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On the Volume Conservation of the Immersed Boundary Method

Published online by Cambridge University Press:  20 August 2015

Boyce E. Griffith
Boyce E. Griffith*
Affiliation:
Leon H. Charney Division of Cardiology, New York University School of Medicine, 550 First Avenue, New York, New York 10016, USA
*
*Corresponding author.Email:[email protected]
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Abstract

The immersed boundary (IB) method is an approach to problems of fluid-structure interaction in which an elastic structure is immersed in a viscous incompressible fluid. The IB formulation of such problems uses a Lagrangian description of the structure and an Eulerian description of the fluid. It is well known that some versions of the IB method can suffer from poor volume conservation. Methods have been introduced to improve the volume-conservation properties of the IB method, but they either have been fairly specialized, or have used complex, nonstandard Eulerian finite-difference discretizations. In this paper, we use quasi-static and dynamic benchmark problems to investigate the effect of the choice of Eulerian discretization on the volume-conservation properties of a formally second-order accurate IB method. We consider both collocated and staggered-grid discretization methods. For the tests considered herein, the staggered-grid IB scheme generally yields at least a modest improvement in volume conservation when compared to cell-centered methods, and in many cases considered in this work, the spurious volume changes exhibited by the staggered-grid IB method are more than an order of magnitude smaller than those of the collocated schemes. We also compare the performance of cell-centered schemes that use either exact or approximate projection methods. We find that the volume-conservation properties of approximate projection IB methods depend strongly on the formulation of the projection method. When used with the IB method, we find that pressure-free approximate projection methods can yield extremely poor volume conservation, whereas pressure-increment approximate projection methods yield volume conservation that is nearly identical to that of a cell-centered exact projection method.

Keywords

65M0665M2065R1074F1076D05Immersed boundary methodfluid-structure interactioncollocated discretizationstaggered-grid discretizationexact projection methodapproximate projection methodvolume conservation
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 2 , August 2012 , pp. 401 - 432
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Peskin, C. S.. The immersed boundary method. Acta Numer., 11: 479–517, 2002.Google Scholar
[2]Peskin, C. S. and Printz, B. F.. Improved volume conservation in the computation of flows with immersed elastic boundaries. J. Comput. Phys., 105(1): 33–46, 1993.Google Scholar
[3]Cortez, R. and Minion, M. L.. The blob projection method for immersed boundary problems. J. Comput. Phys., 161(2): 428–453, 2000.Google Scholar
[4]Kim, Y. and Lai., M.-C.Simulating the dynamics of inextensible vesicles by the penalty immersed boundary method. J. Comput. Phys., 229(12): 4840–4853, 2010.Google Scholar
[5]Kim, Y., Lai, M.-C., and Peskin, C. S.. Numerical simulations of two-dimensional foam by the immersed boundary method. J. Comput. Phys., 229(13): 5194–5207, 2010.Google Scholar
[6]Shoele, K. and Zhu, Q.. Flow-induced vibrations of a deformable ring. J. Fluid Mech., 650: 343–362, 2010.Google Scholar
[7]Newren, E. P.. Enhancing the immersed boundary method: stability, volume conservation, and implicit solvers. PhD thesis, University of Utah, 2007.Google Scholar
[8]Stockie, J. A.. Modelling and simulation of porous immersed boundaries. Comput. Struct., 87(11–12): 701–709, 2009.Google Scholar
[9]Harlow, F. H. and Welch, J. E.. Numerical calculation of time-dependent viscous incompresible flow of fluid with free surface. Phys. Fluid, 8(12): 2182–2189, 1965.Google Scholar
[10]Rider, W. J., Greenough, J. A., and Kamm, J. R.. Accurate monotonicity- and extrema-preserving methods through adaptive nonlinear hybridizations. J. Comput. Phys., 225(2): 1827–1848, 2007.Google Scholar
[11]Colella, P. and Woodward, P. R.. The piecewise parabolic method (PPM) for gas-dynamical simulations. J. Comput. Phys., 54(1): 174–201, 1984.Google Scholar
[12]Chorin, A. J.. Numerical solution of the Navier-Stokes equations. Math. Comput., 22(104): 745–762, 1968.Google Scholar
[13]Chorin, A. J.. On the convergence of discrete approximations to the Navier-Stokes equations. Math. Comput., 23(106): 341–353, 1969.Google Scholar
[14]Kim, J. and Moin, P.. Application of a fractional-step method to incompressible Navier-Stokes equations. J. Comput. Phys., 59(2): 308–323, 1985.Google Scholar
[15]Bell, J. B., Colella, P., and Glaz, H. M.. A second-order projection method for the incompressible Navier-Stokes equations. J. Comput. Phys., 85(2): 257–283, 1989.Google Scholar
[16]Almgren, A. S., Bell, J. B., and Szymczak, W. G.. A numerical method for the incompressible Navier-Stokes equations based on an approximate projection. SIAM J. Sci. Comput., 17(2): 358–369, 1996.CrossRefGoogle Scholar
[17]Almgren, A. S., Bell, J. B., Colella, P., Howell, L. H., and Welcome, M. L.. A conservative adaptive projection method for the variable density incompressible Navier-Stokes equations. J. Comput. Phys., 142(1): 1–46, 1998.Google Scholar
[18]Almgren, A. S., Bell, J. B., and Crutchfield, W. Y.. Approximate projection methods: Part I. Inviscid analysis. SIAM J. Sci. Comput., 22(4): 1139–1159, 2000.Google Scholar
[19]Martin, D. F. and Colella, P.. A cell-centered adaptive projection method for the incompressible Euler equations. J. Comput. Phys., 163(2): 271–312, 2000.Google Scholar
[20]Brown, D. L., Cortez, R., and Minion, M. L.. Accurate projection methods for the incompressible Navier-Stokes equations. J. Comput. Phys., 168(2): 464–499, 2001.Google Scholar
[21]Martin, D. F., Colella, P., and Graves, D.. A cell-centered adaptive projection method for the incompressible Navier-Stokes equations in three dimensions. J. Comput. Phys., 227(3): 1863–1886, 2008.Google Scholar
[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(1): 75–105, 2005.Google Scholar
[23]Griffith, B. E., Hornung, R. D., McQueen, D. M., and Peskin, C. S.. An adaptive, formally second order accurate version of the immersed boundary method. J. Comput. Phys., 223(1): 10–49, 2007.Google Scholar
[24]Newren, E. P., Fogelson, A. L., Guy, R. D., and Kirby, R. M.. Unconditionally stable discretizations of the immersed boundary equations. J. Comput. Phys., 222(2): 702–719, 2007.Google Scholar
[25]Griffith, B. E., Luo, X., McQueen, D. M., and Peskin., C. S.Simulating the fluid dynamics of natural and prosthetic heart valves using the immersed boundary method. Int. J. Appl. Mech., 1(1): 137–177, 2009.Google Scholar
[26]Griffith, B. E., Hornung, R. D., McQueen, D. M., and Peskin, C. S.. Parallel and adaptive simulation of cardiac fluid dynamics. In Parashar, M. and Li, X., editors, Advanced Computational Infrastructures for Parallel and Distributed Adaptive Applications. John Wiley and Sons, Hoboken, NJ, USA, 2009.Google Scholar
[27]Griffith, B. E.. An accurate and efficient method for the incompressible Navier-Stokes equations using the projection method as a preconditioner. J. Comput. Phys., 228(20): 7565–7595, 2009.Google Scholar
[28]Roma, A. M., Peskin, C. S., and Berger, M. J.. An adaptive version of the immersed boundary method. J. Comput. Phys., 153(2): 509–534, 1999.Google Scholar
[29]Griffith, B. E.. Immersed boundary model of aortic heart valve dynamics with physiological driving and loading conditions. Int. J. Numer. Meth. Biomed. Eng., To appear.Google Scholar
[30] IBAMR: An adaptive and distributed-memory parallel implementation of the immersed boundary method. http://ibamr.googlecode.com.Google Scholar
[31] SAMRAI: Structured Adaptive Mesh Refinement Application Infrastructure. http://www.llnl.gov/CASC/SAMRAI.Google Scholar
[32]Hornung, R. D. and Kohn, S. R.. Managing application complexity in the SAMRAI object-oriented framework. Concurrency Comput. Pract. Ex., 14(5): 347–368, 2002.Google Scholar
[33]Hornung, R. D., Wissink, A. M., and Kohn, S. R.. Managing complex data and geometry in parallel structured AMR applications. Eng. Comput., 22(3–4): 181–195, 2006.Google Scholar
[34]Balay, S., Buschelman, K., Gropp, W. D., Kaushik, D., Knepley, M. G., McInnes, L. C., Smith, B. F., and Zhang, H.. PETSc Web page, 2009. http://www.mcs.anl.gov/petsc.Google Scholar
[35]Balay, S., Buschelman, K., Eijkhout, V., Gropp, W. D., Kaushik, D., Knepley, M. G., McInnes, L. C., Smith, B. F., and Zhang, H.. PETSc users manual. Technical Report ANL-95/11 - Revision 3.0.0, Argonne National Laboratory, 2008.Google Scholar
[36]Balay, S., Eijkhout, V., Gropp, W. D., McInnes, L. C., and Smith, B. F.. Efficient management of parallelism in object oriented numerical software libraries. In Arge, E., Bruaset, A. M., and Langtangen, H. P., editors, Modern Software Tools in Scientific Computing, pages 163–202. Birkhäuser Press, 1997.Google Scholar
[37]hypre: High performance preconditioners. http://www.llnl.gov/CASC/hypre.Google Scholar
[38]Falgout, R. D. and Yang, U. M.. hypre: a library of high performance preconditioners. In Sloot, P. M. A., Tan, C. J. K., Dongarra, J. J., and Hoekstra, A. G., editors, Computational Science - ICCS 2002 Part III, volume 2331 of Lecture Notes in Computer Science, pages 632–641. Springer-Verlag, 2002. Also available as LLNL Technical Report UCRL-JC-146175.Google Scholar
[39]Boffi, D., Gastaldi, L., Heltai, L., and Peskin, C. S.. On the hyper-elastic formulation of the immersed boundary method. Comput. Meth. Appl. Mech. Engrg., 197(25–28): 2210–2231, 2008.CrossRefGoogle Scholar
[40]Griffith, B. E. and Luo, X.. Hybrid finite difference/finite element version of the immersed boundary method. Submitted, preprint available from http://www.cims.nyu.edu/~griffith.Google Scholar
[41]Tóth, G. and Roe, P. L.. Divergence- and curl-preserving prolongation and restriction formulas. J. Comput. Phys., 180(2): 736–750, 2002.Google Scholar