def op_template(self):
q = make_vector_field('q', self.len_q)
fluxes = self.flux(q)
# Boundary conditions
q_bc_stressfree = BoundarizeOperator(self.boundaryconditions_tag['stressfree'])(q)
q_bc_stressfree_null = BoundarizeOperator(self.boundaryconditions_tag['stressfree'])(0)
q_bc_fixed = BoundarizeOperator(self.boundaryconditions_tag['fixed'])(q)
q_bc_fixed_null = BoundarizeOperator(self.boundaryconditions_tag['fixed'])(0)
all_tags_and_bcs = [
(self.boundaryconditions_tag['stressfree'], q_bc_stressfree, q_bc_stressfree_null),
(self.boundaryconditions_tag['fixed'], q_bc_fixed, q_bc_fixed_null)
]
bdry_tag_state_flux = [(tag, bc, self.bdry_flux(bc, bc_null, tag))
for tag, bc, bc_null in all_tags_and_bcs]
# Entire operator
nabla = make_nabla(self.dimensions)
result = (numpy.dot(nabla, fluxes) + InverseMassOperator()
* (self.flux_num(q, fluxes, bdry_tag_state_flux)))
result = self.add_sources(result)
return result
def local_derivatives(self, w=None):
"""Template for the spatial derivatives of the relevant components of E and H"""
e, h = self.split_eh(self.field_placeholder(w))
def e_curl(field):
return self.space_cross_e(nabla, field)
def h_curl(field):
return self.space_cross_h(nabla, field)
from hedge.optemplate import make_nabla
from hedge.tools import join_fields, count_subset
nabla = make_nabla(self.dimensions)
if self.current is not None:
from hedge.optemplate import make_vector_field
j = make_vector_field("j",
count_subset(self.get_eh_subset()[:3]))
else:
j = 0
# in conservation form: u_t + A u_x = 0
return join_fields(
(j - h_curl(h)),
e_curl(e)
)
def test_tri_diff_mat():
"""Check differentiation matrix along the coordinate axes on a disk
Uses sines as the function to differentiate.
"""
from hedge.mesh.generator import make_disk_mesh
from hedge.discretization.local import TriangleDiscretization
from math import sin, cos, sqrt
from hedge.optemplate import make_nabla
nabla = make_nabla(2)
for coord in [0, 1]:
mesh = make_disk_mesh()
discr = discr_class(make_disk_mesh(), TriangleDiscretization(4), debug=discr_class.noninteractive_debug_flags())
f = discr.interpolate_volume_function(lambda x, el: sin(3 * x[coord]))
df = discr.interpolate_volume_function(lambda x, el: 3 * cos(3 * x[coord]))
df_num = nabla[coord].apply(discr, f)
# discr.visualize_vtk("diff-err.vtk",
# [("f", f), ("df", df), ("df_num", df_num), ("error", error)])
linf_error = la.norm(df_num - df, numpy.Inf)
print linf_error
assert linf_error < 4e-5
def test_tri_diff_mat():
"""Check differentiation matrix along the coordinate axes on a disk
Uses sines as the function to differentiate.
"""
from hedge.mesh.generator import make_disk_mesh
from hedge.discretization.local import TriangleDiscretization
from math import sin, cos
from hedge.optemplate import make_nabla
nabla = make_nabla(2)
for coord in [0, 1]:
discr = discr_class(make_disk_mesh(),
TriangleDiscretization(4),
debug=discr_class.noninteractive_debug_flags())
f = discr.interpolate_volume_function(lambda x, el: sin(3 * x[coord]))
df = discr.interpolate_volume_function(
lambda x, el: 3 * cos(3 * x[coord]))
df_num = nabla[coord].apply(discr, f)
#discr.visualize_vtk("diff-err.vtk",
#[("f", f), ("df", df), ("df_num", df_num), ("error", error)])
linf_error = la.norm(df_num - df, numpy.Inf)
print linf_error
assert linf_error < 4e-5
def op_template(self):
dim = self.dimensions
q = make_vector_field('q', self.len_q)
f2 = make_vector_field('f2', self.len_f2)
w = join_fields(q, f2)
dim_subset = (True,) * dim + (False,) * (3-dim)
def pad_vec(v, subset):
result = numpy.zeros((3,), dtype=object)
result[numpy.array(subset, dtype=bool)] = v
return result
sigma = pad_vec(make_vector_field('sigma', dim),dim_subset)
alpha = pad_vec(make_vector_field('alpha', dim),dim_subset)
kappa = pad_vec(make_vector_field('kappa', dim),dim_subset)
fluxes = self.flux(w, kappa)
# Boundary conditions
q_bc_stressfree = BoundarizeOperator(self.boundaryconditions_tag['stressfree'])(q)
q_bc_stressfree_null = BoundarizeOperator(self.boundaryconditions_tag['stressfree'])(0)
q_bc_fixed = BoundarizeOperator(self.boundaryconditions_tag['fixed'])(q)
q_bc_fixed_null = BoundarizeOperator(self.boundaryconditions_tag['fixed'])(0)
all_tags_and_bcs = [
(self.boundaryconditions_tag['stressfree'], q_bc_stressfree, q_bc_stressfree_null),
(self.boundaryconditions_tag['fixed'], q_bc_fixed, q_bc_fixed_null)
]
bdry_tag_state_flux = [(tag, bc, self.bdry_flux(bc, bc_null, tag))
for tag, bc, bc_null in all_tags_and_bcs]
# Entire operator
nabla = make_nabla(dim)
res_q = (numpy.dot(nabla, fluxes) + InverseMassOperator()
* (self.flux_num(q, fluxes, bdry_tag_state_flux)))
res_q = self.add_sources(res_q)
F2 = self.F2(w)
P = self.P(q)
v = self.v(q)
if dim == 1:
F = [P[0],v[0]]
elif dim == 2:
F = [P[0],P[2],v[0],v[1],
P[2],P[1],v[1],v[0]]
elif dim == 3:
F = [P[0],P[5],P[4],v[0],v[2],v[1],
P[5],P[1],P[3],v[1],v[2],v[0],
P[4],P[3],P[2],v[2],v[1],v[0]]
else:
raise ValueError("Invalid dimension")
res_f2 = numpy.zeros(self.len_f2, dtype=object)
for i in range(dim):
for j in range(dim*2):
res_f2[i*dim*2+j] = (-1)*F2[i*dim*2+j]*alpha[i]-sigma[i]/kappa[i]*(F2[i*dim*2+j]+F[i*dim*2+j])
return join_fields(res_q, res_f2)
def op_template(self):
from hedge.mesh import TAG_ALL
from hedge.optemplate import Field, BoundaryPair, \
make_nabla, InverseMassOperator, get_flux_operator
u = Field("u")
bc = Field("bc")
nabla = make_nabla(self.dimensions)
flux_op = get_flux_operator(self.flux())
return nabla * u - InverseMassOperator()(
flux_op(u) + flux_op(BoundaryPair(u, bc, TAG_ALL)))
def op_template(self):
from hedge.mesh import TAG_ALL
from hedge.optemplate import Field, BoundaryPair, \
make_nabla, InverseMassOperator, get_flux_operator
u = Field("u")
bc = Field("bc")
nabla = make_nabla(self.dimensions)
flux_op = get_flux_operator(self.flux())
return nabla*u - InverseMassOperator()(
flux_op(u) +
flux_op(BoundaryPair(u, bc, TAG_ALL)))
def op_template(self):
from hedge.optemplate import Field, BoundaryPair, \
get_flux_operator, make_nabla, InverseMassOperator
u = Field("u")
bc_in = Field("bc_in")
nabla = make_nabla(self.dimensions)
m_inv = InverseMassOperator()
flux_op = get_flux_operator(self.flux())
return (-numpy.dot(self.v, nabla * u) + m_inv(
flux_op(u) + flux_op(BoundaryPair(u, bc_in, self.inflow_tag))))
def op_template(self, w=None):
"""The full operator template - the high level description of
the nonlinear mechanics operator.
Combines the relevant operator templates for spatial
derivatives, flux, boundary conditions etc.
NOTE: Only boundary conditions allowed currently are homogenous
dirichlet and neumann, and I'm not sure dirichlet is done
properly
"""
from hedge.optemplate import InverseMassOperator, Field, \
make_vector_field
from hedge.tools import join_fields
w = self.field_placeholder(w)
u,v = self.split_vars(w)
from hedge.optemplate import make_nabla
nabla = make_nabla(self.dimensions)
ij = 0
F = [0,]*9
for i in range(self.dimensions):
for j in range(self.dimensions):
F[ij] = nabla[j](u[i])
if i == j:
F[ij] = F[ij] + 1
ij = ij + 1
w = join_fields(u,v,F)
flux_w = w
from hedge.optemplate import BoundaryPair, get_flux_operator
flux_op = get_flux_operator(self.flux(self.beta, is_dirich=False))
d_flux_op = get_flux_operator(self.flux(self.beta, is_dirich=True))
from hedge.optemplate import make_normal, BoundarizeOperator
dir_normal = make_normal(self.dirichlet_tag, self.dimensions)
dir_bc = self.dirichlet_bc(w)
return self.local_derivatives(w) \
- (flux_op(flux_w) +
d_flux_op(BoundaryPair(flux_w, dir_bc, self.dirichlet_tag))
)
def max_eigenvalue(self, t, fields=None, discr=None):
"""Return the wave speed in the material"""
u,v = self.split_vars(fields)
from hedge.optemplate import make_nabla
nabla = make_nabla(self.dimensions)
dim = self.dimensions
F = [0,]*dim*dim
ij = 0
for i in range(self.dimensions):
for j in range(self.dimensions):
F[ij] = nabla[i]*u[j]
if i == j:
F[ij] = F[ij] + 1
ij = ij + 1
speed = self.material.celerity(F, self.dimensions)
max_speed = discr.nodewise_max(speed)
return max_speed * self.beta
def op_template(self):
from hedge.optemplate import Field, BoundaryPair, \
get_flux_operator, make_nabla, InverseMassOperator
u = Field("u")
bc_in = Field("bc_in")
nabla = make_nabla(self.dimensions)
m_inv = InverseMassOperator()
flux_op = get_flux_operator(self.flux())
return (
-numpy.dot(self.v, nabla*u)
+ m_inv(
flux_op(u)
+ flux_op(BoundaryPair(u, bc_in, self.inflow_tag))))
def calculate_piola(self,u=None):
from hedge.optemplate import make_nabla
from hedge.optemplate import make_vector_field
u = make_vector_field('u', 3)
nabla = make_nabla(self.dimensions)
ij = 0
F = [0,]*9
for i in range(self.dimensions):
for j in range(self.dimensions):
F[ij] = nabla[i](u[j])
if i == j:
F[ij] = F[ij] + 1
ij = ij + 1
return make_obj_array(self.material.stress(F, self.dimensions))
def local_derivatives(self, w=None):
"""Template for the spatial derivatives of the relevant components of
:math:`E` and :math:`H`
"""
e, h = self.split_eh(self.field_placeholder(w))
def e_curl(field):
return self.space_cross_e(nabla, field)
def h_curl(field):
return self.space_cross_h(nabla, field)
from hedge.optemplate import make_nabla
from hedge.tools import join_fields
nabla = make_nabla(self.dimensions)
# in conservation form: u_t + A u_x = 0
return join_fields((self.current - h_curl(h)), e_curl(e))
def op_template(self):
from hedge.mesh import TAG_ALL
from hedge.optemplate import make_sym_vector, BoundaryPair, \
get_flux_operator, make_nabla, InverseMassOperator
nabla = make_nabla(self.dimensions)
m_inv = InverseMassOperator()
v = make_sym_vector("v", self.arg_count)
bc = make_sym_vector("bc", self.arg_count)
local_op_result = 0
idx = 0
for i, i_enabled in enumerate(self.subset):
if i_enabled and i < self.dimensions:
local_op_result += nabla[i] * v[idx]
idx += 1
flux_op = get_flux_operator(self.flux())
return local_op_result - m_inv(
flux_op(v) + flux_op(BoundaryPair(v, bc, TAG_ALL)))
def local_derivatives(self, w=None):
"""Template for the spatial derivatives of the relevant components of
:math:`E` and :math:`H`
"""
e, h = self.split_eh(self.field_placeholder(w))
def e_curl(field):
return self.space_cross_e(nabla, field)
def h_curl(field):
return self.space_cross_h(nabla, field)
from hedge.optemplate import make_nabla
from hedge.tools import join_fields
nabla = make_nabla(self.dimensions)
# in conservation form: u_t + A u_x = 0
return join_fields(
(self.current - h_curl(h)),
e_curl(e)
)
def op_template(self):
from hedge.mesh import TAG_ALL
from hedge.optemplate import make_vector_field, BoundaryPair, \
get_flux_operator, make_nabla, InverseMassOperator
nabla = make_nabla(self.dimensions)
m_inv = InverseMassOperator()
v = make_vector_field("v", self.arg_count)
bc = make_vector_field("bc", self.arg_count)
local_op_result = 0
idx = 0
for i, i_enabled in enumerate(self.subset):
if i_enabled and i < self.dimensions:
local_op_result += nabla[i]*v[idx]
idx += 1
flux_op = get_flux_operator(self.flux())
return local_op_result - m_inv(
flux_op(v) +
flux_op(BoundaryPair(v, bc, TAG_ALL)))
def op_template(self):
from hedge.tools import join_fields
from hedge.optemplate import Field, make_vector_field, BoundaryPair, \
BoundarizeOperator, make_normal, get_flux_operator, \
make_nabla, InverseMassOperator
w = make_vector_field("w", self.component_count)
e, h, phi = self.split_ehphi(w)
rho = Field("rho")
# local part ----------------------------------------------------------
nabla = make_nabla(self.maxwell_op.dimensions)
# in conservation form: u_t + A u_x = 0
# we're describing the A u_x part, the sign gets reversed
# below.
max_local_op = join_fields(self.maxwell_op.local_derivatives(w), 0)
c = self.maxwell_op.c
chi = self.chi
hyp_local_operator = max_local_op + join_fields(
c * chi * (nabla * phi),
0 * h,
c * chi * numpy.dot(nabla, e) -
c * chi * rho / self.maxwell_op.epsilon
# sign gets reversed below, so this is actually
# the decay it advertises to be.
+ self.phi_decay * phi)
# BCs -----------------------------------------------------------------
pec_tag = self.maxwell_op.pec_tag
pec_e = BoundarizeOperator(pec_tag)(e)
pec_h = BoundarizeOperator(pec_tag)(h)
pec_phi = BoundarizeOperator(pec_tag)(phi)
pec_n = make_normal(pec_tag, self.maxwell_op.dimensions)
bc = "prev"
print "HYP CLEAN BC", bc
if bc == "char":
# see hedge/doc/maxima/eclean.mac for derivation
pec_bc = join_fields(
-pec_e + 3 / 2 * pec_n * numpy.dot(pec_n, pec_e) +
1 / 2 * pec_phi * pec_n, pec_h,
1 / 2 * (pec_phi + numpy.dot(pec_n, pec_e)))
elif bc == "invent":
# see hedge/doc/maxima/eclean.mac for derivation
pec_bc = join_fields(-pec_e + 2 * pec_n * numpy.dot(pec_n, pec_e),
pec_h, pec_phi)
elif bc == "munz":
# Munz et al
pec_bc = join_fields(-pec_e, pec_h,
pec_phi - numpy.dot(pec_n, pec_e))
elif bc == "prev":
# previous condition
pec_bc = join_fields(-pec_e + 2 * pec_n * numpy.dot(pec_n, pec_e),
pec_h, -pec_phi)
# assemble operator ---------------------------------------------------
flux_op = get_flux_operator(self.flux())
return -hyp_local_operator + InverseMassOperator()(
flux_op(w) + flux_op(BoundaryPair(w, pec_bc, pec_tag)))
def op_template(self, with_sensor=False):
from hedge.optemplate import \
Field, \
make_sym_vector, \
BoundaryPair, \
get_flux_operator, \
make_nabla, \
InverseMassOperator, \
BoundarizeOperator
d = self.dimensions
w = make_sym_vector("w", d+1)
u = w[0]
v = w[1:]
from hedge.tools import join_fields
flux_w = join_fields(self.c, w)
# {{{ boundary conditions
from hedge.tools import join_fields
# Dirichlet
dir_c = BoundarizeOperator(self.dirichlet_tag) * self.c
dir_u = BoundarizeOperator(self.dirichlet_tag) * u
dir_v = BoundarizeOperator(self.dirichlet_tag) * v
dir_bc = join_fields(dir_c, -dir_u, dir_v)
# Neumann
neu_c = BoundarizeOperator(self.neumann_tag) * self.c
neu_u = BoundarizeOperator(self.neumann_tag) * u
neu_v = BoundarizeOperator(self.neumann_tag) * v
neu_bc = join_fields(neu_c, neu_u, -neu_v)
# Radiation
from hedge.optemplate import make_normal
rad_normal = make_normal(self.radiation_tag, d)
rad_c = BoundarizeOperator(self.radiation_tag) * self.c
rad_u = BoundarizeOperator(self.radiation_tag) * u
rad_v = BoundarizeOperator(self.radiation_tag) * v
rad_bc = join_fields(
rad_c,
0.5*(rad_u - self.time_sign*np.dot(rad_normal, rad_v)),
0.5*rad_normal*(np.dot(rad_normal, rad_v) - self.time_sign*rad_u)
)
# }}}
# {{{ diffusion -------------------------------------------------------
from pytools.obj_array import with_object_array_or_scalar
def make_diffusion(arg):
if with_sensor or (
self.diffusion_coeff is not None and self.diffusion_coeff != 0):
if self.diffusion_coeff is None:
diffusion_coeff = 0
else:
diffusion_coeff = self.diffusion_coeff
if with_sensor:
diffusion_coeff += Field("sensor")
from hedge.second_order import SecondDerivativeTarget
# strong_form here allows the reuse the value of grad u.
grad_tgt = SecondDerivativeTarget(
self.dimensions, strong_form=True,
operand=arg)
self.diffusion_scheme.grad(grad_tgt, bc_getter=None,
dirichlet_tags=[], neumann_tags=[])
div_tgt = SecondDerivativeTarget(
self.dimensions, strong_form=False,
operand=diffusion_coeff*grad_tgt.minv_all)
self.diffusion_scheme.div(div_tgt,
bc_getter=None,
dirichlet_tags=[], neumann_tags=[])
return div_tgt.minv_all
else:
return 0
# }}}
# entire operator -----------------------------------------------------
nabla = make_nabla(d)
flux_op = get_flux_operator(self.flux())
return (
- join_fields(
- self.time_sign*self.c*np.dot(nabla, v) - make_diffusion(u)
+ self.source,
-self.time_sign*self.c*(nabla*u) - with_object_array_or_scalar(
make_diffusion, v)
)
+
InverseMassOperator() * (
flux_op(flux_w)
+ flux_op(BoundaryPair(flux_w, dir_bc, self.dirichlet_tag))
+ flux_op(BoundaryPair(flux_w, neu_bc, self.neumann_tag))
+ flux_op(BoundaryPair(flux_w, rad_bc, self.radiation_tag))
))
def op_template(self):
from hedge.optemplate import \
make_sym_vector, \
BoundaryPair, \
get_flux_operator, \
make_nabla, \
InverseMassOperator, \
BoundarizeOperator
d = self.dimensions
w = make_sym_vector("w", d+1)
u = w[0]
v = w[1:]
# boundary conditions -------------------------------------------------
from hedge.tools import join_fields
# dirichlet BCs -------------------------------------------------------
from hedge.optemplate import normal, Field
dir_u = BoundarizeOperator(self.dirichlet_tag) * u
dir_v = BoundarizeOperator(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 = Field("dir_bc_u")
dir_bc = join_fields(2*dir_g - dir_u, dir_v)
else:
dir_bc = join_fields(-dir_u, dir_v)
# neumann BCs ---------------------------------------------------------
neu_u = BoundarizeOperator(self.neumann_tag) * u
neu_v = BoundarizeOperator(self.neumann_tag) * v
neu_bc = join_fields(neu_u, -neu_v)
# radiation BCs -------------------------------------------------------
rad_normal = normal(self.radiation_tag, d)
rad_u = BoundarizeOperator(self.radiation_tag) * u
rad_v = BoundarizeOperator(self.radiation_tag) * v
rad_bc = 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)
)
# entire operator -----------------------------------------------------
nabla = make_nabla(d)
flux_op = get_flux_operator(self.flux())
from hedge.tools import join_fields
result = (
- join_fields(
-self.c*np.dot(nabla, v),
-self.c*(nabla*u)
)
+
InverseMassOperator() * (
flux_op(w)
+ flux_op(BoundaryPair(w, dir_bc, self.dirichlet_tag))
+ flux_op(BoundaryPair(w, neu_bc, self.neumann_tag))
+ flux_op(BoundaryPair(w, rad_bc, self.radiation_tag))
))
result[0] += self.source_f
return result
def op_template(self, with_sensor):
from hedge.optemplate import (Field, make_stiffness_t, make_nabla,
InverseMassOperator,
ElementwiseMaxOperator,
get_flux_operator)
from hedge.optemplate.operators import (
QuadratureGridUpsampler, QuadratureInteriorFacesGridUpsampler)
to_quad = QuadratureGridUpsampler("quad")
to_if_quad = QuadratureInteriorFacesGridUpsampler("quad")
u = Field("u")
u0 = Field("u0")
# boundary conditions -------------------------------------------------
minv_st = make_stiffness_t(self.dimensions)
nabla = make_nabla(self.dimensions)
m_inv = InverseMassOperator()
def flux(u):
return u**2 / 2
#return u0*u
emax_u = self.characteristic_velocity_optemplate(u)
from hedge.flux.tools import make_lax_friedrichs_flux
from pytools.obj_array import make_obj_array
num_flux = make_lax_friedrichs_flux(
#u0,
to_if_quad(emax_u),
make_obj_array([to_if_quad(u)]),
[make_obj_array([flux(to_if_quad(u))])],
[],
strong=False)[0]
from hedge.second_order import SecondDerivativeTarget
if self.viscosity is not None or with_sensor:
viscosity_coeff = 0
if with_sensor:
viscosity_coeff += Field("sensor")
if isinstance(self.viscosity, float):
viscosity_coeff += self.viscosity
elif self.viscosity is None:
pass
else:
raise TypeError("unsupported type of viscosity coefficient")
# strong_form here allows IPDG to reuse the value of grad u.
grad_tgt = SecondDerivativeTarget(self.dimensions,
strong_form=True,
operand=u)
self.viscosity_scheme.grad(grad_tgt,
bc_getter=None,
dirichlet_tags=[],
neumann_tags=[])
div_tgt = SecondDerivativeTarget(self.dimensions,
strong_form=False,
operand=viscosity_coeff *
grad_tgt.minv_all)
self.viscosity_scheme.div(div_tgt,
bc_getter=None,
dirichlet_tags=[],
neumann_tags=[])
viscosity_bit = div_tgt.minv_all
else:
viscosity_bit = 0
return m_inv((minv_st[0](flux(to_quad(u)))) - num_flux) \
+ viscosity_bit
def test_2d_gauss_theorem():
"""Verify Gauss's theorem explicitly on a mesh"""
from hedge.mesh.generator import make_disk_mesh
from math import sin, cos, sqrt, exp, pi
from numpy import dot
mesh = make_disk_mesh()
order = 2
discr = discr_class(mesh, order=order, debug=discr_class.noninteractive_debug_flags())
ref_discr = discr_class(mesh, order=order)
from hedge.flux import make_normal, FluxScalarPlaceholder
normal = make_normal(discr.dimensions)
flux_f_ph = FluxScalarPlaceholder(0)
one_sided_x = flux_f_ph.int * normal[0]
one_sided_y = flux_f_ph.int * normal[1]
def f1(x, el):
return sin(3 * x[0]) + cos(3 * x[1])
def f2(x, el):
return sin(2 * x[0]) + cos(x[1])
from hedge.discretization import ones_on_volume
ones = ones_on_volume(discr)
f1_v = discr.interpolate_volume_function(f1)
f2_v = discr.interpolate_volume_function(f2)
from hedge.optemplate import BoundaryPair, Field, make_nabla, get_flux_operator
nabla = make_nabla(discr.dimensions)
diff_optp = nabla[0] * Field("f1") + nabla[1] * Field("f2")
divergence = nabla[0].apply(discr, f1_v) + nabla[1].apply(discr, f2_v)
int_div = discr.integral(divergence)
flux_optp = get_flux_operator(one_sided_x)(BoundaryPair(Field("f1"), Field("fz"))) + get_flux_operator(one_sided_y)(
BoundaryPair(Field("f2"), Field("fz"))
)
from hedge.mesh import TAG_ALL
bdry_val = discr.compile(flux_optp)(f1=f1_v, f2=f2_v, fz=discr.boundary_zeros(TAG_ALL))
ref_bdry_val = ref_discr.compile(flux_optp)(f1=f1_v, f2=f2_v, fz=discr.boundary_zeros(TAG_ALL))
boundary_int = dot(bdry_val, ones)
if False:
from hedge.visualization import SiloVisualizer
vis = SiloVisualizer(discr)
visf = vis.make_file("test")
from hedge.tools import make_obj_array
from hedge.mesh import TAG_ALL
vis.add_data(
visf,
[
("bdry", bdry_val),
("ref_bdry", ref_bdry_val),
("div", divergence),
("f", make_obj_array([f1_v, f2_v])),
("n", discr.volumize_boundary_field(discr.boundary_normals(TAG_ALL), TAG_ALL)),
],
expressions=[("bdiff", "bdry-ref_bdry")],
)
# print abs(boundary_int-int_div)
assert abs(boundary_int - int_div) < 5e-15
def op_template(self):
from hedge.optemplate import \
make_vector_field, \
BoundaryPair, \
get_flux_operator, \
make_nabla, \
InverseMassOperator, \
BoundarizeOperator
d = self.dimensions
w = make_vector_field("w", d + 1)
u = w[0]
v = w[1:]
# boundary conditions -------------------------------------------------
from hedge.tools import join_fields
# dirichlet BCs -------------------------------------------------------
from hedge.optemplate import make_normal, Field
dir_normal = make_normal(self.dirichlet_tag, d)
dir_u = BoundarizeOperator(self.dirichlet_tag) * u
dir_v = BoundarizeOperator(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 = Field("dir_bc_u")
dir_bc = join_fields(2 * dir_g - dir_u, dir_v)
else:
dir_bc = join_fields(-dir_u, dir_v)
# neumann BCs ---------------------------------------------------------
neu_u = BoundarizeOperator(self.neumann_tag) * u
neu_v = BoundarizeOperator(self.neumann_tag) * v
neu_bc = join_fields(neu_u, -neu_v)
# radiation BCs -------------------------------------------------------
from hedge.optemplate import make_normal
rad_normal = make_normal(self.radiation_tag, d)
rad_u = BoundarizeOperator(self.radiation_tag) * u
rad_v = BoundarizeOperator(self.radiation_tag) * v
rad_bc = join_fields(
0.5 * (rad_u - self.sign * numpy.dot(rad_normal, rad_v)), 0.5 *
rad_normal * (numpy.dot(rad_normal, rad_v) - self.sign * rad_u))
# entire operator -----------------------------------------------------
nabla = make_nabla(d)
flux_op = get_flux_operator(self.flux())
from hedge.tools import join_fields
result = (
-join_fields(-self.c * numpy.dot(nabla, v), -self.c * (nabla * u))
+ InverseMassOperator() *
(flux_op(w) + flux_op(BoundaryPair(w, dir_bc, self.dirichlet_tag))
+ flux_op(BoundaryPair(w, neu_bc, self.neumann_tag)) + flux_op(
BoundaryPair(w, rad_bc, self.radiation_tag))))
if self.source_f is not None:
result[0] += Field("source_u")
return result
def op_template(self, with_sensor):
from hedge.optemplate import (
Field,
make_stiffness_t,
make_nabla,
InverseMassOperator,
ElementwiseMaxOperator,
get_flux_operator)
from hedge.optemplate.operators import (
QuadratureGridUpsampler,
QuadratureInteriorFacesGridUpsampler)
to_quad = QuadratureGridUpsampler("quad")
to_if_quad = QuadratureInteriorFacesGridUpsampler("quad")
u = Field("u")
u0 = Field("u0")
# boundary conditions -------------------------------------------------
minv_st = make_stiffness_t(self.dimensions)
nabla = make_nabla(self.dimensions)
m_inv = InverseMassOperator()
def flux(u):
return u**2/2
#return u0*u
emax_u = self.characteristic_velocity_optemplate(u)
from hedge.flux.tools import make_lax_friedrichs_flux
from pytools.obj_array import make_obj_array
num_flux = make_lax_friedrichs_flux(
#u0,
to_if_quad(emax_u),
make_obj_array([to_if_quad(u)]),
[make_obj_array([flux(to_if_quad(u))])],
[], strong=False)[0]
from hedge.second_order import SecondDerivativeTarget
if self.viscosity is not None or with_sensor:
viscosity_coeff = 0
if with_sensor:
viscosity_coeff += Field("sensor")
if isinstance(self.viscosity, float):
viscosity_coeff += self.viscosity
elif self.viscosity is None:
pass
else:
raise TypeError("unsupported type of viscosity coefficient")
# strong_form here allows IPDG to reuse the value of grad u.
grad_tgt = SecondDerivativeTarget(
self.dimensions, strong_form=True,
operand=u)
self.viscosity_scheme.grad(grad_tgt, bc_getter=None,
dirichlet_tags=[], neumann_tags=[])
div_tgt = SecondDerivativeTarget(
self.dimensions, strong_form=False,
operand=viscosity_coeff*grad_tgt.minv_all)
self.viscosity_scheme.div(div_tgt,
bc_getter=None,
dirichlet_tags=[], neumann_tags=[])
viscosity_bit = div_tgt.minv_all
else:
viscosity_bit = 0
return m_inv((minv_st[0](flux(to_quad(u)))) - num_flux) \
+ viscosity_bit
def op_template(self):
from hedge.tools import join_fields
from hedge.optemplate import Field, make_vector_field, BoundaryPair, \
BoundarizeOperator, make_normal, get_flux_operator, \
make_nabla, InverseMassOperator
w = make_vector_field("w", self.component_count)
e, h, phi = self.split_ehphi(w)
rho = Field("rho")
# local part ----------------------------------------------------------
nabla = make_nabla(self.maxwell_op.dimensions)
# in conservation form: u_t + A u_x = 0
# we're describing the A u_x part, the sign gets reversed
# below.
max_local_op = join_fields(self.maxwell_op.local_derivatives(w), 0)
c = self.maxwell_op.c
chi = self.chi
hyp_local_operator = max_local_op + join_fields(
c*chi*(nabla*phi),
0*h,
c*chi*numpy.dot(nabla, e)
- c*chi*rho/self.maxwell_op.epsilon
# sign gets reversed below, so this is actually
# the decay it advertises to be.
+ self.phi_decay*phi
)
# BCs -----------------------------------------------------------------
pec_tag = self.maxwell_op.pec_tag
pec_e = BoundarizeOperator(pec_tag)(e)
pec_h = BoundarizeOperator(pec_tag)(h)
pec_phi = BoundarizeOperator(pec_tag)(phi)
pec_n = make_normal(pec_tag, self.maxwell_op.dimensions)
bc = "prev"
print "HYP CLEAN BC", bc
if bc == "char":
# see hedge/doc/maxima/eclean.mac for derivation
pec_bc = join_fields(
-pec_e
+ 3/2 * pec_n * numpy.dot(pec_n, pec_e)
+ 1/2 * pec_phi * pec_n ,
pec_h,
1/2*(pec_phi+numpy.dot(pec_n, pec_e))
)
elif bc == "invent":
# see hedge/doc/maxima/eclean.mac for derivation
pec_bc = join_fields(
-pec_e
+ 2 * pec_n * numpy.dot(pec_n, pec_e),
pec_h,
pec_phi)
elif bc == "munz":
# Munz et al
pec_bc = join_fields(
-pec_e,
pec_h,
pec_phi-numpy.dot(pec_n, pec_e))
elif bc == "prev":
# previous condition
pec_bc = join_fields(
-pec_e
+2*pec_n * numpy.dot(pec_n, pec_e),
pec_h,
-pec_phi)
# assemble operator ---------------------------------------------------
flux_op = get_flux_operator(self.flux())
return -hyp_local_operator + InverseMassOperator()(
flux_op(w)
+ flux_op(BoundaryPair(w, pec_bc, pec_tag)))
def op_template(self, with_sensor=False):
from hedge.optemplate import \
Field, \
make_vector_field, \
BoundaryPair, \
get_flux_operator, \
make_nabla, \
InverseMassOperator, \
BoundarizeOperator
d = self.dimensions
w = make_vector_field("w", d + 1)
u = w[0]
v = w[1:]
from hedge.tools import join_fields
c = Field("c")
flux_w = join_fields(c, w)
# {{{ boundary conditions
from hedge.flux import make_normal
normal = make_normal(d)
from hedge.tools import join_fields
# Dirichlet
dir_c = BoundarizeOperator(self.dirichlet_tag) * c
dir_u = BoundarizeOperator(self.dirichlet_tag) * u
dir_v = BoundarizeOperator(self.dirichlet_tag) * v
dir_bc = join_fields(dir_c, -dir_u, dir_v)
# Neumann
neu_c = BoundarizeOperator(self.neumann_tag) * c
neu_u = BoundarizeOperator(self.neumann_tag) * u
neu_v = BoundarizeOperator(self.neumann_tag) * v
neu_bc = join_fields(neu_c, neu_u, -neu_v)
# Radiation
from hedge.optemplate import make_normal
rad_normal = make_normal(self.radiation_tag, d)
rad_c = BoundarizeOperator(self.radiation_tag) * c
rad_u = BoundarizeOperator(self.radiation_tag) * u
rad_v = BoundarizeOperator(self.radiation_tag) * v
rad_bc = join_fields(
rad_c,
0.5 * (rad_u - self.time_sign * numpy.dot(rad_normal, rad_v)),
0.5 * rad_normal *
(numpy.dot(rad_normal, rad_v) - self.time_sign * rad_u))
# }}}
# {{{ diffusion -------------------------------------------------------
from pytools.obj_array import with_object_array_or_scalar
def make_diffusion(arg):
if with_sensor or (self.diffusion_coeff is not None
and self.diffusion_coeff != 0):
if self.diffusion_coeff is None:
diffusion_coeff = 0
else:
diffusion_coeff = self.diffusion_coeff
if with_sensor:
diffusion_coeff += Field("sensor")
from hedge.second_order import SecondDerivativeTarget
# strong_form here allows the reuse the value of grad u.
grad_tgt = SecondDerivativeTarget(
self.dimensions, strong_form=True, operand=arg)
self.diffusion_scheme.grad(
grad_tgt,
bc_getter=None,
dirichlet_tags=[],
neumann_tags=[])
div_tgt = SecondDerivativeTarget(
self.dimensions,
strong_form=False,
operand=diffusion_coeff * grad_tgt.minv_all)
self.diffusion_scheme.div(
div_tgt,
bc_getter=None,
dirichlet_tags=[],
neumann_tags=[])
return div_tgt.minv_all
else:
return 0
# }}}
# entire operator -----------------------------------------------------
nabla = make_nabla(d)
flux_op = get_flux_operator(self.flux())
return (
-join_fields(
-self.time_sign * c * numpy.dot(nabla, v) - make_diffusion(u),
-self.time_sign * c *
(nabla * u) - with_object_array_or_scalar(make_diffusion, v)) +
InverseMassOperator() * (flux_op(flux_w) + flux_op(
BoundaryPair(flux_w, dir_bc, self.dirichlet_tag)) + flux_op(
BoundaryPair(flux_w, neu_bc, self.neumann_tag)) + flux_op(
BoundaryPair(flux_w, rad_bc, self.radiation_tag))))
def test_2d_gauss_theorem():
"""Verify Gauss's theorem explicitly on a mesh"""
from hedge.mesh.generator import make_disk_mesh
from math import sin, cos
from numpy import dot
mesh = make_disk_mesh()
order = 2
discr = discr_class(mesh,
order=order,
debug=discr_class.noninteractive_debug_flags())
ref_discr = discr_class(mesh, order=order)
from hedge.flux import make_normal, FluxScalarPlaceholder
normal = make_normal(discr.dimensions)
flux_f_ph = FluxScalarPlaceholder(0)
one_sided_x = flux_f_ph.int * normal[0]
one_sided_y = flux_f_ph.int * normal[1]
def f1(x, el):
return sin(3 * x[0]) + cos(3 * x[1])
def f2(x, el):
return sin(2 * x[0]) + cos(x[1])
from hedge.discretization import ones_on_volume
ones = ones_on_volume(discr)
f1_v = discr.interpolate_volume_function(f1)
f2_v = discr.interpolate_volume_function(f2)
from hedge.optemplate import BoundaryPair, Field, make_nabla, \
get_flux_operator
nabla = make_nabla(discr.dimensions)
divergence = nabla[0].apply(discr, f1_v) + nabla[1].apply(discr, f2_v)
int_div = discr.integral(divergence)
flux_optp = (
get_flux_operator(one_sided_x)(BoundaryPair(Field("f1"),
Field("fz"))) +
get_flux_operator(one_sided_y)(BoundaryPair(Field("f2"), Field("fz"))))
from hedge.mesh import TAG_ALL
bdry_val = discr.compile(flux_optp)(f1=f1_v,
f2=f2_v,
fz=discr.boundary_zeros(TAG_ALL))
ref_bdry_val = ref_discr.compile(flux_optp)(
f1=f1_v, f2=f2_v, fz=discr.boundary_zeros(TAG_ALL))
boundary_int = dot(bdry_val, ones)
if False:
from hedge.visualization import SiloVisualizer
vis = SiloVisualizer(discr)
visf = vis.make_file("test")
from hedge.tools import make_obj_array
from hedge.mesh import TAG_ALL
vis.add_data(visf, [
("bdry", bdry_val),
("ref_bdry", ref_bdry_val),
("div", divergence),
("f", make_obj_array([f1_v, f2_v])),
("n",
discr.volumize_boundary_field(discr.boundary_normals(TAG_ALL),
TAG_ALL)),
],
expressions=[("bdiff", "bdry-ref_bdry")])
#print abs(boundary_int-int_div)
assert abs(boundary_int - int_div) < 5e-15