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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。
Aiming at the problem of iterative method to solve the problem of slow convergence of linear equations in the meshless Galerkin method, a gridless Galerkin parallel algorithm based on coupled GPU and preconditioned conjugate gradient method is proposed. In the overall stiffness matrix, the overall penalty The stiffness matrix can be parallelly combined and assembled simultaneously to obtain the diagonal preconditioned conjugate matrix, which effectively saves the storage space and computation time of the GPU. By adopting the tetrahedral integral background grid, the proposed algorithm improves the performance of the three-dimensional complex geometry Adaptability of the problem. The feasibility of the proposed algorithm is verified by two three-dimensional examples. Compared with the conjugate gradient method, the preconditioned conjugate gradient method can reduce the number of iterations by a maximum of 1686 times and the maximum iteration time by 1003 times. At the same time, Compared with the number of threads and the number of nodes, when the number of threads is 64, the speedup can reach the maximum, and the speedup of the preconditioned conjugate gradient method can be increased 4.5 times compared with the conjugate gradient method. Yoke gradient method to achieve the maximum speed up to 88.5 times.