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Two efficient methods for a multifrequency solution of the Helmholtz equation

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Computing and Visualization in Science

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In this paper, two efficient methods for the multifrequency analysis for the Helmholtz equation are compared. The first approach is based on the direct collocation method with a subsequent approximation of the matrix by the Adaptive Cross Approximation method. Using the so-called Fourier scheme, the elements of the matrices for a series of frequencies can be computed efficiently.

In the second approach an indirect Galerkin type method with piecewise linear ansatz and test functions is combined with a special source simulation technique. This combination allows a rather accurate and systematic approximation of acoustical results in major parts of the considered frequency range, leading to a significant reduction of the computer time needed to calculate complete frequency spectra. A representative example demonstrates how the two proposed approaches can be used. Both procedures turn out to be very promising steps towards a more efficient calculation of complex sound radiation problems.

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References

  1. Bebendorf, M.: Approximation of boundary element matrices. Numer. Math. 86(4), 565–589 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bebendorf, M. Rjasanow, S.: Adaptive low-rank approximation of collocation matrices. Computing, 70(1), 1–24 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chen, G. Zhou, J.: Boundary element methods. Academic Press Ltd., London (1992)

    MATH  Google Scholar 

  4. Cheng, H., Greengard, L. Rokhlin, V.: A fast adaptive multipole algorithm in three dimensions. J. Comput. Phys. 155(2), 468–498 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Colton, D.L., Kress, R.: Integral equation methods in scattering theory. Krieger Publishing Company, Malabar (1992)

  6. Hackbusch, W.: A sparse matrix arithmetic based on H-matrices. I. Introduction to H-matrices. Computing 62(2), 89–108 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Hackbusch, W., Khoromskij, B.N.: A sparse H-matrix arithmetic. II. Application to multi-dimensional problems. Computing 64(1), 21–47 (2000)

    MATH  MathSciNet  Google Scholar 

  8. Hackbusch, W. Nowak, Z.P.: On the fast matrix multiplication in the boundary element method by panel clustering. Numer. Math. 54(4), 463–491 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  9. Köhl, M., Rjasanow, S.: Multifrequency analysis for the Helmholtz equation, Comput. Mech. 32(4–6),234–239 (2003)

    Article  MATH  Google Scholar 

  10. Stolper, M.: Computing and compression of the boundary element matrices for the Helmholtz equation. J. Numer. Math. 12(1), 55–75 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  11. von Estorff, O., Zaleski, O.: Efficient acoustic calculations by the BEM and frequency interpolated transfer functions. Eng. Anal. Bound. Elem. 27(7), 683–694 (2003)

    Article  MATH  Google Scholar 

  12. Hamdi M.A.: Formulation variationelle par equations integrales pour la resolution de léquation de Helmholtz avec des conditions aux limites mixtes. C.R. Acad. Sc. Paris 292, 17–20 (1981)

    MATH  MathSciNet  Google Scholar 

  13. Coyette J.P., Van der Peer I.J.: Acoustic Boundary Elements. 17th International Seminar on Modal Analysis, Leuven (1992)

  14. Coyette J.P., Fyfe K.R.: Solution of elasto-acoustic problems using a variational finite element/boundary element technique Proc. of Winter Annual Meeting of ASME: 15–25, San Francisco (1989)

  15. von Estorff O.(Ed.): Boundary elements in acoustics - advances and applications, WIT Press, Southampton (2000)

  16. Tournour, M., Cremers, L., Guisset, P., Augusztinovicz, F., Márki, F.: Inverse numerical acoustics based on acoustic transfer vectors. Proc. of the Seventh International Congress on Sound and Vibration: 2069–2076, Garmisch-Partenkirchen (2000)

  17. Cremers, L., Tournour, M., C.F.: McCulloch. panel acoustic contribution analysis based on acoustic transfer vectors. Inter Noise 2001, The Haughe.

  18. Morse, P.M., Freshbach, H.: Methods of theoretical physics. McGraw-Hill, New York (1953)

  19. Milton, Abramowitz, Stegun (Eds.).: Handbook of Mathematical Functions. Dover Publications, New York (1970)

    Google Scholar 

  20. Zaleski, O., Cremers, L., von Estorff, O.: Zur Anwendung akustischer übertragungsfunktionen bei der Berechnung von Schallfeldern mit der Boundary-Elemente-Methode. Tagungsband DAGA 2001, Hamburg (2001)

  21. von Estorff, O., Zaleski, O.: Effektive Berechnung von Schallfeldern mittels akustischer Transfervektoren. 19th CAD-FEM Users' Meeting 2001, Potsdam (2001)

  22. Rausch, M., Kaltenbacher, M., Landes, H., Lerch, R., Anger, J., Gerth, J. Precise, Numeri-cal simulation of load-controlled noise of power transformers. Proc. of the Seventh International Congress on Sound and Vibration: 1895–1902 Garmisch-Partenkirchen (2000)

  23. Ochmann, M.: Fast algorithms for the calculation of sound fields in interior spaces. Acustica 86, 191 (2000)

    Google Scholar 

  24. Tournour, M., Rossion, J.-P., Bricteux, L., McCulloch, C.F.: Getting useful FEM and BEM Vibro-acoustic solutions faster, using new solution methodologies. Proc. of the ISMA 2002, Leuven (2002)

  25. Wagner, M.; Pinsky, P.M., Malhotra, M.: A Multiple-Frequency Partial-Field Method for Exterior Acoustics Based on Padé via Lanczos Approximants. Proceedings of the ASME International Mechanical Engineering Congress and Exposition, New York (2001)

  26. Seybert, A.F., Wu, T.W.: Acoustic Modeling: Boundary Element Methods. Encyclo-pedia of Acoustics, Chapter 15, Malcolm J. Crocker, ed., John Wiley and Sons, Inc., New York 173–184 (1997)

  27. Tröndle, G., Jäger, M., Antes, H.: Efficient Calculation of Acoustic Fields by Bound-ary Element Method. In Boundary Element Topics, W. Wendland (Ed.), Springer, Heidelberg, 9–29 (1997)

  28. LMS International. User's Manual SYSNOISE Rev 5.5. Leuven (2000)

  29. Zaleski, O., von Karstedt, W.-Ch., von Estorff, O.: Zur Modellierung mit Boundary Elementen und Finiten Elementen bei Schallabstrahlungsberechnungen. 1. Deutsch-sprachige Anwender-Konferenz SYSNOISE, Bühlerhöhe (1999)

  30. Schenk, H.A.: Improved Integral Formulation for Acoustic Radiation Problems. J. Ac. Soc. Am. 44 (1968)

  31. Klemenz, M., von Estorff, O., Heckl, M.: Schallabstrahlung eines Katalysators mit der BEM und dem Rayleigh-Verfahren. Tagungsband DAGA'95, Saarbrücken (1995)

  32. von Estorff, O., Tylkowski, B., Scarth, Ph.: Sound Radiation of Two Engine Blocks by the BEM and the Rayleigh-Method. Proc. of the 2nd Worldwide SYSNOISE Users Meeting, Leuven (1995)

  33. Benthien, G.W., Schenck, H.A.: Structural-Acoustic, Coupling, Chapter 6, Boundary Element Methods in Acoustics, Computational Mechanics Publications, Elsevier Ap-plied Science, Editors R.D. Ciskowski and C.A. Brebbia (1991)

  34. von Estorff, O., Coyette, J.P., Guisset, P.: Neuere Entwicklungen zur Berechnung der Schallabstrahlung mit Boundary Elementen. Tagungsband des XXIII. Internationalen FE-Kongresses (IKOSS), Baden-Baden (1994)

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Authors and Affiliations

  1. Modelling and Computation, Hamburg University of Technology, Denickestr. 17, D-21073, Hamburg, Germany

    Otto von Estorff & Olgierd Zaleski

  2. Department of Mathematics, Saarland University, P.O.B. 15 11 50, D-66041, Saarbrücken, Germany

    Sergej Rjasanow & Mirjam Stolper

Authors
  1. Otto von Estorff
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  2. Sergej Rjasanow
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  3. Mirjam Stolper
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  4. Olgierd Zaleski
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Correspondence to Otto von Estorff.

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Communicated by: O. Steinbach

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Estorff, O.v., Rjasanow, S., Stolper, M. et al. Two efficient methods for a multifrequency solution of the Helmholtz equation. Comput. Visual Sci. 8, 159–167 (2005). https://doi.org/10.1007/s00791-005-0004-7

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