A localized mixed-hybrid method for imposing interfacial constraints in the extended finite element method (XFEM)

International Journal for Numerical Methods in Engineering

Volumn 79, Issue 6, 2009, Pages 733-752

  Zilian, A ^a   Fries, T P ^b  

^a TECHNICAL UNIVERSITY OF BRAUNSCHWEIG   (Germany)

^b RWTH AACHEN UNIVERSITY   (Germany)

Author keywords

Embedded interfaces; Extended finite elements; Interfacial constraints; Mixed method

Indexed keywords

APPROXIMATION SPACES; APRIORI; AUXILIARY VARIABLES; CONVERGENCE BEHAVIORS; DIRICHLET; ELLIPTIC PROBLEM; EMBEDDED INTERFACES; EXTENDED FINITE ELEMENT METHOD; EXTENDED FINITE ELEMENTS; INTERFACE CONDITIONS; INTERFACE CONSTRAINTS; INTERFACIAL CONSTRAINTS; MATERIAL INTERFACES; MIXED METHOD; MIXED-HYBRID FORMULATIONS; MIXED-HYBRID METHODS; MODEL EQUATIONS; PENALTY METHODS; TWO-DIMENSIONAL; VECTOR VALUED; WEAK DISCONTINUITY; WEAK FORM;

CRACK INITIATION; LAGRANGE MULTIPLIERS;

FINITE ELEMENT METHOD;

EID: 68549094269     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.2596     Document Type: Article

References (27)

1. Sukumar N, Chopp DL, Moës N, Belytschko T. Modeling holes and inclusions by level sets in the extended finite-element method. Computer Methods in Applied Mechanics and Engineering 2001; 190:6183-6200.
2. Moës N, Dolbow JE, Belytschko T. A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering 1999; 46(1):131-150.
3. Fries TP. The intrinsic XFEM for two-fluid flows. International Journal for Numerical Methods in Fluids 2008. DOI: 10.1002/fld.1901.
4. Walhorn E, Kölke A, Hübner B, Dinkler D. Fluid-structure coupling within a monolithic model involving free surface flows. Computers and Structures 2005; 83(25-26):2100-2111.
5. Dolbow JE, Fried E, Ji H. A numerical strategy for investigating the kinetic response of stimulus-responsive hydrogels. International Journal for Numerical Methods in Engineering 2005; 194:4447-4480.
6. Dolbow JE, Merle R. Solving thermal and phase change problems with the extended finite element method. Computational Mechanics 2002; 28(5):339-350.
7. Chessa J, Smolinski P, Belytschko T. The extended finite element method (XFEM) for Stefan problems.International Journal for Numerical Methods in Engineering 2002; 53(8):1959-1977.
8. Ji H, Dolbow JE. On strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method. International Journal for Numerical Methods in Engineering 2004; 61:2508-2535.
9. Moës N, Béchet E, Tourbier M. Imposing Dirichlet boundary conditions in the extended finite element method. International Journal for Numerical Methods in Engineering 2006; 67:1641-1669.
10. Mourad HM, Dolbow JE, Harari I. A bubble-stabilized finite element method for Dirichlet constraints on embedded interfaces. International Journal for Numerical Methods in Engineering 2007; 69:772-793.
11. Kölke A. Modellierung und Diskretisierung bewegter Diskontinuitäten in randgekoppelten Mehrfeldsystemen. Ph.D. Thesis, Technische Universität Braunschweig, 2005.
12. Zilian A, Legay A. The enriched space-time finite element method (EST) for simultaneous solution of fluid- structure interaction. International Journal for Numerical Methods in Engineering 2008; 75:305-334.
13. Fernández-Méndez S, Huerta A. Imposing essential boundary conditions in mesh-free methods. Computer Methods in Applied Mechanics and Engineering 2004; 193:1257-1275.
14. Wohlmuth B. Discretization Methods and Iterative Solvers Based on Domain Decomposition. Springer: Berlin, 2001.
15. Hansbo A, Hansbo P. An unfitted finite element method, based on Nitsche's method, for elliptic interface problems. Computer Methods in Applied Mechanics and Engineering 2002; 191:5537-5552.
16. Becker R, Hansbo P, Stenberg R. A finite element method for domain decomposition with non-matching grids. Mathematical Modelling and Numerical Analysis 2003; 37:209-225.
17. Hansbo A, Hansbo P. A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Computer Methods in Applied Mechanics and Engineering 2004; 193:3523-3540.
18. Hansbo P, Lovadina C, Perugia I, Sangalli G. A Lagrange multiplier method for the finite element solution of elliptic interface problems using non-matching meshes. Numerische Mathematik 2005; 100:91-115.
19. Nitsche J. Ü ber ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 1971; 36:9-15.
20. Mergheim J, Kuhl E, Steinmann P. A hybrid discontinuous Galerkin/interface method for the computational modeling of failure. Communications in Numerical Methods in Engineering 2004; 20:511-519.
21. Arnold DN. An interior penalty finite element method with discontinuous elements. SIAM Journal on Numerical Analysis 1982; 19(4):742-760.
22. Arnold DN, Brezzi F, Cockburn B, Marini LD. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM Journal on Numerical Analysis 2002; 39(5):1749-1779.
23. Cockburn B, Karniadakis GE, Shu C-W. Discontinuous Galerkin Methods. Theory, Computation and Applications. Springer: Berlin, 2000.
24. Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics. Springer: New York, 1991.
25. Sethian JA. A fast marching level set method for monotonically advancing fronts. Proceedings of the National Academy of Sciences 1996; 93:1591-1595.
26. Fries TP. A corrected XFEM approximation without problems in blending elements. International Journal for Numerical Methods in Engineering 2008; 75:503-532.
27. Xue W-M, Karlovitz LA, Atluri SN. On the existence and stability conditions for mixed-hybrid finite element solutions based on Reissner's variational principle. Internatioal Journal of Solids and Structures 1985; 21:97-116.