中文摘要：

针对迭代法求解无网格Galerkin法中线性方程组收敛速度慢的问题,提出了一种耦合GPU和预处理共轭梯度法的无网格Galerkin法并行算法,在对其总体刚度矩阵、总体惩罚刚度矩阵进行并行联合组装的同时即可得到对角预处理共轭矩阵,有效地节省了GPU的存储空间和计算时间;通过采用四面体积分背景网格,提高了所提算法对三维复杂几何形状问题的适应性。通过2个三维算例验证了所提算法的可行性,且预处理共轭梯度法与共轭梯度法相比,其迭代次数最大可减少1686倍,最大的迭代时间可节省1003倍;同时探讨了加速比与线程数和节点个数之间的关系,当线程数为64时其加速比可达到最大,且预处理共轭梯度法的加速比与共轭梯度法相比可增大4.5倍,预处理共轭梯度法的加速比最大达到了88.5倍。

英文摘要：

Aiming at the convergence problem of solving linear equations by using iterative method in element-free Galerkin method, the parallel algorithm of element-free Galerkin method is presented by coupling GPU and Preconditioning Conjugate Gradient（PCG） method. The diagonal preconditioning conjugate matrix can be achieved while general stiffness matrix is concurrently assembled together with general penalty stiffness matrix, and it can effectively save the storage space and computation time of GPU. By using tetrahedral cell to carry out integral calculation, the adaptability of proposed algorithm is improved to complex three dimensional geometry problems. Two three-dimension numerical examples prove the feasibility of proposed algorithm. And compared with the Conjugate Gradient（CG） method, the iteration time of PCG method can reduce to 1686 times, the iteration time can save 1003 times. The relation between speedup and thread count, number of node is discussed. When the number of threads is 64 the speedup can achieve the maximum value, and the acceleration of PCG method can increase 4.5 times than one of CG method. The maximum speedup of PCG method reaches 88.5 times.