def absorbing_bc(self, w):
"""Construct part of the flux operator template for 1st order
absorbing boundary conditions.
"""
absorb_normal = sym.normal(self.absorb_tag, self.dimensions)
e, h = self.split_eh(w)
if self.fixed_material:
epsilon = self.epsilon
mu = self.mu
absorb_Z = (mu/epsilon)**0.5 # noqa: N806
absorb_Y = 1/absorb_Z # noqa: N806
absorb_e = sym.cse(sym.project("vol", self.absorb_tag)(e))
absorb_h = sym.cse(sym.project("vol", self.absorb_tag)(h))
bc = flat_obj_array(
absorb_e + 1/2*(self.space_cross_h(absorb_normal, self.space_cross_e(
absorb_normal, absorb_e))
- absorb_Z*self.space_cross_h(absorb_normal, absorb_h)),
absorb_h + 1/2*(
self.space_cross_e(absorb_normal, self.space_cross_h(
absorb_normal, absorb_h))
+ absorb_Y*self.space_cross_e(absorb_normal, absorb_e)))
return bc
def simple_mpi_communication_entrypoint():
cl_ctx = cl.create_some_context()
queue = cl.CommandQueue(cl_ctx)
actx = PyOpenCLArrayContext(queue)
from meshmode.distributed import MPIMeshDistributor, get_partition_by_pymetis
from meshmode.mesh import BTAG_ALL
from mpi4py import MPI
comm = MPI.COMM_WORLD
num_parts = comm.Get_size()
mesh_dist = MPIMeshDistributor(comm)
if mesh_dist.is_mananger_rank():
from meshmode.mesh.generation import generate_regular_rect_mesh
mesh = generate_regular_rect_mesh(a=(-1, ) * 2,
b=(1, ) * 2,
nelements_per_axis=(2, ) * 2)
part_per_element = get_partition_by_pymetis(mesh, num_parts)
local_mesh = mesh_dist.send_mesh_parts(mesh, part_per_element,
num_parts)
else:
local_mesh = mesh_dist.receive_mesh_part()
vol_discr = DiscretizationCollection(actx,
local_mesh,
order=5,
mpi_communicator=comm)
sym_x = sym.nodes(local_mesh.dim)
myfunc_symb = sym.sin(np.dot(sym_x, [2, 3]))
myfunc = bind(vol_discr, myfunc_symb)(actx)
sym_all_faces_func = sym.cse(
sym.project("vol", "all_faces")(sym.var("myfunc")))
sym_int_faces_func = sym.cse(
sym.project("vol", "int_faces")(sym.var("myfunc")))
sym_bdry_faces_func = sym.cse(
sym.project(BTAG_ALL,
"all_faces")(sym.project("vol",
BTAG_ALL)(sym.var("myfunc"))))
bound_face_swap = bind(
vol_discr,
sym.project("int_faces", "all_faces")(
sym.OppositeInteriorFaceSwap("int_faces")(sym_int_faces_func)) -
(sym_all_faces_func - sym_bdry_faces_func))
hopefully_zero = bound_face_swap(myfunc=myfunc)
error = actx.np.linalg.norm(hopefully_zero, ord=np.inf)
print(__file__)
with np.printoptions(threshold=100000000, suppress=True):
logger.debug(hopefully_zero)
logger.info("error: %.5e", error)
assert error < 1e-14
def pmc_bc(self, w):
"Construct part of the flux operator template for PMC boundary conditions"
e, h = self.split_eh(w)
pmc_e = sym.cse(sym.project("vol", self.pmc_tag)(e))
pmc_h = sym.cse(sym.project("vol", self.pmc_tag)(h))
return flat_obj_array(pmc_e, -pmc_h)
def get_strong_wave_op_with_discr_direct(actx, dims=2, order=4):
from meshmode.mesh.generation import generate_regular_rect_mesh
mesh = generate_regular_rect_mesh(a=(-0.5, ) * dims,
b=(0.5, ) * dims,
n=(16, ) * dims)
logger.debug("%d elements", mesh.nelements)
discr = DGDiscretizationWithBoundaries(actx, mesh, order=order)
source_center = np.array([0.1, 0.22, 0.33])[:dims]
source_width = 0.05
source_omega = 3
sym_x = sym.nodes(mesh.dim)
sym_source_center_dist = sym_x - source_center
sym_t = sym.ScalarVariable("t")
from meshmode.mesh import BTAG_ALL
c = -0.1
sign = -1
w = sym.make_sym_array("w", dims + 1)
u = w[0]
v = w[1:]
source_f = (
sym.sin(source_omega * sym_t) *
sym.exp(-np.dot(sym_source_center_dist, sym_source_center_dist) /
source_width**2))
rad_normal = sym.normal(BTAG_ALL, dims)
rad_u = sym.cse(sym.project("vol", BTAG_ALL)(u))
rad_v = sym.cse(sym.project("vol", BTAG_ALL)(v))
rad_bc = sym.cse(
flat_obj_array(
0.5 * (rad_u - sign * np.dot(rad_normal, rad_v)),
0.5 * rad_normal * (np.dot(rad_normal, rad_v) - sign * rad_u)),
"rad_bc")
sym_operator = (
-flat_obj_array(-c * np.dot(sym.nabla(dims), v) - source_f, -c *
(sym.nabla(dims) * u)) + sym.InverseMassOperator()(
sym.FaceMassOperator()
(dg_flux(c, sym.int_tpair(w)) +
dg_flux(c, sym.bv_tpair(BTAG_ALL, w, rad_bc)))))
return (sym_operator, discr)
def sym_operator(self):
from grudge.dof_desc import DOFDesc, DD_VOLUME, DTAG_VOLUME_ALL
from meshmode.mesh import BTAG_ALL
from meshmode.discretization.connection import FACE_RESTR_ALL
u = sym.var("u")
def flux(pair):
return sym.project(pair.dd, face_dd)(self.flux(pair))
face_dd = DOFDesc(FACE_RESTR_ALL, self.quad_tag)
boundary_dd = DOFDesc(BTAG_ALL, self.quad_tag)
quad_dd = DOFDesc(DTAG_VOLUME_ALL, self.quad_tag)
to_quad = sym.project(DD_VOLUME, quad_dd)
stiff_t_op = sym.stiffness_t(self.ambient_dim,
dd_in=quad_dd,
dd_out=DD_VOLUME)
quad_v = to_quad(self.v)
quad_u = to_quad(u)
return sym.InverseMassOperator()(
sum(stiff_t_op[n](quad_u * quad_v[n])
for n in range(self.ambient_dim)) -
sym.FaceMassOperator(face_dd, DD_VOLUME)(
flux(sym.int_tpair(u, self.quad_tag)) +
flux(sym.bv_tpair(boundary_dd, u, self.inflow_u))
# FIXME: Add back support for inflow/outflow tags
#+ flux(sym.bv_tpair(self.inflow_tag, u, bc_in))
#+ flux(sym.bv_tpair(self.outflow_tag, u, bc_out))
))
def sym_operator(self):
u = sym.var("u")
def flux(pair):
return sym.project(pair.dd, face_dd)(self.flux(pair))
face_dd = sym.DOFDesc(sym.FACE_RESTR_ALL, self.quad_tag)
boundary_dd = sym.DOFDesc(sym.BTAG_ALL, self.quad_tag)
quad_dd = sym.DOFDesc(sym.DTAG_VOLUME_ALL, self.quad_tag)
to_quad = sym.project(sym.DD_VOLUME, quad_dd)
stiff_t_op = sym.stiffness_t(self.ambient_dim,
dd_in=quad_dd, dd_out=sym.DD_VOLUME)
quad_v = to_quad(self.v)
quad_u = to_quad(u)
return sym.InverseMassOperator()(
sum(stiff_t_op[n](quad_u * quad_v[n])
for n in range(self.ambient_dim))
- sym.FaceMassOperator(face_dd, sym.DD_VOLUME)(
flux(sym.int_tpair(u, self.quad_tag))
+ flux(sym.bv_tpair(boundary_dd, u, self.inflow_u))
# FIXME: Add back support for inflow/outflow tags
#+ flux(sym.bv_tpair(self.inflow_tag, u, bc_in))
#+ flux(sym.bv_tpair(self.outflow_tag, u, bc_out))
))
def sym_operator(self):
from grudge.dof_desc import DOFDesc, DD_VOLUME, DTAG_VOLUME_ALL
from meshmode.discretization.connection import FACE_RESTR_ALL
u = sym.var("u")
def flux(pair):
return sym.project(pair.dd, face_dd)(self.flux(pair))
face_dd = DOFDesc(FACE_RESTR_ALL, self.quad_tag)
quad_dd = DOFDesc(DTAG_VOLUME_ALL, self.quad_tag)
to_quad = sym.project(DD_VOLUME, quad_dd)
stiff_t_op = sym.stiffness_t(self.ambient_dim,
dd_in=quad_dd,
dd_out=DD_VOLUME)
quad_v = to_quad(self.v)
quad_u = to_quad(u)
return sym.InverseMassOperator()(
sum(stiff_t_op[n](quad_u * quad_v[n])
for n in range(self.ambient_dim)) -
sym.FaceMassOperator(face_dd, DD_VOLUME)
(flux(sym.int_tpair(u, self.quad_tag))))
def sym_operator(self):
d = self.ambient_dim
w = sym.make_sym_array("w", d + 1)
u = w[0]
v = w[1:]
# boundary conditions -------------------------------------------------
# dirichlet BCs -------------------------------------------------------
dir_u = sym.cse(sym.project("vol", self.dirichlet_tag)(u))
dir_v = sym.cse(sym.project("vol", self.dirichlet_tag)(v))
if self.dirichlet_bc_f:
# FIXME
from warnings import warn
warn("Inhomogeneous Dirichlet conditions on the wave equation "
"are still having issues.")
dir_g = sym.var("dir_bc_u")
dir_bc = flat_obj_array(2 * dir_g - dir_u, dir_v)
else:
dir_bc = flat_obj_array(-dir_u, dir_v)
dir_bc = sym.cse(dir_bc, "dir_bc")
# neumann BCs ---------------------------------------------------------
neu_u = sym.cse(sym.project("vol", self.neumann_tag)(u))
neu_v = sym.cse(sym.project("vol", self.neumann_tag)(v))
neu_bc = sym.cse(flat_obj_array(neu_u, -neu_v), "neu_bc")
# radiation BCs -------------------------------------------------------
rad_normal = sym.normal(self.radiation_tag, d)
rad_u = sym.cse(sym.project("vol", self.radiation_tag)(u))
rad_v = sym.cse(sym.project("vol", self.radiation_tag)(v))
rad_bc = sym.cse(
flat_obj_array(
0.5 * (rad_u - self.sign * np.dot(rad_normal, rad_v)), 0.5 *
rad_normal * (np.dot(rad_normal, rad_v) - self.sign * rad_u)),
"rad_bc")
# entire operator -----------------------------------------------------
def flux(pair):
return sym.project(pair.dd, "all_faces")(self.flux(pair))
result = sym.InverseMassOperator()(flat_obj_array(
-self.c * np.dot(sym.stiffness_t(self.ambient_dim), v), -self.c *
(sym.stiffness_t(self.ambient_dim) * u)) - sym.FaceMassOperator()(
flux(sym.int_tpair(w)) +
flux(sym.bv_tpair(self.dirichlet_tag, w, dir_bc)) +
flux(sym.bv_tpair(self.neumann_tag, w, neu_bc)) +
flux(sym.bv_tpair(self.radiation_tag, w, rad_bc))))
result[0] += self.source_f
return result
def dg_flux(c, tpair):
u = tpair[0]
v = tpair[1:]
dims = len(v)
normal = sym.normal(tpair.dd, dims)
flux_weak = flat_obj_array(np.dot(v.avg, normal), u.avg * normal)
flux_weak -= (1 if c > 0 else -1) * flat_obj_array(
0.5 * (u.int - u.ext), 0.5 * (normal * np.dot(normal, v.int - v.ext)))
flux_strong = flat_obj_array(np.dot(v.int, normal),
u.int * normal) - flux_weak
return sym.project(tpair.dd, "all_faces")(c * flux_strong)
def test_2d_gauss_theorem(actx_factory):
"""Verify Gauss's theorem explicitly on a mesh"""
pytest.importorskip("meshpy")
from meshpy.geometry import make_circle, GeometryBuilder
from meshpy.triangle import MeshInfo, build
geob = GeometryBuilder()
geob.add_geometry(*make_circle(1))
mesh_info = MeshInfo()
geob.set(mesh_info)
mesh_info = build(mesh_info)
from meshmode.mesh.io import from_meshpy
mesh = from_meshpy(mesh_info, order=1)
actx = actx_factory()
discr = DGDiscretizationWithBoundaries(actx, mesh, order=2)
def f(x):
return flat_obj_array(
sym.sin(3*x[0])+sym.cos(3*x[1]),
sym.sin(2*x[0])+sym.cos(x[1]))
gauss_err = bind(discr,
sym.integral((
sym.nabla(2) * f(sym.nodes(2))
).sum())
- # noqa: W504
sym.integral(
sym.project("vol", sym.BTAG_ALL)(f(sym.nodes(2)))
.dot(sym.normal(sym.BTAG_ALL, 2)),
dd=sym.BTAG_ALL)
)(actx)
assert abs(gauss_err) < 1e-13
def sym_operator(self):
u = sym.var("u")
def flux(pair):
return sym.project(pair.dd, face_dd)(self.flux(pair))
face_dd = sym.DOFDesc(sym.FACE_RESTR_ALL, self.quad_tag)
quad_dd = sym.DOFDesc(sym.DTAG_VOLUME_ALL, self.quad_tag)
to_quad = sym.project(sym.DD_VOLUME, quad_dd)
stiff_t_op = sym.stiffness_t(self.ambient_dim,
dd_in=quad_dd,
dd_out=sym.DD_VOLUME)
quad_v = to_quad(self.v)
quad_u = to_quad(u)
return sym.InverseMassOperator()(
sum(stiff_t_op[n](quad_u * quad_v[n])
for n in range(self.ambient_dim)) -
sym.FaceMassOperator(face_dd, sym.DD_VOLUME)
(flux(sym.int_tpair(u, self.quad_tag))))
def test_surface_divergence_theorem(actx_factory, mesh_name, visualize=False):
r"""Check the surface divergence theorem.
.. math::
\int_Sigma \phi \nabla_i f_i =
\int_\Sigma \nabla_i \phi f_i +
\int_\Sigma \kappa \phi f_i n_i +
\int_{\partial \Sigma} \phi f_i m_i
where :math:`n_i` is the surface normal and :class:`m_i` is the
face normal (which should be orthogonal to both the surface normal
and the face tangent).
"""
actx = actx_factory()
# {{{ cases
if mesh_name == "2-1-ellipse":
from mesh_data import EllipseMeshBuilder
builder = EllipseMeshBuilder(radius=3.1, aspect_ratio=2.0)
elif mesh_name == "spheroid":
from mesh_data import SpheroidMeshBuilder
builder = SpheroidMeshBuilder()
elif mesh_name == "circle":
from mesh_data import EllipseMeshBuilder
builder = EllipseMeshBuilder(radius=1.0, aspect_ratio=1.0)
elif mesh_name == "starfish":
from mesh_data import StarfishMeshBuilder
builder = StarfishMeshBuilder()
elif mesh_name == "sphere":
from mesh_data import SphereMeshBuilder
builder = SphereMeshBuilder(radius=1.0, mesh_order=16)
else:
raise ValueError("unknown mesh name: %s" % mesh_name)
# }}}
# {{{ convergene
def f(x):
return flat_obj_array(
sym.sin(3 * x[1]) + sym.cos(3 * x[0]) + 1.0,
sym.sin(2 * x[0]) + sym.cos(x[1]),
3.0 * sym.cos(x[0] / 2) + sym.cos(x[1]),
)[:ambient_dim]
from pytools.convergence import EOCRecorder
eoc_global = EOCRecorder()
eoc_local = EOCRecorder()
theta = np.pi / 3.33
ambient_dim = builder.ambient_dim
if ambient_dim == 2:
mesh_rotation = np.array([
[np.cos(theta), -np.sin(theta)],
[np.sin(theta), np.cos(theta)],
])
else:
mesh_rotation = np.array([
[1.0, 0.0, 0.0],
[0.0, np.cos(theta), -np.sin(theta)],
[0.0, np.sin(theta), np.cos(theta)],
])
mesh_offset = np.array([0.33, -0.21, 0.0])[:ambient_dim]
for i, resolution in enumerate(builder.resolutions):
from meshmode.mesh.processing import affine_map
from meshmode.discretization.connection import FACE_RESTR_ALL
mesh = builder.get_mesh(resolution, builder.mesh_order)
mesh = affine_map(mesh, A=mesh_rotation, b=mesh_offset)
from meshmode.discretization.poly_element import \
QuadratureSimplexGroupFactory
discr = DiscretizationCollection(actx,
mesh,
order=builder.order,
discr_tag_to_group_factory={
"product":
QuadratureSimplexGroupFactory(
2 * builder.order)
})
volume = discr.discr_from_dd(dof_desc.DD_VOLUME)
logger.info("ndofs: %d", volume.ndofs)
logger.info("nelements: %d", volume.mesh.nelements)
dd = dof_desc.DD_VOLUME
dq = dd.with_discr_tag("product")
df = dof_desc.as_dofdesc(FACE_RESTR_ALL)
ambient_dim = discr.ambient_dim
dim = discr.dim
# variables
sym_f = f(sym.nodes(ambient_dim, dd=dd))
sym_f_quad = f(sym.nodes(ambient_dim, dd=dq))
sym_kappa = sym.summed_curvature(ambient_dim, dim=dim, dd=dq)
sym_normal = sym.surface_normal(ambient_dim, dim=dim,
dd=dq).as_vector()
sym_face_normal = sym.normal(df, ambient_dim, dim=dim - 1)
sym_face_f = sym.project(dd, df)(sym_f)
# operators
sym_stiff = sum(
sym.StiffnessOperator(d)(f) for d, f in enumerate(sym_f))
sym_stiff_t = sum(
sym.StiffnessTOperator(d)(f) for d, f in enumerate(sym_f))
sym_k = sym.MassOperator(dq,
dd)(sym_kappa * sym_f_quad.dot(sym_normal))
sym_flux = sym.FaceMassOperator()(sym_face_f.dot(sym_face_normal))
# sum everything up
sym_op_global = sym.NodalSum(dd)(sym_stiff - (sym_stiff_t + sym_k))
sym_op_local = sym.ElementwiseSumOperator(dd)(sym_stiff -
(sym_stiff_t + sym_k +
sym_flux))
# evaluate
op_global = bind(discr, sym_op_global)(actx)
op_local = bind(discr, sym_op_local)(actx)
err_global = abs(op_global)
err_local = bind(discr, sym.norm(np.inf, sym.var("x")))(actx,
x=op_local)
logger.info("errors: global %.5e local %.5e", err_global, err_local)
# compute max element size
h_max = bind(
discr,
sym.h_max_from_volume(discr.ambient_dim, dim=discr.dim,
dd=dd))(actx)
eoc_global.add_data_point(h_max, err_global)
eoc_local.add_data_point(h_max, err_local)
if visualize:
from grudge.shortcuts import make_visualizer
vis = make_visualizer(discr, vis_order=builder.order)
filename = f"surface_divergence_theorem_{mesh_name}_{i:04d}.vtu"
vis.write_vtk_file(filename, [("r", actx.np.log10(op_local))],
overwrite=True)
# }}}
order = min(builder.order, builder.mesh_order) - 0.5
logger.info("\n%s", str(eoc_global))
logger.info("\n%s", str(eoc_local))
assert eoc_global.max_error() < 1.0e-12 \
or eoc_global.order_estimate() > order - 0.5
assert eoc_local.max_error() < 1.0e-12 \
or eoc_local.order_estimate() > order - 0.5
def test_face_normal_surface(actx_factory, mesh_name):
"""Check that face normals are orthogonal to the surface normal"""
actx = actx_factory()
# {{{ geometry
if mesh_name == "2-1-ellipse":
from mesh_data import EllipseMeshBuilder
builder = EllipseMeshBuilder(radius=3.1, aspect_ratio=2.0)
elif mesh_name == "spheroid":
from mesh_data import SpheroidMeshBuilder
builder = SpheroidMeshBuilder()
else:
raise ValueError("unknown mesh name: %s" % mesh_name)
mesh = builder.get_mesh(builder.resolutions[0], builder.mesh_order)
discr = DiscretizationCollection(actx, mesh, order=builder.order)
volume_discr = discr.discr_from_dd(dof_desc.DD_VOLUME)
logger.info("ndofs: %d", volume_discr.ndofs)
logger.info("nelements: %d", volume_discr.mesh.nelements)
# }}}
# {{{ symbolic
from meshmode.discretization.connection import FACE_RESTR_INTERIOR
dv = dof_desc.DD_VOLUME
df = dof_desc.as_dofdesc(FACE_RESTR_INTERIOR)
ambient_dim = mesh.ambient_dim
dim = mesh.dim
sym_surf_normal = sym.project(dv,
df)(sym.surface_normal(ambient_dim,
dim=dim,
dd=dv).as_vector())
sym_surf_normal = sym_surf_normal / sym.sqrt(sum(sym_surf_normal**2))
sym_face_normal_i = sym.normal(df, ambient_dim, dim=dim - 1)
sym_face_normal_e = sym.OppositeInteriorFaceSwap(df)(sym_face_normal_i)
if mesh.ambient_dim == 3:
# NOTE: there's only one face tangent in 3d
sym_face_tangent = (
sym.pseudoscalar(ambient_dim, dim - 1, dd=df) /
sym.area_element(ambient_dim, dim - 1, dd=df)).as_vector()
# }}}
# {{{ checks
def _eval_error(x):
return bind(discr, sym.norm(np.inf, sym.var("x", dd=df), dd=df))(actx,
x=x)
rtol = 1.0e-14
surf_normal = bind(discr, sym_surf_normal)(actx)
face_normal_i = bind(discr, sym_face_normal_i)(actx)
face_normal_e = bind(discr, sym_face_normal_e)(actx)
# check interpolated surface normal is orthogonal to face normal
error = _eval_error(surf_normal.dot(face_normal_i))
logger.info("error[n_dot_i]: %.5e", error)
assert error < rtol
# check angle between two neighboring elements
error = _eval_error(face_normal_i.dot(face_normal_e) + 1.0)
logger.info("error[i_dot_e]: %.5e", error)
assert error > rtol
# check orthogonality with face tangent
if ambient_dim == 3:
face_tangent = bind(discr, sym_face_tangent)(actx)
error = _eval_error(face_tangent.dot(face_normal_i))
logger.info("error[t_dot_i]: %.5e", error)
assert error < 5 * rtol
def flux(self, u):
from grudge.dof_desc import DD_VOLUME
surf_v = sym.project(DD_VOLUME, u.dd.with_discr_tag(None))(self.v)
return surface_advection_weak_flux(self.flux_type, u, surf_v)
def flux(pair):
return sym.project(pair.dd, face_dd)(self.flux(pair))
def flux(pair):
return sym.project(pair.dd, "all_faces")(self.flux(pair))
def flux(self, u):
surf_v = sym.project(sym.DD_VOLUME, u.dd.with_qtag(None))(self.v)
return surface_advection_weak_flux(self.flux_type, u, surf_v)
def flux(self, u):
surf_v = sym.project(sym.DD_VOLUME, u.dd)(self.v)
return advection_weak_flux(self.flux_type, u, surf_v)
def flux(self, u):
from grudge.dof_desc import DD_VOLUME
surf_v = sym.project(DD_VOLUME, u.dd)(self.v)
return advection_weak_flux(self.flux_type, u, surf_v)
