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 get_topologies(self, coord):
"""
Returns topologies at coordinates (coord). By a topology, it is
understood a characteristic function (ones and zeros at coord)
of individual inclusions (square, circle, etc.) usually defined in
input file.
"""
inclusions = self.conf['inclusions']
params = self.conf['params']
positions = self.conf['positions']
dim = coord.shape[0]
topos = []
# periodically enlarge coordinates
N = np.array(coord.shape[1:])
cop = np.empty(np.hstack([dim, 3*N]))
mapY = np.array([-1, 0, 1])
for dd in np.arange(dim):
Nshape = np.ones(dim, dtype=np.int32)
Nshape[dd] = 3*N[dd]
Nrep = 3*N
Nrep[dd] = 1
vec = np.repeat(mapY*self.Y[dd], N[dd])
Ymat = np.tile(np.reshape(vec, Nshape), Nrep)
cop[dd] = np.tile(coord[dd], 3*np.ones(dim)) + Ymat
mapY = np.array([-1, 0, 1])
Yiter = mapY[np.newaxis]
for ii in np.arange(1, dim):
Yiter = np.vstack([np.repeat(Yiter, 3, axis=1),
np.tile(mapY, Yiter.shape[1])])
for ii, kind in enumerate(inclusions):
if kind in inclusion_keys['cube']:
param = np.array(params[ii], dtype=np.float64)
pos = np.array(positions[ii], dtype=np.float64)
topos.append(np.zeros(cop.shape[1:]))
for Ycoef in Yiter.T:
Ym = self.Y*Ycoef
topo_loc = np.ones(cop.shape[1:])
for dd in np.arange(dim):
topo_loc *= ((cop[dd]-pos[dd]+Ym[dd]) > -param[dd]/2)
topo_loc *= ((cop[dd]-pos[dd]+Ym[dd]) <= param[dd]/2)
topos[ii] += topo_loc
topos[ii] = decrease(topos[ii], N)
elif kind in inclusion_keys['ball']:
pos = np.array(positions[ii], dtype=np.float64)
topos.append(np.zeros(cop.shape[1:]))
for Ycoef in Yiter.T:
Ym = self.Y*Ycoef
topo_loc = np.ones(cop.shape[1:])
norm2 = 0. # square of norm
for dd in np.arange(dim):
norm2 += (cop[dd]-pos[dd]-Ym[dd])**2
topos[ii] += (norm2**0.5 < params[ii]/2)
topos[ii] = decrease(topos[ii], N)
elif kind in inclusion_keys['pyramid']:
param = np.array(params[ii], dtype=np.float64)
pos = np.array(positions[ii], dtype=np.float64)
topos.append(np.zeros(cop.shape[1:]))
for Ycoef in Yiter.T:
Ym = self.Y*Ycoef
topo_loc = np.ones(cop.shape[1:])
shp = cop[0].shape
tri = lambda t, h: np.maximum(1-np.abs(t)/h, np.zeros(shp))
for dd in np.arange(dim):
topo_loc *= tri(cop[dd]-pos[dd]+2*Ym[dd], param[dd]/2.)
topos[ii] += topo_loc
topos[ii] = decrease(topos[ii], N)
elif kind == 'otherwise':
topos.append(np.ones(coord.shape[1:]))
for jj in np.arange(len(topos)-1):
topos[ii] -= topos[jj]
if not (topos[ii] >= 0).all():
raise NotImplementedError("Overlapping inclusions!")
elif kind == 'all':
topos.append(np.ones(coord.shape[1:]))
else:
msg = "Inclusion (%s) is not implemented." % (kind)
raise NotImplementedError(msg)
return topos
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_topologies(self, coord):
"""
Returns topologies at coordinates (coord). By a topology, it is
understood a characteristic function (ones and zeros at coord)
of individual inclusions (square, circle, etc.) usually defined in
input file.
"""
inclusions = self.conf['inclusions']
params = self.conf['params']
positions = self.conf['positions']
dim = coord.shape[0]
topos = []
# periodically enlarge coordinates
N = np.array(coord.shape[1:])
cop = np.empty(np.hstack([dim, 3 * N]))
mapY = np.array([-1, 0, 1])
for dd in np.arange(dim):
Nshape = np.ones(dim, dtype=np.int32)
Nshape[dd] = 3 * N[dd]
Nrep = 3 * N
Nrep[dd] = 1
vec = np.repeat(mapY * self.Y[dd], N[dd])
Ymat = np.tile(np.reshape(vec, Nshape), Nrep)
cop[dd] = np.tile(coord[dd], 3 * np.ones(dim)) + Ymat
mapY = np.array([-1, 0, 1])
Yiter = mapY[np.newaxis]
for ii in np.arange(1, dim):
Yiter = np.vstack(
[np.repeat(Yiter, 3, axis=1),
np.tile(mapY, Yiter.shape[1])])
for ii, kind in enumerate(inclusions):
if kind in inclusion_keys['cube']:
param = np.array(params[ii], dtype=np.float64)
pos = np.array(positions[ii], dtype=np.float64)
topos.append(np.zeros(cop.shape[1:]))
for Ycoef in Yiter.T:
Ym = self.Y * Ycoef
topo_loc = np.ones(cop.shape[1:])
for dd in np.arange(dim):
topo_loc *= ((cop[dd] - pos[dd] + Ym[dd]) >
-param[dd] / 2)
topo_loc *= ((cop[dd] - pos[dd] + Ym[dd]) <=
param[dd] / 2)
topos[ii] += topo_loc
topos[ii] = decrease(topos[ii], N)
elif kind in inclusion_keys['ball']:
pos = np.array(positions[ii], dtype=np.float64)
topos.append(np.zeros(cop.shape[1:]))
for Ycoef in Yiter.T:
Ym = self.Y * Ycoef
topo_loc = np.ones(cop.shape[1:])
norm2 = 0. # square of norm
for dd in np.arange(dim):
norm2 += (cop[dd] - pos[dd] - Ym[dd])**2
topos[ii] += (norm2**0.5 < params[ii] / 2)
topos[ii] = decrease(topos[ii], N)
elif kind in inclusion_keys['pyramid']:
param = np.array(params[ii], dtype=np.float64)
pos = np.array(positions[ii], dtype=np.float64)
topos.append(np.zeros(cop.shape[1:]))
for Ycoef in Yiter.T:
Ym = self.Y * Ycoef
topo_loc = np.ones(cop.shape[1:])
shp = cop[0].shape
tri = lambda t, h: np.maximum(1 - np.abs(t) / h,
np.zeros(shp))
for dd in np.arange(dim):
topo_loc *= tri(cop[dd] - pos[dd] + 2 * Ym[dd],
param[dd] / 2.)
topos[ii] += topo_loc
topos[ii] = decrease(topos[ii], N)
elif kind == 'otherwise':
topos.append(np.ones(coord.shape[1:]))
for jj in np.arange(len(topos) - 1):
topos[ii] -= topos[jj]
if not (topos[ii] >= 0).all():
raise NotImplementedError("Overlapping inclusions!")
elif kind == 'all':
topos.append(np.ones(coord.shape[1:]))
else:
msg = "Inclusion (%s) is not implemented." % (kind)
raise NotImplementedError(msg)
return topos
