Model Reduction Applied to Square to Rectangular Martensitic Transformations Using Proper Orthogonal Decomposition

TL;DR

This paper applies proper orthogonal decomposition to reduce the complexity of modeling square to rectangular phase transformations in ferroelastic materials, enabling efficient simulation of nonlinear thermo-mechanical dynamics.

Contribution

It introduces a POD-based model reduction approach specifically for ferroelastic phase transformations, improving computational efficiency while maintaining accuracy.

Findings

Low-dimensional models accurately replicate full PDE simulations.

POD effectively captures essential dynamics of phase transformations.

Reduced models facilitate faster simulations of ferroelastic behavior.

Abstract

Model reduction using the proper orthogonal decomposition (POD) method is applied to the dynamics of ferroelastic patches to study the first order square to rectangular phase transformations. Governing equations for the system dynamics are constructed by using the Landau-Ginzburg theory and are solved numerically. By using the POD method, a set of empirical orthogonal basis functions is first constructed, then the system is projected onto the subspace spanned by a small set of basis functions determined by the associated singular values. The performance of the low dimensional model is verified by simulating nonlinear thermo-mechanical waves and square to rectangular transformations in a ferroelastic patch. Comparison between numerical results obtained from the original PDE model and the low dimensional one is carried out.