def test_bessel(ctx_factory):
cl_ctx = ctx_factory()
queue = cl.CommandQueue(cl_ctx)
actx = PyOpenCLArrayContext(queue)
dims = 2
from meshmode.mesh.generation import generate_regular_rect_mesh
mesh = generate_regular_rect_mesh(a=(0.1, ) * dims,
b=(1.0, ) * dims,
n=(8, ) * dims)
discr = DGDiscretizationWithBoundaries(actx, mesh, order=3)
nodes = sym.nodes(dims)
r = sym.cse(sym.sqrt(nodes[0]**2 + nodes[1]**2))
# https://dlmf.nist.gov/10.6.1
n = 3
bessel_zero = (sym.bessel_j(n + 1, r) + sym.bessel_j(n - 1, r) -
2 * n / r * sym.bessel_j(n, r))
z = bind(discr, sym.norm(2, bessel_zero))(actx)
assert z < 1e-15
def max_eigenvalue_expr(self):
"""Return the largest eigenvalue of Maxwell's equations as a hyperbolic
system.
"""
from math import sqrt
if self.fixed_material:
return 1/sqrt(self.epsilon*self.mu) # a number
else:
import grudge.symbolic as sym
return sym.NodalMax("vol")(
1 / sym.sqrt(self.epsilon * self.mu)
)
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 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 test_bessel(actx_factory):
actx = actx_factory()
dims = 2
mesh = mgen.generate_regular_rect_mesh(a=(0.1, ) * dims,
b=(1.0, ) * dims,
nelements_per_axis=(8, ) * dims)
discr = DiscretizationCollection(actx, mesh, order=3)
nodes = sym.nodes(dims)
r = sym.cse(sym.sqrt(nodes[0]**2 + nodes[1]**2))
# https://dlmf.nist.gov/10.6.1
n = 3
bessel_zero = (sym.bessel_j(n + 1, r) + sym.bessel_j(n - 1, r) -
2 * n / r * sym.bessel_j(n, r))
z = bind(discr, sym.norm(2, bessel_zero))(actx)
assert z < 1e-15
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
