Mixed function space with dolfinx - how to implement an integral constraint

[bagla0]

Hi @dokken and @francesco-ballarin
Thank you for your previous support.
I was understanding the domain without decomposition tutorial (neumann BC using nullspace) and relating with mine in Constrains using null space in dolfinx - dolfinx - FEniCS Project. I have general queries related to its application:

1. Does bringing nullspace vector means we are identifying the rigid body motion related to coefficient matrix A (obtained from weak form)?

2. For the application of constraining the linear system of equation (Ax=b) other than rigid body motion constraints, does nullspace still works in alternative to lagrange multiplier method for dolfinx?

3. Can you provide/comment on the nullspace vector for Periodic homogenization of linear elastic materials — Numerical tours of continuum mechanics using FEniCS master documentation
   where,

the fluctuating field was averaged to zero over entire domain \int v dx =0 (2 constraints for 2 dimensions) and third constraint ( \int x_1 v_2- x_2 v_1 dx =0) where v_1 and v_2 are two components of fluctuating field with x_1, x_2 as spatial coordinates.

For this case, I tried the nullspace vectors previously as below,

@bagla0

# Refering neumann BC example without domain decomposition
# Assemble the discrete system

# Nullspace Implement
c1 = dolfinx.fem.Function(V)
c1.interpolate(lambda x: np.array([[1],[0]])) # Nullspace related to Constraint $v1$ averaged to zero.
C1 = c1.vector
#
c2 = dolfinx.fem.Function(V)
c2.interpolate(lambda x: np.array([[0],[1]])) # Nullspace related to Constraint $v[1]$ averaged to zero.
C2 = c2.vector
#
c3 = dolfinx.fem.Function(V)
c3.interpolate(lambda x: np.array((-x[1],x[0])) # Nullspace related to Constraint $-x2*v[0]+v[1]*x1$ averaged to zero.
C3 = c3.vector

# Orthonormalize vectors
dolfinx.la.orthonormalize([C1, C2, C3])
nullspace = petsc4py.PETSc.NullSpace().create(vectors=[C1, C2, C3], comm=mesh.comm)

# Set the nullspace
A.setNullSpace(nullspace)

I made the above attempt previously for nullspace vectors, which you said wrong. Can you here provide the correct nullspace vectors through telling how we look for it (MWE)?

4. What is the process of identifying the nullspace vectors (line-2)?
   
   
   
   Suppose, if we would be having pure Dirichlet BC in poisson equation, then, what would be the nullspace vector in 17. Singular Poisson — FEniCS Project?

I am sorry if I am unable to put right questions. Kindly guide me with whether nullspace works here. I understand some questions might be repeated, sorry for that.

Thanking You