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A hybrid domain decomposition method and its applications to contact problems.

机译：混合域分解方法及其在接触问题中的应用。

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摘要

Our goal is to solve nonlinear contact problems. We consider bodies in contact with each other divided into subdomains, which in turn are unions of elements. The contact surface between the bodies is unknown a priori, and we have a nonpenetration condition between the bodies, which is essentially an inequality constraint. We choose to use an active set method to solve such problems, which has both outer iterations in which the active set is updated, and inner iterations in which a (linear) minimization problem is solved on the current active face. In the first part of this dissertation, we review the basics of domain decomposition methods. In the second part, we consider how to solve the inner minimization problems. Using an approach based purely on FETI algorithms with only Lagrange multi- pliers as unknowns, as has been developed by the engineering community, does not lead to a scalable algorithm with respect to the number of subdomains in each body. We prove that such an algorithm has a condition number estimate which depends linearly on the number of subdomains across a body; numerical experiments suggest that this is the best possible bound. We also consider a new method based on the saddle point formulation of the FETI methods with both displacement vectors and Lagrange multipliers as unknowns. The resulting system is solved with a block-diagonal preconditioner which combines the one-level FETI and the BDDC methods. This approach allows the use of inexact solvers. We show that this new method is scalable with respect to the number of subdomains, and that its convergence rate depends only logarithmically on the number of degrees of freedom of the subdomains and bodies. In the last part of this dissertation, a model contact problem is solved by two approaches. The first one is a nonlinear algorithm which combines an active set method and the new method of Chapter 4. We also present a novel way of finding an initial active set. The second one uses the SMALBE algorithm, developed by Dostal et al. We show that the former approach has advantages over the latter.
机译：我们的目标是解决非线性接触问题。我们将彼此接触的物体划分为子域，而子域又是元素的并集。物体之间的接触表面是先验未知的，我们在物体之间具有非穿透条件，这实际上是一个不等式约束。我们选择使用活动集方法来解决此类问题，该方法既有更新活动集的外部迭代，也有解决当前活动面上的（线性）最小化问题的内部迭代。在本文的第一部分，我们回顾了域分解方法的基础。在第二部分中，我们考虑如何解决内部最小化问题。正如工程界开发的那样，使用仅基于FETI算法且仅将Lagrange乘积作为未知数的方法，就无法得出关于每个主体中子域数量的可伸缩算法。我们证明了这种算法具有条件数估计值，该条件数估计值线性依赖于整个身体的子域数量。数值实验表明，这是最好的界限。我们还考虑了基于FETI方法的鞍点公式化的新方法，其中位移矢量和拉格朗日乘数均为未知数。最终的系统使用块对角预处理器解决，该预处理器结合了一级FETI和BDDC方法。这种方法允许使用不精确的求解器。我们表明，该新方法相对于子域的数量是可扩展的，并且其收敛速度仅在对数上取决于子域和主体的自由度的数量。在本文的最后，通过两种方法解决了模型接触问题。第一个是将活动集方法和第四章的新方法结合在一起的非线性算法。我们还提出了一种寻找初始活动集的新颖方法。第二种使用由Dostal等人开发的SMALBE算法。我们表明，前一种方法比后者具有优势。

著录项

  • 作者

    Lee, Jungho.;

  • 作者单位

    New York University.;

  • 授予单位 New York University.;
  • 学科 Mathematics.;Computer Science.
  • 学位 Ph.D.
  • 年度 2009
  • 页码 146 p.
  • 总页数 146
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

  • 入库时间 2022-08-17 11:38:26

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