Alternating direction method of multiplier for a unilateral contact problem in electro-elastostatics. (English) Zbl 1370.74142
Summary: We study an alternating direction method of multiplier (ADMM) applied to a unilateral frictional contact problem between an electro-elastic material and an electrically non conductive foundation. The frictional contact is modeled by the Tresca friction law. The resulting coupled problem is non symmetric and non coercive. By eliminating the electric potential, we obtain a symmetric and coercive problem which can be reformulated as a convex minimization problem. We then apply an alternating direction method of multiplier for the numerical approximation. To avoid explicit matrices inverse (due to the elimination of the electric potential) we use a preconditioned conjugate gradient algorithm as an inner solver. Numerical experiments are proposed to illustrate the efficiency of the proposed approach.
MSC:
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |
| 74M10 | Friction in solid mechanics |
Keywords:
electro-elastostatics; contact; Tresca friction; augmented Lagrangian; decomposition methodSoftware:
KOKO Mesh GeneratorReferences:
| [1] | Ikeda, T., Fundamentals of Piezoelectricity (1990), Oxford University Press: Oxford University Press Oxford |
| [2] | Bartra, R. C.; Yang, J. S., Saint-Venant’s principle in linear piezoelectricity, J. Elasticity, 38, 209-218 (1995) · Zbl 0828.73061 |
| [3] | Sofonea, M.; Essoufi, E. H., Piezoelectric contact problem with slip dependent coefficient of friction, Math. Model. Anal., 9, 229-242 (2004) · Zbl 1092.74029 |
| [4] | Essoufi, E. H.; Benkhira, E. H.; Fakhar, R., Analysis and numerical approximation of an electroelastic frictional contact problem, Adv. Appl. Math. Mech., 2, 3, 355-378 (2010) · Zbl 1262.74017 |
| [5] | Matei, A., A Variational approach for an electroelastic unilateral contact problem, Math. Model. Anal., 14, 3, 323-334 (2009) · Zbl 1294.74034 |
| [6] | Hüeber, S.; Matei, A.; Wohlmuth, B. I., A mixed variational formulation and an optimal a priori error estimate for a frictional contact problem in elasto-piezoelectricity, Bull. Math. Soc. Sci. Math. Roumanie, 48, 96, 209-232 (2005) · Zbl 1105.74028 |
| [7] | Hüeber, S.; Matei, A.; Wohlmuth, B., A contact problem for electro-elastic materials, ZAMM Z. Angew. Math. Mech., 93, 10-11 (2013) · Zbl 1432.74169 |
| [8] | Barboteu, M.; Sofonea, M., Modeling and analysis of the unilateral contact of piezoelectric body with conductive support, J. Math. Anal. Appl., 358, 110-124 (2009) · Zbl 1168.74039 |
| [9] | Essoufi, E. H.; Fakhar, R.; Koko, J., A decomposition method for a unilateral contact problem with tresca friction arising in electro-elastostatics, Numer. Funct. Anal. Optim., 36, 1533-1558 (2015) · Zbl 1333.74081 |
| [10] | Barboteu, M.; Fernndez, J. R.; Ouafik, Y., Numerical analysis of two frictionless elastic-piezoelectric contact problems, J. Math. Anal. Appl., 339, 2, 905-917 (2008) · Zbl 1127.74028 |
| [11] | Kabbaj, M.; Essoufi, E. H., Frictional contact problem in dynamic electroelasticity, Glas. Mat., 43, 63, 137-158 (2008) · Zbl 1159.35327 |
| [12] | Ekeland, I.; Temam, R., (Convex Analysis and Variational Problems. Convex Analysis and Variational Problems, Classics in Applied Mathematics (1999), SIAM: SIAM Philadelphia) · Zbl 0939.49002 |
| [13] | Haslinger, J.; Kucera, R.; Dostal, Z., An algorithm for the numerical realization of 3D contact problems with Coulomb friction, J. Comput. Appl. Math., 164-165, 387-408 (2004) · Zbl 1107.74328 |
| [14] | Bertsekas, D., Nonlinear Programming (1999), Athena Scientific · Zbl 0935.90037 |
| [15] | Fortin, M.; Glowinski, R., Augmented Lagrangian Methods: Application to the Numerical Solution of Boundary-Value Problems (1983), North-Holland: North-Holland Amsterdam · Zbl 0525.65045 |
| [16] | Glowinski, R.; Le Tallec, P., (Augmented Lagrangian and Operator-splitting Methods in Nonlinear Mechanics. Augmented Lagrangian and Operator-splitting Methods in Nonlinear Mechanics, Stud. Appl. Math. (1989), SIAM: SIAM Philadelphia) · Zbl 0698.73001 |
| [17] | Glowinski, R.; Marocco, A., Sur l’approximation par éléments finis d’ordre un, et la résolution par pénalisation-dualité, d’une classe de problèmes de Dirichlet non linéaires, RAIRO Anal. Num., 9, 2, 41-76 (1975) · Zbl 0368.65053 |
| [18] | Koko, J., Uzawa block relaxation method for the unilateral contact problem, J. Comput. Appl. Math., 235, 2343-2356 (2011) · Zbl 1260.74031 |
| [19] | Mandel, J., On block diagonal and Schur complement preconditioning, Numer. Math., 58, 79-93 (1990) · Zbl 0687.65036 |
| [20] | Cocu, M., Existence of solutions of Signorini problems with friction, Internat. J. Engrg. Sci., 22, 567-575 (1984) · Zbl 0554.73096 |
| [21] | Dostál, Z.; Haslinger, J.; Kučera, R., Implementation of the fixed point method in contact problems with Coulomb friction based on a dual splitting type technique, J. Comput. Appl. Math., 140, 245-256 (2002) · Zbl 1134.74418 |
| [22] | Haslinger, J.; Dostál, Z.; Kučera, R., On the splitting type algorithm for the numerical realization of the contact problems with Coulomb friction, Comput. Methods Appl. Mech. Engrg., 191, 2261-2281 (2002) · Zbl 1131.74344 |
| [23] | Kikuchi, N.; Oden, J. T., (Contact problems in Elasticity: A study of Variational Inequalities and Finite Element Methods. Contact problems in Elasticity: A study of Variational Inequalities and Finite Element Methods, Stud. Appl. Math. (1988), SIAM: SIAM Philadelphia) · Zbl 0685.73002 |
| [24] | Koko, J., Fast MATLAB assembly of FEM matrices in 2D and 3D using cell-array approach, Int. J. Model. Simul. Sci. Comput., 7, 2 (2016), (in press) |
| [25] | Koko, J., A MATLAB mesh generator for two-dimensional finite element method, Appl. Math. Comput., 250, 650-664 (2015) · Zbl 1328.65245 |
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.
