Different material properties in subdomains

dolfinx
1

Hi,

If I use only one material property in the domain, it’s like the following

# Elasticity parameters
E = PETSc.ScalarType(1.0e0)
nu = PETSc.ScalarType(0.4999999)
mu = fem.Constant(domain, E/(2*(1 + nu)))

Now I want to set different Young’s modulus E to two subdomains Omega_0 and Omega_1. It’s the following way to set E:

# Young Modulus E1 and E2 for two different materials
E_0 = 1e4
E_1 = 50e4
def Omega_0(x):
    return x[0]<= 1

def Omega_1(x):
    return x[0] >= 1

E = fem.Function(V)

cells_0 = mesh.locate_entities(domain, domain.topology.dim, Omega_0)
cells_1 = mesh.locate_entities(domain, domain.topology.dim, Omega_1)

E.x.array[cells_0] = np.full_like(cells_0, E_0, dtype=PETSc.ScalarType)
E.x.array[cells_1] = np.full_like(cells_0, E_1, dtype=PETSc.ScalarType)

Could I ask how to set mu, which as a function of E in the two subdomains?
Could I write in the following way?

mu = E/(2*(1 + nu))

Thanks for your help

2

If I write

mu = E/(2*(1 + nu))

Then it’s a tuple, which is not supported for writing the residual form. How could I transfer it to available type?

3

Hi, I see that you’re following along with this tutorial? Is your function space V a DG0 space? The main difference that I see between your code (where several lines seem to be missing) and that example code is that you’re trying to use your Young’s modulus E to compute mu with several other PETSc Scalar instances. I would try defining a DG0 function mu and directly set its values on Omega_0 and Omega_1 with your known values for E_0, E_1 and nu:

mu = fem.Function(V)

cells_0 = mesh.locate_entities(domain, domain.topology.dim, Omega_0)
cells_1 = mesh.locate_entities(domain, domain.topology.dim, Omega_1)

mu.x.array[cells_0] = np.full_like(cells_0, E_0/(2*(1 + nu)), dtype=PETSc.ScalarType)
mu.x.array[cells_1] = np.full_like(cells_0, E_1/(2*(1 + nu)), dtype=PETSc.ScalarType)
4

Yes, I followed with the tutorial mentioned above, the space V here is a CG2 space, I am thinking if want to get the residual, and the residual R should be split into 2 parts: the first part R_0 in subdomain Omega_0 and the other part R_1 in Omega_1, then R = R_0 + R_1. Does that make sense?

P2 = ufl.VectorElement("CG", domain.ufl_cell(), 2)
P1 = ufl.FiniteElement("CG", domain.ufl_cell(), 1)

state_space = fem.FunctionSpace(domain, P2*P1)
V, _ = state_space.sub(0).collapse()
V0, _ = V.sub(0).collapse() #
5

Yes, I followed with the tutorial mentioned above, the space V here is a CG2 space

Why are you using a CG2 space for your material properties if you just want to set them as different constants in 2 parts of your domain?

I am thinking if want to get the residual, and the residual R should be split into 2 parts: the first part R_0 in subdomain Omega_0 and the other part R_1 in Omega_1, then R = R_0 + R_1. Does that make sense?

I’m not sure what you’re wanting to do here. Without seeing your full code in a single code block (preferably as a minimal working example) it’s hard for anyone to figure out the context of what you’re trying to do.

6

Hi,

Thanks a lot for your suggestion, I set DG0 space for E, and now problem is solved.

Could I ask why it should be DG0?

7

If you’re not familiar with Continuous Galerkin (CG) and Discontinuous Galerkin (DG) spaces you should really read up on how they are defined. Very briefly, DG0 is a space of discontinuous piecewise-constant functions. Since you’re trying to just apply constant material properties to different parts of your domain, DG0 contains exactly what you’re trying to do.

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