Embedded + Autodiff

[JordiManyer]

So here are some notes on the implementation of Autodiff for embedded triangulations. A similar discussion on this can be found in here.

Overview of the problem:

We distinguish two cases:

1. When differentiating the boundary, triangulations change. I.e some interior cells become cut, some cuts cells become interior, etc...
2. The above does not happen, i.e all interior, cut and exterior cells remain the same. In this case, the domain is modified by moving around the cuts within their cut cell. I.e the number and shape of the sub-cells stays the same, but the area of the sub-cells is different.

Case 1:

• Since the amount/type of interior/exterior/cut cells changes, quadrature points are basically created and/or destroyed.
• Therefore, there is no 1-1 correspondence between quadrature points in the original and updated measure.
• This makes it (I think) impossible to propagate dual numbers.

Case 2:

• Quadratures in the sub-cells reference space are the same. Differentiability is given by a change in the geometric maps going from the sub-cell reference spaces to their parent cut cell.
• Duality can be embedded within those geometric maps, which are produced from the values of our level-set.

[JordiManyer]

Current state:

• We are currently only considering Case 2, for the above reasons. This is also what theoretical papers do (see here), which is very reasonable since otherwise things are not differentiable.
• We can ensure we always fall under Case 2 by avoiding φh(p) = 0 for all mesh nodes p. During level-set updates, we can always add artificial advection (order ε) in the direction of movement that ensure we do not move our level set right on top of a mesh node.

We have the code working for simplicial meshes. Moving to QUAD/HEX meshes, however, requires some modification of the cutting algorithm:

• Currently , QUADs are simplexified then cut using a marching tetrahedron algorithm. This creates bad cuts when the level set is not linear, because our FE space is locally Q1 (instead of P1, which is what the algorithm assumes).

• To remedy this, we need to implement a marching cubes algorithm. This would allow us to cut QUADs directly, instead of having to go through a mesh of simplices first. (TODO)