Exact 3D scattering solutions for spherical symmetric scatterers

Abstract

In this paper, exact solutions to the problem of acoustic scattering by elastic spherical symmetric scatterers are developed. The scatterer may consist of an arbitrary number of fluid and solid layers, and scattering with single Neumann conditions (replacing Neumann-to-Neumann conditions) is added. The solution is obtained by separation of variables, resulting in an infinite series which must be truncated for numerical evaluation. The implemented numerical solution is exact in the sense that numerical error is solely due to round-off errors, which will be shown using the symbolic toolbox in MATLAB. A system of benchmark problems is proposed for future reference. Numerical examples are presented, including comparisons with reference solutions, far-field patterns and near-field plots of the benchmark problems, and time-dependent solutions obtained by Fourier transformation.

IgA-BEM for 3D Helmholtz problems on multi-patch domains using B-spline tailored numerical integration

“…The Helmholtz equation is of particular interest in acoustic because the solution u = u(x, κ), x ∈ Ω of (1) can be interpreted as the inverse Fourier transform [24] of the time-dependent scattered pressure field p = p(x , t) generated by the scatterer in a given fluid domain, which fulfills the wave equation,…”

“…The product σ f Bj,d is a new spline that can be expressed in term of another B-spline basis defined in a suitable local "product" spline space of bi-degree p + d. Generalizing the idea developed in [19,11] for the one-dimensional case, the coefficients of σ f Bj,d in the B-spline basis of the product space can be written as G j Λ, with Λ defined by QI as specified in (24). In the current bivariate setting the matrix G j is defined as…”

Isogeometric analysis of acoustic scattering using infinite elements

“…Rigid scattering on a sphere and elastic scattering on a spherical shell are investigated in the following. These problems possess analytic solutions [35] and are for this reason often used to verify numerical methods in acoustic scattering, e.g. [14,30,36,31,37,38].…”

“…The third and final benchmark is a further extension with ASI conditions on both sides of the spherical shell (Neumann-Neumann conditions on both surfaces of the shell, NNBC). All of these benchmarks have analytic solutions [35] (see figs. In order to compare C 0 FEM and IGA on the scattering problem, we shall transform the NURBS mesh to a C 0 FEM mesh.…”

ADITU: A mesh-free formulation for the solution of Helmholtz equation in bounded and unbounded domains

“…The radiated pressure is expressed as a series of Hankel and Legendre functions; the coefficients of the series are determined using the known radiation pattern on the sphere. Some researchers have already been developing models to obtain the exact solutions of the scattering by elastic spheres [32].…”