A Fast Multipole Boundary Element Method for Three-Dimensional Half-Space Acoustic Wave Problems Over an Impedance Plane

Haijun Wu, Yijun Liu, Weikang Jiang, Wenbo Lu
2015 International Journal of Computational Methods  
A high-frequency fast multipole boundary element method (FMBEM) based on the Burton-Miller formulation is proposed for three-dimensional acoustic wave problems over an infinite plane with impedance boundary conditions. The Green's function for the sound propagation over an impedance plane is employed explicitly in the boundary integral equation (BIE). To deal with the integral appearing in the half-space Green's function, the downward pass in the FMBEM is divided into two parts to compute
more » ... ts to compute contributions from the real domain to the real and image domains, respectively. A piecewise analytical method is proposed to compute the moment-to-local (M2L) translator from the real domain to the image domain accurately. An algorithm based on the multi-level tree structure is designed to compute the M2L translators efficiently. Correspondingly, the direct coefficient can also be computed efficiently by taking advantage of the algorithm of the efficient M2L. A flexible generalized minimal residual (fGMRES) is applied to accelerating the solution when the convergence is very slow. Numerical examples are presented to demonstrate the accuracy and efficiency of the developed FMBEM. Good solutions and high acceleration ratios compared with the conventional boundary element method clearly show the potential of the FMBEM for large-scale 3D acoustic wave problems over an infinite impedance plane which are of practical significance. Keywords: Fast multipole boundary element method; impedance plane; acoustic wave problems; half-space domain. 1350090-1 Int. J. Comput. Methods Downloaded from www.worldscientific.com by SHANGHAI JIAO TONG UNIVERSITY on 02/05/14. For personal use only. H. Wu et al. studied and used extensively for the numerical solutions of radiation and scattering problems for full-space acoustics. The half-space BEM which has a significant practical importance has also been investigated [Seybert and Soenarko (1988); ; Brick and Ochmann (2008) ]. For the half-space BEM in acoustic simulations, the discretization of the infinite plane is removed and only the boundaries of the structure need to be discretized. The main challenge in the half-space BEM is to find a proper Green's function satisfying the half-space boundary condition. Except for the perfectly rigid or soft infinite plane [Seybert and Soenarko (1988) ], no elementary expression is available to the half-space Green's function for problems over a general infinite impedance plane. The general half-space Green's function (a solution due to a point source over an infinite impedance plane) which involves a Sommerfeld integral is not practically useful when incorporated in the BEM. Much research has been devoted to finding a simple expression of the half-space Green's function and an efficient way to compute it. A theoretical analysis of the half-space Green's function along an impedance plane was presented by Wenzel [1974], where the Green's function was separated into three parts as incident, reflected and radiated wave. Thomasson [1976] found that the exact solution of waves from a point source over an impedance boundary given by Ingard [1951] can be rewritten in terms of a single integral along a steepestdescent contour and a Hankel function which can be easily integrated numerically. An asymptotic solution of the scalar wave field due to a point source above a locally reacting plane surface was obtained by a modified saddle point method [Kawai et al. (1982) ], which was proved more accurate than Thomasson's method [Thomasson (1976)]. Based on a technique in which the solution of the Helmholtz equation is expressed as one fold integral with an integrand identified as the solution of the heat conduction equation for an auxiliary problem, Li et al. [1994] used a new method to derive the Green's function for wave propagation above an impedance ground. Later on, Li and White [1996] developed a very simple method for the efficient computation of the sound field in the near region above an impedance ground. Ochmann [2004] proposed a method called complex equivalent source method to derive the half-space Green's function which is valid for the infinite plane with mass-like as well as spring-like impedance. By converting the Sommerfeld integral into two series representations, Koh and Yook [2006] proposed an exact closed-form expression of the Sommerfeld integral in the general half-space Green's function. Chen and Waubke [2010] reformulated the expression proposed by to make it suitable for two-dimensional half-space problems over an infinite plane with spring-like impedance. Even though some of the approaches introduced above are efficient in computing the half-space Green's function over an infinite impedance plane compared with the direct integration method, they are still time-consuming to be applied in the BEM for solving practical and large-scale problems. It is well-known that the fast multipole method (FMM) [Greengard and Rokhlin (1987) ] can be applied to reduce the operation counts for the BEM in the matrix-vector multiplication. Many fast multipole BEM (FMBEM) approaches 1350090-2 Int. J. Comput. Methods Downloaded from www.worldscientific.com by SHANGHAI JIAO TONG UNIVERSITY on 02/05/14. For personal use only.
doi:10.1142/s0219876213500904 fatcat:xe47jevvvbdnxnbbarxvrggjda