clearer names
This commit is contained in:
Martin Diehl
parent
4970f22e13
commit
15c3cab549
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@ -21,21 +21,21 @@ def Cauchy(P,F):
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return symmetric(sigma)
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def deviatoric_part(x):
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def deviatoric_part(T):
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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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T : numpy.array of shape (:,3,3) or (3,3)
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Tensor of which the deviatoric part is computed.
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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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return T - np.eye(3)*spherical_part(T) if np.shape(T) == (3,3) else \
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T - np.einsum('ijk,i->ijk',np.broadcast_to(np.eye(3),[T.shape[0],3,3]),spherical_part(T))
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def eigenvalues(x):
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def eigenvalues(T_sym):
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"""
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Return the eigenvalues, i.e. principal components, of a symmetric tensor.
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@ -44,14 +44,14 @@ def eigenvalues(x):
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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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T_sym : numpy.array of shape (:,3,3) or (3,3)
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Symmetric tensor of which the eigenvalues are computed.
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"""
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return np.linalg.eigvalsh(symmetric(x))
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return np.linalg.eigvalsh(symmetric(T_sym))
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def eigenvectors(x,RHS=False):
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def eigenvectors(T_sym,RHS=False):
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"""
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Return eigenvectors of a symmetric tensor.
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@ -59,47 +59,47 @@ def eigenvectors(x,RHS=False):
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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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T_sym : 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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RHS: bool, optional
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Enforce right-handed coordinate system. Default is False.
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"""
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(u,v) = np.linalg.eigh(symmetric(x))
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(u,v) = np.linalg.eigh(symmetric(T_sym))
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if RHS:
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if np.shape(x) == (3,3):
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if np.shape(T_sym) == (3,3):
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if np.linalg.det(v) < 0.0: v[:,2] *= -1.0
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else:
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v[np.linalg.det(v) < 0.0,:,2] *= -1.0
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return v
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def left_stretch(x):
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def left_stretch(T):
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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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T : numpy.array of shape (:,3,3) or (3,3)
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Tensor of which the left stretch is computed.
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"""
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return __polar_decomposition(x,'V')[0]
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return __polar_decomposition(T,'V')[0]
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def maximum_shear(x):
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def maximum_shear(T_sym):
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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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T_sym : 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 = 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 = eigenvalues(T_sym)
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return (w[0] - w[2])*0.5 if np.shape(T_sym) == (3,3) else \
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(w[:,0] - w[:,2])*0.5
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@ -147,53 +147,54 @@ def PK2(P,F):
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S = np.einsum('ijk,ikl->ijl',np.linalg.inv(F),P)
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return symmetric(S)
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def right_stretch(x):
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def right_stretch(T):
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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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T : numpy.array of shape (:,3,3) or (3,3)
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Tensor of which the right stretch is computed.
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"""
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return __polar_decomposition(x,'U')[0]
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return __polar_decomposition(T,'U')[0]
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def rotational_part(x):
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def rotational_part(T):
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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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T : numpy.array of shape (:,3,3) or (3,3)
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Tensor of which the rotational part is computed.
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"""
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return __polar_decomposition(x,'R')[0]
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return __polar_decomposition(T,'R')[0]
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def spherical_part(x,tensor=False):
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def spherical_part(T,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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T : numpy.array of shape (:,3,3) or (3,3)
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Tensor of which the hydrostatic part is computed.
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tensor : bool, optional
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Map spherical part onto identity tensor. Default is false
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"""
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if x.shape == (3,3):
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sph = np.trace(x)/3.0
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if T.shape == (3,3):
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sph = np.trace(T)/3.0
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return sph if not tensor else np.eye(3)*sph
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else:
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sph = np.trace(x,axis1=1,axis2=2)/3.0
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sph = np.trace(T,axis1=1,axis2=2)/3.0
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if not tensor:
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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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return np.einsum('ijk,i->ijk',np.broadcast_to(np.eye(3),(T.shape[0],3,3)),sph)
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def strain_tensor(F,t,m):
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@ -234,73 +235,73 @@ def strain_tensor(F,t,m):
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eps
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def symmetric(x):
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def symmetric(T):
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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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T : numpy.array of shape (:,3,3) or (3,3)
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Tensor of which the symmetrized values are computed.
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"""
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return (x+transpose(x))*0.5
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return (T+transpose(T))*0.5
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def transpose(x):
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def transpose(T):
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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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T : numpy.array of shape (:,3,3) or (3,3)
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Tensor of which the transpose is computed.
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"""
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return x.T if np.shape(x) == (3,3) else \
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np.transpose(x,(0,2,1))
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return T.T if np.shape(T) == (3,3) else \
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np.transpose(T,(0,2,1))
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def __polar_decomposition(x,requested):
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def __polar_decomposition(T,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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T : 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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‘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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u, s, vh = np.linalg.svd(T)
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R = np.dot(u,vh) if np.shape(T) == (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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if 'V' in 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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output.append(np.dot(T,R.T) if np.shape(T) == (3,3) else np.einsum('ijk,ilk->ijl',T,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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output.append(np.dot(R.T,T) if np.shape(T) == (3,3) else np.einsum('ikj,ikl->ijl',R,T))
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return tuple(output)
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def __Mises(x,s):
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def __Mises(T_sym,s):
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"""
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Base equation for Mises equivalent of a stres or strain 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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T_sym : numpy.array of shape (:,3,3) or (3,3)
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Symmetric tensor of which the von Mises equivalent is computed.
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s : float
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Scaling factor (2/3 for strain, 3/2 for stress).
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"""
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d = deviatoric_part(x)
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return np.sqrt(s*(np.sum(d**2.0))) if np.shape(x) == (3,3) else \
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d = deviatoric_part(T_sym)
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return np.sqrt(s*(np.sum(d**2.0))) if np.shape(T_sym) == (3,3) else \
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np.sqrt(s*np.einsum('ijk->i',d**2.0))
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