Source code for felupe.constitution._kinematics

# -*- coding: utf-8 -*-
"""
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|   |___ |   |___ |   |    |  |_|  ||   |_| ||   |___ 
|    ___||    ___||   |___ |       ||    ___||    ___|
|   |    |   |___ |       ||       ||   |    |   |___ 
|___|    |_______||_______||_______||___|    |_______|

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
from einsumt import 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, 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 ]