# A CUDA ACCELERATED NUMERICAL INTEGRATION OF ELASTOPLASTIC FINITE ELEMENTS RESIDUALS

## Palavras-chave:

Finite Element Assembly, Residual Numerical Integration, Elastoplasticity, CUDA## Resumo

Finite Element Method (FEM) is a numerical technique to approximate partial differential

equations. It has been widely used to approximate solutions of physical problems in different fields

of research. The numerical simulation challenging engineering problems with small error require fine

meshes and leads to high computational cost. To overcome this difficulty parallel computing is becoming

a mainstream tool. Among the techniques available to improve the performance of this type of computa-

tional application is the execution of the algorithm using Graphics Processing Unit (GPU) programming.

Although GPU was originally developed for graphics processing, it has been used in the last years as a

general purpose machine with high parallelism power through the availability of libraries such as CUDA

or OpenGL. The purpose of this research is to develop an efficient algorithm for the evaluation of the

finite element residual and Jacobian matrix. We target the particular variational formulation of an elasto-

plastic problem with associative plasticity but will try to show that the approach can be extended to other

fields and problems. The presented strategy for the calculation of the residual and tangent matrix rely

on several computational ingredients such as gathering and scattering operations, sparse matrix mul-

tiplications, and a parallel coloring procedure for assembly process. The verification of the nonlinear

approximated solution includes comparison with regular CPU implementation in terms of numerical re-

sults and execution efficiency. For residual computations of elasto-plasticity, the GPU outperforms the

CPU by a factor of up to 10 (details of the architecture are given in the paper).