Meaning of the following boundary conditions w.r.t. a 3D mesh (Static Linear elasticity)

Question

Hi, Can someone pls. explain the physical meaning of the following Boundary conditions w.r.t. a 3D cube mesh (imported from gmsh):

(Source: Pg. 67 of:
http://fenicsproject.org/pub/documents/fenics/presentations/logg-geilo-2006-03/logg-geilo-2006.pdf )

u = u0 on Γ0 ⊂ ∂Ω,
σ(u)nˆ = 0 on ∂Ω \ Γ0,

Also, if my cube (from gmsh) has a hole, some suggestions on which physical surfaces to use for applying the above BC will be very helpful. I have been doing a lot of trial n error and I always get stuck in something or other. Thanks!

Answer 1

Hello,

The link you provided does not seem to work for me.

The boundary conditions (they are called dirichlet boundary conditions) mean that you want to assign a specific displacement to the boundary Gamma_0. An example for this in elasticity would be a beam that is stuck in a wall at one end. In that case u_0 = 0 on that end.

The other boundary conditions are homogenous neumann conditions. Physically, they are equivalent to traction forces on the boundary. Usually, they are equal to zero, which simply means that the boundary has no forces acting on it, which is the case in many applications.

Dirichlet boundary conditions have to be applied by the user for whatever part of the boundary you need them. Neumann conditions that are zero don't have to be applied by the user, they appear automatically whereever there is no other boundary condition.

So for example, if you want to fix one side of the cube of a wall and apply a constant displacement to the opposite side (stretching the cube), then you would set u=0 on the first side, u=1 on the opposite side, and simply do nothing for the other faces.

Unfortunately, I didn't quite get your last question about the hole? Do you want to know which of the two boundary conditions you should apply to the hole. That depends on what your application is, i.e. whether you want to stick something through the hole and pull on it or whether you simply don't want to do anything to
the hole. Or does your question concern the specific way to implement boundary conditions in fenics?

regards

Comments

Hello multigrid202,

The link should work now. Pls. open it and see Pg. 67. What I want to achieve is kind of this (but for a cube with a hole instead of a gear.

Also Pls. tell me what is the meaning of " on ∂Ω \ Γ0 " in the second condition. Does this imply homogeneous Neumann BC on entire boundary except Gamma_0.

So I have my cube with hole. Now do I need to define a separate physical surface on the holes at both ends too? And a physical surface on the circumference of the cylindrical hole?

I agree that the first condition wants me to fix 1 side to zero. There after the code should work fine unless there's something wrong with my .geo file. It would be really helpful if you can help me with this.

---

$$\partial \Omega $$ denotes the domain boundary. If A and B are sets of any kind, then B\A means "all elements from B, except for the elements in A. So yes, you are correct about the way that you understand this notation.

I general, more than one boundary condition per boundary does not make sense (especially not a mixture of neumann and dirichlet).

If there are no dirichlet displacements around your hole, then you don't need to do anything. Homogenous Neumann conditions will automatically appear whereever there are no dirichlet conditions. This has nothing to do with fenics, it follows from the mathematical theory.

Unfortunately, I didn't understand your last question. Let me know if you need more help to get the code running. But please post the .geo file and the fenics code that you want to use :)

regards