Mortar methods generalizations and fixes

[kinnala]

This branch contains my work related to mortar methods.

[kinnala]

I'm still trying to figure out what changes are needed in order to support more general cases (such as contact problems where the boundaries do not initially match) in the future.

[gdmcbain]

Indeed... I guess just trying increasingly trickier examples might reveal what works and what else might be needed?

Is one still fairly simple example that does have a discretely nonmatching interface the insulted wire, with the wire and insulation meshed independently at different mesh-sizes so that the two sides of the interface are different polygonal approximations of the one geometric circle?

[kinnala]

Thanks for the idea of circular mortar boundary. I'll try to support it also.

[kinnala]

Now it's on a really ugly temporary state but I think the idea is now there. See the latest commit message for some spoilers. I'll continue on this next week.

[kinnala]

Fixes #215 .

[kinnala]

Instead of having explicit interface meshes, I've decided to use so-called mortar mappings that use similar interface meshes underneath. Then the syntax becomes quite nice since you only need to initialize two MortarBasis objects like this:

mb = [
    MortarBasis(m, e, mapping = mortar[0], intorder=2),
    MortarBasis(M, E, mapping = mortar[1], intorder=2)
]

where mortar[0] and mortar[1] correspond to the different sides of the interface. Here (m,e) and (M,E) are the two meshes and their elements.

[kinnala]

This is still in very raw state. I'll continue tomorrow. First I create a new example with "real" contact problem and then I tidy up the code.

[kinnala]

[kinnala]

von Mises stress when calculating with TriP2 and Quad2 elements.

[kinnala]

Remaining work:

• Separate MortarMapping code into a separate file and delete mesh_mortar.py
• Make creation of MortarMapping more robust to various edge cases
• Add tests

[kinnala]

Circular mortar boundary is not yet working/tested but it should be as simple as adding a new constructor to MortarPair with slightly different definition for the projection (functions proj and param).

[kinnala]

I'm still writing ex04.rst but it will take a while because even writing down the equations for contact problems takes some meditation.

[kinnala]

Here are the docs https://kinnala.github.io/scikit-fem-docs/examples/ex04.html

[gdmcbain]

I don't necessarily understand all of this, but it's impressive!

I might try to reproduce the solution for the insulated wire ex17 using this, then I might understand it better. The confusing part is how the bilinear and linear forms for the mortar matrices relate to the operators on the two meshes, probably because I'm less familiar with elasticity than heat transfer and fluid mechanics.

[kinnala]

I might try to reproduce the solution for the insulated wire ex17 using this

We would need to first add support for other projections than onto a predefined linear function.

In principle, it should be as easy as using different proj and param (local functions used in MortarPair.init_2D). In the case of insulated wire, you should probably do the projection via arctan of the angle of the point in polar coordinates.

In the literature, there are also ways to generalize this (such as closest point search) but I have not looked into it. I just know that it is possible.

My next goal is to create MortarPair.init_3D for mortaring 3D problems. It is somewhat more involved to create the integration mesh. My current plan is to use libsupermesh (https://bitbucket.org/libsupermesh/libsupermesh/src) and possibly create Python bindings for it. The idea is same: project points into d-1 dimensional domain, create integration mesh (or "supermesh", in MortarPair.init_2D it is simply one call to np.unique), generate d point quadrature rule in the integration mesh and map them into the global coordinate system to check that the orientation of the points matches over the global mortaring interface. If orientation does not match, perturb index columns in the auxiliary meshes so that they match.