def test_values_quadratic_basis():
"""
tests that the polynomial pieces and the numerical splines evaluate to the same value over all the
subtriangles.
"""
X, Y = sy.symbols('X Y')
triangle = np.array([[0, 0], [1, 0], [0, 1]])
S = SplineSpace(triangle, 2)
for basis_num in range(12):
b_num = S.basis()[basis_num]
b_sym = polynomial_pieces(triangle,
KNOT_MULTIPLICITIES_QUADRATIC[basis_num])
subtriangles = ps12_sub_triangles(triangle)
for k in range(12):
t = subtriangles[k]
points = sample_triangle(t, 1)
num_b_sym = sy.lambdify([X, Y], b_sym[k])
for p in points:
np.testing.assert_almost_equal(b_num(p), num_b_sym(p[0], p[1]))
def __init__(self, triangle, knot_multiplicities):
self.triangle = triangle
self.knot_multiplicities = knot_multiplicities + (10 - len(knot_multiplicities)) * [0]
self.knot_indices = [i for i in range(len(knot_multiplicities)) if knot_multiplicities[i] != 0]
self.face_indices = KNOT_CONFIGURATION_TO_FACE_INDICES[tuple(self.knot_indices)]
self.degree = sum(knot_multiplicities) - 3
self.polynomial_pieces = polynomial_pieces(triangle, knot_multiplicities)
def test_polynomial_pieces_quadratic_basis():
X, Y = symbols('X Y')
triangle = np.array([[0, 0], [1, 0], [0, 1]])
# knot multiplicities corresponding to the first quadratic basis function
multiplicities = KNOT_MULTIPLICITIES_QUADRATIC[0]
expected_polynomials = [
1.0 * (-2.0 * X - 2.0 * Y + 1)**2, 1.0 * (-2.0 * X - 2.0 * Y + 1)**2,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
]
computed_polynomials = polynomial_pieces(triangle, multiplicities)
for e, c in zip(expected_polynomials, computed_polynomials):
np.testing.assert_almost_equal(simplify(e - c), 0)
def test_laplacian_quadratic_basis_functions():
X, Y = sy.symbols('X Y')
triangle = np.array([[0, 0], [1, 0], [0, 1]])
S = SplineSpace(triangle, 2)
points = sample_triangle(triangle, 10)
for basis_num in range(12):
b_num = S.basis()[basis_num]
b_sym = polynomial_pieces(triangle,
KNOT_MULTIPLICITIES_QUADRATIC[basis_num])
for p in points:
b = barycentric_coordinates(triangle, p)
k = determine_sub_triangle(b)[0]
b_lapl = sy.diff(b_sym[k], X, X) + sy.diff(b_sym[k], Y, Y)
num_b_sym = sy.lambdify([X, Y], b_lapl)
np.testing.assert_almost_equal(b_num.lapl(p),
num_b_sym(p[0], p[1]))
def test_integration_second_basis_second_triangle():
x, y = sp.symbols('X Y')
k = 2
triangle = np.array([[0, 0], [1, 0], [0, 1]])
b_sym = polynomial_pieces(triangle, KNOT_MULTIPLICITIES_QUADRATIC[2])[k]
b_num = SplineSpace(triangle, 2).basis()[2]
triangle_2 = ps12_sub_triangles(triangle)[k]
expected_integral = 0.0117187
b, w = gaussian_quadrature_data(2)
b_sym_num = sp.lambdify((x, y), b_sym)
def f(p):
return b_num(p.T)
computed_integral = gaussian_quadrature(triangle_2, b_num, b, w)
computed_integral = quadpy.triangle.integrate(f, triangle_2,
quadpy.triangle.SevenPoint())
np.testing.assert_almost_equal(computed_integral, expected_integral)