def get_operator(self, ambient_dim):
sign = +1 if self.side in [+1, "scat"] else -1
knl = self.knl_class(ambient_dim) # noqa: pylint:disable=E1102
if self.bc_type == "dirichlet":
from pytential.symbolic.pde.scalar import DirichletOperator
op = DirichletOperator(knl,
sign,
use_l2_weighting=True,
kernel_arguments=self.knl_sym_kwargs)
elif self.bc_type == "neumann":
from pytential.symbolic.pde.scalar import NeumannOperator
op = NeumannOperator(knl,
sign,
use_l2_weighting=True,
use_improved_operator=False,
kernel_arguments=self.knl_sym_kwargs)
elif self.bc_type == "clamped_plate":
from pytential.symbolic.pde.scalar import BiharmonicClampedPlateOperator
op = BiharmonicClampedPlateOperator(knl, sign)
else:
raise ValueError(f"unknown bc_type: '{self.bc_type}'")
return op
def run_int_eq_test(cl_ctx, queue, case, resolution, visualize):
mesh = case.get_mesh(resolution, case.target_order)
print("%d elements" % mesh.nelements)
from pytential.qbx import QBXLayerPotentialSource
from meshmode.discretization import Discretization
from meshmode.discretization.poly_element import \
InterpolatoryQuadratureSimplexGroupFactory
pre_density_discr = Discretization(
cl_ctx, mesh,
InterpolatoryQuadratureSimplexGroupFactory(case.target_order))
source_order = 4*case.target_order
refiner_extra_kwargs = {}
qbx_lpot_kwargs = {}
if case.fmm_backend is None:
qbx_lpot_kwargs["fmm_order"] = False
else:
if hasattr(case, "fmm_tol"):
from sumpy.expansion.level_to_order import SimpleExpansionOrderFinder
qbx_lpot_kwargs["fmm_level_to_order"] = SimpleExpansionOrderFinder(
case.fmm_tol)
elif hasattr(case, "fmm_order"):
qbx_lpot_kwargs["fmm_order"] = case.fmm_order
else:
qbx_lpot_kwargs["fmm_order"] = case.qbx_order + 5
qbx = QBXLayerPotentialSource(
pre_density_discr,
fine_order=source_order,
qbx_order=case.qbx_order,
_box_extent_norm=getattr(case, "box_extent_norm", None),
_from_sep_smaller_crit=getattr(case, "from_sep_smaller_crit", None),
_from_sep_smaller_min_nsources_cumul=30,
fmm_backend=case.fmm_backend, **qbx_lpot_kwargs)
if case.use_refinement:
if case.k != 0 and getattr(case, "refine_on_helmholtz_k", True):
refiner_extra_kwargs["kernel_length_scale"] = 5/case.k
if hasattr(case, "scaled_max_curvature_threshold"):
refiner_extra_kwargs["_scaled_max_curvature_threshold"] = \
case.scaled_max_curvature_threshold
if hasattr(case, "expansion_disturbance_tolerance"):
refiner_extra_kwargs["_expansion_disturbance_tolerance"] = \
case.expansion_disturbance_tolerance
if hasattr(case, "refinement_maxiter"):
refiner_extra_kwargs["maxiter"] = case.refinement_maxiter
#refiner_extra_kwargs["visualize"] = True
print("%d elements before refinement" % pre_density_discr.mesh.nelements)
qbx, _ = qbx.with_refinement(**refiner_extra_kwargs)
print("%d stage-1 elements after refinement"
% qbx.density_discr.mesh.nelements)
print("%d stage-2 elements after refinement"
% qbx.stage2_density_discr.mesh.nelements)
print("quad stage-2 elements have %d nodes"
% qbx.quad_stage2_density_discr.groups[0].nunit_nodes)
density_discr = qbx.density_discr
if hasattr(case, "visualize_geometry") and case.visualize_geometry:
bdry_normals = bind(
density_discr, sym.normal(mesh.ambient_dim)
)(queue).as_vector(dtype=object)
bdry_vis = make_visualizer(queue, density_discr, case.target_order)
bdry_vis.write_vtk_file("geometry.vtu", [
("normals", bdry_normals)
])
# {{{ plot geometry
if 0:
if mesh.ambient_dim == 2:
# show geometry, centers, normals
nodes_h = density_discr.nodes().get(queue=queue)
pt.plot(nodes_h[0], nodes_h[1], "x-")
normal = bind(density_discr, sym.normal(2))(queue).as_vector(np.object)
pt.quiver(nodes_h[0], nodes_h[1],
normal[0].get(queue), normal[1].get(queue))
pt.gca().set_aspect("equal")
pt.show()
elif mesh.ambient_dim == 3:
bdry_vis = make_visualizer(queue, density_discr, case.target_order+3)
bdry_normals = bind(density_discr, sym.normal(3))(queue)\
.as_vector(dtype=object)
bdry_vis.write_vtk_file("pre-solve-source-%s.vtu" % resolution, [
("bdry_normals", bdry_normals),
])
else:
raise ValueError("invalid mesh dim")
# }}}
# {{{ set up operator
from pytential.symbolic.pde.scalar import (
DirichletOperator,
NeumannOperator)
from sumpy.kernel import LaplaceKernel, HelmholtzKernel
if case.k:
knl = HelmholtzKernel(mesh.ambient_dim)
knl_kwargs = {"k": sym.var("k")}
concrete_knl_kwargs = {"k": case.k}
else:
knl = LaplaceKernel(mesh.ambient_dim)
knl_kwargs = {}
concrete_knl_kwargs = {}
if knl.is_complex_valued:
dtype = np.complex128
else:
dtype = np.float64
loc_sign = +1 if case.prob_side in [+1, "scat"] else -1
if case.bc_type == "dirichlet":
op = DirichletOperator(knl, loc_sign, use_l2_weighting=True,
kernel_arguments=knl_kwargs)
elif case.bc_type == "neumann":
op = NeumannOperator(knl, loc_sign, use_l2_weighting=True,
use_improved_operator=False, kernel_arguments=knl_kwargs)
else:
assert False
op_u = op.operator(sym.var("u"))
# }}}
# {{{ set up test data
if case.prob_side == -1:
test_src_geo_radius = case.outer_radius
test_tgt_geo_radius = case.inner_radius
elif case.prob_side == +1:
test_src_geo_radius = case.inner_radius
test_tgt_geo_radius = case.outer_radius
elif case.prob_side == "scat":
test_src_geo_radius = case.outer_radius
test_tgt_geo_radius = case.outer_radius
else:
raise ValueError("unknown problem_side")
point_sources = make_circular_point_group(
mesh.ambient_dim, 10, test_src_geo_radius,
func=lambda x: x**1.5)
test_targets = make_circular_point_group(
mesh.ambient_dim, 20, test_tgt_geo_radius)
np.random.seed(22)
source_charges = np.random.randn(point_sources.shape[1])
source_charges[-1] = -np.sum(source_charges[:-1])
source_charges = source_charges.astype(dtype)
assert np.sum(source_charges) < 1e-15
source_charges_dev = cl.array.to_device(queue, source_charges)
# }}}
# {{{ establish BCs
from pytential.source import PointPotentialSource
from pytential.target import PointsTarget
point_source = PointPotentialSource(cl_ctx, point_sources)
pot_src = sym.IntG(
# FIXME: qbx_forced_limit--really?
knl, sym.var("charges"), qbx_forced_limit=None, **knl_kwargs)
test_direct = bind((point_source, PointsTarget(test_targets)), pot_src)(
queue, charges=source_charges_dev, **concrete_knl_kwargs)
if case.bc_type == "dirichlet":
bc = bind((point_source, density_discr), pot_src)(
queue, charges=source_charges_dev, **concrete_knl_kwargs)
elif case.bc_type == "neumann":
bc = bind(
(point_source, density_discr),
sym.normal_derivative(
qbx.ambient_dim, pot_src, where=sym.DEFAULT_TARGET)
)(queue, charges=source_charges_dev, **concrete_knl_kwargs)
# }}}
# {{{ solve
bound_op = bind(qbx, op_u)
rhs = bind(density_discr, op.prepare_rhs(sym.var("bc")))(queue, bc=bc)
try:
from pytential.solve import gmres
gmres_result = gmres(
bound_op.scipy_op(queue, "u", dtype, **concrete_knl_kwargs),
rhs,
tol=case.gmres_tol,
progress=True,
hard_failure=True,
stall_iterations=50, no_progress_factor=1.05)
except QBXTargetAssociationFailedException as e:
bdry_vis = make_visualizer(queue, density_discr, case.target_order+3)
bdry_vis.write_vtk_file("failed-targets-%s.vtu" % resolution, [
("failed_targets", e.failed_target_flags),
])
raise
print("gmres state:", gmres_result.state)
weighted_u = gmres_result.solution
# }}}
# {{{ build matrix for spectrum check
if 0:
from sumpy.tools import build_matrix
mat = build_matrix(
bound_op.scipy_op(
queue, arg_name="u", dtype=dtype, k=case.k))
w, v = la.eig(mat)
if 0:
pt.imshow(np.log10(1e-20+np.abs(mat)))
pt.colorbar()
pt.show()
#assert abs(s[-1]) < 1e-13, "h
#assert abs(s[-2]) > 1e-7
#from pudb import set_trace; set_trace()
# }}}
if case.prob_side != "scat":
# {{{ error check
points_target = PointsTarget(test_targets)
bound_tgt_op = bind((qbx, points_target),
op.representation(sym.var("u")))
test_via_bdry = bound_tgt_op(queue, u=weighted_u, k=case.k)
err = test_via_bdry - test_direct
err = err.get()
test_direct = test_direct.get()
test_via_bdry = test_via_bdry.get()
# {{{ remove effect of net source charge
if case.k == 0 and case.bc_type == "neumann" and loc_sign == -1:
# remove constant offset in interior Laplace Neumann error
tgt_ones = np.ones_like(test_direct)
tgt_ones = tgt_ones/la.norm(tgt_ones)
err = err - np.vdot(tgt_ones, err)*tgt_ones
# }}}
rel_err_2 = la.norm(err)/la.norm(test_direct)
rel_err_inf = la.norm(err, np.inf)/la.norm(test_direct, np.inf)
# }}}
print("rel_err_2: %g rel_err_inf: %g" % (rel_err_2, rel_err_inf))
else:
rel_err_2 = None
rel_err_inf = None
# {{{ test gradient
if case.check_gradient and case.prob_side != "scat":
bound_grad_op = bind((qbx, points_target),
op.representation(
sym.var("u"),
map_potentials=lambda pot: sym.grad(mesh.ambient_dim, pot),
qbx_forced_limit=None))
#print(bound_t_deriv_op.code)
grad_from_src = bound_grad_op(
queue, u=weighted_u, **concrete_knl_kwargs)
grad_ref = (bind(
(point_source, points_target),
sym.grad(mesh.ambient_dim, pot_src)
)(queue, charges=source_charges_dev, **concrete_knl_kwargs)
)
grad_err = (grad_from_src - grad_ref)
rel_grad_err_inf = (
la.norm(grad_err[0].get(), np.inf)
/ la.norm(grad_ref[0].get(), np.inf))
print("rel_grad_err_inf: %g" % rel_grad_err_inf)
# }}}
# {{{ test tangential derivative
if case.check_tangential_deriv and case.prob_side != "scat":
bound_t_deriv_op = bind(qbx,
op.representation(
sym.var("u"),
map_potentials=lambda pot: sym.tangential_derivative(2, pot),
qbx_forced_limit=loc_sign))
#print(bound_t_deriv_op.code)
tang_deriv_from_src = bound_t_deriv_op(
queue, u=weighted_u, **concrete_knl_kwargs).as_scalar().get()
tang_deriv_ref = (bind(
(point_source, density_discr),
sym.tangential_derivative(2, pot_src)
)(queue, charges=source_charges_dev, **concrete_knl_kwargs)
.as_scalar().get())
if 0:
pt.plot(tang_deriv_ref.real)
pt.plot(tang_deriv_from_src.real)
pt.show()
td_err = (tang_deriv_from_src - tang_deriv_ref)
rel_td_err_inf = la.norm(td_err, np.inf)/la.norm(tang_deriv_ref, np.inf)
print("rel_td_err_inf: %g" % rel_td_err_inf)
else:
rel_td_err_inf = None
# }}}
# {{{ any-D file plotting
if visualize:
bdry_vis = make_visualizer(queue, density_discr, case.target_order+3)
bdry_normals = bind(density_discr, sym.normal(qbx.ambient_dim))(queue)\
.as_vector(dtype=object)
sym_sqrt_j = sym.sqrt_jac_q_weight(density_discr.ambient_dim)
u = bind(density_discr, sym.var("u")/sym_sqrt_j)(queue, u=weighted_u)
bdry_vis.write_vtk_file("source-%s.vtu" % resolution, [
("u", u),
("bc", bc),
#("bdry_normals", bdry_normals),
])
from sumpy.visualization import make_field_plotter_from_bbox # noqa
from meshmode.mesh.processing import find_bounding_box
vis_grid_spacing = (0.1, 0.1, 0.1)[:qbx.ambient_dim]
if hasattr(case, "vis_grid_spacing"):
vis_grid_spacing = case.vis_grid_spacing
vis_extend_factor = 0.2
if hasattr(case, "vis_extend_factor"):
vis_grid_spacing = case.vis_grid_spacing
fplot = make_field_plotter_from_bbox(
find_bounding_box(mesh),
h=vis_grid_spacing,
extend_factor=vis_extend_factor)
qbx_tgt_tol = qbx.copy(target_association_tolerance=0.15)
from pytential.target import PointsTarget
try:
solved_pot = bind(
(qbx_tgt_tol, PointsTarget(fplot.points)),
op.representation(sym.var("u"))
)(queue, u=weighted_u, k=case.k)
except QBXTargetAssociationFailedException as e:
fplot.write_vtk_file(
"failed-targets.vts",
[
("failed_targets", e.failed_target_flags.get(queue))
])
raise
from sumpy.kernel import LaplaceKernel
ones_density = density_discr.zeros(queue)
ones_density.fill(1)
indicator = bind(
(qbx_tgt_tol, PointsTarget(fplot.points)),
-sym.D(LaplaceKernel(density_discr.ambient_dim),
sym.var("sigma"),
qbx_forced_limit=None))(
queue, sigma=ones_density).get()
solved_pot = solved_pot.get()
true_pot = bind((point_source, PointsTarget(fplot.points)), pot_src)(
queue, charges=source_charges_dev, **concrete_knl_kwargs).get()
#fplot.show_scalar_in_mayavi(solved_pot.real, max_val=5)
if case.prob_side == "scat":
fplot.write_vtk_file(
"potential-%s.vts" % resolution,
[
("pot_scattered", solved_pot),
("pot_incoming", -true_pot),
("indicator", indicator),
]
)
else:
fplot.write_vtk_file(
"potential-%s.vts" % resolution,
[
("solved_pot", solved_pot),
("true_pot", true_pot),
("indicator", indicator),
]
)
# }}}
class Result(Record):
pass
return Result(
h_max=qbx.h_max,
rel_err_2=rel_err_2,
rel_err_inf=rel_err_inf,
rel_td_err_inf=rel_td_err_inf,
gmres_result=gmres_result)
concrete_knl_kwargs = {"k": case.k}
else:
knl = LaplaceKernel(mesh.ambient_dim)
knl_kwargs = {}
concrete_knl_kwargs = {}
if knl.is_complex_valued:
dtype = np.complex128
else:
dtype = np.float64
loc_sign = +1 if case.prob_side in [+1, "scat"] else -1
if case.bc_type == "dirichlet":
op = DirichletOperator(knl,
loc_sign,
use_l2_weighting=True,
kernel_arguments=knl_kwargs)
elif case.bc_type == "neumann":
op = NeumannOperator(knl,
loc_sign,
use_l2_weighting=True,
use_improved_operator=False,
kernel_arguments=knl_kwargs)
else:
assert False
op_u = op.operator(sym.var("u"))
# }}}
# {{{ set up test data
def main():
import logging
logging.basicConfig(level=logging.INFO)
ctx = cl.create_some_context()
queue = cl.CommandQueue(ctx)
mesh = generate_gmsh(
FileSource("circle.step"), 2, order=mesh_order,
force_ambient_dim=2,
other_options=["-string", "Mesh.CharacteristicLengthMax = %g;" % h]
)
logger.info("%d elements" % mesh.nelements)
# {{{ discretizations and connections
vol_discr = Discretization(ctx, mesh,
InterpolatoryQuadratureSimplexGroupFactory(vol_quad_order))
ovsmp_vol_discr = Discretization(ctx, mesh,
InterpolatoryQuadratureSimplexGroupFactory(vol_ovsmp_quad_order))
from meshmode.discretization.connection import (
make_boundary_restriction, make_same_mesh_connection)
bdry_mesh, bdry_discr, bdry_connection = make_boundary_restriction(
queue, vol_discr,
InterpolatoryQuadratureSimplexGroupFactory(bdry_quad_order))
vol_to_ovsmp_vol = make_same_mesh_connection(
queue, ovsmp_vol_discr, vol_discr)
# }}}
# {{{ visualizers
vol_vis = make_visualizer(queue, vol_discr, 20)
bdry_vis = make_visualizer(queue, bdry_discr, 20)
# }}}
vol_x = vol_discr.nodes().with_queue(queue)
ovsmp_vol_x = ovsmp_vol_discr.nodes().with_queue(queue)
rhs = rhs_func(vol_x[0], vol_x[1])
poisson_true_sol = sol_func(vol_x[0], vol_x[1])
vol_vis.write_vtk_file("volume.vtu", [("f", rhs)])
bdry_normals = bind(bdry_discr, p.normal())(queue).as_vector(dtype=object)
bdry_vis.write_vtk_file("boundary.vtu", [
("normals", bdry_normals)
])
bdry_nodes = bdry_discr.nodes().with_queue(queue)
bdry_f = rhs_func(bdry_nodes[0], bdry_nodes[1])
bdry_f_2 = bdry_connection(queue, rhs)
bdry_vis.write_vtk_file("y.vtu", [("f", bdry_f_2)])
if 0:
vol_vis.show_scalar_in_mayavi(rhs, do_show=False)
bdry_vis.show_scalar_in_mayavi(bdry_f - bdry_f_2, line_width=10,
do_show=False)
import mayavi.mlab as mlab
mlab.colorbar()
mlab.show()
# {{{ compute volume potential
from sumpy.qbx import LayerPotential
from sumpy.expansion.local import LineTaylorLocalExpansion
def get_kernel():
from sumpy.symbolic import pymbolic_real_norm_2
from pymbolic.primitives import (make_sym_vector, Variable as var)
r = pymbolic_real_norm_2(make_sym_vector("d", 3))
expr = var("log")(r)
scaling = 1/(2*var("pi"))
from sumpy.kernel import ExpressionKernel
return ExpressionKernel(
dim=3,
expression=expr,
scaling=scaling,
is_complex_valued=False)
laplace_2d_in_3d_kernel = get_kernel()
layer_pot = LayerPotential(ctx, [
LineTaylorLocalExpansion(laplace_2d_in_3d_kernel,
order=vol_qbx_order)])
targets = cl.array.zeros(queue, (3,) + vol_x.shape[1:], vol_x.dtype)
targets[:2] = vol_x
center_dist = np.min(
cl.clmath.sqrt(
bind(vol_discr, p.area_element())(queue)).get())
centers = make_obj_array([ci.copy().reshape(vol_discr.nnodes) for ci in targets])
centers[2][:] = center_dist
sources = cl.array.zeros(queue, (3,) + ovsmp_vol_x.shape[1:], ovsmp_vol_x.dtype)
sources[:2] = ovsmp_vol_x
ovsmp_rhs = vol_to_ovsmp_vol(queue, rhs)
ovsmp_vol_weights = bind(ovsmp_vol_discr, p.area_element() * p.QWeight())(queue)
evt, (vol_pot,) = layer_pot(
queue,
targets=targets.reshape(3, vol_discr.nnodes),
centers=centers,
sources=sources.reshape(3, ovsmp_vol_discr.nnodes),
strengths=(
(ovsmp_vol_weights*ovsmp_rhs).reshape(ovsmp_vol_discr.nnodes),)
)
vol_pot_bdry = bdry_connection(queue, vol_pot)
# }}}
# {{{ solve bvp
from sumpy.kernel import LaplaceKernel
from pytential.symbolic.pde.scalar import DirichletOperator
op = DirichletOperator(LaplaceKernel(2), -1, use_l2_weighting=True)
sym_sigma = sym.var("sigma")
op_sigma = op.operator(sym_sigma)
from pytential.qbx import QBXLayerPotentialSource
qbx = QBXLayerPotentialSource(
bdry_discr, fine_order=bdry_ovsmp_quad_order, qbx_order=qbx_order,
fmm_order=fmm_order
)
bound_op = bind(qbx, op_sigma)
poisson_bc = poisson_bc_func(bdry_nodes[0], bdry_nodes[1])
bvp_bc = poisson_bc - vol_pot_bdry
bdry_f = rhs_func(bdry_nodes[0], bdry_nodes[1])
bvp_rhs = bind(bdry_discr, op.prepare_rhs(sym.var("bc")))(queue, bc=bvp_bc)
from pytential.solve import gmres
gmres_result = gmres(
bound_op.scipy_op(queue, "sigma"),
bvp_rhs, tol=1e-14, progress=True,
hard_failure=False)
sigma = gmres_result.solution
print("gmres state:", gmres_result.state)
# }}}
bvp_sol = bind(
(qbx, vol_discr),
op.representation(sym_sigma))(queue, sigma=sigma)
poisson_sol = bvp_sol + vol_pot
poisson_err = poisson_sol-poisson_true_sol
rel_err = (
norm(vol_discr, queue, poisson_err)
/
norm(vol_discr, queue, poisson_true_sol))
bdry_vis.write_vtk_file("poisson-boundary.vtu", [
("vol_pot_bdry", vol_pot_bdry),
("sigma", sigma),
])
vol_vis.write_vtk_file("poisson-volume.vtu", [
("bvp_sol", bvp_sol),
("poisson_sol", poisson_sol),
("poisson_true_sol", poisson_true_sol),
("poisson_err", poisson_err),
("vol_pot", vol_pot),
("rhs", rhs),
])
print("h = %s" % h)
print("mesh_order = %s" % mesh_order)
print("vol_quad_order = %s" % vol_quad_order)
print("vol_ovsmp_quad_order = %s" % vol_ovsmp_quad_order)
print("bdry_quad_order = %s" % bdry_quad_order)
print("bdry_ovsmp_quad_order = %s" % bdry_ovsmp_quad_order)
print("qbx_order = %s" % qbx_order)
print("vol_qbx_order = %s" % vol_qbx_order)
print("fmm_order = %s" % fmm_order)
print()
print("rel err: %g" % rel_err)
def run_test(cl_ctx, queue):
q_order = 5
qbx_order = q_order
fmm_backend = "sumpy"
mesh = get_ellipse_mesh(20, 40, mesh_order=5)
a = 1
b = 1 / 40
if 0:
from meshmode.mesh.visualization import draw_curve
import matplotlib.pyplot as plt
draw_curve(mesh)
plt.axes().set_aspect('equal')
plt.show()
from pytential.qbx import QBXLayerPotentialSource
from meshmode.discretization import Discretization
from meshmode.discretization.poly_element import \
InterpolatoryQuadratureSimplexGroupFactory
pre_density_discr = Discretization(
cl_ctx, mesh, InterpolatoryQuadratureSimplexGroupFactory(q_order))
refiner_extra_kwargs = {
# "_expansion_disturbance_tolerance": 0.05,
"_scaled_max_curvature_threshold": 1,
"maxiter": 10,
}
qbx, _ = QBXLayerPotentialSource(
pre_density_discr,
fine_order=4 * q_order,
qbx_order=qbx_order,
fmm_backend=fmm_backend,
fmm_order=qbx_order + 5,
).with_refinement(**refiner_extra_kwargs)
if 1:
print("%d stage-1 elements after refinement" %
qbx.density_discr.mesh.nelements)
print("%d stage-2 elements after refinement" %
qbx.stage2_density_discr.mesh.nelements)
print("quad stage-2 elements have %d nodes" %
qbx.quad_stage2_density_discr.groups[0].nunit_nodes)
def reference_solu(rvec):
# a harmonic function
x, y = rvec
return 2.1 * x * y + (x**2 - y**2) * 0.5 + x
bvals = reference_solu(qbx.density_discr.nodes().with_queue(queue))
from pytential.symbolic.pde.scalar import DirichletOperator
from sumpy.kernel import LaplaceKernel
from pytential import sym, bind
op = DirichletOperator(LaplaceKernel(2), -1)
bound_op = bind(qbx.copy(target_association_tolerance=0.5),
op.operator(sym.var('sigma')))
rhs = bind(qbx.density_discr, op.prepare_rhs(sym.var("bc")))(queue,
bc=bvals)
from pytential.solve import gmres
gmres_result = gmres(bound_op.scipy_op(queue, "sigma", dtype=np.float64),
rhs,
tol=1e-12,
progress=True,
hard_failure=True,
stall_iterations=50,
no_progress_factor=1.05)
from sumpy.visualization import FieldPlotter
from pytential.target import PointsTarget
pltsize = b * 1.5
fplot = FieldPlotter(np.array([-1 + pltsize * 0.5, 0]),
extent=pltsize * 1.05,
npoints=500)
plt_targets = cl.array.to_device(queue, fplot.points)
interior_pts = (fplot.points[0]**2 / a**2 +
fplot.points[1]**2 / b**2) < 0.99
exact_vals = reference_solu(fplot.points)
out_errs = []
for assotol in [0.05]:
qbx_stick_out = qbx.copy(target_association_tolerance=0.05)
vol_solution = bind((qbx_stick_out, PointsTarget(plt_targets)),
op.representation(sym.var('sigma')))(
queue, sigma=gmres_result.solution).get()
interior_error_linf = (
np.linalg.norm(np.abs(vol_solution - exact_vals)[interior_pts],
ord=np.inf) /
np.linalg.norm(exact_vals[interior_pts], ord=np.inf))
interior_error_l2 = (np.linalg.norm(
np.abs(vol_solution - exact_vals)[interior_pts], ord=2) /
np.linalg.norm(exact_vals[interior_pts], ord=2))
print("\nassotol = %f" % assotol)
print("L_inf Error = %e " % interior_error_linf)
print("L_2 Error = %e " % interior_error_l2)
out_errs.append(
("error-%f" % assotol, np.abs(vol_solution - exact_vals)))
if 1:
fplot.write_vtk_file("results.vts", out_errs)
def main():
import logging
logging.basicConfig(level=logging.INFO)
ctx = cl.create_some_context()
queue = cl.CommandQueue(ctx)
if 1:
ext = 0.5
mesh = generate_regular_rect_mesh(a=(-ext / 2, -ext / 2),
b=(ext / 2, ext / 2),
n=(int(ext / h), int(ext / h)))
else:
mesh = generate_gmsh(FileSource("circle.step"),
2,
order=mesh_order,
force_ambient_dim=2,
other_options=[
"-string",
"Mesh.CharacteristicLengthMax = %g;" % h
])
logger.info("%d elements" % mesh.nelements)
# {{{ discretizations and connections
vol_discr = Discretization(
ctx, mesh, InterpolatoryQuadratureSimplexGroupFactory(vol_quad_order))
ovsmp_vol_discr = Discretization(
ctx, mesh,
InterpolatoryQuadratureSimplexGroupFactory(vol_ovsmp_quad_order))
from meshmode.mesh import BTAG_ALL
from meshmode.discretization.connection import (make_face_restriction,
make_same_mesh_connection)
bdry_connection = make_face_restriction(
vol_discr, InterpolatoryQuadratureSimplexGroupFactory(bdry_quad_order),
BTAG_ALL)
bdry_discr = bdry_connection.to_discr
vol_to_ovsmp_vol = make_same_mesh_connection(ovsmp_vol_discr, vol_discr)
# }}}
# {{{ visualizers
vol_vis = make_visualizer(queue, vol_discr, 20)
bdry_vis = make_visualizer(queue, bdry_discr, 20)
# }}}
vol_x = vol_discr.nodes().with_queue(queue)
ovsmp_vol_x = ovsmp_vol_discr.nodes().with_queue(queue)
rhs = rhs_func(vol_x[0], vol_x[1])
poisson_true_sol = sol_func(vol_x[0], vol_x[1])
vol_vis.write_vtk_file("volume.vtu", [("f", rhs)])
bdry_normals = bind(bdry_discr, p.normal(
mesh.ambient_dim))(queue).as_vector(dtype=object)
bdry_vis.write_vtk_file("boundary.vtu", [("normals", bdry_normals)])
bdry_nodes = bdry_discr.nodes().with_queue(queue)
bdry_f = rhs_func(bdry_nodes[0], bdry_nodes[1])
bdry_f_2 = bdry_connection(queue, rhs)
bdry_vis.write_vtk_file("y.vtu", [("f", bdry_f_2)])
if 0:
vol_vis.show_scalar_in_mayavi(rhs, do_show=False)
bdry_vis.show_scalar_in_mayavi(bdry_f - bdry_f_2,
line_width=10,
do_show=False)
import mayavi.mlab as mlab
mlab.colorbar()
mlab.show()
# {{{ compute volume potential
from sumpy.qbx import LayerPotential
from sumpy.expansion.local import LineTaylorLocalExpansion
def get_kernel():
from sumpy.symbolic import pymbolic_real_norm_2
from pymbolic.primitives import make_sym_vector
from pymbolic import var
d = make_sym_vector("d", 3)
r = pymbolic_real_norm_2(d[:-1])
# r3d = pymbolic_real_norm_2(d)
#expr = var("log")(r3d)
log = var("log")
sqrt = var("sqrt")
a = d[-1]
expr = log(r)
expr = log(sqrt(r**2 + a**2))
expr = log(sqrt(r + a**2))
#expr = log(sqrt(r**2 + a**2))-a**2/2/(r**2+a**2)
#expr = 2*log(sqrt(r**2 + a**2))
scaling = 1 / (2 * var("pi"))
from sumpy.kernel import ExpressionKernel
return ExpressionKernel(dim=3,
expression=expr,
global_scaling_const=scaling,
is_complex_valued=False)
laplace_2d_in_3d_kernel = get_kernel()
layer_pot = LayerPotential(
ctx, [LineTaylorLocalExpansion(laplace_2d_in_3d_kernel, order=0)])
targets = cl.array.zeros(queue, (3, ) + vol_x.shape[1:], vol_x.dtype)
targets[:2] = vol_x
center_dist = 0.125 * np.min(
cl.clmath.sqrt(
bind(vol_discr, p.area_element(mesh.ambient_dim,
mesh.dim))(queue)).get())
centers = make_obj_array(
[ci.copy().reshape(vol_discr.nnodes) for ci in targets])
centers[2][:] = center_dist
print(center_dist)
sources = cl.array.zeros(queue, (3, ) + ovsmp_vol_x.shape[1:],
ovsmp_vol_x.dtype)
sources[:2] = ovsmp_vol_x
ovsmp_rhs = vol_to_ovsmp_vol(queue, rhs)
ovsmp_vol_weights = bind(
ovsmp_vol_discr,
p.area_element(mesh.ambient_dim, mesh.dim) * p.QWeight())(queue)
print("volume: %d source nodes, %d target nodes" %
(ovsmp_vol_discr.nnodes, vol_discr.nnodes))
evt, (vol_pot, ) = layer_pot(
queue,
targets=targets.reshape(3, vol_discr.nnodes),
centers=centers,
sources=sources.reshape(3, ovsmp_vol_discr.nnodes),
strengths=((ovsmp_vol_weights * ovsmp_rhs).reshape(
ovsmp_vol_discr.nnodes), ),
expansion_radii=np.zeros(vol_discr.nnodes),
)
vol_pot_bdry = bdry_connection(queue, vol_pot)
# }}}
# {{{ solve bvp
from sumpy.kernel import LaplaceKernel
from pytential.symbolic.pde.scalar import DirichletOperator
op = DirichletOperator(LaplaceKernel(2), -1, use_l2_weighting=True)
sym_sigma = sym.var("sigma")
op_sigma = op.operator(sym_sigma)
from pytential.qbx import QBXLayerPotentialSource
qbx = QBXLayerPotentialSource(
bdry_discr,
fine_order=bdry_ovsmp_quad_order,
qbx_order=qbx_order,
fmm_order=fmm_order,
)
bound_op = bind(qbx, op_sigma)
poisson_bc = poisson_bc_func(bdry_nodes[0], bdry_nodes[1])
bvp_bc = poisson_bc - vol_pot_bdry
bdry_f = rhs_func(bdry_nodes[0], bdry_nodes[1])
bvp_rhs = bind(bdry_discr, op.prepare_rhs(sym.var("bc")))(queue, bc=bvp_bc)
from pytential.solve import gmres
gmres_result = gmres(bound_op.scipy_op(queue, "sigma", dtype=np.float64),
bvp_rhs,
tol=1e-14,
progress=True,
hard_failure=False)
sigma = gmres_result.solution
print("gmres state:", gmres_result.state)
# }}}
bvp_sol = bind((qbx, vol_discr), op.representation(sym_sigma))(queue,
sigma=sigma)
poisson_sol = bvp_sol + vol_pot
poisson_err = poisson_sol - poisson_true_sol
rel_err = (norm(vol_discr, queue, poisson_err) /
norm(vol_discr, queue, poisson_true_sol))
bdry_vis.write_vtk_file("poisson-boundary.vtu", [
("vol_pot_bdry", vol_pot_bdry),
("sigma", sigma),
])
vol_vis.write_vtk_file("poisson-volume.vtu", [
("bvp_sol", bvp_sol),
("poisson_sol", poisson_sol),
("poisson_true_sol", poisson_true_sol),
("poisson_err", poisson_err),
("vol_pot", vol_pot),
("rhs", rhs),
])
print("h = %s" % h)
print("mesh_order = %s" % mesh_order)
print("vol_quad_order = %s" % vol_quad_order)
print("vol_ovsmp_quad_order = %s" % vol_ovsmp_quad_order)
print("bdry_quad_order = %s" % bdry_quad_order)
print("bdry_ovsmp_quad_order = %s" % bdry_ovsmp_quad_order)
print("qbx_order = %s" % qbx_order)
#print("vol_qbx_order = %s" % vol_qbx_order)
print("fmm_order = %s" % fmm_order)
print()
print("rel err: %g" % rel_err)
def run_int_eq_test(
cl_ctx, queue, curve_f, nelements, qbx_order, bc_type, loc_sign, k,
target_order, source_order):
mesh = make_curve_mesh(curve_f,
np.linspace(0, 1, nelements+1),
target_order)
if 0:
from pytential.visualization import show_mesh
show_mesh(mesh)
pt.gca().set_aspect("equal")
pt.show()
from pytential.qbx import QBXLayerPotentialSource
from meshmode.discretization import Discretization
from meshmode.discretization.poly_element import \
InterpolatoryQuadratureSimplexGroupFactory
density_discr = Discretization(
cl_ctx, mesh, InterpolatoryQuadratureSimplexGroupFactory(target_order))
if source_order is None:
source_order = 4*target_order
qbx = QBXLayerPotentialSource(
density_discr, fine_order=source_order, qbx_order=qbx_order,
# Don't use FMM for now
fmm_order=False)
# {{{ set up operator
from pytential.symbolic.pde.scalar import (
DirichletOperator,
NeumannOperator)
from sumpy.kernel import LaplaceKernel, HelmholtzKernel, AxisTargetDerivative
if k:
knl = HelmholtzKernel(2)
knl_kwargs = {"k": k}
else:
knl = LaplaceKernel(2)
knl_kwargs = {}
if knl.is_complex_valued:
dtype = np.complex128
else:
dtype = np.float64
if bc_type == "dirichlet":
op = DirichletOperator((knl, knl_kwargs), loc_sign, use_l2_weighting=True)
elif bc_type == "neumann":
op = NeumannOperator((knl, knl_kwargs), loc_sign, use_l2_weighting=True,
use_improved_operator=False)
else:
assert False
op_u = op.operator(sym.var("u"))
# }}}
# {{{ set up test data
inner_radius = 0.1
outer_radius = 2
if loc_sign < 0:
test_src_geo_radius = outer_radius
test_tgt_geo_radius = inner_radius
else:
test_src_geo_radius = inner_radius
test_tgt_geo_radius = outer_radius
point_sources = make_circular_point_group(10, test_src_geo_radius,
func=lambda x: x**1.5)
test_targets = make_circular_point_group(20, test_tgt_geo_radius)
np.random.seed(22)
source_charges = np.random.randn(point_sources.shape[1])
source_charges[-1] = -np.sum(source_charges[:-1])
source_charges = source_charges.astype(dtype)
assert np.sum(source_charges) < 1e-15
# }}}
if 0:
# show geometry, centers, normals
nodes_h = density_discr.nodes().get(queue=queue)
pt.plot(nodes_h[0], nodes_h[1], "x-")
normal = bind(density_discr, sym.normal())(queue).as_vector(np.object)
pt.quiver(nodes_h[0], nodes_h[1], normal[0].get(queue), normal[1].get(queue))
pt.gca().set_aspect("equal")
pt.show()
# {{{ establish BCs
from sumpy.p2p import P2P
pot_p2p = P2P(cl_ctx,
[knl], exclude_self=False, value_dtypes=dtype)
evt, (test_direct,) = pot_p2p(
queue, test_targets, point_sources, [source_charges],
out_host=False, **knl_kwargs)
nodes = density_discr.nodes()
evt, (src_pot,) = pot_p2p(
queue, nodes, point_sources, [source_charges],
**knl_kwargs)
grad_p2p = P2P(cl_ctx,
[AxisTargetDerivative(0, knl), AxisTargetDerivative(1, knl)],
exclude_self=False, value_dtypes=dtype)
evt, (src_grad0, src_grad1) = grad_p2p(
queue, nodes, point_sources, [source_charges],
**knl_kwargs)
if bc_type == "dirichlet":
bc = src_pot
elif bc_type == "neumann":
normal = bind(density_discr, sym.normal())(queue).as_vector(np.object)
bc = (src_grad0*normal[0] + src_grad1*normal[1])
# }}}
# {{{ solve
bound_op = bind(qbx, op_u)
rhs = bind(density_discr, op.prepare_rhs(sym.var("bc")))(queue, bc=bc)
from pytential.solve import gmres
gmres_result = gmres(
bound_op.scipy_op(queue, "u", k=k),
rhs, tol=1e-14, progress=True,
hard_failure=False)
u = gmres_result.solution
print("gmres state:", gmres_result.state)
if 0:
# {{{ build matrix for spectrum check
from sumpy.tools import build_matrix
mat = build_matrix(bound_op.scipy_op("u"))
w, v = la.eig(mat)
if 0:
pt.imshow(np.log10(1e-20+np.abs(mat)))
pt.colorbar()
pt.show()
#assert abs(s[-1]) < 1e-13, "h
#assert abs(s[-2]) > 1e-7
#from pudb import set_trace; set_trace()
# }}}
# }}}
# {{{ error check
from pytential.target import PointsTarget
bound_tgt_op = bind((qbx, PointsTarget(test_targets)),
op.representation(sym.var("u")))
test_via_bdry = bound_tgt_op(queue, u=u, k=k)
err = test_direct-test_via_bdry
err = err.get()
test_direct = test_direct.get()
test_via_bdry = test_via_bdry.get()
# {{{ remove effect of net source charge
if k == 0 and bc_type == "neumann" and loc_sign == -1:
# remove constant offset in interior Laplace Neumann error
tgt_ones = np.ones_like(test_direct)
tgt_ones = tgt_ones/la.norm(tgt_ones)
err = err - np.vdot(tgt_ones, err)*tgt_ones
# }}}
rel_err_2 = la.norm(err)/la.norm(test_direct)
rel_err_inf = la.norm(err, np.inf)/la.norm(test_direct, np.inf)
# }}}
print("rel_err_2: %g rel_err_inf: %g" % (rel_err_2, rel_err_inf))
# {{{ test tangential derivative
bound_t_deriv_op = bind(qbx,
op.representation(
sym.var("u"), map_potentials=sym.tangential_derivative,
qbx_forced_limit=loc_sign))
#print(bound_t_deriv_op.code)
tang_deriv_from_src = bound_t_deriv_op(queue, u=u).as_scalar().get()
tangent = bind(
density_discr,
sym.pseudoscalar()/sym.area_element())(queue).as_vector(np.object)
tang_deriv_ref = (src_grad0 * tangent[0] + src_grad1 * tangent[1]).get()
if 0:
pt.plot(tang_deriv_ref.real)
pt.plot(tang_deriv_from_src.real)
pt.show()
td_err = tang_deriv_from_src - tang_deriv_ref
rel_td_err_inf = la.norm(td_err, np.inf)/la.norm(tang_deriv_ref, np.inf)
print("rel_td_err_inf: %g" % rel_td_err_inf)
# }}}
# {{{ plotting
if 0:
fplot = FieldPlotter(np.zeros(2),
extent=1.25*2*max(test_src_geo_radius, test_tgt_geo_radius),
npoints=200)
#pt.plot(u)
#pt.show()
evt, (fld_from_src,) = pot_p2p(
queue, fplot.points, point_sources, [source_charges],
**knl_kwargs)
fld_from_bdry = bind(
(qbx, PointsTarget(fplot.points)),
op.representation(sym.var("u"))
)(queue, u=u, k=k)
fld_from_src = fld_from_src.get()
fld_from_bdry = fld_from_bdry.get()
nodes = density_discr.nodes().get(queue=queue)
def prep():
pt.plot(point_sources[0], point_sources[1], "o",
label="Monopole 'Point Charges'")
pt.plot(test_targets[0], test_targets[1], "v",
label="Observation Points")
pt.plot(nodes[0], nodes[1], "k-",
label=r"$\Gamma$")
from matplotlib.cm import get_cmap
cmap = get_cmap()
cmap._init()
if 0:
cmap._lut[(cmap.N*99)//100:, -1] = 0 # make last percent transparent?
prep()
if 1:
pt.subplot(131)
pt.title("Field error (loc_sign=%s)" % loc_sign)
log_err = np.log10(1e-20+np.abs(fld_from_src-fld_from_bdry))
log_err = np.minimum(-3, log_err)
fplot.show_scalar_in_matplotlib(log_err, cmap=cmap)
#from matplotlib.colors import Normalize
#im.set_norm(Normalize(vmin=-6, vmax=1))
cb = pt.colorbar(shrink=0.9)
cb.set_label(r"$\log_{10}(\mathdefault{Error})$")
if 1:
pt.subplot(132)
prep()
pt.title("Source Field")
fplot.show_scalar_in_matplotlib(
fld_from_src.real, max_val=3)
pt.colorbar(shrink=0.9)
if 1:
pt.subplot(133)
prep()
pt.title("Solved Field")
fplot.show_scalar_in_matplotlib(
fld_from_bdry.real, max_val=3)
pt.colorbar(shrink=0.9)
# total field
#fplot.show_scalar_in_matplotlib(
#fld_from_src.real+fld_from_bdry.real, max_val=0.1)
#pt.colorbar()
pt.legend(loc="best", prop=dict(size=15))
from matplotlib.ticker import NullFormatter
pt.gca().xaxis.set_major_formatter(NullFormatter())
pt.gca().yaxis.set_major_formatter(NullFormatter())
pt.gca().set_aspect("equal")
if 0:
border_factor_top = 0.9
border_factor = 0.3
xl, xh = pt.xlim()
xhsize = 0.5*(xh-xl)
pt.xlim(xl-border_factor*xhsize, xh+border_factor*xhsize)
yl, yh = pt.ylim()
yhsize = 0.5*(yh-yl)
pt.ylim(yl-border_factor_top*yhsize, yh+border_factor*yhsize)
#pt.savefig("helmholtz.pdf", dpi=600)
pt.show()
# }}}
class Result(Record):
pass
return Result(
rel_err_2=rel_err_2,
rel_err_inf=rel_err_inf,
rel_td_err_inf=rel_td_err_inf,
gmres_result=gmres_result)