Algebraic multigrid methods for dual mortar finite element formulations in contact mechanics

• Published in: International journal for numerical methods in engineering Vol. 114; no. 4; pp. 399 - 430
• Main Authors: Wiesner, T. A., Popp, A., Gee, M. W., Wall, W. A.
• Format: Journal Article
• Language: English
• Published: Bognor Regis Wiley Subscription Services, Inc 27.04.2018
• Subjects: Algebra; algebraic multigrid methods; Contact problems; dual mortar approach; Finite element method; iterative linear solvers; Iterative methods; Lagrange multiplier; Linear operators; Linear systems; Mathematical analysis; Mathematical models; mortar methods; Multigrid methods; Operators (mathematics); Preconditioning; Saddle points; Solvers; structural contact problems
• ISSN: 0029-5981; 1097-0207; 1097-0207
• DOI: 10.1002/nme.5748

Summary This paper proposes novel strategies to enable multigrid preconditioners within iterative solvers for linear systems arising from contact problems based on mortar finite element formulations. The so‐called dual mortar approach that is exclusively employed here allows for an easy condensation of the discrete Lagrange multipliers. Therefore, it has the advantage over standard mortar methods that linear systems with a saddle‐point structure are avoided, which generally require special preconditioning techniques. However, even with the dual mortar approach, the resulting linear systems turn out to be hard to solve using iterative linear solvers. A basic analysis of the mathematical properties of the linear operators reveals why the naive application of standard iterative solvers shows instabilities and provides new insights of how contact modeling affects the corresponding linear systems. This information is used to develop new strategies that make multigrid methods efficient preconditioners for the class of contact problems based on dual mortar methods. It is worth mentioning that these strategies primarily adapt the input of the multigrid preconditioners in a way that no contact‐specific enhancements are necessary in the multigrid algorithms. This makes the implementation comparably easy. With the proposed method, we are able to solve large contact problems, which is an important step toward the application of dual mortar–based contact formulations in the industry. Numerical results are presented illustrating the performance of the presented algebraic multigrid method.