No Normal Flow Boundary Condition

[hgpeterson]

Hello,

I'm wondering how one can implement a no-normal-flow boundary condition (u ⋅ n = 0) on a general mesh? It looks like the boundary condition was written out in the Darcy equation tutorial, but in the end, the lines

uD = VectorValue(0.0,0.0)
U = TrialFESpace(V,uD)

just set u = 0 on the boundary. For this square geometry, it would be simple to set u ⋅ n = u₂ = 0, but what about for a general mesh with the normal vector n a function of space?

Thanks.

[amartinhuertas]

You can define uD as a Julia function that given the coordinates of a point in the boundary, returns a VectorValue instance with the result of evaluating the point.

Note that for Hdiv-conforming spaces, e.g., Raviart-Thomas spaces, the implementation of such conditions is relatively easy as these can be imposed strongly.

When you have other types of spaces (not sure if this is the case), such as, e.g., H1-conforming spaces discretized with Lagrangian FEs, the imposition of such conditions is more involved. In such a case, you may use, e.g., weak Nitsche boundary conditions, or Lagrange multipliers.

[hgpeterson]

Thanks for the response! Apologies for the long delay; only recently got the time to take a look at this again.

I see now why this boundary condition is easier in the RT basis. I am solving a form of the Stokes equations, so I am working with the second case you mentioned. I'd like to avoid having to weakly impose the boundary condition, if possible. Does Gridap currently have a way of computing the normal vectors of a mesh and adding the u ⋅ n constraints to the linear system? Something like what deal.II does here.

[JordiManyer]

Yeah, you can use the FESpaceWithLinearConstraints, but no way around tinkering with the code. You can have a look at this slack thread https://julialang.slack.com/archives/C02AK382QSG/p1749623151793159

[hgpeterson]

I see, so it looks like with FESpaceWithLinearConstraints if I manually compute the constraints from my u ⋅ n = 0 boundary condition, I can add them to the space I'm working with (assuming I can keep track of all the indices properly). I'll give it a try, thank you!

[ThomasHaine]

@hgpeterson I'm also interested in applying u ⋅ n = 0 boundary conditions for fluids with impermeable stress-free walls. Did you get FESpaceWithLinearConstraints to work? Do you have a mwe?

[hgpeterson]

Hi Tom, I haven't gotten to it yet, but I'll post here if I get something working. If there's enough interest, maybe it would be worth creating an issue? The limited documentation on FESpaceWithLinearConstraints does seem to be something multiple people have brought up.

[JordiManyer]

If you go through the actual code file, there is a big block of comments that might help you out.