129 lines
2.9 KiB
Python
129 lines
2.9 KiB
Python
"""
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Tensor mathematics.
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All routines operate on numpy.ndarrays of shape (...,3,3).
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"""
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import numpy as _np
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def deviatoric(T: _np.ndarray) -> _np.ndarray:
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"""
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Calculate deviatoric part of a tensor.
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Parameters
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----------
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T : numpy.ndarray, shape (...,3,3)
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Tensor of which the deviatoric part is computed.
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Returns
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-------
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T' : numpy.ndarray, shape (...,3,3)
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Deviatoric part of T.
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"""
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return T - spherical(T,tensor=True)
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def eigenvalues(T_sym: _np.ndarray) -> _np.ndarray:
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"""
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Eigenvalues, i.e. principal components, of a symmetric tensor.
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Parameters
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----------
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T_sym : numpy.ndarray, shape (...,3,3)
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Symmetric tensor of which the eigenvalues are computed.
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Returns
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-------
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lambda : numpy.ndarray, shape (...,3)
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Eigenvalues of T_sym sorted in ascending order, each repeated
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according to its multiplicity.
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"""
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return _np.linalg.eigvalsh(symmetric(T_sym))
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def eigenvectors(T_sym: _np.ndarray, RHS: bool = False) -> _np.ndarray:
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"""
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Eigenvectors of a symmetric tensor.
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Parameters
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----------
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T_sym : numpy.ndarray, shape (...,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. Defaults to False.
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Returns
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-------
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x : numpy.ndarray, shape (...,3,3)
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Eigenvectors of T_sym sorted in ascending order of their
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associated eigenvalues.
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"""
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(u,v) = _np.linalg.eigh(symmetric(T_sym))
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if RHS:
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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 spherical(T: _np.ndarray, tensor: bool = True) -> _np.ndarray:
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"""
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Calculate spherical part of a tensor.
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Parameters
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----------
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T : numpy.ndarray, shape (...,3,3)
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Tensor of which the spherical part is computed.
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tensor : bool, optional
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Map spherical part onto identity tensor. Defaults to True.
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Returns
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-------
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p : numpy.ndarray, shape (...,3,3)
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unless tensor == False: shape (...,)
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Spherical part of tensor T. p is an isotropic tensor.
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"""
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sph = _np.trace(T,axis2=-2,axis1=-1)/3.0
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return _np.einsum('...jk,...',_np.eye(3),sph) if tensor else sph
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def symmetric(T: _np.ndarray) -> _np.ndarray:
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"""
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Symmetrize tensor.
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Parameters
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----------
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T : numpy.ndarray, shape (...,3,3)
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Tensor of which the symmetrized values are computed.
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Returns
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-------
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T_sym : numpy.ndarray, shape (...,3,3)
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Symmetrized tensor T.
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"""
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return (T+transpose(T))*0.5
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def transpose(T: _np.ndarray) -> _np.ndarray:
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"""
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Transpose tensor.
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Parameters
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----------
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T : numpy.ndarray, shape (...,3,3)
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Tensor of which the transpose is computed.
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Returns
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-------
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T.T : numpy.ndarray, shape (...,3,3)
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Transpose of tensor T.
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"""
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return _np.swapaxes(T,axis2=-2,axis1=-1)
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