等几何分析的多重网格共轭梯度法

刘石; 陈德祥; 冯永新; 徐自力; 郑李坤

doi:10.3879/j.issn.1000-0887.2014.06.005

等几何分析的多重网格共轭梯度法

doi: 10.3879/j.issn.1000-0887.2014.06.005

¹广东电网公司电力科学研究院, 广州 510080;²西安交通大学航天航空学院; 机械结构强度与振动国家重点实验室, 西安 710049

基金项目: 国家重点基础研究发展计划（973计划）(2011CB706505)；国家自然科学基金(51275385)

详细信息

作者简介:
刘石(1974—)，男，湖北大冶人，高级工程师，博士(E-mail: 13925041516@139.com)

中图分类号: O241.82
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- 被引次数: 0
出版历程
- 收稿日期: 2013-12-03
- 修回日期: 2014-05-04
- 刊出日期: 2014-06-11

A Multigrid Preconditioned Conjugate Gradient Method for Isogeometric Analysis

¹Electric Power Research Institute, Guangdong Power Grid Corporation, Guangzhou 510080, P.R.China;²State Key Laboratory for Strength and Vibration of Mechanical Structures;School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, P.R.China

Funds: The National Basic Research Program of China (973 Program)(2011CB706505)；The National Natural Science Foundation of China(51275385)

摘要

摘要: 提高NURBS基函数阶数可以提高等几何分析的精度，同时也会降低多重网格迭代收敛速度.将共轭梯度法与多重网格方法相结合，提出了一种提高收敛速度的方法，该方法用共轭梯度法作为基础迭代算法，用多重网格进行预处理.对Poisson（泊松）方程分别用多重网格方法和多重网格共轭梯度法进行了求解，计算结果表明：等几何分析中采用高阶NURBS基函数处理三维问题时，多重网格共轭梯度法比多重网格法的收敛速度更快.
- 等几何分析 /
- 多重网格 /
- 共轭梯度法 /
- Poisson方程 /
- 迭代算法 /
- NURBS
Abstract: Accuracy of the isogeometric analysis can be improved through increase of the order of the NURBS basis function, but convergence of the multigrid will be slowed down at the same time. A method which combined the multigrid technique and preconditioned conjugate gradient iteration was proposed to accelerate the multigrid convergence. In the proposed method, the conjugate gradient part serves as the primary iteration, while the multigrid part serves as the preconditioner. The Poisson’s equation was solved with the multigrid method and multigrid preconditioned conjugate gradient method repectively for comparison. The results show that the multigrid preconditioned conjugate gradient method converges faster than the multigrid method especially in the cases of high-order NURBS basis functions or 3-dimensional problems.
- isogeometric analysis /
- multigrid /
- conjugate gradient /
- Poisson’s equation /
- iterative algorithm /
- NURBS

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参考文献(14)

[1]	Hughes T J R, Cottrell J A, Bazilevs Y. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement[J].Computer Methods in Applied Mechanics and Engineering,2005,194(39/41): 4135-4195.
[2]	陈德祥, 徐自力, 刘石, 冯永新. 求解Stokes方程的最小二乘等几何分析方法[J]. 西安交通大学学报, 2013,47(5): 51-55.(CHEN De-xiang, XU Zi-li, LIU Shi, FENG Yong-xin. Least squares isogeometric analysis for Stokes equation[J].Journal of Xi’an Jiaotong University,2013,47(5): 51-55.(in Chinese))
[3]	Evans J A, Bazilevs Y, Babuska I, Hughes T J R.n -widths, sup-infs, and optimality ratios for thek -version of the isogeometric finite element method[J].Computer Methods in Applied Mechanics and Engineering,2009,198(21/26): 1726-1741.
[4]	Lipton S, Evans J A, Bazilevs Y, Elguedj T, Hughes T J R. Robustness of isogeometric structural discretizations under severe mesh distortion[J].Computer Methods in Applied Mechanics and Engineering,2010,199(5/8): 357-373.
[5]	陶文铨. 计算传热学的近代进展[M]. 北京: 科学出版社, 2000.(TAO Wen-quan.Advances in Computational Heat Transfer[M]. Beijing: Science Press, 2000.(in Chinese))
[6]	Kumar D S, Kumar K S, Das M K. A fine grid solution for a lid-driven cavity flow using multigrid method[J].Engineering Applications of Computational Fluid Mechanics,2009,3(3): 336-354.
[7]	Trottenberg U, Oosterlee C W, Schuller A.Multigrid [M]. Academic Press, 2000.
[8]	Briggs W L, Henson V E, McCormick S F.A Multigrid Tutorial [M]. Philadelphia, PA, USA: Society for Industrial and Applied Mathematics, 2000.
[9]	Sampath R S, Biros G. A parallel geometric multigrid method for finite elements on octree meshes[J].SIAM Journal on Scientific Computing,2010,32(3): 1361-1392.
[10]	Gahalaut K P S, Kraus J K, Tomar S K. Multigrid methods for isogeometric discretization[J].Computer Methods in Applied Mechanics and Engineering,2013,253: 413-425.
[11]	Saad Y.Iterative Methods for Sparse Linear Systems [M]. SIAM, 2003.
[12]	Tatebe O. The multigrid preconditioned conjugate gradient method[C]//Melson N D.The Sixth Copper Mountain Conference on Multigrid Methods.Copper Mountain: NASA, 1993: 621-634.
[13]	Cohen E, Lyche T, Riesenfeld R. Discrete B-splines and subdivision techniques in computer-aided geometric design and computer graphics[J].Computer Graphics and Image Processing,1980,14(2): 87-111.
[14]	de Falco C, Reali A, Vazquez R. GeoPDEs: a research tool for isogeometric analysis of PDEs[J].Advances in Engineering Software,2011,42(12): 1020-1034.