An assessment of numerical techniques to find energy-minimizing microstructures associated with nonconvex potentials

Microstructural patterns emerge ubiquitously during phase transformations, deformation twinning, or crystal plasticity. Challenges are the prediction of such microstructural patterns and the resulting effective material behavior. Mathematically, the experimentally observed patterns are energy-minimizing sequences produced by an underlying non-(quasi)convex strain energy. Therefore, identifying the microstructure and effective response is linked to finding the quasiconvex, relaxed energy. Due to its nonlocal nature, quasiconvexification has traditionally been limited to (semi-)analytical techniques or has been dealt with by numerical techniques such as the finite element method (FEM). Both have been restricted to primarily simple material models. We here contrast three numerical techniques—FEM, a Fourier-based spectral formulation, and a meshless maximum-entropy (max-ent) method. We demonstrate their performance by minimizing the energy of a representative volume element for both hyperelasticity and finite-strain phase transformations. Unlike FEM, which fails to converge in most scenarios, the Fourier-based spectral formulation (FFT) scheme captures microstructures of intriguingly high resolution, whereas max-ent is superior at approximating the relaxed energy. None of the methods are capable of accurately predicting both microstructures and relaxed energy; yet, both FFT and max-ent show significant advantages over FEM. Numerical errors are explained by the energy associated with microstructural interfaces in the numerical techniques compared here.