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Mortar contact formulations for deformable-deformable contact: past contributions and new extensions for enriched and embedded interface formulations. (English) Zbl 1239.74070

Summary: The past 10-15 years have seen important extensions of the mortar method, a technique for joining dissimilar grids popularized by the domain decomposition community, to the more general problem of contact and impact interactions in finite element analysis. This development has taken place largely in response to several long-standing problems in computational contact mechanics: lack of robustness in solution of the nonlinear and nonsmooth equations of evolution; degradation of spatial convergence rates in problems involving nonconforming meshes on interfaces; lack of a variationally consistent technique for stress recovery on interfaces; and so on. This survey paper summarizes some of the major steps in development of mortar contact formulations. It begins with a basic summary of the mortaring idea in the context of tied contact, it discusses key concepts required for the extension of these methods to large deformation, large sliding formulations of contact-impact, and it previews new results where lessons learned from mortar contact formulations can be extended to a much broader class of interface mechanics applications, considering in particular enriched interface formulations and embedded interface approaches to fluid-structure interaction.

MSC:

74M15 Contact in solid mechanics
74G15 Numerical approximation of solutions of equilibrium problems in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs

Software:

Nike2D
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References:

[1] Bernardi, C.; Maday, Y.; Patera, A., A new nonconforming approach to domain decomposition: the mortar element method, (Brezia, H.; Lions, J., Nonlinear Partial Differential Equations and their Applications (1992), Pitman and Wiley), 13-51 · Zbl 0797.65094
[2] Kim, C.; Lazarov, R.; Pasciak, J.; Vassilevski, P., Multiplier spaces for the mortar finite element method in three dimensions, SIAM J. Numer. Anal., 38, 519-538 (2001) · Zbl 1006.65129
[3] Wohlmuth, B., Discretization Methods and Iterative Solvers Based on Domain Decomposition (2001), Springer-Verlag: Springer-Verlag Heidelberg · Zbl 0966.65097
[4] Strang, G.; Fix, G., An Analysis of the Finite Element Method (1973), Prentice-Hall: Prentice-Hall Englewood Cliffs, New Jersey · Zbl 0278.65116
[5] Hughes, T., The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (2000), Dover · Zbl 1191.74002
[6] Laursen, T., Computational Contact and Impact Mechanics: Fundamentals of Modeling Interfacial Phenomena in Nonlinear Finite Element Analysis (2003), Springer: Springer Berlin
[7] Belgacem, F.; Maday, Y., The mortar element method for three dimensional finite elements, Rairo - Math. Model. Numer. Anal., 31, 289-302 (1997) · Zbl 0868.65082
[8] Seshayer, P.; Suri, M., hp Submeshing via non-conforming finite element methods, Comput. Methods Appl. Mech. Engrg., 189, 1011-1030 (2000) · Zbl 0971.65101
[9] Belgacem, F.; Hild, P.; Laborde, P., Approximation of the unilateral contact problem by the mortar finite element method, CR Acad. Sci., 324, 123-127 (1991) · Zbl 0872.65057
[10] Hild, P., Numerical implementation of two nonconforming finite element methods for unilateral contact, Comput. Methods Appl. Mech. Engrg., 184, 99-123 (2000) · Zbl 1009.74062
[11] McDevitt, T.; Laursen, T., A mortar-finite element formulation for frictional contact problems, Int. J. Numer. Methods Engrg., 48, 1525-1547 (2000) · Zbl 0972.74067
[12] J. Hallquist, NIKE2D: A Vectorized Implicit, Finite Deformation Finite Element Code for Analyzing the Static and Dynamic Response of 2-D Solids with Interactive Rezoning and Graphics, Technical Report UCID-19677, Lawrence Livermore National Laboratory, University of California, 1986 (Revision 1).; J. Hallquist, NIKE2D: A Vectorized Implicit, Finite Deformation Finite Element Code for Analyzing the Static and Dynamic Response of 2-D Solids with Interactive Rezoning and Graphics, Technical Report UCID-19677, Lawrence Livermore National Laboratory, University of California, 1986 (Revision 1).
[13] Simo, J.; Wriggers, P.; Taylor, R., A perturbed Lagrangian formulation for the finite element solution of contact problems, Comput. Methods Appl. Mech. Engrg., 50, 163-180 (1985) · Zbl 0552.73097
[14] Papadopoulos, P.; Taylor, R., A mixed formulation for the finite element solution of contact problems, Comput. Methods Appl. Mech. Engrg., 94, 373-389 (1992) · Zbl 0743.73029
[15] Zavarise, G.; Wriggers, P., A segment-to-segment contact strategy, Math. Comput. Modell., 28, 497-515 (1998) · Zbl 1002.74564
[16] Puso, M.; Laursen, T., A mortar segment-to-segment contact method for large deformation solid mechanics, Comput. Methods Appl. Mech. Engrg., 193, 601-629 (2004) · Zbl 1060.74636
[17] Hueber, S.; Wohlmuth, B., A primal-dual active set strategy for non-linear multibody contact problems, Comput. Methods Appl. Mech. Engrg., 194, 3147-3166 (2005) · Zbl 1093.74056
[18] Puso, M., A 3D mortar method for solid mechanics, Int. J. Numer. Methods Engrg., 59, 601-629 (2004) · Zbl 1060.74636
[19] Flemisch, B.; Puso, M.; Wohlmuth, B., A new dual mortar method for curved interfaces: 2d elasticity, Int. J. Numer. Methods Engrg., 63, 813-832 (2005) · Zbl 1084.74050
[20] Popp, A.; Gitterle, M.; Gee, M.; Wall, W., A dual mortar approach for 3d finite deformation contact with consistent linearization, Int. J. Numer. Methods Engrg., 83, 1428-1465 (2010) · Zbl 1202.74183
[21] Puso, M.; Laursen, T.; Solberg, J., A segment-to-segment mortar contact method for quadratic elements and large deformations, Comput. Methods Appl. Mech. Engrg., 197, 556-566 (2008) · Zbl 1169.74627
[22] Luenberger, D., Linear and Nonlinear Programming (1984), Addison-Wesley: Addison-Wesley Reading, Massachusetts
[23] Puso, M.; Laursen, T., A mortar segment-to-segment contact method for large deformation solid mechanics, Comput. Methods Appl. Mech. Engrg., 193, 601-629 (2004) · Zbl 1060.74636
[24] Puso, M.; Laursen, T., A mortar segment-to-segment frictional contact method for large deformations, Comput. Methods Appl. Mech. Engrg., 193, 601-629 (2004) · Zbl 1060.74636
[25] Fischer, K.; Wriggers, P., Frictionless 2d contact formulations for finite deformations based on the mortar method, Comput. Mech., 36, 226-244 (2005) · Zbl 1102.74033
[26] Fischer, K.; Wriggers, P., Mortar based frictional contact formulation for higher order interpolations using the moving friction cone, Comput. Methods Appl. Mech. Engrg., 195, 5020-5036 (2006) · Zbl 1118.74047
[27] Yang, B.; Laursen, T., A mortar segment-to-segment frictional contact method for large deformations, Int. J. Numer. Methods Engrg., 62, 1183-1225 (2005) · Zbl 1161.74497
[28] Kikuchi, N.; Oden, J., Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods (1988), SIAM: SIAM Philadelphia · Zbl 0685.73002
[29] Laursen, T.; Simo, J., On the formulation and numerical treatment of finite deformation frictional contact problems, (Wriggers, P.; Wagner, W., Nonlinear Computational Mechanics - State of the Art (1991), Springer-Verlag: Springer-Verlag Berlin), 716-736
[30] Foley, J., Computer Graphics Principles and Practice (1997), Addison-Wesley: Addison-Wesley Reading
[31] Cowper, G., Gaussian quadrature formulas for triangles, Int. J. Numer. Methods Engrg., 7, 405-408 (1973) · Zbl 0265.65013
[32] Yang, B.; Laursen, T., A large deformation mortar formulation of self contact with finite sliding, Comput. Methods Appl. Mech. Engrg., 197 (2008) · Zbl 1169.74513
[33] Yang, B.; Laursen, T., A contact searching algorithm including bounding volume trees applied to finite sliding mortar formulations, Comput. Mech., 41, 189-205 (2008) · Zbl 1162.74481
[34] Yang, B.; Laursen, T., A mortar-finite element approach to lubricated contact problems, Comput. Methods Appl. Mech. Engrg., 198, 3656-3669 (2009) · Zbl 1230.74204
[35] Sanders, J.; Dolbow, J.; Laursen, T., On methods for stabilizing constraints over enriched interfaces in elasticity, Int. J. Numer. Methods Engrg., 78, 1009-1036 (2009) · Zbl 1183.74313
[36] Moës, N.; Béchet, E.; Tourbier, M., Imposing Dirichlet boundary conditions in the extended finite element method, Int. J. Numer. Methods Engrg., 67, 1641-1669 (2006) · Zbl 1113.74072
[37] Nitsche, J., Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilrä umen, die keinen Randbedingungen unterworfen sind, Comput. Methods Appl. Mech. Engrg., 191, 1122-1145 (1971)
[38] Becker, R.; Hansbo, P.; Stenberg, R., A finite element method for domain decomposition with non-matching grids, ESAIM - Math. Model. Numer. Anal., 37, 209-225 (2003) · Zbl 1047.65099
[39] Wriggers, P.; Zavarise, G., A formulation for frictionless contact problems using a weak form introduced by Nitsche, Comput. Mech., 41, 407-420 (2008) · Zbl 1162.74419
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