"""
Tensor mathematics.
All routines operate on numpy.ndarrays of shape (...,3,3).
"""
import numpy as _np
[docs]
def deviatoric(T: _np.ndarray) -> _np.ndarray:
r"""
Calculate deviatoric part of a tensor.
Parameters
----------
T : numpy.ndarray, shape (...,3,3)
Tensor of which the deviatoric part is computed.
Returns
-------
T' : numpy.ndarray, shape (...,3,3)
Deviatoric part of T.
See Also
--------
spherical : Calculate spherical part of a tensor.
Notes
-----
The deviatoric part of a tensor is defined as:
.. math::
\vb{T}' = \vb{T} - \vb{I}_\text{p},
where :math:`\vb{I}_\text{p}` is the spherical
part of the tensor mapped onto identity.
"""
return T - spherical(T,tensor=True)
[docs]
def eigenvalues(T_sym: _np.ndarray) -> _np.ndarray:
r"""
Calculate eigenvalues of a symmetric tensor.
Parameters
----------
T_sym : numpy.ndarray, shape (...,3,3)
Symmetric tensor of which the eigenvalues are computed.
Returns
-------
lambda : numpy.ndarray, shape (...,3)
Eigenvalues of T_sym sorted in ascending order, each repeated
according to its multiplicity.
See Also
--------
eigenvectors : Calculate eigenvectors of a symmetric tensor.
Notes
-----
The eigenvalues :math:`\lambda` are defined from:
.. math::
\vb{T}_\text{sym} \vb{v}= \lambda \vb{v},
where :math:`\vb{v}` are the eigenvectors.
"""
return _np.linalg.eigvalsh(symmetric(T_sym))
[docs]
def eigenvectors(T_sym: _np.ndarray,
RHS: bool = False) -> _np.ndarray:
r"""
Calculate eigenvectors of a symmetric tensor.
Parameters
----------
T_sym : numpy.ndarray, shape (...,3,3)
Symmetric tensor of which the eigenvectors are computed.
RHS : bool, optional
Enforce right-handed coordinate system. Defaults to False.
Returns
-------
v : numpy.ndarray, shape (...,3,3)
Eigenvectors of T_sym sorted in ascending order of their
associated eigenvalues.
See Also
--------
eigenvalues : Calculate eigenvalues of a symmetric tensor.
Notes
-----
The eigenvectors :math:`\vb{v}` are defined from:
.. math::
\vb{T}_\text{sym} \vb{v}= \lambda \vb{v},
where :math:`\lambda` are the eigenvalues.
"""
_,v = _np.linalg.eigh(symmetric(T_sym))
if RHS: v[_np.linalg.det(v) < 0.0,:,2] *= -1.0
return v
[docs]
def spherical(T: _np.ndarray,
tensor: bool = True) -> _np.ndarray:
r"""
Calculate spherical part of a tensor.
Parameters
----------
T : numpy.ndarray, shape (...,3,3)
Tensor of which the spherical part is computed.
tensor : bool, optional
Map spherical part onto identity tensor. Defaults to True.
Returns
-------
I_p/p : numpy.ndarray, shape (...,3,3) / shape (...,)
Spherical part of tensor T. I_p is an isotropic tensor.
See Also
--------
deviatoric : Calculate deviatoric part of a tensor.
Notes
-----
The spherical part of a tensor is defined as:
.. math::
p &= \text{trace}( \vb{T} )/3 \\
\vb{I}_p &= \vb{I} \, p
"""
sph = _np.trace(T,axis2=-2,axis1=-1)/3.0
return _np.einsum('...jk,...',_np.eye(3),sph) if tensor else sph
[docs]
def symmetric(T: _np.ndarray) -> _np.ndarray:
r"""
Symmetrize tensor.
Parameters
----------
T : numpy.ndarray, shape (...,3,3)
Tensor of which the symmetrized values are computed.
Returns
-------
T_sym : numpy.ndarray, shape (...,3,3)
Symmetrized tensor T.
Notes
-----
The symmetrization of a second-order tensor is defined as:
.. math::
\vb{T}_\text{sym} = \left( \vb{T}+ \vb{T}^\text{T} \right)/2
"""
return (T+transpose(T))*0.5
[docs]
def transpose(T: _np.ndarray) -> _np.ndarray:
r"""
Transpose tensor.
Parameters
----------
T : numpy.ndarray, shape (...,3,3)
Tensor of which the transpose is computed.
Returns
-------
T.T : numpy.ndarray, shape (...,3,3)
Transpose of tensor T.
Notes
-----
The transpose of a second-order tensor is defined as:
.. math::
T_{ij}^\text{T} = T_{ji}
"""
return _np.swapaxes(T,axis2=-2,axis1=-1)
