Three-dimensional matched interface and boundary (MIB) method for treating geometric singularities

Journal of Computational Physics

Volumn 227, Issue 1, 2007, Pages 602-632

  Yu, Sining ^a   Wei, G W ^a,b  

^a MICHIGAN STATE UNIVERSITY   (United States)

^b Michigan State University   (United States)

Author keywords

Discontinuous coefficients; Elliptic equations; Geometric singularity; Immersed boundary method; Immersed interface method; Material interface

Indexed keywords

COMPUTATIONAL FLUID DYNAMICS; FINITE DIFFERENCE METHOD; INTERFACES (MATERIALS); ITERATIVE METHODS; LINEAR ALGEBRA; LINEAR EQUATIONS; TURBULENT FLOW; VAN DER WAALS FORCES;

BOUNDARY METHOD; COMPLEX INTERFACE; DISCONTINUOUS COEFFICIENTS; ELLIPTIC EQUATIONS; GEOMETRIC SINGULARITY; IMMERSED BOUNDARY METHODS; IMMERSED INTERFACE METHOD; JUMP CONDITIONS; MATERIAL INTERFACES; SMOOTH INTERFACE;

MESH GENERATION;

EID: 35348917599     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2007.08.003     Document Type: Article

References (75)

1. Adams L., and Li Z.L. The immersed interface/multigrid methods for interface problems. SIAM J. Sci. Comput. 24 (2002) 463-479
2. Babuška I. The finite element method for elliptic equations with discontinuous coefficients. Computing 5 (1970) 207-213
3. Babuška I., and Sauter S.A. Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave number?. SIAM J. Numer. Anal. 34 (1997) 2392-2423
4. Baetke F., Werner H., and Wengle H. Numerical-simulation of turbulent-flow over surface-mounted obstacles with sharp edges and corners. J. Wind Engng. Indust. Aerodyn. 35 (1990) 129-147
5. Ben Ameur H., Burger M., and Hackl B. Level set methods for geometric inverse problems in linear elasticity. Inverse Probl. 20 (2004) 673-696
6. Berthelsen P.A. A decomposed immersed interface method for variable coefficient elliptic equations with non-smooth and discontinuous solutions. J. Comput. Phys. 197 (2004) 364-386
7. Biros G., Ying L.X., and Zorin D. A fast solver for the Stokes equations with distributed forces in complex geometries. J. Comput. Phys. 193 (2004) 317-348
8. Bramble J., and King J. A finite element method for interface problems in domains with smooth boundaries and interfaces. Adv. Comput. Math. 6 (1996) 109-138
9. Cai W., and Deng S.Z. An upwinding embedded boundary method for Maxwell's equations in media with material interfaces: 2D case. J. Comput. Phys. 190 (2003) 159-183
10. Caorsi S., Pastorino M., and Raffetto M. Electromagnetic scattering by a conducting strip with a multilayer elliptic dielectric coating. IEEE T. Electromag. Compat. 41 (1999) 335-343
11. Cendes Z.J., Shenton D.N., and Shahnasser H. Magnetic field computation using Delaney triangulation and complementary finite element methods. IEEE Ttans. Magn. 19 (1983) 2251-2554
12. Deng S.Z., Ito K., and Li Z.L. Three-dimensional elliptic solvers for interface problems and applications. J. Comput. Phys. 184 (2003) 215-243
13. Dumett M.A., and Keener J.P. An immersed interface method for solving anisotropic elliptic boundary value problems in three dimensions. SIAM J. Sci. Comput. 25 (2003) 348-367
14. Fadlun E.A., Verzicco R., Orlandi P., and Mohd-Yusof J. Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations. J. Comput. Phys. 161 (2000) 30-60
15. Fedkiw R.P., Aslam T., Merriman B., and Osher S. A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys. 152 (1999) 457-492
16. Fogelson A.L., and Keener J.P. Immersed interface methods for Neumann and related problems in two and three dimensions. SIAM J. Sci. Comput. 22 (2000) 1630-1654
17. Fornberg B. Calculation of weights in finite difference formulas. SIAM Rev. 40 (1998) 685-691
18. Francois M., and Shyy W. Computations of drop dynamics with the immersed boundary method, part 2: Drop impact and heat transfer. Numer. Heat Trans. Part B - Fund. 44 (2003) 119-143
19. Francois M., Uzgoren E., Jackson J., and Shyy W. Multigrid computations with the immersed boundary technique for multiphase flows. Int. J. Numer. Methods Heat Fluid Flow 14 (2004) 98-115
20. Geng W.H., Yu S.N., and Wei G.W. Treatment of charge singularities in the implicit solvent models. J. Chem. Phys. 127 (2007) 114106
21. Gibou F., and Fedkiw R.P. A forth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem. J. Comput. Phys. 202 (2005) 577-601
22. Griffith B.E., and Peskin C.S. On the order of accuracy of the immersed boundary method: higher order convergence rates for sufficiently smooth problems. J. Comput. Phys. 208 (2005) 75-105
23. Hadley G.R. High-accuracy finite-difference equations for dielectric waveguide analysis I: Uniform regions and dielectric interfaces. J. Lightwave Technol. 20 (2002) 1210-1218
24. Hesthaven J.S. High-order accurate methods in time-domain computational electromagnetics. A review. Adv. Imaging Electron Phys. 127 (2003) 59-123
25. Horikis T.P., and Kath W.L. Modal analysis of circular Bragg fibers with arbitrary index profiles. Optics Lett. 31 (2006) 3417-3419
26. Hou S.M., and Liu X.-D. A numerical method for solving variable coefficient elliptic equation with interfaces. J. Comput. Phys. 202 (2005) 411-445
27. Hou T.Y., Li Z.L., Osher S., and Zhao H. A hybrid method for moving interface problems with application to the Hele-Shaw flow. J. Comput. Phys. 134 (1997) 236-252
28. Huang H., and Li Z.L. Convergence analysis of the immersed interface method. IMA J. Numer. Anal. 19 (1999) 583-608
29. Hunter J.K., Li Z.L., and Zhao H. Reactive autophobic spreading of drops. J. Comput. Phys. 183 (2002) 335-366
30. Iaccarino G., and Verzicco R. Immersed boundary technique for turbulent flow simulations. Appl. Mech. Rev. 56 (2003) 331-347
31. Jin S., and Wang X.L. Robust numerical simulation of porosity evolution in chemical vapor infiltration II. Two-dimensional anisotropic fronts. J. Comput. Phys. 179 (2002) 557-577
32. Johansen H., and Colella P. A Cartesian grid embedding boundary method for Poisson's equation on irregular domains. J. Comput. Phys. 147 (1998) 60-85
33. Kafafy R., Lin T., Lin Y., and Wang J. Three-dimensional immersed finite element methods for electric field simulation in composite materials. Int. J. Numer. Methods Eng. 64 (2005) 940-972
34. Kandilarov J.D. Immersed interface method for a reaction-diffusion equation with a moving own concentrated source. Lecture Notes Comput. Sci. 2542 (2003) 506-513
35. Lai M.C., and Peskin C.S. An immersed boundary method with formal second-order accuracy and reduced numerical viscosity. J. Comput. Phys. 160 (2000) 705-719
36. Lee L., and LeVeque R.J. An immersed interface method for incompressible Navier-Stokes equations. SIAM J. Sci. Comput. 25 (2003) 832-856
37. LeVeque R.J., and Li Z.L. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31 (1994) 1019-1044
38. Li Z.L. A fast iterative algorithm for elliptic interface problems. SIAM J. Numer. Anal. 35 (1998) 230-254
39. Li Z.L., and Ito K. Maximum principle preserving schemes for interface problems with discontinuous coefficients. SIAM J. Sci. Comput. 23 (2001) 339-361
40. Li Z.L., Lin T., and Wu X.H. New Cartesian grid methods for interface problems using the finite element formulation. Numer. Math. 96 (2003) 61-98
41. Li Z.L., and Lubkin S.R. Numerical analysis of interfacial two-dimensional Stokes flow with discontinuous viscosity and variable surface tension. Int. J. Numer. Methods Fluid 37 (2001) 525-540
42. Li Z.L., Wang W.-C., Chern I.-L., and Lai M.-C. New formulations for interface problems in polar coordinates. SIAM J. Sci. Comput. 25 (2003) 224-245
43. Linnick M.N., and Fasel H.F. A high-order immersed interface method for simulating unsteady incompressible flows on irregular domains. J. Comput. Phys. 204 (2005) 157-192
44. Liu W.K., Liu Y., Farrell D., Zhang L., Wang X., Fukui Y., Patankar N., Zhang Y., Bajaj C., Chen X., and Hsu H. Immersed finite element method and its applications to biological systems. Comput. Methods Appl. Mech. Eng. 195 (2006) 1722-1749
45. Liu X.D., Fedkiw R.P., and Kang M. A boundary condition capturing method for Poisson's equation on irregular domains. J. Comput. Phys. 160 (2000) 151-178
46. Lombard B., and Piraux J. How to incorporate the spring-mass conditions in finite-difference schemes. SIAM J. Sci. Comput. 24 (2003) 1379-1407
47. Macak E.B., Munz W.D., and Rodenburg J.M. Plasma-surface interaction at sharp edges and corners during ion-assisted physical vapor deposition. Part I: Edge-related effects and their influence on coating morphology and composition. J. Appl. Phys. 94 (2003) 2829-2836
48. MacKerell Jr. A.D., Bashford D., Bellott M., Dunbrack J.D., Evanseck M.J., Field M.J., Fischer S., Gao J., Guo H., Ha S., Joseph-McCarthy D., Kuchnir L., Kuczera K., Lau F.T.K., Mattos C., Michnick S., Ngo T., Nguyen D.T., Prodhom B., Reiher W.E., Roux B., Schlenkrich M., Smith J.C., Stote R., Straub J., Watanabe M., Wiorkiewicz-Kuczera J., Yin D., and Karplus M. All-atom empirical potential for molecular modeling and dynamics studies of proteins. J. Phys. Chem. 102 (1998) 3586-3616
49. Mayo A. The fast solution of Poisson's and the biharmonic equations on irregular regions. SIAM J. Numer. Anal. 21 (1984) 285-299
50. Mayo A., and Greenbaum A. Fourth order accurate evaluation of integrals in potential theory on exterior 3D regions. J. Comput. Phys. 220 (2007) 900-914
51. McKenney A., Greengard L., and Mayo A. A fast Poisson solver for complex geometries. J. Comput. Phys. 118 (1995) 348-355
52. Miniowitz R., and Webb J.P. Covariant-projection quadrilateral elements for the analysis of wave-guides with sharp edges. IEEE Trans. Microwave Theory Techn. 39 (1991) 501-505
53. Mittal R., and Iaccarino G. Immersed boundary methods. Annu. Rev. Fluid Mech. 37 (2005) 236-261
54. Oevermann M., and Klein R. A Cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces. J. Comput. Phys. 219 (2006) 749-769
55. Pan G., Tong M., and Gilbert B. Multiwavelet based moment method under discrete Sobolev-type norm. Microwave Opt. Techn. Lett. 40 (2004) 47-50
56. Pantic-Tanner Z., Savage J.Z., Tanner D.R., and Peterson A.F. Two-dimensional singular vector elements for finite-element analysis. IEEE Trans. Microwave Theory Techn. 46 (1998) 178-184
57. Peskin C.S. Numerical analysis of blood flow in heart. J. Comput. Phys. 25 (1977) 220-252
58. Peskin C.S. Lectures on mathematical aspects of physiology. Lectures Appl. Math. 19 (1981) 69-107
59. Peskin C.S., and McQueen D.M. A 3-dimensional computational method for blood-flow in the heart. 1. Immersed elastic fibers in a viscous incompressible fluid. J. Comput. Phys. 81 (1989) 372-405
60. Sanner M.F., Olson A.J., and Spehner J.C. Reduced surface: an efficient way to compute molecular surfaces. Biopolymers 38 (1996) 305-320
61. Schulz M., and Steinebach G. Two-dimensional modelling of the river Rhine. J. Comput. Appl. Math. 145 (2002) 11-20
62. Sethian J.A., and Wiegmann A. Structural boundary design via level set and immersed interface methods. J. Comput. Phys. 163 (2000) 489-528
63. Tornberg A.K., and Engquist B. Numerical approximations of singular source terms in differential equations. J. Comput. Phys. 200 (2004) 462-488
64. van Rens B.J.E., Brekelmans W.A.M., and Baaijens F.P.T. Modelling friction near sharp edges using a Eulerian reference frame: application to aluminum extrusion. Int. J. Numer. Methods Engng. 54 (2002) 453-471
65. Voorde J.V., Vierendeels J., and Dick E. Flow simulations in rotary volumetric pumps and compressors with the fictitious domain method. J. Comput. Appl. Math. 168 (2004) 491-499
66. Walther J.H., and Morgenthal G. An immersed interface method for the vortex-in-cell algorithm. J. Turbul. 3 (2002) Art. No. 039
67. Wang Z.J., and Srinivasan K. An adaptive Cartesian grid generation method for 'dirty' geometry. Int. J. Numer. Methods Fluids 39 (2002) 703-717
68. Wiegmann A., and Bube K.P. The explicit-jump immersed interface method: finite difference methods for PDEs with piecewise smooth solutions. SIAM J. Numer. Anal. 37 (2000) 827-862
69. Yu S.N., Zhou Y.C., and Wei G.W. Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces. J. Comput. Phys. 224 (2007) 729-756
70. Yu S.N., Geng W.H., and Wei G.W. Treatment of geometric singularities in the implicit solvent models. J. Chem. Phys. 126 (2007) 244108
71. Zhao S., and Wei G.W. High order FDTD methods via derivative matching for Maxwell's equations with material interfaces. J. Comput. Phys. 200 (2004) 60-103
72. Zhao S., Wei G.W., and Xiang Y. DSC analysis of free-edged beams by an iteratively matched boundary method. J. Sound Vib. 284 (2005) 487-493
73. Zhou Y.C., Zhao S., Feig M., and Wei G.W. High order matched interface and boundary (MIB) schemes for elliptic equations with discontinuous coefficients and singular sources. J. Comput. Phys. 213 (2006) 1-30
74. Zhou Y.C., and Wei G.W. On the fictitious-domain and interpolation formulations of the matched interface and boundary (MIB) method. J. Comput. Phys. 219 (2006) 228-246
75. Y.C. Zhou, M. Feig, G.W. Wei, Highly accurate biomolecular electrostatics in continuum dielectric environments, J. Comput. Chem., in press.