J-ADMM for a multi-contact problem in electro-elastostatics. (English) Zbl 1540.74095
Summary: In this work we apply a regularized alternating direction method of multiplier of Jacobi type (J-ADMM) to a frictional multi-contact (both unilateral and bilateral) problem between an electro-elastic material and a rigid non conductive foundations. The frictional contact is modelled by the Coulomb friction law. The resulted problem is non symmetric and non coercive. By dissociating the electric potential from the mechanical field, we obtain a symmetric and coercive problem which can be reformulated as a convex minimization problem. We then apply a J-ADMM for the numerical approximation to the resulting perturbed elastic problem. The resolution of obtained sub-problems is based on convex dualities. Numerical experiments are proposed to illustrate the efficiency of the proposed approach.
© 2021 Wiley-VCH GmbH
© 2021 Wiley-VCH GmbH
MSC:
| 74M15 | Contact in solid mechanics |
| 74M10 | Friction in solid mechanics |
| 74F15 | Electromagnetic effects in solid mechanics |
| 74S99 | Numerical and other methods in solid mechanics |
Keywords:
alternating direction method; Jacobi multiplier; unilateral/bilateral contact; Coulomb friction; convex minimization problem; convex dualitySoftware:
KOKO Mesh GeneratorReferences:
| [1] | 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 |
| [2] | Barboteu, M., Fernández, 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 |
| [3] | Bartra, R.C., Yang, J.S.: Saint‐Venants principale in linear piezoelectricity. J. Elast.38, 209-218 (1995) · Zbl 0828.73061 |
| [4] | Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, Cham, second edition, (2017) · Zbl 1359.26003 |
| [5] | Bertsekas, D.: Nonlinear Programming. Athena Scientific, (1999) · Zbl 0935.90037 |
| [6] | Börgens, E., Kanzow, C.: Regularized Jacobi‐type ADMM‐methods for a class of separable convex optimization problems in Hilbert spaces. Comput. Optim. Appl.73(3), 755-790 (2017) · Zbl 1423.90176 |
| [7] | Baiz, O., Benaissa, H., Moutawakil, D.E., Fakhar, R.: Variational and numerical analysis of a quasistatic thermo‐electro‐visco‐elastic frictional contact problem. ZAMM‐J. Appl. Math. Mech./Z. Angew. Math. Mech.99(3), e201800138 (2019) · Zbl 07805144 |
| [8] | Cai, X., Han, D., Yuan, X.: The direct extension of ADMM for three‐block separable convex minimization models is convergent when one function is strongly convex. Optim. Online 229, 230 (2014). |
| [9] | Chen, C., He, B., Ye, Y., Yuan, X.: The direct extension of ADMM for multi‐block convex minimization problems is not necessarily convergent. Math. Program.155(1‐2), 57-79 (2016) · Zbl 1332.90193 |
| [10] | Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North‐Holland, Amsterdam (1978) · Zbl 0383.65058 |
| [11] | Cocu, M.: Existence of solutions of Signorini problems with friction. Internat. J. Eng. Sci.22, 567-575 (1984) · Zbl 0554.73096 |
| [12] | Deng, W., Lai, M.J., Peng, Z., Yin, W.: Parallel multi‐block ADMM with \(o(1/k)\) convergence. J. Sci. Comput.71(2), 712-736 (2017) · Zbl 1398.65121 |
| [13] | 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 |
| [14] | Eckstein, J.: Augmented Lagrangian and alternating direction methods for convex optimization: a tutorial and some illustrative computational results, Rutcor Research Report RRR, 32‐2012 (2012) |
| [15] | Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. Classics in Applied Mathematics, SIAM, Philadelphia (1999) · Zbl 0939.49002 |
| [16] | 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 |
| [17] | 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 |
| [18] | Essoufi, E.‐H., Koko, J., Zafrar, A.: Alternating direction method of multiplier for a unilateral contact problem in electro‐elastostatics. Comput. Math. Appl.73(8), 1789-1802 (2017) · Zbl 1370.74142 |
| [19] | Fortin, M., Glowinski, R.: Augmented Lagrangian Methods: Application to the Numerical Solution of Boundary‐Value Problems. North‐Holland, Amsterdam (1983) · Zbl 0525.65045 |
| [20] | Glowinski, R., Le Tallec, P.: Augmented Lagrangian and Operator‐Splitting Methods in Nonlinear Mechanics. Studies in Applied Mathematics, SIAM, Philadelphia (1989) · Zbl 0698.73001 |
| [21] | 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 |
| [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. Eng.191, 2261-2281 (2002) · Zbl 1131.74344 |
| [23] | 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 |
| [24] | 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. Roumanie48(96), 209-232 (2005) · Zbl 1105.74028 |
| [25] | Hüeber, S., Matei, A., Wohlmuth, B.: A contact problem for electro‐elastic materials. ZAMM‐J. Appl. Math. Mech./Z. Angew. Math. Mech.93(10‐11), (2013) · Zbl 1432.74169 |
| [26] | Ikeda, T.: Fundamentals of Piezoelectricity. Oxford University Press, Oxford (1990) |
| [27] | Jiang, W.W., Gao, X.W., Xu, B.B., Lv, J.: Analysis of piezoelectric problems using zonal free element method. Eng. Anal. Bound Elem.127, 40-52 (2021) · Zbl 1464.74183 |
| [28] | Koko, J.: A MATLAB mesh generator for two‐dimensional finite element method. Appl. Math. Comput.250, 650-664 (2015) · Zbl 1328.65245 |
| [29] | Koko, J.: Uzawa block relaxation method for the unilateral contact problem. J. Comput. Appl Math.235, 2343-2356 (2011) · Zbl 1260.74031 |
| [30] | 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) |
| [31] | Kabbaj, M., Essoufi, E.H.: Frictional contact problem in dynamic electroelasticity. Glas. Mat.43(63), 137-158 (2008) · Zbl 1159.35327 |
| [32] | Kikuchi, N., Oden, J.T.: Contact problems in Elasticity: A study of Variational Inequalities and Finite Element Methods. Studies in Applied Mathematics. SIAM, Philadelphia (1988) · Zbl 0685.73002 |
| [33] | Li, C., Zhou, Y.T.: Fundamental solutions and frictionless contact problem in a semi‐infinite space of 2D hexagonal piezoelectric QCs. ZAMM‐J. Appl. Math. Mech./Z. Angew. Math. Mech.99(5), e201800132 (2019) · Zbl 07806633 |
| [34] | Matei, A.: A Variational approach for an electro‐elastic unilateral contact problem. Math. Model. Anal.14(3), 323-334 (2009) · Zbl 1294.74034 |
| [35] | Matei, A., Sofonea, M.: A mixed variational formulation for a piezoelectric frictional contact problem. IMA J. Appl. Math.82(2), 334-354 (2017) · Zbl 1408.74042 |
| [36] | Mindlin, R.D.: Elasticity, piezoelasticity and crystal lattice dynamics. J. Elast.4, 217-280 (1972) |
| [37] | Sofonea, M., Essoufi, E.H.: Piezoelectric contact problem with slip dependent coefficient of friction. Math. Model. Anal.9, 229-242 (2004) · Zbl 1092.74029 |
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.
