiac tissue, as well as the papillary muscles, and the net force on the tissue is close to zero.

See https://onlinelibrary-wiley-com.stanford.idm.oclc.org/doi/full/10.1002/cnm.70119 for a more complete discussion of this problem.

Solution

Implement the solution proposed in https://onlinelibrary-wiley-com.stanford.idm.oclc.org/doi/full/10.1002/cnm.70119. Essentially, we "add distributed forces acting on the structure to model the attachment of valves to the cardiac walls". This amounts to computing the net force on a URIS valve, then distributing it over a particular region of the tissue domain. Clear equations are provided in the paper. Moreover, they propose an explicit treatment of this extra forcing term and do not observe stability issues. This should make it more straightforward to implement in svMultiPhysics.

Alternatives considered

None. I am not aware of any other proposed solutions to this problem.

Additional context

No response

Code of Conduct

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[aabrown100-git]

@michelebucelli if you are available, we would obviously love your input and advice on implementing this!

[michelebucelli]

The main thing I found to be important and I'd like to reinforce is that the coupling should be treated explicitly (as you already say in the description).

I did try to make it work implicitly too, updating the valve-to-structure forces within the Newton iterations, and I had computed some of the derivative of valve-to-structure force terms to be included in the Jacobian. In simple examples (e.g. the sphere test in the paper), this didn't make much of a difference in terms of the solution, while requiring more Newton iterations for convergence. In more complicated examples, this prevented the Newton solver from converging altogether. I figure that this is because of the significant nonlinearity introduced by the coupling (which also introduces a non-local dependence between solution variables).

Additionally, explicit coupling should make the implementation relatively straightforward.

On the other hand, it is worth noting that using explicit coupling will introduce a time discretization error of order $O(\Delta t)$. I had considered using some higher order extrapolation of the valve-to-structure forces, but initial experiments did not work as expected (the accuracy order seemed to remain the same). If that's something you may need, we can discuss this aspect further.

[aabrown100-git]

@michelebucelli thanks so much for this input, very helpful! I don't think the time discretization error is a big deal in our current expected use cases.

I plan on attempting this next month after finishing some current projects.