A discontinuous-Galerkin-based immersed boundary method

International Journal for Numerical Methods in Engineering

Volumn 76, Issue 4, 2008, Pages 427-454

  Lew, Adrián J ^a   Buscaglia, Gustavo C ^b,c  

^a Stanford University   (United States)

^b UNIVERSITY OF SÃO PAULO   (Brazil)

^c CENTRO ATÓMICO BARILOCHE   (Argentina)

Author keywords

Boundary locking; Cartesian grids; Dirichlet conditions; Discontinuous Galerkin method; Immersed boundary; Immersed finite element method; Interfaces

Indexed keywords

BOUNDARY VALUE PROBLEMS; COMPUTATIONAL FLUID DYNAMICS; FINITE ELEMENT METHOD; GALERKIN METHODS; NUMERICAL ANALYSIS; TURBULENT FLOW;

BOUNDARY LOCKING; CARTESIAN GRIDS; DIRICHLET CONDITIONS; DISCONTINUOUS-GALERKIN METHOD; IMMERSED BOUNDARY; IMMERSED FINITE ELEMENT METHOD; INTERFACES;

BOUNDARY ELEMENT METHOD;

EID: 57849164037     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.2312     Document Type: Article

References (77)

1. Steger J, Dougherty F, Benek J. A chimera grid scheme (multiple overset body-conforming mesh system for finite difference adaptation to complex aircraft configuration). Advances in Grid Generation, Proceedings of the Applied Mechanics, Bioengineering and Fluids Engineering Conference, Houston, U.S.A., 1983; 59-69.
2. Tilch R, Tabbal A, Zhu M, Dekker F, Löner R. Combination of body-fitted and embedded grids for external vehicle aerodynamics. Forty-fifth Aerospace Sciences Meeting and Exhibit. AIAA: New York, 2007.
3. Belytschko T, Parimi C, Moes N, Sukumar N, Usui S. Structured extended finite element methods for solids defined by implicit surfaces. International Journal for Numerical Methods in Engineering 2003; 56:609-635.
4. Peskin C. Flow patterns around heart valves: a numerical method. Journal of Computational Physics 1972; 10:252-271.
5. Harlow F, Welch J. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Physics of Fluids 1965; 8:2182-2189.
6. Unverdi S, Tryggvason G. A front-tracking method for viscous, incompressible, multi-fluid flows. Journal of Computational Physics 1992; 100:25-37.
7. Tryggvason G, Bunner B, Esmaeeli A, Juric D, Al-Rawahi N, Tauber W, Han J, Nas S, Jan Y-J. A front-tracking method for the computations of multiphase flow. Journal of Computational Physics 2001; 169:708-759.
8. Sousa FS, Mangiavacchi N, Nonato LG, Castelo A, Tomé M, Ferreira V, Cuminato J, McKee S. A front-tracking method for simulation of 3D multifluid flows with free surfaces. Journal of Computational Physics 2004; 198:469-499.
9. Hirt C, Nichols B. Volume of fluid (VOF) method for the dynamics of free boundaries. Journal of Computational Physics 1981; 39:201-225.
10. Renardy YY, Renardy M, Cristini V. A new volume-of-fluid formulation for surfactants and simulations of drop deformation under shear at a low viscosity ratio. European Journal of Mechanics B - Fluids 2002; 21:49-59.
11. Renardy Y, Renardy M. PROST: a parabolic reconstruction of surface tension for the volume-of-fluid method. Journal of Computational Physics 2002; 183:400-421.
12. Ginzburg I, Wittum G. Two-phase flows on interface refined grids modeled with VOF, staggered finite volumes, and spline interpolants. Journal of Computational Physics 2001; 166:302-335.
13. Osher S, Sethian JA. Front propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics 1988; 79:12-49.
14. Sethian JA. Fast marching methods and level set methods for propagating interfaces. Computational Fluid Mechanics. von Karman Institute Lecture Series, 1998. http://math.berkeley.edu/~sethian/2006/papers/ sethian.vonkarman1.pdf.
15. Sethian JA. Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision and Materials Science (2nd edn). Cambridge University Press: Cambridge, 1999.
16. Sethian JA, Smereka P. Level set methods for fluid interfaces. Annual Review of Fluid Mechanics 2003; 35:341-372.
17. Osher S, Fedkiw R. Level Set Methods and Dynamic Implicit Surfaces. Applied Mathematical Sciences, vol. 153. Springer: Berlin, 2003.
18. Sussman M, Smereka P, Osher S. A level set approach for computing solutions to incompressible two-phase flow. Journal of Computational Physics 1994; 114:146-159.
19. Chang YC, Hou TY, Merriman B, Osher S. A level set formulation of Eulerian interface capturing methods for incompressible fluid flows. Journal of Computational Physics 1996; 124:449-464.
20. Fedkiw RP, Sapiro G, Shu C. Shock capturing, level sets, and PDE based methods in computer vision and image processing: a review of Osher's contributions. Journal of Computational Physics 2003; 185:309-341.
21. Mut F, Buscaglia GC, Dari EA. New mass-conserving algorithm for level set redistancing on unstructured meshes. Journal of Applied Mechanics 2006; 73:1011-1016.
22. Babuška I. The finite element method with Lagrangian multipliers. Numerische Mathematik 1973; 20:179-192.
23. Pitkäranta J. Boundary subspaces for the finite element method with Lagrange multipliers. Numerische Mathematik 1979; 33:273-289.
24. Babuška I, Banerjee U, Osborn J. Survey of meshless and generalized finite element methods: a unified approach. Acta Numerica 2003; 12:1-125.
25. Ferziger J, Peric M. Computational Methods for Fluid Dynamics. Springer: Berlin, 1999.
26. Peskin C. Numerical analysis of blood flow in the heart. Journal of Computational Physics 1977; 25:220-252.
27. Goldstein D, Handler R, Sirovich L. Modeling a no-slip flow boundary with an external flow field. Journal of Computational Physics 1993; 105:354-366.
28. Stockie J, Wetton B. Analysis of stiffness in the immersed boundary method and implications for time-stepping schemes. Journal of Computational Physics 1999; 154:41-64.
29. Lai M-C, Peskin C. An immersed boundary method with formal second-order accuracy and reduced numerical viscosity. Journal of Computational Physics 2000; 160:705-719.
30. Tornberg AK, Engquist B. Regularization techniques for numerical approximations of PDEs with singularities. Journal of Scientific Computing 2003; 19(1-3):527-552.
31. Tornberg A, Engquist B. Numerical approximations of singular source terms in differential equations. Journal of Computational Physics 2004; 200:462-488.
32. Engquist B, Tornberg A, Tsai R. Discretization of Dirac delta functions in level set methods. Journal of Computational Physics 2005; 207:28-51.
33. Li Z, Lai M-C. The immersed interface method for the Navier-Stokes equations with singular forces. Journal of Computational Physics 2001; 171:822-842.
34. Smereka P. The numerical approximation of a delta function with application to level set methods. Journal of Computational Physics 2006; 211:77-90.
35. Zhang L, Gerstenberger A, Wang X, Liu W. Immersed finite element method. Computer Methods in Applied Mechanics and Engineering 2004; 193:2051-2067.
36. Liu Y, Zhang L, Wang X, Liu WK. Coupling of Navier-Stokes equations with protein molecular dynamics and its application to hemodynamics. International Journal for Numerical Methods in Fluids 2004; 46:1237-1252.
37. Chessa J, Smolinsky P, Belytschko T. The extended finite element method (xfem) for solidification problems. International Journal for Numerical Methods in Engineering 2002; 53:1959-1977.
38. Fernández-Méndez S, Huerta A. Imposing essential boundary conditions in mesh-free methods. Computer Methods in Applied Mechanics and Engineering 2004; 193:1257-1275.
39. Lamballais E, Silvestrini JH. Direct numerical simulation of interactions between a mixing layer and a wake around a cylinder. Journal of Turbulence 2002; 3(28):1-21.
40. 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.
41. Angot P, Bruneau Ch-H, Fabrie P. A penalization method to take into account obstacles in incompressible viscous flows. Numerische Mathematik 1999; 81:497-520.
42. Leveque R, Li Z. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM Journal on Numerical Analysis 1994; 31:1019-1044.
43. Lee L, Leveque R. An immersed interface method for incompressible Navier-Stokes equations. SIAM Journal on Scientific Computing 2003; 25:832-856.
44. Ito K, Li Z, Kyei Y. Higher-order, Cartesian grid based finite difference schemes for elliptic equations on irregular domains. SIAM Journal on Scientific Computing 2005; 27:346-367.
45. Scardovelli R, Zaleski S. Direct numerical simulation of free-surface and interfacial flow. Annual Review of Fluid Mechanics 1999; 31:567-603.
46. Mittal R, Iaccarino G. Immersed boundary methods. Annual Review of Fluid Mechanics 2005; 37:239-261.
47. Tseng Y-J, Ferziger J. A ghost-cell immersed boundary method for flow in complex geometry. Journal of Computational Physics 2003; 192:593-623.
48. Stenberg R. On some techniques for approximating boundary conditions in the finite element method. Journal of Computational and Applied Mathematics 1995; 63:139-148.
49. Brenner S, Scott LR. The Mathematical Theory of Finite Element Methods (2nd edn). Texts in Applied Mathematics. Springer: Berlin, 2002.
50. Glowinski R, Pan T-W, Periaux J. A fictitious domain method for Dirichlet problems and applications. Computer Methods in Applied Mechanics and Engineering 1994; 111:283-303.
51. Glowinski R, Pan T-W, Hesla T, Joseph D. A distributed Lagrange multiplier/fictitious domain method for particulate flows. International Journal of Multiphase Flow 99; 25:755-794.
52. Girault V, Glowinski R. Error analysis of a fictitious domain method applied to a Dirichlet problem. Japan Journal of Industrial and Applied Mathematics 1995; 12:487-514.
53. Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. Springer: Heidelberg, 1991.
54. Ji H, Dolbow J. 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.
55. Barbosa H, Hughes TJR. The finite element method with Lagrange multipliers on the boundary: circumventing the Babuška-Brezzi condition. Computer Methods in Applied Mechanics and Engineering 1991; 85:109-128.
56. Mourad H, Dolbow J, Harari I. A bubble-stabilized finite element for Dirichlet constraints on embedded interfaces. International Journal for Numerical Methods in Engineering 2007; 69:772-793.
57. Hughes TJR, Franca L, Balestra M. A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuška-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Computer Methods in Applied Mechanics and Engineering 1986; 59:85-99.
58. Douglas J, Wang J. An absolutely stabilized finite element method for the Stokes problem. Mathematics of Computation 1989; 52:495-508.
59. TenEyck A, Lew A. Discontinuous Galerkin methods for nonlinear elasticity. International Journal for Numerical Methods in Engineering 2006; 67:1204-1243.
60. Li F, Shu CW. Locally divergence-free discontinuous Galerkin methods for MHD equations. Journal of Scientific Computing 2005; 22-23:413-442.
61. Lew A, Neff P, Sulsky D, Ortiz M. Optimal BV estimates for a discontinuous Galerkin method in linear elasticity. Applied Mathematics Research Express 2004; 3:73-106.
62. Arnold D, Brezzi F, Cockburn B, Marini L. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM Journal on Numerical Analysis 2002; 39(5):1749-1779.
63. Hemker P, Hoffmann W, van Raalte M. Discontinuous Galerkin discretization with embedded boundary conditions. Computer Methods in Applied Mechanics and Engineering 2003; 3:135-158.
64. van Raalte M. A feasibility study for discontinuous Galerkin discretization with embedded Dirichlet boundary condition. Applied Numerical Mathematics 2004; 51:361-383.
65. Engwer C, Peter B. A discontinuous Galerkin method for simulations in complex domains. Technical Report IWR 5707, Universität Heidelberg, 2005 (Available from: http://www.ub.uni-heidelberg.de/archiv/ 5707/).
66. Dolbow J, Moës N, Belytschko T. Discontinuous enrichment in finite elements with a partition of unity method. Finite Elements in Analysis and Design 2000; 36:235-260.
67. Chessa J, Belytschko T. A local space-time discontinuous finite element method. Computer Methods in Applied Mechanics and Engineering 2006; 195:1325-1343.
68. Coppola-Owen A, Codina R. Improving Eulerian two-phase flow finite element approximation with discontinuous gradient pressure shape functions. International Journal for Numerical Methods in Fluids 2005; 49:1287-1304.
69. Bassi F, Rebay S. High-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. Journal of Computational Physics 1997; 131:267-279.
70. Storti M, Crivelli L, Idelsohn S. Making curved interfaces straight in phase-change problems. International Journal for Numerical Methods in Engineering 1987; 24:375-392.
71. Brezzi F, Manzini G, Marini D, Pietra P, Russo A. Discontinuous finite elements for diffusion problems. In Atti Convegno in onore di F. Brioschi. Accademia di Scienze e Lettere: Milan, 1999; 197-217.
72. Brezzi F, Manzini G, Marini D, Pietra P, Russo A. Discontinuous Galerkin approximations for elliptic problems. Numerical Methods for Partial Differential Equations 2000; 16:365-378.
73. Idelsohn S, Nigro N, Storti M, Buscaglia G. A Petrov-Galerkin formulation for advection-reaction-diffusion problems. Computer Methods in Applied Mechanics and Engineering 1996; 136:27-46.
74. Codina R. Comparison of some finite element methods for solving the diffusion-convection-reaction equations. Computer Methods in Applied Mechanics and Engineering 1998; 156:185-210.
75. Franca L, Madureira A, Valentin F. Towards multiscale functions: enriching finite element spaces with local but not bubble-like functions. Computer Methods in Applied Mechanics and Engineering 2005; 194:3006-3021.
76. Li J. Uniform convergence of discontinuous finite element methods for singularly perturbed reaction-diffusion problems. Computers and Mathematics with Applications 2002; 44:231-240.
77. Hughes TJR, Scovazzi G, Bochev P, Buffa A. A multiscale discontinuous Galerkin method with the computational structure of a continuous Galerkin method. Computer Methods in Applied Mechanics and Engineering 2006; 195:2761-2787.