def facet_reference_edge_vectors(L, tablename, cellname):
celltype = getattr(basix.CellType, cellname)
topology = basix.topology(celltype)
geometry = basix.geometry(celltype)
triangle_edges = basix.topology(basix.CellType.triangle)[1]
quadrilateral_edges = basix.topology(basix.CellType.quadrilateral)[1]
if len(topology) != 4:
raise ValueError("Can only get facet edges for 3D cells.")
edge_vectors = []
for facet in topology[-2]:
if len(facet) == 3:
edge_vectors += [
geometry[facet[j]] - geometry[facet[i]]
for i, j in triangle_edges
]
elif len(facet) == 4:
edge_vectors += [
geometry[facet[j]] - geometry[facet[i]]
for i, j in quadrilateral_edges
]
else:
raise ValueError(
"Only triangular and quadrilateral faces supported.")
out = numpy.array(edge_vectors)
return L.ArrayDecl("static const double", f"{cellname}_{tablename}",
out.shape, out)
def test_outward_normals(cell):
cell_type = getattr(basix.CellType, cell)
normals = basix.cell.facet_outward_normals(cell_type)
facets = basix.topology(cell_type)[-2]
geometry = basix.geometry(cell_type)
midpoint = sum(geometry) / len(geometry)
for normal, facet in zip(normals, facets):
assert np.dot(geometry[facet[0]] - midpoint, normal) > 0
def reference_edge_vectors(L, tablename, cellname):
celltype = getattr(basix.CellType, cellname)
topology = basix.topology(celltype)
geometry = basix.geometry(celltype)
edge_vectors = [geometry[j] - geometry[i] for i, j in topology[1]]
out = numpy.array(edge_vectors[cellname])
return L.ArrayDecl("static const double", f"{cellname}_{tablename}",
out.shape, out)
def test_normals(cell):
cell_type = getattr(basix.CellType, cell)
normals = basix.cell.facet_normals(cell_type)
facets = basix.topology(cell_type)[-2]
geometry = basix.geometry(cell_type)
for normal, facet in zip(normals, facets):
assert np.isclose(np.linalg.norm(normal), 1)
for v in facet[1:]:
tangent = geometry[v] - geometry[facet[0]]
assert np.isclose(np.dot(tangent, normal), 0)
def map_facet_points(points, facet, cellname):
geom = basix.geometry(basix_cells[cellname])
facet_vertices = [
geom[i] for i in basix.topology(basix_cells[cellname])[-2][facet]
]
return [
facet_vertices[0] + sum((i - facet_vertices[0]) * j
for i, j in zip(facet_vertices[1:], p))
for p in points
]
def map_facet_points(points, facet, cellname):
"""Map points from a reference facet to a physical facet."""
geom = basix.geometry(basix.cell.string_to_type(cellname))
facet_vertices = [
geom[i]
for i in basix.topology(basix.cell.string_to_type(cellname))[-2][facet]
]
return [
facet_vertices[0] + sum((i - facet_vertices[0]) * j
for i, j in zip(facet_vertices[1:], p))
for p in points
]
def sympy_lagrange(celltype, n):
x = sympy.Symbol("x")
y = sympy.Symbol("y")
z = sympy.Symbol("z")
from sympy import S
topology = basix.topology(celltype)
geometry = S(basix.geometry(celltype).astype(int))
pt = []
for dim, entities in enumerate(topology):
for ent in entities:
entity_geom = [geometry[t, :] for t in ent]
if (dim == 0):
pt += [entity_geom[0]]
elif (dim == 1):
for i in range(n - 1):
pt += [
entity_geom[0] + sympy.Rational(i + 1, n) *
(entity_geom[1] - entity_geom[0])
]
elif (dim == 2):
for i in range(n - 2):
for j in range(n - 2 - i):
pt += [
entity_geom[0] + sympy.Rational(i + 1, n) *
(entity_geom[2] - entity_geom[0]) +
sympy.Rational(j + 1, n) *
(entity_geom[1] - entity_geom[0])
]
elif (dim == 3):
for i in range(n - 3):
for j in range(n - 3 - i):
for k in range(n - 3 - i - j):
pt += [
entity_geom[0] + sympy.Rational(i + 1, n) *
(entity_geom[3] - entity_geom[0]) +
sympy.Rational(j + 1, n) *
(entity_geom[2] - entity_geom[0]) +
sympy.Rational(k + 1, n) *
(entity_geom[1] - entity_geom[0])
]
funcs = []
if celltype == basix.CellType.interval:
for i in range(n + 1):
funcs += [x**i]
mat = numpy.empty((len(pt), len(funcs)), dtype=object)
for i, f in enumerate(funcs):
for j, p in enumerate(pt):
mat[i, j] = f.subs([(x, p[0])])
elif celltype == basix.CellType.triangle:
for i in range(n + 1):
for j in range(n + 1 - i):
funcs += [x**j * y**i]
mat = numpy.empty((len(pt), len(funcs)), dtype=object)
for i, f in enumerate(funcs):
for j, p in enumerate(pt):
mat[i, j] = f.subs([(x, p[0]), (y, p[1])])
elif celltype == basix.CellType.tetrahedron:
for i in range(n + 1):
for j in range(n + 1 - i):
for k in range(n + 1 - i - j):
funcs += [x**j * y**i * z**k]
mat = numpy.empty((len(pt), len(funcs)), dtype=object)
for i, f in enumerate(funcs):
for j, p in enumerate(pt):
mat[i, j] = f.subs([(x, p[0]), (y, p[1]), (z, p[2])])
mat = sympy.Matrix(mat)
mat = mat.inv()
g = []
for r in range(mat.shape[0]):
g += [sum([v * funcs[i] for i, v in enumerate(mat.row(r))])]
return g
def sympy_rt(celltype, n):
x = sympy.Symbol("x")
y = sympy.Symbol("y")
z = sympy.Symbol("z")
from sympy import S
topology = basix.topology(celltype)
geometry = S(basix.geometry(celltype).astype(int))
dummy = [
sympy.Symbol("DUMMY1"),
sympy.Symbol("DUMMY2"),
sympy.Symbol("DUMMY3")
]
funcs = []
if celltype == basix.CellType.triangle:
tdim = 2
for i in range(n):
for j in range(n - i):
for d in range(2):
funcs += [[x**j * y**i if k == d else 0 for k in range(2)]]
for i in range(n):
funcs.append([x**(n - i) * y**i, x**(n - 1 - i) * y**(i + 1)])
mat = numpy.empty((len(funcs), len(funcs)), dtype=object)
# edge normals
for i, f in enumerate(funcs):
if n == 1:
edge_basis = [sympy.Integer(1)]
else:
edge_basis = sympy_lagrange(basix.CellType.interval, n - 1)
edge_basis = [a.subs(x, dummy[0]) for a in edge_basis]
j = 0
for edge in topology[1]:
edge_geom = [geometry[t, :] for t in edge]
tangent = edge_geom[1] - edge_geom[0]
norm = sympy.sqrt(sum(i**2 for i in tangent))
tangent = [i / norm for i in tangent]
normal = [-tangent[1], tangent[0]]
param = [(1 - dummy[0]) * a + dummy[0] * b
for a, b in zip(edge_geom[0], edge_geom[1])]
for g in edge_basis:
integrand = sum((f_i * v_i) for f_i, v_i in zip(f, normal))
integrand = integrand.subs(x, param[0]).subs(y, param[1])
integrand *= g * norm
mat[i, j] = integrand.integrate((dummy[0], 0, 1))
j += 1
# interior dofs
if n > 1:
for i, f in enumerate(funcs):
if n == 2:
face_basis = [sympy.Integer(1)]
else:
face_basis = sympy_lagrange(basix.CellType.triangle, n - 2)
j = n * 3
for g in face_basis:
for vec in [(1, 0), (0, 1)]:
integrand = sum(
(f_i * v_i) for f_i, v_i in zip(f, vec)) * g
mat[i, j] = integrand.integrate(
(x, 0, 1 - y)).integrate((y, 0, 1))
j += 1
elif celltype == basix.CellType.tetrahedron:
tdim = 3
for i in range(n):
for j in range(n - i):
for k in range(n - i - j):
for d in range(3):
funcs += [[
x**k * y**j * z**i if m == d else 0
for m in range(3)
]]
for j in range(n):
for k in range(n - j):
p = x**(n - 1 - j - k) * y**j * z**k
funcs.append((x * p, y * p, z * p))
mat = numpy.empty((len(funcs), len(funcs)), dtype=object)
# face normals
for i, f in enumerate(funcs):
if n == 1:
face_basis = [sympy.Integer(1)]
else:
face_basis = sympy_lagrange(basix.CellType.triangle, n - 1)
face_basis = [
a.subs(x, dummy[0]).subs(y, dummy[1]) for a in face_basis
]
j = 0
for face in topology[2]:
face_geom = [geometry[t, :] for t in face]
axes = [
face_geom[1] - face_geom[0], face_geom[2] - face_geom[0]
]
normal = [
axes[0][1] * axes[1][2] - axes[0][2] * axes[1][1],
axes[0][2] * axes[1][0] - axes[0][0] * axes[1][2],
axes[0][0] * axes[1][1] - axes[0][1] * axes[1][0]
]
norm = sympy.sqrt(sum(i**2 for i in normal))
normal = [k / norm for k in normal]
param = [
a + dummy[0] * b + dummy[1] * c
for a, b, c in zip(face_geom[0], *axes)
]
for g in face_basis:
integrand = sum(f_i * v_i for f_i, v_i in zip(f, normal))
integrand = integrand.subs(x, param[0]).subs(
y, param[1]).subs(z, param[2])
integrand *= g * norm
mat[i, j] = integrand.integrate(
(dummy[0], 0, 1 - dummy[1])).integrate(
(dummy[1], 0, 1))
j += 1
assert j == 2 * n * (n + 1)
if n > 1:
for i, f in enumerate(funcs):
if n == 2:
interior_basis = [sympy.Integer(1)]
else:
interior_basis = sympy_lagrange(basix.CellType.tetrahedron,
n - 2)
j = 2 * n * (n + 1)
for g in interior_basis:
for vec in [(1, 0, 0), (0, 1, 0), (0, 0, 1)]:
integrand = sum(f_i * v_i for f_i, v_i in zip(f, vec))
integrand *= g
mat[i, j] = integrand.integrate(
(x, 0, 1 - y - z)).integrate(
(y, 0, 1 - z)).integrate((z, 0, 1))
j += 1
mat = sympy.Matrix(mat)
mat = mat.inv()
g = []
for r in range(mat.shape[0]):
row = []
for dim in range(tdim):
row.append(
sum([v * funcs[i][dim] for i, v in enumerate(mat.row(r))]))
g.append(row)
return g
def sympy_nedelec(celltype, n):
x = sympy.Symbol("x")
y = sympy.Symbol("y")
z = sympy.Symbol("z")
from sympy import S
topology = basix.topology(celltype)
geometry = S(basix.geometry(celltype).astype(int))
dummy = [
sympy.Symbol("DUMMY1"),
sympy.Symbol("DUMMY2"),
sympy.Symbol("DUMMY3")
]
funcs = []
if celltype == basix.CellType.triangle:
tdim = 2
for i in range(n + 1):
for j in range(n + 1 - i):
for d in range(2):
funcs += [[x**j * y**i if k == d else 0 for k in range(2)]]
mat = numpy.empty((len(funcs), len(funcs)), dtype=object)
# edge tangents
edge_basis = sympy_lagrange(basix.CellType.interval, n)
edge_basis = [a.subs(x, dummy[0]) for a in edge_basis]
for i, f in enumerate(funcs):
j = 0
for edge in topology[1]:
edge_geom = [geometry[t, :] for t in edge]
tangent = edge_geom[1] - edge_geom[0]
norm = sympy.sqrt(sum(i**2 for i in tangent))
tangent = [i / norm for i in tangent]
param = [(1 - dummy[0]) * a + dummy[0] * b
for a, b in zip(edge_geom[0], edge_geom[1])]
for g in edge_basis:
integrand = sum(
(f_i * v_i) for f_i, v_i in zip(f, tangent))
integrand = integrand.subs(x, param[0]).subs(y, param[1])
integrand *= g * norm
mat[i, j] = integrand.integrate((dummy[0], 0, 1))
j += 1
# interior dofs
if n > 1:
face_basis = sympy_rt(basix.CellType.triangle, n - 1)
for i, f in enumerate(funcs):
j = (n + 1) * 3
for g in face_basis:
integrand = sum((f_i * v_i) for f_i, v_i in zip(f, g))
mat[i, j] = integrand.integrate((x, 0, 1 - y)).integrate(
(y, 0, 1))
j += 1
elif celltype == basix.CellType.tetrahedron:
tdim = 3
for i in range(n + 1):
for j in range(n + 1 - i):
for k in range(n + 1 - i - j):
for d in range(3):
funcs += [[
x**k * y**j * z**i if m == d else 0
for m in range(3)
]]
mat = numpy.empty((len(funcs), len(funcs)), dtype=object)
# edge tangents
edge_basis = sympy_lagrange(basix.CellType.interval, n)
edge_basis = [a.subs(x, dummy[0]) for a in edge_basis]
for i, f in enumerate(funcs):
j = 0
for edge in topology[1]:
edge_geom = [geometry[t, :] for t in edge]
tangent = edge_geom[1] - edge_geom[0]
norm = sympy.sqrt(sum(i**2 for i in tangent))
tangent = [i / norm for i in tangent]
param = [(1 - dummy[0]) * a + dummy[0] * b
for a, b in zip(edge_geom[0], edge_geom[1])]
for g in edge_basis:
integrand = sum(
(f_i * v_i) for f_i, v_i in zip(f, tangent))
integrand = integrand.subs(x, param[0]).subs(
y, param[1]).subs(z, param[2])
integrand *= g * norm
mat[i, j] = integrand.integrate((dummy[0], 0, 1))
j += 1
# face dofs
if n > 1:
face_basis = sympy_rt(basix.CellType.triangle, n - 1)
face_basis = [[b.subs(x, dummy[0]).subs(y, dummy[1]) for b in a]
for a in face_basis]
for i, f in enumerate(funcs):
j = (n + 1) * 6
for face in topology[2]:
face_geom = [geometry[t, :] for t in face]
axes = [
face_geom[1] - face_geom[0],
face_geom[2] - face_geom[0]
]
norm = sympy.sqrt(
sum(i**2 for i in [
axes[0][1] * axes[1][2] -
axes[0][2] * axes[1][1], axes[0][2] * axes[1][0] -
axes[0][0] * axes[1][2], axes[0][0] * axes[1][1] -
axes[0][1] * axes[1][0]
]))
scaled_axes = []
for a in axes:
scaled_axes.append([k / norm for k in a])
param = [
a + dummy[0] * b + dummy[1] * c
for a, b, c in zip(face_geom[0], *axes)
]
this_face_basis = [[
a[0] * b + a[1] * c for b, c in zip(*scaled_axes)
] for a in face_basis]
for g in this_face_basis:
integrand = sum(f_i * v_i for f_i, v_i in zip(f, g))
integrand = integrand.subs(x, param[0]).subs(
y, param[1]).subs(z, param[2])
integrand *= norm
mat[i, j] = integrand.integrate(
(dummy[0], 0, 1 - dummy[1])).integrate(
(dummy[1], 0, 1))
j += 1
# interior dofs
if n > 2:
interior_basis = sympy_rt(basix.CellType.tetrahedron, n - 2)
for i, f in enumerate(funcs):
j = (n + 1) * 6 + 4 * (n - 1) * (n + 1)
for g in interior_basis:
integrand = sum(f_i * v_i for f_i, v_i in zip(f, g))
mat[i, j] = integrand.integrate(
(x, 0, 1 - y - z)).integrate((y, 0, 1 - z)).integrate(
(z, 0, 1))
j += 1
mat = sympy.Matrix(mat)
mat = mat.inv()
g = []
for r in range(mat.shape[0]):
row = []
for dim in range(tdim):
row.append(
sum([v * funcs[i][dim] for i, v in enumerate(mat.row(r))]))
g.append(row)
return g
def sympy_nedelec(celltype, n):
x = sympy.Symbol("x")
y = sympy.Symbol("y")
z = sympy.Symbol("z")
from sympy import S
topology = basix.topology(celltype)
geometry = S(basix.geometry(celltype).astype(int))
dummy = [sympy.Symbol("DUMMY1"), sympy.Symbol("DUMMY2"), sympy.Symbol("DUMMY3")]
funcs = []
if celltype == basix.CellType.triangle:
tdim = 2
for i in range(n):
for j in range(n - i):
for d in range(2):
funcs += [[x**j * y**i if k == d else 0 for k in range(2)]]
for i in range(n):
funcs += [[x ** (n - 1 - i) * y ** (i + 1),
-x ** (n - i) * y ** i]]
mat = numpy.empty((len(funcs), len(funcs)), dtype=object)
# edge tangents
if n == 1:
edge_basis = [sympy.Integer(1)]
else:
edge_basis = sympy_lagrange(basix.CellType.interval, n - 1)
edge_basis = [a.subs(x, dummy[0]) for a in edge_basis]
for i, f in enumerate(funcs):
j = 0
for edge in topology[1]:
edge_geom = [geometry[t, :] for t in edge]
tangent = edge_geom[1] - edge_geom[0]
norm = sympy.sqrt(sum(i ** 2 for i in tangent))
tangent = [i / norm for i in tangent]
param = [(1 - dummy[0]) * a + dummy[0] * b for a, b in zip(edge_geom[0], edge_geom[1])]
for g in edge_basis:
integrand = sum((f_i * v_i) for f_i, v_i in zip(f, tangent))
integrand = integrand.subs(x, param[0]).subs(y, param[1])
integrand *= g * norm
mat[i, j] = integrand.integrate((dummy[0], 0, 1))
j += 1
# interior dofs
if n > 1:
if n == 2:
face_basis = [sympy.Integer(1)]
else:
face_basis = sympy_lagrange(basix.CellType.triangle, n - 2)
for i, f in enumerate(funcs):
j = n * 3
for g in face_basis:
for vec in [(1, 0), (0, 1)]:
integrand = sum((f_i * v_i) for f_i, v_i in zip(f, vec)) * g
mat[i, j] = integrand.integrate((x, 0, 1 - y)).integrate((y, 0, 1))
j += 1
elif celltype == basix.CellType.tetrahedron:
tdim = 3
for i in range(n):
for j in range(n - i):
for k in range(n - i - j):
for d in range(3):
funcs += [[x**k * y**j * z**i if m == d else 0 for m in range(3)]]
if n == 1:
funcs += [[y, -x, sympy.Integer(0)], [z, sympy.Integer(0), -x], [sympy.Integer(0), z, -y]]
elif n == 2:
funcs += [
[y ** 2, -x * y, sympy.Integer(0)],
[x * y, -x ** 2, sympy.Integer(0)],
[z * y, -z * x, sympy.Integer(0)],
[sympy.Integer(0), y * z, -y ** 2],
[sympy.Integer(0), z ** 2, -z * y],
[sympy.Integer(0), x * z, -x * y],
[x * z, sympy.Integer(0), -x ** 2],
[z ** 2, sympy.Integer(0), -z * x],
]
elif n == 3:
funcs += [
[x ** 2 * y, -x ** 3, sympy.Integer(0)],
[x ** 2 * z, sympy.Integer(0), -x ** 3],
[sympy.Integer(0), x ** 2 * z, -x ** 2 * y],
[x * y ** 2, -x ** 2 * y, sympy.Integer(0)],
[2 * x * y * z, -x ** 2 * z, -x ** 2 * y],
[sympy.Integer(0), x * y * z, -x * y ** 2],
[x * z ** 2, sympy.Integer(0), -x ** 2 * z],
[sympy.Integer(0), x * z ** 2, -x * y * z],
[y ** 3, -x * y ** 2, sympy.Integer(0)],
[9 * y ** 2 * z, -4 * x * y * z, -5 * x * y ** 2],
[sympy.Integer(0), y ** 2 * z, -y ** 3],
[9 * y * z ** 2, -5 * x * z ** 2, -4 * x * y * z],
[sympy.Integer(0), y * z ** 2, -y ** 2 * z],
[z ** 3, sympy.Integer(0), -x * z ** 2],
[sympy.Integer(0), z ** 3, -y * z ** 2],
]
else:
raise NotImplementedError
mat = numpy.empty((len(funcs), len(funcs)), dtype=object)
# edge tangents
if n == 1:
edge_basis = [sympy.Integer(1)]
else:
edge_basis = sympy_lagrange(basix.CellType.interval, n - 1)
edge_basis = [a.subs(x, dummy[0]) for a in edge_basis]
for i, f in enumerate(funcs):
j = 0
for edge in topology[1]:
edge_geom = [geometry[t, :] for t in edge]
tangent = edge_geom[1] - edge_geom[0]
norm = sympy.sqrt(sum(i ** 2 for i in tangent))
tangent = [i / norm for i in tangent]
param = [(1 - dummy[0]) * a + dummy[0] * b for a, b in zip(edge_geom[0], edge_geom[1])]
for g in edge_basis:
integrand = sum((f_i * v_i) for f_i, v_i in zip(f, tangent))
integrand = integrand.subs(x, param[0]).subs(y, param[1]).subs(z, param[2])
integrand *= g * norm
mat[i, j] = integrand.integrate((dummy[0], 0, 1))
j += 1
# face dofs
if n > 1:
def dot(a, b):
return sum(i * j for i, j in zip(a, b))
def cross(a, b):
assert len(a) == 3 and len(b) == 3
return [a[1] * b[2] - a[2] * b[1],
a[2] * b[0] - a[0] * b[2],
a[0] * b[1] - a[1] * b[0]]
if n == 2:
face_basis = [sympy.Integer(1)]
else:
face_basis = sympy_lagrange(basix.CellType.triangle, n - 2)
face_basis = [a.subs(x, dummy[0]).subs(y, dummy[1]) for a in face_basis]
for i, f in enumerate(funcs):
j = n * 6
for face in topology[2]:
face_geom = [geometry[t, :] for t in face]
axes = [face_geom[1] - face_geom[0], face_geom[2] - face_geom[0]]
norm = sympy.sqrt(sum(i**2 for i in cross(axes[0], axes[1])))
scaled_axes = []
for a in axes:
scaled_axes.append([k / norm for k in a])
param = [a + dummy[0] * b + dummy[1] * c for a, b, c in zip(face_geom[0], *axes)]
for g in face_basis:
for vec in scaled_axes:
integrand = dot(vec, f)
integrand = integrand.subs(x, param[0]).subs(y, param[1]).subs(z, param[2])
integrand *= g * norm
mat[i, j] = integrand.integrate((dummy[0], 0, 1 - dummy[1])).integrate((dummy[1], 0, 1))
j += 1
# interior dofs
if n > 2:
if n == 3:
interior_basis = [sympy.Integer(1)]
else:
interior_basis = sympy_lagrange(basix.CellType.tetrahedron, n - 3)
for i, f in enumerate(funcs):
j = n * 6 + 4 * n * (n - 1)
for g in interior_basis:
for vec in [(1, 0, 0), (0, 1, 0), (0, 0, 1)]:
integrand = sum(f_i * v_i for f_i, v_i in zip(f, vec))
integrand *= g
mat[i, j] = integrand.integrate((x, 0, 1 - y - z)).integrate((y, 0, 1 - z)).integrate((z, 0, 1))
j += 1
mat = sympy.Matrix(mat)
mat = mat.inv()
g = []
for r in range(mat.shape[0]):
row = []
for dim in range(tdim):
row.append(sum([v * funcs[i][dim] for i, v in enumerate(mat.row(r))]))
g.append(row)
return g
def reference_cell_vertices(cellname):
"""Get the vertices of a reference cell."""
return basix.geometry(basix.cell.string_to_type(cellname))
def reference_geometry(self):
"""Get the geometry of the reference element."""
return basix.geometry(self.element.cell_type)
def reference_cell_vertices(cellname):
return basix.geometry(basix_cells[cellname])
def reference_geometry(self):
return basix.geometry(self.element.cell_type)
if len(p) == 3:
return 100 * p[0] + 30 * p[1]
def to_y(p):
if len(p) == 1:
return 120
if len(p) == 2:
return 120 - 100 * p[1]
if len(p) == 3:
return 120 - 100 * p[2] - 40 * p[1]
for shape in ["interval", "triangle", "tetrahedron",
"quadrilateral", "hexahedron", "prism", "pyramid"]:
cell_type = getattr(basix.CellType, shape)
geometry = basix.geometry(cell_type)
topology = basix.topology(cell_type)
yadd = 0
width = 100
if shape == "interval":
yadd = -100
if shape == "hexahedron":
yadd = 40
width = 140
if shape == "pyramid":
width = 140
if shape == "prism":
yadd = 40
svg = ""
# The facets of a tetrahedron are triangular, so we create a quadrature
# rule on a triangle. We use an order 3 rule so that we can integrate the
# basis functions in our space exactly.
points, weights = basix.make_quadrature(CellType.triangle, 3)
# Next, we must map the quadrature points to our facet. We use the function
# `geometry` to get the coordinates of the vertices of the tetrahedron, and
# we use `sub_entity_connectivity` to see which vertices are adjacent to
# our facet. We get the sub-entity connectivity using the indices 2 (facets
# of 3D cells have dimension 2), 0 (vertices have dimension 0), and 0 (the
# index of the facet we chose to use).
#
# Using this information, we can map the quadrature points to the facet.
vertices = basix.geometry(CellType.tetrahedron)
facet = basix.cell.sub_entity_connectivity(CellType.tetrahedron)[2][0][0]
mapped_points = np.array([
vertices[facet[0]] * (1 - x - y) + vertices[facet[1]] * x +
vertices[facet[2]] * y for x, y in points
])
# We now compute the normal derivative of the fifth basis function at
# the quadrature points. First, we use `facet_outward_normals` to get
# the normal vector to the facet.
#
# We then tabulate the basis functions of our space at the quadrature
# points. We pass 1 in as the first argument, as we want the derivatives
# of the basis functions. The result of tabulation will be an array of
# size 4 by number of quadrature points by number of degrees of freedom.
# To get the data that we want, we use the indices `1:` (to get the