Computational Mechanics

, Volume 49, Issue 2, pp 243–257 | Cite as

A Nitsche embedded mesh method

  • Jessica D. Sanders
  • Tod A. Laursen
  • Michael A. Puso
Original Paper


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 Baaijens (Int J Numer Methods Eng 35:743–761, 2001)] 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.


Embedded mesh Nitsche’s method Interfaces 


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  1. 1.
    Baaijens FPT (2001) A fictitious domain/mortar element method for fluid-structure interaction. Int J Numer Methods Eng 35: 743–761MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Babuška I (1973) The finite element method with penalty. Math Comput 27: 221–228MATHGoogle Scholar
  3. 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: 109–128MathSciNetMATHCrossRefGoogle Scholar
  4. 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: 931–954MATHCrossRefGoogle Scholar
  5. 5.
    Ben Dhia H, Rateau G (2005) The arlequin method as a flexible engineering design tool. Int J Numer Methods Eng 62: 1442–1462MATHCrossRefGoogle Scholar
  6. 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): 655–660MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Dolbow JE, Harari I (2009) An efficient finite element method for embedded interface problems. Int J Numer Methods Eng 78: 229–252MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Fernández-Méndez S, Huerta A (2004) Imposing essential boundary conditions in mesh-free methods. Comput Methods Appl Mech Eng 193: 1257–1275MATHCrossRefGoogle Scholar
  9. 9.
    Fritz A, Hüeber S, Wohlmuth BI (2004) A comparison of mortar and nitsche techniques for linear elasticity. CALCOLO 41: 115–137MathSciNetMATHCrossRefGoogle Scholar
  10. 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: 1699–1714MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Glowinski R, Pan T, Périaux J (1994) A fictitious domain method for dirichlet problems and applications. Comput Methods Appl Mech Eng 111: 283–303MATHCrossRefGoogle Scholar
  12. 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 519–546Google Scholar
  13. 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: 5537–5552MathSciNetMATHCrossRefGoogle Scholar
  14. 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: 209–225MATHCrossRefGoogle Scholar
  15. 15.
    Houzeaux G, Codina R (2003) A chimera method based on dirichlet/neumann(robin) coupling for navier stokes. Comput Methods Appl Mech Eng 192: 3343–3377MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Lew AJ, Buscaglia GC (2008) A discontinuous-Galerkin based immersed boundary method. Int J Numer Methods Eng 76(4): 427–454MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46: 131–150MATHCrossRefGoogle Scholar
  18. 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 36Google Scholar
  19. 19.
    Noh WF (1964) A time-dependent, two space dimensional, coupled eulerian–lagrangian code. Methods Comput PhysGoogle Scholar
  20. 20.
    Sanders JD, Dolbow JE, Laursen TA (2009) On methods for stabilizing constraints over enriched interfaces in elasticity. Int J Numer Methods Eng 78: 1009–1036MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Stenberg R (1995) On some techniques for approximating boundary conditions in the finite element method. J Comput Appl Math 63: 139–148MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Wriggers P, Zavarise G (2008) A formulation for frictionless contact problems using a weak form introduced by nitsche. Comput Mech 41: 407–420MATHCrossRefGoogle Scholar
  23. 23.
    Zhang L, Gerstenberger A, Wang X, Liu W (2004) Immersed finite element method. Comput Methods Appl Mech Eng 193(21–22): 2051–2067MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Jessica D. Sanders
    • 1
  • Tod A. Laursen
    • 2
  • Michael A. Puso
    • 1
  1. 1.Lawrence Livermore National LaboratoryLivermoreUSA
  2. 2.Khalifa University of Science, Technology and ResearchAbu DhabiUAE

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