无网格 Galerkin 法 GPU 加速并行计算及其应用

摘要

针对无网格 Galerkin 法计算耗时的问题，采用逐节点对法来组装刚度矩阵、共轭梯度法求解基于 CSR 格式存储的稀疏线性方程组，提出了一种利用罚函数法施加本质边界条件的 EFG 法 GPU 加速并行算法，给出了刚度矩阵和惩罚刚度矩阵的统一格式，以及 GPU 加速并行算法的流程图。编写了基于 CUDA 构架平台的 GPU 程序，且在 NVIDIA GeForce GTX 660显卡上通过数值算例对所提算法进行了性能测试与分析比较，探讨了影响加速比的因素。算例结果验证了所提算法的可行性，并在满足计算精度的前提下，其加速比最大可达17倍；同时线性方程组的求解对加速比起决定性影响。%In order to reduce the computing time of Element-Free Galerkin(EFG)method,a GPU accele-ration parallel algorithm of EFG method that essential boundary condition is imposed by penalty function method is proposed,in which stiffness matrix is assembled by node pair-wise approach,and sparse linear equations based on CSR format is solved by conjugate gradient methods.The unified format of stiffness matrix and penalty stiffness matrix was derived,and the flow chart of the parallel algorithm was provided.The GPU codes were programmed on CUDA,and algorithm testing was finished on the device of NVIDIA GeForce GTX 660 by numerical examples.The factors of affecting speedup ratio were discussed.The example results verified the feasibility of the proposed algorithm.The maximum speedup ratio was up to 17 times on the premise that the calculating accuracy is met,and to solve linear equations is the major factor in the speedup.