×

zbMATH — the first resource for mathematics

An efficient finite element method for embedded interface problems. (English) Zbl 1183.76803
Summary: A stabilized finite element method based on the Nitsche technique for enforcing constraints leads to an efficient computational procedure for embedded interface problems. We consider cases in which the jump of a field across the interface is given, as well as cases in which the primary field on the interface is given. The finite element mesh need not be aligned with the interface geometry. We present closed-form analytical expressions for interfacial stabilization terms and simple procedures for accurate flux evaluations. Representative numerical examples demonstrate the effectiveness of the proposed methodology.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Moës, A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering 46 (1) pp 131– (1999) · Zbl 0955.74066
[2] Daux, Arbitrary branched and intersecting cracks with the extended finite element method, International Journal for Numerical Methods in Engineering 48 (12) pp 1741– (2000) · Zbl 0989.74066
[3] Hansbo, An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems, Computer Methods in Applied Mechanics and Engineering 191 pp 5537– (2002) · Zbl 1035.65125
[4] Zhang, Immersed finite element method, Computer Methods in Applied Mechanics and Engineering 193 pp 2051– (2004) · Zbl 1067.76576
[5] Moës, A computational approach to handle complex microstructure geometries, Computer Methods in Applied Mechanics and Engineering 192 pp 3163– (2003) · Zbl 1054.74056
[6] Merle, Solving thermal and phase change problems with the extended finite element method, Computational Mechanics 28 (5) pp 339– (2002) · Zbl 1073.76589
[7] Chessa, The extended finite element method for solidification problems, International Journal for Numerical Methods in Engineering 53 (14) pp 1959– (2002)
[8] Dolbow, A numerical strategy for investigating the kinetic response of stimulus-responsive hydrogels, Computer Methods in Applied Mechanics and Engineering 194 pp 4447– (2005) · Zbl 1094.76039
[9] Ji, A hybrid extended finite element level set method for modeling phase transformations, International Journal for Numerical Methods in Engineering 54 pp 1209– (2002) · Zbl 1098.76572
[10] Duddu, A combined extended finite element and level set method for biofilm growth, International Journal for Numerical Methods in Engineering 74 (5) pp 848– (2008) · Zbl 1195.74169
[11] Nitsche, 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 pp 9– (19701971) · Zbl 0229.65079
[12] Gracie, Blending in the extended finite element method by discontinuous Galerkin and assumed strain methods, International Journal for Numerical Methods in Engineering 74 (11) pp 1645– (2008) · Zbl 1195.74175
[13] Fries, A corrected XFEM approximation without problems in blending elements, International Journal for Numerical Methods in Engineering 75 (5) pp 503– (2008) · Zbl 1195.74173
[14] Ji, On strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method, International Journal for Numerical Methods in Engineering 61 (14) pp 2508– (2004) · Zbl 1075.74651
[15] Moës, Imposing essential boundary conditions in the extended finite element method, International Journal for Numerical Methods in Engineering 67 (12) pp 1641– (2006)
[16] Mourad, A bubble-stabilized finite element method for Dirichlet constraints on embedded interfaces, International Journal for Numerical Methods in Engineering 69 pp 772– (2007) · Zbl 1194.65136
[17] Dolbow, Residual-free bubbles for embedded Dirichlet problems, Computer Methods in Applied Mechanics and Engineering 197 (14) pp 3751– (2008) · Zbl 1197.65180
[18] Carey, Approximate boundary-flux calculations, Computer Methods in Applied Mechanics and Engineering 50 pp 107– (1985) · Zbl 0546.73057
[19] Areias, A comment on the article ’A finite element method for simulation of strong and weak discontinuities in solid mechanics’ by A. Hansbo and P. Hansbo, Computer Methods in Applied Mechanics and Engineering 195 (9-12) pp 1275– (2006)
[20] Song, A method for dynamic crack and shear band propagation with phantom nodes, International Journal for Numerical Methods in Engineering 67 (6) pp 868– (2006) · Zbl 1113.74078
[21] Melenk, The partition of unity method: basic theory and applications, Computer Methods in Applied Mechanics and Engineering 139 pp 289– (1996) · Zbl 0881.65099
[22] Liu, A boundary condition capturing method for Poisson’s equation on irregular domains, Journal of Computational Physics 160 pp 151– (2000) · Zbl 0958.65105
[23] Barbosa, The finite element method with Lagrange multipliers on the boundary: circumventing the Babuška-Brezzi condition, Computer Methods in Applied Mechanics and Engineering 85 (1) pp 109– (1991) · Zbl 0764.73077
[24] Stenberg, On some techniques for approximating boundary conditions in the finite element method, Journal of Computational and Applied Mathematics 63 pp 139– (1995) · Zbl 0856.65130
[25] Arnold, A stable finite element for the Stokes equations, Calcolo 21 (4) pp 337– (1984) · Zbl 0593.76039
[26] Fernandez-Mendez, Imposing essential boundary conditions in mesh-free methods, Computer Methods in Applied Mechanics and Engineering 193 pp 1257– (2004)
[27] Wagner, Hierarchical enrichment for bridging scales and mesh-free boundary conditions, International Journal for Numerical Methods in Engineering 50 (3) pp 507– (2001) · Zbl 1006.76073
[28] Vaughan, A comparison of the extended finite element method with the immersed interface method for elliptic equations with discontinuous coefficients and singular sources, Communications in Applied Mathematics and Computational Science 1 pp 207– (2006) · Zbl 1153.65373
[29] Beale, Correction to: a comparison of the extended finite element method with the immersed interface method for elliptic equations with discontinuous coefficients and singular sources, Communications in Applied Mathematics and Computational Science 3 pp 95– (2008) · Zbl 1175.65134
[30] Mourad, An assumed gradient finite element method for the level set equation, International Journal for Numerical Methods in Engineering 64 pp 1009– (2005) · Zbl 1114.65115
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.