def derivative_ex(dirichl_space, neumann_space, ep_in, ep_ex, kappa,
operator_assembler):
"""
Construct the system matrix and RHS grid functions using derivative formulation with interior values.
"""
phi_id = sparse.identity(dirichl_space, dirichl_space, dirichl_space)
dph_id = sparse.identity(neumann_space, neumann_space, neumann_space)
ep = ep_ex / ep_in
dF = laplace.double_layer(dirichl_space,
dirichl_space,
dirichl_space,
assembler=operator_assembler)
dP = modified_helmholtz.double_layer(dirichl_space,
dirichl_space,
dirichl_space,
kappa,
assembler=operator_assembler)
B = 1 / ep * dF - dP
F = laplace.single_layer(neumann_space,
dirichl_space,
dirichl_space,
assembler=operator_assembler)
P = modified_helmholtz.single_layer(neumann_space,
dirichl_space,
dirichl_space,
kappa,
assembler=operator_assembler)
A = F - P
ddF = laplace.hypersingular(dirichl_space,
neumann_space,
neumann_space,
assembler=operator_assembler)
ddP = modified_helmholtz.hypersingular(dirichl_space,
neumann_space,
neumann_space,
kappa,
assembler=operator_assembler)
D = 1 / ep * (ddP - ddF)
dF0 = laplace.adjoint_double_layer(neumann_space,
neumann_space,
neumann_space,
assembler=operator_assembler)
dP0 = modified_helmholtz.adjoint_double_layer(neumann_space,
neumann_space,
neumann_space,
kappa,
assembler=operator_assembler)
C = dF0 - 1.0 / ep * dP0
A_sys = bempp.api.BlockedOperator(2, 2)
A_sys[0, 0] = (0.5 * (1.0 + (1.0 / ep)) * phi_id) + B
A_sys[0, 1] = -A
A_sys[1, 0] = D
A_sys[1, 1] = (0.5 * (1.0 + (1.0 / ep)) * dph_id) - C
return A_sys
def test_multitrace_single_layer_agrees_with_standard_single_layer_dual_lin(
self):
"""Multitrace SLP agrees with standard SLP."""
parameters = bempp.api.common.global_parameters()
parameters.assembly.boundary_operator_assembly_type = 'dense'
from bempp.api.operators.boundary.modified_helmholtz import \
multitrace_operator, single_layer
multitrace = multitrace_operator(
self._grid, WAVE_NUMBER, parameters=parameters)
actual = multitrace[0, 1]
expected = single_layer(
self._lin_space, self._lin_space, self._lin_space, WAVE_NUMBER,
parameters=parameters)
mat_expected = bempp.api.as_matrix(expected.weak_form())
mat_actual = bempp.api.as_matrix(actual.weak_form())
diff_norm = np.linalg.norm(
mat_expected - mat_actual) / np.linalg.norm(mat_actual)
self.assertAlmostEqual(diff_norm, 0)
def direct(dirichl_space, neumann_space, q, x_q, ep_in, ep_out, kappa):
from bempp.api.operators.boundary import sparse, laplace, modified_helmholtz
identity = sparse.identity(dirichl_space, dirichl_space, dirichl_space)
slp_in = laplace.single_layer(neumann_space, dirichl_space, dirichl_space)
dlp_in = laplace.double_layer(dirichl_space, dirichl_space, dirichl_space)
slp_out = modified_helmholtz.single_layer(neumann_space, dirichl_space, dirichl_space, kappa)
dlp_out = modified_helmholtz.double_layer(dirichl_space, dirichl_space, dirichl_space, kappa)
# Matrix Assembly
blocked = bempp.api.BlockedOperator(2, 2)
blocked[0, 0] = 0.5*identity + dlp_in
blocked[0, 1] = -slp_in
blocked[1, 0] = 0.5*identity - dlp_out
blocked[1, 1] = (ep_in/ep_out)*slp_out
#A = blocked.strong_form()
A = blocked
@bempp.api.real_callable
def charges_fun(x, n, domain_index, result):
nrm = np.sqrt((x[0]-x_q[:,0])**2 + (x[1]-x_q[:,1])**2 + (x[2]-x_q[:,2])**2)
aux = np.sum(q/nrm)
result[0] = aux/(4*np.pi*ep_in)
@bempp.api.real_callable
def zero(x, n, domain_index, result):
result[0] = 0
rhs_1 = bempp.api.GridFunction(dirichl_space, fun=charges_fun)
rhs_2 = bempp.api.GridFunction(neumann_space, fun=zero)
return A, rhs_1, rhs_2
def mod_helm_multitrace_scaled(dirichl_space, neumann_space, kappa,
scaling_factors, operator_assembler):
A = bempp.api.BlockedOperator(2, 2)
A[0, 0] = scaling_factors[0][0] * (-1.0) * modified_helmholtz.double_layer(
dirichl_space,
dirichl_space,
dirichl_space,
kappa,
assembler=operator_assembler)
A[0, 1] = scaling_factors[0][1] * modified_helmholtz.single_layer(
neumann_space,
dirichl_space,
dirichl_space,
kappa,
assembler=operator_assembler)
A[1, 0] = scaling_factors[1][0] * modified_helmholtz.hypersingular(
dirichl_space,
neumann_space,
neumann_space,
kappa,
assembler=operator_assembler)
A[1, 1] = scaling_factors[1][1] * modified_helmholtz.adjoint_double_layer(
neumann_space,
neumann_space,
neumann_space,
kappa,
assembler=operator_assembler)
return A
def test_modified_helmholtz_single_layer(
default_parameters, helpers, precision, device_interface
):
"""Test dense assembler for modified Helmholtz."""
from bempp.api import function_space
from bempp.api.operators.boundary.modified_helmholtz import single_layer
grid = helpers.load_grid("sphere")
space = function_space(grid, "P", 1)
discrete_op = single_layer(
space,
space,
space,
OMEGA,
assembler="dense",
precision=precision,
device_interface=device_interface,
parameters=default_parameters,
).weak_form()
# Bempp 3 assembles modified Helmholtz as complex type, so cast to real.
expected = _np.real(
helpers.load_npy_data("modified_helmholtz_single_layer_boundary")
)
_np.testing.assert_allclose(
discrete_op.A, expected, rtol=helpers.default_tolerance(precision)
)
def block_diagonal_precon_direct_external(dirichl_space,
neumann_space,
ep_in,
ep_ex,
kappa,
permuted_rows=False):
from scipy.sparse import diags, bmat
from scipy.sparse.linalg import aslinearoperator
from bempp.api.operators.boundary import sparse, laplace, modified_helmholtz
# block-diagonal preconditioner
identity = sparse.identity(dirichl_space, dirichl_space, dirichl_space)
identity_diag = identity.weak_form().to_sparse().diagonal()
slp_in_diag = laplace.single_layer(
neumann_space,
dirichl_space,
dirichl_space,
assembler="only_diagonal_part").weak_form().get_diagonal()
dlp_in_diag = laplace.double_layer(
dirichl_space,
dirichl_space,
dirichl_space,
assembler="only_diagonal_part").weak_form().get_diagonal()
slp_out_diag = modified_helmholtz.single_layer(
neumann_space,
dirichl_space,
dirichl_space,
kappa,
assembler="only_diagonal_part").weak_form().get_diagonal()
dlp_out_diag = modified_helmholtz.double_layer(
neumann_space,
dirichl_space,
dirichl_space,
kappa,
assembler="only_diagonal_part").weak_form().get_diagonal()
if permuted_rows:
diag11 = 0.5 * identity_diag + dlp_in_diag
diag12 = -(ep_ex / ep_in) * slp_in_diag
diag21 = 0.5 * identity_diag - dlp_out_diag
diag22 = slp_out_diag
else:
diag11 = 0.5 * identity_diag - dlp_out_diag
diag12 = slp_out_diag
diag21 = 0.5 * identity_diag + dlp_in_diag
diag22 = -(ep_ex / ep_in) * slp_in_diag
d_aux = 1 / (diag22 - diag21 * diag12 / diag11)
diag11_inv = 1 / diag11 + 1 / diag11 * diag12 * d_aux * diag21 / diag11
diag12_inv = -1 / diag11 * diag12 * d_aux
diag21_inv = -d_aux * diag21 / diag11
diag22_inv = d_aux
block_mat_precond = bmat([[diags(diag11_inv),
diags(diag12_inv)],
[diags(diag21_inv),
diags(diag22_inv)]]).tocsr()
return aslinearoperator(block_mat_precond)
def alpha_beta_single_blocked_operator(dirichl_space, neumann_space, ep_in,
ep_ex, kappa, alpha, beta,
operator_assembler):
dlp_in = laplace.double_layer(dirichl_space,
dirichl_space,
dirichl_space,
assembler=operator_assembler)
slp_in = laplace.single_layer(neumann_space,
dirichl_space,
dirichl_space,
assembler=operator_assembler)
hlp_in = laplace.hypersingular(dirichl_space,
neumann_space,
neumann_space,
assembler=operator_assembler)
adlp_in = laplace.adjoint_double_layer(neumann_space,
neumann_space,
neumann_space,
assembler=operator_assembler)
dlp_out = modified_helmholtz.double_layer(dirichl_space,
dirichl_space,
dirichl_space,
kappa,
assembler=operator_assembler)
slp_out = modified_helmholtz.single_layer(neumann_space,
dirichl_space,
dirichl_space,
kappa,
assembler=operator_assembler)
hlp_out = modified_helmholtz.hypersingular(dirichl_space,
neumann_space,
neumann_space,
kappa,
assembler=operator_assembler)
adlp_out = modified_helmholtz.adjoint_double_layer(
neumann_space,
neumann_space,
neumann_space,
kappa,
assembler=operator_assembler)
phi_identity = sparse.identity(dirichl_space, dirichl_space, dirichl_space)
dph_identity = sparse.identity(neumann_space, neumann_space, neumann_space)
ep = ep_ex / ep_in
A = bempp.api.BlockedOperator(2, 2)
A[0, 0] = (-0.5 * (1 + alpha)) * phi_identity + (alpha * dlp_out) - dlp_in
A[0, 1] = slp_in - ((alpha / ep) * slp_out)
A[1, 0] = hlp_in - (beta * hlp_out)
A[1,
1] = (-0.5 *
(1 +
(beta / ep))) * dph_identity + adlp_in - ((beta / ep) * adlp_out)
return A
def juffer(dirichl_space, neumann_space, ep_in, ep_ex, kappa,
operator_assembler):
phi_id = sparse.identity(dirichl_space, dirichl_space, dirichl_space)
dph_id = sparse.identity(neumann_space, neumann_space, neumann_space)
ep = ep_ex / ep_in
dF = laplace.double_layer(dirichl_space,
dirichl_space,
dirichl_space,
assembler=operator_assembler)
dP = modified_helmholtz.double_layer(dirichl_space,
dirichl_space,
dirichl_space,
kappa,
assembler=operator_assembler)
L1 = (ep * dP) - dF
F = laplace.single_layer(neumann_space,
dirichl_space,
dirichl_space,
assembler=operator_assembler)
P = modified_helmholtz.single_layer(neumann_space,
dirichl_space,
dirichl_space,
kappa,
assembler=operator_assembler)
L2 = F - P
ddF = laplace.hypersingular(dirichl_space,
neumann_space,
neumann_space,
assembler=operator_assembler)
ddP = modified_helmholtz.hypersingular(dirichl_space,
neumann_space,
neumann_space,
kappa,
assembler=operator_assembler)
L3 = ddP - ddF
dF0 = laplace.adjoint_double_layer(neumann_space,
neumann_space,
neumann_space,
assembler=operator_assembler)
dP0 = modified_helmholtz.adjoint_double_layer(neumann_space,
neumann_space,
neumann_space,
kappa,
assembler=operator_assembler)
L4 = dF0 - ((1.0 / ep) * dP0)
A = bempp.api.BlockedOperator(2, 2)
A[0, 0] = (0.5 * (1.0 + ep) * phi_id) - L1
A[0, 1] = (-1.0) * L2
A[1, 0] = L3 # Sign change due to bempp definition
A[1, 1] = (0.5 * (1.0 + (1.0 / ep)) * dph_id) - L4
return A
def test_modified_helmholtz_single_layer_evaluator(default_parameters, helpers,
precision,
device_interface):
"""Test dense evaluator for modified Helmholtz slp."""
from bempp.api import function_space
from bempp.api.operators.boundary.modified_helmholtz import single_layer
grid = helpers.load_grid("sphere")
space = function_space(grid, "P", 1)
discrete_op = single_layer(
space,
space,
space,
OMEGA,
assembler="dense_evaluator",
precision=precision,
device_interface=device_interface,
parameters=default_parameters,
).weak_form()
mat = single_layer(
space,
space,
space,
OMEGA,
assembler="dense",
precision=precision,
device_interface=device_interface,
parameters=default_parameters,
).weak_form()
x = _np.random.RandomState(0).randn(space.global_dof_count)
actual = discrete_op @ x
expected = mat @ x
if precision == "single":
tol = 1e-4
else:
tol = 1e-12
_np.testing.assert_allclose(actual, expected, rtol=tol)
def modHelmMultitrace(dirichl_space, neumann_space, kappa):
from bempp.api.operators.boundary import modified_helmholtz
A = bempp.api.BlockedOperator(2, 2)
A[0, 0] = (-1.0)*modified_helmholtz.double_layer(dirichl_space, dirichl_space, dirichl_space, kappa)
A[0, 1] = modified_helmholtz.single_layer(neumann_space, dirichl_space, dirichl_space, kappa)
A[1, 0] = modified_helmholtz.hypersingular(dirichl_space, neumann_space, neumann_space, kappa)
A[1, 1] = modified_helmholtz.adjoint_double_layer(neumann_space, neumann_space, neumann_space, kappa)
return A
def lu(dirichl_space, neumann_space, ep_in, ep_ex, kappa, operator_assembler):
dlp_in = laplace.double_layer(dirichl_space,
dirichl_space,
dirichl_space,
assembler=operator_assembler)
slp_in = laplace.single_layer(neumann_space,
dirichl_space,
dirichl_space,
assembler=operator_assembler)
hlp_in = laplace.hypersingular(dirichl_space,
neumann_space,
neumann_space,
assembler=operator_assembler)
adlp_in = laplace.adjoint_double_layer(neumann_space,
neumann_space,
neumann_space,
assembler=operator_assembler)
dlp_ex = modified_helmholtz.double_layer(dirichl_space,
dirichl_space,
dirichl_space,
kappa,
assembler=operator_assembler)
slp_ex = modified_helmholtz.single_layer(neumann_space,
dirichl_space,
dirichl_space,
kappa,
assembler=operator_assembler)
hlp_ex = modified_helmholtz.hypersingular(dirichl_space,
neumann_space,
neumann_space,
kappa,
assembler=operator_assembler)
adlp_ex = modified_helmholtz.adjoint_double_layer(
neumann_space,
neumann_space,
neumann_space,
kappa,
assembler=operator_assembler)
phi_identity = sparse.identity(dirichl_space, dirichl_space, dirichl_space)
dph_identity = sparse.identity(neumann_space, neumann_space, neumann_space)
ep = ep_ex / ep_in
A = bempp.api.BlockedOperator(2, 2)
A[0, 0] = (0.5 *
(1 + (1.0 / ep))) * phi_identity - dlp_ex + (1.0 / ep) * dlp_in
A[0, 1] = slp_ex - slp_in
A[1, 0] = (1.0 / ep) * hlp_ex - (1.0 / ep) * hlp_in
A[1,
1] = (0.5 * (1 +
(1.0 / ep))) * dph_identity + (1.0 / ep) * adlp_ex - adlp_in
return A
def juffer(dirichl_space, neumann_space, q, x_q, ep_in, ep_ex, kappa):
from bempp.api.operators.boundary import sparse, laplace, modified_helmholtz
phi_id = sparse.identity(dirichl_space, dirichl_space, dirichl_space)
dph_id = sparse.identity(neumann_space, neumann_space, neumann_space)
ep = ep_ex / ep_in
dF = laplace.double_layer(dirichl_space, dirichl_space, dirichl_space)
dP = modified_helmholtz.double_layer(dirichl_space, dirichl_space,
dirichl_space, kappa)
L1 = (ep * dP) - dF
F = laplace.single_layer(neumann_space, dirichl_space, dirichl_space)
P = modified_helmholtz.single_layer(neumann_space, dirichl_space,
dirichl_space, kappa)
L2 = F - P
ddF = laplace.hypersingular(dirichl_space, neumann_space, neumann_space)
ddP = modified_helmholtz.hypersingular(dirichl_space, neumann_space,
neumann_space, kappa)
L3 = ddP - ddF
dF0 = laplace.adjoint_double_layer(neumann_space, neumann_space,
neumann_space)
dP0 = modified_helmholtz.adjoint_double_layer(neumann_space, neumann_space,
neumann_space, kappa)
L4 = dF0 - ((1.0 / ep) * dP0)
A = bempp.api.BlockedOperator(2, 2)
A[0, 0] = (0.5 * (1.0 + ep) * phi_id) - L1
A[0, 1] = (-1.0) * L2
A[1, 0] = L3 # Cambio de signo por definicion de bempp
A[1, 1] = (0.5 * (1.0 + (1.0 / ep)) * dph_id) - L4
@bempp.api.real_callable
def d_green_func(x, n, domain_index, result):
nrm = np.sqrt((x[0] - x_q[:, 0])**2 + (x[1] - x_q[:, 1])**2 +
(x[2] - x_q[:, 2])**2)
const = -1. / (4. * np.pi * ep_in)
result[:] = const * np.sum(q * np.dot(x - x_q, n) / (nrm**3))
@bempp.api.real_callable
def green_func(x, n, domain_index, result):
nrm = np.sqrt((x[0] - x_q[:, 0])**2 + (x[1] - x_q[:, 1])**2 +
(x[2] - x_q[:, 2])**2)
result[:] = np.sum(q / nrm) / (4. * np.pi * ep_in)
rhs_1 = bempp.api.GridFunction(dirichl_space, fun=green_func)
rhs_2 = bempp.api.GridFunction(dirichl_space, fun=d_green_func)
return A, rhs_1, rhs_2
def muller_internal(dirichl_space, neumann_space, ep_in, ep_ex, kappa,
operator_assembler):
dlp_in = laplace.double_layer(dirichl_space,
dirichl_space,
dirichl_space,
assembler=operator_assembler)
slp_in = laplace.single_layer(neumann_space,
dirichl_space,
dirichl_space,
assembler=operator_assembler)
hlp_in = laplace.hypersingular(dirichl_space,
neumann_space,
neumann_space,
assembler=operator_assembler)
adlp_in = laplace.adjoint_double_layer(neumann_space,
neumann_space,
neumann_space,
assembler=operator_assembler)
dlp_ex = modified_helmholtz.double_layer(dirichl_space,
dirichl_space,
dirichl_space,
kappa,
assembler=operator_assembler)
slp_ex = modified_helmholtz.single_layer(neumann_space,
dirichl_space,
dirichl_space,
kappa,
assembler=operator_assembler)
hlp_ex = modified_helmholtz.hypersingular(dirichl_space,
neumann_space,
neumann_space,
kappa,
assembler=operator_assembler)
adlp_ex = modified_helmholtz.adjoint_double_layer(
neumann_space,
neumann_space,
neumann_space,
kappa,
assembler=operator_assembler)
phi_identity = sparse.identity(dirichl_space, dirichl_space, dirichl_space)
dph_identity = sparse.identity(neumann_space, neumann_space, neumann_space)
ep = ep_ex / ep_in
A = bempp.api.BlockedOperator(2, 2)
A[0, 0] = phi_identity + dlp_in - dlp_ex
A[0, 1] = -slp_in + ((1.0 / ep) * slp_ex)
A[1, 0] = -hlp_in + (ep * hlp_ex)
A[1, 1] = dph_identity - adlp_in + adlp_ex
return A
def direct(dirichl_space, neumann_space, assembler, q, x_q):
"""
Construct the system matrix and RHS grid functions using direct formulation.
"""
ep_in = PARAMS.ep_in
ep_ex = PARAMS.ep_ex
kappa = PARAMS.kappa
identity = sparse.identity(dirichl_space, dirichl_space, dirichl_space)
slp_in = laplace.single_layer(neumann_space,
dirichl_space,
dirichl_space,
assembler=assembler)
dlp_in = laplace.double_layer(dirichl_space,
dirichl_space,
dirichl_space,
assembler=assembler)
slp_out = modified_helmholtz.single_layer(neumann_space,
dirichl_space,
dirichl_space,
kappa,
assembler=assembler)
dlp_out = modified_helmholtz.double_layer(dirichl_space,
dirichl_space,
dirichl_space,
kappa,
assembler=assembler)
A_sys = bempp.api.BlockedOperator(2, 2)
A_sys[0, 0] = 0.5 * identity + dlp_in
A_sys[0, 1] = -slp_in
A_sys[1, 0] = 0.5 * identity - dlp_out
A_sys[1, 1] = (ep_in / ep_ex) * slp_out
@bempp.api.callable(vectorized=True)
def rhs1_fun(x, n, domain_index, result):
import exafmm.laplace as _laplace
sources = _laplace.init_sources(x_q, q)
targets = _laplace.init_targets(x.T)
fmm = _laplace.LaplaceFmm(p=10, ncrit=500, filename='.rhs.tmp')
tree = _laplace.setup(sources, targets, fmm)
values = _laplace.evaluate(tree, fmm)
os.remove('.rhs.tmp')
result[:] = values[:, 0] / ep_in
@bempp.api.real_callable
def rhs2_fun(x, n, domain_index, result):
result[0] = 0
rhs1 = bempp.api.GridFunction(dirichl_space, fun=rhs1_fun)
rhs2 = bempp.api.GridFunction(neumann_space, fun=rhs2_fun)
return A_sys, rhs1, rhs2
def block_diagonal(dirichl_space, neumann_space, A):
"""
Compute the block-diagonal preconditioner for system matrix A.
Suitable for direct formulation.
"""
ep_in = PARAMS.ep_in
ep_ex = PARAMS.ep_ex
kappa = PARAMS.kappa
identity = sparse.identity(dirichl_space, dirichl_space, dirichl_space)
identity_diag = identity.weak_form().A.diagonal()
slp_in_diag = laplace.single_layer(
neumann_space,
dirichl_space,
dirichl_space,
assembler="only_diagonal_part").weak_form().A
dlp_in_diag = laplace.double_layer(
dirichl_space,
dirichl_space,
dirichl_space,
assembler="only_diagonal_part").weak_form().A
slp_out_diag = modified_helmholtz.single_layer(
neumann_space,
dirichl_space,
dirichl_space,
kappa,
assembler="only_diagonal_part").weak_form().A
dlp_out_diag = modified_helmholtz.double_layer(
neumann_space,
dirichl_space,
dirichl_space,
kappa,
assembler="only_diagonal_part").weak_form().A
diag11 = .5 * identity_diag + dlp_in_diag
diag12 = -slp_in_diag
diag21 = .5 * identity_diag - dlp_out_diag
diag22 = (ep_in / ep_ex) * slp_out_diag
d_aux = 1 / (diag22 - diag21 * diag12 / diag11)
diag11_inv = 1 / diag11 + 1 / diag11 * diag12 * d_aux * diag21 / diag11
diag12_inv = -1 / diag11 * diag12 * d_aux
diag21_inv = -d_aux * diag21 / diag11
diag22_inv = d_aux
block_mat_precond = bmat([[diags(diag11_inv),
diags(diag12_inv)],
[diags(diag21_inv),
diags(diag22_inv)]]).tocsr()
return aslinearoperator(block_mat_precond)
def direct(dirichl_space,
neumann_space,
ep_in,
ep_out,
kappa,
operator_assembler,
permute_rows=False):
identity = sparse.identity(dirichl_space, dirichl_space, dirichl_space)
slp_in = laplace.single_layer(neumann_space,
dirichl_space,
dirichl_space,
assembler=operator_assembler)
dlp_in = laplace.double_layer(dirichl_space,
dirichl_space,
dirichl_space,
assembler=operator_assembler)
slp_out = modified_helmholtz.single_layer(neumann_space,
dirichl_space,
dirichl_space,
kappa,
assembler=operator_assembler)
dlp_out = modified_helmholtz.double_layer(dirichl_space,
dirichl_space,
dirichl_space,
kappa,
assembler=operator_assembler)
A = bempp.api.BlockedOperator(2, 2)
if permute_rows: #Use permuted rows formulation
A[0, 0] = 0.5 * identity - dlp_out
A[0, 1] = (ep_in / ep_out) * slp_out
A[1, 0] = 0.5 * identity + dlp_in
A[1, 1] = -slp_in
else: #Normal direct formulation
A[0, 0] = 0.5 * identity + dlp_in
A[0, 1] = -slp_in
A[1, 0] = 0.5 * identity - dlp_out
A[1, 1] = (ep_in / ep_out) * slp_out
return A
def test_slp_hyp_pair_linear_space(self):
"""SLP and HYP pair on linear spaces."""
lin_space = self._lin_space
parameters = bempp.api.common.global_parameters()
parameters.assembly.boundary_operator_assembly_type = 'dense'
parameters.quadrature.double_singular = 9
parameters.quadrature.near.double_order = 9
parameters.quadrature.medium.double_order = 9
parameters.quadrature.far.double_order = 9
from bempp.api.operators.boundary.modified_helmholtz import \
single_layer_and_hypersingular_pair, single_layer, hypersingular
actual_ops_dual = single_layer_and_hypersingular_pair(
self._grid, WAVE_NUMBER, spaces='linear', parameters=parameters)
expected_slp = bempp.api.as_matrix(single_layer(
lin_space, lin_space, lin_space, WAVE_NUMBER,
parameters=parameters).weak_form())
actual_slp = bempp.api.as_matrix(actual_ops_dual[0].weak_form())
expected_hyp = bempp.api.as_matrix(hypersingular(
lin_space, lin_space, lin_space, WAVE_NUMBER,
parameters=parameters).weak_form())
actual_hyp = bempp.api.as_matrix(actual_ops_dual[1].weak_form())
diff_norm_slp = np.linalg.norm(
expected_slp - actual_slp) / np.linalg.norm(actual_slp)
diff_norm_hyp = np.linalg.norm(
expected_hyp - actual_hyp) / np.linalg.norm(actual_hyp)
self.assertAlmostEqual(diff_norm_slp, 0, 6)
self.assertAlmostEqual(diff_norm_hyp, 0, 6)
def block_diagonal_precon_juffer(dirichl_space, neumann_space, ep_in, ep_ex,
kappa):
from scipy.sparse import diags, bmat
from scipy.sparse.linalg import factorized, LinearOperator
from bempp.api.operators.boundary import sparse, laplace, modified_helmholtz
phi_id = sparse.identity(dirichl_space, dirichl_space,
dirichl_space).weak_form().A.diagonal()
dph_id = sparse.identity(neumann_space, neumann_space,
neumann_space).weak_form().A.diagonal()
ep = ep_ex / ep_in
dF = laplace.double_layer(dirichl_space,
dirichl_space,
dirichl_space,
assembler="only_diagonal_part").weak_form().A
dP = modified_helmholtz.double_layer(
dirichl_space,
dirichl_space,
dirichl_space,
kappa,
assembler="only_diagonal_part").weak_form().A
L1 = (ep * dP) - dF
F = laplace.single_layer(neumann_space,
dirichl_space,
dirichl_space,
assembler="only_diagonal_part").weak_form().A
P = modified_helmholtz.single_layer(
neumann_space,
dirichl_space,
dirichl_space,
kappa,
assembler="only_diagonal_part").weak_form().A
L2 = F - P
ddF = laplace.hypersingular(dirichl_space,
neumann_space,
neumann_space,
assembler="only_diagonal_part").weak_form().A
ddP = modified_helmholtz.hypersingular(
dirichl_space,
neumann_space,
neumann_space,
kappa,
assembler="only_diagonal_part").weak_form().A
L3 = ddP - ddF
dF0 = laplace.adjoint_double_layer(
neumann_space,
neumann_space,
neumann_space,
assembler="only_diagonal_part").weak_form().A
dP0 = modified_helmholtz.adjoint_double_layer(
neumann_space,
neumann_space,
neumann_space,
kappa,
assembler="only_diagonal_part").weak_form().A
L4 = dF0 - ((1.0 / ep) * dP0)
diag11 = diags((0.5 * (1.0 + ep) * phi_id) - L1)
diag12 = diags((-1.0) * L2)
diag21 = diags(L3)
diag22 = diags((0.5 * (1.0 + (1.0 / ep)) * dph_id) - L4)
block_mat_precond = bmat([[diag11, diag12],
[diag21, diag22]]).tocsr() # csr_matrix
solve = factorized(
block_mat_precond
) # a callable for solving a sparse linear system (treat it as an inverse)
precond = LinearOperator(matvec=solve,
dtype='float64',
shape=block_mat_precond.shape)
return precond
def stern_asc(neumann_space_in, dirichl_space_ex, neumann_space_ex, ep_in,
ep_ex, q, x_q, kappa):
# Functions to proyect the carges potential to the boundary with constants
def d_green_func(x, n, domain_index, result):
const = -1. / (4. * np.pi * ep_in)
result[:] = const * np.sum(q * np.dot(x - x_q, n) /
(np.linalg.norm(x - x_q, axis=1)**3))
print "\nProjecting charges over surface..."
charged_grid_fun = bempp.api.GridFunction(neumann_space_in,
fun=d_green_func)
rhs = np.concatenate([
charged_grid_fun.coefficients,
np.zeros(dirichl_space_ex.global_dof_count),
np.zeros(neumann_space_ex.global_dof_count)
])
print "Defining operators..."
# OPERATORS FOR INTERNAL SURFACE
from bempp.api.operators.boundary import sparse, laplace, modified_helmholtz
idn_in = sparse.identity(neumann_space_in, neumann_space_in,
neumann_space_in)
adj_1T1 = laplace.adjoint_double_layer(neumann_space_in, neumann_space_in,
neumann_space_in)
hyp_2T1 = laplace.hypersingular(dirichl_space_ex, neumann_space_in,
neumann_space_in)
adj_2T1 = laplace.adjoint_double_layer(neumann_space_ex, neumann_space_in,
neumann_space_in)
# OPERATORS FOR EXTERNAL SURFACE
idn_ex = sparse.identity(dirichl_space_ex, dirichl_space_ex,
dirichl_space_ex)
slp_1T2 = laplace.single_layer(neumann_space_in, dirichl_space_ex,
dirichl_space_ex)
dlp_2T2 = laplace.double_layer(dirichl_space_ex, dirichl_space_ex,
dirichl_space_ex)
slp_2T2 = laplace.single_layer(neumann_space_ex, dirichl_space_ex,
dirichl_space_ex)
dlp_ex = modified_helmholtz.double_layer(dirichl_space_ex,
dirichl_space_ex,
dirichl_space_ex, kappa)
slp_ex = modified_helmholtz.single_layer(neumann_space_ex,
dirichl_space_ex,
dirichl_space_ex, kappa)
ep = (ep_in / ep_ex)
print "Creating operators..."
# Matrix Assemble
blocked = bempp.api.BlockedOperator(3, 3)
blocked[0, 0] = idn_in + (ep - 1.) * adj_1T1
blocked[0, 1] = -hyp_2T1
blocked[0, 2] = -adj_2T1
blocked[1, 0] = ep * slp_1T2
blocked[1, 1] = .5 * idn_ex + dlp_2T2
blocked[1, 2] = -slp_2T2
#blocked[2, 0] = 0
blocked[2, 1] = .5 * idn_ex - dlp_ex
blocked[2, 2] = slp_ex
A = blocked.weak_form()
return A, rhs
neumann_space = bempp.api.function_space(grid, "DP", 0)
dual_to_dir_s = bempp.api.function_space(grid, "DP", 0)
# Se crea u_s y su derivada normal du_s en la frontera
u_s = bempp.api.GridFunction(dirichl_space, fun=u_s_G)
du_s = bempp.api.GridFunction(neumann_space, fun=du_s_G)
# Se crean los operadores asociados al sistema, que dependen de los espacios de las soluciones
# identity : I : matriz identidad
# dlp_in : K_in : Double-Layer operator para la region interior
# slp_in : V_in : Single-Layer operator para la region interior
# _out : Mismos operadores pero para la región exterior, con k=kappa=0.125
identity = sparse.identity( dirichl_space, dirichl_space, dual_to_dir_s)
slp_in = laplace.single_layer(neumann_space, dirichl_space, dual_to_dir_s)
dlp_in = laplace.double_layer(dirichl_space, dirichl_space, dual_to_dir_s)
slp_out = modified_helmholtz.single_layer(neumann_space, dirichl_space, dual_to_dir_s, k)
dlp_out = modified_helmholtz.double_layer(dirichl_space, dirichl_space, dual_to_dir_s, k)
# 4.1 Solución armónica ---------------------------------------------------------------
# Dada por V_in du_s = (1/2+K_in)u_h = -(1/2+K_in)u_s (BC)
sol, info,it_count = bempp.api.linalg.gmres( slp_in, -(dlp_in+0.5*identity)*u_s , return_iteration_count=True, tol=1e-4)
print("The linear system for du_h was solved in {0} iterations".format(it_count))
u_h = -u_s
du_h = sol
# 4.2 Solución regular -----------------------------------------------------------------
def stern_formulation(dirichl_space_in, neumann_space_in, dirichl_space_ex,
neumann_space_ex, ep_in, ep_ex, q, x_q, kappa):
# Functions to proyect the carges potential to the boundary with constants
def green_func(x, n, domain_index, result):
result[:] = np.sum(
q / np.linalg.norm(x - x_q, axis=1)) / (4. * np.pi * ep_in)
print "\nProjecting charges over surface..."
charged_grid_fun = bempp.api.GridFunction(dirichl_space_in, fun=green_func)
rhs = np.concatenate([
charged_grid_fun.coefficients,
np.zeros(neumann_space_in.global_dof_count),
np.zeros(dirichl_space_ex.global_dof_count),
np.zeros(neumann_space_ex.global_dof_count)
])
print "Defining operators..."
# OPERATOR FOR INTERNAL SURFACE
from bempp.api.operators.boundary import sparse, laplace, modified_helmholtz
idn_in = sparse.identity(dirichl_space_in, dirichl_space_in,
dirichl_space_in)
# Ec 1
slp_in = laplace.single_layer(neumann_space_in, dirichl_space_in,
dirichl_space_in)
dlp_in = laplace.double_layer(dirichl_space_in, dirichl_space_in,
dirichl_space_in)
# Ec 2
# adj_1T1 = laplace.single_layer(neumann_space_in, dirichl_space_in, dirichl_space_in)
# adj_1T1 = laplace.double_layer(dirichl_space_in, dirichl_space_in, dirichl_space_in)
slp_2T1 = laplace.single_layer(neumann_space_ex, dirichl_space_in,
dirichl_space_in)
dlp_2T1 = laplace.double_layer(dirichl_space_ex, dirichl_space_in,
dirichl_space_in)
# OPERATOR FOR EXTERNAL SURFACE
idn_ex = sparse.identity(dirichl_space_ex, dirichl_space_ex,
dirichl_space_ex)
# Internal Boudary
slp_1T2 = laplace.single_layer(neumann_space_in, dirichl_space_ex,
dirichl_space_ex)
# dlp_1T2 = laplace.double_layer(dirichl_space_in, dirichl_space_ex, dirichl_space_ex)
slp_2T2 = laplace.single_layer(neumann_space_ex, dirichl_space_ex,
dirichl_space_ex)
dlp_2T2 = laplace.double_layer(dirichl_space_ex, dirichl_space_ex,
dirichl_space_ex)
# External Boundary
slp_ex = modified_helmholtz.single_layer(neumann_space_ex,
dirichl_space_ex,
dirichl_space_ex, kappa)
dlp_ex = modified_helmholtz.double_layer(dirichl_space_ex,
dirichl_space_ex,
dirichl_space_ex, kappa)
ep = (ep_in / ep_ex)
print "Creating operators..."
# Matrix Assemble
blocked = bempp.api.BlockedOperator(4, 4)
blocked[0, 0] = .5 * idn_in + dlp_in
blocked[0, 1] = -slp_in
# blocked[0, 2] = 0
# blocked[0, 3] = 0
# Original formulation
blocked[1, 0] = .5 * idn_in - dlp_in # dlp_in = dlp_1T1
blocked[1, 1] = ep * slp_in # slp_in = slp_1T1
blocked[1, 2] = dlp_2T1
blocked[1, 3] = -slp_2T1
# blocked[2, 0] = -dlp_1T2 ## eliminar**
blocked[2, 1] = ep * slp_1T2
blocked[2, 2] = .5 * idn_ex + dlp_2T2
blocked[2, 3] = -slp_2T2
# blocked[3, 0] = 0
# blocked[3, 1] = 0
blocked[3, 2] = .5 * idn_ex - dlp_ex
blocked[3, 3] = slp_ex
A = blocked.strong_form()
return A, rhs
def derivative_in(dirichl_space, neumann_space, assembler, q, x_q):
"""
Construct the system matrix and RHS grid functions using derivative formulation with interior values.
"""
ep_in = PARAMS.ep_in
ep_ex = PARAMS.ep_ex
kappa = PARAMS.kappa
phi_id = sparse.identity(dirichl_space, dirichl_space, dirichl_space)
dph_id = sparse.identity(neumann_space, neumann_space, neumann_space)
ep = ep_ex / ep_in
dF = laplace.double_layer(dirichl_space,
dirichl_space,
dirichl_space,
assembler=assembler)
dP = modified_helmholtz.double_layer(dirichl_space,
dirichl_space,
dirichl_space,
kappa,
assembler=assembler)
L1 = (ep * dP) - dF
F = laplace.single_layer(neumann_space,
dirichl_space,
dirichl_space,
assembler=assembler)
P = modified_helmholtz.single_layer(neumann_space,
dirichl_space,
dirichl_space,
kappa,
assembler=assembler)
L2 = F - P
ddF = laplace.hypersingular(dirichl_space,
neumann_space,
neumann_space,
assembler=assembler)
ddP = modified_helmholtz.hypersingular(dirichl_space,
neumann_space,
neumann_space,
kappa,
assembler=assembler)
L3 = ddP - ddF
dF0 = laplace.adjoint_double_layer(neumann_space,
neumann_space,
neumann_space,
assembler=assembler)
dP0 = modified_helmholtz.adjoint_double_layer(neumann_space,
neumann_space,
neumann_space,
kappa,
assembler=assembler)
L4 = dF0 - ((1.0 / ep) * dP0)
A_sys = bempp.api.BlockedOperator(2, 2)
A_sys[0, 0] = (0.5 * (1.0 + ep) * phi_id) - L1
A_sys[0, 1] = (-1.0) * L2
A_sys[1, 0] = L3
A_sys[1, 1] = (0.5 * (1.0 + (1.0 / ep)) * dph_id) - L4
@bempp.api.callable(vectorized=True)
def rhs1_fun(x, n, domain_index, result):
import exafmm.laplace as _laplace
sources = _laplace.init_sources(x_q, q)
targets = _laplace.init_targets(x.T)
fmm = _laplace.LaplaceFmm(p=10, ncrit=500, filename='.rhs.tmp')
tree = _laplace.setup(sources, targets, fmm)
values = _laplace.evaluate(tree, fmm)
os.remove('.rhs.tmp')
result[:] = values[:, 0] / ep_in
@bempp.api.callable(vectorized=True)
def rhs2_fun(x, n, domain_index, result):
import exafmm.laplace as _laplace
sources = _laplace.init_sources(x_q, q)
targets = _laplace.init_targets(x.T)
fmm = _laplace.LaplaceFmm(p=10, ncrit=500, filename='.rhs.tmp')
tree = _laplace.setup(sources, targets, fmm)
values = _laplace.evaluate(tree, fmm)
os.remove('.rhs.tmp')
result[:] = np.sum(values[:, 1:] * n.T, axis=1) / ep_in
rhs1 = bempp.api.GridFunction(dirichl_space, fun=rhs1_fun)
rhs2 = bempp.api.GridFunction(neumann_space, fun=rhs2_fun)
return A_sys, rhs1, rhs2
def first_kind_external(dirichl_space, neumann_space, ep_in, ep_ex, kappa,
operator_assembler):
dlp_in = laplace.double_layer(dirichl_space,
dirichl_space,
dirichl_space,
assembler=operator_assembler)
slp_in = laplace.single_layer(neumann_space,
dirichl_space,
dirichl_space,
assembler=operator_assembler)
hlp_in = laplace.hypersingular(dirichl_space,
neumann_space,
neumann_space,
assembler=operator_assembler)
adlp_in = laplace.adjoint_double_layer(neumann_space,
neumann_space,
neumann_space,
assembler=operator_assembler)
dlp_ex = modified_helmholtz.double_layer(dirichl_space,
dirichl_space,
dirichl_space,
kappa,
assembler=operator_assembler)
slp_ex = modified_helmholtz.single_layer(neumann_space,
dirichl_space,
dirichl_space,
kappa,
assembler=operator_assembler)
hlp_ex = modified_helmholtz.hypersingular(dirichl_space,
neumann_space,
neumann_space,
kappa,
assembler=operator_assembler)
adlp_ex = modified_helmholtz.adjoint_double_layer(
neumann_space,
neumann_space,
neumann_space,
kappa,
assembler=operator_assembler)
ep = ep_ex / ep_in
A = bempp.api.BlockedOperator(2, 2)
A[0, 0] = (-1.0 * dlp_ex) - dlp_in
A[0, 1] = slp_ex + (ep * slp_in)
A[1, 0] = hlp_ex + ((1.0 / ep) * hlp_in)
A[1, 1] = adlp_ex + adlp_in
calderon_int_scal = bempp.api.BlockedOperator(2, 2)
calderon_int_scal[0, 0] = -1.0 * dlp_in
calderon_int_scal[0, 1] = ep * slp_in
calderon_int_scal[1, 0] = (1.0 / ep) * hlp_in
calderon_int_scal[1, 1] = adlp_in
calderon_ext = bempp.api.BlockedOperator(2, 2)
calderon_ext[0, 0] = -1.0 * dlp_ex
calderon_ext[0, 1] = slp_ex
calderon_ext[1, 0] = hlp_ex
calderon_ext[1, 1] = adlp_ex
return A, calderon_int_scal, calderon_ext
def block_diagonal_precon_alpha_beta(dirichl_space, neumann_space, ep_in,
ep_ex, kappa, alpha, beta):
from scipy.sparse import diags, bmat
from scipy.sparse.linalg import factorized, LinearOperator
from bempp.api.operators.boundary import sparse, laplace, modified_helmholtz
slp_in_diag = laplace.single_layer(
neumann_space,
dirichl_space,
dirichl_space,
assembler="only_diagonal_part").weak_form().A
dlp_in_diag = laplace.double_layer(
dirichl_space,
dirichl_space,
dirichl_space,
assembler="only_diagonal_part").weak_form().A
hlp_in_diag = laplace.hypersingular(
dirichl_space,
neumann_space,
neumann_space,
assembler="only_diagonal_part").weak_form().A
adlp_in_diag = laplace.adjoint_double_layer(
neumann_space,
neumann_space,
neumann_space,
assembler="only_diagonal_part").weak_form().A
slp_out_diag = modified_helmholtz.single_layer(
neumann_space,
dirichl_space,
dirichl_space,
kappa,
assembler="only_diagonal_part").weak_form().A
dlp_out_diag = modified_helmholtz.double_layer(
dirichl_space,
dirichl_space,
dirichl_space,
kappa,
assembler="only_diagonal_part").weak_form().A
hlp_out_diag = modified_helmholtz.hypersingular(
dirichl_space,
neumann_space,
neumann_space,
kappa,
assembler="only_diagonal_part").weak_form().A
adlp_out_diag = modified_helmholtz.adjoint_double_layer(
neumann_space,
neumann_space,
neumann_space,
kappa,
assembler="only_diagonal_part").weak_form().A
phi_identity_diag = sparse.identity(
dirichl_space, dirichl_space, dirichl_space).weak_form().A.diagonal()
dph_identity_diag = sparse.identity(
neumann_space, neumann_space, neumann_space).weak_form().A.diagonal()
ep = ep_ex / ep_in
diag11 = diags((-0.5 * (1 + alpha)) * phi_identity_diag +
(alpha * dlp_out_diag) - dlp_in_diag)
diag12 = diags(slp_in_diag - ((alpha / ep) * slp_out_diag))
diag21 = diags(hlp_in_diag - (beta * hlp_out_diag))
diag22 = diags((-0.5 * (1 + (beta / ep))) * dph_identity_diag +
adlp_in_diag - ((beta / ep) * adlp_out_diag))
block_mat_precond = bmat([[diag11, diag12],
[diag21, diag22]]).tocsr() # csr_matrix
solve = factorized(
block_mat_precond
) # a callable for solving a sparse linear system (treat it as an inverse)
precond = LinearOperator(matvec=solve,
dtype='float64',
shape=block_mat_precond.shape)
return precond
