def get_weights_circ(r, Nbar, Y):
"""
it evaluates integral weights,
which are used for upper-lower bounds calculation,
with constant circular inclusion
Parameters
----------
r - the parameter determining the size of inclusion
Nbar - no. of points of regular grid where the weights are evaluated
Y - the size of periodic unit cell
Returns
-------
Wphi - integral weights at regular grid sizing Nbar
"""
d = np.size(Y)
ZN2l = Grid.get_ZNl(Nbar)
meas_puc = np.prod(Y)
circ = 0
for m in np.arange(d):
Nshape = np.ones(d)
Nshape[m] = Nbar[m]
Nrep = np.copy(Nbar)
Nrep[m] = 1
xi_p2 = np.tile(np.reshape((ZN2l[m] / Y[m])**2, Nshape), Nrep)
circ += xi_p2
circ = circ**0.5
ind = tuple(np.round(Nbar / 2))
circ[ind] = 1.
Wphi = r**2 * sp.jn(1, 2 * np.pi * circ * r) / (circ * r)
Wphi[ind] = np.pi * r**2
Wphi = Wphi / meas_puc
return Wphi
def get_weights_circ(r, Nbar, Y):
"""
it evaluates integral weights,
which are used for upper-lower bounds calculation,
with constant circular inclusion
Parameters
----------
r - the parameter determining the size of inclusion
Nbar - no. of points of regular grid where the weights are evaluated
Y - the size of periodic unit cell
Returns
-------
Wphi - integral weights at regular grid sizing Nbar
"""
d = np.size(Y)
ZN2l = Grid.get_ZNl(Nbar)
meas_puc = np.prod(Y)
circ = 0
for m in np.arange(d):
Nshape = np.ones(d)
Nshape[m] = Nbar[m]
Nrep = np.copy(Nbar)
Nrep[m] = 1
xi_p2 = np.tile(np.reshape((ZN2l[m]/Y[m])**2, Nshape), Nrep)
circ += xi_p2
circ = circ**0.5
ind = tuple(np.round(Nbar/2))
circ[ind] = 1.
Wphi = r**2 * sp.jn(1, 2*np.pi*circ*r) / (circ*r)
Wphi[ind] = np.pi*r**2
Wphi = Wphi / meas_puc
return Wphi
def get_A_GaNi(self, N, primaldual='primal'):
"""
Returns stiffness matrix for a scheme with trapezoidal quadrature rule.
"""
coord = Grid.get_coordinates(N, self.Y)
A = self.evaluate(coord)
if primaldual is 'dual':
A = A.inv()
return A
def get_A_GaNi(self, N, primaldual='primal'):
"""
Returns stiffness matrix for a scheme with trapezoidal quadrature rule.
"""
coord = Grid.get_coordinates(N, self.Y)
A = self.evaluate(coord)
if primaldual is 'dual':
A = A.inv()
return A
def get_A_Ga(self, Nbar, primaldual='primal', order=None, P=None):
""" Returns stiffness matrix for scheme with exact integration."""
if order is None and 'order' in self.conf:
order = self.conf['order']
if order is None:
shape_funs = self.get_shape_functions(Nbar)
val = np.zeros(self.conf['vals'][0].shape + shape_funs[0].shape)
for ii in range(len(self.conf['inclusions'])):
if primaldual is 'primal':
Aincl = self.conf['vals'][ii]
elif primaldual is 'dual':
Aincl = np.linalg.inv(self.conf['vals'][ii])
val += np.einsum('ij...,k...->ijk...', Aincl, shape_funs[ii])
return Matrix(name='A_Ga', val=val, Fourier=False)
else:
if P is None and 'P' in self.conf:
P = self.conf['P']
coord = Grid.get_coordinates(P, self.Y)
vals = self.evaluate(coord)
dim = vals.d
if primaldual is 'dual':
vals = vals.inv()
h = self.Y/P
if order in [0, 'constant']:
Wraw = get_weights_con(h, Nbar, self.Y)
elif order in [1, 'bilinear']:
Wraw = get_weights_lin(h, Nbar, self.Y)
Aapp = np.zeros(np.hstack([dim, dim, Nbar]))
for m in np.arange(dim):
for n in np.arange(dim):
hAM0 = DFT.fftnc(vals[m, n], P)
if np.allclose(P, Nbar):
hAM = hAM0
elif np.all(np.greater_equal(P, Nbar)):
hAM = decrease(hAM0, Nbar)
elif np.all(np.less(P, Nbar)):
factor = np.ceil(np.array(Nbar, dtype=np.float64) / P)
hAM0per = np.tile(hAM0, 2*factor-1)
hAM = decrease(hAM0per, Nbar)
else:
raise ValueError()
pNbar = np.prod(Nbar)
""" if DFT is normalized in accordance with articles there
should be np.prod(M) instead of np.prod(Nbar)"""
Aapp[m, n] = np.real(pNbar*DFT.ifftnc(Wraw*hAM, Nbar))
name = 'A_Ga_o%d_P%d' % (order, P.max())
return Matrix(name=name, val=Aapp, Fourier=False)
def savefig(self, fname='material.pdf', N=50*np.ones(2)):
import pylab as pl
pl.figure(num=None, figsize=(3,3), dpi=1000)
coord = Grid.get_coordinates(N, self.Y)
vals = self.evaluate(coord)[0, 0]
pl.pcolor(coord[0], coord[1], -vals)
pl.xlim([-self.Y[0]/2, self.Y[0]/2])
pl.ylim([-self.Y[1]/2, self.Y[1]/2])
pl.xlabel(r'coordinate $x_1$')
pl.ylabel(r'coordinate $x_2$')
pl.savefig(fname, pad_inches=0.02, bbox_inches='tight')
pl.close()
def savefig(self, fname='material.pdf', N=50 * np.ones(2)):
import pylab as pl
pl.figure(num=None, figsize=(3, 3), dpi=1000)
coord = Grid.get_coordinates(N, self.Y)
vals = self.evaluate(coord)[0, 0]
pl.pcolor(coord[0], coord[1], -vals)
pl.xlim([-self.Y[0] / 2, self.Y[0] / 2])
pl.ylim([-self.Y[1] / 2, self.Y[1] / 2])
pl.xlabel(r'coordinate $x_1$')
pl.ylabel(r'coordinate $x_2$')
pl.savefig(fname, pad_inches=0.02, bbox_inches='tight')
pl.close()
def get_shift_inclusion(N, h, Y):
N = np.array(N, dtype=np.int32)
Y = np.array(Y, dtype=np.float64)
dim = N.size
ZN = Grid.get_ZNl(N)
SS = np.ones(N, dtype=np.complex128)
for ii in np.arange(dim):
Nshape = np.ones(dim)
Nshape[ii] = N[ii]
Nrep = N
Nrep[ii] = 1
SS *= np.tile(np.reshape(np.exp(-2*np.pi*1j*(h[ii]*ZN[ii]/Y[ii])),
Nshape), Nrep)
return SS
def get_shift_inclusion(N, h, Y):
N = np.array(N, dtype=np.int32)
Y = np.array(Y, dtype=np.float64)
dim = N.size
ZN = Grid.get_ZNl(N)
SS = np.ones(N, dtype=np.complex128)
for ii in np.arange(dim):
Nshape = np.ones(dim)
Nshape[ii] = N[ii]
Nrep = N
Nrep[ii] = 1
SS *= np.tile(
np.reshape(np.exp(-2 * np.pi * 1j * (h[ii] * ZN[ii] / Y[ii])),
Nshape), Nrep)
return SS
def curl_norm(e, Y):
"""
it calculates curl-based norm,
it controls that the fields are curl-free with zero mean as
it is required of electric fields
Parameters
----------
e - electric field
Y - the size of periodic unit cell
Returns
-------
curlnorm - curl-based norm
"""
N = np.array(np.shape(e[0]))
d = np.size(N)
xil = Grid.get_xil(N, Y)
xiM = []
Fe = []
for m in np.arange(d):
Nshape = np.ones(d)
Nshape[m] = N[m]
Nrep = np.copy(N)
Nrep[m] = 1
xiM.append(np.tile(np.reshape(xil[m], Nshape), Nrep))
Fe.append(DFT.fftnc(e[m], N)/np.prod(N))
if d == 2:
Fe.append(np.zeros(N))
xiM.append(np.zeros(N))
ind_mean = tuple(np.fix(N/2))
curl = []
e0 = []
for m in np.arange(3):
j = (m+1) % 3
k = (j+1) % 3
curl.append(xiM[j]*Fe[k]-xiM[k]*Fe[j])
e0.append(np.real(Fe[m][ind_mean]))
curl = np.array(curl)
curlnorm = np.real(np.sum(curl[:]*np.conj(curl[:])))
curlnorm = (curlnorm/np.prod(N))**0.5
norm_e0 = np.linalg.norm(e0)
if norm_e0 > 1e-10:
curlnorm = curlnorm/norm_e0
return curlnorm
def matrix(self):
"""
This function returns the object as a matrix of DFT or iDFT resp.
"""
prodN = np.prod(self.N)
proddN = self.d*prodN
DTM = np.zeros(np.hstack([self.N, self.N]), dtype=np.complex128)
ZNl = Grid.get_ZNl(self.N)
if not self.inverse:
coef = 1./np.prod(self.N)
else:
coef = np.prod(self.N)
for nx in ZNl[0]:
for ny in ZNl[1]:
IM = np.zeros(np.array(self.N), dtype=np.float64)
IM[nx, ny] = 1
DTM[nx, ny, :, :] = coef*DFT.fftnc(IM, self.N)
import numpy.matlib as npmatlib
DTMd = npmatlib.zeros([proddN, proddN], dtype=np.complex128)
for ii in np.arange(self.d):
DTMd[prodN*ii:prodN**(ii+1), prodN*ii:prodN**(ii+1)] = DTM
return DTMd
def scalar(N, Y, centered=True, NyqNul=True):
"""
Assembly of discrete kernels in Fourier space for scalar elliptic problems.
Parameters
----------
N : numpy.ndarray
no. of discretization points
Y : numpy.ndarray
size of periodic unit cell
Returns
-------
G1l : numpy.ndarray
discrete kernel in Fourier space; provides projection
on curl-free fields with zero mean
G2l : numpy.ndarray
discrete kernel in Fourier space; provides projection
on divergence-free fields with zero mean
"""
d = np.size(N)
N = np.array(N)
if NyqNul:
Nred = get_Nodd(N)
else:
Nred = N
xi = Grid.get_xil(Nred, Y)
xi2 = []
for m in np.arange(d):
xi2.append(xi[m]**2)
G0l = np.zeros(np.hstack([d, d, Nred]))
G1l = np.zeros(np.hstack([d, d, Nred]))
G2l = np.zeros(np.hstack([d, d, Nred]))
num = np.zeros(np.hstack([d, d, Nred]))
denom = np.zeros(Nred)
ind_center = tuple(np.fix(np.array(Nred)/2))
for m in np.arange(d): # diagonal components
Nshape = np.ones(d)
Nshape[m] = Nred[m]
Nrep = np.copy(Nred)
Nrep[m] = 1
a = np.reshape(xi2[m], Nshape)
num[m][m] = np.tile(a, Nrep) # numerator
denom = denom + num[m][m]
G0l[m, m][ind_center] = 1
for m in np.arange(d): # upper diagonal components
for n in np.arange(m+1, d):
NshapeM = np.ones(d)
NshapeM[m] = Nred[m]
NrepM = np.copy(Nred)
NrepM[m] = 1
NshapeN = np.ones(d)
NshapeN[n] = Nred[n]
NrepN = np.copy(Nred)
NrepN[n] = 1
num[m][n] = np.tile(np.reshape(xi[m], NshapeM), NrepM) \
* np.tile(np.reshape(xi[n], NshapeN), NrepN)
# avoiding a division by zero
denom[ind_center] = 1
# calculation of projections
for m in np.arange(d):
for n in np.arange(m, d):
G1l[m][n] = num[m][n]/denom
G2l[m][n] = (m == n)*np.ones(Nred) - G1l[m][n]
G2l[m][n][ind_center] = 0
# symmetrization
for m in np.arange(1, d):
for n in np.arange(m):
G1l[m][n] = G1l[n][m]
G2l[m][n] = G2l[n][m]
if not centered:
for m in np.arange(d):
for n in np.arange(d):
G1l[m][n] = np.fft.ifftshift(G1l[m][n])
G2l[m][n] = np.fft.ifftshift(G2l[m][n])
G0l = Matrix(name='hG0', val=G0l, Fourier=True)
G1l = Matrix(name='hG1', val=G1l, Fourier=True)
G2l = Matrix(name='hG2', val=G2l, Fourier=True)
if NyqNul:
G0l = G0l.enlarge(N)
G1l = G1l.enlarge(N)
G2l = G2l.enlarge(N)
return G0l, G1l, G2l
def elasticity(N, Y, centered=True, NyqNul=True):
"""
Projection matrix on a space of admissible strain fields
INPUT =
N : ndarray of e.g. stiffness coefficients
d : dimension; d = 2
D : dimension in engineering notation; D = 3
Y : the size of periodic unit cell
OUTPUT =
G1h,G1s,G2h,G2s : projection matrices of size DxDxN
"""
xi = Grid.get_xil(N, Y)
N = np.array(N)
d = N.size
D = d*(d+1)/2
if NyqNul:
Nred = get_Nodd(N)
else:
Nred = N
xi2 = []
for ii in np.arange(d):
xi2.append(xi[ii]**2)
num = np.zeros(np.hstack([d, d, Nred]))
norm2_xi = np.zeros(Nred)
for mm in np.arange(d): # diagonal components
Nshape = np.ones(d)
Nshape[mm] = Nred[mm]
Nrep = np.copy(Nred)
Nrep[mm] = 1
num[mm][mm] = np.tile(np.reshape(xi2[mm], Nshape), Nrep) # numerator
norm2_xi += num[mm][mm]
norm4_xi = norm2_xi**2
ind_center = tuple(Nred/2)
# avoid division by zero
norm2_xi[ind_center] = 1
norm4_xi[ind_center] = 1
for m in np.arange(d): # upper diagonal components
for n in np.arange(m+1, d):
NshapeM = np.ones(d)
NshapeM[m] = Nred[m]
NrepM = np.copy(Nred)
NrepM[m] = 1
NshapeN = np.ones(d)
NshapeN[n] = Nred[n]
NrepN = np.copy(Nred)
NrepN[n] = 1
num[m][n] = np.tile(np.reshape(xi[m], NshapeM), NrepM) \
* np.tile(np.reshape(xi[n], NshapeN), NrepN)
# G1h = np.zeros([D,D]).tolist()
G1h = np.zeros(np.hstack([D, D, Nred]))
G1s = np.zeros(np.hstack([D, D, Nred]))
IS0 = np.zeros(np.hstack([D, D, Nred]))
mean = np.zeros(np.hstack([D, D, Nred]))
Lamh = np.zeros(np.hstack([D, D, Nred]))
S = np.zeros(np.hstack([D, D, Nred]))
W = np.zeros(np.hstack([D, D, Nred]))
WT = np.zeros(np.hstack([D, D, Nred]))
for m in np.arange(d):
S[m][m] = 2*num[m][m]/norm2_xi
for n in np.arange(d):
G1h[m][n] = num[m][m]*num[n][n]/norm4_xi
Lamh[m][n] = np.ones(Nred)/d
Lamh[m][n][ind_center] = 0
for m in np.arange(D):
IS0[m][m] = np.ones(Nred)
IS0[m][m][ind_center] = 0
mean[m][m][ind_center] = 1
if d == 2:
S[0][2] = 2**0.5*num[0][1]/norm2_xi
S[1][2] = 2**0.5*num[0][1]/norm2_xi
S[2][2] = np.ones(Nred)
S[2][2][ind_center] = 0
G1h[0][2] = 2**0.5*num[0][0]*num[0][1]/norm4_xi
G1h[1][2] = 2**0.5*num[0][1]*num[1][1]/norm4_xi
G1h[2][2] = 2*num[0][0]*num[1][1]/norm4_xi
for m in np.arange(d):
for n in np.arange(d):
W[m][n] = num[m][m]/norm2_xi
W[2][m] = 2**.5*num[0][1]/norm2_xi
elif d == 3:
for m in np.arange(d):
S[m+3][m+3] = 1 - num[m][m]/norm2_xi
S[m+3][m+3][ind_center] = 0
for m in np.arange(d):
for n in np.arange(m+1, d):
S[m+3][n+3] = num[m][n]/norm2_xi
G1h[m+3][n+3] = num[m][m]*num[n][n]/norm4_xi
for m in np.arange(d):
for n in np.arange(d):
ind = sp.setdiff1d(np.arange(d), [n])
S[m][n+3] = (0 == (m == n))*2**.5*num[ind[0]][ind[1]]/norm2_xi
G1h[m][n+3] = 2**.5*num[m][m]*num[ind[0]][ind[1]]/norm4_xi
W[m][n] = num[m][m]/norm2_xi
W[n+3][m] = 2**.5*num[ind[0]][ind[1]]/norm2_xi
for m in np.arange(d):
for n in np.arange(d):
ind_m = sp.setdiff1d(np.arange(d), [m])
ind_n = sp.setdiff1d(np.arange(d), [n])
G1h[m+3][n+3] = 2*num[ind_m[0]][ind_m[1]] \
* num[ind_n[0]][ind_n[1]] / norm4_xi
# symmetrization
for n in np.arange(D):
for m in np.arange(n+1, D):
S[m][n] = S[n][m]
G1h[m][n] = G1h[n][m]
for m in np.arange(D):
for n in np.arange(D):
G1s[m][n] = S[m][n] - 2*G1h[m][n]
WT[m][n] = W[n][m]
G2h = 1./(d-1)*(d*Lamh + G1h - W - WT)
G2s = IS0 - G1h - G1s - G2h
if not centered:
for m in np.arange(d):
for n in np.arange(d):
G1h[m][n] = np.fft.ifftshift(G1h[m][n])
G1s[m][n] = np.fft.ifftshift(G1s[m][n])
G2h[m][n] = np.fft.ifftshift(G2h[m][n])
G2s[m][n] = np.fft.ifftshift(G2s[m][n])
G0 = Matrix(name='hG1', val=mean, Fourier=True)
G1h = Matrix(name='hG1', val=G1h, Fourier=True)
G1s = Matrix(name='hG1', val=G1s, Fourier=True)
G2h = Matrix(name='hG1', val=G2h, Fourier=True)
G2s = Matrix(name='hG1', val=G2s, Fourier=True)
if NyqNul:
G0 = G0.enlarge(N)
G1h = G1h.enlarge(N)
G1s = G1s.enlarge(N)
G2h = G2h.enlarge(N)
G2s = G2s.enlarge(N)
return mean, G1h, G1s, G2h, G2s
def get_A_Ga(self, Nbar, primaldual='primal', order=-1, P=None):
"""
Returns stiffness matrix for scheme with exact integration.
"""
if order == -1:
if 'order' in self.conf:
order = self.conf['order']
else:
raise ValueError('The material order is undefined!')
elif order not in [None, 'exact', 0, 1]:
raise ValueError('Wrong material order (%s)!' % str(order))
if order in [None, 'exact']:
shape_funs = self.get_shape_functions(Nbar)
val = np.zeros(self.conf['vals'][0].shape + shape_funs[0].shape)
for ii in range(len(self.conf['inclusions'])):
if primaldual is 'primal':
Aincl = self.conf['vals'][ii]
elif primaldual is 'dual':
Aincl = np.linalg.inv(self.conf['vals'][ii])
val += np.einsum('ij...,k...->ijk...', Aincl, shape_funs[ii])
name = 'A_Ga'
else:
if P is None and 'P' in self.conf:
P = self.conf['P']
coord = Grid.get_coordinates(P, self.Y)
vals = self.evaluate(coord)
dim = vals.d
if primaldual is 'dual':
vals = vals.inv()
h = self.Y / P
if order in [0, 'constant']:
Wraw = get_weights_con(h, Nbar, self.Y)
elif order in [1, 'bilinear']:
Wraw = get_weights_lin(h, Nbar, self.Y)
val = np.zeros(np.hstack([dim, dim, Nbar]))
for m in np.arange(dim):
for n in np.arange(dim):
hAM0 = DFT.fftnc(vals[m, n], P)
if np.allclose(P, Nbar):
hAM = hAM0
elif np.all(np.greater_equal(P, Nbar)):
hAM = decrease(hAM0, Nbar)
elif np.all(np.less(P, Nbar)):
factor = np.ceil(np.array(Nbar, dtype=np.float64) / P)
hAM0per = np.tile(hAM0, 2 * factor - 1)
hAM = decrease(hAM0per, Nbar)
else:
raise ValueError()
pNbar = np.prod(Nbar)
""" if DFT is normalized in accordance with articles there
should be np.prod(M) instead of np.prod(Nbar)"""
val[m, n] = np.real(pNbar * DFT.ifftnc(Wraw * hAM, Nbar))
name = 'A_Ga_o%d_P%d' % (order, P.max())
return Matrix(name=name, val=val, Fourier=False)
def get_A_GaNi(self, N, primaldual='primal'):
coord = Grid.get_coordinates(N, self.Y)
A = self.evaluate(coord)
if primaldual is 'dual':
A = A.inv()
return A
