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A Nitsche embedded mesh method. (English) Zbl 1366.74075
Summary: A new technique for treating the mechanical interactions of overlapping finite element meshes is proposed. Numerous names have been applied to related approaches, here we refer to such techniques as embedded mesh methods. Such methods are useful for numerous applications e.g., fluid-solid interaction with a superposed meshed solid on an Eulerian background fluid grid or solid-solid interaction with a superposed meshed particle on a matrix background mesh etc. In this work we consider the interaction of two elastic domains: one mesh is the foreground and defines the surface of interaction, the other is a background mesh and is often a grid. We first employ a classical mortar type approach (see F. P. T. Baaijens [Int. J. Numer. Methods Fluids 35, No. 7, 743–761 (2001; Zbl 0979.76044)]) to impose constraints on the interface. It turns out that this approach will work well except in special cases. In fact, many related approaches can exhibit mesh locking under certain conditions. This motivates the proposed version of Nitsche’s method which is shown to eliminate the locking phenomenon in example problems.

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
74A50 Structured surfaces and interfaces, coexistent phases
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[1] Baaijens FPT (2001) A fictitious domain/mortar element method for fluid-structure interaction. Int J Numer Methods Eng 35: 743–761 · Zbl 0979.76044
[2] Babuška I (1973) The finite element method with penalty. Math Comput 27: 221–228 · Zbl 0299.65057
[3] Barbosa JC, Hughes TRJ (1991) The finite element method with lagrange multipliers on the boundary: circumventing the babuška-brezzi condition. Comput Methods Appl Mech Eng 85: 109–128 · Zbl 0764.73077
[4] Bechét E, Moës N, Wohlmuth B (2009) A stable lagrange multiplier space for stiff interface conditions within the extended finite element method. Int J Numer Methods Eng 78: 931–954 · Zbl 1183.74259
[5] Ben Dhia H, Rateau G (2005) The arlequin method as a flexible engineering design tool. Int J Numer Methods Eng 62: 1442–1462 · Zbl 1084.74049
[6] Brezzi F, Lions JL, Pironneau O (2001) Analysis of a Chimera method. Comptes Rendus De L’académie des Sciences Serie I Mathematics 332(7): 655–660 · Zbl 0988.65117
[7] Dolbow JE, Harari I (2009) An efficient finite element method for embedded interface problems. Int J Numer Methods Eng 78: 229–252 · Zbl 1183.76803
[8] Fernández-Méndez S, Huerta A (2004) Imposing essential boundary conditions in mesh-free methods. Comput Methods Appl Mech Eng 193: 1257–1275 · Zbl 1060.74665
[9] Fritz A, Hüeber S, Wohlmuth BI (2004) A comparison of mortar and nitsche techniques for linear elasticity. CALCOLO 41: 115–137 · Zbl 1099.65123
[10] Gerstenberger A, Wall WA (2008) An extended finite element method/lagrange multiplier based approach for fluid-structure interaction. Comput Methods Appl Mech Eng 197: 1699–1714 · Zbl 1194.76117
[11] Glowinski R, Pan T, Périaux J (1994) A fictitious domain method for dirichlet problems and applications. Comput Methods Appl Mech Eng 111: 283–303 · Zbl 0845.73078
[12] Griebel M, Schweitzer MA (2002) A particle-partition of unity method. Part V: boundary conditions. In: Geometric analysis and nonlinear partial differential equations. Springer, Berlin, pp 519–546
[13] Hansbo A, Hansbo P (2002) An unfitted finite element method, based on Nitsche’s method, for elliptical interface problems. Comput Methods Appl Mech Eng 191: 5537–5552 · Zbl 1035.65125
[14] Hansbo A, Hansbo P, Larson MG (2003) A finite element method on composite grids based on Nitsche’s method. ESAIM Math Model Numer Anal 37: 209–225 · Zbl 1047.65099
[15] Houzeaux G, Codina R (2003) A chimera method based on dirichlet/neumann(robin) coupling for navier stokes. Comput Methods Appl Mech Eng 192: 3343–3377 · Zbl 1054.76049
[16] Lew AJ, Buscaglia GC (2008) A discontinuous-Galerkin based immersed boundary method. Int J Numer Methods Eng 76(4): 427–454 · Zbl 1195.76258
[17] Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46: 131–150 · Zbl 0955.74066
[18] Nitsche J (1971) Über ein Variationsprinzip zur Lösung von Dirichlet-problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 36 · Zbl 0229.65079
[19] Noh WF (1964) A time-dependent, two space dimensional, coupled eulerian–lagrangian code. Methods Comput Phys
[20] Sanders JD, Dolbow JE, Laursen TA (2009) On methods for stabilizing constraints over enriched interfaces in elasticity. Int J Numer Methods Eng 78: 1009–1036 · Zbl 1183.74313
[21] Stenberg R (1995) On some techniques for approximating boundary conditions in the finite element method. J Comput Appl Math 63: 139–148 · Zbl 0856.65130
[22] Wriggers P, Zavarise G (2008) A formulation for frictionless contact problems using a weak form introduced by nitsche. Comput Mech 41: 407–420 · Zbl 1162.74419
[23] Zhang L, Gerstenberger A, Wang X, Liu W (2004) Immersed finite element method. Comput Methods Appl Mech Eng 193(21–22): 2051–2067 · Zbl 1067.76576
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