cratepy.tensor.tensoroperations.spectral_decomposition

spectral_decomposition(x, is_real_if_close=False)

Perform spectral decomposition of symmetric second-order tensor.

The computational implementation of the spectral decomposition follows the Appendix A of Computational Methods for Plasticity [1].

[1]

de Souza Neto, E. A., Peri, D., and Owen, D. R. J. (2008). Computational Methods for Plasticity. John Wiley & Sons, Ltd, Chichester, UK (see here)

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Parameters:

• x (numpy.ndarray (2d)) – Second-order tensor (square array) whose eigenvalues and eigenvectors are computed.

• is_real_if_close (bool, default=False) – If True, then drop imaginary parts of eigenvalues and eigenvectors if these are close to zero (tolerance with respect to machine epsilon for input type) and convert to real type.

Returns:

• eigenvals (numpy.ndarray (1d)) – Eigenvalues of second-order tensor sorted in descending order.

• eigenvectors (numpy.ndarray (2d)) – Eigenvectors of second-order tensor stored columnwise according with eigenvalues.

• eig_multiplicity (dict) – Multiplicity (item, int) of the eigenvalue stored at given index (key, str).

• eigenprojections (list[tuple]) – Eigenprojections of second-order tensor stored as tuples (item) as (eigenvalue, eigenprojection) and sorted in descending order of eigenvalues. Only available for 2x2 and 3x3 second-order tensors, otherwise an empty list is returned.