Efficient and flexible MATLAB implementation of 2D and 3D elastoplastic problems

TL;DR

This paper presents a highly efficient, fully vectorized MATLAB implementation for solving 2D and 3D elastoplastic problems, applicable to various models and yield criteria, with optimized assembly of stiffness matrices for improved computational performance.

Contribution

The paper introduces a novel vectorized approach for assembling stiffness matrices in MATLAB, significantly enhancing efficiency in elastoplastic simulations across multiple models and finite element types.

Findings

Assembly times scale linearly with the number of integration points in plastic phases.

The proposed method outperforms existing MATLAB codes in efficiency.

Plasticity assembly times are comparable to elastic assembly times.

Abstract

We propose an effective and flexible way to implement 2D and 3D elastoplastic problems in MATLAB using fully vectorized codes. Our technique is applied to a broad class of the problems including perfect plasticity or plasticity with hardening and several yield criteria. The problems are formulated in terms of displacements, discretized by the implicit Euler method in time and the finite element method in space, and solved by the semismooth Newton method. We discuss in detail selected models with the von Mises and Prager-Drucker yield criteria and four types of finite elements. The related codes are available for download. A particular interest is devoted to the assembling of tangential stiffness matrices. Since these matrices are repeatedly constructed in each Newton iteration and in each time step, we propose another vectorized assembling than current ones known for the elastic stiffness matrices. The main idea is based on a construction of two large and sparse matrices representing the strain-displacement and tangent operators, respectively, where the former matrix remains fixed and the latter one is updated only at some integration points. Comparisons with other available MATLAB codes show that our technique is also efficient for purely elastic problems. In elastoplasticity, the assembly times are linearly proportional to the number of integration points in a plastic phase and additional times due to plasticity never exceed assembly time of the elastic stiffness matrix.