Sensitivity of the Acoustic Scattering Problem in Prolate Spheroidal Geometry with Respect to Wavenumber and Shape
D. Kourounis, L.N. Gergidis, and A. Charalambopoulos

Source CMES: Computer Modeling in Engineering & Sciences, Vol. 28, No. 3, pp. 185-202, 2008
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Keywords Prolate Spheroid; Acoustic Scattering; Vekua Transformation; Arbitrary Precision;$L^2$-norm Minimization; Mathematical Modeling; Numerical Acoustics; Scientific Computing.
Abstract The sensitivity of analytical solutions of the direct acoustic scattering problem in prolate spheroidal geometry on the wavenumber and shape, is extensively investigated in this work. Using the well known Vekua transformation and the complete set of radiating ``outwards'' eigensolutions of the Helmholtz equation, introduced in our previous work ([\relax \begingroup \catcode `\ 12\relax \catcode `\\12\relax \catcode `\$12\relax \catcode `\&12\relax \catcode `\#12\relax \catcode `\^12\relax \catcode `\_12\relax \catcode `\%12\relax \catcode `\~12\relax \endgroup \relax \cite *{CharalambopoulosDassios:2002}],[\relax \begingroup \catcode `\ 12\relax \catcode `\\12\relax \catcode `\$12\relax \catcode `\&12\relax \catcode `\#12\relax \catcode `\^12\relax \catcode `\_12\relax \catcode `\%12\relax \catcode `\~12\relax \endgroup \relax \cite *{Gergidis:2007}]), the scattered field is expanded in terms of it, detouring so the standard spheroidal wave functions along with their inherent numerical deficiencies. An approach is employed for the determination of the expansion coefficients, which is optimal in the sense, that minimizes the$L^2$ norm of the error related to the satisfaction of the boundary condition on the surface of the scatterer. The study of the conditioning of the matrices involved in the linear systems, the solution of which provides the expansion coefficients, reveals the need for implication of numerical implementations using arbitrary precision arithmetics. Numerical and convergence properties estimations such as condition numbers,$L^2$ and$L^{\infty }$ error norms prove the robustness of the adopted methodology. A study of the dependence of the error with respect to geometrical, physical and numerical parameters is developed. Three dimensional representation of the$L^2$ norm clarifies the distribution of errors on the scatterer's surface.
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