Evaluating strains at nodal points

General
1

Hello,

I am interested in finding the strains at the nodes of the mesh. The following provides the strain values at Gauss points. How can I extend it to find the strains at the nodes?

import dolfin


mesh = dolfin.UnitSquareMesh(10, 10)
V = dolfin.VectorFunctionSpace(mesh, "Lagrange", 2)
u = dolfin.Function(V)
u.interpolate(dolfin.Expression(("2*x[0]*x[0]", "x[1]"), degree=2))

degree = 2
el = dolfin.FiniteElement("Quadrature", mesh.ufl_cell(),
                          degree=degree, quad_scheme="default")
Q = dolfin.FunctionSpace(mesh, el)
qh = dolfin.Function(Q)

p = dolfin.TrialFunction(Q)
q = dolfin.TestFunction(Q)
dx = dolfin.Measure("dx", domain=mesh, metadata={
    "representation": "quadrature",
    "quadrature_degree": degree})
a = dolfin.inner(p, q)*dx
strain = dolfin.sym(dolfin.grad(u))
L = dolfin.inner(strain[0, 0], q)*dx
dolfin.solve(a == L, qh)

Thanks

2

Project the strains into a suitable function space

3

Thanks for the reply, @dokken!

I tried it, but it does not work for some reason.

strain_at_nodes = project(strain, V)

I got the following error:

UFLException: Shapes do not match: Argument id=5287909344 and Sym id=4847112096.

4

The strain is a tensor, thus you need to create a tensor-function space.

Note that even if the projection is unique, you are projecting a discontinuous quantity into a continuous space.

Is there a specific motivation to have the values at the mesh nodes (where they are non-unique?)

5

Thanks again, @dokken!

I see.
I tried the following, and it seems that it worked, but it is not able to convert it to numpy array.

V1 = dolfin.TensorFunctionSpace(mesh, "Lagrange", 2)

strain_at_nodes = project(strain, V1)
e11 = strain_at_nodes[0,0]
e11 = e11.vector().get_local()

AttributeError: ‘Indexed’ object has no attribute ‘vector’

Regarding motivation, I just need a rough estimate of the strain values at the nodes.

6

Consider:

import dolfin


mesh = dolfin.UnitSquareMesh(10, 10)
V = dolfin.VectorFunctionSpace(mesh, "Lagrange", 2)
u = dolfin.Function(V)
u.interpolate(dolfin.Expression(("2*x[0]*x[0]", "x[1]"), degree=2))

degree = 2
el = dolfin.FiniteElement("Quadrature", mesh.ufl_cell(),
                          degree=degree, quad_scheme="default")
Q = dolfin.FunctionSpace(mesh, el)
qh = dolfin.Function(Q)

p = dolfin.TrialFunction(Q)
q = dolfin.TestFunction(Q)
dx = dolfin.Measure("dx", domain=mesh, metadata={
    "representation": "quadrature",
    "quadrature_degree": degree})
a = dolfin.inner(p, q)*dx
strain = dolfin.sym(dolfin.grad(u))

W = dolfin.TensorFunctionSpace(mesh, "Lagrange", 1)
strain_cg_1 = dolfin.project(strain, W)
strain00 = strain_cg_1.sub(0, deepcopy=True)
strain10 = strain_cg_1.sub(2, deepcopy=True)
print(strain00.function_space().tabulate_dof_coordinates())
print(strain00.vector().get_local())

print(strain10.function_space().tabulate_dof_coordinates())
print(strain10.vector().get_local())
1 Like
7

@dokken, Many thanks!