Hostname: page-component-7dd5485656-glrdx Total loading time: 0 Render date: 2025-10-28T12:39:14.991Z Has data issue: false hasContentIssue false
  • English
  • Français

Reference map technique for incompressible fluid–structure interaction

Published online by Cambridge University Press:  30 June 2020

Chris H. Rycroft Open the ORCID record for Chris H. Rycroft [Opens in a new window] ,
Chen-Hung Wu ,
Yue Yu  and
Ken Kamrin Open the ORCID record for Ken Kamrin [Opens in a new window]
Chris H. Rycroft*
Affiliation:
John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02139, USA Mathematics Group, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA
Chen-Hung Wu
Affiliation:
John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02139, USA Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Yue Yu
Affiliation:
Department of Mathematics, Lehigh University, Bethlehem, PA 18015, USA
Ken Kamrin*
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
† Email addresses for correspondence: chr@seas.harvard.edu, kkamrin@mit.edu
† Email addresses for correspondence: chr@seas.harvard.edu, kkamrin@mit.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a general simulation approach for fluid–solid interactions based on the fully Eulerian reference map technique. The approach permits the modelling of one or more finitely deformable continuum solid bodies interacting with a fluid and with each other. A key advantage of this approach is its ease of use, as the solid and fluid are discretized on the same fixed grid, which greatly simplifies the coupling between the phases. We use the method to study a number of illustrative examples involving an incompressible Navier–Stokes fluid interacting with multiple neo-Hookean solids. Our method has several useful features including the ability to model solids with sharp corners and the ability to model actuated solids. The latter permits the simulation of active media such as swimmers, which we demonstrate. The method is validated favourably in the flag-flapping geometry, for which a number of experimental, numerical and analytical studies have been performed. We extend the flapping analysis beyond the thin-flag limit, revealing an additional destabilization mechanism to induce flapping.

JFM classification

Mathematical Foundations: Computational methods

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 898 , 10 September 2020 , A9
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Almgren, A. S., Bell, J. B., Colella, P., Howell, L. H. & Welcome, M. L. 1998 A conservative adaptive projection method for the variable density incompressible Navier–Stokes equations. J. Comput. Phys. 142 (1), 1–46.CrossRefGoogle Scholar
Almgren, A. S., Bell, J. B. & Szymczak, W. G. 1996 A numerical method for the incompressible Navier–Stokes equations based on an approximate projection. SIAM J. Sci. Comput. 17 (2), 358–369.CrossRefGoogle Scholar
Argentina, M. & Mahadevan, L. 2005 Fluid-flow-induced flutter of a flag. Proc. Natl Acad. Sci. USA 102 (6), 1829–1834.CrossRefGoogle ScholarPubMed
Aslam, T. D. 2004 A partial differential equation approach to multidimensional extrapolation. J. Comput. Phys. 193 (1), 349–355.CrossRefGoogle Scholar
Bathe, K.-J. 2007 Proceedings of the 4th MIT Conference on Computational Fluid and Solid Mechanics. Elsevier Science.Google Scholar
Bell, J. B., Colella, P. & Glaz, H. M. 1989 A second-order projection method for the incompressible Navier–Stokes equations. J. Comput. Phys. 85 (2), 257–283.CrossRefGoogle Scholar
Bellotti, T. & Theillard, M. 2019 A coupled level-set and reference map method for interface representation with applications to two-phase flows simulation. J. Comput. Phys. 392, 266–290.CrossRefGoogle Scholar
Belytschko, T., Liu, W. K., Moran, B. & Elkhodary, K. 2013 Nonlinear Finite Elements for Continua and Structures, 2nd edn. Wiley.Google Scholar
Beyer, R. P. & LeVeque, R. J. 1992 Analysis of a one-dimensional model for the immersed boundary method. SIAM J. Numer. Anal. 29 (2), 332–364.CrossRefGoogle Scholar
Brown, D. L., Cortez, R. & Minion, M. L. 2001 Accurate projection methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 168 (2), 464–499.CrossRefGoogle Scholar
Chopp, D. L. 2001 Some improvements of the fast marching method. SIAM J. Sci. Comput. 23 (1), 230–244.CrossRefGoogle Scholar
Chopp, D. L. 2009 Another look at velocity extensions in the level set method. SIAM J. Sci. Comput. 31 (5), 3255–3273.CrossRefGoogle Scholar
Chorin, A. J. 1967 A numerical method for solving incompressible viscous flow problems. J. Comput. Phys. 2 (1), 12–26.CrossRefGoogle Scholar
Chorin, A. J. 1968 Numerical solution of the Navier–Stokes equations. Maths Comput. 22 (104), 745–762.CrossRefGoogle Scholar
Colella, P. 1985 A direct Eulerian MUSCL scheme for gas dynamics. SIAM J. Sci. Stat. Comput. 6 (1), 104–107.CrossRefGoogle Scholar
Colella, P. 1990 Multidimensional upwind methods for hyperbolic conservation laws. J. Comput. Phys. 87 (1), 171–200.CrossRefGoogle Scholar
Connell, B. S. H. & Yue, D. K. P. 2007 Flapping dynamics of a flag in a uniform stream. J. Fluid Mech. 581, 33–67.CrossRefGoogle Scholar
Coquerelle, M. & Cottet, G.-H. 2008 A vortex level set method for the two-way coupling of an incompressible fluid with colliding rigid bodies. J. Comput. Phys. 227 (21), 9121–9137.CrossRefGoogle Scholar
Cottet, G.-H., Maitre, E. & Milcent, T. 2008 Eulerian formulation and level set models for incompressible fluid–structure interaction. ESAIM: Math. Modelling Numer. Anal. 42, 471–492.CrossRefGoogle Scholar
Courant, R., Friedrichs, K. & Lewy, H. 1967 On the partial difference equations of mathematical physics. IBM J. Res. Dev. 11 (2), 215–234.CrossRefGoogle Scholar
Demmel, J. W. 1997 Applied Numerical Linear Algebra. SIAM.CrossRefGoogle Scholar
Dunne, T. 2006 An Eulerian approach to fluid–structure interaction and goal-oriented mesh adaptation. Intl J. Numer. Meth. Fluids 51 (9-10), 1017–1039.CrossRefGoogle Scholar
Dunne, T., Rannacher, R. & Richter, T. 2010 Numerical Simulation of Fluid–Structure Interaction Based on Monolithic Variational Formulations, pp. 1–75. World Scientific.Google Scholar
Engels, T., Kolomenskiy, D., Schneider, K. & Sesterhenn, J. 2015 Numerical simulation of fluid–structure interaction with the volume penalization method. J. Comput. Phys. 281, 96–115.CrossRefGoogle Scholar
Fachinotti, V. D., Cardona, A. & Jetteur, P. 2008 Finite element modelling of inverse design problems in large deformations anisotropic hyperelasticity. Intl J. Numer. Meth. Engng 74 (6), 894–910.CrossRefGoogle Scholar
Fai, T. G., Griffith, B. E., Mori, Y. & Peskin, C. S. 2013 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.CrossRefGoogle Scholar
Francois, M. M., Cummins, S. J., Dendy, E. D., Kothe, D. B., Sicilian, J. M. & Williams, M. W. 2006 A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework. J. Comput. Phys. 213 (1), 141–173.CrossRefGoogle Scholar
Froehle, B. & Persson, P.-O. 2015 Nonlinear elasticity for mesh deformation with high-order discontinuous Galerkin methods for the Navier–Stokes equations on deforming domains. In Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014, pp. 73–85. Springer.CrossRefGoogle Scholar
Gazzola, M., Argentina, M. & Mahadevan, L. 2015 Gait and speed selection in slender inertial swimmers. Proc. Natl Acad. Sci. USA 112 (13), 3874–3879.CrossRefGoogle ScholarPubMed
Gazzola, M., Chatelain, P., van Rees, W. M. & Koumoutsakos, P. 2011 Simulations of single and multiple swimmers with non-divergence free deforming geometries. J. Comput. Phys. 230 (19), 7093–7114.CrossRefGoogle Scholar
Gibou, F. & Fedkiw, R. 2005 A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem. J. Comput. Phys. 202 (2), 577–601.CrossRefGoogle Scholar
Govindjee, S. & Mihalic, P. A. 1996 Computational methods for inverse finite elastostatics. Comput. Meth. Appl. Mech. Engng 136 (1–2), 47–57.CrossRefGoogle Scholar
Griffith, B. E. 2012 On the volume conservation of the immersed boundary method. Commun. Comput. Phys. 12 (2), 401–432.CrossRefGoogle Scholar
Griffith, B. E., Luo, X., McQueen, D. M. & Peskin, C. S. 2009 Simulating the fluid dynamics of natural and prosthetic heat valves using the immersed boundary method. Intl J. Appl. Mech. 1 (1), 137–177.CrossRefGoogle Scholar
Gurtin, M. E., Fried, E. & Anand, L. 2010 The Mechanics and Thermodynamics of Continua. Cambridge University Press.CrossRefGoogle Scholar
Hairer, E., Nørsett, S. P. & Wanner, G. 1993 Solving Ordinary Differential Equations I: Nonstiff Problems. Springer.Google Scholar
Heath, M. T. 2002 Scientific Computing: An Introductory Survery. McGraw-Hill.Google Scholar
Hirt, C. W., Amsden, A. A. & Cook, J. L. 1974 An arbitrary Lagrangian Eulerian computing method for all flow speeds. J. Comput. Phys. 14, 227–253.CrossRefGoogle Scholar
Hoover, W. G. 2006 Smooth Particle Applied Mechanics: The State of the Art. World Scientific.CrossRefGoogle Scholar
Ii, S., Sugiyama, K., Takagi, S. & Matsumoto, Y. 2012 A computational blood flow analysis in a capillary vessel including multiple red blood cells and platelets. J. Biomech. Sci. Engng 7 (1), 72–83.CrossRefGoogle Scholar
Jain, S. S. & Mani, A. 2017 An incompressible Eulerian formulation of soft solids in fluids. In Annual Research Briefs, Center for Turbulence Research, pp. 347–362.Google Scholar
Johnson, C. 2009 Numerical Solution of Partial Differential Equations by the Finite Element Method. Dover.Google Scholar
Kamrin, K.2008 Stochastic and deterministic models for dense granular flow. PhD thesis, MIT.Google Scholar
Kamrin, K. & Nave, J.-C.2009 An Eulerian approach to the simulation of deformable solids: application to finite-strain elasticity. arXiv:0901.3799.Google Scholar
Kamrin, K., Rycroft, C. H. & Nave, J.-C. 2012 Reference map technique for finite-strain elasticity and fluid–solid interaction. J. Mech. Phys. Solids 60 (11), 1952–1969.CrossRefGoogle Scholar
Krishnan, S., Shaqfeh, E. S. G. & Iaccarino, G. 2017 Fully resolved viscoelastic particulate simulations using unstructured grids. J. Comput. Phys. 338, 313–338.CrossRefGoogle Scholar
Kröner, E. 1960 Allgemeine kontinuumstheorie der versetzungen und eigenspannungen. Arch. Rat. Mech. Anal. 4, 273–334.CrossRefGoogle Scholar
Lax, P. & Wendroff, B. 1960 Systems of conservation laws. Commun. Pure Appl. Maths 13, 217–237.CrossRefGoogle Scholar
Lee, E. H. 1969 Elastic plastic deformation at finite strain. Trans. ASME J. Appl. Mech. 36, 1–6.CrossRefGoogle Scholar
LeVeque, R. J. 2002 Finite Volume Methods for Hyperbolic Problems. Cambridge University Press.CrossRefGoogle Scholar
LeVeque, R. J. 2007 Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems. SIAM.CrossRefGoogle Scholar
Lienhard, J. H. 1966 Synopsis of Lift, Drag, and Vortex Frequency Data for Rigid Circular Cylinders, vol. 300. Technical Extension Service, Washington State University.Google Scholar
Lim, S. & Peskin, C. S. 2012 Fluid-mechanical interaction of flexible bacterial flagella by the immersed boundary method. Phys. Rev. E 85, 036307.Google ScholarPubMed
Liu, C. & Walkington, N. J. 2001 An Eulerian description of fluids containing visco-elastic particles. Arch. Rat. Mech. Anal. 159 (3), 229–252.CrossRefGoogle Scholar
Lubliner, J. 2008 Plasticity Theory. Dover.Google Scholar
Maitre, E., Milcent, T., Cottet, G.-H., Raoult, A. & Usson, Y. 2009 Applications of level set methods in computational biophysics. Math. Comput. Model. 49 (11–12), 2161–2169.CrossRefGoogle Scholar
McMillen, T., Williams, T. & Holmes, P. 2008 Nonlinear muscles, passive viscoelasticity and body taper conspire to create neuromechanical phase lags in anguilliform swimmers. PLoS Comput. Biol. 4 (8), e1000157.CrossRefGoogle ScholarPubMed
Milcent, T. & Maitre, E. 2016 Eulerian model of immersed elastic surfaces with full membrane elasticity. Commun. Math. Sci. 14 (3), 857–881.CrossRefGoogle Scholar
Osher, S. & Sethian, J. A. 1988 Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79 (1), 12–49.CrossRefGoogle Scholar
Osher, S. J. & Fedkiw, R. P. 2003 Level Set Methods and Dynamic Implicit Surfaces. Springer.CrossRefGoogle Scholar
Patel, N. K., Bhalla, A. P. S. & Patankar, N. A. 2018 A new constraint-based formulation for hydrodynamically resolved computational neuromechanics of swimming animals. J. Comput. Phys. 375, 684–716.CrossRefGoogle Scholar
Peskin, C. S.1972a Flow patterns around heart valves: a digital computer method for solving the equations of motion. PhD thesis, Albert Einstein College of Medicine.CrossRefGoogle Scholar
Peskin, C. S. 1972b Flow patterns around heart valves: a numerical method. J. Comput. Phys. 10 (2), 252–271.CrossRefGoogle Scholar
Peskin, C. S. 1977 Numerical analysis of blood flow in the heart. J. Comput. Phys. 25 (3), 220–252.CrossRefGoogle Scholar
Peskin, C. S. 2002 The immersed boundary method. Acta Numerica 11, 479–517.CrossRefGoogle Scholar
Plohr, B. J. & Sharp, D. H. 1988 A conservative Eulerian formulation of the equations for elastic flow. Adv. Appl. Maths 9 (4), 481–499.CrossRefGoogle Scholar
Puckett, E. G., Almgren, A. S., Bell, J. B., Marcus, D. L. & Rider, W. J. 1997 A high-order projection method for tracking fluid interfaces in variable density incompressible flows. J. Comput. Phys. 130 (2), 269–282.CrossRefGoogle Scholar
Rabczuk, T., Gracie, R., Song, J.-H. & Belytschko, T. 2010 Immersed particle method for fluid–structure interaction. Intl J. Numer. Meth. Engng 81 (1), 48–71.CrossRefGoogle Scholar
Richardson, L. F. 1911 IX. The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam. Phil. Trans. R. Soc. Lond. A 210 (459–470), 307–357.Google Scholar
Richter, T. 2013 A fully Eulerian formulation for fluid–structure-interaction problems. J. Comput. Phys. 233, 227–240.CrossRefGoogle Scholar
Rugonyi, S. & Bathe, K.-J. 2001 On finite element analysis of fluid flows fully coupled with structural interactions. Comput. Model. Engng Sci. 2, 195–212.Google Scholar
Rycroft, C. H. & Gibou, F. 2012 Simulations of a stretching bar using a plasticity model from the shear transformation zone theory. J. Comput. Phys. 231 (5), 2155–2179.CrossRefGoogle Scholar
Rycroft, C. H., Sui, Y. & Bouchbinder, E. 2015 An Eulerian projection method for quasi-static elastoplasticity. J. Comput. Phys. 300, 136–166.CrossRefGoogle Scholar
Sethian, J. A. 1999 Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision and Materials Science. Cambridge University Press.Google Scholar
Sugiyama, K., Ii, S., Takeuchi, S., Takagi, S. & Matsumoto, Y. 2011 A full Eulerian finite difference approach for solving fluid–structure coupling problems. J. Comput. Phys. 230 (3), 596–627.CrossRefGoogle Scholar
Süli, E. & Mayers, D. 2003 An Introduction to Numerical Analysis. Cambridge University Press.CrossRefGoogle Scholar
Sulsky, D., Chen, Z. & Schreyer, H. L. 1994 A particle method for history-dependent materials. Comput. Meth. Appl. Mech. Engng 118 (1–2), 179–196.CrossRefGoogle Scholar
Sussman, M., Almgren, A. S., Bell, J. B., Colella, P., Howell, L. H. & Welcome, M. L. 1999 An adaptive level set approach for incompressible two-phase flows. J. Comput. Phys. 148 (1), 81–124.CrossRefGoogle Scholar
Sussman, M., Smereka, P. & Osher, S. 1994 A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114 (1), 146–159.CrossRefGoogle Scholar
Takagi, S., Sugiyama, K., Ii, S. & Matsumoto, Y. 2011 A review of full Eulerian methods for fluid–structure interaction problems. Trans. ASME J. Appl. Mech. 79 (1), 010911.Google Scholar
Tannehill, J. C., Anderson, D. A. & Pletcher, R. H. 1997 Computational Fluid Mechanics and Heat Transfer. Taylor & Francis.Google Scholar
Tornberg, A.-K. & Engquist, B. 2003 Regularization techniques for numerical approximation of PDEs with singularities. J. Sci. Comput. 19 (1), 527–552.CrossRefGoogle Scholar
Tornberg, A.-K. & Engquist, B. 2004 Numerical approximations of singular source terms in differential equations. J. Comput. Phys. 200 (2), 462–488.CrossRefGoogle Scholar
Trangenstein, J. A. & Colella, P. 1991 A higher-order Godunov method for modeling finite deformation in elastic-plastic solids. Commun. Pure Appl. Maths 44 (1), 41–100.CrossRefGoogle Scholar
Truesdell, C. 1955 Hypo-elasticity. Indiana Univ. Math. J. 4, 83–133.CrossRefGoogle Scholar
Tytell, E. D., Hsu, C.-Y., Williams, T. L., Cohen, A. H. & Fauci, L. J. 2010 Interactions between internal forces, body stiffness, and fluid environment in a neuromechanical model of lamprey swimming. Proc. Natl Acad. Sci. USA 107 (46), 19832–19837.CrossRefGoogle Scholar
Udaykumar, H. S., Tran, L., Belk, D. M. & Vanden, K. J. 2003 An Eulerian method for computation of multimaterial impact with ENO shock-capturing and sharp interfaces. J. Comput. Phys. 186 (1), 136–177.CrossRefGoogle Scholar
Vadala-Roth, B., Acharya, S., Patankar, N. A., Rossi, S. & Griffith, B. E. 2020 Stabilization approaches for the hyperelastic immersed boundary method for problems of large-deformation incompressible elasticity. Comput. Meth. Appl. Mech. Engng 365, 112978.CrossRefGoogle ScholarPubMed
Valkov, B., Rycroft, C. H. & Kamrin, K. 2015 Eulerian method for multiphase interactions of soft solid bodies in fluids. Trans. ASME J. Appl. Mech. 82 (4), 041011.CrossRefGoogle Scholar
Versteeg, H. K. & Malalasekera, W. 1995 An Introduction to Computational Fluid Dynamics: The Finite Volume Method. Addison-Wesley Longman.Google Scholar
Wang, X. S., Zhang, L. T. & Liu, W. K. 2009 On computational issues of immersed finite element methods. J. Comput. Phys. 228 (7), 2535–2551.CrossRefGoogle Scholar
Watanabe, Y., Suzuki, S., Sugihara, M. & Sueoka, Y. 2002 An experimental study of paper flutter. J. Fluids Struct. 16 (4), 529–542.CrossRefGoogle Scholar
Wick, T. 2013 Fully Eulerian fluid–structure interaction for time-dependent problems. Comput. Meth. Appl. Mech. Engng 255, 14–26.CrossRefGoogle Scholar
Yu, J.-D., Sakai, S. & Sethian, J. A. 2003 A coupled level set projection method applied to ink jet simulation. Interfaces Free Boundaries 5 (4), 459–482.CrossRefGoogle Scholar
Yu, J.-D., Sakai, S. & Sethian, J. A. 2007 Two-phase viscoelastic jetting. J. Comput. Phys. 220 (2), 568–585.CrossRefGoogle Scholar
Zhang, J., Childress, S., Libchaber, A. & Shelley, M. 2000 Flexible filaments in a flowing soap film as a model for one-dimensional flags in a two-dimensional wind. Nature 408 (6814), 835–839.CrossRefGoogle Scholar
Zhu, L. & Peskin, C. S. 2002 Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method. J. Comput. Phys. 179 (2), 452–468.CrossRefGoogle Scholar
Zienkiewicz, O. C. & Taylor, R. L. 1967 The Finite Element Method for Solid and Structural Mechanics. McGraw-Hill.Google Scholar

Rycroft et al. supplementary movie 1

See movie captions pdf
Download Rycroft et al. supplementary movie 1(Video)
Video 8.2 MB

Rycroft et al. supplementary movie 2

See movie captions pdf
Download Rycroft et al. supplementary movie 2(Video)
Video 4.6 MB

Rycroft et al. supplementary movie 3

See movie captions pdf
Download Rycroft et al. supplementary movie 3(Video)
Video 4.4 MB

Rycroft et al. supplementary movie 4

See movie captions pdf

Download Rycroft et al. supplementary movie 4(Video)
Video 5.1 MB

Rycroft et al. supplementary movie 5

See movie captions pdf

Download Rycroft et al. supplementary movie 5(Video)
Video 1.6 MB

Rycroft et al. supplementary movie 6

See movie captions pdf
Download Rycroft et al. supplementary movie 6(Video)
Video 9.8 MB

Rycroft et al. supplementary movie 7

See movie captions pdf.
Download Rycroft et al. supplementary movie 7(Video)
Video 9.8 MB

Rycroft et al. supplementary movie 8

See movie captions pdf.
Download Rycroft et al. supplementary movie 8(Video)
Video 8.6 MB

Rycroft et al. supplementary movie 9

See movie captions pdf
Download Rycroft et al. supplementary movie 9(Video)
Video 9.4 MB
Supplementary material: PDF

Rycroft et al. supplementary material

Captions for movies 1-9

Download Rycroft et al. supplementary material(PDF)
PDF 78.9 KB