Numerical Studies of Vanka-Type Smoothers in Computational Solid Mechanics
Numerical Studies of Vanka-Type Smoothers in Computational Solid Mechanics
复制标题
计算固体力学中 Vanka 型平滑器的数值研究
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
S. Turek
中科院分区：
文献类型：
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作者：
Hilmar Wobker;S. Turek
本文在计算固体力学的背景下，对vanka型多网格平滑器进行了研究。这些平滑器最初是为了解决计算流体动力学（CFD）领域中出现的鞍点系统，特别是不可压缩流动问题而开发的。当处理（几乎）不可压缩的固体时，出现了类似的方程组，因此有理由采用CSM的“Vanka思想”。虽然在CFD文献中有大量关于Vanka平滑的研究，但很少有出版物描述了其在固体力学问题中的应用。通过这篇论文，我们希望为缩小这一差距做出贡献。我们描述并比较了四种不同的类vanka平滑器，其中两种面向稳定的等阶Q1/Q1有限元对。通过不同的测试配置，我们评估了平滑器能够处理几乎不可压缩材料和各向异性网格所产生的数值困难的程度。一方面，我们证明了所有vanka -smooth的效率在很大程度上取决于适当的参数选择。另一方面，我们证明，只有其中一些能够稳健地处理更关键的情况。此外，我们还说明了用外克雷洛夫空间方法包围多重网格方案如何影响求解器的整体性能，并将所有的检验扩展到非线性有限变形情况。
In this paper multigrid smoothers of Vanka-type are studied in the con- text of Computational Solid Mechanics (CSM).These smoothers were originally de- veloped to solve saddle-point systems arising in the field of Computational Fluid Dynamics (CFD), particularly for incompressible flow problems. When treating (nearly) incompressible solids, similar equation systems arise so that it is reason- able to adopt the 'Vanka idea' for CSM. While there exist numerous studies about Vanka smoothers in the CFD literature, only few publications describe applications to solid mechanical problems. With this paper we want to contribute to close this gap. We depict and compare four different Vanka-like smoothers, two of them are oriented towards the stabilised equal-order Q1/Q1 finite element pair. By means of different test configurations we assess how far the smoothers are able to handle the numerical difficulties that arise for nearly incompressible material and anisotropic meshes. On the one hand, we show that the efficiency of all Vanka-smoothers heav- ily depends on the proper parameter choice. On the other hand, we demonstrate that only some of them are able to robustly deal with more critical situations. Fur- thermore, we illustrate how the enclosure of the multigrid scheme by an outer Krylov space method influences the overall solver performance, and we extend all our examinations to the nonlinear finite deformation case.