## A decomposition method for a unilateral contact problem with Tresca friction arising in electro-elastostatics.(English)Zbl 1333.74081

Summary: This article is concerned with the numerical modeling of unilateral contact problems in an electro-elastic material with Tresca friction law and electrical conductivity condition. First, we prove the existence and uniqueness of the weak solution of the model. Rather than deriving a solution method for the full coupled problem, we present and study a successive iterative (decomposition) method. The idea is to solve successively a displacement subproblem and an electric potential subproblem in block Gauss-Seidel fashion. The displacement subproblem leads to a constraint non-differentiable (convex) minimization problem for which we propose an augmented Lagrangian algorithm. The electric potential unknown is computed explicitly using the Riesz’s representation theorem. The convergence of the iterative decomposition method is proved. Some numerical experiments are carried out to illustrate the performances of the proposed algorithm.

### MSC:

 74S05 Finite element methods applied to problems in solid mechanics 74B05 Classical linear elasticity 65N22 Numerical solution of discretized equations for boundary value problems involving PDEs 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
Full Text:

### References:

 [1] DOI: 10.1016/j.jmaa.2009.04.030 · Zbl 1168.74039 [2] DOI: 10.1007/BF00042498 · Zbl 0828.73061 [3] Essoufi E. H., Adv. Appl. Math. Mech 2 (3) pp 355– (2010) [4] Fortin M., Augmented Lagrangian Methods: Application to the Numerical Solution of Boundary-Value Problems (1983) · Zbl 0525.65045 [5] DOI: 10.1137/1.9781611970838 [6] Glowinski R., RAIRO Anal. Num 9 (2) pp 41– (1975) [7] Ikeda T., Fundamentals of Piezoelectricity (1990) [8] DOI: 10.1137/1.9781611970845 [9] Koko J., Calcul Scientifique Avec Matlab (2009) [10] DOI: 10.1016/j.cma.2008.08.011 · Zbl 1228.74086 [11] DOI: 10.1016/j.aml.2009.03.021 · Zbl 1171.74391 [12] DOI: 10.1016/j.cam.2010.10.032 · Zbl 1260.74031 [13] Lerguet Z., Electronic Journal of Differential Equations 2007 (170) pp 1– (2007) [14] DOI: 10.1016/S0895-7177(98)00105-8 · Zbl 1126.74392 [15] Migórski S., Discrete Continuous Dynam. Syst. Ser 1 pp 117– (2008) · Zbl 1140.74020 [16] DOI: 10.1007/BF00045712 [17] Sofonea M., A Math. Model. Anal 9 pp 229– (2004) [18] DOI: 10.1016/0020-7225(63)90027-2
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.