import numpy as nm
from sfepy.linalg import dot_sequences
from sfepy.terms.terms import Term, terms
from sfepy.terms.terms_multilinear import ETermBase
[docs]class PiezoCouplingTerm(Term):
r"""
Piezoelectric coupling term. Can be evaluated.
:Definition:
.. math::
\int_{\Omega} g_{kij}\ e_{ij}(\ul{v}) \nabla_k p\\
\int_{\Omega} g_{kij}\ e_{ij}(\ul{u}) \nabla_k q
:Arguments 1:
- material: :math:`g_{kij}`
- virtual/parameter_v: :math:`\ul{v}`
- state/parameter_s: :math:`p`
:Arguments 2:
- material : :math:`g_{kij}`
- state : :math:`\ul{u}`
- virtual : :math:`q`
"""
name = 'dw_piezo_coupling'
arg_types = (('material', 'virtual', 'state'),
('material', 'state', 'virtual'),
('material', 'parameter_v', 'parameter_s'))
arg_shapes = {'material' : 'D, S',
'virtual/grad' : ('D', None), 'state/grad' : 1,
'virtual/div' : (1, None), 'state/div' : 'D',
'parameter_v' : 'D', 'parameter_s' : 1}
modes = ('grad', 'div', 'eval')
[docs] def get_fargs(self, mat, vvar, svar,
mode=None, term_mode=None, diff_var=None, **kwargs):
if self.mode == 'grad':
qp_var, qp_name = svar, 'grad'
else:
qp_var, qp_name = vvar, 'cauchy_strain'
vvg, _ = self.get_mapping(vvar)
if mode == 'weak':
aux = nm.array([0], ndmin=4, dtype=nm.float64)
if diff_var is None:
# grad or strain according to mode.
val_qp = self.get(qp_var, qp_name)
fmode = 0
else:
val_qp = aux
fmode = 1
if self.mode == 'grad':
strain, grad = aux, val_qp
else:
strain, grad = val_qp, aux
fmode += 2
return strain, grad, mat, vvg, fmode
elif mode == 'eval':
strain = self.get(vvar, 'cauchy_strain')
grad = self.get(svar, 'grad')
return strain, grad, mat, vvg
else:
raise ValueError('unsupported evaluation mode in %s! (%s)'
% (self.name, mode))
[docs] def get_eval_shape(self, mat, vvar, svar,
mode=None, term_mode=None, diff_var=None, **kwargs):
n_el, n_qp, dim, n_en, n_c = self.get_data_shape(vvar)
return (n_el, 1, 1, 1), vvar.dtype
[docs] def set_arg_types( self ):
self.function = {
'grad' : terms.dw_piezo_coupling,
'div' : terms.dw_piezo_coupling,
'eval' : terms.d_piezo_coupling,
}[self.mode]
[docs]class SDPiezoCouplingTerm(ETermBase):
r"""
Sensitivity (shape derivative) of the piezoelectric coupling term.
:Definition:
.. math::
\int_{\Omega} \hat{g}_{kij}\ e_{ij}(\ul{u}) \nabla_k p
.. math::
\hat{g}_{kij} = g_{kij}(\nabla \cdot \ul{\Vcal})
- g_{kil}{\partial \Vcal_j \over \partial x_l}
- g_{lij}{\partial \Vcal_k \over \partial x_l}
:Arguments:
- material : :math:`g_{kij}`
- parameter_u : :math:`\ul{u}`
- parameter_p : :math:`p`
- parameter_mv : :math:`\ul{\Vcal}`
"""
name = 'ev_sd_piezo_coupling'
arg_types = ('material', 'parameter_u', 'parameter_p', 'parameter_mv')
arg_shapes = {'material': 'D, S', 'parameter_u': 'D', 'parameter_p': 1,
'parameter_mv': 'D'}
geometries = ['2_3', '2_4', '3_4', '3_8']
[docs] def get_function(self, mat, par_u, par_p, par_mv,
mode=None, term_mode=None, diff_var=None, **kwargs):
grad_mv = self.get(par_mv, 'grad')
div_mv = self.get(par_mv, 'div')
nel, nqp, dim, _ = mat.shape
remap = nm.array([0, 2, 2, 1] if dim == 2 else [0, 3, 4, 3, 1, 5, 4, 5, 2])
mat_f = mat[:, :, :, remap].reshape((nel, nqp, dim, dim, dim))
## grad_mv = [v11, v21; v12, v22]
sa_mat_f = mat_f * div_mv[..., nm.newaxis]
sa_mat_f -= nm.einsum('qpkil,qplj->qpkij', mat_f, grad_mv,
optimize='greedy')
sa_mat_f -= nm.einsum('qplij,qplk->qpkij', mat_f, grad_mv,
optimize='greedy')
return self.make_function(
'kij,i.j,0.k', (sa_mat_f, 'material'), par_u, par_p,
diff_var=diff_var
)
[docs]class PiezoStressTerm(Term):
r"""
Evaluate piezoelectric stress tensor.
It is given in the usual vector form exploiting symmetry: in 3D it has 6
components with the indices ordered as :math:`[11, 22, 33, 12, 13, 23]`, in
2D it has 3 components with the indices ordered as :math:`[11, 22, 12]`.
Supports 'eval', 'el_avg' and 'qp' evaluation modes.
:Definition:
.. math::
\int_{\Omega} g_{kij} \nabla_k p
:Arguments:
- material : :math:`g_{kij}`
- parameter : :math:`p`
"""
name = 'ev_piezo_stress'
arg_types = ('material', 'parameter')
arg_shapes = {'material' : 'D, S', 'parameter' : '1'}
[docs] @staticmethod
def function(out, val_qp, vg, fmode):
if fmode == 2:
out[:] = val_qp
status = 0
else:
status = vg.integrate(out, val_qp, fmode)
return status
[docs] def get_fargs(self, mat, parameter,
mode=None, term_mode=None, diff_var=None, **kwargs):
vg, _ = self.get_mapping(parameter)
grad = self.get(parameter, 'grad')
val_qp = dot_sequences(mat, grad, mode='ATB')
fmode = {'eval' : 0, 'el_avg' : 1, 'qp' : 2}.get(mode, 1)
return val_qp, vg, fmode
[docs] def get_eval_shape(self, mat, parameter,
mode=None, term_mode=None, diff_var=None, **kwargs):
n_el, n_qp, dim, n_en, n_c = self.get_data_shape(parameter)
if mode != 'qp':
n_qp = 1
return (n_el, n_qp, dim * (dim + 1) // 2, 1), parameter.dtype
[docs]class PiezoStrainTerm(PiezoStressTerm):
r"""
Evaluate piezoelectric strain tensor.
It is given in the usual vector form exploiting symmetry: in 3D it has 6
components with the indices ordered as :math:`[11, 22, 33, 12, 13, 23]`, in
2D it has 3 components with the indices ordered as :math:`[11, 22, 12]`.
Supports 'eval', 'el_avg' and 'qp' evaluation modes.
:Definition:
.. math::
\int_{\Omega} g_{kij} e_{ij}(\ul{u})
:Arguments:
- material : :math:`g_{kij}`
- parameter : :math:`\ul{u}`
"""
name = 'ev_piezo_strain'
arg_shapes = {'material' : 'D, S', 'parameter' : 'D'}
[docs] def get_fargs(self, mat, parameter,
mode=None, term_mode=None, diff_var=None, **kwargs):
vg, _ = self.get_mapping(parameter)
strain = self.get(parameter, 'cauchy_strain')
val_qp = dot_sequences(mat, strain, mode='AB')
fmode = {'eval': 0, 'el_avg': 1, 'qp': 2}.get(mode, 1)
return val_qp, vg, fmode
[docs] def get_eval_shape(self, mat, parameter,
mode=None, term_mode=None, diff_var=None, **kwargs):
n_el, n_qp, dim, n_en, n_c = self.get_data_shape(parameter)
if mode != 'qp':
n_qp = 1
return (n_el, n_qp, dim, 1), parameter.dtype