Advertisement

Computational Mechanics

, Volume 53, Issue 1, pp 29–43 | Cite as

Flapping and contact FSI computations with the fluid–solid interface-tracking/interface-capturing technique and mesh adaptivity

  • Thomas WickEmail author
Original Paper

Abstract

The fluid–solid interface-tracking/interface-capturing technique (FSITICT) with arbitrary Lagrangian–Eulerian interface-tracking and Eulerian interface-capturing is applied to computations of fluid–structure interaction problems with flapping and contact. The two-dimensional model with contacting flaps is intended to represent a valve problem from biomechanics. The FSITICT is complemented with local mesh adaptivity, which significantly increases the performance of the interface-capturing component of the method. The test computations presented demonstrate how our approach works.

Keywords

Finite elements Fluid–structure interaction Monolithic formulation Arbitrary Lagrangian–Eulerian approach  Fully Eulerian approach 

Mathematics Subject Classificaion

65N30 65M60 74F10 

Notes

Acknowledgments

I thank Prof. R. Rannacher (Heidelberg) and Dr. med. J. Mizerski (Warsaw) for initiating this project during my PhD studies.

References

  1. 1.
    Akin JE, Tezduyar T, Ungor M (2007) Computation of flow problems with the mixed interface-tracking/interface-capturing technique (MITICT). Comput Fluids 36:2–11CrossRefzbMATHGoogle Scholar
  2. 2.
    Asterino M, Gerbeau JF, Pantz O, Traoré KF (2009) Fluid-structure interaction and multi-body contact: application to aortic valves. Comput Methods Appl Mech Eng 198:3603–3612CrossRefGoogle Scholar
  3. 3.
    Bangerth W, Heister T, Kanschat G (2012) Differential Equations Analysis Library.Google Scholar
  4. 4.
    Becker R, Rannacher R (1996) A feed-back approach to error control in finite element methods: basic analysis and examples. East-West J Numer Math 4:237–264zbMATHMathSciNetGoogle Scholar
  5. 5.
    Belytschko T, Parimi C, Moes N, Sukumar N, Usui S (2003) Structured extended finite element methods for solids defined by implicit surfaces. Int J Numer Methods Eng 56:609–635CrossRefzbMATHGoogle Scholar
  6. 6.
    Berenger J (1994) A perfectly matched layer for the absorption of electromagnetic waves. J Comput Phys 114.Google Scholar
  7. 7.
    Brooks A, Hughes T (1982) Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput Methods Appl Mech Eng 32(1–3):199–259CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Bukac M, Canic S, Glowinski R, Tambaca J, Quaini A (2012) Fluid-structure interaction in blood ow capturing non-zero longitudinal structure displacement. J Comput Phys. http://dx.doi.org/10.1016/j.jcp.2012.08.033
  9. 9.
    Ciarlet PG (1984) Mathematical elasticity. Volume 1: three dimensional elasticity. North-HollandGoogle Scholar
  10. 10.
    Ciarlet PG (1987) The finite element method for elliptic problems, 2. pr. edn. North-Holland, Amsterdam [u.a.].Google Scholar
  11. 11.
    Cottet GH, Maitre E, Mileent T (2008) Eulerian formulation and level set models for incompressible fluid-structure interaction. Math Model Numer Anal 42:471–492CrossRefzbMATHGoogle Scholar
  12. 12.
    Cruchaga M, Celentano D, Tezduyar T (2007) A numerical model based on the mixed interface-tracking/ interface-capturing technique (MITICT). Int J Numer Methods Fluids 54:1021–1030CrossRefzbMATHGoogle Scholar
  13. 13.
    Donéa J, Fasoli-Stella P, Giuliani S (1977) Lagrangian and Eulerian finite element techniques for transient fluid-structure interaction problems. In: Trans. 4th Int. Conf. on Structural Mechanics in Reactor Technology, p. Paper B1/2Google Scholar
  14. 14.
    Dunne T (2006) An Eulerian approach to fluid-structure interaction and goal-oriented mesh adaption. Int J Numer Methods Fluids 51:1017–1039CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Fernández F, Moubachir M (2005) A Newton method using exact Jacobians for solving fluid-structure coupling. Comput Struct 83:127–142CrossRefGoogle Scholar
  16. 16.
    Formaggia L, Gerbeau JF, Nobile F, Quarteroni A (2001) On the coupling of 3d and 1d Navier-Stokes equations for flow problems in compliant vessels. Comput Methods Appl Mech Eng 191:561–582CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Formaggia L, Quarteroni A, Veneziani A (2009) Cardiovascular mathematics: modeling and simulation of the circulatory system. Springer, Italia, MilanoCrossRefGoogle Scholar
  18. 18.
    Fung Y (1984) Biodynamics: circulation, first ed. edn. Springer, Berlin.Google Scholar
  19. 19.
    Gazzola F, Squassina M (2006) Global solutions and finite time blow up for damped semilinear wave equations. Ann I H Poincaré 23:185–207CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Gil AJ, Carreno AA, Bonet J, Hassan O (2010) The immersed structural potential method for haemodynamic applications. J Comput Phys 229:8613–8641CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Girault V, Raviart PA (1986) Finite element method for the Navier-Stokes equations. Number 5 in computer series in computational mathematics. Springer, Berlin.Google Scholar
  22. 22.
    He P, Qiao R (2011) A full-Eulerian solid level set method for simulation of fluid-structure interactions. Microfluid Nanofluid 11:557–567CrossRefGoogle Scholar
  23. 23.
    Heywood JG, Rannacher R, Turek S (1996) Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations. Int J Numer Methods Fluids 22:325–352CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Hirt C, Amsden A, Cook J (1974) An arbitrary Lagrangian-Eulerian computing method for all flow speeds. J Comput Phys 14:227–253CrossRefzbMATHGoogle Scholar
  25. 25.
    Hughes T, Liu W, Zimmermann T (1981) Lagrangian-Eulerian finite element formulation for incompressible viscous flows. Comput Methods Appl Mech Eng 29:329–349CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Johnson A, Tezduyar T (1999) Advanced mesh generation and update methods for 3D flow simulations. Comp Mech 23:130–143CrossRefzbMATHGoogle Scholar
  27. 27.
    Johnson A, Tezduyar T (2001) Methods for 3D computation of fluid-object interactions in spatially-periodic flows. Comput Methods Appl Mech Eng 190:3201–3221CrossRefzbMATHGoogle Scholar
  28. 28.
    Johnson AA, Tezduyar T (1996) 3D simulation of fluid-particle interactions with the number of particles reaching 100. Comput Methods Appl Mech Eng 145:301–321CrossRefMathSciNetGoogle Scholar
  29. 29.
    Johnson AA, Tezduyar T (1996) Simulation of multiple spheres falling in a liquid-filled tube. Comput Methods Appl Mech Eng 134:351–373CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Moes N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46:131–150CrossRefzbMATHGoogle Scholar
  31. 31.
    Moghadam ME, Bazilevs Y, Hsia TY, Vignon-Clementel IE, Marsden AL (2011) A comparison of outlet boundary treatments for prevention of backflow divergence with relevance to blood flow simulations. Comput Mech 48:277–291CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Noh W (1964) A time-dependent two-space-dimensional coupled Eulerian-Lagrangian code, Methods Comput Phys, vol 3, 31st edn. Academic Press, New YorkGoogle Scholar
  33. 33.
    Quarteroni A (2006) What mathematics can do for the simulation of blood circulation. Tech. rep, MOX Institute, MilanoGoogle Scholar
  34. 34.
    Rannacher R (1986) On the stabilization of the Crank-Nicolson scheme for long time calculations. PreprintGoogle Scholar
  35. 35.
    Richter T (2012) A fully Eulerian formulation for fluid-structure interaction problems. J Comput Phys 233:227–240CrossRefGoogle Scholar
  36. 36.
    Richter T, Wick T (2010) Finite elements for fluid-structure interaction in ALE and fully Eulerian coordinates. Comput Methods Appl Mech Eng 199:2633–2642CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Santos NDD, Gerbeau JF, Bourgat J (2008) A partitioned fluid-structure algorithm for elastic thin valves with contact. Comput Methods Appl Mech Eng 197(19–20):1750–1761CrossRefzbMATHGoogle Scholar
  38. 38.
    Sathe S, Tezduyar T (2008) Modeling of fluid-structure interactions with the space-time finite elements: contact problems. Comput Mech 43:51–60CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Sugiyama K, Li S, Takeuchi S, Takagi S, Matsumato Y (2011) A full Eulerian finite difference approach for solving fluid-structure interacion. J Comput Phys 230:596–627CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Takagi S, Sugiyama K, Matsumato Y (2012) A review of full Eulerian mehtods for fluid structure interaction problems. J Appl Mech 79(1):010911CrossRefGoogle Scholar
  41. 41.
    Takizawa K, Tezduyar T (2012) Computational methods for parachute fluid-structure interactions. Arch Comput Methods Eng 19:125–169CrossRefMathSciNetGoogle Scholar
  42. 42.
    Takizawa K, Wright S, Moorman C, Tezduyar T (2011) Fluid-structure interaction modeling of parachute clusters. Int J Numer Methods Fluids 65:286–307CrossRefzbMATHGoogle Scholar
  43. 43.
    Takizawa K, Fritze M, Montes D, Spielman T, Tezduyar T (2012) Fluid-structure interaction modeling of ringsail parachutes with disreefing and modified geometric porosity. Comput Mech 50:835–854CrossRefzbMATHGoogle Scholar
  44. 44.
    Takizawa K, Henicke B, Puntel A, Kostov N, Tezduyar T (2012) Space-time techiques for computational aerodynamics modeling of flapping wings of an actual locust. Comput Mech 50:743–760CrossRefzbMATHGoogle Scholar
  45. 45.
    Takizawa K, Henicke B, Puntel A, Spielman T, Tezduyar T (2012) Space-time techiques for the aerodynamics of flapping wings. J Appl Mech 79:010903CrossRefGoogle Scholar
  46. 46.
    Takizawa K, Kostov N, Puntel A, Henicke B, Tezduyar T (2012) Space-time computational analysis of bio-inspired flapping-wing aerodynamics of a micro aerial vehicle. Comput Mech 50:761–778CrossRefzbMATHGoogle Scholar
  47. 47.
    Takizawa K, Spielman T, Tezduyar T (2011) Space-time FSI modeling and dynamical analysis of spacecraft parachutes and parachute clusters. Comput Mech 48:345–364CrossRefzbMATHGoogle Scholar
  48. 48.
    Tezduyar T (1992) Stabilized finite element formulations for incompressible flow computations. Adv Appl Mech 28:1–44CrossRefzbMATHMathSciNetGoogle Scholar
  49. 49.
    Tezduyar T (2001) Finite element methods for flow problems with moving boundaries and interfaces. Arch Comput Methods Eng 8(2):83–130CrossRefzbMATHMathSciNetGoogle Scholar
  50. 50.
    Tezduyar T (2003) Computation of moving boundaries and interfaces and stabilization parameters. Int J Numer Methods Fluids 43:555–575CrossRefzbMATHMathSciNetGoogle Scholar
  51. 51.
    Tezduyar T (2006) Interface-tracking and interface-capturing techniques for finite element computation of moving boundaries and interfaces. Comput Methods Appl Mech Eng 195:2983–3000CrossRefzbMATHMathSciNetGoogle Scholar
  52. 52.
    Tezduyar T, Aliabadi S (2000) EDICT for 3D computation of two-fluid interfaces. Comput Methods Appl Mech Eng 190:403–410CrossRefzbMATHGoogle Scholar
  53. 53.
    Tezduyar T, Sathe S (2007) Modeling of fluid-structure interactions with the space-time finite elements: solution techniques. Int J Numer Methods Fluids 54:855–900CrossRefzbMATHMathSciNetGoogle Scholar
  54. 54.
    Tezduyar T, Behr M, Liou J (1992) A new strategy for finite element computations involving moving boundaries and interfaces–the deforming-spatial-domain/space-time procedure: I. The concept and the preliminary numerical tests. Comput Methods Appl Mech Eng 94:339–351CrossRefzbMATHMathSciNetGoogle Scholar
  55. 55.
    Tezduyar T, Behr M, Mittal S, Liou J (1992) A new strategy for finite element computations involving moving boundaries and interfaces—the deforming-spatial-domain/space-time procedure: II. Computation of free-surface flows, two-liquid flows, and flows with drifting cylinders. Comput Methods Appl Mech Eng 94:353–371CrossRefzbMATHMathSciNetGoogle Scholar
  56. 56.
    Tezduyar T, Aliabadi S, Behr M (1998) Enhanced-discretization interface-capturing technique (EDICT) for computation of unsteady flows with interfaces. Comput Methods Appl Mech Eng 155:235–248CrossRefzbMATHGoogle Scholar
  57. 57.
    Tezduyar T, Sathe S, Keedy R, Stein K (2006) Space-time finite element techniques for computation of fluid-structure interactions. Comput Methods Appl Mech Eng 195:2002–2027CrossRefzbMATHMathSciNetGoogle Scholar
  58. 58.
    Tezduyar T, Sathe S, Stein K (2006) Solution techniques for the fully discretized equations in computation of fluid-structure interaction with space-time formulations. Comput Methods Appl Mech Eng 195(41–43):5743–5753CrossRefzbMATHMathSciNetGoogle Scholar
  59. 59.
    Tezduyar T, Takizawa K, Moorman C, Wright S, Christopher J (2010) Space-time finite element computation of complex fluid-structure interaction. Int J Numer Meth Fluids 64:1201–1218CrossRefzbMATHGoogle Scholar
  60. 60.
    Wick T (2011) Adaptive finite element simulation of fluid-structure interaction with application to Heart-Valve Dynamics. Ph.D. thesis, University of HeidelbergGoogle Scholar
  61. 61.
    Wick T (2011) Fluid-structure interactions using different mesh motion techniques. Comput Struct 89(13–14):1456–1467CrossRefGoogle Scholar
  62. 62.
    Wick T (2012) Coupling of fully Eulerian with arbitrary Lagrangian-Eulerian coordinates for fluid-structure interaction. PreprintGoogle Scholar
  63. 63.
    Wick T (2012) Fully Eulerian fluid-structure interaction for time-dependent problems. Comput Methods Appl Mech Eng 255:14–26. doi: 10.1016/j.cma.2012.11.009 CrossRefMathSciNetGoogle Scholar
  64. 64.
    Wick T (2012) Goal-oriented mesh adaptivity for fluid-structure interaction with application to heart-valve settings. Arch Mech Eng 59(6):73–99MathSciNetGoogle Scholar
  65. 65.
    Wick T (2013) Coupling of fully Eulerian and arbitrary Lagrangian-Eulerian methods for fluid-structure interaction computations. Comput Mech. doi: 10.1007/s00466-013-0866-3 MathSciNetGoogle Scholar
  66. 66.
    Wick T (2013) Solving monolithic fluid-structure interaction problems in arbitrary Lagrangian Eulerian coordinates with the deal.ii library. Arch Numerl Software 1, 1–19. http://www.archnumsoft.org
  67. 67.
    Zhao H, Freund J, Moser R (2008) A fixed-mesh method for incompressible flow-structure systems with finite solid deformations. J Comput Phys 227(6):3114–3140CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.The Institute for Computational Engineering and SciencesThe University of Texas at AustinAustinUSA
  2. 2.Institute of Applied MathematicsHeidelberg UniversityHeidelbergGermany

Personalised recommendations