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Keywords:
mortar method; Lagrange multipliers; inf-sup stable pairs; stabilized formulation
mortar method; Lagrange multipliers; inf-sup stable pairs; stabilized formulation
Summary:
The mortar method is a powerful technique to enforce constraints between non-conforming discretizations by introducing a set of Lagrange multipliers on the connecting interface. Usually, the multipliers are not obtained explicitly because they can be eliminated with the aid of the so-called mortar interpolation operator. However, their explicit computation becomes essential when the contact constraint is governed by some non-linear law, and in this situation it is necessary to guarantee that discrete spaces of the primary variables and multipliers are inf-sup stable. In this work, we investigate the issue of inf-sup stability when using various families of piecewise linear and piecewise constant multipliers. The focus is on the role of the mesh resolution and the enforcement of boundary conditions, which are important factors in practical applications. Then, we develop a stabilized formulation for piecewise-constant multipliers inspired by the framework of minimal stabilization. The effectiveness of the proposed approach is demonstrated through numerical benchmarks and examples.
References:
[1] Ao, J., Zhou, M., Zhang, B.: A dual mortar embedded mesh method for internal interface problems with strong discontinuities. Int. J. Numer. Methods Eng. 123 (2022), 5652-5681. DOI 10.1002/nme.7082 | MR 4509716 | Zbl 07769291
[2] Babuška, I.: The finite element method with Lagrangian multipliers. Numer. Math. 20 (1973), 179-192. DOI 10.1007/BF01436561 | MR 0359352 | Zbl 0258.65108
[3] Belgacem, F. Ben: The mortar finite element method with Lagrange multipliers. Numer. Math. 84 (1999), 173-197. DOI 10.1007/s002110050468 | MR 1730018 | Zbl 0944.65114
[4] Bernardi, C., Maday, Y., Patera, A. T.: Domain decomposition by the mortar element method. Asymptotic and Numerical Methods for Partial Differential Equations with Critical Parameters Kluwer Academic Publishers, Dordrecht (1993), 269-286. DOI 10.1007/978-94-011-1810-1_17 | MR 1222428 | Zbl 0799.65124
[5] Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics 44. Springer, Berlin (2013). DOI 10.1007/978-3-642-36519-5 | MR 3097958 | Zbl 1277.65092
[6] Burman, E.: Projection stabilization of Lagrange multipliers for the imposition of constraints on interfaces and boundaries. Numer. Methods Partial Differ. Equations 30 (2014), 567-592. DOI 10.1002/num.21829 | MR 3163976 | Zbl 1288.65153
[7] Chapelle, D., Bathe, K.-J.: The inf-sup test. Comput. Struct. 47 (1993), 537-545. DOI 10.1016/0045-7949(93)90340-J | MR 1224095 | Zbl 0780.73074
[8] El-Abbasi, N., Bathe, K.-J.: Stability and patch test performance of contact discretizations and a new solution algorithm. Comput. Struct. 79 (2001), 1473-1486. DOI 10.1016/S0045-7949(01)00048-7
[9] Elman, H. C., Silvester, D. J., Wathen, A. J.: Iterative Methods for Problems in Computational Fluid Dynamics. Report No. 96/19. Oxford University Computing Laboratory, Oxford (1996).
[10] Elman, H. C., Silvester, D. J., Wathen, A. J.: Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2014). DOI 10.1093/acprof:oso/9780199678792.001.0001 | MR 3235759 | Zbl 1304.76002
[11] Farah, P., Popp, A., Wall, W. A.: Segment-based vs. element-based integration for mortar methods in computational contact mechanics. Comput. Mech. 55 (2015), 209-228. DOI 10.1007/s00466-014-1093-2 | MR 3295001 | Zbl 1311.74120
[12] Franceschini, A., Castelletto, N., White, J. A., Tchelepi, H. A.: Algebraically stabilized Lagrange multiplier method for frictional contact mechanics with hydraulically active fractures. Comput. Methods Appl. Mech. Eng. 368 (2020), Article ID 113161, 32 pages. DOI 10.1016/j.cma.2020.113161 | MR 4106669 | Zbl 1506.74400
[13] Franceschini, A., Castelletto, N., White, J. A., Tchelepi, H. A.: Scalable preconditioning for the stabilized contact mechanics problem. J. Comput. Phys. 459 (2022), Article ID 111150, 32 pages. DOI 10.1016/j.jcp.2022.111150 | MR 4401969 | Zbl 1560.74066
[14] Franceschini, A., Ferronato, M., Janna, C., Teatini, P.: A novel Lagrangian approach for the stable numerical simulation of fault and fracture mechanics. J. Comput. Phys. 314 (2016), 503-521. DOI 10.1016/j.jcp.2016.03.032 | MR 3484946 | Zbl 1349.74321
[15] Gustafsson, T., RÃ¥back, P., Videman, J.: Mortaring for linear elasticity using mixed and stabilized finite elements. Comput. Methods Appl. Mech. Eng. 404 (2023), Article ID 115796, 13 pages. DOI 10.1016/j.cma.2022.115796 | MR 4520467 | Zbl 1550.74366
[16] Hauret, P., Tallec, P. Le: A discontinuous stabilized mortar method for general 3D elastic problems. Comput. Methods Appl. Mech. Eng. 196 (2007), 4881-4900. DOI 10.1016/j.cma.2007.06.014 | MR 2355734 | Zbl 1173.74424
[17] Heintz, P., Hansbo, P.: Stabilized Lagrange multiplier methods for bilateral elastic contact with friction. Comput. Methods Appl. Mech. Eng. 195 (2006), 4323-4333. DOI 10.1016/j.cma.2005.09.008 | MR 2229843 | Zbl 1123.74045
[18] Lamichhane, B. P.: Higher Order Mortar Finite Elements with Dual Lagrange Multiplier Spaces and Applications. Universität Stuttgart, Stuttgart (2006). Zbl 1196.65012
[19] Moretto, D., Franceschini, A., Ferronato, M.: A novel mortar method integration using radial basis functions. Comput. Math. Appl. 198 (2025), 38-58. DOI 10.1016/j.camwa.2025.08.008 | MR 4945430
[20] Pitkäranta, J.: Local stability conditions for the Babuška method of Lagrange multipliers. Math. Comput. 35 (1980), 1113-1129. DOI 10.1090/S0025-5718-1980-0583490-9 | MR 0583490 | Zbl 0473.65072
[21] Popp, A.: Mortar Methods for Computational Contact Mechanics and General Interface Problems: Ph.D. Thesis. Technische Universität München, München (2012).
[22] A. Popp, M. Gitterle, M., W. Gee, W. A. Wall: A dual mortar approach for 3D finite deformation contact with consistent linearization. Int. J. Numer. Methods Eng. 83 (2010), 1428-1465. DOI 10.1002/nme.2866 | MR 2722505 | Zbl 1202.74183
[23] Popp, A., Wall, W. A.: Dual mortar methods for computational contact mechanics: Overview and recent developments. GAMM Mitt. 37 (2014), 66-84. DOI 10.1002/gamm.201410004 | MR 3234156 | Zbl 1308.74119
[24] Popp, A., Wohlmuth, B. I., Gee, M. W., Wall, W. A.: Dual quadratic mortar finite element methods for 3D finite deformation contact. SIAM J. Sci. Comput. 34 (2012), B421--B446. DOI 10.1137/110848190 | MR 2970413 | Zbl 1250.74020
[25] Puso, M. A., Laursen, T. A.: Mesh tying on curved interfaces in 3D. Eng. Comput. (Bradf.) 20 (2003), 305-319. DOI 10.1108/02644400310467225 | Zbl 1045.65107
[26] Puso, M. A., Laursen, T. A.: A mortar segment-to-segment frictional contact method for large deformations. Comput. Methods Appl. Mech. Eng. 193 (2004), 4891-4913. DOI 10.1016/j.cma.2004.06.001 | MR 2097761 | Zbl 1112.74535
[27] Puso, M. A., Solberg, J. M.: A dual pass mortar approach for unbiased constraints and self-contact. Comput. Methods Appl. Mech. Eng. 367 (2020), Article ID 113092, 32 pages. DOI 10.1016/j.cma.2020.113092 | MR 4099873 | Zbl 1442.74142
[28] Sanders, J., Puso, M. A.: An embedded mesh method for treating overlapping finite element meshes. Int. J. Numer. Methods Eng. 91 (2012), 289-305. DOI 10.1002/nme.4265 | MR 2947862 | Zbl 1246.74064
[29] Tur, M., Fuenmayor, F. J., Wriggers, P.: A mortar-based frictional contact formulation for large deformations using Lagrange multipliers. Comput. Methods Appl. Mech. Eng. 198 (2009), 2860-2873. DOI 10.1016/j.cma.2009.04.007 | MR 2567849 | Zbl 1229.74141
[30] Wohlmuth, B. I.: Discretization techniques based on domain decomposition. Discretization Methods and Iterative Solvers Based on Domain Decomposition Lecture Notes in Computational Science and Engineering 17. Springer, Berlin (2001), 1-84. DOI 10.1007/978-3-642-56767-4_1
[31] Wohlmuth, B.: Variationally consistent discretization schemes and numerical algorithms for contact problems. Acta Numerica 20 (2011), 569-734. DOI 10.1017/S0962492911000079 | MR 2805157 | Zbl 1432.74176
[32] Wriggers, P.: Computational contact mechanics. Comput. Mech. 32 (2003), 141. DOI 10.1007/s00466-003-0472-x
[33] Yang, B., Laursen, T. A., Meng, X.: Two dimensional mortar contact methods for large deformation frictional sliding. Int. J. Numer. Methods Eng. 62 (2005), 1183-1225. DOI 10.1002/nme.1222 | MR 2120292 | Zbl 1161.74497
[34] Zhou, M., Zhang, B., Chen, T., Peng, C., Fang, H.: A three-field dual mortar method for elastic problems with nonconforming mesh. Comput. Methods Appl. Mech. Eng. 362 (2020), Article ID 112870, 24 pages. DOI 10.1016/j.cma.2020.112870 | MR 4059410 | Zbl 1439.74482
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