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Created
Mon, Oct 28, 13:19
Size
5 MB
Mime Type
application/octet-stream
Expires
Wed, Oct 30, 13:19 (2 d)
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chunks
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Chunks
Handle
hqawP_QBKoke
Attached To
R6746 RationalROMPy
Total Chunks
2
Completed Chunks
2
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End
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0
4194304
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4194304
5618816
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This file is larger than 256 KB, so syntax highlighting was skipped.
diff --git a/examples/Data/.gitkeep b/examples/Data/.gitkeep
deleted file mode 100644
index e69de29..0000000
diff --git a/examples/Data/airfoil.xml b/examples/Data/airfoil.xml
new file mode 100644
index 0000000..d2bbb5d
--- /dev/null
+++ b/examples/Data/airfoil.xml
@@ -0,0 +1,14704 @@
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diff --git a/examples/Data/crack_coarse.xml b/examples/Data/crack_coarse.xml
new file mode 100644
index 0000000..8b53349
--- /dev/null
+++ b/examples/Data/crack_coarse.xml
@@ -0,0 +1,7505 @@
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diff --git a/examples/Data/crack_fine.xml b/examples/Data/crack_fine.xml
new file mode 100644
index 0000000..99d8708
--- /dev/null
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+ </cells>
+ </mesh>
+</dolfin>
diff --git a/examples/Python/HelmholtzAirfoilTaylor.py b/examples/Python/HelmholtzAirfoilTaylor.py
new file mode 100644
index 0000000..de4c530
--- /dev/null
+++ b/examples/Python/HelmholtzAirfoilTaylor.py
@@ -0,0 +1,257 @@
+#!/usr/bin/env python3
+# -*- coding: utf-8 -*-
+
+import fenics as fen
+import numpy as np
+import sympy as sp
+from context import FenicsHelmholtzScatteringEngine as HFS
+from context import FenicsHelmholtzScatteringAugmentedEngine as HFSA
+from context import FenicsHSEngine as HS
+from context import FenicsHSAugmentedEngine as HSA
+from context import ROMApproximantTaylorPade as TP
+from context import ROMApproximantTaylorRB as TRB
+from context import ROMApproximantLagrangePade as LP
+from context import ROMApproximantLagrangeRB as LRB
+from context import ROMApproximantSweeper as Sweeper
+from matplotlib import pyplot as plt
+plt.set_cmap('jet')
+from operator import itemgetter
+def subdict(d, ks):
+ return dict(zip(ks, itemgetter(*ks)(d)))
+
+####################
+
+test = "solve"
+#test = "Taylor"
+#test = "Lagrange"
+test = "TaylorSweep"
+#test = "LagrangeSweep"
+
+ttype = "simple"
+#ttype = "augmentedI"
+#ttype = "augmentedM"
+
+plotSamples = 'ALL'
+#plotSamples = []
+
+k0 = 10
+kLeft, kRight = 8 + 1.j, 12 + 1.j
+ktar = 12
+ktars = np.linspace(8, 12, 33) - .5j
+ktars = np.linspace(12.125, 12.5, 4) - .5j
+
+####################
+
+PI = np.pi
+R = 2
+def Dboundary(x, on_boundary):
+ return on_boundary and (x[0]**2+x[1]**2)**.5 < .95 * R
+
+kappa = 10
+theta = PI * -30 / 180.
+mu = 1.1
+epsilon = .1
+
+x, y = sp.symbols('x[0] x[1]', real=True)
+phiex = kappa * (x * np.cos(theta) + y * np.sin(theta))
+u0ex = - sp.exp(1.j * phiex)
+
+checkReal = x**2-x+y**2
+rhoroot = ((x**2+y**2)/((x-1)**2+y**2))**.25
+phiroot1 = sp.atan(-y/(x**2-x+y**2)) / 2
+phiroot2 = sp.atan(-y/(x**2-x+y**2)) / 2 - PI * sp.sign(-y/(x**2-x+y**2)) / 2
+kappam1 = (((rhoroot*sp.cos(phiroot1)+.5)**2.+(rhoroot*sp.sin(phiroot1))**2.)/
+ ((rhoroot*sp.cos(phiroot1)-1.)**2.+(rhoroot*sp.sin(phiroot1))**2.))**.5 - mu
+kappam2 = (((rhoroot*sp.cos(phiroot2)+.5)**2.+(rhoroot*sp.sin(phiroot2))**2.)/
+ ((rhoroot*sp.cos(phiroot2)-1.)**2.+(rhoroot*sp.sin(phiroot2))**2.))**.5 - mu
+Heps1 = .9 * .5 * (1 + kappam1/epsilon + sp.sin(PI*kappam1/epsilon) / PI) + .1
+Heps2 = .9 * .5 * (1 + kappam2/epsilon + sp.sin(PI*kappam2/epsilon) / PI) + .1
+
+mesh = fen.Mesh('../Data/airfoil.xml')
+
+plt.figure()
+fen.plot(mesh, title = 'mesh')
+
+a = fen.Expression(('{0}>=0 ? ({2}>=-{1} ? ({2}<={1} ? {4} : 1) : .1) : '
+ '({3}>=-{1} ? ({3}<={1} ? {5} : 1) : .1)')\
+ .format(sp.printing.ccode(checkReal), str(epsilon), sp.printing.ccode(kappam1),
+ sp.printing.ccode(kappam2), sp.printing.ccode(Heps1),
+ sp.printing.ccode(Heps2)), degree = 4)
+
+DirichletTerm = [sp.printing.ccode(sp.simplify(sp.re(u0ex))), sp.printing.ccode(sp.simplify(sp.im(u0ex)))]
+
+if ttype == "simple":
+ solver = HFS(mesh = mesh, wavenumber = k0, diffusivity = a,
+ forcingTerm = 0, FEDegree = 3, DirichletBoundary = Dboundary,
+ RobinBoundary = 'rest', DirichletDatum = DirichletTerm)
+ plotter = HS(solver.V)
+else:
+ if ttype[-1] == "I": constraintType = "IDENTITY"
+ else: constraintType = "MASS"
+ solver = HFSA(mesh = mesh, wavenumber = k0, diffusivity = a,
+ forcingTerm = 0, FEDegree = 3, DirichletBoundary = Dboundary,
+ RobinBoundary = 'rest', DirichletDatum = DirichletTerm,
+ constraintType = constraintType)
+ plotter = HSA(solver.V, d = 2)
+
+baseRe, baseIm = DirichletTerm
+baseRe = fen.project(fen.Expression(baseRe, degree = 4), solver.V)
+baseIm = fen.project(fen.Expression(baseIm, degree = 4), solver.V)
+uinc = np.array(baseRe.vector()) + 1.j * np.array(baseIm.vector())
+if ttype[: -1] == "augmented":
+ uinc = [uinc, kappa * uinc]
+
+if test == "solve":
+ aF = fen.interpolate(a, solver.V)
+ av = aF.vector()
+ uh = solver.solve()
+ print(plotter.norm(uh, kappa))
+ if ttype == "simple":
+ uhtot = uh - uinc
+ else:
+ uhtot = [x - y for x, y in zip(uh, uinc)]
+ print(plotter.norm(uhtot, kappa))
+ plotter.plot(av, what = 'Real', name = 'a')
+ plt.gcf().savefig('./RetreatPoster/diff.eps', format='eps', dpi=1000)
+ plotter.plot(uhtot - uh, what = 'Real', name = 'u_inc')
+ plt.gcf().savefig('./RetreatPoster/dbc.eps', format='eps', dpi=1000)
+ plotter.plot(uh, what = 'ABS')
+ plt.gcf().savefig('./RetreatPoster/u0.eps', format='eps', dpi=1000)
+ plotter.plot(uhtot, what = 'ABS', name = 'u + u_inc')
+ plt.gcf().savefig('./RetreatPoster/utot0.eps', format='eps', dpi=1000)
+
+elif test in ["Taylor", "Lagrange"]:
+ if test == "Taylor":
+ params = {'N':8, 'M':7, 'R':8, 'E':8, 'sampleType':'Krylov', 'POD':True}
+
+ approxPade = TP(solver, plotter, k0 = k0, plotDer = plotSamples,
+ approxParameters = subdict(params, ['N', 'M', 'E', 'sampleType', 'POD']))
+ approxRB = TRB(solver, plotter, k0 = k0,
+ approxParameters = subdict(params, ['R', 'E', 'sampleType', 'POD']))
+ else:
+ params = {'N':8, 'M':7, 'R':9, 'S':9, 'polyBasis':'CHEBYSHEV',
+ 'nodesType':'CHEBYSHEV','POD':True}
+
+ approxPade = LP(solver, plotter, ks = [kLeft, kRight], w = kappa, plotSnap = plotSamples,
+ approxParameters = subdict(params, ['N', 'M', 'S', 'polyBasis', 'POD']))
+ approxRB = LRB(solver, plotter, ks = [kLeft, kRight], w = kappa,
+ approxParameters = subdict(params, ['R', 'S', 'nodesType', 'POD']))
+
+ PadeErr, solNorm = approxPade.approxError(ktar), approxPade.HFNorm(ktar)
+ RBErr = approxRB.approxError(ktar)
+ print('SolNorm:\t{}\nErrPade:\t{}\nErrRelPade:\t{}\nErrRB:\t\t{}\nErrRelRB:\t{}'.format(
+ solNorm, PadeErr, np.divide(PadeErr, solNorm), RBErr, np.divide(RBErr, solNorm)))
+
+ print('\nPoles Pade'':')
+ print(approxPade.getPoles(True))
+ print('\nPoles RB:')
+ print(approxRB.getPoles(True))
+
+ uHF = approxPade.getHF(ktar)
+ plotter.plot(uHF, name = 'u_ex')
+
+ approxPade.plotApp(ktar, name = 'u_Pade''')
+ approxPade.plotErr(ktar, name = 'errPade''')
+ approxRB.plotApp(ktar, name = 'u_RB')
+ approxRB.plotErr(ktar, name = 'errRB')
+
+elif test in ["TaylorSweep", "LagrangeSweep"]:
+ if test == "TaylorSweep":
+ shift = 2
+ nsets = 4
+ stride = 3
+ Emax = stride * (nsets - 1) + shift + 1
+ params = {'Emax':Emax, 'sampleType':'Krylov', 'POD':True}
+ paramsSetsPade = [None] * nsets
+ paramsSetsRB = [None] * nsets
+ for i in range(nsets):
+ paramsSetsPade[i] = {'N':stride*i+shift+1, 'M':stride*i+shift,
+ 'E':stride*i+shift+1}
+ paramsSetsRB[i] = {'E':stride*i+shift,'R':stride*i+shift+1}
+ approxPade = TP(solver, plotter, k0 = kappa, approxParameters = params)
+ approxRB = TRB(solver, plotter, k0 = kappa, approxParameters = params)
+ else:
+ kLeft, kRight = 8, 12
+ shift = 3
+ nsets = 3
+ stride = 3
+ Smax = stride * (nsets - 1) + shift + 2
+ paramsPade = {'S':Smax, 'polyBasis':'CHEBYSHEV', 'POD':True}
+ paramsRB = {'S':Smax, 'nodesType':'CHEBYSHEV', 'POD':True}
+ paramsSetsPade = [None] * nsets
+ paramsSetsRB = [None] * nsets
+ for i in range(nsets):
+ paramsSetsPade[i] = {'N': stride * i + shift + 1, 'M': stride * i + shift,
+ 'S': stride * i + shift + 2}
+ paramsSetsRB[i] = {'R': stride * i + shift + 1, 'S': stride * i + shift + 2}
+ approxPade = LP(solver, plotter, ks = [kLeft, kRight], w = kappa,
+ approxParameters = paramsPade)
+ approxRB = LRB(solver, plotter, ks = [kLeft, kRight], w = kappa,
+ approxParameters = paramsRB)
+
+ sweeper = Sweeper.ROMApproximantSweeper(ktars = ktars, mostExpensive = 'Approx')
+
+ sweeper.ROMEngine = approxPade
+ sweeper.params = paramsSetsPade
+ filenamePade = sweeper.sweep('../Data/HelmholtzAirfoil' + test[: -5] + 'Pade.dat')
+
+ sweeper.ROMEngine = approxRB
+ sweeper.params = paramsSetsRB
+ filenameRB = sweeper.sweep('../Data/HelmholtzAirfoil' + test[: -5] + 'RB.dat')
+
+ for i in range(nsets):
+ if test == "TaylorSweep":
+ val = stride*i+shift+1
+ PadeOutput = sweeper.read(filenamePade, {'E':val},
+ ['kRe', 'HFNorm', 'AppNorm', 'ErrNorm'])
+ RBOutput = sweeper.read(filenameRB, {'E':val},
+ ['kRe', 'AppNorm', 'ErrNorm'])
+ let = 'E'
+ else:
+ val = stride*i+shift+2
+ PadeOutput = sweeper.read(filenamePade, {'S':val},
+ ['kRe', 'HFNorm', 'AppNorm', 'ErrNorm'])
+ RBOutput = sweeper.read(filenameRB, {'S':val},
+ ['kRe', 'AppNorm', 'ErrNorm'])
+ let = 'S'
+
+ ktarsF = PadeOutput['kRe']
+ solNormF = PadeOutput['HFNorm']
+ PadektarsF = PadeOutput['kRe']
+ PadeNormF = PadeOutput['AppNorm']
+ PadeErrorF = PadeOutput['ErrNorm']
+ RBktarsF = RBOutput['kRe']
+ RBNormF = RBOutput['AppNorm']
+ RBErrorF = RBOutput['ErrNorm']
+
+ plt.figure(figsize = (10, 5))
+ plt.plot(ktarsF, solNormF, 'k-', label='Sol norm')
+ plt.plot(PadektarsF, PadeNormF, 'b--',
+ label='Pade'' norm, {1} = {0}'.format(val, let))
+ plt.plot(RBktarsF, RBNormF, 'g--',
+ label='RB norm, {1} = {0}'.format(val, let))
+ plt.legend()
+ plt.grid()
+ p = plt.legend(loc = 'upper left')
+ plt.savefig('./RetreatPoster/normA' + str(i) + '.eps', format='eps', dpi=1000)
+
+ plt.figure(figsize = (10, 5))
+ plt.semilogy(PadektarsF, PadeErrorF, 'b',
+ label='Pade'' error, {1} = {0}'.format(val, let))
+ plt.semilogy(RBktarsF, RBErrorF, 'g',
+ label='RB error, {1} = {0}'.format(val, let))
+ plt.legend()
+ plt.grid()
+ p = plt.legend(loc = 'lower right')
+ plt.savefig('./RetreatPoster/errorA' + str(i) + '.eps', format='eps', dpi=1000)
+
+ plt.figure(figsize = (10, 5))
+ plt.semilogy(ktarsF, np.divide(PadeErrorF, solNormF), 'b',
+ label='Pade'' relative error, {1} = {0}'.format(val, let))
+ plt.semilogy(RBktarsF, np.divide(RBErrorF, solNormF), 'g',
+ label='RB relative error, {1} = {0}'.format(val, let))
+ plt.legend()
+ plt.grid()
+ p = plt.legend(loc = 'lower right')
+ plt.savefig('./RetreatPoster/errorAR' + str(i) + '.eps', format='eps', dpi=1000)
+
\ No newline at end of file
diff --git a/examples/Python/HelmholtzLagrangeApproximantsSweep.py b/examples/Python/HelmholtzLagrangeApproximantsSweep.py
index 6c7ffbe..6c92979 100644
--- a/examples/Python/HelmholtzLagrangeApproximantsSweep.py
+++ b/examples/Python/HelmholtzLagrangeApproximantsSweep.py
@@ -1,100 +1,100 @@
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
# Example homogeneous Dirichlet forcing wave SWEEP
from __future__ import print_function
import fenics as fen
import numpy as np
import sympy as sp
from context import FenicsHelmholtzEngine as HFEngine
from context import FenicsHSEngine as HSEngine
from context import ROMApproximantLagrangePade as Pade
from context import ROMApproximantLagrangeRB as RB
from context import ROMApproximantSweeper as Sweeper
PI = np.pi
nu = 12**.5
theta = PI / 3
npoints = 31
ktars = np.linspace(0, 21, npoints)
x, y = sp.symbols('x[0] x[1]', real=True)
wex = 16/PI**4 * x * y * (x - PI) * (y - PI)
phiex = nu * (x * np.cos(theta) + y * np.sin(theta))
uex = wex * sp.exp(-1.j * phiex)
fex = - uex.diff(x, 2) - uex.diff(y, 2) - nu**2 * uex
nx = ny = 10
mesh = fen.RectangleMesh(fen.Point(0,0), fen.Point(PI,PI), nx, ny)
forcingTerm = [sp.printing.ccode(sp.simplify(sp.re(fex))), sp.printing.ccode(sp.simplify(sp.im(fex)))]
solver = HFEngine(mesh = mesh, wavenumber = nu, forcingTerm = forcingTerm, FEDegree = 3,
DirichletBoundary = 'all', DirichletDatum = 0)
plotter = HSEngine(solver.V)
shift = 3
nsets = 5
stride = 3
Smax = stride * (nsets - 1) + shift + 2
paramsPade = {'S':Smax, 'polyBasis':'CHEBYSHEV', 'POD':True}
paramsRB = {'S':Smax, 'nodesType':'CHEBYSHEV', 'POD':True}
paramsSetsPade = [None] * nsets
paramsSetsRB = [None] * nsets
for i in range(nsets):
paramsSetsPade[i] = {'N': stride * i + shift + 1, 'M': stride * i + shift,
'S': stride * i + shift + 2}
paramsSetsRB[i] = {'R': stride * i + shift + 1, 'S': stride * i + shift + 2}
appPade = Pade(solver, plotter, ks = [4 + .5j, 14 + .5j],
w = nu, approxParameters = paramsPade)
appRB = RB(solver, plotter, ks = [4 + .5j, 14 + .5j],
w = nu, approxParameters = paramsRB)
sweeper = Sweeper.ROMApproximantSweeper(ktars = ktars, mostExpensive = 'Approx')
sweeper.ROMEngine = appPade
sweeper.params = paramsSetsPade
filenamePade = sweeper.sweep('../Data/HelmholtzBubbleLagrangePade.dat', outputs = 'ALL')
sweeper.ROMEngine = appRB
sweeper.params = paramsSetsRB
filenameRB = sweeper.sweep('../Data/HelmholtzBubbleLagrangeRB.dat', outputs = 'ALL')
####################
from matplotlib import pyplot as plt
plt.jet()
for i in range(nsets):
nSamples = stride*i+shift+2
PadeOutput = sweeper.read(filenamePade, {'S':nSamples},
- ['kRe', 'HFNorm', 'AppNorm', 'AppError'])
+ ['kRe', 'HFNorm', 'AppNorm', 'ErrNorm'])
RBOutput = sweeper.read(filenameRB, {'S':nSamples},
- ['kRe', 'AppNorm', 'AppError'])
+ ['kRe', 'AppNorm', 'ErrNorm'])
ktarsF = PadeOutput['kRe']
solNormF = PadeOutput['HFNorm']
PadektarsF = PadeOutput['kRe']
PadeNormF = PadeOutput['AppNorm']
- PadeErrorF = PadeOutput['AppError']
+ PadeErrorF = PadeOutput['ErrNorm']
RBktarsF = RBOutput['kRe']
RBNormF = RBOutput['AppNorm']
- RBErrorF = RBOutput['AppError']
+ RBErrorF = RBOutput['ErrNorm']
plt.figure()
plt.semilogy(ktarsF, solNormF, 'k-', label='Sol norm')
plt.semilogy(PadektarsF, PadeNormF, 'b.--', label='Pade'' norm, S = {}'.format(nSamples))
plt.semilogy(RBktarsF, RBNormF, 'g.--', label='RB norm, S = {}'.format(nSamples))
plt.legend()
plt.grid()
plt.figure()
plt.semilogy(PadektarsF, PadeErrorF, 'b', label='Pade'' error, S = {}'.format(nSamples))
plt.semilogy(RBktarsF, RBErrorF, 'g', label='RB error, S = {}'.format(nSamples))
plt.legend()
plt.grid()
plt.figure()
plt.semilogy(ktarsF, PadeErrorF / solNormF, 'b', label='Pade'' relative error, S = {}'.format(nSamples))
plt.semilogy(RBktarsF, RBErrorF / solNormF, 'g', label='RB relative error, S = {}'.format(nSamples))
plt.legend()
plt.grid()
diff --git a/examples/Python/HelmholtzMembraneTaylor.py b/examples/Python/HelmholtzMembraneTaylor.py
new file mode 100644
index 0000000..c848a40
--- /dev/null
+++ b/examples/Python/HelmholtzMembraneTaylor.py
@@ -0,0 +1,141 @@
+#!/usr/bin/env python3
+# -*- coding: utf-8 -*-
+
+import fenics as fen
+import numpy as np
+import sympy as sp
+from context import FenicsHelmholtzEngine as HF
+from context import FenicsHSEngine as HS
+from context import ROMApproximantTaylorPade as TP
+from context import ROMApproximantSweeper as Sweeper
+from matplotlib import pyplot as plt
+plt.jet()
+
+####################
+
+test = "solve"
+#test = "Taylor"
+test = "TaylorSweep"
+
+ttype = "simple"
+
+plotSamples = 'ABS'
+#plotSamples = []
+
+z0 = 100 + 1.j
+ktar = 97.5**.5
+ktars = np.sort(np.concatenate((np.linspace(78, 122, 101), np.linspace(100.9, 101.6, 5))))
+ktars = np.array([z0])
+
+####################
+
+def boundaryNeumann(x, on_boundary):
+ return on_boundary and x[1] > .25 and x[0] > 0.995 and x[0] < 1.005
+
+PI = np.pi
+nu = 10
+
+x0, y0 = .5, .5
+A, B = 1., 0
+Rr, Ri = .1, .1
+x, y = sp.symbols('x[0] x[1]', real=True)
+fex = (A * sp.exp(- ((x - x0)**2 + (y - y0)**2) / 2 / Rr**2)
+ + 1.j * B * sp.exp(- ((x - x0)**2 + (y - y0)**2) / 2 / Ri**2))
+
+meshname = '../Data/crack_coarse.xml'
+#meshname = '../Data/crack_fine.xml'
+mesh = fen.Mesh(meshname)
+forcingTerm = [sp.printing.ccode(sp.simplify(sp.re(fex))), sp.printing.ccode(sp.simplify(sp.im(fex)))]
+
+plt.figure()
+fen.plot(mesh, title = 'mesh')
+#plt.gcf().savefig('./Membrane/mesh.eps', format='eps', dpi=1000)
+
+solver = HF(mesh = mesh, wavenumber = ktar, forcingTerm = forcingTerm, FEDegree = 3,
+ NeumannBoundary = boundaryNeumann, DirichletBoundary = 'rest')
+plotter = HS(solver.V)
+
+def crack_disp(u):
+ uRe = fen.Function(solver.V)
+ uIm = fen.Function(solver.V)
+ uRe.vector()[:] = np.array(np.real(u), dtype = float)
+ uIm.vector()[:] = np.array(np.imag(u), dtype = float)
+ return fen.assemble((uRe * uRe + uIm * uIm) * solver.ds(0)) ** .5
+solver.functional = crack_disp
+
+if test == "solve":
+ uh = solver.solve()
+ plotter.plot(uh, what = 'REAL')
+ #plt.gcf().savefig('./Membrane/u0.eps', format='eps', dpi=1000)
+ print(solver.functional(uh))
+
+elif test == "Taylor":
+ params = {'N':6, 'M':5, 'E':6, 'sampleType':'Arnoldi', 'POD':True}
+
+ approxPade = TP(solver, plotter, k0 = z0, w = nu, plotDer = plotSamples,
+ approxParameters = params)
+
+ PadeErr, solNorm = approxPade.approxError(ktar), approxPade.HFNorm(ktar)
+ print('SolNorm:\t{}\nErrPade:\t{}\nErrRelPade:\t{}'.format(
+ solNorm, PadeErr, np.divide(PadeErr, solNorm)))
+ print('\nPoles Pade'':')
+ print(approxPade.getPoles(True))
+
+ uHF = approxPade.getHF(ktar)
+ plotter.plot(uHF, name = 'u_ex')
+ approxPade.plotApp(ktar, name = 'u_Pade''')
+ approxPade.plotErr(ktar, name = 'errPade''')
+
+elif test == "TaylorSweep":
+ shift = 2
+ nsets = 3
+ stride = 3
+ Emax = stride * (nsets - 1) + shift + 1
+ params = {'Emax':Emax, 'sampleType':'Arnoldi', 'POD':True}
+ paramsSetsPade = [None] * nsets
+ for i in range(nsets):
+ paramsSetsPade[i] = {'N':stride*i+shift+1, 'M':stride*i+shift,
+ 'E':stride*i+shift+1}
+ approxPade = TP(solver, plotter, k0 = z0, approxParameters = params)
+
+ sweeper = Sweeper.ROMApproximantSweeper(ktars = ktars, mostExpensive = 'Approx')
+ sweeper.ROMEngine = approxPade
+ sweeper.params = paramsSetsPade
+ filenamePade = sweeper.sweep('../Data/HelmholtzMembraneTP.dat', outputs = 'ALL')
+
+ for i in range(nsets):
+ val = stride*i+shift + 1
+ PadeOutput = sweeper.read(filenamePade, {'E':val},
+ ['kRe', 'HFNorm', 'AppNorm', 'ErrNorm'])
+
+ ktarsF = PadeOutput['kRe']
+ solNormF = PadeOutput['HFNorm']
+ PadektarsF = PadeOutput['kRe']
+ PadeNormF = PadeOutput['AppNorm']
+ PadeErrorF = PadeOutput['ErrNorm']
+
+ plt.figure(figsize = (10, 5))
+ plt.semilogy(ktarsF, solNormF, 'k-', label='Sol norm')
+ plt.semilogy(PadektarsF, PadeNormF, 'b--',
+ label='Pade'' norm, E = {0}'.format(val))
+ plt.legend()
+ plt.grid()
+ p = plt.legend(loc = 'upper left')
+ #plt.savefig('./Membrane/normA' + str(i) + '.eps', format='eps', dpi=1000)
+
+ plt.figure(figsize = (10, 5))
+ plt.semilogy(PadektarsF, PadeErrorF, 'b',
+ label='Pade'' error, E = {0}'.format(val))
+ plt.legend()
+ plt.grid()
+ p = plt.legend(loc = 'upper left')
+ #plt.savefig('./Membrane/errorA' + str(i) + '.eps', format='eps', dpi=1000)
+
+ plt.figure(figsize = (10, 5))
+ plt.semilogy(ktarsF, np.divide(PadeErrorF, solNormF), 'b',
+ label='Pade'' relative error, E = {0}'.format(val))
+ plt.legend()
+ plt.grid()
+ p = plt.legend(loc = 'upper left')
+ #plt.savefig('./Membrane/errorAR' + str(i) + '.eps', format='eps', dpi=1000)
+
\ No newline at end of file
diff --git a/examples/Python/HelmholtzPadeTaylorApproximant.py b/examples/Python/HelmholtzPadeTaylorApproximant.py
index 082e30c..70342fd 100644
--- a/examples/Python/HelmholtzPadeTaylorApproximant.py
+++ b/examples/Python/HelmholtzPadeTaylorApproximant.py
@@ -1,199 +1,199 @@
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
# Example homogeneous Dirichlet forcing wave
from __future__ import print_function
import fenics as fen
import numpy as np
import sympy as sp
from context import FenicsHelmholtzEngine as HFEngine
from context import FenicsHelmholtzScatteringEngine as HFSEngine
from context import FenicsHelmholtzScatteringAugmentedEngine as HFSAEngine
from context import FenicsHSEngine as HSEngine
from context import FenicsHSAugmentedEngine as HSAEngine
from context import ROMApproximantTaylorPade as Pade
PI = np.pi
-testNo = 4
+testNo = 3
if testNo == 1:
params = {'N':4, 'M':3, 'E':4, 'sampleType':'Krylov', 'POD':True}
nu = 12**.5
theta = PI / 3
z0 = 12+.5j
ztar = 11
x, y = sp.symbols('x[0] x[1]', real=True)
wex = 16/PI**4 * x * y * (x - PI) * (y - PI)
phiex = nu * (x * np.cos(theta) + y * np.sin(theta))
uex = wex * sp.exp(-1.j * phiex)
fex = - uex.diff(x, 2) - uex.diff(y, 2) - nu**2 * uex
nx = ny = 40
mesh = fen.RectangleMesh(fen.Point(0,0), fen.Point(PI,PI), nx, ny)
forcingTerm = [sp.printing.ccode(sp.simplify(sp.re(fex))), sp.printing.ccode(sp.simplify(sp.im(fex)))]
solver = HFEngine(mesh = mesh, wavenumber = z0**.5, forcingTerm = forcingTerm, FEDegree = 3,
DirichletBoundary = 'all', DirichletDatum = 0)
plotter = HSEngine(solver.V)
approx = Pade(solver, plotter, k0 = z0, w = np.real(z0**.5),
approxParameters = params)
approx.plotApp(ztar, name = 'u_Pade''')
approx.plotHF(ztar, name = 'u_HF')
approx.plotErr(ztar, name = 'err')
appErr, solNorm = approx.approxError(ztar), approx.HFNorm(ztar)
print(appErr, solNorm, appErr/solNorm)
print(approx.getPoles(True))
############
elif testNo == 2:
params = {'N':8, 'M':7, 'E':8, 'sampleType':'Arnoldi', 'POD':True}
ztar = 3.95 ** 2
n1 = 2**.5
n2 = 3**.5
kappa = 4
theta = PI * 75 / 180
d1, d2 = np.cos(theta), np.sin(theta)
K1 = kappa * n1 * d1
if kappa * n2 >= K1:
K2 = ((kappa*n2)**2 - K1**2)**.5
else:
K2 = 1.j * (K1**2 - (kappa*n2)**2)**.5
R = (kappa * n1 * d2 - K2) / (kappa * n1 * d2 + K2)
T = R + 1
x, y = sp.symbols('x[0] x[1]', real=True)
uex1 = T*sp.exp(1.j*(K1*x+K2*y))
uex2 = sp.exp(1.j*kappa*n1*(d1*x+d2*y)) + R*sp.exp(1.j*kappa*n1*(d1*x-d2*y))
# Exact solution
uexRe = fen.Expression('x[1]>=0 ? {0} : {1}'.format(\
sp.printing.ccode(sp.re(uex1)), sp.printing.ccode(sp.re(uex2))), degree=4)
uexIm = fen.Expression('x[1]>=0 ? {0} : {1}'.format(\
sp.printing.ccode(sp.im(uex1)), sp.printing.ccode(sp.im(uex2))), degree=4)
# wavenumber term
nRe = fen.Expression('x[1]<0 ? n1r : n2r',
n1r = n1.real, n2r = n2.real, degree=4)
nIm = fen.Expression('x[1]<0 ? n1i : n2i',
n1i = n1.imag, n2i = n2.imag, degree=4)
# Create mesh and define function space
nx = ny = 50
mesh = fen.RectangleMesh(fen.Point(-PI/2,-PI/2), fen.Point(PI/2,PI/2), nx, ny)
solver = HFEngine(mesh = mesh, wavenumber = kappa, refractionIndex = (nRe, nIm),
forcingTerm = 0, FEDegree = 3, DirichletBoundary = 'all',
DirichletDatum = (uexRe, uexIm))
plotter = HSEngine(solver.V)
approx = Pade(solver, plotter, k0 = kappa ** 2,
w = kappa, approxParameters = params,
plotDer = True)
approx.plotApp(ztar, name = 'u_Pade''')
approx.plotHF(ztar, name = 'u_HF')
approx.plotErr(ztar, name = 'err')
appErr, solNorm = approx.approxError(ztar), approx.HFNorm(ztar)
print(appErr, solNorm, appErr/solNorm)
print(approx.getPoles(True))
############
elif testNo == 3:
def Dboundary(x, on_boundary):
return on_boundary and (fen.near(x[0], 0) or fen.near(x[0], PI))
A = 10
kappa = 2
theta = PI * 30 / 180
x, y = sp.symbols('x[0] x[1]', real=True)
wex = 4/PI**2 * x * (PI - x)
phiex = kappa * (x * np.cos(theta) + y * np.sin(theta))
uex = wex * sp.exp(-1.j * phiex)
fex = - uex.diff(x, 2) - uex.diff(y, 2) - kappa**2 * uex
nx = ny = 50
mesh = fen.RectangleMesh(fen.Point(0,0), fen.Point(PI,PI), nx, ny)
forcingTerm = [sp.printing.ccode(sp.simplify(sp.re(fex))), sp.printing.ccode(sp.simplify(sp.im(fex)))]
params = {'N':8, 'M':7, 'E':8, 'sampleType':'Krylov', 'POD':True}
solver = HFSEngine(mesh = mesh, wavenumber = kappa, forcingTerm = forcingTerm,
FEDegree = 3, DirichletBoundary = Dboundary,
RobinBoundary = 'rest', DirichletDatum = 0)
plotter = HSEngine(solver.V)
approx = Pade(solver, plotter, k0 = kappa,
approxParameters = params,
plotDer = True)
approx.setupApprox()
print(np.roots(approx.Q[::-1]) + kappa)
ktar = 1.8 - .3j
approx.plotApp(ktar, name = 'u_Pade''')
approx.plotHF(ktar, name = 'u_HF')
approx.plotErr(ktar, name = 'err')
appErr, solNorm = approx.approxError(ktar), approx.HFNorm(ktar)
print(appErr, solNorm, appErr/solNorm)
print(approx.getPoles(True))
############
elif testNo == 4:
def Dboundary(x, on_boundary):
return on_boundary and (fen.near(x[0], 0) or fen.near(x[0], PI))
A = 10
kappa = 2
theta = PI * 30 / 180
x, y = sp.symbols('x[0] x[1]', real=True)
wex = 4/PI**2 * x * (PI - x)
phiex = kappa * (x * np.cos(theta) + y * np.sin(theta))
uex = wex * sp.exp(-1.j * phiex)
fex = - uex.diff(x, 2) - uex.diff(y, 2) - kappa**2 * uex
nx = ny = 20
mesh = fen.RectangleMesh(fen.Point(0,0), fen.Point(PI,PI), nx, ny)
forcingTerm = [sp.printing.ccode(sp.simplify(sp.re(fex))), sp.printing.ccode(sp.simplify(sp.im(fex)))]
params = {'N':8, 'M':7, 'E':8, 'sampleType':'Arnoldi', 'POD':True}
solver = HFSAEngine(mesh = mesh, wavenumber = kappa, forcingTerm = forcingTerm,
FEDegree = 3, DirichletBoundary = Dboundary,
RobinBoundary = 'rest', DirichletDatum = 0,
constraintType = 'IDENTITY')
plotter = HSAEngine(solver.V, 2)
approx = Pade(solver, plotter, k0 = kappa,
approxParameters = params,
plotDer = True)
approx.setupApprox()
print(np.roots(approx.Q[::-1]) + kappa)
ktar = 1.8 - .3j
approx.plotApp(ktar, name = 'u_Pade''')
approx.plotHF(ktar, name = 'u_HF')
approx.plotErr(ktar, name = 'err')
appErr, solNorm = approx.approxError(ktar, kappa), approx.HFNorm(ktar, kappa)
print(appErr, solNorm, np.divide(appErr, solNorm))
print(approx.getPoles(True))
diff --git a/examples/Python/HelmholtzRBTaylorApproximant.py b/examples/Python/HelmholtzRBTaylorApproximant.py
index 9a0293b..9ed1d08 100644
--- a/examples/Python/HelmholtzRBTaylorApproximant.py
+++ b/examples/Python/HelmholtzRBTaylorApproximant.py
@@ -1,137 +1,137 @@
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
# Example homogeneous Dirichlet forcing wave
from __future__ import print_function
import fenics as fen
import numpy as np
import sympy as sp
from context import FenicsHelmholtzEngine as HFEngine
from context import FenicsHelmholtzScatteringEngine as HFSEngine
from context import FenicsHelmholtzScatteringAugmentedEngine as HFSAEngine
from context import FenicsHSEngine as HSEngine
from context import FenicsHSAugmentedEngine as HSAEngine
from context import ROMApproximantTaylorRB as RB
PI = np.pi
testNo = 3
if testNo == 1:
nu = 12**.5
theta = PI / 3
z0 = 12+.5j
ztar = 11
x, y = sp.symbols('x[0] x[1]', real=True)
wex = 16/PI**4 * x * y * (x - PI) * (y - PI)
phiex = nu * (x * np.cos(theta) + y * np.sin(theta))
uex = wex * sp.exp(-1.j * phiex)
fex = - uex.diff(x, 2) - uex.diff(y, 2) - nu**2 * uex
params = {'E':5, 'POD':True, 'sampleType':'ARNOLDI'}
nx = ny = 40
mesh = fen.RectangleMesh(fen.Point(0,0), fen.Point(PI,PI), nx, ny)
forcingTerm = [sp.printing.ccode(sp.simplify(sp.re(fex))), sp.printing.ccode(sp.simplify(sp.im(fex)))]
solver = HFEngine(mesh = mesh, wavenumber = z0**.5, forcingTerm = forcingTerm, FEDegree = 3,
DirichletBoundary = 'all', DirichletDatum = 0)
plotter = HSEngine(solver.V)
approx = RB(solver, plotter, k0 = z0, w = np.real(z0**.5),
approxParameters = params)
approx.plotApp(ztar, name = 'u_RB')
approx.plotHF(ztar, name = 'u_HF')
approx.plotErr(ztar, name = 'err')
appErr, solNorm = approx.approxError(ztar), approx.HFNorm(ztar)
print(appErr, solNorm, appErr/solNorm)
print(approx.getPoles(True))
############
elif testNo == 2:
def Dboundary(x, on_boundary):
return on_boundary and (fen.near(x[0], 0) or fen.near(x[0], PI))
A = 10
kappa = 2
theta = PI * 30 / 180
x, y = sp.symbols('x[0] x[1]', real=True)
wex = 4/PI**2 * x * (PI - x)
phiex = kappa * (x * np.cos(theta) + y * np.sin(theta))
uex = wex * sp.exp(-1.j * phiex)
fex = - uex.diff(x, 2) - uex.diff(y, 2) - kappa**2 * uex
nx = ny = 20
mesh = fen.RectangleMesh(fen.Point(0,0), fen.Point(PI,PI), nx, ny)
forcingTerm = [sp.printing.ccode(sp.simplify(sp.re(fex))), sp.printing.ccode(sp.simplify(sp.im(fex)))]
params = {'E':8, 'POD':True}
solver = HFSEngine(mesh = mesh, wavenumber = kappa, forcingTerm = forcingTerm,
FEDegree = 3, DirichletBoundary = Dboundary,
RobinBoundary = 'rest', DirichletDatum = 0)
plotter = HSEngine(solver.V)
approx = RB(solver, plotter, k0 = kappa,
approxParameters = params,
- plotDer = False)
+ plotDer = True)
ktar = 1.8 - .2j
approx.plotApp(ktar, name = 'u_RB')
approx.plotHF(ktar, name = 'u_HF')
approx.plotErr(ktar, name = 'err')
appErr, solNorm = approx.approxError(ktar), approx.HFNorm(ktar)
print(appErr, solNorm, appErr/solNorm)
print(approx.getPoles(True))
############
elif testNo == 3:
def Dboundary(x, on_boundary):
return on_boundary and (fen.near(x[0], 0) or fen.near(x[0], PI))
A = 10
kappa = 2
theta = PI * 30 / 180
x, y = sp.symbols('x[0] x[1]', real=True)
wex = 4/PI**2 * x * (PI - x)
phiex = kappa * (x * np.cos(theta) + y * np.sin(theta))
uex = wex * sp.exp(-1.j * phiex)
fex = - uex.diff(x, 2) - uex.diff(y, 2) - kappa**2 * uex
nx = ny = 20
mesh = fen.RectangleMesh(fen.Point(0,0), fen.Point(PI,PI), nx, ny)
forcingTerm = [sp.printing.ccode(sp.simplify(sp.re(fex))), sp.printing.ccode(sp.simplify(sp.im(fex)))]
- params = {'E':8, 'POD':True}
+ params = {'E':8, 'sampleType':'Arnoldi', 'POD':True}
solver = HFSAEngine(mesh = mesh, wavenumber = kappa, forcingTerm = forcingTerm,
FEDegree = 3, DirichletBoundary = Dboundary,
RobinBoundary = 'rest', DirichletDatum = 0,
constraintType = 'MASS')
plotter = HSAEngine(solver.V, 2)
approx = RB(solver, plotter, k0 = kappa,
approxParameters = params,
- plotDer = False)
+ plotDer = True)
ktar = 1.8 - .2j
approx.plotApp(ktar, name = 'u_RB')
approx.plotHF(ktar, name = 'u_HF')
approx.plotErr(ktar, name = 'err')
appErr, solNorm = approx.approxError(ktar, kappa), approx.HFNorm(ktar, kappa)
print(appErr, solNorm, np.divide(appErr, solNorm))
print(approx.getPoles(True))
diff --git a/examples/Python/HelmholtzSolver.py b/examples/Python/HelmholtzSolver.py
index c476a53..3bec98a 100644
--- a/examples/Python/HelmholtzSolver.py
+++ b/examples/Python/HelmholtzSolver.py
@@ -1,199 +1,191 @@
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
from __future__ import print_function
import fenics as fen
import numpy as np
import sympy as sp
from context import FenicsHelmholtzEngine as HF
from context import FenicsHelmholtzScatteringEngine as HFS
from context import FenicsHelmholtzScatteringAugmentedEngine as HFSA
from context import FenicsHSEngine as HS
from context import FenicsHSAugmentedEngine as HSA
testNo = 4
if testNo == 1:
PI = np.pi
def boundary(x, on_boundary):
return on_boundary
nu = 12 ** .5
theta = PI / 3
x, y = sp.symbols('x[0] x[1]', real=True)
wex = 16/PI**4 * x * y * (x - PI) * (y - PI)
phiex = nu * (x * np.cos(theta) + y * np.sin(theta))
uex = wex * sp.exp(-1.j * phiex)
fex = - uex.diff(x, 2) - uex.diff(y, 2) - nu**2 * uex
nx = ny = 40
mesh = fen.RectangleMesh(fen.Point(0,0), fen.Point(PI, PI), nx, ny)
forcingTerm = [sp.printing.ccode(sp.simplify(sp.re(fex))), sp.printing.ccode(sp.simplify(sp.im(fex)))]
solver = HF(mesh = mesh, wavenumber = nu + 1.j, forcingTerm = forcingTerm, FEDegree = 3,\
DirichletBoundary = boundary, DirichletDatum = 0)
uh = solver.solve()
plotter = HS(solver.V)
print(plotter.norm(uh, np.real(nu)))
plotter.plot(uh)
###########
elif testNo == 2:
def boundary(x, on_boundary):
return on_boundary
PI = np.pi
n1 = 4**.5
n2 = 1**.5
kappa = 3
theta = PI * 70 / 180
d1, d2 = np.cos(theta), np.sin(theta)
K1 = kappa * n1 * d1
if kappa * n2 >= K1:
K2 = ((kappa*n2)**2 - K1**2)**.5
else:
K2 = 1.j * (K1**2 - (kappa*n2)**2)**.5
R = (kappa * n1 * d2 - K2) / (kappa * n1 * d2 + K2)
T = R + 1
x, y = sp.symbols('x[0] x[1]', real=True)
uex1 = T*sp.exp(1.j*(K1*x+K2*y))
uex2 = sp.exp(1.j*kappa*n1*(d1*x+d2*y)) + R*sp.exp(1.j*kappa*n1*(d1*x-d2*y))
# Exact solution
uexRe = fen.Expression('x[1]>=0 ? {0} : {1}'.format(\
sp.printing.ccode(sp.re(uex1)), sp.printing.ccode(sp.re(uex2))), degree=4)
uexIm = fen.Expression('x[1]>=0 ? {0} : {1}'.format(\
sp.printing.ccode(sp.im(uex1)), sp.printing.ccode(sp.im(uex2))), degree=4)
# refraction index
n2Re = fen.Expression('x[1]<0 ? n1r : n2r',
n1r = n1.real, n2r = n2.real, degree=4)
n2Im = fen.Expression('x[1]<0 ? n1i : n2i',
n1i = n1.imag, n2i = n2.imag, degree=4)
# Create mesh and define function space
nx = ny = 50
mesh = fen.RectangleMesh(fen.Point(-PI/2,-PI/2), fen.Point(PI/2,PI/2), nx, ny)
solver = HF(mesh = mesh, wavenumber = kappa, refractionIndex = (n2Re, n2Im),\
forcingTerm = 0, FEDegree = 3, DirichletBoundary = boundary,\
DirichletDatum = (uexRe, uexIm))
uh = solver.solve()
plotter = HS(solver.V)
print(plotter.norm(uh, kappa))
plotter.plot(uh)
###########
elif testNo == 3:
import mshr
from matplotlib import pyplot as plt
PI = np.pi
R = 5
def Dboundary(x, on_boundary):
return on_boundary and (x[0]**2+x[1]**2)**.5 < .95 * R
A = 10
kappa = 12**.5
theta = - PI * 90 / 180
x, y = sp.symbols('x[0] x[1]', real=True)
phiex = kappa * (x * np.cos(theta) + y * np.sin(theta))
u0ex = - A * sp.exp(1.j * phiex)
- npoints = 50
+ npoints = 20
scatterer = mshr.Polygon([fen.Point(-1, -.5),
fen.Point(1, -.5),
fen.Point(1, .5),
fen.Point(.8, .5),
fen.Point(.8, -.3),
fen.Point(-.8, -.3),
fen.Point(-.8, .5),
fen.Point(-1, .5),
])
mesh = mshr.generate_mesh(mshr.Circle(fen.Point(0, 0), R) - scatterer, npoints)
plt.jet()
plt.figure()
fen.plot(mesh)
DirichletTerm = [sp.printing.ccode(sp.simplify(sp.re(u0ex))), sp.printing.ccode(sp.simplify(sp.im(u0ex)))]
solver = HFS(mesh = mesh, wavenumber = kappa, forcingTerm = 0, FEDegree = 3,\
DirichletBoundary = Dboundary, RobinBoundary = 'rest',\
DirichletDatum = DirichletTerm)
- baseRe, baseIm = DirichletTerm
- baseRe = fen.project(fen.Expression(baseRe, degree = 4), solver.V)
- baseIm = fen.project(fen.Expression(baseIm, degree = 4), solver.V)
- uinc = np.array(baseRe.vector()) + 1.j * np.array(baseIm.vector())
-
+ uinc = solver.liftDirichletData()
uh = solver.solve()
plotter = HS(solver.V)
print(plotter.norm(uh, kappa))
print(plotter.norm(uh - uinc, kappa))
plotter.plot(uh)
- plotter.plot(uh - uinc)
+ plotter.plot(uh - uinc, name = 'u_tot')
###########
elif testNo == 4:
import mshr
from matplotlib import pyplot as plt
PI = np.pi
R = 5
def Dboundary(x, on_boundary):
return on_boundary and (x[0]**2+x[1]**2)**.5 < .95 * R
A = 10
kappa = 12**.5
theta = - PI * 90 / 180
x, y = sp.symbols('x[0] x[1]', real=True)
phiex = kappa * (x * np.cos(theta) + y * np.sin(theta))
u0ex = - A * sp.exp(1.j * phiex)
npoints = 40
scatterer = mshr.Polygon([fen.Point(-1, -.5),
fen.Point(1, -.5),
fen.Point(1, .5),
fen.Point(.8, .5),
fen.Point(.8, -.3),
fen.Point(-.8, -.3),
fen.Point(-.8, .5),
fen.Point(-1, .5),
])
mesh = mshr.generate_mesh(mshr.Circle(fen.Point(0, 0), R) - scatterer, npoints)
plt.jet()
plt.figure()
fen.plot(mesh)
DirichletTerm = [sp.printing.ccode(sp.simplify(sp.re(u0ex))), sp.printing.ccode(sp.simplify(sp.im(u0ex)))]
solver = HFSA(mesh = mesh, wavenumber = kappa, forcingTerm = 0, FEDegree = 3,\
DirichletBoundary = Dboundary, RobinBoundary = 'rest',\
DirichletDatum = DirichletTerm, constraintType = 'MASS')
- baseRe, baseIm = DirichletTerm
- baseRe = fen.project(fen.Expression(baseRe, degree = 4), solver.V)
- baseIm = fen.project(fen.Expression(baseIm, degree = 4), solver.V)
- uinc = np.array(baseRe.vector()) + 1.j * np.array(baseIm.vector())
- uinc = np.concatenate((uinc, kappa * uinc))
-
+ uinc = solver.liftDirichletData()
+ uinc[1] = kappa * uinc[0]
uh = solver.solve()
plotter = HSA(solver.V, 2)
print(plotter.norm(uh, kappa))
print(plotter.norm(uh - uinc, kappa))
plotter.plot(uh)
- plotter.plot(uh - uinc)
\ No newline at end of file
+ plotter.plot(uh - uinc, name = 'u_tot')
\ No newline at end of file
diff --git a/examples/Python/HelmholtzSolver_FF.py b/examples/Python/HelmholtzSolver_FF.py
new file mode 100644
index 0000000..0e1cedf
--- /dev/null
+++ b/examples/Python/HelmholtzSolver_FF.py
@@ -0,0 +1,134 @@
+#!/usr/bin/env python3
+# -*- coding: utf-8 -*-
+
+from __future__ import print_function
+import numpy as np
+from context import FreeFemHelmholtzEngine as HF
+from context import FreeFemHelmholtzScatteringEngine as HFS
+from context import FreeFemHelmholtzScatteringAugmentedEngine as HFSA
+from context import FreeFemHSEngine as HS
+from context import FreeFemHSAugmentedEngine as HSA
+
+PI = np.pi
+testNo = 1
+
+if testNo == 1:
+
+ V = ("int n = 40;\n"
+ "real nu = 12^.5, theta = pi / 3;\n"
+ "mesh Th = square(n, n, [pi * x, pi * y]);\n"
+ "load \"Element_P3\";\n"
+ "fespace V(Th, P3);")
+
+ f = ("16 / pi^4 * exp(-1i * nu*(cos(theta) * x + sin(theta) * y)) * ( 2i *"
+ " nu *cos(theta) * (2 * x * y^2 - 2 * pi * x * y - pi * y^2 + pi^2 * "
+ "y) + 2i * nu * sin(theta) * (2 *x^2 * y - pi * x^2 - 2 * pi * x * y "
+ "+ pi^2 * x) - (2 * y^2 - 2 * pi * y + 2 * x^2 - 2 * pi * x))")
+
+ solver = HF(V, 12**.5 + 1.j, forcingTerm = f, DirichletBoundary = "1,2,3,4")
+
+ uh = solver.solve()
+ plotter = HS(solver.V)
+ print(plotter.norm(uh, 12**.5))
+ plotter.plot(uh, figspecs = "fill = 1, value = 1", save = True)
+
+###########
+elif testNo == 2:
+
+ kappa = 3
+ theta = np.pi * 70 / 180
+ k1 = kappa * 2 * np.cos(theta)
+ if kappa > k1:
+ k2 = (kappa**2. - k1**2.)**.5
+ else:
+ k2 = 1.j * (k1**2. - kappa**2.)**.5
+ R = (kappa * 2 * np.sin(theta) - k2) / (kappa * 2 * np.sin(theta) + k2)
+ T = R + 1
+ V = ("int n = 50;\n"
+ "real kappa = {}, theta = {};\n"
+ "mesh Th = square(n, n, [pi * (x - .5), pi * (y - .5)]);\n"
+ "load \"Element_P3\";\n"
+ "fespace V(Th, P3);").format(kappa, theta)
+ n = "(y >= 0 ? 1 : 2)"
+ u0 = ("(y >= 0 ? {0}*exp(1.i*({1}*x+{2}*y)) : exp(2.i*{3}*({4}*x+{5}*y)) + "
+ "{6}*exp(2.i*{3}*({4}*x-{5}*y)))".format(T, k1, k2, kappa,
+ np.cos(theta), np.sin(theta), R))
+
+ solver = HF(V, kappa, refractionIndex = n, DirichletBoundary = "1,2,3,4", DirichletDatum = u0)
+
+ uh = solver.solve()
+ plotter = HS(solver.V)
+ print(plotter.norm(uh, kappa))
+ plotter.plot(uh, figspecs = "fill = 1, value = 1")
+
+
+###########
+elif testNo == 3:
+
+ A = 10
+ kappa = 12**.5
+ theta = - PI * 90 / 180
+
+ V = ("int n = 10, R = 5;\n"
+ "real kappa = {}, theta = {}, A = {};\n"
+ "border S1(t = 0, 1){{x = 2*t-1.; y = -.5; label = 1;}}\n"
+ "border S2(t = 0, 1){{x = 1.; y = -.5+t; label = 1;}}\n"
+ "border S3(t = 0, 1){{x = 1-.2*t; y = .5; label = 1;}}\n"
+ "border S4(t = 0, 1){{x = .8; y = .5-.8*t; label = 1;}}\n"
+ "border S5(t = 0, 1){{x = .8-1.6*t; y = -.3; label = 1;}}\n"
+ "border S6(t = 0, 1){{x = -.8; y = -.3+.8*t; label = 1;}}\n"
+ "border S7(t = 0, 1){{x = -.8-.2*t; y = .5; label = 1;}}\n"
+ "border S8(t = 0, 1){{x = -1.; y = .5-t; label = 1;}}\n"
+ "border Inf(t = 0, 1){{x = R * cos(2 * pi * t); y = R * sin(2 * pi * t); label = 2;}}\n"
+ "mesh Th = buildmesh(S1(-4*n) + S2(-3*n) + S3(-n) + S4(-2*n) + S5(-4*n) + S6(-2*n) + S7(-n) + S8(-3*n) + Inf(10*n));\n"
+ "load \"Element_P3\";\n"
+ "fespace V(Th, P3);").format(kappa, theta, A)
+ u0 = "- {0} * exp(1.i * {1} * (cos({2}) * x + sin({2}) * y))".format(A, kappa, theta)
+
+
+ solver = HFS(V, kappa, DirichletBoundary = "1", DirichletDatum = u0, RobinBoundary = "2")
+
+ uinc = solver.liftDirichletData()
+ uh = solver.solve()
+ plotter = HS(solver.V)
+ plotter.plotmesh()
+ print(plotter.norm(uh, kappa))
+ plotter.plot(uh, figspecs = 'fill = 1, value = 1')
+ print(plotter.norm(uh - uinc, kappa))
+ plotter.plot(uh - uinc, name = 'u_tot', figspecs = 'fill = 1, value = 1')
+
+###########
+elif testNo == 4:
+
+ A = 10
+ kappa = 12**.5
+ theta = - PI * 90 / 180
+
+ V = ("int n = 5, R = 5;\n"
+ "real kappa = {}, theta = {}, A = {};\n"
+ "border S1(t = 0, 1){{x = 2*t-1.; y = -.5; label = 1;}}\n"
+ "border S2(t = 0, 1){{x = 1.; y = -.5+t; label = 1;}}\n"
+ "border S3(t = 0, 1){{x = 1-.2*t; y = .5; label = 1;}}\n"
+ "border S4(t = 0, 1){{x = .8; y = .5-.8*t; label = 1;}}\n"
+ "border S5(t = 0, 1){{x = .8-1.6*t; y = -.3; label = 1;}}\n"
+ "border S6(t = 0, 1){{x = -.8; y = -.3+.8*t; label = 1;}}\n"
+ "border S7(t = 0, 1){{x = -.8-.2*t; y = .5; label = 1;}}\n"
+ "border S8(t = 0, 1){{x = -1.; y = .5-t; label = 1;}}\n"
+ "border Inf(t = 0, 1){{x = R * cos(2 * pi * t); y = R * sin(2 * pi * t); label = 2;}}\n"
+ "mesh Th = buildmesh(S1(-4*n) + S2(-3*n) + S3(-n) + S4(-2*n) + S5(-4*n) + S6(-2*n) + S7(-n) + S8(-3*n) + Inf(10*n));\n"
+ "load \"Element_P3\";\n"
+ "fespace V(Th, P3);").format(kappa, theta, A)
+ u0 = "- {0} * exp(1.i * {1} * (cos({2}) * x + sin({2}) * y))".format(A, kappa, theta)
+
+
+ solver = HFSA(V, kappa, DirichletBoundary = "1", DirichletDatum = u0, RobinBoundary = "2")
+
+ uinc = solver.liftDirichletData()
+ uinc[1] = kappa * uinc[0]
+ uh = solver.solve()
+ plotter = HSA(solver.V, 2)
+ plotter.plotmesh()
+ print(plotter.norm(uh, kappa))
+ print(plotter.norm(uh - uinc, kappa))
+ plotter.plot(uh, figspecs = 'fill = 1, value = 1')
+ plotter.plot(uh - uinc, name = 'u_tot', figspecs = 'fill = 1, value = 1')
diff --git a/examples/Python/HelmholtzTaylorApproximantsSweep.py b/examples/Python/HelmholtzTaylorApproximantsSweep.py
index 537a5c2..103afa0 100644
--- a/examples/Python/HelmholtzTaylorApproximantsSweep.py
+++ b/examples/Python/HelmholtzTaylorApproximantsSweep.py
@@ -1,100 +1,100 @@
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
# Example homogeneous Dirichlet forcing wave SWEEP
from __future__ import print_function
import fenics as fen
import numpy as np
import sympy as sp
from context import FenicsHelmholtzEngine as HFEngine
from context import FenicsHSEngine as HSEngine
from context import ROMApproximantTaylorPade as Pade
from context import ROMApproximantTaylorRB as RB
from context import ROMApproximantSweeper as Sweeper
PI = np.pi
nu = 12**.5
theta = PI / 3
z0 = 12 + .5j
npoints = 31
ktars = np.linspace(7, 16, npoints)
x, y = sp.symbols('x[0] x[1]', real=True)
wex = 16/PI**4 * x * y * (x - PI) * (y - PI)
phiex = nu * (x * np.cos(theta) + y * np.sin(theta))
uex = wex * sp.exp(-1.j * phiex)
fex = - uex.diff(x, 2) - uex.diff(y, 2) - nu**2 * uex
nx = ny = 10
mesh = fen.RectangleMesh(fen.Point(0,0), fen.Point(PI,PI), nx, ny)
forcingTerm = [sp.printing.ccode(sp.simplify(sp.re(fex))), sp.printing.ccode(sp.simplify(sp.im(fex)))]
solver = HFEngine(mesh = mesh, wavenumber = nu, forcingTerm = forcingTerm, FEDegree = 3,
DirichletBoundary = 'all', DirichletDatum = 0)
plotter = HSEngine(solver.V)
shift = 5
nsets = 5
stride = 2
Emax = stride * (nsets - 1) + shift + 2
params = {'Emax':Emax, 'sampleType':'ARNOLDI', 'POD':True}
paramsSetsPade = [None] * nsets
paramsSetsRB = [None] * nsets
for i in range(nsets):
paramsSetsPade[i] = {'N':stride*i+shift+1, 'M':stride*i+shift,
'E':stride*i+shift+1}
paramsSetsRB[i] = {'E':stride*i+shift+1,'R':stride*i+shift+2}
appPade = Pade(solver, plotter, k0 = z0, w = np.real(z0**.5),
approxParameters = params)
appRB = RB(solver, plotter, k0 = z0, w = np.real(z0**.5),
approxParameters = params)
sweeper = Sweeper.ROMApproximantSweeper(ktars = ktars, mostExpensive = 'Approx')
sweeper.ROMEngine = appPade
sweeper.params = paramsSetsPade
filenamePade = sweeper.sweep('../Data/HelmholtzBubbleTaylorPade.dat', outputs = 'ALL')
sweeper.ROMEngine = appRB
sweeper.params = paramsSetsRB
filenameRB = sweeper.sweep('../Data/HelmholtzBubbleTaylorRB.dat', outputs = 'ALL')
####################
from matplotlib import pyplot as plt
plt.jet()
for i in range(nsets):
nDerivatives = stride*i+shift+1
PadeOutput = sweeper.read(filenamePade, {'E':nDerivatives},
- ['kRe', 'HFNorm', 'AppNorm', 'AppError'])
+ ['kRe', 'HFNorm', 'AppNorm', 'ErrNorm'])
RBOutput = sweeper.read(filenameRB, {'E':nDerivatives},
- ['kRe', 'AppNorm', 'AppError'])
+ ['kRe', 'AppNorm', 'ErrNorm'])
ktarsF = PadeOutput['kRe']
solNormF = PadeOutput['HFNorm']
PadektarsF = PadeOutput['kRe']
PadeNormF = PadeOutput['AppNorm']
- PadeErrorF = PadeOutput['AppError']
+ PadeErrorF = PadeOutput['ErrNorm']
RBktarsF = RBOutput['kRe']
RBNormF = RBOutput['AppNorm']
- RBErrorF = RBOutput['AppError']
+ RBErrorF = RBOutput['ErrNorm']
plt.figure()
plt.semilogy(ktarsF, solNormF, 'k-', label='Sol norm')
plt.semilogy(PadektarsF, PadeNormF, 'b.--', label='Pade'' norm, E = {}'.format(nDerivatives))
plt.semilogy(RBktarsF, RBNormF, 'g.--', label='RB norm, E = {}'.format(nDerivatives))
plt.legend()
plt.grid()
plt.figure()
plt.semilogy(PadektarsF, PadeErrorF, 'b', label='Pade'' error, E = {}'.format(nDerivatives))
plt.semilogy(RBktarsF, RBErrorF, 'g', label='RB error, E = {}'.format(nDerivatives))
plt.legend()
plt.grid()
plt.figure()
plt.semilogy(ktarsF, PadeErrorF / solNormF, 'b', label='Pade'' relative error, E = {}'.format(nDerivatives))
plt.semilogy(RBktarsF, RBErrorF / solNormF, 'g', label='RB relative error, E = {}'.format(nDerivatives))
plt.legend()
plt.grid()
\ No newline at end of file
diff --git a/examples/Python/context.py b/examples/Python/context.py
index 049df6a..0089ef8 100644
--- a/examples/Python/context.py
+++ b/examples/Python/context.py
@@ -1,16 +1,16 @@
# -*- coding: utf-8 -*-
import sys, os
module_path = os.path.abspath(os.path.join(os.path.dirname(__file__), '..', '..', 'main'))
if module_path not in sys.path:
sys.path.append(module_path)
import utilities, ROMApproximantSweeper, FreeFemConversionTools
-from FenicsHelmholtzEngine import FenicsHelmholtzEngine, FenicsHelmholtzScatteringEngine, FenicsHelmholtzScatteringAugmentedEngine
from FenicsHSEngine import FenicsHSEngine, FenicsHSAugmentedEngine
-from FreeFemHSEngine import FreeFemHSEngine#, FreeFemHSAugmentedEngine
-#from FreeFemHelmholtzEngine import FenicsHelmholtzEngine, FenicsHelmholtzScatteringEngine, FenicsHelmholtzScatteringAugmentedEngine
+from FenicsHelmholtzEngine import FenicsHelmholtzEngine, FenicsHelmholtzScatteringEngine, FenicsHelmholtzScatteringAugmentedEngine
+from FreeFemHSEngine import FreeFemHSEngine, FreeFemHSAugmentedEngine
+from FreeFemHelmholtzEngine import FreeFemHelmholtzEngine, FreeFemHelmholtzScatteringEngine, FreeFemHelmholtzScatteringAugmentedEngine
from ROMApproximantLagrangePade import ROMApproximantLagrangePade
from ROMApproximantLagrangeRB import ROMApproximantLagrangeRB
from ROMApproximantTaylorPade import ROMApproximantTaylorPade
from ROMApproximantTaylorRB import ROMApproximantTaylorRB
diff --git a/examples/Python/test.py b/examples/Python/test.py
deleted file mode 100644
index aecd2d3..0000000
--- a/examples/Python/test.py
+++ /dev/null
@@ -1,20 +0,0 @@
-#!/usr/bin/env python3
-# -*- coding: utf-8 -*-
-
-from __future__ import print_function
-import numpy as np
-from context import FreeFemHSEngine as FFHSE
-
-V = ("int n = 5;\n"
- "mesh Th = square(n, n, [pi * x, pi * y]);\n"
- "load \"Element_P3\";\n"
- "fespace V(Th, P3);")
-
-np.random.seed(420)
-u = np.random.rand(256) + 1.j * np.random.rand(256)
-
-engine = FFHSE(V, 2, "Th", "V<complex>")
-
-nm = engine.norm(u, 'H1')
-
-engine.plot(u, figspecs = "fill = 1, value = 1")
\ No newline at end of file
diff --git a/main/FenicsHSEngine.py b/main/FenicsHSEngine.py
index 301da21..243fe78 100644
--- a/main/FenicsHSEngine.py
+++ b/main/FenicsHSEngine.py
@@ -1,202 +1,182 @@
#!/usr/bin/python
import numpy as np
+import utilities
from matplotlib import pyplot as plt
import fenics as fen
class FenicsHSEngine:
"""
Fenics-based Hilbert space engine.
"""
def __init__(self, V:"Fenics FE space"):
self.V = V
def name(self) -> str:
"""Class label."""
return self.__class__.__name__
def norm(self, u:"numpy 1D array",
normType : "numpy 2D array, number or str" = "H10") -> float:
"""
Compute general norm of complex-valued function with given dofs.
Args:
u: numpy complex array with function dofs.
normType(optional): Target norm identifier. If matrix, target norm
is one induced by normType. If number, target norm is weighted
H^1 norm with given weight. If string, must be recognizable by
Fenics norm command. Defaults to 'H10'.
Returns:
Norm of the function (non-negative).
"""
if type(normType).__name__[-6:] == "matrix":
return np.abs(u.dot(normType.dot(u).conj())) ** .5
if isinstance(normType, (int, float)):
return (FenicsHSEngine.norm(self, u, "H10")**2
+ (normType * FenicsHSEngine.norm(self, u, "L2"))**2)**.5
uRe = fen.Function(self.V)
uIm = fen.Function(self.V)
uRe.vector()[:] = np.array(np.real(u), dtype = float)
uIm.vector()[:] = np.array(np.imag(u), dtype = float)
return (fen.norm(uRe, normType)**2 + fen.norm(uIm, normType)**2)**.5
- def plot(self, u:"numpy 1D array", name : str = "u", **figspecs):
+ def plot(self, u:"numpy 1D array", name : str = "u",
+ what : "str or list" = 'all', **figspecs):
"""
Do some nice plots of the complex-valued function with given dofs.
Args:
u: numpy complex array with function dofs.
name(optional): Name to be shown as title of the plots. Defaults to
'u'.
+ what(optional): Which plots to do. If list, can contain 'ABS',
+ 'PHASE', 'REAL', 'IMAG'. If str, same plus wildcard 'ALL'.
+ Defaults to 'ALL'.
figspecs(optional key args): Optional arguments for matplotlib
figure creation.
"""
- if 'figsize' not in figspecs.keys(): figspecs['figsize'] = (13, 3)
+ if isinstance(what, (str,)):
+ if what.upper() == 'ALL':
+ what = ['ABS', 'PHASE', 'REAL', 'IMAG']
+ else:
+ what = [what]
+ what = utilities.purgeList(what, ['ABS', 'PHASE', 'REAL', 'IMAG'],
+ listname = self.name() + ".what")
- uRe = fen.Function(self.V)
- uIm = fen.Function(self.V)
- uRe.vector()[:] = np.array(np.real(u), dtype = float)
- uIm.vector()[:] = np.array(np.imag(u), dtype = float)
+ if len(what) == 0: return
+ if 'figsize' not in figspecs.keys():
+ figspecs['figsize'] = (13. * len(what) / 4, 3)
+
+ subplotcode = 100 + len(what) * 10
+
+ if not what == ['IMAG']:
+ uRe = fen.Function(self.V)
+ uRe.vector()[:] = np.array(np.real(u), dtype = float)
+ if not what == ['REAL']:
+ uIm = fen.Function(self.V)
+ uIm.vector()[:] = np.array(np.imag(u), dtype = float)
- uAbs = fen.project(np.power(np.power(uRe, 2) + np.power(uIm, 2),
- .5), self.V)
- uPhase = fen.project(fen.atan(np.divide(uIm, uRe)), self.V)
- combined_data_ReIm = np.concatenate([np.real(u), np.imag(u)])
- _min, _max = np.amin(combined_data_ReIm), np.amax(combined_data_ReIm)
+ if 'REAL' in what and 'IMAG' in what:
+ combined_data_ReIm = np.concatenate([np.real(u), np.imag(u)])
+ _min = np.amin(combined_data_ReIm)
+ _max = np.amax(combined_data_ReIm)
+ pars = {'vmin' : _min, 'vmax' : _max}
+ else:
+ pars = {}
plt.figure(**figspecs)
-
plt.jet()
- plt.subplot(141)
- p = fen.plot(uAbs, title = "|{0}|".format(name))
- plt.colorbar(p)
-
- plt.subplot(142)
- p = fen.plot(uPhase, title = "phase({0})".format(name))
- plt.colorbar(p)
-
- plt.subplot(143)
- p = fen.plot(uRe, title = "Re({0})".format(name),
- vmin = _min, vmax = _max)
- plt.colorbar(p)
-
- plt.subplot(144)
- p = fen.plot(uIm, title = "Im({0})".format(name),
- vmin = _min, vmax = _max)
- plt.colorbar(p)
- plt.tight_layout()
- plt.show()
-
+ if 'ABS' in what:
+ uAbs = fen.project(np.power(np.power(uRe, 2) + np.power(uIm, 2),
+ .5), self.V)
+ subplotcode = subplotcode + 1
+ plt.subplot(subplotcode)
+ p = fen.plot(uAbs, title = "|{0}|".format(name))
+ plt.colorbar(p)
+ if 'PHASE' in what:
+ uPhase = fen.project(fen.atan(np.divide(uIm, uRe)), self.V)
+ subplotcode = subplotcode + 1
+ plt.subplot(subplotcode)
+ p = fen.plot(uPhase, title = "phase({0})".format(name))
+ plt.colorbar(p)
+ if 'REAL' in what:
+ subplotcode = subplotcode + 1
+ plt.subplot(subplotcode)
+ p = fen.plot(uRe, title = "Re({0})".format(name), **pars)
+ plt.colorbar(p)
+ if 'IMAG' in what:
+ subplotcode = subplotcode + 1
+ plt.subplot(subplotcode)
+ p = fen.plot(uIm, title = "Im({0})".format(name), **pars)
+ plt.colorbar(p)
+
class FenicsHSAugmentedEngine(FenicsHSEngine):
"""
Fenics-based Hilbert space engine for augmented spaces.
"""
- def __init__(self, V:"Fenics FE space", d : int = 1):
+ def __init__(self, V:"Fenics FE space", d : int = 2):
self.d = d
FenicsHSEngine.__init__(self, V)
- def getActualSize(self, u:"numpy 1D array"):
- """
- Compute size of unaugmented vector batches.
-
- Args:
- u: numpy complex array with function dofs.
-
- Returns:
- Batch size.
- """
- N = int(len(u) / self.d)
- if not np.isclose(len(u), self.d * N):
- raise Exception(("Input vector lenght must be multiple of"
- "augmentation dimension d."))
- return N
-
- def norm(self, u:"numpy 1D array",
+ def norm(self, u:"numpy 1D array or list",
normType : "numpy 2D array, number or str" = "H10")\
-> "numpy 1D array":
"""
Compute general norm of complex-valued function with given dofs.
Args:
u: numpy complex array with function dofs.
normType(optional): Target norm identifier. If matrix, target norm
is one induced by normType. If number, target norm is weighted
H^1 norm with given weight. If string, must be recognizable by
Fenics norm command. Defaults to 'H10'.
Returns:
Norms of the function (non-negative).
"""
- N = self.getActualSize(u)
if isinstance(normType, (int, float)):
+ u = utilities.listify(u, self.d)
norms = [None] * self.d
for j in range(self.d):
- uj = u[j * N : (j + 1) * N]
- norms[j] = FenicsHSEngine.norm(self, uj, normType)
+ norms[j] = FenicsHSEngine.norm(self, u[j], normType)
return norms
- else:
- if (type(normType).__name__[-6:] == "matrix"
- and normType.shape[0] % N == 0):
- N = normType.shape[0]
+ elif type(normType).__name__[-6:] == "matrix":
+ u = utilities.vectorify(u, self.d)
+ if normType.shape[0] % u[0] == 0:
+ N = int(normType.shape[0] / u[0])
else:
raise Exception("Energy matrix dimension mismatch.")
- return FenicsHSEngine.norm(self, u[: N], normType)
+ return FenicsHSEngine.norm(self, np.concatenate(u[: N]), normType)
+ raise Exception("Norm type not recognized.")
- def plot(self, u:"numpy 1D array", name : str = "u", **figspecs):
+ def plot(self, u:"numpy 1D array", name : str = "u",
+ what : "str or list" = 'all', **figspecs) -> list:
"""
Do some nice plots of the complex-valued function with given dofs.
Args:
u: numpy complex array with function dofs.
name(optional): Name to be shown as title of the plots. Defaults to
'u'.
+ what(optional): Which plots to do. If list, can contain 'ABS',
+ 'PHASE', 'REAL', 'IMAG'. If str, same plus wildcard 'ALL'.
+ Defaults to 'ALL'.
figspecs(optional key args): Optional arguments for matplotlib
figure creation.
+
+ Returns:
+ List of pyplot figure handles.
"""
- if 'figsize' not in figspecs.keys(): figspecs['figsize'] = (13, 3)
-
- N = self.getActualSize(u)
+ u = utilities.listify(u, self.d)
for j in range(self.d):
- uj = u[j * N : (j + 1) * N]
-
- uRe = fen.Function(self.V)
- uIm = fen.Function(self.V)
- uRe.vector()[:] = np.array(np.real(uj), dtype = float)
- uIm.vector()[:] = np.array(np.imag(uj), dtype = float)
-
- uAbs = fen.project(np.power(np.power(uRe, 2) + np.power(uIm, 2),
- .5), self.V)
- uPhase = fen.project(fen.atan(np.divide(uIm, uRe)), self.V)
- combined_data_ReIm = np.concatenate([np.real(uj), np.imag(uj)])
- _min = np.amin(combined_data_ReIm)
- _max = np.amax(combined_data_ReIm)
-
- plt.figure(**figspecs)
-
- plt.jet()
-
- plt.subplot(141)
- p = fen.plot(uAbs, title = "|{0}|, comp. {1}".format(name, j))
- plt.colorbar(p)
-
- plt.subplot(142)
- p = fen.plot(uPhase,
- title = "phase({0}, comp. {1})".format(name, j))
- plt.colorbar(p)
+ FenicsHSEngine.plot(self, u[j], what = what,
+ name = "{0}, comp. {1}".format(name, j),
+ **figspecs)
- plt.subplot(143)
- p = fen.plot(uRe, title = "Re({0}, comp. {1})".format(name, j),
- vmin = _min, vmax = _max)
- plt.colorbar(p)
-
- plt.subplot(144)
- p = fen.plot(uIm, title = "Im({0}, comp. {1})".format(name, j),
- vmin = _min, vmax = _max)
- plt.colorbar(p)
- plt.tight_layout()
- plt.show()
+
\ No newline at end of file
diff --git a/main/FenicsHelmholtzEngine.py b/main/FenicsHelmholtzEngine.py
index 205bf9f..416a224 100644
--- a/main/FenicsHelmholtzEngine.py
+++ b/main/FenicsHelmholtzEngine.py
@@ -1,954 +1,866 @@
#!/usr/bin/python
from copy import copy
import warnings
import numpy as np
import sympy as sp
import sympy.printing as sprint
import scipy.sparse as scsp
-import scipy.sparse.linalg as spla
import fenics as fen
PI = np.pi
fenZERO = fen.Constant(0)
fenZEROC = [fenZERO, fenZERO]
class DomainError(Exception):
"""
Domain error exception.
Args:
value: Human readable string describing the exception.
Attributes:
value: Human readable string describing the exception.
"""
def __init__(self, value:str):
self.value = value
def __str__(self):
return self.value
def CustomExpressionParser(value:"expression",
degree:int) -> "Fenics Expression":
"""
Numerical and Fenics expressions parser.
Args:
value: Expression to be parsed. Accepts 2-tuples composed of real and
imaginary parts. Available elementary formats are numbers, strings
and Fenics Expression's.
degree: Degree of Fenics FE interpolant of expression.
Returns:
Fenics FE interpolant of expression.
"""
try:
if isinstance(value, (list, tuple)):
if len(value) == 1:
return CustomExpressionParser(value[0])
elif len(value) != 2:
raise Exception("Parsing error")
if isinstance(value[0], (int, float, complex)):
if any([isinstance(y, complex) for y in value]):
raise Exception("Parsing error")
valueRe = fen.Constant(value[0])
valueIm = fen.Constant(value[1])
elif isinstance(value[0], str):
x, y, z, x0, x1, x2 = sp.symbols("x y z x[0] x[1] x[2]",
real=True)
xyDict = {"x": x, "y": y, "z": z}
valueReParsed = value[0].replace("x[0]", "x")\
.replace("x[1]", "y")\
.replace("x[2]", "z")
valueImParsed = value[1].replace("x[0]", "x")\
.replace("x[1]", "y")\
.replace("x[2]", "z")
valueReSym = sp.sympify(valueReParsed, locals=xyDict)\
.subs([(x, x0), (y, x1), (z, x2)])
valueImSym = sp.sympify(valueImParsed, locals=xyDict)\
.subs([(x, x0), (y, x1), (z, x2)])
valueReStr = sprint.ccode(valueReSym).rpartition("\n")[-1]
valueImStr = sprint.ccode(valueImSym).rpartition("\n")[-1]
valueRe = fen.Expression(valueReStr, degree = degree)
valueIm = fen.Expression(valueImStr, degree = degree)
else:
valueRe = value[0]
valueIm = value[1]
else:
if isinstance(value, (int, float, complex)):
valueRe = fen.Constant(np.real(value))
valueIm = fen.Constant(np.imag(value))
elif isinstance(value, str):
x, y, z, x0, x1, x2 = sp.symbols("x y z x[0] x[1] x[2]",
real=True)
xyDict = {"x": x, "y": y, "z": z}
valueParsed = value.replace("x[0]", "x").replace("x[1]", "y")\
.replace("x[2]", "z")
valueSym = sp.sympify(valueParsed, locals=xyDict)\
.subs([(x, x0), (y, x1), (z, x2)])
valueReStr = sprint.ccode(sp.re(valueSym)).rpartition("\n")[-1]
valueImStr = sprint.ccode(sp.im(valueSym)).rpartition("\n")[-1]
valueImStr = sprint.ccode(valueImSym).rpartition("\n")[-1]
valueRe = fen.Expression(valueReStr, degree = degree)
valueIm = fen.Expression(valueImStr, degree = degree)
else:
valueRe = value
valueIm = fenZERO
except:
raise Exception("Parsing error")
return copy(valueRe), copy(valueIm)
class FenicsHelmholtzEngine:
"""
Fenics-based solver for Helmholtz problems.
- \nabla \cdot (a \nabla u) - k^2 * n**2 * u = f in \Omega
u = u0 on \Gamma_D
\partial_nu = g1 on \Gamma_N
\partial_nu + h u = g2 on \Gamma_R
Args:
mesh: Domain of Helmholtz problem.
FEDegree: FE degree.
wavenumber2: Value of k^2.
diffusivity(optional): Value of a. Defaults to identically 1.
refractionIndex(optional): Value of n. Defaults to identically 1.
forcingTerm(optional): Value of f. Defaults to identically 0.
DirichletBoundary(optional): Function flagging Dirichlet boundary as
True, in Fenics format. Keywords 'ALL', 'NONE' and 'REST' are
accepted. Defaults to False everywhere.
NeumannBoundary(optional): Function flagging Neumann boundary as True,
in Fenics format. Keywords 'ALL', 'NONE' and 'REST' are accepted.
Defaults to False everywhere.
RobinBoundary(optional): Function flagging Robin boundary as True, in
Fenics format. Keywords 'ALL', 'NONE' and 'REST' are accepted.
Defaults to False everywhere.
DirichletDatum(optional): Value of u0. Defaults to identically 0.
NeumannDatum(optional): Value of g1. Defaults to identically 0.
RobinDatum(optional): Value of h. Defaults to identically 0.
RobinDatum2(optional): Value of g2. Defaults to identically 0.
Attributes:
FEDegree: FE degree.
wavenumber2: Copy of processed wavenumber2 parameter.
diffusivity: Copy of processed diffusivity parameter.
refractionIndex: Copy of processed refractionIndex parameter.
forcingTerm: Copy of processed forcingTerm parameter.
DirichletBoundary: Copy of processed DirichletBoundary parameter.
NeumannBoundary: Copy of processed NeumannBoundary parameter.
RobinBoundary: Copy of processed RobinBoundary parameter.
DirichletDatum: Copy of processed DirichletDatum parameter.
NeumannDatum: Copy of processed NeumannDatum parameter.
RobinDatum: Copy of processed RobinDatum parameter.
RobinDatum2: Copy of processed RobinDatum2 parameter.
LU: Whether to use LU solver for computation of derivatives.
V: Real FE space.
u: Helmholtz problem solution as numpy complex vector.
v1: Generic trial functions for variational form evaluation.
v2: Generic test functions for variational form evaluation.
ds: Boundary measure 2-tuple (resp. for Neumann and Robin boundaries).
nRe,nIm: Real and imaginary parts of n.
n2Re,n2Im: Real and imaginary parts of n^2.
fRe,fIm: Real and imaginary parts of f.
u0Re,u0Im: Real and imaginary parts of u0.
g1Re,g1Im: Real and imaginary parts of g1.
hRe,hIm: Real and imaginary parts of h.
g2Re,g2Im: Real and imaginary parts of g2.
DirichletBCRe,DirichletBCIm: Fenics BC manager for real and imaginary
parts of Dirichlet data.
A*: Scipy sparse array representation (in CSC format) of A*.
- b*Re,b*Im: Numpy array representation of real and imaginary parts of
- b*.
- invA: Numpy LU solver for A.
+ b*: Numpy array representation of b*.
"""
def __init__(self, mesh:"Fenics mesh",
FEDegree:int,
wavenumber:complex,
diffusivity : "custom expression" = 1,
refractionIndex : "custom expression" = 1,
forcingTerm : "custom expression" = fenZEROC,
DirichletBoundary : "bool function or str" = None,
NeumannBoundary : "bool function or str" = None,
RobinBoundary : "bool function or str" = None,
DirichletDatum : "custom expression" = fenZEROC,
NeumannDatum : "custom expression" = fenZEROC,
RobinDatum : "custom expression" = fenZEROC,
RobinDatum2 : "custom expression" = fenZEROC):
- self.FEDegree: FEDegree
+ self.FEDegree = FEDegree
# process boundaries
boundariesList = [DirichletBoundary, NeumannBoundary, RobinBoundary]
for i in range(3):
if isinstance(boundariesList[i], str):
boundariesList[i] = boundariesList[i].upper()
if boundariesList[i] == "NONE": boundariesList[i] = None
DirichletBoundary, NeumannBoundary, RobinBoundary = boundariesList
if boundariesList.count(None) == 3:
raise DomainError("At least one boundary must be prescribed.")
if boundariesList.count("ALL") + boundariesList.count("REST") >= 2:
raise DomainError("Only one keyword 'ALL'/'REST' can be used.")
def DirichletB(x, on_boundary):
if DirichletBoundary is None:
return False
elif DirichletBoundary == "ALL":
return on_boundary
elif DirichletBoundary == "REST":
return (on_boundary
and not self.NeumannBoundary(x, on_boundary)
and not self.RobinBoundary(x, on_boundary))
else:
return DirichletBoundary(x, on_boundary)
def NeumannB(x, on_boundary):
if NeumannBoundary is None:
return False
elif NeumannBoundary == "ALL":
return on_boundary
elif NeumannBoundary == "REST":
return (on_boundary
and not self.DirichletBoundary(x, on_boundary)
and not self.RobinBoundary(x, on_boundary))
else:
return NeumannBoundary(x, on_boundary)
def RobinB(x, on_boundary):
if RobinBoundary is None:
return False
elif RobinBoundary == "ALL":
return on_boundary
elif RobinBoundary == "REST":
return (on_boundary
and not self.DirichletBoundary(x, on_boundary)
and not self.NeumannBoundary(x, on_boundary))
else:
return RobinBoundary(x, on_boundary)
self.DirichletBoundary = copy(DirichletB)
self.NeumannBoundary = copy(NeumannB)
self.RobinBoundary = copy(RobinB)
# create boundary measure
class NBoundary(fen.SubDomain):
def inside(self, x, on_boundary):
return NeumannB(x, on_boundary)
class RBoundary(fen.SubDomain):
def inside(self, x, on_boundary):
return RobinB(x, on_boundary)
boundary_markers = fen.FacetFunction("size_t", mesh)
NBoundary().mark(boundary_markers, 0)
RBoundary().mark(boundary_markers, 1)
self.ds = fen.Measure("ds", domain=mesh,
subdomain_data=boundary_markers)
self.V = fen.FunctionSpace(mesh, "P", FEDegree)
self.v1 = fen.TrialFunction(self.V)
self.v2 = fen.TestFunction(self.V)
self.wavenumber2 = wavenumber ** 2
self.diffusivity = diffusivity
self.refractionIndex = refractionIndex
self.forcingTerm = forcingTerm
self.DirichletDatum = DirichletDatum
self.NeumannDatum = NeumannDatum
self.RobinDatum = RobinDatum
self.RobinDatum2 = RobinDatum2
+
+ functional = lambda self, u: 0.
def energyNormMatrix(self, w:float) -> "CSC sparse matrix":
"""
Sparse matrix (in CSR format) assoociated to w-weighted H10 norm.
Args:
w: Weight.
Returns:
Sparse matrix in CSR format.
"""
normMatFen = fen.assemble((
fen.dot(fen.grad(self.v1), fen.grad(self.v2))
+ w**2 * fen.dot(self.v1, self.v2)
) * fen.dx)
normMat = fen.as_backend_type(normMatFen).mat()
return scsp.csr_matrix(normMat.getValuesCSR()[::-1],
shape = normMat.size)
def problemData(self):
"""List of HF problem data."""
dataDict = {}
dataDict["forcingTerm"] = self.forcingTerm
dataDict["DirichletDatum"] = self.DirichletDatum
dataDict["NeumannDatum"] = self.NeumannDatum
dataDict["RobinDatum2"] = self.RobinDatum2
return dataDict
def setProblemData(self, dataDict:dict, k2:complex):
"""
Set problem data.
Args:
dataDict: Dictionary of problem data.
k2: Parameter value.
"""
self.wavenumber2 = k2
self.forcingTerm = dataDict["forcingTerm"]
self.DirichletDatum = dataDict["DirichletDatum"]
self.NeumannDatum = dataDict["NeumannDatum"]
self.RobinDatum2 = dataDict["RobinDatum2"]
-
- def getLSBlocks(self) -> ("numpy 2D array" * 3, "numpy 1D array"):
- """
- Get blocks of linear system.
-
- Returns:
- Blocks of system (\sum_{j=0}^J k^j A_j = b)
- """
- self.assembleA()
- self.assembleb()
- return [self.AL + self.AR, - self.AM], [self.b]
+ if hasattr(self, "A0"): del self.A0
+ if hasattr(self, "A1"): del self.A1
+ if hasattr(self, "b0"): del self.b0
def liftDirichletData(self):
"""Lift Dirichlet datum."""
solLRe = fen.interpolate(self.u0Re, self.V)
solLIm = fen.interpolate(self.u0Im, self.V)
return np.array(solLRe.vector()) + 1.j * np.array(solLIm.vector())
- def setupDerivativeComputation(self, j:int, up:"numpy 1D array",
- upp : "numpy 1D array" = None):
- """
- Setup problem data to compute solution derivatives.
-
- Args:
- j: Derivative index.
- up: Adjusted previous derivative.
- upp: Adjusted pre-previous derivative.
- """
- upRe = fen.Function(self.V)
- upIm = fen.Function(self.V)
- upRe.vector()[:] = np.array(np.real(up), dtype = float)
- upIm.vector()[:] = np.array(np.imag(up), dtype = float)
- upRe, upIm = self.n2Re * upRe - self.n2Im * upIm,\
- self.n2Re * upIm + self.n2Im * upRe
- self.forcingTerm = [upRe, upIm]
- self.DirichletDatum = fenZEROC
- self.NeumannDatum = fenZEROC
- self.RobinDatum2 = fenZEROC
-
- @property
- def wavenumber2(self):
- """Value of k^2."""
- return self._wavenumber2
- @wavenumber2.setter
- def wavenumber2(self, wavenumber2):
- if hasattr(self, "A"): del self.A
- if hasattr(self, "u"): del self.u
- self._wavenumber2 = wavenumber2
-
@property
def diffusivity(self):
"""Value of a. Its assignment changes aRe and aIm."""
return self._diffusivity
@diffusivity.setter
def diffusivity(self, diffusivity):
- if hasattr(self, "A"): del self.A
- if hasattr(self, "AL"): del self.AL
- if hasattr(self, "u"): del self.u
+ if hasattr(self, "A0"): del self.A0
self.aRe, self.aIm = CustomExpressionParser(diffusivity,
degree = self.FEDegree)
self._diffusivity = copy(diffusivity)
@property
def refractionIndex(self):
"""Value of n. Its assignment changes nRe, nIm, n2Re and n2Im."""
return self._refractionIndex
@refractionIndex.setter
def refractionIndex(self, refractionIndex):
- if hasattr(self, "A"): del self.A
- if hasattr(self, "AM"): del self.AM
- if hasattr(self, "u"): del self.u
+ if hasattr(self, "A1"): del self.A1
self.nRe, self.nIm = CustomExpressionParser(refractionIndex,
degree = self.FEDegree)
self.n2Re = self.nRe*self.nRe - self.nIm*self.nIm
self.n2Im = 2 * self.nRe * self.nIm
self._refractionIndex = copy(refractionIndex)
@property
def forcingTerm(self):
"""Value of f. Its assignment changes fRe and fIm."""
return self._forcingTerm
@forcingTerm.setter
def forcingTerm(self, forcingTerm):
- if hasattr(self, "b"): del self.b
- if hasattr(self, "bF"): del self.bF
- if hasattr(self, "u"): del self.u
+ if hasattr(self, "b0"): del self.b0
self.fRe, self.fIm = CustomExpressionParser(forcingTerm,
degree = self.FEDegree)
self._forcingTerm = copy(forcingTerm)
@property
def DirichletDatum(self):
"""
Value of u0. Its assignment changes u0Re, u0Im, DirichletBCRe and
DirichletBCIm.
"""
return self._DirichletDatum
@DirichletDatum.setter
def DirichletDatum(self, DirichletDatum):
- if hasattr(self, "b"): del self.b
- if hasattr(self, "u"): del self.u
+ if hasattr(self, "b0"): del self.b0
self.u0Re, self.u0Im = CustomExpressionParser(DirichletDatum,
degree = self.FEDegree)
self.DirichletBCRe = fen.DirichletBC(self.V, self.u0Re,
self.DirichletBoundary)
self.DirichletBCIm = fen.DirichletBC(self.V, self.u0Im,
self.DirichletBoundary)
self.DirichletBC0 = fen.DirichletBC(self.V, fenZERO,
self.DirichletBoundary)
self._DirichletDatum = copy(DirichletDatum)
@property
def NeumannDatum(self):
"""Value of g1. Its assignment changes g1Re and g1Im."""
return self._NeumannDatum
@NeumannDatum.setter
def NeumannDatum(self, NeumannDatum):
- if hasattr(self, "b"): del self.b
- if hasattr(self, "bN"): del self.bN
- if hasattr(self, "u"): del self.u
+ if hasattr(self, "b0"): del self.b0
self.g1Re, self.g1Im = CustomExpressionParser(NeumannDatum,
degree = self.FEDegree)
self._NeumannDatum = copy(NeumannDatum)
@property
def RobinDatum(self):
"""Value of h. Its assignment changes hRe and hIm."""
return self._RobinDatum
@RobinDatum.setter
def RobinDatum(self, RobinDatum):
- if hasattr(self, "A"): del self.A
- if hasattr(self, "AR"): del self.AR
- if hasattr(self, "u"): del self.u
+ if hasattr(self, "A0"): del self.A0
self.hRe, self.hIm = CustomExpressionParser(RobinDatum,
degree = self.FEDegree)
self._RobinDatum = copy(RobinDatum)
@property
def RobinDatum2(self):
"""Value of g2. Its assignment changes g2Re and g2Im."""
return self._RobinDatum2
@RobinDatum2.setter
def RobinDatum2(self, RobinDatum2):
- if hasattr(self, "b"): del self.b
- if hasattr(self, "bR"): del self.bR
- if hasattr(self, "u"): del self.u
+ if hasattr(self, "b0"): del self.b0
self.g2Re, self.g2Im = CustomExpressionParser(RobinDatum2,
degree = self.FEDegree)
self._RobinDatum2 = copy(RobinDatum2)
- def assembleA(self, Rfact : complex = 1.):
- """Assemble matrix of linear system."""
- if not hasattr(self, "A"):
- if not hasattr(self, "AL"):
- aLRe = self.aRe * fen.dot(fen.grad(self.v1),
- fen.grad(self.v2)) * fen.dx
- aLIm = self.aIm * fen.dot(fen.grad(self.v1),
- fen.grad(self.v2)) * fen.dx
- ALRe = fen.assemble(aLRe)
- ALIm = fen.assemble(aLIm)
- self.DirichletBC0.apply(ALRe)
- self.DirichletBC0.zero(ALIm)
- ALReMat = fen.as_backend_type(ALRe).mat()
- ALRer, ALRec, ALRev = ALReMat.getValuesCSR()
- ALImr, ALImc, ALImv = fen.as_backend_type(
- ALIm).mat().getValuesCSR()
- self.AL = (scsp.csr_matrix((ALRev, ALRec, ALRer),
- shape = ALReMat.size)
- + 1.j * scsp.csr_matrix((ALImv, ALImc, ALImr),
- shape = ALReMat.size))
- if not hasattr(self, "AM"):
- aMRe = self.n2Re * fen.dot(self.v1, self.v2) * fen.dx
- aMIm = self.n2Im * fen.dot(self.v1, self.v2) * fen.dx
- AMRe = fen.assemble(aMRe)
- AMIm = fen.assemble(aMIm)
- self.DirichletBC0.zero(AMRe)
- self.DirichletBC0.zero(AMIm)
- AMRer, AMRec, AMRev = fen.as_backend_type(
- AMRe).mat().getValuesCSR()
- AMImr, AMImc, AMImv = fen.as_backend_type(
- AMIm).mat().getValuesCSR()
- self.AM = (scsp.csr_matrix((AMRev, AMRec, AMRer),
- shape = self.AL.shape)
- + 1.j * scsp.csr_matrix((AMImv, AMImc, AMImr),
- shape = self.AL.shape))
- if not hasattr(self, "AR"):
- aRRe = self.hRe * fen.dot(self.v1, self.v2) * self.ds(1)
- aRIm = self.hIm * fen.dot(self.v1, self.v2) * self.ds(1)
- ARRe = fen.assemble(aRRe)
- ARIm = fen.assemble(aRIm)
- self.DirichletBC0.zero(ARRe)
- self.DirichletBC0.zero(ARIm)
- ARRer, ARRec, ARRev = fen.as_backend_type(
- ARRe).mat().getValuesCSR()
- ARImr, ARImc, ARImv = fen.as_backend_type(
- ARIm).mat().getValuesCSR()
- self.AR = (scsp.csr_matrix((ARRev, ARRec, ARRer),
- shape = self.AL.shape)
- + 1.j * scsp.csr_matrix((ARImv, ARImc, ARImr),
- shape = self.AL.shape))
-
- self.A = self.AL - self.wavenumber2 * self.AM + Rfact * self.AR
- if hasattr(self, "invA"): del self.invA
+ def assembleA(self):
+ """Assemble matrix blocks of linear system."""
+ if not hasattr(self, "A0"):
+ a0Re = (self.aRe * fen.dot(fen.grad(self.v1),
+ fen.grad(self.v2)) * fen.dx
+ + self.hRe * fen.dot(self.v1, self.v2) * self.ds(1))
+ a0Im = (self.aIm * fen.dot(fen.grad(self.v1),
+ fen.grad(self.v2)) * fen.dx
+ + self.hIm * fen.dot(self.v1, self.v2) * self.ds(1))
+ A0Re = fen.assemble(a0Re)
+ A0Im = fen.assemble(a0Im)
+ self.DirichletBC0.apply(A0Re)
+ self.DirichletBC0.zero(A0Im)
+ A0ReMat = fen.as_backend_type(A0Re).mat()
+ A0Rer, A0Rec, A0Rev = A0ReMat.getValuesCSR()
+ A0Imr, A0Imc, A0Imv = fen.as_backend_type(
+ A0Im).mat().getValuesCSR()
+ self.A0 = (scsp.csr_matrix((A0Rev, A0Rec, A0Rer),
+ shape = A0ReMat.size)
+ + 1.j * scsp.csr_matrix((A0Imv, A0Imc, A0Imr),
+ shape = A0ReMat.size))
+ if not hasattr(self, "A1"):
+ a1Re = - self.n2Re * fen.dot(self.v1, self.v2) * fen.dx
+ a1Im = - self.n2Im * fen.dot(self.v1, self.v2) * fen.dx
+ A1Re = fen.assemble(a1Re)
+ A1Im = fen.assemble(a1Im)
+ self.DirichletBC0.zero(A1Re)
+ self.DirichletBC0.zero(A1Im)
+ A1Rer, A1Rec, A1Rev = fen.as_backend_type(A1Re).mat()\
+ .getValuesCSR()
+ A1Imr, A1Imc, A1Imv = fen.as_backend_type(A1Im).mat()\
+ .getValuesCSR()
+ self.A1 = (scsp.csr_matrix((A1Rev, A1Rec, A1Rer),
+ shape = self.A0.shape)
+ + 1.j * scsp.csr_matrix((A1Imv, A1Imc, A1Imr),
+ shape = self.A0.shape))
def assembleb(self):
+ """Assemble RHS blocks of linear system."""
+ if not hasattr(self, "b0"):
+ L0Re = (fen.dot(self.fRe, self.v2) * fen.dx
+ + fen.dot(self.g1Re, self.v2) * self.ds(0)
+ + fen.dot(self.g2Re, self.v2) * self.ds(1))
+ L0Im = (fen.dot(self.fIm, self.v2) * fen.dx
+ + fen.dot(self.g1Im, self.v2) * self.ds(0)
+ + fen.dot(self.g2Im, self.v2) * self.ds(1))
+ b0Re = fen.assemble(L0Re)
+ b0Im = fen.assemble(L0Im)
+ self.DirichletBCRe.apply(b0Re)
+ self.DirichletBCIm.apply(b0Im)
+ self.b0 = np.array(b0Re.array()[:] + 1.j * b0Im.array()[:],
+ dtype = np.complex)
+
+ parDegree = 1
+
+ def As(self) -> list:
+ """Matrix blocks of linear system."""
+ self.assembleA()
+ return [self.A0, self.A1]
+
+ def bs(self) -> list:
+ """RHS blocks of linear system."""
+ self.assembleb()
+ return [self.b0]
+
+ def A(self) -> "CSR scipy matrix":
+ """Assemble matrix of linear system."""
+ self.assembleA()
+ return self.A0 + self.wavenumber2 * self.A1
+
+ def b(self) -> "numpy 1D array":
"""Assemble RHS of linear system."""
- if not hasattr(self, "b"):
- if not hasattr(self, "bF"):
- LFRe = fen.dot(self.fRe, self.v2) * fen.dx
- LFIm = fen.dot(self.fIm, self.v2) * fen.dx
- bFRe = fen.assemble(LFRe)
- bFIm = fen.assemble(LFIm)
- self.DirichletBCRe.apply(bFRe)
- self.DirichletBCIm.apply(bFIm)
- self.bF = np.array(bFRe.array()[:] + 1.j * bFIm.array()[:],
- dtype = np.complex)
- if not hasattr(self, "bN"):
- LNRe = fen.dot(self.g1Re, self.v2) * self.ds(0)
- LNIm = fen.dot(self.g1Im, self.v2) * self.ds(0)
- bNRe = fen.assemble(LNRe)
- bNIm = fen.assemble(LNIm)
- self.DirichletBC0.apply(bNRe)
- self.DirichletBC0.apply(bNIm)
- self.bN = np.array(bNRe.array()[:] + 1.j * bNIm.array()[:],
- dtype = np.complex)
- if not hasattr(self, "bR"):
- LRRe = fen.dot(self.g2Re, self.v2) * self.ds(1)
- LRIm = fen.dot(self.g2Im, self.v2) * self.ds(1)
- bRRe = fen.assemble(LRRe)
- bRIm = fen.assemble(LRIm)
- self.DirichletBC0.apply(bRRe)
- self.DirichletBC0.apply(bRIm)
- self.bR = np.array(bRRe.array()[:] + 1.j * bRIm.array()[:],
- dtype = np.complex)
-
- self.b = self.bF + self.bN + self.bR
-
- def buildLU(self):
- """Build LU decomposition of A."""
- if not hasattr(self, "A"): self.assembleA()
- if not hasattr(self, "invA"):
- warnings.simplefilter("ignore", scsp.SparseEfficiencyWarning)
- self.invA = spla.splu(self.A)
- warnings.simplefilter("default", scsp.SparseEfficiencyWarning)
-
- def solve(self, LU : bool = False)\
- -> ("Fenics function", "Fenics function"):
+ self.assembleb()
+ return self.b0
+
+ def solve(self) -> "numpy 1D array":
"""
Find solution of linear system.
- Args:
- LU(optional): Whether to use a LU solver for the system. Defaults
- to False.
+ Returns:
+ Solution vector.
+ """
+ return scsp.linalg.spsolve(self.A(), self.b())
+ def getLSBlocks(self) -> (list, list):
+ """
+ Get blocks of linear system.
+
Returns:
- Real and imaginary parts of solution.
+ Blocks of LS (\sum_{j=0}^J k^j A_j = \sum_{j=0}^K k^j b_j).
"""
- if not hasattr(self, "u"):
- if not hasattr(self, "A"): self.assembleA()
- if not hasattr(self, "b"): self.assembleb()
- if LU:
- self.buildLU()
- self.u = self.invA.solve(self.b)
- else:
- self.u = spla.spsolve(self.A, self.b)
- self.__solved = True
- return self.u
+ return self.As(), self.bs()
class FenicsHelmholtzScatteringEngine(FenicsHelmholtzEngine):
"""
Fenics-based solver for Helmholtz scattering problems with Robin ABC.
- \nabla \cdot (a \nabla u) - k^2 * n**2 * u = f in \Omega
u = u0 on \Gamma_D
\partial_nu = g1 on \Gamma_N
\partial_nu - i k u = g2 on \Gamma_R
Args:
mesh: Domain of Helmholtz problem.
FEDegree: FE degree.
wavenumber: Value of k.
diffusivity(optional): Value of a. Defaults to identically 1.
refractionIndex(optional): Value of n. Defaults to identically 1.
forcingTerm(optional): Value of f. Defaults to identically 0.
DirichletBoundary(optional): Function flagging Dirichlet boundary as
True, in Fenics format. Keywords 'ALL', 'NONE' and 'REST' are
accepted. Defaults to False everywhere.
NeumannBoundary(optional): Function flagging Neumann boundary as True,
in Fenics format. Keywords 'ALL', 'NONE' and 'REST' are accepted.
Defaults to False everywhere.
RobinBoundary(optional): Function flagging Robin boundary as True, in
Fenics format. Keywords 'ALL', 'NONE' and 'REST' are accepted.
Defaults to False everywhere.
DirichletDatum(optional): Value of u0. Defaults to identically 0.
NeumannDatum(optional): Value of g1. Defaults to identically 0.
RobinDatum2(optional): Value of g2. Defaults to identically 0.
Attributes:
FEDegree: FE degree.
wavenumber: Copy of processed wavenumber parameter.
diffusivity: Copy of processed diffusivity parameter.
refractionIndex: Copy of processed refractionIndex parameter.
forcingTerm: Copy of processed forcingTerm parameter.
DirichletBoundary: Copy of processed DirichletBoundary parameter.
NeumannBoundary: Copy of processed NeumannBoundary parameter.
RobinBoundary: Copy of processed RobinBoundary parameter.
DirichletDatum: Copy of processed DirichletDatum parameter.
NeumannDatum: Copy of processed NeumannDatum parameter.
RobinDatum: Value of h, i.e. (- i * k).
RobinDatum2: Copy of processed RobinDatum2 parameter.
V: Real FE space.
u: Helmholtz problem solution as numpy complex vector.
v1: Generic trial functions for variational form evaluation.
v2: Generic test functions for variational form evaluation.
ds: Boundary measure 2-tuple (resp. for Neumann and Robin boundaries).
nRe,nIm: Real and imaginary parts of n.
n2Re,n2Im: Real and imaginary parts of n^2.
fRe,fIm: Real and imaginary parts of f.
u0Re,u0Im: Real and imaginary parts of u0.
g1Re,g1Im: Real and imaginary parts of g1.
hRe,hIm: Real and imaginary parts of h.
g2Re,g2Im: Real and imaginary parts of g2.
DirichletBCRe,DirichletBCIm: Fenics BC manager for real and imaginary
parts of Dirichlet data.
A*: Scipy sparse array representation (in CSC format) of A*.
- b*Re,b*Im: Numpy array representation of real and imaginary parts of
- b*.
- solRe,solIm: Real and imaginary parts of Helmholtz problem solution.
- invA: Numpy LU solver for A.
+ b*: Numpy array representation of b*.
"""
def __init__(self, mesh:"Fenics mesh",
FEDegree:int,
wavenumber:"custom complex",
diffusivity : "custom expression" = 1,
refractionIndex : "custom expression" = 1,
forcingTerm : "custom expression" = fenZEROC,
DirichletBoundary : "bool function or str" = None,
NeumannBoundary : "bool function or str" = None,
RobinBoundary : "bool function or str" = None,
DirichletDatum : "custom expression" = fenZEROC,
NeumannDatum : "custom expression" = fenZEROC,
RobinDatum2 : "custom expression" = fenZEROC):
self.wavenumber = wavenumber
FenicsHelmholtzEngine.__init__(self, mesh = mesh,
FEDegree = FEDegree,
wavenumber = wavenumber,
diffusivity = diffusivity,
refractionIndex = refractionIndex,
forcingTerm = forcingTerm,
DirichletBoundary = DirichletBoundary,
NeumannBoundary = NeumannBoundary,
RobinBoundary = RobinBoundary,
DirichletDatum = DirichletDatum,
NeumannDatum = NeumannDatum,
RobinDatum = -1.j * wavenumber,
RobinDatum2 = RobinDatum2)
def setProblemData(self, dataDict:dict, k:complex):
"""
Set problem data.
Args:
dataDict: Dictionary of problem data.
k: Parameter value.
"""
self.wavenumber = k
self.forcingTerm = dataDict["forcingTerm"]
self.DirichletDatum = dataDict["DirichletDatum"]
self.NeumannDatum = dataDict["NeumannDatum"]
self.RobinDatum2 = dataDict["RobinDatum2"]
+ if hasattr(self, "A0"): del self.A0
+ if hasattr(self, "A1"): del self.A1
+ if hasattr(self, "A2"): del self.A2
+ if hasattr(self, "b0"): del self.b0
- def getLSBlocks(self) -> ("numpy 2D array" * 3, "numpy 1D array"):
- """
- Get blocks of linear system.
-
- Returns:
- Blocks of system (\sum_{j=0}^J k^j A_j = b)
- """
- self.assembleA()
- self.assembleb()
- return [self.AL, - 1.j * self.AR, - self.AM], [self.b]
-
- def setupDerivativeComputation(self, j:int, up:"numpy 1D array",
- upp : "numpy 1D array"):
- """
- Setup problem data to compute solution derivatives.
-
- Args:
- j: Derivative index.
- up: Adjusted previous derivative.
- upp: Adjusted pre-previous derivative.
- """
- upRe = fen.Function(self.V)
- upIm = fen.Function(self.V)
- ftRe = fen.Function(self.V)
- ftIm = fen.Function(self.V)
- upRe.vector()[:] = np.array(np.real(up), dtype=float)
- upIm.vector()[:] = np.array(np.imag(up), dtype=float)
- ft0 = 2 * self.wavenumber * up + upp
- ftRe.vector()[:] = np.array(np.real(ft0), dtype=float)
- ftIm.vector()[:] = np.array(np.imag(ft0), dtype=float)
- ftRe, ftIm = self.n2Re * ftRe - self.n2Im * ftIm,\
- self.n2Re * ftIm + self.n2Im * ftRe
- self.forcingTerm = [ftRe, ftIm]
- self.DirichletDatum = fenZEROC
- self.NeumannDatum = fenZEROC
- self.RobinDatum2 = [-upIm, upRe]
-
@property
def wavenumber2(self):
"""Value of k^2."""
return self._wavenumber2
@wavenumber2.setter
def wavenumber2(self, wavenumber2):
- FenicsHelmholtzEngine.wavenumber2.fset(self, wavenumber2)
+ self._wavenumber2 = wavenumber2
self._wavenumber = self.wavenumber2 ** .5
if (not hasattr(self, "h")
or not np.isclose(self.h, -1.j * self.wavenumber, 1e-14)):
self.RobinDatum = -1.j * self.wavenumber
@property
def wavenumber(self):
"""Value of k."""
return self._wavenumber
@wavenumber.setter
def wavenumber(self, wavenumber):
self.wavenumber2 = wavenumber ** 2.
@property
def RobinDatum(self):
"""
Value of h. Its assignment changes hRe and hIm (and maybe kRe, kIm, k
and z).
"""
return super().RobinDatum
@RobinDatum.setter
def RobinDatum(self, RobinDatum):
- if hasattr(self, "A"): del self.A
- if hasattr(self, "solRe"): del self.solRe, self.solIm
self.h = RobinDatum
self._RobinDatum = copy(RobinDatum)
if (not hasattr(self, "wavenumber")
or not np.isclose(self.h, -1.j * self.wavenumber, 1e-14)):
self.wavenumber = 1.j * self.h
def assembleA(self):
+ """Assemble matrix blocks of linear system."""
+ if not hasattr(self, "A0"):
+ a0Re = self.aRe * fen.dot(fen.grad(self.v1),
+ fen.grad(self.v2)) * fen.dx
+ a0Im = self.aIm * fen.dot(fen.grad(self.v1),
+ fen.grad(self.v2)) * fen.dx
+ A0Re = fen.assemble(a0Re)
+ A0Im = fen.assemble(a0Im)
+ self.DirichletBC0.apply(A0Re)
+ self.DirichletBC0.zero(A0Im)
+ A0ReMat = fen.as_backend_type(A0Re).mat()
+ A0Rer, A0Rec, A0Rev = A0ReMat.getValuesCSR()
+ A0Imr, A0Imc, A0Imv = fen.as_backend_type(
+ A0Im).mat().getValuesCSR()
+ self.A0 = (scsp.csr_matrix((A0Rev, A0Rec, A0Rer),
+ shape = A0ReMat.size)
+ + 1.j * scsp.csr_matrix((A0Imv, A0Imc, A0Imr),
+ shape = A0ReMat.size))
+ if hasattr(self, "A1"):
+ aux = copy(self.A1)
+ else:
+ aux = None
+ if not hasattr(self, "A2"):
+ if aux is not None:
+ del self.A1
+ FenicsHelmholtzEngine.assembleA(self)
+ self.A2 = copy(self.A1)
+ if aux is None:
+ a1 = fen.dot(self.v1, self.v2) * self.ds(1)
+ A1 = fen.assemble(a1)
+ self.DirichletBC0.zero(A1)
+ A1r, A1c, A1v = fen.as_backend_type(A1).mat().getValuesCSR()
+ self.A1 = - 1.j * scsp.csr_matrix((A1v, A1c, A1r),
+ shape = self.A0.shape)
+ else:
+ self.A1 = aux
+
+ parDegree = 2
+
+ def As(self) -> list:
+ """Matrix blocks of linear system."""
+ self.assembleA()
+ return [self.A0, self.A1, self.A2]
+
+ def A(self) -> "CSR scipy matrix":
"""Assemble matrix of linear system."""
- if not hasattr(self, "A"):
- if not hasattr(self, "AR"):
- aR = fen.dot(self.v1, self.v2) * self.ds(1)
- AR = fen.assemble(aR)
- self.DirichletBC0.zero(AR)
- AR_mat = fen.as_backend_type(AR).mat().transpose()
- self.AR = scsp.csc_matrix(AR_mat.getValuesCSR()[::-1],
- shape = AR_mat.size)
- FenicsHelmholtzEngine.assembleA(self, self.h)
-
+ self.assembleA()
+ return self.A0 + self.wavenumber * self.A1 + self.wavenumber2 * self.A2
+
-class FenicsHelmholtzScatteringAugmentedEngine(FenicsHelmholtzScatteringEngine):
+class FenicsHelmholtzScatteringAugmentedEngine(
+ FenicsHelmholtzScatteringEngine):
"""
Fenics-based solver for Helmholtz scattering problems with Robin ABC.
- \nabla \cdot (a \nabla u) - k^2 * n**2 * u = f in \Omega
u = u0 on \Gamma_D
\partial_nu = g1 on \Gamma_N
\partial_nu - i k u = g2 on \Gamma_R
Linear dependence on k is achieved by introducing an additional variable.
Args:
mesh: Domain of Helmholtz problem.
FEDegree: FE degree.
wavenumber: Value of k.
diffusivity(optional): Value of a. Defaults to identically 1.
refractionIndex(optional): Value of n. Defaults to identically 1.
forcingTerm(optional): Value of f. Defaults to identically 0.
DirichletBoundary(optional): Function flagging Dirichlet boundary as
True, in Fenics format. Keywords 'ALL', 'NONE' and 'REST' are
accepted. Defaults to False everywhere.
NeumannBoundary(optional): Function flagging Neumann boundary as True,
in Fenics format. Keywords 'ALL', 'NONE' and 'REST' are accepted.
Defaults to False everywhere.
RobinBoundary(optional): Function flagging Robin boundary as True, in
Fenics format. Keywords 'ALL', 'NONE' and 'REST' are accepted.
Defaults to False everywhere.
DirichletDatum(optional): Value of u0. Defaults to identically 0.
NeumannDatum(optional): Value of g1. Defaults to identically 0.
RobinDatum2(optional): Value of g2. Defaults to identically 0.
constraintType(optional): Type of augmentation. Keywords 'IDENTITY' and
'MASS' are accepted. Defaults to 'IDENTITY'.
Attributes:
FEDegree: FE degree.
wavenumber: Copy of processed wavenumber parameter.
diffusivity: Copy of processed diffusivity parameter.
refractionIndex: Copy of processed refractionIndex parameter.
forcingTerm: Copy of processed forcingTerm parameter.
DirichletBoundary: Copy of processed DirichletBoundary parameter.
NeumannBoundary: Copy of processed NeumannBoundary parameter.
RobinBoundary: Copy of processed RobinBoundary parameter.
DirichletDatum: Copy of processed DirichletDatum parameter.
NeumannDatum: Copy of processed NeumannDatum parameter.
RobinDatum: Value of h, i.e. (- i * k).
RobinDatum2: Copy of processed RobinDatum2 parameter.
constraintType: Type of augmentation.
V: Real FE space.
u: Helmholtz problem solution as numpy complex vector.
v1: Generic trial functions for variational form evaluation.
v2: Generic test functions for variational form evaluation.
ds: Boundary measure 2-tuple (resp. for Neumann and Robin boundaries).
nRe,nIm: Real and imaginary parts of n.
n2Re,n2Im: Real and imaginary parts of n^2.
fRe,fIm: Real and imaginary parts of f.
u0Re,u0Im: Real and imaginary parts of u0.
g1Re,g1Im: Real and imaginary parts of g1.
hRe,hIm: Real and imaginary parts of h.
g2Re,g2Im: Real and imaginary parts of g2.
DirichletBCRe,DirichletBCIm: Fenics BC manager for real and imaginary
parts of Dirichlet data.
A*: Scipy sparse array representation (in CSC format) of A*.
- b*Re,b*Im: Numpy array representation of real and imaginary parts of
- b*.
- solRe,solIm: Real and imaginary parts of Helmholtz problem solution.
- invA: Numpy LU solver for A.
+ b*: Numpy array representation of b*.
"""
def __init__(self, mesh:"Fenics mesh",
FEDegree:int,
wavenumber:"custom complex",
diffusivity : "custom expression" = 1,
refractionIndex : "custom expression" = 1,
forcingTerm : "custom expression" = fenZEROC,
DirichletBoundary : "bool function or str" = None,
NeumannBoundary : "bool function or str" = None,
RobinBoundary : "bool function or str" = None,
DirichletDatum : "custom expression" = fenZEROC,
NeumannDatum : "custom expression" = fenZEROC,
RobinDatum2 : "custom expression" = fenZEROC,
constraintType : str = "IDENTITY"):
- self.wavenumber = wavenumber
self.constraintType = constraintType
FenicsHelmholtzScatteringEngine.__init__(self, mesh = mesh,
- FEDegree = FEDegree,
- wavenumber = wavenumber,
- diffusivity = diffusivity,
- refractionIndex = refractionIndex,
- forcingTerm = forcingTerm,
- DirichletBoundary = DirichletBoundary,
- NeumannBoundary = NeumannBoundary,
- RobinBoundary = RobinBoundary,
- DirichletDatum = DirichletDatum,
- NeumannDatum = NeumannDatum,
- RobinDatum2 = RobinDatum2)
+ FEDegree = FEDegree,
+ wavenumber = wavenumber,
+ diffusivity = diffusivity,
+ refractionIndex = refractionIndex,
+ forcingTerm = forcingTerm,
+ DirichletBoundary = DirichletBoundary,
+ NeumannBoundary = NeumannBoundary,
+ RobinBoundary = RobinBoundary,
+ DirichletDatum = DirichletDatum,
+ NeumannDatum = NeumannDatum,
+ RobinDatum2 = RobinDatum2)
def energyNormMatrix(self, w:float) -> "CSC sparse matrix":
"""
Sparse matrix (in CSR format) assoociated to w-weighted H10 norm.
Args:
w: Weight.
Returns:
Sparse matrix in CSR format.
"""
normMatFen = FenicsHelmholtzEngine.energyNormMatrix(self, w)
return scsp.block_diag((normMatFen, 1/w * normMatFen))
def liftDirichletData(self):
"""Lift Dirichlet datum."""
lift1 = FenicsHelmholtzScatteringEngine.liftDirichletData(self)
- return np.pad(lift1, (0, len(lift1)), 'constant')
+ return np.array([lift1, np.zeros_like(lift1)])
+
+ def setProblemData(self, dataDict:dict, k:complex):
+ """
+ Set problem data.
+
+ Args:
+ dataDict: Dictionary of problem data.
+ k: Parameter value.
+ """
+ self.wavenumber = k
+ self.forcingTerm = dataDict["forcingTerm"]
+ self.DirichletDatum = dataDict["DirichletDatum"]
+ self.NeumannDatum = dataDict["NeumannDatum"]
+ self.RobinDatum2 = dataDict["RobinDatum2"]
+ if hasattr(self, "A0"): del self.A0
+ if hasattr(self, "A1"): del self.A1
+ if hasattr(self, "b0"): del self.b0
@property
def constraintType(self):
"""Value of constraintType."""
return self._constraintType
@constraintType.setter
def constraintType(self, constraintType):
try:
constraintType = constraintType.upper().strip().replace(" ","")
if constraintType not in ["IDENTITY", "MASS"]: raise
self._constraintType = constraintType
except:
warnings.warn(("Prescribed constraintType not recognized. "
- "Overriding to 'KRYLOV'."))
+ "Overriding to 'IDENTITY'."))
self.constraintType = "IDENTITY"
- def getLSBlocks(self) -> (list, list):
- """
- Get blocks of linear system.
-
- Returns:
- Blocks of system (\sum_{j=0}^J k^j A_j = b)
- """
- self.assembleA()
- self.assembleb()
- return [self.A0, - self.A1], [self.b]
-
- def setupDerivativeComputation(self, j:int, up:"numpy 1D array",
- upp : "numpy 1D array"):
- """
- Setup problem data to compute solution derivatives.
-
- Args:
- j: Derivative index.
- up: Adjusted previous derivative.
- upp: Adjusted pre-previous derivative.
- """
- up1Re = fen.Function(self.V)
- up1Im = fen.Function(self.V)
- up2Re = fen.Function(self.V)
- up2Im = fen.Function(self.V)
- lup = int(len(up) / 2)
- up1Re.vector()[:] = np.array(np.real(up[: lup]), dtype=float)
- up1Im.vector()[:] = np.array(np.imag(up[: lup]), dtype=float)
- up2Re.vector()[:] = np.array(np.real(up[lup :]), dtype=float)
- up2Im.vector()[:] = np.array(np.imag(up[lup :]), dtype=float)
- if self.constraintType == "IDENTITY":
- b2 = up[: lup]
- else:
- self.forcingTerm = [up1Re, up1Im]
- self.DirichletDatum = fenZEROC
- self.NeumannDatum = fenZEROC
- self.RobinDatum2 = fenZEROC
- self.assembleb()
- b2 = self.b[lup :]
- self.forcingTerm = [up2Re, up2Im]
- self.DirichletDatum = fenZEROC
- self.NeumannDatum = fenZEROC
- self.RobinDatum2 = [-up1Im, up1Re]
- self.assembleb()
- self.b[lup :] = b2
-
def assembleA(self):
- """Assemble matrix of linear system."""
- if not hasattr(self, "A"):
- if hasattr(self, "A0") and not hasattr(self, "AL"):
- del self.A0
- if (hasattr(self, "A1")
- and not (hasattr(self, "AM") and hasattr(self, "AR"))):
- del self.A1
+ """Assemble matrix blocks of linear system."""
+ builtA0 = hasattr(self, "A0")
+ builtA1 = hasattr(self, "A1")
+ if not (builtA0 and builtA1):
+ if builtA1 and self.constraintType == "IDENTITY":
+ self.A2 = None
FenicsHelmholtzScatteringEngine.assembleA(self)
if self.constraintType == "IDENTITY":
- block1 = scsp.eye(self.AL.shape[0])
- block2 = block1
+ if not builtA0:
+ block = scsp.eye(self.A0.shape[0])
+ else:
+ block = scsp.eye(self.A1.shape[0])
else: #MASS
- block1 = copy(self.AM)
+ block = - self.A2
I = list(self.DirichletBC0.get_boundary_values().keys())
warnings.simplefilter("ignore", scsp.SparseEfficiencyWarning)
- block1[I, I] = 1.
+ block[I, I] = 1.
warnings.simplefilter("default", scsp.SparseEfficiencyWarning)
- block2 = self.AM
- self.A0 = scsp.block_diag((self.AL, block1))
- if not hasattr(self, "A1"):
- self.A1 = scsp.bmat([[1.j * self.AR, self.AM], [block2, None]])
-
- self.A = self.A0 - self.wavenumber * self.A1
- if hasattr(self, "invA"): del self.invA
+ if not builtA0:
+ self.A0 = scsp.block_diag((self.A0, block))
+ if not builtA1:
+ self.A1 = scsp.bmat([[self.A1, self.A2], [- block, None]])
+ if hasattr(self, 'A2'): del self.A2
def assembleb(self):
"""Assemble RHS of linear system."""
- if not hasattr(self, "b"):
+ if not hasattr(self, "b0"):
FenicsHelmholtzScatteringEngine.assembleb(self)
- self.b = np.pad(self.b, (0, self.b.shape[0]), 'constant')
+ self.b0 = np.pad(self.b0, (0, self.b0.shape[0]), 'constant')
+
+ parDegree = 1
+
+ def As(self) -> list:
+ """Matrix blocks of linear system."""
+ self.assembleA()
+ return [self.A0, self.A1]
+
+ def A(self) -> "CSR scipy matrix":
+ """Assemble matrix of linear system."""
+ self.assembleA()
+ return self.A0 + self.wavenumber * self.A1
+
+ def solve(self) -> ("numpy 1D array", "numpy 1D array"):
+ """
+ Find solution of linear system.
+
+ Returns:
+ Solution vector.
+ """
+ u = FenicsHelmholtzScatteringEngine.solve(self)
+ lu2 = int(len(u) / 2)
+ return np.array([u[: lu2], u[lu2 :]])
diff --git a/main/FreeFemConversionTools.py b/main/FreeFemConversionTools.py
index f5f79e1..b3e2ac2 100644
--- a/main/FreeFemConversionTools.py
+++ b/main/FreeFemConversionTools.py
@@ -1,101 +1,103 @@
#!/usr/bin/python
#import warnings
import numpy as np
import scipy.sparse as sp
import os
+def prod2(v1 : str = "u", v2 : str = "v", compl : bool = True) -> str:
+ if compl:
+ return "({} * conj({}))".format(v1, v2)
+ else:
+ return "({} * {})".format(v1, v2)
+
def diff(var : str = "u", dim : int = 2) -> str:
- dstr = "abs(dx({}))^2".format(var)
+ dstr = "(abs(dx({}))^2".format(var)
if dim > 1:
dstr = dstr + " + abs(dy({}))^2".format(var)
if dim > 2:
dstr = dstr + " + abs(dz({}))^2".format(var)
- return dstr
+ return dstr + ")"
def diff2(v1 : str = "u", v2 : str = "v", dim : int = 2,
compl : bool = True) -> str:
- def term2(string:str):
- global compl
- if compl:
- return "conj({})".format(string)
- else:
- return string
- dstr = "dx({}) * {}".format(v1, term2("dx({})".format(v2)))
+ dstr = prod2("(dx({})".format(v1), "dx({})".format(v2), compl)
if dim > 1:
- dstr = dstr + " + dy({}) * {}".format(v1, term2("dy({})".format(v2)))
+ dstr = dstr + " + " + prod2("dy({})".format(v1),
+ "dy({})".format(v2), compl)
if dim > 2:
- dstr = dstr + " + dz({}) * {}".format(v1, term2("dz({})".format(v2)))
- return dstr
+ dstr = dstr + " + " + prod2("dz({})".format(v1),
+ "dz({})".format(v2), compl)
+ return dstr + ")"
def getNewFilename(prefix : str = "", extension : str = "dat") -> str:
n = len(extension) + 1
filename = "{}{}.{}".format(prefix, np.random.randint(0, 10), extension)
while os.path.exists(filename):
filename = filename[: - n] + "{}.{}".format(np.random.randint(10),
extension)
return filename
def ffvec2npvec(filename:str) -> "numpy 1D array":
with open(filename) as f:
a = f.readline()
v = np.zeros(int(a), dtype = np.complex)
i = 0
for line in f:
for x in line[: -1].split():
if "(" in x:
y = x[1:-1].split(",")
v[i] = float(y[0]) + 1.j * float(y[1])
else:
v[i] = complex(x)
i = i + 1
return v
def ffmat2spmat(filename:str) -> "scipy coo matrix":
with open(filename) as f:
a = f.readline()[: -1]
while a[0] == "#":
a = f.readline()[: -1]
n, m, sym, nnz = [int(x) for x in a.split()]
row = np.zeros(nnz)
col = np.zeros(nnz)
data = np.zeros(nnz, dtype = np.complex)
for i, line in enumerate(f):
x = line[: -1].split()
row[i] = int(x[0]) - 1
col[i] = int(x[1]) - 1
if "(" in x[2]:
y = x[2][1:-1].split(",")
data[i] = float(y[0]) + 1.j * float(y[1])
else:
data[i] = complex(x)
return sp.coo_matrix((data, (row, col)), shape = (n, m))
def npvec2ffvec(v:"numpy 1D array") -> str:
filename = getNewFilename("v", "tmp")
with open(filename, 'w', buffering = 1) as f:
f.write(str(len(v)) + '\n')
for x in v:
if v.dtype == np.complex:
f.write("({0},{1})\n".format(np.real(x), np.imag(x)))
else:
f.write(str(x) + '\n')
return filename
def spmat2ffmat(m:"scipy sparse matrix") -> str:
mEff = m.tocoo()
filename = getNewFilename("m", "tmp")
with open(filename, 'w', buffering = 1) as f:
f.write(("# Sparse Matrix (Morse)\n# first line: n m (is symmetic) "
"nbcoef\n# after for each nonzero coefficient: i j a_ij "
"where (i,j) \in {1,...,n}x{1,...,m}\n"))
f.write("{} {} 0 {}\n".format(mEff.shape[0], mEff.shape[1], mEff.nnz))
for i in range(mEff.nnz):
r = mEff.row[i]
c = mEff.col[i]
x = mEff.data[i]
if mEff.dtype == np.complex:
f.write("{} {} ({},{})\n".format(r, c, np.real(x), np.imag(x)))
else:
f.write("{} {} {}\n".format(r, c, x))
return filename
diff --git a/main/FreeFemHSEngine.py b/main/FreeFemHSEngine.py
index 927ebc7..f27cfe5 100644
--- a/main/FreeFemHSEngine.py
+++ b/main/FreeFemHSEngine.py
@@ -1,113 +1,290 @@
#!/usr/bin/python
import os
import subprocess
import numpy as np
+import utilities
#from matplotlib import pyplot as plt
import FreeFemConversionTools as FFCT
class FreeFemHSEngine:
"""
FreeFem++-based Hilbert space engine.
"""
def __init__(self, V:str, compl : bool = True, dim : int = 2,
meshname : str = "Th", Vname : str = "V"):
if compl:
self.compl = "<complex>"
else:
self.compl = ""
self.V = V
self.dim = dim
self.meshname = meshname
self.Vname = Vname
def name(self) -> str:
"""Class label."""
return self.__class__.__name__
def norm(self, u:"numpy 1D array",
normType : "numpy 2D array, number or str" = "H10") -> float:
"""
Compute general norm of complex-valued function with given dofs.
Args:
u: numpy complex array with function dofs.
normType(optional): Target norm identifier. If matrix, target norm
is one induced by normType. If number, target norm is weighted
H^1 norm with given weight. If string, must be among 'H1',
'L2', and 'H10'. Defaults to 'H10'.
Returns:
Norm of the function (non-negative).
"""
if type(normType).__name__[-6:] == "matrix":
return np.abs(u.dot(normType.dot(u).conj())) ** .5
filename = FFCT.getNewFilename("temp", "edp")
ufile = FFCT.npvec2ffvec(u)
if normType == 'L2':
coeff1 = 0.
coeff2 = 1.
else:
coeff1 = 1.
if isinstance(normType, (int, float)):
coeff2 = normType ** 2.
elif normType == 'H10':
coeff2 = 0.
elif normType == 'H1':
coeff2 = 1.
else:
raise Exception("Norm type not recognized.")
with open(filename, 'w') as f:
f.write(self.V + "\n")
f.write("ifstream fin(\"{}\");\n".format(ufile))
f.write("{}{} u;\n".format(self.Vname, self.compl))
f.write("fin >> u[];\n")
- f.write("cout << sqrt(")
+ f.write("cout.precision(15);")
+ f.write("cout.scientific << sqrt(")
if np.isclose(coeff1, 0.):
f.write("0")
else:
f.write("{} * int{}d({})({})".format(coeff1, self.dim,
self.meshname,
FFCT.diff("u", self.dim)))
if np.isclose(coeff2, 0.):
f.write(" + 0)/n")
else:
- f.write(" + {} * int{}d({})(abs(u)^2));\n".format(coeff1,
+ f.write(" + {} * int{}d({})(abs(u)^2));\n".format(
+ coeff2,
self.dim,
self.meshname))
bashCommand = "FreeFem++ -v 0 {}".format(filename)
process = subprocess.Popen(bashCommand.split(), stdout=subprocess.PIPE)
output, error = process.communicate()
+ try:
+ nrm = np.float(output)
+ except:
+ raise Exception(("Error was encountered in the execution of "
+ "FreeFem++ code:\n{}\nSee files {} and {}.")\
+ .format(output.decode('utf-8'), filename, ufile))
os.remove(filename)
os.remove(ufile)
- return float(output)
-
- def plot(self, u:"numpy 1D array", name : str = "u", figspecs : str = ""):
+ return nrm
+
+ def plot(self, u:"numpy 1D array", name : str = "u", save : bool = False,
+ what : "str or list" = "ALL", figspecs : str = ""):
"""
Do some nice plots of the complex-valued function with given dofs.
Args:
u: numpy complex array with function dofs.
name(optional): Name to be shown as title of the plots. Defaults to
'u'.
+ save(optional): Whether to save plot(s). Defaults to False.
figspecs(optional key args): Optional arguments for FreeFem++
figure creation.
"""
+ if isinstance(what, (str,)):
+ if what.upper() != 'ALL':
+ return self.plot(u, name, save, [what], **figspecs)
+ else:
+ what = utilities.purgeList(what, ['ABS', 'PHASE', 'REAL', 'IMAG'],
+ listname = self.name() + ".what")
+ if len(what) == 0: return
+
filename = FFCT.getNewFilename("temp", "edp")
ufile = FFCT.npvec2ffvec(u)
figspecs = figspecs.strip()
if len(figspecs) > 0 and figspecs[0] != ",":
figspecs = ", " + figspecs
+
+
with open(filename, 'w') as f:
f.write(self.V + "\n")
f.write("ifstream fin(\"{}\");\n".format(ufile))
f.write("{}{} u;\n".format(self.Vname, self.compl))
f.write("fin >> u[];\n")
- f.write("plot(u, wait = 1, cmm=\"{}\"{});".format(name, figspecs))
+ if isinstance(what, (str,)):
+ outspecs = ""
+ if save:
+ outspecs = (", ps = \""
+ + FFCT.getNewFilename("fig", "eps") + "\"")
+ f.write("plot(u, wait = 1, cmm=\"{}\"{}{});\n".format(
+ name, figspecs, outspecs))
+ else:
+ for i, w in enumerate(what):
+ if i == 0:
+ VnameEff = self.Vname
+ else:
+ VnameEff = ""
+ if w == 'ABS':
+ outspecs = ""
+ if save:
+ outspecs = (", ps = \""
+ + FFCT.getNewFilename("abs", "eps")
+ + "\"")
+ f.write("{} v = abs(u);\n".format(VnameEff))
+ f.write("plot(v,wait = 1, cmm=\"|{}|\"{}{});\n".format(
+ name, figspecs, outspecs))
+ if w == 'PHASE':
+ outspecs = ""
+ if save:
+ outspecs = (", ps = \""
+ + FFCT.getNewFilename("ph", "eps")
+ + "\"")
+ f.write("{} v = arg(u);\n".format(VnameEff))
+ f.write("plot(v,wait = 1, cmm=\"ph{}\"{}{});\n".format(
+ name, figspecs, outspecs))
+ if w == 'REAL':
+ outspecs = ""
+ if save:
+ outspecs = (", ps = \""
+ + FFCT.getNewFilename("re", "eps")
+ + "\"")
+ f.write("{} v = real(u);\n".format(VnameEff))
+ f.write("plot(v,wait = 1, cmm=\"Re{}\"{}{});\n".format(
+ name, figspecs, outspecs))
+ if w == 'IMAG':
+ outspecs = ""
+ if save:
+ outspecs = (", ps = \""
+ + FFCT.getNewFilename("im", "eps")
+ + "\"")
+ f.write("{} v = imag(u);\n".format(VnameEff))
+ f.write("plot(v,wait = 1, cmm=\"Im{}\"{}{});\n".format(
+ name, figspecs, outspecs))
+
bashCommand = "FreeFem++ -v 0 {}".format(filename)
process = subprocess.Popen(bashCommand.split(), stdout=subprocess.PIPE)
output, error = process.communicate()
+ if len(output) > 0:
+ raise Exception(("Error was encountered in the execution of "
+ "FreeFem++ code:\n{}\nSee files {} and {}.")\
+ .format(output.decode('utf-8'), filename, ufile))
os.remove(filename)
os.remove(ufile)
+
+ def plotmesh(self, name : str = "u", save : bool = False,
+ figspecs : str = ""):
+ """
+ Do a nice plot of the mesh.
+
+ Args:
+ name(optional): Name to be shown as title of the plots. Defaults to
+ 'u'.
+ save(optional): Whether to save plot(s). Defaults to False.
+ figspecs(optional key args): Optional arguments for FreeFem++
+ figure creation.
+ """
+ filename = FFCT.getNewFilename("temp", "edp")
+
+ figspecs = figspecs.strip()
+ if len(figspecs) > 0 and figspecs[0] != ",":
+ figspecs = ", " + figspecs
+
+ outspecs = ""
+ if save:
+ outspecs = ", ps = \"" + FFCT.getNewFilename("msh", "eps") + "\""
+
+ with open(filename, 'w') as f:
+ f.write(self.V + "\n")
+ f.write("plot({}, wait = 1, cmm=\"{}\"{});\n".format(
+ self.meshname, name, figspecs, outspecs))
+ bashCommand = "FreeFem++ -v 0 {}".format(filename)
+ process = subprocess.Popen(bashCommand.split(), stdout=subprocess.PIPE)
+ output, error = process.communicate()
+ if len(output) > 0:
+ raise Exception(("Error was encountered in the execution of "
+ "FreeFem++ code:\n{}\nSee file {}.")\
+ .format(output.decode('utf-8'), filename))
+ os.remove(filename)
+
+
+class FreeFemHSAugmentedEngine(FreeFemHSEngine):
+ """
+ FreeFem-based Hilbert space engine for augmented spaces.
+ """
+ def __init__(self, V:"FreeFem FE space", d : int = 2):
+ self.d = d
+ FreeFemHSEngine.__init__(self, V)
+
+ def norm(self, u:"numpy 1D array or list",
+ normType : "numpy 2D array, number or str" = "H10")\
+ -> "numpy 1D array":
+ """
+ Compute general norm of complex-valued function with given dofs.
+
+ Args:
+ u: numpy complex array with function dofs.
+ normType(optional): Target norm identifier. If matrix, target norm
+ is one induced by normType. If number, target norm is weighted
+ H^1 norm with given weight. If string, must be recognizable by
+ FreeFem norm command. Defaults to 'H10'.
+
+ Returns:
+ Norms of the function (non-negative).
+ """
+ if isinstance(normType, (int, float)):
+ u = utilities.listify(u, self.d)
+ norms = [None] * self.d
+ for j in range(self.d):
+ norms[j] = FreeFemHSEngine.norm(self, u[j], normType)
+ return norms
+ elif type(normType).__name__[-6:] == "matrix":
+ u = utilities.vectorify(u, self.d)
+ if normType.shape[0] % u[0] == 0:
+ N = int(normType.shape[0] / u[0])
+ else:
+ raise Exception("Energy matrix dimension mismatch.")
+ return FreeFemHSEngine.norm(self, np.concatenate(u[: N]), normType)
+ raise Exception("Norm type not recognized.")
+
+ def plot(self, u:"numpy 1D array", name : str = "u", save : bool = False,
+ what : "str or list" = 'all', figspecs : str = "") -> list:
+ """
+ Do some nice plots of the complex-valued function with given dofs.
+
+ Args:
+ u: numpy complex array with function dofs.
+ name(optional): Name to be shown as title of the plots. Defaults to
+ 'u'.
+ save(optional): Whether to save plot(s). Defaults to False.
+ what(optional): Which plots to do. If list, can contain 'ABS',
+ 'PHASE', 'REAL', 'IMAG'. If str, same plus wildcard 'ALL'.
+ Defaults to 'ALL'.
+ figspecs(optional key args): Optional arguments for matplotlib
+ figure creation.
+
+ Returns:
+ List of pyplot figure handles.
+ """
+ u = utilities.listify(u, self.d)
+
+ for j in range(self.d):
+ FreeFemHSEngine.plot(self, u[j], what = what, save = save,
+ name = "{0}, comp. {1}".format(name, j),
+ figspecs = figspecs)
+
+
diff --git a/main/FreeFemHelmholtzEngine.py b/main/FreeFemHelmholtzEngine.py
index af02fae..dd83c28 100644
--- a/main/FreeFemHelmholtzEngine.py
+++ b/main/FreeFemHelmholtzEngine.py
@@ -1,402 +1,734 @@
#!/usr/bin/python
-from copy import copy
-import os
import warnings
+import os
import subprocess
import numpy as np
import scipy.sparse as scsp
-import scipy.sparse.linalg as spla
-import fenics as fen
import FreeFemConversionTools as FFCT
PI = np.pi
-fenZERO = fen.Constant(0)
-fenZEROC = [fenZERO, fenZERO]
class DomainError(Exception):
"""
Domain error exception.
Args:
value: Human readable string describing the exception.
Attributes:
value: Human readable string describing the exception.
"""
def __init__(self, value:str):
self.value = value
def __str__(self):
return self.value
class FreeFemHelmholtzEngine:
"""
FreeFem++-based solver for Helmholtz problems.
- \nabla \cdot (a \nabla u) - k^2 * n**2 * u = f in \Omega
u = u0 on \Gamma_D
\partial_nu = g1 on \Gamma_N
\partial_nu + h u = g2 on \Gamma_R
Args:
Attributes:
"""
def __init__(self, V:str,
wavenumber:complex,
compl : bool = True,
dim : int = 2,
meshname : str = "Th",
Vname : str = "V",
diffusivity : str = "1",
refractionIndex : str = "1",
forcingTerm : str = "0",
DirichletBoundary : str = "",
NeumannBoundary : str = "",
RobinBoundary : str = "",
DirichletDatum : str = "0",
NeumannDatum : str = "0",
RobinDatum : str = "0",
RobinDatum2 : str = "0"):
# process boundaries
boundariesList = [DirichletBoundary, NeumannBoundary, RobinBoundary]
for i in range(3):
- boundariesList[i] = boundariesList[i].strip().upper()
+ boundariesList[i] = boundariesList[i].strip()
DirichletBoundary, NeumannBoundary, RobinBoundary = boundariesList
if boundariesList.count("") == 3:
raise DomainError("At least one boundary must be prescribed.")
self.DirichletBoundary = DirichletBoundary
self.NeumannBoundary = NeumannBoundary
self.RobinBoundary = RobinBoundary
if compl:
self.compl = "<complex>"
else:
self.compl = ""
self.V = V
self.dim = dim
self.meshname = meshname
self.Vname = Vname
- self.wavenumber2 = wavenumber ** 2
+ self.wavenumber2 = wavenumber ** 2.
self.diffusivity = diffusivity
self.refractionIndex = refractionIndex
self.forcingTerm = forcingTerm
self.DirichletDatum = DirichletDatum
self.NeumannDatum = NeumannDatum
self.RobinDatum = RobinDatum
self.RobinDatum2 = RobinDatum2
+ functional = lambda self, u: 0.
+
def energyNormMatrix(self, w:float) -> "CSC sparse matrix":
"""
Sparse matrix (in CSR format) assoociated to w-weighted H10 norm.
Args:
w: Weight.
Returns:
Sparse matrix in CSR format.
"""
filename = FFCT.getNewFilename("temp", "edp")
Mfile = FFCT.getNewFilename("m", "tmp")
with open(filename, 'w') as f:
f.write(self.V + "\n")
f.write("{}{} u, v;\n".format(self.Vname, self.compl))
- f.write("varf energy(u, v) = int{}d({})({})".format(self.dim,
- self.meshname,
- FFCT.diff(dim = self.dim,
- compl = not self.compl == "")))
- f.write(" + {} * int{}d({})(abs(u)^2);\n".format(w ** 2,
- self.dim,
- self.meshname))
+ f.write("varf energy(u, v) = int{}d({})({})".format(
+ self.dim,
+ self.meshname,
+ FFCT.diff2(dim = self.dim,
+ compl = not self.compl == "")))
+ f.write(" + int{}d({})({} * {});\n".format(
+ self.dim,
+ self.meshname,
+ w ** 2,
+ FFCT.prod2(compl = not self.compl == "")))
f.write("matrix{0} M = energy({1}, {1});\n".format(self.compl,
self.Vname))
f.write("ofstream Mout(\"{}\");\n".format(Mfile))
f.write("Mout << M;\n")
bashCommand = "FreeFem++ -v 0 {}".format(filename)
process = subprocess.Popen(bashCommand.split(), stdout=subprocess.PIPE)
output, error = process.communicate()
+ if len(output) > 0:
+ raise Exception(("Error was encountered in the execution of "
+ "FreeFem++ code:\n{}\nSee file {}.")\
+ .format(output.decode('utf-8'), filename))
M = FFCT.ffmat2spmat(Mfile)
os.remove(filename)
os.remove(Mfile)
return M.tocsr()
def problemData(self):
"""List of HF problem data."""
dataDict = {}
dataDict["forcingTerm"] = self.forcingTerm
dataDict["DirichletDatum"] = self.DirichletDatum
dataDict["NeumannDatum"] = self.NeumannDatum
dataDict["RobinDatum2"] = self.RobinDatum2
return dataDict
def setProblemData(self, dataDict:dict, k2:complex):
"""
Set problem data.
Args:
dataDict: Dictionary of problem data.
k2: Parameter value.
"""
self.wavenumber2 = k2
self.forcingTerm = dataDict["forcingTerm"]
self.DirichletDatum = dataDict["DirichletDatum"]
self.NeumannDatum = dataDict["NeumannDatum"]
self.RobinDatum2 = dataDict["RobinDatum2"]
-
- def getLSBlocks(self) -> ("numpy 2D array" * 3, "numpy 1D array"):
- """
- Get blocks of linear system.
-
- Returns:
- Blocks of system (\sum_{j=0}^J k^j A_j = b)
- """
- self.assembleA()
- self.assembleb()
- return [self.AL + self.AR, - self.AM], [self.b]
+ if hasattr(self, "A0"): del self.A0
+ if hasattr(self, "A1"): del self.A1
+ if hasattr(self, "b0"): del self.b0
def liftDirichletData(self):
"""Lift Dirichlet datum."""
- raise Exception("Not implemented.")
- return
-
- def setupDerivativeComputation(self, j:int, up:"numpy 1D array",
- upp : "numpy 1D array" = None):
- """
- Setup problem data to compute solution derivatives.
-
- Args:
- j: Derivative index.
- up: Adjusted previous derivative.
- upp: Adjusted pre-previous derivative.
- """
- raise Exception("Not implemented.")
- upRe = fen.Function(self.V)
- upIm = fen.Function(self.V)
- upRe.vector()[:] = np.array(np.real(up), dtype = float)
- upIm.vector()[:] = np.array(np.imag(up), dtype = float)
- upRe, upIm = self.n2Re * upRe - self.n2Im * upIm,\
- self.n2Re * upIm + self.n2Im * upRe
- self.forcingTerm = [upRe, upIm]
- self.DirichletDatum = fenZEROC
- self.NeumannDatum = fenZEROC
- self.RobinDatum2 = fenZEROC
-
- @property
- def wavenumber2(self):
- """Value of k^2."""
- return self._wavenumber2
- @wavenumber2.setter
- def wavenumber2(self, wavenumber2):
- if hasattr(self, "A"): del self.A
- if hasattr(self, "u"): del self.u
- self._wavenumber2 = wavenumber2
+ filename = FFCT.getNewFilename("temp", "edp")
+ u0file = FFCT.getNewFilename("u0", "tmp")
+ with open(filename, 'w') as f:
+ f.write(self.V + "\n")
+ f.write("{}{} u0 = {};\n".format(self.Vname, self.compl,
+ self.DirichletDatum.replace('j', 'i')))
+ f.write("ofstream u0out(\"{}\");\n".format(u0file))
+ f.write("u0out << u0[];\n")
+ bashCommand = "FreeFem++ -v 0 {}".format(filename)
+ process = subprocess.Popen(bashCommand.split(), stdout=subprocess.PIPE)
+ output, error = process.communicate()
+ if len(output) > 0:
+ raise Exception(("Error was encountered in the execution of "
+ "FreeFem++ code:\n{}\nSee file {}.")\
+ .format(output.decode('utf-8'), filename))
+ u0 = FFCT.ffvec2npvec(u0file)
+ os.remove(filename)
+ os.remove(u0file)
+ return u0
@property
def diffusivity(self):
- """Value of a. Its assignment changes aRe and aIm."""
+ """Value of a."""
return self._diffusivity
@diffusivity.setter
def diffusivity(self, diffusivity):
- if hasattr(self, "A"): del self.A
- if hasattr(self, "AL"): del self.AL
- if hasattr(self, "u"): del self.u
- self.a = diffusivity
- self._diffusivity = copy(diffusivity)
+ if hasattr(self, "A0"): del self.A0
+ self._diffusivity = diffusivity
@property
def refractionIndex(self):
- """Value of n. Its assignment changes nRe, nIm, n2Re and n2Im."""
+ """Value of n."""
return self._refractionIndex
@refractionIndex.setter
def refractionIndex(self, refractionIndex):
- if hasattr(self, "A"): del self.A
- if hasattr(self, "AM"): del self.AM
- if hasattr(self, "u"): del self.u
- self.n = refractionIndex
- self._refractionIndex = copy(refractionIndex)
+ if hasattr(self, "A1"): del self.A1
+ self._refractionIndex = refractionIndex
@property
def forcingTerm(self):
- """Value of f. Its assignment changes fRe and fIm."""
+ """Value of f."""
return self._forcingTerm
@forcingTerm.setter
def forcingTerm(self, forcingTerm):
- if hasattr(self, "b"): del self.b
- if hasattr(self, "bF"): del self.bF
- if hasattr(self, "u"): del self.u
- self.f = forcingTerm
- self._forcingTerm = copy(forcingTerm)
+ if hasattr(self, "b0"): del self.b0
+ self._forcingTerm = forcingTerm
@property
def DirichletDatum(self):
- """
- Value of u0. Its assignment changes u0Re, u0Im, DirichletBCRe and
- DirichletBCIm.
- """
+ """Value of u0."""
return self._DirichletDatum
@DirichletDatum.setter
def DirichletDatum(self, DirichletDatum):
- if hasattr(self, "b"): del self.b
- if hasattr(self, "u"): del self.u
- self.u0 = DirichletDatum
- self._DirichletDatum = copy(DirichletDatum)
+ if hasattr(self, "b0"): del self.b0
+ self._DirichletDatum = DirichletDatum
@property
def NeumannDatum(self):
- """Value of g1. Its assignment changes g1Re and g1Im."""
+ """Value of g1."""
return self._NeumannDatum
@NeumannDatum.setter
def NeumannDatum(self, NeumannDatum):
- if hasattr(self, "b"): del self.b
- if hasattr(self, "bN"): del self.bN
- if hasattr(self, "u"): del self.u
- self.g1 = NeumannDatum
- self._NeumannDatum = copy(NeumannDatum)
+ if hasattr(self, "b0"): del self.b0
+ self._NeumannDatum = NeumannDatum
@property
def RobinDatum(self):
- """Value of h. Its assignment changes hRe and hIm."""
+ """Value of h."""
return self._RobinDatum
@RobinDatum.setter
def RobinDatum(self, RobinDatum):
- if hasattr(self, "A"): del self.A
- if hasattr(self, "AR"): del self.AR
- if hasattr(self, "u"): del self.u
- self.h = RobinDatum
- self._RobinDatum = copy(RobinDatum)
+ if hasattr(self, "A0"): del self.A0
+ self._RobinDatum = RobinDatum
@property
def RobinDatum2(self):
- """Value of g2. Its assignment changes g2Re and g2Im."""
+ """Value of g2."""
return self._RobinDatum2
@RobinDatum2.setter
def RobinDatum2(self, RobinDatum2):
- if hasattr(self, "b"): del self.b
- if hasattr(self, "bR"): del self.bR
- if hasattr(self, "u"): del self.u
- self.g2 = RobinDatum2
- self._RobinDatum2 = copy(RobinDatum2)
+ if hasattr(self, "b0"): del self.b0
+ self._RobinDatum2 = RobinDatum2
+
+ def assembleA(self):
+ """Assemble matrix blocks of linear system."""
+ if not (hasattr(self, "A0") and hasattr(self, "A1")):
+ filename = FFCT.getNewFilename("temp", "edp")
+ with open(filename, 'w') as f:
+ f.write(self.V + "\n")
+ f.write("{}{} u, v;\n".format(self.Vname, self.compl))
+ if not hasattr(self, "A0"):
+ A0file = FFCT.getNewFilename("A0", "tmp")
+ f.write("varf a0(u, v) = int{}d({})({} * {})".format(
+ self.dim,
+ self.meshname,
+ self.diffusivity.replace('j', 'i'),
+ FFCT.diff2(dim = self.dim,
+ compl = not self.compl == "")))
+ if len(self.RobinBoundary) > 0:
+ f.write(" + int{}d({}, {})({} * {})".format(
+ self.dim - 1,
+ self.meshname,
+ self.RobinBoundary,
+ self.RobinDatum.replace('j', 'i'),
+ FFCT.prod2(compl = not self.compl == "")))
+ if len(self.DirichletBoundary) > 0:
+ f.write(" + on({}, u = 0.);\n".format(
+ self.DirichletBoundary))
+ else:
+ f.write(";\n")
+ f.write("matrix{0} A0 = a0({1},{1});\n".format(self.compl,
+ self.Vname))
+ f.write("ofstream A0out(\"{}\");\n".format(A0file))
+ f.write("A0out << A0;\n")
+ if not hasattr(self, "A1"):
+ A1file = FFCT.getNewFilename("A1", "tmp")
+ f.write("varf a1(u, v) = - int{}d({})(({})^2 * {})".format(
+ self.dim,
+ self.meshname,
+ self.refractionIndex.replace('j', 'i'),
+ FFCT.prod2(compl = not self.compl == "")))
+ if len(self.DirichletBoundary) > 0:
+ f.write(" + on({}, u = 0.);\n".format(
+ self.DirichletBoundary))
+ else:
+ f.write(";\n")
+ f.write("matrix{0} A1 = a1({1},{1}, tgv = 0.);\n".format(
+ self.compl, self.Vname))
+ f.write("ofstream A1out(\"{}\");\n".format(A1file))
+ f.write("A1out << A1;\n")
+ bashCommand = "FreeFem++ -v 0 {}".format(filename)
+ process = subprocess.Popen(bashCommand.split(),
+ stdout = subprocess.PIPE)
+ output, error = process.communicate()
+ if len(output) > 0:
+ raise Exception(("Error was encountered in the execution of "
+ "FreeFem++ code:\n{}\nSee file {}.")\
+ .format(output.decode('utf-8'), filename))
+ if not hasattr(self, "A0"):
+ self.A0 = FFCT.ffmat2spmat(A0file)
+ os.remove(A0file)
+ if not hasattr(self, "A1"):
+ self.A1 = FFCT.ffmat2spmat(A1file)
+ os.remove(A1file)
+ os.remove(filename)
- def assembleA(self, Rfact : complex = 1.):
+ def assembleb(self):
+ """Assemble RHS blocks of linear system."""
+ if not hasattr(self, "b0"):
+ filename = FFCT.getNewFilename("temp", "edp")
+ b0file = FFCT.getNewFilename("b0", "tmp")
+ if self.compl == "":
+ complEff = "real"
+ else:
+ complEff = "complex"
+ with open(filename, 'w') as f:
+ f.write(self.V + "\n")
+ f.write("{}{} u, v;\n".format(self.Vname, self.compl))
+ f.write("varf b0(u, v) = int{}d({})({})".format(
+ self.dim,
+ self.meshname,
+ FFCT.prod2(self.forcingTerm.replace('j', 'i'),
+ compl = not self.compl == "")))
+ if len(self.NeumannBoundary) > 0:
+ f.write(" + int{}d({}, {})({})".format(
+ self.dim - 1,
+ self.meshname,
+ self.NeumannBoundary,
+ FFCT.prod2(self.NeumannDatum.replace('j', 'i'),
+ compl = not self.compl == "")))
+ if len(self.RobinBoundary) > 0:
+ f.write(" + int{}d({}, {})({})".format(
+ self.dim - 1,
+ self.meshname,
+ self.RobinBoundary,
+ FFCT.prod2(self.RobinDatum2.replace('j', 'i'),
+ compl = not self.compl == "")))
+ if len(self.DirichletBoundary) > 0:
+ f.write(" + on({}, u = {});\n".format(
+ self.DirichletBoundary, self.DirichletDatum.replace('j', 'i')))
+ else:
+ f.write(";\n")
+ f.write("{}[int] B0 = b0(0, {});\n".format(complEff,
+ self.Vname))
+ f.write("ofstream b0out(\"{}\");\n".format(b0file))
+ f.write("b0out << B0;\n")
+ bashCommand = "FreeFem++ -v 0 {}".format(filename)
+ process = subprocess.Popen(bashCommand.split(),
+ stdout = subprocess.PIPE)
+ output, error = process.communicate()
+ if len(output) > 0:
+ raise Exception(("Error was encountered in the execution of "
+ "FreeFem++ code:\n{}\nSee file {}.")\
+ .format(output.decode('utf-8'), filename))
+ self.b0 = FFCT.ffvec2npvec(b0file)
+ os.remove(b0file)
+ os.remove(filename)
+
+ parDegree = 1
+
+ def As(self) -> list:
+ """Matrix blocks of linear system."""
+ self.assembleA()
+ return [self.A0, self.A1]
+
+ def bs(self) -> list:
+ """RHS blocks of linear system."""
+ self.assembleb()
+ return [self.b0]
+
+ def A(self) -> "CSR scipy matrix":
"""Assemble matrix of linear system."""
- raise Exception("Not implemented.")
- if not hasattr(self, "A"):
- if not hasattr(self, "AL"):
- aLRe = self.aRe * fen.dot(fen.grad(self.v1),
- fen.grad(self.v2)) * fen.dx
- aLIm = self.aIm * fen.dot(fen.grad(self.v1),
- fen.grad(self.v2)) * fen.dx
- ALRe = fen.assemble(aLRe)
- ALIm = fen.assemble(aLIm)
- self.DirichletBC0.apply(ALRe)
- self.DirichletBC0.zero(ALIm)
- ALReMat = fen.as_backend_type(ALRe).mat()
- ALRer, ALRec, ALRev = ALReMat.getValuesCSR()
- ALImr, ALImc, ALImv = fen.as_backend_type(
- ALIm).mat().getValuesCSR()
- self.AL = (scsp.csr_matrix((ALRev, ALRec, ALRer),
- shape = ALReMat.size)
- + 1.j * scsp.csr_matrix((ALImv, ALImc, ALImr),
- shape = ALReMat.size))
- if not hasattr(self, "AM"):
- aMRe = self.n2Re * fen.dot(self.v1, self.v2) * fen.dx
- aMIm = self.n2Im * fen.dot(self.v1, self.v2) * fen.dx
- AMRe = fen.assemble(aMRe)
- AMIm = fen.assemble(aMIm)
- self.DirichletBC0.zero(AMRe)
- self.DirichletBC0.zero(AMIm)
- AMRer, AMRec, AMRev = fen.as_backend_type(
- AMRe).mat().getValuesCSR()
- AMImr, AMImc, AMImv = fen.as_backend_type(
- AMIm).mat().getValuesCSR()
- self.AM = (scsp.csr_matrix((AMRev, AMRec, AMRer),
- shape = self.AL.shape)
- + 1.j * scsp.csr_matrix((AMImv, AMImc, AMImr),
- shape = self.AL.shape))
- if not hasattr(self, "AR"):
- aRRe = self.hRe * fen.dot(self.v1, self.v2) * self.ds(1)
- aRIm = self.hIm * fen.dot(self.v1, self.v2) * self.ds(1)
- ARRe = fen.assemble(aRRe)
- ARIm = fen.assemble(aRIm)
- self.DirichletBC0.zero(ARRe)
- self.DirichletBC0.zero(ARIm)
- ARRer, ARRec, ARRev = fen.as_backend_type(
- ARRe).mat().getValuesCSR()
- ARImr, ARImc, ARImv = fen.as_backend_type(
- ARIm).mat().getValuesCSR()
- self.AR = (scsp.csr_matrix((ARRev, ARRec, ARRer),
- shape = self.AL.shape)
- + 1.j * scsp.csr_matrix((ARImv, ARImc, ARImr),
- shape = self.AL.shape))
-
- self.A = self.AL - self.wavenumber2 * self.AM + Rfact * self.AR
- if hasattr(self, "invA"): del self.invA
+ self.assembleA()
+ return self.A0 + self.wavenumber2 * self.A1
- def assembleb(self):
+ def b(self) -> "numpy 1D array":
"""Assemble RHS of linear system."""
- raise Exception("Not implemented.")
- if not hasattr(self, "b"):
- if not hasattr(self, "bF"):
- LFRe = fen.dot(self.fRe, self.v2) * fen.dx
- LFIm = fen.dot(self.fIm, self.v2) * fen.dx
- bFRe = fen.assemble(LFRe)
- bFIm = fen.assemble(LFIm)
- self.DirichletBCRe.apply(bFRe)
- self.DirichletBCIm.apply(bFIm)
- self.bF = np.array(bFRe.array()[:] + 1.j * bFIm.array()[:],
- dtype = np.complex)
- if not hasattr(self, "bN"):
- LNRe = fen.dot(self.g1Re, self.v2) * self.ds(0)
- LNIm = fen.dot(self.g1Im, self.v2) * self.ds(0)
- bNRe = fen.assemble(LNRe)
- bNIm = fen.assemble(LNIm)
- self.DirichletBC0.apply(bNRe)
- self.DirichletBC0.apply(bNIm)
- self.bN = np.array(bNRe.array()[:] + 1.j * bNIm.array()[:],
- dtype = np.complex)
- if not hasattr(self, "bR"):
- LRRe = fen.dot(self.g2Re, self.v2) * self.ds(1)
- LRIm = fen.dot(self.g2Im, self.v2) * self.ds(1)
- bRRe = fen.assemble(LRRe)
- bRIm = fen.assemble(LRIm)
- self.DirichletBC0.apply(bRRe)
- self.DirichletBC0.apply(bRIm)
- self.bR = np.array(bRRe.array()[:] + 1.j * bRIm.array()[:],
- dtype = np.complex)
-
- self.b = self.bF + self.bN + self.bR
-
- def buildLU(self):
- """Build LU decomposition of A."""
- raise Exception("Not implemented.")
- if not hasattr(self, "A"): self.assembleA()
- if not hasattr(self, "invA"):
- warnings.simplefilter("ignore", scsp.SparseEfficiencyWarning)
- self.invA = spla.splu(self.A)
- warnings.simplefilter("default", scsp.SparseEfficiencyWarning)
-
- def solve(self, LU : bool = False)\
- -> ("Fenics function", "Fenics function"):
+ self.assembleb()
+ return self.b0
+
+ def solve(self) -> "numpy 1D array":
"""
Find solution of linear system.
+ Returns:
+ Solution vector.
+ """
+ return scsp.linalg.spsolve(self.A(), self.b())
+
+ def getLSBlocks(self) -> (list, list):
+ """
+ Get blocks of linear system.
+
+ Returns:
+ Blocks of LS (\sum_{j=0}^J k^j A_j = \sum_{j=0}^K k^j b_j).
+ """
+ return self.As(), self.bs()
+
+
+class FreeFemHelmholtzScatteringEngine(FreeFemHelmholtzEngine):
+ """
+ FreeFem++-based solver for Helmholtz scattering problems with Robin ABC.
+ - \nabla \cdot (a \nabla u) - k^2 * n**2 * u = f in \Omega
+ u = u0 on \Gamma_D
+ \partial_nu = g1 on \Gamma_N
+ \partial_nu - i k u = g2 on \Gamma_R
+
+ Args:
+
+ Attributes:
+ """
+
+ def __init__(self, V:str,
+ wavenumber:complex,
+ compl : bool = True,
+ dim : int = 2,
+ meshname : str = "Th",
+ Vname : str = "V",
+ diffusivity : str = "1",
+ refractionIndex : str = "1",
+ forcingTerm : str = "0",
+ DirichletBoundary : str = "",
+ NeumannBoundary : str = "",
+ RobinBoundary : str = "",
+ DirichletDatum : str = "0",
+ NeumannDatum : str = "0",
+ RobinDatum2 : str = "0"):
+ self.wavenumber = wavenumber
+ FreeFemHelmholtzEngine.__init__(self, V = V,
+ wavenumber = wavenumber,
+ compl = compl,
+ dim = dim,
+ meshname = meshname,
+ Vname = Vname,
+ diffusivity = diffusivity,
+ refractionIndex = refractionIndex,
+ forcingTerm = forcingTerm,
+ DirichletBoundary = DirichletBoundary,
+ NeumannBoundary = NeumannBoundary,
+ RobinBoundary = RobinBoundary,
+ DirichletDatum = DirichletDatum,
+ NeumannDatum = NeumannDatum,
+ RobinDatum = -1.j * wavenumber,
+ RobinDatum2 = RobinDatum2)
+ if len(self.RobinBoundary) == 0:
+ raise Exception(("Empty Robin boundary not allowed in scattering "
+ "engine. Use standard engine instead."))
+
+ def setProblemData(self, dataDict:dict, k:complex):
+ """
+ Set problem data.
+
Args:
- LU(optional): Whether to use a LU solver for the system. Defaults
- to False.
+ dataDict: Dictionary of problem data.
+ k: Parameter value.
+ """
+ self.wavenumber = k
+ self.forcingTerm = dataDict["forcingTerm"]
+ self.DirichletDatum = dataDict["DirichletDatum"]
+ self.NeumannDatum = dataDict["NeumannDatum"]
+ self.RobinDatum2 = dataDict["RobinDatum2"]
+ if hasattr(self, "A0"): del self.A0
+ if hasattr(self, "A1"): del self.A1
+ if hasattr(self, "A2"): del self.A2
+ if hasattr(self, "b0"): del self.b0
+ @property
+ def wavenumber2(self):
+ """Value of k^2."""
+ return self._wavenumber2
+ @wavenumber2.setter
+ def wavenumber2(self, wavenumber2):
+ self._wavenumber2 = wavenumber2
+ self._wavenumber = self.wavenumber2 ** .5
+ self.RobinDatum = "{}".format(-1.j*self.wavenumber)
+
+ @property
+ def wavenumber(self):
+ """Value of k."""
+ return self._wavenumber
+ @wavenumber.setter
+ def wavenumber(self, wavenumber):
+ self.wavenumber2 = wavenumber ** 2.
+
+ @property
+ def RobinDatum(self):
+ """Value of h."""
+ return super().RobinDatum
+ @RobinDatum.setter
+ def RobinDatum(self, RobinDatum):
+ try:
+ self._RobinDatum = "{}".format(complex(RobinDatum))
+ except:
+ raise Exception(("Value of RobinDatum not recognized. Must be "
+ "complex number of str convertible to complex."))
+ if not (hasattr(self, 'wavenumber')
+ and np.isclose(self.wavenumber, 1.j * complex(self.RobinDatum))):
+ self.wavenumber = 1.j * complex(self.RobinDatum)
+
+ def assembleA(self):
+ """Assemble matrix blocks of linear system."""
+ if not (hasattr(self, "A0") and hasattr(self, "A1")
+ and hasattr(self, "A2")):
+ filename = FFCT.getNewFilename("temp", "edp")
+ with open(filename, 'w') as f:
+ f.write(self.V + "\n")
+ f.write("{}{} u, v;\n".format(self.Vname, self.compl))
+ if not hasattr(self, "A0"):
+ A0file = FFCT.getNewFilename("A0", "tmp")
+ f.write("varf a0(u, v) = int{}d({})({} * {})".format(
+ self.dim,
+ self.meshname,
+ self.diffusivity.replace('j', 'i'),
+ FFCT.diff2(dim = self.dim,
+ compl = not self.compl == "")))
+ if len(self.DirichletBoundary) > 0:
+ f.write(" + on({}, u = 0.);\n".format(
+ self.DirichletBoundary))
+ else:
+ f.write(";\n")
+ f.write("matrix{0} A0 = a0({1},{1});\n".format(self.compl,
+ self.Vname))
+ f.write("ofstream A0out(\"{}\");\n".format(A0file))
+ f.write("A0out << A0;\n")
+ if not hasattr(self, "A1"):
+ A1file = FFCT.getNewFilename("A1", "tmp")
+ f.write("varf a1(u, v) = int{}d({}, {})({} * {})".format(
+ self.dim - 1,
+ self.meshname,
+ self.RobinBoundary,
+ self.RobinDatum.replace('j', 'i'),
+ FFCT.prod2(compl = not self.compl == "")))
+ if len(self.DirichletBoundary) > 0:
+ f.write(" + on({}, u = 0.);\n".format(
+ self.DirichletBoundary))
+ else:
+ f.write(";\n")
+ f.write("matrix{0} A1 = a1({1},{1}, tgv = 0.);\n".format(
+ self.compl, self.Vname))
+ f.write("ofstream A1out(\"{}\");\n".format(A1file))
+ f.write("A1out << A1;\n")
+ if not hasattr(self, "A2"):
+ A2file = FFCT.getNewFilename("A2", "tmp")
+ f.write("varf a2(u, v) = - int{}d({})({} * {})".format(
+ self.dim,
+ self.meshname,
+ self.refractionIndex.replace('j', 'i'),
+ FFCT.prod2(compl = not self.compl == "")))
+ if len(self.DirichletBoundary) > 0:
+ f.write(" + on({}, u = 0.);\n".format(
+ self.DirichletBoundary))
+ else:
+ f.write(";\n")
+ f.write("matrix{0} A2 = a2({1},{1}, tgv = 0.);\n".format(
+ self.compl, self.Vname))
+ f.write("ofstream A2out(\"{}\");\n".format(A2file))
+ f.write("A2out << A2;\n")
+ bashCommand = "FreeFem++ -v 0 {}".format(filename)
+ process = subprocess.Popen(bashCommand.split(),
+ stdout = subprocess.PIPE)
+ output, error = process.communicate()
+ if len(output) > 0:
+ raise Exception(("Error was encountered in the execution of "
+ "FreeFem++ code:\n{}\nSee file {}.")\
+ .format(output.decode('utf-8'), filename))
+ if not hasattr(self, "A0"):
+ self.A0 = FFCT.ffmat2spmat(A0file)
+ os.remove(A0file)
+ if not hasattr(self, "A1"):
+ self.A1 = FFCT.ffmat2spmat(A1file)
+ os.remove(A1file)
+ if not hasattr(self, "A2"):
+ self.A2 = FFCT.ffmat2spmat(A2file)
+ os.remove(A2file)
+ os.remove(filename)
+
+ def As(self) -> list:
+ """Matrix blocks of linear system."""
+ self.assembleA()
+ return [self.A0, self.A1, self.A2]
+
+ def A(self) -> "CSR scipy matrix":
+ """Assemble matrix of linear system."""
+ self.assembleA()
+ return self.A0 + self.wavenumber * self.A1 + self.wavenumber2 * self.A2
+
+
+class FreeFemHelmholtzScatteringAugmentedEngine(
+ FreeFemHelmholtzScatteringEngine):
+ """
+ FreeFem++-based solver for Helmholtz scattering problems with Robin ABC.
+ - \nabla \cdot (a \nabla u) - k^2 * n**2 * u = f in \Omega
+ u = u0 on \Gamma_D
+ \partial_nu = g1 on \Gamma_N
+ \partial_nu - i k u = g2 on \Gamma_R
+ Linear dependence on k is achieved by introducing an additional variable.
+
+ Args:
+
+ Attributes:
+ """
+
+ def __init__(self, V:str,
+ wavenumber:complex,
+ compl : bool = True,
+ dim : int = 2,
+ meshname : str = "Th",
+ Vname : str = "V",
+ diffusivity : str = "1",
+ refractionIndex : str = "1",
+ forcingTerm : str = "0",
+ DirichletBoundary : str = "",
+ NeumannBoundary : str = "",
+ RobinBoundary : str = "",
+ DirichletDatum : str = "0",
+ NeumannDatum : str = "0",
+ RobinDatum2 : str = "0",
+ constraintType : str = "IDENTITY"):
+ self.constraintType = constraintType
+ FreeFemHelmholtzScatteringEngine.__init__(self, V = V,
+ wavenumber = wavenumber,
+ compl = compl,
+ dim = dim,
+ meshname = meshname,
+ Vname = Vname,
+ diffusivity = diffusivity,
+ refractionIndex = refractionIndex,
+ forcingTerm = forcingTerm,
+ DirichletBoundary = DirichletBoundary,
+ NeumannBoundary = NeumannBoundary,
+ RobinBoundary = RobinBoundary,
+ DirichletDatum = DirichletDatum,
+ NeumannDatum = NeumannDatum,
+ RobinDatum2 = RobinDatum2)
+
+ def energyNormMatrix(self, w:float) -> "CSC sparse matrix":
+ """
+ Sparse matrix (in CSR format) assoociated to w-weighted H10 norm.
+
+ Args:
+ w: Weight.
+
Returns:
- Real and imaginary parts of solution.
+ Sparse matrix in CSR format.
"""
- raise Exception("Not implemented.")
- if not hasattr(self, "u"):
- if not hasattr(self, "A"): self.assembleA()
- if not hasattr(self, "b"): self.assembleb()
- if LU:
- self.buildLU()
- self.u = self.invA.solve(self.b)
- else:
- self.u = spla.spsolve(self.A, self.b)
- self.__solved = True
- return self.u
+ normMatFen = FreeFemHelmholtzEngine.energyNormMatrix(self, w)
+ return scsp.block_diag((normMatFen, 1/w * normMatFen))
+
+ def liftDirichletData(self):
+ """Lift Dirichlet datum."""
+ lift1 = FreeFemHelmholtzScatteringEngine.liftDirichletData(self)
+ return np.array([lift1, np.zeros_like(lift1)])
+
+ def setProblemData(self, dataDict:dict, k:complex):
+ """
+ Set problem data.
+
+ Args:
+ dataDict: Dictionary of problem data.
+ k: Parameter value.
+ """
+ self.wavenumber = k
+ self.forcingTerm = dataDict["forcingTerm"]
+ self.DirichletDatum = dataDict["DirichletDatum"]
+ self.NeumannDatum = dataDict["NeumannDatum"]
+ self.RobinDatum2 = dataDict["RobinDatum2"]
+ if hasattr(self, "A0"): del self.A0
+ if hasattr(self, "A1"): del self.A1
+ if hasattr(self, "b0"): del self.b0
+
+ @property
+ def constraintType(self):
+ """Value of constraintType."""
+ return self._constraintType
+ @constraintType.setter
+ def constraintType(self, constraintType):
+ try:
+ constraintType = constraintType.upper().strip().replace(" ","")
+ if constraintType not in ["IDENTITY", "MASS"]: raise
+ self._constraintType = constraintType
+ except:
+ warnings.warn(("Prescribed constraintType not recognized. "
+ "Overriding to 'IDENTITY'."))
+ self.constraintType = "IDENTITY"
+
+ def assembleA(self):
+ """Assemble matrix blocks of linear system."""
+ builtA0 = hasattr(self, "A0")
+ builtA1 = hasattr(self, "A1")
+ if not (builtA0 and builtA1):
+ if builtA1 and self.constraintType == "IDENTITY":
+ self.A2 = None
+ FreeFemHelmholtzScatteringEngine.assembleA(self)
+ if self.constraintType == "IDENTITY":
+ if not builtA0:
+ block = scsp.eye(self.A0.shape[0])
+ else:
+ block = scsp.eye(self.A1.shape[0])
+ else: #MASS
+ block = - self.A2
+ I = list(self.DirichletBC0.get_boundary_values().keys())
+ warnings.simplefilter("ignore", scsp.SparseEfficiencyWarning)
+ block[I, I] = self.A0[I[0], I[0]]
+ warnings.simplefilter("default", scsp.SparseEfficiencyWarning)
+ if not builtA0:
+ self.A0 = scsp.block_diag((self.A0, block))
+ if not builtA1:
+ self.A1 = scsp.bmat([[self.A1, self.A2], [- block, None]])
+ if hasattr(self, 'A2'): del self.A2
+
+ def assembleb(self):
+ """Assemble RHS of linear system."""
+ if not hasattr(self, "b0"):
+ FreeFemHelmholtzScatteringEngine.assembleb(self)
+ self.b0 = np.pad(self.b0, (0, self.b0.shape[0]), 'constant')
+
+ parDegree = 1
+
+ def As(self) -> list:
+ """Matrix blocks of linear system."""
+ self.assembleA()
+ return [self.A0, self.A1]
+
+ def A(self) -> "CSR scipy matrix":
+ """Assemble matrix of linear system."""
+ self.assembleA()
+ return self.A0 + self.wavenumber * self.A1
+
+ def solve(self) -> ("numpy 1D array", "numpy 1D array"):
+ """
+ Find solution of linear system.
+
+ Returns:
+ Solution vector.
+ """
+ u = FreeFemHelmholtzScatteringEngine.solve(self)
+ lu2 = int(len(u) / 2)
+ return np.array([u[: lu2], u[lu2 :]])
diff --git a/main/ROMApproximant.py b/main/ROMApproximant.py
index c0069d3..a397648 100644
--- a/main/ROMApproximant.py
+++ b/main/ROMApproximant.py
@@ -1,307 +1,333 @@
#!/usr/bin/python
from abc import abstractmethod
import numpy as np
import utilities
+from copy import copy
+import scipy.sparse.linalg as spla
class ROMApproximant:
"""
ROM approximant computation for parametric problems.
Args:
HFEngine: HF problem solver. Should have members:
- energyNormMatrix: assemble sparse matrix (in CSC format)
associated to weighted H10 norm;
- problemData: list of HF problem data (excluding k);
- setProblemData: set HF problem data and k to prescribed values;
- - solve: get HF solution as complex numpy vector.
+ - getLSBlocks: get algebraic system blocks.
HSEngine: Hilbert space general purpose engine. Should have members:
- norm: compute Hilbert norm of expression represented as complex
numpy vector;
- plot: plot Hilbert expression represented as complex numpy vector.
k0(optional): Default parameter. Defaults to 0.
w(optional): Weight for norm computation. If None, set to Re(k0).
Defaults to None.
approxParameters(optional): Dictionary containing values for main
parameters of approximant. Recognized keys are:
- 'POD': whether to compute POD of snapshots; defaults to True.
Defaults to empty dict.
Attributes:
HFEngine: HF problem solver.
HSEngine: Hilbert space general purpose engine.
k0: Default parameter.
w: Weight for norm computation.
approxParameters: Dictionary containing values for main parameters of
approximant. Recognized keys are:
- 'POD': whether to compute POD of snapshots.
initialHFData: HF problem initial data.
energyNormMatrix: Sparse matrix (in CSC format) assoociated to
w-weighted H10 norm.
uHF: High fidelity solution with wavenumber lastSolvedHF as numpy
complex vector.
lastSolvedHF: Wavenumber corresponding to last computed high fidelity
solution.
uApp: Last evaluated approximant as numpy complex vector.
lastApproxParameters: List of parameters corresponding to last
computed approximant.
"""
def __init__(self, HFEngine:'HF solver', HSEngine:'HS engine',
k0 : complex = 0, w : float = None,
approxParameters : dict = {}):
self.HFEngine = HFEngine
self.HSEngine = HSEngine
self.initialHFData = self.HFEngine.problemData()
self.k0 = k0
if w is None:
w = np.real(self.k0)
self.w = w
self.approxParameters = approxParameters
self.energyNormMatrix = self.HFEngine.energyNormMatrix(self.w)
+ self.As, self.bs = self.HFEngine.getLSBlocks()
def name(self) -> str:
"""Approximant label."""
return self.__class__.__name__
def parameterList(self) -> list:
"""
List containing keys which are allowed in approxParameters.
Returns:
List of strings.
"""
return ["POD"]
@property
def approxParameters(self):
"""Value of approximant parameters."""
return self._approxParameters
@approxParameters.setter
def approxParameters(self, approxParams):
if not hasattr(self, "approxParameters"):
self._approxParameters = {}
approxParameters = utilities.purgeDict(approxParams,
- ROMApproximant.parameterList(self),
- dictname = "approxParameters")
+ ROMApproximant.parameterList(self),
+ dictname = self.name() + ".approxParameters")
keyList = list(approxParameters.keys())
if "POD" in keyList:
self.POD = approxParameters["POD"]
elif hasattr(self, "POD"):
self.POD = self.POD
else:
self.POD = True
@property
def POD(self):
"""Value of POD."""
return self._POD
@POD.setter
def POD(self, POD:bool):
if hasattr(self, "POD"): PODold = self.POD
else: PODold = -1
self._POD = POD
self._approxParameters["POD"] = self.POD
if PODold != self.POD:
self.resetSamples()
def solveHF(self, k : complex = None):
"""
Find high fidelity solution with original parameters and arbitrary
wavenumber.
Args:
k: Target wavenumber.
"""
if k is None: k = self.k0
if (not hasattr(self, "lastSolvedHF")
or not np.isclose(self.lastSolvedHF, k)):
- self.HFEngine.setProblemData(self.initialHFData, k)
- self.uHF = self.HFEngine.solve(False)
+ A, b = self.buildLS(k)
+ self.uHF = spla.spsolve(A, b)
+ return self.uHF
+
+ def buildLS(self, k:complex, As : list = None, bs : list = None):
+ """
+ Build linear system from polynomial blocks.
+
+ Args:
+ k: Parameter value.
+ As: Matrix blocks. If None, set to self.As. Defaults to None.
+ bs: RHS blocks. If None, set to self.bs. Defaults to None.
+
+ Returns:
+ Matrix and RHS of system.
+ """
+ if As is None: As = self.As
+ if bs is None: bs = self.bs
+ A = copy(As[0])
+ b = copy(bs[0])
+ for j in range(1, len(As)):
+ A = A + np.power(k, j) * As[j]
+ for j in range(1, len(bs)):
+ b = b + np.power(k, j) * bs[j]
+ return A, b
@abstractmethod
def resetSamples(self):
"""Reset samples. (ABSTRACT)"""
pass
@abstractmethod
def setupApprox(self):
"""
Setup approximant. (ABSTRACT)
Any specialization should include something like
self.computeDerivatives()
if not self.checkComputedApprox():
...
self.lastApproxParameters = copy(self.approxParameters)
"""
pass
def checkComputedApprox(self) -> bool:
"""
Check if setup of new approximant is not needed.
Returns:
True if new setup is not needed. False otherwise.
"""
return (hasattr(self, "lastApproxParameters")
and self.approxParameters == self.lastApproxParameters)
@abstractmethod
def evalApprox(self, k:complex) -> ("numpy 1D array"):
"""
Evaluate approximant at arbitrary wavenumber. (ABSTRACT)
Any specialization should include something like
self.setupApprox()
self.uApp = ...
Args:
k: Target wavenumber.
Returns:
Approximant as numpy complex vector.
"""
pass
@abstractmethod
def getPoles(self, centered : bool = False) -> "numpy 1D array":
"""
Obtain approximant poles.
Args:
centered(optional): Whether to return pole positions relative to
approximation center. Defaults to False.
Returns:
Numpy complex vector of poles.
"""
pass
def HFNorm(self, k:complex, normType : "number or str" = None) -> float:
"""
Compute norm of HF solution at arbitrary wavenumber.
Args:
k: Target wavenumber.
normType(optional): Target norm identifier. If number, target norm
is weighted H^1 norm with given weight. If string, must be
recognizable by Fenics norm command. If None, set to w.
Defaults to None.
Returns:
Target norm of HFsolution.
"""
self.solveHF(k)
if normType is None: normType = self.w
return self.HSEngine.norm(self.uHF, normType)
def approxNorm(self, k:complex, normType : "number or str" = None)-> float:
"""
Compute norm of (translated) approximant at arbitrary wavenumber.
Args:
k: Target wavenumber.
normType(optional): Target norm identifier. If number, target norm
is weighted H^1 norm with given weight. If string, must be
recognizable by Fenics norm command. If None, set to w.
Defaults to None.
Returns:
Target norm of approximant.
"""
self.evalApprox(k)
if normType is None: normType = self.w
return self.HSEngine.norm(self.uApp, normType)
def approxError(self, k:complex, normType : "number or str" = None)\
-> float:
"""
Compute norm of approximant at arbitrary wavenumber.
Args:
k: Target wavenumber.
normType(optional): Target norm identifier. If number, target norm
is weighted H^1 norm with given weight. If string, must be
recognizable by Fenics norm command. If None, set to w.
Defaults to None.
Returns:
Target norm of (approximant - HFsolution).
"""
self.evalApprox(k)
self.solveHF(k)
self.evalApprox(k)
if normType is None: normType = self.w
return self.HSEngine.norm(self.uApp - self.uHF, normType)
def getHF(self, k:complex) -> "numpy 1D array":
"""
Get HF solution at arbitrary wavenumber.
Args:
k: Target wavenumber.
Returns:
HFsolution as numpy complex vector.
"""
self.solveHF(k)
return self.uHF
def getApp(self, k:complex) -> "numpy 1D array":
"""
Get approximant at arbitrary wavenumber.
Args:
k: Target wavenumber.
Returns:
Approximant as numpy complex vector.
"""
self.evalApprox(k)
return self.uApp
def plotHF(self, k:complex, name : str = "u", **figspecs):
"""
Do some nice plots of the HF solution at arbitrary wavenumber.
Args:
k: Target wavenumber.
name(optional): Name to be shown as title of the plots. Defaults to
'u'.
figspecs(optional key args): Optional arguments for matplotlib
figure creation.
"""
self.solveHF(k)
self.HSEngine.plot(self.uHF, name, **figspecs)
def plotApp(self, k:complex, name : str = "u", **figspecs):
"""
Do some nice plots of approximant at arbitrary wavenumber.
Args:
k: Target wavenumber.
name(optional): Name to be shown as title of the plots. Defaults to
'u'.
figspecs(optional key args): Optional arguments for matplotlib
figure creation.
"""
self.evalApprox(k)
self.HSEngine.plot(self.uApp, name, **figspecs)
def plotErr(self, k:complex, name : str = "u", **figspecs):
"""
Do some nice plots of approximation error at arbitrary wavenumber.
Args:
k: Target wavenumber.
name(optional): Name to be shown as title of the plots. Defaults to
'u'.
figspecs(optional key args): Optional arguments for matplotlib
figure creation.
"""
self.evalApprox(k)
self.solveHF(k)
self.HSEngine.plot(self.uApp - self.uHF, name, **figspecs)
diff --git a/main/ROMApproximantLagrange.py b/main/ROMApproximantLagrange.py
index b1da41f..3d3c81d 100644
--- a/main/ROMApproximantLagrange.py
+++ b/main/ROMApproximantLagrange.py
@@ -1,193 +1,193 @@
#!/usr/bin/python
import numpy as np
import utilities
from ROMApproximant import ROMApproximant
class ROMApproximantLagrange(ROMApproximant):
"""
ROM Lagrange interpolant computation for parametric problems.
Args:
HFEngine: HF problem solver. Should have members:
- energyNormMatrix: assemble sparse matrix (in CSC format)
associated to weighted H10 norm;
- problemData: list of HF problem data (excluding k);
- setProblemData: set HF problem data and k to prescribed values;
- - solve: get HF solution as complex numpy vector.
+ - getLSBlocks: get algebraic system blocks.
HSEngine: Hilbert space general purpose engine. Should have members:
- norm: compute Hilbert norm of expression represented as complex
numpy vector;
- plot: plot Hilbert expression represented as complex numpy vector.
ks(optional): Array of snapshot parameters. Defaults to np.array([0]).
w(optional): Weight for norm computation. If None, set to
Re(np.mean(ks)). Defaults to None.
approxParameters(optional): Dictionary containing values for main
parameters of approximant. Recognized keys are:
- 'POD': whether to compute POD of snapshots; defaults to True;
- 'S': total number of samples current approximant relies upon.
Defaults to empty dict.
- plotSnapshots(optional): Whether to plot snapshots of the Helmholtz
- solution map. Defaults to False.
+ plotSnap(optional): What to plot of snapshots of the Helmholtz
+ solution map. See plot method in HSEngine. Defaults to [].
Attributes:
HFEngine: HF problem solver.
HSEngine: Hilbert space general purpose engine.
solSnapshots: Array whose columns are FE dofs of snapshots of solution
map.
k0: Default parameter.
ks: Array of snapshot parameters.
w: Weight for norm computation.
approxParameters: Dictionary containing values for main parameters of
approximant. Recognized keys are:
- 'POD': whether to compute POD of snapshots;
- 'S': total number of snapshots current approximant relies upon.
S: Number of solution snapshots over which current approximant is
based upon.
- plotSnapshots: Whether to plot snapshots of the Helmholtz solution map.
+ plotSnap: What to plot of snapshots of the Helmholtz solution map.
POD: Whether to compute POD of snapshots.
energyNormMatrix: Sparse matrix (in CSC format) assoociated to
w-weighted H10 norm.
uHF: High fidelity solution with wavenumber lastSolvedHF as numpy
complex vector.
lastSolvedHF: Wavenumber corresponding to last computed high fidelity
solution.
uApp: Last evaluated approximant as numpy complex vector.
lastApproxParameters: List of parameters corresponding to last
computed approximant.
"""
extraApproxParameters = ["S"]
def __init__(self, HFEngine:'HF solver', HSEngine:'HS engine',
ks : "Numpy 1D array" = np.array([0]), w : float = None,
- approxParameters : dict = {}, plotSnapshots : bool = False):
+ approxParameters : dict = {}, plotSnap : list = []):
self.ks = ks
ROMApproximant.__init__(self, HFEngine, HSEngine,
np.mean(ks), w, approxParameters)
if w is None:
w = np.mean(self.ksRe)
self.w = w
- self.plotSnapshots = plotSnapshots
+ self.plotSnap = plotSnap
def parameterList(self) -> list:
"""
List containing keys which are allowed in approxParameters.
Returns:
List of strings.
"""
return (ROMApproximant.parameterList(self)
+ ROMApproximantLagrange.extraApproxParameters)
@property
def ks(self):
"""Value of ks. Its assignment may reset snapshots."""
return self._ks
@ks.setter
def ks(self, ks):
if hasattr(self, 'ks'):
ksOld = self.ks
else:
ksOld = None
self._ks = np.array(ks)
if (ksOld is None or len(self.ks) != len(ksOld)
or not np.allclose(self.ks, ksOld, 1e-14)):
self.resetSamples()
@property
def ksRe(self):
"""Real part of ks."""
return np.real(self.ks)
@property
def ksIm(self):
"""Imaginary part of ks."""
return np.imag(self.ks)
@property
def zs(self):
"""Square of ks."""
return np.power(self.ks, 2.)
@property
def approxParameters(self):
"""Value of approximant parameters. Its assignment may change S."""
return self._approxParameters
@approxParameters.setter
def approxParameters(self, approxParams):
if not hasattr(self, "approxParameters"):
self._approxParameters = {}
approxParameters = utilities.purgeDict(approxParams,
ROMApproximantLagrange.parameterList(self),
- dictname = "approxParameters")
+ dictname = self.name() + ".approxParameters")
keyList = list(approxParameters.keys())
approxParametersCopy = utilities.purgeDict(approxParameters,
ROMApproximantLagrange.extraApproxParameters,
True, True)
ROMApproximant.approxParameters.fset(self, approxParametersCopy)
if "S" in keyList:
self.S = approxParameters["S"]
elif hasattr(self, "S"):
self.S = self.S
else:
self.S = 2
@property
def S(self):
"""Value of S."""
return self._S
@S.setter
def S(self, S):
if S <= 0: raise ArithmeticError("S must be positive.")
if hasattr(self, "S"): Sold = self.S
else: Sold = -1
self._S = S
self._approxParameters["S"] = self.S
if Sold != self.S:
self.resetSamples()
def resetSamples(self):
"""Reset samples. (ABSTRACT)"""
self.solSnapshots = None
self.RPOD = None
def computeSnapshots(self):
"""
Compute snapshots of solution map.
"""
if self.solSnapshots is None:
for j, k in enumerate(self.ks):
self.solveHF(k)
- if self.plotSnapshots:
- self.HSEngine.plot(self.uHF, name = "u({:.4f})".format(k))
+ self.HSEngine.plot(self.uHF, name = "u({:.4f})".format(k),
+ what = self.plotSnap)
self.manageSnapshots(self.uHF, j)
def manageSnapshots(self, u:"numpy 1D array", pos:int):
"""
Store snapshots of solution map.
Args:
u: solution derivative as numpy complex vector;
pos: Derivative index.
"""
if pos == 0:
self.solSnapshots = np.empty((u.shape[0], self.S),
dtype = np.complex)
self.solSnapshots[:, pos] = u
if self.POD:
if pos == 0:
self.RPOD = np.eye(self.S, dtype = np.complex)
beta = 1
for j in range(2):
nu = self.solSnapshots[:, pos].conj().dot(
self.energyNormMatrix.dot(self.solSnapshots[:, : pos])).conj()
self.RPOD[: pos, pos] = self.RPOD[: pos, pos] + beta * nu
eta = (self.solSnapshots[:, pos]
- self.solSnapshots[:, : pos].dot(nu))
beta = eta.conj().dot(self.energyNormMatrix.dot(eta))**.5
self.solSnapshots[:, pos] = eta / beta
self.RPOD[pos, pos] = beta * self.RPOD[pos, pos]
diff --git a/main/ROMApproximantLagrangePade.py b/main/ROMApproximantLagrangePade.py
index 457c2c1..5442806 100644
--- a/main/ROMApproximantLagrangePade.py
+++ b/main/ROMApproximantLagrangePade.py
@@ -1,453 +1,453 @@
#!/usr/bin/python
from __future__ import print_function
from copy import copy
import warnings
import numpy as np
import utilities
from ROMApproximantLagrange import ROMApproximantLagrange
PI = np.pi
class ROMApproximantLagrangePade(ROMApproximantLagrange):
"""
ROM Lagrange Pade' interpolant computation for parametric problems.
Args:
HFEngine: HF problem solver. Should have members:
- energyNormMatrix: assemble sparse matrix (in CSC format)
associated to weighted H10 norm;
- problemData: list of HF problem data (excluding k);
- setProblemData: set HF problem data and k to prescribed values;
- - solve: get HF solution as complex numpy vector.
+ - getLSBlocks: get algebraic system blocks.
HSEngine: Hilbert space general purpose engine. Should have members:
- norm: compute Hilbert norm of expression represented as complex
numpy vector;
- plot: plot Hilbert expression represented as complex numpy vector.
ks(optional): Array of snapshot parameters. Defaults to np.array([0]).
ws(optional): Array of snapshots weights. Defaults to uniform = 1.
w(optional): Weight for norm computation. If None, set to
Re(np.mean(ks)). Defaults to None.
approxParameters(optional): Dictionary containing values for main
parameters of approximant. Recognized keys are:
- 'POD': whether to compute POD of snapshots; defaults to True;
- 'S': total number of samples current approximant relies upon;
defaults to 2;
- 'M': degree of Pade' interpolant numerator; defaults to 0;
- 'N': degree of Pade' interpolant denominator; defaults to 0.
- 'polyBasis': label of polynomial basis for LS problem; available
values are:
- 'CHEBYSHEV': Chebyshev nodes and weights;
- 'GAUSSLEGENDRE': Gauss-Legendre nodes and weights;
defaults to 'CHEBYSHEV'.
Defaults to empty dict.
- plotSnapshots(optional): Whether to plot snapshots of the Helmholtz
- solution map. Defaults to False.
+ plotSnap(optional): What to plot of snapshots of the Helmholtz
+ solution map. See plot method in HSEngine. Defaults to [].
verboseRobust(optional): Verbosity level for robustness-related
parameter modifications. Defaults to 1.
cleanupParameters(optional): Parameter values for pole cleanup.
Available fields are:
- 'boolCondition'(bool function handle): if evaluation on pole
returns False, pole is removed; defaults to always True;
- 'residueCheck'(bool): whether to apply residue check; defaults to
False;
- 'residueNPoints'(int): number of sample points to be used in
residue estimation; defaults to 2;
- 'residueRadius'(float): sample radius to be used in residue
estimation by Cauchy formula; defaults to 1e-5;
- 'residueTol'(float): target tolerance to be used in residue
estimation; defaults to 1e-4.
Attributes:
HFEngine: HF problem solver.
HSEngine: Hilbert space general purpose engine.
solSnapshots: Array whose columns are FE dofs of snapshots of solution
map.
k0: Default parameter.
ks: Array of snapshot parameters.
ws: Array of snapshots weights.
w: Weight for norm computation.
approxParameters: Dictionary containing values for main parameters of
approximant. Recognized keys are:
- 'POD': whether to compute POD of snapshots;
- 'S': total number of samples current approximant relies upon;
- 'M': degree of Pade' interpolant numerator;
- 'N': degree of Pade' interpolant denominator;
- 'polyBasis': label of polynomial basis for LS problem.
M: Numerator degree of approximant.
N: Denominator degree of approximant.
S: Number of solution snapshots over which current approximant is
based upon.
polyBasis: Polynomial basis for LS problem.
- plotSnapshots: Whether to plot snapshots of the Helmholtz solution map.
+ plotSnap: What to plot of snapshots of the Helmholtz solution map.
POD: Whether to compute POD of snapshots.
verboseRobust: Verbosity level for robustness-related parameter
modifications.
cleanupParameters; Parameter values for pole cleanup. Available fields
are:
- 'boolCondition': if evaluation on pole returns False, pole is
removed;
- 'residueCheck': whether to apply residue check;
- 'residueNPoints': number of sample points to be used in residue
estimation;
- 'residueRadius': sample radius to be used in residue estimation
by Cauchy formula;
- 'residueTol': target tolerance to be used in residue estimation.
energyNormMatrix: Sparse matrix (in CSC format) assoociated to
w-weighted H10 norm.
uHF: High fidelity solution with wavenumber lastSolvedHF as numpy
complex vector.
lastSolvedHF: Wavenumber corresponding to last computed high fidelity
solution.
Q: Numpy 1D vector containing complex coefficients of approximant
denominator.
P: Numpy 2D vector whose columns are FE dofs of coefficients of
approximant numerator.
uApp: Last evaluated approximant as numpy complex vector.
lastApproxParameters: List of parameters corresponding to last
computed approximant.
"""
extraApproxParameters = ["M", "N", "polyBasis"]
polyBasisParameters = ["CHEBYSHEV", "GAUSSLEGENDRE"]
def __init__(self, HFEngine:'HF solver', HSEngine:'HS engine',
ks : "Numpy 1D array" = np.array([0]),
ws : "Numpy 1D array" = None, w : float = None,
- approxParameters : dict = {}, plotSnapshots : bool = False,
+ approxParameters : dict = {}, plotSnap : list = [],
verboseRobust : int = 1, cleanupParameters : dict = {}):
ROMApproximantLagrange.__init__(self, HFEngine, HSEngine, ks, w,
- approxParameters, plotSnapshots)
+ approxParameters, plotSnap)
if ws == None:
ws = np.ones(np.size(self.ks))
self.ws = ws
self.verboseRobust = verboseRobust
self.cleanupParameters = cleanupParameters
def parameterList(self) -> list:
"""
List containing keys which are allowed in approxParameters.
Returns:
List of strings.
"""
return (ROMApproximantLagrange.parameterList(self)
+ ROMApproximantLagrangePade.extraApproxParameters)
@property
def k0(self):
"""Dummy center of approximant (i.e. k0)."""
self.k0 = np.mean(self.ks)
return self._k0
@k0.setter
def k0(self, k0:bool):
self._k0 = k0
@property
def approxParameters(self):
"""
Value of approximant parameters. Its assignment may change M, N and S.
"""
return self._approxParameters
@approxParameters.setter
def approxParameters(self, approxParams):
if not hasattr(self, "approxParameters"):
self._approxParameters = {}
approxParameters = utilities.purgeDict(approxParams,
ROMApproximantLagrangePade.parameterList(self),
- dictname = "approxParameters")
+ dictname = self.name() + ".approxParameters")
keyList = list(approxParameters.keys())
if "M" in keyList:
self.M = 0 #to avoid warnings if M and S are changed simultaneously
if "N" in keyList:
self.N = 0 #to avoid warnings if N and S are changed simultaneously
approxParametersCopy = utilities.purgeDict(approxParameters,
ROMApproximantLagrangePade.extraApproxParameters,
True, True)
ROMApproximantLagrange.approxParameters.fset(self,approxParametersCopy)
if "M" in keyList:
self.M = approxParameters["M"]
elif hasattr(self, "M"):
self.M = self.M
else:
self.M = 0
if "N" in keyList:
self.N = approxParameters["N"]
elif hasattr(self, "N"):
self.N = self.N
else:
self.N = 0
if "polyBasis" in keyList:
self.polyBasis = approxParameters["polyBasis"]
elif hasattr(self, "polyBasis"):
self.polyBasis = self.polyBasis
else:
self.polyBasis = "CHEBYSHEV"
@property
def M(self):
"""Value of M. Its assignment may change S."""
return self._M
@M.setter
def M(self, M):
if M < 0: raise ArithmeticError("M must be non-negative.")
self._M = M
self._approxParameters["M"] = self.M
if hasattr(self, "S") and self.S < self.M + 1:
warnings.warn("Prescribed S is too small. Updating S to M + 1.")
self.S = self.M + 1
@property
def N(self):
"""Value of N. Its assignment may change S."""
return self._N
@N.setter
def N(self, N):
if N < 0: raise ArithmeticError("N must be non-negative.")
self._N = N
self._approxParameters["N"] = self.N
if hasattr(self, "S") and self.S < self.N + 1:
warnings.warn("Prescribed S is too small. Updating S to N + 1.")
self.S = self.N + 1
@property
def S(self):
"""Value of S."""
return self._S
@S.setter
def S(self, S):
if S <= 0: raise ArithmeticError("S must be positive.")
if hasattr(self, "S"): Sold = self.S
else: Sold = -1
vals, label = [0] * 2, {0:"M", 1:"N"}
if hasattr(self, "M"): vals[0] = self.M
if hasattr(self, "N"): vals[1] = self.N
idxmax = np.argmax(vals)
if vals[idxmax] + 1 > S:
warnings.warn("Prescribed S is too small. Updating S to {} + 1."\
.format(label[idxmax]))
self.Emax = vals[idxmax] + 1
else:
self._S = S
self._approxParameters["S"] = self.S
if Sold != self.S:
self.resetSamples()
@property
def polyBasis(self):
"""Value of polyBasis."""
return self._polyBasis
@polyBasis.setter
def polyBasis(self, polyBasis):
if hasattr(self, "polyBasis"): polyBasisold = self.polyBasis
else: polyBasisold = -1
try:
polyBasis = polyBasis.upper().strip().replace(" ","")
if polyBasis not in self.polyBasisParameters: raise
except:
warnings.warn(("Prescribed polyBasis not recognized. Overriding to"
" 'CHEBYSHEV'."))
polyBasis = "CHEBYSHEV"
self._polyBasis = polyBasis
self._approxParameters["polyBasis"] = self.polyBasis
if polyBasisold != self.polyBasis:
self.resetSamples()
@property
def cleanupParameters(self):
"""Value of cleanupParameters."""
return self._cleanupParameters
@cleanupParameters.setter
def cleanupParameters(self, cleanupParameters):
allowedCleanupKeys = ['boolCondition','residueCheck','residueNPoints',
'residueRadius','residueTol']
cleanupKeys = cleanupParameters.keys()
if 'boolCondition' not in cleanupKeys:
cleanupParameters['boolCondition'] = lambda x : True
if 'residueCheck' not in cleanupKeys:
cleanupParameters['residueCheck'] = False
if 'residueNPoints' not in cleanupKeys:
cleanupParameters['residueNPoints'] = 2
if 'residueRadius' not in cleanupKeys:
cleanupParameters['residueRadius'] = 1e-5
if 'residueTol' not in cleanupKeys:
cleanupParameters['residueTol'] = 1e-4
cleanupParameters['boolCondition'] = np.vectorize(
cleanupParameters['boolCondition'])
self._cleanupParameters = {key : cleanupParameters[key]
for key in allowedCleanupKeys}
def setupApprox(self):
"""
Compute Pade' interpolant.
SVD-based robust eigenvalue management.
"""
S0 = self.S
M1 = self.M + 1
N1 = self.N + 1
if self.solSnapshots is None:
self.approxRadius = np.max(np.abs(self.k0 - self.ks))
if self.polyBasis == "CHEBYSHEV":
nodes, weights = np.polynomial.chebyshev.chebgauss(S0)
self.ws = weights * 2. / PI
elif self.polyBasis == "GAUSSLEGENDRE":
nodes, weights = np.polynomial.legendre.leggauss(S0)
self.ws = weights
self.ks = nodes * self.approxRadius + self.k0
self.ws = self.ws[:, None]
self.computeSnapshots()
if not self.checkComputedApprox():
Phi = np.zeros((S0, max(M1, N1)), dtype = np.complex)
Phi[:, 0] = np.ones((S0,)) / 2**.5
for j in range(1, max(M1, N1)):
c = np.zeros((j + 1,))
c[-1] = 1.
if self.polyBasis == "CHEBYSHEV":
Phi[:, j] = np.polynomial.chebyshev.chebval(
self.radiusPade(self.ks), c)
elif self.polyBasis == "GAUSSLEGENDRE":
Phi[:, j] = (j + .5) ** .5 * np.polynomial.legendre.legval(
self.radiusPade(self.ks), c)
if self.POD:
II = np.array(np.arange(0, S0**3, S0)
+ np.kron(np.arange(S0), np.ones(S0)),
dtype = np.int)
Rtall = np.zeros(S0**3, dtype = np.complex)
Rtall[II] = np.reshape(self.RPOD.T, (S0**2,))
Rtall = np.reshape(Rtall, (S0, S0**2)).T
B = Rtall.dot(Phi[:, : N1])
Z = copy(B)
Z = np.reshape(Z.T, (S0 * (N1), S0)).T
for j in range(2):
Z = Z - Phi[:, : M1].dot(Phi[:, : M1].conj().T.dot(
np.multiply(self.ws, Z)))
Z = np.reshape(Z.T, (N1, S0**2)).T
else:
ker = self.solSnapshots.conj().T.dot(self.energyNormMat.dot(
self.solSnapshots))
WPhi = np.reshape(np.multiply(self.ws, Phi[:, : M1]).T,
(S0 * M1)).conj()[:, None]
Y = np.multiply(WPhi, np.kron(np.ones((M1, 1)), Phi[:, : N1]))
Ylarge = np.reshape(Y.T, (M1 * N1, S0)).T
B = np.reshape(ker.dot(Ylarge).T, (N1, M1 * S0)).T
D = np.multiply(self.ws, np.diag(ker)[:, None])
D = Phi[:, : N1].conj().T.dot(np.multiply(D, Phi[:, : N1]))
Z = B.conj().T.dot(Y) - D
_, ev, eV = np.linalg.svd(Z, full_matrices=False)
eV = eV.conj().T
phi = eV[:, np.argmin(ev)]
if self.POD:
c = Phi[:, : M1].conj().T.dot(np.multiply(self.ws,
np.reshape(B.dot(phi).T, (S0, S0)).T))
else:
c = np.reshape(Y.dot(phi).T, (M1, S0))
polybase = np.zeros((max(M1, N1), max(M1, N1)))
polybase[0, 0] = 1
polybase[1, 1] = 1
for j in range(2, max(M1, N1)):
if self.polyBasis == "CHEBYSHEV":
polybase[1 : j + 1, j] = 2 * polybase[: j, j - 1]
polybase[: j + 1, j] = (polybase[: j + 1, j]
- polybase[: j + 1, j - 2])
elif self.polyBasis == "GAUSSLEGENDRE":
polybase[1 : j + 1, j] = ((2 * j - 1) / j
* polybase[: j, j - 1])
polybase[: j + 1, j] = (polybase[: j + 1, j]
- ((j - 1) / j * polybase[: j + 1, j - 2]))
if self.polyBasis == "CHEBYSHEV":
polybase[0, 0] = .5 ** .5
elif self.polyBasis == "GAUSSLEGENDRE":
polybase = np.multiply(np.power(
np.arange(.5, max(M1, N1)), .5), polybase)
self.P = self.solSnapshots.dot(polybase[: M1, : M1].dot(c).T)
self.Q = polybase[: N1, : N1].dot(phi)
self.approxParameters = {"N" : self.N, "M" : self.M, "S" : S0}
self.lastApproxParameters = copy(self.approxParameters)
def _cleanup(self):
"""Cleanup Pade' denominator by removing unwanted poles."""
if self.N == 0: return
poles = np.roots(self.Q[::-1]) + self.k0
NpolesOld = len(poles)
poles = poles[self.cleanupParameters['boolCondition'](poles)]
if self.cleanupParameters['residueCheck']:
resR = self.cleanupParameters['residueRadius']
resTol = self.cleanupParameters['residueTol']
residues = np.zeros_like(poles)
for l, pole in enumerate(poles):
resV = np.zeros(self.solDerivatives.shape[0],
dtype = np.complex)
NPoints = self.cleanupParameters['residueNPoints']
for theta in 2 * PI * np.arange(NPoints) / NPoints:
deltapole = resR * np.exp(1.j * theta)
ksample = pole + deltapole
self.solveHF(ksample)
resV = deltapole / NPoints * (resV + self.uHF)
residues[l] = np.abs(resV.dot(
self.energyNormMatrix.dot(resV).conj())) ** .5
poles = poles[residues >= resTol]
NpolesNew = len(poles)
if NpolesOld > NpolesNew:
if self.verboseRobust >= 1:
sSing = "s" * (NpolesOld - NpolesNew > 1)
print(("Identified {0} pole{1} to be removed.\n"
"Reducing N from {2} to {3}.").format(
NpolesOld - NpolesNew, sSing, NpolesOld, NpolesNew))
self.N = NpolesNew
newQ = np.polyfit(np.append(poles - self.k0, 0.),
np.append(np.zeros(NpolesNew), 1.), NpolesNew)
self.Q = newQ[::-1] / np.linalg.norm(newQ)
def radiusPade(self, k:complex):
"""
Compute translated radius to be plugged into Pade' approximant.
Args:
k: Target wavenumber.
Returns:
Translated radius to be plugged into Pade' approximant.
"""
return (k - self.k0) / self.approxRadius
def evalApprox(self, k:complex) -> ("Fenics function", "Fenics function"):
"""
Evaluate Pade' approximant at arbitrary wavenumber.
Args:
k: Target wavenumber.
Returns:
Real and imaginary parts of approximant.
"""
self.setupApprox()
powerlist = np.power(self.radiusPade(k), range(max(self.M, self.N) + 1))
self.uApp = (self.P.dot(powerlist[:self.M + 1])
/ self.Q.dot(powerlist[:self.N + 1]))
return self.uApp
def getPoles(self, centered : bool = False) -> "numpy 1D array":
"""
Obtain approximant poles.
Args:
centered(optional): Whether to return pole positions relative to
approximation center. Defaults to False.
Returns:
Numpy complex vector of poles.
"""
self.setupApprox()
return np.roots(self.Q[::-1]) * self.approxRadius + self.k0 * centered
\ No newline at end of file
diff --git a/main/ROMApproximantLagrangeRB.py b/main/ROMApproximantLagrangeRB.py
index b67574a..f9a8764 100644
--- a/main/ROMApproximantLagrangeRB.py
+++ b/main/ROMApproximantLagrangeRB.py
@@ -1,307 +1,306 @@
#!/usr/bin/python
from copy import copy
import warnings
import numpy as np
import scipy as sp
import utilities
from ROMApproximantLagrange import ROMApproximantLagrange
PI = np.pi
class ROMApproximantLagrangeRB(ROMApproximantLagrange):
"""
ROM RB approximant computation for parametric problems.
Args:
HFEngine: HF problem solver. Should have members:
- energyNormMatrix: assemble sparse matrix (in CSC format)
associated to weighted H10 norm;
- problemData: list of HF problem data (excluding k);
- setProblemData: set HF problem data and k to prescribed values;
- - solve: get HF solution as complex numpy vector.
+ - getLSBlocks: get algebraic system blocks;
+ - liftDirichletData: perform lifting of Dirichlet BC as numpy
+ complex vector.
HSEngine: Hilbert space general purpose engine. Should have members:
- norm: compute Hilbert norm of expression represented as complex
numpy vector;
- plot: plot Hilbert expression represented as complex numpy vector.
ks(optional): Array of snapshot parameters. Defaults to np.array([0]).
ws(optional): Array of snapshots weights. Defaults to uniform = 1.
w(optional): Weight for norm computation. If None, set to
Re(np.mean(ks)). Defaults to None.
approxParameters(optional): Dictionary containing values for main
parameters of approximant. Recognized keys are:
- 'POD': whether to compute POD of snapshots; defaults to True;
- 'S': total number of samples current approximant relies upon;
defaults to 2;
- 'R': rank for Galerkin projection; defaults to S;
- 'nodesType': sampling node type; available values are:
- 'CHEBYSHEV': Chebyshev nodes;
- 'GAUSSLEGENDRE': Gauss-Legendre nodes;
- 'MANUAL': manual selection;
defaults to 'MANUAL'.
Defaults to empty dict.
- plotSnapshots(optional): Whether to plot snapshots of the Helmholtz
- solution map. Defaults to False.
+ plotSnap(optional): What to plot of snapshots of the Helmholtz
+ solution map. See plot method in HSEngine. Defaults to [].
Attributes:
HFEngine: HF problem solver.
HSEngine: Hilbert space general purpose engine.
solSnapshots: Array whose columns are FE dofs of snapshots of solution
map.
k0: Default parameter.
ks: Array of snapshot parameters.
ws: Array of snapshots weights.
w: Weight for norm computation.
approxRadius: Dummy radius of approximant (i.e. distance from k0 to
farthest sample point).
approxParameters: Dictionary containing values for main parameters of
approximant. Recognized keys are:
- 'POD': whether to compute POD of snapshots;
- 'S': total number of samples current approximant relies upon;
- 'R': rank for Galerkin projection;
- 'nodesType': sampling node type.
S: Number of solution snapshots over which current approximant is
based upon.
R: Rank for Galerkin projection.
nodesType: Sampling node type.
- plotSnapshots: Whether to plot snapshots of the Helmholtz solution map.
+ plotSnap: What to plot of snapshots of the Helmholtz solution map.
POD: Whether to compute POD of snapshots.
energyNormMatrix: Sparse matrix (in CSC format) assoociated to
w-weighted H10 norm.
uHF: High fidelity solution with wavenumber lastSolvedHF as numpy
complex vector.
lastSolvedHF: Wavenumber corresponding to last computed high fidelity
solution.
uApp: Last evaluated approximant as numpy complex vector.
lastApproxParameters: List of parameters corresponding to last
computed approximant.
"""
extraApproxParameters = ["R", "nodesType"]
nodesTypeParameters = ["CHEBYSHEV", "GAUSSLEGENDRE", "MANUAL"]
def __init__(self, HFEngine:'HF solver', HSEngine:'HS engine',
ks : "Numpy 1D array" = np.array([0]),
ws : "Numpy 1D array" = None, w : float = None,
- approxParameters : dict = {}, plotSnapshots : bool = False):
+ approxParameters : dict = {}, plotSnap : list = []):
ROMApproximantLagrange.__init__(self, HFEngine, HSEngine, ks, w,
- approxParameters, plotSnapshots)
+ approxParameters, plotSnap)
if ws == None:
ws = np.ones(np.size(self.ks))
self.ws = ws
self.solLifting = self.HFEngine.liftDirichletData()
@property
def k0(self):
"""Dummy center of approximant (i.e. k0)."""
self.k0 = np.mean(self.ks)
return self._k0
@k0.setter
def k0(self, k0:bool):
self._k0 = k0
def resetSamples(self):
"""Reset samples."""
ROMApproximantLagrange.resetSamples(self)
self.projMat = None
def parameterList(self) -> list:
"""
List containing keys which are allowed in approxParameters.
Returns:
List of strings.
"""
return (ROMApproximantLagrange.parameterList(self)
+ ROMApproximantLagrangeRB.extraApproxParameters)
@property
def approxParameters(self):
"""
Value of approximant parameters. Its assignment may change M, N and S.
"""
return self._approxParameters
@approxParameters.setter
def approxParameters(self, approxParams):
if not hasattr(self, "approxParameters"):
self._approxParameters = {}
approxParameters = utilities.purgeDict(approxParams,
ROMApproximantLagrangeRB.parameterList(self),
- dictname = "approxParameters")
+ dictname = self.name() + ".approxParameters")
keyList = list(approxParameters.keys())
approxParametersCopy = utilities.purgeDict(approxParameters,
ROMApproximantLagrangeRB.extraApproxParameters,
True, True)
ROMApproximantLagrange.approxParameters.fset(self,approxParametersCopy)
if "R" in keyList:
self.R = approxParameters["R"]
elif hasattr(self, "R"):
self.R = self.R
else:
self.R = self.S
if "nodesType" in keyList:
self.nodesType = approxParameters["nodesType"]
elif hasattr(self, "nodesType"):
self.nodesType = self.nodesType
else:
self.nodesType = "MANUAL"
@property
def R(self):
"""Value of R. Its assignment may change S."""
return self._R
@R.setter
def R(self, R):
if R < 0: raise ArithmeticError("R must be non-negative.")
self._R = R
self._approxParameters["R"] = self.R
if hasattr(self, "S") and self.S < self.R:
warnings.warn("Prescribed S is too small. Updating S to R.")
self.S = self.R
@property
def nodesType(self):
"""Value of nodesType."""
return self._nodesType
@nodesType.setter
def nodesType(self, nodesType):
if hasattr(self, "nodesType"): nodesTypeold = self.nodesType
else: nodesTypeold = -1
try:
nodesType = nodesType.upper().strip().replace(" ","")
if nodesType not in self.nodesTypeParameters: raise
except:
warnings.warn(("Prescribed nodesType not recognized. Overriding to"
" 'MANUAL'."))
nodesType = "MANUAL"
self._nodesType = nodesType
self._approxParameters["nodesType"] = self.nodesType
if nodesTypeold != self.nodesType:
self.resetSamples()
def manageSnapshots(self, u:"2-tuple of Fenics function", pos:int):
"""
Post-process snapshots of solution map.
Any specialization should include something like
self.solSnapshots[:, pos] = (np.array(u[0].vector()[:])
+ 1.j * np.array(u[1].vector()[:]))
Args:
u: 2-tuple containing real and imaginary parts of FE dofs of
snapshot.
pos: Derivative index.
"""
- if pos == 0:
- self.As, self.bs = self.HFEngine.getLSBlocks()
- u = u - self.solLifting
- return ROMApproximantLagrange.manageSnapshots(self, u, pos)
+ return ROMApproximantLagrange.manageSnapshots(self, u - self.solLifting, pos)
def setupApprox(self):
"""
Compute RB projection matrix.
See ``Householder triangularization of a quasimatrix'', L.Trefethen,
2008 for QR algorithm.
"""
if self.solSnapshots is None:
if self.nodesType == "MANUAL" and len(self.ks) != self.S:
warnings.warn(("Number of prescribed nodes different from S. "
"Overriding S to len(ks)"))
self.S = len(self.ks)
self.approxRadius = np.max(np.abs(self.k0 - self.ks))
if self.nodesType != "MANUAL":
if self.nodesType == "CHEBYSHEV":
nodes, weights = np.polynomial.chebyshev.chebgauss(self.S)
self.ws = weights * 2. / PI
elif self.nodesType == "GAUSSLEGENDRE":
nodes, weights = np.polynomial.legendre.leggauss(self.S)
self.ws = weights
self.ks = nodes * self.approxRadius + self.k0
self.ws = self.ws[:, None]
self.computeSnapshots()
if not self.checkComputedApprox():
if self.POD:
U, _, _ = np.linalg.svd(self.RPOD, full_matrices = False)
self.projMat = self.solSnapshots.dot(U[:, : self.R])
else:
self.projMat = self.solSnapshots[:, : self.R]
self.assembleReducedSystem()
self.lastApproxParameters = copy(self.approxParameters)
def assembleReducedSystem(self):
"""Build affine blocks of RB linear system through projections."""
projMatH = self.projMat.T.conjugate()
self.ARBs = [None] * len(self.As)
self.bRBs = [None] * max(len(self.As), len(self.bs))
for j in range(len(self.As)):
self.ARBs[j] = projMatH.dot(self.As[j].dot(self.projMat))
if j < len(self.bs):
self.bRBs[j] = projMatH.dot(self.bs[j]
- self.As[j].dot(self.solLifting))
else:
self.bRBs[j] = - projMatH.dot(self.As[j].dot(self.solLifting))
for j in range(len(self.As), len(self.bs)):
self.bRBs[j] = projMatH.dot(self.bs[j])
def solveReducedSystem(self, k:complex) -> "Numpy 1D array":
"""
Solve RB linear system.
Args:
k: Target wavenumber.
Returns:
Solution of RB linear system.
"""
self.setupApprox()
ARBk = self.ARBs[0][:self.R,:self.R]
bRBk = self.bRBs[0][:self.R]
for j in range(1, len(self.ARBs)):
ARBk = ARBk + np.power(k, j) * self.ARBs[j][:self.R, :self.R]
for j in range(1, len(self.bRBs)):
bRBk = bRBk + np.power(k, j) * self.bRBs[j][:self.R]
return self.projMat[:, :self.R].dot(np.linalg.solve(ARBk, bRBk))
def evalApprox(self, k:complex) -> ("Fenics function", "Fenics function"):
"""
Evaluate RB approximant at arbitrary wavenumber.
Args:
k: Target wavenumber.
Returns:
Real and imaginary parts of approximant.
"""
self.setupApprox()
self.uApp = self.solLifting + self.solveReducedSystem(k)
return self.uApp
def getPoles(self, centered : bool = False) -> "numpy 1D array":
"""
Obtain approximant poles.
Args:
centered(optional): Whether to return pole positions relative to
approximation center. Defaults to False.
Returns:
Numpy complex vector of poles.
"""
self.setupApprox()
A = np.eye(self.ARBs[0].shape[0] * (len(self.ARBs) - 1),
dtype = np.complex)
B = np.zeros_like(A)
A[: self.ARBs[0].shape[0], : self.ARBs[0].shape[0]] = - self.ARBs[0]
for j in range(len(self.ARBs) - 1):
Aj = self.ARBs[j + 1]
B[: Aj.shape[0], j * Aj.shape[0] : (j + 1) * Aj.shape[0]] = Aj
II = np.arange(self.ARBs[0].shape[0],
self.ARBs[0].shape[0] * (len(self.ARBs) - 1))
B[II, II - self.ARBs[0].shape[0]] = 1.
try:
return sp.linalg.eigvals(A, B) - self.k0 * (not centered)
except:
warnings.warn("Generalized eig algorithm did not converge.")
x = np.empty(A.shape[0])
x[:] = np.NaN
return x
diff --git a/main/ROMApproximantSweeper.py b/main/ROMApproximantSweeper.py
index 0378b1d..fb0baad 100644
--- a/main/ROMApproximantSweeper.py
+++ b/main/ROMApproximantSweeper.py
@@ -1,257 +1,272 @@
#!/usr/bin/python
import os
import csv
import warnings
import numpy as np
from context import utilities
class ROMApproximantSweeper:
"""
ROM approximant sweeper.
Args:
ROMEngine(optional): ROMApproximant class. Defaults to None.
ktars(optional): Array of parameter values to sweep. Defaults to empty
array.
params(optional): List of parameter settings (each as a dict) to
explore. Defaults to single empty set.
mostExpensive(optional): String containing label of most expensive
step, to be executed fewer times. Allowed options are 'HF' and
'Approx'. Defaults to 'HF'.
Attributes:
ROMEngine: ROMApproximant class.
ktars: Array of parameter values to sweep.
params: List of parameter settings (each as a dict) to explore.
mostExpensive: String containing label of most expensive step, to be
executed fewer times.
"""
- allowedOutputs = ["HFNorm", "AppNorm", "AppError"]
+ allowedOutputs = ["HFNorm", "HFFunc", "AppNorm", "AppFunc", "ErrNorm", "ErrFunc"]
def __init__(self, ROMEngine : 'ROMApproximant' = None,
ktars : 'numpy 1D array' = np.array([]),
params : "List of dicts" = [{}],
mostExpensive : str = "HF"):
self.ROMEngine = ROMEngine
self.ktars = ktars
self.params = params
self.mostExpensive = mostExpensive
def name(self) -> str:
"""Approximant label."""
return self.__class__.__name__
@property
def mostExpensive(self):
"""Value of mostExpensive."""
return self._mostExpensive
@mostExpensive.setter
def mostExpensive(self, mostExpensive:str):
mostExpensive = mostExpensive.upper()
if mostExpensive not in ["HF", "APPROX"]:
warnings.warn(("Value of mostExpensive not recognized. Overriding "
"to 'HF'."))
mostExpensive = "HF"
self._mostExpensive = mostExpensive
def checkValues(self) -> bool:
"""Check if sweep can be performed."""
if self.ROMEngine is None:
warnings.warn("ROMEngine is missing. Aborting.")
return False
if len(self.ktars) == 0:
warnings.warn("Empty target parameter vector. Aborting.")
return False
if len(self.params) == 0:
warnings.warn("Empty method parameters vector. Aborting.")
return False
return True
def sweep(self, filename : str = "./out", outputs : list = [],
verbose : int = 1):
if not self.checkValues(): return
try:
if outputs.upper() == "ALL":
outputs = self.allowedOutputs + ["poles"]
except:
if len(outputs) == 0:
outputs = self.allowedOutputs
- outputs = utilities.purgeList(outputs,
- self.allowedOutputs + ["poles"],
- listname = "outputs list")
+ outputs = utilities.purgeList(outputs, self.allowedOutputs + ["poles"],
+ listname = self.name() + ".outputs")
poles = ("poles" in outputs)
if len(outputs) == 0:
warnings.warn("Empty outputs. Aborting.")
return
outParList = self.ROMEngine.parameterList()
Nparams = len(self.params)
allowedParams = self.ROMEngine.parameterList()
while os.path.exists(filename):
filename = filename + "{}".format(np.random.randint(10))
append_write = "w"
initial_row = (outParList + ["kRe", "kIm"]
+ [x for x in self.allowedOutputs if x in outputs]
+ ["type"] + ["poles"] * poles)
with open(filename, append_write, buffering = 1) as fout:
writer = csv.writer(fout, delimiter=",")
writer.writerow(initial_row)
if self.mostExpensive == "HF":
outerSet = self.ktars
innerSet = self.params
elif self.mostExpensive == "APPROX":
outerSet = self.params
innerSet = self.ktars
for outerIdx, outerPar in enumerate(outerSet):
if self.mostExpensive == "HF":
i, ktar = outerIdx, outerPar
elif self.mostExpensive == "APPROX":
j, par = outerIdx, outerPar
self.ROMEngine.approxParameters = {k: par[k] for k in\
par.keys() & allowedParams}
self.ROMEngine.setupApprox()
for innerIdx, innerPar in enumerate(innerSet):
if self.mostExpensive == "APPROX":
i, ktar = innerIdx, innerPar
elif self.mostExpensive == "HF":
j, par = innerIdx, innerPar
self.ROMEngine.approxParameters = {k: par[k] for k in\
par.keys() & allowedParams}
self.ROMEngine.setupApprox()
if verbose >= 1:
print("Set {}/{}\tk_{} = {:.10f}{}"\
.format(j+1, Nparams, i, ktar, " " * 15),
end="\r")
outData = []
if "HFNorm" in outputs:
- outData = outData + [self.ROMEngine.HFNorm(ktar)]
+ val = self.ROMEngine.HFNorm(ktar)
+ if isinstance(val, (list,)): val = val[0]
+ outData = outData + [val]
+ if "HFFunc" in outputs:
+ outData = outData +[self.ROMEngine.HFEngine.functional(
+ self.ROMEngine.solveHF(ktar))]
if "AppNorm" in outputs:
- outData = outData + [self.ROMEngine.approxNorm(ktar)]
- if "AppError" in outputs:
- outData = outData + [self.ROMEngine.approxError(ktar)]
+ val = self.ROMEngine.approxNorm(ktar)
+ if isinstance(val, (list,)): val = val[0]
+ outData = outData + [val]
+ if "AppFunc" in outputs:
+ outData = outData +[self.ROMEngine.HFEngine.functional(
+ self.ROMEngine.evalApprox(ktar))]
+ if "ErrNorm" in outputs:
+ val = self.ROMEngine.approxError(ktar)
+ if isinstance(val, (list,)): val = val[0]
+ outData = outData + [val]
+ if "ErrFunc" in outputs:
+ outData = outData +[self.ROMEngine.HFEngine.functional(
+ self.ROMEngine.evalApprox(ktar)
+ - self.ROMEngine.solveHF(ktar))]
writeData = []
for parn in outParList:
writeData = (writeData
+ [self.ROMEngine.approxParameters[parn]])
writeData = (writeData + [ktar.real, ktar.imag]
+ outData + [self.ROMEngine.name()])
if poles:
writeData = writeData + list(self.ROMEngine.getPoles())
writer.writerow(str(x) for x in writeData)
if verbose >= 1:
if self.mostExpensive == "APPROX":
print("Set {}/{}\tdone{}".format(j+1, Nparams, " "*25))
elif self.mostExpensive == "HF":
print("Point k_{} = {:.10f}\tdone{}".format(i, ktar,
" " * 25))
self.filename = filename
return self.filename
def read(self, filename:str, restrictions : dict = {},
outputs : list = []) -> dict:
"""
Execute a query on a custom format CSV.
Args:
filename: CSV filename.
restrictions(optional): Parameter configurations to output.
Defaults to empty dictionary, i.e. output all.
outputs(optional): Values to output. Defaults to empty list, i.e.
no output.
Returns:
Dictionary of desired results, with a key for each entry of
outputs, and a numpy 1D array as corresponding value.
"""
def pairKFromCSV(filename:str, kTarget:complex) -> "2-tuple of str":
"""
Find complex point in CSV closer to a prescribed value.
Args:
filename: CSV filename.
zTarget: Target complex value.
Returns:
Strings containing real and imaginary part of desired value, in
the same format as in the CSV file.
"""
ktarsF = np.array([], dtype = complex)
kRetarsF = np.array([], dtype = complex)
kImtarsF = np.array([], dtype = complex)
with open(filename, 'r') as f:
reader = csv.reader(f, delimiter=',')
header = next(reader)
kReindex = header.index('kRe')
kImindex = header.index('kIm')
for row in reader:
try:
if row[kReindex] not in [" ", ""]:
kRetarsF = np.append(kRetarsF, row[kReindex])
kImtarsF = np.append(kImtarsF, row[kImindex])
ktarsF = np.append(ktarsF, float(row[kReindex])
+ 1.j * float(row[kImindex]))
except:
pass
optimalIndex = np.argmin(np.abs(ktarsF - kTarget))
return [kRetarsF[optimalIndex], kImtarsF[optimalIndex]]
with open(filename, 'r') as f:
reader = csv.reader(f, delimiter=',')
header = next(reader)
restrIndices, outputIndices, outputData = {}, {}, {}
for key in restrictions.keys():
try:
restrIndices[key] = header.index(key)
if not isinstance(restrictions[key], list):
restrictions[key] = [restrictions[key]]
restrictions[key] = [str(x) for x in restrictions[key]]
except:
warnings.warn("Ignoring key {} from restrictions".format(
key))
if 'kRe' in restrIndices.keys() or 'kIm' in restrIndices.keys():
if 'kRe' not in restrIndices.keys():
restrIndices['kRe'] = header.index('kRe')
restrictions['kRe'] = [0.] * len(restrictions['kIm'])
elif 'kIm' not in restrIndices.keys():
restrIndices['kIm'] = header.index('kIm')
restrictions['kIm'] = [0.] * len(restrictions['kRe'])
elif len(restrictions['kRe']) != len(restrictions['kIm']):
raise Exception(("The lists of values for kRe and kIm "
"must have the same length."))
for i in range(len(restrictions['kRe'])):
k = (1.0 * float(restrictions['kRe'][i])
+ 1.j * float(restrictions['kIm'][i]))
restrictions['kRe'][i], restrictions['kIm'][i] =\
pairKFromCSV(filename, k)
for key in outputs:
try:
outputIndices[key] = header.index(key)
outputData[key] = np.array([])
except:
warnings.warn("Ignoring key {} from outputs".format(key))
for row in reader:
if all([row[restrIndices[key]] in restrictions[key]\
for key in restrictions.keys()]):
for key in outputIndices.keys():
try:
val = row[outputIndices[key]]
val = int(val)
except:
try:
val = float(val)
except:
val = np.nan
finally:
outputData[key] = np.append(outputData[key], val)
return outputData
\ No newline at end of file
diff --git a/main/ROMApproximantTaylor.py b/main/ROMApproximantTaylor.py
index f554993..2e339d8 100644
--- a/main/ROMApproximantTaylor.py
+++ b/main/ROMApproximantTaylor.py
@@ -1,264 +1,275 @@
#!/usr/bin/python
import warnings
import numpy as np
+from scipy.special import comb
import utilities
+import scipy.sparse.linalg as spla
from ROMApproximant import ROMApproximant
-
class ROMApproximantTaylor(ROMApproximant):
"""
ROM single-point approximant computation for parametric problems.
Args:
HFEngine: HF problem solver. Should have members:
- energyNormMatrix: assemble sparse matrix (in CSC format)
associated to weighted H10 norm;
- problemData: list of HF problem data (excluding k);
- setProblemData: set HF problem data and k to prescribed values;
- - setupDerivativeComputation: setup HF problem data to compute j-th
- solution derivative at k0;
- - solve: get HF solution as complex numpy vector.
+ - getLSBlocks: get algebraic system blocks.
HSEngine: Hilbert space general purpose engine. Should have members:
- norm: compute Hilbert norm of expression represented as complex
numpy vector;
- plot: plot Hilbert expression represented as complex numpy vector.
k0(optional): Default parameter. Defaults to 0.
w(optional): Weight for norm computation. If None, set to Re(k0).
Defaults to None.
approxParameters(optional): Dictionary containing values for main
parameters of approximant. Recognized keys are:
- 'POD': whether to compute POD of snapshots; defaults to True;
- 'E': total number of derivatives current approximant relies upon;
defaults to Emax;
- 'Emax': total number of derivatives of solution map to be
computed; defaults to E;
- 'sampleType': label of sampling type; available values are:
- 'ARNOLDI': orthogonalization of solution derivatives through
Arnoldi algorithm;
- 'KRYLOV': standard computation of solution derivatives.
Defaults to 'KRYLOV'.
Defaults to empty dict.
- plotDer(optional): Whether to plot derivatives of the Helmholtz
- solution map. Defaults to False.
+ plotDer(optional): What to plot of derivatives of the Helmholtz
+ solution map. See plot method in HSEngine. Defaults to [].
Attributes:
HFEngine: HF problem solver.
HSEngine: Hilbert space general purpose engine.
solDerivatives: Array whose columns are FE dofs of solution
derivatives.
k0: Default parameter.
w: Weight for norm computation.
approxParameters: Dictionary containing values for main parameters of
approximant. Recognized keys are:
- 'POD': whether to compute POD of snapshots;
- 'E': total number of derivatives current approximant relies upon;
- 'Emax': total number of derivatives of solution map to be
computed;
- 'sampleType': label of sampling type.
E: Number of solution derivatives over which current approximant is
based upon.
Emax: Total number of solution derivatives to be computed.
sampleType: Label of sampling type.
- plotDer: Whether to plot derivatives of the Helmholtz solution map.
+ plotDer: What to plot of derivatives of the Helmholtz solution map.
initialHFData: HF problem initial data.
energyNormMatrix: Sparse matrix (in CSC format) assoociated to
w-weighted H10 norm.
uHF: High fidelity solution with wavenumber lastSolvedHF as numpy
complex vector.
lastSolvedHF: Wavenumber corresponding to last computed high fidelity
solution.
uApp: Last evaluated approximant as numpy complex vector.
lastApproxParameters: List of parameters corresponding to last
computed approximant.
"""
extraApproxParameters = ["E", "Emax", "sampleType"]
def __init__(self, HFEngine:'HF solver', HSEngine:'HS engine',
k0 : complex = 0, w : float = None,
- approxParameters : dict = {}, plotDer : bool = False):
+ approxParameters : dict = {}, plotDer : list = []):
self.plotDer = plotDer
ROMApproximant.__init__(self, HFEngine, HSEngine,
k0, w, approxParameters)
def parameterList(self) -> list:
"""
List containing keys which are allowed in approxParameters.
Returns:
List of strings.
"""
return (ROMApproximant.parameterList(self)
+ ROMApproximantTaylor.extraApproxParameters)
@property
def approxParameters(self):
"""
Value of approximant parameters. Its assignment may change E and Emax.
"""
return self._approxParameters
@approxParameters.setter
def approxParameters(self, approxParams):
if not hasattr(self, "approxParameters"):
self._approxParameters = {}
approxParameters = utilities.purgeDict(approxParams,
- ROMApproximantTaylor.parameterList(self),
- dictname = "approxParameters")
+ ROMApproximantTaylor.parameterList(self),
+ dictname = self.name() + ".approxParameters")
keyList = list(approxParameters.keys())
approxParametersCopy = utilities.purgeDict(approxParameters,
ROMApproximantTaylor.extraApproxParameters,
True, True)
ROMApproximant.approxParameters.fset(self, approxParametersCopy)
if "E" in keyList:
self._E = approxParameters["E"]
self._approxParameters["E"] = self.E
if "Emax" in keyList:
self.Emax = approxParameters["Emax"]
else:
if not hasattr(self, "Emax"):
self.Emax = self.E
else:
self.Emax = self.Emax
else:
if "Emax" in keyList:
self._E = approxParameters["Emax"]
self._approxParameters["E"] = self.E
self.Emax = self.E
else:
if not (hasattr(self, "Emax") and hasattr(self, "E")):
raise Exception("At least one of E and Emax must be set.")
if "sampleType" in keyList:
self.sampleType = approxParameters["sampleType"]
elif hasattr(self, "sampleType"):
self.sampleType = self.sampleType
else:
self.sampleType = "KRYLOV"
@property
def E(self):
"""Value of E. Its assignment may change Emax."""
return self._E
@E.setter
def E(self, E):
if E < 0: raise ArithmeticError("E must be non-negative.")
self._E = E
self._approxParameters["E"] = self.E
if hasattr(self, "Emax") and self.Emax < self.E:
warnings.warn("Prescribed E is too large. Updating Emax to E.")
self.Emax = self.E
@property
def Emax(self):
"""Value of Emax. Its assignment may reset computed derivatives."""
return self._Emax
@Emax.setter
def Emax(self, Emax):
if Emax < 0: raise ArithmeticError("Emax must be non-negative.")
if hasattr(self, "Emax"): EmaxOld = self.Emax
else: EmaxOld = -1
self._Emax = Emax
if hasattr(self, "E") and self.Emax < self.E:
warnings.warn("Prescribed Emax is too small. Updating Emax to E.")
self.Emax = self.E
else:
self._approxParameters["Emax"] = self.Emax
if EmaxOld >= self.Emax and self.solDerivatives is not None:
self.solDerivatives = self.solDerivatives[:, : self.Emax + 1]
if hasattr(self, "HArnoldi"):
self.HArnoldi = self.HArnoldi[: self.Emax + 1,
: self.Emax + 1]
self.RArnoldi = self.RArnoldi[: self.Emax + 1,
: self.Emax + 1]
else:
self.resetSamples()
@property
def sampleType(self):
"""Value of sampleType."""
return self._sampleType
@sampleType.setter
def sampleType(self, sampleType):
if hasattr(self, "sampleType"): sampleTypeOld = self.sampleType
else: sampleTypeOld = -1
try:
sampleType = sampleType.upper().strip().replace(" ","")
if sampleType not in ["ARNOLDI", "KRYLOV"]: raise
if sampleType == "ARNOLDI" and not self.POD:
warnings.warn(("Prescribed sampleType not compatible with POD "
"value. Overriding to 'KRYLOV'."))
sampleType = "KRYLOV"
+ if sampleType == "ARNOLDI" and self.HFEngine.parDegree > 1:
+ warnings.warn(("Arnoldi sampling not allowed with par degree "
+ "higher than 1. Overriding to 'KRYLOV'."))
+ sampleType = "KRYLOV"
self._sampleType = sampleType
except:
warnings.warn(("Prescribed sampleType not recognized. Overriding "
"to 'KRYLOV'."))
- self.sampleType = "KRYLOV"
+ self._sampleType = "KRYLOV"
self._approxParameters["sampleType"] = self.sampleType
if sampleTypeOld != self.sampleType:
self.resetSamples()
def resetSamples(self):
"""Reset samples."""
self.solDerivatives = None
if hasattr(self, "HArnoldi"):
del self.HArnoldi
if hasattr(self, "RArnoldi"):
del self.RArnoldi
def computeDerivatives(self):
- """
- Compute derivatives of solution map starting from order 0.
- """
+ """Compute derivatives of solution map starting from order 0."""
if self.solDerivatives is None:
- up = np.zeros(self.HSEngine.V.dim())
- upp = up
+ lAs = len(self.As)
+ lbs = len(self.bs)
+ uOlds = [np.zeros(self.HSEngine.V.dim())] * (lAs - 1)
for j in range(self.Emax + 1):
if j == 0:
self.HFEngine.setProblemData(self.initialHFData, self.k0)
+ bs = self.bs
else:
- self.HFEngine.setupDerivativeComputation(j, up, upp)
- uj = self.HFEngine.solve(True)
- if self.plotDer:
- self.HSEngine.plot(uj, name = "u_{0}".format(j))
- upp = up
- up = self.manageDerivatives(uj, j)
+ bs = [np.zeros_like(uOlds[0])] * max(lAs - 1, lbs - j)
+ for ii in range(lbs - j):
+ bs[ii] = self.bs[j + ii]
+ for ii in range(lAs - 1):
+ for jj in range(1, min(j + 1, lAs - ii)):
+ bs[ii] =(bs[ii]
+ - comb(jj + ii, jj, exact = False)
+ * self.As[jj + ii].dot(uOlds[lAs - jj - 1]))
+ A, b = self.buildLS(self.k0, bs = bs)
+ uj = spla.spsolve(A, b)
+ self.HSEngine.plot(uj, name = "u_{0}".format(j),
+ what = self.plotDer)
+ del uOlds[0]
+ uOlds = uOlds + [self.manageDerivatives(uj, j)]
def manageDerivatives(self, u:"numpy 1D array",
pos:int) -> ("numpy 1D array"):
"""
Store derivatives of solution map.
Args:
u: solution derivative as numpy complex vector;
pos: Derivative index.
Returns:
Adjusted solution derivative as numpy complex vector.
"""
if pos == 0:
self.solDerivatives = np.empty((u.shape[0], self.Emax + 1),
dtype = np.complex)
self.solDerivatives[:, pos] = u
if self.sampleType == "ARNOLDI":
if pos == 0:
self.HArnoldi = np.zeros((self.Emax + 1, self.Emax + 1),
dtype = np.complex)
self.RArnoldi = np.zeros((self.Emax + 1, self.Emax + 1),
dtype = np.complex)
self.HArnoldi[: pos, pos] = self.solDerivatives[:, pos].conj().dot(
self.energyNormMatrix.dot(self.solDerivatives[:, : pos])
).conj()
self.solDerivatives[:, pos] = (self.solDerivatives[:, pos]
- self.solDerivatives[:, : pos].dot(
self.HArnoldi[: pos, pos]))
self.HArnoldi[pos, pos] = (self.solDerivatives[:, pos].conj().dot(
self.energyNormMatrix.dot(self.solDerivatives[:, pos])))**.5
self.solDerivatives[:, pos] = (self.solDerivatives[:, pos]
/ self.HArnoldi[pos, pos])
if pos == 0:
self.RArnoldi[pos, pos] = self.HArnoldi[pos, pos]
else:
self.RArnoldi[:pos+1, pos] = self.HArnoldi[: pos+1, 1 : pos+1]\
.dot(self.RArnoldi[: pos, pos - 1])
return self.solDerivatives[:, pos]
diff --git a/main/ROMApproximantTaylorPade.py b/main/ROMApproximantTaylorPade.py
index e5b6d87..eeee6fd 100644
--- a/main/ROMApproximantTaylorPade.py
+++ b/main/ROMApproximantTaylorPade.py
@@ -1,552 +1,552 @@
#!/usr/bin/python
from __future__ import print_function
from copy import copy
import warnings
import numpy as np
import utilities
from ROMApproximantTaylor import ROMApproximantTaylor
PI = np.pi
class ROMApproximantTaylorPade(ROMApproximantTaylor):
"""
ROM single-point fast Pade' approximant computation for parametric
problems.
Args:
HFEngine: HF problem solver. Should have members:
- energyNormMatrix: assemble sparse matrix (in CSC format)
associated to weighted H10 norm;
- problemData: list of HF problem data (excluding k);
- setProblemData: set HF problem data and k to prescribed values;
- setupDerivativeComputation: setup HF problem data to compute j-th
solution derivative at k0;
- solve: get HF solution as complex numpy vector.
HSEngine: Hilbert space general purpose engine. Should have members:
- norm: compute Hilbert norm of expression represented as complex
numpy vector;
- plot: plot Hilbert expression represented as complex numpy vector.
k0(optional): Default parameter. Defaults to 0.
w(optional): Weight for norm computation. If None, set to Re(k0).
Defaults to None.
approxParameters(optional): Dictionary containing values for main
parameters of approximant. Recognized keys are:
- 'POD': whether to compute POD of snapshots; defaults to True;
- 'rho': weight for computation of original Pade' approximant;
defaults to inf, i.e. fast approximant;
- 'M': degree of Pade' approximant numerator; defaults to 0;
- 'N': degree of Pade' approximant denominator; defaults to 0;
- 'E': total number of derivatives current approximant relies upon;
defaults to Emax;
- 'Emax': total number of derivatives of solution map to be
computed; defaults to E;
- 'robustTol': tolerance for robust Pade' denominator management;
defaults to 0;
- 'sampleType': label of sampling type; available values are:
- 'ARNOLDI': orthogonalization of solution derivatives through
Arnoldi algorithm;
- 'KRYLOV': standard computation of solution derivatives.
Defaults to 'KRYLOV'.
Defaults to empty dict.
plotDer(optional): Whether to plot derivatives of the Helmholtz
solution map. Defaults to False.
equilibration(optional): Whether to apply equilibration to Gram matrix.
Defaults to False.
verboseRobust(optional): Verbosity level for robustness-related
parameter modifications. Defaults to 1.
cleanupParameters(optional): Parameter values for pole cleanup.
Available fields are:
- 'boolCondition'(bool function handle): if evaluation on pole
returns False, pole is removed; defaults to always True;
- 'residueCheck'(bool): whether to apply residue check; defaults to
False;
- 'residueNPoints'(int): number of sample points to be used in
residue estimation; defaults to 2;
- 'residueRadius'(float): sample radius to be used in residue
estimation by Cauchy formula; defaults to 1e-5;
- 'residueTol'(float): target tolerance to be used in residue
estimation; defaults to 1e-4.
Attributes:
HFEngine: HF problem solver.
HSEngine: Hilbert space general purpose engine.
solDerivatives: Array whose columns are FE dofs of solution
derivatives.
k0: Default parameter.
w: Weight for norm computation.
approxParameters: Dictionary containing values for main parameters of
approximant. Recognized keys are:
- 'POD': whether to compute POD of snapshots;
- 'rho': weight for computation of original Pade' approximant;
defaults to inf, i.e. fast approximant;
- 'M': degree of Pade' approximant numerator;
- 'N': degree of Pade' approximant denominator;
- 'E': total number of derivatives current approximant relies upon;
- 'Emax': total number of derivatives of solution map to be
computed;
- 'robustTol': tolerance for robust Pade' denominator management;
- 'sampleType': label of sampling type.
M: Numerator degree of approximant.
N: Denominator degree of approximant.
E: Number of solution derivatives over which current approximant is
based upon.
Emax: Total number of solution derivatives to be computed.
robustTol: tolerance for robust Pade' denominator management.
sampleType: Label of sampling type.
plotDer: Whether to plot derivatives of the Helmholtz solution map.
initialHFData: HF problem initial data.
equilibration: Whether to apply equilibration to Gram matrix.
verboseRobust: Verbosity level for robustness-related parameter
modifications.
cleanupParameters; Parameter values for pole cleanup. Available fields
are:
- 'boolCondition': if evaluation on pole returns False, pole is
removed;
- 'residueCheck': whether to apply residue check;
- 'residueNPoints': number of sample points to be used in residue
estimation;
- 'residueRadius': sample radius to be used in residue estimation
by Cauchy formula;
- 'residueTol': target tolerance to be used in residue estimation.
energyNormMatrix: Sparse matrix (in CSC format) assoociated to
w-weighted H10 norm.
uHF: High fidelity solution with wavenumber lastSolvedHF as numpy
complex vector.
lastSolvedHF: Wavenumber corresponding to last computed high fidelity
solution.
G: Square Numpy 2D vector of size (N+1) corresponding to Pade'
denominator matrix (see paper).
equilPowers: Numpy 1D (column) array containing equilibration powers.
Does not exist if equilibration is disabled.
Q: Numpy 1D vector containing complex coefficients of approximant
denominator.
P: Numpy 2D vector whose columns are FE dofs of coefficients of
approximant numerator.
uApp: Last evaluated approximant as numpy complex vector.
lastApproxParameters: List of parameters corresponding to last
computed approximant.
"""
extraApproxParameters = ["M", "N", "robustTol", "rho"]
def __init__(self, HFEngine:'HF solver', HSEngine:'HS engine',
k0 : complex = 0, w : float = None,
- approxParameters : dict = {}, plotDer : bool = False,
+ approxParameters : dict = {}, plotDer : list = [],
equilibration : bool = False, verboseRobust : int = 1,
cleanupParameters : dict = {}):
self.equilibration = equilibration
self.verboseRobust = verboseRobust
self.cleanupParameters = cleanupParameters
ROMApproximantTaylor.__init__(self, HFEngine, HSEngine, k0, w,
approxParameters, plotDer)
def parameterList(self) -> list:
"""
List containing keys which are allowed in approxParameters.
Returns:
List of strings.
"""
return (ROMApproximantTaylor.parameterList(self)
+ ROMApproximantTaylorPade.extraApproxParameters)
@property
def approxParameters(self):
"""
Value of approximant parameters. Its assignment may change M, N, E,
Emax and robustTol.
"""
return self._approxParameters
@approxParameters.setter
def approxParameters(self, approxParams):
if not hasattr(self, "approxParameters"):
self._approxParameters = {}
approxParameters = utilities.purgeDict(approxParams,
ROMApproximantTaylorPade.parameterList(self),
- dictname = "approxParameters")
+ dictname = self.name() + ".approxParameters")
keyList = list(approxParameters.keys())
if "M" in keyList:
self.M = 0 #to avoid warnings if M and E are changed simultaneously
if "N" in keyList:
self.N = 0 #to avoid warnings if N and E are changed simultaneously
approxParametersCopy = utilities.purgeDict(approxParameters,
ROMApproximantTaylorPade.extraApproxParameters,
True, True)
ROMApproximantTaylor.approxParameters.fset(self, approxParametersCopy)
if "rho" in keyList:
self.rho = approxParameters["rho"]
elif hasattr(self, "rho"):
self.rho = self.rho
else:
self.rho = np.inf
if "M" in keyList:
self.M = approxParameters["M"]
elif hasattr(self, "M"):
self.M = self.M
else:
self.M = 0
if "N" in keyList:
self.N = approxParameters["N"]
elif hasattr(self, "N"):
self.N = self.N
else:
self.N = 0
if "robustTol" in keyList:
self.robustTol = approxParameters["robustTol"]
elif hasattr(self, "robustTol"):
self.robustTol = self.robustTol
else:
self.robustTol = 0
@property
def rho(self):
"""Value of rho."""
return self._rho
@rho.setter
def rho(self, rho):
self._rho = np.abs(rho)
self._approxParameters["rho"] = self.rho
@property
def M(self):
"""Value of M. Its assignment may change Emax and E."""
return self._M
@M.setter
def M(self, M):
if M < 0: raise ArithmeticError("M must be non-negative.")
self._M = M
self._approxParameters["M"] = self.M
if hasattr(self, "Emax") and self.Emax < self.M:
warnings.warn("Prescribed Emax is too small. Updating Emax to M.")
self.Emax = self.M
if hasattr(self, "E") and self.E < self.M:
warnings.warn("Prescribed E is too small. Updating E to M.")
self.E = self.M
@property
def N(self):
"""Value of N. Its assignment may change Emax."""
return self._N
@N.setter
def N(self, N):
if N < 0: raise ArithmeticError("N must be non-negative.")
self._N = N
self._approxParameters["N"] = self.N
if hasattr(self, "Emax") and self.Emax < self.N:
warnings.warn("Prescribed Emax is too small. Updating Emax to N.")
self.Emax = self.N
if hasattr(self, "E") and self.E < self.N:
warnings.warn("Prescribed E is too small. Updating E to N.")
self.E = self.N
@property
def robustTol(self):
"""Value of tolerance for robust Pade' denominator management."""
return self._robustTol
@robustTol.setter
def robustTol(self, robustTol):
if robustTol < 0.:
warnings.warn(("Overriding prescribed negative robustness "
"tolerance to 0."))
robustTol = 0.
self._robustTol = robustTol
self._approxParameters["robustTol"] = self.robustTol
@property
def E(self):
"""Value of E. Its assignment may change Emax."""
return self._E
@E.setter
def E(self, E):
if E < 0: raise ArithmeticError("E must be non-negative.")
self._E = E
if hasattr(self, "M") and self.M > self.E:
warnings.warn("Prescribed E is too small. Updating E to M.")
self.E = self.M
if hasattr(self, "N") and self.N > self.E:
warnings.warn("Prescribed E is too small. Updating E to N.")
self.E = self.N
self._approxParameters["E"] = self.E
if hasattr(self, "Emax") and self.Emax < self.E:
warnings.warn("Prescribed Emax is too small. Updating Emax to E.")
self.Emax = self.E
@property
def Emax(self):
"""Value of Emax. Its assignment may reset computed derivatives."""
return self._Emax
@Emax.setter
def Emax(self, Emax):
if Emax < 0: raise ArithmeticError("Emax must be non-negative.")
if hasattr(self, "Emax"): EmaxOld = self.Emax
else: EmaxOld = -1
vals, label = [0] * 3, {0:"E", 1:"M", 2:"N"}
if hasattr(self, "E"): vals[0] = self.E
if hasattr(self, "M"): vals[1] = self.M
if hasattr(self, "N"): vals[2] = self.N
idxmax = np.argmax(vals)
if vals[idxmax] > Emax:
warnings.warn("Prescribed Emax is too small. Updating Emax to {}."\
.format(label[idxmax]))
self.Emax = vals[idxmax]
else:
self._Emax = Emax
self._approxParameters["Emax"] = Emax
if (EmaxOld > self.Emax and self.solDerivatives is not None
and self.HArnoldi is not None and self.RArnoldi is not None):
self.solDerivatives = self.solDerivatives[:, : self.Emax + 1]
if self.sampleType == "ARNOLDI":
self.HArnoldi = self.HArnoldi[: self.Emax + 1,
: self.Emax + 1]
self.RArnoldi = self.RArnoldi[: self.Emax + 1,
: self.Emax + 1]
else:
self.resetSamples()
@property
def cleanupParameters(self):
"""Value of cleanupParameters."""
return self._cleanupParameters
@cleanupParameters.setter
def cleanupParameters(self, cleanupParameters):
allowedCleanupKeys = ['boolCondition','residueCheck','residueNPoints',
'residueRadius','residueTol']
cleanupKeys = cleanupParameters.keys()
if 'boolCondition' not in cleanupKeys:
cleanupParameters['boolCondition'] = lambda x : True
if 'residueCheck' not in cleanupKeys:
cleanupParameters['residueCheck'] = False
if 'residueNPoints' not in cleanupKeys:
cleanupParameters['residueNPoints'] = 2
if 'residueRadius' not in cleanupKeys:
cleanupParameters['residueRadius'] = 1e-5
if 'residueTol' not in cleanupKeys:
cleanupParameters['residueTol'] = 1e-4
cleanupParameters['boolCondition'] = np.vectorize(
cleanupParameters['boolCondition'])
self._cleanupParameters = {key : cleanupParameters[key]
for key in allowedCleanupKeys}
def setupApprox(self):
"""
Compute Pade' approximant.
See Section 4 in Fast Pade' paper.
SVD-based robust eigenvalue management.
"""
self.computeDerivatives()
if not self.checkComputedApprox():
while True:
if self.POD:
if self.sampleType == "ARNOLDI":
ev, eV = self.findeveVGArnoldi()
else:
ev, eV = self.findeveVGQR()
else:
self.buildG()
ev, eV = np.linalg.eigh(self.G)
ts = self.robustTol * np.linalg.norm(ev)
Nnew = np.sum(np.abs(ev) >= ts)
diff = self.N - Nnew
if diff <= 0: break
Enew = self.E - diff
Mnew = min(self.M, Enew)
if Mnew == self.M:
strM = ""
else:
strM = ", M from {0} to {1},".format(self.M, Mnew)
if self.verboseRobust >= 1:
print(("Smallest {0} eigenvalues below tolerance.\n"
"Reducing N from {1} to {2}{5} and E from {3} to "
"{4}.").format(diff + 1, self.N, Nnew,
self.E, Enew, strM))
newParameters = {"N" : Nnew, "M" : Mnew, "E" : Enew}
self.approxParameters = newParameters
self.Q = eV[::-1, 0]
if self.equilibration:
self.Q = np.multiply(self.equilPowers.flatten()[::-1], self.Q)
self._cleanup()
QToeplitz = np.zeros((self.Emax + 1, self.M + 1),
dtype = np.complex)
for i in range(len(self.Q)):
rng = np.arange(self.M + 1 - i)
QToeplitz[rng, rng + i] = self.Q[i]
if self.sampleType == "ARNOLDI":
QToeplitz = self.RArnoldi.dot(QToeplitz)
self.P = self.solDerivatives.dot(QToeplitz)
self.lastApproxParameters = copy(self.approxParameters)
def buildG(self):
"""Assemble Pade' denominator matrix."""
if self.N == 0:
self.G = np.array([[1]], dtype = np.complex)
return
self.computeDerivatives()
if np.isinf(self.rho):
lgSize = self.N + 1
else:
lgSize = self.E + self.N - self.M
Gshift = self.E - lgSize + 1
largeG = np.zeros((lgSize, lgSize), dtype=np.complex)
largeG = self.solDerivatives[:, Gshift:self.E+1].conj().T.dot(
self.energyNormMatrix.dot(self.solDerivatives[:, Gshift:self.E+1]))
if np.isinf(self.rho):
self.G = largeG
else:
self.G = np.zeros((self.N + 1, self.N + 1), dtype=np.complex)
for k in range(self.E - self.M):
self.G = (self.G + self.rho ** (2 * k)
* largeG[k : k+self.N+1, k : k+self.N+1])
if self.equilibration:
Gd = np.diag(self.G)
gamma = np.average(np.abs(np.divide(Gd[1:], Gd[:-1])),
weights = np.power(1.2, np.arange(self.N)))**.5
self.equilPowers = np.power(gamma, np.arange(self.N + 1)[:, None])
self.G = np.multiply(self.equilPowers,
np.multiply(self.equilPowers, self.G).T).T
def _cleanup(self):
"""Cleanup Pade' denominator by removing unwanted poles."""
if self.N == 0: return
poles = np.roots(self.Q[::-1]) + self.k0
NpolesOld = len(poles)
poles = poles[self.cleanupParameters['boolCondition'](poles)]
if self.cleanupParameters['residueCheck']:
resR = self.cleanupParameters['residueRadius']
resTol = self.cleanupParameters['residueTol']
residues = np.zeros_like(poles)
for l, pole in enumerate(poles):
resV = np.zeros(self.solDerivatives.shape[0],
dtype = np.complex)
NPoints = self.cleanupParameters['residueNPoints']
for theta in 2 * PI * np.arange(NPoints) / NPoints:
deltapole = resR * np.exp(1.j * theta)
ksample = pole + deltapole
self.solveHF(ksample)
resV = deltapole / NPoints * (resV + self.uHF)
residues[l] = np.abs(resV.dot(
self.energyNormMatrix.dot(resV).conj())) ** .5
poles = poles[residues >= resTol]
NpolesNew = len(poles)
if NpolesOld > NpolesNew:
if self.verboseRobust >= 1:
sSing = "s" * (NpolesOld - NpolesNew > 1)
print(("Identified {0} pole{1} to be removed.\n"
"Reducing N from {2} to {3}.").format(
NpolesOld - NpolesNew, sSing, NpolesOld, NpolesNew))
self.N = NpolesNew
newQ = np.polyfit(np.append(poles - self.k0, 0.),
np.append(np.zeros(NpolesNew), 1.), NpolesNew)
self.Q = newQ[::-1] / np.linalg.norm(newQ)
def findeveVGQR(self):
"""
Compute eigenvalues and eigenvectors of Pade' denominator matrix
through SVD of R factor. See ``Householder triangularization of a
quasimatrix'', L.Trefethen, 2008 for QR algorithm.
Returns:
Eigenvalues in ascending order and corresponding eigenvector
matrix.
"""
self.computeDerivatives()
A = copy(self.solDerivatives[:, (self.E - self.N) : (self.E + 1)])
M = self.energyNormMatrix
E = np.zeros_like(A, dtype = np.complex)
R = np.zeros((self.N + 1, self.N + 1), dtype = np.complex)
for k in range(self.N + 1):
E[:, k] = (np.random.randn(E.shape[0])
+ 1.j * np.random.randn(E.shape[0]))
if k > 0:
for times in range(2):
E[:, k] = E[:, k] - E[:, :k].dot(
(E[:, k].conj().dot(M.dot(E[:, :k]))).conj())
E[:, k] = E[:, k] / (E[:, k].conj().dot(M.dot(E[:, k]))) ** .5
R[k, k] = np.abs(A[:, k].conj().dot(M.dot(A[:, k]))) ** .5
alpha = E[:, k].conj().dot(M.dot(A[:, k]))
if np.isclose(np.abs(alpha), 0.): s = 1.
else: s = - alpha / np.abs(alpha)
E[:, k] = s * E[:, k]
v = R[k, k] * E[:, k] - A[:, k]
for times in range(2):
v = v - E[:, :k].dot((v.conj().dot(M.dot(E[:, :k]))).conj())
sigma = np.abs(v.conj().dot(M.dot(v))) ** .5
if np.isclose(sigma, 0.): v = E[:, k]
else: v = v / sigma
J = np.arange(k + 1, self.N + 1)
vtQ = v.conj().dot(M.dot(A[:, J]))
A[:, J] = A[:, J] - 2 * np.outer(v, vtQ)
R[k, J] = E[:, k].conj().dot(M.dot(A[:, J]))
A[:, J] = A[:, J] - np.outer(E[:, k], R[k, J])
if self.equilibration:
Rd = np.diag(R)
gamma = np.average(np.abs(np.divide(Rd[1:], Rd[:-1])),
weights = np.power(1.2, np.arange(self.N)))**.5
self.equilPowers = np.power(gamma, np.arange(self.N + 1)[:, None])
R = np.multiply(self.equilPowers, R.T).T
_, s, V = np.linalg.svd(R, full_matrices = False)
return s[::-1], V.conj().T[:, ::-1]
def findeveVGArnoldi(self):
"""
Compute eigenvalues and eigenvectors of Pade' denominator matrix
through SVD of R factor of solution derivatives orthogonalized by
Arnoldi algorithm.
Returns:
Eigenvalues in ascending order and corresponding eigenvector
matrix.
"""
self.computeDerivatives()
if self.sampleType != "ARNOLDI":
raise Exception(("Eigensolver different from 'ARNOLDI'."
"Arnoldi eigenproblem solver cannot be called."))
R = self.RArnoldi[: self.E + 1, self.E - self.N : self.E + 1]
if self.equilibration:
Rd = np.diag(R)
gamma = np.average(np.abs(np.divide(Rd[1:], Rd[:-1])),
weights = np.power(1.2, np.arange(self.N)))**.5
self.equilPowers = np.power(gamma, np.arange(self.N + 1)[:, None])
R = np.multiply(self.equilPowers, R.T).T
_, s, V = np.linalg.svd(R, full_matrices = False)
return s[::-1], V.conj().T[:, ::-1]
def evalApprox(self, k:complex) -> ("Fenics function", "Fenics function"):
"""
Evaluate Pade' approximant at arbitrary wavenumber.
Args:
k: Target wavenumber.
Returns:
Approximant as numpy complex vector.
"""
self.setupApprox()
powerlist = np.power(k - self.k0, range(max(self.M, self.N) + 1))
self.uApp = (self.P.dot(powerlist[:self.M + 1])
/ self.Q.dot(powerlist[:self.N + 1]))
return self.uApp
def getPoles(self, centered : bool = False) -> "numpy 1D array":
"""
Obtain approximant poles.
Args:
centered(optional): Whether to return pole positions relative to
approximation center. Defaults to False.
Returns:
Numpy complex vector of poles.
"""
self.setupApprox()
return np.roots(self.Q[::-1]) + self.k0 * centered
\ No newline at end of file
diff --git a/main/ROMApproximantTaylorRB.py b/main/ROMApproximantTaylorRB.py
index f3898ec..195e2f4 100644
--- a/main/ROMApproximantTaylorRB.py
+++ b/main/ROMApproximantTaylorRB.py
@@ -1,308 +1,311 @@
#!/usr/bin/python
from copy import copy
import warnings
import numpy as np
import scipy as sp
import utilities
from ROMApproximantTaylor import ROMApproximantTaylor
class ROMApproximantTaylorRB(ROMApproximantTaylor):
"""
ROM single-point fast RB approximant computation for parametric problems
with polynomial dependence up to degree 2.
Args:
HFEngine: HF problem solver. Should have members:
- energyNormMatrix: sparse matrix (in CSC format) associated to
w-weighted H10 norm;
- problemData: list of HF problem data (excluding k);
- setProblemData: set HF problem data and k to prescribed values;
- getLSBlocks: get blocks of HF linear system as sparse matrices in
CSC format;
- liftDirichletData: perform lifting of Dirichlet BC as numpy
complex vector;
- setupDerivativeComputation: setup HF problem data to compute j-th
solution derivative at k0;
- solve: get HF solution as complex numpy vector.
HSEngine: Hilbert space general purpose engine. Should have members:
- norm: compute Hilbert norm of expression represented as complex
numpy vector;
- plot: plot Hilbert expression represented as complex numpy vector.
k0(optional): Default parameter. Defaults to 0.
w(optional): Weight for norm computation. If None, set to Re(k0).
Defaults to None.
approxParameters(optional): Dictionary containing values for main
parameters of approximant. Recognized keys are:
- 'POD': whether to compute POD of snapshots; defaults to True;
- 'R': rank for Galerkin projection; defaults to E + 1;
- 'E': total number of derivatives current approximant relies upon;
defaults to Emax;
- 'Emax': total number of derivatives of solution map to be
computed; defaults to E;
- 'sampleType': label of sampling type; available values are:
- 'ARNOLDI': orthogonalization of solution derivatives through
Arnoldi algorithm;
- 'KRYLOV': standard computation of solution derivatives.
Defaults to 'KRYLOV'.
Defaults to empty dict.
plotDer(optional): Whether to plot derivatives of the Helmholtz
solution map. Defaults to False.
Attributes:
HFEngine: HF problem solver.
HSEngine: Hilbert space general purpose engine.
solDerivatives: Array whose columns are FE dofs of solution
derivatives.
k0: Default parameter.
w: Weight for norm computation.
approxParameters: Dictionary containing values for main parameters of
approximant. Recognized keys are:
- 'POD': whether to compute POD of snapshots;
- 'R': rank for Galerkin projection;
- 'E': total number of derivatives current approximant relies upon;
- 'Emax': total number of derivatives of solution map to be
computed;
- 'sampleType': label of sampling type.
R: Rank for Galerkin projection.
E: Number of solution derivatives over which current approximant is
based upon.
Emax: Total number of solution derivatives to be computed.
sampleType: Label of sampling type, i.e. 'KRYLOV'.
plotDer: Whether to plot derivatives of the Helmholtz solution map.
energyNormMatrix: Sparse matrix (in CSC format) associated to
w-weighted H10 norm.
uHF: High fidelity solution with wavenumber lastSolvedHF as numpy
complex vector.
lastSolvedHF: Wavenumber corresponding to last computed high fidelity
solution.
uApp: Last evaluated approximant as numpy complex vector.
lastApproxParameters: List of parameters corresponding to last
computed approximant.
solLifting: Numpy complex vector with lifting of real part of Dirichlet
boundary datum.
projMat: Numpy matrix representing projection onto RB space.
projMat: Numpy matrix representing projection onto RB space.
As: List of sparse matrices (in CSC format) representing coefficients
of linear system matrix wrt k.
bs: List of numpy vectors representing coefficients of linear system
RHS wrt k.
ARBs: List of sparse matrices (in CSC format) representing RB
coefficients of linear system matrix wrt k.
bRBs: List of numpy vectors representing RB coefficients of linear
system RHS wrt k.
"""
extraApproxParameters = ["R"]
def __init__(self, HFEngine:'HF solver', HSEngine:'HS engine',
k0 : complex = 0, w : float = None,
- approxParameters : dict = {}, plotDer : bool = False):
+ approxParameters : dict = {}, plotDer : list = []):
ROMApproximantTaylor.__init__(self, HFEngine, HSEngine, k0, w,
approxParameters, plotDer)
self.solLifting = self.HFEngine.liftDirichletData()
def resetSamples(self):
"""Reset samples."""
ROMApproximantTaylor.resetSamples(self)
self.projMat = None
def parameterList(self) -> list:
"""
List containing keys which are allowed in approxParameters.
Returns:
List of strings.
"""
return (ROMApproximantTaylor.parameterList(self)
+ ROMApproximantTaylorRB.extraApproxParameters)
@property
def approxParameters(self):
"""
Value of approximant parameters. Its assignment may change M, N and S.
"""
return self._approxParameters
@approxParameters.setter
def approxParameters(self, approxParams):
if not hasattr(self, "approxParameters"):
self._approxParameters = {}
approxParameters = utilities.purgeDict(approxParams,
- ROMApproximantTaylorRB.parameterList(self),
- dictname = "approxParameters")
+ ROMApproximantTaylorRB.parameterList(self),
+ dictname = self.name() + ".approxParameters")
keyList = list(approxParameters.keys())
approxParametersCopy = utilities.purgeDict(approxParameters,
ROMApproximantTaylorRB.extraApproxParameters,
True, True)
ROMApproximantTaylor.approxParameters.fset(self, approxParametersCopy)
if "R" in keyList:
self.R = approxParameters["R"]
elif hasattr(self, "R"):
self.R = self.R
else:
self.R = self.E + 1
@property
def R(self):
"""Value of R. Its assignment may change S."""
return self._R
@R.setter
def R(self, R):
if R < 0: raise ArithmeticError("R must be non-negative.")
self._R = R
self._approxParameters["R"] = self.R
if hasattr(self, "E") and self.E + 1 < self.R:
warnings.warn("Prescribed E is too small. Updating E to R - 1.")
self.E = self.R - 1
def manageDerivatives(self, u:"2-tuple of Fenics function", pos:int)\
-> ("Fenics function", "Fenics function"):
"""
Store derivatives of solution map. Subtract lifting for order 0.
Args:
u: 2-tuple containing real and imaginary parts of FE dofs of
derivative.
pos: Derivative index.
Returns:
Real and imaginary parts of derivative (possibly adjusted).
"""
if pos == 0:
- self.As, self.bs = self.HFEngine.getLSBlocks()
+ #self.As, self.bs = self.HFEngine.getLSBlocks()
u = u - self.solLifting
- return ROMApproximantTaylor.manageDerivatives(self, u, pos)
+ v = ROMApproximantTaylor.manageDerivatives(self, u, pos)
+ if pos == 0:
+ return v + self.solLifting
+ return v
def setupApprox(self):
"""
Setup RB system. For usage of correlation matrix in POD see Section
6.3.1 in 'Reduced Basis Methods for PDEs. An introduction', A.
Quarteroni, A. Manzoni, F. Negri, Springer, 2016.
"""
need2Setup = (self.solDerivatives is None) or (self.projMat is None)
self.computeDerivatives()
if need2Setup:
if self.POD and not self.sampleType == "ARNOLDI":
A = copy(self.solDerivatives)
M = self.energyNormMatrix
V = np.zeros_like(A, dtype = np.complex)
Q = np.zeros_like(A, dtype = np.complex)
R = np.zeros((A.shape[1], A.shape[1]), dtype = np.complex)
for k in range(A.shape[1]):
Q[:, k] = (np.random.randn(Q.shape[0])
+ 1.j * np.random.randn(Q.shape[0]))
if k > 0:
for times in range(2):
Q[:, k] = Q[:, k] - Q[:, :k].dot(
(Q[:, k].conj().dot(M.dot(Q[:, :k]))).conj())
Q[:, k] = Q[:, k]/(Q[:, k].conj().dot(M.dot(Q[:, k])))**.5
R[k, k] = np.abs(A[:, k].conj().dot(M.dot(A[:, k]))) ** .5
alpha = Q[:, k].conj().dot(M.dot(A[:, k]))
if np.isclose(np.abs(alpha), 0.): s = 1.
else: s = - alpha / np.abs(alpha)
Q[:, k] = s * Q[:, k]
v = R[k, k] * Q[:, k] - A[:, k]
for times in range(2):
v = v - Q[:, :k].dot((v.conj().dot(M.dot(Q[:, :k]))
).conj())
sigma = np.abs(v.conj().dot(M.dot(v))) ** .5
if np.isclose(sigma, 0.): v = Q[:, k]
else: v = v / sigma
V[:, k] = v
J = np.arange(k + 1, A.shape[1])
vtQ = v.conj().dot(M.dot(A[:, J]))
A[:, J] = A[:, J] - 2 * np.outer(v, vtQ)
R[k, J] = Q[:, k].conj().dot(M.dot(A[:, J]))
A[:, J] = A[:, J] - np.outer(Q[:, k], R[k, J])
for k in range(A.shape[1] - 1, -1, -1):
v = V[:, k]
J = np.arange(k, A.shape[1])
vtQ = v.conj().dot(M.dot(Q[:, J]))
Q[:, J] = Q[:, J] - 2 * np.outer(v, vtQ)
self.projMatQ = Q
self.projMatR = R
if self.POD and not self.sampleType == "ARNOLDI":
U, _, _ = np.linalg.svd(self.projMatR[: self.R, : self.R])
self.projMat = self.projMatQ[:, : self.R].dot(U)
else:
self.projMat = self.solDerivatives[:, : self.R]
self.assembleReducedSystem()
self.lastApproxParameters = copy(self.approxParameters)
def assembleReducedSystem(self):
"""Build affine blocks of RB linear system through projections."""
projMatH = self.projMat.T.conjugate()
self.ARBs = [None] * len(self.As)
self.bRBs = [None] * max(len(self.As), len(self.bs))
for j in range(len(self.As)):
self.ARBs[j] = projMatH.dot(self.As[j].dot(self.projMat))
if j < len(self.bs):
self.bRBs[j] = projMatH.dot(self.bs[j]
- self.As[j].dot(self.solLifting))
else:
self.bRBs[j] = - projMatH.dot(self.As[j].dot(self.solLifting))
for j in range(len(self.As), len(self.bs)):
self.bRBs[j] = projMatH.dot(self.bs[j])
def solveReducedSystem(self, k:complex) -> "Numpy 1D array":
"""
Solve RB linear system.
Args:
k: Target wavenumber.
Returns:
Solution of RB linear system.
"""
self.setupApprox()
ARBk = self.ARBs[0][: self.R, : self.R]
bRBk = self.bRBs[0][: self.R]
for j in range(1, len(self.ARBs)):
ARBk = ARBk + np.power(k, j) * self.ARBs[j][:self.R, :self.R]
for j in range(1, len(self.bRBs)):
bRBk = bRBk + np.power(k, j) * self.bRBs[j][:self.R]
return self.projMat[:, :self.R].dot(np.linalg.solve(ARBk, bRBk))
def evalApprox(self, k:complex) -> ("Fenics function", "Fenics function"):
"""
Evaluate RB approximant at arbitrary wavenumber.
Args:
k: Target wavenumber.
Returns:
Real and imaginary parts of approximant.
"""
self.setupApprox()
self.uApp = self.solLifting + self.solveReducedSystem(k)
return self.uApp
def getPoles(self, centered : bool = False) -> "numpy 1D array":
"""
Obtain approximant poles.
Args:
centered(optional): Whether to return pole positions relative to
approximation center. Defaults to False.
Returns:
Numpy complex vector of poles.
"""
self.setupApprox()
A = np.eye(self.ARBs[0].shape[0] * (len(self.ARBs) - 1),
dtype = np.complex)
B = np.zeros_like(A)
A[: self.ARBs[0].shape[0], : self.ARBs[0].shape[0]] = - self.ARBs[0]
for j in range(len(self.ARBs) - 1):
Aj = self.ARBs[j + 1]
B[: Aj.shape[0], j * Aj.shape[0] : (j + 1) * Aj.shape[0]] = Aj
II = np.arange(self.ARBs[0].shape[0],
self.ARBs[0].shape[0] * (len(self.ARBs) - 1))
B[II, II - self.ARBs[0].shape[0]] = 1.
try:
return sp.linalg.eigvals(A, B) - self.k0 * (not centered)
except:
warnings.warn("Generalized eig algorithm did not converge.")
x = np.empty(A.shape[0])
x[:] = np.NaN
return x
diff --git a/main/utilities.py b/main/utilities.py
index f405d07..c6465d4 100644
--- a/main/utilities.py
+++ b/main/utilities.py
@@ -1,90 +1,151 @@
#!/usr/bin/python
import warnings
import numpy as np
#from copy import copy
def findDictStrKey(key, keyList):
for akey in keyList:
if isinstance(key, str) and key.lower() == akey.lower():
return akey
return None
def purgeList(lst:list, allowedEntries : list = [], silent : bool = False,
complement : bool = False, listname : str = ""):
if listname != "":
listname = " in " + listname
- allowedEntriesSet = frozenset(allowedEntries)
lstcp = []
for x in lst:
- if (x in allowedEntriesSet) != complement:
- lstcp = lstcp + [x]
- elif not silent:
- warnings.warn("Ignoring entry {0}{1}.".format(x, listname))
+ ax = findDictStrKey(x, allowedEntries)
+ if (ax is None) != complement:
+ if not silent:
+ warnings.warn("Ignoring entry {0}{1}.".format(x, listname))
+ else:
+ lstcp = lstcp + [ax]
return lstcp
def purgeDict(dct:dict, allowedKeys : list = [], silent : bool = False,
complement : bool = False, dictname : str = ""):
if dictname != "":
dictname = " in " + dictname
dctcp = {}
for key in dct.keys():
akey = findDictStrKey(key, allowedKeys)
if (akey is None) != complement:
if not silent:
warnings.warn("Ignoring key {0}{2} with value {1}."\
.format(key, dct[key], dictname))
else:
if akey is None:
akey = key
dctcp[akey] = dct[key]
return dctcp
prime_v = [] #memoization vector
def squareResonances(a:int, b:int, zero_terms : bool = True):
spectrum = []
a = max(int(np.floor(a)), 0)
b = max(int(np.ceil(b)), 0)
global prime_v
if len(prime_v) == 0:
prime_v = [2, 3]
if a > prime_v[-1]:
for i in range(prime_v[-1], a, 2):
get_next_prime_factor(i)
for i in range(a, b + 1):
spectrum = spectrum + [i] * count_square_sums(i, zero_terms)
return np.array(spectrum)
def get_next_prime_factor(n):
global prime_v
for x in prime_v:
if n % x == 0:
return x
if prime_v[-1] < n:
prime_v = prime_v + [n]
return n
def prime_factorize(n):
factors = []
number = n
while number > 1:
factor = get_next_prime_factor(number)
factors.append(factor)
number = int(number / factor)
if n < -1:
factors[0] = -factors[0]
return list(factors)
def count_square_sums(n, zero_terms : bool = True):
if n < 2: return (n + 1) * zero_terms
factors = prime_factorize(n)
funique, fcounts = np.unique(factors, return_counts = True)
count = 1
for fac, con in zip(funique, fcounts): #using number theory magic
if fac % 4 == 1:
count = count * (con + 1)
elif fac % 4 == 3 and con % 2 == 1:
return 0
return count + (2 * zero_terms - 1) * (int(n ** .5) ** 2 == n)
+
+def listify(u:"numpy 1D array or list", d:int):
+ """
+ Convert numpy array into list-like format.
+
+ Args:
+ u: numpy complex array with function dofs or list of such.
+ d: secondary dimension.
+
+ Returns:
+ List-like numpy array of numpy arrays.
+ """
+ if isinstance(u, (list,)) or len(u) == d:
+ if len(u) != d:
+ raise Exception(("Input list lenght must be equal to "
+ "augmentation dimension d."))
+ N = len(u[0])
+ for x in u[1 :]:
+ if len(x) != N:
+ raise Exception(("Elements of input list must all have "
+ "the same lenght."))
+ return np.array(u)
+ else:
+ N = int(len(u) / d)
+ if not np.isclose(len(u), d * N):
+ raise Exception(("Input vector lenght must be multiple of"
+ "augmentation dimension d."))
+ v = [None] * N
+ for i in range(N):
+ v[i] = u[i * N : (i + 1) * N]
+ return np.array(v)
+
+def vectorify(u:"numpy 1D array or list", d:int):
+ """
+ Convert list-like object into numpy array.
+
+ Args:
+ u: numpy complex array with function dofs or list of such.
+ d: secondary dimension.
+
+ Returns:
+ Numpy arrays.
+ """
+ if isinstance(u, (list,)) or len(u) == d:
+ if len(u) != d:
+ raise Exception(("Input list lenght must be equal to "
+ "augmentation dimension d."))
+ N = len(u[0])
+ for x in u[1 :]:
+ if len(x) != N:
+ raise Exception(("Elements of input list must all have "
+ "the same lenght."))
+ return np.concatenate(u)
+ else:
+ N = int(len(u) / d)
+ if not np.isclose(len(u), d * N):
+ raise Exception(("Input vector lenght must be multiple of"
+ "augmentation dimension d."))
+ return u
+
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