非线性消去算法在跨音速全速势方程计算中的应用

doi:10.12288/szjs.s2021-0770

数值计算与计算机应用 ›› 2022, Vol. 43 ›› Issue (4): 425-446.DOI: 10.12288/szjs.s2021-0770

• 论文 • 上一篇

非线性消去算法在跨音速全速势方程计算中的应用

周佳敏¹, 刘璐璐¹, 余瀚^2,3

1. 南京理工大学, 数学与统计学院, 南京 210094;
2. 南京邮电大学, 江苏省大数据安全与智能处理重点实验室, 南京 210023;
3. 苏州科达科技股份有限公司, 苏州 215011

收稿日期:2021-07-16 发布日期:2022-12-08
基金资助:
国家自然科学基金(11901296), 江苏省自然科学基金(BK20180450), 中国博士后科学基金项目(2021M692367)和江苏省博士后科研项目(2021K475C)资助.

PDF ( 19 ) 引用

周佳敏, 刘璐璐, 余瀚. 非线性消去算法在跨音速全速势方程计算中的应用[J]. 数值计算与计算机应用, 2022, 43(4): 425-446.

Zhou Jiamin, Liu Lulu, Yu Han. APPLICATION OF NONLINEAR ELIMINATION IN SOLVING THE TRANSONIC FULL POTENTIAL EQUATION[J]. Journal on Numerica Methods and Computer Applications, 2022, 43(4): 425-446.

APPLICATION OF NONLINEAR ELIMINATION IN SOLVING THE TRANSONIC FULL POTENTIAL EQUATION

Zhou Jiamin¹, Liu Lulu¹, Yu Han^2,3

1. School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, China;
2. Jiangsu Key Laboratory of Big Data Security & Intelligent Processing, Nanjing University of Posts and Telecommunications, Nanjing 210023, China;
3. Suzhou Keda Technology Company Ltd., Suzhou 215011, China

Received:2021-07-16 Published:2022-12-08

基于非线性消去技术,构造了跨音速全速势方程的并行非线性求解器.首先详细阐述了跨音速全速势方程及其有限差分格式.其次,借助非线性消去技术,通过隐式消去局部强非线性的方式,改善牛顿迭代法的全局收敛性质,从而在激波存在的情况下达到加速收敛的目的.最后,研究了马赫数、网格大小、处理器个数以及局部子问题求解精度等参数对算法收敛情况和总计算时间的影响.数值结果表明,提出的右侧非线性预条件算法在求解全速势方程时具有很好的鲁棒性和可扩展性.

关键词： 非线性消去, 全速势方程, 不平衡非线性, 并行计算

Based on nonlinear elimination, a parallel nonlinear solver is developed to simulate the flows for the transonic full potential equation. Firstly, we present the transonic full potential equation and its finite difference discretization in detail. Secondly, the local high nonlinearities are removed implicitly by nonlinear elimination in order to improve the global convergence performance of the Newton iterative method, so that fast convergence can be obtained even when the shock occurs. Finally, we study the effects of the Mach number, the mesh size, the number of processors, and the relative tolerance for the local subproblems on the convergence performance and the total execution time. Numerical results demonstrate that the right nonlinear preconditioned algorithm performs well to solve the full potential equation in terms of robustness and scaling.

Key words: nonlinear elimination, the full potential equation, unbalanced nonlinearity, parallel computing

MR(2010)主题分类:

分享此文:

[1] Knoll D A, Keyes D E. Jacobian-free Newton-Krylov methods:A survey of approaches and applications[J]. Journal of Computational Physics, 2004, 193(2):357-397.
[2] Cai X C, Gropp W D, Keyes D E, Melvin R G, Young D P. Parallel Newton-Krylov-Schwarz algorithms for the transonic full potential equation[J]. SIAM Journal on Scientific Computing, 1996, 19(1):246-265.
[3] Cai X C, Li X F. Inexact Newton methods with restricted additive Schwarz based nonlinear elimination for problems with high local nonlinearity[J]. SIAM Journal on Scientific Computing, 2013, 33(2):746-762.
[4] Knoll D A, Rider W J. A multigrid preconditioned Newton-Krylov method[J]. SIAM Journal on Scientific Computing, 1999, 21(2)
[5] 曾祥州. 微波热传导方程的Newton-Krylov-Schwarz算法研究[D]. PhD thesis, 湖南大学, 2018.
[6] Cai X C, Keyes D E. Nonlinearly preconditioned inexact Newton algorithms[J]. SIAM Journal on Scientific Computing, 2002, 24(1):183-200.
[7] Liu L, Keyes D E. Field-split preconditioned inexact Newton algorithms[J]. SIAM Journal on Scientific Computing, 2015, 37(3):1388-1409.
[8] Hwang F N, Lin H L. Two-level nonlinear elimination based preconditioners for inexact Newton methods with application in shocked duct flow calculation[J]. Electronic Transactions on Numerical Analysis, 2010, 37:239-251.
[9] Liu L, Keyes D E, Krause R. A note on adaptive nonlinear preconditioning techniques[J]. SIAM Journal on Scientific Computing, 2018, 40(2):A1171-A1186.
[10] Yang H, Yang C, Sun S. Active-set reduced-space methods with nonlinear elimination for two-phase flow problems in porous media[J]. SIAM Journal on Scientific Computing, 2016, 38(4):B593-B618.
[11] Yang H, Sun S, Yang C. Nonlinearly preconditioned semismooth Newton methods for variational inequality solution of two-phase flow in porous media[J]. Journal of Computational Physics, 2016, 332:1-20.
[12] Lanzkron P J, Rose D J, Wilkes J T. An analysis of approximate nonlinear elimination[J]. SIAM Journal on Scientific Computing, 1996, 19(1):538-559.
[13] Yang H, Hwang F N, Cai X C. Nonlinear preconditioning techniques for full-space LagrangeNewton solution of PDE-constrained optimization problems[J]. SIAM Journal on Scientific Computing, 2016, 38(5):A2756-A2778.
[14] Luo L, Shiu W S, Chen R, Cai X C. A nonlinear elimination preconditioned inexact Newton method for blood flow problems in human artery with stenosis[J]. Journal of Computational Physics, 2019, 399:108926.
[15] Young D P, Melvin R G, Bieterman M B, Johnson F T, Samant S S, Bussoletti J E. A locally refined rectangular grid finite element method:application to computational fluid dynamics and computational physics[J]. Journal of Computational Physics, 1991, 92(1):1-66.
[16] Lyu F X, Xiao T H, Yu X Q. A fast and automatic full-potential finite volume solver on Cartesian grids for unconventional configurations[J].中国航空学报:英文版, 2017, 030(003):951-963.
[17] 吕凡熹, 肖天航, 余雄庆.基于自适应直角网格的二维全速势方程有限体积解法[J].计算力学学报, 2016,3: 424-430.
[18] Jameson A. Transonic potential flow calculations using conservation form[C]. In Proc. 2nd AIAA Computational Fluid Dynamics Conference, pages 148-155, 1975.
[19] Holst T L. Transonic flow computations using nonlinear potential methods[J]. Progress In Aerospace Sciences, 2000, 36(1):1-61.
[20] Ngoc H. A numerical approach for assessing flow compressibility and transonic effect on airfoil aerodynamics[J]. Journal of Mechanical Science and Technology, 2020, 34(5):1-7.
[21] Hwang F N, Su Y C, Cai X C. A parallel adaptive nonlinear elimination preconditioned inexact Newton method for transonic full potential equation[J]. Computers & Fluids, 2015, 110:96-107.
[22] Young D P, Huffman W P, Melvin R G, Hilmes C L, Johnson F T. Nonlinear elimination in aerodynamic analysis and design optimization[G]. In Large-scale PDE-constrained optimization, pages 17-43. Springer, 2003.
[23] Dennis J E, Schnabel R B. Numerical methods for unconstrained optimization and nonlinear equations[M]. SIAM, Philadelphia, 1996.
[24] Dembo R S, Steihaug E T. Inexact Newton methods[J]. SIAM Journal on Numerical Analysis, 1982, 19(2):400-408.
[25] Flueck A J, Chiang H D. Solving the nonlinear power flow equations with an inexact Newton method using GMRES[J]. IEEE Transactions on Power Systems, 2002, 13(2):267-273.
[26] Eisenstat S C, Walker H F. Globally convergent inexact Newton methods[J]. SIAM Journal on Optimization, 1994, 4(2):393-422.
[27] Saad Y, Schultz M H. GMRES:A generalized minimal residual algorithm for solving nonsymmetric linear systems[J]. SIAM Journal on Scientific and Statistical Computing, 1986, 7(3):856-869.
[28] Balay S, Abhyankar S, Adams M F, Brown J, Brune P, Buschelman K, Dalcin L, A. D, Eijkhout V, Gropp W D, D. K, Kaushik D, Knepley M G, A. M D, McInnes L C, T. M R, T. M, Rupp K, P. S, Smith B F, Zampini S, Zhang H, Zhang H. PETSc web page http://www.mcs.anl.gov/petsc[M]. 2021.

[1]	杜玉龙, 徐凯文, 赵昆磊, 袁礼. 基于PHG平台的非结构四面体网格欧拉方程间断有限元并行求解器[J]. 数值计算与计算机应用, 2021, 42(2): 155-168.
[2]	邓维山, 徐进. 一种泊松-玻尔兹曼方程稳定算法的高效有限元并行实现[J]. 数值计算与计算机应用, 2018, 39(2): 91-110.
[3]	吴长茂, 杨超, 尹亮, 刘芳芳, 孙乔, 李力刚. 基于CPU-MIC异构众核环境的行星流体动力学数值模拟[J]. 数值计算与计算机应用, 2017, 38(3): 197-214.
[4]	宋梦召, 冯仰德, 聂宁明, 王武. 基于空位跃迁的KMC并行实现[J]. 数值计算与计算机应用, 2017, 38(2): 130-142.
[5]	刘辉, 冷伟, 崔涛. 并行自适应有限元计算中的负载平衡研究[J]. 数值计算与计算机应用, 2015, 36(3): 166-184.
[6]	刘辉, 崔涛, 冷伟. hp自适应有限元计算中一种新的自适应策略[J]. 数值计算与计算机应用, 2015, 36(2): 100-112.
[7]	吴立飞, 杨晓忠, 张帆. 非线性Leland方程的一种并行本性差分方法[J]. 数值计算与计算机应用, 2014, 35(1): 69-80.
[8]	段班祥, 朱小平, 张爱萍. 解线性互补问题的并行交替迭代算法[J]. 数值计算与计算机应用, 2011, 32(3): 183-195.
[9]	王顺绪, 戴华. 二次特征值问题的并行Jacobi-Davidson方法及其应用[J]. 数值计算与计算机应用, 2008, 29(4): 313-320.
[10]	姜金荣,迟学斌,陆忠华,刘洪利. 在MM5中实现IKAWA参数自适应选择[J]. 数值计算与计算机应用, 2007, 28(3): 215-220.
[11]	李永刚,欧阳洁,肖曼玉. 基于时间分解求解时间依赖问题的并行算法研究[J]. 数值计算与计算机应用, 2007, 28(1): 27-37.
[12]	余胜生,文元桥,周敬利. 隧道算法的分布式并行计算模型[J]. 数值计算与计算机应用, 2006, 27(4): 299-306.
[13]	曹建文,刘洋,孙家昶,姚继锋,潘峰. 大规模油藏数值模拟软件并行计算技术及在Beowulf系统上的应用进展[J]. 数值计算与计算机应用, 2006, 27(2): 86-95.
[14]	吕桂霞,马富明. 二维热传导方程有限差分区域分解算法[J]. 数值计算与计算机应用, 2006, 27(2): 96-105.
[15]	盛志强,刘兴平,崔霞. 对抛物方程使用新显格式的区域分解算法[J]. 数值计算与计算机应用, 2005, 26(4): 249-261.

阅读次数

全文

摘要

非线性消去算法在跨音速全速势方程计算中的应用

APPLICATION OF NONLINEAR ELIMINATION IN SOLVING THE TRANSONIC FULL POTENTIAL EQUATION

扫码分享