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more systematic names and extended docstrings

This commit is contained in:
Martin Diehl 2020-11-16 01:01:32 +01:00
parent 6f81f5278d
commit b893967b68
5 changed files with 182 additions and 71 deletions
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2
python/damask/_result.py
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@ -1024,7 +1024,7 @@ class Result:
@staticmethod
def _add_stretch_tensor(F,t):
return {
'data': (mechanics.left_stretch if t.upper() == 'V' else mechanics.right_stretch)(F['data']),
'data': (mechanics.stretch_left if t.upper() == 'V' else mechanics.stretch_right)(F['data']),
'label': f"{t}({F['label']})",
'meta': {
'Unit': F['meta']['Unit'],

186
python/damask/mechanics.py
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@ -5,18 +5,58 @@ from . import tensor
import numpy as _np
def Cauchy(P,F):
def Cauchy_Green_deformation_left(F):
"""
Return Cauchy stress calculated from first Piola-Kirchhoff stress and deformation gradient.
Resulting tensor is symmetrized as the Cauchy stress needs to be symmetric.
Calculate left Cauchy-Green deformation tensor (Finger deformation tensor).
Parameters
----------
F : numpy.ndarray of shape (...,3,3)
Deformation gradient.
Returns
-------
B : numpy.ndarray of shape (...,3,3)
Left Cauchy-Green deformation tensor.
"""
return _np.matmul(F,tensor.transpose(F))
def Cauchy_Green_deformation_right(F):
"""
Calculate right Cauchy-Green deformation tensor.
Parameters
----------
F : numpy.ndarray of shape (...,3,3)
Deformation gradient.
Returns
-------
C : numpy.ndarray of shape (...,3,3)
Right Cauchy-Green deformation tensor.
"""
return _np.matmul(tensor.transpose(F),F)
def Cauchy(P,F):
"""
Calculate the Cauchy (true) stress.
Resulting tensor is symmetrized as the Cauchy stress needs to be symmetric.
Parameters
----------
P : numpy.ndarray of shape (...,3,3)
First Piola-Kirchhoff stress.
F : numpy.ndarray of shape (...,3,3)
Deformation gradient.
Returns
-------
sigma : numpy.ndarray of shape (...,3,3)
Cauchy stress.
"""
sigma = _np.einsum('...,...ij,...kj->...ik',1.0/_np.linalg.det(F),P,F)
@ -25,39 +65,36 @@ def Cauchy(P,F):
def deviatoric_part(T):
"""
Return deviatoric part of a tensor.
Calculate deviatoric part of a tensor.
Parameters
----------
T : numpy.ndarray of shape (...,3,3)
Tensor of which the deviatoric part is computed.
Returns
-------
T' : numpy.ndarray of shape (...,3,3)
Deviatoric part of T.
"""
return T - _np.einsum('...ij,...->...ij',_np.eye(3),spherical_part(T))
def left_stretch(T):
"""
Return the left stretch of a tensor.
Parameters
----------
T : numpy.ndarray of shape (...,3,3)
Tensor of which the left stretch is computed.
"""
return _polar_decomposition(T,'V')[0]
def maximum_shear(T_sym):
"""
Return the maximum shear component of a symmetric tensor.
Calculate the maximum shear component of a symmetric tensor.
Parameters
----------
T_sym : numpy.ndarray of shape (...,3,3)
Symmetric tensor of which the maximum shear is computed.
Returns
-------
gamma_max : numpy.ndarray of shape (...)
Maximum shear of T_sym.
"""
w = tensor.eigenvalues(T_sym)
return (w[...,0] - w[...,2])*0.5
@ -65,33 +102,46 @@ def maximum_shear(T_sym):
def Mises_strain(epsilon):
"""
Return the Mises equivalent of a strain tensor.
Calculate the Mises equivalent of a strain tensor.
Parameters
----------
epsilon : numpy.ndarray of shape (...,3,3)
Symmetric strain tensor of which the von Mises equivalent is computed.
Returns
-------
epsilon_vM : numpy.ndarray of shape (...)
Von Mises equivalent strain of epsilon.
"""
return _Mises(epsilon,2.0/3.0)
def Mises_stress(sigma):
"""
Return the Mises equivalent of a stress tensor.
Calculate the Mises equivalent of a stress tensor.
Parameters
----------
sigma : numpy.ndarray of shape (...,3,3)
Symmetric stress tensor of which the von Mises equivalent is computed.
Returns
-------
sigma_vM : numpy.ndarray of shape (...)
Von Mises equivalent stress of sigma.
"""
return _Mises(sigma,3.0/2.0)
def PK2(P,F):
"""
Calculate second Piola-Kirchhoff stress from first Piola-Kirchhoff stress and deformation gradient.
Calculate the second Piola-Kirchhoff stress.
Resulting tensor is symmetrized as the second Piola-Kirchhoff stress
needs to be symmetric.
Parameters
----------
@ -100,47 +150,51 @@ def PK2(P,F):
F : numpy.ndarray of shape (...,3,3)
Deformation gradient.
Returns
-------
S : numpy.ndarray of shape (...,3,3)
Second Piola-Kirchhoff stress.
"""
S = _np.einsum('...jk,...kl->...jl',_np.linalg.inv(F),P)
return tensor.symmetric(S)
def right_stretch(T):
"""
Return the right stretch of a tensor.
Parameters
----------
T : numpy.ndarray of shape (...,3,3)
Tensor of which the right stretch is computed.
"""
return _polar_decomposition(T,'U')[0]
def rotational_part(T):
"""
Return the rotational part of a tensor.
Calculate the rotational part of a tensor.
Parameters
----------
T : numpy.ndarray of shape (...,3,3)
Tensor of which the rotational part is computed.
Returns
-------
R : numpy.ndarray of shape (...,3,3)
Rotational part.
"""
return _polar_decomposition(T,'R')[0]
def spherical_part(T,tensor=False):
"""
Return spherical (hydrostatic) part of a tensor.
Calculate spherical (hydrostatic) part of a tensor.
Parameters
----------
T : numpy.ndarray of shape (...,3,3)
Tensor of which the hydrostatic part is computed.
tensor : bool, optional
Map spherical part onto identity tensor. Default is false
Map spherical part onto identity tensor. Defaults to false
Returns
-------
p : numpy.ndarray of shape (...)
unless tensor == True: shape (...,3,3)
Spherical part of tensor T, e.g. the hydrostatic part/pressure
of a stress tensor.
"""
sph = _np.trace(T,axis2=-2,axis1=-1)/3.0
@ -149,7 +203,7 @@ def spherical_part(T,tensor=False):
def strain_tensor(F,t,m):
"""
Return strain tensor calculated from deformation gradient.
Calculate strain tensor from deformation gradient.
For details refer to https://en.wikipedia.org/wiki/Finite_strain_theory and
https://de.wikipedia.org/wiki/Verzerrungstensor
@ -159,17 +213,21 @@ def strain_tensor(F,t,m):
F : numpy.ndarray of shape (...,3,3)
Deformation gradient.
t : {‘V’, ‘U’}
Type of the polar decomposition, ‘V’ for left stretch tensor and ‘U’ for right stretch tensor.
Type of the polar decomposition, ‘V’ for left stretch tensor
and ‘U’ for right stretch tensor.
m : float
Order of the strain.
Returns
-------
epsilon : numpy.ndarray of shape (...,3,3)
Strain of F.
"""
if t == 'V':
B = _np.matmul(F,tensor.transpose(F))
w,n = _np.linalg.eigh(B)
w,n = _np.linalg.eigh(Cauchy_Green_deformation_left(F))
elif t == 'U':
C = _np.matmul(tensor.transpose(F),F)
w,n = _np.linalg.eigh(C)
w,n = _np.linalg.eigh(Cauchy_Green_deformation_right(F))
if m > 0.0:
eps = 1.0/(2.0*abs(m)) * (+ _np.matmul(n,_np.einsum('...j,...kj->...jk',w**m,n))
@ -184,9 +242,45 @@ def strain_tensor(F,t,m):
return eps
def stretch_left(T):
"""
Calculate left stretch of a tensor.
Parameters
----------
T : numpy.ndarray of shape (...,3,3)
Tensor of which the left stretch is computed.
Returns
-------
V : numpy.ndarray of shape (...,3,3)
Left stretch tensor from Polar decomposition of T.
"""
return _polar_decomposition(T,'V')[0]
def stretch_right(T):
"""
Calculate right stretch of a tensor.
Parameters
----------
T : numpy.ndarray of shape (...,3,3)
Tensor of which the right stretch is computed.
Returns
-------
U : numpy.ndarray of shape (...,3,3)
Left stretch tensor from Polar decomposition of T.
"""
return _polar_decomposition(T,'U')[0]
def _polar_decomposition(T,requested):
"""
Singular value decomposition.
Perform singular value decomposition.
Parameters
----------
@ -197,7 +291,7 @@ def _polar_decomposition(T,requested):
‘V’ for left stretch tensor and ‘U’ for right stretch tensor.
"""
u, s, vh = _np.linalg.svd(T)
u, _, vh = _np.linalg.svd(T)
R = _np.einsum('...ij,...jk->...ik',u,vh)
output = []

37
python/damask/tensor.py
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@ -13,58 +13,75 @@ import numpy as _np
def symmetric(T):
"""
Return the symmetrized tensor.
Symmetrize tensor.
Parameters
----------
T : numpy.ndarray of shape (...,3,3)
Tensor of which the symmetrized values are computed.
Returns
-------
T_sym : numpy.ndarray of shape (...,3,3)
Symmetrized tensor T.
"""
return (T+transpose(T))*0.5
def transpose(T):
"""
Return the transpose of a tensor.
Transpose tensor.
Parameters
----------
T : numpy.ndarray of shape (...,3,3)
Tensor of which the transpose is computed.
Returns
-------
T.T : numpy.ndarray of shape (...,3,3)
Transpose of tensor T.
"""
return _np.swapaxes(T,axis2=-2,axis1=-1)
def eigenvalues(T_sym):
"""
Return the eigenvalues, i.e. principal components, of a symmetric tensor.
The eigenvalues are sorted in ascending order, each repeated according to
its multiplicity.
Eigenvalues, i.e. principal components, of a symmetric tensor.
Parameters
----------
T_sym : numpy.ndarray of shape (...,3,3)
Symmetric tensor of which the eigenvalues are computed.
Returns
-------
lambda : numpy.ndarray of shape (...,3)
Eigenvalues of T_sym sorted in ascending order, each repeated
according to its multiplicity.
"""
return _np.linalg.eigvalsh(symmetric(T_sym))
def eigenvectors(T_sym,RHS=False):
"""
Return eigenvectors of a symmetric tensor.
The eigenvalues are sorted in ascending order of their associated eigenvalues.
Eigenvectors of a symmetric tensor.
Parameters
----------
T_sym : numpy.ndarray of shape (...,3,3)
Symmetric tensor of which the eigenvectors are computed.
RHS: bool, optional
Enforce right-handed coordinate system. Default is False.
Enforce right-handed coordinate system. Defaults to False.
Returns
-------
x : numpy.ndarray of shape (...,3,3)
Eigenvectors of T_sym sorted in ascending order of their
associated eigenvalues.
"""
(u,v) = _np.linalg.eigh(symmetric(T_sym))

4
python/tests/test_Result.py
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@ -292,7 +292,7 @@ class TestResult:
default.add_stretch_tensor('F','U')
loc = {'F': default.get_dataset_location('F'),
'U(F)': default.get_dataset_location('U(F)')}
in_memory = mechanics.right_stretch(default.read_dataset(loc['F'],0))
in_memory = mechanics.stretch_right(default.read_dataset(loc['F'],0))
in_file = default.read_dataset(loc['U(F)'],0)
assert np.allclose(in_memory,in_file)
@ -300,7 +300,7 @@ class TestResult:
default.add_stretch_tensor('F','V')
loc = {'F': default.get_dataset_location('F'),
'V(F)': default.get_dataset_location('V(F)')}
in_memory = mechanics.left_stretch(default.read_dataset(loc['F'],0))
in_memory = mechanics.stretch_left(default.read_dataset(loc['F'],0))
in_file = default.read_dataset(loc['V(F)'],0)
assert np.allclose(in_memory,in_file)

24
python/tests/test_mechanics.py
View File

@ -17,10 +17,6 @@ def eigenvalues(T_sym):
return np.linalg.eigvalsh(symmetric(T_sym))
def left_stretch(T):
return polar_decomposition(T,'V')[0]
def maximum_shear(T_sym):
w = eigenvalues(T_sym)
return (w[0] - w[2])*0.5
@ -39,10 +35,6 @@ def PK2(P,F):
return symmetric(S)
def right_stretch(T):
return polar_decomposition(T,'U')[0]
def rotational_part(T):
return polar_decomposition(T,'R')[0]
@ -73,6 +65,14 @@ def strain_tensor(F,t,m):
return eps
def stretch_left(T):
return polar_decomposition(T,'V')[0]
def stretch_right(T):
return polar_decomposition(T,'U')[0]
def symmetric(T):
return (T+T.T)*0.5
@ -113,13 +113,13 @@ class TestMechanics:
assert np.allclose(single(test_data_flat[i]),v)
@pytest.mark.parametrize('vectorized,single',[(mechanics.deviatoric_part, deviatoric_part),
(mechanics.left_stretch , left_stretch ),
(mechanics.maximum_shear , maximum_shear ),
(mechanics.Mises_strain , Mises_strain ),
(mechanics.Mises_stress , Mises_stress ),
(mechanics.right_stretch , right_stretch ),
(mechanics.rotational_part, rotational_part),
(mechanics.spherical_part , spherical_part ),
(mechanics.stretch_left , stretch_left ),
(mechanics.stretch_right , stretch_right ),
])
def test_vectorize_1_arg(self,vectorized,single):
epsilon = np.random.rand(self.n,3,3)
@ -166,8 +166,8 @@ class TestMechanics:
"""F = RU = VR."""
F = np.broadcast_to(np.eye(3),[self.n,3,3])*np.random.rand(self.n,3,3)
R = mechanics.rotational_part(F)
V = mechanics.left_stretch(F)
U = mechanics.right_stretch(F)
V = mechanics.stretch_left(F)
U = mechanics.stretch_right(F)
assert np.allclose(np.matmul(R,U),
np.matmul(V,R))