def test_displacement_discontinuity_derivative():
bf = basis_from_degree(1)
msh = simple_line_mesh(2)
qs = QuadStrategy(msh, 10, 10, 10, 10)
apply_to_elements(msh, "basis", bf, non_gen = True)
apply_to_elements(msh, "continuous", True, non_gen = True)
apply_to_elements(msh, "qs", qs, non_gen = True)
init_dofs(msh)
k_rh = RegularizedHypersingularKernel(1.0, 0.25)
# basis function should be equal to x_hat
k = 1
i = 1
qi = qs.get_simple()
strength = ConstantBasis([1.0, 1.0])
k_rh.set_interior_data(np.array([-2.0, 0.0]), np.array([0.0, 1.0]))
basis = bf.get_gradient_basis()
result = single_integral(msh.elements[k].mapping.eval, k_rh, strength,
basis, qi, 0, i)
np.testing.assert_almost_equal(result[1][1], 0.193011, 4)
np.testing.assert_almost_equal(result[0][0], -0.0191957, 4)
i = 0
result = single_integral(msh.elements[k].mapping.eval, k_rh, strength,
basis, qi, 0, i)
np.testing.assert_almost_equal(result[1][1], -0.193011, 4)
np.testing.assert_almost_equal(result[0][0], 0.0191957, 4)
strlocnorm = [((1.0, 0.0), np.array([-2.0, 0.0]), np.zeros(2))]
rhs = point_source_rhs(msh, strlocnorm, k_rh)
def test_improved_hypersingular():
mesh = simple_line_mesh(1, (0, 0), (1, 0))
mapping = PolynomialMapping(mesh.elements[0])
degree = 7
bf = gll_basis(degree)
one = ConstantBasis(np.ones(2))
quad_info_exact = gauss(100)
a = 0.01
b = 0.05
mapped_ay = map_singular_pt(a, 0.0, 1.0)
mapped_by = map_distance_to_interval(b, 0.0, 1.0)
quad_deg = degree + 4
moments_xa0_r0 = legendre.legendre_integrals(quad_deg)
moments_xa0_r2 = modify_divide_r2(
quad_deg, moments_xa0_r0, mapped_ay, mapped_by, mu_2_0(mapped_ay, mapped_by), mu_2_1(mapped_ay, mapped_by)
)
moments_xa0_r4 = modify_divide_r2(
quad_deg, moments_xa0_r2, mapped_ay, mapped_by, mu_4_0(mapped_ay, mapped_by), mu_4_1(mapped_ay, mapped_by)
)
moments_xa0_r6 = modify_divide_r2(
quad_deg, moments_xa0_r4, mapped_ay, mapped_by, mu_6_0(mapped_ay, mapped_by), mu_6_1(mapped_ay, mapped_by)
)
moments_xa1_r6 = modify_times_x_minus_a(len(moments_xa0_r6) - 2, moments_xa0_r6, mapped_ay)
moments_xa2_r6 = modify_times_x_minus_a(len(moments_xa1_r6) - 2, moments_xa1_r6, mapped_ay)
moments_xa3_r6 = modify_times_x_minus_a(len(moments_xa2_r6) - 2, moments_xa2_r6, mapped_ay)
moments_xa4_r6 = modify_times_x_minus_a(len(moments_xa3_r6) - 2, moments_xa3_r6, mapped_ay)
est = [[0, 0], [0, 0]]
x, w = recursive_quad(moments_xa0_r6[: degree + 1])
x, w = map_pts_wts(x, w, 0.0, 1.0)
w = map_weights_by_inv_power(w, 6.0, 0.0, 1.0) * ((-b / 2) ** 2) * ((-2 * b) ** 2)
est[1][1] = w[0]
x, w = recursive_quad(moments_xa1_r6[: degree + 1])
x, w = map_pts_wts(x, w, 0.0, 1.0)
w = map_weights_by_inv_power(w, 6.0, 0.0, 1.0) * ((-b / 2) ** 2) * ((-2 * b) ** 1)
est[0][1] = w[0]
est[1][0] = w[0]
x, w = recursive_quad(moments_xa2_r6[: degree + 1])
x, w = map_pts_wts(x, w, 0.0, 1.0)
w = map_weights_by_inv_power(w, 6.0, 0.0, 1.0) * ((-b / 2) ** 2) * ((-2 * b) ** 0)
est[0][0] = w[0]
# qi = QuadratureInfo(a, x, w)
kernelv1 = HypersingularPart1V1()
kernelv2 = HypersingularPart1V2()
pt = [a, b]
normal = [0.0, 1.0]
kernelv1.set_interior_data(pt, normal)
kernelv2.set_interior_data(pt, normal)
integrate = lambda qi, k: single_integral(mapping.eval, k, one, bf, qi, 0, 0)
exact = integrate(quad_info_exact, kernelv1)
np.testing.assert_almost_equal(np.array(exact), np.array(est), 10)
def single_integral_wrapper(map_eval, kernel, basis, quad_info, which_fnc):
"""
A wrapper so that single integral has the interface expected by the
interior point computation functions.
"""
return single_integral(map_eval, kernel, one,
basis, quad_info, 0, which_fnc)
def test_M_integral_diff_dof():
msh = simple_line_mesh(2)
q = quadrature.gauss(2)
bf = basis_funcs.basis_from_degree(1)
kernel = MassMatrixKernel(0, 0)
M_local = single_integral(msh.elements[0].mapping.eval, kernel, bf,
bf, q, 0, 1)
# integral of (1-x)*x from 0 to 1
np.testing.assert_almost_equal(M_local[0][0], 1.0 / 6.0)
def test_M_integral_same_dof_with_jacobian():
msh = simple_line_mesh(4)
q = quadrature.gauss(2)
bf = basis_funcs.basis_from_degree(1)
kernel = MassMatrixKernel(0, 0)
M_local = single_integral(msh.elements[0].mapping.eval, kernel, bf,
bf, q, 0, 0)
# Element size divided by two so the M value should be divided by two
np.testing.assert_almost_equal(M_local[0][0], 1.0 / 6.0)
def test_telles_quad_strategy():
msh = simple_line_mesh(4)
tqs = TellesQuadStrategy(4)
qi = tqs.get_interior_quadrature(msh.elements[3], [0.0, 0.0])
kernel = TractionKernel(1.0, 0.25)
one = ConstantBasis((1.0, 1.0))
est = single_integral(msh.elements[3].mapping.eval,
kernel, one, one, qi, 0, 0)
exact = -0.0367726
np.testing.assert_almost_equal(exact, est[0][1])
def test_plot_single_integral_kernel():
msh = simple_line_mesh(2)
x = np.linspace(0.0, 1, 1000)
kernel = TractionKernel(1.0, 0.25)
bf = basis_funcs.basis_from_degree(8)
bf1 = ConstantBasis((1.0, 1.0))
k = np.zeros_like(x)
k_int = np.zeros_like(x)
for (i, x_val) in enumerate(x):
q = quadrature.piessens(16, x_val, 16)
kernel.set_interior_data([x_val, 0.0], [0.0,1.0])
kd = kernel.get_interior_integral_data([1.0, 0.0], [0.0, 1.0])
k[i] = kernel._call(kd, 0, 1)
k_int[i] = single_integral(msh.elements[1].mapping.eval, kernel, bf1,
bf, q, 0, 4)[0][1]
def test_fast_lobatto():
N = 15
mesh = simple_line_mesh(1, (0, 0), (1, 0))
mapping = PolynomialMapping(mesh.elements[0])
kernel = TractionKernel(1.0, 0.25)
bf = gll_basis(N)
one = ConstantBasis(np.ones(2))
quad_info_old = lobatto(N + 1)
pt = [0.0, -5.0]
normal = [0.0, 1.0]
kernel.set_interior_data(pt, normal)
est_slow = single_integral(mapping.eval, kernel, one, bf, quad_info_old, 0, 0)
est_fast = aligned_single_integral(mapping.eval, kernel, bf, quad_info_old, 0)
np.testing.assert_almost_equal(np.array(est_slow), np.array(est_fast))
def _element_mass(matrix, e_k):
# Because the term is identical (just replace u by t) for both
# integral equations, this function does not care about the BC type
bc_basis = e_k.bc.basis
if type(bc_basis) is ZeroBasis:
return
q_info = e_k.qs.get_nonsingular_minpts()
kernel = MassMatrixKernel(0, 0)
for i in range(e_k.basis.n_fncs):
for j in range(bc_basis.n_fncs):
M_local = single_integral(e_k.mapping.eval,
kernel,
e_k.basis,
bc_basis,
q_info,
i, j)
matrix[e_k.dofs[0, i], e_k.dofs[0, j]] += M_local[0][0]
matrix[e_k.dofs[1, i], e_k.dofs[1, j]] += M_local[1][1]
def test_gauss_displacement_xy():
mesh = simple_line_mesh(1, (0, 0), (1, 0))
mapping = PolynomialMapping(mesh.elements[0])
kernel = TractionKernel(1.0, 0.25)
degree = 4
bf = gll_basis(degree)
one = ConstantBasis(np.ones(2))
quad_info_exact = gauss(100)
quad_info = lobatto(degree + 1)
# Testing GLL basis and quadrature combination on the nonsingular part
# of the displacement kernel.
pt = [0.5, 1000.0]
normal = [0.0, 1.0]
kernel.set_interior_data(pt, normal)
integrate = lambda qi: single_integral(mapping.eval, kernel, one, bf, qi, 0, 0)
exact = integrate(quad_info_exact)
est = integrate(quad_info)
np.testing.assert_almost_equal(exact[0][1], est[0][1], 16)
def test_rl_integral():
mesh = simple_line_mesh(1, (0, 0), (1, 0))
mapping = PolynomialMapping(mesh.elements[0])
kernel = TractionKernel(1.0, 0.25)
bf = gll_basis(4)
one = ConstantBasis(np.ones(2))
quad_info_old = lobatto(5)
x_bf = np.array(bf.nodes)
x_q = np.array(quad_info_old.x)
distance = 5.0
quad_info_new = rl_quad(5, 0.0, distance)
# This one is comparing lobatto quadrature and recursive legendre quadrature
# on the TractionKernel which is 1/r singular.
pt = [0.0, -distance]
normal = [0.0, 1.0]
# exact = 0.2473475767
exact = 0.00053055607635
integrate = lambda qi: single_integral(mapping.eval, kernel, one, bf, qi, 0, 0)
aligned_integrate = lambda qi: aligned_single_integral(mapping.eval, kernel, bf, qi, 0)
kernel.set_interior_data(pt, normal)
est_gauss = integrate(quad_info_old)
np.testing.assert_almost_equal(est_gauss[0][0], exact)
est_gauss_fast = aligned_integrate(quad_info_old)
np.testing.assert_almost_equal(est_gauss_fast[0][0], exact)
# This stuff doesn't work yet
est_new = integrate(quad_info_new)
np.testing.assert_almost_equal(est_new[0][0], exact, 6)
est_new_fast = aligned_integrate(quad_info_new)
np.testing.assert_almost_equal(est_new_fast[0][0], exact, 6)
def _element_pair(matrix, e_k, e_l, which_kernels, rhs_or_matrix):
# Determine whether to use the boundary condition or the solution basis
# as the inner basis function
if rhs_or_matrix == "rhs":
init_e_l_basis = e_l.bc.basis
else:
init_e_l_basis = e_l.basis
# If either of the bases are the zero basis, then don't compute
# anything
if type(e_k.basis) is ZeroBasis or \
type(init_e_l_basis) is ZeroBasis:
return
# Determine which kernel and which bases to use
kernel, factor = which_kernels[e_k.bc.type][e_l.bc.type][rhs_or_matrix]
if kernel is None:
return
# Decide whether to use the basis or its gradient
e_k_basis, e_k_pt_srcs = _choose_basis(e_k.basis, kernel.test_gradient)
e_l_basis, e_l_pt_srcs = _choose_basis(init_e_l_basis, kernel.soln_gradient)
# Now that we might have taken a derivative, check for ZeroBases again.
if type(e_k_basis) is ZeroBasis:
return
# Determine what quadrature formula to use
quad_outer, quad_inner = e_k.qs.get_quadrature(
kernel.singularity_type, e_k, e_l)
# Handle point sources
# Filter out the point sources that we can safely ignore
# many of these will be ignored because there will be an equal and
# opposite point source contribution from the adjacent element.
# TODO!
# Currently, I ignore all point source on the test function side
# of the problem, because these should be handled by the continuity
# of the displacement field. I should probably think about this
# a bit more...
e_k_pt_srcs = [[],[]]
# This probably explains some of the problems I was having with the
# constant traction crack problem.
# I also ignore all point source if we are dealing
if rhs_or_matrix == "matrix":
e_l_pt_srcs = [[],[]]
# Loop over point sources and integrate!
# All cross multiplications are necessary.
# the pt sources are tuples like ((node, str_x, str_y), local_dof)
# for e_k_pt in e_k_pt_srcs:
# phys_pt_k = e_k.mapping.get_physical_point(e_k_pt[0][0])
# kernel.set_interior_data(phys_pt_k,
# for j in range(e_l.basis.n_fncs):
# pass
# for e_l_pt in e_k_pt_srcs:
# phys_pt_l = e_l.mapping.get_physical_point(e_l_pt[0][0])
e_l_pt_left = None
e_l_pt_right = None
if e_l_pt_srcs != [[],[]] and (e_l.neighbors_left == [] or e_l.neighbors_left[0].bc.type != 'traction'):
e_l_pt_left = e_l_pt_srcs[0][0]
if e_l_pt_srcs != [[],[]] and (e_l.neighbors_right == [] or e_l.neighbors_right[0].bc.type != 'traction'):
e_l_pt_right = e_l_pt_srcs[0][1]
for i in range(e_k.basis.n_fncs):
for pt_src_idx, e_l_pt in enumerate([e_l_pt_left, e_l_pt_right]):
if e_l_pt is None:
continue
e_l_dof = e_l_pt_srcs[1][pt_src_idx]
phys_pt_l = e_l.mapping.get_physical_point(e_l_pt[0])
normal_l = e_l.mapping.get_normal(e_l_pt[0])
strength = ConstantBasis([e_l_pt[1], e_l_pt[2]])
kernel.set_interior_data(phys_pt_l, normal_l)
integral = single_integral(e_k.mapping.eval,
kernel,
e_k_basis,
strength,
quad_outer,
i, 0)
for idx1 in range(2):
for idx2 in range(2):
matrix[e_k.dofs[idx1, i], e_l.dofs[idx2, e_l_dof]] += \
factor * integral[idx1][idx2]
if type(e_l_basis) is ZeroBasis:
return
# Handle the integration of pairs of basis functions
for i in range(e_k.basis.n_fncs):
for j in range(e_l.basis.n_fncs):
integral = \
double_integral(e_k.mapping.eval, e_l.mapping.eval,
kernel, e_k_basis, e_l_basis,
quad_outer, quad_inner, i, j)
# Add the integrated term in the appropriate location
for idx1 in range(2):
for idx2 in range(2):
matrix[e_k.dofs[idx1, i], e_l.dofs[idx2, j]] +=\
factor * integral[idx1][idx2]
