def test_integration_second_basis_model_problem_rhs_subdomain_integration():
k = 2
triangle = np.array([[0, 0], [1, 0], [0, 1]])
b_num = SplineSpace(triangle, 2).basis()[2]
triangle_2 = ps12_sub_triangles(triangle)[k]
expected_integral = 188.32704191224434
def f(p):
"""
The exact source term for the model problem for the biharmonic equation.
"""
x, y = p[:, 0], p[:, 1]
return 8 * np.pi**4 * (
np.sin(2 * np.pi * x) - 8 * np.sin(4 * np.pi * x) +
25 * np.sin(2 * np.pi * (x - 2 * y)) -
32 * np.sin(4 * np.pi * (x - y)) - np.sin(2 * np.pi * y) + 8 *
(np.sin(4 * np.pi * y) + np.sin(2 * np.pi * (-x + y))) +
25 * np.sin(4 * np.pi * x - 2 * np.pi * y))
def integrand(p):
return b_num(p.T) * f(p.T)
sub_triangles = ps12_sub_triangles(triangle_2)
computed_integral = 0
for sub_triangle in sub_triangles:
computed_integral += quadpy.triangle.integrate(
integrand, sub_triangle, quadpy.triangle.XiaoGimbutas(12))
np.testing.assert_almost_equal(computed_integral, expected_integral)
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, mesh):
"""
Initializes a Composite C^1 spline space over the given mesh.
:param mesh:
"""
self.mesh = mesh
self.dimension = 3*len(mesh.vertices) + len(mesh.edges)
self.local_to_global_map, self.dof_to_edge_map, self.dof_to_vertex_map = local_to_global(mesh.vertices, mesh.triangles)
self.local_spline_spaces = [SplineSpace(mesh.vertices[triangle], degree=2) for triangle in mesh.triangles]
self.local_spline_bases = [S.hermite_basis() for S in self.local_spline_spaces]
self._construct_basis_to_triangle_map()
self._construct_global_to_local_map()
self._construct_interior_and_boundary_dofs()
self.basis = [self._construct_global_basis_function(i) for i in range(self.dimension)]
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)