The quantities on the new grid are the integrals of those on the old grid.

[DSTxj]

"""
g = Function(U)
new_mesh = Mesh(g)
"""
Hello, I'm here again to ask a question. It's about the issue from the previous grid replication. For a two-dimensional grid, when I created a new grid in the form of grid replication and moved the coordinates of the new grid, I then calculated the unit exterior normal vector n_vec1 = FacetNormal(new_mesh) on the new grid, as well as the area area_int2 = FacetArea(mesh2) of each face. I would like to ask if I can perform the inner product of these two quantities of the new grid on the old grid? For example, inner(n_vec1 *area_int2 )*dx(old_mesh) Because I found that these two expressions cannot be interpolated onto the old grid.

[dham]

This isn't going to work for the simple reason that interpolating facet quantities never works. Interpolation basically depends on the ability to evaluate the source quantity at any point in space. Facet quantities are not defined on the interior of the cells, so the point evaluation fails.

If you can provide the maths of what you are ultimately attempting to compute, there may be an alternative formulation that is actually computable. We may also be able to do something in this case based on the fact that the quantities share a topology.

[DSTxj]

"inner(Hn_newarea_new,T)*dS(interface_id)"

[DSTxj]

[dham]

I don't understand how this even makes mathematical sense. The normals on the old mesh aren't even defined at the same points in space as those on the new mesh. Rather than just quoting code, can you provide a minimal mathematical example of what you are trying to express?

[DSTxj]

I plan to calculate the moment of inertia A as given in the figure using the external unit normal vector and the area of the interface element, and then use the trapezoidal integration formula to calculate the time-weighted normal vector in equation (3.12). It is easy to see that this results in the need for two time layers to correspond to the unit normal vectors of the grid.

[dham]

OK, I think you won't get this to work via the route you are currently going down because of the way that Firedrake abstracts the integrals: this approach basically requires quantities defined on different meshes to appear in the same integral and we can't do that.

What I think would work, but will be rather involved as it's right at the experimental edge of what Firedrake can do, is to use spacetime finite element on the problem. What you would do is extrude your 2D mesh to form 3D wedges, with the third dimension representing time. You would then solve/set the "top" coordinate values to the new mesh values, and solve for the new state variables over the space-time slab. This approach is, for example, used in https://doi.org/10.1137/25M1756673. The time weighted normals will then just naturally be the normals to the side facets of the space-time mesh.

The implementation approach used in that paper was slightly different and I'm not sure it will do what you want. The extruded mesh approach was pursued by a masters student last year and unfortunately his code didn't get merged, but it is availabe here: https://github.com/RexZhann/m4r_project/tree/main/fest

[DSTxj]

I plan to calculate the moment of inertia A as given in the figure using the external unit normal vector and the area of the interface element, and then use the trapezoidal integration formula to calculate the time-weighted normal vector in equation (3.12). It is easy to see that this results in the need for two time layers to correspond to the unit normal vectors of the grid.