from __future__ import division, absolute_import, print_function
__copyright__ = "Copyright (C) 2010-2013 Andreas Kloeckner"
__license__ = """
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in
all copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
THE SOFTWARE.
"""
__doc__ = """
.. autoclass:: L2WeightedPDEOperator
.. autoclass:: DirichletOperator
.. autoclass:: NeumannOperator
.. autoclass:: BiharmonicClampedPlateOperator
"""
from pytential import sym
from pytential.symbolic.primitives import (
cse,
sqrt_jac_q_weight, QWeight, area_element)
from sumpy.kernel import DirectionalSourceDerivative
import numpy as np # noqa
# {{{ L^2 weighting
[docs]class L2WeightedPDEOperator(object):
def __init__(self, kernel, use_l2_weighting):
self.kernel = kernel
self.use_l2_weighting = use_l2_weighting
if not use_l2_weighting:
from warnings import warn
warn("should use L2 weighting in %s" % type(self).__name__,
stacklevel=3)
def get_weight(self, dofdesc=None):
if self.use_l2_weighting:
return cse(area_element(self.kernel.dim, dofdesc=dofdesc)
* QWeight(dofdesc=dofdesc))
else:
return 1
def get_sqrt_weight(self, dofdesc=None):
if self.use_l2_weighting:
return sqrt_jac_q_weight(self.kernel.dim, dofdesc=dofdesc)
else:
return 1
def prepare_rhs(self, b):
return self.get_sqrt_weight()*b
def get_density_var(self, name):
"""
Returns a symbolic variable/array corresponding to the density with the
given name.
"""
return sym.var(name)
# }}}
# {{{ dirichlet
[docs]class DirichletOperator(L2WeightedPDEOperator):
"""IE operator and field representation for solving Dirichlet boundary
value problems with scalar kernels (e.g.
:class:`sumpy.kernel.LaplaceKernel`,
:class:`sumpy.kernel.HelmholtzKernel`, :class:`sumpy.kernel.YukawaKernel`)
.. note ::
When testing this as a potential matcher, note that it can only
access potentials that come from charge distributions having *no* net
charge. (This is true at least in 2D.)
.. automethod:: is_unique_only_up_to_constant
.. automethod:: representation
.. automethod:: operator
"""
def __init__(self, kernel, loc_sign, alpha=None, use_l2_weighting=False,
kernel_arguments=None):
"""
:arg loc_sign: +1 for exterior, -1 for interior
:arg alpha: the coefficient for the combined-field representation
Set to 0 for Laplace.
"""
assert loc_sign in [-1, 1]
from sumpy.kernel import Kernel, LaplaceKernel
assert isinstance(kernel, Kernel)
if kernel_arguments is None:
kernel_arguments = {}
self.kernel_arguments = kernel_arguments
self.loc_sign = loc_sign
L2WeightedPDEOperator.__init__(self, kernel, use_l2_weighting)
if alpha is None:
if isinstance(self.kernel, LaplaceKernel):
alpha = 0
else:
alpha = 1j
self.alpha = alpha
[docs] def is_unique_only_up_to_constant(self):
# No ones matrix needed in Helmholtz case, cf. Hackbusch Lemma 8.5.3.
from sumpy.kernel import LaplaceKernel
return isinstance(self.kernel, LaplaceKernel) and self.loc_sign > 0
[docs] def representation(self,
u, map_potentials=None, qbx_forced_limit=None):
sqrt_w = self.get_sqrt_weight()
inv_sqrt_w_u = cse(u/sqrt_w)
if map_potentials is None:
def map_potentials(x): # pylint:disable=function-redefined
return x
def S(density): # noqa
return sym.S(self.kernel, density,
kernel_arguments=self.kernel_arguments,
qbx_forced_limit=qbx_forced_limit)
def D(density): # noqa
return sym.D(self.kernel, density,
kernel_arguments=self.kernel_arguments,
qbx_forced_limit=qbx_forced_limit)
return (
self.alpha*map_potentials(S(inv_sqrt_w_u))
- map_potentials(D(inv_sqrt_w_u)))
[docs] def operator(self, u):
sqrt_w = self.get_sqrt_weight()
inv_sqrt_w_u = cse(u/sqrt_w)
if self.is_unique_only_up_to_constant():
# The exterior Dirichlet operator in this representation
# has a nullspace. The mean of the density must be matched
# to the desired solution separately. As is, this operator
# returns a mean that is not well-specified.
#
# See Hackbusch, http://books.google.com/books?id=Ssnf7SZB0ZMC
# Theorem 8.2.18b
amb_dim = self.kernel.dim
ones_contribution = (
sym.Ones() * sym.mean(amb_dim, amb_dim-1, inv_sqrt_w_u))
else:
ones_contribution = 0
return (-self.loc_sign*0.5*u
+ sqrt_w*(
self.alpha*sym.S(self.kernel, inv_sqrt_w_u,
qbx_forced_limit=+1, kernel_arguments=self.kernel_arguments)
- sym.D(self.kernel, inv_sqrt_w_u,
qbx_forced_limit="avg",
kernel_arguments=self.kernel_arguments)
+ ones_contribution))
# }}}
# {{{ neumann
[docs]class NeumannOperator(L2WeightedPDEOperator):
"""IE operator and field representation for solving Dirichlet boundary
value problems with scalar kernels (e.g.
:class:`sumpy.kernel.LaplaceKernel`,
:class:`sumpy.kernel.HelmholtzKernel`, :class:`sumpy.kernel.YukawaKernel`)
.. note ::
When testing this as a potential matcher, note that it can only
access potentials that come from charge distributions having *no* net
charge. (This is true at least in 2D.)
.. automethod:: is_unique_only_up_to_constant
.. automethod:: representation
.. automethod:: operator
"""
def __init__(self, kernel, loc_sign, alpha=None,
use_improved_operator=True,
laplace_kernel=0, use_l2_weighting=False,
kernel_arguments=None):
"""
:arg loc_sign: +1 for exterior, -1 for interior
:arg alpha: the coefficient for the combined-field representation
Set to 0 for Laplace.
:arg use_improved_operator: Whether to use the least singular
operator available
"""
assert loc_sign in [-1, 1]
from sumpy.kernel import Kernel, LaplaceKernel
assert isinstance(kernel, Kernel)
if kernel_arguments is None:
kernel_arguments = {}
self.kernel_arguments = kernel_arguments
self.loc_sign = loc_sign
self.laplace_kernel = LaplaceKernel(kernel.dim)
L2WeightedPDEOperator.__init__(self, kernel, use_l2_weighting)
if alpha is None:
if isinstance(self.kernel, LaplaceKernel):
alpha = 0
else:
alpha = 1j
self.alpha = alpha
self.use_improved_operator = use_improved_operator
[docs] def is_unique_only_up_to_constant(self):
from sumpy.kernel import LaplaceKernel
return isinstance(self.kernel, LaplaceKernel) and self.loc_sign < 0
[docs] def representation(self, u, map_potentials=None, qbx_forced_limit=None,
**kwargs):
sqrt_w = self.get_sqrt_weight()
inv_sqrt_w_u = cse(u/sqrt_w)
if map_potentials is None:
def map_potentials(x): # pylint:disable=function-redefined
return x
kwargs["qbx_forced_limit"] = qbx_forced_limit
kwargs["kernel_arguments"] = self.kernel_arguments
return (
map_potentials(
sym.S(self.kernel, inv_sqrt_w_u, **kwargs))
- self.alpha
* map_potentials(
sym.D(self.kernel,
sym.S(self.laplace_kernel, inv_sqrt_w_u),
**kwargs)))
[docs] def operator(self, u):
from sumpy.kernel import HelmholtzKernel, LaplaceKernel
sqrt_w = self.get_sqrt_weight()
inv_sqrt_w_u = cse(u/sqrt_w)
knl = self.kernel
lknl = self.laplace_kernel
knl_kwargs = {}
knl_kwargs["kernel_arguments"] = self.kernel_arguments
DpS0u = sym.Dp(knl, # noqa
cse(sym.S(lknl, inv_sqrt_w_u)),
**knl_kwargs)
if self.use_improved_operator:
Dp0S0u = -0.25*u + sym.Sp( # noqa
lknl, # noqa
sym.Sp(lknl, inv_sqrt_w_u, qbx_forced_limit="avg"),
qbx_forced_limit="avg")
if isinstance(self.kernel, HelmholtzKernel):
DpS0u = ( # noqa
sym.Dp(knl - lknl, # noqa
cse(sym.S(lknl, inv_sqrt_w_u, qbx_forced_limit=+1)),
qbx_forced_limit=+1, **knl_kwargs)
+ Dp0S0u)
elif isinstance(knl, LaplaceKernel):
DpS0u = Dp0S0u # noqa
else:
raise ValueError("no improved operator for %s known"
% self.kernel)
if self.is_unique_only_up_to_constant():
# The interior Neumann operator in this representation
# has a nullspace. The mean of the density must be matched
# to the desired solution separately. As is, this operator
# returns a mean that is not well-specified.
amb_dim = self.kernel.dim
ones_contribution = (
sym.Ones() * sym.mean(amb_dim, amb_dim-1, inv_sqrt_w_u))
else:
ones_contribution = 0
return (-self.loc_sign*0.5*u
+ sqrt_w*(
sym.Sp(knl, inv_sqrt_w_u, qbx_forced_limit="avg", **knl_kwargs)
- self.alpha*DpS0u
+ ones_contribution
))
[docs]class BiharmonicClampedPlateOperator:
r"""IE operator and field representation for solving clamped plate Biharmonic
equation where,
.. math::
\begin{align*}
\Delta^2 u &= 0 \text{ on } D \\
u &= g_1 \text{ on } \delta D \\
\frac{\partial u}{\partial \nu} &= g_2 \text{ on } \delta D.
\end{align*}
This operator assumes that the boundary data :math:`g_1, g_2` are
represented as column vectors and vertically stacked.
.. note :: This operator supports only interior problem.
Ref: Farkas, Peter. Mathematical foundations for fast algorithms for the
biharmonic equation. Technical Report 765, Department of Computer Science,
Yale University, 1990.
.. automethod:: representation
.. automethod:: operator
"""
def __init__(self, knl, loc_sign):
self.knl = knl
self.loc_sign = loc_sign
if loc_sign != -1:
raise ValueError("loc_sign!=-1 is not supported.")
def prepare_rhs(self, b):
return b
def get_density_var(self, name):
"""
Returns a symbolic variable of length 2 corresponding to the density with
the given name.
"""
return sym.make_sym_vector(name, 2)
[docs] def representation(self,
sigma, map_potentials=None, qbx_forced_limit=None):
if map_potentials is None:
def map_potentials(x): # pylint:disable=function-redefined
return x
def dv(knl):
return DirectionalSourceDerivative(knl, "normal_dir")
def dt(knl):
return DirectionalSourceDerivative(knl, "tangent_dir")
normal_dir = sym.normal(2).as_vector()
tangent_dir = np.array([-normal_dir[1], normal_dir[0]])
k1 = map_potentials(sym.S(dv(dv(dv(self.knl))), sigma[0],
kernel_arguments={"normal_dir": normal_dir},
qbx_forced_limit=qbx_forced_limit)) + \
3*map_potentials(sym.S(dv(dt(dt(self.knl))), sigma[0],
kernel_arguments={"normal_dir": normal_dir,
"tangent_dir": tangent_dir},
qbx_forced_limit=qbx_forced_limit))
k2 = -map_potentials(sym.S(dv(dv(self.knl)), sigma[1],
kernel_arguments={"normal_dir": normal_dir},
qbx_forced_limit=qbx_forced_limit)) + \
map_potentials(sym.S(dt(dt(self.knl)), sigma[1],
kernel_arguments={"tangent_dir": tangent_dir},
qbx_forced_limit=qbx_forced_limit))
return k1 + k2
[docs] def operator(self, sigma):
"""
Returns the two second kind integral equations.
"""
rep = self.representation(sigma, qbx_forced_limit='avg')
rep_diff = sym.normal_derivative(2, rep)
int_eq1 = sigma[0]/2 + rep
int_eq2 = -sym.mean_curvature(2)*sigma[0] + sigma[1]/2 + rep_diff
return np.array([int_eq1, int_eq2])
# }}}
# vim: foldmethod=marker