def pml_local_op(self, w):
sub_e, sub_h, sub_p, sub_q = self.split_ehpq(w)
e_subset = self.get_eh_subset()[0:3]
h_subset = self.get_eh_subset()[3:6]
dim_subset = (True,) * self.dimensions + (False,) * (3-self.dimensions)
def pad_vec(v, subset):
result = numpy.zeros((3,), dtype=object)
result[numpy.array(subset, dtype=bool)] = v
return result
from hedge.optemplate import make_vector_field
sig = pad_vec(
make_vector_field("sigma", self.dimensions),
dim_subset)
sig_prime = pad_vec(
make_vector_field("sigma_prime", self.dimensions),
dim_subset)
if self.add_decay:
tau = pad_vec(
make_vector_field("tau", self.dimensions),
dim_subset)
else:
tau = numpy.zeros((3,))
e = pad_vec(sub_e, e_subset)
h = pad_vec(sub_h, h_subset)
p = pad_vec(sub_p, dim_subset)
q = pad_vec(sub_q, dim_subset)
rhs = numpy.zeros(12, dtype=object)
for mx in range(3):
my = (mx+1) % 3
mz = (mx+2) % 3
from hedge.tools.mathematics import levi_civita
assert levi_civita((mx,my,mz)) == 1
rhs[mx] += -sig[my]/self.epsilon*(2*e[mx]+p[mx]) - 2*tau[my]/self.epsilon*e[mx]
rhs[my] += -sig[mx]/self.epsilon*(2*e[my]+p[my]) - 2*tau[mx]/self.epsilon*e[my]
rhs[3+mz] += 1/(self.epsilon*self.mu) * (
sig_prime[mx] * q[mx] - sig_prime[my] * q[my])
rhs[6+mx] += sig[my]/self.epsilon*e[mx]
rhs[6+my] += sig[mx]/self.epsilon*e[my]
rhs[9+mx] += -sig[mx]/self.epsilon*q[mx] - (e[my] + e[mz])
from hedge.tools import full_to_subset_indices
sub_idx = full_to_subset_indices(e_subset+h_subset+dim_subset+dim_subset)
return rhs[sub_idx]
def field_placeholder(self, w=None):
"A placeholder for u and v."
fld_cnt = self.dimensions*2
if w is None:
from hedge.optemplate import make_vector_field
w = make_vector_field("w", fld_cnt)
return w
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 apply_diff_tensor(v):
if isinstance(self.diffusion_tensor, numpy.ndarray):
sym_diff_tensor = self.diffusion_tensor
else:
sym_diff_tensor = (make_vector_field(
"diffusion", self.dimensions**2)
.reshape(self.dimensions, self.dimensions))
return numpy.dot(sym_diff_tensor, v)
def field_placeholder(self, w=None):
"A placeholder for E and H."
from hedge.tools import count_subset
fld_cnt = count_subset(self.get_eh_subset())
if w is None:
from hedge.optemplate import make_vector_field
w = make_vector_field("w", fld_cnt)
return w
def op_template(self, w=None):
from hedge.tools import count_subset
fld_cnt = count_subset(self.get_eh_subset())
if w is None:
from hedge.optemplate import make_vector_field
w = make_vector_field("w", fld_cnt+2*self.dimensions)
from hedge.tools import join_fields
return join_fields(
MaxwellOperator.op_template(self, w[:fld_cnt]),
numpy.zeros((2*self.dimensions,), dtype=object)
) + self.pml_local_op(w)
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 bind_characteristic_velocity(self, discr):
from hedge.optemplate.operators import (
ElementwiseMaxOperator)
from hedge.optemplate import make_vector_field
velocity_vec = make_vector_field("v", self.dimensions)
velocity = ElementwiseMaxOperator()(
numpy.dot(velocity_vec, velocity_vec)**0.5)
from hedge.optemplate import Field
compiled = discr.compile(velocity)
def do(t, u):
return compiled(v=self.advec_v.volume_interpolant(t, discr))
return do
def dirichlet_bc(self, w=None):
"""
Flux term for dirichlet (displacement) boundary conditions
"""
u, v = self.split_vars(self.field_placeholder(w))
if self.dirichlet_bc_data is not None:
from hedge.optemplate import make_vector_field
dir_field = cse(
-make_vector_field("dirichlet_bc", 3))
else:
dir_field = make_obj_array([0,]*3)
from hedge.tools import join_fields
return join_fields(dir_field, [0]*3, [0,]*9)
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 incident_bc(self, w=None):
"Flux terms for incident boundary conditions"
# NOTE: Untested for inhomogeneous materials, but would usually be
# physically meaningless anyway (are there exceptions to this?)
e, h = self.split_eh(self.field_placeholder(w))
if not self.fixed_material:
from warnings import warn
if self.incident_tag != hedge.mesh.TAG_NONE:
warn("Incident boundary conditions assume homogeneous"+
" background material, results may be unphysical")
from hedge.tools import count_subset
from hedge.tools import join_fields
fld_cnt = count_subset(self.get_eh_subset())
if self.incident_bc_data is not None:
from hedge.optemplate import make_vector_field
inc_field = cse(
-make_vector_field("incident_bc", fld_cnt))
else:
inc_field = make_obj_array([0]*fld_cnt)
return inc_field
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):
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):
# {{{ operator preliminaries ------------------------------------------
from hedge.optemplate import (Field, BoundaryPair, get_flux_operator,
make_stiffness_t, InverseMassOperator, make_vector_field,
ElementwiseMaxOperator, BoundarizeOperator)
from hedge.tools.symbolic import make_common_subexpression as cse
from hedge.optemplate.operators import (
QuadratureGridUpsampler,
QuadratureInteriorFacesGridUpsampler)
to_quad = QuadratureGridUpsampler("quad")
to_if_quad = QuadratureInteriorFacesGridUpsampler("quad")
from hedge.tools import join_fields, \
ptwise_dot
u = Field("u")
v = make_vector_field("v", self.dimensions)
c = ElementwiseMaxOperator()(ptwise_dot(1, 1, v, v))
quad_u = cse(to_quad(u))
quad_v = cse(to_quad(v))
w = join_fields(u, v, c)
quad_face_w = to_if_quad(w)
# }}}
# {{{ boundary conditions ---------------------------------------------
from hedge.mesh import TAG_ALL
bc_c = to_quad(BoundarizeOperator(TAG_ALL)(c))
bc_u = to_quad(Field("bc_u"))
bc_v = to_quad(BoundarizeOperator(TAG_ALL)(v))
if self.bc_u_f is "None":
bc_w = join_fields(0, bc_v, bc_c)
else:
bc_w = join_fields(bc_u, bc_v, bc_c)
minv_st = make_stiffness_t(self.dimensions)
m_inv = InverseMassOperator()
flux_op = get_flux_operator(self.flux())
# }}}
# {{{ diffusion -------------------------------------------------------
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 IPDG to reuse the value of grad u.
grad_tgt = SecondDerivativeTarget(
self.dimensions, strong_form=True,
operand=u)
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=[])
diffusion_part = div_tgt.minv_all
else:
diffusion_part = 0
# }}}
to_quad = QuadratureGridUpsampler("quad")
quad_u = cse(to_quad(u))
quad_v = cse(to_quad(v))
return m_inv(numpy.dot(minv_st, cse(quad_v*quad_u))
- (flux_op(quad_face_w)
+ flux_op(BoundaryPair(quad_face_w, bc_w, TAG_ALL)))) \
+ diffusion_part
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=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 f_bar(self):
from hedge.optemplate import make_vector_field
return make_vector_field("f_bar", len(self.method))
