Abstract
The immersed boundary (IB) method is an approach to fluid-structure interaction that uses Lagrangian variables to describe the deformations and resulting forces of the structure and Eulerian variables to describe the motion and forces of the fluid. Explicit time stepping schemes for the IB method require solvers only for Eulerian equations, for which fast Cartesian grid solution methods are available. Such methods are relatively straightforward to develop and are widely used in practice but often require very small time steps to maintain stability. Implicit-time IB methods permit the stable use of large time steps, but efficient implementations of such methods require significantly more complex solvers that effectively treat both Lagrangian and Eulerian variables simultaneously. Several different approaches to solving the coupled Lagrangian-Eulerian equations have been proposed, but a complete understanding of this problem is still emerging. This paper presents a geometric multigrid method for an implicit-time discretization of the IB equations. This multigrid scheme uses a generalization of box relaxation that is shown to handle problems in which the physical stiffness of the structure is very large. Numerical examples are provided to illustrate the effectiveness and efficiency of the algorithms described herein. These tests show that using multigrid as a preconditioner for a Krylov method yields improvements in both robustness and efficiency as compared to using multigrid as a solver. They also demonstrate that with a time step 100–1000 times larger than that permitted by an explicit IB method, the multigrid-preconditioned implicit IB method is approximately 50–200 times more efficient than the explicit method.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Boffi, D., Gastaldi, L., Heltai, L., Peskin, C.S.: On the hyper-elastic formulation of the immersed boundary method. Comput. Methods Appl. Mech. Engrg 197(25-28), 2210–2231 (2008)
Brandt, A., Dinar, N.: Multi-grid solutions to elliptic flow problems. In: Institute for Computer Applications in Science and Engineering. NASA Langley Research Center, Hampton Va (1979)
Briggs, W., Henson, V., McCormick, S.: A Multigrid Tutorial, 2nd edn. SIAM, Philadelphia (2000). http://lccn.loc.gov/00024103
Ceniceros, H.D., Fisher, J.E.: A fast, robust, and non-stiff immersed boundary method. J. Comput. Phys. 230(12), 5133–5153 (2011)
Ceniceros, H.D., Fisher, J.E., Roma, A.M.: Efficient solutions to robust, semi-implicit discretizations of the immersed boundary method. J. Comput. Phys. 228(19), 7137–7158 (2009)
Elman, H.C.: Multigrid and krylov subspace methods for the discrete stokes equations. Internat. J. Numer. Methods Fluids 22(8), 755–770 (1996)
Fai, T., Griffith, B., Mori, Y., Peskin, C.: Immersed boundary method for variable viscosity and variable density problems using fast constant-coefficient linear solvers I: Numerical method and results. SIAM J. Sci. Comput. 35(5), B1132–B1161 (2013)
Gong, Z., Huang, H., Lu, C.: Stability analysis of the immersed boundary method for a two-dimensional membrane with bending rigidity. Commun. Comput. Phys. 3, 704–723 (2008)
Griffith, B.E.: IBAMR: An adaptive and distributed-memory parallel implementation of the immersed boundary method. http://ibamr.googlecode.com
Griffith, B.E.: On the volume conservation of the immersed boundary method. Commun. Comput. Phys. 12, 401–432 (2012)
Griffith, B.E., Hornung, R.D., McQueen, D.M., Peskin, C.S.: An adaptive, formally second order accurate version of the immersed boundary method. J. Comput. Phys. 223(1), 10–49 (2007)
Griffith, B.E., 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(1), 75–105 (2005)
Guy, R.D., Philip, B.: A multigrid method for a model of the implicit immersed boundary equations. Commun. Comput. Phys. 12, 378–400 (2012)
Hou, T.Y., Shi, Z.: An efficient semi-implicit immersed boundary method for the navier-stokes equations. J. Comput. Phys. 227(20), 8968–8991 (2008)
Hou, T.Y., Shi, Z.: Removing the stiffness of elastic force from the immersed boundary method for the 2d stokes equations. J. Comput. Phys. 227(21), 9138–9169 (2008). Special Issue Celebrating Tony Leonard’s 70th Birthday
Huang, W.X., Sung, H.J.: An immersed boundary method for fluidflexible structure interaction. Comput. Methods Appl. Mech. Eng. 198(3336), 2650–2661 (2009)
Le, D., White, J., Peraire, J., Lim, K., Khoo, B.: An implicit immersed boundary method for three-dimensional fluid-membrane interactions. J. Comput. Phys. 228(22), 8427–8445 (2009)
Lee, L., LeVeque, R.J.: An immersed interface method for incompressible navier–stokes equations. SIAM J. Sci. Comput. 25(3), 832–856 (2003)
Linden, J., Lonsdale, G., Steckel, B., Stben, K.: Multigrid for the steady-state incompressible navier-stokes equations: A survey. In: Dwoyer, D., Hussaini, M., Voigt, R. (eds.) 11th International Conference on Numerical Methods in Fluid Dynamics, Lecture Notes in Physics, vol. 323, pp. 57–68. Springer Berlin, Heidelberg (1989)
Mayo, A.A., Peskin, C.S.: An implicit numerical method for fluid dynamics problems with immersed elastic boundaries. In: Fluid dynamics in biology (Seattle, WA, 1991), Contemp. Math. 141, 261–277 RI, Amer. Math. Soc., Providence (1993)
Mittal, R., Iaccarino, G.: Immersed boundary methods. Annu. Rev. Fluid Mech. 37, 239–261 (2005)
Mori, Y., Peskin, C.S.: Implicit second-order immersed boundary methods with boundary mass. Comput. Methods Appl. Mech. Eng. 197(25-28), 2049–2067 (2008). Immersed Boundary Method and Its Extensions
Newren, E.P., Fogelson, A.L., Guy, R.D., Kirby, R.M.: Unconditionally stable discretizations of the immersed boundary equations. J. Comput. Phys. 222(2), 702–719 (2007)
Newren, E.P., Fogelson, A.L., Guy, R.D., Kirby, R.M.: A comparison of implicit solvers for the immersed boundary equations. Comput. Methods Appl. Mech. Eng. 197(25-28), 2290–2304 (2008). Immersed Boundary Method and Its Extensions
Niestegge, A., Witsch, K.: Analysis of a multigrid strokes solver. Appl. Math. Comput. 35(3), 291–303 (1990)
Oosterlee, C.W., Gaspar, F.J.: Multigrid relaxation methods for systems of saddle point type. Appl. Numer. Math 58(12), 1933–1950 (2008)
Oosterlee, C.W., Washio, T.: An evaluation of parallel multigrid as a solver and a preconditioner for singularly perturbed problems. SIAM. J. Sci. Comput. 19, 87–110 (1998)
Peskin, C.S.: Numerical analysis of blood flow in the heart. J. Comput. Phys. 25(3), 220–252 (1977)
Peskin, C.S.: The immersed boundary method. Acta Numer. 11(-1), 479–517 (2002)
Saad, Y., Schultz, M.H.: Gmres: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM. J. Sci. Stat. Comput. 7(3), 856–869 (1986)
Stockie, J.M., Wetton, B.R.: Analysis of stiffness in the immersed boundary method and implications for time-stepping schemes. J. Comput. Phys. 154(1), 41–64 (1999)
Strychalski, W., Guy, R.D.: Viscoelastic immersed boundary methods for zero reynolds number flow. Commun. Comput. Phys. 12, 462–478 (2012)
Sussman, M., Almgren, A.S., Bell, J.B., Colella, P., Howell, L.H., Welcome, M.L.: An adaptive level set approach for incompressible two-phase flows. J. Comput. Phys. 148(1), 81–124 (1999)
Tatebe, O.: The multigrid preconditioned conjugate gradient method. In: Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods, pp. 621–634 (1993)
Trottenberg, U., Oosterlee, C.W., Schüller, A.: Multigrid. Academic Press, San Diego (2001). http://lccn.loc.gov/00103940
Tu, C., Peskin, C.S.: Stability and instability in the computation of flows with moving immersed boundaries: A comparison of three methods. SIAM. J. Sci. Stat. Comput. 13(6), 1361–1376 (1992)
Vanka, S.P.: Block-implicit multigrid solution of navier-stokes equations in primitive variables. J. Comput. Phys. 65(1), 138 (1986)
Wittum, G.: On the convergence of multi-grid methods with transforming smoothers. Numer. Math 57(1), 15 (1990)
Wright, G.B., Guy, R.D., Fogelson, A.L.: An efficient and robust method for simulating two-phase gel dynamics. SIAM J. Sci. Comput. 30, 2535–2565 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Guy, R.D., Philip, B. & Griffith, B.E. Geometric multigrid for an implicit-time immersed boundary method. Adv Comput Math 41, 635–662 (2015). https://doi.org/10.1007/s10444-014-9380-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-014-9380-1
Keywords
- Fluid-structure interaction
- Immersed boundary method
- Krylov methods
- Multigrid solvers
- Multigrid preconditioners