outsourcing tensor math to mechanics class
strain calculation is generalize to arbitrary order and simplified: No need for svd, F^T F/F F^T does the job.
This commit is contained in:
Martin Diehl
parent
e51f6cee72
commit
b31de5d0f6
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@ -360,7 +360,7 @@ class DADF5():
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def cell_coordinates(self):
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def cell_coordinates(self):
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"""Initial coordinates of the cell centers."""
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"""Return initial coordinates of the cell centers."""
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if self.structured:
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if self.structured:
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delta = self.size/self.grid*0.5
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delta = self.size/self.grid*0.5
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z, y, x = np.meshgrid(np.linspace(delta[2],self.size[2]-delta[2],self.grid[2]),
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z, y, x = np.meshgrid(np.linspace(delta[2],self.size[2]-delta[2],self.grid[2]),
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@ -375,62 +375,72 @@ class DADF5():
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def add_Cauchy(self,P='P',F='F'):
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def add_Cauchy(self,P='P',F='F'):
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"""
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"""
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Adds Cauchy stress calculated from 1st Piola-Kirchhoff stress and deformation gradient.
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Add Cauchy stress calculated from 1. Piola-Kirchhoff stress and deformation gradient.
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Resulting tensor is symmetrized as the Cauchy stress should be symmetric.
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Parameters
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----------
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P : str, optional
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Label of the dataset containing the 1. Piola-Kirchhoff stress. Default value is ‘P’.
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F : str, optional
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Label of the dataset containing the deformation gradient. Default value is ‘F’.
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"""
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"""
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def Cauchy(F,P):
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def __add_Cauchy(F,P):
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return {
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'data' : mechanics.Cauchy(F['data'],P['data']),
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return {
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'label' : 'sigma',
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'data': mechanics.Cauchy(F['data'],P['data']),
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'meta' : {
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'label': 'sigma',
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'Unit' : P['meta']['Unit'],
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'meta': {
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'Description' : 'Cauchy stress calculated from {} ({}) '.format(P['label'],P['meta']['Description'])+\
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'Unit': P['meta']['Unit'],
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'and deformation gradient {} ({})'.format(F['label'],F['meta']['Description']),
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'Description': 'Cauchy stress calculated from {} ({}) '.format(P['label'],P['meta']['Description'])+\
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'Creator' : 'dadf5.py:add_Cauchy v{}'.format(version)
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'and deformation gradient {} ({})'.format(F['label'],F['meta']['Description']),
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'Creator': 'dadf5.py:add_Cauchy v{}'.format(version)
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}
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}
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}
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}
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requested = [{'label':F,'arg':'F'},
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requested = [{'label':F,'arg':'F'},
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{'label':P,'arg':'P'} ]
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{'label':P,'arg':'P'} ]
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self.__add_generic_pointwise(Cauchy,requested)
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self.__add_generic_pointwise(__add_Cauchy,requested)
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def add_Mises(self,x):
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def add_Mises(self,x):
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"""Adds the equivalent Mises stress or strain of a tensor."""
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"""
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def Mises(x):
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Add the equivalent Mises stress or strain of a symmetric tensor.
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if x['meta']['Unit'] == 'Pa': #ToDo: Should we use this? Then add_Cauchy and add_strain_tensors also should perform sanity checks
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Parameters
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t = 'stress'
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----------
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elif x['meta']['Unit'] == '1':
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x : str
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t = 'strain'
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Label of the dataset containing a symmetric stress or strain tensor
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else:
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"""
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print(x['meta']['Unit'])
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def __add_Mises(x):
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raise ValueError
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return {
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return {
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'data' : mechanics.Mises_strain(x['data']) if t=='strain' else mechanics.Mises_stress(x['data']),
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'data': mechanics.Mises_strain(x) if t=='strain' else mechanics.Mises_stress(x),
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'label' : '{}_vM'.format(x['label']),
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'label': '{}_vM'.format(x['label']),
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'meta' : {
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'meta': {
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'Unit' : x['meta']['Unit'],
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'Unit': x['meta']['Unit'],
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'Description' : 'Mises equivalent {} of {} ({})'.format(t,x['label'],x['meta']['Description']),
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'Description': 'Mises equivalent {} of {} ({})'.format(t,x['label'],x['meta']['Description']),
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'Creator' : 'dadf5.py:add_Mises_stress v{}'.format(version)
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'Creator': 'dadf5.py:add_Mises v{}'.format(version)
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}
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}
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}
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}
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requested = [{'label':x,'arg':'x'}]
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requested = [{'label':x,'arg':'x'}]
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self.__add_generic_pointwise(Mises,requested)
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self.__add_generic_pointwise(__add_Mises,requested)
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def add_norm(self,x,ord=None):
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def add_norm(self,x,ord=None):
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"""
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"""
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Adds norm of vector or tensor.
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Add the norm of vector or tensor.
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See numpy.linalg.norm manual for details.
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Parameters
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----------
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x : str
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Label of the dataset containing a vector or tensor.
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ord : {non-zero int, inf, -inf, ‘fro’, ‘nuc’}, optional
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Order of the norm. inf means numpy’s inf object. For details refer to numpy.linalg.norm.
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"""
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"""
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def norm(x,ord):
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def __add_norm(x,ord):
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o = ord
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o = ord
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if len(x['data'].shape) == 2:
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if len(x['data'].shape) == 2:
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@ -444,220 +454,258 @@ class DADF5():
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else:
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else:
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raise ValueError
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raise ValueError
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return {
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return {
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'data' : np.linalg.norm(x['data'],ord=o,axis=axis,keepdims=True),
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'data': np.linalg.norm(x['data'],ord=o,axis=axis,keepdims=True),
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'label' : '|{}|_{}'.format(x['label'],o),
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'label': '|{}|_{}'.format(x['label'],o),
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'meta' : {
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'meta': {
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'Unit' : x['meta']['Unit'],
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'Unit': x['meta']['Unit'],
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'Description' : '{}-Norm of {} {} ({})'.format(ord,t,x['label'],x['meta']['Description']),
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'Description': '{}-Norm of {} {} ({})'.format(ord,t,x['label'],x['meta']['Description']),
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'Creator' : 'dadf5.py:add_norm v{}'.format(version)
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'Creator': 'dadf5.py:add_norm v{}'.format(version)
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}
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}
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}
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}
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requested = [{'label':x,'arg':'x'}]
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requested = [{'label':x,'arg':'x'}]
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self.__add_generic_pointwise(norm,requested,{'ord':ord})
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self.__add_generic_pointwise(__add_norm,requested,{'ord':ord})
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def add_absolute(self,x):
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def add_absolute(self,x):
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"""Adds absolute value."""
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"""
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def absolute(x):
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Add absolute value.
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return {
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Parameters
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'data' : np.abs(x['data']),
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----------
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'label' : '|{}|'.format(x['label']),
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x : str
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'meta' : {
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Label of the dataset containing a scalar, vector, or tensor.
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'Unit' : x['meta']['Unit'],
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"""
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'Description' : 'Absolute value of {} ({})'.format(x['label'],x['meta']['Description']),
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def __add_absolute(x):
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'Creator' : 'dadf5.py:add_abs v{}'.format(version)
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return {
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'data': np.abs(x['data']),
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'label': '|{}|'.format(x['label']),
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'meta': {
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'Unit': x['meta']['Unit'],
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'Description': 'Absolute value of {} ({})'.format(x['label'],x['meta']['Description']),
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'Creator': 'dadf5.py:add_abs v{}'.format(version)
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}
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}
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}
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}
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requested = [{'label':x,'arg':'x'}]
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requested = [{'label':x,'arg':'x'}]
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self.__add_generic_pointwise(absolute,requested)
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self.__add_generic_pointwise(__add_absolute,requested)
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def add_determinant(self,x):
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def add_determinant(self,x):
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"""Adds the determinant component of a tensor."""
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"""
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def determinant(x):
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Add the determinant of a tensor.
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return {
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Parameters
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'data' : np.linalg.det(x['data']),
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----------
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'label' : 'det({})'.format(x['label']),
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x : str
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'meta' : {
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Label of the dataset containing a tensor.
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'Unit' : x['meta']['Unit'],
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"""
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'Description' : 'Determinant of tensor {} ({})'.format(x['label'],x['meta']['Description']),
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def __add_determinant(x):
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'Creator' : 'dadf5.py:add_determinant v{}'.format(version)
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return {
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'data': np.linalg.det(x['data']),
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'label': 'det({})'.format(x['label']),
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'meta': {
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'Unit': x['meta']['Unit'],
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'Description': 'Determinant of tensor {} ({})'.format(x['label'],x['meta']['Description']),
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'Creator': 'dadf5.py:add_determinant v{}'.format(version)
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}
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}
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}
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}
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requested = [{'label':x,'arg':'x'}]
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requested = [{'label':x,'arg':'x'}]
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self.__add_generic_pointwise(determinant,requested)
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self.__add_generic_pointwise(__add_determinant,requested)
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def add_spherical(self,x):
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def add_spherical(self,x):
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"""Adds the spherical component of a tensor."""
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"""
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def spherical(x):
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Add the spherical (hydrostatic) part of a tensor.
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Parameters
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----------
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x : str
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Label of the dataset containing a tensor.
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"""
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def __add_spherical(x):
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if not np.all(np.array(x['data'].shape[1:]) == np.array([3,3])):
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if not np.all(np.array(x['data'].shape[1:]) == np.array([3,3])):
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raise ValueError
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raise ValueError
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return {
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return {
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'data' : np.trace(x['data'],axis1=1,axis2=2)/3.0,
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'data': mechanics.spherical_part(x),
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'label' : 'sph({})'.format(x['label']),
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'label': 'p_{}'.format(x['label']),
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'meta' : {
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'meta': {
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'Unit' : x['meta']['Unit'],
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'Unit': x['meta']['Unit'],
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'Description' : 'Spherical component of tensor {} ({})'.format(x['label'],x['meta']['Description']),
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'Description': 'Spherical component of tensor {} ({})'.format(x['label'],x['meta']['Description']),
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'Creator' : 'dadf5.py:add_spherical v{}'.format(version)
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'Creator': 'dadf5.py:add_spherical v{}'.format(version)
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}
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}
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}
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}
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requested = [{'label':x,'arg':'x'}]
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requested = [{'label':x,'arg':'x'}]
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self.__add_generic_pointwise(spherical,requested)
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self.__add_generic_pointwise(__add_spherical,requested)
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def add_deviator(self,x):
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def add_deviator(self,x):
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"""Adds the deviator of a tensor."""
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"""
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def deviator(x):
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Add the deviatoric part of a tensor.
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Parameters
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----------
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x : str
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Label of the dataset containing a tensor.
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"""
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def __add_deviator(x):
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if not np.all(np.array(x['data'].shape[1:]) == np.array([3,3])):
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if not np.all(np.array(x['data'].shape[1:]) == np.array([3,3])):
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raise ValueError
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raise ValueError
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return {
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return {
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'data' : mechanics.deviator(x['data']),
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'data': mechanics.deviator(x['data']),
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'label' : 'dev({})'.format(x['label']),
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'label': 's_{}'.format(x['label']),
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'meta' : {
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'meta': {
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'Unit' : x['meta']['Unit'],
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'Unit': x['meta']['Unit'],
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'Description' : 'Deviator of tensor {} ({})'.format(x['label'],x['meta']['Description']),
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'Description': 'Deviator of tensor {} ({})'.format(x['label'],x['meta']['Description']),
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'Creator' : 'dadf5.py:add_deviator v{}'.format(version)
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'Creator': 'dadf5.py:add_deviator v{}'.format(version)
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}
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}
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}
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}
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requested = [{'label':x,'arg':'x'}]
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requested = [{'label':x,'arg':'x'}]
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self.__add_generic_pointwise(deviator,requested)
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self.__add_generic_pointwise(__add_deviator,requested)
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def add_calculation(self,formula,label,unit='n/a',description=None,vectorized=True):
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def add_calculation(self,formula,label,unit='n/a',description=None,vectorized=True):
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"""
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"""
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General formula.
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Add result of a general formula.
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Works currently only for vectorized expressions
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Parameters
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----------
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formula : str
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Formula, refer to datasets by ‘#Label#‘.
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label : str
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Label of the dataset containing the result of the calculation.
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unit : str, optional
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Physical unit of the result.
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description : str, optional
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Human readable description of the result.
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vectorized : bool, optional
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Indicate whether the formula is written in vectorized form.
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"""
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"""
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if vectorized is not True:
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if vectorized is not True:
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raise NotImplementedError
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raise NotImplementedError
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def calculation(**kwargs):
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def __add_calculation(**kwargs):
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formula = kwargs['formula']
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formula = kwargs['formula']
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for d in re.findall(r'#(.*?)#',formula):
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for d in re.findall(r'#(.*?)#',formula):
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formula = formula.replace('#{}#'.format(d),"kwargs['{}']['data']".format(d))
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formula = formula.replace('#{}#'.format(d),"kwargs['{}']['data']".format(d))
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return {
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return {
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'data' : eval(formula),
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'data': eval(formula),
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'label' : kwargs['label'],
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'label': kwargs['label'],
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'meta' : {
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'meta': {
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'Unit' : kwargs['unit'],
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'Unit': kwargs['unit'],
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'Description' : '{} (formula: {})'.format(kwargs['description'],kwargs['formula']),
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'Description': '{} (formula: {})'.format(kwargs['description'],kwargs['formula']),
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'Creator' : 'dadf5.py:add_calculation v{}'.format(version)
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'Creator': 'dadf5.py:add_calculation v{}'.format(version)
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}
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}
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}
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}
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requested = [{'label':d,'arg':d} for d in set(re.findall(r'#(.*?)#',formula))] # datasets used in the formula
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requested = [{'label':d,'arg':d} for d in set(re.findall(r'#(.*?)#',formula))] # datasets used in the formula
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pass_through = {'formula':formula,'label':label,'unit':unit,'description':description}
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pass_through = {'formula':formula,'label':label,'unit':unit,'description':description}
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self.__add_generic_pointwise(calculation,requested,pass_through)
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self.__add_generic_pointwise(__add_calculation,requested,pass_through)
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def add_strain_tensor(self,t,ord,defgrad='F'):
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def add_strain_tensor(self,F='F',t='U',ord=0):
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"""
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"""
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Adds the a strain tensor.
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Add strain tensor calculated from a deformation gradient.
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Albrecht Bertram: Elasticity and Plasticity of Large Deformations An Introduction (3rd Edition, 2012), p. 102.
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For details refer to damask.mechanics.strain_tensor
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Parameters
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----------
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F : str, optional
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Label of the dataset containing the deformation gradient. Default value is ‘F’.
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t : {‘V’, ‘U’}, optional
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Type of the polar decomposition, ‘V’ for right stretch tensor and ‘U’ for left stretch tensor.
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Defaults value is ‘U’.
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ord : float, optional
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Order of the strain calculation. Default value is ‘0.0’.
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"""
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"""
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def strain_tensor(defgrad,t,ord):
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def __add_strain_tensor(F,t,ord):
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operator = {
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return {
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'V#ln': lambda V: np.log(V),
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'data': mechanics.strain_tensor(F['data'],t,ord),
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'U#ln': lambda U: np.log(U),
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'label': 'epsilon_{}^{}({})'.format(t,ord,F['label']),
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'V#Biot': lambda V: np.broadcast_to(np.ones(3),[V.shape[0],3]) - 1.0/V,
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'meta': {
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'U#Biot': lambda U: U - np.broadcast_to(np.ones(3),[U.shape[0],3]),
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'Unit': F['meta']['Unit'],
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'V#Green':lambda V: np.broadcast_to(np.ones(3),[V.shape[0],3]) - 1.0/V**2,
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'Description': 'Strain tensor of {} ({})'.format(F['label'],F['meta']['Description']),
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'U#Green':lambda U: U**2 - np.broadcast_to(np.ones(3),[U.shape[0],3]),
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'Creator': 'dadf5.py:add_strain_tensor v{}'.format(version)
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}
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if t.lower() in ['l','left']:
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stretch = 'V'
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elif t.lower() in ['r','right']:
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stretch = 'U'
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else:
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raise KeyError
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(U,S,Vh) = np.linalg.svd(defgrad['data']) # singular value decomposition
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|
||||||
R_inv = np.transpose(np.matmul(U,Vh),(0,2,1)) # transposed rotation of polar decomposition
|
|
||||||
s = np.matmul(R_inv,defgrad['data']) if stretch == 'U' else \
|
|
||||||
np.matmul(defgrad['data'],R_inv) # compute either left or right stretch
|
|
||||||
(D,V) = np.linalg.eigh((s+np.transpose(s,(0,2,1)))*.5) # eigen decomposition (of symmetric(ed) matrix)
|
|
||||||
|
|
||||||
d = operator[stretch+'#'+{0:'ln',1:'Biot',2:'Green'}[ord]](D)
|
|
||||||
a = np.matmul(V,np.einsum('ij,ikj->ijk',d,V))
|
|
||||||
|
|
||||||
return {
|
|
||||||
'data' : a,
|
|
||||||
'label' : 'epsilon_{}^{}({})'.format(stretch,ord,defgrad['label']),
|
|
||||||
'meta' : {
|
|
||||||
'Unit' : defgrad['meta']['Unit'],
|
|
||||||
'Description' : 'Strain tensor of {} ({})'.format(defgrad['label'],defgrad['meta']['Description']),
|
|
||||||
'Creator' : 'dadf5.py:add_strain_tensor v{}'.format(version)
|
|
||||||
}
|
}
|
||||||
}
|
}
|
||||||
|
|
||||||
requested = [{'label':defgrad,'arg':'defgrad'}]
|
requested = [{'label':F,'arg':'F'}]
|
||||||
|
|
||||||
self.__add_generic_pointwise(strain_tensor,requested,{'t':t,'ord':ord})
|
self.__add_generic_pointwise(__add_strain_tensor,requested,{'t':t,'ord':ord})
|
||||||
|
|
||||||
|
|
||||||
def add_principal_components(self,x):
|
def add_principal_components(self,x):
|
||||||
"""Adds principal components of symmetric tensor."""
|
"""
|
||||||
def principal_components(x):
|
Add principal components of symmetric tensor.
|
||||||
|
|
||||||
return {
|
The principal components are sorted in descending order, each repeated according to its multiplicity.
|
||||||
'data' : np.linalg.eigvalsh((x['data']+np.transpose(x['data'],(0,2,1)))*.5)[:,::-1], # eigenvalues (of symmetric(ed) matrix)
|
|
||||||
'label' : 'lambda_{}'.format(x['label']),
|
Parameters
|
||||||
'meta' : {
|
----------
|
||||||
'Unit' : x['meta']['Unit'],
|
x : str
|
||||||
'Description' : 'Pricipal components of {} ({})'.format(x['label'],x['meta']['Description']),
|
Label of the dataset containing a symmetric tensor.
|
||||||
'Creator' : 'dadf5.py:add_principal_components v{}'.format(version)
|
"""
|
||||||
|
def __add_principal_components(x):
|
||||||
|
|
||||||
|
return {
|
||||||
|
'data': mechanics.principal_components(x),
|
||||||
|
'label': 'lambda_{}'.format(x['label']),
|
||||||
|
'meta': {
|
||||||
|
'Unit': x['meta']['Unit'],
|
||||||
|
'Description': 'Pricipal components of {} ({})'.format(x['label'],x['meta']['Description']),
|
||||||
|
'Creator': 'dadf5.py:add_principal_components v{}'.format(version)
|
||||||
}
|
}
|
||||||
}
|
}
|
||||||
|
|
||||||
requested = [{'label':x,'arg':'x'}]
|
requested = [{'label':x,'arg':'x'}]
|
||||||
|
|
||||||
self.__add_generic_pointwise(principal_components,requested)
|
self.__add_generic_pointwise(__add_principal_components,requested)
|
||||||
|
|
||||||
|
|
||||||
def add_maximum_shear(self,x):
|
def add_maximum_shear(self,x):
|
||||||
"""Adds maximum shear components of symmetric tensor."""
|
"""
|
||||||
def maximum_shear(x):
|
Add maximum shear components of symmetric tensor.
|
||||||
|
|
||||||
w = np.linalg.eigvalsh((x['data']+np.transpose(x['data'],(0,2,1)))*.5) # eigenvalues (of symmetric(ed) matrix)
|
Parameters
|
||||||
|
----------
|
||||||
|
x : str
|
||||||
|
Label of the dataset containing a symmetric tensor.
|
||||||
|
"""
|
||||||
|
def __add_maximum_shear(x):
|
||||||
|
|
||||||
return {
|
return {
|
||||||
'data' : (w[:,2] - w[:,0])*0.5,
|
'data': mechanics.maximum_shear(x),
|
||||||
'label' : 'max_shear({})'.format(x['label']),
|
'label': 'max_shear({})'.format(x['label']),
|
||||||
'meta' : {
|
'meta': {
|
||||||
'Unit' : x['meta']['Unit'],
|
'Unit': x['meta']['Unit'],
|
||||||
'Description' : 'Maximum shear component of of {} ({})'.format(x['label'],x['meta']['Description']),
|
'Description': 'Maximum shear component of of {} ({})'.format(x['label'],x['meta']['Description']),
|
||||||
'Creator' : 'dadf5.py:add_maximum_shear v{}'.format(version)
|
'Creator': 'dadf5.py:add_maximum_shear v{}'.format(version)
|
||||||
}
|
}
|
||||||
}
|
}
|
||||||
|
|
||||||
requested = [{'label':x,'arg':'x'}]
|
requested = [{'label':x,'arg':'x'}]
|
||||||
|
|
||||||
self.__add_generic_pointwise(maximum_shear,requested)
|
self.__add_generic_pointwise(__add_maximum_shear,requested)
|
||||||
|
|
||||||
|
|
||||||
def __add_generic_pointwise(self,func,datasets_requested,extra_args={}):
|
def __add_generic_pointwise(self,func,datasets_requested,extra_args={}):
|
||||||
|
|
@ -666,13 +714,12 @@ class DADF5():
|
||||||
|
|
||||||
Parameters
|
Parameters
|
||||||
----------
|
----------
|
||||||
func : function
|
func : function
|
||||||
Function that calculates a new dataset from one or more datasets per HDF5 group.
|
Function that calculates a new dataset from one or more datasets per HDF5 group.
|
||||||
datasets_requested : list of dictionaries
|
datasets_requested : list of dictionaries
|
||||||
Details of the datasets to be used: label (in HDF5 file) and arg (argument to which the data is parsed in func).
|
Details of the datasets to be used: label (in HDF5 file) and arg (argument to which the data is parsed in func).
|
||||||
extra_args : dictionary, optional
|
extra_args : dictionary, optional
|
||||||
Any extra arguments parsed to func.
|
Any extra arguments parsed to func.
|
||||||
|
|
||||||
"""
|
"""
|
||||||
def job(args):
|
def job(args):
|
||||||
"""Call function with input data + extra arguments, returns results + group."""
|
"""Call function with input data + extra arguments, returns results + group."""
|
||||||
|
|
|
||||||
|
|
@ -2,84 +2,166 @@ import numpy as np
|
||||||
|
|
||||||
def Cauchy(F,P):
|
def Cauchy(F,P):
|
||||||
"""
|
"""
|
||||||
Calculate Cauchy stress from 1st Piola-Kirchhoff stress and deformation gradient.
|
Return Cauchy stress calculated from 1. Piola-Kirchhoff stress and deformation gradient.
|
||||||
|
|
||||||
Resulting tensor is symmetrized as the Cauchy stress needs to be symmetric.
|
Resulting tensor is symmetrized as the Cauchy stress needs to be symmetric.
|
||||||
|
|
||||||
Parameters
|
Parameters
|
||||||
----------
|
----------
|
||||||
F : numpy.array of shape (x,3,3) or (3,3)
|
F : numpy.array of shape (x,3,3) or (3,3)
|
||||||
Deformation gradient.
|
Deformation gradient.
|
||||||
P : numpy.array of shape (x,3,3) or (3,3)
|
P : numpy.array of shape (x,3,3) or (3,3)
|
||||||
1st Piola-Kirchhoff.
|
1. Piola-Kirchhoff stress.
|
||||||
"""
|
"""
|
||||||
if np.shape(F) == np.shape(P) == (3,3):
|
if np.shape(F) == np.shape(P) == (3,3):
|
||||||
sigma = 1.0/np.linalg.det(F) * np.dot(F,P)
|
sigma = 1.0/np.linalg.det(F) * np.dot(F,P)
|
||||||
return (sigma+sigma.T)*0.5
|
|
||||||
else:
|
else:
|
||||||
sigma = np.einsum('i,ijk,ilk->ijl',1.0/np.linalg.det(F),P,F)
|
sigma = np.einsum('i,ijk,ilk->ijl',1.0/np.linalg.det(F),P,F)
|
||||||
return (sigma + np.transpose(sigma,(0,2,1)))*0.5
|
return symmetric(sigma)
|
||||||
|
|
||||||
|
|
||||||
|
def strain_tensor(F,t,ord):
|
||||||
|
"""
|
||||||
|
Return strain tensor calculated from deformation gradient.
|
||||||
|
|
||||||
|
For details refer to Albrecht Bertram: Elasticity and Plasticity of Large Deformations:
|
||||||
|
An Introduction (3rd Edition, 2012), p. 102.
|
||||||
|
|
||||||
|
Parameters
|
||||||
|
----------
|
||||||
|
F : numpy.array of shape (x,3,3) or (3,3)
|
||||||
|
Deformation gradient.
|
||||||
|
t : {‘V’, ‘U’}
|
||||||
|
Type of the polar decomposition, ‘V’ for right stretch tensor and ‘U’ for left stretch tensor.
|
||||||
|
ord : float
|
||||||
|
Order of the strain
|
||||||
|
"""
|
||||||
|
F_expanded = F if len(F.shape) == 3 else F.reshape(1,3,3)
|
||||||
|
|
||||||
|
if t == 'U':
|
||||||
|
B = np.matmul(F_expanded,transpose(F_expanded))
|
||||||
|
U,n = np.linalg.eigh(symmetric(B))
|
||||||
|
l = np.log(U) if ord == 0 else U**ord - np.broadcast_to(np.ones(3),[U.shape[0],3])
|
||||||
|
elif t == 'V':
|
||||||
|
C = np.matmul(transpose(F_expanded),F_expanded)
|
||||||
|
V,n = np.linalg.eigh(symmetric(C))
|
||||||
|
l = np.log(V) if ord == 0 else np.broadcast_to(np.ones(3),[V.shape[0],3]) - 1.0/V**ord
|
||||||
|
|
||||||
|
epsilon = np.matmul(n,np.einsum('ij,ikj->ijk',l,n))
|
||||||
|
|
||||||
|
return epsilon.reshape((3,3)) if np.shape(F) == (3,3) else \
|
||||||
|
epsilon
|
||||||
|
|
||||||
|
|
||||||
def deviatoric_part(x):
|
def deviatoric_part(x):
|
||||||
"""
|
"""
|
||||||
Calculate deviatoric part of a tensor.
|
Return deviatoric part of a tensor.
|
||||||
|
|
||||||
Parameters
|
Parameters
|
||||||
----------
|
----------
|
||||||
x : numpy.array of shape (x,3,3) or (3,3)
|
x : numpy.array of shape (x,3,3) or (3,3)
|
||||||
Tensor.
|
Tensor.
|
||||||
"""
|
"""
|
||||||
if np.shape(x) == (3,3):
|
return x - np.eye(3)*spherical_part(x) if np.shape(x) == (3,3) else \
|
||||||
return x - np.eye(3)*np.trace(x)/3.0
|
x - np.einsum('ijk,i->ijk',np.broadcast_to(np.eye(3),[x.shape[0],3,3]),spherical_part(x))
|
||||||
else:
|
|
||||||
return x - np.einsum('ijk,i->ijk',np.broadcast_to(np.eye(3),[x.shape[0],3,3]),np.trace(x,axis1=1,axis2=2)/3.0)
|
|
||||||
|
|
||||||
|
|
||||||
def spherical_part(x):
|
def spherical_part(x):
|
||||||
"""
|
"""
|
||||||
Calculate spherical(hydrostatic) part of a tensor.
|
Return spherical (hydrostatic) part of a tensor.
|
||||||
|
|
||||||
A single scalar is returned, i.e. the hydrostatic part is not mapped on the 3rd order identity matrix.
|
A single scalar is returned, i.e. the hydrostatic part is not mapped on the 3rd order identity
|
||||||
|
matrix.
|
||||||
|
|
||||||
Parameters
|
Parameters
|
||||||
----------
|
----------
|
||||||
x : numpy.array of shape (x,3,3) or (3,3)
|
x : numpy.array of shape (x,3,3) or (3,3)
|
||||||
Tensor.
|
Tensor.
|
||||||
"""
|
"""
|
||||||
if np.shape(x) == (3,3):
|
|
||||||
return np.trace(x)/3.0
|
return np.trace(x)/3.0 if np.shape(x) == (3,3) else \
|
||||||
else:
|
np.trace(x,axis1=1,axis2=2)/3.0
|
||||||
return np.trace(x,axis1=1,axis2=2)/3.0
|
|
||||||
|
|
||||||
|
|
||||||
def Mises_stress(sigma):
|
def Mises_stress(sigma):
|
||||||
"""
|
"""
|
||||||
Calculate the Mises equivalent of a stress tensor.
|
Return the Mises equivalent of a stress tensor.
|
||||||
|
|
||||||
Parameters
|
Parameters
|
||||||
----------
|
----------
|
||||||
sigma : numpy.array of shape (x,3,3) or (3,3)
|
sigma : numpy.array of shape (x,3,3) or (3,3)
|
||||||
Symmetric stress tensor.
|
Symmetric stress tensor.
|
||||||
"""
|
"""
|
||||||
s = deviatoric_part(sigma)
|
s = deviatoric_part(sigma)
|
||||||
if np.shape(sigma) == (3,3):
|
return np.sqrt(3.0/2.0*np.trace(s)) if np.shape(sigma) == (3,3) else \
|
||||||
return np.sqrt(3.0/2.0*np.trace(s))
|
np.sqrt(3.0/2.0*np.einsum('ijk->i',s))
|
||||||
else:
|
|
||||||
return np.sqrt(np.einsum('ijk->i',s)*3.0/2.0)
|
|
||||||
|
|
||||||
|
|
||||||
def Mises_strain(epsilon):
|
def Mises_strain(epsilon):
|
||||||
"""
|
"""
|
||||||
Calculate the Mises equivalent of a strain tensor.
|
Return the Mises equivalent of a strain tensor.
|
||||||
|
|
||||||
Parameters
|
Parameters
|
||||||
----------
|
----------
|
||||||
sigma : numpy.array of shape (x,3,3) or (3,3)
|
epsilon : numpy.array of shape (x,3,3) or (3,3)
|
||||||
Symmetric strain tensor.
|
Symmetric strain tensor.
|
||||||
"""
|
"""
|
||||||
s = deviatoric_part(epsilon)
|
s = deviatoric_part(epsilon)
|
||||||
if np.shape(epsilon) == (3,3):
|
return np.sqrt(2.0/3.0*np.trace(s)) if np.shape(epsilon) == (3,3) else \
|
||||||
return np.sqrt(2.0/3.0*np.trace(s))
|
np.sqrt(2.0/3.0*np.einsum('ijk->i',s))
|
||||||
else:
|
|
||||||
return np.sqrt(2.0/3.0*np.einsum('ijk->i',s))
|
|
||||||
|
def symmetric(x):
|
||||||
|
"""
|
||||||
|
Return the symmetrized tensor.
|
||||||
|
|
||||||
|
Parameters
|
||||||
|
----------
|
||||||
|
x : numpy.array of shape (x,3,3) or (3,3)
|
||||||
|
Tensor.
|
||||||
|
"""
|
||||||
|
return (x+transpose(x))*0.5
|
||||||
|
|
||||||
|
|
||||||
|
def maximum_shear(x):
|
||||||
|
"""
|
||||||
|
Return the maximum shear component of a symmetric tensor.
|
||||||
|
|
||||||
|
Parameters
|
||||||
|
----------
|
||||||
|
x : numpy.array of shape (x,3,3) or (3,3)
|
||||||
|
Symmetric tensor.
|
||||||
|
"""
|
||||||
|
w = np.linalg.eigvalsh(symmetric(x)) # eigenvalues in ascending order
|
||||||
|
return (w[2] - w[0])*0.5 if np.shape(epsilon) == (3,3) else \
|
||||||
|
(w[:,2] - w[:,0])*0.5
|
||||||
|
|
||||||
|
|
||||||
|
def principal_components(x):
|
||||||
|
"""
|
||||||
|
Return the principal components of a symmetric tensor.
|
||||||
|
|
||||||
|
The principal components (eigenvalues) are sorted in descending order, each repeated according to
|
||||||
|
its multiplicity.
|
||||||
|
|
||||||
|
Parameters
|
||||||
|
----------
|
||||||
|
x : numpy.array of shape (x,3,3) or (3,3)
|
||||||
|
Symmetric tensor.
|
||||||
|
"""
|
||||||
|
w = np.linalg.eigvalsh(symmetric(x)) # eigenvalues in ascending order
|
||||||
|
return w[::-1] if np.shape(epsilon) == (3,3) else \
|
||||||
|
w[:,::-1]
|
||||||
|
|
||||||
|
|
||||||
|
def transpose(x):
|
||||||
|
"""
|
||||||
|
Return the transpose of a tensor.
|
||||||
|
|
||||||
|
Parameters
|
||||||
|
----------
|
||||||
|
x : numpy.array of shape (x,3,3) or (3,3)
|
||||||
|
Tensor.
|
||||||
|
"""
|
||||||
|
return x.T if np.shape(x) == (3,3) else \
|
||||||
|
np.transpose(x,(0,2,1))
|
||||||
|
|
|
||||||
Loading…
Reference in New Issue