Advertisement

Archives of Computational Methods in Engineering

, Volume 14, Issue 4, pp 343–381 | Cite as

Trefftz-Based Methods for Time-Harmonic Acoustics

  • B. Pluymers
  • B. van Hal
  • D. Vandepitte
  • W. Desmet
Article

Abstract

Over the last decade, Computer Aided Engineering (CAE) tools have become essential in the assessment and optimization of the acoustic characteristics of products and processes. The possibility of evaluating these characteristics on virtual prototypes at almost any stage of the design process reduces the need for very expensive and time consuming physical prototype testing. However, despite their steady improvements and extensions, CAE techniques are still primarily used by analysis specialists. In order to turn them into easy-to-use, versatile tools that are also easily accessible for designers, several bottlenecks have to be resolved. The latter include, amongst others, the lack of efficient numerical techniques for solving system-level functional performance models in a wide frequency range. This paper reviews the CAE modelling techniques which can be used for the analysis of time-harmonic acoustic problems and focusses on techniques which have the Trefftz approach as baseline methodology. The basic properties of the different methods are highlighted and their strengths and limitations are discussed. Furthermore, an overview is given of the state-of-the-art of the extensions and the enhancements which have been recently investigated to enlarge the application range of the different techniques. Specific attention is paid to one very promising Trefftz-based technique, which is the so-called wave based method. This method has all the necessary attributes for putting a next step in the evolution towards truly virtual product design.

Keywords

Boundary Element Method Helmholtz Equation Acoustic Problem Boundary Integral Formulation Helmholtz Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Morse P, Ingard K (1968) Theoretical acoustics. McGraw-Hill, New York Google Scholar
  2. 2.
    Pierce A (1981) Acoustics: an introduction to its physical principles and applications. McGraw-Hill series in mechanical engineering. McGraw-Hill, New York Google Scholar
  3. 3.
    Zienkiewicz OC, Taylor RL, Zhu JZ, Nithiarasu P (2005) The finite element method—the three volume set, 6th edn. Butterworth-Heinemann, London Google Scholar
  4. 4.
    Ihlenburg F (1998) Finite element analysis of acoustic scattering. Applied Mathematical Sciences, vol 132. Springer, New York MATHGoogle Scholar
  5. 5.
    Brebbia CA, Telles JCF, Wrobel LCL (1984) Boundary element techniques: theory and applications in engineering. Springer, New York MATHGoogle Scholar
  6. 6.
    Kirkup S (1998) The boundary element method in acoustics. Integrated Sound Software Google Scholar
  7. 7.
    Von Estorff O (2000) Boundary elements in acoustics: advances and applications. WIT Press Google Scholar
  8. 8.
    Zienkiewicz OC (2000) Achievements and some unsolved problems of the finite element method. Int J Num Methods Eng 47:9–28 MATHMathSciNetGoogle Scholar
  9. 9.
    Proceedings of the fifth world congress on computational mechanics, Vienna, Austria, 2002 Google Scholar
  10. 10.
    Proceedings of the European congress on computational methods in applied sciences and engineering, Jyvaskyla, Finland, 2004 Google Scholar
  11. 11.
    Proceedings of the 2004 international conference on noise and vibration engineering, Leuven, Belgium, 2004 Google Scholar
  12. 12.
    Proceedings of the 13th international congress on sound and vibration, Vienna, Austria, 2006 Google Scholar
  13. 13.
    Proceedings of the 2006 international conference on noise and vibration engineering, Leuven, Belgium, 2006 Google Scholar
  14. 14.
    Desmet W (2002) Mid-frequency vibro-acoustic modelling: challenges and potential solutions. In: Proceedings of the 2002 international conference on noise and vibration engineering, Leuven, Belgium, pp 835–862 Google Scholar
  15. 15.
    Harari I (2006) A survey of finite element methods for time-harmonic acoustics. Comput Methods Appl Mech Eng 195:1594–1607 MATHMathSciNetGoogle Scholar
  16. 16.
    Thompson LL (2006) A review of finite element methods for time-harmonic acoustics. J Acoust Soc Am 119:1315–1330 Google Scholar
  17. 17.
    Trefftz E (1926) Ein Gegenstück zum Ritzschen Verfahren. In: Proceedings of the 2nd international congress on applied mechanics, Zürich, Switzerland, pp 131–137 Google Scholar
  18. 18.
    Jirousek J, Wróblewski A (1996) T-elements: state of the art and future trends. Arch Comput Methods Eng 3:323–434 Google Scholar
  19. 19.
    Kita E, Kamiya N (1995) Trefftz method: an overview. Adv Eng Softw 24:3–12 MATHGoogle Scholar
  20. 20.
    Desmet W, Van Hal B, Sas P, Vandepitte D (2002) A computationally efficient prediction technique for the steady-state dynamic analysis of coupled vibro-acoustic systems. Adv Eng Softw 33:527–540 MATHGoogle Scholar
  21. 21.
    Van Hal B, Desmet W, Vandepitte D, Sas P (2003) A coupled finite element—wave based approach for the steady-state dynamic analysis of acoustic systems. J Comput Acoust 11:255–283 MathSciNetGoogle Scholar
  22. 22.
    Pluymers B, Desmet W, Vandepitte D, Sas P (2004) Application of an efficient wave based prediction technique for the analysis of vibro-acoustic radiation problems. J Comput Appl Math 168:353–364 MATHMathSciNetGoogle Scholar
  23. 23.
    Pluymers B, Desmet W, Vandepitte D, Sas P (2005) On the use of a wave based prediction technique for steady-state structural-acoustic radiation analysis. J Comput Model Eng Sci 7(2):173–184 MATHGoogle Scholar
  24. 24.
    Van Hal B, Desmet W, Vandepitte D (2005) Hybrid finite element–wave based method for steady-state interior structural-acoustic problems. Comput Struct 83:167–180 Google Scholar
  25. 25.
    Colton D, Kress R (1998) Inverse acoustic and electromagnetic scattering theory, 2nd edn. Springer, New York MATHGoogle Scholar
  26. 26.
    Desmet W (2006) Boundary element method in acoustics. Course notes ISAAC17: seminar on advanced techniques in applied and numerical acoustics, Leuven, Belgium, September 2006 Google Scholar
  27. 27.
    Wolf JP, Song C (1996) Finite element modelling of unbounded media. Wiley, Chichester MATHGoogle Scholar
  28. 28.
    Thompson LL, Pinsky PM (2004) Acoustics. Encyclopedia of computational mechanics Google Scholar
  29. 29.
    Shirron JJ, Babuška I (1998) A comparison of approximate boundary conditions and infinite element methods for exterior Helmholtz problems. Comput Methods Appl Mech Eng 164:121–139 MATHGoogle Scholar
  30. 30.
    Givoli D (2004) High-order local non-reflecting boundary conditions: a review. Wave Motion 39:319–326 MATHMathSciNetGoogle Scholar
  31. 31.
    Keller JB, Givoli D (1989) Exact non-reflecting boundary conditions. J Comput Phys 82:172–192 MATHMathSciNetGoogle Scholar
  32. 32.
    Harari I, Patlashenko I, Givoli D (1998) Dirichlet-to-Neumann maps for unbounded wave guides. J Comput Phys 143:200–223 MATHMathSciNetGoogle Scholar
  33. 33.
    Nicholls DP, Nigam N (2004) Exact non-reflecting boundary conditions on general domains. J Comput Phys 194:278–303 MATHMathSciNetGoogle Scholar
  34. 34.
    Bettess P (1992) Infinite elements. Penshaw Google Scholar
  35. 35.
    Gerdes K (2000) A review of infinite element methods for exterior Helmholtz problems. J Comput Acoust 8:43–62 MathSciNetGoogle Scholar
  36. 36.
    Burnett DS, Holford RL (1998) Prolate and oblate spheroidal acoustic infinite elements. Comput Methods Appl Mech Eng 158:117–141 MATHMathSciNetGoogle Scholar
  37. 37.
    Burnett DS, Holford RL (1998) An ellipsoidal acoustic infinite element. Comput Methods Appl Mech Eng 164:49–76 MATHMathSciNetGoogle Scholar
  38. 38.
    Astley RJ, Macaulay GJ, Coyette JP (1994) Mapped wave envelope elements for acoustical radiation and scattering. J Sound Vib 170:97–118 MATHGoogle Scholar
  39. 39.
    Astley RJ, Macaulay GJ, Coyette JP, Cremers L (1998) Three-dimensional wave-envelope elements of variable order for acoustic radiation and scattering. Part I. Formulation in the frequency domain. J Acoust Soc Am 103:49–63 Google Scholar
  40. 40.
    Astley RJ, Coyette JP (2001) The performance of spheroidal infinite elements. Int J Numer Methods Eng 52:1379–1396 MATHGoogle Scholar
  41. 41.
    Berenger J-P (1994) A perfectly matched layer for the absorption of electromagnetic waves. J Comput Phys 114:185–200 MATHMathSciNetGoogle Scholar
  42. 42.
    Collino F, Monk P (1998) The perfectly matched layer in curvilinear coordinates. SIAM J Sci Comput 19:2061–2090 MATHMathSciNetGoogle Scholar
  43. 43.
    Bouillard P, Ihlenburg F (1999) Error estimation and adaptivity for the finite element method in acoustics: 2D and 3D applications. Comput Methods Appl Mech Eng 176:147–163 MATHGoogle Scholar
  44. 44.
    Bartsch G, Wulf C (2003) Adaptive multigrid for Helmholtz problems. J Comput Acoust 11:341–350 MathSciNetGoogle Scholar
  45. 45.
    Harari I, Avraham D (1997) High-order finite element methods for acoustic problems. J Comput Acoust 5:33–51 Google Scholar
  46. 46.
    Dey S, Datta DK, Shirron JJ, Shephard MS (2006) p-Version FEM for structural acoustics with a posteriori error estimation. Comput Methods Appl Mech Eng 195:1946–1957 MATHGoogle Scholar
  47. 47.
    Burnett DS (2004) 3D Structural acoustic modelling with hp-adaptive finite elements. In: Proceedings of the XXI international congress of theoretical and applied mechanics ICTAM, Warsaw, Poland Google Scholar
  48. 48.
    Ainsworth M, Oden TJ (1997) Posteriori error estimation in finite element analysis. Comput Methods Appl Mech Eng 142:1–88 MATHMathSciNetGoogle Scholar
  49. 49.
    Zienkiewicz OC (2006) The background of error estimation and adaptivity in finite element computations. Comput Methods Appl Mech Eng 195:207–213 MATHMathSciNetGoogle Scholar
  50. 50.
    Oden JT, Prudhomme S, Demkowicz L (2005) A posteriori error estimation for acoustic wave propagation. Arch Comput Methods Eng 12:343–390 MATHMathSciNetGoogle Scholar
  51. 51.
    Thompson LL, Kunthong P (2005) A residual based variational method for reducing dispersion error in finite element methods. In: Proceedings of the 2005 ASME international mechanical engineering congress and exposition, Orlando, Florida, USA, paper IMECE2005-80551 Google Scholar
  52. 52.
    Guddati MN, Yue B (2004) Modified integration rules for reducing dispersion error in finite element methods. Comput Methods Appl Mech Eng 193:275–287 MATHGoogle Scholar
  53. 53.
    Duff I (1998) Direct methods. Technical Report RAL-TR-1998-054, Rutherford Appleton Laboratory, Oxon, USA Google Scholar
  54. 54.
    Saad Y (2003) Iterative methods for sparse linear systems, 2nd edn. SIAM, Philadelphia MATHGoogle Scholar
  55. 55.
    Craig RR Jr (1981) Structural dynamics: an introduction to computer methods. Wiley, New York Google Scholar
  56. 56.
    Bennighof JK, Kaplan MF (1998) Frequency window implementation of adaptive multi-level substructuring. J Vib Acoust 120:409–418 Google Scholar
  57. 57.
    Bennighof JK, Kaplan MF, Muller MB, Kim M (2000) Meeting the NVH computational challenge: automated multi-level substructuring. In: Proceedings of the international modal analysis conference XVIII, San Antonio, USA, pp 909–915 Google Scholar
  58. 58.
    Kropp A, Heiserer D (2003) Efficient broadband vibro-acoustic analysis of passenger car bodies using an FE-based component mode synthesis approach. J Comput Acoust 11:139–157 Google Scholar
  59. 59.
    Stryczek R, Kropp A, Wegner S (2004) Vibro-acoustic computations in the mid-frequency range: efficiency, evaluation and validation. In: Proceedings of the international conference on noise and vibration engineering ISMA2004, Leuven, Belgium, pp 1603–1612 Google Scholar
  60. 60.
    Després B (1991) Méthodes de décomposition de domaine pour les problémes de propagation d’ondes en régime harmonique. PhD thesis, Paris IX Dauphine, Paris, France Google Scholar
  61. 61.
    Cai X, Casarin M, Elliott F Jr, Widlund O (1998) Overlapping Schwartz algorithms for solving Helmholtz’s equation. Contemp Math 218:391–399 MathSciNetGoogle Scholar
  62. 62.
    Farhat C, Macedo A, Lesoinne M (2000) A two-level domain decomposition method for the iterative solution of high frequency exterior Helmholtz problems. Numer Math 85:283–308 MATHMathSciNetGoogle Scholar
  63. 63.
    Farhat C, Roux F-X (1991) A method of finite element tearing and interconnecting and its parallel solution algorithm. Int J Numer Methods Eng 32:1205–1227 MATHGoogle Scholar
  64. 64.
    Farhat C, Macedo A, Lesoinne M, Roux F-X, Magoules F, de La Bourdonnaie A (2000) Two-level domain decomposition methods with Lagrange multipliers for the fast iterative solution of acoustic scattering problems. Comput Methods Appl Mech Eng 184:213–239 MATHGoogle Scholar
  65. 65.
    Datta DK, Dey S, Shirron JJ (2005) Scalable three-dimensional acoustics using hp-finite/infinite elements and FETI-DP. In: Proceedings of the 16th international domain decomposition conference, New York, USA Google Scholar
  66. 66.
    Magoules F (2006) Decomposition of domains in acoustics. http://www.iecn.u-nancy.fr/~magoules/pai/greenwich/aeroacoustics2/
  67. 67.
    Mandel J (2002) An iterative substructuring method for coupled fluid-solid acoustic problems. J Comput Phys 177:95–116 MATHMathSciNetGoogle Scholar
  68. 68.
    Harari I, Magoules F (2004) Numerical investigations of stabilized finite element computations for acoustics. Wave Motion 39:339–349 MATHMathSciNetGoogle Scholar
  69. 69.
    Harari I, Hughes TJR (1992) Galerkin/least-squares finite element methods for the reduced wave equation with non-reflecting boundary conditions in unbounded domains. Comput Methods Appl Mech Eng 98:411–454 MATHMathSciNetGoogle Scholar
  70. 70.
    Thompson LL, Pinsky PM (1995) A Galerkin least squares finite element method for the two-dimensional Helmholtz equation. Int J Numer Methods Eng 38:371–397 MATHMathSciNetGoogle Scholar
  71. 71.
    Franca LP, Dutra do Carmo EG (1989) Galerkin gradient least-squares method. Comput Methods Appl Mech Eng 74:41–54 MATHMathSciNetGoogle Scholar
  72. 72.
    Harari I (1997) Reducing spurious dispersion, anisotropy and reflection in finite element analysis of time-harmonic acoustics. Comput Methods Appl Mech Eng 140:39–58 MATHMathSciNetGoogle Scholar
  73. 73.
    Babus̆ka I, Ihlenburg F, Paik ET, Sauter SA (1995) A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution. Comput Methods Appl Mech Eng 128:325–359 MathSciNetGoogle Scholar
  74. 74.
    Babus̆ka I, Sauter SA (1997) Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM J Numer Anal 34:2392–2423 MathSciNetGoogle Scholar
  75. 75.
    Yue B, Guddati MN (2003) Highly accurate local mesh-dependent augmented Galerkin (L-MAG) FEM for simulation of time-harmonic wave propagation. In: Proceedings of the 16th ASCE engineering mechanics conference, Seattle, USA Google Scholar
  76. 76.
    Melenk JM (1995) On generalized finite element methods. PhD thesis, University of Maryland at College Park Google Scholar
  77. 77.
    Babus̆ka I, Melenk JM (1997) The partition of unity method. Int J Numer Methods Eng 40:727–758 MathSciNetGoogle Scholar
  78. 78.
    Melenk JM, Babus̆ka I (1996) The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech Eng 139:289–314 MATHMathSciNetGoogle Scholar
  79. 79.
    Laghrouche O, Bettess P (2000) Short wave modelling using special finite elements. J Comput Acoust 8:189–210 MathSciNetGoogle Scholar
  80. 80.
    Strouboulis T, Babus̆ka I, Copps K (2000) The design and analysis of the generalized finite element method. Comput Methods Appl Mech Eng 181:43–69 MATHMathSciNetGoogle Scholar
  81. 81.
    Strouboulis T, Copps K, Babus̆ka I (2001) The generalized finite element method. Comput Methods Appl Mech Eng 190:4081–4193 MATHMathSciNetGoogle Scholar
  82. 82.
    Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P (1996) Meshless methods: an overview and recent developments. Comput Methods Appl Mech Eng 139:3–47 MATHGoogle Scholar
  83. 83.
    Bouillard P, Suleau S (1998) Element-free Galerkin solutions for Helmholtz problems: formulation and numerical assessment of the polution effect. Comput Methods Appl Mech Eng 162:317–335 MATHGoogle Scholar
  84. 84.
    Suleau S, Deraemaeker A, Bouillard P (2000) Dispersion and pollution of meshless solutions for the Helmholtz equation. Comput Methods Appl Mech Eng 190:639–657 MATHMathSciNetGoogle Scholar
  85. 85.
    Lacroix V, Bouillard P, Villon P (2003) An iterative defect-correction type meshless method for acoustics. Int J Numer Methods Eng 57:2131–2146 MATHGoogle Scholar
  86. 86.
    Lancaster P, Salkauskas K (1981) Surfaces generated by moving least squares methods. Math Comput 37:141–158 MATHMathSciNetGoogle Scholar
  87. 87.
    Hughes TJR (1995) Multiscale phenomena: Green’s functions, Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput Methods Appl Mech Eng 127:387–401 MATHGoogle Scholar
  88. 88.
    Hughes TJR, Feijoo GR, Mazzei L, Quincy J-B (1998) The variational multiscale method—a paradigm for computational mechanics. Comput Methods Appl Mech Eng 166:3–24 MATHMathSciNetGoogle Scholar
  89. 89.
    Franca L, Farhat C, Macedo A, Lesoinne M (1997) Residual-free bubbles for Helmholtz equation. Int J Numer Methods Eng 40:4003–4009 MATHMathSciNetGoogle Scholar
  90. 90.
    Brezzie F, Franca LP, Hughes TJR, Russo A (1997) b= g. Comput Methods Appl Mech Eng 145:329–339 Google Scholar
  91. 91.
    Cipolla JL (1999) Subgrid modeling in a Galerkin method for the Helmholtz equation. Comput Methods Appl Mech Eng 177:35–49 MATHMathSciNetGoogle Scholar
  92. 92.
    Oberai AA, Pinsky PM (1998) A multiscale finite element method for the Helmholtz equation. Comput Methods Appl Mech Eng 154:281–297 MATHMathSciNetGoogle Scholar
  93. 93.
    Farhat C, Harari I, Franca LP (2001) The discontinuous enrichment method. Comput Methods Appl Mech Eng 190:6455–6479 MATHMathSciNetGoogle Scholar
  94. 94.
    Farhat C, Harari I, Hetmaniuk U (2003) The discontinuous enrichment method for multiscale analysis. Comput Methods Appl Mech Eng 192:3195–3209 MATHMathSciNetGoogle Scholar
  95. 95.
    Marburg S (2002) Six boundary elements per wavelength: is that enough? J Comput Acoust 10:25–51 Google Scholar
  96. 96.
    Harari I, Hughes TJR (1992) A cost comparison of boundary element and finite element methods for problems of time-harmonic acoustics. Comput Methods Appl Mech Eng 98:77–102 Google Scholar
  97. 97.
    Herrin DW, Martinus F, Wu TW, Seybert AF (2006) An assessment of the high frequency boundary element and Rayleigh integral approximations. Appl Acoust 67:819–833 Google Scholar
  98. 98.
    Nagy AB, Fiala P, Marki F, Augustinovicz F, Degrande G, Jacobs S, Brassenx D (2006) Prediction of interior noise in buildings generated by underground rail traffic. J Sound Vib 293:680–690 Google Scholar
  99. 99.
    DeBiesme FX, Verheij JW, Verbeek G (2003) Lumped parameter BEM for faster calculations of sound radiation from vibrating structures. In: Proceedings of the tenth international congress on sound and vibration, Stockholm, Sweden, pp 1515–1522 Google Scholar
  100. 100.
    DeBiesme FX, Verheij JW (2004) Speed gains of the lumped parameter boundary element method for vibroacoustics. In: Proceedings of the international conference on noise and vibration engineering ISMA2004, Leuven, Belgium, pp 3765–3778 Google Scholar
  101. 101.
    Fahnline JB, Koopman GH (1996) A lumped parameter model for the acoustic power output from a vibrating structure. J Acoust Soc Am 100:3539–3547 Google Scholar
  102. 102.
    Perrey-Debain E, Trevelyan J, Bettess P (2004) Wave boundary elements: a theoretical overview presenting applications in scattering of short waves. J Eng Anal Bound Elem 28:131–141 MATHGoogle Scholar
  103. 103.
    Perrey-Debain E, Laghrouche O, Bettess P, Trevelyan J (2004) Plane wave basis finite elements and boundary elements for three dimensional wave scattering. Phil Trans R Soc A 362:561–577 MATHMathSciNetGoogle Scholar
  104. 104.
    Chandler-Wilde NS, Langdon S, Ritter L (2004) A high-wavenumber boundary-element method for an acoustic scattering problem. Phil Trans R Soc A 362:647–671 MATHMathSciNetGoogle Scholar
  105. 105.
    Arden S, Chandler-Wilde NS, Langdon S (2005) A collocation method for high frequency scattering by convex polygons. Technical Report Numerical Analysis Report 7/05, Reading University Google Scholar
  106. 106.
    Huybrechs D, Vandewalle S (2006) On the evaluation of highly oscillatory integrals by analytic continuation. SIAM J Numer Anal 44:1026–1048 MATHMathSciNetGoogle Scholar
  107. 107.
    Huybrechs D, Vandewalle S (2005) The construction of cubature rules for multivariate highly oscillatory integrals. Technical Report Technisch rapport TW-442, KU Leuven Google Scholar
  108. 108.
    Huybrechs D, Vandewalle S (2006) A sparse discretisation for integral equation formulations of high frequency scattering problems. Technical Report Technisch rapport TW-447, KU Leuven Google Scholar
  109. 109.
    Huybrechs D (2006) Multiscale and hybrid methods for the solution of oscillatory integral equations. PhD thesis, KU Leuven, Leuven, Belgium. http://www.cs.kuleuven.be/~daan/phd/thesis.pdf
  110. 110.
    Sakuma T, Yasuda Y (2002) Fast multipole boundary element method for large-scale steady-state sound field analysis. Part I: setup and validation. Acustica 88:513–525 Google Scholar
  111. 111.
    Yasuda Y, Sakuma T (2002) Fast multipole boundary element method for large-scale steady-state sound field analysis. Part I: examination of numerical items. Acustica 89:28–38 Google Scholar
  112. 112.
    Schneider S (2003) Application of fast methods for acoustic scattering and radiation problems. J Comput Acoust 11:387–401 Google Scholar
  113. 113.
    Fischer M, Gauger U, Gaul L (2004) A multipole Galerkin boundary element method for acoustics. Eng Anal Bound Elem 28:155–162 MATHGoogle Scholar
  114. 114.
    Yasuda Y, Sakuma T (2005) An effective setting of hierarchical cell structure for the fast multipole boundary element method. J Comput Acoust 13:47–70 MATHGoogle Scholar
  115. 115.
    Yasuda Y, Sakuma T (2005) A technique for plane-symmetric sound field analysis in fast multipole boundary element method. J Comput Acoust 13:71–86 MATHGoogle Scholar
  116. 116.
    Fischer M, Gaul L (2005) Fast BEM-FEM mortar coupling for acoustic-structure interaction. Int J Numer Methods Eng 62:1677–1690 MATHGoogle Scholar
  117. 117.
    Cheung YK, Jin WG, Zienkiewicz OC (1991) Solution of Helmholtz problems by Trefftz method. Int J Numer Methods Eng 32:63–78 MATHGoogle Scholar
  118. 118.
    Herrera I (1984) Boundary methods: an algebraic theory. Pitman, London MATHGoogle Scholar
  119. 119.
    Masson P, Redon E, Priou J-P, Gervais Y (1994) The application of the Trefftz method for acoustics. In: Proceedings of the third international congress on sound and vibration, Montreal, Canada, pp 1809–1816 Google Scholar
  120. 120.
    Sladek J, Sladek V, Van Keer R (2002) Global and local Trefftz boundary integral formulations for sound vibration. Adv Eng Softw 33:469–476 MATHGoogle Scholar
  121. 121.
    Zielinski AP, Zienkiewicz OC (1985) Generalized finite element analysis with T-complete boundary solution functions. Int J Numer Methods Eng 21:509–528 MATHGoogle Scholar
  122. 122.
    Zielinski AP, Herrera I (1987) Trefftz method: fitting boundary conditions. Int J Numer Methods Eng 24:871–891 MathSciNetGoogle Scholar
  123. 123.
    Qin Q-H (2000) The Trefftz finite and boundary element method. WIT Press, Southampton MATHGoogle Scholar
  124. 124.
    Kompis V, Stiavnicky M (2005) Trefftz functions in FEM, BEM and meshless methods. In: Proceedings of the 10th international conference on numerical methods for continuum mechanics & the 4th international workshop on Trefftz methods, Zilina, Slovak Republic Google Scholar
  125. 125.
    Ochmann M (1995) The source simulation technique for acoustic radiation problems. Acustica 81:512–527 MATHGoogle Scholar
  126. 126.
    Cessenat O, Deprés B (1998) Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem. SIAM J Numer Anal 35:255–299 MATHMathSciNetGoogle Scholar
  127. 127.
    Kansa EJ (1990) Multiquadrics: a scattered data approximation scheme with applications to computational fluid-dynamics-II solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput Math Appl 19:147–161 MathSciNetGoogle Scholar
  128. 128.
    Van Hal B (2004) Automation and performance optimization of the wave based method for interior structural-acoustic problems. PhD thesis, KU Leuven, Leuven, Belgium. http://www.mech.kuleuven.be/dept/resources/docs/vanhal.pdf
  129. 129.
    Farhat C, Harari I, Hetmaniuk U (2003) A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency range. Comput Methods Appl Mech Eng 192:1389–1419 MATHMathSciNetGoogle Scholar
  130. 130.
    Tezaur R, Farhat C (2006) Three-dimensional discontinuous Galerkin elements with plane waves and Lagrange multipliers for the solution of mid-frequency Helmholtz problems. Int J Numer Methods Eng 66:796–815 MATHMathSciNetGoogle Scholar
  131. 131.
    Jirousek J, Wróblewski A (1994) Least-squares T-elements: equivalent FE and BE forms of substructure oriented boundary solution approach. Commun Numer Methods Eng 10:21–32 MATHGoogle Scholar
  132. 132.
    Stojek M (1996) Finite T-elements for the Poisson and Helmholtz equations. PhD thesis, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland Google Scholar
  133. 133.
    Stojek M (1998) Least-squares Trefftz-type elements for the Helmholtz equation. Int J Numer Methods Eng 41:831–849 MATHMathSciNetGoogle Scholar
  134. 134.
    Monk P, Wang D-Q (1999) A least-squares method for the Helmholtz equation. Comput Methods Appl Mech Eng 175:121–136 MATHMathSciNetGoogle Scholar
  135. 135.
    Riou H, Ladeveze P, Rouch P (2004) Extension of the variational theory of complex rays to shells for medium-frequency vibrations. J Sound Vib 272:341–360 Google Scholar
  136. 136.
    Desmet W (1998) A wave based prediction technique for coupled vibro-acoustic analysis. PhD thesis, KU Leuven, Leuven, Belgium. http://people.mech.kuleuven.ac.be/~wdesmet/desmet_phd_thesis.pdf
  137. 137.
    Pluymers B, Vanmaele C, Desmet W, Vandepitte D, Sas P (2004) Application of a novel wave based prediction technique for acoustic cavity analysis. In: Proceedings of the 30th German convention on acoustics (DAGA) together with the 7th congrès Francais d’acoustique (CFA), Strasbourg, France, pp 313–314 Google Scholar
  138. 138.
    Pluymers B (2006) Wave based modelling methods for steady-state vibro-acoustics. PhD thesis, KU Leuven, Leuven, Belgium. http://people.mech.kuleuven.be/~bpluymer/docs/thesis_bpluymers.pdf
  139. 139.
    Pluymers B, Vanmaele C, Desmet W, Vandepitte D (2005) Application of a hybrid finite element—Trefftz approach for acoustic analysis. In: Proceedings of the 10th international conference on numerical methods for continuum mechanics & the 4th international workshop on Trefftz methods, Zilina, Slovak Republic Google Scholar
  140. 140.
    Van Genechten B, Pluymers B, Vanmaele C, Vandepitte D, Desmet W (2006) On the coupling of wave based models with modally reduced finite element models for structural-acoustic analysis. In: Proceedings of the international conference on noise and vibration engineering ISMA2006, Leuven, Belgium, pp 2383–2404 Google Scholar
  141. 141.
    Benamou JD, Depres B (1997) A domain decomposition method for the Helmholtz equation and related optimal control problems. J Comput Phys 136:68–82 MATHMathSciNetGoogle Scholar
  142. 142.
    Koopmann GH, Song L, Fahnline JB (1989) A method for computing acoustic fields based on the principle of wave superposition. J Acoust Soc Am 86:2433–2438 Google Scholar
  143. 143.
    Stupfel B, Lavie A, Decarpigny JH (1988) Combined integral equation formulation and null-field method for the exterior acoustic problem. J Acoust Soc Am 83:927–941 Google Scholar
  144. 144.
    Reutskiy SY (2006) The method of fundamental solutions for Helmholtz eigenvalue problems in simply and multiply connected domains. Eng Anal Bound Elem 30:150–159 Google Scholar
  145. 145.
    Ochmann M (1999) The full-field equations for acoustic radiation and scattering. J Acoust Soc Am 105:2574–2584 Google Scholar
  146. 146.
    Ochmann M (2004) The complex equivalent source method for sound propagation over an impedance plane. J Acoust Soc Am 116:3304–3311 Google Scholar
  147. 147.
    Pavic G (2005) An engineering technique for the computation of sound radiation by vibrating bodies using substitute sources. Acustica 91:1–16 Google Scholar
  148. 148.
    Pavic G (2005) A technique for the computation of sound radiation by vibrating bodies using multipole substitute sources. In: Proceedings of the international congress on noise and vibration emerging methods (NOVEM2005), Saint-Raphael, France Google Scholar
  149. 149.
    Bouchet L, Loyau T, Hamzaoui N, Boisson C (2000) Calculation of acoustic radiation using equivalent-sphere methods. J Acoust Soc Am 107:2387–2397 Google Scholar
  150. 150.
    Reboul E, Perret-Liaudet J, Le Bot A (2005) Vibroacoustic prediction of mechanisms using a hybrid method. In: Proceedings of the international congress on noise and vibration emerging methods (NOVEM2005), Saint-Raphael, France Google Scholar
  151. 151.
    Herrin DW, Wu TW, Seybert AF (2004) The energy source simulation method. J Sound Vib 278:135–153 Google Scholar
  152. 152.
    Chen W (2002) Symmetric boundary knot method. Eng Anal Bound Elem 26:489–494 MATHGoogle Scholar
  153. 153.
    Goldbert MA, Chen CS (1999) The MFS for potential, Helmholtz and diffusion problems. In: Golberg MA (ed) Boundary integral methods: numerical and mathematical aspects. WIT Press and Computational Mechanics Publ, Boston Southampton, Chapter 4 Google Scholar
  154. 154.
    Alves CJS, Valtchev SS (2005) Numerical comparison of two meshfree methods for acoustic wave scattering. Eng Anal Bound Elem 29:371–382 Google Scholar
  155. 155.
    Limic N (1981) Galerkin-Petrov method for Helmholtz equation on exterior problems. Glas Math 36:245–260 MathSciNetGoogle Scholar
  156. 156.
    Fairweather G, Kargeorghis A (1998) The MFS for elliptic BVPs. Adv Comput Math 9:69–95 MATHMathSciNetGoogle Scholar
  157. 157.
    Chen CS, Golberg MA, Schaback RS (2002) Transformation of domain effects to the boundary. In: Rashed YF (ed) Recent developments of the dual reciprocity method using compactly supported radial basis functions. WIT Press, Southampton, Chapter 8 Google Scholar
  158. 158.
    Huttunen T, Gamallo P, Astley RJ (2006) Comparison of two wave element methods for the Helmholtz problem. Comput Methods Appl Mech Eng Google Scholar
  159. 159.
    Sas P, Desmet W, Van Hal B, Pluymers B (2001) On the use of a wave based prediction technique for vehicle interior acoustics. In: Proceedings of the Styrian noise, vibration & harshness congress: integrated vehicle acoustics and comfort, Graz, Austria, pp 175–185 Google Scholar

Copyright information

© CIMNE, Barcelona, Spain 2007

Authors and Affiliations

  • B. Pluymers
    • 1
  • B. van Hal
    • 1
  • D. Vandepitte
    • 1
  • W. Desmet
    • 1
  1. 1.Department of Mechanical EngineeringKatholieke Universiteit LeuvenHeverleeBelgium

Personalised recommendations