Computational Mechanics

, Volume 53, Issue 4, pp 549–560 | Cite as

Sound transmission through a poroelastic layered panel

  • Loris Nagler
  • Ping Rong
  • Martin SchanzEmail author
  • Otto von Estorff
Original Paper


Multi-layered panels are often used to improve the acoustics in cars, airplanes, rooms, etc. For such an application these panels include porous and/or fibrous layers. The proposed numerical method is an approach to simulate the acoustical behavior of such multi-layered panels. The model assumes plate-like structures and, hence, combines plate theories for the different layers. The poroelastic layer is modelled with a recently developed plate theory. This theory uses a series expansion in thickness direction with subsequent analytical integration in this direction to reduce the three dimensions to two. The same idea is used to model either air gaps or fibrous layers. The latter are modeled as equivalent fluid and can be handled like an air gap, i.e., a kind of ‘air plate’ is used. The coupling of the layers is done by using the series expansion to express the continuity conditions on the surfaces of the plates. The final system is solved with finite elements, where domain decomposition techniques in combination with preconditioned iterative solvers are applied to solve the final system of equations. In a large frequency range, the comparison with measurements shows very good agreement. From the numerical solution process it can be concluded that different preconditioners for the different layers are necessary. A reuse of the Krylov subspace of the iterative solvers pays if several excitations have to be computed but not that much in the loop over the frequencies.


Poroelasticity Plate Iterative solver with recycling subspace Multilayered panels Sound transmission 



The authors gratefully acknowledge the financial support by a common project of the Austrian Science Fund (FWF, Grant No P18481-N13) and the German Research Foundation (DFG, Grant No ES 70/4-1). The material investigations of the porous blankets have been made by Rieter Management AG, Winterthur, Switzerland.


  1. 1.
    Allard JF (1993) Propagation of sound in porous media. Elsevier Applied Science, LondonCrossRefGoogle Scholar
  2. 2.
    Allard JF, Champoux Y, Depollier C (1987) Modelization of layered sound absorbing materials with transfer matrices. J Acoust Soc Am 82(5):1792–1796CrossRefGoogle Scholar
  3. 3.
    Arnold DN, Falk RS, Winther R (1997) Preconditioning discrete approximations of the Reissner-Mindlin plate model. Math Mod Numer Anal 31(4):517–557zbMATHMathSciNetGoogle Scholar
  4. 4.
    Atalla N, Hamdi MA, Panneton R (2001) Enhanced weak integral formulation for the mixed (u, p) poroelastic equations. J Acoust Soc Am 109(6):3065–3068CrossRefGoogle Scholar
  5. 5.
    Balay S, Brown J, Buschelman K, Gropp WD, Kaushik D, Knepley MG, McInnes LC, Smith BF, Zhang H (2012) PETSc web page.
  6. 6.
    Biermann J, von Estorff O (2011) Beschleunigung von Multifrequenz-FE-Analysen durch die Verwendung von Gleichungsloesern mit Subspace Recycling. In: The 37th German annual conference on acoustics (DAGA)Google Scholar
  7. 7.
    Biot MA (1941) General theory of three-dimensional consolidation. J Appl Phys 12:155–164CrossRefzbMATHGoogle Scholar
  8. 8.
    Biot MA (1956) Theory of propagation of elastic waves in a fluid-saturated porous solid. I/II. Low/higher frequency range. J Acoust Soc Am 28(2):168–191CrossRefMathSciNetGoogle Scholar
  9. 9.
    de Sturler E (1996) Nested Krylov methods based on GCR. J Comput Appl Math 67(1):15–41CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Dreyer D, Petersen S, von Estorff O (2006) Effectiveness and robustness of improved infinite elements for exterior acoustics. Comput Methods Appl Mech Eng 195:3591–3607CrossRefzbMATHGoogle Scholar
  11. 11.
    Fahy F (2001) Foundations of engineering acoustics. Academic Press, San DiegoGoogle Scholar
  12. 12.
    Gaidamour J, Henon P (2008) A parallel direct/iterative solver based on a Schur complement approach. In: 11th IEEE international conference on computational science and engineering, computational science and engineering, pp 98–105Google Scholar
  13. 13.
    Ihlenburg F (1998) Finite element analysis of acoustic scattering, applied mathematical sciences, vol 132. Springer, HamburgCrossRefGoogle Scholar
  14. 14.
    ISO 140-3 (1995) Acoustics, measurement of sound insulation in buildings and of building elements—laboratory measurements of airborne sound insulation of building elementsGoogle Scholar
  15. 15.
    Jacobsen F, Roisin T (2000) The coherence of reverberant sound fields. J Acoust Soc Am 108(1):204–210CrossRefGoogle Scholar
  16. 16.
    Kang YJ, Bolton JS (1995) Finite element modeling of isotropic elastic porous materials coupled with acoustical finite elements. J Acoust Soc Am 98:635–643CrossRefGoogle Scholar
  17. 17.
    Kienzler R (2002) On consistent plate theories. Arch Appl Mech 72:229–247CrossRefzbMATHGoogle Scholar
  18. 18.
    Kienzler R (2004) On consistent second-order plate theories. In: Kienzler R, Altenbach H, Ott I (eds) Theories of plates and shells: critical review and new applications, lecture notes in applied and computational mechanics, vol 16. Springer-Verlag, Berlin, pp 85–96CrossRefGoogle Scholar
  19. 19.
    Kirk BS, Peterson JW, Stogner RH, Carey GF (2006) libMesh: a C++ library for parallel adaptive mesh refinement/coarsening simulations. Eng Comput 22(3–4):237–254. Google Scholar
  20. 20.
    Nagler L (2011) Simulation of sound transmission through poroelastic plate-like structures, computation in engineering and science, vol 10. Verlag der Technischen Universität Graz, Graz Google Scholar
  21. 21.
    Nagler L, Schanz M (2010) An extendable poroelastic plate formulation in dynamics. Arch Appl Mech 80:1177–1195CrossRefzbMATHGoogle Scholar
  22. 22.
    Parks ML, de Sturler E, Mackey G, Johnson DD, Maiti S (2006) Recycling Krylov subspaces for sequences of linear systems. SIAM J Sci Comput 28(5):1651–1674CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Rong P, von Estorff O, Nagler L, Schanz M (2013) An acoustical finite shell element for the simulation of air layers. J Comput Acoustics. doi: 10.1142/S0218396X13500148
  24. 24.
    Saad Y, Suchomel B (2002) Arms: an algebraic recursive multilevel solver for general sparse linear systems. Numer Linear Algebra Appl 9:359–378CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Wandel M (2006) Validation of A400M FE-Model for primary insulation and lining material. Technischer Bericht/Airbus-Akustikabteilung, HamburgGoogle Scholar
  26. 26.
    Zienkiewicz O, Taylor R (2000) The finite element method, 5th edn. Butterworth Heinemann, OxfordzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Loris Nagler
    • 1
  • Ping Rong
    • 2
  • Martin Schanz
    • 1
    Email author
  • Otto von Estorff
    • 3
  1. 1.Institute of Applied MechanicsGraz University of TechnologyGrazAustria
  2. 2.Department of Acoustics and VibrationsBMW GroupMunichGermany
  3. 3.Institute of Modelling and ComputationHamburg University of TechnologyHamburgGermany

Personalised recommendations