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Arabian Journal for Science and Engineering

, Volume 41, Issue 2, pp 611–622 | Cite as

Adaptive FEM with Domain Decomposition Method for Partitioned-Based Fluid–Structure Interaction

  • Aizat AbasEmail author
  • Razi Abdul-Rahman
Research Article - Mechanical Engineering

Abstract

A numerical assessment on the solution performance of adaptive finite element methods for fluid–structure interaction (FSI) is presented in this paper. The partitioned-based approach involving separate fluid and structure solvers is considered for adaptive finite element computation with triangular mesh. The performance of the hp-adaptivity, in which error in energy norm is reduced by way of mesh refinement (h) and polynomial order extension (p), is of particular interest. In addition, parallel solution process based on the domain decomposition method is also assessed due to the complexity of two way coupling scheme. Based on its numerical convergence, the hp-adaptive procedure for the FSI problem is shown to yield the fastest convergence as generally expected. Moreover, our domain decomposition parallelization scheme shows reduction in the computation time by up to order 2. Subsequently, the convergence performance is highly dependent on the aspect ratios of the triangular elements. The solution performance strongly suggests the viability of the parallel hp-adaptive method in partitioned-based FSI formulation.

Keywords

Fluid–structure interaction Adaptive finite element method Arbitrary Lagrangian–Eulerian Error estimator Domain decomposition 

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References

  1. 1.
    Gu M., Xiang H., Lin Z.: Flutter- and buffeting-based selection for long-span bridges. J. Wind Eng. Ind. Aerodyn. 80(3), 373–382 (1999)CrossRefGoogle Scholar
  2. 2.
    Quaini A., Quarteroni A.: A semi implicit approach for fluid structure interaction based on an algebraic fractional step method. Math. Models Methods Appl. Sci. 17(6), 957–983 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Sigrist J.F., Broc D.: Fluid–structure interaction effects modeling for the modal analysis of a steam generator tube bundle. J. Press. Vessel Technol. 131(3), 031302 (2009)CrossRefGoogle Scholar
  4. 4.
    Turek S., Hron J., Razzaq M., Wobker H., Schäfer M.: Numerical benchmarking of fluid–structure interaction: a comparison of different discretization and solution approaches. In: Bungartz, H.J., Mehl, M., Schäfer, M. (eds) Fluid Structure Interaction II, pp. 413–424. Springer, Berlin (2010)Google Scholar
  5. 5.
    Ryzhakov P., Rossi R., Idelsohn S., Oñate E.: A monolithic Lagrangian approach for fluid–structure interaction problems. Comput. Mech. 46, 883–899 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Degroote J., Bathe K.J., Vierendeels J.: Performance of a new partitioned procedure versus a monolithic procedure in fluid–structure interaction. Comput. Struct. 87(11–12), 793–801 (2009)CrossRefGoogle Scholar
  7. 7.
    Grétarsson J.T., Kwatra N., Fedkiw R.: Numerically stable fluid–structure interactions between compressible flow and solid structures. J. Comput. Phys. 230(8), 3062–3084 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Causin P., Gerbeau J., Nobile F.: Added-mass effect in the design of partitioned algorithms for fluid–structure problems. Comput. Methods Appl. Mech. Eng. 194(4244), 4506–4527 (2005). doi: 10.1016/j.cma.2004.12.005 zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Degroote J., Bruggeman P., Haelterman R., Vierendeels J.: Stability of a coupling technique for partitioned solvers in {FSI} applications. Comput. Struct. 86(2324), 2224–2234 (2008). doi: 10.1016/j.compstruc.2008.05.005 CrossRefGoogle Scholar
  10. 10.
    Vierendeels, J.; Degroote, J.; Annerel, S.; Haelterman, R.: Stability issues in partitioned fsi calculations. In: Bungartz H.J., Mehl M., Schäfer M. (eds.) Fluid Structure Interaction II, Lecture Notes in Computational Science and Engineering, vol. 73, pp. 83–102. Springer, Berlin (2010). doi: 10.1007/978-3-642-14206-2-4
  11. 11.
    Dunne T.: An Eulerian approach to fluid–structure interaction and goal-oriented mesh adaptation. Int. J. Numer. Methods Fluids 51, 1017–1039 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Hoffman J., Jansson J., Stöckli M.: Unified continuum modeling of fluid–structure interaction. Math. Models Methods Appl. Sci. 21(3), 491–513 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Fernandez M.A., Gerbeau J.F., Grandmont C.: A projection semi-implicit scheme for the coupling of an elastic structure with an incompressible fluid. Int. J. Numer. Methods Eng. 69, 794–821 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Schwab C.: p- and hp- Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics. Clarendon Press, Oxford (1998)zbMATHGoogle Scholar
  15. 15.
    Oden, J.T.: Modeling and computational analysis of multiscale phenomena in fluid–structure interaction problems. Tech. Rep. AD-A248 723/9, Texas Inst. for Computational Mechanics, Austin (1992)Google Scholar
  16. 16.
    Richter T., Wick T.: Finite elements for fluid–structure interaction in ALE and fully Eulerian coordinates. Comput. Methods Appl. Mech. Eng. 199(41-44), 2633–2642 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Van der Zee, K.G.: Goal-adaptive discretization of fluid–structure interaction. Ph.D. thesis, TU Delft, Delft University of Technology (2009)Google Scholar
  18. 18.
    Abas M.A., Abdul-Rahman R.: Finite element method for fluid structure interaction with hp-adaptivity. Int. J. Comput. Theory Eng. 5(4), 663–667 (2013)CrossRefGoogle Scholar
  19. 19.
    Haghighat A., Binesh S.: Domain decomposition algorithm for coupling of finite element and boundary element methods. Arab. J. Sci. Eng. 39(5), 3489–3497 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Cermak M., Hapla V., Hork D., Merta M., Markopoulos A.: Total-FETI domain decomposition method for solution of elasto-plastic problems. Adv. Eng. Softw. 84, 48–54 (2005). doi: 10.1016/j.advengsoft.2014.12.011 CrossRefGoogle Scholar
  21. 21.
    Gosselet, P.; Rixen, D.; Roux, F.X.; Spillane, N.: Simultaneous-FETI and Block-FETI: robust domain decomposition with multiple search directions. Int. J. Numer. Methods Eng., p. online preview (2015). https://hal.archives-ouvertes.fr/hal-01056928
  22. 22.
    Maday Y., Magouls F.: Optimal convergence properties of the FETI domain decomposition method stiffness matrix of a floating structure. Int. J. Numer. Methods Fluids 55(1), 1–14 (2007)zbMATHCrossRefGoogle Scholar
  23. 23.
    Ballmann J., Behr M., Brix K., Dahmen W., Hohn C., Massjung R., Melian S., Müller S., Schieffer G.: Parallel and adaptive methods for fluid–structure-interactions. In: Schröder, W. (ed.) Summary of Flow Modulation and Fluid–Structure Interaction Findings, pp. 265–294. Springer, Berlin (2010)CrossRefGoogle Scholar
  24. 24.
    Paszyński M., Demkowicz L.: Parallel fully automatic hp-adaptive 3d finite element package. Eng. Comput. 22(3–4), 255–276 (2006)CrossRefGoogle Scholar
  25. 25.
    Arjmandi S., Lotfi V.: Comparison of three efficient methods for computing mode shapes of fluid–structure interaction systems. Arab. J. Sci. Eng. 38(4), 787–803 (2013). doi: 10.1007/s13369-012-0523-8 MathSciNetCrossRefGoogle Scholar
  26. 26.
    Belytschko T., Liu W.K., Moran B.: Nonlinear Finite Elements for Continua and Structures. 1st edn. Wiley, New York (2000)zbMATHGoogle Scholar
  27. 27.
    Che Sidik N., Salehi M.: Eulerian–Lagrangian numerical scheme for contaminant removal from different cavity shapes. Arab. J. Sci. Eng. 39(4), 3181–3189 (2014). doi: 10.1007/s13369-013-0886-5 CrossRefGoogle Scholar
  28. 28.
    Hron, J.; Turek, S.: A monolithic FEM/Multigrid solver for an ALE formulation of fluid–structure interaction with applications in biomechanics (2006). doi: 10.1007/3-540-34596-5_7
  29. 29.
    Hron, J.; Turek, S.: A monolithic FEM solver for an ALE formulation of fluid–structure interaction with configuration for numerical benchmarking. Ergeb. Angew. Math. Univ. (2006). http://books.google.com.my/books?id=py0rPwAACAAJ
  30. 30.
    Tallec P.L., Mouro J.: Fluid structure interaction with large structural displacements. Comput. Methods Appl. Mech. Eng. 190(2425), 3039–3067 (2001). doi: 10.1016/S0045-7825(00)00381-9 zbMATHCrossRefGoogle Scholar
  31. 31.
    Melenk J.M., Wohlmuth B.I.: On residual-based a posteriori error estimation in hp-fem. Adv. Comput. Math. 15(1-4), 311–331 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Galdi G., Rannacher R.: Fundamental Trends in Fluid–Structure Interaction. Contemporary Challenges in Mathematical Fluid Dynamics and Its Applications. World Scientific, Singapore (2010)CrossRefGoogle Scholar
  33. 33.
    Donea, J.; Huerta, A.; Ponthot, J.P.; Rodríguez-Ferran, A.: Arbitrary Lagrangian–Eulerian methods. In: Encyclopedia of Computational Mechanics, pp. 413–437. Wiley, New York (2004)Google Scholar
  34. 34.
    Segeth K.: A review of some a posteriori error estimates for adaptive finite element methods. Math. Comput. Simul. 80(8), 1589–1600 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Schober M., Kasper M.: Comparison of hp-adaptive methods in finite element electromagnetic wave propagation. COMPEL Int. J. Comput. Math. Electr. Electron. Eng. 26(2), 431–446 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Devine, K.D.; Flaherty, J.E.: Parallel adaptive hp-refinement techniques for conservation laws. Appl. Numer. Math. 20(4), 367–386 (1996). doi: 10.1016/0168-9274(95)00103-4, http://www.sciencedirect.com/science/article/pii/0168927495001034
  37. 37.
    Hoffman J., Jansson J., de Abreu R.V., Degirmenci N.C., Jansson N., Müller K., Nazarov M., Spühler J.H.: Unicorn: parallel adaptive finite element simulation of turbulent flow and fluid–structure interaction for deforming domains and complex geometry. Comput. Fluids 80, 310–319 (2012)CrossRefGoogle Scholar
  38. 38.
    Hussain, M.; Ahmad, M.; Abid, M.; Khokhar, A.: Implementation of 2D parallel ALE mesh generation technique in FSI problems using OpenMP. In: Proceedings of the 7th International Conference on Frontiers of Information Technology, FIT ’09, pp. 18:1–18:6. ACM, New York, NY, USA (2009)Google Scholar
  39. 39.
    Oden J.T., Patra A.: A parallel adaptive strategy for hp finite element computations. Comput. Methods Appl. Mech. Eng. 121(1–4), 449–470 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Brzobohat T., Dostl Z., Kozubek T., Kov P., Markopoulos A.: Cholesky decomposition with fixing nodes to stable computation of a generalized inverse of the stiffness matrix of a floating structure. Int. J. Numer. Methods Eng. 88(5), 493–509 (2011)CrossRefGoogle Scholar
  41. 41.
    Davis, T.A.: User Guide for CHOLMOD: A Sparse Cholesky Factorization and Modification Package. Technical Report, Dept. of Computer and Information Science and Engineering, University of Florida (2009)Google Scholar
  42. 42.
    Karypis G., Kumar V.: A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20(1), 359–392 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  43. 43.
    Agouzal A., Lipnikov K., Vassilevski Y.: On optimal convergence rate of finite element solutions of boundary value problems on adaptive anisotropic meshes. Math. Comput. Simul. 81, 1949–1961 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    Foster I.: Designing and Building Parallel Programs: Concepts and Tools for Parallel Software Engineering. Addison-Wesley Longman Publishing Co., Inc., Boston (1995)zbMATHGoogle Scholar

Copyright information

© King Fahd University of Petroleum & Minerals 2015

Authors and Affiliations

  1. 1.School of Mechanical EngineeringUniversiti Sains MalaysiaNibong TebalMalaysia

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