def main(write_output=True):
cl_ctx = cl.create_some_context()
queue = cl.CommandQueue(cl_ctx)
from meshmode.mesh.generation import generate_warped_rect_mesh
mesh = generate_warped_rect_mesh(dim=2, order=4, n=6)
discr = DGDiscretizationWithBoundaries(cl_ctx, mesh, order=4)
sym_op = sym.normal(sym.BTAG_ALL, mesh.dim)
#sym_op = sym.nodes(mesh.dim, where=sym.BTAG_ALL)
print(sym.pretty(sym_op))
op = bind(discr, sym_op)
print()
print(op.eval_code)
vec = op(queue)
vis = shortcuts.make_visualizer(discr, 4)
vis.write_vtk_file("geo.vtu", [
])
bvis = shortcuts.make_boundary_visualizer(discr, 4)
bvis.write_vtk_file("bgeo.vtu", [
("normals", vec)
])
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 main(write_output=True):
cl_ctx = cl.create_some_context()
queue = cl.CommandQueue(cl_ctx)
actx = PyOpenCLArrayContext(queue)
from meshmode.mesh import BTAG_ALL
from meshmode.mesh.generation import generate_warped_rect_mesh
mesh = generate_warped_rect_mesh(dim=2, order=4, nelements_side=6)
discr = DiscretizationCollection(actx, mesh, order=4)
sym_op = sym.normal(BTAG_ALL, mesh.dim)
# sym_op = sym.nodes(mesh.dim, dd=BTAG_ALL)
print(sym.pretty(sym_op))
op = bind(discr, sym_op)
print()
print(op.eval_code)
vec = op(actx)
vis = shortcuts.make_visualizer(discr)
vis.write_vtk_file("geo.vtu", [])
bvis = shortcuts.make_boundary_visualizer(discr)
bvis.write_vtk_file("bgeo.vtu", [("normals", vec)])
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.interp("vol", self.dirichlet_tag)(u))
dir_v = sym.cse(sym.interp("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.Field("dir_bc_u")
dir_bc = join_fields(2 * dir_g - dir_u, dir_v)
else:
dir_bc = join_fields(-dir_u, dir_v)
dir_bc = sym.cse(dir_bc, "dir_bc")
# neumann BCs ---------------------------------------------------------
neu_u = sym.cse(sym.interp("vol", self.neumann_tag)(u))
neu_v = sym.cse(sym.interp("vol", self.neumann_tag)(v))
neu_bc = sym.cse(join_fields(neu_u, -neu_v), "neu_bc")
# radiation BCs -------------------------------------------------------
rad_normal = sym.normal(self.radiation_tag, d)
rad_u = sym.cse(sym.interp("vol", self.radiation_tag)(u))
rad_v = sym.cse(sym.interp("vol", self.radiation_tag)(v))
rad_bc = sym.cse(
join_fields(
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.interp(pair.dd, "all_faces")(self.flux(pair))
result = sym.InverseMassOperator()(
join_fields(-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 normal(self, dd):
surface_discr = self.discr_from_dd(dd)
actx = surface_discr._setup_actx
return freeze(
bind(self,
sym.normal(dd, surface_discr.ambient_dim, surface_discr.dim),
local_only=True)
(array_context=actx))
def flux(self, w):
"""The numerical flux for variable coefficients.
:param flux_type: can be in [0,1] for anything between central and upwind,
or "lf" for Lax-Friedrichs.
As per Hesthaven and Warburton page 433.
"""
normal = sym.normal(w.dd, self.dimensions)
if self.fixed_material:
e, h = self.split_eh(w)
epsilon = self.epsilon
mu = self.mu
Z_int = (mu/epsilon)**0.5 # noqa: N806
Y_int = 1/Z_int # noqa: N806
Z_ext = (mu/epsilon)**0.5 # noqa: N806
Y_ext = 1/Z_ext # noqa: N806
if self.flux_type == "lf":
# if self.fixed_material:
# max_c = (self.epsilon*self.mu)**(-0.5)
return flat_obj_array(
# flux e,
1/2*(
-self.space_cross_h(normal, h.ext-h.int)
# multiplication by epsilon undoes material divisor below
#-max_c*(epsilon*e.int - epsilon*e.ext)
),
# flux h
1/2*(
self.space_cross_e(normal, e.ext-e.int)
# multiplication by mu undoes material divisor below
#-max_c*(mu*h.int - mu*h.ext)
))
elif isinstance(self.flux_type, (int, float)):
# see doc/maxima/maxwell.mac
return flat_obj_array(
# flux e,
(
-1/(Z_int+Z_ext)*self.space_cross_h(normal,
Z_ext*(h.ext-h.int)
- self.flux_type*self.space_cross_e(normal, e.ext-e.int))
),
# flux h
(
1/(Y_int + Y_ext)*self.space_cross_e(normal,
Y_ext*(e.ext-e.int)
+ self.flux_type*self.space_cross_h(normal, h.ext-h.int))
),
)
else:
raise ValueError("maxwell: invalid flux_type (%s)"
% self.flux_type)
def v_dot_n_tpair(velocity, dd=None):
if dd is None:
dd = sym.DOFDesc(sym.FACE_RESTR_INTERIOR)
ambient_dim = len(velocity)
normal = sym.normal(dd.with_qtag(None), ambient_dim, dim=ambient_dim - 2)
return sym.int_tpair(velocity.dot(normal),
qtag=dd.quadrature_tag,
from_dd=dd.with_qtag(None))
def normal(dcoll, dd):
"""Get unit normal to specified surface discretization, *dd*.
:arg dd: a :class:`~grudge.dof_desc.DOFDesc` as the surface discretization.
:returns: an object array of :class:`~meshmode.dof_array.DOFArray`.
"""
surface_discr = dcoll.discr_from_dd(dd)
actx = surface_discr._setup_actx
return freeze(
bind(dcoll,
sym.normal(dd, surface_discr.ambient_dim, surface_discr.dim),
local_only=True)(array_context=actx))
def get_strong_wave_op_with_discr_direct(cl_ctx, 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(cl_ctx, 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.interp("vol", BTAG_ALL)(u))
rad_v = sym.cse(sym.interp("vol", BTAG_ALL)(v))
rad_bc = sym.cse(
sym.join_fields(
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 = (
-sym.join_fields(-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 v_dot_n_tpair(velocity, dd=None):
from grudge.dof_desc import DOFDesc
from meshmode.discretization.connection import FACE_RESTR_INTERIOR
if dd is None:
dd = DOFDesc(FACE_RESTR_INTERIOR)
ambient_dim = len(velocity)
normal = sym.normal(dd.with_discr_tag(None),
ambient_dim,
dim=ambient_dim - 2)
return sym.int_tpair(velocity.dot(normal),
qtag=dd.discretization_tag,
from_dd=dd.with_discr_tag(None))
def flux(self, w):
u = w[0]
v = w[1:]
normal = sym.normal(w.dd, self.ambient_dim)
flux_weak = flat_obj_array(np.dot(v.avg, normal), u.avg * normal)
if self.flux_type == "central":
return -self.c * flux_weak
elif self.flux_type == "upwind":
return -self.c * (flux_weak + self.sign * flat_obj_array(
0.5 * (u.int - u.ext), 0.5 *
(normal * np.dot(normal, v.int - v.ext))))
else:
raise ValueError("invalid flux type '%s'" % self.flux_type)
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 dg_flux(c, tpair):
u = tpair[0]
v = tpair[1:]
dims = len(v)
normal = sym.normal(tpair.dd, dims)
flux_weak = sym.join_fields(np.dot(v.avg, normal), u.avg * normal)
flux_weak -= (1 if c > 0 else -1) * sym.join_fields(
0.5 * (u.int - u.ext), 0.5 * (normal * np.dot(normal, v.int - v.ext)))
flux_strong = sym.join_fields(np.dot(v.int, normal),
u.int * normal) - flux_weak
return sym.interp(tpair.dd, "all_faces")(c * flux_strong)
def test_empty_boundary(actx_factory):
# https://github.com/inducer/grudge/issues/54
from meshmode.mesh import BTAG_NONE
actx = actx_factory()
dim = 2
mesh = mgen.generate_regular_rect_mesh(a=(-0.5, ) * dim,
b=(0.5, ) * dim,
nelements_per_axis=(8, ) * dim,
order=4)
discr = DiscretizationCollection(actx, mesh, order=4)
normal = bind(discr, sym.normal(BTAG_NONE, dim, dim=dim - 1))(actx)
from meshmode.dof_array import DOFArray
for component in normal:
assert isinstance(component, DOFArray)
assert len(component) == len(discr.discr_from_dd(BTAG_NONE).groups)
def weak_flux(self, u):
normal = sym.normal(u. dd, self.ambient_dim)
v_dot_normal = sym.cse(self.v.dot(normal), "v_dot_normal")
norm_v = sym.sqrt((self.v**2).sum())
if self.flux_type == "central":
return u.avg*v_dot_normal
elif self.flux_type == "lf":
return u.avg*v_dot_normal + 0.5*norm_v*(u.int - u.ext)
elif self.flux_type == "upwind":
return (
v_dot_normal * sym.If(
sym.Comparison(v_dot_normal, ">", 0),
u.int, # outflow
u.ext, # inflow
))
else:
raise ValueError("invalid flux type")
def advection_weak_flux(flux_type, u, velocity):
normal = sym.normal(u.dd, len(velocity))
v_dot_n = sym.cse(velocity.dot(normal), "v_dot_normal")
flux_type = flux_type.lower()
if flux_type == "central":
return u.avg * v_dot_n
elif flux_type == "lf":
norm_v = sym.sqrt((velocity**2).sum())
return u.avg * v_dot_n + 0.5 * norm_v * (u.int - u.ext)
elif flux_type == "upwind":
u_upwind = sym.If(
sym.Comparison(v_dot_n, ">", 0),
u.int, # outflow
u.ext # inflow
)
return u_upwind * v_dot_n
else:
raise ValueError("flux `{}` is not implemented".format(flux_type))
def flux(self, w):
c = w[0]
u = w[1]
v = w[2:]
normal = sym.normal(w.dd, self.ambient_dim)
if self.flux_type == "central":
return -0.5 * flat_obj_array(
np.dot(v.int * c.int + v.ext * c.ext, normal),
(u.int * c.int + u.ext * c.ext) * normal)
elif self.flux_type == "upwind":
return -0.5 * flat_obj_array(
np.dot(normal, c.ext * v.ext + c.int * v.int) + c.ext * u.ext -
c.int * u.int,
normal * (np.dot(normal, c.ext * v.ext - c.int * v.int) +
c.ext * u.ext + c.int * u.int))
else:
raise ValueError("invalid flux type '%s'" % self.flux_type)
def flux(self, w):
u = w[0]
v = w[1:]
normal = sym.normal(w.dd, self.ambient_dim)
flux_weak = flat_obj_array(np.dot(v.avg, normal), u.avg * normal)
if self.flux_type == "central":
pass
elif self.flux_type == "upwind":
# see doc/notes/grudge-notes.tm
flux_weak -= self.sign * flat_obj_array(
0.5 * (u.int - u.ext), 0.5 *
(normal * np.dot(normal, v.int - v.ext)))
else:
raise ValueError("invalid flux type '%s'" % self.flux_type)
flux_strong = flat_obj_array(np.dot(v.int, normal),
u.int * normal) - flux_weak
return self.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 main(ctx_factory, dim=2, order=4, product_tag=None, visualize=False):
cl_ctx = ctx_factory()
queue = cl.CommandQueue(cl_ctx)
actx = PyOpenCLArrayContext(queue)
# {{{ parameters
# sphere radius
radius = 1.0
# sphere resolution
resolution = 64 if dim == 2 else 1
# cfl
dt_factor = 2.0
# final time
final_time = np.pi
# velocity field
sym_x = sym.nodes(dim)
c = make_obj_array([-sym_x[1], sym_x[0], 0.0])[:dim]
# flux
flux_type = "lf"
# }}}
# {{{ discretization
if dim == 2:
from meshmode.mesh.generation import make_curve_mesh, ellipse
mesh = make_curve_mesh(lambda t: radius * ellipse(1.0, t),
np.linspace(0.0, 1.0, resolution + 1), order)
elif dim == 3:
from meshmode.mesh.generation import generate_icosphere
mesh = generate_icosphere(radius,
order=4 * order,
uniform_refinement_rounds=resolution)
else:
raise ValueError("unsupported dimension")
discr_tag_to_group_factory = {}
if product_tag == "none":
product_tag = None
else:
product_tag = dof_desc.DISCR_TAG_QUAD
from meshmode.discretization.poly_element import \
PolynomialWarpAndBlendGroupFactory, \
QuadratureSimplexGroupFactory
discr_tag_to_group_factory[dof_desc.DISCR_TAG_BASE] = \
PolynomialWarpAndBlendGroupFactory(order)
if product_tag:
discr_tag_to_group_factory[product_tag] = \
QuadratureSimplexGroupFactory(order=4*order)
from grudge import DiscretizationCollection
discr = DiscretizationCollection(
actx, mesh, discr_tag_to_group_factory=discr_tag_to_group_factory)
volume_discr = discr.discr_from_dd(dof_desc.DD_VOLUME)
logger.info("ndofs: %d", volume_discr.ndofs)
logger.info("nelements: %d", volume_discr.mesh.nelements)
# }}}
# {{{ symbolic operators
def f_initial_condition(x):
return x[0]
from grudge.models.advection import SurfaceAdvectionOperator
op = SurfaceAdvectionOperator(c, flux_type=flux_type, quad_tag=product_tag)
bound_op = bind(discr, op.sym_operator())
u0 = bind(discr, f_initial_condition(sym_x))(actx, t=0)
def rhs(t, u):
return bound_op(actx, t=t, u=u)
# check velocity is tangential
sym_normal = sym.surface_normal(dim, dim=dim - 1,
dd=dof_desc.DD_VOLUME).as_vector()
error = bind(discr, sym.norm(2, c.dot(sym_normal)))(actx)
logger.info("u_dot_n: %.5e", error)
# }}}
# {{{ time stepping
# compute time step
h_min = bind(discr, sym.h_max_from_volume(discr.ambient_dim,
dim=discr.dim))(actx)
dt = dt_factor * h_min / order**2
nsteps = int(final_time // dt) + 1
dt = final_time / nsteps + 1.0e-15
logger.info("dt: %.5e", dt)
logger.info("nsteps: %d", nsteps)
from grudge.shortcuts import set_up_rk4
dt_stepper = set_up_rk4("u", dt, u0, rhs)
plot = Plotter(actx, discr, order, visualize=visualize)
norm = bind(discr, sym.norm(2, sym.var("u")))
norm_u = norm(actx, u=u0)
step = 0
event = dt_stepper.StateComputed(0.0, 0, 0, u0)
plot(event, "fld-surface-%04d" % 0)
if visualize and dim == 3:
from grudge.shortcuts import make_visualizer
vis = make_visualizer(discr)
vis.write_vtk_file("fld-surface-velocity.vtu",
[("u", bind(discr, c)(actx)),
("n", bind(discr, sym_normal)(actx))],
overwrite=True)
df = dof_desc.DOFDesc(FACE_RESTR_INTERIOR)
face_discr = discr.connection_from_dds(dof_desc.DD_VOLUME, df).to_discr
face_normal = bind(
discr, sym.normal(df, face_discr.ambient_dim,
dim=face_discr.dim))(actx)
from meshmode.discretization.visualization import make_visualizer
vis = make_visualizer(actx, face_discr)
vis.write_vtk_file("fld-surface-face-normals.vtu",
[("n", face_normal)],
overwrite=True)
for event in dt_stepper.run(t_end=final_time, max_steps=nsteps + 1):
if not isinstance(event, dt_stepper.StateComputed):
continue
step += 1
if step % 10 == 0:
norm_u = norm(actx, u=event.state_component)
plot(event, "fld-surface-%04d" % step)
logger.info("[%04d] t = %.5f |u| = %.5e", step, event.t, norm_u)
plot(event, "fld-surface-%04d" % step)
def 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):
normal = sym.normal(u.dd, self.ambient_dim)
v_dot_normal = sym.cse(self.v.dot(normal), "v_dot_normal")
return u.int * v_dot_normal - self.weak_flux(u)
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