def grad(X):
if X.shape==(1,):
shape=(X.dim,)
else:
shape=X.shape+(X.dim,)
name='grad({0})'.format(X.name[:10])
gX=Tensor(name=name, shape=shape, N=X.N,
Fourier=True, fft_form=X.fft_form)
if X.Fourier:
FX=X
else:
F=DFT(N=X.N, fft_form=X.fft_form) # TODO:change to X.fourier()
FX=F(X)
dim=len(X.N)
freq=Grid.get_freq(X.N, X.Y, fft_form=X.fft_form)
strfreq='xyz'
coef=2*np.pi*1j
val=np.empty((X.dim,)+X.shape+X.N_fft, dtype=np.complex)
for ii in range(X.dim):
mul_str='{0},...{1}->...{1}'.format(strfreq[ii], strfreq[:dim])
val[ii]=np.einsum(mul_str, coef*freq[ii], FX.val, dtype=np.complex)
if X.shape==(1,):
gX.val=np.squeeze(val)
else:
gX.val=np.moveaxis(val, 0, X.order)
if not X.Fourier:
iF=DFT(N=X.N, inverse=True, fft_form=gX.fft_form)
gX=iF(gX)
gX.name='grad({0})'.format(X.name[:10])
return gX
def scalar(N, Y, fft_form=fft_form_default):
dim = np.size(N)
N = np.array(N, dtype=np.int)
xi = Grid.get_freq(N, Y, fft_form=fft_form)
N_fft=tuple(xi[i].size for i in range(dim))
hGrad = np.zeros((dim,)+N_fft) # zero initialize
for ind in itertools.product(*[range(n) for n in N_fft]):
for i in range(dim):
hGrad[i][ind] = xi[i][ind[i]]
kok= np.einsum('i...,j...->ij...', hGrad, hGrad).real
k2 = np.einsum('i...,i...', hGrad, hGrad).real
ind_center=mean_index(N, fft_form=fft_form)
k2[ind_center]=1.
G0lval=np.zeros_like(kok)
Ival=np.zeros_like(kok)
for ii in range(dim): # diagonal components
G0lval[ii, ii][ind_center] = 1
Ival[ii, ii] = 1
G1l=Tensor(name='G1', val=kok/k2, order=2, N=N, Y=Y, multype=21, Fourier=True, fft_form=fft_form)
G0l=Tensor(name='G1', val=G0lval, order=2, N=N, Y=Y, multype=21, Fourier=True, fft_form=fft_form)
I = Tensor(name='I', val=Ival, order=2, N=N, Y=Y, multype=21, Fourier=True, fft_form=fft_form)
G2l=I-G1l-G0l
return G0l, G1l, G2l
def div(X):
if X.shape==(1,):
shape=()
else:
shape=X.shape[:-1]
assert(X.shape[-1]==X.dim)
assert(X.order==1)
dX=Tensor(shape=shape, N=X.N, Fourier=True, fft_form=X.fft_form)
if X.Fourier:
FX=X
else:
F=DFT(N=X.N, fft_form=X.fft_form)
FX=F(X)
dim=len(X.N)
freq=Grid.get_freq(X.N, X.Y, fft_form=FX.fft_form)
strfreq='xyz'
coef=2*np.pi*1j
for ii in range(X.dim):
mul_str='{0},...{1}->...{1}'.format(strfreq[ii], strfreq[:dim])
dX.val+=np.einsum(mul_str, coef*freq[ii], FX.val[ii], dtype=np.complex)
if not X.Fourier:
iF=DFT(N=X.N, inverse=True, fft_form=dX.fft_form)
dX=iF(dX)
dX.name='div({0})'.format(X.name[:10])
return dX
def div(X):
if X.shape == (1, ):
shape = ()
else:
shape = X.shape[:-1]
assert (X.shape[-1] == X.dim)
assert (X.order == 1)
dX = Tensor(shape=shape, N=X.N, Fourier=True, fft_form=X.fft_form)
if X.Fourier:
FX = X
else:
F = DFT(N=X.N, fft_form=X.fft_form)
FX = F(X)
dim = len(X.N)
freq = Grid.get_freq(X.N, X.Y, fft_form=FX.fft_form)
strfreq = 'xyz'
coef = 2 * np.pi * 1j
for ii in range(X.dim):
mul_str = '{0},...{1}->...{1}'.format(strfreq[ii], strfreq[:dim])
dX.val += np.einsum(mul_str,
coef * freq[ii],
FX.val[ii],
dtype=np.complex)
if not X.Fourier:
iF = DFT(N=X.N, inverse=True, fft_form=dX.fft_form)
dX = iF(dX)
dX.name = 'div({0})'.format(X.name[:10])
return dX
def scalar(N, Y, fft_form=fft_form_default):
dim = np.size(N)
N = np.array(N, dtype=np.int)
xi = Grid.get_freq(N, Y, fft_form=fft_form)
N_fft=tuple(xi[i].size for i in range(dim))
hGrad = np.zeros((dim,)+N_fft) # zero initialize
for ind in itertools.product(*[list(range(n)) for n in N_fft]):
for i in range(dim):
hGrad[i][ind] = xi[i][ind[i]]
kok= np.einsum('i...,j...->ij...', hGrad, hGrad).real
k2 = np.einsum('i...,i...', hGrad, hGrad).real
ind_center=mean_index(N, fft_form=fft_form)
k2[ind_center]=1.
G0lval=np.zeros_like(kok)
Ival=np.zeros_like(kok)
for ii in range(dim): # diagonal components
G0lval[ii, ii][ind_center] = 1
Ival[ii, ii] = 1
G1l=Tensor(name='G1', val=kok/k2, order=2, N=N, Y=Y, multype=21, Fourier=True, fft_form=fft_form)
G0l=Tensor(name='G1', val=G0lval, order=2, N=N, Y=Y, multype=21, Fourier=True, fft_form=fft_form)
I = Tensor(name='I', val=Ival, order=2, N=N, Y=Y, multype=21, Fourier=True, fft_form=fft_form)
G2l=I-G1l-G0l
return G0l, G1l, G2l
def grad(X):
if X.shape==(1,):
shape=(X.dim,)
else:
shape=X.shape+(X.dim,)
name='grad({0})'.format(X.name[:10])
gX=Tensor(name=name, shape=shape, N=X.N,
Fourier=True, fft_form=X.fft_form)
if X.Fourier:
FX=X
else:
F=DFT(N=X.N, fft_form=X.fft_form) # TODO:change to X.fourier()
FX=F(X)
dim=len(X.N)
freq=Grid.get_freq(X.N, X.Y, fft_form=X.fft_form)
strfreq='xyz'
coef=2*np.pi*1j
val=np.empty((X.dim,)+X.shape+X.N_fft, dtype=np.complex)
for ii in range(X.dim):
mul_str='{0},...{1}->...{1}'.format(strfreq[ii], strfreq[:dim])
val[ii]=np.einsum(mul_str, coef*freq[ii], FX.val, dtype=np.complex)
if X.shape==(1,):
gX.val=np.squeeze(val)
else:
gX.val=np.moveaxis(val, 0, X.order)
if not X.Fourier:
iF=DFT(N=X.N, inverse=True, fft_form=gX.fft_form)
gX=iF(gX)
gX.name='grad({0})'.format(X.name[:10])
return gX
def potential(X, small_strain=False):
if X.Fourier:
FX = X
else:
F = DFT(N=X.N, fft_form=X.fft_form)
FX = F(X)
freq = Grid.get_freq(X.N, X.Y, fft_form=FX.fft_form)
if X.order == 1:
assert (X.dim == X.shape[0])
iX = Tensor(name='potential({0})'.format(X.name[:10]),
shape=(1, ),
N=X.N,
Fourier=True,
fft_form=FX.fft_form)
iX.val[0] = potential_scalar(FX.val,
freq=freq,
mean_index=FX.mean_index())
elif X.order == 2:
assert (X.dim == X.shape[0])
assert (X.dim == X.shape[1])
iX = Tensor(name='potential({0})'.format(X.name[:10]),
shape=(X.dim, ),
N=X.N,
Fourier=True,
fft_form=FX.fft_form)
if not small_strain:
for ii in range(X.dim):
iX.val[ii] = potential_scalar(FX.val[ii],
freq=freq,
mean_index=FX.mean_index())
else:
assert ((X - X.transpose()).norm() < 1e-14) # symmetricity
omeg = FX.zeros_like() # non-symmetric part of the gradient
gomeg = Tensor(name='potential({0})'.format(X.name[:10]),
shape=FX.shape + (X.dim, ),
N=X.N,
Fourier=True)
grad_ep = grad(FX) # gradient of strain
gomeg.val = np.einsum('ikj...->ijk...', grad_ep.val) - np.einsum(
'jki...->ijk...', grad_ep.val)
for ij in itertools.product(list(range(X.dim)), repeat=2):
omeg.val[ij] = potential_scalar(gomeg.val[ij],
freq=freq,
mean_index=FX.mean_index())
gradu = FX + omeg
iX = potential(gradu, small_strain=False)
if X.Fourier:
return iX
else:
iF = DFT(N=X.N, inverse=True, fft_form=FX.fft_form)
return iF(iX)
def elasticity_large_deformation(N, Y, fft_form=fft_form_default):
N = np.array(N, dtype=np.int)
dim = N.size
assert(dim==3)
freq = Grid.get_freq(N, Y, fft_form=fft_form)
N_fft=tuple(freq[i].size for i in range(dim))
Ghat = np.zeros(np.hstack([dim*np.ones(4, dtype=np.int), N_fft]))
delta = lambda i,j: np.float(i==j)
for i, j, k, l in itertools.product(range(dim), repeat=4):
for x, y, z in np.ndindex(*N_fft):
q = np.array([freq[0][x], freq[1][y], freq[2][z]])
if not q.dot(q) == 0:
Ghat[i,j,k,l,x,y,z] = delta(i,k)*q[j]*q[l] / (q.dot(q))
Ghat_tensor = Tensor(name='Ghat', val=Ghat, order=4, N=N, multype=42,
Fourier=True, fft_form=fft_form)
return Ghat_tensor
def elasticity_large_deformation(N, Y, fft_form=fft_form_default):
N = np.array(N, dtype=np.int)
dim = N.size
assert(dim==3)
freq = Grid.get_freq(N, Y, fft_form=fft_form)
N_fft=tuple(freq[i].size for i in range(dim))
Ghat = np.zeros(np.hstack([dim*np.ones(4, dtype=np.int), N_fft]))
delta = lambda i,j: np.float(i==j)
for i, j, k, l in itertools.product(list(range(dim)), repeat=4):
for x, y, z in np.ndindex(*N_fft):
q = np.array([freq[0][x], freq[1][y], freq[2][z]])
if not q.dot(q) == 0:
Ghat[i,j,k,l,x,y,z] = delta(i,k)*q[j]*q[l] / (q.dot(q))
Ghat_tensor = Tensor(name='Ghat', val=Ghat, order=4, N=N, multype=42,
Fourier=True, fft_form=fft_form)
return Ghat_tensor
def potential(X, small_strain=False):
if X.Fourier:
FX=X
else:
F=DFT(N=X.N, fft_form=X.fft_form)
FX=F(X)
freq=Grid.get_freq(X.N, X.Y, fft_form=FX.fft_form)
if X.order==1:
assert(X.dim==X.shape[0])
iX=Tensor(name='potential({0})'.format(X.name[:10]), shape=(1,), N=X.N,
Fourier=True, fft_form=FX.fft_form)
iX.val[0]=potential_scalar(FX.val, freq=freq, mean_index=FX.mean_index())
elif X.order==2:
assert(X.dim==X.shape[0])
assert(X.dim==X.shape[1])
iX=Tensor(name='potential({0})'.format(X.name[:10]), shape=(X.dim,), N=X.N,
Fourier=True, fft_form=FX.fft_form)
if not small_strain:
for ii in range(X.dim):
iX.val[ii]=potential_scalar(FX.val[ii], freq=freq, mean_index=FX.mean_index())
else:
assert((X-X.transpose()).norm()<1e-14) # symmetricity
omeg=FX.zeros_like() # non-symmetric part of the gradient
gomeg=Tensor(name='potential({0})'.format(X.name[:10]),
shape=FX.shape+(X.dim,), N=X.N, Fourier=True)
grad_ep=grad(FX) # gradient of strain
gomeg.val=np.einsum('ikj...->ijk...', grad_ep.val)-np.einsum('jki...->ijk...', grad_ep.val)
for ij in itertools.product(range(X.dim), repeat=2):
omeg.val[ij]=potential_scalar(gomeg.val[ij], freq=freq, mean_index=FX.mean_index())
gradu=FX+omeg
iX=potential(gradu, small_strain=False)
if X.Fourier:
return iX
else:
iF=DFT(N=X.N, inverse=True, fft_form=FX.fft_form)
return iF(iX)
