def discr_rel_error(actx, discr, x, xref, p=None):
from pytential import norm
ref_norm = actx.to_numpy(norm(discr, xref, p=p))
if ref_norm < 1.0e-14:
ref_norm = 1
diff_norm = actx.to_numpy(norm(discr, x - xref, p=p))
return diff_norm / ref_norm
stall_iterations=0,
hard_failure=True)
result = bind(places, sym.real(sym_result))(
actx, sigma=actx.np.real(result.solution),
**solution.context)
ref_result = solution.exact(actx, density_discr)
# }}}
from pytential import norm
h_max = actx.to_numpy(
bind(places, sym.h_max(places.ambient_dim))(actx)
)
error = actx.to_numpy(
norm(density_discr, result - ref_result, p=2)
/ norm(density_discr, ref_result, p=2)
)
eoc.add_data_point(h_max, error)
logger.info("resolution %3d h_max %.5e rel_error %.5e",
resolution, h_max, error)
if not visualize:
continue
from meshmode.discretization.visualization import make_visualizer
vis = make_visualizer(actx, density_discr, case.target_order)
filename = f"beltrami_{case.name}_{solution.name}_{resolution}.vtu"
vis.write_vtk_file(filename, [
def test_sanity_balls(ctx_getter, src_file, dim, mesh_order,
visualize=False):
pytest.importorskip("pytential")
logging.basicConfig(level=logging.INFO)
ctx = ctx_getter()
queue = cl.CommandQueue(ctx)
from pytools.convergence import EOCRecorder
vol_eoc_rec = EOCRecorder()
surf_eoc_rec = EOCRecorder()
# overkill
quad_order = mesh_order
from pytential import bind, sym
for h in [0.2, 0.14, 0.1]:
from meshmode.mesh.io import generate_gmsh, FileSource
mesh = generate_gmsh(
FileSource(src_file), dim, order=mesh_order,
other_options=["-string", "Mesh.CharacteristicLengthMax = %g;" % h],
force_ambient_dim=dim)
logger.info("%d elements" % mesh.nelements)
# {{{ discretizations and connections
from meshmode.discretization import Discretization
from meshmode.discretization.poly_element import \
InterpolatoryQuadratureSimplexGroupFactory
vol_discr = Discretization(ctx, mesh,
InterpolatoryQuadratureSimplexGroupFactory(quad_order))
from meshmode.discretization.connection import make_boundary_restriction
bdry_mesh, bdry_discr, bdry_connection = make_boundary_restriction(
queue, vol_discr,
InterpolatoryQuadratureSimplexGroupFactory(quad_order))
# }}}
# {{{ visualizers
from meshmode.discretization.visualization import make_visualizer
vol_vis = make_visualizer(queue, vol_discr, 20)
bdry_vis = make_visualizer(queue, bdry_discr, 20)
# }}}
from math import gamma
true_surf = 2*np.pi**(dim/2)/gamma(dim/2)
true_vol = true_surf/dim
vol_x = vol_discr.nodes().with_queue(queue)
vol_one = vol_x[0].copy()
vol_one.fill(1)
from pytential import norm, integral # noqa
comp_vol = integral(vol_discr, queue, vol_one)
rel_vol_err = abs(true_vol - comp_vol) / true_vol
vol_eoc_rec.add_data_point(h, rel_vol_err)
print("VOL", true_vol, comp_vol)
bdry_x = bdry_discr.nodes().with_queue(queue)
bdry_one_exact = bdry_x[0].copy()
bdry_one_exact.fill(1)
bdry_one = bdry_connection(queue, vol_one).with_queue(queue)
intp_err = norm(bdry_discr, queue, bdry_one-bdry_one_exact)
assert intp_err < 1e-14
comp_surf = integral(bdry_discr, queue, bdry_one)
rel_surf_err = abs(true_surf - comp_surf) / true_surf
surf_eoc_rec.add_data_point(h, rel_surf_err)
print("SURF", true_surf, comp_surf)
if visualize:
vol_vis.write_vtk_file("volume-h=%g.vtu" % h, [
("f", vol_one),
("area_el", bind(vol_discr, sym.area_element())(queue)),
])
bdry_vis.write_vtk_file("boundary-h=%g.vtu" % h, [("f", bdry_one)])
# {{{ check normals point outward
normal_outward_check = bind(bdry_discr,
sym.normal() | sym.Nodes(),
)(queue).as_scalar() > 0
assert normal_outward_check.get().all(), normal_outward_check.get()
# }}}
print("---------------------------------")
print("VOLUME")
print("---------------------------------")
print(vol_eoc_rec)
assert vol_eoc_rec.order_estimate() >= mesh_order
print("---------------------------------")
print("SURFACE")
print("---------------------------------")
print(surf_eoc_rec)
assert surf_eoc_rec.order_estimate() >= mesh_order
def scat_norm(f):
return norm(qbx, queue, f, p=np.inf)
def scat_norm(f):
return norm(qbx, queue, f, p=np.inf)
def test_ellipse_eigenvalues(ctx_factory,
ellipse_aspect,
mode_nr,
qbx_order,
force_direct,
visualize=False):
logging.basicConfig(level=logging.INFO)
print("ellipse_aspect: %s, mode_nr: %d, qbx_order: %d" %
(ellipse_aspect, mode_nr, qbx_order))
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()
if ellipse_aspect != 1:
nelements_values = [60, 100, 150, 200]
else:
nelements_values = [30, 70]
# See
#
# [1] G. J. Rodin and O. Steinbach, "Boundary Element Preconditioners
# for Problems Defined on Slender Domains", SIAM Journal on Scientific
# Computing, Vol. 24, No. 4, pg. 1450, 2003.
# https://dx.doi.org/10.1137/S1064827500372067
for nelements in nelements_values:
mesh = make_curve_mesh(partial(ellipse, ellipse_aspect),
np.linspace(0, 1, nelements + 1), target_order)
fmm_order = 12
if force_direct:
fmm_order = False
pre_density_discr = Discretization(
actx, mesh,
InterpolatoryQuadratureSimplexGroupFactory(target_order))
qbx = QBXLayerPotentialSource(
pre_density_discr,
4 * target_order,
qbx_order,
fmm_order=fmm_order,
_expansions_in_tree_have_extent=True,
)
places = GeometryCollection(qbx)
density_discr = places.get_discretization(places.auto_source.geometry)
from meshmode.dof_array import thaw, flatten
nodes = thaw(actx, density_discr.nodes())
if visualize:
# plot geometry, centers, normals
centers = bind(places, sym.expansion_centers(qbx.ambient_dim,
+1))(actx)
normals = bind(places,
sym.normal(qbx.ambient_dim))(actx).as_vector(object)
nodes_h = np.array(
[actx.to_numpy(axis) for axis in flatten(nodes)])
centers_h = np.array(
[actx.to_numpy(axis) for axis in flatten(centers)])
normals_h = np.array(
[actx.to_numpy(axis) for axis in flatten(normals)])
pt.plot(nodes_h[0], nodes_h[1], "x-")
pt.plot(centers_h[0], centers_h[1], "o")
pt.quiver(nodes_h[0], nodes_h[1], normals_h[0], normals_h[1])
pt.gca().set_aspect("equal")
pt.show()
angle = actx.np.arctan2(nodes[1] * ellipse_aspect, nodes[0])
ellipse_fraction = ((1 - ellipse_aspect) /
(1 + ellipse_aspect))**mode_nr
# (2.6) in [1]
J = actx.np.sqrt( # noqa
actx.np.sin(angle)**2 +
(1 / ellipse_aspect)**2 * actx.np.cos(angle)**2)
from sumpy.kernel import LaplaceKernel
lap_knl = LaplaceKernel(2)
# {{{ single layer
sigma_sym = sym.var("sigma")
s_sigma_op = sym.S(lap_knl, sigma_sym, qbx_forced_limit=+1)
sigma = actx.np.cos(mode_nr * angle) / J
s_sigma = bind(places, s_sigma_op)(actx, sigma=sigma)
# SIGN BINGO! :)
s_eigval = 1 / (2 * mode_nr) * (1 + (-1)**mode_nr * ellipse_fraction)
# (2.12) in [1]
s_sigma_ref = s_eigval * J * sigma
if 0:
#pt.plot(s_sigma.get(), label="result")
#pt.plot(s_sigma_ref.get(), label="ref")
pt.plot(actx.to_numpy(flatten(s_sigma_ref - s_sigma)), label="err")
pt.legend()
pt.show()
h_max = bind(places, sym.h_max(qbx.ambient_dim))(actx)
s_err = (norm(density_discr, s_sigma - s_sigma_ref) /
norm(density_discr, s_sigma_ref))
s_eoc_rec.add_data_point(h_max, s_err)
# }}}
# {{{ double layer
d_sigma_op = sym.D(lap_knl, sigma_sym, qbx_forced_limit="avg")
sigma = actx.np.cos(mode_nr * angle)
d_sigma = bind(places, d_sigma_op)(actx, sigma=sigma)
# SIGN BINGO! :)
d_eigval = -(-1)**mode_nr * 1 / 2 * ellipse_fraction
d_sigma_ref = d_eigval * sigma
if 0:
pt.plot(actx.to_numpy(flatten(d_sigma)), label="result")
pt.plot(actx.to_numpy(flatten(d_sigma_ref)), label="ref")
pt.legend()
pt.show()
if ellipse_aspect == 1:
d_ref_norm = norm(density_discr, sigma)
else:
d_ref_norm = norm(density_discr, d_sigma_ref)
d_err = (norm(density_discr, d_sigma - d_sigma_ref) / d_ref_norm)
d_eoc_rec.add_data_point(h_max, d_err)
# }}}
if ellipse_aspect == 1:
# {{{ S'
sp_sigma_op = sym.Sp(lap_knl,
sym.var("sigma"),
qbx_forced_limit="avg")
sigma = actx.np.cos(mode_nr * angle)
sp_sigma = bind(places, sp_sigma_op)(actx, sigma=sigma)
sp_eigval = 0
sp_sigma_ref = sp_eigval * sigma
sp_err = (norm(density_discr, sp_sigma - sp_sigma_ref) /
norm(density_discr, sigma))
sp_eoc_rec.add_data_point(h_max, sp_err)
# }}}
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 - 1.5
if ellipse_aspect == 1:
print("Errors for S':")
print(sp_eoc_rec)
required_order = qbx_order
assert sp_eoc_rec.order_estimate() > required_order - 1.5
def test_sanity_balls(ctx_getter, src_file, dim, mesh_order, visualize=False):
pytest.importorskip("pytential")
logging.basicConfig(level=logging.INFO)
ctx = ctx_getter()
queue = cl.CommandQueue(ctx)
from pytools.convergence import EOCRecorder
vol_eoc_rec = EOCRecorder()
surf_eoc_rec = EOCRecorder()
# overkill
quad_order = mesh_order
from pytential import bind, sym
for h in [0.2, 0.14, 0.1]:
from meshmode.mesh.io import generate_gmsh, FileSource
mesh = generate_gmsh(FileSource(src_file),
dim,
order=mesh_order,
other_options=[
"-string",
"Mesh.CharacteristicLengthMax = %g;" % h
],
force_ambient_dim=dim)
logger.info("%d elements" % mesh.nelements)
# {{{ discretizations and connections
from meshmode.discretization import Discretization
vol_discr = Discretization(
ctx, mesh, InterpolatoryQuadratureSimplexGroupFactory(quad_order))
from meshmode.discretization.connection import make_face_restriction
bdry_connection = make_face_restriction(
vol_discr, InterpolatoryQuadratureSimplexGroupFactory(quad_order),
BTAG_ALL)
bdry_discr = bdry_connection.to_discr
# }}}
# {{{ visualizers
from meshmode.discretization.visualization import make_visualizer
vol_vis = make_visualizer(queue, vol_discr, 20)
bdry_vis = make_visualizer(queue, bdry_discr, 20)
# }}}
from math import gamma
true_surf = 2 * np.pi**(dim / 2) / gamma(dim / 2)
true_vol = true_surf / dim
vol_x = vol_discr.nodes().with_queue(queue)
vol_one = vol_x[0].copy()
vol_one.fill(1)
from pytential import norm, integral # noqa
comp_vol = integral(vol_discr, queue, vol_one)
rel_vol_err = abs(true_vol - comp_vol) / true_vol
vol_eoc_rec.add_data_point(h, rel_vol_err)
print("VOL", true_vol, comp_vol)
bdry_x = bdry_discr.nodes().with_queue(queue)
bdry_one_exact = bdry_x[0].copy()
bdry_one_exact.fill(1)
bdry_one = bdry_connection(queue, vol_one).with_queue(queue)
intp_err = norm(bdry_discr, queue, bdry_one - bdry_one_exact)
assert intp_err < 1e-14
comp_surf = integral(bdry_discr, queue, bdry_one)
rel_surf_err = abs(true_surf - comp_surf) / true_surf
surf_eoc_rec.add_data_point(h, rel_surf_err)
print("SURF", true_surf, comp_surf)
if visualize:
vol_vis.write_vtk_file("volume-h=%g.vtu" % h, [
("f", vol_one),
("area_el", bind(vol_discr, sym.area_element())(queue)),
])
bdry_vis.write_vtk_file("boundary-h=%g.vtu" % h, [("f", bdry_one)])
# {{{ check normals point outward
normal_outward_check = bind(
bdry_discr,
sym.normal(mesh.ambient_dim) | sym.nodes(mesh.ambient_dim),
)(queue).as_scalar() > 0
assert normal_outward_check.get().all(), normal_outward_check.get()
# }}}
print("---------------------------------")
print("VOLUME")
print("---------------------------------")
print(vol_eoc_rec)
assert vol_eoc_rec.order_estimate() >= mesh_order
print("---------------------------------")
print("SURFACE")
print("---------------------------------")
print(surf_eoc_rec)
assert surf_eoc_rec.order_estimate() >= mesh_order
def test_3d_jump_relations(ctx_factory, relation, visualize=False):
# logging.basicConfig(level=logging.INFO)
cl_ctx = ctx_factory()
queue = cl.CommandQueue(cl_ctx)
if relation == "div_s":
target_order = 3
else:
target_order = 4
qbx_order = target_order
from pytools.convergence import EOCRecorder
eoc_rec = EOCRecorder()
for nel_factor in [6, 10, 14]:
from meshmode.mesh.generation import generate_torus
mesh = generate_torus(
5, 2, order=target_order,
n_outer=2*nel_factor, n_inner=nel_factor)
from meshmode.discretization import Discretization
from meshmode.discretization.poly_element import \
InterpolatoryQuadratureSimplexGroupFactory
pre_discr = Discretization(
cl_ctx, mesh,
InterpolatoryQuadratureSimplexGroupFactory(3))
from pytential.qbx import QBXLayerPotentialSource
qbx, _ = QBXLayerPotentialSource(
pre_discr, fine_order=4*target_order,
qbx_order=qbx_order,
fmm_order=qbx_order + 5,
fmm_backend="fmmlib"
).with_refinement()
from sumpy.kernel import LaplaceKernel
knl = LaplaceKernel(3)
def nxcurlS(qbx_forced_limit):
return sym.n_cross(sym.curl(sym.S(
knl,
sym.cse(sym.tangential_to_xyz(density_sym), "jxyz"),
qbx_forced_limit=qbx_forced_limit)))
x, y, z = qbx.density_discr.nodes().with_queue(queue)
m = cl.clmath
if relation == "nxcurls":
density_sym = sym.make_sym_vector("density", 2)
jump_identity_sym = (
nxcurlS(+1)
- (nxcurlS("avg") + 0.5*sym.tangential_to_xyz(density_sym)))
# The tangential coordinate system is element-local, so we can't just
# conjure up some globally smooth functions, interpret their values
# in the tangential coordinate system, and be done. Instead, generate
# an XYZ function and project it.
density = bind(
qbx,
sym.xyz_to_tangential(sym.make_sym_vector("jxyz", 3)))(
queue,
jxyz=sym.make_obj_array([
m.cos(0.5*x) * m.cos(0.5*y) * m.cos(0.5*z),
m.sin(0.5*x) * m.cos(0.5*y) * m.sin(0.5*z),
m.sin(0.5*x) * m.cos(0.5*y) * m.cos(0.5*z),
]))
elif relation == "sp":
density = m.cos(2*x) * m.cos(2*y) * m.cos(z)
density_sym = sym.var("density")
jump_identity_sym = (
sym.Sp(knl, density_sym, qbx_forced_limit=+1)
- (sym.Sp(knl, density_sym, qbx_forced_limit="avg")
- 0.5*density_sym))
elif relation == "div_s":
density = m.cos(2*x) * m.cos(2*y) * m.cos(z)
density_sym = sym.var("density")
jump_identity_sym = (
sym.div(sym.S(knl, sym.normal(3).as_vector()*density_sym,
qbx_forced_limit="avg"))
+ sym.D(knl, density_sym, qbx_forced_limit="avg"))
else:
raise ValueError("unexpected value of 'relation': %s" % relation)
bound_jump_identity = bind(qbx, jump_identity_sym)
jump_identity = bound_jump_identity(queue, density=density)
err = (
norm(qbx, queue, jump_identity, np.inf)
/ norm(qbx, queue, density, np.inf))
print("ERROR", qbx.h_max, err)
eoc_rec.add_data_point(qbx.h_max, err)
# {{{ visualization
if visualize and relation == "nxcurls":
nxcurlS_ext = bind(qbx, nxcurlS(+1))(queue, density=density)
nxcurlS_avg = bind(qbx, nxcurlS("avg"))(queue, density=density)
jtxyz = bind(qbx, sym.tangential_to_xyz(density_sym))(
queue, density=density)
from meshmode.discretization.visualization import make_visualizer
bdry_vis = make_visualizer(queue, qbx.density_discr, target_order+3)
bdry_normals = bind(qbx, sym.normal(3))(queue)\
.as_vector(dtype=object)
bdry_vis.write_vtk_file("source-%s.vtu" % nel_factor, [
("jt", jtxyz),
("nxcurlS_ext", nxcurlS_ext),
("nxcurlS_avg", nxcurlS_avg),
("bdry_normals", bdry_normals),
])
if visualize and relation == "sp":
sp_ext = bind(qbx, sym.Sp(knl, density_sym, qbx_forced_limit=+1))(
queue, density=density)
sp_avg = bind(qbx, sym.Sp(knl, density_sym, qbx_forced_limit="avg"))(
queue, density=density)
from meshmode.discretization.visualization import make_visualizer
bdry_vis = make_visualizer(queue, qbx.density_discr, target_order+3)
bdry_normals = bind(qbx, sym.normal(3))(queue)\
.as_vector(dtype=object)
bdry_vis.write_vtk_file("source-%s.vtu" % nel_factor, [
("density", density),
("sp_ext", sp_ext),
("sp_avg", sp_avg),
("bdry_normals", bdry_normals),
])
# }}}
print(eoc_rec)
assert eoc_rec.order_estimate() >= qbx_order - 1.5
def rel_err(comp, ref):
return actx.to_numpy(
norm(density_discr, comp - ref) / norm(density_discr, ref))
def get_green_error(geometry_getter,
lpot_kwargs,
center,
k,
vis_error_filename=None,
vis_order=TARGET_ORDER):
"""Return the Green identity error for a geometry.
The density function for the Green error is the on-surface restriction of
the potential due to a source charge in the exterior of the geometry, whose
location is specified. The error is reported relative to the norm of the
density.
Params:
geometry_getter: Geometry getter
lpot_kwargs: Constructor args to QBXLayerPotentialSource
center: Center of source charge used to obtain the constructed density
k: Helmholtz parameter
Returns:
A dictionary containing Green identity errors in l^2 and l^infty norm
"""
context = cl.create_some_context(interactive=False)
queue = cl.CommandQueue(context)
lpot_source = geometry_getter(queue, lpot_kwargs)
d = lpot_source.ambient_dim
u_sym = sym.var("u")
dn_u_sym = sym.var("dn_u")
from sumpy.kernel import LaplaceKernel, HelmholtzKernel
lap_k_sym = LaplaceKernel(d)
if k == 0:
k_sym = lap_k_sym
knl_kwargs = {}
else:
k_sym = HelmholtzKernel(d)
knl_kwargs = {"k": sym.var("k")}
S_part = (sym.S(k_sym, dn_u_sym, qbx_forced_limit=-1, **knl_kwargs))
D_part = (sym.D(k_sym, u_sym, qbx_forced_limit="avg", **knl_kwargs))
sym_op = S_part - D_part - 0.5 * u_sym
density_discr = lpot_source.density_discr
# {{{ compute values of a solution to the PDE
nodes_host = density_discr.nodes().get(queue)
normal = bind(density_discr, sym.normal(d))(queue).as_vector(np.object)
normal_host = [normal[j].get() for j in range(d)]
if k != 0:
if d == 2:
angle = 0.3
wave_vec = np.array([np.cos(angle), np.sin(angle)])
u = np.exp(1j * k * np.tensordot(wave_vec, nodes_host, axes=1))
grad_u = 1j * k * wave_vec[:, np.newaxis] * u
else:
diff = nodes_host - center[:, np.newaxis]
r = la.norm(diff, axis=0)
u = np.exp(1j * k * r) / r
grad_u = diff * (1j * k * u / r - u / r**2)
else:
diff = nodes_host - center[:, np.newaxis]
dist_squared = np.sum(diff**2, axis=0)
dist = np.sqrt(dist_squared)
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 = cl.array.to_device(queue, u)
dn_u_dev = cl.array.to_device(queue, dn_u)
grad_u_dev = cl.array.to_device(queue, grad_u)
bound_op = bind(lpot_source, sym_op)
error = bound_op(queue, u=u_dev, dn_u=dn_u_dev, grad_u=grad_u_dev, k=k)
scaling_l2 = 1 / norm(density_discr, queue, u_dev, p=2)
scaling_linf = 1 / norm(density_discr, queue, u_dev, p="inf")
if vis_error_filename is not None:
from meshmode.discretization.visualization import make_visualizer
bdry_vis = make_visualizer(queue, lpot_source.density_discr, vis_order)
bdry_vis.write_vtk_file(vis_error_filename, [
("green_zero", error),
("u_dev", u_dev),
])
err_l2 = scaling_l2 * norm(density_discr, queue, error, p=2)
err_linf = scaling_linf * norm(density_discr, queue, error, p="inf")
return dict(err_l2=err_l2, err_linf=err_linf)
def get_green_error(lpot_source, fmm_order, qbx_order, k=0, time=None):
queue = cl.CommandQueue(lpot_source.cl_context)
lpot_source = lpot_source.copy(
qbx_order=qbx_order,
fmm_level_to_order=(False if fmm_order is False else
lambda *args: fmm_order))
d = lpot_source.ambient_dim
u_sym = sym.var("u")
dn_u_sym = sym.var("dn_u")
from sumpy.kernel import LaplaceKernel, HelmholtzKernel
lap_k_sym = LaplaceKernel(d)
if k == 0:
k_sym = lap_k_sym
knl_kwargs = {}
else:
k_sym = HelmholtzKernel(d)
knl_kwargs = {"k": sym.var("k")}
S_part = ( # noqa: N806
sym.S(k_sym, dn_u_sym, qbx_forced_limit=-1, **knl_kwargs))
D_part = ( # noqa: N806
sym.D(k_sym, u_sym, qbx_forced_limit="avg", **knl_kwargs))
density_discr = lpot_source.density_discr
# {{{ compute values of a solution to the PDE
nodes_host = density_discr.nodes().get(queue)
normal = bind(density_discr, sym.normal(d))(queue).as_vector(np.object)
normal_host = [normal[j].get() for j in range(d)]
if k != 0:
if d == 2:
angle = 0.3
wave_vec = np.array([np.cos(angle), np.sin(angle)])
u = np.exp(1j * k * np.tensordot(wave_vec, nodes_host, axes=1))
grad_u = 1j * k * wave_vec[:, np.newaxis] * u
else:
center = np.array([3, 1, 2])
diff = nodes_host - center[:, np.newaxis]
r = la.norm(diff, axis=0)
u = np.exp(1j * k * r) / r
grad_u = diff * (1j * k * u / r - u / r**2)
else:
center = np.array([2, 1, 2])[:d]
diff = nodes_host - center[:, np.newaxis]
dist_squared = np.sum(diff**2, axis=0)
dist = np.sqrt(dist_squared)
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 = cl.array.to_device(queue, u)
dn_u_dev = cl.array.to_device(queue, dn_u)
grad_u_dev = cl.array.to_device(queue, grad_u)
bound_S_part = bind(lpot_source, S_part) # noqa: N806
bound_D_part = bind(lpot_source, D_part) # noqa: N806
from time import time as curr_time
t_start = curr_time()
S_result = bound_S_part( # noqa: N806
queue, u=u_dev, dn_u=dn_u_dev, grad_u=grad_u_dev, k=k)
t_end = curr_time()
if time is not None:
time[0] = t_end - t_start
D_result = bound_D_part( # noqa: N806
queue, u=u_dev, dn_u=dn_u_dev, grad_u=grad_u_dev, k=k)
scaling_l2 = 1 / norm(density_discr, queue, u_dev, p=2)
scaling_linf = 1 / norm(density_discr, queue, u_dev, p="inf")
error = S_result - D_result - 0.5 * u_dev
return (scaling_l2 * norm(density_discr, queue, error, p=2),
scaling_linf * norm(density_discr, queue, error, p="inf"))
def rel_err(comp, ref):
return (
norm(density_discr, queue, comp - ref)
/ norm(density_discr, queue, ref))
def obs_norm(f):
return norm(obs_discr, queue, f, p=np.inf)
def test_sanity_balls(actx_factory,
src_file,
dim,
mesh_order,
visualize=False):
pytest.importorskip("pytential")
logging.basicConfig(level=logging.INFO)
actx = actx_factory()
from pytools.convergence import EOCRecorder
vol_eoc_rec = EOCRecorder()
surf_eoc_rec = EOCRecorder()
# overkill
quad_order = mesh_order
from pytential import bind, sym
for h in [0.2, 0.1, 0.05]:
from meshmode.mesh.io import generate_gmsh, FileSource
mesh = generate_gmsh(FileSource(src_file),
dim,
order=mesh_order,
other_options=[
"-string",
"Mesh.CharacteristicLengthMax = %g;" % h
],
force_ambient_dim=dim,
target_unit="MM")
logger.info("%d elements", mesh.nelements)
# {{{ discretizations and connections
from meshmode.discretization import Discretization
vol_discr = Discretization(
actx, mesh, InterpolatoryQuadratureSimplexGroupFactory(quad_order))
from meshmode.discretization.connection import make_face_restriction
bdry_connection = make_face_restriction(
actx, vol_discr,
InterpolatoryQuadratureSimplexGroupFactory(quad_order), BTAG_ALL)
bdry_discr = bdry_connection.to_discr
# }}}
from math import gamma
true_surf = 2 * np.pi**(dim / 2) / gamma(dim / 2)
true_vol = true_surf / dim
vol_x = thaw(vol_discr.nodes(), actx)
vol_one = vol_x[0] * 0 + 1
from pytential import norm, integral # noqa
comp_vol = integral(vol_discr, vol_one)
rel_vol_err = abs(true_vol - comp_vol) / true_vol
vol_eoc_rec.add_data_point(h, rel_vol_err)
print("VOL", true_vol, comp_vol)
bdry_x = thaw(bdry_discr.nodes(), actx)
bdry_one_exact = bdry_x[0] * 0 + 1
bdry_one = bdry_connection(vol_one)
intp_err = norm(bdry_discr, bdry_one - bdry_one_exact)
assert intp_err < 1e-14
comp_surf = integral(bdry_discr, bdry_one)
rel_surf_err = abs(true_surf - comp_surf) / true_surf
surf_eoc_rec.add_data_point(h, rel_surf_err)
print("SURF", true_surf, comp_surf)
if visualize:
from meshmode.discretization.visualization import make_visualizer
vol_vis = make_visualizer(actx, vol_discr, 7)
bdry_vis = make_visualizer(actx, bdry_discr, 7)
name = src_file.split("-")[0]
vol_vis.write_vtk_file(f"sanity_balls_volume_{name}_{h:g}.vtu", [
("f", vol_one),
("area_el",
bind(vol_discr,
sym.area_element(mesh.ambient_dim,
mesh.ambient_dim))(actx)),
])
bdry_vis.write_vtk_file(f"sanity_balls_boundary_{name}_{h:g}.vtu",
[("f", bdry_one)])
# {{{ check normals point outward
normal_outward_check = bind(
bdry_discr,
sym.normal(mesh.ambient_dim) | sym.nodes(mesh.ambient_dim),
)(actx).as_scalar()
normal_outward_check = flatten_to_numpy(actx, normal_outward_check > 0)
assert normal_outward_check.all(), normal_outward_check
# }}}
print("---------------------------------")
print("VOLUME")
print("---------------------------------")
print(vol_eoc_rec)
assert vol_eoc_rec.order_estimate() >= mesh_order
print("---------------------------------")
print("SURFACE")
print("---------------------------------")
print(surf_eoc_rec)
assert surf_eoc_rec.order_estimate() >= mesh_order
def test_ellipse_eigenvalues(ctx_getter, ellipse_aspect, mode_nr, qbx_order):
logging.basicConfig(level=logging.INFO)
print("ellipse_aspect: %s, mode_nr: %d, qbx_order: %d" % (
ellipse_aspect, mode_nr, qbx_order))
cl_ctx = ctx_getter()
queue = cl.CommandQueue(cl_ctx)
target_order = 7
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()
if ellipse_aspect != 1:
nelements_values = [60, 100, 150, 200]
else:
nelements_values = [30, 70]
# See
#
# [1] G. J. Rodin and O. Steinbach, "Boundary Element Preconditioners
# for Problems Defined on Slender Domains", SIAM Journal on Scientific
# Computing, Vol. 24, No. 4, pg. 1450, 2003.
# http://dx.doi.org/10.1137/S1064827500372067
for nelements in nelements_values:
mesh = make_curve_mesh(partial(ellipse, ellipse_aspect),
np.linspace(0, 1, nelements+1),
target_order)
fmm_order = qbx_order
if fmm_order > 3:
# FIXME: for now
fmm_order = False
density_discr = Discretization(
cl_ctx, mesh,
InterpolatoryQuadratureSimplexGroupFactory(target_order))
qbx = QBXLayerPotentialSource(density_discr, 4*target_order,
qbx_order, fmm_order=fmm_order)
nodes = density_discr.nodes().with_queue(queue)
if 0:
# plot geometry, centers, normals
centers = qbx.centers(density_discr, 1)
nodes_h = nodes.get()
centers_h = [centers[0].get(), centers[1].get()]
pt.plot(nodes_h[0], nodes_h[1], "x-")
pt.plot(centers_h[0], centers_h[1], "o")
normal = bind(qbx, sym.normal())(queue).as_vector(np.object)
pt.quiver(nodes_h[0], nodes_h[1],
normal[0].get(), normal[1].get())
pt.gca().set_aspect("equal")
pt.show()
angle = cl.clmath.atan2(nodes[1]*ellipse_aspect, nodes[0])
ellipse_fraction = ((1-ellipse_aspect)/(1+ellipse_aspect))**mode_nr
# (2.6) in [1]
J = cl.clmath.sqrt( # noqa
cl.clmath.sin(angle)**2
+ (1/ellipse_aspect)**2 * cl.clmath.cos(angle)**2)
# {{{ single layer
sigma = cl.clmath.cos(mode_nr*angle)/J
s_sigma_op = bind(qbx, sym.S(0, sym.var("sigma")))
s_sigma = s_sigma_op(queue=queue, sigma=sigma)
# SIGN BINGO! :)
s_eigval = 1/(2*mode_nr) * (1 + (-1)**mode_nr * ellipse_fraction)
# (2.12) in [1]
s_sigma_ref = s_eigval*J*sigma
if 0:
#pt.plot(s_sigma.get(), label="result")
#pt.plot(s_sigma_ref.get(), label="ref")
pt.plot((s_sigma_ref-s_sigma).get(), label="err")
pt.legend()
pt.show()
s_err = (
norm(density_discr, queue, s_sigma - s_sigma_ref)
/
norm(density_discr, queue, s_sigma_ref))
s_eoc_rec.add_data_point(1/nelements, s_err)
# }}}
# {{{ double layer
sigma = cl.clmath.cos(mode_nr*angle)
d_sigma_op = bind(qbx, sym.D(0, sym.var("sigma")))
d_sigma = d_sigma_op(queue=queue, sigma=sigma)
# SIGN BINGO! :)
d_eigval = -(-1)**mode_nr * 1/2*ellipse_fraction
d_sigma_ref = d_eigval*sigma
if 0:
pt.plot(d_sigma.get(), label="result")
pt.plot(d_sigma_ref.get(), label="ref")
pt.legend()
pt.show()
if ellipse_aspect == 1:
d_ref_norm = norm(density_discr, queue, sigma)
else:
d_ref_norm = norm(density_discr, queue, d_sigma_ref)
d_err = (
norm(density_discr, queue, d_sigma - d_sigma_ref)
/
d_ref_norm)
d_eoc_rec.add_data_point(1/nelements, d_err)
# }}}
if ellipse_aspect == 1:
# {{{ S'
sigma = cl.clmath.cos(mode_nr*angle)
sp_sigma_op = bind(qbx, sym.Sp(0, sym.var("sigma")))
sp_sigma = sp_sigma_op(queue=queue, sigma=sigma)
sp_eigval = 0
sp_sigma_ref = sp_eigval*sigma
sp_err = (
norm(density_discr, queue, sp_sigma - sp_sigma_ref)
/
norm(density_discr, queue, sigma))
sp_eoc_rec.add_data_point(1/nelements, sp_err)
# }}}
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 - 1.5
if ellipse_aspect == 1:
print("Errors for S':")
print(sp_eoc_rec)
required_order = qbx_order
assert sp_eoc_rec.order_estimate() > required_order - 1.5
def test_3d_jump_relations(ctx_factory, relation, visualize=False):
# logging.basicConfig(level=logging.INFO)
cl_ctx = ctx_factory()
queue = cl.CommandQueue(cl_ctx)
actx = PyOpenCLArrayContext(queue)
if relation == "div_s":
target_order = 3
else:
target_order = 4
qbx_order = target_order
from pytools.convergence import EOCRecorder
eoc_rec = EOCRecorder()
for nel_factor in [6, 10, 14]:
from meshmode.mesh.generation import generate_torus
mesh = generate_torus(
5, 2, order=target_order,
n_major=2*nel_factor, n_minor=nel_factor)
from meshmode.discretization import Discretization
from meshmode.discretization.poly_element import \
InterpolatoryQuadratureSimplexGroupFactory
pre_discr = Discretization(
actx, mesh,
InterpolatoryQuadratureSimplexGroupFactory(3))
from pytential.qbx import QBXLayerPotentialSource
qbx = QBXLayerPotentialSource(
pre_discr, fine_order=4*target_order,
qbx_order=qbx_order,
fmm_order=qbx_order + 5,
fmm_backend="fmmlib"
)
places = GeometryCollection(qbx)
density_discr = places.get_discretization(places.auto_source.geometry)
from sumpy.kernel import LaplaceKernel
knl = LaplaceKernel(3)
def nxcurlS(qbx_forced_limit):
return sym.n_cross(sym.curl(sym.S(
knl,
sym.cse(sym.tangential_to_xyz(density_sym), "jxyz"),
qbx_forced_limit=qbx_forced_limit)))
from meshmode.dof_array import thaw
x, y, z = thaw(actx, density_discr.nodes())
m = actx.np
if relation == "nxcurls":
density_sym = sym.make_sym_vector("density", 2)
jump_identity_sym = (
nxcurlS(+1)
- (nxcurlS("avg") + 0.5*sym.tangential_to_xyz(density_sym)))
# The tangential coordinate system is element-local, so we can't just
# conjure up some globally smooth functions, interpret their values
# in the tangential coordinate system, and be done. Instead, generate
# an XYZ function and project it.
density = bind(places,
sym.xyz_to_tangential(sym.make_sym_vector("jxyz", 3)))(
actx,
jxyz=sym.make_obj_array([
m.cos(0.5*x) * m.cos(0.5*y) * m.cos(0.5*z),
m.sin(0.5*x) * m.cos(0.5*y) * m.sin(0.5*z),
m.sin(0.5*x) * m.cos(0.5*y) * m.cos(0.5*z),
]))
elif relation == "sp":
density = m.cos(2*x) * m.cos(2*y) * m.cos(z)
density_sym = sym.var("density")
jump_identity_sym = (
sym.Sp(knl, density_sym, qbx_forced_limit=+1)
- (sym.Sp(knl, density_sym, qbx_forced_limit="avg")
- 0.5*density_sym))
elif relation == "div_s":
density = m.cos(2*x) * m.cos(2*y) * m.cos(z)
density_sym = sym.var("density")
jump_identity_sym = (
sym.div(sym.S(knl, sym.normal(3).as_vector()*density_sym,
qbx_forced_limit="avg"))
+ sym.D(knl, density_sym, qbx_forced_limit="avg"))
else:
raise ValueError("unexpected value of 'relation': %s" % relation)
bound_jump_identity = bind(places, jump_identity_sym)
jump_identity = bound_jump_identity(actx, density=density)
h_max = bind(places, sym.h_max(qbx.ambient_dim))(actx)
err = (
norm(density_discr, jump_identity, np.inf)
/ norm(density_discr, density, np.inf))
print("ERROR", h_max, err)
eoc_rec.add_data_point(h_max, err)
# {{{ visualization
if visualize and relation == "nxcurls":
nxcurlS_ext = bind(places, nxcurlS(+1))(actx, density=density)
nxcurlS_avg = bind(places, nxcurlS("avg"))(actx, density=density)
jtxyz = bind(places, sym.tangential_to_xyz(density_sym))(
actx, density=density)
from meshmode.discretization.visualization import make_visualizer
bdry_vis = make_visualizer(actx, qbx.density_discr, target_order+3)
bdry_normals = bind(places, sym.normal(3))(actx)\
.as_vector(dtype=object)
bdry_vis.write_vtk_file("source-%s.vtu" % nel_factor, [
("jt", jtxyz),
("nxcurlS_ext", nxcurlS_ext),
("nxcurlS_avg", nxcurlS_avg),
("bdry_normals", bdry_normals),
])
if visualize and relation == "sp":
op = sym.Sp(knl, density_sym, qbx_forced_limit=+1)
sp_ext = bind(places, op)(actx, density=density)
op = sym.Sp(knl, density_sym, qbx_forced_limit="avg")
sp_avg = bind(places, op)(actx, density=density)
from meshmode.discretization.visualization import make_visualizer
bdry_vis = make_visualizer(actx, qbx.density_discr, target_order+3)
bdry_normals = bind(places,
sym.normal(3))(actx).as_vector(dtype=object)
bdry_vis.write_vtk_file("source-%s.vtu" % nel_factor, [
("density", density),
("sp_ext", sp_ext),
("sp_avg", sp_avg),
("bdry_normals", bdry_normals),
])
# }}}
print(eoc_rec)
assert eoc_rec.order_estimate() >= qbx_order - 1.5
def test_identities(ctx_getter, zero_op_name, curve_name, curve_f, qbx_order, k):
cl_ctx = ctx_getter()
queue = cl.CommandQueue(cl_ctx)
# prevent cache 'splosion
from sympy.core.cache import clear_cache
clear_cache()
target_order = 7
u_sym = sym.var("u")
grad_u_sym = sym.VectorVariable("grad_u")
dn_u_sym = sym.var("dn_u")
if k == 0:
k_sym = 0
else:
k_sym = "k"
zero_op_table = {
"green":
sym.S(k_sym, dn_u_sym) - sym.D(k_sym, u_sym) - 0.5*u_sym,
"green_grad":
d1.nabla * d1(sym.S(k_sym, dn_u_sym))
- d2.nabla * d2(sym.D(k_sym, u_sym))
- 0.5*grad_u_sym,
# only for k==0:
"zero_calderon":
-sym.Dp(0, sym.S(0, u_sym))
- 0.25*u_sym + sym.Sp(0, sym.Sp(0, u_sym))
}
order_table = {
"green": qbx_order,
"green_grad": qbx_order-1,
"zero_calderon": qbx_order-1,
}
zero_op = zero_op_table[zero_op_name]
from pytools.convergence import EOCRecorder
eoc_rec = EOCRecorder()
for nelements in [30, 50, 70]:
mesh = make_curve_mesh(curve_f,
np.linspace(0, 1, nelements+1),
target_order)
from meshmode.discretization import Discretization
from meshmode.discretization.poly_element import \
InterpolatoryQuadratureSimplexGroupFactory
from pytential.qbx import QBXLayerPotentialSource
density_discr = Discretization(
cl_ctx, mesh,
InterpolatoryQuadratureSimplexGroupFactory(target_order))
qbx = QBXLayerPotentialSource(density_discr, 4*target_order,
qbx_order,
# Don't use FMM for now
fmm_order=False)
# {{{ compute values of a solution to the PDE
nodes_host = density_discr.nodes().get(queue)
normal = bind(density_discr, sym.normal())(queue).as_vector(np.object)
normal_host = [normal[0].get(), normal[1].get()]
if k != 0:
angle = 0.3
wave_vec = np.array([np.cos(angle), np.sin(angle)])
u = np.exp(1j*k*np.tensordot(wave_vec, nodes_host, axes=1))
grad_u = 1j*k*wave_vec[:, np.newaxis]*u
else:
center = np.array([3, 1])
diff = nodes_host - center[:, np.newaxis]
dist_squared = np.sum(diff**2, axis=0)
dist = np.sqrt(dist_squared)
u = np.log(dist)
grad_u = diff/dist_squared
dn_u = normal_host[0]*grad_u[0] + normal_host[1]*grad_u[1]
# }}}
u_dev = cl.array.to_device(queue, u)
dn_u_dev = cl.array.to_device(queue, dn_u)
grad_u_dev = cl.array.to_device(queue, grad_u)
key = (qbx_order, curve_name, nelements, zero_op_name)
bound_op = bind(qbx, zero_op)
error = bound_op(
queue, u=u_dev, dn_u=dn_u_dev, grad_u=grad_u_dev, k=k)
if 0:
pt.plot(error)
pt.show()
l2_error_norm = norm(density_discr, queue, error)
print(key, l2_error_norm)
eoc_rec.add_data_point(1/nelements, l2_error_norm)
print(eoc_rec)
tgt_order = order_table[zero_op_name]
assert eoc_rec.order_estimate() > tgt_order - 1.3
def main():
import logging
logging.basicConfig(level=logging.INFO)
ctx = cl.create_some_context()
queue = cl.CommandQueue(ctx)
mesh = generate_gmsh(
FileSource("circle.step"), 2, order=mesh_order,
force_ambient_dim=2,
other_options=["-string", "Mesh.CharacteristicLengthMax = %g;" % h]
)
logger.info("%d elements" % mesh.nelements)
# {{{ discretizations and connections
vol_discr = Discretization(ctx, mesh,
InterpolatoryQuadratureSimplexGroupFactory(vol_quad_order))
ovsmp_vol_discr = Discretization(ctx, mesh,
InterpolatoryQuadratureSimplexGroupFactory(vol_ovsmp_quad_order))
from meshmode.discretization.connection import (
make_boundary_restriction, make_same_mesh_connection)
bdry_mesh, bdry_discr, bdry_connection = make_boundary_restriction(
queue, vol_discr,
InterpolatoryQuadratureSimplexGroupFactory(bdry_quad_order))
vol_to_ovsmp_vol = make_same_mesh_connection(
queue, ovsmp_vol_discr, vol_discr)
# }}}
# {{{ visualizers
vol_vis = make_visualizer(queue, vol_discr, 20)
bdry_vis = make_visualizer(queue, bdry_discr, 20)
# }}}
vol_x = vol_discr.nodes().with_queue(queue)
ovsmp_vol_x = ovsmp_vol_discr.nodes().with_queue(queue)
rhs = rhs_func(vol_x[0], vol_x[1])
poisson_true_sol = sol_func(vol_x[0], vol_x[1])
vol_vis.write_vtk_file("volume.vtu", [("f", rhs)])
bdry_normals = bind(bdry_discr, p.normal())(queue).as_vector(dtype=object)
bdry_vis.write_vtk_file("boundary.vtu", [
("normals", bdry_normals)
])
bdry_nodes = bdry_discr.nodes().with_queue(queue)
bdry_f = rhs_func(bdry_nodes[0], bdry_nodes[1])
bdry_f_2 = bdry_connection(queue, rhs)
bdry_vis.write_vtk_file("y.vtu", [("f", bdry_f_2)])
if 0:
vol_vis.show_scalar_in_mayavi(rhs, do_show=False)
bdry_vis.show_scalar_in_mayavi(bdry_f - bdry_f_2, line_width=10,
do_show=False)
import mayavi.mlab as mlab
mlab.colorbar()
mlab.show()
# {{{ compute volume potential
from sumpy.qbx import LayerPotential
from sumpy.expansion.local import LineTaylorLocalExpansion
def get_kernel():
from sumpy.symbolic import pymbolic_real_norm_2
from pymbolic.primitives import (make_sym_vector, Variable as var)
r = pymbolic_real_norm_2(make_sym_vector("d", 3))
expr = var("log")(r)
scaling = 1/(2*var("pi"))
from sumpy.kernel import ExpressionKernel
return ExpressionKernel(
dim=3,
expression=expr,
scaling=scaling,
is_complex_valued=False)
laplace_2d_in_3d_kernel = get_kernel()
layer_pot = LayerPotential(ctx, [
LineTaylorLocalExpansion(laplace_2d_in_3d_kernel,
order=vol_qbx_order)])
targets = cl.array.zeros(queue, (3,) + vol_x.shape[1:], vol_x.dtype)
targets[:2] = vol_x
center_dist = np.min(
cl.clmath.sqrt(
bind(vol_discr, p.area_element())(queue)).get())
centers = make_obj_array([ci.copy().reshape(vol_discr.nnodes) for ci in targets])
centers[2][:] = center_dist
sources = cl.array.zeros(queue, (3,) + ovsmp_vol_x.shape[1:], ovsmp_vol_x.dtype)
sources[:2] = ovsmp_vol_x
ovsmp_rhs = vol_to_ovsmp_vol(queue, rhs)
ovsmp_vol_weights = bind(ovsmp_vol_discr, p.area_element() * p.QWeight())(queue)
evt, (vol_pot,) = layer_pot(
queue,
targets=targets.reshape(3, vol_discr.nnodes),
centers=centers,
sources=sources.reshape(3, ovsmp_vol_discr.nnodes),
strengths=(
(ovsmp_vol_weights*ovsmp_rhs).reshape(ovsmp_vol_discr.nnodes),)
)
vol_pot_bdry = bdry_connection(queue, vol_pot)
# }}}
# {{{ solve bvp
from sumpy.kernel import LaplaceKernel
from pytential.symbolic.pde.scalar import DirichletOperator
op = DirichletOperator(LaplaceKernel(2), -1, use_l2_weighting=True)
sym_sigma = sym.var("sigma")
op_sigma = op.operator(sym_sigma)
from pytential.qbx import QBXLayerPotentialSource
qbx = QBXLayerPotentialSource(
bdry_discr, fine_order=bdry_ovsmp_quad_order, qbx_order=qbx_order,
fmm_order=fmm_order
)
bound_op = bind(qbx, op_sigma)
poisson_bc = poisson_bc_func(bdry_nodes[0], bdry_nodes[1])
bvp_bc = poisson_bc - vol_pot_bdry
bdry_f = rhs_func(bdry_nodes[0], bdry_nodes[1])
bvp_rhs = bind(bdry_discr, op.prepare_rhs(sym.var("bc")))(queue, bc=bvp_bc)
from pytential.solve import gmres
gmres_result = gmres(
bound_op.scipy_op(queue, "sigma"),
bvp_rhs, tol=1e-14, progress=True,
hard_failure=False)
sigma = gmres_result.solution
print("gmres state:", gmres_result.state)
# }}}
bvp_sol = bind(
(qbx, vol_discr),
op.representation(sym_sigma))(queue, sigma=sigma)
poisson_sol = bvp_sol + vol_pot
poisson_err = poisson_sol-poisson_true_sol
rel_err = (
norm(vol_discr, queue, poisson_err)
/
norm(vol_discr, queue, poisson_true_sol))
bdry_vis.write_vtk_file("poisson-boundary.vtu", [
("vol_pot_bdry", vol_pot_bdry),
("sigma", sigma),
])
vol_vis.write_vtk_file("poisson-volume.vtu", [
("bvp_sol", bvp_sol),
("poisson_sol", poisson_sol),
("poisson_true_sol", poisson_true_sol),
("poisson_err", poisson_err),
("vol_pot", vol_pot),
("rhs", rhs),
])
print("h = %s" % h)
print("mesh_order = %s" % mesh_order)
print("vol_quad_order = %s" % vol_quad_order)
print("vol_ovsmp_quad_order = %s" % vol_ovsmp_quad_order)
print("bdry_quad_order = %s" % bdry_quad_order)
print("bdry_ovsmp_quad_order = %s" % bdry_ovsmp_quad_order)
print("qbx_order = %s" % qbx_order)
print("vol_qbx_order = %s" % vol_qbx_order)
print("fmm_order = %s" % fmm_order)
print()
print("rel err: %g" % rel_err)
def main():
import logging
logging.basicConfig(level=logging.INFO)
ctx = cl.create_some_context()
queue = cl.CommandQueue(ctx)
if 1:
ext = 0.5
mesh = generate_regular_rect_mesh(a=(-ext / 2, -ext / 2),
b=(ext / 2, ext / 2),
n=(int(ext / h), int(ext / h)))
else:
mesh = generate_gmsh(FileSource("circle.step"),
2,
order=mesh_order,
force_ambient_dim=2,
other_options=[
"-string",
"Mesh.CharacteristicLengthMax = %g;" % h
])
logger.info("%d elements" % mesh.nelements)
# {{{ discretizations and connections
vol_discr = Discretization(
ctx, mesh, InterpolatoryQuadratureSimplexGroupFactory(vol_quad_order))
ovsmp_vol_discr = Discretization(
ctx, mesh,
InterpolatoryQuadratureSimplexGroupFactory(vol_ovsmp_quad_order))
from meshmode.mesh import BTAG_ALL
from meshmode.discretization.connection import (make_face_restriction,
make_same_mesh_connection)
bdry_connection = make_face_restriction(
vol_discr, InterpolatoryQuadratureSimplexGroupFactory(bdry_quad_order),
BTAG_ALL)
bdry_discr = bdry_connection.to_discr
vol_to_ovsmp_vol = make_same_mesh_connection(ovsmp_vol_discr, vol_discr)
# }}}
# {{{ visualizers
vol_vis = make_visualizer(queue, vol_discr, 20)
bdry_vis = make_visualizer(queue, bdry_discr, 20)
# }}}
vol_x = vol_discr.nodes().with_queue(queue)
ovsmp_vol_x = ovsmp_vol_discr.nodes().with_queue(queue)
rhs = rhs_func(vol_x[0], vol_x[1])
poisson_true_sol = sol_func(vol_x[0], vol_x[1])
vol_vis.write_vtk_file("volume.vtu", [("f", rhs)])
bdry_normals = bind(bdry_discr, p.normal(
mesh.ambient_dim))(queue).as_vector(dtype=object)
bdry_vis.write_vtk_file("boundary.vtu", [("normals", bdry_normals)])
bdry_nodes = bdry_discr.nodes().with_queue(queue)
bdry_f = rhs_func(bdry_nodes[0], bdry_nodes[1])
bdry_f_2 = bdry_connection(queue, rhs)
bdry_vis.write_vtk_file("y.vtu", [("f", bdry_f_2)])
if 0:
vol_vis.show_scalar_in_mayavi(rhs, do_show=False)
bdry_vis.show_scalar_in_mayavi(bdry_f - bdry_f_2,
line_width=10,
do_show=False)
import mayavi.mlab as mlab
mlab.colorbar()
mlab.show()
# {{{ compute volume potential
from sumpy.qbx import LayerPotential
from sumpy.expansion.local import LineTaylorLocalExpansion
def get_kernel():
from sumpy.symbolic import pymbolic_real_norm_2
from pymbolic.primitives import make_sym_vector
from pymbolic import var
d = make_sym_vector("d", 3)
r = pymbolic_real_norm_2(d[:-1])
# r3d = pymbolic_real_norm_2(d)
#expr = var("log")(r3d)
log = var("log")
sqrt = var("sqrt")
a = d[-1]
expr = log(r)
expr = log(sqrt(r**2 + a**2))
expr = log(sqrt(r + a**2))
#expr = log(sqrt(r**2 + a**2))-a**2/2/(r**2+a**2)
#expr = 2*log(sqrt(r**2 + a**2))
scaling = 1 / (2 * var("pi"))
from sumpy.kernel import ExpressionKernel
return ExpressionKernel(dim=3,
expression=expr,
global_scaling_const=scaling,
is_complex_valued=False)
laplace_2d_in_3d_kernel = get_kernel()
layer_pot = LayerPotential(
ctx, [LineTaylorLocalExpansion(laplace_2d_in_3d_kernel, order=0)])
targets = cl.array.zeros(queue, (3, ) + vol_x.shape[1:], vol_x.dtype)
targets[:2] = vol_x
center_dist = 0.125 * np.min(
cl.clmath.sqrt(
bind(vol_discr, p.area_element(mesh.ambient_dim,
mesh.dim))(queue)).get())
centers = make_obj_array(
[ci.copy().reshape(vol_discr.nnodes) for ci in targets])
centers[2][:] = center_dist
print(center_dist)
sources = cl.array.zeros(queue, (3, ) + ovsmp_vol_x.shape[1:],
ovsmp_vol_x.dtype)
sources[:2] = ovsmp_vol_x
ovsmp_rhs = vol_to_ovsmp_vol(queue, rhs)
ovsmp_vol_weights = bind(
ovsmp_vol_discr,
p.area_element(mesh.ambient_dim, mesh.dim) * p.QWeight())(queue)
print("volume: %d source nodes, %d target nodes" %
(ovsmp_vol_discr.nnodes, vol_discr.nnodes))
evt, (vol_pot, ) = layer_pot(
queue,
targets=targets.reshape(3, vol_discr.nnodes),
centers=centers,
sources=sources.reshape(3, ovsmp_vol_discr.nnodes),
strengths=((ovsmp_vol_weights * ovsmp_rhs).reshape(
ovsmp_vol_discr.nnodes), ),
expansion_radii=np.zeros(vol_discr.nnodes),
)
vol_pot_bdry = bdry_connection(queue, vol_pot)
# }}}
# {{{ solve bvp
from sumpy.kernel import LaplaceKernel
from pytential.symbolic.pde.scalar import DirichletOperator
op = DirichletOperator(LaplaceKernel(2), -1, use_l2_weighting=True)
sym_sigma = sym.var("sigma")
op_sigma = op.operator(sym_sigma)
from pytential.qbx import QBXLayerPotentialSource
qbx = QBXLayerPotentialSource(
bdry_discr,
fine_order=bdry_ovsmp_quad_order,
qbx_order=qbx_order,
fmm_order=fmm_order,
)
bound_op = bind(qbx, op_sigma)
poisson_bc = poisson_bc_func(bdry_nodes[0], bdry_nodes[1])
bvp_bc = poisson_bc - vol_pot_bdry
bdry_f = rhs_func(bdry_nodes[0], bdry_nodes[1])
bvp_rhs = bind(bdry_discr, op.prepare_rhs(sym.var("bc")))(queue, bc=bvp_bc)
from pytential.solve import gmres
gmres_result = gmres(bound_op.scipy_op(queue, "sigma", dtype=np.float64),
bvp_rhs,
tol=1e-14,
progress=True,
hard_failure=False)
sigma = gmres_result.solution
print("gmres state:", gmres_result.state)
# }}}
bvp_sol = bind((qbx, vol_discr), op.representation(sym_sigma))(queue,
sigma=sigma)
poisson_sol = bvp_sol + vol_pot
poisson_err = poisson_sol - poisson_true_sol
rel_err = (norm(vol_discr, queue, poisson_err) /
norm(vol_discr, queue, poisson_true_sol))
bdry_vis.write_vtk_file("poisson-boundary.vtu", [
("vol_pot_bdry", vol_pot_bdry),
("sigma", sigma),
])
vol_vis.write_vtk_file("poisson-volume.vtu", [
("bvp_sol", bvp_sol),
("poisson_sol", poisson_sol),
("poisson_true_sol", poisson_true_sol),
("poisson_err", poisson_err),
("vol_pot", vol_pot),
("rhs", rhs),
])
print("h = %s" % h)
print("mesh_order = %s" % mesh_order)
print("vol_quad_order = %s" % vol_quad_order)
print("vol_ovsmp_quad_order = %s" % vol_ovsmp_quad_order)
print("bdry_quad_order = %s" % bdry_quad_order)
print("bdry_ovsmp_quad_order = %s" % bdry_ovsmp_quad_order)
print("qbx_order = %s" % qbx_order)
#print("vol_qbx_order = %s" % vol_qbx_order)
print("fmm_order = %s" % fmm_order)
print()
print("rel err: %g" % rel_err)
def rel_err(comp, ref):
return (norm(density_discr, comp - ref) / norm(density_discr, ref))
def test_3d_jump_relations(actx_factory, relation, visualize=False):
# logging.basicConfig(level=logging.INFO)
actx = actx_factory()
if relation == "div_s":
target_order = 3
else:
target_order = 4
qbx_order = target_order
if relation == "sp":
resolutions = [10, 14, 18]
else:
resolutions = [6, 10, 14]
from pytools.convergence import EOCRecorder
eoc_rec = EOCRecorder()
for nel_factor in resolutions:
from meshmode.mesh.generation import generate_torus
mesh = generate_torus(
5,
2,
n_major=2 * nel_factor,
n_minor=nel_factor,
order=target_order,
)
from meshmode.discretization import Discretization
from meshmode.discretization.poly_element import \
InterpolatoryQuadratureSimplexGroupFactory
pre_density_discr = Discretization(
actx, mesh,
InterpolatoryQuadratureSimplexGroupFactory(target_order))
from pytential.qbx import QBXLayerPotentialSource
qbx = QBXLayerPotentialSource(pre_density_discr,
fine_order=5 * target_order,
qbx_order=qbx_order,
fmm_order=qbx_order + 5,
fmm_backend="fmmlib")
places = GeometryCollection(qbx)
density_discr = places.get_discretization(places.auto_source.geometry)
from sumpy.kernel import LaplaceKernel
knl = LaplaceKernel(places.ambient_dim)
def nxcurlS(qbx_forced_limit):
sigma_sym = sym.cse(sym.tangential_to_xyz(density_sym), "jxyz")
return sym.n_cross(
sym.curl(
sym.S(knl, sigma_sym, qbx_forced_limit=qbx_forced_limit)))
x, y, z = thaw(density_discr.nodes(), actx)
if relation == "nxcurls":
density_sym = sym.make_sym_vector("density", 2)
jump_identity_sym = (nxcurlS(+1) - nxcurlS("avg") -
0.5 * sym.tangential_to_xyz(density_sym))
# The tangential coordinate system is element-local, so we can't just
# conjure up some globally smooth functions, interpret their values
# in the tangential coordinate system, and be done. Instead, generate
# an XYZ function and project it.
jxyz = sym.make_obj_array([
actx.np.cos(0.5 * x) * actx.np.cos(0.5 * y) *
actx.np.cos(0.5 * z),
actx.np.sin(0.5 * x) * actx.np.cos(0.5 * y) *
actx.np.sin(0.5 * z),
actx.np.sin(0.5 * x) * actx.np.cos(0.5 * y) *
actx.np.cos(0.5 * z),
])
density = bind(
places,
sym.xyz_to_tangential(sym.make_sym_vector("jxyz",
3)))(actx, jxyz=jxyz)
elif relation == "sp":
density_sym = sym.var("density")
jump_identity_sym = (
0.5 * density_sym +
sym.Sp(knl, density_sym, qbx_forced_limit=+1) -
sym.Sp(knl, density_sym, qbx_forced_limit="avg"))
density = actx.np.cos(2 * x) * actx.np.cos(2 * y) * actx.np.cos(z)
elif relation == "div_s":
density_sym = sym.var("density")
sigma_sym = sym.normal(
places.ambient_dim).as_vector() * density_sym
jump_identity_sym = (
sym.div(sym.S(knl, sigma_sym, qbx_forced_limit="avg")) +
sym.D(knl, density_sym, qbx_forced_limit="avg"))
density = actx.np.cos(2 * x) * actx.np.cos(2 * y) * actx.np.cos(z)
else:
raise ValueError(f"unexpected value of 'relation': '{relation}'")
bound_jump_identity = bind(places, jump_identity_sym)
jump_identity = bound_jump_identity(actx, density=density)
h_max = actx.to_numpy(
bind(places, sym.h_max(places.ambient_dim))(actx))
err = actx.to_numpy(
norm(density_discr, jump_identity, np.inf) /
norm(density_discr, density, np.inf))
eoc_rec.add_data_point(h_max, err)
logging.info("error: nel %d h_max %.5e %.5e", nel_factor, h_max, err)
# {{{ visualization
if not visualize:
continue
from meshmode.discretization.visualization import make_visualizer
vis = make_visualizer(actx, density_discr, target_order)
normals = bind(places,
sym.normal(places.ambient_dim).as_vector())(actx)
error = actx.np.log10(actx.np.abs(jump_identity) + 1.0e-15)
if relation == "nxcurls":
nxcurlS_ext = bind(places, nxcurlS(+1))(actx, density=density)
nxcurlS_avg = bind(places, nxcurlS("avg"))(actx, density=density)
jtxyz = bind(places,
sym.tangential_to_xyz(density_sym))(actx,
density=density)
vis.write_vtk_file(f"source-nxcurls-{nel_factor:03d}.vtu", [
("jt", jtxyz),
("nxcurlS_ext", nxcurlS_ext),
("nxcurlS_avg", nxcurlS_avg),
("bdry_normals", normals),
("error", error),
])
elif relation == "sp":
op = sym.Sp(knl, density_sym, qbx_forced_limit=+1)
sp_ext = bind(places, op)(actx, density=density)
op = sym.Sp(knl, density_sym, qbx_forced_limit="avg")
sp_avg = bind(places, op)(actx, density=density)
vis.write_vtk_file(f"source-sp-{nel_factor:03d}.vtu", [
("density", density),
("sp_ext", sp_ext),
("sp_avg", sp_avg),
("bdry_normals", normals),
("error", error),
])
elif relation == "div_s":
vis.write_vtk_file(f"source-div-{nel_factor:03d}.vtu", [
("density", density),
("bdry_normals", normals),
("error", error),
])
# }}}
logger.info("\n%s", str(eoc_rec))
assert eoc_rec.order_estimate() >= qbx_order - 1.5
def scat_norm(f):
return norm(density_discr, f, p=np.inf)
grad_u_dev = cl.array.to_device(queue, grad_u)
key = (case.qbx_order, case.geometry.mesh_name, resolution,
case.expr.zero_op_name)
bound_op = bind(qbx, case.expr.get_zero_op(k_sym, **knl_kwargs))
error = bound_op(queue,
u=u_dev,
dn_u=dn_u_dev,
grad_u=grad_u_dev,
k=case.k)
if 0:
pt.plot(error)
pt.show()
linf_error_norm = norm(density_discr, queue, error, p=np.inf)
print("--->", key, linf_error_norm)
h_max = bind(qbx, sym.h_max(qbx.ambient_dim))(queue)
eoc_rec.add_data_point(h_max, linf_error_norm)
if visualize:
from meshmode.discretization.visualization import make_visualizer
bdry_vis = make_visualizer(queue, density_discr, target_order)
bdry_normals = bind(density_discr, sym.normal(mesh.ambient_dim))(queue)\
.as_vector(dtype=object)
bdry_vis.write_vtk_file("source-%s.vtu" % resolution, [
("u", u_dev),
("bdry_normals", bdry_normals),
def obs_norm(f):
return norm(obs_discr, queue, f, p=np.inf)
def test_identity_convergence(ctx_getter, case, visualize=False):
logging.basicConfig(level=logging.INFO)
case.check()
cl_ctx = ctx_getter()
queue = cl.CommandQueue(cl_ctx)
# prevent cache 'splosion
from sympy.core.cache import clear_cache
clear_cache()
target_order = 8
from pytools.convergence import EOCRecorder
eoc_rec = EOCRecorder()
for resolution in (
getattr(case, "resolutions", None)
or case.geometry.resolutions
):
mesh = case.geometry.get_mesh(resolution, target_order)
if mesh is None:
break
d = mesh.ambient_dim
k = case.k
lap_k_sym = LaplaceKernel(d)
if k == 0:
k_sym = lap_k_sym
knl_kwargs = {}
else:
k_sym = HelmholtzKernel(d)
knl_kwargs = {"k": sym.var("k")}
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))
refiner_extra_kwargs = {}
if case.k != 0:
refiner_extra_kwargs["kernel_length_scale"] = 5/case.k
qbx, _ = QBXLayerPotentialSource(
pre_density_discr, 4*target_order,
case.qbx_order,
fmm_order=case.fmm_order,
fmm_backend=case.fmm_backend,
_expansions_in_tree_have_extent=True,
_expansion_stick_out_factor=getattr(
case, "_expansion_stick_out_factor", 0),
).with_refinement(**refiner_extra_kwargs)
density_discr = qbx.density_discr
# {{{ compute values of a solution to the PDE
nodes_host = density_discr.nodes().get(queue)
normal = bind(density_discr, sym.normal(d))(queue).as_vector(np.object)
normal_host = [normal[j].get() for j in range(d)]
if k != 0:
if d == 2:
angle = 0.3
wave_vec = np.array([np.cos(angle), np.sin(angle)])
u = np.exp(1j*k*np.tensordot(wave_vec, nodes_host, axes=1))
grad_u = 1j*k*wave_vec[:, np.newaxis]*u
elif d == 3:
center = np.array([3, 1, 2])
diff = nodes_host - center[:, np.newaxis]
r = la.norm(diff, axis=0)
u = np.exp(1j*k*r) / r
grad_u = diff * (1j*k*u/r - u/r**2)
else:
raise ValueError("invalid dim")
else:
center = np.array([3, 1, 2])[:d]
diff = nodes_host - center[:, np.newaxis]
dist_squared = np.sum(diff**2, axis=0)
dist = np.sqrt(dist_squared)
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 = cl.array.to_device(queue, u)
dn_u_dev = cl.array.to_device(queue, dn_u)
grad_u_dev = cl.array.to_device(queue, grad_u)
key = (case.qbx_order, case.geometry.mesh_name, resolution,
case.expr.zero_op_name)
bound_op = bind(qbx, case.expr.get_zero_op(k_sym, **knl_kwargs))
error = bound_op(
queue, u=u_dev, dn_u=dn_u_dev, grad_u=grad_u_dev, k=case.k)
if 0:
pt.plot(error)
pt.show()
linf_error_norm = norm(density_discr, queue, error, p=np.inf)
print("--->", key, linf_error_norm)
eoc_rec.add_data_point(qbx.h_max, linf_error_norm)
if visualize:
from meshmode.discretization.visualization import make_visualizer
bdry_vis = make_visualizer(queue, density_discr, target_order)
bdry_normals = bind(density_discr, sym.normal(mesh.ambient_dim))(queue)\
.as_vector(dtype=object)
bdry_vis.write_vtk_file("source-%s.vtu" % resolution, [
("u", u_dev),
("bdry_normals", bdry_normals),
("error", error),
])
print(eoc_rec)
tgt_order = case.qbx_order - case.expr.order_drop
assert eoc_rec.order_estimate() > tgt_order - 1.6
