def test_map_vtk_to_dolfin(vtk, dolfin, cell_type):
p = perm_vtk(cell_type, len(vtk))
cell_p = np.array(vtk)[p]
assert (cell_p == dolfin).all()
p = np.argsort(perm_vtk(cell_type, len(vtk)))
cell_p = np.array(dolfin)[p]
assert (cell_p == vtk).all()
def xtest_read_write_p2_mesh(tempdir, encoding):
pygmsh = pytest.importorskip("pygmsh")
if MPI.COMM_WORLD.rank == 0:
geom = pygmsh.opencascade.Geometry()
geom.add_ball([0.0, 0.0, 0.0], 1.0, char_length=0.4)
pygmsh_mesh = pygmsh.generate_mesh(
geom, mesh_file_type="vtk", extra_gmsh_arguments=["-order", "2"])
cells, x = pygmsh_mesh.cells[-1].data, pygmsh_mesh.points
pygmsh_cell = pygmsh_mesh.cells[-1].type
cell_type, gdim, num_nodes = MPI.COMM_WORLD.bcast(
[pygmsh_cell, x.shape[1], cells.shape[1]], root=0)
else:
cell_type, gdim, num_nodes = MPI.COMM_WORLD.bcast([None, None, None],
root=0)
cells, x = np.empty([0, num_nodes]), np.empty([0, gdim])
domain = ufl_mesh_from_gmsh(cell_type, gdim)
cell_type = cpp.mesh.to_type(str(domain.ufl_cell()))
cells = cells[:, perm_vtk(cell_type, cells.shape[1])]
mesh = create_mesh(MPI.COMM_WORLD, cells, x, domain)
filename = os.path.join(tempdir, "tet10_mesh.xdmf")
with XDMFFile(mesh.mpi_comm(), filename, "w", encoding=encoding) as xdmf:
xdmf.write_mesh(mesh)
with XDMFFile(mesh.mpi_comm(), filename, "r", encoding=encoding) as xdmf:
mesh2 = xdmf.read_mesh()
assert mesh.topology.index_map(0).size_global == mesh2.topology.index_map(
0).size_global
dim = mesh.topology.dim
assert mesh.topology.index_map(
dim).size_global == mesh2.topology.index_map(dim).size_global
def test_fourth_order_quad(L, H, Z):
"""Test by comparing integration of z+x*y against sympy/scipy integration
of a quad element. Z>0 implies curved element.
*---------* 20-21-22-23-24-41--42--43--44
| | | | |
| | 15 16 17 18 19 37 38 39 40
| | | | |
| | 10 11 12 13 14 33 34 35 36
| | | | |
| | 5 6 7 8 9 29 30 31 32
| | | | |
*---------* 0--1--2--3--4--25--26--27--28
"""
points = np.array([[0, 0, 0], [L / 4, 0, 0], [L / 2, 0, 0], # 0 1 2
[3 * L / 4, 0, 0], [L, 0, 0], # 3 4
[0, H / 4, -Z / 3], [L / 4, H / 4, -Z / 3], [L / 2, H / 4, -Z / 3], # 5 6 7
[3 * L / 4, H / 4, -Z / 3], [L, H / 4, -Z / 3], # 8 9
[0, H / 2, 0], [L / 4, H / 2, 0], [L / 2, H / 2, 0], # 10 11 12
[3 * L / 4, H / 2, 0], [L, H / 2, 0], # 13 14
[0, (3 / 4) * H, 0], [L / 4, (3 / 4) * H, 0], # 15 16
[L / 2, (3 / 4) * H, 0], [3 * L / 4, (3 / 4) * H, 0], # 17 18
[L, (3 / 4) * H, 0], [0, H, Z], [L / 4, H, Z], # 19 20 21
[L / 2, H, Z], [3 * L / 4, H, Z], [L, H, Z], # 22 23 24
[(5 / 4) * L, 0, 0], [(6 / 4) * L, 0, 0], # 25 26
[(7 / 4) * L, 0, 0], [2 * L, 0, 0], # 27 28
[(5 / 4) * L, H / 4, -Z / 3], [(6 / 4) * L, H / 4, -Z / 3], # 29 30
[(7 / 4) * L, H / 4, -Z / 3], [2 * L, H / 4, -Z / 3], # 31 32
[(5 / 4) * L, H / 2, 0], [(6 / 4) * L, H / 2, 0], # 33 34
[(7 / 4) * L, H / 2, 0], [2 * L, H / 2, 0], # 35 36
[(5 / 4) * L, 3 / 4 * H, 0], # 37
[(6 / 4) * L, 3 / 4 * H, 0], # 38
[(7 / 4) * L, 3 / 4 * H, 0], [2 * L, 3 / 4 * H, 0], # 39 40
[(5 / 4) * L, H, Z], [(6 / 4) * L, H, Z], # 41 42
[(7 / 4) * L, H, Z], [2 * L, H, Z]]) # 43 44
# VTK ordering
cells = np.array([[0, 4, 24, 20, 1, 2, 3, 9, 14, 19, 21, 22, 23, 5, 10, 15, 6, 7, 8, 11, 12, 13, 16, 17, 18],
[4, 28, 44, 24, 25, 26, 27, 32, 36, 40, 41, 42, 43, 9, 14, 19,
29, 30, 31, 33, 34, 35, 37, 38, 39]])
cells = cells[:, perm_vtk(CellType.quadrilateral, cells.shape[1])]
cell = ufl.Cell("quadrilateral", geometric_dimension=points.shape[1])
domain = ufl.Mesh(ufl.VectorElement("Lagrange", cell, 4))
mesh = create_mesh(MPI.COMM_WORLD, cells, points, domain)
def e2(x):
return x[2] + x[0] * x[1]
V = FunctionSpace(mesh, ("CG", 4))
u = Function(V)
u.interpolate(e2)
intu = assemble_scalar(u * dx(mesh))
intu = mesh.mpi_comm().allreduce(intu, op=MPI.SUM)
nodes = [0, 5, 10, 15, 20]
ref = sympy_scipy(points, nodes, 2 * L, H)
assert ref == pytest.approx(intu, rel=1e-5)
def xtest_third_order_tri():
# *---*---*---* 3--11--10--2
# | \ | | \ |
# * * * * 8 7 15 13
# | \ | | \ |
# * * * * 9 14 6 12
# | \ | | \ |
# *---*---*---* 0--4---5---1
for H in (1.0, 2.0):
for Z in (0.0, 0.5):
L = 1
points = np.array([
[0, 0, 0],
[L, 0, 0],
[L, H, Z],
[0, H, Z], # 0, 1, 2, 3
[L / 3, 0, 0],
[2 * L / 3, 0, 0], # 4, 5
[2 * L / 3, H / 3, 0],
[L / 3, 2 * H / 3, 0], # 6, 7
[0, 2 * H / 3, 0],
[0, H / 3, 0], # 8, 9
[2 * L / 3, H, Z],
[L / 3, H, Z], # 10, 11
[L, H / 3, 0],
[L, 2 * H / 3, 0], # 12, 13
[L / 3, H / 3, 0], # 14
[2 * L / 3, 2 * H / 3, 0]
]) # 15
cells = np.array([[0, 1, 3, 4, 5, 6, 7, 8, 9, 14],
[1, 2, 3, 12, 13, 10, 11, 7, 6, 15]])
cells = cells[:, perm_vtk(CellType.triangle, cells.shape[1])]
mesh = Mesh(MPI.COMM_WORLD,
CellType.triangle,
points,
cells, [],
degree=3)
def e2(x):
return x[2] + x[0] * x[1]
degree = mesh.geometry.dofmap_layout().degree()
# Interpolate function
V = FunctionSpace(mesh, ("CG", degree))
u = Function(V)
u.interpolate(e2)
intu = assemble_scalar(u * dx(metadata={"quadrature_degree": 40}))
intu = mesh.mpi_comm().allreduce(intu, op=MPI.SUM)
nodes = [0, 9, 8, 3]
ref = sympy_scipy(points, nodes, L, H)
assert ref == pytest.approx(intu, rel=1e-6)
def test_third_order_tri():
# *---*---*---* 3--11--10--2
# | \ | | \ |
# * * * * 8 7 15 13
# | \ | | \ |
# * * * * 9 14 6 12
# | \ | | \ |
# *---*---*---* 0--4---5---1
for H in (1.0, 2.0):
for Z in (0.0, 0.5):
L = 1
points = np.array([
[0, 0, 0],
[L, 0, 0],
[L, H, Z],
[0, H, Z], # 0, 1, 2, 3
[L / 3, 0, 0],
[2 * L / 3, 0, 0], # 4, 5
[2 * L / 3, H / 3, 0],
[L / 3, 2 * H / 3, 0], # 6, 7
[0, 2 * H / 3, 0],
[0, H / 3, 0], # 8, 9
[2 * L / 3, H, Z],
[L / 3, H, Z], # 10, 11
[L, H / 3, 0],
[L, 2 * H / 3, 0], # 12, 13
[L / 3, H / 3, 0], # 14
[2 * L / 3, 2 * H / 3, 0]
]) # 15
cells = np.array([[0, 1, 3, 4, 5, 6, 7, 8, 9, 14],
[1, 2, 3, 12, 13, 10, 11, 7, 6, 15]])
cells = cells[:, perm_vtk(CellType.triangle, cells.shape[1])]
cell = ufl.Cell("triangle", geometric_dimension=points.shape[1])
domain = ufl.Mesh(ufl.VectorElement("Lagrange", cell, 3))
mesh = create_mesh(MPI.COMM_WORLD, cells, points, domain)
def e2(x):
return x[2] + x[0] * x[1]
# Interpolate function
V = FunctionSpace(mesh, ("Lagrange", 3))
u = Function(V)
u.interpolate(e2)
intu = assemble_scalar(u * dx(metadata={"quadrature_degree": 40}))
intu = mesh.mpi_comm().allreduce(intu, op=MPI.SUM)
nodes = [0, 9, 8, 3]
ref = sympy_scipy(points, nodes, L, H)
assert ref == pytest.approx(intu, rel=1e-6)
def test_third_order_quad(L, H, Z):
"""Test by comparing integration of z+x*y against sympy/scipy integration
of a quad element. Z>0 implies curved element.
*---------* 3--8--9--2-22-23-17
| | | | |
| | 11 14 15 7 26 27 21
| | | | |
| | 10 12 13 6 24 25 20
| | | | |
*---------* 0--4--5--1-18-19-16
"""
points = np.array([[0, 0, 0], [L, 0, 0], [L, H, Z], [0, H, Z], # 0 1 2 3
[L / 3, 0, 0], [2 * L / 3, 0, 0], # 4 5
[L, H / 3, 0], [L, 2 * H / 3, 0], # 6 7
[L / 3, H, Z], [2 * L / 3, H, Z], # 8 9
[0, H / 3, 0], [0, 2 * H / 3, 0], # 10 11
[L / 3, H / 3, 0], [2 * L / 3, H / 3, 0], # 12 13
[L / 3, 2 * H / 3, 0], [2 * L / 3, 2 * H / 3, 0], # 14 15
[2 * L, 0, 0], [2 * L, H, Z], # 16 17
[4 * L / 3, 0, 0], [5 * L / 3, 0, 0], # 18 19
[2 * L, H / 3, 0], [2 * L, 2 * H / 3, 0], # 20 21
[4 * L / 3, H, Z], [5 * L / 3, H, Z], # 22 23
[4 * L / 3, H / 3, 0], [5 * L / 3, H / 3, 0], # 24 25
[4 * L / 3, 2 * H / 3, 0], [5 * L / 3, 2 * H / 3, 0]]) # 26 27
# Change to multiple cells when matthews dof-maps work for quads
cells = np.array([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15],
[1, 16, 17, 2, 18, 19, 20, 21, 22, 23, 6, 7, 24, 25, 26, 27]])
cells = cells[:, perm_vtk(CellType.quadrilateral, cells.shape[1])]
cell = ufl.Cell("quadrilateral", geometric_dimension=points.shape[1])
domain = ufl.Mesh(ufl.VectorElement("Lagrange", cell, 3))
mesh = create_mesh(MPI.COMM_WORLD, cells, points, domain)
def e2(x):
return x[2] + x[0] * x[1]
# Interpolate function
V = FunctionSpace(mesh, ("CG", 3))
u = Function(V)
u.interpolate(e2)
intu = assemble_scalar(u * dx(mesh))
intu = mesh.mpi_comm().allreduce(intu, op=MPI.SUM)
nodes = [0, 3, 10, 11]
ref = sympy_scipy(points, nodes, 2 * L, H)
assert ref == pytest.approx(intu, rel=1e-6)
def test_fourth_order_tri():
L = 1
# *--*--*--*--* 3-21-20-19--2
# | \ | | \ |
# * * * * * 10 9 24 23 18
# | \ | | \ |
# * * * * * 11 15 8 22 17
# | \ | | \ |
# * * * * * 12 13 14 7 16
# | \ | | \ |
# *--*--*--*--* 0--4--5--6--1
for H in (1.0, 2.0):
for Z in (0.0, 0.5):
points = np.array(
[[0, 0, 0], [L, 0, 0], [L, H, Z], [0, H, Z], # 0, 1, 2, 3
[L / 4, 0, 0], [L / 2, 0, 0], [3 * L / 4, 0, 0], # 4, 5, 6
[3 / 4 * L, H / 4, Z / 2], [L / 2, H / 2, 0], # 7, 8
[L / 4, 3 * H / 4, 0], [0, 3 * H / 4, 0], # 9, 10
[0, H / 2, 0], [0, H / 4, Z / 2], # 11, 12
[L / 4, H / 4, Z / 2], [L / 2, H / 4, Z / 2], [L / 4, H / 2, 0], # 13, 14, 15
[L, H / 4, Z / 2], [L, H / 2, 0], [L, 3 * H / 4, 0], # 16, 17, 18
[3 * L / 4, H, Z], [L / 2, H, Z], [L / 4, H, Z], # 19, 20, 21
[3 * L / 4, H / 2, 0], [3 * L / 4, 3 * H / 4, 0], # 22, 23
[L / 2, 3 * H / 4, 0]] # 24
)
cells = np.array([[0, 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15],
[1, 2, 3, 16, 17, 18, 19, 20, 21, 9, 8, 7, 22, 23, 24]])
cells = cells[:, perm_vtk(CellType.triangle, cells.shape[1])]
cell = ufl.Cell("triangle", geometric_dimension=points.shape[1])
domain = ufl.Mesh(ufl.VectorElement("Lagrange", cell, 4))
mesh = create_mesh(MPI.COMM_WORLD, cells, points, domain)
def e2(x):
return x[2] + x[0] * x[1]
# Interpolate function
V = FunctionSpace(mesh, ("Lagrange", 4))
u = Function(V)
u.interpolate(e2)
intu = assemble_scalar(u * dx(metadata={"quadrature_degree": 50}))
intu = mesh.mpi_comm().allreduce(intu, op=MPI.SUM)
nodes = [0, 3, 10, 11, 12]
ref = sympy_scipy(points, nodes, L, H)
assert ref == pytest.approx(intu, rel=1e-4)
def test_gmsh_input_quad(order):
pygmsh = pytest.importorskip("pygmsh")
# Parameterize test if gmsh gets wider support
R = 1
res = 0.2 if order == 2 else 0.2
algorithm = 2 if order == 2 else 5
element = "quad{0:d}".format(int((order + 1)**2))
geo = pygmsh.opencascade.Geometry()
geo.add_raw_code("Mesh.ElementOrder={0:d};".format(order))
geo.add_ball([0, 0, 0], R, char_length=res)
geo.add_raw_code("Recombine Surface {1};")
geo.add_raw_code("Mesh.Algorithm = {0:d};".format(algorithm))
msh = pygmsh.generate_mesh(geo, verbose=True, dim=2)
if order > 2:
# Quads order > 3 have a gmsh specific ordering, and has to be
# re-mapped
msh_to_dolfin = np.array(
[0, 3, 11, 10, 1, 2, 6, 7, 4, 9, 12, 15, 5, 8, 13, 14])
cells = np.zeros(msh.cells_dict[element].shape)
for i in range(len(cells)):
for j in range(len(msh_to_dolfin)):
cells[i, j] = msh.cells_dict[element][i, msh_to_dolfin[j]]
else:
# XDMF does not support higher order quads
cells = msh.cells_dict[
element][:,
perm_vtk(CellType.quadrilateral, msh.cells_dict[element].
shape[1])]
mesh = Mesh(MPI.COMM_WORLD,
CellType.quadrilateral,
msh.points,
cells, [],
degree=order)
surface = assemble_scalar(1 * dx(mesh))
assert mesh.mpi_comm().allreduce(surface, op=MPI.SUM) == pytest.approx(
4 * np.pi * R * R, rel=1e-5)
def test_second_order_tri():
# Test second order mesh by computing volume of two cells
# *-----*-----* 3----6-----2
# | \ | | \ |
# | \ | | \ |
# * * * 7 8 5
# | \ | | \ |
# | \ | | \ |
# *-----*-----* 0----4-----1
for H in (1.0, 2.0):
for Z in (0.0, 0.5):
L = 1
points = np.array([[0, 0, 0], [L, 0, 0], [L, H, Z], [0, H, Z],
[L / 2, 0, 0], [L, H / 2, 0], [L / 2, H, Z],
[0, H / 2, 0], [L / 2, H / 2, 0]])
cells = np.array([[0, 1, 3, 4, 8, 7], [1, 2, 3, 5, 6, 8]])
cells = cells[:, perm_vtk(CellType.triangle, cells.shape[1])]
mesh = Mesh(MPI.COMM_WORLD,
CellType.triangle,
points,
cells, [],
degree=2)
def e2(x):
return x[2] + x[0] * x[1]
# Interpolate function
V = FunctionSpace(mesh, ("CG", 2))
u = Function(V)
u.interpolate(e2)
intu = assemble_scalar(
u * dx(mesh, metadata={"quadrature_degree": 20}))
intu = mesh.mpi_comm().allreduce(intu, op=MPI.SUM)
nodes = [0, 3, 7]
ref = sympy_scipy(points, nodes, L, H)
assert ref == pytest.approx(intu, rel=1e-6)
def test_quadrilateral_mesh_vtk(order):
random.seed(13)
points = []
points += [[i / order, 0] for i in range(order + 1)]
for j in range(1, order):
points += [[i / order + 0.1, j / order] for i in range(order + 1)]
points += [[j / order, 1] for j in range(order + 1)]
def coord_to_vertex(x, y):
return (order + 1) * y + x
# Make the cell, following https://blog.kitware.com/modeling-arbitrary-order-lagrange-finite-elements-in-the-visualization-toolkit/ # noqa: E501
cell = [
coord_to_vertex(i, j)
for i, j in [(0, 0), (order, 0), (order, order), (0, order)]
]
if order > 1:
for i in range(1, order):
cell.append(coord_to_vertex(i, 0))
for i in range(1, order):
cell.append(coord_to_vertex(order, i))
for i in range(1, order):
cell.append(coord_to_vertex(i, order))
for i in range(1, order):
cell.append(coord_to_vertex(0, i))
for j in range(1, order):
for i in range(1, order):
cell.append(coord_to_vertex(i, j))
cell = np.array(cell)[perm_vtk(CellType.quadrilateral, len(cell))]
domain = ufl.Mesh(
ufl.VectorElement("Q", ufl.Cell("quadrilateral",
geometric_dimension=2), order))
check_cell_volume(points, cell, domain, 1)
def test_second_order_quad(L, H, Z):
""" Test by comparing integration of z+x*y against sympy/scipy
integration of a quad element. Z>0 implies curved element.
*-----* 3--6--2
| | | |
| | 7 8 5
| | | |
*-----* 0--4--1
"""
points = np.array([[0, 0, 0], [L, 0, 0], [L, H, Z], [0, H, Z],
[L / 2, 0, 0], [L, H / 2, 0], [L / 2, H, Z],
[0, H / 2, 0], [L / 2, H / 2, 0], [2 * L, 0, 0],
[2 * L, H, Z]])
cells = np.array([[0, 1, 2, 3, 4, 5, 6, 7, 8]])
cells = cells[:, perm_vtk(CellType.quadrilateral, cells.shape[1])]
mesh = Mesh(MPI.COMM_WORLD,
CellType.quadrilateral,
points,
cells, [],
degree=2)
def e2(x):
return x[2] + x[0] * x[1]
# Interpolate function
V = FunctionSpace(mesh, ("CG", 2))
u = Function(V)
u.interpolate(e2)
intu = assemble_scalar(u * dx(mesh))
intu = mesh.mpi_comm().allreduce(intu, op=MPI.SUM)
nodes = [0, 3, 7]
ref = sympy_scipy(points, nodes, L, H)
assert ref == pytest.approx(intu, rel=1e-6)
def test_triangle_mesh_vtk(order):
points = []
points += [[i / order, 0] for i in range(order + 1)]
for j in range(1, order):
points += [[i / order + 0.1, j / order] for i in range(order + 1 - j)]
points += [[0, 1]]
def coord_to_vertex(x, y):
return y * (2 * order + 3 - y) // 2 + x
# Make the cell, following https://blog.kitware.com/modeling-arbitrary-order-lagrange-finite-elements-in-the-visualization-toolkit/ # noqa: E501
cell = [coord_to_vertex(i, j) for i, j in [(0, 0), (order, 0), (0, order)]]
if order > 1:
for i in range(1, order):
cell.append(coord_to_vertex(i, 0))
for i in range(1, order):
cell.append(coord_to_vertex(order - i, i))
for i in range(1, order):
cell.append(coord_to_vertex(0, order - i))
if order == 3:
cell.append(coord_to_vertex(1, 1))
elif order > 3:
cell.append(coord_to_vertex(1, 1))
cell.append(coord_to_vertex(order - 2, 1))
cell.append(coord_to_vertex(1, order - 2))
if order > 4:
raise NotImplementedError
cell = np.array(cell)[perm_vtk(CellType.triangle, len(cell))]
domain = ufl.Mesh(
ufl.VectorElement("Lagrange",
ufl.Cell("triangle", geometric_dimension=2), order))
check_cell_volume(points, cell, domain, 0.5)
def test_nth_order_triangle(order):
num_nodes = (order + 1) * (order + 2) / 2
cells = np.array([range(int(num_nodes))])
cells = cells[:, perm_vtk(CellType.triangle, cells.shape[1])]
if order == 1:
points = np.array([[0.00000, 0.00000, 0.00000],
[1.00000, 0.00000, 0.00000],
[0.00000, 1.00000, 0.00000]])
elif order == 2:
points = np.array([[0.00000, 0.00000, 0.00000],
[1.00000, 0.00000, 0.00000],
[0.00000, 1.00000, 0.00000],
[0.50000, 0.00000, 0.00000],
[0.50000, 0.50000, -0.25000],
[0.00000, 0.50000, -0.25000]])
elif order == 3:
points = np.array([[0.00000, 0.00000, 0.00000],
[1.00000, 0.00000, 0.00000],
[0.00000, 1.00000, 0.00000],
[0.33333, 0.00000, 0.00000],
[0.66667, 0.00000, 0.00000],
[0.66667, 0.33333, -0.11111],
[0.33333, 0.66667, 0.11111],
[0.00000, 0.66667, 0.11111],
[0.00000, 0.33333, -0.11111],
[0.33333, 0.33333, -0.11111]])
elif order == 4:
points = np.array([[0.00000, 0.00000, 0.00000],
[1.00000, 0.00000, 0.00000],
[0.00000, 1.00000, 0.00000],
[0.25000, 0.00000, 0.00000],
[0.50000, 0.00000, 0.00000],
[0.75000, 0.00000, 0.00000],
[0.75000, 0.25000, -0.06250],
[0.50000, 0.50000, 0.06250],
[0.25000, 0.75000, -0.06250],
[0.00000, 0.75000, -0.06250],
[0.00000, 0.50000, 0.06250],
[0.00000, 0.25000, -0.06250],
[0.25000, 0.25000, -0.06250],
[0.50000, 0.25000, -0.06250],
[0.25000, 0.50000, 0.06250]])
elif order == 5:
points = np.array([[0.00000, 0.00000, 0.00000],
[1.00000, 0.00000, 0.00000],
[0.00000, 1.00000, 0.00000],
[0.20000, 0.00000, 0.00000],
[0.40000, 0.00000, 0.00000],
[0.60000, 0.00000, 0.00000],
[0.80000, 0.00000, 0.00000],
[0.80000, 0.20000, -0.04000],
[0.60000, 0.40000, 0.04000],
[0.40000, 0.60000, -0.04000],
[0.20000, 0.80000, 0.04000],
[0.00000, 0.80000, 0.04000],
[0.00000, 0.60000, -0.04000],
[0.00000, 0.40000, 0.04000],
[0.00000, 0.20000, -0.04000],
[0.20000, 0.20000, -0.04000],
[0.60000, 0.20000, -0.04000],
[0.20000, 0.60000, -0.04000],
[0.40000, 0.20000, -0.04000],
[0.40000, 0.40000, 0.04000],
[0.20000, 0.40000, 0.04000]])
elif order == 6:
points = np.array([[0.00000, 0.00000, 0.00000],
[1.00000, 0.00000, 0.00000],
[0.00000, 1.00000, 0.00000],
[0.16667, 0.00000, 0.00000],
[0.33333, 0.00000, 0.00000],
[0.50000, 0.00000, 0.00000],
[0.66667, 0.00000, 0.00000],
[0.83333, 0.00000, 0.00000],
[0.83333, 0.16667, -0.00463],
[0.66667, 0.33333, 0.00463],
[0.50000, 0.50000, -0.00463],
[0.33333, 0.66667, 0.00463],
[0.16667, 0.83333, -0.00463],
[0.00000, 0.83333, -0.00463],
[0.00000, 0.66667, 0.00463],
[0.00000, 0.50000, -0.00463],
[0.00000, 0.33333, 0.00463],
[0.00000, 0.16667, -0.00463],
[0.16667, 0.16667, -0.00463],
[0.66667, 0.16667, -0.00463],
[0.16667, 0.66667, 0.00463],
[0.33333, 0.16667, -0.00463],
[0.50000, 0.16667, -0.00463],
[0.50000, 0.33333, 0.00463],
[0.33333, 0.50000, -0.00463],
[0.16667, 0.50000, -0.00463],
[0.16667, 0.33333, 0.00463],
[0.33333, 0.33333, 0.00463]])
elif order == 7:
points = np.array([[0.00000, 0.00000, 0.00000],
[1.00000, 0.00000, 0.00000],
[0.00000, 1.00000, 0.00000],
[0.14286, 0.00000, 0.00000],
[0.28571, 0.00000, 0.00000],
[0.42857, 0.00000, 0.00000],
[0.57143, 0.00000, 0.00000],
[0.71429, 0.00000, 0.00000],
[0.85714, 0.00000, 0.00000],
[0.85714, 0.14286, -0.02041],
[0.71429, 0.28571, 0.02041],
[0.57143, 0.42857, -0.02041],
[0.42857, 0.57143, 0.02041],
[0.28571, 0.71429, -0.02041],
[0.14286, 0.85714, 0.02041],
[0.00000, 0.85714, 0.02041],
[0.00000, 0.71429, -0.02041],
[0.00000, 0.57143, 0.02041],
[0.00000, 0.42857, -0.02041],
[0.00000, 0.28571, 0.02041],
[0.00000, 0.14286, -0.02041],
[0.14286, 0.14286, -0.02041],
[0.71429, 0.14286, -0.02041],
[0.14286, 0.71429, -0.02041],
[0.28571, 0.14286, -0.02041],
[0.42857, 0.14286, -0.02041],
[0.57143, 0.14286, -0.02041],
[0.57143, 0.28571, 0.02041],
[0.42857, 0.42857, -0.02041],
[0.28571, 0.57143, 0.02041],
[0.14286, 0.57143, 0.02041],
[0.14286, 0.42857, -0.02041],
[0.14286, 0.28571, 0.02041],
[0.28571, 0.28571, 0.02041],
[0.42857, 0.28571, 0.02041],
[0.28571, 0.42857, -0.02041]])
# Higher order tests are too slow
elif order == 8:
points = np.array([[0.00000, 0.00000, 0.00000],
[1.00000, 0.00000, 0.00000],
[0.00000, 1.00000, 0.00000],
[0.12500, 0.00000, 0.00000],
[0.25000, 0.00000, 0.00000],
[0.37500, 0.00000, 0.00000],
[0.50000, 0.00000, 0.00000],
[0.62500, 0.00000, 0.00000],
[0.75000, 0.00000, 0.00000],
[0.87500, 0.00000, 0.00000],
[0.87500, 0.12500, -0.00195],
[0.75000, 0.25000, 0.00195],
[0.62500, 0.37500, -0.00195],
[0.50000, 0.50000, 0.00195],
[0.37500, 0.62500, -0.00195],
[0.25000, 0.75000, 0.00195],
[0.12500, 0.87500, -0.00195],
[0.00000, 0.87500, -0.00195],
[0.00000, 0.75000, 0.00195],
[0.00000, 0.62500, -0.00195],
[0.00000, 0.50000, 0.00195],
[0.00000, 0.37500, -0.00195],
[0.00000, 0.25000, 0.00195],
[0.00000, 0.12500, -0.00195],
[0.12500, 0.12500, -0.00195],
[0.75000, 0.12500, -0.00195],
[0.12500, 0.75000, 0.00195],
[0.25000, 0.12500, -0.00195],
[0.37500, 0.12500, -0.00195],
[0.50000, 0.12500, -0.00195],
[0.62500, 0.12500, -0.00195],
[0.62500, 0.25000, 0.00195],
[0.50000, 0.37500, -0.00195],
[0.37500, 0.50000, 0.00195],
[0.25000, 0.62500, -0.00195],
[0.12500, 0.62500, -0.00195],
[0.12500, 0.50000, 0.00195],
[0.12500, 0.37500, -0.00195],
[0.12500, 0.25000, 0.00195],
[0.25000, 0.25000, 0.00195],
[0.50000, 0.25000, 0.00195],
[0.25000, 0.50000, 0.00195],
[0.37500, 0.25000, 0.00195],
[0.37500, 0.37500, -0.00195],
[0.25000, 0.37500, -0.00195]])
cell = ufl.Cell("triangle", geometric_dimension=points.shape[1])
domain = ufl.Mesh(ufl.VectorElement("Lagrange", cell, order))
mesh = create_mesh(MPI.COMM_WORLD, cells, points, domain)
# Find nodes corresponding to y axis
nodes = []
for j in range(points.shape[0]):
if np.isclose(points[j][0], 0):
nodes.append(j)
def e2(x):
return x[2] + x[0] * x[1]
# For solution to be in functionspace
V = FunctionSpace(mesh, ("CG", max(2, order)))
u = Function(V)
u.interpolate(e2)
quad_order = 30
intu = assemble_scalar(u * dx(metadata={"quadrature_degree": quad_order}))
intu = mesh.mpi_comm().allreduce(intu, op=MPI.SUM)
ref = scipy_one_cell(points, nodes)
assert ref == pytest.approx(intu, rel=3e-3)
def test_hexahedron_mesh_vtk(order):
if order > 2:
pytest.xfail(
"VTK permutation for order > 2 hexahedra not implemented in DOLFINX."
)
random.seed(13)
points = []
points += [[i / order, j / order, 0] for j in range(order + 1)
for i in range(order + 1)]
for k in range(1, order):
points += [[i / order, j / order + 0.1, k / order]
for j in range(order + 1) for i in range(order + 1)]
points += [[i / order, j / order, 1] for j in range(order + 1)
for i in range(order + 1)]
def coord_to_vertex(x, y, z):
return (order + 1)**2 * z + (order + 1) * y + x
# Make the cell, following https://blog.kitware.com/modeling-arbitrary-order-lagrange-finite-elements-in-the-visualization-toolkit/ # noqa: E501
cell = [
coord_to_vertex(x, y, z)
for x, y, z in [(0, 0, 0), (order, 0, 0), (
order, order,
0), (0, order, 0), (0, 0,
order), (order, 0,
order), (order, order,
order), (0, order, order)]
]
if order > 1:
for i in range(1, order):
cell.append(coord_to_vertex(i, 0, 0))
for i in range(1, order):
cell.append(coord_to_vertex(order, i, 0))
for i in range(1, order):
cell.append(coord_to_vertex(i, order, 0))
for i in range(1, order):
cell.append(coord_to_vertex(0, i, 0))
for i in range(1, order):
cell.append(coord_to_vertex(i, 0, order))
for i in range(1, order):
cell.append(coord_to_vertex(order, i, order))
for i in range(1, order):
cell.append(coord_to_vertex(i, order, order))
for i in range(1, order):
cell.append(coord_to_vertex(0, i, order))
for i in range(1, order):
cell.append(coord_to_vertex(0, 0, i))
for i in range(1, order):
cell.append(coord_to_vertex(order, 0, i))
for i in range(1, order):
cell.append(coord_to_vertex(order, order, i))
for i in range(1, order):
cell.append(coord_to_vertex(0, order, i))
# The ordering of faces does not match documentation.
# See https://gitlab.kitware.com/vtk/vtk/uploads/a0dc0173a41d3cf6b03a9266c0e23688/image.png
# The edge flip in this like however has been fixed in VTK so we follow the main documentation link for edges
for j in range(1, order):
for i in range(1, order):
cell.append(coord_to_vertex(0, i, j))
for j in range(1, order):
for i in range(1, order):
cell.append(coord_to_vertex(order, i, j))
for j in range(1, order):
for i in range(1, order):
cell.append(coord_to_vertex(i, 0, j))
for j in range(1, order):
for i in range(1, order):
cell.append(coord_to_vertex(i, order, j))
for j in range(1, order):
for i in range(1, order):
cell.append(coord_to_vertex(i, j, 0))
for j in range(1, order):
for i in range(1, order):
cell.append(coord_to_vertex(i, j, order))
for k in range(1, order):
for j in range(1, order):
for i in range(1, order):
cell.append(coord_to_vertex(i, j, k))
cell = np.array(cell)[perm_vtk(CellType.hexahedron, len(cell))]
domain = ufl.Mesh(
ufl.VectorElement("Q", ufl.Cell("hexahedron", geometric_dimension=3),
order))
check_cell_volume(points, cell, domain, 1)
def test_tetrahedron_mesh_vtk(order):
if order > 3:
pytest.xfail(
"VTK permutation for order > 3 tetrahedra not implemented in DOLFINX."
)
points = []
points += [[i / order, j / order, 0] for j in range(order + 1)
for i in range(order + 1 - j)]
for k in range(1, order):
points += [[i / order, j / order + 0.1, k / order]
for j in range(order + 1 - k)
for i in range(order + 1 - k - j)]
points += [[0, 0, 1]]
def coord_to_vertex(x, y, z):
return z * (3 * order**2 - 3 * order * z + 12 * order + z**2 - 6 * z +
11) // 6 + y * (2 * (order - z) + 3 - y) // 2 + x
# Make the cell, following https://blog.kitware.com/modeling-arbitrary-order-lagrange-finite-elements-in-the-visualization-toolkit/ # noqa: E501
cell = [
coord_to_vertex(x, y, z)
for x, y, z in [(0, 0, 0), (order, 0, 0), (0, order, 0), (0, 0, order)]
]
if order > 1:
for i in range(1, order):
cell.append(coord_to_vertex(i, 0, 0))
for i in range(1, order):
cell.append(coord_to_vertex(order - i, i, 0))
for i in range(1, order):
cell.append(coord_to_vertex(0, order - i, 0))
for i in range(1, order):
cell.append(coord_to_vertex(0, 0, i))
for i in range(1, order):
cell.append(coord_to_vertex(order - i, 0, i))
for i in range(1, order):
cell.append(coord_to_vertex(0, order - i, i))
if order == 3:
# The ordering of faces does not match documentation.
# See https://gitlab.kitware.com/vtk/vtk/uploads/a0dc0173a41d3cf6b03a9266c0e23688/image.png
cell.append(coord_to_vertex(1, 0, 1))
cell.append(coord_to_vertex(1, 1, 1))
cell.append(coord_to_vertex(0, 1, 1))
cell.append(coord_to_vertex(1, 1, 0))
elif order == 4:
# The ordering of faces does not match documentation.
# See https://gitlab.kitware.com/vtk/vtk/uploads/a0dc0173a41d3cf6b03a9266c0e23688/image.png
cell.append(coord_to_vertex(1, 0, 1))
cell.append(coord_to_vertex(2, 0, 1))
cell.append(coord_to_vertex(1, 0, 2))
cell.append(coord_to_vertex(1, 2, 1))
cell.append(coord_to_vertex(1, 1, 2))
cell.append(coord_to_vertex(2, 1, 1))
cell.append(coord_to_vertex(0, 1, 1))
cell.append(coord_to_vertex(0, 1, 2))
cell.append(coord_to_vertex(0, 2, 1))
cell.append(coord_to_vertex(1, 1, 0))
cell.append(coord_to_vertex(1, 2, 0))
cell.append(coord_to_vertex(2, 1, 0))
cell.append(coord_to_vertex(1, 1, 1))
elif order > 4:
raise NotImplementedError
if False:
for j in range(1, order):
for i in range(1, order - j):
cell.append(coord_to_vertex(i, 0, j))
for j in range(1, order):
for i in range(1, order - j):
cell.append(coord_to_vertex(0, i, j))
for j in range(1, order):
for i in range(1, order - j):
cell.append(coord_to_vertex(i, j, 0))
for j in range(1, order):
for i in range(1, order - j):
cell.append(coord_to_vertex(order - i - j, i, j))
for k in range(1, order):
for j in range(1, order - k):
for i in range(1, order - j - k):
cell.append(coord_to_vertex(i, j, k))
cell = np.array(cell)[perm_vtk(CellType.tetrahedron, len(cell))]
domain = ufl.Mesh(
ufl.VectorElement("Lagrange",
ufl.Cell("tetrahedron", geometric_dimension=3),
order))
check_cell_volume(points, cell, domain, 1 / 6)
