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# Copyright (c) 2020 Chris Richardson
# FEniCS Project
# SPDX-License-Identifier: MIT
import sympy
import basix
import numpy as np
def P_interval(n, x):
from sympy import S
r = [sympy.sqrt(p + sympy.Rational(1, 2))
* sympy.legendre(p, x * S(2) - S(1))
for p in range(n + 1)]
return(r)
def test_symbolic_interval():
n = 7
nderiv = 7
x = sympy.Symbol("x")
w = P_interval(n, x)
cell = basix.CellType.interval
pts0 = basix.create_lattice(cell, 10, basix.LatticeType.equispaced, True)
wtab = basix.tabulate_polynomial_set(cell, n, nderiv, pts0)
wd = [w[i] for i in range(n + 1)]
for k in range(nderiv + 1):
wsym = np.zeros_like(wtab[k])
for i in range(n + 1):
for j, p in enumerate(pts0):
wsym[j, i] = wd[i].subs(x, p[0])
wd[i] = sympy.diff(wd[i], x)
assert(np.isclose(wtab[k], wsym).all())
def test_symbolic_quad():
n = 2
nderiv = 2
idx = basix.index
x = sympy.Symbol("x")
wx = P_interval(n, x)
y = sympy.Symbol("y")
wy = P_interval(n, y)
w = []
for i in range(n + 1):
for j in range(n + 1):
w += [wx[i] * wy[j]]
m = (n + 1)**2
cell = basix.CellType.quadrilateral
pts0 = basix.create_lattice(cell, 2, basix.LatticeType.equispaced, True)
wtab = basix.tabulate_polynomial_set(cell, n, nderiv, pts0)
for kx in range(nderiv):
for ky in range(0, nderiv - kx):
wsym = np.zeros_like(wtab[0])
for i in range(m):
wd = sympy.diff(w[i], x, kx, y, ky)
for j, p in enumerate(pts0):
wsym[j, i] = wd.subs([(x, p[0]), (y, p[1])])
print(kx, ky)
print(wtab[idx(kx, ky)])
print()
print(wsym)
assert(np.isclose(wtab[idx(kx, ky)], wsym).all())
def test_symbolic_triangle():
n = 5
nderiv = 4
idx = basix.index
from sympy import S
m = (n + 1) * (n + 2) // 2
x = sympy.Symbol("x")
y = sympy.Symbol("y")
x0 = x * S(2) - S(1)
y0 = y * S(2) - S(1)
w = [None] * m
zeta = (S(2) * x0 + y0 + S(1)) / (S(1) - y0)
for p in range(n + 1):
for q in range(n - p + 1):
w[idx(p, q)] = sympy.sqrt(S(2 * p + 1) * S(p + q + 1) / S(2)) \
* sympy.cancel(sympy.legendre(p, zeta)
* ((S(1) - y0) / S(2))**p) \
* sympy.jacobi(S(q), S(2 * p + 1), S(0), y0)
np.set_printoptions(linewidth=200)
cell = basix.CellType.triangle
pts0 = basix.create_lattice(cell, 3, basix.LatticeType.equispaced, True)
wtab = basix.tabulate_polynomial_set(cell, n, nderiv, pts0)
for kx in range(nderiv):
for ky in range(0, nderiv - kx):
wsym = np.zeros_like(wtab[0])
for i in range(m):
wd = sympy.diff(w[i], x, kx, y, ky)
for j, p in enumerate(pts0):
wsym[j, i] = wd.subs([(x, p[0]), (y, p[1])])
assert(np.isclose(wtab[idx(kx, ky)], wsym).all())
def test_symbolic_tetrahedron():
n = 4
nderiv = 4
idx = basix.index
from sympy import S
m = (n + 1) * (n + 2) * (n + 3) // 6
x = sympy.Symbol("x")
y = sympy.Symbol("y")
z = sympy.Symbol("z")
x0 = x * S(2) - S(1)
y0 = y * S(2) - S(1)
z0 = z * S(2) - S(1)
w = [None] * m
np.set_printoptions(linewidth=200, suppress=True, precision=3)
zeta = S(2) * (S(1) + x0) / (y0 + z0) + S(1)
xi = S(2) * (S(1) + y0) / (S(1) - z0) - S(1)
for p in range(n + 1):
for q in range(n - p + 1):
for r in range(n - p - q + 1):
w[idx(p, q, r)] = sympy.cancel(
sympy.legendre(p, zeta) * ((y0 + z0) / S(2))**p)
w[idx(p, q, r)] *= sympy.cancel(
sympy.jacobi(S(q), S(2 * p + 1), 0, xi)
* ((S(1) - z0) / S(2))**q)
w[idx(p, q, r)] *= sympy.jacobi(S(r),
S(2 * p + 2 * q + 2), 0, z0)
w[idx(p, q, r)] *= sympy.sqrt(
S((2 * p + 1) * (p + q + 1)
* (2 * p + 2 * q + 2 * r + 3))) / S(2)
cell = basix.CellType.tetrahedron
pts0 = basix.create_lattice(cell, 2, basix.LatticeType.equispaced, True)
wtab = basix.tabulate_polynomial_set(cell, n, nderiv, pts0)
for k in range(nderiv + 1):
for q in range(k + 1):
for kx in range(q + 1):
ky = q - kx
kz = k - q
print((kx, ky, kz))
wsym = np.zeros_like(wtab[0])
for i in range(m):
wd = sympy.diff(w[i], x, kx, y, ky, z, kz)
for j, p in enumerate(pts0):
wsym[j, i] = wd.subs([(x, p[0]),
(y, p[1]),
(z, p[2])])
assert(np.isclose(wtab[idx(kx, ky, kz)], wsym).all())
def test_symbolic_pyramid():
np.set_printoptions(linewidth=200, suppress=True, precision=2)
n = 3
nderiv = 3
idx = basix.index
def pyr_idx(p, q, r):
rv = n - r + 1
r0 = r * (n + 1) * (n - r + 2) + (2 * r - 1) * (r - 1) * r // 6
idx = r0 + p * rv + q
return idx
from sympy import S
m = (n + 1) * (n + 2) * (2 * n + 3) // 6
x = sympy.Symbol("x")
y = sympy.Symbol("y")
z = sympy.Symbol("z")
x0 = x * S(2) - S(1)
y0 = y * S(2) - S(1)
z0 = z * S(2) - S(1)
w = [None] * m
zetax = (S(2) * x0 + S(1) + z0) / (S(1) - z0)
zetay = (S(2) * y0 + S(1) + z0) / (S(1) - z0)
for r in range(n + 1):
for p in range(n - r + 1):
for q in range(n - r + 1):
w[pyr_idx(p, q, r)] = sympy.cancel(
sympy.legendre(p, zetax) * ((S(1) - z0) / S(2))**p)
w[pyr_idx(p, q, r)] *= sympy.cancel(
sympy.legendre(q, zetay) * ((S(1) - z0) / S(2))**q)
w[pyr_idx(p, q, r)] *= \
sympy.jacobi(r, S(2 * p + 2 * q + 2), S(0), z0)
w[pyr_idx(p, q, r)] *= \
sympy.sqrt(S((2 * q + 1) * (2 * p + 1)
* (2 * p + 2 * q + 2 * r + 3)) / S(8))
cell = basix.CellType.pyramid
pts0 = basix.create_lattice(cell, 1, basix.LatticeType.equispaced, True)
wtab = basix.tabulate_polynomial_set(cell, n, nderiv, pts0)
for k in range(nderiv + 1):
for q in range(k + 1):
for kx in range(q + 1):
ky = q - kx
kz = k - q
print((kx, ky, kz))
wsym = np.zeros_like(wtab[0])
for i in range(m):
wd = sympy.diff(w[i], x, kx, y, ky, z, kz)
for j, p in enumerate(pts0):
wsym[j, i] = wd.subs([(x, p[0]),
(y, p[1]),
(z, p[2])])
print(wd)
print(w[idx(kx, ky, kz)])
print(wsym)
assert(np.isclose(wtab[idx(kx, ky, kz)], wsym).all())
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