Periodic Boundary Condition + Constant

[mtrautner]

I'm looking to enforce a constraint like u[x,1] = u[x,0] + f0(t) and u[0,y] = u[1,y] + f1(t) for some given functions f0 and f1 for all x and y on a 2D square. It seems like dolfinx_mpc is the closest to achieving such a constraint. Do you have an idea of which functions I might modify to make this happen? Thanks.

[jorgensd]

I did make a try at this a while back. It should work with very minor modifications, like:

https://github.com/jorgensd/dolfinx_mpc/blob/main/cpp/lifting.h#L260

Is there. It is only in lifting the constant becomes relevant, thus only minor modifications is needed. I'll see if I can find the mathematical derivation i did on this (late next week).

[mtrautner]

Okay thanks. If I have this right, the commented out lines 260~264 of what you linked subtract a constant times some matrix values from relevant entities of the linear system, so the idea would be to instead just add a constant in this spot?

[RFurze]

Were you able to get this modification working? I'm looking to implement precisely this kind of constraint in what sounds like a similar multiscale model.

[jorgensd]

I've been thinking more about this, and I am in the progress of designing a version that should support this.
To try to not modify the existing workflow too much, I'm suggesting the following design rule:

• A slave cannot be a DirichletBC, so all current constructors will still keep these inputs as they are.
  • If one wants this behavior, one should create the constraint $u_i = g$ with MPC rather than DirichletBC.
• However, if $u_i = \alpha u_j + u_k, u_j = g$,$i\neq j$, $j\neq k$, $i \neq k$ this should be treated as $u_i \alpha g + u_k$.

Mathematically, it breaks into the following problem:

$$ A x = b $$

where $x$ is the unconstrained unknown, relating to the MPC as:

$$ K \bar x + \bar g $$

Inserting this, we would get

$$ A K \bar x + A \bar g = b $$

Symmetrizing and shuffling, we get:

$$ K^TAK \bar x = b - K^T A \bar g $$

which is very similar to what we currently do in dolfinx_mpc.apply_lifting (there $\bar g$ is equal to the Dirichlet value when at a Dirichlet dof).
So I'm thinking the following API change:

MultiPointConstraint will take in a new, optional list of DirichletBCs.
What they will do effectively is to zero out the coefficients in front of the given degree of freedom
MultiPointConstraint will also take in a new array called constants, which would be equivalent to a d_i in the constraint u_i = d_i.
In apply_lifting, the contributions from DirichletBC dofs and the extra constants would be accumulated.