Layered Solid Element in fenicsx?

dolfinx
1

I have a problem of a composite plate where the properties keep changing in every layer. I know I can create multiple sub-domains and apply the properties accordingly.
But if I have too many layers (>200), the number of elements become too large and so does the computational time.

I came across a layered solid element like SOLID185 in ansys where inside a single element I can have multiple integration layers through the thickness.

How can I implement this in fenicsx??

Thanks!

2

I would suggest using a standard function space (P1) on hexahedra, and use a high order quadrature space for your material (maybe with custom points) to specify an «overintegration» per element to capture the material variation.

3

How do I define a custom quadrature space??

The below code does not let me create an element using custom quadrature. (dolfinx = ‘0.8.0’)

Q_element = basix.ufl.quadrature_element(
    domain.basix_cell(),
    value_shape=(6, 6),
    scheme="gauss_jacobi",degree = deg_quad)
Q = fem.functionspace(domain, Q_element)

But I can specify my custom points here. What does this actually do??

ufl.Measure("dx" , domain = domain , metadata={"quadrature_scheme": "custom", "quadrature_points": [[1.0, 0.5], ...], "quadrature_weights": [1.0, ...]})
4

Here is a minimal example:

import dolfinx.fem.petsc
from mpi4py import MPI
import dolfinx
import basix.ufl
import ufl

mesh = dolfinx.mesh.create_unit_cube(
    MPI.COMM_WORLD, 10, 10, 10, cell_type=dolfinx.mesh.CellType.hexahedron
)

q_points, q_weights = basix.make_quadrature(
    mesh.basix_cell(), 5, basix.QuadratureType.gll
)

el = basix.ufl.quadrature_element(mesh.basix_cell(), points=q_points, weights=q_weights)
Q = dolfinx.fem.functionspace(mesh, el)
q = dolfinx.fem.Function(Q)
q.interpolate(lambda x: x[0] + x[1] + x[2])

V = dolfinx.fem.functionspace(mesh, ("CG", 1))
u = ufl.TrialFunction(V)
v = ufl.TestFunction(V)
a = q * ufl.inner(ufl.grad(u), ufl.grad(v)) * ufl.dx
dolfinx.fem.petsc.assemble_matrix(dolfinx.fem.form(a))


import matplotlib.pyplot as plt

ax = plt.figure().add_subplot(projection="3d")
ax.scatter(q_points[:, 0], q_points[:, 1], q_points[:, 2])
plt.savefig("gll_points.png")

you can make value_shape whatever tensor the data you have is.

As long as you have a quadrature element in the form, you do not need to change the integration measure, as it picks up the degree when compiling the form.

5

Thank you! This works for me.

I have one more doubt

  1. For a complex case, where the number of layers in my composite plate model are not constant everywhere. Eg. region A has 10 layers, B has 4 layers, C has 7 layers. Can I define different quad points for each region? say ( 2 * 2 * 10 ) for A, ( 2 * 2 * 4 ) for B, ( 2 * 2 * 7 ) for C.
6

It is not straightforward to do that at the moment.
What you could do is to split the integrals into regions, depending on the number of layers in each region, and have a quadrature space describing the material coefficient in each of those layers with its own quadrature rule.

7

Thanks a lot! I’ll try that