求解椭圆型变分不等式的三种数值方法分析及应用
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摘要
椭圆型变分不等式问题在非线性问题中扮演着重要的角色,同时也是研究力学,物理与工程中许多自由边界问题的重要方法之一。随着数值方法的深入发展和计算机进行数值计算时运行速度的快速提高,变分不等式的数值求解不仅成为可能,而且可以模拟解决很多实际的问题。
在广大学者研究成果的基础上,论文首先采用对偶方法和罚方法求解一类变分不等式问题(P),并对两种算法的优劣性进行了比较。其次,将对偶理论应用到典型实例圆柱管道中的Bingham流体问题中,为数值求解带来了方便。最后,对于近年来学者广泛关注的圆柱体棒弹塑性挠度问题进行了研究。
全文共分为五章。第一章主要概述了椭圆型变分不等式及其研究现状,并说明了课题的来源和意义。第二章介绍了论文所涉及的一些重要的基本概念和结论,为论文要研究的内容奠定了理论上的基础。
第三章首先针对一类变分不等式问题(P)分别采用对偶方法和罚方法进行分析,给出求解的公式,并证明算法的收敛性。其次根据数值算例,分析方法中各参数对数值结果的影响,并比较了对偶方法和罚方法数值求解变分不等式问题(P)的优劣性。
第四章将对偶理论应用到Bingham流体问题中,首先将由该问题导出的变分不等式通过对偶理论进行了转化。其次对于转化后的对偶问题直接采用松弛法进行求解。最后给出了数值算例,并对数值结果与相对误差进行了分析,验证了方法的可行性。
第五章研究近年来学者广泛关注的圆柱体棒弹塑性挠度问题,由该问题所导出的变分不等式问题实际上是问题(P)的一个特例,采用带投影的松弛法这一重要工具进行数值求解,给出了数值算例,体现了方法的灵活性。
The elliptic variational inequality problem plays an important role in nonlinear problems. At the same time, it is one of the important ways to study many free boundary problems in mechanics, physics and engineering. With in-depth development of numerical methods and the rapid increase of the computer speed in numerical calculation, it is possible that we can solve variational inequalities by numerical methods and solve many practical problems by simulation.
On the basis of the research results of the majority of academic, the paper firstly solves a class of variational inequality problem (P) using the dual method and penalty method, as well as the pros and cons of the two algorithms were compared. Secondly, the duality theory will be applied to typical examples of the Bingham fluid problem in cylindrical pipe, which takes the advantage for the numerical solution. Finally, for the elastic-plastic torsion problem of a cylinder bar widespread concerned by scholars in recent years,the paper conducts a study.
Full-text is divided into five chapters. The first chapter mainly outlines the elliptic variational inequalities and their status and shows the source and significance of the subject. The second chapter describes some of the important basic concepts and conclusions involved by the papers, which have laid a theoretical foundation for the studied contents in the paper.
The third chapter firstly conducts analysis for a class of variational inequalities problem (P) using dual methods and penalty methods, giving the formula to solve and proving convergence of the algorithms. Secondly, according to numerical examples the paper does an analysis for various parameters in methods on the impact of the numerical results and comparison of the pros and cons of the dual method and penalty method on numerical solving variational inequality problem (P) .
The fourth chapter puts duality theory into the Bingham fluid problems. Firstly the variational inequality problem derived by the problem is conducted a conversion through the dual theory. Secondly, the transformation dual problem is solved directly using relaxation method. Finally, numerical examples are given, and numerical results and relative error is analyzed to verify the feasibility of the method.
The fifth chapter, for the elastic-plastic torsion problem of a cylinder bar widespread concerned by scholars in recent years, the paper conducts a study. The variational inequality problem derived by the problem actually is a special case of the problem (P). The numerical solution of the variational inequality problem will be solved using this important tool--the relaxation method with a projection, and numerical examples are given, reflecting the flexibility of the method.
在广大学者研究成果的基础上,论文首先采用对偶方法和罚方法求解一类变分不等式问题(P),并对两种算法的优劣性进行了比较。其次,将对偶理论应用到典型实例圆柱管道中的Bingham流体问题中,为数值求解带来了方便。最后,对于近年来学者广泛关注的圆柱体棒弹塑性挠度问题进行了研究。
全文共分为五章。第一章主要概述了椭圆型变分不等式及其研究现状,并说明了课题的来源和意义。第二章介绍了论文所涉及的一些重要的基本概念和结论,为论文要研究的内容奠定了理论上的基础。
第三章首先针对一类变分不等式问题(P)分别采用对偶方法和罚方法进行分析,给出求解的公式,并证明算法的收敛性。其次根据数值算例,分析方法中各参数对数值结果的影响,并比较了对偶方法和罚方法数值求解变分不等式问题(P)的优劣性。
第四章将对偶理论应用到Bingham流体问题中,首先将由该问题导出的变分不等式通过对偶理论进行了转化。其次对于转化后的对偶问题直接采用松弛法进行求解。最后给出了数值算例,并对数值结果与相对误差进行了分析,验证了方法的可行性。
第五章研究近年来学者广泛关注的圆柱体棒弹塑性挠度问题,由该问题所导出的变分不等式问题实际上是问题(P)的一个特例,采用带投影的松弛法这一重要工具进行数值求解,给出了数值算例,体现了方法的灵活性。
The elliptic variational inequality problem plays an important role in nonlinear problems. At the same time, it is one of the important ways to study many free boundary problems in mechanics, physics and engineering. With in-depth development of numerical methods and the rapid increase of the computer speed in numerical calculation, it is possible that we can solve variational inequalities by numerical methods and solve many practical problems by simulation.
On the basis of the research results of the majority of academic, the paper firstly solves a class of variational inequality problem (P) using the dual method and penalty method, as well as the pros and cons of the two algorithms were compared. Secondly, the duality theory will be applied to typical examples of the Bingham fluid problem in cylindrical pipe, which takes the advantage for the numerical solution. Finally, for the elastic-plastic torsion problem of a cylinder bar widespread concerned by scholars in recent years,the paper conducts a study.
Full-text is divided into five chapters. The first chapter mainly outlines the elliptic variational inequalities and their status and shows the source and significance of the subject. The second chapter describes some of the important basic concepts and conclusions involved by the papers, which have laid a theoretical foundation for the studied contents in the paper.
The third chapter firstly conducts analysis for a class of variational inequalities problem (P) using dual methods and penalty methods, giving the formula to solve and proving convergence of the algorithms. Secondly, according to numerical examples the paper does an analysis for various parameters in methods on the impact of the numerical results and comparison of the pros and cons of the dual method and penalty method on numerical solving variational inequality problem (P) .
The fourth chapter puts duality theory into the Bingham fluid problems. Firstly the variational inequality problem derived by the problem is conducted a conversion through the dual theory. Secondly, the transformation dual problem is solved directly using relaxation method. Finally, numerical examples are given, and numerical results and relative error is analyzed to verify the feasibility of the method.
The fifth chapter, for the elastic-plastic torsion problem of a cylinder bar widespread concerned by scholars in recent years, the paper conducts a study. The variational inequality problem derived by the problem actually is a special case of the problem (P). The numerical solution of the variational inequality problem will be solved using this important tool--the relaxation method with a projection, and numerical examples are given, reflecting the flexibility of the method.
引文
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5 Z..George, I.Peter. Finite Elasto-Plastic Analysis of Torsion Problems Using Different Spin Tensors. International Journal of Plasticity,1992:271-314
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7薛莲.变分不等式的数值方法的研究. [浙江大学理学博士学位论文]. 2004:1-10
8 N.Kikuchi. Convergence of a Penalty-Finite Element Approximation for an Obstacle Problem. Numer.Math., 1981:105-120
9 Bnouhachem, Abdella. Some New Extragradient Iterative Methods for Variational Inequalities.Nonlinear Analysis, Theory, Methods and Application, 2009,70:1321- 1329
10严侃.求解变分不等式的一些新算法. [上海大学理学硕士学位论文]. 2008: 4-7
11 M.Tao. Comparison of Two Kinds of Approximate Proximal Point Algorithms for Monotone Variational Inequalities. Journal of Southeast University,2008, 24(4):537- 540
12 J.K.Djoko.Discontinuous Galerkin Finite Element Methods for Variational Inequali- ties of First and Second Kinds. Numerical Methods for Partial Differential Equations, 2008, 24(1):296-311
13 S.Shahram. Iterative Algorithms for Finding Common Solutions of Variational Inequalities and Systems of Equilibrium Problems and Fixed Points of Families andSemigroups of Nonexpansive Mappings. Nonlinear Analysis, Theory, Methods and Applications, 2009, 70(12):4195-4208
14 M. Yeganeh, R. Naghdabadi. Axial Effects Investigation in Fixed-End Circular Bars under Torsion with a Finite Deformation Model Based on Logarithmicstrain. International Jour- nal of Mechanical Sciences , 2006,48:75-84
15方长杰,陈胜.广义混合变分不等式的近似点投影算法.应用数学, 2008,03:587-595
16 A.Caboussat, R.Glowinski. A Two-Grids Projection Algorithm for Obstacle Problems. Computers and Mathematics with Application, 2005,50:171-178
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26 L.H. Wang. On the Quadratic Finite Element Approximation to the Obstacle Problem. Numer. Math, 2001
27王烈衡,王光辉.弹性接触问题的一种新的混合变分形式.计算数学, 1999, 21(2):237-244
28王烈衡.弹性接触问题的对偶混合有限元分析.计算数学, 1999,21(4):483-494
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31张传宝.求解变分不等式问题的一个投影算法.枣庄学院学报, 2005,22(2):1-3
32孙敏,张传宝.一种新的求解变分不等式问题的外梯度投影算法.曲阜师范大学学报, 2005,31(3):27-29
33拂晓婴,周武. Hilbert空间中一类拟变分不等式问题.电子科技大学学报, 2005, 34(5):717-719
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36 B.Jiang.A Priori Error Analysis of the Mortar Finite Element Method for Variational Inequalities.Numerical Methods for Partial Differential Equations,2008,24:476-503
37陈明飞,徐林荣.二阶椭圆型微分方程谱方法的后验误差估计.高等学校计算数学学报, 2005,27:60-68
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2 R.Glowinski. Numerical Methods for Nonlinear Variational Problems. New York: Springer-Verlag,1984
3韩渭敏,程晓良.变分不等式简介.北京:高等教育出版社, 2007
4邵新慧.求解鞍点问题的一般加速超松弛方法.数值计算与计算机应用, 2006
5 Z..George, I.Peter. Finite Elasto-Plastic Analysis of Torsion Problems Using Different Spin Tensors. International Journal of Plasticity,1992:271-314
6唐恒永.求解随机凸规划概率约束问题的对偶算法.高等学校计算数学学报, 2008,02
7薛莲.变分不等式的数值方法的研究. [浙江大学理学博士学位论文]. 2004:1-10
8 N.Kikuchi. Convergence of a Penalty-Finite Element Approximation for an Obstacle Problem. Numer.Math., 1981:105-120
9 Bnouhachem, Abdella. Some New Extragradient Iterative Methods for Variational Inequalities.Nonlinear Analysis, Theory, Methods and Application, 2009,70:1321- 1329
10严侃.求解变分不等式的一些新算法. [上海大学理学硕士学位论文]. 2008: 4-7
11 M.Tao. Comparison of Two Kinds of Approximate Proximal Point Algorithms for Monotone Variational Inequalities. Journal of Southeast University,2008, 24(4):537- 540
12 J.K.Djoko.Discontinuous Galerkin Finite Element Methods for Variational Inequali- ties of First and Second Kinds. Numerical Methods for Partial Differential Equations, 2008, 24(1):296-311
13 S.Shahram. Iterative Algorithms for Finding Common Solutions of Variational Inequalities and Systems of Equilibrium Problems and Fixed Points of Families andSemigroups of Nonexpansive Mappings. Nonlinear Analysis, Theory, Methods and Applications, 2009, 70(12):4195-4208
14 M. Yeganeh, R. Naghdabadi. Axial Effects Investigation in Fixed-End Circular Bars under Torsion with a Finite Deformation Model Based on Logarithmicstrain. International Jour- nal of Mechanical Sciences , 2006,48:75-84
15方长杰,陈胜.广义混合变分不等式的近似点投影算法.应用数学, 2008,03:587-595
16 A.Caboussat, R.Glowinski. A Two-Grids Projection Algorithm for Obstacle Problems. Computers and Mathematics with Application, 2005,50:171-178
17屈彪.求解拟变分不等式的一种投影算法.应用数学学报, 2008,05:922-928
18 R.Glowinski, Y.Kuznetzov, T. W.Pan. A Penalty Newton Conjugate Gradient Method for the Solution of Obstacle Problems. C R Acad Sei Paris, Sér.I, 2003,336:435-440
19丁方允,张欣,丁睿.摩擦问题中的边界混合变分不等.应用数学和力学, 1999, 20(2):201-210
20 W.Han, M.Sofonea.Quasistatic Contact Problems in Viscoelasticity and Viscop- lasticity.American Mathematical Society and International Press, 2002
21 N.Kikuchi. Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. Phiadelphia: SIAM, 1988
22 D.Kinderlehrer, G. Stampacchia. An Introduction to Variational Inequalities and Their Applications. New York: Academic Press, 1980
23 P. D. Panagiotopoulos. Inequality Problems in Mechanics and Applications. Boston: Birkh?user, 1985
24 J. F. Rodrigues.Obstacle Problems in Mathematical Physics. Amsterdam: North- Holland, 1987
25王烈衡.摩擦接触问题正则化方法的注记.计算数学, 1998,20(3):300-304
26 L.H. Wang. On the Quadratic Finite Element Approximation to the Obstacle Problem. Numer. Math, 2001
27王烈衡,王光辉.弹性接触问题的一种新的混合变分形式.计算数学, 1999, 21(2):237-244
28王烈衡.弹性接触问题的对偶混合有限元分析.计算数学, 1999,21(4):483-494
29孔祥言.高等渗流力学.合肥:中国科学技术大学出版社,1999:408-415
30李林珂,丁协平.解变分不等式的超梯度Mann迭代算法.四川师范大学学报(自然科学版), 2005,28(5):514-517
31张传宝.求解变分不等式问题的一个投影算法.枣庄学院学报, 2005,22(2):1-3
32孙敏,张传宝.一种新的求解变分不等式问题的外梯度投影算法.曲阜师范大学学报, 2005,31(3):27-29
33拂晓婴,周武. Hilbert空间中一类拟变分不等式问题.电子科技大学学报, 2005, 34(5):717-719
34丁协平.自反Banach空间内混合非线性似变分不等式解的算法.应用数学和力学, 1998,19(6):489-496
35桂海莲.混合变分不等式在弹性有摩擦接触边界元法中的应用. [燕山大学理学硕士学位论文]. 2006:48-55
36 B.Jiang.A Priori Error Analysis of the Mortar Finite Element Method for Variational Inequalities.Numerical Methods for Partial Differential Equations,2008,24:476-503
37陈明飞,徐林荣.二阶椭圆型微分方程谱方法的后验误差估计.高等学校计算数学学报, 2005,27:60-68
38冯康.数值计算方法.北京:国防工业出版社, 1978
39冯康,石钟慈.弹性结构的数学理论.北京:科学出版社, 1981
40胡建伟,汤怀民.微分方程数值方法.北京:科学出版社, 1999
41李开泰,黄艾香,黄庆怀.有限元方法及其应用.西安:西安交通大学出版社, 1984
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