An adaptive multigrid iterative approach for frictional contact problems. (English) Zbl 1323.74092
Summary: The objective of this paper is the construction of a robust strategy towards adaptively solving Signorini’s frictional contact problems. The frictional contact problem between a linearly elastic body and rigid foundation is formulated as a classical boundary value problem for the elastic body but associated with special inequality conditions on the contact surface. A new iterative approach is presented to solve the problem on a given mesh. In the first iteration the candidate nodes are assumed to be in micro-slip contact and then proceeding to update the contact status according to the actual displacements and stresses obtained at the end of each increment. An efficient multigrid method is developed to solve the discrete problems of different iterations.
The proposed iterative procedure is integrated with an error indicator and automatic grid generator to construct an adaptive multigrid method. Numerical results on the convergence rates, automatically generated grid sequence, contact stresses and strains as well as two parametric studies are presented to prove the efficiency of the proposal.
The proposed iterative procedure is integrated with an error indicator and automatic grid generator to construct an adaptive multigrid method. Numerical results on the convergence rates, automatically generated grid sequence, contact stresses and strains as well as two parametric studies are presented to prove the efficiency of the proposal.
MSC:
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 74M15 | Contact in solid mechanics |
| 74M10 | Friction in solid mechanics |
| 74B05 | Classical linear elasticity |
| 65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |
Software:
UGReferences:
| [1] | Duvaut, Inequalities in Mechanics and Physics (1976) · doi:10.1007/978-3-642-66165-5 |
| [2] | Baiocchi, Variational and Quasivariational Inequalities (1984) |
| [3] | Kikuchi, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods (1988) · Zbl 0685.73002 · doi:10.1137/1.9781611970845 |
| [4] | Oden, Nonlocal and nonlinear friction laws and variational principles for contact problems in elasticity, Journal of Applied Mechanics 50 pp 67– (1983) · Zbl 0515.73121 |
| [5] | Böhm, A comparison of different contact algorithms with applications, Computers and Structures 26 (1/2) pp 207– (1987) |
| [6] | Eterovic, On the treatment of inequality constraints arising from contact conditions in finite element analysis, Computers and Structures 40 (2) pp 203– (1991) |
| [7] | Refaat, A novel finite element approach to frictional contact problems, International Journal for Numerical Methods in Engineering 39 pp 3889– (1996) · Zbl 0885.73087 |
| [8] | Paris, An incremental procedure for friction contact problems with the boundary element method, Engineering Analysis with Boundary Elements 6 (4) pp 202– (1989) |
| [9] | Heyliger, A mixed computational algorithm for plane elastic contact problems, Computers and Structures 26 (4) pp 621– (1987) · Zbl 0612.73117 |
| [10] | Mahmoud, An adaptive incremental approach for the solution of convex programming models, Mathematics and Computers in Simulation 35 pp 501– (1993) · Zbl 0808.90109 |
| [11] | Kornhuber, Adaptive Monotone Multigrid Methods for Nonlinear Variational Problems (1997) · Zbl 0879.65041 |
| [12] | Buscaglia, An adaptive finite elements approach for frictionless contact problems, International Journal for Numerical Methods in Engineering 50 pp 395– (2001) · Zbl 1006.74084 |
| [13] | Mohamed, An adaptive multigrid solver for contact problems, Journal of Engineering and Applied Sciences 48 pp 11– (2001) |
| [14] | Iontcheva, Monotone multigrid methods based on element agglomeration coarsening away from the contact boundary for the Signorini’s problem, Numerical Linear Algebra with Applications 11 pp 189– (2004) · Zbl 1164.74554 |
| [15] | Oden, Toward a universal h-p adaptive finite element strategy. Part 2: posteriori error estimates, Computer Methods in Applied Mechanics and Engineering 77 pp 113– (1989) |
| [16] | Oden JT Prudhomme S Goal-oriented error estimation and adaptivity for the finite element method 1999 1 32 |
| [17] | Bank, Technical Report (1985) |
| [18] | Hemker, On the structure of an adaptive multilevel algorithm, BIT 20 pp 289– (1980) · Zbl 0441.65075 |
| [19] | Mohamed, Automatic mesh refinement and data structure for muligrid finite element technique, Computers and Structures 65 (6) pp 985– (1997) |
| [20] | Brandt, Multilevel adaptive solutions to boundary value problems, Mathematics of Computation 31 pp 333– (1977) · Zbl 0373.65054 · doi:10.1090/S0025-5718-1977-0431719-X |
| [21] | Lecture Notes in Mathematics, in: Muligrid Methods (1982) |
| [22] | Multigrid Methods (1987) · Zbl 0659.65094 |
| [23] | Krause RH Monotone multigrid methods for signorini’s problem with friction 2001 |
| [24] | Bastian, UG-a flexible software toolbox for solving partial differential equations, Computing and Visualization in Science 1 pp 27– (1997) · Zbl 0970.65129 |
| [25] | Rocca, Existence and approximation of a solution to quasistatic signorini problem with local friction, Journal of Engineering Sciences 39 pp 1233– (2001) · Zbl 1210.74126 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
