Integral equation methods with unique solution for all wavenumbers applied to acoustic radiation
Keywords:
acoustic radiation, boundary element method, irregular frequenciesAbstract
The acoustic exterior Neumann problem is solved using an easy process based upon the boundary element method and able to eliminate effects of irregular frequencies in time harmonic domain. This technique is performed as follows: (i) two computations are done around the characteristic frequency, decreased and increased by a small imaginary part; (ii) average between pressures at these two frequencies ensures unique solution for all wavenumbers. This method is numerically tested for an infinite cylinder, an axisymmetric cylinder, a sphere and a three-dimensional cat’s eye structure. This work highlights ease and efficiency of the technique under consideration to remove the irregular frequencies effects.
Downloads
References
Abramowitz M., Stegun I. A., Handbook of Mathematical Functions with Formulas, Graphs,
and Mathematical Tables, Dover Publications, New York, 1964.
Alves C. J. S., « The method of fundamental solutions for scattering and radiation problems »,
Engng Anal Bound Elem, vol. 27, p. 759-769, 2003.
BenthienW., Schenck A., « Nonexistence and nonuniqueness problems associated with integral
equation methods in acoustics », Comput. Struct., vol. 65, n 3, p. 295-305, 1997.
Brebbia C. A., Boundary Element Methods in Acoustics, Springer, 1991.
Burton A. J., Miller G. F., « The application of integral equation methods to the numerical
solution of some exterior boundary-value problems », Proc. Roy. Soc. London, vol. A323,
p. 201-210, 1971.
Chen I. L., « Using the method of fundamental solutions in conjunction with the degenerate
kernel in cylindrical acoustic problems », J. Chin. Inst. Eng., vol. 29, n 3, p. 445-457, 2006.
Chen Z. S., Hofstetter G., Mang H. A., « A symmetric Galerkin formulation of the boundary
element method for acoustic radiation and scattering », J. Comput. Acoust., vol. 5, p. 219-
, 1997.
Copley L. G., « Fundamental Results Concerning Integral Representations in Acoustic Radiation
», J. Acoust. Soc. Am., vol. 44, p. 41-58, 1968.
Courant R., Hilbert D., Methods of Mathematical Physics, vol. 1, Wiley, New York, 1953.
Fairweather G., Karageorghis A., Martin P. A., « The method of fundamental solutions for
scattering and radiation problems », Eng. Anal. Boundary Elem., vol. 27, p. 759-769, 2003.
Jones D. S., « Integral equations for the exterior acoustic problem », Q. J. Mech appl. Math,
vol. XXVII, p. 129-142, 1974.
Juhl P., « A numerical study of the coefficient matrix of the boundary element method near
characteristic frequencies », J. Sound Vib., vol. 175, p. 39-50, 1994.
Koopmann G. H., Song L., Fahnline J. B., « A method for computing acoustic fields based
on the principle of wave superposition », J. Acoust. Soc. Am., vol. 86, n 6, p. 2433-2438,
December, 1989.
Lavie A., Modélisation du rayonnement ou de la diffraction acoustique par une méthode mixte
équations intégrales-champ nul, PhD thesis, Université des Sciences et Techniques de Lille,
Leblanc A., Ing R. K., Lavie A., « A Wave Superposition Method Based on Monopole Sources
with Unique Solution for AllWave Numbers », Acta Acustica united with Acustica, vol. 96,
p. 125-130, 2010.
Makarov S. N., Ochmann M., « An iterative solver of the Helmholtz integral equation for highfrequency
acoustic scattering », J. Acoust. Soc. Am., vol. 103, p. 742-750, February, 1998.
Marburg S., Amini S., « Cat’s eye radiation with boundary elements: comparative study on
treatment of irregular frequencies », J. Comput. Acoust., vol. 13, n 1, p. 21-45, 2005.
Marburg S., Nolte B., Acoustics of Noise Propagation in Fluids: Finite and Boundary Element
Methods, Springer, 2008.
Marburg S., Schneider S., « Performance of iterative solvers for acoustic problems. Part I.
Solvers and effect of diagonal preconditioning », Eng. Anal. Boundary Elem., vol. 27, n 7,
p. 727 - 750, 2003. Special issue on Acoustics.
Marschall R. A., « Boundary element solution of a body’s exterior acoustic field near its internal
eigenvalues », J. Comput. Acoust., vol. 1, p. 335-353, 1993.
Meyer W. L., Bell W. A., Zinn B. T., Stalybrass M. P., « Boundary integral solutions of three
dimensional acoustic radiation problems », J. Sound Vib., vol. 59, p. 245-262, July, 1978.
Schenck H. A., « Improved integral formulation for acoustic radiation problems », J. Acoust.
Soc. Am., vol. 44, p. 41-58, 1968.
Schneider S., Marburg S., « Performance of iterative solvers for acoustic problems. Part II.
Acceleration by ILU-type preconditioner », Eng. Anal. Boundary Elem., vol. 27, n 7, p. 751
- 757, 2003. Special issue on Acoustics.
Segalman D. J., Lobitz D. W., SuperCHIEF: a modified CHIEF method, Technical Report n
SAND-90-1266, SANDIA Labs, 1990.
Segalman D. J., Lobitz D.W., « A method to overcome computational difficulties in the exterior
acoustics problem », J. Acoust. Soc. Am., vol. 91, n 4, p. 1855-1861, 1992.
Seybert A. F., Rengarajan T. K., « The use of CHIEF to obtain unique solutions for acoustic
radiation using boundary-integral equations », J. Acoust. Soc. Am., vol. 81, n S1, p. S99-
S99, 1987.
Seybert A. F., Soenarko B., Rizzo F. J., Shippy D. J., « An advanced computational method for
radiation and scattering of acoustic waves in three dimensions », J. Acoust. Soc. Am., vol.
, p. 362-368, 1985.
Stupfel B., Lavie A., Decarpigny J.-N., « Combined integral equation and null-field method for
the exterior acoustic problem », J. Acoust. Soc. Am., vol. 83, p. 927-941, 1988.
Waterman P. C., « New Formulation of Acoustic Scattering », J. Acoust. Soc. Am, vol. 45, n 6,
p. 1417-1429, 1969.
Wilcox C. H., « A generalization of theorems of Rellich and Atkinson », Proc. Am. Math. Soc.,
vol. 7, p. 271-276, 1956.
Wilton D. T., Mathews I. C., Jeans R. A., « A clarification of nonexistence problems with the
superposition method », J. Acoust. Soc. Am, vol. 94, n 3, p. 1676-1680, September, 1993.
Wu T. W., Seybert A. F., « A weighted residual formulation for the CHIEF method in acoustics
», J. Acoust. Soc. Am., vol. 90, n 3, p. 1608-1614, 1991.
Yang S. A., « An integral equation approach to three-dimensional acoustic radiation and scattering
problems », J. Acoust. Soc. Am., vol. 116, p. 1372-1380, September, 2004.
Zienkiewicz O. C., The finite element method in engineering science, McGraw-Hill, Maidenhead,
