Custom normal in weak form

General
1

Hello,

I am currently doing unfitted methods. In order to guarantee stability for higher order finite elements, the following normal gradient stabilization is proposed

s(w, v) = c h (nh · ∇w, nh · ∇v)Th ,

to be added to the standard weak form, where \mathcal{T}_h and where the normal is the one to the surface of the level set. is the set of all cells cut by the geometry. I am currently not able to implement this, in the sense that it is not clear to me how to pass a custom normal to a weak form in FEniCSx. Of course, simply doing something like

dx_uncut = ufl.dx(subdomain_data=celltags, domain=mesh)
higher_order_stab = beta * h * inner(dot(n, grad(u)), dot(n, grad(v))) * dx_uncut(cut_cell_tag)

gives an error.

2

You would have to define the custom normal as a function in some function space.

As the integral is over a cell, it boils down to evaluating a function at some given points in the cell (quadrature points). How are you planning to Express the normal as a function of the level set function?

3

Hello @dokken,

Yes I will compute the normal with the gradient of the level set. So you would say I have to define the normal as a dolfinx.fem.Function and then use this function in the weak form?

4

If you can simply define it as the gradient of the level set, you can use grad(phi) directly in the variational form.