On the foundations of the stochastic immersed boundary method

Computer Methods in Applied Mechanics and Engineering

Volumn 197, Issue 25-28, 2008, Pages 2232-2249

  Kramer, Peter R ^a   Peskin, Charles S ^b   Atzberger, Paul J ^c  

^a RENSSELAER POLYTECHNIC INSTITUTE   (United States)

^b NEW YORK UNIVERSITY   (United States)

^c UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Fluctuation dissipation theorem; Langevin; Non equilibrium; Thermal fluctuations

Indexed keywords

BODY FLUIDS; BOUNDARY ELEMENT METHOD; COMPUTER SIMULATION; DEGREES OF FREEDOM (MECHANICS); THERMAL GRADIENTS;

IMMERSED BOUNDARY METHOD; LANGEVIN-TYPE DYNAMICS; THERMAL FORCING;

FLUID STRUCTURE INTERACTION;

BODY FLUIDS; BOUNDARY ELEMENT METHOD; COMPUTER SIMULATION; DEGREES OF FREEDOM (MECHANICS); FLUID STRUCTURE INTERACTION; THERMAL GRADIENTS;

EID: 41849147912     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2007.11.010     Document Type: Article

References (72)

1. Peskin C.S. The immersed boundary method. Acta Numer. 11 (2002) 479-517
2. P.R. Kramer, C.S. Peskin, Incorporating thermal fluctuations into the immersed boundary method, in: K.J. Bathe (Ed.), Computational Fluid and Solid Mechanics 2003, vol. 2, Elsevier Science Ltd., Oxford, UK, 2003, pp. 1755-1758 (Proceedings of the Second MIT Conference on Computational Fluid and Solid Mechanics, June 17-20, 2003).
3. Kramer P.R., and Majda A.J. Stochastic mode reduction for the immersed boundary method. SIAM J. Appl. Math. 64 2 (2003/2004) 369-400 (electronic)
4. Atzberger P.J., Kramer P.R., and Peskin C.S. A stochastic immersed boundary method for fluid-structure dynamics at microscopic length scales. J. Comput. Phys. 224 2 (2007) 1255-1292
5. Atzberger P.J. A note on the correspondence of an immersed boundary method incorporating thermal fluctuations with Stokesian-Brownian dynamics. Physica D 226 2 (2007) 144-150 10.1016/j.physd.2006.11.013
6. J.F. Brady, G. Bossis, Stokesian dynamics, in: Annual review of fluid mechanics, vol. 20 of Annual Reviews in Fluid Mechanics, Annual Reviews, Palo Alto, CA, 1988, pp. 111-157.
7. Sierou A., and Brady J.F. Accelerated Stokesian dynamics simulations. J. Fluid Mech. 448 (2001) 115-146
8. Banchio A.J., and Brady J.F. Accelerated Stokesian dynamics: Brownian motion. J. Chem. Phys. 118 22 (2003) 10323-10332
9. Ermak D.L., and McCammon J.A. Brownian dynamics with hydrodynamic interactions. J. Chem. Phys. 69 4 (1978) 1352-1360
10. Schlick T. Molecular modeling and simulation: an interdisciplinary guide. Interdisciplinary Applied Mathematics vol. 21 (2002), Springer-Verlag, Berlin
11. Avalos J.B., and Mackie A.D. Dissipative particle dynamics with energy conservation. Europhys. Lett. 40 2 (1997) 141-146
12. Español P. Dissipative particle dynamics with energy conservation. Europhys. Lett. 40 6 (1997) 631-636
13. Español P., and Warren P. Statistical mechanics of dissipative particle dynamics. Europhys. Lett. 30 4 (1995) 191-196
14. Hoogerbrugge P.J., and Koelman J.M.V.A. Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics. Europhys. Lett. 19 3 (1992) 155-160
15. Marsh C.A., and Yeomans J.M. Dissipative particle dynamics: the equilibrium for finite time steps. Europhys. Lett. 37 8 (1997) 511-516
16. Pagonabarraga I., Hagen M.H.J., and Frenkel D. Self-consistent dissipative particle dynamics algorithm. Europhys. Lett. 42 4 (1998) 377-382
17. Groot R.D., and Warren P.B. Dissipative particle dynamics: bridging the gap between atomistic and mesoscopic simulation. J. Chem. Phys. 107 11 (1997) 4423-4435
18. Den Otter W.K., and Clarke J.H.R. A new algorithm for dissipative particle dynamics. Europhys. Lett. 53 4 (2001) 426-431
19. Nikunen P., Karttunen M., and Vattulainen I. How would you integrate the equations of motion in dissipative particle dynamics simulations?. Comput. Phys. Commun. 153 3 (2003) 407-423
20. Ladd A.J.C. Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 1. Theoretical foundation. J. Fluid Mech. 271 (1994) 285-309
21. Ladd A.J.C., and Verberg R. Lattice-Boltzmann simulations of particle-fluid suspensions. J. Stat. Phys. 104 5-6 (2001) 1191-1251
22. Inoue Y., Chen Y., and Ohashi H. Development of a simulation model for solid objects suspended in a fluctuating fluid. J. Stat. Phys. 107 1/2 (2002) 85-100
23. Sharma N., and Patankar N.A. Direct numerical simulation of the Brownian motion of particles by using fluctuating hydrodynamic equations. J. Comput. Phys. 201 (2004) 466-486
24. Öttinger H.C., and Rabin Y. Diffusion equation versus coupled Langevin equations approach to hydrodynamics of dilute polymer solutions. J. Rheology 33 5 (1989) 725-743
25. P.J. Atzberger, P.R. Kramer, Error analysis of a stochastic immersed boundary method incorporating thermal fluctuations, Math. Comput. Simulat., submitted for publication.
26. Kubo R., Toda M., and Hashitsume N. Statistical Physics II. second ed. (1991), Springer-Verlag, Berlin (Section 4, nonequilibrium statistical mechanics)
27. Kubo R. The fluctuation-dissipation theorem. Rep. Prog. Phys. 29 (1966) 255-284
28. Fox R.F., and Uhlenbeck G.E. Contributions to non-equilibrium thermodynamics. I. Theory of hydrodynamical fluctuations. Phys. Fluids 13 8 (1970) 1893-1902
29. Pathria R.K. Statistical Mechanics (1972), Pergamon Press, Oxford (Section 13.4)
30. Nelson E. Dynamical Theories of Brownian Motion (1967), Princeton University Press, Princeton, NJ (Chapter 9)
31. Kramer P. Brownian motion. In: Scott A. (Ed). Encyclopedia of Nonlinear Science (2005), Routledge & Kegan Paul, London and New York 78-81
32. Reichl L.E. A Modern Course in Statistical Physics. second ed. (1998), John Wiley & Sons Inc., New York (Chapters S4.3.1, 5.E, 7)
33. Oster G., and Peskin C.S. Dynamics of osmotic fluid flow. In: Karalis T. (Ed). Swelling Mechanics: From Clays to Living Cells and Tissues (1992), Springer-Verlag, Berlin 731-742
34. Brady J.F. Brownian motion, hydrodynamics, and the osmotic pressure. J. Chem. Phys. 98 4 (1993) 3335-3341
35. Peskin C.S., and McQueen D.M. A general method for the computer simulation of biological systems interacting with fluids. In: Ellington C.P., and Pedley T.J. (Eds). Biological Fluid Dynamics (1995), The Company of Biologists Limited, Cambridge, UK 265-276
36. J.M. Deutch, I. Oppenheim, The concept of Brownian motion in modern statistical mechanics, in: Brownian Motion, vol. 83 of Faraday Discussions of Chemical Society, The Faraday Division of the Royal Society of Chemistry, The Royal Society of Chemistry, London, 1987, pp. 1-20.
37. Glowinski R., Pan T.W., Hesla T.I., Joseph D.D., and Périaux J. A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow. J. Comput. Phys. 169 2 (2001) 363-426
38. Patankar N.A., Singh P., Joseph D.D., Glowinski R., and Pan T.-W. A new formulation of the distributed Lagrange multiplier/fictitious domain method for particulate flows. Int. J. Multiphase Flow 26 (2000) 1509-1524
39. Karrila S.J., Fuentes Y.O., and Kim S. Parallel computational strategies for hydrodynamic interactions between rigid particles of arbitrary shape in a viscous fluid. J. Rheol. 33 6 (1989) 913-947
40. Tran-Cong T., and Phan-Thien N. Stokes problems of multiparticle systems: a numerical method for arbitrary flows. Phys. Fluids A 1 3 (1989) 453-461
41. Atzberger P.J. Velocity correlations of a thermally fluctuating Brownian particle: a novel model of the hydrodynamic coupling. Phys. Lett. A 351 4-5 (2006) 225-230
42. Español P. Stochastic differential equations for non-linear hydrodynamics. Physica A 248 1-2 (1998) 77-96
43. Majda A.J., and Kramer P.R. Simplified models for turbulent diffusion: theory, numerical modelling, and physical phenomena. Phys. Rep. 314 4-5 (1999) 237-574
44. Yaglom A.M. Correlation theory of stationary and related random functions. Basic Results vol. I (1987), Springer-Verlag, Berlin
45. B. Øksendal, Stochastic Differential Equations, An Introduction with Applications, fifth ed., Universitext, Springer-Verlag, Berlin, 1998 (Chapter 4).
46. V. I. Arnold, Mathematical methods of classical mechanics, second ed., vol. 60 in Graduate Texts in Mathematics, Springer-Verlag, New York, 1989 (Sections 13 and 15).
47. Beard D.A., and Schlick T. Computational modeling predicts the structure and dynamics of chromatin fiber. Structure 9 2 (2001) 105-114
48. Tritton D.J. Physical fluid dynamics. second ed. (1988), Clarendon Press, Oxford (Chapter 5.8)
49. Landau L.D., and Lifshitz E.M. Fluid mechanics. Course of Theoretical Physics. second ed. vol. 6 (1987), Pergamon Press, Oxford (Sections 10, 20, 50)
50. Batchelor G.K. An introduction to fluid dynamics (1967), Cambridge University Press, Cambridge, UK (Appendix 1)
51. Hairer E., and Lubich C. Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J. Numer. Anal. 38 2 (2000) 414-441
52. García-Archilla B., Sanz-Serna J.M., and Skeel R.D. Long-time-step methods for oscillatory differential equations. SIAM J. Sci. Comput. 20 3 (1999) 930-963 (electronic)
53. Kassam A.-K., and Trefethen L.N. Fourth-order time-stepping for stiff PDEs. SIAM J. Sci. Comput. 26 4 (2005) 1214-1233 (electronic)
54. DelSole T. A theory for the forcing and dissipation in stochastic turbulence models. J. Atmos. Sci. 58 (2001) 3762-3775
55. Farrell B.F., and Ioannou P.J. Structural stability of turbulent jets. J. Atmos. Sci. 60 (2003) 2101-2118
56. Majda A.J., Timofeyev I., and Vanden Eijnden E. A mathematical framework for stochastic climate models. Commun. Pure Appl. Math. 54 8 (2001) 891-974
57. Majda A., Timofeyev I., and Vanden-Eijnden E. A priori tests of a stochastic mode reduction strategy. Physica D 170 3-4 (2002) 206-252
58. Majda A., Timofeyev I., and Vanden-Eijnden E. Systematic strategies for stochastic mode reduction in climate. J. Atmos. Sci. 60 14 (2003) 1705-1722
59. Majda A., Timofeyev I., and Vanden-Eijnden E. Stochastic models for selected slow variables in large deterministic systems. Nonlinearity 19 4 (2006) 769-794
60. Franzke C., Majda A.J., and Vanden-Eijnden E. Low-order stochastic mode reduction for a realistic barotropic model climate. J. Atmosph. Sci. 62 6 (2005) 1722-1745
61. Bocquet L., and Piasecki J. Microscopic derivation of non-Markovian thermalization of a Brownian particle. J. Stat. Phys. 87 5-6 (1997) 1005-1035
62. Hauge E.H., and Martin-Löf A. Fluctuating hydrodynamics and Brownian motion. J. Stat. Phys. 7 (1973) 259-281
63. Masters A.J. Time-scale separations and the validity of the Smoluchowski, Fokker-Planck and Langevin equations as applied to concentrated particle suspensions. Mol. Phys. 57 2 (1986) 303-317
64. Bocquet L. From a stochastic to a microscopic approach to Brownian motion. Acta Phys. Polon. B 29 6 (1998) 1551-1564
65. Concentrated Colloidal Dispersions, Faraday Discussions of the Chemical Society, The Faraday Division of the Royal Society of Chemistry, vol. 76, London, 1983, p. 234.
66. Roux J.-N. Brownian particles at different times scales: a new derivation of the Smoluchowski equation. Physica A 188 4 (1992) 526-552
67. P.L. Auer (Ed.), Macromolecules, John Gamble Kirkwood Collected Works, Gordon and Breach Science Publishers, New York, 1967.
68. Ahlrichs P., and Dünweg B. Simulation of a single polymer chain in solution by combining lattice Boltzmann and molecular dynamics. J. Chem. Phys. 111 17 (1999) 8225-8239
69. R. Kupferman, Osmotic Pressure in Confined Polymer Solutions, notes, October 4, 1996.
70. Lee L., and LeVeque R.J. An immersed interface method for incompressible Navier-Stokes equations. SIAM J. Sci. Comput. 25 (2003) 832-856. http://epubs.siam.org/sam-bin/dbq/article/41406
71. Zhang L., Gerstenberger A., Wang X., and Liu W.K. Immersed finite element method. Comput. Methods Appl. Mech. Engrg. 193 21-22 (2004) 2051-2067
72. Liu W.K., Liu Y., Farrell D., Zhang L., Wang X.S., Fukui Y., Patankar N., Zhang Y., Bajaj C., Lee J., Hong J., Chen X., and Hsu H. Immersed finite element method and its applications to biological systems. Comput. Methods Appl. Mech. Engrg. 195 13-16 (2006) 1722-1749