Accelerating Finite Element Assembly on a GPU

Abstract

In this paper, a parallel strategy for assembly of finite element matrices on graphics processing unit (GPU) is presented. Considering the limited memory size of a GPU, the proposed strategy doesn’t store the elemental matrices into memory but performs on-the-fly computation and stores the data directly into a global stiffness matrix, reducing memory requirement and preventing overhead due to a separate assembly step. The global stiffness matrix is stored in compressed sparse row (CSR) storage format, commonly used by GPU-accelerated linear solver libraries. However, the assembly of elemental matrices directly into a sparse storage format requires prior knowledge of locations of nonzeros. The current work presents an efficient strategy to pre-compute indices for assembly into CSR sparse storage format. The proposed strategy has been implemented on both CPU and GPU. The performance characteristic of the proposed finite element solver is measured by solving large-scale three-dimensional (3D) elasticity problem involving a maximum of 4.7 million degrees of freedom (DOFs). A comparison is made with the standard assembly implementation in Eigen C++ library, which first stores the nonzero values in the form of triplets and then assembles into CSR sparse format. For the finest mesh with 4.7 million DOFs, the proposed CPU-based assembly strategy achieves 9.3× speedup over Eigen library. The computation of indices for assembly into CSR format takes 15.7 s on CPU and 2.4 s on GPU for 4.7 million DOFs. The computation of elemental matrices and their assembly, implemented on GPU as a single compute kernel, is found to be up to 24.3× faster than optimized CPU implementation. In terms of wall-clock time, the GPU-accelerated finite element solver is found to have up to 4× speedup over CPU solver.