def convolve_field_sym_tensor( fvars, pvars, var_name, dim, iel, ts ):
r"""
.. math::
\int_0^t f^{ij}(t-s) p_{ij}(s) ds
Notes
-----
- t is given by step
- f: fvars
field variables, defined in a micro domain, have shape [step][fmf dims]
- p: pvars
sym. tensor point variables, a scalar in a point of
macro-domain, FMField style, have shape [dim, dim][var_name][n_step][var
dims]
"""
step0 = max( 0, ts.step - fvars[0,0][var_name].steps[-1] )
val = nm.zeros_like( fvars[0,0][var_name][0] )
for ik in xrange( step0, ts.step + 1 ):
## print ' ', ik, ts.step-ik
for ir in range( dim ):
for ic in range( dim ):
ii = coor_to_sym( ir, ic, dim )
vf = fvars[ir,ic][var_name][ts.step-ik]
vp = pvars[ik][iel,0,ii,0]
val += vf * vp * ts.dt
return val
def compute_cat_sym_sym(coef, iw_dir):
"""
Christoffel acoustic tensor (part) of elasticity tensor dimension.
"""
dim = iw_dir.shape[0]
cat = nm.zeros((dim, dim), dtype=nm.float64)
for ii in range(dim):
for ij in range(dim):
ir = coor_to_sym(ii, ij, dim)
for ik in range(dim):
for il in range(dim):
ic = coor_to_sym(ik, il, dim)
cat[ii,ik] += coef[ir,ic] * iw_dir[ij] * iw_dir[il]
return cat
def convolve_field_sym_tensor(fvars, pvars, var_name, dim, iel, ts):
r"""
.. math::
\int_0^t f^{ij}(t-s) p_{ij}(s) ds
Notes
-----
- t is given by step
- f: fvars
field variables, defined in a micro domain, have shape [step][fmf dims]
- p: pvars
sym. tensor point variables, a scalar in a point of
macro-domain, FMField style, have shape [dim, dim][var_name][n_step][var
dims]
"""
step0 = max(0, ts.step - fvars[0, 0][var_name].steps[-1])
val = nm.zeros_like(fvars[0, 0][var_name][0])
for ik in xrange(step0, ts.step + 1):
## print ' ', ik, ts.step-ik
for ir in range(dim):
for ic in range(dim):
ii = coor_to_sym(ir, ic, dim)
vf = fvars[ir, ic][var_name][ts.step - ik]
vp = pvars[ik][iel, 0, ii, 0]
val += vf * vp * ts.dt
return val
def compute_cat_sym_sym(coef, iw_dir):
"""
Christoffel acoustic tensor (part) of elasticity tensor dimension.
"""
dim = iw_dir.shape[0]
cat = nm.zeros((dim, dim), dtype=nm.float64)
for ii in range(dim):
for ij in range(dim):
ir = coor_to_sym(ii, ij, dim)
for ik in range(dim):
for il in range(dim):
ic = coor_to_sym(ik, il, dim)
cat[ii, ik] += coef[ir, ic] * iw_dir[ij] * iw_dir[il]
return cat
def add_strain_rs(corrs_rs, strain, vu, dim, iel, out=None):
if out is None:
out = nm.zeros_like(corrs_rs[0, 0][vu][0])
for ir in range(dim):
for ic in range(dim):
ii = coor_to_sym(ir, ic, dim)
out += corrs_rs[ir, ic][vu].data * strain[iel, 0, ii, 0]
return out
def add_strain_rs( corrs_rs, strain, vu, dim, iel, out = None ):
if out is None:
out = nm.zeros_like( corrs_rs[0,0][vu][0] )
for ir in range( dim ):
for ic in range( dim ):
ii = coor_to_sym( ir, ic, dim )
out += corrs_rs[ir,ic][vu].data * strain[iel,0,ii,0]
return out
def compute_cat( coefs, iw_dir, mode = 'simple' ):
r"""Compute Christoffel acoustic tensor (cat) given the incident wave
direction (unit vector).
- if mode == 'simple', coefs.elastic is the elasticity tensor C and
cat := \Gamma_{ik} = C_{ijkl} n_j n_l
- if mode == 'piezo', coefs.elastic, .coupling, .dielectric are the
elasticity, piezo-coupling and dielectric tensors C, G, D and
cat := H_{ik} = \Gamma_{ik} + \frac{1}{\xi} \gamma_i \gamma_j, where
\gamma_i = G_{kij} n_j n_k,
\xi = D_{kl} n_k n_l
"""
dim = iw_dir.shape[0]
cat = nm.zeros( (dim, dim), dtype = nm.float64 )
mtx_c = coefs.elastic
for ii in range( dim ):
for ij in range( dim ):
ir = coor_to_sym( ii, ij, dim )
for ik in range( dim ):
for il in range( dim ):
ic = coor_to_sym( ik, il, dim )
cat[ii,ik] += mtx_c[ir,ic] * iw_dir[ij] * iw_dir[il]
# print cat
if mode =='piezo':
xi = nm.dot( nm.dot( coefs.dielectric, iw_dir ), iw_dir )
# print xi
gamma = nm.zeros( (dim,), dtype = nm.float64 )
mtx_g = coefs.coupling
for ii in range( dim ):
for ij in range( dim ):
ir = coor_to_sym( ii, ij, dim )
for ik in range( dim ):
gamma[ii] += mtx_g[ik,ir] * iw_dir[ij] * iw_dir[ik]
# print gamma
cat += nm.outer( gamma, gamma ) / xi
return cat
def compute_cat_dim_sym(coef, iw_dir):
"""
Christoffel acoustic tensor part of piezo-coupling tensor dimension.
"""
dim = iw_dir.shape[0]
cat = nm.zeros((dim,), dtype=nm.float64)
for ii in range(dim):
for ij in range(dim):
ir = coor_to_sym(ii, ij, dim)
for ik in range(dim):
cat[ii] += coef[ik,ir] * iw_dir[ij] * iw_dir[ik]
return cat
def compute_cat_dim_sym(coef, iw_dir):
"""
Christoffel acoustic tensor part of piezo-coupling tensor dimension.
"""
dim = iw_dir.shape[0]
cat = nm.zeros((dim, ), dtype=nm.float64)
for ii in range(dim):
for ij in range(dim):
ir = coor_to_sym(ii, ij, dim)
for ik in range(dim):
cat[ii] += coef[ik, ir] * iw_dir[ij] * iw_dir[ik]
return cat
def compute_micro_u(corrs, strain, vu, dim, out=None):
r"""
Micro displacements.
.. math::
\bm{u}^1 = \bm{\chi}^{ij}\, e_{ij}^x(\bm{u}^0)
"""
if out is None:
out = nm.zeros_like(corrs[vu + '_00'])
for ir in range(dim):
for ic in range(dim):
ii = coor_to_sym(ir, ic, dim)
out += corrs[vu + '_%d%d' % (ir, ic)] * strain[ii]
return out
def compute_micro_u( corrs, strain, vu, dim, out = None ):
r"""
Micro displacements.
.. math::
\bm{u}^1 = \bm{\chi}^{ij}\, e_{ij}^x(\bm{u}^0)
"""
if out is None:
out = nm.zeros_like( corrs[vu+'_00'] )
for ir in range( dim ):
for ic in range( dim ):
ii = coor_to_sym( ir, ic, dim )
out += corrs[vu+'_%d%d' % (ir, ic)] * strain[ii]
return out
def compute_u_from_macro(strain, coor, iel, centre=None):
r"""
Macro-induced displacements.
.. math::
e_{ij}^x(\bm{u})\,(y_j - y_j^c)
"""
n_nod, dim = coor.shape
if centre is None:
centre = nm.zeros((dim, ), dtype=nm.float64)
n_nod, dim = coor.shape
um = nm.zeros((n_nod * dim, ), dtype=nm.float64)
for ir in range(dim):
for ic in range(dim):
ii = coor_to_sym(ir, ic, dim)
um[ir::dim] += strain[iel, 0, ii, 0] * (coor[:, ic] - centre[ic])
return um
def compute_u_from_macro(strain, coor, iel, centre=None):
r"""
Macro-induced displacements.
.. math::
e_{ij}^x(\bm{u})\,(y_j - y_j^c)
"""
n_nod, dim = coor.shape
if centre is None:
centre = nm.zeros((dim,), dtype=nm.float64)
n_nod, dim = coor.shape
um = nm.zeros((n_nod * dim,), dtype=nm.float64)
for ir in range(dim):
for ic in range(dim):
ii = coor_to_sym(ir, ic, dim)
um[ir::dim] += strain[iel,0,ii,0] * (coor[:,ic] - centre[ic])
return um
def recovery_micro_dfc(pb, corrs, macro):
eps0 = macro['eps0']
mesh = pb.domain.mesh
regions = pb.domain.regions
dim = mesh.dim
Yms_map = regions['Yms'].get_entities(0)
Ym_map = regions['Ym'].get_entities(0)
gl = '_' + list(corrs.keys())[0].split('_')[-1]
u1 = -corrs['corrs_p' + gl]['u'] * macro['press'][Yms_map, :]
phi = -corrs['corrs_p' + gl]['r'] * macro['press'][Ym_map, :]
for ii in range(2):
u1 += corrs['corrs_k%d' % ii + gl]['u'] * macro['phi'][ii]
phi += corrs['corrs_k%d' % ii + gl]['r'] * macro['phi'][ii]
for ii in range(dim):
for jj in range(dim):
kk = coor_to_sym(ii, jj, dim)
phi += corrs['corrs_rs' + gl]['r_%d%d' % (ii, jj)]\
* nm.expand_dims(macro['strain'][Ym_map, kk], axis=1)
u1 += corrs['corrs_rs' + gl]['u_%d%d' % (ii, jj)]\
* nm.expand_dims(macro['strain'][Yms_map, kk], axis=1)
u = macro['u'][Yms_map, :] + eps0 * u1
mvar = pb.create_variables(['u', 'r', 'svar'])
e_mac_Yms = [None] * macro['strain'].shape[1]
for ii in range(dim):
for jj in range(dim):
kk = coor_to_sym(ii, jj, dim)
mvar['svar'].set_data(macro['strain'][:, kk])
mac_e_Yms = pb.evaluate('ev_volume_integrate.i2.Yms(svar)',
mode='el_avg',
var_dict={'svar': mvar['svar']})
e_mac_Yms[kk] = mac_e_Yms.squeeze()
e_mac_Yms = nm.vstack(e_mac_Yms).T[:, nm.newaxis, :, nm.newaxis]
mvar['r'].set_data(phi)
E_mic = pb.evaluate(
'ev_grad.i2.Ym(r)', mode='el_avg', var_dict={'r': mvar['r']}) / eps0
mvar['u'].set_data(u1)
e_mic = pb.evaluate('ev_cauchy_strain.i2.Yms(u)',
mode='el_avg',
var_dict={'u': mvar['u']})
e_mic += e_mac_Yms
out = {
'u0': (macro['u'][Yms_map, :], 'u', 'p'), # macro displacement
'u1': (u1, 'u', 'p'), # local displacement corrections, see eq. (58)
'u': (u, 'u', 'p'), # total displacement
'e_mic': (e_mic, 'u', 'c'), # micro strain field, see eq. (58)
'phi': (phi, 'r', 'p'), # electric potential, see eq. (57)
'E_mic': (E_mic, 'r', 'c'), # electric field, see eq. (58)
}
return data_to_struct(out)
def recovery_micro(pb, corrs, macro):
eps0 = macro['eps0']
mesh = pb.domain.mesh
regions = pb.domain.regions
dim = mesh.dim
Ymc_map = regions['Ymc'].get_entities(0)
Ym_map = regions['Ym'].get_entities(0)
# deformation
u1, phi = 0, 0
for ii in range(2):
u1 += corrs['corrs_k%d' % ii]['u'] * macro['phi'][ii]
phi += corrs['corrs_k%d' % ii]['r'] * macro['phi'][ii]
for ii in range(dim):
for jj in range(dim):
kk = coor_to_sym(ii, jj, dim)
phi += corrs['corrs_rs']['r_%d%d' % (ii, jj)]\
* nm.expand_dims(macro['strain'][Ym_map, kk], axis=1)
u1 += corrs['corrs_rs']['u_%d%d' % (ii, jj)]\
* nm.expand_dims(macro['strain'][Ymc_map, kk], axis=1)
u = macro['u'][Ymc_map, :] + eps0 * u1
mvar = pb.create_variables(['u', 'r', 'svar'])
e_mac_Ymc = [None] * macro['strain'].shape[1]
for ii in range(dim):
for jj in range(dim):
kk = coor_to_sym(ii, jj, dim)
mvar['svar'].set_data(macro['strain'][:, kk])
mac_e_Ymc = pb.evaluate('ev_integrate.i2.Ymc(svar)',
mode='el_avg',
var_dict={'svar': mvar['svar']})
e_mac_Ymc[kk] = mac_e_Ymc.squeeze()
e_mac_Ymc = nm.vstack(e_mac_Ymc).T[:, nm.newaxis, :, nm.newaxis]
mvar['r'].set_data(phi)
E_mic = pb.evaluate(
'ev_grad.i2.Ym(r)', mode='el_avg', var_dict={'r': mvar['r']}) / eps0
mvar['u'].set_data(u1)
e_mic = pb.evaluate('ev_cauchy_strain.i2.Ymc(u)',
mode='el_avg',
var_dict={'u': mvar['u']})
e_mic += e_mac_Ymc
out = {
'u0': (macro['u'][Ymc_map, :], 'u', 'p'),
'u': (u, 'u', 'p'),
'u1': (u1, 'u', 'p'),
'e_mic': (e_mic, 'u', 'c'),
'phi': (phi, 'r', 'p'),
'E_mic': (E_mic, 'r', 'c'),
}
out_struct = {}
for k, v in out.items():
out_struct[k] = Struct(name='output_data',
mode='cell' if v[2] == 'c' else 'vertex',
data=v[0],
var_name=v[1],
dofs=None)
return out_struct
def recovery_micro(pb, corrs, macro):
eps0 = macro['eps0']
mesh = pb.domain.mesh
regions = pb.domain.regions
dim = mesh.dim
Ymc_map = regions['Ymc'].get_entities(0)
Ym_map = regions['Ym'].get_entities(0)
# deformation
u1, phi = 0, 0
for ii in range(2):
u1 += corrs['corrs_k%d' % ii]['u'] * macro['phi'][ii]
phi += corrs['corrs_k%d' % ii]['r'] * macro['phi'][ii]
for ii in range(dim):
for jj in range(dim):
kk = coor_to_sym(ii, jj, dim)
phi += corrs['corrs_rs']['r_%d%d' % (ii, jj)]\
* nm.expand_dims(macro['strain'][Ym_map, kk], axis=1)
u1 += corrs['corrs_rs']['u_%d%d' % (ii, jj)]\
* nm.expand_dims(macro['strain'][Ymc_map, kk], axis=1)
u = macro['u'][Ymc_map, :] + eps0 * u1
mvar = pb.create_variables(['u', 'r', 'svar'])
e_mac_Ymc = [None] * macro['strain'].shape[1]
for ii in range(dim):
for jj in range(dim):
kk = coor_to_sym(ii, jj, dim)
mvar['svar'].set_data(macro['strain'][:, kk])
mac_e_Ymc = pb.evaluate('ev_volume_integrate.i2.Ymc(svar)',
mode='el_avg',
var_dict={'svar': mvar['svar']})
e_mac_Ymc[kk] = mac_e_Ymc.squeeze()
e_mac_Ymc = nm.vstack(e_mac_Ymc).T[:, nm.newaxis, :, nm.newaxis]
mvar['r'].set_data(phi)
E_mic = pb.evaluate('ev_grad.i2.Ym(r)',
mode='el_avg',
var_dict={'r': mvar['r']}) / eps0
mvar['u'].set_data(u1)
e_mic = pb.evaluate('ev_cauchy_strain.i2.Ymc(u)',
mode='el_avg',
var_dict={'u': mvar['u']})
e_mic += e_mac_Ymc
out = {
'u0': (macro['u'][Ymc_map, :], 'u', 'p'),
'u': (u, 'u', 'p'),
'u1': (u1, 'u', 'p'),
'e_mic': (e_mic, 'u', 'c'),
'phi': (phi, 'r', 'p'),
'E_mic': (E_mic, 'r', 'c'),
}
out_struct = {}
for k, v in out.items():
out_struct[k] = Struct(name='output_data',
mode='cell' if v[2] == 'c' else 'vertex',
data=v[0],
var_name=v[1],
dofs=None)
return out_struct
