An efficient and highly accurate solver for multi-body acoustic scattering problems involving rotationally symmetric scatterers

Abstract

a b s t r a c tA numerical method for solving the equations modeling acoustic scattering in three dimensions is presented. The method is capable of handling several dozen scatterers, each of which is several wave-lengths long, on a personal work station. Even for geometries involving cavities, solutions accurate to seven digits or better were obtained. The method relies on a Boundary Integral Equation formulation of the scattering problem, discretized using a high-order accurate Nyström method. A hybrid iterative/direct solver is used in which a local scattering matrix for each body is computed, and then GMRES, accelerated by the Fast Multipole Method, is used to handle reflections between the scatterers. The main limitation of the method described is that it currently applies only to scattering bodies that are rotationally symmetric.

Efficient numerical solution of acoustic scattering from doubly-periodic arrays of axisymmetric objects

“…Analytic series for this kernel are presented by Conway-Cohl [24], and evaluation methods for them in the BIE setting, where the target may come very close to the source, are given in [74,41,42,43]. However, in the MFS setting, the target (ρ, z) is separated from the source (ρ , z ) by a distance of at least a few times the quadrature node spacing (see Sec.…”

“…(Recent work also shows that several digits of accuracy is possible with the MFS in corner domains [44,60].) We note that recently some technical challenges of high-order BIE on axisymmetric surfaces have been solved [74,41,42,43], but not in the case of transmission boundary conditions that we address.…”

Fast Multipole Method as a Matrix-Free Hierarchical Low-Rank Approximation

“…Scientific applications of FMM for boundary integrals include acoustics [95,57], biomolecular electrostatics [103], electromagnetics [33,41], fluid dynamics for Euler [94] and Stokes [86] flows, geomechanics [90], and seismology [21,93]. Application areas of FMM for discrete volume integrals are astrophysics [14], Brownian dynamics [73], classical molecular dynamics [82], density functional theory [88], vortex dynamics [104], and force directed graph layout [105].…”

Robust integral formulations for electromagnetic scattering from three-dimensional cavities

“…There are various numerical technicalities associated with implementing body-of-revolution integral equation solvers, and we do not seek to review the substantial literature here. We instead refer the reader to [23,24,28,34,39] and the references therein. A concise overview of the discretization and resulting solver is given in Section 5.…”

“…A concise overview of the discretization and resulting solver is given in Section 5. Similar high-order techniques have been applied to solve the Helmholtz equation on surfaces of revolution [23,24,34,39] and the full Maxwell equations (for closed-cavity resonance problems) in [25].Due to the applicability of cavity scattering in physics and engineering, there has been much work dedicated to both the mathematical and numerical aspects of the problem. The well-posedness of the (forward) scattering problem is discussed in [1,2] in the case of the two-dimensional problem, and in [3] for the three-dimensional case.…”