Why the coupling blocks in the stiffness matrix are zero?

mesh
1

In a 2D elastic problem (in-plane), I found my stiffness and mass matrices for a 2D rectangle in which a homogeneous Dirichlet is applied on its left. Here: How to apply Dirichlet boundary conditions using a matrix free method - FEniCS Q&A , it is explained that when a homogeneous DC boundary conditions is applied, the diagonal entries for the relevant dofs are set to 1, and all off-diagonal entries in the same rows are set to zero. First, It would be great if someone can explain why should all the dofs on the diagonal be 1 and the rest zero, applying Dirichlet for a discrete chain gives me something else.
Second, here are first 36 dofs of my stiffness matrix:

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image1667×519 4.75 KB

and here is my mesh:

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image794×702 76.6 KB

Why do the yellow blocks which are showing the coupling of the dofs on the boundary and those on their right side, are zero sub-matrices? Clearly, the degrees of freedom on the boundary are not disconnected to those on their right side.

Thank you,

2

As you have not supplied your implementation, it is hard to pinpoint the coupling issue. I suspect that you are using assemble_system.
See: How to implement the equivalence of `assemble_system` in DOLFINx - #2 by dokken
for an explaination of why the diagonal dofs in legacy dolfin is not equal to one.

3

I am sorry for that Jorgen. Here is how I assembled it:

Blockquote
k_form = inner(sigma(du),eps(u_))*dx
l_form = Constant(1.0)*u_[0]*dx
K = PETScMatrix()
b = PETScVector()
e,o=assemble_system(k_form, l_form, bc, A_tensor=K, b_tensor=b)
K= e.array()

and this is my boundary conditions:

Blockquote
class boundary(SubDomain):
def inside(self, x, on_boundary):
return x[0] < DOLFIN_EPS/100 and on_boundary

Blockquote
BCS=boundary()
bc = DirichletBC(V, Constant((0.,0.)), BCS)

4

When you call assemble_system, it applies boundary conditions in a symmetric fashion. This means that instead of only setting the bc rows to an identity/multiple of identity row.
One instead uses lifting, which results in a Matrix where all bc entries are discoupled, and is instead added in the rhs ( b-=Ag) where g is equal to the DirichletBC at respective dofs, 0 otherwise.