Computational modeling of microstructure

TL;DR

This work addresses computing metastable microstructure and phase transformations in martensitic and ferromagnetic crystals through an energy-based model E(y) = ∫Ω φ(grad y(x), theta(x)) dx plus interfacial terms, with wells on SO(3) at high temperature and on martensitic variants Ui at low temperature. It develops finite-element discretizations and proves convergence and stability results, including an energy-error bound E(y_h) ≤ C h^{1/2} and convergence of laminate volume fractions to the prescribed mix, supported by a projection-based decomposition of gradients. A thin-film quasi-static transformation model is introduced, incorporating a total-variation surface energy and a nucleation mechanism that triggers phase changes as theta varies, with an asymptotic form u ≈ y + b x3. The computational framework uses a P1-P0 finite-element discretization, a jump-based energy for the gradient split, and a probabilistic nucleation scheme updated per time step, solved via a Polak-Ribière conjugate gradient method; the approach enables robust simulation of microstructure patterns and phase-boundary evolution in thin films.

Abstract

Many materials such as martensitic or ferromagnetic crystals are observed to be in metastable states exhibiting a fine-scale, structured spatial oscillation called microstructure; and hysteresis is observed as the temperature, boundary forces, or external magnetic field changes. We have developed a numerical analysis of microstructure and used this theory to construct numerical methods that have been used to compute approximations to the deformation of crystals with microstructure.