def test_assembly_into_quadrature_function():
"""Test assembly into a Quadrature function.
This test evaluates a UFL Expression into a Quadrature function space by
evaluating the Expression on all cells of the mesh, and then inserting the
evaluated values into a PETSc Vector constructed from a matching Quadrature
function space.
Concretely, we consider the evaluation of:
e = B*(K(T)))**2 * grad(T)
where
K = 1/(A + B*T)
where A and B are Constants and T is a Coefficient on a P2 finite element
space with T = x + 2*y.
The result is compared with interpolating the analytical expression of e
directly into the Quadrature space.
In parallel, each process evaluates the Expression on both local cells and
ghost cells so that no parallel communication is required after insertion
into the vector.
"""
mesh = dolfinx.UnitSquareMesh(MPI.COMM_WORLD, 3, 6)
quadrature_degree = 2
quadrature_points = basix.make_quadrature(basix.CellType.triangle, quadrature_degree)
Q_element = ufl.VectorElement("Quadrature", ufl.triangle, quadrature_degree, quad_scheme="default")
Q = dolfinx.FunctionSpace(mesh, Q_element)
def T_exact(x):
return x[0] + 2.0 * x[1]
P2 = dolfinx.FunctionSpace(mesh, ("P", 2))
T = dolfinx.Function(P2)
T.interpolate(T_exact)
A = dolfinx.Constant(mesh, 1.0)
B = dolfinx.Constant(mesh, 2.0)
K = 1.0 / (A + B * T)
e = B * K**2 * ufl.grad(T)
e_expr = dolfinx.Expression(e, quadrature_points)
map_c = mesh.topology.index_map(mesh.topology.dim)
num_cells = map_c.size_local + map_c.num_ghosts
cells = np.arange(0, num_cells, dtype=np.int32)
e_eval = e_expr.eval(cells)
# Assemble into Function
e_Q = dolfinx.Function(Q)
with e_Q.vector.localForm() as e_Q_local:
e_Q_local.setBlockSize(e_Q.function_space.dofmap.bs)
e_Q_local.setValuesBlocked(Q.dofmap.list.array, e_eval, addv=PETSc.InsertMode.INSERT)
def e_exact(x):
T = x[0] + 2.0 * x[1]
K = 1.0 / (A.value + B.value * T)
grad_T = np.zeros((2, x.shape[1]))
grad_T[0, :] = 1.0
grad_T[1, :] = 2.0
e = B.value * K**2 * grad_T
return e
e_exact_Q = dolfinx.Function(Q)
e_exact_Q.interpolate(e_exact)
assert np.isclose((e_exact_Q.vector - e_Q.vector).norm(), 0.0)
def test_assembly_into_quadrature_function():
"""Test assembly into a Quadrature function.
This test evaluates a UFL Expression into a Quadrature function space by
evaluating the Expression on all cells of the mesh, and then inserting the
evaluated values into a PETSc Vector constructed from a matching Quadrature
function space.
Concretely, we consider the evaluation of:
e = B*(K(T)))**2 * grad(T)
where
K = 1/(A + B*T)
where A and B are Constants and T is a Coefficient on a P2 finite element
space with T = x + 2*y.
The result is compared with interpolating the analytical expression of e
directly into the Quadrature space.
In parallel, each process evaluates the Expression on both local cells and
ghost cells so that no parallel communication is required after insertion
into the vector.
"""
mesh = dolfinx.UnitSquareMesh(MPI.COMM_WORLD, 3, 6)
quadrature_degree = 2
quadrature_points, wts = basix.make_quadrature("default",
basix.CellType.triangle,
quadrature_degree)
Q_element = ufl.VectorElement("Quadrature",
ufl.triangle,
quadrature_degree,
quad_scheme="default")
Q = dolfinx.FunctionSpace(mesh, Q_element)
def T_exact(x):
return x[0] + 2.0 * x[1]
P2 = dolfinx.FunctionSpace(mesh, ("P", 2))
T = dolfinx.Function(P2)
T.interpolate(T_exact)
A = dolfinx.Constant(mesh, 1.0)
B = dolfinx.Constant(mesh, 2.0)
K = 1.0 / (A + B * T)
e = B * K**2 * ufl.grad(T)
e_expr = dolfinx.Expression(e, quadrature_points)
map_c = mesh.topology.index_map(mesh.topology.dim)
num_cells = map_c.size_local + map_c.num_ghosts
cells = np.arange(0, num_cells, dtype=np.int32)
e_eval = e_expr.eval(cells)
# Assemble into Function
e_Q = dolfinx.Function(Q)
with e_Q.vector.localForm() as e_Q_local:
e_Q_local.setBlockSize(e_Q.function_space.dofmap.bs)
e_Q_local.setValuesBlocked(Q.dofmap.list.array,
e_eval,
addv=PETSc.InsertMode.INSERT)
def e_exact(x):
T = x[0] + 2.0 * x[1]
K = 1.0 / (A.value + B.value * T)
grad_T = np.zeros((2, x.shape[1]))
grad_T[0, :] = 1.0
grad_T[1, :] = 2.0
e = B.value * K**2 * grad_T
return e
# FIXME: Below is only for testing purposes,
# never to be used in user code!
#
# Replace when interpolation into Quadrature element works.
coord_dofs = mesh.geometry.dofmap
x_g = mesh.geometry.x
tdim = mesh.topology.dim
Q_dofs = Q.dofmap.list.array.reshape(num_cells, quadrature_points.shape[0])
bs = Q.dofmap.bs
Q_dofs_unrolled = bs * np.repeat(Q_dofs, bs).reshape(-1,
bs) + np.arange(bs)
Q_dofs_unrolled = Q_dofs_unrolled.reshape(
-1, bs * quadrature_points.shape[0]).astype(Q_dofs.dtype)
with e_Q.vector.localForm() as local:
e_exact_eval = np.zeros_like(local.array)
for cell in range(num_cells):
xg = x_g[coord_dofs.links(cell), :tdim]
x = mesh.geometry.cmap.push_forward(quadrature_points, xg)
e_exact_eval[Q_dofs_unrolled[cell]] = e_exact(x.T).T.flatten()
assert np.allclose(local.array, e_exact_eval)
def test_simple_evaluation():
"""Test evaluation of UFL Expression.
This test evaluates a UFL Expression on cells of the mesh and compares the
result with an analytical expression.
For a function f(x, y) = 3*(x^2 + 2*y^2) the result is compared with the
exact gradient:
grad f(x, y) = 3*[2*x, 4*y].
(x^2 + 2*y^2) is first interpolated into a P2 finite element space. The
scaling by a constant factor of 3 and the gradient is calculated using code
generated by FFCX. The analytical solution is found by evaluating the
spatial coordinates as an Expression using UFL/FFCX and passing the result
to a numpy function that calculates the exact gradient.
"""
mesh = dolfinx.generation.UnitSquareMesh(MPI.COMM_WORLD, 3, 3)
P2 = dolfinx.FunctionSpace(mesh, ("P", 2))
# NOTE: The scaling by a constant factor of 3.0 to get f(x, y) is
# implemented within the UFL Expression. This is to check that the
# Constants are being set up correctly.
def exact_expr(x):
return x[0] ** 2 + 2.0 * x[1] ** 2
# Unused, but remains for clarity.
def f(x):
return 3 * (x[0] ** 2 + 2.0 * x[1] ** 2)
def exact_grad_f(x):
values = np.zeros_like(x)
values[:, 0::2] = 2 * x[:, 0::2]
values[:, 1::2] = 4 * x[:, 1::2]
values *= 3.0
return values
expr = dolfinx.Function(P2)
expr.interpolate(exact_expr)
ufl_grad_f = dolfinx.Constant(mesh, 3.0) * ufl.grad(expr)
points = np.array([[0.0, 0.0], [1.0, 0.0], [0.0, 1.0]])
grad_f_expr = dolfinx.Expression(ufl_grad_f, points)
assert grad_f_expr.num_points == points.shape[0]
assert grad_f_expr.value_size == 2
# NOTE: Cell numbering is process local.
map_c = mesh.topology.index_map(mesh.topology.dim)
num_cells = map_c.size_local + map_c.num_ghosts
cells = np.arange(0, num_cells, dtype=np.int32)
grad_f_evaluated = grad_f_expr.eval(cells)
assert grad_f_evaluated.shape[0] == cells.shape[0]
assert grad_f_evaluated.shape[1] == grad_f_expr.value_size * grad_f_expr.num_points
# Evaluate points in global space
ufl_x = ufl.SpatialCoordinate(mesh)
x_expr = dolfinx.Expression(ufl_x, points)
assert x_expr.num_points == points.shape[0]
assert x_expr.value_size == 2
x_evaluated = x_expr.eval(cells)
assert x_evaluated.shape[0] == cells.shape[0]
assert x_evaluated.shape[1] == x_expr.num_points * x_expr.value_size
# Evaluate exact gradient using global points
grad_f_exact = exact_grad_f(x_evaluated)
assert np.allclose(grad_f_evaluated, grad_f_exact)
def test_expression():
"""
Test UFL expression evaluation
"""
mesh = dolfinx.UnitSquareMesh(MPI.COMM_WORLD, 10, 10)
V = dolfinx.FunctionSpace(mesh, ("CG", 2))
def f(x):
return 2 * x[0]**2 + x[1]**2
def gradf(x):
return numpy.asarray([4 * x[0], 2 * x[1]])
u = dolfinx.Function(V)
u.interpolate(f)
u.x.scatter_forward()
grad_u = ufl.grad(u)
points = numpy.array([[0.15, 0.3, 0], [0.953, 0.81, 0]])
gdim = mesh.geometry.dim
tdim = mesh.topology.dim
bb = dolfinx.geometry.BoundingBoxTree(mesh, mesh.topology.dim)
# Find colliding cells on proc
closest_cell = []
local_map = []
for i, p in enumerate(points):
cells = dolfinx.geometry.compute_collisions_point(bb, p)
if len(cells) > 0:
actual_cells = dolfinx.geometry.select_colliding_cells(
mesh, cells, p, 1)
if len(actual_cells) > 0:
local_map.append(i)
closest_cell.append(actual_cells[0])
num_dofs_x = mesh.geometry.dofmap.links(
0).size # NOTE: Assumes same cell geometry in whole mesh
t_imap = mesh.topology.index_map(tdim)
num_cells = t_imap.size_local + t_imap.num_ghosts
x = mesh.geometry.x
x_dofs = mesh.geometry.dofmap.array.reshape(num_cells, num_dofs_x)
cell_geometry = numpy.zeros((num_dofs_x, gdim), dtype=numpy.float64)
points_ref = numpy.zeros((len(local_map), tdim))
# Map cells on process back to reference element
for i, cell in enumerate(closest_cell):
cell_geometry[:] = x[x_dofs[cell], :gdim]
point_ref = mesh.geometry.cmap.pull_back(
points[local_map[i]][:gdim].reshape(1, -1), cell_geometry)
points_ref[i] = point_ref
# Eval using Expression
expr = dolfinx.Expression(grad_u, points_ref)
grad_u_at_x = expr.eval(closest_cell).reshape(len(closest_cell),
points_ref.shape[0], gdim)
# Compare solutions
for i, cell in enumerate(closest_cell):
point = points[local_map[i]]
assert numpy.allclose(grad_u_at_x[i, i], gradf(point))