from __future__ import absolute_import
import numpy as nm
from sfepy.base.base import assert_, Struct
from sfepy.terms.terms import Term, terms
from sfepy.terms.terms_hyperelastic_base import\
HyperElasticBase, HyperElasticFamilyData
[docs]
class HyperElasticTLFamilyData(HyperElasticFamilyData):
"""
Family data for TL formulation.
"""
family_function = staticmethod(terms.dq_finite_strain_tl)
cache_name = 'tl_common'
data_names = ('mtx_f', 'det_f', 'sym_c', 'tr_c', 'in2_c', 'sym_inv_c',
'green_strain')
[docs]
class HyperElasticTLBase(HyperElasticBase):
"""
Base class for all hyperelastic terms in TL formulation family.
The subclasses should have the following static method attributes:
- `stress_function()` (the stress)
- `tan_mod_function()` (the tangent modulus)
The common (family) data are cached in the evaluate cache of state
variable.
"""
weak_function = staticmethod(terms.dw_he_rtm)
hyperelastic_mode = 0
get_family_data = HyperElasticTLFamilyData()
[docs]
class NeoHookeanTLTerm(HyperElasticTLBase):
r"""
Hyperelastic neo-Hookean term. Effective stress
:math:`S_{ij} = \mu J^{-\frac{2}{3}}(\delta_{ij} -
\frac{1}{3}C_{kk}C_{ij}^{-1})`.
:Definition:
.. math::
\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v})
:Arguments:
- material : :math:`\mu`
- virtual : :math:`\ul{v}`
- state : :math:`\ul{u}`
"""
name = 'dw_tl_he_neohook'
family_data_names = ['det_f', 'tr_c', 'sym_inv_c']
stress_function = staticmethod(terms.dq_tl_he_stress_neohook)
tan_mod_function = staticmethod(terms.dq_tl_he_tan_mod_neohook)
[docs]
class GenYeohTLTerm(HyperElasticTLBase):
r"""
Hyperelastic generalized Yeoh term [1]. Effective stress
:math:`S_{ij} = 2 p K (I_1 - 3)^{p-1} J^{-\frac{2}{3}}(\delta{ij} -
\frac{1}{3}C_{kk}C{ij}^{-1})`.
:Definition:
.. math::
\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v})
:Arguments:
- material : :math:`p, K`
- virtual : :math:`\ul{v}`
- state : :math:`\ul{u}`
[1] Travis W. Hohenberger, Richard J. Windslow, Nicola M. Pugno,
James J. C. Busfield. Aconstitutive Model For Both Lowand High
Strain Nonlinearities In Highly Filled Elastomers And
Implementation With User-Defined Material Subroutines In
Abaqus. Rubber Chemistry And Technology, Vol. 92, No. 4,
Pp. 653-686 (2019)
"""
name = 'dw_tl_he_genyeoh'
family_data_names = ['det_f', 'tr_c', 'sym_inv_c']
arg_shapes = {'material' : '1, 2',
'virtual' : ('D', 'state'), 'state' : 'D'}
geometries = ['3_4', '3_8']
@staticmethod
def _get_single_stress(i_1, i_3, c_inv, coef, exp):
_bracket = (i_3**(-1 / 3) * i_1 - 3.)
_bracket_pow = _bracket**(exp - 1) \
if (_bracket > 0. or exp >= 1.) else 1.
out = 2 * exp * coef * _bracket_pow \
* i_3**(-1 / 3) * (nm.eye(3) - i_1 * c_inv / 3)
return out
[docs]
def stress_function(self, out, mat, *fargs, **kwargs):
det_f, tr_c, inv_c = fargs
mat = HyperElasticBase.tile_mat(mat, det_f.shape[0])
coef, exp = mat[:, :, :, :1], mat[:, :, :, 1:]
i_3 = det_f**2
ident = nm.array([1., 1, 1, 0, 0, 0])
n_cells, n_qps, _, _ = out.shape
for cell in range(n_cells):
for qp in range(n_qps):
_inv_c = inv_c[cell, qp]
_c_inv = nm.array([
[_inv_c[0], _inv_c[3], _inv_c[4]],
[_inv_c[3], _inv_c[1], _inv_c[5]],
[_inv_c[4], _inv_c[5], _inv_c[2]],
])[:, :, 0]
_val = self._get_single_stress(
tr_c[cell, qp, 0, 0], det_f[cell, qp, 0, 0]**2, _c_inv,
coef[cell, qp, 0, 0], exp[cell, qp, 0, 0])
out[cell, qp, :, 0] = [
_val[0, 0], _val[1, 1], _val[2, 2],
_val[0, 1], _val[0, 2], _val[1, 2]]
return out
@staticmethod
def _get_single_tan_mod(i_1, i_3, c_inv, coef, exp):
krond = nm.eye(3)
_bracket = i_3**(-1/3) * i_1 - 3
_bracket_p1 = _bracket**(exp - 1) \
if (_bracket > 0. or exp >= 1.) else 1.
_bracket_p2 = _bracket**(exp - 2) \
if (_bracket > 0. or exp >= 2.) else 0.
tan_mod = nm.zeros((3, 3, 3, 3))
for ii, jj, kk, ll in zip(*[ind.flatten()
for ind in nm.indices(tan_mod.shape)]):
tan_mod[ii, jj, kk, ll] = 4 / 3 * exp * coef * (
3 * (exp - 1) * _bracket_p2 * i_3**(-2/3)
*krond[ii, jj] * krond[kk, ll]
-(
(exp - 1) * _bracket_p2 * i_1 * i_3**(-2/3)
+_bracket_p1 * i_3**(-1/3)
) * (krond[ii, jj] * c_inv[kk, ll]
+c_inv[ii, jj] * krond[kk, ll])
+(
(exp - 1) * _bracket_p2 * i_1**2 * i_3**(-2/3)
+_bracket_p1 * i_3**(-1/3) * i_1
) / 3 * c_inv[ii, jj] * c_inv[kk, ll]
+.5 * _bracket_p1 * i_3**(-1/3) * i_1 * (
c_inv[ii, kk] * c_inv[jj, ll]
+c_inv[ii, ll] * c_inv[jj, kk])
)
return tan_mod
[docs]
def tan_mod_function(self, out, mat, *fargs, **kwargs):
det_f, tr_c, inv_c = fargs
mat = HyperElasticBase.tile_mat(mat, det_f.shape[0])
coef, exp = mat[:, :, :, :1], mat[:, :, :, 1:]
n_cells, n_qps, _, _ = out.shape
for cell in range(n_cells):
for qp in range(n_qps):
_inv_c = inv_c[cell, qp, :, 0]
_c_inv = nm.array([
[_inv_c[0], _inv_c[3], _inv_c[4]],
[_inv_c[3], _inv_c[1], _inv_c[5]],
[_inv_c[4], _inv_c[5], _inv_c[2]],
])
_dh = self._get_single_tan_mod(
tr_c[cell, qp, 0, 0], det_f[cell, qp, 0, 0]**2, _c_inv,
coef[cell, qp, 0, 0], exp[cell, qp, 0, 0],
)
out[cell, qp] = nm.array([
[_dh[0, 0, 0, 0], _dh[0, 0, 1, 1], _dh[0, 0, 2, 2],
_dh[0, 0, 0, 1], _dh[0, 0, 0, 2], _dh[0, 0, 1, 2]],
[_dh[1, 1, 0, 0], _dh[1, 1, 1, 1], _dh[1, 1, 2, 2],
_dh[1, 1, 0, 1], _dh[1, 1, 0, 2], _dh[1, 1, 1, 2]],
[_dh[2, 2, 0, 0], _dh[2, 2, 1, 1], _dh[2, 2, 2, 2],
_dh[2, 2, 0, 1], _dh[2, 2, 0, 2], _dh[2, 2, 1, 2]],
[_dh[0, 1, 0, 0], _dh[0, 1, 1, 1], _dh[0, 1, 2, 2],
_dh[0, 1, 0, 1], _dh[0, 1, 0, 2], _dh[0, 1, 1, 2]],
[_dh[0, 2, 0, 0], _dh[0, 2, 1, 1], _dh[0, 2, 2, 2],
_dh[0, 2, 0, 1], _dh[0, 2, 0, 2], _dh[0, 2, 1, 2]],
[_dh[1, 2, 0, 0], _dh[1, 2, 1, 1], _dh[1, 2, 2, 2],
_dh[1, 2, 0, 1], _dh[1, 2, 0, 2], _dh[1, 2, 1, 2]],
])
[docs]
class OgdenTLTerm(HyperElasticTLBase):
r"""
Single term of the hyperelastic Ogden model [1] with the strain energy
density
.. math::
W = \frac{\mu}{\alpha} \, \left(
\lambda_1^{\alpha} + \lambda_2^{\alpha} + \lambda_3^{\alpha}
- 3 \right) \; ,
where :math:`\lambda_k, k=1, 2, 3` are the principal stretches, whose
squares are the principal values of the right Cauchy-Green deformation
tensor :math:`\mathbf{C}`.
Effective stress (2nd Piola-Kirchhoff) is [2]
.. math::
S_{ij} = 2 \, \frac{\partial W}{\partial C_{ij}} =
\sum_{k=1}^3 S^{(k)} \, N^{(k)}_i \, N^{(k)}_j \; ,
where the principal stresses are
.. math::
S^{(k)} = J^{-2/3} \, \left(
\mu \, \bar\lambda^{\alpha - 2}
-\sum_{j=1}^3 \frac{\mu}{3}
\frac{\lambda_j^{\alpha}}{\lambda_k^2} \right) \; ,
\quad k = 1, 2, 3 \; .
and :math:`\mathbf{N}^{(k)}`, :math:`k=1, 2, 3` are the eigenvectors of
:math:`\mathbf{C}`.
:Definition:
.. math::
\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v})
:Arguments:
- material : :math:`p, K`
- virtual : :math:`\ul{v}`
- state : :math:`\ul{u}`
[1] Ogden, R. W. Large deformation isotropic elasticity - on the
correlation of theory and experiment for incompressible rubberlike solids.
Proceedings of the Royal Society A, Vol. 326, No. 1567, Pp. 565-584 (1972),
DOI `10.1098/rspa.1972.0026 <https://doi.org/10.1098/rspa.1972.0026>`_.
[2] Steinmann, P., Hossain, M., Possart, G. Hyperelastic models for
rubber-like materials: Consistent tangent operators and suitability for
Treloar's data. Archive of Applied Mechanics, Vol. 82, No. 9, Pp. 1183-1217
(2012), DOI `10.1007/s00419-012-0610-z
<https://dx.doi.org/10.1007/s00419-012-0610-z>`_.
"""
name = 'dw_tl_he_ogden'
family_data_names = ['det_f', 'sym_c', 'tr_c', 'sym_inv_c']
arg_shapes = {'material' : '1, 2',
'virtual' : ('D', 'state'), 'state' : 'D'}
geometries = ['3_4', '3_8']
@staticmethod
def _get_single_stress(lbds, nks, det_f, coef, exp):
a_p = sum(lbds**exp)
s_k = [
coef * det_f**(-2. / 3) *(
lbdi**(exp - 2) - a_p / lbdi**2 / 3)
for lbdi, ni in zip(lbds, nks.T)]
out = nm.sum(
[ski * nm.outer(nki, nki) for ski, nki in zip(s_k, nks.T)], axis=0)
return out
[docs]
def stress_function(self, out, mat, *fargs, **kwargs):
det_f, sym_c, _, _ = fargs
mat = HyperElasticBase.tile_mat(mat, det_f.shape[0])
coef, exp = mat[:, :, :, :1], mat[:, :, :, 1:]
# compute principal stretches and directions
c_mats = sym_c[:, :, [[0, 3, 4], [3, 1, 5], [4, 5, 2]], 0]
lbds, nks = nm.linalg.eigh(c_mats)
lbds = lbds**.5
# evaluate stress
n_cells, n_qps, _, _ = out.shape
for cell in range(n_cells):
for qp in range(n_qps):
_val = self._get_single_stress(
lbds[cell, qp], nks[cell, qp], det_f[cell, qp, 0, 0],
coef[cell, qp, 0, 0], exp[cell, qp, 0, 0])
out[cell, qp, :, 0] = [
_val[0, 0], _val[1, 1], _val[2, 2],
_val[0, 1], _val[0, 2], _val[1, 2]]
return out
@staticmethod
def _get_single_tan_mod(total_lbds, nks, det_f, coef, exp):
lbds = det_f**(-1/3) * total_lbds
_bracket = [
(det_f**(-1 / 3) * lbdi)**(exp - 2)
-sum([1. / 3 * lbdj**exp / lbdi**2 for lbdj in lbds])
for lbdi in lbds]
s_k = nm.array([
coef * det_f**(-2 / 3) * _bracket_ii
for lbdi, _bracket_ii in zip(lbds, _bracket)])
dj_dlbd = [det_f / lbdj for lbdj in lbds]
dlbd_dlbd = det_f**(-1/3) * nm.array([[
nm.eye(3)[ii, jj] - lbds[ii] / lbds[jj] / 3
for jj in range(3)] for ii, dj_dlbdi in enumerate(dj_dlbd)])
dsk_dlbd_1 = coef * nm.array([[
-2 / 3 * det_f**(-5 / 3) * dj_dlbd[ii] * lbds[kk]**(exp - 2)
+det_f**(-2/3) * (exp - 2) * lbds[kk]**(exp - 3) * dlbd_dlbd[kk, ii]
for ii in range(3)] for kk in range(3)])
dsk_dlbd_2 = coef / 3 * nm.array([[
-2 / 3 * det_f**(-5 / 3) * dj_dlbd[ii] * nm.sum([
lbds[jj]**exp / lbds[kk]**2
for jj in range(3)])
+det_f**(-2/3) * nm.sum([
exp * lbds[jj]**(exp - 1) / lbds[kk]**2 * dlbd_dlbd[jj, ii]
-2 * lbds[jj]**exp / lbds[kk]**3 * dlbd_dlbd[kk, ii]
for jj in range(3)])
for ii in range(3)] for kk in range(3)])
dsk_dlbd = dsk_dlbd_1 - dsk_dlbd_2
tan_mod = nm.zeros((3, 3, 3, 3))
for mm, nn, pp, qq in zip(*[ind.flatten()
for ind in nm.indices(tan_mod.shape)]):
for ii, jj in zip(*[ind.flatten() for ind in nm.indices((3, 3))]):
tan_mod[mm, nn, pp, qq] += 1. / lbds[jj] * dsk_dlbd[ii, jj] * (
nks[ii, mm] * nks[ii, nn] * nks[jj, pp] * nks[jj, qq])
if ii != jj:
if lbds[ii] != lbds[jj]:
tan_mod[mm, nn, pp, qq] += (s_k[jj] - s_k[ii]) / \
(lbds[jj]**2 - lbds[ii]**2) * (
nks[ii, mm] * nks[jj, nn] * nks[ii, pp]
*nks[jj, qq]
+nks[ii, mm] * nks[jj, nn] * nks[jj, pp]
*nks[ii, qq])
else:
_val = 0.5 * (
dsk_dlbd[ii, ii] - dsk_dlbd[jj, ii]) / lbds[ii]
tan_mod[mm, nn, pp, qq] += _val * (
nks[ii, mm] * nks[jj, nn] * nks[ii, pp]
*nks[jj, qq]
+nks[ii, mm] * nks[jj, nn] * nks[jj, pp]
*nks[ii, qq])
return tan_mod
[docs]
def tan_mod_function(self, out, mat, *fargs, **kwargs):
det_f, sym_c, tr_c, inv_c = fargs
mat = HyperElasticBase.tile_mat(mat, det_f.shape[0])
coef, exp = mat[:, :, :, :1], mat[:, :, :, 1:]
# compute principal stretches and directions
c_mats = sym_c[:, :, [[0, 3, 4], [3, 1, 5], [4, 5, 2]], 0]
lbds, nks = nm.linalg.eigh(c_mats)
lbds = lbds**.5
n_cells, n_qps, _, _ = out.shape
for cell in range(n_cells):
for qp in range(n_qps):
_dh = self._get_single_tan_mod(
lbds[cell, qp], nks[cell, qp].T, det_f[cell, qp, 0, 0],
coef[cell, qp, 0, 0], exp[cell, qp, 0, 0],
)
out[cell, qp] = nm.array([
[_dh[0, 0, 0, 0], _dh[0, 0, 1, 1], _dh[0, 0, 2, 2],
_dh[0, 0, 0, 1], _dh[0, 0, 0, 2], _dh[0, 0, 1, 2]],
[_dh[1, 1, 0, 0], _dh[1, 1, 1, 1], _dh[1, 1, 2, 2],
_dh[1, 1, 0, 1], _dh[1, 1, 0, 2], _dh[1, 1, 1, 2]],
[_dh[2, 2, 0, 0], _dh[2, 2, 1, 1], _dh[2, 2, 2, 2],
_dh[2, 2, 0, 1], _dh[2, 2, 0, 2], _dh[2, 2, 1, 2]],
[_dh[0, 1, 0, 0], _dh[0, 1, 1, 1], _dh[0, 1, 2, 2],
_dh[0, 1, 0, 1], _dh[0, 1, 0, 2], _dh[0, 1, 1, 2]],
[_dh[0, 2, 0, 0], _dh[0, 2, 1, 1], _dh[0, 2, 2, 2],
_dh[0, 2, 0, 1], _dh[0, 2, 0, 2], _dh[0, 2, 1, 2]],
[_dh[1, 2, 0, 0], _dh[1, 2, 1, 1], _dh[1, 2, 2, 2],
_dh[1, 2, 0, 1], _dh[1, 2, 0, 2], _dh[1, 2, 1, 2]],
])
[docs]
class MooneyRivlinTLTerm(HyperElasticTLBase):
r"""
Hyperelastic Mooney-Rivlin term. Effective stress
:math:`S_{ij} = \kappa J^{-\frac{4}{3}} (C_{kk} \delta_{ij} - C_{ij}
- \frac{2}{3 } I_2 C_{ij}^{-1})`.
:Definition:
.. math::
\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v})
:Arguments:
- material : :math:`\kappa`
- virtual : :math:`\ul{v}`
- state : :math:`\ul{u}`
"""
name = 'dw_tl_he_mooney_rivlin'
family_data_names = ['det_f', 'tr_c', 'sym_inv_c', 'sym_c', 'in2_c']
stress_function = staticmethod(terms.dq_tl_he_stress_mooney_rivlin)
tan_mod_function = staticmethod(terms.dq_tl_he_tan_mod_mooney_rivlin)
[docs]
class BulkPenaltyTLTerm(HyperElasticTLBase):
r"""
Hyperelastic bulk penalty term. Stress
:math:`S_{ij} = K(J-1)\; J C_{ij}^{-1}`.
:Definition:
.. math::
\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v})
:Arguments:
- material : :math:`K`
- virtual : :math:`\ul{v}`
- state : :math:`\ul{u}`
"""
name = 'dw_tl_bulk_penalty'
family_data_names = ['det_f', 'sym_inv_c']
stress_function = staticmethod(terms.dq_tl_he_stress_bulk)
tan_mod_function = staticmethod(terms.dq_tl_he_tan_mod_bulk)
[docs]
class BulkActiveTLTerm(HyperElasticTLBase):
r"""
Hyperelastic bulk active term. Stress :math:`S_{ij} = A J C_{ij}^{-1}`,
where :math:`A` is the activation in :math:`[0, F_{\rm max}]`.
:Definition:
.. math::
\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v})
:Arguments:
- material : :math:`A`
- virtual : :math:`\ul{v}`
- state : :math:`\ul{u}`
"""
name = 'dw_tl_bulk_active'
family_data_names = ['det_f', 'sym_inv_c']
stress_function = staticmethod(terms.dq_tl_he_stress_bulk_active)
tan_mod_function = staticmethod(terms.dq_tl_he_tan_mod_bulk_active)
[docs]
class BulkPressureTLTerm(HyperElasticTLBase):
r"""
Hyperelastic bulk pressure term. Stress
:math:`S_{ij} = -p J C_{ij}^{-1}`.
:Definition:
.. math::
\int_{\Omega} S_{ij}(p) \delta E_{ij}(\ul{u};\ul{v})
:Arguments:
- virtual : :math:`\ul{v}`
- state : :math:`\ul{u}`
- state_p : :math:`p`
"""
name = 'dw_tl_bulk_pressure'
arg_types = ('virtual', 'state', 'state_p')
arg_geometry_types = {('state_p', None) : {'facet_extra' : 'facet'}}
arg_shapes = {'virtual' : ('D', 'state'), 'state' : 'D', 'state_p' : 1}
family_data_names = ['det_f', 'sym_inv_c']
weak_function = staticmethod(terms.dw_he_rtm)
weak_dp_function = staticmethod(terms.dw_tl_volume)
stress_function = staticmethod(terms.dq_tl_stress_bulk_pressure)
tan_mod_u_function = staticmethod(terms.dq_tl_tan_mod_bulk_pressure_u)
[docs]
def compute_data(self, family_data, mode, **kwargs):
det_f, sym_inv_c = family_data.det_f, family_data.sym_inv_c
p_qp = family_data.p_qp
if mode == 0:
out = nm.empty_like(sym_inv_c)
fun = self.stress_function
elif mode == 1:
shape = list(sym_inv_c.shape)
shape[-1] = shape[-2]
out = nm.empty(shape, dtype=nm.float64)
fun = self.tan_mod_u_function
else:
raise ValueError('bad mode! (%d)' % mode)
fun(out, p_qp, det_f, sym_inv_c)
return out
[docs]
def get_fargs(self, virtual, state, state_p,
mode=None, term_mode=None, diff_var=None, **kwargs):
vgv, _ = self.get_mapping(state)
name = state.name
fd = self.get_family_data(state, self.region, self.integral,
self.geometry_types[name],
self.arg_steps[name],
self.arg_derivatives[name])
fd.p_qp = self.get(state_p, 'val')
if mode == 'weak':
if diff_var != state_p.name:
if diff_var is None:
stress = self.compute_data(fd, 0, **kwargs)
self.stress_cache = stress
tan_mod = nm.array([0], ndmin=4, dtype=nm.float64)
fmode = 0
else:
stress = self.stress_cache
if stress is None:
stress = self.compute_data(fd, 0, **kwargs)
tan_mod = self.compute_data(fd, 1, **kwargs)
fmode = 1
fargs = (self.weak_function,
stress, tan_mod, fd.mtx_f, fd.det_f, vgv, fmode, 0)
else:
vgs, _ = self.get_mapping(state_p)
fargs = (self.weak_dp_function,
fd.mtx_f, fd.sym_inv_c, fd.det_f, vgs, vgv, 1, -1)
return fargs
elif mode == 'el_avg':
if term_mode == 'strain':
out_qp = fd.green_strain
elif term_mode == 'stress':
out_qp = self.compute_data(fd, 0, **kwargs)
else:
raise ValueError('unsupported term mode in %s! (%s)'
% (self.name, term_mode))
return self.integrate, out_qp, vgv, 1
else:
raise ValueError('unsupported evaluation mode in %s! (%s)'
% (self.name, mode))
[docs]
def get_eval_shape(self, virtual, state, state_p,
mode=None, term_mode=None, diff_var=None, **kwargs):
n_el, n_qp, dim, n_en, n_c = self.get_data_shape(state)
sym = (dim + 1) * dim // 2
return (n_el, 1, sym, 1), state.dtype
[docs]
class VolumeTLTerm(HyperElasticTLBase):
r"""
Volume term (weak form) in the total Lagrangian formulation.
:Definition:
.. math::
\begin{array}{l}
\int_{\Omega} q J(\ul{u}) \\
\mbox{volume mode: vector for } K \from \Ical_h: \int_{T_K}
J(\ul{u}) \\
\mbox{rel\_volume mode: vector for } K \from \Ical_h:
\int_{T_K} J(\ul{u}) / \int_{T_K} 1
\end{array}
:Arguments:
- virtual : :math:`q`
- state : :math:`\ul{u}`
"""
name = 'dw_tl_volume'
arg_types = ('virtual', 'state')
arg_geometry_types = {('virtual', None) : {'facet_extra' : 'facet'}}
arg_shapes = {'virtual' : (1, None), 'state' : 'D'}
family_data_names = ['mtx_f', 'det_f', 'sym_inv_c']
function = staticmethod(terms.dw_tl_volume)
[docs]
def get_fargs(self, virtual, state,
mode=None, term_mode=None, diff_var=None, **kwargs):
vgs, _ = self.get_mapping(virtual)
vgv, _ = self.get_mapping(state)
name = state.name
fd = self.get_family_data(state, self.region, self.integral,
self.geometry_types[name],
self.arg_steps[name],
self.arg_derivatives[name])
if mode == 'weak':
if diff_var is None:
fmode = 0
else:
fmode = 1
elif (mode == 'eval') or (mode == 'el_avg'):
if term_mode == 'volume':
fmode = 2
elif term_mode == 'rel_volume':
fmode = 3
else:
raise ValueError('unsupported term evaluation mode in %s! (%s)'
% (self.name, term_mode))
else:
raise ValueError('unsupported evaluation mode in %s! (%s)'
% (self.name, mode))
return fd.mtx_f, fd.sym_inv_c, fd.det_f, vgs, vgv, 0, fmode
[docs]
def get_eval_shape(self, virtual, state,
mode=None, term_mode=None, diff_var=None, **kwargs):
n_el, n_qp, dim, n_en, n_c = self.get_data_shape(state)
return (n_el, 1, 1, 1), state.dtype
[docs]
class DiffusionTLTerm(HyperElasticTLBase):
r"""
Diffusion term in the total Lagrangian formulation with
linearized deformation-dependent permeability
:math:`\ull{K}(\ul{u}) = J \ull{F}^{-1} \ull{k} f(J) \ull{F}^{-T}`,
where :math:`\ul{u}` relates to the previous time step :math:`(n-1)`
and
:math:`f(J) = \max\left(0, \left(1 + \frac{(J - 1)}{N_f}\right)\right)^2`
expresses the dependence on volume compression/expansion.
:Definition:
.. math::
\int_{\Omega} \ull{K}(\ul{u}^{(n-1)}) : \pdiff{q}{\ul{X}}
\pdiff{p}{\ul{X}}
:Arguments:
- material_1 : :math:`\ull{k}`
- material_2 : :math:`N_f`
- virtual : :math:`q`
- state : :math:`p`
- parameter : :math:`\ul{u}^{(n-1)}`
"""
name = 'dw_tl_diffusion'
arg_types = ('material_1', 'material_2', 'virtual', 'state', 'parameter')
arg_shapes = {'material_1' : 'D, D', 'material_2' : '1, 1',
'virtual' : (1, 'state'), 'state' : 1, 'parameter' : 'D'}
family_data_names = ['mtx_f', 'det_f']
function = staticmethod(terms.dw_tl_diffusion)
[docs]
def get_fargs(self, perm, ref_porosity, virtual, state, parameter,
mode=None, term_mode=None, diff_var=None, **kwargs):
vgv, _ = self.get_mapping(parameter)
name = parameter.name
fd = self.get_family_data(parameter, self.region, self.integral,
self.geometry_types[name],
self.arg_steps[name],
self.arg_derivatives[name])
grad = self.get(state, 'grad')
if mode == 'weak':
if diff_var is None:
fmode = 0
else:
fmode = 1
elif mode == 'el_avg':
if term_mode == 'diffusion_velocity':
fmode = 2
else:
raise ValueError('unsupported term evaluation mode in %s! (%s)'
% (self.name, term_mode))
else:
raise ValueError('unsupported evaluation mode in %s! (%s)'
% (self.name, mode))
return grad, perm, ref_porosity, fd.mtx_f, fd.det_f, vgv, fmode
[docs]
def get_eval_shape(self, perm, ref_porosity, virtual, state, parameter,
mode=None, term_mode=None, diff_var=None, **kwargs):
n_el, n_qp, dim, n_en, n_c = self.get_data_shape(state)
return (n_el, 1, dim, 1), state.dtype
[docs]
class HyperElasticSurfaceTLFamilyData(HyperElasticFamilyData):
"""
Family data for TL formulation applicable for surface terms.
"""
family_function = staticmethod(terms.dq_tl_finite_strain_surface)
cache_name = 'tl_surface_common'
data_names = ('mtx_f', 'det_f', 'inv_f')
def __call__(self, state, region, integral, geometry_type,
step=0, derivative=None):
sg, _ = state.field.get_mapping(region,
integral, geometry_type[0],
get_saved=True)
sd = state.field.extra_data[f'sd_{region.name}']
vec = state(step=step, derivative=derivative)
st_shape = state.get_data_shape(integral, geometry_type[0],
region.name)
data = self.init_data_struct(st_shape, name='surface_family_data')
fargs = tuple([getattr(data, k) for k in self.data_names])
fargs = fargs + (vec, sg, sd.fis, state.field.econn)
fargs = Term.translate_fargs_mapping(self.family_function,
list(fargs))
self.family_function(*fargs)
return data
[docs]
class HyperElasticSurfaceTLBase(HyperElasticTLBase):
"""
Base class for all hyperelastic surface terms in TL formulation family.
"""
get_family_data = HyperElasticSurfaceTLFamilyData()
[docs]
class SurfaceFluxTLTerm(HyperElasticSurfaceTLBase):
r"""
Surface flux term in the total Lagrangian formulation, consistent with
:class:`DiffusionTLTerm`.
:Definition:
.. math::
\int_{\Gamma} \ul{\nu} \cdot \ull{K}(\ul{u}^{(n-1)}) \pdiff{p}{\ul{X}}
:Arguments:
- material_1 : :math:`\ull{k}`
- material_2 : :math:`N_f`
- parameter_1 : :math:`p`
- parameter_2 : :math:`\ul{u}^{(n-1)}`
"""
name = 'ev_tl_surface_flux'
arg_types = ('material_1', 'material_2', 'parameter_1', 'parameter_2')
arg_shapes = {'material_1' : 'D, D', 'material_2' : '1, 1',
'parameter_1' : 1, 'parameter_2' : 'D'}
family_data_names = ['det_f', 'inv_f']
integration = 'facet_extra'
function = staticmethod(terms.d_tl_surface_flux)
[docs]
def get_fargs(self, perm, ref_porosity, pressure, displacement,
mode=None, term_mode=None, diff_var=None, **kwargs):
sg, _ = self.get_mapping(displacement)
name = displacement.name
fd = self.get_family_data(displacement, self.region, self.integral,
self.geometry_types[name],
self.arg_steps[name],
self.arg_derivatives[name])
grad = self.get(pressure, 'grad')
fmode = {'eval' : 0, 'el_avg' : 1}.get(mode, 0)
return grad, perm, ref_porosity, fd.inv_f, fd.det_f, sg, fmode
[docs]
def get_eval_shape(self, perm, ref_porosity, pressure, displacement,
mode=None, term_mode=None, diff_var=None, **kwargs):
n_fa, n_qp, dim, n_en, n_c = self.get_data_shape(displacement)
return (n_fa, 1, 1, 1), pressure.dtype
[docs]
class SurfaceTractionTLTerm(HyperElasticSurfaceTLBase):
r"""
Surface traction term in the total Lagrangian formulation, expressed
using :math:`\ul{\nu}`, the outward unit normal vector w.r.t. the
undeformed surface, :math:`\ull{F}(\ul{u})`, the deformation gradient,
:math:`J = \det(\ull{F})`, and :math:`\ull{\sigma}` a given traction,
often equal to a given pressure, i.e.
:math:`\ull{\sigma} = \pi \ull{I}`.
:Definition:
.. math::
\int_{\Gamma} \ul{\nu} \cdot \ull{F}^{-1} \cdot \ull{\sigma} \cdot
\ul{v} J
:Arguments:
- material : :math:`\ull{\sigma}`
- virtual : :math:`\ul{v}`
- state : :math:`\ul{u}`
"""
name = 'dw_tl_surface_traction'
arg_types = ('opt_material', 'virtual', 'state')
arg_shapes = [{'opt_material' : 'D, D', 'virtual' : ('D', 'state'),
'state' : 'D'},
{'opt_material' : None}]
family_data_names = ['det_f', 'inv_f']
integration = 'facet_extra'
function = staticmethod(terms.dw_tl_surface_traction)
[docs]
def get_fargs(self, mat, virtual, state,
mode=None, term_mode=None, diff_var=None, **kwargs):
sg, _ = self.get_mapping(virtual)
sd = virtual.field.extra_data[f'sd_{self.region.name}']
bf = virtual.field.eval_basis(sd.bkey, 0, self.integral)
name = state.name
fd = self.get_family_data(state, self.region, self.integral,
self.geometry_types[name],
self.arg_steps[name],
self.arg_derivatives[name])
if mat is None:
eye = nm.eye(sg.dim, dtype=nm.float64)
mat = nm.tile(eye, ((1, sg.n_qp, 1, 1)))
if diff_var is None:
fmode = 0
else:
fmode = 1
return mat, fd.det_f, fd.inv_f, bf, sg, sd.fis, fmode
[docs]
class VolumeSurfaceTLTerm(HyperElasticSurfaceTLBase):
r"""
Volume of a :math:`D`-dimensional domain, using a surface integral in the
total Lagrangian formulation, expressed using :math:`\ul{\nu}`, the outward
unit normal vector w.r.t. the undeformed surface, :math:`\ull{F}(\ul{u})`,
the deformation gradient, and :math:`J = \det(\ull{F})`. Uses the
approximation of :math:`\ul{u}` for the deformed surface coordinates
:math:`\ul{x}`.
:Definition:
.. math::
1 / D \int_{\Gamma} \ul{\nu} \cdot \ull{F}^{-1} \cdot \ul{x} J
:Arguments:
- parameter : :math:`\ul{u}`
"""
name = 'ev_tl_volume_surface'
arg_types = ('parameter',)
arg_shapes = {'parameter' : 'D'}
family_data_names = ['det_f', 'inv_f']
integration = 'facet_extra'
function = staticmethod(terms.d_tl_volume_surface)
[docs]
def get_fargs(self, parameter,
mode=None, term_mode=None, diff_var=None, **kwargs):
sg, _ = self.get_mapping(parameter)
sd = parameter.field.extra_data[f'sd_{self.region.name}']
bf = parameter.field.eval_basis(sd.bkey, 0, self.integral)
name = parameter.name
fd = self.get_family_data(parameter, self.region, self.integral,
self.geometry_types[name],
self.arg_steps[name],
self.arg_derivatives[name])
asc = nm.ascontiguousarray
coors0 = parameter.field.get_coor()
coors = asc(coors0 + parameter().reshape(coors0.shape))
return coors, fd.det_f, fd.inv_f, bf, sg, asc(sd.econn)
[docs]
def get_eval_shape(self, parameter,
mode=None, term_mode=None, diff_var=None, **kwargs):
n_el, n_qp, dim, n_en, n_c = self.get_data_shape(parameter)
return (n_el, 1, 1, 1), parameter.dtype