# -*- coding: utf-8 -*-
"""
This file is part of FElupe.
FElupe is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
FElupe is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with FElupe. If not, see <http://www.gnu.org/licenses/>.
"""
import numpy as np
try:
from einsumt import einsumt
except ModuleNotFoundError:
from numpy import einsum as einsumt
from ..math import cdya_ik, cdya_il, det, dot, dya, identity, inv, transpose
[docs]
class LineChange:
r"""Line Change.
.. math::
d\boldsymbol{x} = \boldsymbol{F} d\boldsymbol{X}
Gradient:
.. math::
\frac{\partial \boldsymbol{F}}{\partial \boldsymbol{F}} =
\boldsymbol{I} \overset{ik}{\otimes} \boldsymbol{I}
"""
def __init__(self, parallel=False):
self.parallel = parallel
[docs]
def function(self, extract):
"""Line change.
Arguments
---------
extract : list of ndarray
List of extracted field values with Deformation gradient as first
item.
Returns
-------
F : ndarray
Deformation gradient
"""
return extract
[docs]
def gradient(self, extract, parallel=None):
"""Gradient of line change.
Arguments
---------
extract : list of ndarray
List of extracted field values with Deformation gradient as first
item.
Returns
-------
ndarray
Gradient of line change
"""
F = extract[0]
if parallel is None:
parallel = self.parallel
Eye = identity(F)
return [cdya_ik(Eye, Eye, parallel=parallel)]
[docs]
class AreaChange:
r"""Area Change.
.. math::
d\boldsymbol{a} = J \boldsymbol{F}^{-T} d\boldsymbol{A}
Gradient:
.. math::
\frac{\partial J \boldsymbol{F}^{-T}}{\partial \boldsymbol{F}} =
J \left( \boldsymbol{F}^{-T} \otimes \boldsymbol{F}^{-T}
- \boldsymbol{F}^{-T} \overset{il}{\otimes} \boldsymbol{F}^{-T} \right)
"""
def __init__(self, parallel=False):
self.parallel = parallel
[docs]
def function(self, extract, N=None, parallel=None):
"""Area change.
Arguments
---------
extract : list of ndarray
List of extracted field values with Deformation gradient as first
item.
N : ndarray or None, optional
Area normal vector (default is None)
Returns
-------
ndarray
Cofactor matrix of the deformation gradient
"""
F = extract[0]
J = det(F)
Fs = J * transpose(inv(F, J))
if parallel is None:
parallel = self.parallel
if N is None:
return [Fs]
else:
return [dot(Fs, N, mode=(2, 1), parallel=parallel)]
[docs]
def gradient(self, extract, N=None, parallel=None):
"""Gradient of area change.
Arguments
---------
extract : list of ndarray
List of extracted field values with Deformation gradient as first
item.
N : ndarray or None, optional
Area normal vector (default is None)
Returns
-------
ndarray
Gradient of cofactor matrix of the deformation gradient
"""
F = extract[0]
J = det(F)
if parallel is None:
parallel = self.parallel
dJdF = self.function([F])[0]
dFsdF = (
dya(dJdF, dJdF, parallel=parallel) - cdya_il(dJdF, dJdF, parallel=parallel)
) / J
if parallel:
einsum = einsumt
else:
einsum = np.einsum
if N is None:
return [dFsdF]
else:
return [einsum("ijkl...,j...->ikl...", dFsdF, N)]
[docs]
class VolumeChange:
r"""Volume Change.
.. math::
d\boldsymbol{v} = \text{det}(\boldsymbol{F}) d\boldsymbol{V}
Gradient and hessian (equivalent to gradient of area change):
.. math::
\frac{\partial J}{\partial \boldsymbol{F}} &= J \boldsymbol{F}^{-T}
\frac{\partial^2 J}{\partial \boldsymbol{F}\ \partial \boldsymbol{F}} &=
J \left( \boldsymbol{F}^{-T} \otimes \boldsymbol{F}^{-T} -
\boldsymbol{F}^{-T} \overset{il}{\otimes} \boldsymbol{F}^{-T} \right)
"""
def __init__(self, parallel=False):
self.parallel = parallel
[docs]
def function(self, extract):
"""Gradient of volume change.
Arguments
---------
extract : list of ndarray
List of extracted field values with Deformation gradient as first
item.
Returns
-------
J : ndarray
Determinant of the deformation gradient
"""
F = extract[0]
return [det(F)]
[docs]
def gradient(self, extract):
"""Gradient of volume change.
Arguments
---------
F : ndarray
Deformation gradient
Returns
-------
ndarray
Gradient of the determinant of the deformation gradient
"""
F = extract[0]
J = self.function([F])[0]
return [J * transpose(inv(F, J))]
[docs]
def hessian(self, extract, parallel=None):
"""Hessian of volume change.
Arguments
---------
extract : list of ndarray
List of extracted field values with Deformation gradient as first
item.
Returns
-------
ndarray
Hessian of the determinant of the deformation gradient
"""
F = extract[0]
if parallel is None:
parallel = self.parallel
J = self.function([F])[0]
dJdF = self.gradient([F])[0]
return [
(
dya(dJdF, dJdF, parallel=parallel)
- cdya_il(dJdF, dJdF, parallel=parallel)
)
/ J
]
