zbMATH — the first resource for mathematics

A robust Nitsche’s formulation for interface problems. (English) Zbl 1253.74096
Summary: We propose a novel weighting for the interfacial consistency terms arising in a Nitsche variational form. We demonstrate through numerical analysis and extensive numerical evidence that the choice of the weighting parameter has a great bearing on the stability of the method. Consequently, we propose a weighting that results in an estimate for the stabilization parameter such that the method remains well behaved in varied settings; ranging from the configuration of embedded interfaces resulting in arbitrarily small elements to such cases where a large contrast in material properties exists. An important consequence of this weighting is that the bulk as well as the interfacial fields remain well behaved in the presence of (a) elements with arbitrarily small volume fractions, (b) large material heterogeneities and (c) both large heterogeneities as well as arbitrarily small elements. We then highlight the accuracy and efficiency of the proposed formulation through numerical examples, focusing particular attention on interfacial quantities of interest.

74S05 Finite element methods applied to problems in solid mechanics
74A50 Structured surfaces and interfaces, coexistent phases
Full Text: DOI
[1] Arnold, D.; Brezzi, F.; Cockburn, B.; Marini, L., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. numer. anal., 1749-1779, (2002) · Zbl 1008.65080
[2] Babuška, I., The finite element method with Lagrange multipliers, Numer. math., 20, 3, 179-192, (1973) · Zbl 0258.65108
[3] Brezzi, F., Existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers, RAIRO-oper. res., 8, 2, 129-151, (1974) · Zbl 0338.90047
[4] Nitsche, J., Über ein variationsprinzip zur Lösung von Dirichlet-problemen bei verwendung von teilräumen, die keine randbedingungen untervorfen sind, Abh. math. univ. Hamburg, 36, 1, 9-15, (1971) · Zbl 0229.65079
[5] T.A. Laursen, M.A. Puso, J. Sanders, Mortar contact formulations for deformable-deformable contact: Past contributions and new extensions for enriched and embedded interface formulations, Comput. Methods Appl. Mech. Engrg. in press, <http://dx.doi.org/10.1016/j.cma.2010.09.006>. · Zbl 1239.74070
[6] J. Sanders, T. Laursen, M. Puso, A nitsche embedded mesh method, Comput. Mech. <http://dx.doi.org/10.1007/s00466-011-0641-2>. · Zbl 1366.74075
[7] Hansbo, A.; Hansbo, P., An unfitted finite element method, based on nitsche’s method, for elliptic interface problems, Comput. methods appl. mech. engrg., 191, 47-48, 5537-5552, (2002) · Zbl 1035.65125
[8] Hansbo, A.; Hansbo, P., A finite element method for the simulation of strong and weak discontinuities in solid mechanics, Comput. methods appl. mech. engrg., 193, 33-35, 3523-3540, (2004) · Zbl 1068.74076
[9] Sanders, J.D.; Dolbow, J.E.; Laursen, T.A., On the methods for stabilizing constraints over enriched interfaces in elasticity, Int. J. numer. methods engrg., 78, 9, 1009-1036, (2009) · Zbl 1183.74313
[10] Griebel, M.; Schweitzer, M.A., A particle-partition of unity method—part V: boundary conditions, (), 517-540 · Zbl 1033.65102
[11] Mourad, H.M.; Dolbow, J.; Harari, I., A bubble-stabilized finite element method for Dirichlet constraints on embedded interfaces, Int. J. numer. methods engrg., 69, 4, 772-793, (2007) · Zbl 1194.65136
[12] Dolbow, J.E.; Franca, L.P., Residual-free bubbles for embedded Dirichlet problems, Comput. methods appl. mech. engrg., 197, 45-48, 3751-3759, (2008) · Zbl 1197.65180
[13] Dolbow, J.E.; Harari, I., An efficient finite element method for embedded interface problems, Int. J. numer. methods engrg., 78, 2, 229-252, (2009) · Zbl 1183.76803
[14] M. Hautefeuille, C. Annavarapu, J.E. Dolbow, Robust imposition of dirichlet boundary conditions on embedded surfaces, Int. J. Numer. Methods Engrg. doi.10.1137/050634736. · Zbl 1242.76124
[15] C. Annavarapu, M. Hautefeuille, J.E. Dolbow, Stable imposition of stiff constraints in explicit dynamics for embedded finite element methods, Int. J. Numer. Methods Engrg. in revision. URL http://www.duke.edu/∼mh186/TransientEmbedded.pdf. · Zbl 1352.74314
[16] Lew, A.J.; Negri, M., Optimal convergence of a discontinuous-Galerkin-based immersed boundary method, ESAIM: mathematical modelling and numerical analysis, 45, 04, 651-674, (2011) · Zbl 1269.65108
[17] E. Burman, P. Hansbo, Fictitious domain finite element methods using cut elements: Ii. a stabilized nitsche method, Applied Numerical Mathematics:10.1016/j.apnum.2011.01.008. · Zbl 1316.65099
[18] Zunino, P., Discontinuous Galerkin methods based on weighted interior penalties for second order pdes with non-smooth coefficients, J. sci. comput., 38, 1, 99-126, (2009) · Zbl 1203.65263
[19] Burman, E.; Zunino, P., A domain decomposition method based on weighted interior penalties for advection – diffusion – reaction problems, SIAM J. numer. anal., 44, 4, 1612-1638, (2006) · Zbl 1125.65113
[20] Ern, A.; Stephansen, A.F.; Zunino, P., A discontinuous Galerkin method with weighted averages for advection – diffusion equations with locally small and anisotropic diffusivity, IMA J. numer. anal., 29, 2, 235-256, (2009) · Zbl 1165.65074
[21] Cai, Z.; Ye, X.; Zhang, S., Discontinuous Galerkin finite element methods for interface problems: a priori and a posteriori error estimations, SIAM J. numer. anal., 49, 5, 1761-1787, (2011) · Zbl 1232.65142
[22] Zunino, P.; Cattaneo, L.; Colciago, C.M., An unfitted interface penalty method for the numerical approximation of contrast problems, Appl. numer. math., 61, 1059-1076, (2011) · Zbl 1232.65152
[23] Simone, A., Partition of unity-based discontinuous elements for interface phenomena: computational issues, Commun. numer. methods engrg., 20, 6, 465-478, (2004) · Zbl 1058.74082
[24] Béchet, E.; Moës, N.; Wohlmuth, B., A stable Lagrange multiplier space for stiff interface conditions within the extended finite element method, Int. J. numer. methods engrg., 78, 8, 931-954, (2009) · Zbl 1183.74259
[25] Stenberg, R., On some techniques for approximating boundary conditions in the finite element method, J. comput. appl. math., 63, 1-3, 139-148, (1995) · Zbl 0856.65130
[26] Fernández-Méndez, S.; Huerta, A., Imposing Dirichlet boundary conditions in mesh-free method, Comput. methods appl. mech. engrg., 193, 12-14, 1257-1275, (2004) · Zbl 1060.74665
[27] Juntunen, M.; Stenberg, R., Nitsches method for general boundary conditions, Math. comput., 78, 267, 1353-1374, (2009) · Zbl 1198.65223
[28] Barbosa, H.J.C.; Hughes, T.J.R., The finite element method with Lagrange multipliers on the boundary: circumventing the babuska-Brezzi condition, Cmame, 85, 1, 109-128, (1991) · Zbl 0764.73077
[29] Embar, A.; Dolbow, J.E.; Harari, I., Imposing Dirichlet boundary conditions with nitsche’s method and spline-based finite elements, Int. J. numer. methods engrg., 83, 7, 877-898, (2010) · Zbl 1197.74178
[30] Ji, H.; Dolbow, J.E., On strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method, Int. J. numer. methods engrg., 61, 14, 2508-2535, (2004) · Zbl 1075.74651
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.