Static condensation of linear elasticity

This demo solves a Cook’s plane stress elasticity test in a mixed space formulation. The test is a sloped cantilever under upward traction force at free end. Static condensation of internal (stress) degrees-of-freedom is demonstrated.
from pathlib import Path

import cffi
import numba
import numba.core.typing.cffi_utils as cffi_support
import numpy as np

import ufl
from dolfinx import geometry
from dolfinx.cpp.fem import Form_complex128, Form_float64
from dolfinx.fem import (Function, FunctionSpace, IntegralType, dirichletbc,
                         form, locate_dofs_topological)
from dolfinx.fem.petsc import (apply_lifting, assemble_matrix, assemble_vector,
                               set_bc)
from dolfinx.io import XDMFFile
from dolfinx.jit import ffcx_jit
from dolfinx.mesh import locate_entities_boundary, meshtags

from mpi4py import MPI
from petsc4py import PETSc

infile = XDMFFile(MPI.COMM_WORLD, Path(Path(__file__).parent, "data", "cooks_tri_mesh.xdmf"),
                  "r", encoding=XDMFFile.Encoding.ASCII)
msh = infile.read_mesh(name="Grid")
infile.close()

# Stress (Se) and displacement (Ue) elements
Se = ufl.TensorElement("DG", msh.ufl_cell(), 1, symmetry=True)
Ue = ufl.VectorElement("Lagrange", msh.ufl_cell(), 2)

S = FunctionSpace(msh, Se)
U = FunctionSpace(msh, Ue)

# Get local dofmap sizes for later local tensor tabulations
Ssize = S.element.space_dimension
Usize = U.element.space_dimension

sigma, tau = ufl.TrialFunction(S), ufl.TestFunction(S)
u, v = ufl.TrialFunction(U), ufl.TestFunction(U)

# Locate all facets at the free end and assign them value 1. Sort the
# facet indices (requirement for constructing MeshTags)
free_end_facets = np.sort(locate_entities_boundary(msh, 1, lambda x: np.isclose(x[0], 48.0)))
mt = meshtags(msh, 1, free_end_facets, 1)

ds = ufl.Measure("ds", subdomain_data=mt)

# Homogeneous boundary condition in displacement
u_bc = Function(U)
u_bc.x.array[:] = 0.0

# Displacement BC is applied to the left side
left_facets = locate_entities_boundary(msh, 1, lambda x: np.isclose(x[0], 0.0))
bdofs = locate_dofs_topological(U, 1, left_facets)
bc = dirichletbc(u_bc, bdofs)

# Elastic stiffness tensor and Poisson ratio
E, nu = 1.0, 1.0 / 3.0


def sigma_u(u):
    """Constitutive relation for stress-strain. Assuming plane-stress in XY"""
    eps = 0.5 * (ufl.grad(u) + ufl.grad(u).T)
    sigma = E / (1. - nu ** 2) * ((1. - nu) * eps + nu * ufl.Identity(2) * ufl.tr(eps))
    return sigma


a00 = ufl.inner(sigma, tau) * ufl.dx
a10 = - ufl.inner(sigma, ufl.grad(v)) * ufl.dx
a01 = - ufl.inner(sigma_u(u), tau) * ufl.dx

f = ufl.as_vector([0.0, 1.0 / 16])
b1 = form(- ufl.inner(f, v) * ds(1))

# JIT compile individual blocks tabulation kernels
nptype = "complex128" if np.issubdtype(PETSc.ScalarType, np.complexfloating) else "float64"
ffcxtype = "double _Complex" if np.issubdtype(PETSc.ScalarType, np.complexfloating) else "double"
ufcx_form00, _, _ = ffcx_jit(msh.comm, a00, form_compiler_params={"scalar_type": ffcxtype})
kernel00 = getattr(ufcx_form00.integrals(0)[0], f"tabulate_tensor_{nptype}")
ufcx_form01, _, _ = ffcx_jit(msh.comm, a01, form_compiler_params={"scalar_type": ffcxtype})
kernel01 = getattr(ufcx_form01.integrals(0)[0], f"tabulate_tensor_{nptype}")
ufcx_form10, _, _ = ffcx_jit(msh.comm, a10, form_compiler_params={"scalar_type": ffcxtype})
kernel10 = getattr(ufcx_form10.integrals(0)[0], f"tabulate_tensor_{nptype}")

ffi = cffi.FFI()
cffi_support.register_type(ffi.typeof('double _Complex'), numba.types.complex128)
c_signature = numba.types.void(
    numba.types.CPointer(numba.typeof(PETSc.ScalarType())),
    numba.types.CPointer(numba.typeof(PETSc.ScalarType())),
    numba.types.CPointer(numba.typeof(PETSc.ScalarType())),
    numba.types.CPointer(numba.types.double),
    numba.types.CPointer(numba.types.int32),
    numba.types.CPointer(numba.types.uint8))


@numba.cfunc(c_signature, nopython=True)
def tabulate_condensed_tensor_A(A_, w_, c_, coords_, entity_local_index, permutation=ffi.NULL):
    # Prepare target condensed local element tensor
    A = numba.carray(A_, (Usize, Usize), dtype=PETSc.ScalarType)

    # Tabulate all sub blocks locally
    A00 = np.zeros((Ssize, Ssize), dtype=PETSc.ScalarType)
    kernel00(ffi.from_buffer(A00), w_, c_, coords_, entity_local_index, permutation)

    A01 = np.zeros((Ssize, Usize), dtype=PETSc.ScalarType)
    kernel01(ffi.from_buffer(A01), w_, c_, coords_, entity_local_index, permutation)

    A10 = np.zeros((Usize, Ssize), dtype=PETSc.ScalarType)
    kernel10(ffi.from_buffer(A10), w_, c_, coords_, entity_local_index, permutation)

    # A = - A10 * A00^{-1} * A01
    A[:, :] = - A10 @ np.linalg.solve(A00, A01)


# Prepare a Form with a condensed tabulation kernel
Form = Form_float64 if PETSc.ScalarType == np.float64 else Form_complex128

integrals = {IntegralType.cell: ([(-1, tabulate_condensed_tensor_A.address)], None)}
a_cond = Form([U._cpp_object, U._cpp_object], integrals, [], [], False, None)

A_cond = assemble_matrix(a_cond, bcs=[bc])
A_cond.assemble()

b = assemble_vector(b1)
apply_lifting(b, [a_cond], bcs=[[bc]])
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
set_bc(b, [bc])

uc = Function(U)
solver = PETSc.KSP().create(A_cond.getComm())
solver.setOperators(A_cond)
solver.solve(b, uc.vector)

# Pure displacement based formulation
a = form(- ufl.inner(sigma_u(u), ufl.grad(v)) * ufl.dx)
A = assemble_matrix(a, bcs=[bc])
A.assemble()

# Create bounding box for function evaluation
bb_tree = geometry.BoundingBoxTree(msh, 2)

# Check against standard table value
p = np.array([48.0, 52.0, 0.0], dtype=np.float64)
cell_candidates = geometry.compute_collisions(bb_tree, p)
cells = geometry.compute_colliding_cells(msh, cell_candidates, p)

uc.x.scatter_forward()
if len(cells) > 0:
    value = uc.eval(p, cells[0])
    print(value[1])
    assert np.isclose(value[1], 23.95, rtol=1.e-2)

# Check the equality of displacement based and mixed condensed global
# matrices, i.e. check that condensation is exact
assert np.isclose((A - A_cond).norm(), 0.0)