大规模有限元系统的GPU加速计算研究

Solving large finite element system by GPU computation

研究了GPU(Graphics Processing Units)计算应用于有限元方法中的总刚计算和组装、稀疏矩阵与向量乘积运算、线性方程组求解问题,并基于CUDA(Compute Unified Device Architecture)平台利用GTX295GPU进行程序实现和测试。系统总刚采用CSR(Compressed Sparse Row)压缩格式存放于GPU显存中,用单元染色方法实现总刚并行计算组装,用共轭梯度迭代法求解大规模线性方程组。对300万自由度以内的空间桁架和平面问题算例,GPU有限元计算分别获得最高9.5倍和6.5倍的计算加速比,并且加速比随系统自由度的增加而近似线性增加,GFLOP/s峰值也有近10倍的增加。

Some techniques for applying GPU（Graphics Processing Units） computation in FEM（Finite El- ement Method） were investigated in this paper, which include element stiffness matrix parallel calcula- tion and global stiffness matrix assembly method, unstructured sparse matrix-vector multiplication and large-scale linear system solving method. A FEM code was implemented by using CUDA（Compute Uni- fied Device Architecture） platform and tested on nVidia GeForce GPU device. The system stiffness ma- trix was stored in the graphics memory in CSR（Compressed Sparse Row） format,and assembled via element coloring. Conjugate gradient method was used to solve FEM linear system iteratively. For the truss and 2D examples, the GPU-based FEM code gained speedups up to 9. 5x and 6.5x, respectively. It is found that the GPU speedup values are roughly linear with system DOFs（Degree Of Freedoms）,and the peak values of GFLOP/s increase approximately 10 times when comparing with those of CPU＇s.