Unconditionally stable, second-order accurate schemes for solid state phase transformations driven by mechano-chemical spinodal decomposition

Abstract

We consider solid state phase transformations that are caused by free energy densities with domains of non-convexity in strain-composition space; we refer to the non-convex domains as mechano-chemical spinodals. The non-convexity with respect to composition and strain causes segregation into phases with different crystal structures. We work on an existing model that couples the classical Cahn-Hilliard model with Toupin's theory of gradient elasticity at finite strains. Both systems are represented by fourth-order, nonlinear, partial differential equations. The goal of this work is to develop unconditionally stable, second-order accurate time-integration schemes, motivated by the need to carry out large scale computations of dynamically evolving microstructures in three dimensions. We also introduce reduced formulations naturally derived from these proposed schemes for faster computations that are still second-order accurate. Although our method is developed and analyzed here for a specific class of mechano-chemical problems, one can readily apply the same method to develop unconditionally stable, second-order accurate schemes for any problems for which free energy density functions are multivariate polynomials of solution components and component gradients. Apart from an analysis and construction of methods, we present a suite of numerical results that demonstrate the schemes in action.

Unconditionally stable, second-order schemes for gradient-regularized, non-convex, finite-strain elasticity modeling martensitic phase transformations

“…The Gonzalez-type scheme (GS) failed to converge with Δ = 10 −3 ∕0.5 at = 0.038 after a few timesteps at which more than 10 nonlinear iterations were required. favorable convergence behavior of the Taylor-series based schemes was observed in Sagiyama et al [15] for related first-order problems, where it was compared to the Backward Euler scheme.…”

“…are to be defined as functions of ℎ, to make the space-time discrete weak form (16) and (17) unconditionally stable and second-order accurate. We propose two schemes: variations of the Gonzalez scheme [12] and the Taylor-series scheme [15]. The former is robust and easy to implement, while the latter tends to have better convergence properties and has a symmetric tangent.…”

“…To obtain solutions representing fine twin microstructures in our three-dimensional problems, we also adopt the dynamic relaxation technique, but, instead of using a convex-splitting method, we propose and use two unconditionally stable, secondorder schemes designed for the IBVPs derived from non-convex gradient elasticity; one is a straightforward application of the idea proposed by Gonzalez [12] and the other is based on a Taylor-series expansion of the free energy density functions introduced in [15]. In this work we focus on the development and comparison of these two accurate schemes, and detailed study of the model parameters and features of the resulting microstructures will appear elsewhere.…”

A graph theoretic framework for representation, exploration and analysis on computed states of physical systems

“…The three tetragonal crystal structures in Figure 4 occur in the martensitic microstructures of Figure 5, with each tetragonal variant being represented by the same color in the two figures. A more detailed exposition of this problem from the mathematical and computational points of view has been laid out elsewhere [20,15,21,17]. Using isogeometric analytic methods, described in the preceding references, for the ease of representing finite-dimensional functions of high-order continuity, we have obtained numerical solutions to boundary value problems posed on the system of equations (9-12e).…”

Machine learning materials physics: Integrable deep neural networks enable scale bridging by learning free energy functions

“…As such, it becomes necessary to use a fitting process to create a faithful mathematical model of the free energy. The free energy density can be a function of multiple variables, including composition, temperature, strain, and order parameters, thus leading to a high-dimensional function [6,7]. Additionally, realistic data for the free energy can contain regions with rapid fluctuations, along with other regions with very gradual slopes [8].…”