A Hybrid Domain Decomposition Method and its Applications to Contact Problems TR2009-924
A Hybrid Domain Decomposition Method and its Applications to Contact Problems TR2009-924
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一种混合域分解方法及其在接触问题中的应用TR2009-924
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发表时间：
2009
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通讯作者：
Jungho Lee
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作者：
Jungho Lee
我们的目标是解决非线性接触问题。我们把相互接触的物体划分为子域，而子域又是元素的结合。物体之间的接触面是先验未知的，我们有物体之间的非侵彻条件，这本质上是一个不等式约束。我们选择使用活动集方法来解决这类问题，该方法既有更新活动集的外部迭代，也有在当前活动面上解决（线性）最小化问题的内部迭代。在本文的第一部分，我们回顾了区域分解方法的基础。在第二部分，我们考虑如何解决内部最小化问题。使用纯基于FETI算法的方法，仅将拉格朗日多钳子作为未知数，正如工程社区所开发的那样，不会导致关于每个主体中子域数量的可扩展算法。证明了该算法具有一个条件数估计，该估计与物体上的子域数线性相关；数值实验表明，这是可能的最佳边界。我们还考虑了一种新的基于鞍点公式的FETI方法，其中位移矢量和拉格朗日乘子都是未知数。用一阶FETI和BDDC相结合的块对角预调节器对系统进行求解。这种方法允许使用非精确求解器。我们证明了这种新方法相对于子域的数量是可扩展的，并且它的收敛速度仅依赖于子域和体的自由度的对数数量。在论文的最后一部分，采用两种方法求解模型接触问题。第一个是将活动集方法与第四章的新方法相结合的非线性算法。我们还提出了一种寻找初始活动集的新方法。第二种方法使用了由Dostal等人开发的smallbe算法。我们表明，前一种方法比后者有优势。
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.