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Simulating an Elastic Ring with Bend and Twist by an Adaptive Generalized Immersed Boundary Method

Published online by Cambridge University Press:  20 August 2015

Boyce E. Griffith  and
Sookkyung Lim
Boyce E. Griffith*
Affiliation:
Leon H. Charney Division of Cardiology, Department of Medicine, New York University School of Medicine, 550 First Avenue, New York, New York 10016, USA
Sookkyung Lim*
Affiliation:
Department of Mathematical Sciences, University of Cincinnati, 839 Old Chemistry Building, Cincinnati, Ohio 45221, USA
*
Email:boyce.griffith@nyumc.org
Corresponding author.Email:limsk@math.uc.edu
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Abstract

Many problems involving the interaction of an elastic structure and a viscous fluid can be solved by the immersed boundary (IB) method. In the IB approach to such problems, the elastic forces generated by the immersed structure are applied to the surrounding fluid, and the motion of the immersed structure is determined by the local motion of the fluid. Recently, the IB method has been extended to treat more general elasticity models that include both positional and rotational degrees of freedom. For such models, force and torque must both be applied to the fluid. The positional degrees of freedom of the immersed structure move according to the local linear velocity of the fluid, whereas the rotational degrees of freedom move according to the local angular velocity. This paper introduces a spatially adaptive, formally second-order accurate version of this generalized immersed boundary method. We use this adaptive scheme to simulate the dynamics of an elastic ring immersed in fluid. To describe the elasticity of the ring, we use an unconstrained version of Kirchhoff rod theory. We demonstrate empirically that our numerical scheme yields essentially second-order convergence rates when applied to such problems. We also study dynamical instabilities of such fluid-structure systems, and we compare numerical results produced by our method to classical analytic results from elastic rod theory.

Keywords

65M0665M5074S2076D05Immersed boundary methodKirchhoff rod theoryadaptive mesh refinement
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 2 , August 2012 , pp. 433 - 461
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Peskin, C. S.. Flow patterns around heart valves: A digital computer method for solving the equations of motion. PhD thesis, Albert Einstein College of Medicine, 1972.Google Scholar
[2]Peskin, C. S.. Numerical analysis of blood flow in the heart. J. Comput. Phys., 25(3):220–252, 1977.Google Scholar
[3]Peskin, C. S.. The immersed boundary method. Acta Numer., 11:479–517, 2002.CrossRefGoogle Scholar
[4]Lim, S., Ferent, A., Wang, X. S., and Peskin, C. S.. Dynamics of a closed rod with twist and bend in fluid. SIAM J. Sci. Comput., 31(1):273–302, 2008.Google Scholar
[5]Lim, S.. Dynamics of an open elastic rod with intrinsic curvature and twist in a viscous fluid. Phys. Fluid, 22(2):024104, 2010.Google Scholar
[6]Lu, C.-L. and Perkins, N. C.. Nonlinear spatial equilibria and stability of cables under uniaxial torque and thrust. J. Appl. Mech., 61(4):879–886, 1994.Google Scholar
[7]Goyal, S., Perkins, N. C., and Lee, C. L.. Nonlinear dynamics and loop formation in Kirchhoff rods with implications to the mechanics of DNA and cables. J. Comput. Phys., 209(1):371–389, 2005.CrossRefGoogle Scholar
[8]Goyal, S., Lillian, T., Blumberg, S., Meiners, J.-C., Meyhöfer, E., and Perkins, N. C.. Intrinsic curvature of DNA influences LacR-mediated looping. Biophys. J., 93(12):4342–4359, 2007.Google Scholar
[9]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
[10]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
[11]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
[12]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
[13]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
[14]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
[15]Griffith, B. E.. On the volume conservation of the immersed boundary method. Commun. Comput. Phys. In press.Google Scholar
[16]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
[17]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.Google Scholar
[18]Coleman, B. D. and Swigon, D.. Theory of supercoiled elastic rings with self contact and its application to DNA plasmids. J. Elasticity, 60(3):171–221, 2000.Google Scholar
[19]Coleman, B. D., Swigon, D., and Tobias, I.. Elastic stability of DNA configurations: II. Super-coiled plasmids with self contact. Phys. Rev. E, 61(1):759–770, 2000.Google Scholar
[20]Biton, Y. Y., Coleman, B. D., and Swigon, D.. On bifurcations of equilibria of intrinsically curved, electrically charged, rod-like structures that model DNA molecules in solution. J. Elasticity, 87(2–3):187–210, 2007.Google Scholar
[21]Goriely, A.. Twisted elastic rings and the rediscoveries of Michell’s instability. J. Elasticity, 84(3):281–299, 2006.Google Scholar
[22]Michell, J. H.. On the stability of a bent and twisted wire. Messeng. Math., 19:181–184, 1890.Google Scholar
[23]Tobias, I. and Olson, W. K.. The effect of intrinsic curvature on supercoiling: predictions of elasticity theory. Biopolymers, 33(4):639–646, 1993.Google Scholar
[24]Tobias, I., Coleman, B. D., and Lembo, M.. A class of exact dynamical solutions in the elastic rod model of DNA with implications for the theory of fluctuations in the torsional motion of plasmids. J. Chem. Phys., 105(6):2517–2526, 1996.Google Scholar
[25]Qian, H. and White, J. H.. Terminal twist induced continuous writhe of a circular rod with intrinsic curvature. J. Biomol. Struct. Dyn., 16(3):663–669, 1998.Google Scholar
[26]Haijun, Z. and Zhong-can, O.-Y.. Spontaneous curvature-induced dynamical instability of Kirchhoff filaments: application to DNA kink deformations. J. Chem. Phys., 110(2):1247–1251, 1999.Google Scholar
[27]Manning, R. S. and Hoffman, K. A.. Stability of n-covered circles for elastic rods with constant planar intrinsic curvature. J. Elasticity, 62(1):1–23, 2001.Google Scholar
[28]Laundon, C. H. and Griffith, J. D.. Curved helix segments can uniquely orient the topology of supertwisted DNA. Cell, 52(4):545–549,1988.Google Scholar
[29]Bauer, W. R., Lund, R. A., and White, J. H.. Twist and writhe of a DNA loop containing intrinsic bends. Proc. Natl. Acad. Sci. USA, 90(3):833–837, 1993.Google Scholar
[30]Bringley, T. T. and Peskin, C. S.. Validation of a simple method for representing spheres and slender bodies in an immersed boundary method for Stokes flow on an unbounded domain. J. Comput. Phys., 227(11):5397–5425, 2008.Google Scholar
[31]Antman, S. S.. Nonlinear Problems of Elasticity. Springer-Verlag, 1995.Google Scholar
[32]Berger, M. J. and Colella, P.. Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys., 82(1):64–84, 1989.Google Scholar
[33]Minion, M. L.. A projection method for locally refined grids. J. Comput. Phys., 127(1):158–178, 1996.Google Scholar
[34]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.CrossRefGoogle Scholar
[35]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
[36]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.CrossRefGoogle Scholar
[37]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
[38]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
[39]Saad, Y.. A flexible inner-outer preconditioned GMRES algorithm. SIAM J. Sci. Comput., 14(2):461–469, 1993.Google Scholar
[40]Berger, M. J. and Rigoutsos, I.. An algorithm for point clustering and grid generation. IEEE Trans. Syst. Man. Cybern., 21(5):1278–1286, 1991.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
[42] IBAMR: An adaptive and distributed-memory parallel implementation of the immersed boundary method. http://ibamr.googlecode.com.Google Scholar
[43] SAMRAI: Structured Adaptive Mesh Refinement Application Infrastructure. http://www.llnl.gov/CASC/SAMRAI.Google Scholar
[44]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
[45]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
[46]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
[47]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
[48]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
[49]McCormick, S. F.. Multilevel Adaptive Methods for Partial Differential Equations. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1989.Google Scholar
[50]McCormick, S. F., McKay, S. M., and Thomas, J. W.. Computational complexity of the fast adaptive composite grid (FAC) method. Applied Numerical Mathematics, 6(3):315–327, 1989.Google Scholar
[51]McCormick, S. F.. Multilevel Projection Methods for Partial Differential Equations. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1992.Google Scholar
[52]hypre: High performance preconditioners. http://www.llnl.gov/CASC/hypre.Google Scholar
[53]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
[54]Goriely, A. and Tabor, M.. The nonlinear dynamics of filaments. Nonlinear Dynam., 21(1):101–133, 2000.Google Scholar
[55]Chouaieb, N., Goriely, A., and Maddocks, J. H.. Helices. Proc. Natl. Acad. Sci. USA, 103(25):9398–9403, 2006.Google Scholar
[56]Le Bret, M.. Twist and writhing in short circular DNAs according to first-order elasticity. Biopolymers, 23(10):1835–1867, 1984.Google Scholar
[57]Macnab, R. M.. Flagella and motility. In Neidhardt, F. C., Curtiss, R. III, Ingraham, J. L., Lin, E. C. C., Low, K. B., Magasanik, B., Reznikoff, W. S., Riley, M., Schaechter, M., and Umbarger, H. E., editors, Escherichia coli and Salmonella: Cellular and Molecular Biology, pages 123–145. American Society for Microbiology, Washington, DC, second edition, 1996.Google Scholar