def scalar(N, Y, NyqNul=True, tensor=True, fft_form=fft_form_default):
"""
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
"""
if fft_form in ['r']:
fft_form_r=True
fft_form=0
else:
fft_form_r=False
d = np.size(N)
N = np.array(N, dtype=np.int)
if NyqNul:
Nred = get_Nodd(N)
else:
Nred = N
xi = Grid.get_xil(Nred, Y, fft_form=fft_form)
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 = mean_index(Nred, fft_form=fft_form)
for m in np.arange(d): # diagonal components
Nshape = np.ones(d, dtype=np.int)
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, dtype=np.int)
NshapeM[m] = Nred[m]
NrepM = np.copy(Nred)
NrepM[m] = 1
NshapeN = np.ones(d, dtype=np.int)
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 tensor:
G0l = Tensor(name='hG0', val=G0l, order=2, N=N, multype=21, Fourier=True, fft_form=fft_form)
G1l = Tensor(name='hG1', val=G1l, order=2, N=N, multype=21, Fourier=True, fft_form=fft_form)
G2l = Tensor(name='hG2', val=G2l, order=2, N=N, multype=21, Fourier=True, fft_form=fft_form)
else:
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)
if fft_form_r:
for tensor in [G0l, G1l, G2l]:
tensor.set_fft_form(fft_form='r')
tensor.val/=np.prod(tensor.N)
return G0l, G1l, G2l
def scalar(N, Y, NyqNul=True, tensor=True, fft_form=fft_form_default):
"""
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
"""
if fft_form in ['r']:
fft_form_r = True
fft_form = 0
else:
fft_form_r = False
d = np.size(N)
N = np.array(N, dtype=np.int)
if NyqNul:
Nred = get_Nodd(N)
else:
Nred = N
xi = Grid.get_xil(Nred, Y, fft_form=fft_form)
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 = mean_index(Nred, fft_form=fft_form)
for m in np.arange(d): # diagonal components
Nshape = np.ones(d, dtype=np.int)
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, dtype=np.int)
NshapeM[m] = Nred[m]
NrepM = np.copy(Nred)
NrepM[m] = 1
NshapeN = np.ones(d, dtype=np.int)
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 tensor:
G0l = Tensor(name='hG0',
val=G0l,
order=2,
N=N,
multype=21,
Fourier=True,
fft_form=fft_form)
G1l = Tensor(name='hG1',
val=G1l,
order=2,
N=N,
multype=21,
Fourier=True,
fft_form=fft_form)
G2l = Tensor(name='hG2',
val=G2l,
order=2,
N=N,
multype=21,
Fourier=True,
fft_form=fft_form)
else:
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)
if fft_form_r:
for tensor in [G0l, G1l, G2l]:
tensor.set_fft_form(fft_form='r')
tensor.val /= np.prod(tensor.N)
return G0l, G1l, G2l
def elasticity(N, Y, NyqNul=True, tensor=True, fft_form=fft_form_default):
"""
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
"""
if fft_form in ['r']:
fft_form_r=True
fft_form=0
else:
fft_form_r=False
xi = Grid.get_xil(N, Y, fft_form=fft_form)
N = np.array(N, dtype=np.int)
d = N.size
D = int(d*(d+1)/2)
if NyqNul:
Nred = get_Nodd(N)
else:
Nred = N
xi2 = []
for ii in range(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, dtype=np.int)
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 = mean_index(Nred, fft_form=fft_form)
# 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, dtype=np.int)
NshapeM[m] = Nred[m]
NrepM = np.copy(Nred)
NrepM[m] = 1
NshapeN = np.ones(d, dtype=np.int)
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 tensor:
G0 = Tensor(name='hG0', val=mean, order=2, N=N, Fourier=True, multype=21, fft_form=fft_form)
G1h = Tensor(name='hG1h', val=G1h, order=2, N=N, Fourier=True, multype=21, fft_form=fft_form)
G1s = Tensor(name='hG1s', val=G1s, order=2, N=N, Fourier=True, multype=21, fft_form=fft_form)
G2h = Tensor(name='hG2h', val=G2h, order=2, N=N, Fourier=True, multype=21, fft_form=fft_form)
G2s = Tensor(name='hG2s', val=G2s, order=2, N=N, Fourier=True, multype=21, fft_form=fft_form)
else:
G0 = Matrix(name='hG0', val=mean, Fourier=True)
G1h = Matrix(name='hG1h', val=G1h, Fourier=True)
G1s = Matrix(name='hG1s', val=G1s, Fourier=True)
G2h = Matrix(name='hG2h', val=G2h, Fourier=True)
G2s = Matrix(name='hG2s', 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)
if fft_form_r:
for tensor in [G0, G1h, G1s, G2h, G2s]:
tensor.set_fft_form(fft_form='r')
tensor.val=1./np.prod(tensor.N)*tensor.val
return G0, G1h, G1s, G2h, G2s
def elasticity(N, Y, NyqNul=True, tensor=True, fft_form=fft_form_default):
"""
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
"""
if fft_form in ['r']:
fft_form_r = True
fft_form = 0
else:
fft_form_r = False
xi = Grid.get_xil(N, Y, fft_form=fft_form)
N = np.array(N, dtype=np.int)
d = N.size
D = int(d * (d + 1) / 2)
if NyqNul:
Nred = get_Nodd(N)
else:
Nred = N
xi2 = []
for ii in range(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, dtype=np.int)
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 = mean_index(Nred, fft_form=fft_form)
# 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, dtype=np.int)
NshapeM[m] = Nred[m]
NrepM = np.copy(Nred)
NrepM[m] = 1
NshapeN = np.ones(d, dtype=np.int)
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 tensor:
G0 = Tensor(name='hG0',
val=mean,
order=2,
N=N,
Fourier=True,
multype=21,
fft_form=fft_form)
G1h = Tensor(name='hG1h',
val=G1h,
order=2,
N=N,
Fourier=True,
multype=21,
fft_form=fft_form)
G1s = Tensor(name='hG1s',
val=G1s,
order=2,
N=N,
Fourier=True,
multype=21,
fft_form=fft_form)
G2h = Tensor(name='hG2h',
val=G2h,
order=2,
N=N,
Fourier=True,
multype=21,
fft_form=fft_form)
G2s = Tensor(name='hG2s',
val=G2s,
order=2,
N=N,
Fourier=True,
multype=21,
fft_form=fft_form)
else:
G0 = Matrix(name='hG0', val=mean, Fourier=True)
G1h = Matrix(name='hG1h', val=G1h, Fourier=True)
G1s = Matrix(name='hG1s', val=G1s, Fourier=True)
G2h = Matrix(name='hG2h', val=G2h, Fourier=True)
G2s = Matrix(name='hG2s', 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)
if fft_form_r:
for tensor in [G0, G1h, G1s, G2h, G2s]:
tensor.set_fft_form(fft_form='r')
tensor.val = 1. / np.prod(tensor.N) * tensor.val
return G0, G1h, G1s, G2h, G2s