  Citation: Frost, M.; Valdman, J. Vectorized MATLAB Implementation of the Incremental Minimization Principle for Rate-Independent Dissipative Solids Using FEM: A Constitutive Model of Shape Memory Alloys. Mathematics 2022, 10, 4412. https://doi.org/10.3390/ math10234412 Academic Editor: Fernando Simoes Received: 26 October 2022 Accepted: 18 November 2022 Published: 23 November 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). mathematics Article Vectorized MATLAB Implementation of the Incremental Minimization Principle for Rate-Independent Dissipative Solids Using FEM: A Constitutive Model of Shape Memory Alloys Miroslav Frost 1, * and Jan Valdman 2,3 1 Institute of Thermomechanics, Czech Academy of Sciences, Dolejškova 5, CZ-18200 Prague, Czech Republic 2 Institute of Information Theory and Automation, Czech Academy of Sciences, Pod Vodárenskou vˇ eží 4, CZ-18200 Prague, Czech Republic 3 Faculty of Information Technology, Czech Technical University in Prague, Thákurova 9, CZ-16000 Prague, Czech Republic * Correspondence: mfrost@it.cas.cz; Tel.: +420-266-053-051 Abstract: The incremental energy minimization principle provides a compact variational formulation for evolutionary boundary problems based on constitutive models of rate-independent dissipative solids. In this work, we develop and implement a versatile computational tool for the resolution of these problems via the finite element method (FEM). The implementation is coded in the MATLAB programming language and benefits from vector operations, allowing all local energy contributions to be evaluated over all degrees of freedom at once. The monolithic solution scheme combined with gradient-based optimization methods is applied to the inherently nonlinear, non-smooth convex minimization problem. An advanced constitutive model for shape memory alloys, which features a strongly coupled rate-independent dissipation function and several constraints on internal variables, is implemented as a benchmark example. Numerical simulations demonstrate the capabilities of the computational tool, which is suited for the rapid development and testing of advanced constitutive laws of rate-independent dissipative solids. Keywords: vectorized FEM implementation; incremental minimization principle; dissipative solids; shape memory alloys MSC: 74A15; 74S05 1. Introduction New experimental techniques enable more thorough investigation of the complex response of materials to mechanical loading, which opens space for the development of more elaborate material models and physical simulations. On the macroscopic (continuum thermodynamics) level of modeling, such development involves deducing more complex constitutive laws which complement the fundamental balance laws and side conditions (boundary and initial) so that the response of a material body in time to external stimuli can be determined via solving evolutionary boundary value problems. For the development of complex constitutive laws characterizing the materials and material systems, various thermodynamic frameworks have been developed in the literature [1–5]. They allow for the formulation of a wide range of models in a very concise and consistent way. The incremental energy minimization approach can be considered a compact variational formulation of the evolutionary boundary value problem for models of rate-independent dissipative solids which were derived within such frameworks [6–9]. Let us note that rate-independent processes are invariant under a change in time scale [10]. Mathematics 2022, 10, 4412. https://doi.org/10.3390/math10234412 https://www.mdpi.com/journal/mathematics