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laplace_disk_gaussian.py
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Fri, May 31, 05:17

laplace_disk_gaussian.py
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# Copyright (C) 2018 by the RROMPy authors
#
# This file is part of RROMPy.
#
# RROMPy is free software: you can redistribute it and/or modify
# it under the terms of the GNU Lesser General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# RROMPy is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU Lesser General Public License for more details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with RROMPy. If not, see <http://www.gnu.org/licenses/>.
#
import numpy as np
from numpy.polynomial.polynomial import polyfit as fit
import fenics as fen
from rrompy.utilities.base.types import Np1D, Tuple, FenExpr, paramVal
from .laplace_base_problem_engine import LaplaceBaseProblemEngine
from rrompy.solver.fenics import fenZERO, fenONE
from rrompy.utilities.base import verbosityManager as vbMng
from rrompy.utilities.poly_fitting.polynomial import (
hashDerivativeToIdx as hashD, hashIdxToDerivative as hashI)
from rrompy.solver.fenics import fenics2Vector
__all__ = ['LaplaceDiskGaussian']
class LaplaceDiskGaussian(LaplaceBaseProblemEngine):
"""
Solver for disk Laplace problems with parametric forcing term center.
- \Delta u = C exp(-.5 * ||\cdot - (mu, 0)||^2) in \Omega = B(0, 5)
u = 0 on \partial\Omega.
"""
def __init__(self, n:int, mu0 : paramVal = [0.],
degree_threshold : int = np.inf, verbosity : int = 10,
timestamp : bool = True):
super().__init__(mu0 = mu0, degree_threshold = degree_threshold,
verbosity = verbosity, timestamp = timestamp)
self.nbs = 19
self.forcingTermMu = np.nan
import mshr
mesh = mshr.generate_mesh(mshr.Circle(fen.Point(0., 0.), 5.), 3 * n)
self.V = fen.FunctionSpace(mesh, "P", 1)
def getForcingTerm(self, mu : paramVal = []) -> Tuple[FenExpr, FenExpr]:
"""Compute forcing term."""
mu = self.checkParameter(mu)
vbMng(self, "INIT", ("Assembling base expression for forcing term "
"at {}.").format(mu), 25)
x, y = fen.SpatialCoordinate(self.V.mesh())[:]
C = np.exp(-.5 * mu(0, 0) ** 2.)
CR, CI = np.real(C), np.imag(C)
f0 = (2 * np.pi) ** -.5 * fen.exp(-.5 * (x ** 2. + y ** 2.))
muR, muI = np.real(mu(0, 0)), np.imag(mu(0, 0))
f1R = fen.exp(muR * x) * fen.cos(muI * x)
f1I = fen.exp(muR * x) * fen.sin(muI * x)
forcingTerm = [f0 * (CR * f1R - CI * f1I) + fenZERO,
f0 * (CR * f1I + CI * f1R) + fenZERO]
vbMng(self, "DEL", "Done assembling base expression.", 25)
return forcingTerm
def b(self, mu : paramVal = [], der : int = 0,
homogeneized : bool = False) -> Np1D:
"""Assemble (derivative of) RHS of linear system."""
mu = self.checkParameter(mu)
if not hasattr(der, "__len__"): der = [der] * self.npar
derI = hashD(der)
nbsTot = self.nbsH if homogeneized else self.nbs
bs = self.bsH if homogeneized else self.bs
if homogeneized and self.mu0 != self.mu0BC:
self.liftDirichletData(self.mu0)
bDEIMCoeffs = None
for j in range(derI, nbsTot):
if bs[j] is None:
vbMng(self, "INIT", "Assembling forcing term b{}.".format(j),
20)
if bDEIMCoeffs is None:
bDEIM = np.empty((self.nbs, self.spacedim()),
dtype = np.complex)
muDEIM = 3. * np.linspace(0., 1., self.nbs // 2 + 1) ** 2.
muDEIM = np.concatenate((-muDEIM[:0:-1], muDEIM))
for jj, muD in enumerate(muDEIM):
fRe, fIm = self.getForcingTerm(muD)
parsRe = self.iterReduceQuadratureDegree(zip([fRe],
["forcingTerm{}Real".format(jj)]))
parsIm = self.iterReduceQuadratureDegree(zip([fIm],
["forcingTerm{}Imag".format(jj)]))
LR = fen.dot(fRe, self.v) * fen.dx
LI = fen.dot(fIm, self.v) * fen.dx
DBC0 = fen.DirichletBC(self.V, fenZERO,
self.DirichletBoundary)
bDEIM[jj] = (fenics2Vector(LR, parsRe, DBC0, 1)
+ 1.j * fenics2Vector(LI, parsIm, DBC0, 1))
bDEIMCoeffs = (fit(muDEIM / 3., bDEIM, self.nbs - 1).T
* np.power(3., - np.arange(self.nbs))).T
b = bDEIMCoeffs[j]
if homogeneized:
Ader = self.A(0, hashI(j, self.npar))
b -= Ader.dot(self.liftedDirichletDatum)
if homogeneized:
self.bsH[j] = b
else:
self.bs[j] = b
vbMng(self, "DEL", "Done assembling forcing term.", 20)
return self._assembleb(mu, der, derI, homogeneized)

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