def build_nullspace(V):
"""Function to build PETSc nullspace for 3D elasticity"""
# Create list of vectors for null space
index_map = V.dofmap.index_map
bs = V.dofmap.index_map_bs
ns = [la.create_petsc_vector(index_map, bs) for i in range(6)]
with ExitStack() as stack:
vec_local = [stack.enter_context(x.localForm()) for x in ns]
basis = [np.asarray(x) for x in vec_local]
# Get dof indices for each subspace (x, y and z dofs)
dofs = [V.sub(i).dofmap.list.array for i in range(3)]
# Build translational nullspace basis
for i in range(3):
basis[i][dofs[i]] = 1.0
# Build rotational nullspace basis
x = V.tabulate_dof_coordinates()
dofs_block = V.dofmap.list.array
x0, x1, x2 = x[dofs_block, 0], x[dofs_block, 1], x[dofs_block, 2]
basis[3][dofs[0]] = -x1
basis[3][dofs[1]] = x0
basis[4][dofs[0]] = x2
basis[4][dofs[2]] = -x0
basis[5][dofs[2]] = x1
basis[5][dofs[1]] = -x2
la.orthonormalize(ns)
assert la.is_orthonormal(ns)
return PETSc.NullSpace().create(vectors=ns)
def test_nullspace_orthogonal(mesh, degree):
"""Test that null spaces orthogonalisation"""
V = VectorFunctionSpace(mesh, ('Lagrange', degree))
nullspace = build_elastic_nullspace(V)
assert not la.is_orthonormal(nullspace)
la.orthonormalize(nullspace)
assert la.is_orthonormal(nullspace)
def test_nullspace_check(mesh, degree):
V = VectorFunctionSpace(mesh, ('Lagrange', degree))
u, v = TrialFunction(V), TestFunction(V)
E, nu = 2.0e2, 0.3
mu = E / (2.0 * (1.0 + nu))
lmbda = E * nu / ((1.0 + nu) * (1.0 - 2.0 * nu))
def sigma(w, gdim):
return 2.0 * mu * ufl.sym(grad(w)) + lmbda * ufl.tr(
grad(w)) * ufl.Identity(gdim)
a = form(inner(sigma(u, mesh.geometry.dim), grad(v)) * dx)
# Assemble matrix and create compatible vector
A = assemble_matrix(a)
A.assemble()
# Create null space basis and test
nullspace = build_elastic_nullspace(V)
la.orthonormalize(nullspace)
ns = PETSc.NullSpace().create(vectors=nullspace)
assert ns.test(A)
# Create incorrect null space basis and test
nullspace = build_broken_elastic_nullspace(V)
la.orthonormalize(nullspace)
ns = PETSc.NullSpace().create(vectors=nullspace)
assert not ns.test(A)
def rigid_motions_nullspace(V: _fem.FunctionSpace):
"""
Function to build nullspace for 2D/3D elasticity.
Parameters:
===========
V
The function space
"""
_x = _fem.Function(V)
# Get geometric dim
gdim = V.mesh.geometry.dim
assert gdim == 2 or gdim == 3
# Set dimension of nullspace
dim = 3 if gdim == 2 else 6
# Create list of vectors for null space
nullspace_basis = [_x.vector.copy() for _ in range(dim)]
with ExitStack() as stack:
vec_local = [stack.enter_context(x.localForm()) for x in nullspace_basis]
basis = [np.asarray(x) for x in vec_local]
dofs = [V.sub(i).dofmap.list.array for i in range(gdim)]
# Build translational null space basis
for i in range(gdim):
basis[i][dofs[i]] = 1.0
# Build rotational null space basis
x = V.tabulate_dof_coordinates()
dofs_block = V.dofmap.list.array
x0, x1, x2 = x[dofs_block, 0], x[dofs_block, 1], x[dofs_block, 2]
if gdim == 2:
basis[2][dofs[0]] = -x1
basis[2][dofs[1]] = x0
elif gdim == 3:
basis[3][dofs[0]] = -x1
basis[3][dofs[1]] = x0
basis[4][dofs[0]] = x2
basis[4][dofs[2]] = -x0
basis[5][dofs[2]] = x1
basis[5][dofs[1]] = -x2
_la.orthonormalize(nullspace_basis)
assert _la.is_orthonormal(nullspace_basis)
return PETSc.NullSpace().create(vectors=nullspace_basis)
def build_nullspace(V, x):
"""Function to build null space for 2D elasticity"""
# Create list of vectors for null space
ns = [x.copy() for i in range(3)]
with ExitStack() as stack:
vec_local = [stack.enter_context(x.localForm()) for x in ns]
basis = [np.asarray(x) for x in vec_local]
# Build null space basis
dofs = [V.sub(i).dofmap.list.array for i in range(2)]
for i in range(2):
basis[i][dofs[i]] = 1.0
x = V.tabulate_dof_coordinates()
basis[2][dofs[0]] = -x[dofs[0], 1]
basis[2][dofs[1]] = x[dofs[1], 0]
la.orthonormalize(ns)
return ns