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# Copyright (c) 2021 Matthew Scroggs
# FEniCS Project
# SPDX-License-Identifier: MIT
import numpy as np
import pytest
import sympy
import basix
one = sympy.Integer(1)
x = sympy.Symbol("x")
y = sympy.Symbol("y")
z = sympy.Symbol("z")
@pytest.mark.parametrize("degree", range(6))
@pytest.mark.parametrize(
"cell_type",
[
basix.CellType.interval,
basix.CellType.triangle,
basix.CellType.quadrilateral,
basix.CellType.tetrahedron,
basix.CellType.hexahedron,
basix.CellType.prism,
basix.CellType.pyramid,
],
)
def test_legendre(cell_type, degree):
points, weights = basix.make_quadrature(cell_type, 2 * degree)
polys = basix.tabulate_polynomials(basix.PolynomialType.legendre, cell_type, degree, points)
matrix = np.empty((polys.shape[0], polys.shape[0]))
for i, col_i in enumerate(polys):
for j, col_j in enumerate(polys):
matrix[i, j] = sum(col_i * col_j * weights)
assert np.allclose(matrix, np.identity(polys.shape[0]))
def evaluate(function, pt):
if len(pt) == 1:
return function.subs(x, pt[0])
elif len(pt) == 2:
return function.subs(x, pt[0]).subs(y, pt[1])
elif len(pt) == 3:
return function.subs(x, pt[0]).subs(y, pt[1]).subs(z, pt[2])
@pytest.mark.parametrize(
"cell_type, ptype, functions, degree",
[
[basix.CellType.interval, basix.PolynomialType.legendre, [one, x], 1],
[basix.CellType.interval, basix.PolynomialType.legendre, [one, x, x**2], 2],
[basix.CellType.interval, basix.PolynomialType.legendre, [one, x, x**2, x**3], 3],
[basix.CellType.triangle, basix.PolynomialType.legendre, [one, y, x], 1],
[basix.CellType.triangle, basix.PolynomialType.legendre, [one, y, x, y**2, x * y, x**2], 2],
[
basix.CellType.triangle,
basix.PolynomialType.legendre,
[one, y, x, y**2, x * y, x**2, y**3, x * y**2, x**2 * y, x**3],
3,
],
[basix.CellType.tetrahedron, basix.PolynomialType.legendre, [one], 0],
[basix.CellType.tetrahedron, basix.PolynomialType.legendre, [one, z, y, x], 1],
[
basix.CellType.tetrahedron,
basix.PolynomialType.legendre,
[one, z, y, x, z**2, y * z, x * z, y**2, x * y, x**2],
2,
],
[
basix.CellType.tetrahedron,
basix.PolynomialType.legendre,
[
one,
z,
y,
x,
z**2,
y * z,
x * z,
y**2,
x * y,
x**2,
z**3,
y * z**2,
x * z**2,
y**2 * z,
x * y * z,
x**2 * z,
y**3,
x * y**2,
x**2 * y,
x**3,
],
3,
],
[basix.CellType.quadrilateral, basix.PolynomialType.legendre, [one, y, x, x * y], 1],
[
basix.CellType.quadrilateral,
basix.PolynomialType.legendre,
[one, y, y**2, x, x * y, x * y**2, x**2, x**2 * y, x**2 * y**2],
2,
],
[
basix.CellType.hexahedron,
basix.PolynomialType.legendre,
[one, z, y, y * z, x, x * z, x * y, x * y * z],
1,
],
[basix.CellType.prism, basix.PolynomialType.legendre, [one, z, y, y * z, x, x * z], 1],
[
basix.CellType.prism,
basix.PolynomialType.legendre,
[
one,
z,
z**2,
y,
y * z,
y * z**2,
x,
x * z,
x * z**2,
y**2,
y**2 * z,
y**2 * z**2,
x * y,
x * y * z,
x * y * z**2,
x**2,
x**2 * z,
x**2 * z**2,
],
2,
],
[basix.CellType.pyramid, basix.PolynomialType.legendre, [one], 0],
[basix.CellType.pyramid, basix.PolynomialType.lagrange, [one], 0],
[
basix.CellType.pyramid,
basix.PolynomialType.lagrange,
[one, 2 * y + z - 1, 2 * x + z - 1, (2 * x + z - 1) * (2 * y + z - 1) / (1 - z), z],
1,
],
[
basix.CellType.pyramid,
basix.PolynomialType.legendre,
[one, z, y / (1 - z), y, x / (1 - z), x, x * y / (1 - z) ** 2, x * y / (1 - z)],
1,
],
],
)
def test_order(cell_type, ptype, functions, degree):
points, weights = basix.make_quadrature(cell_type, 2 * degree)
polys = basix.tabulate_polynomials(ptype, cell_type, degree, points)
assert len(functions) == polys.shape[0]
eval_points = basix.create_lattice(cell_type, 10, basix.LatticeType.equispaced, False)
eval_polys = basix.tabulate_polynomials(ptype, cell_type, degree, eval_points)
for n, function in enumerate(functions):
expected_eval = [float(evaluate(function, i)) for i in eval_points]
# Using n polynomials
# The monomial should NOT be exactly represented using this many
coeffs = []
values = np.array([evaluate(function, i) for i in points])
coeffs = [sum(values * polys[p, :] * weights) for p in range(n)]
actual_eval = [float(sum(coeffs * p[:n])) for p in eval_polys.T]
assert not np.allclose(expected_eval, actual_eval)
# Using n+1 polynomials
# The monomial should be exactly represented using this many
coeffs = []
values = np.array([evaluate(function, i) for i in points])
for p in range(n + 1):
coeffs.append(sum(values * polys[p, :] * weights))
actual_eval = [float(sum(coeffs * p[: n + 1])) for p in eval_polys.T]
assert np.allclose(expected_eval, actual_eval)
@pytest.mark.parametrize("degree", range(8))
def test_not_nan_pyramid(degree):
points = np.array([[0.0, 0.0, 1.0]])
values = basix.tabulate_polynomials(
basix.PolynomialType.legendre, basix.CellType.pyramid, degree, points
)[:, 0]
for i in values:
assert not np.isnan(i)
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