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_sym_vector
sig = pad_vec(
make_sym_vector("sigma", self.dimensions),
dim_subset)
sig_prime = pad_vec(
make_sym_vector("sigma_prime", self.dimensions),
dim_subset)
if self.add_decay:
tau = pad_vec(
make_sym_vector("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 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_sym_vector
sig = pad_vec(make_sym_vector("sigma", self.dimensions), dim_subset)
sig_prime = pad_vec(make_sym_vector("sigma_prime", self.dimensions),
dim_subset)
if self.add_decay:
tau = pad_vec(make_sym_vector("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 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_sym_vector
w = make_sym_vector("w", fld_cnt)
return w
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_sym_vector
w = make_sym_vector("w", fld_cnt)
return w
def apply_diff_tensor(v):
if isinstance(self.diffusion_tensor, np.ndarray):
sym_diff_tensor = self.diffusion_tensor
else:
sym_diff_tensor = make_sym_vector("diffusion", self.dimensions ** 2).reshape(
self.dimensions, self.dimensions
)
return np.dot(sym_diff_tensor, v)
def apply_diff_tensor(v):
if isinstance(self.diffusion_tensor, np.ndarray):
sym_diff_tensor = self.diffusion_tensor
else:
sym_diff_tensor = (make_sym_vector("diffusion",
self.dimensions**2).reshape(
self.dimensions,
self.dimensions))
return np.dot(sym_diff_tensor, v)
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_sym_vector
w = make_sym_vector("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, 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_sym_vector
w = make_sym_vector("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_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 bind_characteristic_velocity(self, discr):
from hedge.optemplate.operators import (ElementwiseMaxOperator)
from hedge.optemplate import make_sym_vector
velocity_vec = make_sym_vector("v", self.dimensions)
velocity = ElementwiseMaxOperator()(numpy.dot(velocity_vec,
velocity_vec)**0.5)
compiled = discr.compile(velocity)
def do(t, u):
return compiled(v=self.advec_v.volume_interpolant(t, discr))
return do
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 bind_characteristic_velocity(self, discr):
from hedge.optemplate.operators import (
ElementwiseMaxOperator)
from hedge.optemplate import make_sym_vector
velocity_vec = make_sym_vector("v", self.dimensions)
velocity = ElementwiseMaxOperator()(
numpy.dot(velocity_vec, velocity_vec)**0.5)
compiled = discr.compile(velocity)
def do(t, u):
return compiled(v=self.advec_v.volume_interpolant(t, discr))
return do
def op_template(self, with_sensor=False):
# {{{ operator preliminaries ------------------------------------------
from hedge.optemplate import (Field, BoundaryPair, get_flux_operator,
make_stiffness_t, InverseMassOperator, make_sym_vector,
ElementwiseMaxOperator, BoundarizeOperator)
from hedge.optemplate.primitives 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_sym_vector("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 f_bar(self):
from hedge.optemplate import make_sym_vector
return make_sym_vector("f_bar", len(self.method))
def op_template(self, with_sensor=False):
# {{{ operator preliminaries ------------------------------------------
from hedge.optemplate import (Field, BoundaryPair, get_flux_operator,
make_stiffness_t, InverseMassOperator,
make_sym_vector, ElementwiseMaxOperator,
BoundarizeOperator)
from hedge.optemplate.primitives 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_sym_vector("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 f_bar(self):
from hedge.optemplate import make_sym_vector
return make_sym_vector("f_bar", len(self.method))
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