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Coupling the Curvilinear Immersed Boundary Method with Rotation-Free Finite Elements for Simulating Fluid–Structure Interaction: Concepts and Applications

  • Anvar GilmanovEmail author
  • Henryk Stolarski
  • Fotis Sotiropoulos
Chapter
  • 78 Downloads
Part of the Computational Methods in Engineering & the Sciences book series (CMES)

Abstract

The sharp interface curvilinear immersed boundary (CURVIB) method coupled with a rotation-free finite element (FE) method for thin shells provides a powerful framework for simulating fluid–structure interaction (FSI) problems for geometrically complex, arbitrarily deformable structures. The CURVIB and FE solvers are coupled together on the flexible solid–fluid interfaces, which contain the structural nodal positions, displacements, velocities, and loads calculated at each time level and exchanged between the flow and structural solvers. Loose and strong coupling FSI schemes are employed, enhanced by the Aitken acceleration technique to ensure robust coupling and fast convergence, especially for low mass ratio problems. Large-eddy simulation (LES) of turbulent flow FSI problems employ the dynamic Smagorinsky subgrid scale model with a wall model for reconstructing velocity boundary conditions near the immersed boundaries. In this chapter, the CURVIB-FE FSI algorithm is reviewed and its capabilities are demonstrated via a series of examples involving thin flexible structures undergoing very large deformations. The inverted flag problem is employed to validate the method, and the problem of a tri-leaflet aortic valve in an anatomic aorta is employed to demonstrate its potential in complex cardiovascular flow applications.

Keywords

Immersed boundary methods CURVIB Fluid–structure interaction Rotation-free finite element shell model 

Notes

Acknowledgements

We acknowledge the financial support through a grant from the Lillehei Heart Institute at the University of Minnesota, NSF grants IIP-1318201 and CBET-1509071, and US Department of Energy grant (DE-EE 0005482). Computational resources have been provided by the Minnesota Supercomputing Institute and supercomputers of Saint Anthony Falls Laboratory, University of Minnesota.

References

  1. Angelidis D, Chawdhary S, Sotiropoulos F (2016) Unstructured Cartesian refinement with sharp interface immersed boundary method for 3D unsteady incompressible flows. J Comput Phys 325:272–300MathSciNetzbMATHGoogle Scholar
  2. Baek H, Karniadakis GE (2012) A convergence study of a new partitioned fluid–structure interaction algorithm based on fictitious mass and damping. J Comput Phys 231:629–652MathSciNetzbMATHGoogle Scholar
  3. Barker AT, Cai X (2010) Scalable parallel methods for monolithic coupling in fluid–structure interaction with application to blood flow modeling. J Comput Phys 229:642–659MathSciNetzbMATHGoogle Scholar
  4. Bazilevs Y, Hsu M-C, Scott MA (2012) Isogeometric fluid–structure interaction analysis with emphasis on non-matching discretizations, and with application to wind turbines. Comput Methods Appl Mech Eng 249:28–41MathSciNetzbMATHGoogle Scholar
  5. Borazjani I (2013) Fluid–structure interaction, immersed boundary-finite element method simulations of bio-prosthetic heart valves. Comput Methods Appl Mech Eng 257:103–116MathSciNetzbMATHGoogle Scholar
  6. Borazjani I, Ge L, Sotiropoulos F (2008) Curvilinear immersed boundary method for simulating fluid–structure interaction with complex 3d rigid bodies. J Comput Phys 227(16):7587–7620MathSciNetzbMATHGoogle Scholar
  7. Calderer A, Kang S, Sotiropoulos F (2014) Level set immersed boundary method for coupled simulation of air/water interaction with complex floating structures. J Comput Phys 277:201–227MathSciNetzbMATHGoogle Scholar
  8. Carmody CJ, Burriesci G, Howard IC, Patterson EA (2006) An approach to the simulation of fluid–structure interaction in the aortic valve. J Biomech 39:158–169Google Scholar
  9. Dettmer W, Períc D (2006) A computational framework for fluid–structure interaction: finite element formulation and applications. Comput Methods Appl Mech Eng 195(41):5754–5779Google Scholar
  10. Donea J, Giuliani S, Halleux JP (1982) An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid–structure interactions. Comput Methods Appl Mech Eng 33:689–723zbMATHGoogle Scholar
  11. Farhat C, Lakshminarayan VK (2014) An ALE formulation of embedded boundary methods for tracking boundary layers in turbulent fluid–structure interaction problems. J Comput Phys 263:53–70MathSciNetzbMATHGoogle Scholar
  12. Felippa CA, Park KC, Farhat C (2001) Partitioned analysis of coupled mechanical systems. Comput Methods Appl Mech Eng 190(24–25):3247–3270zbMATHGoogle Scholar
  13. Fernandez MA, Gerbeau J-F, Grandmont C (2007) A projection semi-implicit scheme for the coupling of an elastic structure with an incompressible fluid. Int J Numer Methods Eng 69:794–821MathSciNetzbMATHGoogle Scholar
  14. Gal E, Levy R (2006) Geometrically nonlinear analysis of shell structures using a flat triangular shell finite element. Arch. Comput. Methods Eng. 13:331–388zbMATHGoogle Scholar
  15. Ge L, Sotiropoulos F (2007) A numerical method for solving the 3 D unsteady incompressible Navier-Stokes equations in curvilinear domains with complex immersed boundaries. J Comput Phys 225:1782–1809MathSciNetzbMATHGoogle Scholar
  16. Ge L, Sotiropoulos F (2010) Direction and magnitude of blood flow shear stresses on the leaflets of aortic valves: is there a link with valve calcification? J Biomech Eng 132Google Scholar
  17. Germano M, Piomelli U, Moin P, Cabot WH (1991) A dynamic subgrid-scale eddy viscosity model. Phys Fluids 3(7):1760–1765zbMATHGoogle Scholar
  18. Gilmanov A, Sotiropoulos F (2005) A hybrid Cartesian/immersed boundary method for simulating flows with 3D, geometrically complex, moving bodies. J Comput Phys 207(2):457zbMATHGoogle Scholar
  19. Gilmanov A, Sotiropoulos F, Balaras E (2003) A general reconstruction algorithm for simulating flows with complex 3D immersed boundaries on Cartesian grids. J Comput Phys 191:660–669zbMATHGoogle Scholar
  20. Gilmanov A, Le Bao T, Sotiropoulos F (2015) A numerical approach for simulating fluid-structure interaction of flexible thin shells undergoing arbitrarily large deformations in complex domains. J Comput Phys 300:814–843MathSciNetzbMATHGoogle Scholar
  21. Gilmanov A, Stolarski H, Sotiropoulos F (2016) Non-linear rotation-free shell finite-element models for aortic heart valves. J Biomech 50:56–62Google Scholar
  22. Gilmanov A, Stolarski H, Sotiropoulos F (2018) Flow-structure interaction simulations of the aortic heart valve at physiologic conditions: the role of tissue constitutive model. J Biomech Eng 140:1003–1012Google Scholar
  23. Griffith BE, Luo X, McQueen DM, Peskin CS (2009) Simulating the fluid dynamics of natural and prosthetic heart valves using the immersed boundary method. Int J Appl Mech 1(01):137–177Google Scholar
  24. Hirt CW, Amsden AA, Cook JL (1974) An arbitrary Lagrangian-Eulerian computing method for all flow speeds. J Comput Phys 14(3):227–253zbMATHGoogle Scholar
  25. Hunt JCR, Wray AA, Moin P (1988) Eddies, streams, and convergence zones in turbulent flows. In: Proceedings of the 1988 summer program, Stanford N.A.S.A. Centre for Turbulence Research, CTR-S88, vol 736, pp 193–208Google Scholar
  26. Kamensky D, Hsu MC, Schillinger D, Evans JA, Aggarwal A, Bazilevs Y, Sacks MS, Hughes TJR (2015) An immersogeometric variational framework for fluid–structure interaction: application to bio prosthetic heart valves. Comput Methods Appl Mech Eng 284:1005–1053zbMATHGoogle Scholar
  27. Kang S, Lightbody A, Hill C, Sotiropoulos F (2011) High-resolution numerical simulation of turbulence in natural water ways. Adv Water Resour 34:98–113Google Scholar
  28. Kang S, Borazjani I, Colby J, Sotiropoulos F (2012) Numerical simulation of 3d flow past a real-life marine hydrokinetic turbine. Adv Water Resour 39:33–43Google Scholar
  29. Kang S, Yang X, Sotiropoulos F (2014) On the onset of wake meandering for an axial flow turbine in a turbulent open channel flow. J Fluid Mech 744:376–403Google Scholar
  30. Khosronejad A, Sotiropoulos F (2014) Numerical simulation of sand waves in a turbulent open channel flow. J Fluid Mech 753:150–216Google Scholar
  31. Kim D, Cossé J, Cerdeira CH, Gharib M (2013) Flapping dynamics of an inverted flag. J Fluid Mech 736Google Scholar
  32. Küttler U, Wall W (2008) Fixed-point fluid–structure interaction solvers with dynamic relaxation. Comput Mech 43:61–72zbMATHGoogle Scholar
  33. Le TB, Sotiropoulos F (2013) Fluid–structure interaction of an aortic heart valve prosthesis driven by an animated anatomic left ventricle. J Comput Phys 244:41–62MathSciNetzbMATHGoogle Scholar
  34. Le DV, White J, Peraire J, Lim KM, Khoo BC (2009) An implicit immersed boundary method for three-dimensional fluid–membrane interactions. J Comput Phys 228(22):8427–8445MathSciNetzbMATHGoogle Scholar
  35. Luo H, Mittal R, Zheng X, Bielamowicz SA, Walsh RJ, Hahn JK (2008) An immersed-boundary method for flow–structure interaction in biological systems with application to phonation. J Comput Phys 227:9303–9332MathSciNetzbMATHGoogle Scholar
  36. Luo H, Yin B, Dai H, Doyle JF (2010) A 3d computational study of the flow–structure interaction in flapping flight. Technical Report. In: 48th AIAA aerospace sciences meeting including the new horizons forum and aerospace exposition, 4–7 Jan 2010, Orlando, FloridaGoogle Scholar
  37. Macosko CW (1994) Rheology: principles, measurements, and applications. Wiley VCH, New YorkGoogle Scholar
  38. May-Newman K, Yin F (1998) A constitutive law for mitral valve tissue. ASME J Biomech Eng 120(1):38–47Google Scholar
  39. Mittal R, Iaccarino G (2005) Immersed boundary methods. Annu Rev Fluid Mech 37:239–261MathSciNetzbMATHGoogle Scholar
  40. New TH, Tsovolos D (2012) Vortex behaviour and velocity characteristics of jets issuing from hybrid inclined elliptic nozzles. Flow Turbul Combust 89(4):601–625Google Scholar
  41. Newmark N (1959) A method of computation for structural dynamics. J Eng Mech Div 85:67–94Google Scholar
  42. Sacks MS, Schoen FJ, Mayer JE Jr (2009) Bioengineering challenges for heart valve tissue engineering. Annu Rev Biomed Eng 11:289–313Google Scholar
  43. Smith IM, Griffith DV (2004) Programming the finite element method. Willey, New YorkGoogle Scholar
  44. Sotiropoulos F, Yang X (2014) Immersed boundary methods for simulating fluid–structure interaction. Prog Aerosp Sci 65:1–21Google Scholar
  45. Stolarski H, Belytschko T, Lee S-H (1995) A review of shell finite elements and co-rotational theories. Comput. Mech. Adv. 2:125–212MathSciNetzbMATHGoogle Scholar
  46. Stolarski H, Gilmanov A, Sotiropoulos F (2013) Non-linear rotation-free 3-node shell finite-element formulation. Int J Numer Methods Eng 95:740–770zbMATHGoogle Scholar
  47. Tepole AB, Kabari H, Bletzinger K-U, Kuhl E (2015) Isogeometric Kirchhoff-Love shell formulations for biological membranes. Comput Methods Appl Mech Eng 293:328–347MathSciNetzbMATHGoogle Scholar
  48. Tian FB, Dai H, Luo H, Doyle JF, Rousseau B (2014) Fluid–structure interaction involving large deformations: 3d simulations and applications to biological systems. J Comput Phys 258:451–469MathSciNetzbMATHGoogle Scholar
  49. Timoshenko S, Woinowsky-Krieger S (1959) Theory of plates and shells. McGraw–Hill, New YorkGoogle Scholar
  50. Vanella M, Rabenold P, Balaras E (2010) A direct-forcing embedded-boundary method with adaptive mesh refinement for fluid-structure interaction problems. J Comput Phys 229(18):6427–6449MathSciNetzbMATHGoogle Scholar
  51. Wang M, Moin P (2002) Dynamic wall modeling for large-eddy simulation of complex turbulent flows. Phys Fluids 14:2043–2051MathSciNetzbMATHGoogle Scholar
  52. Wiens JK, Stockie JM (2015) An efficient parallel immersed boundary algorithm using a pseudo-compressible fluid solver. J Comput Phys 281:917–941MathSciNetzbMATHGoogle Scholar
  53. Zheng X, Xue Q, Mittal R, Beilamowicz S (2010) A coupled sharp-interface immersed boundary-finite element method for flow–structure interaction with application to human phonation. J Biomech Eng 132:111003Google Scholar
  54. Zhu L, Peskin CS (2002) Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method. J Comput Phys 179(2):452–468MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  • Anvar Gilmanov
    • 1
    Email author
  • Henryk Stolarski
    • 1
  • Fotis Sotiropoulos
    • 2
  1. 1.University of MinnesotaMinneapolisUSA
  2. 2.Stony Brook UniversityStony BrookUSA

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