How to access normal components to create a Neumann boundary condition ?

Question

Dear All,

Assume we have a cube with a cubic hole at the centre (i.e. two concentric cubes) for which we would like to solve Laplace equation with the following applied boundary conditions (BC):
1) a u=0 Dirichlet BC on the surface of the external cube.
2) On the inner cube surface, we would like to apply Neumann BC using 3 known functions gx(x,y,z), gy(x,y,z) and gz(x,y,z), such that the applied boundary condition is : g(x,y,z) = gx(x,y,z)nx + gy(x,y,z)ny + gz(x,y,z)*nz where n = (nx, ny, nz) is the normal on each node (x,y,z) of the inner cubic surface mesh.
How would one access the normal components and write the corresponding Neumann boundary condition ?

Thanks a lot for any help.

Answer 1

You have to
-) make the known functions gx, gy, gz available as dolfin Expressions or Functions
-) create a dolfin SubDomain that describes the inner cube boundary and mark a FacetFunction with it, so that the inner cube boundary corresponds to some integer value, e.g. 1 (see this demo).
Then creating the Neumann condition you desire can accomplished with the following code (assuming you have already defined the mesh, the function space V and the FacetFunction boundaries).

ds = Measure('ds', domain=mesh, subdomain_data=boundaries)
n = FacetNormal(mesh)
v = TestFunction(V)

L = (gx*n[0] + gy*n[1] + gz*n[2])*v*ds(1)

Now L can be used as a right hand side for the equation you want to solve (or added to an additional right hand side, something like fvdx) and will then act as the Neumann boundary condition.