【作者】 赵万帅；

【导师】 徐玉民；

【作者基本信息】 燕山大学 ， 概率论与数理统计， 2010， 硕士

【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。更多还原

【Abstract】 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.更多还原