# FEM: Possible to have boundary conditions "inside" the domain?

I work on geological problems and I use the Finite Element Method. But this question can be applied on other classical mechanical problems.

I work on implicit 3D surfaces (which represent the limits between two geological layers aka two media).

I have to impose displacement on these surfaces (Dirichlet Condition). The condition that I impose on surfaces are transfered on the surrounding nodes. I make the assumption that I can use the " classical" way to apply Dirichlet Condition (penalty method in my case) on these nodes, that are NOT on the the boundaries.

My question is: do you think this assumption is valid? Have you got some references about the subject? As I understand the problem I try to apply Boundary Conditions not on the boundaries...

Thank you in advance.

Best.

EDIT

We try to solve this problem: \left\{ \begin{aligned} \sigma_{ij,j} + F_{i} &= 0 & &\text{in the domain \Omega}&\\ u_i&= q_{i}& &\text{on the boundary \Gamma_{q}}&\\ \sigma_{ij}n_{j} &= h_{i}& &\text{on the boundary \Gamma_{h}}&\\ u_i &= b_{i}& &\text{punctually in the domain \Omega}&\\ \end{aligned} \right.

We run the FEA on a mesh. We basically try to impose Dirichlet Condition on the nodes which are inside the domain $\Omega$ (fourth line). We constrain the displacement value of $u_i$ to $b_i$ for these nodes.

• Can you explain in mathematical terms what the condition is you want to impose? Oct 1, 2015 at 11:17
• I have edited the question. Oct 1, 2015 at 13:34
• Yes, it's possible to prescribe "boundary conditions" in the domain and that's one of the greatest advantages of FEM. We can do that because in FEM our "domains" are the finite elements. Dec 5, 2018 at 15:38

The solution of the equation you are looking for is in the space $H^1$ of functions that have one weak derivative, but in 2d and 3d, this does not imply that the solution is in fact continuous. As a consequence, it is not possible to define the value of a solution at individual points, and equally consequentially, it is mathematically not possible to prescribe (boundary) values at individual points.

But I suspect that you're not actually interested in prescribing values for the displacement at individual points, but for all points along a line (in 2d) or surface (in 3d), and this is perfectly valid. (Mathematically speaking, this is so because the trace operator is well defined on $H^1$). At the discrete level, this is then equivalent to prescribing the values on all nodes that lie on this line/surface.

Conceptually, you can think of this as a crack in your domain that is occupied by an infinitely thin device whose displacement you can manipulate and that attached to the two sides of the crack.

• OK thank you very much for these informations ! Oct 5, 2015 at 12:15
• Revisiting this post, I have seen many papers (e.g. mdpi.com/2073-4360/14/8/1559/htm) applying only one Dirichlet boundary condition in 2D elasticity problem (plates). I am trying to do the same but I ended up with a stiffness matrix that is not a positive definite matrix. Does anyone know how they (also commercial packages) were able to solve the problem this way? Oct 10, 2022 at 20:11
• @ProfessorP.CosmoKlunk Does the matrix have zero eigenvalues, or in fact eigenvalues of both signs? Oct 10, 2022 at 21:16
• @WolfgangBangerth after applying the Dirichlet boundary conditions on the matrix (as you show on the deal.II's videos) the negative one is -4.12e-17 (can I consider it as zero?). The other ones are close to zero (8.08e-17) and positive. Oct 10, 2022 at 23:51
• -4e-17 is small only if it is small compared to the other eigenvalues. If the other eigenvalues are of the order +4e-15, then it is not small, for example. If the other eigenvalues are of the order +4e3, then it is small. Oct 11, 2022 at 3:31

Well you can model slip between geologic faults (a discontinuity) using constraint equations which can be thought of as loading at the interface instead of the boundary. You simply specify the relative slip and solve the problem. E.g., see the top-left figure in https://bitbucket.org/stali/defmod/wiki/Gallery

If that is indeed what you want then I would suggest that you use Lagrange multipliers instead of the penalty method. Off course you can use any constraint equation you want but it should make physical sense.

• Thank your for suggestions but my initial question is more like "Is FEA, which is based on variational formulation of boundary values problem, still a valid tool to solve a constrained equation with constraints on values inside the domain (which is no longer a boundary value problem)?" EDIT: I take a look to tour code it's pretty interesting Oct 1, 2015 at 14:37
• Yes it is. In theory you can think of the fault interface as a boundary (there are coincident nodes). Oct 1, 2015 at 14:54
• Ok but the surfaces on which I impose the boundary conditions are horizons. (interface between two layers) Oct 1, 2015 at 15:10