Volume 1, Issue 1
Numerical Studies of Vanka-Type Smoothers in Computational Solid Mechanics

Adv. Appl. Math. Mech., 1 (2009), pp. 29-55.

Published online: 2009-01

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• Abstract

In this paper multigrid smoothers of Vanka-type are studied in the context of Computational Solid Mechanics (CSM). These smoothers were originally developed 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 reasonable 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 closing this gap. We depict and compare four different Vanka-like smoothers, two of them are oriented towards the stabilised equal-order $Q_1/Q_1$ 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 heavily 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. Furthermore, 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.

• Keywords

Coupled multigrid, Vanka smoother, linear and finite elasticity, nearly incompressible material, saddle point systems, finite elements.

• AMS Subject Headings

65N22, 65N55, 65H10, 65F50, 74S05

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@Article{AAMM-1-29, author = {Hilmar Wobker , and Turek , Stefan}, title = {Numerical Studies of Vanka-Type Smoothers in Computational Solid Mechanics}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2009}, volume = {1}, number = {1}, pages = {29--55}, abstract = {

In this paper multigrid smoothers of Vanka-type are studied in the context of Computational Solid Mechanics (CSM). These smoothers were originally developed 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 reasonable 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 closing this gap. We depict and compare four different Vanka-like smoothers, two of them are oriented towards the stabilised equal-order $Q_1/Q_1$ 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 heavily 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. Furthermore, 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.

}, issn = {2075-1354}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/aamm/207.html} }
TY - JOUR T1 - Numerical Studies of Vanka-Type Smoothers in Computational Solid Mechanics AU - Hilmar Wobker , AU - Turek , Stefan JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 29 EP - 55 PY - 2009 DA - 2009/01 SN - 1 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/aamm/207.html KW - Coupled multigrid, Vanka smoother, linear and finite elasticity, nearly incompressible material, saddle point systems, finite elements. AB -

In this paper multigrid smoothers of Vanka-type are studied in the context of Computational Solid Mechanics (CSM). These smoothers were originally developed 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 reasonable 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 closing this gap. We depict and compare four different Vanka-like smoothers, two of them are oriented towards the stabilised equal-order $Q_1/Q_1$ 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 heavily 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. Furthermore, 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.

Hilmar Wobker & Stefan Turek. (1970). Numerical Studies of Vanka-Type Smoothers in Computational Solid Mechanics. Advances in Applied Mathematics and Mechanics. 1 (1). 29-55. doi:
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