def get_observation_mesh(self, target_order):
from meshmode.mesh.generation import generate_icosphere
if self.is_interior:
return generate_icosphere(5, target_order)
else:
return generate_icosphere(0.5, target_order)
def get_observation_mesh(self, target_order):
from meshmode.mesh.generation import generate_icosphere
if self.is_interior:
return generate_icosphere(5, target_order)
else:
return generate_icosphere(0.5, target_order)
def test_mesh_without_vertices(ctx_factory):
ctx = ctx_factory()
queue = cl.CommandQueue(ctx)
# create a mesh
from meshmode.mesh.generation import generate_icosphere
mesh = generate_icosphere(r=1.0, order=4)
# create one without the vertices
from meshmode.mesh import Mesh
grp, = mesh.groups
groups = [
grp.copy(nodes=grp.nodes, vertex_indices=None) for grp in mesh.groups
]
mesh = Mesh(None, groups, is_conforming=False)
# try refining it
from meshmode.mesh.refinement import refine_uniformly
mesh = refine_uniformly(mesh, 1)
# make sure the world doesn't end
from meshmode.discretization import Discretization
from meshmode.discretization.poly_element import \
InterpolatoryQuadratureSimplexGroupFactory as GroupFactory
discr = Discretization(ctx, mesh, GroupFactory(4))
discr.nodes().with_queue(queue)
from meshmode.discretization.visualization import make_visualizer
make_visualizer(queue, discr, 4)
def test_mesh_without_vertices(actx_factory):
actx = actx_factory()
# create a mesh
mesh = mgen.generate_icosphere(r=1.0, order=4)
# create one without the vertices
grp, = mesh.groups
groups = [
grp.copy(nodes=grp.nodes, vertex_indices=None) for grp in mesh.groups
]
mesh = Mesh(None, groups, is_conforming=False)
# try refining it
from meshmode.mesh.refinement import refine_uniformly
mesh = refine_uniformly(mesh, 1)
# make sure the world doesn't end
from meshmode.discretization import Discretization
discr = Discretization(actx, mesh,
InterpolatoryQuadratureSimplexGroupFactory(4))
thaw(discr.nodes(), actx)
from meshmode.discretization.visualization import make_visualizer
make_visualizer(actx, discr, 4)
def get_mesh(self, resolution, target_order):
from meshmode.mesh.generation import generate_icosphere
from meshmode.mesh.refinement import refine_uniformly
mesh = refine_uniformly(generate_icosphere(1, target_order),
resolution)
return mesh
def get_mesh(self, resolution, target_order):
from meshmode.mesh.generation import generate_icosphere
from meshmode.mesh.refinement import refine_uniformly
mesh = refine_uniformly(
generate_icosphere(1, target_order),
resolution)
return mesh
def test_conformity_of_uniform_mesh(refinement_rounds):
mesh = mgen.generate_icosphere(r=1.0,
order=4,
uniform_refinement_rounds=refinement_rounds)
assert mesh.is_conforming
from meshmode.mesh import is_boundary_tag_empty, BTAG_ALL
assert is_boundary_tag_empty(mesh, BTAG_ALL)
def get_unit_sphere_with_ref_mean_curvature(cl_ctx):
order = 8
from meshmode.mesh.generation import generate_icosphere
mesh = generate_icosphere(1, order)
discr = Discretization(cl_ctx, mesh,
InterpolatoryQuadratureSimplexGroupFactory(order))
return discr, 1
def get_sphere_mesh(refinement_increment, target_order):
from meshmode.mesh.generation import generate_icosphere
mesh = generate_icosphere(1, target_order)
from meshmode.mesh.refinement import Refiner
refiner = Refiner(mesh)
for i in range(refinement_increment):
flags = np.ones(mesh.nelements, dtype=bool)
refiner.refine(flags)
mesh = refiner.get_current_mesh()
return mesh
def test_mesh_with_interior_unit_nodes(actx_factory, ambient_dim):
actx = actx_factory()
# NOTE: smaller orders or coarser meshes make the cases fail the
# node_vertex_consistency test; the default warp_and_blend_nodes have
# nodes at the vertices, so they pass for much smaller tolerances
order = 8
nelements = 32
n_minor = 2 * nelements
uniform_refinement_rounds = 4
import modepy as mp
if ambient_dim == 2:
unit_nodes = mp.LegendreGaussQuadrature(order,
force_dim_axis=True).nodes
mesh = mgen.make_curve_mesh(partial(mgen.ellipse, 2.0),
np.linspace(0.0, 1.0, nelements + 1),
order=order,
unit_nodes=unit_nodes)
elif ambient_dim == 3:
unit_nodes = mp.VioreanuRokhlinSimplexQuadrature(order, 2).nodes
mesh = mgen.generate_torus(4.0,
2.0,
n_major=2 * n_minor,
n_minor=n_minor,
order=order,
unit_nodes=unit_nodes)
mesh = mgen.generate_icosphere(
1.0,
uniform_refinement_rounds=uniform_refinement_rounds,
order=order,
unit_nodes=unit_nodes)
else:
raise ValueError(f"unsupported dimension: '{ambient_dim}'")
assert mesh.facial_adjacency_groups
assert mesh.nodal_adjacency
from meshmode.discretization import Discretization
from meshmode.discretization.poly_element import QuadratureSimplexGroupFactory
discr = Discretization(actx, mesh, QuadratureSimplexGroupFactory(order))
from meshmode.discretization.connection import make_face_restriction
conn = make_face_restriction(
actx,
discr,
group_factory=QuadratureSimplexGroupFactory(order),
boundary_tag=FACE_RESTR_ALL)
assert conn
def get_sphere_mesh(refinement_increment, target_order):
from meshmode.mesh.generation import generate_icosphere
mesh = generate_icosphere(1, target_order)
from meshmode.mesh.refinement import Refiner
refiner = Refiner(mesh)
for i in range(refinement_increment):
flags = np.ones(mesh.nelements, dtype=bool)
refiner.refine(flags)
mesh = refiner.get_current_mesh()
return mesh
def test_copy_visualizer(actx_factory, ambient_dim, visualize=True):
actx = actx_factory()
target_order = 4
if ambient_dim == 2:
nelements = 128
mesh = mgen.make_curve_mesh(partial(mgen.ellipse, 1.0),
np.linspace(0.0, 1.0, nelements + 1),
target_order)
elif ambient_dim == 3:
mesh = mgen.generate_icosphere(1.0,
target_order,
uniform_refinement_rounds=2)
else:
raise ValueError(f"unsupported dimension: {ambient_dim}")
from meshmode.mesh.processing import affine_map
translated_mesh = affine_map(mesh,
b=np.array([2.5, 0.0, 0.0][:ambient_dim]))
from meshmode.discretization import Discretization
discr = Discretization(actx, mesh,
PolynomialWarpAndBlendGroupFactory(target_order))
translated_discr = Discretization(
actx, translated_mesh,
PolynomialWarpAndBlendGroupFactory(target_order))
from meshmode.discretization.visualization import make_visualizer
vis = make_visualizer(actx, discr, target_order, force_equidistant=True)
assert vis._vtk_connectivity
assert vis._vtk_lagrange_connectivity
translated_vis = vis.copy_with_same_connectivity(actx, translated_discr)
assert translated_vis._cached_vtk_connectivity is not None
assert translated_vis._cached_vtk_lagrange_connectivity is not None
assert translated_vis._vtk_connectivity \
is vis._vtk_connectivity
assert translated_vis._vtk_lagrange_connectivity \
is vis._vtk_lagrange_connectivity
if not visualize:
return
vis.write_vtk_file(f"visualizer_copy_{ambient_dim}d_orig.vtu", [],
overwrite=True)
translated_vis.write_vtk_file(
f"visualizer_copy_{ambient_dim}d_translated.vtu", [], overwrite=True)
def test_is_affine_group_check(mesh_name):
order = 4
nelements = 16
if mesh_name.startswith("box"):
dim = int(mesh_name[-2])
is_affine = True
mesh = mgen.generate_regular_rect_mesh(
a=(-0.5, ) * dim,
b=(0.5, ) * dim,
nelements_per_axis=(nelements, ) * dim,
order=order)
elif mesh_name.startswith("warped_box"):
dim = int(mesh_name[-2])
is_affine = False
mesh = mgen.generate_warped_rect_mesh(dim,
order,
nelements_side=nelements)
elif mesh_name == "cross_warped_box":
dim = 2
is_affine = False
mesh = _generate_cross_warped_rect_mesh(dim, order, nelements)
elif mesh_name == "circle":
is_affine = False
mesh = mgen.make_curve_mesh(lambda t: mgen.ellipse(1.0, t),
np.linspace(0.0, 1.0, nelements + 1),
order=order)
elif mesh_name == "ellipse":
is_affine = False
mesh = mgen.make_curve_mesh(lambda t: mgen.ellipse(2.0, t),
np.linspace(0.0, 1.0, nelements + 1),
order=order)
elif mesh_name == "sphere":
is_affine = False
mesh = mgen.generate_icosphere(r=1.0, order=order)
elif mesh_name == "torus":
is_affine = False
mesh = mgen.generate_torus(10.0, 2.0, order=order)
else:
raise ValueError(f"unknown mesh name: {mesh_name}")
assert all(grp.is_affine for grp in mesh.groups) == is_affine
def test_refine_surfaces(actx_factory, mesh_name, visualize=False):
if mesh_name == "torus":
mesh = mgen.generate_torus(10, 1, 40, 4, order=4)
elif mesh_name == "icosphere":
mesh = mgen.generate_icosphere(1, order=4)
else:
raise ValueError(f"invalid mesh name '{mesh_name}'")
if visualize:
actx = actx_factory()
from meshmode.mesh.visualization import vtk_visualize_mesh
vtk_visualize_mesh(actx, mesh, "surface.vtu")
# check for absence of node-vertex consistency error
from meshmode.mesh.refinement import refine_uniformly
refined_mesh = refine_uniformly(mesh, 1)
if visualize:
actx = actx_factory()
from meshmode.mesh.visualization import vtk_visualize_mesh
vtk_visualize_mesh(actx, refined_mesh, "surface-refined.vtu")
def get_sphere_lpot_source(queue):
from meshmode.discretization import Discretization
from meshmode.discretization.poly_element import (
InterpolatoryQuadratureSimplexGroupFactory)
target_order = 8
from meshmode.mesh.generation import generate_icosphere
mesh = generate_icosphere(1, target_order)
pre_density_discr = Discretization(
queue.context, mesh,
InterpolatoryQuadratureSimplexGroupFactory(target_order))
from pytential.qbx import QBXLayerPotentialSource
lpot_source = QBXLayerPotentialSource(pre_density_discr,
qbx_order=5,
fmm_order=10,
fine_order=4 * target_order,
fmm_backend="fmmlib")
lpot_source, _ = lpot_source.with_refinement()
return lpot_source
def get_mesh(self, resolution, mesh_order):
from meshmode.mesh.generation import generate_icosphere
return generate_icosphere(self.radius,
order=mesh_order,
uniform_refinement_rounds=resolution)
def test_sphere_eigenvalues(ctx_factory, mode_m, mode_n, qbx_order,
fmm_backend):
logging.basicConfig(level=logging.INFO)
special = pytest.importorskip("scipy.special")
cl_ctx = ctx_factory()
queue = cl.CommandQueue(cl_ctx)
actx = PyOpenCLArrayContext(queue)
target_order = 8
from meshmode.discretization import Discretization
from meshmode.discretization.poly_element import \
InterpolatoryQuadratureSimplexGroupFactory
from pytential.qbx import QBXLayerPotentialSource
from pytools.convergence import EOCRecorder
s_eoc_rec = EOCRecorder()
d_eoc_rec = EOCRecorder()
sp_eoc_rec = EOCRecorder()
dp_eoc_rec = EOCRecorder()
def rel_err(comp, ref):
return (norm(density_discr, comp - ref) / norm(density_discr, ref))
for nrefinements in [0, 1]:
from meshmode.mesh.generation import generate_icosphere
mesh = generate_icosphere(1, target_order)
from meshmode.mesh.refinement import Refiner
refiner = Refiner(mesh)
for i in range(nrefinements):
flags = np.ones(mesh.nelements, dtype=bool)
refiner.refine(flags)
mesh = refiner.get_current_mesh()
pre_density_discr = Discretization(
actx, mesh,
InterpolatoryQuadratureSimplexGroupFactory(target_order))
qbx = QBXLayerPotentialSource(
pre_density_discr,
4 * target_order,
qbx_order,
fmm_order=6,
fmm_backend=fmm_backend,
)
places = GeometryCollection(qbx)
from meshmode.dof_array import flatten, unflatten, thaw
density_discr = places.get_discretization(places.auto_source.geometry)
nodes = thaw(actx, density_discr.nodes())
r = actx.np.sqrt(nodes[0] * nodes[0] + nodes[1] * nodes[1] +
nodes[2] * nodes[2])
phi = actx.np.arccos(nodes[2] / r)
theta = actx.np.arctan2(nodes[0], nodes[1])
ymn = unflatten(
actx, density_discr,
actx.from_numpy(
special.sph_harm(mode_m, mode_n, actx.to_numpy(flatten(theta)),
actx.to_numpy(flatten(phi)))))
from sumpy.kernel import LaplaceKernel
lap_knl = LaplaceKernel(3)
# {{{ single layer
s_sigma_op = bind(
places, sym.S(lap_knl, sym.var("sigma"), qbx_forced_limit=+1))
s_sigma = s_sigma_op(actx, sigma=ymn)
s_eigval = 1 / (2 * mode_n + 1)
h_max = bind(places, sym.h_max(qbx.ambient_dim))(actx)
s_eoc_rec.add_data_point(h_max, rel_err(s_sigma, s_eigval * ymn))
# }}}
# {{{ double layer
d_sigma_op = bind(
places, sym.D(lap_knl, sym.var("sigma"), qbx_forced_limit="avg"))
d_sigma = d_sigma_op(actx, sigma=ymn)
d_eigval = -1 / (2 * (2 * mode_n + 1))
d_eoc_rec.add_data_point(h_max, rel_err(d_sigma, d_eigval * ymn))
# }}}
# {{{ S'
sp_sigma_op = bind(
places, sym.Sp(lap_knl, sym.var("sigma"), qbx_forced_limit="avg"))
sp_sigma = sp_sigma_op(actx, sigma=ymn)
sp_eigval = -1 / (2 * (2 * mode_n + 1))
sp_eoc_rec.add_data_point(h_max, rel_err(sp_sigma, sp_eigval * ymn))
# }}}
# {{{ D'
dp_sigma_op = bind(
places, sym.Dp(lap_knl, sym.var("sigma"), qbx_forced_limit="avg"))
dp_sigma = dp_sigma_op(actx, sigma=ymn)
dp_eigval = -(mode_n * (mode_n + 1)) / (2 * mode_n + 1)
dp_eoc_rec.add_data_point(h_max, rel_err(dp_sigma, dp_eigval * ymn))
# }}}
print("Errors for S:")
print(s_eoc_rec)
required_order = qbx_order + 1
assert s_eoc_rec.order_estimate() > required_order - 1.5
print("Errors for D:")
print(d_eoc_rec)
required_order = qbx_order
assert d_eoc_rec.order_estimate() > required_order - 0.5
print("Errors for S':")
print(sp_eoc_rec)
required_order = qbx_order
assert sp_eoc_rec.order_estimate() > required_order - 1.5
print("Errors for D':")
print(dp_eoc_rec)
required_order = qbx_order
assert dp_eoc_rec.order_estimate() > required_order - 1.5
def run(actx, *,
ambient_dim: int = 3,
resolution: int = None,
target_order: int = 4,
tmax: float = 1.0,
timestep: float = 1.0e-2,
group_factory_name: str = "warp_and_blend",
visualize: bool = True):
if ambient_dim not in (2, 3):
raise ValueError(f"unsupported dimension: {ambient_dim}")
mesh_order = target_order
radius = 1.0
# {{{ geometry
# {{{ element groups
import modepy as mp
import meshmode.discretization.poly_element as poly
# NOTE: picking the same unit nodes for the mesh and the discr saves
# a bit of work when reconstructing after a time step
if group_factory_name == "warp_and_blend":
group_factory_cls = poly.PolynomialWarpAndBlendGroupFactory
unit_nodes = mp.warp_and_blend_nodes(ambient_dim - 1, mesh_order)
elif group_factory_name == "quadrature":
group_factory_cls = poly.InterpolatoryQuadratureSimplexGroupFactory
if ambient_dim == 2:
unit_nodes = mp.LegendreGaussQuadrature(
mesh_order, force_dim_axis=True).nodes
else:
unit_nodes = mp.VioreanuRokhlinSimplexQuadrature(mesh_order, 2).nodes
else:
raise ValueError(f"unknown group factory: '{group_factory_name}'")
# }}}
# {{{ discretization
import meshmode.mesh.generation as gen
if ambient_dim == 2:
nelements = 8192 if resolution is None else resolution
mesh = gen.make_curve_mesh(
lambda t: radius * gen.ellipse(1.0, t),
np.linspace(0.0, 1.0, nelements + 1),
order=mesh_order,
unit_nodes=unit_nodes)
else:
nrounds = 4 if resolution is None else resolution
mesh = gen.generate_icosphere(radius,
uniform_refinement_rounds=nrounds,
order=mesh_order,
unit_nodes=unit_nodes)
from meshmode.discretization import Discretization
discr0 = Discretization(actx, mesh, group_factory_cls(target_order))
logger.info("ndofs: %d", discr0.ndofs)
logger.info("nelements: %d", discr0.mesh.nelements)
# }}}
if visualize:
from meshmode.discretization.visualization import make_visualizer
vis = make_visualizer(actx, discr0,
vis_order=target_order,
# NOTE: setting this to True will add some unnecessary
# resampling in Discretization.nodes for the vis_discr underneath
force_equidistant=False)
# }}}
# {{{ ode
def velocity_field(nodes, alpha=1.0):
return make_obj_array([
alpha * nodes[0], -alpha * nodes[1], 0.0 * nodes[0]
][:ambient_dim])
def source(t, x):
discr = reconstruct_discr_from_nodes(actx, discr0, x)
u = velocity_field(thaw(discr.nodes(), actx))
# {{{
# NOTE: these are just here because this was at some point used to
# profile some more operators (turned out well!)
from meshmode.discretization import num_reference_derivative
x = thaw(discr.nodes()[0], actx)
gradx = sum(
num_reference_derivative(discr, (i,), x)
for i in range(discr.dim))
intx = sum(actx.np.sum(xi * wi) for xi, wi in zip(x, discr.quad_weights()))
assert gradx is not None
assert intx is not None
# }}}
return u
# }}}
# {{{ evolve
maxiter = int(tmax // timestep) + 1
dt = tmax / maxiter + 1.0e-15
x = thaw(discr0.nodes(), actx)
t = 0.0
if visualize:
filename = f"moving-geometry-{0:09d}.vtu"
plot_solution(actx, vis, filename, discr0, t, x)
for n in range(1, maxiter + 1):
x = advance(actx, dt, t, x, source)
t += dt
if visualize:
discr = reconstruct_discr_from_nodes(actx, discr0, x)
vis = make_visualizer(actx, discr, vis_order=target_order)
# vis = vis.copy_with_same_connectivity(actx, discr)
filename = f"moving-geometry-{n:09d}.vtu"
plot_solution(actx, vis, filename, discr, t, x)
logger.info("[%05d/%05d] t = %.5e/%.5e dt = %.5e",
n, maxiter, t, tmax, dt)
pt.ylim([-1.5, 1.5])
filename = "test_proxy_generator_{}d_{:04}.png".format(
ambient_dim, i)
pt.savefig(filename, dpi=300)
pt.clf()
else:
from meshmode.discretization.visualization import make_visualizer
from meshmode.mesh.processing import ( # noqa
affine_map, merge_disjoint_meshes)
from meshmode.discretization import Discretization
from meshmode.discretization.poly_element import \
InterpolatoryQuadratureSimplexGroupFactory
from meshmode.mesh.generation import generate_icosphere
ref_mesh = generate_icosphere(1, generator.nproxy)
# NOTE: this does not plot the actual proxy points
for i in range(srcindices.nblocks):
mesh = affine_map(ref_mesh,
A=(pxyradii[i] * np.eye(ambient_dim)),
b=pxycenters[:, i].reshape(-1))
mesh = merge_disjoint_meshes([mesh, density_discr.mesh])
discr = Discretization(
actx, mesh, InterpolatoryQuadratureSimplexGroupFactory(10))
vis = make_visualizer(actx, discr, 10)
filename = "test_proxy_generator_{}d_{:04}.vtu".format(
ambient_dim, i)
vis.write_vtk_file(filename, [])
def test_harmonic_extension_exterior_3d(ctx_factory):
dim = 3 # noqa: F841
mesh_order = 8
fmm_order = 3
bdry_quad_order = mesh_order
bdry_ovsmp_quad_order = 4 * bdry_quad_order
qbx_order = mesh_order
cl_ctx = ctx_factory()
queue = cl.CommandQueue(cl_ctx)
from meshmode.mesh.generation import generate_icosphere
from meshmode.mesh.refinement import refine_uniformly
radius = 2.5
base_mesh = generate_icosphere(radius, order=mesh_order)
mesh = refine_uniformly(base_mesh, 1)
pre_density_discr = Discretization(
cl_ctx, mesh,
InterpolatoryQuadratureSimplexGroupFactory(bdry_quad_order))
from pytential.qbx import QBXLayerPotentialSource
qbx, _ = QBXLayerPotentialSource(
pre_density_discr,
fine_order=bdry_ovsmp_quad_order,
qbx_order=qbx_order,
fmm_order=fmm_order,
).with_refinement()
density_discr = qbx.density_discr
ntgts = 200
rho = np.random.rand(ntgts) * 3 + radius + 0.05
theta = np.random.rand(ntgts) * 2 * np.pi
phi = (np.random.rand(ntgts) - 0.5) * np.pi
nodes = density_discr.nodes().with_queue(queue)
source = np.array([0, 1, 2])
def test_func(x):
return 1.0 / la.norm(x.get() - source[:, None], axis=0)
f = cl.array.to_device(queue, test_func(nodes))
targets = cl.array.to_device(
queue,
np.array([
rho * np.cos(phi) * np.cos(theta),
rho * np.cos(phi) * np.sin(theta), rho * np.sin(phi)
]))
exact_f = test_func(targets)
ext_f, _ = compute_harmonic_extension(queue,
PointsTarget(targets),
qbx,
density_discr,
f,
loc_sign=1)
assert np.linalg.norm(exact_f -
ext_f.get()) < 1e-3 * np.linalg.norm(exact_f)
def test_proxy_generator(ctx_factory, ndim, factor, visualize=False):
ctx = ctx_factory()
queue = cl.CommandQueue(ctx)
qbx = _build_qbx_discr(queue, ndim=ndim)
srcindices = _build_block_index(qbx.density_discr, factor=factor)
from pytential.linalg.proxy import ProxyGenerator
generator = ProxyGenerator(qbx, ratio=1.1)
proxies, pxyranges, pxycenters, pxyradii = generator(queue, srcindices)
proxies = np.vstack([p.get() for p in proxies])
pxyranges = pxyranges.get()
pxycenters = np.vstack([c.get() for c in pxycenters])
pxyradii = pxyradii.get()
for i in range(srcindices.nblocks):
ipxy = np.s_[pxyranges[i]:pxyranges[i + 1]]
r = la.norm(proxies[:, ipxy] - pxycenters[:, i].reshape(-1, 1), axis=0)
assert np.allclose(r - pxyradii[i], 0.0, atol=1.0e-14)
srcindices = srcindices.get(queue)
if visualize:
if qbx.ambient_dim == 2:
import matplotlib.pyplot as pt
density_nodes = qbx.density_discr.nodes().get(queue)
ci = bind(qbx, sym.expansion_centers(qbx.ambient_dim, -1))(queue)
ci = np.vstack([c.get(queue) for c in ci])
ce = bind(qbx, sym.expansion_centers(qbx.ambient_dim, +1))(queue)
ce = np.vstack([c.get(queue) for c in ce])
r = bind(qbx, sym.expansion_radii(qbx.ambient_dim))(queue).get()
for i in range(srcindices.nblocks):
isrc = srcindices.block_indices(i)
ipxy = np.s_[pxyranges[i]:pxyranges[i + 1]]
pt.figure(figsize=(10, 8))
axis = pt.gca()
for j in isrc:
c = pt.Circle(ci[:, j], r[j], color='k', alpha=0.1)
axis.add_artist(c)
c = pt.Circle(ce[:, j], r[j], color='k', alpha=0.1)
axis.add_artist(c)
pt.plot(density_nodes[0],
density_nodes[1],
'ko',
ms=2.0,
alpha=0.5)
pt.plot(density_nodes[0, srcindices.indices],
density_nodes[1, srcindices.indices],
'o',
ms=2.0)
pt.plot(density_nodes[0, isrc],
density_nodes[1, isrc],
'o',
ms=2.0)
pt.plot(proxies[0, ipxy], proxies[1, ipxy], 'o', ms=2.0)
pt.xlim([-1.5, 1.5])
pt.ylim([-1.5, 1.5])
filename = "test_proxy_generator_{}d_{:04}.png".format(ndim, i)
pt.savefig(filename, dpi=300)
pt.clf()
else:
from meshmode.discretization.visualization import make_visualizer
from meshmode.mesh.processing import ( # noqa
affine_map, merge_disjoint_meshes)
from meshmode.discretization import Discretization
from meshmode.discretization.poly_element import \
InterpolatoryQuadratureSimplexGroupFactory
from meshmode.mesh.generation import generate_icosphere
ref_mesh = generate_icosphere(1, generator.nproxy)
# NOTE: this does not plot the actual proxy points
for i in range(srcindices.nblocks):
mesh = affine_map(ref_mesh,
A=(pxyradii[i] * np.eye(ndim)),
b=pxycenters[:, i].reshape(-1))
mesh = merge_disjoint_meshes([mesh, qbx.density_discr.mesh])
discr = Discretization(
ctx, mesh, InterpolatoryQuadratureSimplexGroupFactory(10))
vis = make_visualizer(queue, discr, 10)
filename = "test_proxy_generator_{}d_{:04}.vtu".format(ndim, i)
vis.write_vtk_file(filename, [])
def test_sphere_eigenvalues(ctx_getter, mode_m, mode_n, qbx_order,
fmm_backend):
logging.basicConfig(level=logging.INFO)
special = pytest.importorskip("scipy.special")
cl_ctx = ctx_getter()
queue = cl.CommandQueue(cl_ctx)
target_order = 8
from meshmode.discretization import Discretization
from meshmode.discretization.poly_element import \
InterpolatoryQuadratureSimplexGroupFactory
from pytential.qbx import QBXLayerPotentialSource
from pytools.convergence import EOCRecorder
s_eoc_rec = EOCRecorder()
d_eoc_rec = EOCRecorder()
sp_eoc_rec = EOCRecorder()
dp_eoc_rec = EOCRecorder()
def rel_err(comp, ref):
return (
norm(density_discr, queue, comp - ref)
/ norm(density_discr, queue, ref))
for nrefinements in [0, 1]:
from meshmode.mesh.generation import generate_icosphere
mesh = generate_icosphere(1, target_order)
from meshmode.mesh.refinement import Refiner
refiner = Refiner(mesh)
for i in range(nrefinements):
flags = np.ones(mesh.nelements, dtype=bool)
refiner.refine(flags)
mesh = refiner.get_current_mesh()
pre_density_discr = Discretization(
cl_ctx, mesh,
InterpolatoryQuadratureSimplexGroupFactory(target_order))
qbx, _ = QBXLayerPotentialSource(
pre_density_discr, 4*target_order,
qbx_order, fmm_order=6,
fmm_backend=fmm_backend,
).with_refinement()
density_discr = qbx.density_discr
nodes = density_discr.nodes().with_queue(queue)
r = cl.clmath.sqrt(nodes[0]**2 + nodes[1]**2 + nodes[2]**2)
phi = cl.clmath.acos(nodes[2]/r)
theta = cl.clmath.atan2(nodes[0], nodes[1])
ymn = cl.array.to_device(queue,
special.sph_harm(mode_m, mode_n, theta.get(), phi.get()))
from sumpy.kernel import LaplaceKernel
lap_knl = LaplaceKernel(3)
# {{{ single layer
s_sigma_op = bind(qbx, sym.S(lap_knl, sym.var("sigma")))
s_sigma = s_sigma_op(queue=queue, sigma=ymn)
s_eigval = 1/(2*mode_n + 1)
s_eoc_rec.add_data_point(qbx.h_max, rel_err(s_sigma, s_eigval*ymn))
# }}}
# {{{ double layer
d_sigma_op = bind(qbx, sym.D(lap_knl, sym.var("sigma")))
d_sigma = d_sigma_op(queue=queue, sigma=ymn)
d_eigval = -1/(2*(2*mode_n + 1))
d_eoc_rec.add_data_point(qbx.h_max, rel_err(d_sigma, d_eigval*ymn))
# }}}
# {{{ S'
sp_sigma_op = bind(qbx, sym.Sp(lap_knl, sym.var("sigma")))
sp_sigma = sp_sigma_op(queue=queue, sigma=ymn)
sp_eigval = -1/(2*(2*mode_n + 1))
sp_eoc_rec.add_data_point(qbx.h_max, rel_err(sp_sigma, sp_eigval*ymn))
# }}}
# {{{ D'
dp_sigma_op = bind(qbx, sym.Dp(lap_knl, sym.var("sigma")))
dp_sigma = dp_sigma_op(queue=queue, sigma=ymn)
dp_eigval = -(mode_n*(mode_n+1))/(2*mode_n + 1)
dp_eoc_rec.add_data_point(qbx.h_max, rel_err(dp_sigma, dp_eigval*ymn))
# }}}
print("Errors for S:")
print(s_eoc_rec)
required_order = qbx_order + 1
assert s_eoc_rec.order_estimate() > required_order - 1.5
print("Errors for D:")
print(d_eoc_rec)
required_order = qbx_order
assert d_eoc_rec.order_estimate() > required_order - 0.5
print("Errors for S':")
print(sp_eoc_rec)
required_order = qbx_order
assert sp_eoc_rec.order_estimate() > required_order - 1.5
print("Errors for D':")
print(dp_eoc_rec)
required_order = qbx_order
assert dp_eoc_rec.order_estimate() > required_order - 1.5
def test_proxy_generator(ctx_factory, ndim, factor, visualize=False):
ctx = ctx_factory()
queue = cl.CommandQueue(ctx)
qbx = _build_qbx_discr(queue, ndim=ndim)
srcindices = _build_block_index(qbx.density_discr,
method='nodes', factor=factor)
from pytential.linalg.proxy import ProxyGenerator
generator = ProxyGenerator(qbx, ratio=1.1)
proxies, pxyranges, pxycenters, pxyradii = generator(queue, srcindices)
proxies = np.vstack([p.get() for p in proxies])
pxyranges = pxyranges.get()
pxycenters = np.vstack([c.get() for c in pxycenters])
pxyradii = pxyradii.get()
for i in range(srcindices.nblocks):
ipxy = np.s_[pxyranges[i]:pxyranges[i + 1]]
r = la.norm(proxies[:, ipxy] - pxycenters[:, i].reshape(-1, 1), axis=0)
assert np.allclose(r - pxyradii[i], 0.0, atol=1.0e-14)
srcindices = srcindices.get(queue)
if visualize:
if qbx.ambient_dim == 2:
import matplotlib.pyplot as pt
from pytential.qbx.utils import get_centers_on_side
density_nodes = qbx.density_discr.nodes().get(queue)
ci = get_centers_on_side(qbx, -1)
ci = np.vstack([c.get(queue) for c in ci])
ce = get_centers_on_side(qbx, +1)
ce = np.vstack([c.get(queue) for c in ce])
r = qbx._expansion_radii("nsources").get(queue)
for i in range(srcindices.nblocks):
isrc = srcindices.block_indices(i)
ipxy = np.s_[pxyranges[i]:pxyranges[i + 1]]
pt.figure(figsize=(10, 8))
axis = pt.gca()
for j in isrc:
c = pt.Circle(ci[:, j], r[j], color='k', alpha=0.1)
axis.add_artist(c)
c = pt.Circle(ce[:, j], r[j], color='k', alpha=0.1)
axis.add_artist(c)
pt.plot(density_nodes[0], density_nodes[1],
'ko', ms=2.0, alpha=0.5)
pt.plot(density_nodes[0, srcindices.indices],
density_nodes[1, srcindices.indices],
'o', ms=2.0)
pt.plot(density_nodes[0, isrc], density_nodes[1, isrc],
'o', ms=2.0)
pt.plot(proxies[0, ipxy], proxies[1, ipxy],
'o', ms=2.0)
pt.xlim([-1.5, 1.5])
pt.ylim([-1.5, 1.5])
filename = "test_proxy_generator_{}d_{:04}.png".format(ndim, i)
pt.savefig(filename, dpi=300)
pt.clf()
else:
from meshmode.discretization.visualization import make_visualizer
from meshmode.mesh.processing import ( # noqa
affine_map, merge_disjoint_meshes)
from meshmode.discretization import Discretization
from meshmode.discretization.poly_element import \
InterpolatoryQuadratureSimplexGroupFactory
from meshmode.mesh.generation import generate_icosphere
ref_mesh = generate_icosphere(1, generator.nproxy)
# NOTE: this does not plot the actual proxy points
for i in range(srcindices.nblocks):
mesh = affine_map(ref_mesh,
A=(pxyradii[i] * np.eye(ndim)),
b=pxycenters[:, i].reshape(-1))
mesh = merge_disjoint_meshes([mesh, qbx.density_discr.mesh])
discr = Discretization(ctx, mesh,
InterpolatoryQuadratureSimplexGroupFactory(10))
vis = make_visualizer(queue, discr, 10)
filename = "test_proxy_generator_{}d_{:04}.vtu".format(ndim, i)
if os.path.isfile(filename):
os.remove(filename)
vis.write_vtk_file(filename, [])
def main(ctx_factory, dim=2, order=4, use_quad=False, visualize=False):
cl_ctx = ctx_factory()
queue = cl.CommandQueue(cl_ctx)
actx = PyOpenCLArrayContext(
queue,
allocator=cl_tools.MemoryPool(cl_tools.ImmediateAllocator(queue)),
force_device_scalars=True,
)
# {{{ parameters
# sphere radius
radius = 1.0
# sphere resolution
resolution = 64 if dim == 2 else 1
# final time
final_time = np.pi
# flux
flux_type = "lf"
# }}}
# {{{ discretization
if dim == 2:
from meshmode.mesh.generation import make_curve_mesh, ellipse
mesh = make_curve_mesh(
lambda t: radius * ellipse(1.0, t),
np.linspace(0.0, 1.0, resolution + 1),
order)
elif dim == 3:
from meshmode.mesh.generation import generate_icosphere
mesh = generate_icosphere(radius, order=4 * order,
uniform_refinement_rounds=resolution)
else:
raise ValueError("unsupported dimension")
discr_tag_to_group_factory = {}
if use_quad:
qtag = dof_desc.DISCR_TAG_QUAD
else:
qtag = None
from meshmode.discretization.poly_element import \
default_simplex_group_factory, \
QuadratureSimplexGroupFactory
discr_tag_to_group_factory[dof_desc.DISCR_TAG_BASE] = \
default_simplex_group_factory(base_dim=dim-1, order=order)
if use_quad:
discr_tag_to_group_factory[qtag] = \
QuadratureSimplexGroupFactory(order=4*order)
from grudge import DiscretizationCollection
dcoll = DiscretizationCollection(
actx, mesh,
discr_tag_to_group_factory=discr_tag_to_group_factory
)
volume_discr = dcoll.discr_from_dd(dof_desc.DD_VOLUME)
logger.info("ndofs: %d", volume_discr.ndofs)
logger.info("nelements: %d", volume_discr.mesh.nelements)
# }}}
# {{{ Surface advection operator
# velocity field
x = thaw(dcoll.nodes(), actx)
c = make_obj_array([-x[1], x[0], 0.0])[:dim]
def f_initial_condition(x):
return x[0]
from grudge.models.advection import SurfaceAdvectionOperator
adv_operator = SurfaceAdvectionOperator(
dcoll,
c,
flux_type=flux_type,
quad_tag=qtag
)
u0 = f_initial_condition(x)
def rhs(t, u):
return adv_operator.operator(t, u)
# check velocity is tangential
from grudge.geometry import normal
surf_normal = normal(actx, dcoll, dd=dof_desc.DD_VOLUME)
error = op.norm(dcoll, c.dot(surf_normal), 2)
logger.info("u_dot_n: %.5e", error)
# }}}
# {{{ time stepping
# FIXME: dt estimate is not necessarily valid for surfaces
dt = actx.to_numpy(
0.45 * adv_operator.estimate_rk4_timestep(actx, dcoll, fields=u0))
nsteps = int(final_time // dt) + 1
logger.info("dt: %.5e", dt)
logger.info("nsteps: %d", nsteps)
from grudge.shortcuts import set_up_rk4
dt_stepper = set_up_rk4("u", dt, u0, rhs)
plot = Plotter(actx, dcoll, order, visualize=visualize)
norm_u = actx.to_numpy(op.norm(dcoll, u0, 2))
step = 0
event = dt_stepper.StateComputed(0.0, 0, 0, u0)
plot(event, "fld-surface-%04d" % 0)
if visualize and dim == 3:
from grudge.shortcuts import make_visualizer
vis = make_visualizer(dcoll)
vis.write_vtk_file(
"fld-surface-velocity.vtu",
[
("u", c),
("n", surf_normal)
],
overwrite=True
)
df = dof_desc.DOFDesc(FACE_RESTR_INTERIOR)
face_discr = dcoll.discr_from_dd(df)
face_normal = thaw(dcoll.normal(dd=df), actx)
from meshmode.discretization.visualization import make_visualizer
vis = make_visualizer(actx, face_discr)
vis.write_vtk_file("fld-surface-face-normals.vtu", [
("n", face_normal)
], overwrite=True)
for event in dt_stepper.run(t_end=final_time, max_steps=nsteps + 1):
if not isinstance(event, dt_stepper.StateComputed):
continue
step += 1
if step % 10 == 0:
norm_u = actx.to_numpy(op.norm(dcoll, event.state_component, 2))
plot(event, "fld-surface-%04d" % step)
logger.info("[%04d] t = %.5f |u| = %.5e", step, event.t, norm_u)
# NOTE: These are here to ensure the solution is bounded for the
# time interval specified
assert norm_u < 3
def main():
# cl.array.to_device(queue, numpy_array)
from meshmode.mesh.io import generate_gmsh, FileSource
from meshmode.mesh.generation import generate_icosphere
from meshmode.mesh.refinement import Refiner
mesh = generate_icosphere(1,target_order)
refinement_increment = 1
refiner = Refiner(mesh)
for i in range(refinement_increment):
flags = np.ones(mesh.nelements, dtype=bool)
refiner.refine(flags)
mesh = refiner.get_current_mesh()
from meshmode.mesh.processing import perform_flips
# Flip elements--gmsh generates inside-out geometry.
mesh = perform_flips(mesh, np.ones(mesh.nelements))
print("%d elements" % mesh.nelements)
from meshmode.mesh.processing import find_bounding_box
bbox_min, bbox_max = find_bounding_box(mesh)
bbox_center = 0.5*(bbox_min+bbox_max)
bbox_size = max(bbox_max-bbox_min) / 2
logger.info("%d elements" % mesh.nelements)
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))
qbx = QBXLayerPotentialSource(density_discr, 4*target_order, qbx_order,
fmm_order=False, fmm_backend="fmmlib")
from pytential.symbolic.pde.maxwell import MuellerAugmentedMFIEOperator
pde_op = MuellerAugmentedMFIEOperator(
omega=1.0,
epss=[1.0, 1.0],
mus=[1.0, 1.0],
)
from pytential import bind, sym
unk = pde_op.make_unknown("sigma")
sym_operator = pde_op.operator(unk)
sym_rhs = pde_op.rhs(
sym.make_sym_vector("Einc", 3),
sym.make_sym_vector("Hinc", 3))
sym_repr = pde_op.representation(1, unk)
if 1:
expr = sym_repr
print(sym.pretty(expr))
print("#"*80)
from pytential.target import PointsTarget
tgt_points=np.zeros((3,1))
tgt_points[0,0] = 100
tgt_points[1,0] = -200
tgt_points[2,0] = 300
bound_op = bind((qbx, PointsTarget(tgt_points)), expr)
print(bound_op.code)
if 1:
def green3e(x,y,z,source,strength,k):
# electric field corresponding to dyadic green's function
# due to monochromatic electric dipole located at "source".
# "strength" is the the intensity of the dipole.
# E = (I + Hess)(exp(ikr)/r) dot (strength)
#
dx = x - source[0]
dy = y - source[1]
dz = z - source[2]
rr = np.sqrt(dx**2 + dy**2 + dz**2)
fout = np.exp(1j*k*rr)/rr
evec = fout*strength
qmat = np.zeros((3,3),dtype=np.complex128)
qmat[0,0]=(2*dx**2-dy**2-dz**2)*(1-1j*k*rr)
qmat[1,1]=(2*dy**2-dz**2-dx**2)*(1-1j*k*rr)
qmat[2,2]=(2*dz**2-dx**2-dy**2)*(1-1j*k*rr)
qmat[0,0]=qmat[0,0]+(-k**2*dx**2*rr**2)
qmat[1,1]=qmat[1,1]+(-k**2*dy**2*rr**2)
qmat[2,2]=qmat[2,2]+(-k**2*dz**2*rr**2)
qmat[0,1]=(3-k**2*rr**2-3*1j*k*rr)*(dx*dy)
qmat[1,2]=(3-k**2*rr**2-3*1j*k*rr)*(dy*dz)
qmat[2,0]=(3-k**2*rr**2-3*1j*k*rr)*(dz*dx)
qmat[1,0]=qmat[0,1]
qmat[2,1]=qmat[1,2]
qmat[0,2]=qmat[2,0]
fout=np.exp(1j*k*rr)/rr**5/k**2
fvec = fout*np.dot(qmat,strength)
evec = evec + fvec
return evec
def green3m(x,y,z,source,strength,k):
# magnetic field corresponding to dyadic green's function
# due to monochromatic electric dipole located at "source".
# "strength" is the the intensity of the dipole.
# H = curl((I + Hess)(exp(ikr)/r) dot (strength)) =
# strength \cross \grad (exp(ikr)/r)
#
dx = x - source[0]
dy = y - source[1]
dz = z - source[2]
rr = np.sqrt(dx**2 + dy**2 + dz**2)
fout=(1-1j*k*rr)*np.exp(1j*k*rr)/rr**3
fvec = np.zeros(3,dtype=np.complex128)
fvec[0] = fout*dx
fvec[1] = fout*dy
fvec[2] = fout*dz
hvec = np.cross(strength,fvec)
return hvec
def dipole3e(x,y,z,source,strength,k):
#
# evalaute electric and magnetic field due
# to monochromatic electric dipole located at "source"
# with intensity "strength"
evec = green3e(x,y,z,source,strength,k)
evec = evec*1j*k
hvec = green3m(x,y,z,source,strength,k)
# print(hvec)
# print(strength)
return evec,hvec
def dipole3m(x,y,z,source,strength,k):
#
# evalaute electric and magnetic field due
# to monochromatic magnetic dipole located at "source"
# with intensity "strength"
evec = green3m(x,y,z,source,strength,k)
hvec = green3e(x,y,z,source,strength,k)
hvec = -hvec*1j*k
return evec,hvec
def dipole3eall(x,y,z,sources,strengths,k):
ns = len(strengths)
evec = np.zeros(3,dtype=np.complex128)
hvec = np.zeros(3,dtype=np.complex128)
for i in range(ns):
evect,hvect = dipole3e(x,y,z,sources[i],strengths[i],k)
evec = evec + evect
hvec = hvec + hvect
nodes = density_discr.nodes().with_queue(queue).get()
source = [0.01,-0.03,0.02]
# source = cl.array.to_device(queue,np.zeros(3))
# source[0] = 0.01
# source[1] =-0.03
# source[2] = 0.02
strength = np.ones(3)
# evec = cl.array.to_device(queue,np.zeros((3,len(nodes[0])),dtype=np.complex128))
# hvec = cl.array.to_device(queue,np.zeros((3,len(nodes[0])),dtype=np.complex128))
evec = np.zeros((3,len(nodes[0])),dtype=np.complex128)
hvec = np.zeros((3,len(nodes[0])),dtype=np.complex128)
for i in range(len(nodes[0])):
evec[:,i],hvec[:,i] = dipole3e(nodes[0][i],nodes[1][i],nodes[2][i],source,strength,k)
print(np.shape(hvec))
print(type(evec))
print(type(hvec))
evec = cl.array.to_device(queue,evec)
hvec = cl.array.to_device(queue,hvec)
bvp_rhs = bind(qbx, sym_rhs)(queue,Einc=evec,Hinc=hvec)
print(np.shape(bvp_rhs))
print(type(bvp_rhs))
# print(bvp_rhs)
1/-1
bound_op = bind(qbx, sym_operator)
from pytential.solve import gmres
if 1:
gmres_result = gmres(
bound_op.scipy_op(queue, "sigma", dtype=np.complex128, k=k),
bvp_rhs, tol=1e-8, progress=True,
stall_iterations=0,
hard_failure=True)
sigma = gmres_result.solution
fld_at_tgt = bind((qbx, PointsTarget(tgt_points)), sym_repr)(queue,
sigma=sigma,k=k)
fld_at_tgt = np.array([
fi.get() for fi in fld_at_tgt
])
print(fld_at_tgt)
fld_exact_e,fld_exact_h = dipole3e(tgt_points[0,0],tgt_points[1,0],tgt_points[2,0],source,strength,k)
print(fld_exact_e)
print(fld_exact_h)
1/0
# }}}
#mlab.figure(bgcolor=(1, 1, 1))
if 1:
from meshmode.discretization.visualization import make_visualizer
bdry_vis = make_visualizer(queue, density_discr, target_order)
bdry_normals = bind(density_discr, sym.normal(3))(queue)\
.as_vector(dtype=object)
bdry_vis.write_vtk_file("source.vtu", [
("sigma", sigma),
("bdry_normals", bdry_normals),
])
fplot = FieldPlotter(bbox_center, extent=2*bbox_size, npoints=(150, 150, 1))
qbx_stick_out = qbx.copy(target_stick_out_factor=0.1)
from pytential.target import PointsTarget
from pytential.qbx import QBXTargetAssociationFailedException
rho_sym = sym.var("rho")
try:
fld_in_vol = bind(
(qbx_stick_out, PointsTarget(fplot.points)),
sym.make_obj_array([
sym.S(pde_op.kernel, rho_sym, k=sym.var("k"),
qbx_forced_limit=None),
sym.d_dx(3, sym.S(pde_op.kernel, rho_sym, k=sym.var("k"),
qbx_forced_limit=None)),
sym.d_dy(3, sym.S(pde_op.kernel, rho_sym, k=sym.var("k"),
qbx_forced_limit=None)),
sym.d_dz(3, sym.S(pde_op.kernel, rho_sym, k=sym.var("k"),
qbx_forced_limit=None)),
])
)(queue, jt=jt, rho=rho, k=k)
except QBXTargetAssociationFailedException as e:
fplot.write_vtk_file(
"failed-targets.vts",
[
("failed_targets", e.failed_target_flags.get(queue))
])
raise
fld_in_vol = sym.make_obj_array(
[fiv.get() for fiv in fld_in_vol])
#fplot.show_scalar_in_mayavi(fld_in_vol.real, max_val=5)
fplot.write_vtk_file(
"potential.vts",
[
("potential", fld_in_vol[0]),
("grad", fld_in_vol[1:]),
]
)
def run_exterior_stokes(
ctx_factory,
*,
ambient_dim,
target_order,
qbx_order,
resolution,
fmm_order=False, # FIXME: FMM is slower than direct evaluation
source_ovsmp=None,
radius=1.5,
mu=1.0,
visualize=False,
_target_association_tolerance=0.05,
_expansions_in_tree_have_extent=True):
cl_ctx = cl.create_some_context()
queue = cl.CommandQueue(cl_ctx)
actx = PyOpenCLArrayContext(queue)
# {{{ geometry
if source_ovsmp is None:
source_ovsmp = 4 if ambient_dim == 2 else 8
places = {}
if ambient_dim == 2:
from meshmode.mesh.generation import make_curve_mesh, ellipse
mesh = make_curve_mesh(lambda t: radius * ellipse(1.0, t),
np.linspace(0.0, 1.0, resolution + 1),
target_order)
elif ambient_dim == 3:
from meshmode.mesh.generation import generate_icosphere
mesh = generate_icosphere(radius,
target_order + 1,
uniform_refinement_rounds=resolution)
else:
raise ValueError(f"unsupported dimension: {ambient_dim}")
pre_density_discr = Discretization(
actx, mesh, InterpolatoryQuadratureSimplexGroupFactory(target_order))
from pytential.qbx import QBXLayerPotentialSource
qbx = QBXLayerPotentialSource(
pre_density_discr,
fine_order=source_ovsmp * target_order,
qbx_order=qbx_order,
fmm_order=fmm_order,
target_association_tolerance=_target_association_tolerance,
_expansions_in_tree_have_extent=_expansions_in_tree_have_extent)
places["source"] = qbx
from extra_int_eq_data import make_source_and_target_points
point_source, point_target = make_source_and_target_points(
side=+1,
inner_radius=0.5 * radius,
outer_radius=2.0 * radius,
ambient_dim=ambient_dim,
)
places["point_source"] = point_source
places["point_target"] = point_target
if visualize:
from sumpy.visualization import make_field_plotter_from_bbox
from meshmode.mesh.processing import find_bounding_box
fplot = make_field_plotter_from_bbox(find_bounding_box(mesh),
h=0.1,
extend_factor=1.0)
mask = np.linalg.norm(fplot.points, ord=2, axis=0) > (radius + 0.25)
from pytential.target import PointsTarget
plot_target = PointsTarget(fplot.points[:, mask].copy())
places["plot_target"] = plot_target
del mask
places = GeometryCollection(places, auto_where="source")
density_discr = places.get_discretization("source")
logger.info("ndofs: %d", density_discr.ndofs)
logger.info("nelements: %d", density_discr.mesh.nelements)
# }}}
# {{{ symbolic
sym_normal = sym.make_sym_vector("normal", ambient_dim)
sym_mu = sym.var("mu")
if ambient_dim == 2:
from pytential.symbolic.stokes import HsiaoKressExteriorStokesOperator
sym_omega = sym.make_sym_vector("omega", ambient_dim)
op = HsiaoKressExteriorStokesOperator(omega=sym_omega)
elif ambient_dim == 3:
from pytential.symbolic.stokes import HebekerExteriorStokesOperator
op = HebekerExteriorStokesOperator()
else:
assert False
sym_sigma = op.get_density_var("sigma")
sym_bc = op.get_density_var("bc")
sym_op = op.operator(sym_sigma, normal=sym_normal, mu=sym_mu)
sym_rhs = op.prepare_rhs(sym_bc, mu=mu)
sym_velocity = op.velocity(sym_sigma, normal=sym_normal, mu=sym_mu)
sym_source_pot = op.stokeslet.apply(sym_sigma,
sym_mu,
qbx_forced_limit=None)
# }}}
# {{{ boundary conditions
normal = bind(places, sym.normal(ambient_dim).as_vector())(actx)
np.random.seed(42)
charges = make_obj_array([
actx.from_numpy(np.random.randn(point_source.ndofs))
for _ in range(ambient_dim)
])
if ambient_dim == 2:
total_charge = make_obj_array([actx.np.sum(c) for c in charges])
omega = bind(places, total_charge * sym.Ones())(actx)
if ambient_dim == 2:
bc_context = {"mu": mu, "omega": omega}
op_context = {"mu": mu, "omega": omega, "normal": normal}
else:
bc_context = {}
op_context = {"mu": mu, "normal": normal}
bc = bind(places, sym_source_pot,
auto_where=("point_source", "source"))(actx,
sigma=charges,
mu=mu)
rhs = bind(places, sym_rhs)(actx, bc=bc, **bc_context)
bound_op = bind(places, sym_op)
# }}}
# {{{ solve
from pytential.solve import gmres
gmres_tol = 1.0e-9
result = gmres(bound_op.scipy_op(actx, "sigma", np.float64, **op_context),
rhs,
x0=rhs,
tol=gmres_tol,
progress=visualize,
stall_iterations=0,
hard_failure=True)
sigma = result.solution
# }}}
# {{{ check velocity at "point_target"
def rnorm2(x, y):
y_norm = actx.np.linalg.norm(y.dot(y), ord=2)
if y_norm < 1.0e-14:
y_norm = 1.0
d = x - y
return actx.np.linalg.norm(d.dot(d), ord=2) / y_norm
ps_velocity = bind(places,
sym_velocity,
auto_where=("source", "point_target"))(actx,
sigma=sigma,
**op_context)
ex_velocity = bind(places,
sym_source_pot,
auto_where=("point_source",
"point_target"))(actx, sigma=charges, mu=mu)
v_error = rnorm2(ps_velocity, ex_velocity)
h_max = bind(places, sym.h_max(ambient_dim))(actx)
logger.info("resolution %4d h_max %.5e error %.5e", resolution, h_max,
v_error)
# }}}}
# {{{ visualize
if not visualize:
return h_max, v_error
from meshmode.discretization.visualization import make_visualizer
vis = make_visualizer(actx, density_discr, target_order)
filename = "stokes_solution_{}d_{}_ovsmp_{}.vtu".format(
ambient_dim, resolution, source_ovsmp)
vis.write_vtk_file(filename, [
("density", sigma),
("bc", bc),
("rhs", rhs),
],
overwrite=True)
# }}}
return h_max, v_error
def test_target_specific_qbx(ctx_factory, op, helmholtz_k, qbx_order):
logging.basicConfig(level=logging.INFO)
cl_ctx = ctx_factory()
queue = cl.CommandQueue(cl_ctx)
actx = PyOpenCLArrayContext(queue)
target_order = 4
fmm_tol = 1e-3
from meshmode.mesh.generation import generate_icosphere
mesh = generate_icosphere(1, target_order)
from meshmode.discretization import Discretization
from meshmode.discretization.poly_element import \
InterpolatoryQuadratureSimplexGroupFactory
from pytential.qbx import QBXLayerPotentialSource
pre_density_discr = Discretization(
actx, mesh,
InterpolatoryQuadratureSimplexGroupFactory(target_order))
from sumpy.expansion.level_to_order import SimpleExpansionOrderFinder
qbx = QBXLayerPotentialSource(
pre_density_discr, 4*target_order,
qbx_order=qbx_order,
fmm_level_to_order=SimpleExpansionOrderFinder(fmm_tol),
fmm_backend="fmmlib",
_expansions_in_tree_have_extent=True,
_expansion_stick_out_factor=0.9,
_use_target_specific_qbx=False,
)
kernel_length_scale = 5 / abs(helmholtz_k) if helmholtz_k else None
places = {
"qbx": qbx,
"qbx_target_specific": qbx.copy(_use_target_specific_qbx=True)
}
from pytential.qbx.refinement import refine_geometry_collection
places = GeometryCollection(places, auto_where="qbx")
places = refine_geometry_collection(places,
kernel_length_scale=kernel_length_scale)
density_discr = places.get_discretization("qbx")
from meshmode.dof_array import thaw
nodes = thaw(actx, density_discr.nodes())
u_dev = actx.np.sin(nodes[0])
if helmholtz_k == 0:
kernel = LaplaceKernel(3)
kernel_kwargs = {}
else:
kernel = HelmholtzKernel(3, allow_evanescent=True)
kernel_kwargs = {"k": sym.var("k")}
u_sym = sym.var("u")
if op == "S":
op = sym.S
elif op == "D":
op = sym.D
elif op == "Sp":
op = sym.Sp
else:
raise ValueError("unknown operator: '%s'" % op)
expr = op(kernel, u_sym, qbx_forced_limit=-1, **kernel_kwargs)
from meshmode.dof_array import flatten
bound_op = bind(places, expr)
pot_ref = actx.to_numpy(flatten(bound_op(actx, u=u_dev, k=helmholtz_k)))
bound_op = bind(places, expr, auto_where="qbx_target_specific")
pot_tsqbx = actx.to_numpy(flatten(bound_op(actx, u=u_dev, k=helmholtz_k)))
assert np.allclose(pot_tsqbx, pot_ref, atol=1e-13, rtol=1e-13)
def test_target_specific_qbx(ctx_getter, op, helmholtz_k, qbx_order):
logging.basicConfig(level=logging.INFO)
cl_ctx = ctx_getter()
queue = cl.CommandQueue(cl_ctx)
target_order = 4
fmm_tol = 1e-3
from meshmode.mesh.generation import generate_icosphere
mesh = generate_icosphere(1, target_order)
from meshmode.discretization import Discretization
from meshmode.discretization.poly_element import \
InterpolatoryQuadratureSimplexGroupFactory
from pytential.qbx import QBXLayerPotentialSource
pre_density_discr = Discretization(
cl_ctx, mesh, InterpolatoryQuadratureSimplexGroupFactory(target_order))
from sumpy.expansion.level_to_order import SimpleExpansionOrderFinder
refiner_extra_kwargs = {}
if helmholtz_k != 0:
refiner_extra_kwargs["kernel_length_scale"] = 5 / abs(helmholtz_k)
qbx, _ = QBXLayerPotentialSource(
pre_density_discr,
4 * target_order,
qbx_order=qbx_order,
fmm_level_to_order=SimpleExpansionOrderFinder(fmm_tol),
fmm_backend="fmmlib",
_expansions_in_tree_have_extent=True,
_expansion_stick_out_factor=0.9,
_use_target_specific_qbx=False,
).with_refinement(**refiner_extra_kwargs)
density_discr = qbx.density_discr
nodes = density_discr.nodes().with_queue(queue)
u_dev = clmath.sin(nodes[0])
if helmholtz_k == 0:
kernel = LaplaceKernel(3)
kernel_kwargs = {}
else:
kernel = HelmholtzKernel(3, allow_evanescent=True)
kernel_kwargs = {"k": sym.var("k")}
u_sym = sym.var("u")
if op == "S":
op = sym.S
elif op == "D":
op = sym.D
elif op == "Sp":
op = sym.Sp
else:
raise ValueError("unknown operator: '%s'" % op)
expr = op(kernel, u_sym, qbx_forced_limit=-1, **kernel_kwargs)
bound_op = bind(qbx, expr)
pot_ref = bound_op(queue, u=u_dev, k=helmholtz_k).get()
qbx = qbx.copy(_use_target_specific_qbx=True)
bound_op = bind(qbx, expr)
pot_tsqbx = bound_op(queue, u=u_dev, k=helmholtz_k).get()
assert np.allclose(pot_tsqbx, pot_ref, atol=1e-13, rtol=1e-13)
def main(ctx_factory, dim=2, order=4, product_tag=None, visualize=False):
cl_ctx = ctx_factory()
queue = cl.CommandQueue(cl_ctx)
actx = PyOpenCLArrayContext(queue)
# {{{ parameters
# sphere radius
radius = 1.0
# sphere resolution
resolution = 64 if dim == 2 else 1
# cfl
dt_factor = 2.0
# final time
final_time = np.pi
# velocity field
sym_x = sym.nodes(dim)
c = make_obj_array([-sym_x[1], sym_x[0], 0.0])[:dim]
# flux
flux_type = "lf"
# }}}
# {{{ discretization
if dim == 2:
from meshmode.mesh.generation import make_curve_mesh, ellipse
mesh = make_curve_mesh(lambda t: radius * ellipse(1.0, t),
np.linspace(0.0, 1.0, resolution + 1), order)
elif dim == 3:
from meshmode.mesh.generation import generate_icosphere
mesh = generate_icosphere(radius,
order=4 * order,
uniform_refinement_rounds=resolution)
else:
raise ValueError("unsupported dimension")
discr_tag_to_group_factory = {}
if product_tag == "none":
product_tag = None
else:
product_tag = dof_desc.DISCR_TAG_QUAD
from meshmode.discretization.poly_element import \
PolynomialWarpAndBlendGroupFactory, \
QuadratureSimplexGroupFactory
discr_tag_to_group_factory[dof_desc.DISCR_TAG_BASE] = \
PolynomialWarpAndBlendGroupFactory(order)
if product_tag:
discr_tag_to_group_factory[product_tag] = \
QuadratureSimplexGroupFactory(order=4*order)
from grudge import DiscretizationCollection
discr = DiscretizationCollection(
actx, mesh, discr_tag_to_group_factory=discr_tag_to_group_factory)
volume_discr = discr.discr_from_dd(dof_desc.DD_VOLUME)
logger.info("ndofs: %d", volume_discr.ndofs)
logger.info("nelements: %d", volume_discr.mesh.nelements)
# }}}
# {{{ symbolic operators
def f_initial_condition(x):
return x[0]
from grudge.models.advection import SurfaceAdvectionOperator
op = SurfaceAdvectionOperator(c, flux_type=flux_type, quad_tag=product_tag)
bound_op = bind(discr, op.sym_operator())
u0 = bind(discr, f_initial_condition(sym_x))(actx, t=0)
def rhs(t, u):
return bound_op(actx, t=t, u=u)
# check velocity is tangential
sym_normal = sym.surface_normal(dim, dim=dim - 1,
dd=dof_desc.DD_VOLUME).as_vector()
error = bind(discr, sym.norm(2, c.dot(sym_normal)))(actx)
logger.info("u_dot_n: %.5e", error)
# }}}
# {{{ time stepping
# compute time step
h_min = bind(discr, sym.h_max_from_volume(discr.ambient_dim,
dim=discr.dim))(actx)
dt = dt_factor * h_min / order**2
nsteps = int(final_time // dt) + 1
dt = final_time / nsteps + 1.0e-15
logger.info("dt: %.5e", dt)
logger.info("nsteps: %d", nsteps)
from grudge.shortcuts import set_up_rk4
dt_stepper = set_up_rk4("u", dt, u0, rhs)
plot = Plotter(actx, discr, order, visualize=visualize)
norm = bind(discr, sym.norm(2, sym.var("u")))
norm_u = norm(actx, u=u0)
step = 0
event = dt_stepper.StateComputed(0.0, 0, 0, u0)
plot(event, "fld-surface-%04d" % 0)
if visualize and dim == 3:
from grudge.shortcuts import make_visualizer
vis = make_visualizer(discr)
vis.write_vtk_file("fld-surface-velocity.vtu",
[("u", bind(discr, c)(actx)),
("n", bind(discr, sym_normal)(actx))],
overwrite=True)
df = dof_desc.DOFDesc(FACE_RESTR_INTERIOR)
face_discr = discr.connection_from_dds(dof_desc.DD_VOLUME, df).to_discr
face_normal = bind(
discr, sym.normal(df, face_discr.ambient_dim,
dim=face_discr.dim))(actx)
from meshmode.discretization.visualization import make_visualizer
vis = make_visualizer(actx, face_discr)
vis.write_vtk_file("fld-surface-face-normals.vtu",
[("n", face_normal)],
overwrite=True)
for event in dt_stepper.run(t_end=final_time, max_steps=nsteps + 1):
if not isinstance(event, dt_stepper.StateComputed):
continue
step += 1
if step % 10 == 0:
norm_u = norm(actx, u=event.state_component)
plot(event, "fld-surface-%04d" % step)
logger.info("[%04d] t = %.5f |u| = %.5e", step, event.t, norm_u)
plot(event, "fld-surface-%04d" % step)
def main():
# cl.array.to_device(queue, numpy_array)
from meshmode.mesh.io import generate_gmsh, FileSource
from meshmode.mesh.generation import generate_icosphere
from meshmode.mesh.refinement import Refiner
mesh = generate_icosphere(1, target_order)
refinement_increment = 1
refiner = Refiner(mesh)
for i in range(refinement_increment):
flags = np.ones(mesh.nelements, dtype=bool)
refiner.refine(flags)
mesh = refiner.get_current_mesh()
from meshmode.mesh.processing import perform_flips
# Flip elements--gmsh generates inside-out geometry.
mesh = perform_flips(mesh, np.ones(mesh.nelements))
print("%d elements" % mesh.nelements)
from meshmode.mesh.processing import find_bounding_box
bbox_min, bbox_max = find_bounding_box(mesh)
bbox_center = 0.5 * (bbox_min + bbox_max)
bbox_size = max(bbox_max - bbox_min) / 2
logger.info("%d elements" % mesh.nelements)
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))
qbx = QBXLayerPotentialSource(density_discr,
4 * target_order,
qbx_order,
fmm_order=False,
fmm_backend="fmmlib")
from pytential.symbolic.pde.maxwell import MuellerAugmentedMFIEOperator
pde_op = MuellerAugmentedMFIEOperator(
omega=1.0,
epss=[1.0, 1.0],
mus=[1.0, 1.0],
)
from pytential import bind, sym
unk = pde_op.make_unknown("sigma")
sym_operator = pde_op.operator(unk)
sym_rhs = pde_op.rhs(sym.make_sym_vector("Einc", 3),
sym.make_sym_vector("Hinc", 3))
sym_repr = pde_op.representation(1, unk)
if 1:
expr = sym_repr
print(sym.pretty(expr))
print("#" * 80)
from pytential.target import PointsTarget
tgt_points = np.zeros((3, 1))
tgt_points[0, 0] = 100
tgt_points[1, 0] = -200
tgt_points[2, 0] = 300
bound_op = bind((qbx, PointsTarget(tgt_points)), expr)
print(bound_op.code)
if 1:
def green3e(x, y, z, source, strength, k):
# electric field corresponding to dyadic green's function
# due to monochromatic electric dipole located at "source".
# "strength" is the the intensity of the dipole.
# E = (I + Hess)(exp(ikr)/r) dot (strength)
#
dx = x - source[0]
dy = y - source[1]
dz = z - source[2]
rr = np.sqrt(dx**2 + dy**2 + dz**2)
fout = np.exp(1j * k * rr) / rr
evec = fout * strength
qmat = np.zeros((3, 3), dtype=np.complex128)
qmat[0, 0] = (2 * dx**2 - dy**2 - dz**2) * (1 - 1j * k * rr)
qmat[1, 1] = (2 * dy**2 - dz**2 - dx**2) * (1 - 1j * k * rr)
qmat[2, 2] = (2 * dz**2 - dx**2 - dy**2) * (1 - 1j * k * rr)
qmat[0, 0] = qmat[0, 0] + (-k**2 * dx**2 * rr**2)
qmat[1, 1] = qmat[1, 1] + (-k**2 * dy**2 * rr**2)
qmat[2, 2] = qmat[2, 2] + (-k**2 * dz**2 * rr**2)
qmat[0, 1] = (3 - k**2 * rr**2 - 3 * 1j * k * rr) * (dx * dy)
qmat[1, 2] = (3 - k**2 * rr**2 - 3 * 1j * k * rr) * (dy * dz)
qmat[2, 0] = (3 - k**2 * rr**2 - 3 * 1j * k * rr) * (dz * dx)
qmat[1, 0] = qmat[0, 1]
qmat[2, 1] = qmat[1, 2]
qmat[0, 2] = qmat[2, 0]
fout = np.exp(1j * k * rr) / rr**5 / k**2
fvec = fout * np.dot(qmat, strength)
evec = evec + fvec
return evec
def green3m(x, y, z, source, strength, k):
# magnetic field corresponding to dyadic green's function
# due to monochromatic electric dipole located at "source".
# "strength" is the the intensity of the dipole.
# H = curl((I + Hess)(exp(ikr)/r) dot (strength)) =
# strength \cross \grad (exp(ikr)/r)
#
dx = x - source[0]
dy = y - source[1]
dz = z - source[2]
rr = np.sqrt(dx**2 + dy**2 + dz**2)
fout = (1 - 1j * k * rr) * np.exp(1j * k * rr) / rr**3
fvec = np.zeros(3, dtype=np.complex128)
fvec[0] = fout * dx
fvec[1] = fout * dy
fvec[2] = fout * dz
hvec = np.cross(strength, fvec)
return hvec
def dipole3e(x, y, z, source, strength, k):
#
# evalaute electric and magnetic field due
# to monochromatic electric dipole located at "source"
# with intensity "strength"
evec = green3e(x, y, z, source, strength, k)
evec = evec * 1j * k
hvec = green3m(x, y, z, source, strength, k)
# print(hvec)
# print(strength)
return evec, hvec
def dipole3m(x, y, z, source, strength, k):
#
# evalaute electric and magnetic field due
# to monochromatic magnetic dipole located at "source"
# with intensity "strength"
evec = green3m(x, y, z, source, strength, k)
hvec = green3e(x, y, z, source, strength, k)
hvec = -hvec * 1j * k
return evec, hvec
def dipole3eall(x, y, z, sources, strengths, k):
ns = len(strengths)
evec = np.zeros(3, dtype=np.complex128)
hvec = np.zeros(3, dtype=np.complex128)
for i in range(ns):
evect, hvect = dipole3e(x, y, z, sources[i], strengths[i], k)
evec = evec + evect
hvec = hvec + hvect
nodes = density_discr.nodes().with_queue(queue).get()
source = [0.01, -0.03, 0.02]
# source = cl.array.to_device(queue,np.zeros(3))
# source[0] = 0.01
# source[1] =-0.03
# source[2] = 0.02
strength = np.ones(3)
# evec = cl.array.to_device(queue,np.zeros((3,len(nodes[0])),dtype=np.complex128))
# hvec = cl.array.to_device(queue,np.zeros((3,len(nodes[0])),dtype=np.complex128))
evec = np.zeros((3, len(nodes[0])), dtype=np.complex128)
hvec = np.zeros((3, len(nodes[0])), dtype=np.complex128)
for i in range(len(nodes[0])):
evec[:, i], hvec[:, i] = dipole3e(nodes[0][i], nodes[1][i],
nodes[2][i], source, strength, k)
print(np.shape(hvec))
print(type(evec))
print(type(hvec))
evec = cl.array.to_device(queue, evec)
hvec = cl.array.to_device(queue, hvec)
bvp_rhs = bind(qbx, sym_rhs)(queue, Einc=evec, Hinc=hvec)
print(np.shape(bvp_rhs))
print(type(bvp_rhs))
# print(bvp_rhs)
1 / -1
bound_op = bind(qbx, sym_operator)
from pytential.solve import gmres
if 1:
gmres_result = gmres(bound_op.scipy_op(queue,
"sigma",
dtype=np.complex128,
k=k),
bvp_rhs,
tol=1e-8,
progress=True,
stall_iterations=0,
hard_failure=True)
sigma = gmres_result.solution
fld_at_tgt = bind((qbx, PointsTarget(tgt_points)),
sym_repr)(queue, sigma=sigma, k=k)
fld_at_tgt = np.array([fi.get() for fi in fld_at_tgt])
print(fld_at_tgt)
fld_exact_e, fld_exact_h = dipole3e(tgt_points[0, 0], tgt_points[1, 0],
tgt_points[2,
0], source, strength, k)
print(fld_exact_e)
print(fld_exact_h)
1 / 0
# }}}
#mlab.figure(bgcolor=(1, 1, 1))
if 1:
from meshmode.discretization.visualization import make_visualizer
bdry_vis = make_visualizer(queue, density_discr, target_order)
bdry_normals = bind(density_discr, sym.normal(3))(queue)\
.as_vector(dtype=object)
bdry_vis.write_vtk_file("source.vtu", [
("sigma", sigma),
("bdry_normals", bdry_normals),
])
fplot = FieldPlotter(bbox_center,
extent=2 * bbox_size,
npoints=(150, 150, 1))
qbx_stick_out = qbx.copy(target_stick_out_factor=0.1)
from pytential.target import PointsTarget
from pytential.qbx import QBXTargetAssociationFailedException
rho_sym = sym.var("rho")
try:
fld_in_vol = bind((qbx_stick_out, PointsTarget(fplot.points)),
sym.make_obj_array([
sym.S(pde_op.kernel,
rho_sym,
k=sym.var("k"),
qbx_forced_limit=None),
sym.d_dx(
3,
sym.S(pde_op.kernel,
rho_sym,
k=sym.var("k"),
qbx_forced_limit=None)),
sym.d_dy(
3,
sym.S(pde_op.kernel,
rho_sym,
k=sym.var("k"),
qbx_forced_limit=None)),
sym.d_dz(
3,
sym.S(pde_op.kernel,
rho_sym,
k=sym.var("k"),
qbx_forced_limit=None)),
]))(queue, jt=jt, rho=rho, k=k)
except QBXTargetAssociationFailedException as e:
fplot.write_vtk_file(
"failed-targets.vts",
[("failed_targets", e.failed_target_flags.get(queue))])
raise
fld_in_vol = sym.make_obj_array([fiv.get() for fiv in fld_in_vol])
#fplot.show_scalar_in_mayavi(fld_in_vol.real, max_val=5)
fplot.write_vtk_file("potential.vts", [
("potential", fld_in_vol[0]),
("grad", fld_in_vol[1:]),
])
