Spectra and pseudospectra in the evaluation of material stability in phase field schemes

TL;DR

This paper investigates how pseudospectra analysis can better predict material stability in phase field models with microstructure, especially for non-normal matrices where eigenvalues are sensitive to perturbations.

Contribution

It introduces pseudospectrum analysis for stability evaluation in phase field schemes, highlighting its importance over traditional spectral analysis for non-normal matrices.

Findings

Pseudospectra can reveal potential instabilities missed by eigenvalue analysis.

Structured perturbations in pseudospectra predict instability in cases where spectral analysis suggests stability.

Application to quasicrystals demonstrates the method's relevance for uncertain material parameters.

Abstract

We consider the dynamics of bodies with "active" microstructure described by vector-valued phase fields. For waves with time-varying amplitude, the associated evolution equation involves a matrix that can be non-normal, depending on the constitutive choices adopted for the microstructural actions associated with the considered phase field. The occurrence of non-normality requires to look at the pseudospectrum of the considered matrix, namely the set of all possible eigenvalues of matrices in a $\varepsilon$-neighborhood of the matrix itself, because the eigenvalues of non-normal matrices can be very sensitive to small perturbations and therefore the spectral analysis alone would not be sufficient to distinguish with certainty between table and unstable behavior. We develop the relevant analyses in the case of quasicrystals for which the values of some constitutive parameters are not known or are uncertain from an experimental point of view, a circumstance suggesting parametric analyses. We find circumstances in which the pseudospectra obtained by means of the so-called structured perturbations predict instability when, instead, the spectral analysis indicates stability.