Optimization of the branching pattern in coherent phase transitions

TL;DR

This paper combines finite element simulations and geometric optimization to identify a new class of branching patterns in martensitic phase transformations, achieving lower energy bounds than previously known models.

Contribution

It introduces a novel, topologically different class of branching patterns and derives a low-dimensional family that better matches simulations, improving energy bounds.

Findings

New branching pattern class identified through simulations

Geometric optimization yields patterns with lower energy bounds

Patterns match simulation results closely

Abstract

Branching can be observed at the austenite-martensite interface of martensitic phase transformations. For a model problem, Kohn and M\"uller studied a branching pattern with optimal scaling of the energy with respect to its parameters. Here, we present finite element simulations that suggest a topologically different class of branching patterns and derive a novel, low dimensional family of patterns. After a geometric optimization within this family, the resulting pattern bears a striking resemblance to our simulation. The novel microstructure admits the same scaling exponents but results in a significantly lower upper energy bound.