Simulation of Phase Combinations in Shape Memory Alloys Patches by Hybrid Optimization Methods

TL;DR

The paper tackles predicting phase combinations in shape memory alloys by solving a non-convex variational problem derived from Landau theory. It introduces a two-stage hybrid optimization: Chebyshev pseudospectral spatial discretization to convert the problem into a nonlinear finite-dimensional minimization, followed by a genetic algorithm for global search and a quasi-Newton refinement with a BFGS Hessian update for local convergence. Numerical demonstrations on 1D wires and 2D patches show robust emergence of coexisting martensite variants and coherent interfaces, captured by an order parameter around the two martensite states. The framework provides a practical tool for SMA design under mechanical loading and offers avenues to incorporate thermal dynamics in future work.

Abstract

In this paper, phase combinations among martensitic variants in shape memory alloys patches and bars are simulated by a hybrid optimization methodology. The mathematical model is based on the Landau theory of phase transformations. Each stable phase is associated with a local minimum of the free energy function, and the phase combinations are simulated by minimizing the bulk energy. At low temperature, the free energy function has double potential wells leading to non-convexity of the optimization problem. The methodology proposed in the present paper is based on an initial estimate of the global solution by a genetic algorithm, followed by a refined quasi-Newton procedure to locally refine the optimum. By combining the local and global search algorithms, the phase combinations are successfully simulated. Numerical experiments are presented for the phase combinations in a SMA patch under several typical mechanical loadings.

Figures (7)

• Figure 1: (a) Sketch of the square to rectangular transformation, (b) The temperature dependency of the free energy function for the transformation.
• Figure 2: Sketch of an one dimensional SMA wire under mechanical loading
• Figure 3: Sketch of a two dimensional SMA patch under mechanical loading
• Figure 4: Convergence of the hybrid optimization method. Left: reduction of the bulk energy in the GA. Right: reduction of the step size in the quasi-Newton method
• Figure 5: Mechanically induced phase combination in a SMA wire.