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Analysis of continuous formulations underlying the computation of time- harmonic acoustics in exterior domains. (English) Zbl 0769.76063
This paper reviews uniqueness properties of the solutions of boundary integral equations and analyses formulations for domain-based computations which impose a relation between the function and its normal derivative on an artificial boundary. The \(DtN\) formulation possesses non-reflective boundary conditions and gives rise to exact solutions but the practical implementation is often truncated. The truncated operator fails to inhibit reflection completely, resulting in loss of uniqueness, but simple expressions that determine a sufficient number of terms in the operator for unique solutions at any given wave number are derived in this paper.

MSC:
76Q05 Hydro- and aero-acoustics
76M15 Boundary element methods applied to problems in fluid mechanics
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[1] Chertock, G., Integral equation methods in sound radiation and scattering from arbitrary structures, NSRDC technical report 3538, (1971), Washington, DC
[2] Kleinman, R.E.; Roach, G.F., Boundary integral equations for the three-dimensional Helmholtz equation, SIAM rev., 16, 214-236, (1974) · Zbl 0253.35023
[3] Harari, I.; Hughes, T.J.R., A cost comparison of boundary element and finite element methods for problems of time-harmonic acoustics, Comput. methods appl. mech. engrg., 97, 77-102, (1991) · Zbl 0775.76095
[4] Copely, L.G., Fundamental results concerning integral representations in acoustic radiation, J. acoust. soc. amer., 44, 28-32, (1968) · Zbl 0162.57204
[5] Kupradze, V.D.; Benster, C.D., (), N.B.S. rep. 2008, (1952), translated by
[6] Lamb, H., Hydrodynamics, (1932), Cambridge Univ. Press Cambridge · JFM 26.0868.02
[7] Burton, A.J.; Miller, G.F., The application of integral equation methods to the numerical solution of some exterior boundary-value problems, (), 201-210 · Zbl 0235.65080
[8] Jones, D.S., Integral equations for the exterior acoustic problem, Quart. J. mech. appl. math., 27, 129-142, (1974) · Zbl 0281.45006
[9] Schenck, H.A., Improved integral formulation for acoustic radiation problems, J. acoust. soc. amer., 44, 41-58, (1968) · Zbl 0187.50302
[10] Cunefare, K.A.; Koopman, G.; Brod, K., A boundary element method for acoustic radiation valid for all wavenumbers, J. acoust. soc. amer., 85, 39-48, (1989)
[11] Hall, H.R.; Bernhard, R.J., Total least squares solutions to acoustic boundary element models, (), 145-152
[12] Kirkup, S.M., Solution of exterior acoustic problems by the boundary element method, () · Zbl 0775.73037
[13] Seybert, A.F.; Rengarajan, T.K., The use of CHIEF to obtain unique solutions for acoustic radiation using boundary integral equations, J. acoust. soc. amer., 81, 1299-1306, (1987)
[14] Bettess, P., Infinite elements, Internat. J. numer. methods engrg., 11, 53-64, (1977) · Zbl 0362.65093
[15] Bayliss, A.; Turkel, E., Radiation boundary conditions for wave-like equations, Comm. pure appl. math., 33, 707-725, (1980) · Zbl 0438.35043
[16] Abboud, N.N., A mixed finite element formulation for the transient and harmonic exterior fluid-structure interaction problem, ()
[17] Givoli, D.; Keller, J.B., A finite element method for large domains, Comput. methods appl. mech. engrg., 76, 41-66, (1989) · Zbl 0687.73065
[18] Courant, R.; Hilbert, D., ()
[19] Stakgold, I., ()
[20] Wilcox, C.H., A generalization of theorems of Rellich and atkinson, (), 271-276 · Zbl 0074.08102
[21] Wilcox, C.H., Scattering theory for the d’Alembert equation in exterior domains, (1975), Springer Berlin · Zbl 0299.35002
[22] Gonsalves, I.R.; Shippy, D.J.; Rizzo, F.J., The direct boundary integral equation method for the three-dimensional elastodynamic transmission problem, Math. comput. modell., 15, 155-164, (1991) · Zbl 0727.73087
[23] Miller, R.D.; Huang, H.; Moyer, E.T.; Uberall, H., The analysis of the radiated and scattered acoustic fields from submerged shell structures using a modal finite element/boundary element formulation, (), 83-94
[24] 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. amer., 77, 362-368, (1985) · Zbl 0574.73038
[25] Amini, S.; Ke, C.; Harris, P.J., Iterative solution of boundary element equations for the exterior Helmholtz problem, (), 123-128
[26] Kleinman, R.E.; Roach, G.F.; Schuetz, L.S.; Shirron, J., An iterative solution to acoustic scattering by rigid objects, J. acoust. soc. amer., 84, 385-391, (1988)
[27] Keller, J.B.; Givoli, D., Exact non-reflecting boundary conditions, J. comput. phys., 82, 172-192, (1989) · Zbl 0671.65094
[28] Hughes, T.J.R., The finite element method: linear static and dynamic finite element analysis, (1987), Prentice Hall Englewood Cliffs, NJ
[29] Givoli, D.; Keller, J.B., Non-reflecting boundary conditions for elastic waves, Wave motion, 12, 261-279, (1990) · Zbl 0708.73012
[30] Morse, P.M.; Feshbach, H., Methods of theoretical physics, (1953), McGraw-Hall New York · Zbl 0051.40603
[31] D. Givoli, Private communication, 1990.
[32] Harari, I.; Hughes, T.J.R., Design and analysis of finite element methods for the Helmholtz equation in exterior domains, Appl. mech. rev., 43, 366-373, (1990)
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