基于SA-AMG的弹塑性有限元计算的并行实现
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摘要
利用增量-牛顿法和光滑聚集代数多重网格(SA-AMG)预条件共轭梯度法(PCG),实现一种弹塑性问题的有限元并行求解方法。在求解过程中,分步施加荷载并循环;在每个循环中,使用牛顿法迭代;在每次迭代中,使用SA-AMG预条件共轭梯度法并行求解线性化后的方程组。基于Trilinos开发相应的并行程序,并在天河二号超级计算机上进行数值实验,验证算法和程序的正确性。分析光滑聚集代数多重网格法的主要参数对计算性能的影响,测试程序的并行性和可扩展性。
This paper used the quasi-Newton method and preconditioned conjugate gradient(PCG) method with smoothed aggregation algebraic multigrid(SA-AMG) were used to solve elastic-plastic problems in parallel. In the solution process, the load was applied step by step and circulated. During each cycle, bewton method was used iteratively. In each iteration, PCG with SA-AMG was utilized to solve linearized equations. The corresponding parallel program was developed based on Trilinos, and the numerical experiments were carried out on Tianhe-2 super computer. Through the experiments, the correctness of the algorithm and program was verified. The influence of the main parameters of SA-AMG on the calculate performance was analyzed, and the parallel performance and scalability of the program were tested.
This paper used the quasi-Newton method and preconditioned conjugate gradient(PCG) method with smoothed aggregation algebraic multigrid(SA-AMG) were used to solve elastic-plastic problems in parallel. In the solution process, the load was applied step by step and circulated. During each cycle, bewton method was used iteratively. In each iteration, PCG with SA-AMG was utilized to solve linearized equations. The corresponding parallel program was developed based on Trilinos, and the numerical experiments were carried out on Tianhe-2 super computer. Through the experiments, the correctness of the algorithm and program was verified. The influence of the main parameters of SA-AMG on the calculate performance was analyzed, and the parallel performance and scalability of the program were tested.
引文
[1] 付朝江, 张武. 结构有限元分析并行处理的研究进展[J]. 力学进展, 2006, 36(3):354-362.
[2] 苗新强. 有限元结构分析多层并行算法研究及应用[D]. 上海:上海交通大学, 2015.
[3] 仲妍. 大型稀疏线性方程组并行求解及预处理技术研究[D]. 长沙:国防科学技术大学, 2010.
[4] 杨玉明. 中厚板壳结构弹塑性并行有限元法[D]. 南京:南京理工大学, 2005.
[5] 茹忠亮, 冯夏庭, 李洪东,等. 大型地下工程三维弹塑性并行有限元分析[J]. 岩石力学与工程学报, 2006, 25(6):1141.
[6] 张友良, 冯夏庭. 岩土工程百万以上自由度有限元并行计算[J]. 岩土力学, 2007, 28(4):684-688.
[7] Bencheva G , Margenov S . Parallel incomplete factorization preconditioning of rotated linear FEM systems[J]. Journal of Computational & Applied Mechanics, 2002, 4(2):105-117.
[8] Saad Y. Iterative methods for sparse linear systems[M]. Siam, 2003.
[9] Rao A R M. MPI-based parallel finite element approaches for implicit nonlinear dynamic analysis employing sparse PCG solvers[J]. Advances in Engineering Software, 2005, 36(3): 181-198.
[10] Mennel U, Arbenz P. A multilevel PCG algorithm for the μ-FE analysis of bone structures[D]. Institute of Computational Science, ETH Zürich, 2006.
[11] 郑超, 张建海. 预处理共轭梯度法在岩土工程有限元中的应用[J]. 岩石力学与工程学报, 2007, 26(S1):2820-2826.
[12] Van P, Brezina M, Mandel J. Convergence of algebraic multigrid based on smoothed aggregation[J]. Numerische Mathematik, 2001, 88(3): 559-579.
[13] 肖映雄. 代数多重网格算法研究及其在固体力学计算中的应用[D]. 湘潭:湘潭大学, 2006.
[14] 武立伟, 张健飞, 张倩. 基于光滑聚集代数多重网格的有限元并行计算实现方法[J]. 计算机辅助工程, 2017, 26(6):16-22.
[15] Tuminaro R S, Tong C. Parallel smoothed aggregation multigrid: Aggregation strategies on massively parallel machines[C]//Supercomputing,ACM/IEEE 2000 Conference.IEEE, 2000: 5-5.
[16] Long K R, Tuminaro R S, Bartlett R A, et al. An overview of Trilinos[R]. Sandia National Laboratories, SAND2003-2927, 2003-08-01.
[2] 苗新强. 有限元结构分析多层并行算法研究及应用[D]. 上海:上海交通大学, 2015.
[3] 仲妍. 大型稀疏线性方程组并行求解及预处理技术研究[D]. 长沙:国防科学技术大学, 2010.
[4] 杨玉明. 中厚板壳结构弹塑性并行有限元法[D]. 南京:南京理工大学, 2005.
[5] 茹忠亮, 冯夏庭, 李洪东,等. 大型地下工程三维弹塑性并行有限元分析[J]. 岩石力学与工程学报, 2006, 25(6):1141.
[6] 张友良, 冯夏庭. 岩土工程百万以上自由度有限元并行计算[J]. 岩土力学, 2007, 28(4):684-688.
[7] Bencheva G , Margenov S . Parallel incomplete factorization preconditioning of rotated linear FEM systems[J]. Journal of Computational & Applied Mechanics, 2002, 4(2):105-117.
[8] Saad Y. Iterative methods for sparse linear systems[M]. Siam, 2003.
[9] Rao A R M. MPI-based parallel finite element approaches for implicit nonlinear dynamic analysis employing sparse PCG solvers[J]. Advances in Engineering Software, 2005, 36(3): 181-198.
[10] Mennel U, Arbenz P. A multilevel PCG algorithm for the μ-FE analysis of bone structures[D]. Institute of Computational Science, ETH Zürich, 2006.
[11] 郑超, 张建海. 预处理共轭梯度法在岩土工程有限元中的应用[J]. 岩石力学与工程学报, 2007, 26(S1):2820-2826.
[12] Van P, Brezina M, Mandel J. Convergence of algebraic multigrid based on smoothed aggregation[J]. Numerische Mathematik, 2001, 88(3): 559-579.
[13] 肖映雄. 代数多重网格算法研究及其在固体力学计算中的应用[D]. 湘潭:湘潭大学, 2006.
[14] 武立伟, 张健飞, 张倩. 基于光滑聚集代数多重网格的有限元并行计算实现方法[J]. 计算机辅助工程, 2017, 26(6):16-22.
[15] Tuminaro R S, Tong C. Parallel smoothed aggregation multigrid: Aggregation strategies on massively parallel machines[C]//Supercomputing,ACM/IEEE 2000 Conference.IEEE, 2000: 5-5.
[16] Long K R, Tuminaro R S, Bartlett R A, et al. An overview of Trilinos[R]. Sandia National Laboratories, SAND2003-2927, 2003-08-01.