internal boundaries in with_boundaries

[gdmcbain]

Internal boundaries were queried in #336 and support added in #340. I wanted to check that these were preserved in serialization #158 #261 under #680, but it appeared whereas they were supported by the old Mesh.define_boundary, this has been deprecated in favour of the (functional rather than mutating) Mesh.with_boundaries which doesn't support them.

The problem is in

scikit-fem/skfem/mesh/mesh.py

Line 180 in 0855dae

**{name: self.facets_satisfying(test, True)

which hardcodes boundaries_only=True.

Minimal example:

m = MeshLine([0, 1, 2]).with_boundaries({"mid0": lambda x: x[0] == 1.0})
m.define_boundary("mid1", lambda x: x[0] == 1.0, boundaries_only=False)

Then m.boundaries is

{'mid0': array([], dtype=int64), 'mid1': array([1])}

[kinnala]

I'm sorry that we are are back in square one regarding the interfaces. Because the new Mesh class was quite a big refactoring, I probably deleted the one test we had with centreline interface to cut some corners.

However, can we take a critical look at the use of these interfaces. I'd like to fully understand the use cases. Now that we have probes, I think the one-dimensional example is not meaningful anymore? How about higher dimensional examples? Is the main use case to integrate over interfaces? Is there something else?

I propose that we use a separate Mesh.interfaces field, in addition to Mesh.subdomains and Mesh.boundaries, to make it clear that it is sets of internal facets. This is mainly because Mesh.boundaries would naturally connect to BoundaryFacetBasis and then Mesh.interfaces would connect to InteriorFacetBasis.

[gdmcbain]

I'd like to fully understand the use cases. Now that we have probes, I think the one-dimensional example is not meaningful anymore? How about higher dimensional examples? Is the main use case to integrate over interfaces? Is there something else?

One use case is a hypersurface source, like a point-source but (n – 1)-dimensional; e.g. the line-load on a Kirchhoff plate mentioned in the original issue. That shows up as a linear form so it's kind of an integral, yes.

That might be the main use, but I'm also wondering about multiphysics problems in which a subprogram might be solved on a subdomain for which an interface is really a boundary. There might be better ways to handle that though.

[kinnala]

I think InteriorFacetBasis can be used for creating mortar coupling between two parts of the mesh if you have an internal boundary.