On the numerical implementation of a Cauchy principal value integral to insure a unique solution for acoustic radiation and scattering

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Abstract
In this paper, the boundary integral formulation proposed by Burton and Miller [Proc. R. Soc. London Ser. A 323, 201–210 (1971)] to insure a unique solution for all frequencies is implemented in an isoparametric element environment. A regularized normal derivative integral equation, originally derived by Maue [Z. Phys. 126, 601–618 (1949)], is used in the formulation to form a linear combination with the conventional Helmholtz integral equation. This regularized normal derivative integral equation converges in the Cauchy principal value sense rather than only in the finite-part sense. The Cauchy principal value integral can be further transformed into an integral that converges in the normal sense. The C° continuous isoparametric elements are used in the formulation. Collocation points are placed inside each element to insure a unique normal direction and continuity of tangential derivatives of the acoustic pressure. Through a systematic collocation point generation scheme, the number of collocation points is always greater than the number of nodal points. The overdetermined system is then solved by a least-squares procedure. Numerical examples are given for several radiation and scattering problems.

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