A GPU-based framework for finite element analysis of elastoplastic problems

Elastoplasticity is observed in a wide range of materials like metals that have real-world applications. The design and optimization process of such materials depends strongly on elastoplastic analysis for the prediction of displacement and stress. However, elastoplastic simulation is computationally expensive and often requires the use of parallel computers in real-world applications like crashworthiness and metal forming. This paper presents a novel parallel framework for finite element analysis of elastoplastic problems on massively parallel Graphics Processing Units (GPUs) architecture. We propose GPU-based parallel algorithms for all expensive steps in elastoplastic analysis, namely the computation of elemental matrices and their assembly, the computation of stress using the well-known radial-return method and the computation of internal force vectors and their assembly. Since GPUs have limited memory, assembly is done directly into a sparse storage format that can be seamlessly integrated with a GPU-based linear solver. The proposed algorithms are optimized for efficient memory access and fine-grain parallelism and prefer computation over data storage and reuse. In the proposed framework, all the computations are performed on the GPU and expensive data transfers to the CPU are avoided to achieve the best performance. Numerical experiments are conducted over three benchmark examples in three dimensions (3D) considering 8-noded hexahedral elements to demonstrate the performance of the proposed framework. The comparison of execution timings with sequential CPU implementation reveals speedups in the range 20.4×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times $$\end{document}–69.7×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times $$\end{document} for computation of elemental matrices and assembly, 47.2×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times $$\end{document}–66.1×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times $$\end{document} for computation of stress using radial-return method, 53.7×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times $$\end{document}–67.3×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times $$\end{document} for computation of internal force vectors and their assembly. A comparison of wall-clock timings shows 1.4×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times $$\end{document} to 7.2×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times $$\end{document} speedup by the proposed GPU implementation. The proposed framework is able to solve up to 5.1 million degrees of freedom (DOFs) elastoplasticity problem on a single GPU.