eigenvalues is more specific name than principal components
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Martin Diehl
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@ -3,9 +3,9 @@ import numpy as np
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def Cauchy(F,P):
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"""
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Return Cauchy stress calculated from 1. Piola-Kirchhoff stress and deformation gradient.
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Resulting tensor is symmetrized as the Cauchy stress needs to be symmetric.
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Parameters
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----------
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F : numpy.array of shape (:,3,3) or (3,3)
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@ -24,7 +24,7 @@ def Cauchy(F,P):
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def PK2(F,P):
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"""
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Return 2. Piola-Kirchhoff stress calculated from 1. Piola-Kirchhoff stress and deformation gradient.
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Parameters
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----------
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F : numpy.array of shape (:,3,3) or (3,3)
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@ -37,16 +37,16 @@ def PK2(F,P):
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S = np.dot(np.linalg.inv(F),P)
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else:
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S = np.einsum('ijk,ikl->ijl',np.linalg.inv(F),P)
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return S
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return symmetric(S)
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def strain_tensor(F,t,m):
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"""
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Return strain tensor calculated from deformation gradient.
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For details refer to https://en.wikipedia.org/wiki/Finite_strain_theory and
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https://de.wikipedia.org/wiki/Verzerrungstensor
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Parameters
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----------
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F : numpy.array of shape (:,3,3) or (3,3)
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@ -64,16 +64,16 @@ def strain_tensor(F,t,m):
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elif t == 'U':
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C = np.matmul(transpose(F_),F_)
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w,n = np.linalg.eigh(C)
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if m > 0.0:
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eps = 1.0/(2.0*abs(m)) * (+ np.matmul(n,np.einsum('ij,ikj->ijk',w**m,n))
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eps = 1.0/(2.0*abs(m)) * (+ np.matmul(n,np.einsum('ij,ikj->ijk',w**m,n))
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- np.broadcast_to(np.eye(3),[F_.shape[0],3,3]))
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elif m < 0.0:
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eps = 1.0/(2.0*abs(m)) * (- np.matmul(n,np.einsum('ij,ikj->ijk',w**m,n))
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+ np.broadcast_to(np.eye(3),[F_.shape[0],3,3]))
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else:
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eps = np.matmul(n,np.einsum('ij,ikj->ijk',0.5*np.log(w),n))
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return eps.reshape((3,3)) if np.shape(F) == (3,3) else \
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eps
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@ -81,7 +81,7 @@ def strain_tensor(F,t,m):
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def deviatoric_part(x):
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"""
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Return deviatoric part of a tensor.
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Parameters
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----------
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x : numpy.array of shape (:,3,3) or (3,3)
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@ -89,13 +89,13 @@ def deviatoric_part(x):
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"""
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return x - np.eye(3)*spherical_part(x) if np.shape(x) == (3,3) else \
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x - np.einsum('ijk,i->ijk',np.broadcast_to(np.eye(3),[x.shape[0],3,3]),spherical_part(x))
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x - np.einsum('ijk,i->ijk',np.broadcast_to(np.eye(3),[x.shape[0],3,3]),spherical_part(x))
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def spherical_part(x,tensor=False):
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"""
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Return spherical (hydrostatic) part of a tensor.
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Parameters
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----------
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x : numpy.array of shape (:,3,3) or (3,3)
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@ -113,12 +113,12 @@ def spherical_part(x,tensor=False):
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return sph
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else:
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return np.einsum('ijk,i->ijk',np.broadcast_to(np.eye(3),(x.shape[0],3,3)),sph)
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def Mises_stress(sigma):
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"""
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Return the Mises equivalent of a stress tensor.
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Parameters
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----------
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sigma : numpy.array of shape (:,3,3) or (3,3)
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@ -128,12 +128,12 @@ def Mises_stress(sigma):
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s = deviatoric_part(sigma)
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return np.sqrt(3.0/2.0*(np.sum(s**2.0))) if np.shape(sigma) == (3,3) else \
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np.sqrt(3.0/2.0*np.einsum('ijk->i',s**2.0))
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def Mises_strain(epsilon):
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"""
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Return the Mises equivalent of a strain tensor.
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Parameters
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----------
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epsilon : numpy.array of shape (:,3,3) or (3,3)
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@ -148,7 +148,7 @@ def Mises_strain(epsilon):
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def symmetric(x):
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"""
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Return the symmetrized tensor.
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Parameters
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----------
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x : numpy.array of shape (:,3,3) or (3,3)
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@ -161,40 +161,54 @@ def symmetric(x):
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def maximum_shear(x):
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"""
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Return the maximum shear component of a symmetric tensor.
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Parameters
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----------
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x : numpy.array of shape (:,3,3) or (3,3)
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Symmetric tensor of which the maximum shear is computed.
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"""
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w = np.linalg.eigvalsh(symmetric(x)) # eigenvalues in ascending order
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return (w[2] - w[0])*0.5 if np.shape(x) == (3,3) else \
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(w[:,2] - w[:,0])*0.5
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def principal_components(x):
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w = eigenvalues(x)
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return (w[0] - w[2])*0.5 if np.shape(x) == (3,3) else \
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(w[:,0] - w[:,2])*0.5
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def eigenvalues(x):
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"""
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Return the principal components of a symmetric tensor.
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The principal components (eigenvalues) are sorted in descending order, each repeated according to
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Return the eigenvalues, i.e. principal components, of a symmetric tensor.
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The eigenvalues are sorted in ascending order, each repeated according to
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its multiplicity.
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Parameters
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----------
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x : numpy.array of shape (:,3,3) or (3,3)
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Symmetric tensor of which the principal compontents are computed.
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Symmetric tensor of which the eigenvalues are computed.
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"""
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w = np.linalg.eigvalsh(symmetric(x)) # eigenvalues in ascending order
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return w[::-1] if np.shape(x) == (3,3) else \
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w[:,::-1]
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return np.linalg.eigvalsh(symmetric(x))
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def eigenvectors(x):
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"""
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Return eigenvectors of a symmetric tensor.
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The eigenvalues are sorted in ascending order of their associated eigenvalues.
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Parameters
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----------
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x : numpy.array of shape (:,3,3) or (3,3)
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Symmetric tensor of which the eigenvectors are computed.
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"""
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(u,v) = np.linalg.eigh(symmetric(x))
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return v
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def transpose(x):
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"""
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Return the transpose of a tensor.
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Parameters
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----------
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x : numpy.array of shape (:,3,3) or (3,3)
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@ -208,7 +222,7 @@ def transpose(x):
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def rotational_part(x):
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"""
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Return the rotational part of a tensor.
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Parameters
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----------
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x : numpy.array of shape (:,3,3) or (3,3)
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@ -221,7 +235,7 @@ def rotational_part(x):
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def left_stretch(x):
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"""
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Return the left stretch of a tensor.
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Parameters
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----------
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x : numpy.array of shape (:,3,3) or (3,3)
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@ -229,12 +243,12 @@ def left_stretch(x):
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"""
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return __polar_decomposition(x,'V')[0]
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def right_stretch(x):
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"""
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Return the right stretch of a tensor.
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Parameters
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----------
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x : numpy.array of shape (:,3,3) or (3,3)
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@ -247,20 +261,20 @@ def right_stretch(x):
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def __polar_decomposition(x,requested):
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"""
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Singular value decomposition.
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Parameters
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----------
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x : numpy.array of shape (:,3,3) or (3,3)
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Tensor of which the singular values are computed.
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requested : iterable of str
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Requested outputs: ‘R’ for the rotation tensor,
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Requested outputs: ‘R’ for the rotation tensor,
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‘V’ for left stretch tensor and ‘U’ for right stretch tensor.
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"""
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u, s, vh = np.linalg.svd(x)
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R = np.dot(u,vh) if np.shape(x) == (3,3) else \
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np.einsum('ijk,ikl->ijl',u,vh)
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output = []
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if 'R' in requested:
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output.append(R)
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@ -268,5 +282,5 @@ def __polar_decomposition(x,requested):
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output.append(np.dot(x,R.T) if np.shape(x) == (3,3) else np.einsum('ijk,ilk->ijl',x,R))
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if 'U' in requested:
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output.append(np.dot(R.T,x) if np.shape(x) == (3,3) else np.einsum('ikj,ikl->ijl',R,x))
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return tuple(output)
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@ -2,10 +2,10 @@ import numpy as np
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from damask import mechanics
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class TestMechanics:
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n = 1000
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c = np.random.randint(n)
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def test_vectorize_Cauchy(self):
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P = np.random.random((self.n,3,3))
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@ -58,10 +58,23 @@ class TestMechanics:
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mechanics.maximum_shear(x[self.c]))
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def test_vectorize_principal_components(self):
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def test_vectorize_eigenvalues(self):
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x = np.random.random((self.n,3,3))
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assert np.allclose(mechanics.principal_components(x)[self.c],
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mechanics.principal_components(x[self.c]))
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assert np.allclose(mechanics.eigenvalues(x)[self.c],
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mechanics.eigenvalues(x[self.c]))
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def test_vectorize_eigenvectors(self):
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x = np.random.random((self.n,3,3))
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assert np.allclose(mechanics.eigenvectors(x)[self.c],
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mechanics.eigenvectors(x[self.c]))
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def test_vectorize_PK2(self):
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F = np.random.random((self.n,3,3))
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P = np.random.random((self.n,3,3))
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assert np.allclose(mechanics.PK2(F,P)[self.c],
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mechanics.PK2(F[self.c],P[self.c]))
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def test_vectorize_transpose(self):
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@ -102,7 +115,14 @@ class TestMechanics:
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U = mechanics.right_stretch(F)
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assert np.allclose(np.matmul(R,U),
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np.matmul(V,R))
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def test_PK2(self):
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"""Ensure 2. Piola-Kirchhoff stress is symmetrized 1. Piola-Kirchhoff stress for no deformation."""
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P = np.random.random((self.n,3,3))
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assert np.allclose(mechanics.PK2(np.broadcast_to(np.eye(3),(self.n,3,3)),P),
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mechanics.symmetric(P))
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def test_strain_tensor_no_rotation(self):
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"""Ensure that left and right stretch give same results for no rotation."""
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@ -110,7 +130,7 @@ class TestMechanics:
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m = np.random.random()*20.0-10.0
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assert np.allclose(mechanics.strain_tensor(F,'U',m),
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mechanics.strain_tensor(F,'V',m))
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def test_strain_tensor_rotation_equivalence(self):
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"""Ensure that left and right strain differ only by a rotation."""
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F = np.broadcast_to(np.eye(3),[self.n,3,3]) + (np.random.random((self.n,3,3))*0.5 - 0.25)
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@ -125,7 +145,7 @@ class TestMechanics:
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m = np.random.random()*2.0 - 1.0
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assert np.allclose(mechanics.strain_tensor(F,t,m),
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0.0)
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def test_rotation_determinant(self):
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"""
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Ensure that the determinant of the rotational part is +- 1.
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@ -186,3 +206,22 @@ class TestMechanics:
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x = np.random.random((self.n,3,3))
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assert np.allclose(mechanics.Mises_stress(x)/mechanics.Mises_strain(x),
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1.5)
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def test_eigenvalues(self):
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"""Ensure that the characteristic polynomial can be solved."""
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A = mechanics.symmetric(np.random.random((self.n,3,3)))
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lambd = mechanics.eigenvalues(A)
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s = np.random.randint(self.n)
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for i in range(3):
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assert np.allclose(np.linalg.det(A[s]-lambd[s,i]*np.eye(3)),.0)
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def test_eigenvalues_and_vectors(self):
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"""Ensure that eigenvalues and -vectors are the solution to the characteristic polynomial."""
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A = mechanics.symmetric(np.random.random((self.n,3,3)))
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lambd = mechanics.eigenvalues(A)
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x = mechanics.eigenvectors(A)
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s = np.random.randint(self.n)
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for i in range(3):
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assert np.allclose(np.dot(A[s]-lambd[s,i]*np.eye(3),x[s,:,i]),.0)
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