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# Copyright (c) 2020 Chris Richardson & Matthew Scroggs
# FEniCS Project
# SPDX-License-Identifier: MIT
import basix
import numpy
import pytest
import sympy
from .test_lagrange import sympy_disc_lagrange
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_disc_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_disc_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_disc_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_disc_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_disc_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 dim in range(tdim):
for r in range(mat.shape[0]):
g += [sum([v * funcs[i][dim] for i, v in enumerate(mat.row(r))])]
return g
@pytest.mark.parametrize("order", [1, 2, 3])
def test_tri(order):
celltype = basix.CellType.triangle
g = sympy_nedelec(celltype, order)
x = sympy.Symbol("x")
y = sympy.Symbol("y")
nedelec = basix.Nedelec("triangle", order)
pts = basix.create_lattice(celltype, 6, basix.LatticeType.equispaced, True)
nderiv = 3
wtab = nedelec.tabulate(nderiv, pts)
for kx in range(nderiv):
for ky in range(0, nderiv - kx):
wsym = numpy.zeros_like(wtab[0])
for i in range(len(g)):
wd = sympy.diff(g[i], x, kx, y, ky)
for j, p in enumerate(pts):
wsym[j, i] = wd.subs([(x, p[0]), (y, p[1])])
assert(numpy.isclose(wtab[basix.index(kx, ky)], wsym).all())
@pytest.mark.parametrize("order", [1, 2, 3])
def test_tet(order):
celltype = basix.CellType.tetrahedron
g = sympy_nedelec(celltype, order)
x = sympy.Symbol("x")
y = sympy.Symbol("y")
z = sympy.Symbol("z")
nedelec = basix.Nedelec("tetrahedron", order)
pts = basix.create_lattice(celltype, 6, basix.LatticeType.equispaced, True)
nderiv = 1
wtab = nedelec.tabulate(nderiv, pts)
for k in range(nderiv + 1):
for q in range(k + 1):
for kx in range(q + 1):
ky = q - kx
kz = k - q
wsym = numpy.zeros_like(wtab[0])
for i in range(len(g)):
wd = sympy.diff(g[i], x, kx, y, ky, z, kz)
for j, p in enumerate(pts):
wsym[j, i] = wd.subs([(x, p[0]),
(y, p[1]),
(z, p[2])])
assert(numpy.isclose(wtab[basix.index(kx, ky, kz)], wsym).all())
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