有限元（Finite element，FE）-无网格Galerkin法（Element-free Galerkin，EFG）耦合能充分发挥有限元和无网格法各自具有的优势，为进一步提高FE-EFG耦合法在大规模工程应用中的计算效率，提出了一种FE-EFG耦合法的图形处理器（Graphic processing unit，GPU）并行加速算法，通过采用局域搜索法搜索EFG区域中节点影响域内的节点或积分点，以及积分点定义域内的节点；利用统一计算架构（Compute unified device architecture，CUDA）特点，在全求解域内引入交叉节点法实现了总体刚度矩阵的并行组装及按行压缩（Compress sparse row，CSR）存储；利用CUDA库函数并结合预条件共轭梯度（Preconditioned conjugate gradient，PCG）法对总体离散方程进行了迭代求解，2个数值算例验证了所提方法的可行性和计算精度，所得结果显示FE-EFG耦合法的计算效率得到显著提高，且其加速比会随计算规模的增加而增大，从而为大规模工程计算提供了一种高效的耦合算法。
The finite element (FE)-element free Galerkin (EFG) coupling method can make full use of the respective advantages of finite element method and element-free Galerkin method. In order to further improve the computational efficiency of FE-EFG coupling method in large-scale engineering application, Graphic processing unit parallel speedup algorithm of coupled FE-EFG method is presented. Nodes or integral points in the nodal influence field, and the nodes in the integral point definition field are found efficiently by using the local search method in the EFG domain. Based on the characteristics of compute unified device architecture (CUDA), the idea of interacting node pair is introduced to parallel assemble the total stiffness matrix and store it by compress sparse row storage format. The preconditioned conjugate gradient is used to solve the total discrete equations based on the CUDA library function. Two numerical examples verified the feasibility and computational accuracy of presented coupling method. The results show that computational efficiency of FE-EFG coupling method is improved remarkably, and its speedup ratio will increase with the increase of computing scale. Thus, a highly efficient coupling algorithm is offered to large-scale engineering computation.
[1] BOKIN M E, BENNETT J A. Shape optimization of three-dimension folded plate structures[J]. AIAA Journal, 1985, 23(11):1804-1810.
[2] BELYTSCHKO T, LU Y Y, GU L. Element-free Galerkin methods[J]. International Journal for Numerical Methods in Engineering, 1994, 37(2):229-256.
[3] BELYTSCHKO T, ORGAN D, KRONGAUZ Y. A coupled finite element-element-free Galerkin method[J]. Computational Mechanics, 1995, 17(3):186-195.
[4] HUERTA A. Enrichment and coupling of the finite element and meshless methods[J]. International Journal for Numerical Methods in Engineering, 2000, 48(11):1615-1636.
[5] XIAO Q Z, DHANASEKAR M. Coupling of FE and EFG using Collocation Approach[J]. Advances in Engineering Software, 2002, 33(s7-10):507-515.
[6] 杨海天,刘岩.一种FEM-EFGM耦合技术及其应用[J].计算力学学报, 2003, 20(5):511-517. YANG Haitian, LIU Yan. A coupled FEM-EFGM technique and its application[J]. Chinese Journal of Computational Mechanics, 2003, 20(5):511-517.
[7] RAO B N, RAHMAN S. A coupled Mesh-Finite element method for fracture analysis of cracks[J]. International Journal of Pressure Vessels & Piping, 2001, 78(9):647-657.
[8] GONG Shuguang, XIE Guilan, ZHANG Jianping, et al. Sensitivity analysis and shape optimization based on FE-EFG coupled method[J]. Research in Engineering Design, 2008, 20(2):117-128.
[9] WANG Zhongjin, YUAN Binxian. Numerical analysis of coupled finite element with element-free Galerkin in sheet flexible-die forming[J]. Transactions of Nonferrous Metals Society of China, 2014, 24(2):462-469.
[10] MARTÍNEZ-FRUTOS J, HERRERO-PÉREZ D. Efficient matrix-free GPU implementation of fixed grid finite element analysis[J]. Finite Elements in Analysis & Design, 2015, 104(1):61-71.
[11] CAI Yong, LI Guangyao, WANG Hu. A parallel nodebased solution scheme for implicit finite element method using GPU[J]. Procedia Engineering, 2013, 61:318-324.
[12] 蔡勇,王琥,李光耀,等. 基于边光滑三角形壳元和统一计算架构的板料成形仿真并行计算方法[J]. 机械工程学报, 2012, 48(6):32-38. CAI Yong, WANG Hu, LI Guangyao, et al. Parallel simulation of sheet metal forming based on EST element and compute unified device architecture[J]. Journal of Mechanical Engineering, 2012, 48(6):32-38.
[13] 龚曙光,刘奇良,卢海山,等. 无网格Galerkin法GPU加速并行计算及其应用[J]. 计算力学学报,2015,32(6):745-751. GONG Shuguang, LIU Qiliang, LU Haishan, et al. Parallel computing and application of Element-Free Galerkin method for GPU acceleration[J]. Chinese Journal of Computational Mechanics, 2015, 32(6):745-751.
[14] 龚曙光,卢海山,张建平,等. 基于交叉节点对无网格Galerkin法的改进算法研究[J]. 工程力学, 2015, 32(8):16-21,28. GONG Shuguang, LU Haishan, ZHANG Jianping, et al. Study on an improved algorithm of element-free Galerkin method based on interacting nodal pairs[J]. Engineering Mechanics, 2015, 32(8):16-21,28.
[15] KARATARAKIS A, METSIS P, PAPADRAKAKIS M. GPU-acceleration of stiffness matrix calculation and efficient initialization of EFG meshless methods[J]. Computer Methods in Applied Mechanics & Engineering, 2013, 258(5):63-80.
