An embedded Dirichlet formulation for 3D continua

International Journal for Numerical Methods in Engineering

Volumn 82, Issue 5, 2010, Pages 537-563

  Gerstenberger, A ^a   Wall, W A ^a  

^a TECHNICAL UNIVERSITY OF MUNICH   (Germany)

Author keywords

Domain decomposition; Embedded Dirichlet conditions; Extended finite element method; Fixed grid; Fluid structure interaction; Hybrid mixed finite elements; Lagrange multiplier

Indexed keywords

DIRICHLET CONDITION; DOMAIN DECOMPOSITIONS; EXTENDED FINITE ELEMENT METHOD; FIXED GRIDS; HYBRID/MIXED FINITE ELEMENT; HYBRID/MIXED FINITE ELEMENTS;

COERCIVE FORCE; DOMAIN DECOMPOSITION METHODS; FLUID STRUCTURE INTERACTION; FREQUENCY MULTIPLYING CIRCUITS; LAGRANGE MULTIPLIERS; NAVIER STOKES EQUATIONS; THREE DIMENSIONAL;

FINITE ELEMENT METHOD;

EID: 77949991774     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.2755     Document Type: Article

References (41)

1. Wall WA, Gerstenberger A, Gamnitzer P, Förster C, Ramm E. Large deformation fluid-structure interaction-advances in ALE methods and new fixed grid approaches. In Fluid-Structure Interaction Modelling Simulation, Optimisation, Bungartz H-J, Schäfer M (eds). Lecture Notes in Computational Science and Engineering, vol. 53. Springer: Berlin, 2006.
2. Gerstenberger A, Wall WA. An extended finite element method/Lagrange multiplier based approach for fluid- structure interaction. Computer Methods in Applied Mechanics and Engineering 2008; 197(19-20):1699-1714.
3. Legay A, Chessa J, Belytschko T. An Eulerian-Lagrangian method for fluid-structure interaction based on level sets. Computer Methods in Applied Mechanics and Engineering 2006; 195(17-18):2070-2087.
4. Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering 1999; 45(5):601-620.
5. Moës N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering 1999; 46(1):131-150.
6. 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(46-47):6183-6200.
7. 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(14): 2508-2535.
8. 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(12):1641-1669.
9. 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(3):305-334.
10. Dolbow J, Franca L. Residual-free bubbles for embedded Dirichlet problems. Computer Methods in Applied Mechanics and Engineering 2008; 197:3751-3759.
11. Mourad HM, Dolbow J, Harari I. A bubble-stabilized finite element method for Dirichlet constraints on embedded interfaces. International Journal for Numerical Methods in Engineering 2007; 69(4):772-793.
12. Sanders JD, Dolbow JE, Laursen TA. On methods for stabilizing constraints over enriched interfaces in elasticity. International Journal for Numerical Methods in Engineering 2009; 78(9):1009-1036.
13. Dolbow J, Harari I. An efficient finite element method for embedded interface problems. International Journal for Numerical Methods in Engineering 2009; 78(2):229-252.
14. Fernández-Méndez S, Huerta A. Imposing essential boundary conditions in mesh-free methods. Computer Methods in Applied Mechanics and Engineering 2004; 193(12-14):1257-1275.
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(47-48):5537-5552.
16. 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(33-35):3523-3540.
17. Mergheim J, Kuhl E, Steinmann P. A hybrid discontinuous Galerkin/interface method for the computational modelling of failure. Communications in Numerical Methods in Engineering 2004; 20(7):511-519.
18. Nitsche J. Ü ber ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abhandlungen aus dem Mathematisches Seminar der Universität 1971; 36:9-15.
19. Hemker P, Hoffmann W, van Raalte M. Discontinuous Galerkin discretization with embedded boundary conditions. Computational Methods in Applied Mathematics 2003; 3:135-158.
20. Lew AJ, Buscaglia GC. A discontinuous-Galerkin-based immersed boundary method. International Journal for Numerical Methods in Engineering 2008; 76(4):427-454.
21. Gee M, Sala M, Siefert C, Hu J, Tuminaro R. Ml 5.0 smoothed aggregation user's guide. SAND2006-2649, 2006.
22. Sala M, Tuminaro R. A new Petrov-Galerkin smoothed aggregation preconditioner for nonsymmetric linear systems. SIAM Journal on Scientific Computing 2008; 31:143-166.
23. Gerstenberger A, Mayer UM, Wall WA. A 3d XFEM/Lagrange multiplier based approach for fluid structure interaction. Eighth World Congress on Computational Mechanics/Fifth European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2008, Venice, July 2008.
24. Zilian A, Fries T-P. A localized mixed-hybrid method for imposing interfacial constraints in the extended finite element method (XFEM). International Journal for Numerical Methods in Engineering 2009; DOI:http://dx.doi.org/10.1002/nme.2596.
25. Daux C, Moës N, Dolbow J, Sukumar N, Belytschko T. Arbitrary cracks and holes with the extended finite element method. International Journal for Numerical Methods in Engineering 2000; 48(12):1741-1760.
26. Legrain G, Moës N, Huerta A. Stability of incompressible formulations enriched with x-FEM. Computer Methods in Applied Mechanics and Engineering 2008; 197(21-24):1835-1849.
27. Behr MA, Franca LP, Tezduyar TE. Stabilized finite element methods for the velocity-pressure-stress formulation of incompressible flows. Computer Methods in Applied Mechanics and Engineering 1993; 104(1):31-48.
28. Brooks AN, Hughes TJ. Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering 1982; 32(1-3):199-259.
29. Hughes TJR, Franca LP, Scovazzi G. Multiscale sand stabilized methods. In Encyclopedia of Computational Mechanics, Stein E, de Borst R, Hughes TJR (eds), vol. 3. Wiley: New York, 2004.
30. Gresho P, Sani R. Incompressible Flow and the Finite Element Method, Volume 2: Isothermal Laminar Flow. Wiley: New York, 2000.
31. Engels H. Numerical Quadrature and Cubature. Academic Press: London, New York, 1980.
32. Peano A. Gauss-Lobatto integration of high precision tetrahedral elements. International Journal for Numerical Methods in Engineering 1982; 18(2):311-313.
33. Mayer UM, Gerstenberger A, Wall WA. Interface handling for three-dimensional higher-order XFEM computations in fluid-structure interaction. International Journal for Numerical Methods in Engineering 2009; 554:4545-5454. DOI: http://dx.doi.org/10.1002/nme.2600.
34. Hamel G. Spiralförmige Bewegungen zäher Flüssigkeiten. Jahresbericht der Deutschen Mathematiker-Vereinigung 1916; 25:34-60.
35. Jeffery GB. The two-dimensional steady motion of a viscous fluid. Philosophical Magazine 1915; 29:455-465.
36. Rosenhead L. The steady two-dimensional radial flow of viscous fluid between two inclined plane walls. Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences 1940; 175(963): 436-467.
37. Corless RM, Assefa D. Jeffery-hamel flow with maple: a case study of integration of elliptic functions in a cas. ISSAC'07: Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation. ACM: New York, NY, U.S.A., 2007.
38. Bagheri B, Scott LR, Zhang S. Implementing, using high-order finite element methods. Finite Elements in Analysis and Design 1994; 16(3-4):175-189.
39. Schäfer M, Turek S. Flow simulation with high-performance computers II. Benchmark Computations of Laminar Flow Around a Cylinder. Notes on Numerical Fluid Mechanics, vol. 52. Vieweg: Braunschweig, 1996; 547-566.
40. Gerstenberger A, Wall WA. Enhancement of fixed-grid methods towards complex fluid-structure interaction applications. International Journal for Numerical Methods in Fluids 2008; 57(9):1227-1248.
41. Wall WA, Gamnitzer P, Gerstenberger A. Fluid-structure interaction approaches on fixed grids based on two different domain decomposition ideas. International Journal of Computational Fluid Dynamics 2008; 22(6): 411-427.