def test_target_association_failure(ctx_factory):
cl_ctx = ctx_factory()
queue = cl.CommandQueue(cl_ctx)
actx = PyOpenCLArrayContext(queue)
# {{{ generate circle
order = 5
nelements = 40
# Make the curve mesh.
curve_f = partial(ellipse, 1)
mesh = make_curve_mesh(curve_f, np.linspace(0, 1, nelements + 1), order)
from meshmode.discretization import Discretization
from meshmode.discretization.poly_element import \
InterpolatoryQuadratureSimplexGroupFactory
factory = InterpolatoryQuadratureSimplexGroupFactory(order)
discr = Discretization(actx, mesh, factory)
lpot_source = QBXLayerPotentialSource(
discr,
qbx_order=order, # not used in target association
fine_order=order)
places = GeometryCollection(lpot_source)
# }}}
# {{{ generate targets and check
close_circle = 0.999 * np.exp(
2j * np.pi * np.linspace(0, 1, 500, endpoint=False))
from pytential.target import PointsTarget
close_circle_target = (PointsTarget(
actx.from_numpy(np.array([close_circle.real, close_circle.imag]))))
targets = ((close_circle_target, 0), )
from pytential.qbx.target_assoc import (TargetAssociationCodeContainer,
associate_targets_to_qbx_centers,
QBXTargetAssociationFailedException
)
from pytential.qbx.utils import TreeCodeContainer
code_container = TargetAssociationCodeContainer(actx,
TreeCodeContainer(actx))
with pytest.raises(QBXTargetAssociationFailedException):
associate_targets_to_qbx_centers(places,
places.auto_source,
code_container.get_wrangler(actx),
targets,
target_association_tolerance=1e-10)
def _visualize_refinement(actx: PyOpenCLArrayContext,
discr,
niter,
stage_nr,
stage_name,
flags,
visualize=False):
if not visualize:
return
if stage_nr not in (1, 2):
raise ValueError("unexpected stage number")
flags = flags.get()
logger.info("for stage %s: splitting %d/%d stage-%d elements", stage_name,
np.sum(flags), discr.mesh.nelements, stage_nr)
from meshmode.discretization.visualization import make_visualizer
vis = make_visualizer(actx, discr, 3)
assert len(flags) == discr.mesh.nelements
flags = flags.astype(bool)
nodes_flags_template = discr.zeros(actx)
nodes_flags = []
for grp in discr.groups:
meg = grp.mesh_el_group
nodes_flags_grp = actx.to_numpy(nodes_flags_template[grp.index])
nodes_flags_grp[flags[meg.element_nr_base:meg.nelements +
meg.element_nr_base]] = 1
nodes_flags.append(actx.from_numpy(nodes_flags_grp))
nodes_flags = DOFArray(actx, tuple(nodes_flags))
vis_data = [
("refine_flags", nodes_flags),
]
if 0:
from pytential import sym, bind
bdry_normals = bind(discr, sym.normal(
discr.ambient_dim))(actx).as_vector(dtype=object)
vis_data.append(("bdry_normals", bdry_normals), )
vis.write_vtk_file(f"refinement-{stage_name}-{niter:03d}.vtu", vis_data)
def test_node_reduction(ctx_factory):
ctx = ctx_factory()
queue = cl.CommandQueue(ctx)
actx = PyOpenCLArrayContext(queue)
# {{{ build discretization
target_order = 4
nelements = 32
mesh = make_curve_mesh(starfish,
np.linspace(0.0, 1.0, nelements + 1),
target_order)
discr = Discretization(actx, mesh,
InterpolatoryQuadratureSimplexGroupFactory(target_order))
# }}}
# {{{ test
# create a shuffled [1, nelements + 1] array
ary = []
el_nr_base = 0
for grp in discr.groups:
x = 1 + np.arange(el_nr_base, grp.nelements)
np.random.shuffle(x)
ary.append(actx.freeze(actx.from_numpy(x.reshape(-1, 1))))
el_nr_base += grp.nelements
from meshmode.dof_array import DOFArray
ary = DOFArray(actx, tuple(ary))
for func, expected in [
(sym.NodeSum, nelements * (nelements + 1) // 2),
(sym.NodeMax, nelements),
(sym.NodeMin, 1),
]:
r = bind(discr, func(sym.var("x")))(actx, x=ary)
assert abs(r - expected) < 1.0e-15, r
def main(curve_fn=starfish, visualize=True):
import logging
logging.basicConfig(level=logging.WARNING) # INFO for more progress info
import pyopencl as cl
cl_ctx = cl.create_some_context()
queue = cl.CommandQueue(cl_ctx)
actx = PyOpenCLArrayContext(queue, force_device_scalars=True)
from meshmode.mesh.generation import make_curve_mesh
mesh = make_curve_mesh(
curve_fn,
np.linspace(0, 1, nelements+1),
target_order)
from pytential.qbx import QBXLayerPotentialSource
from meshmode.discretization import Discretization
from meshmode.discretization.poly_element import \
InterpolatoryQuadratureSimplexGroupFactory
pre_density_discr = Discretization(
actx, mesh, InterpolatoryQuadratureSimplexGroupFactory(target_order))
qbx = QBXLayerPotentialSource(
pre_density_discr, 4*target_order, qbx_order,
fmm_order=qbx_order+3,
target_association_tolerance=0.005,
#fmm_backend="fmmlib",
)
from pytential.target import PointsTarget
fplot = FieldPlotter(np.zeros(2), extent=5, npoints=1000)
targets_dev = actx.from_numpy(fplot.points)
from pytential import GeometryCollection
places = GeometryCollection({
"qbx": qbx,
"targets": PointsTarget(targets_dev),
}, auto_where="qbx")
density_discr = places.get_discretization("qbx")
nodes = thaw(density_discr.nodes(), actx)
angle = actx.np.arctan2(nodes[1], nodes[0])
if k:
kernel = HelmholtzKernel(2)
kernel_kwargs = {"k": sym.var("k")}
else:
kernel = LaplaceKernel(2)
kernel_kwargs = {}
def op(**kwargs):
kwargs.update(kernel_kwargs)
#op = sym.d_dx(sym.S(kernel, sym.var("sigma"), **kwargs))
return sym.D(kernel, sym.var("sigma"), **kwargs)
#op = sym.S(kernel, sym.var("sigma"), qbx_forced_limit=None, **kwargs)
if 0:
from random import randrange
sigma = actx.zeros(density_discr.ndofs, angle.entry_dtype)
for _ in range(5):
sigma[randrange(len(sigma))] = 1
from arraycontext import unflatten
sigma = unflatten(angle, sigma, actx)
else:
sigma = actx.np.cos(mode_nr*angle)
if isinstance(kernel, HelmholtzKernel):
for i, elem in np.ndenumerate(sigma):
sigma[i] = elem.astype(np.complex128)
bound_bdry_op = bind(places, op())
if visualize:
fld_in_vol = actx.to_numpy(
bind(places, op(
source="qbx",
target="targets",
qbx_forced_limit=None))(actx, sigma=sigma, k=k))
if enable_mayavi:
fplot.show_scalar_in_mayavi(fld_in_vol.real, max_val=5)
else:
fplot.write_vtk_file("layerpot-potential.vts", [
("potential", fld_in_vol)
])
if 0:
apply_op = bound_bdry_op.scipy_op(actx, "sigma", np.float64, k=k)
from sumpy.tools import build_matrix
mat = build_matrix(apply_op)
import matplotlib.pyplot as pt
pt.imshow(mat)
pt.colorbar()
pt.show()
if enable_mayavi:
# {{{ plot boundary field
from arraycontext import flatten
fld_on_bdry = actx.to_numpy(
flatten(bound_bdry_op(actx, sigma=sigma, k=k), actx))
nodes_host = actx.to_numpy(
flatten(density_discr.nodes(), actx)
).reshape(density_discr.ambient_dim, -1)
mlab.points3d(nodes_host[0], nodes_host[1],
fld_on_bdry.real, scale_factor=0.03)
mlab.colorbar()
mlab.show()
# {{{ come up with a solution to Maxwell's equations
j_sym = sym.make_sym_vector("j", 3)
jt_sym = sym.make_sym_vector("jt", 2)
rho_sym = sym.var("rho")
from pytential.symbolic.pde.maxwell import (PECChargeCurrentMFIEOperator,
get_sym_maxwell_point_source,
get_sym_maxwell_plane_wave)
mfie = PECChargeCurrentMFIEOperator()
test_source = case.get_source(actx)
calc_patch = CalculusPatch(np.array([-3, 0, 0]), h=0.01)
calc_patch_tgt = PointsTarget(actx.from_numpy(calc_patch.points))
rng = cl.clrandom.PhiloxGenerator(cl_ctx, seed=12)
from pytools.obj_array import make_obj_array
src_j = make_obj_array([
rng.normal(actx.queue, (test_source.ndofs), dtype=np.float64)
for _ in range(3)
])
def eval_inc_field_at(places, source=None, target=None):
if source is None:
source = "test_source"
if use_plane_wave:
# plane wave
return bind(places,
def main(mesh_name="ellipse", visualize=False):
import logging
logging.basicConfig(level=logging.INFO) # INFO for more progress info
cl_ctx = cl.create_some_context()
queue = cl.CommandQueue(cl_ctx)
actx = PyOpenCLArrayContext(queue)
from meshmode.mesh.generation import ellipse, make_curve_mesh
from functools import partial
if mesh_name == "ellipse":
mesh = make_curve_mesh(partial(ellipse, 1),
np.linspace(0, 1, nelements + 1), mesh_order)
elif mesh_name == "ellipse_array":
base_mesh = make_curve_mesh(partial(ellipse, 1),
np.linspace(0, 1, nelements + 1),
mesh_order)
from meshmode.mesh.processing import affine_map, merge_disjoint_meshes
nx = 2
ny = 2
dx = 2 / nx
meshes = [
affine_map(base_mesh,
A=np.diag([dx * 0.25, dx * 0.25]),
b=np.array([dx * (ix - nx / 2), dx * (iy - ny / 2)]))
for ix in range(nx) for iy in range(ny)
]
mesh = merge_disjoint_meshes(meshes, single_group=True)
if visualize:
from meshmode.mesh.visualization import draw_curve
draw_curve(mesh)
import matplotlib.pyplot as plt
plt.show()
else:
raise ValueError(f"unknown mesh name: {mesh_name}")
pre_density_discr = Discretization(
actx, mesh,
InterpolatoryQuadratureSimplexGroupFactory(bdry_quad_order))
from pytential.qbx import (QBXLayerPotentialSource,
QBXTargetAssociationFailedException)
qbx = QBXLayerPotentialSource(pre_density_discr,
fine_order=bdry_ovsmp_quad_order,
qbx_order=qbx_order,
fmm_order=fmm_order)
from sumpy.visualization import FieldPlotter
fplot = FieldPlotter(np.zeros(2), extent=5, npoints=500)
targets = actx.from_numpy(fplot.points)
from pytential import GeometryCollection
places = GeometryCollection(
{
"qbx":
qbx,
"qbx_high_target_assoc_tol":
qbx.copy(target_association_tolerance=0.05),
"targets":
PointsTarget(targets)
},
auto_where="qbx")
density_discr = places.get_discretization("qbx")
# {{{ describe bvp
from sumpy.kernel import LaplaceKernel, HelmholtzKernel
kernel = HelmholtzKernel(2)
sigma_sym = sym.var("sigma")
sqrt_w = sym.sqrt_jac_q_weight(2)
inv_sqrt_w_sigma = sym.cse(sigma_sym / sqrt_w)
# Brakhage-Werner parameter
alpha = 1j
# -1 for interior Dirichlet
# +1 for exterior Dirichlet
loc_sign = +1
k_sym = sym.var("k")
bdry_op_sym = (
-loc_sign * 0.5 * sigma_sym + sqrt_w *
(alpha * sym.S(kernel, inv_sqrt_w_sigma, k=k_sym, qbx_forced_limit=+1)
- sym.D(kernel, inv_sqrt_w_sigma, k=k_sym, qbx_forced_limit="avg")))
# }}}
bound_op = bind(places, bdry_op_sym)
# {{{ fix rhs and solve
from meshmode.dof_array import thaw
nodes = thaw(actx, density_discr.nodes())
k_vec = np.array([2, 1])
k_vec = k * k_vec / la.norm(k_vec, 2)
def u_incoming_func(x):
return actx.np.exp(1j * (x[0] * k_vec[0] + x[1] * k_vec[1]))
bc = -u_incoming_func(nodes)
bvp_rhs = bind(places, sqrt_w * sym.var("bc"))(actx, bc=bc)
from pytential.solve import gmres
gmres_result = gmres(bound_op.scipy_op(actx,
sigma_sym.name,
dtype=np.complex128,
k=k),
bvp_rhs,
tol=1e-8,
progress=True,
stall_iterations=0,
hard_failure=True)
# }}}
# {{{ postprocess/visualize
repr_kwargs = dict(source="qbx_high_target_assoc_tol",
target="targets",
qbx_forced_limit=None)
representation_sym = (
alpha * sym.S(kernel, inv_sqrt_w_sigma, k=k_sym, **repr_kwargs) -
sym.D(kernel, inv_sqrt_w_sigma, k=k_sym, **repr_kwargs))
u_incoming = u_incoming_func(targets)
ones_density = density_discr.zeros(actx)
for elem in ones_density:
elem.fill(1)
indicator = actx.to_numpy(
bind(places, sym.D(LaplaceKernel(2), sigma_sym,
**repr_kwargs))(actx, sigma=ones_density))
try:
fld_in_vol = actx.to_numpy(
bind(places, representation_sym)(actx,
sigma=gmres_result.solution,
k=k))
except QBXTargetAssociationFailedException as e:
fplot.write_vtk_file("helmholtz-dirichlet-failed-targets.vts",
[("failed", e.failed_target_flags.get(queue))])
raise
#fplot.show_scalar_in_mayavi(fld_in_vol.real, max_val=5)
fplot.write_vtk_file("helmholtz-dirichlet-potential.vts", [
("potential", fld_in_vol),
("indicator", indicator),
("u_incoming", actx.to_numpy(u_incoming)),
])
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
elif ambient_dim == 3:
from meshmode.discretization.visualization import make_visualizer
marker = np.empty(density_discr.ndofs)
for i in range(srcindices.nblocks):
isrc = srcindices.block_indices(i)
inbr = nbrindices.block_indices(i)
marker.fill(0.0)
marker[srcindices.indices] = 0.0
marker[isrc] = -42.0
marker[inbr] = +42.0
from meshmode.dof_array import unflatten
marker_dev = unflatten(actx, density_discr,
actx.from_numpy(marker))
vis = make_visualizer(actx, density_discr, 10)
filename = "test_area_query_{}d_{:04}.vtu".format(
ambient_dim, i)
vis.write_vtk_file(filename, [
("marker", marker_dev),
])
# }}}
if __name__ == "__main__":
import sys
if len(sys.argv) > 1:
exec(sys.argv[1])
else:
def test_interpolation(ctx_factory, name, source_discr_stage,
target_granularity):
ctx = ctx_factory()
queue = cl.CommandQueue(ctx)
actx = PyOpenCLArrayContext(queue)
nelements = 32
target_order = 7
qbx_order = 4
where = sym.as_dofdesc("test_interpolation")
from_dd = sym.DOFDescriptor(geometry=where.geometry,
discr_stage=source_discr_stage,
granularity=sym.GRANULARITY_NODE)
to_dd = sym.DOFDescriptor(geometry=where.geometry,
discr_stage=sym.QBX_SOURCE_QUAD_STAGE2,
granularity=target_granularity)
mesh = make_curve_mesh(starfish, np.linspace(0.0, 1.0, nelements + 1),
target_order)
discr = Discretization(
actx, mesh, InterpolatoryQuadratureSimplexGroupFactory(target_order))
from pytential.qbx import QBXLayerPotentialSource
qbx = QBXLayerPotentialSource(discr,
fine_order=4 * target_order,
qbx_order=qbx_order,
fmm_order=False)
from pytential import GeometryCollection
places = GeometryCollection(qbx, auto_where=where)
sigma_sym = sym.var("sigma")
op_sym = sym.sin(sym.interp(from_dd, to_dd, sigma_sym))
bound_op = bind(places, op_sym, auto_where=where)
from meshmode.dof_array import thaw, flatten, unflatten
def discr_and_nodes(stage):
density_discr = places.get_discretization(where.geometry, stage)
return density_discr, np.array([
actx.to_numpy(flatten(axis))
for axis in thaw(actx, density_discr.nodes())
])
_, target_nodes = discr_and_nodes(sym.QBX_SOURCE_QUAD_STAGE2)
source_discr, source_nodes = discr_and_nodes(source_discr_stage)
sigma_target = np.sin(la.norm(target_nodes, axis=0))
sigma_dev = unflatten(actx, source_discr,
actx.from_numpy(la.norm(source_nodes, axis=0)))
sigma_target_interp = actx.to_numpy(
flatten(bound_op(actx, sigma=sigma_dev)))
if name in ("default", "default_explicit", "stage2", "quad"):
error = la.norm(sigma_target_interp -
sigma_target) / la.norm(sigma_target)
assert error < 1.0e-10
elif name in ("stage2_center", ):
assert len(sigma_target_interp) == 2 * len(sigma_target)
else:
raise ValueError(f"unknown test case name: {name}")
if d == 2:
u = np.log(dist)
grad_u = diff/dist_squared
elif d == 3:
u = 1/dist
grad_u = -diff/dist**3
else:
assert False
dn_u = 0
for i in range(d):
dn_u = dn_u + normal_host[i]*grad_u[i]
# }}}
u_dev = unflatten(actx, density_discr, actx.from_numpy(u))
dn_u_dev = unflatten(actx, density_discr, actx.from_numpy(dn_u))
from pytools.obj_array import make_obj_array, obj_array_vectorize
grad_u_dev = unflatten(actx, density_discr,
obj_array_vectorize(actx.from_numpy, make_obj_array(grad_u)))
key = (case.qbx_order, case.geometry.mesh_name, resolution,
case.expr.zero_op_name)
bound_op = bind(places, case.expr.get_zero_op(k_sym, **knl_kwargs))
error = bound_op(
actx, u=u_dev, dn_u=dn_u_dev, grad_u=grad_u_dev, k=case.k)
if 0:
pt.plot(error)
pt.show()
def timing_run(nx, ny, visualize=False):
import logging
logging.basicConfig(level=logging.WARNING) # INFO for more progress info
cl_ctx = cl.create_some_context()
queue = cl.CommandQueue(cl_ctx)
actx = PyOpenCLArrayContext(queue)
mesh = make_mesh(nx=nx, ny=ny, visualize=visualize)
density_discr = Discretization(
actx, mesh,
InterpolatoryQuadratureSimplexGroupFactory(bdry_quad_order))
from pytential.qbx import (QBXLayerPotentialSource,
QBXTargetAssociationFailedException)
qbx = QBXLayerPotentialSource(density_discr,
fine_order=bdry_ovsmp_quad_order,
qbx_order=qbx_order,
fmm_order=fmm_order)
places = {"qbx": qbx}
if visualize:
from sumpy.visualization import FieldPlotter
fplot = FieldPlotter(np.zeros(2), extent=5, npoints=1500)
targets = PointsTarget(actx.from_numpy(fplot.points))
places.update({
"plot-targets":
targets,
"qbx-indicator":
qbx.copy(target_association_tolerance=0.05,
fmm_level_to_order=lambda lev: 7,
qbx_order=2),
"qbx-target-assoc":
qbx.copy(target_association_tolerance=0.1)
})
from pytential import GeometryCollection
places = GeometryCollection(places, auto_where="qbx")
density_discr = places.get_discretization("qbx")
# {{{ describe bvp
from sumpy.kernel import HelmholtzKernel
kernel = HelmholtzKernel(2)
sigma_sym = sym.var("sigma")
sqrt_w = sym.sqrt_jac_q_weight(2)
inv_sqrt_w_sigma = sym.cse(sigma_sym / sqrt_w)
# Brakhage-Werner parameter
alpha = 1j
# -1 for interior Dirichlet
# +1 for exterior Dirichlet
loc_sign = +1
k_sym = sym.var("k")
S_sym = sym.S(kernel, inv_sqrt_w_sigma, k=k_sym, qbx_forced_limit=+1)
D_sym = sym.D(kernel, inv_sqrt_w_sigma, k=k_sym, qbx_forced_limit="avg")
bdry_op_sym = -loc_sign * 0.5 * sigma_sym + sqrt_w * (alpha * S_sym +
D_sym)
# }}}
bound_op = bind(places, bdry_op_sym)
# {{{ fix rhs and solve
mode_nr = 3
from meshmode.dof_array import thaw
nodes = thaw(actx, density_discr.nodes())
angle = actx.np.arctan2(nodes[1], nodes[0])
sigma = actx.np.cos(mode_nr * angle)
# }}}
# {{{ postprocess/visualize
repr_kwargs = dict(k=sym.var("k"), qbx_forced_limit=+1)
sym_op = sym.S(kernel, sym.var("sigma"), **repr_kwargs)
bound_op = bind(places, sym_op)
print("FMM WARM-UP RUN 1: %5d elements" % mesh.nelements)
bound_op(actx, sigma=sigma, k=k)
queue.finish()
print("FMM WARM-UP RUN 2: %5d elements" % mesh.nelements)
bound_op(actx, sigma=sigma, k=k)
queue.finish()
from time import time
t_start = time()
bound_op(actx, sigma=sigma, k=k)
actx.queue.finish()
elapsed = time() - t_start
print("FMM TIMING RUN: %5d elements -> %g s" %
(mesh.nelements, elapsed))
if visualize:
ones_density = density_discr.zeros(queue)
ones_density.fill(1)
indicator = bind(places,
sym_op,
auto_where=("qbx-indicator", "plot-targets"))(
queue, sigma=ones_density).get()
try:
fld_in_vol = bind(places,
sym_op,
auto_where=("qbx-target-assoc",
"plot-targets"))(queue,
sigma=sigma,
k=k).get()
except QBXTargetAssociationFailedException as e:
fplot.write_vtk_file("scaling-study-failed-targets.vts", [
("failed", e.failed_target_flags.get(queue)),
])
raise
fplot.write_vtk_file("scaling-study-potential.vts", [
("potential", fld_in_vol),
("indicator", indicator),
])
return (mesh.nelements, elapsed)
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 main(mesh_name="torus", visualize=False):
import logging
logging.basicConfig(level=logging.WARNING) # INFO for more progress info
cl_ctx = cl.create_some_context()
queue = cl.CommandQueue(cl_ctx)
actx = PyOpenCLArrayContext(queue)
if mesh_name == "torus":
rout = 10
rin = 1
from meshmode.mesh.generation import generate_torus
base_mesh = generate_torus(
rout, rin, 40, 4,
mesh_order)
from meshmode.mesh.processing import affine_map, merge_disjoint_meshes
# nx = 1
# ny = 1
nz = 1
dz = 0
meshes = [
affine_map(
base_mesh,
A=np.diag([1, 1, 1]),
b=np.array([0, 0, iz*dz]))
for iz in range(nz)]
mesh = merge_disjoint_meshes(meshes, single_group=True)
if visualize:
from meshmode.mesh.visualization import draw_curve
draw_curve(mesh)
import matplotlib.pyplot as plt
plt.show()
else:
raise ValueError(f"unknown mesh name: {mesh_name}")
pre_density_discr = Discretization(
actx, mesh,
InterpolatoryQuadratureSimplexGroupFactory(bdry_quad_order))
from pytential.qbx import (
QBXLayerPotentialSource, QBXTargetAssociationFailedException)
qbx = QBXLayerPotentialSource(
pre_density_discr, fine_order=bdry_ovsmp_quad_order, qbx_order=qbx_order,
fmm_order=fmm_order,
)
from sumpy.visualization import FieldPlotter
fplot = FieldPlotter(np.zeros(3), extent=20, npoints=50)
targets = actx.from_numpy(fplot.points)
from pytential import GeometryCollection
places = GeometryCollection({
"qbx": qbx,
"qbx_target_assoc": qbx.copy(target_association_tolerance=0.2),
"targets": PointsTarget(targets)
}, auto_where="qbx")
density_discr = places.get_discretization("qbx")
# {{{ describe bvp
from sumpy.kernel import LaplaceKernel
kernel = LaplaceKernel(3)
sigma_sym = sym.var("sigma")
#sqrt_w = sym.sqrt_jac_q_weight(3)
sqrt_w = 1
inv_sqrt_w_sigma = sym.cse(sigma_sym/sqrt_w)
# -1 for interior Dirichlet
# +1 for exterior Dirichlet
loc_sign = +1
bdry_op_sym = (loc_sign*0.5*sigma_sym
+ sqrt_w*(
sym.S(kernel, inv_sqrt_w_sigma, qbx_forced_limit=+1)
+ sym.D(kernel, inv_sqrt_w_sigma, qbx_forced_limit="avg")
))
# }}}
bound_op = bind(places, bdry_op_sym)
# {{{ fix rhs and solve
from meshmode.dof_array import thaw, flatten, unflatten
nodes = thaw(actx, density_discr.nodes())
source = np.array([rout, 0, 0])
def u_incoming_func(x):
from pytools.obj_array import obj_array_vectorize
x = obj_array_vectorize(actx.to_numpy, flatten(x))
x = np.array(list(x))
# return 1/cl.clmath.sqrt( (x[0] - source[0])**2
# +(x[1] - source[1])**2
# +(x[2] - source[2])**2 )
return 1.0/la.norm(x - source[:, None], axis=0)
bc = unflatten(actx,
density_discr,
actx.from_numpy(u_incoming_func(nodes)))
bvp_rhs = bind(places, sqrt_w*sym.var("bc"))(actx, bc=bc)
from pytential.solve import gmres
gmres_result = gmres(
bound_op.scipy_op(actx, "sigma", dtype=np.float64),
bvp_rhs, tol=1e-14, progress=True,
stall_iterations=0,
hard_failure=True)
sigma = bind(places, sym.var("sigma")/sqrt_w)(
actx, sigma=gmres_result.solution)
# }}}
from meshmode.discretization.visualization import make_visualizer
bdry_vis = make_visualizer(actx, density_discr, 20)
bdry_vis.write_vtk_file("laplace.vtu", [
("sigma", sigma),
])
# {{{ postprocess/visualize
repr_kwargs = dict(
source="qbx_target_assoc",
target="targets",
qbx_forced_limit=None)
representation_sym = (
sym.S(kernel, inv_sqrt_w_sigma, **repr_kwargs)
+ sym.D(kernel, inv_sqrt_w_sigma, **repr_kwargs))
try:
fld_in_vol = actx.to_numpy(
bind(places, representation_sym)(actx, sigma=sigma))
except QBXTargetAssociationFailedException as e:
fplot.write_vtk_file("laplace-dirichlet-3d-failed-targets.vts", [
("failed", e.failed_target_flags.get(queue)),
])
raise
#fplot.show_scalar_in_mayavi(fld_in_vol.real, max_val=5)
fplot.write_vtk_file("laplace-dirichlet-3d-potential.vts", [
("potential", fld_in_vol),
])
def main():
import logging
logging.basicConfig(level=logging.WARNING) # INFO for more progress info
import pyopencl as cl
cl_ctx = cl.create_some_context()
queue = cl.CommandQueue(cl_ctx)
actx = PyOpenCLArrayContext(queue, force_device_scalars=True)
target_order = 16
qbx_order = 3
nelements = 60
mode_nr = 0
k = 0
if k:
kernel = HelmholtzKernel(2)
else:
kernel = LaplaceKernel(2)
mesh = make_curve_mesh(
#lambda t: ellipse(1, t),
starfish,
np.linspace(0, 1, nelements + 1),
target_order)
from pytential.qbx import QBXLayerPotentialSource
from meshmode.discretization import Discretization
from meshmode.discretization.poly_element import \
InterpolatoryQuadratureSimplexGroupFactory
pre_density_discr = Discretization(
actx, mesh, InterpolatoryQuadratureSimplexGroupFactory(target_order))
unaccel_qbx = QBXLayerPotentialSource(
pre_density_discr,
fine_order=2 * target_order,
qbx_order=qbx_order,
fmm_order=False,
target_association_tolerance=.05,
)
from pytential.target import PointsTarget
fplot = FieldPlotter(np.zeros(2), extent=5, npoints=600)
from pytential import GeometryCollection
places = GeometryCollection(
{
"unaccel_qbx": unaccel_qbx,
"qbx": unaccel_qbx.copy(fmm_order=10),
"targets": PointsTarget(actx.freeze(actx.from_numpy(fplot.points)))
},
auto_where=("qbx", "targets"))
density_discr = places.get_discretization("unaccel_qbx")
nodes = thaw(density_discr.nodes(), actx)
angle = actx.np.arctan2(nodes[1], nodes[0])
from pytential import bind, sym
if k:
kernel_kwargs = {"k": sym.var("k")}
else:
kernel_kwargs = {}
def get_op():
kwargs = dict(qbx_forced_limit=None)
kwargs.update(kernel_kwargs)
# return sym.d_dx(2, sym.S(kernel, sym.var("sigma"), **kwargs))
# return sym.D(kernel, sym.var("sigma"), **kwargs)
return sym.S(kernel, sym.var("sigma"), **kwargs)
op = get_op()
sigma = actx.np.cos(mode_nr * angle)
if isinstance(kernel, HelmholtzKernel):
for i, elem in np.ndenumerate(sigma):
sigma[i] = elem.astype(np.complex128)
fld_in_vol = actx.to_numpy(
bind(places, op, auto_where=("unaccel_qbx", "targets"))(actx,
sigma=sigma,
k=k))
fmm_fld_in_vol = actx.to_numpy(
bind(places, op, auto_where=("qbx", "targets"))(actx, sigma=sigma,
k=k))
err = fmm_fld_in_vol - fld_in_vol
try:
import matplotlib
except ImportError:
return
matplotlib.use("Agg")
im = fplot.show_scalar_in_matplotlib(np.log10(np.abs(err) + 1e-17))
from matplotlib.colors import Normalize
im.set_norm(Normalize(vmin=-12, vmax=0))
import matplotlib.pyplot as pt
from matplotlib.ticker import NullFormatter
pt.gca().xaxis.set_major_formatter(NullFormatter())
pt.gca().yaxis.set_major_formatter(NullFormatter())
cb = pt.colorbar(shrink=0.9)
cb.set_label(r"$\log_{10}(\mathrm{Error})$")
pt.savefig("fmm-error-order-%d.pdf" % qbx_order)
def main(mesh_name="ellipsoid"):
import logging
logger = logging.getLogger(__name__)
logging.basicConfig(level=logging.WARNING) # INFO for more progress info
import pyopencl as cl
cl_ctx = cl.create_some_context()
queue = cl.CommandQueue(cl_ctx)
actx = PyOpenCLArrayContext(queue, force_device_scalars=True)
if mesh_name == "ellipsoid":
cad_file_name = "geometries/ellipsoid.step"
h = 0.6
elif mesh_name == "two-cylinders":
cad_file_name = "geometries/two-cylinders-smooth.step"
h = 0.4
else:
raise ValueError("unknown mesh name: %s" % mesh_name)
from meshmode.mesh.io import generate_gmsh, FileSource
mesh = generate_gmsh(
FileSource(cad_file_name),
2,
order=2,
other_options=["-string",
"Mesh.CharacteristicLengthMax = %g;" % h],
target_unit="MM")
from meshmode.mesh.processing import perform_flips
# Flip elements--gmsh generates inside-out geometry.
mesh = perform_flips(mesh, np.ones(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(
actx, mesh, InterpolatoryQuadratureSimplexGroupFactory(target_order))
qbx = QBXLayerPotentialSource(density_discr,
4 * target_order,
qbx_order,
fmm_order=qbx_order + 3,
target_association_tolerance=0.15)
from pytential.target import PointsTarget
fplot = FieldPlotter(bbox_center, extent=3.5 * bbox_size, npoints=150)
from pytential import GeometryCollection
places = GeometryCollection(
{
"qbx": qbx,
"targets": PointsTarget(actx.from_numpy(fplot.points))
},
auto_where="qbx")
density_discr = places.get_discretization("qbx")
nodes = thaw(density_discr.nodes(), actx)
angle = actx.np.arctan2(nodes[1], nodes[0])
if k:
kernel = HelmholtzKernel(3)
else:
kernel = LaplaceKernel(3)
#op = sym.d_dx(sym.S(kernel, sym.var("sigma"), qbx_forced_limit=None))
op = sym.D(kernel, sym.var("sigma"), qbx_forced_limit=None)
#op = sym.S(kernel, sym.var("sigma"), qbx_forced_limit=None)
if 0:
from random import randrange
sigma = actx.zeros(density_discr.ndofs, angle.entry_dtype)
for _ in range(5):
sigma[randrange(len(sigma))] = 1
from arraycontext import unflatten
sigma = unflatten(angle, sigma, actx)
else:
sigma = actx.np.cos(mode_nr * angle)
if isinstance(kernel, HelmholtzKernel):
for i, elem in np.ndenumerate(sigma):
sigma[i] = elem.astype(np.complex128)
fld_in_vol = actx.to_numpy(
bind(places, op, auto_where=("qbx", "targets"))(actx, sigma=sigma,
k=k))
#fplot.show_scalar_in_mayavi(fld_in_vol.real, max_val=5)
fplot.write_vtk_file("layerpot-3d-potential.vts",
[("potential", fld_in_vol)])
bdry_normals = bind(places, sym.normal(
density_discr.ambient_dim))(actx).as_vector(dtype=object)
from meshmode.discretization.visualization import make_visualizer
bdry_vis = make_visualizer(actx, density_discr, target_order)
bdry_vis.write_vtk_file("layerpot-3d-density.vtu", [
("sigma", sigma),
("bdry_normals", bdry_normals),
])
def run_exterior_stokes_2d(
ctx_factory,
nelements,
mesh_order=4,
target_order=4,
qbx_order=4,
fmm_order=False, # FIXME: FMM is slower than direct eval
mu=1,
circle_rad=1.5,
visualize=False):
# This program tests an exterior Stokes flow in 2D using the
# compound representation given in Hsiao & Kress,
# ``On an integral equation for the two-dimensional exterior Stokes problem,''
# Applied Numerical Mathematics 1 (1985).
logging.basicConfig(level=logging.INFO)
cl_ctx = cl.create_some_context()
queue = cl.CommandQueue(cl_ctx)
actx = PyOpenCLArrayContext(queue)
ovsmp_target_order = 4 * target_order
# {{{ geometries
from meshmode.mesh.generation import ( # noqa
make_curve_mesh, starfish, ellipse, drop)
mesh = make_curve_mesh(lambda t: circle_rad * ellipse(1, t),
np.linspace(0, 1, nelements + 1), target_order)
coarse_density_discr = Discretization(
actx, mesh, InterpolatoryQuadratureSimplexGroupFactory(target_order))
from pytential.qbx import QBXLayerPotentialSource
target_association_tolerance = 0.05
qbx = QBXLayerPotentialSource(
coarse_density_discr,
fine_order=ovsmp_target_order,
qbx_order=qbx_order,
fmm_order=fmm_order,
target_association_tolerance=target_association_tolerance,
_expansions_in_tree_have_extent=True,
)
def circle_mask(test_points, radius):
return (test_points[0, :]**2 + test_points[1, :]**2 > radius**2)
def outside_circle(test_points, radius):
mask = circle_mask(test_points, radius)
return np.array([row[mask] for row in test_points])
from pytential.target import PointsTarget
nsamp = 30
eval_points_1d = np.linspace(-3., 3., nsamp)
eval_points = np.zeros((2, len(eval_points_1d)**2))
eval_points[0, :] = np.tile(eval_points_1d, len(eval_points_1d))
eval_points[1, :] = np.repeat(eval_points_1d, len(eval_points_1d))
eval_points = outside_circle(eval_points, radius=circle_rad)
point_targets = PointsTarget(eval_points)
fplot = FieldPlotter(np.zeros(2), extent=6, npoints=100)
plot_targets = PointsTarget(outside_circle(fplot.points,
radius=circle_rad))
places = GeometryCollection({
sym.DEFAULT_SOURCE: qbx,
sym.DEFAULT_TARGET: qbx.density_discr,
"point_target": point_targets,
"plot_target": plot_targets,
})
density_discr = places.get_discretization(sym.DEFAULT_SOURCE)
normal = bind(places, sym.normal(2).as_vector())(actx)
path_length = bind(places, sym.integral(2, 1, 1))(actx)
# }}}
# {{{ describe bvp
from pytential.symbolic.stokes import StressletWrapper, StokesletWrapper
dim = 2
cse = sym.cse
sigma_sym = sym.make_sym_vector("sigma", dim)
meanless_sigma_sym = cse(sigma_sym - sym.mean(2, 1, sigma_sym))
int_sigma = sym.Ones() * sym.integral(2, 1, sigma_sym)
nvec_sym = sym.make_sym_vector("normal", dim)
mu_sym = sym.var("mu")
# -1 for interior Dirichlet
# +1 for exterior Dirichlet
loc_sign = 1
stresslet_obj = StressletWrapper(dim=2)
stokeslet_obj = StokesletWrapper(dim=2)
bdry_op_sym = (-loc_sign * 0.5 * sigma_sym - stresslet_obj.apply(
sigma_sym, nvec_sym, mu_sym, qbx_forced_limit="avg") +
stokeslet_obj.apply(
meanless_sigma_sym, mu_sym, qbx_forced_limit="avg") -
(0.5 / np.pi) * int_sigma)
# }}}
bound_op = bind(places, bdry_op_sym)
# {{{ fix rhs and solve
def fund_soln(x, y, loc, strength):
#with direction (1,0) for point source
r = actx.np.sqrt((x - loc[0])**2 + (y - loc[1])**2)
scaling = strength / (4 * np.pi * mu)
xcomp = (-actx.np.log(r) + (x - loc[0])**2 / r**2) * scaling
ycomp = ((x - loc[0]) * (y - loc[1]) / r**2) * scaling
return [xcomp, ycomp]
def rotlet_soln(x, y, loc):
r = actx.np.sqrt((x - loc[0])**2 + (y - loc[1])**2)
xcomp = -(y - loc[1]) / r**2
ycomp = (x - loc[0]) / r**2
return [xcomp, ycomp]
def fund_and_rot_soln(x, y, loc, strength):
#with direction (1,0) for point source
r = actx.np.sqrt((x - loc[0])**2 + (y - loc[1])**2)
scaling = strength / (4 * np.pi * mu)
xcomp = ((-actx.np.log(r) + (x - loc[0])**2 / r**2) * scaling -
(y - loc[1]) * strength * 0.125 / r**2 + 3.3)
ycomp = (((x - loc[0]) * (y - loc[1]) / r**2) * scaling +
(x - loc[0]) * strength * 0.125 / r**2 + 1.5)
return make_obj_array([xcomp, ycomp])
from meshmode.dof_array import unflatten, flatten, thaw
nodes = flatten(thaw(actx, density_discr.nodes()))
fund_soln_loc = np.array([0.5, -0.2])
strength = 100.
bc = unflatten(
actx, density_discr,
fund_and_rot_soln(nodes[0], nodes[1], fund_soln_loc, strength))
omega_sym = sym.make_sym_vector("omega", dim)
u_A_sym_bdry = stokeslet_obj.apply( # noqa: N806
omega_sym, mu_sym, qbx_forced_limit=1)
from pytential.utils import unflatten_from_numpy
omega = unflatten_from_numpy(
actx, density_discr,
make_obj_array([(strength / path_length) * np.ones(len(nodes[0])),
np.zeros(len(nodes[0]))]))
bvp_rhs = bind(places,
sym.make_sym_vector("bc", dim) + u_A_sym_bdry)(actx,
bc=bc,
mu=mu,
omega=omega)
gmres_result = gmres(bound_op.scipy_op(actx,
"sigma",
np.float64,
mu=mu,
normal=normal),
bvp_rhs,
x0=bvp_rhs,
tol=1e-9,
progress=True,
stall_iterations=0,
hard_failure=True)
# }}}
# {{{ postprocess/visualize
sigma = gmres_result.solution
sigma_int_val_sym = sym.make_sym_vector("sigma_int_val", 2)
int_val = bind(places, sym.integral(2, 1, sigma_sym))(actx, sigma=sigma)
int_val = -int_val / (2 * np.pi)
print("int_val = ", int_val)
u_A_sym_vol = stokeslet_obj.apply( # noqa: N806
omega_sym, mu_sym, qbx_forced_limit=2)
representation_sym = (
-stresslet_obj.apply(sigma_sym, nvec_sym, mu_sym, qbx_forced_limit=2) +
stokeslet_obj.apply(meanless_sigma_sym, mu_sym, qbx_forced_limit=2) -
u_A_sym_vol + sigma_int_val_sym)
where = (sym.DEFAULT_SOURCE, "point_target")
vel = bind(places, representation_sym,
auto_where=where)(actx,
sigma=sigma,
mu=mu,
normal=normal,
sigma_int_val=int_val,
omega=omega)
print("@@@@@@@@")
plot_vel = bind(places,
representation_sym,
auto_where=(sym.DEFAULT_SOURCE,
"plot_target"))(actx,
sigma=sigma,
mu=mu,
normal=normal,
sigma_int_val=int_val,
omega=omega)
def get_obj_array(obj_array):
return make_obj_array([ary.get() for ary in obj_array])
exact_soln = fund_and_rot_soln(actx.from_numpy(eval_points[0]),
actx.from_numpy(eval_points[1]),
fund_soln_loc, strength)
vel = get_obj_array(vel)
err = vel - get_obj_array(exact_soln)
# FIXME: Pointwise relative errors don't make sense!
rel_err = err / (get_obj_array(exact_soln))
if 0:
print("@@@@@@@@")
print("vel[0], err[0], rel_err[0] ***** vel[1], err[1], rel_err[1]: ")
for i in range(len(vel[0])):
print(
"{:15.8e}, {:15.8e}, {:15.8e} ***** {:15.8e}, {:15.8e}, {:15.8e}"
.format(vel[0][i], err[0][i], rel_err[0][i], vel[1][i],
err[1][i], rel_err[1][i]))
print("@@@@@@@@")
l2_err = np.sqrt((6. / (nsamp - 1))**2 * np.sum(err[0] * err[0]) +
(6. / (nsamp - 1))**2 * np.sum(err[1] * err[1]))
l2_rel_err = np.sqrt((6. /
(nsamp - 1))**2 * np.sum(rel_err[0] * rel_err[0]) +
(6. /
(nsamp - 1))**2 * np.sum(rel_err[1] * rel_err[1]))
print("L2 error estimate: ", l2_err)
print("L2 rel error estimate: ", l2_rel_err)
print("max error at sampled points: ", max(abs(err[0])), max(abs(err[1])))
print("max rel error at sampled points: ", max(abs(rel_err[0])),
max(abs(rel_err[1])))
if visualize:
import matplotlib.pyplot as plt
full_pot = np.zeros_like(fplot.points) * float("nan")
mask = circle_mask(fplot.points, radius=circle_rad)
for i, vel in enumerate(plot_vel):
full_pot[i, mask] = vel.get()
plt.quiver(fplot.points[0],
fplot.points[1],
full_pot[0],
full_pot[1],
linewidth=0.1)
plt.savefig("exterior-2d-field.pdf")
# }}}
h_max = bind(places, sym.h_max(qbx.ambient_dim))(actx)
return h_max, l2_err
