underworld.function.tensor module

This module provides functions relating to tensor operations.

All Underworld2 functions return 1d array type objects. For tensor objects, the following convention is used:

Full tensors:

2D:

\[\begin{split}\left[ a_{00}, a_{01}, \\ a_{10}, a_{11} \right]\end{split}\]

3D:

\[\begin{split}\left[ a_{00}, a_{01}, a_{02}, \\ a_{10}, a_{11}, a_{12}, \\ a_{20}, a_{21}, a_{22} \right]\end{split}\]

Symmetric tensors:

2D:

\[\left[ a_{00}, a_{11}, a_{01} \right]\]

3D:

\[\left[ a_{00}, a_{11}, a_{22}, a_{01}, a_{02}, a_{12} \right]\]

Classes

underworld.function.tensor.antisymmetric	This function calculates the anti-symmetric part of a tensor, returning it as a tensor.
underworld.function.tensor.deviatoric	This function calculates the deviatoric stress tensor from the provided symmetric tensor.
underworld.function.tensor.second_invariant	This function calculates the second invariant of (symmetric)tensor provided by the subject function.
underworld.function.tensor.symmetric	This function calculates the symmetric part of a tensor and returns it as a symmetric tensor.

class underworld.function.tensor.antisymmetric(fn, *args, **kwargs)

Bases: underworld.function._function.Function

This function calculates the anti-symmetric part of a tensor, returning it as a tensor. The function generated by this class returns objects of type TensorType.

\[v_{ij} = \tfrac{1}{2} ( u_{ij} - u_{ji} )\]

Parameters:fn (underworld.function.Function) – The function which provides the required tensor. This function must return objects of type TensorType.

class underworld.function.tensor.deviatoric(fn, *args, **kwargs)

Bases: underworld.function._function.Function

This function calculates the deviatoric stress tensor from the provided symmetric tensor. The function generated by this class returns objects of type SymmetricTensorType.

\[\tau_{ij} = \sigma_{ij} - \frac{\sigma_{kk}}{\delta_{ll}}\delta_{ij}\]

Parameters:fn (underworld.function.Function) – The function which provides the required stress symmetric tensor. This function must return objects of type SymmetricTensorType.

class underworld.function.tensor.second_invariant(fn, *args, **kwargs)

Bases: underworld.function._function.Function

This function calculates the second invariant of (symmetric)tensor provided by the subject function. The function generated by this class returns objects of type ScalarType.

\[u = \sqrt{ \tfrac{1}{2} u_{ij} u_{ij} }\]

Parameters:fn (underworld.function.Function) – The function which provides the required tensor. This function must return objects of type TensorType or SymmetricTensorType.

class underworld.function.tensor.symmetric(fn, *args, **kwargs)

Bases: underworld.function._function.Function

This function calculates the symmetric part of a tensor and returns it as a symmetric tensor. The function generated by this class returns objects of type SymmetricTensorType.

\[v_{ij} = \tfrac{1}{2} ( u_{ij} + u_{ji} )\]

Parameters:fn (underworld.function.Function) – The function which provides the required tensor. This function must return objects of type TensorType.