Abstract
This paper reviews the equivalent source method (ESM), an attractive alternative to the standard boundary element method (BEM). The ESM has been developed under different names: method of fundamental solutions, wave superposition method, equivalent source method, etc. However, regardless of the method name, the basic concept is very similar; that is to use auxiliary points called equivalent sources to reconstruct the acoustic pressure for radiation or scattering problems. The strength of the equivalent sources are then determined via various approaches such that the boundary conditions on the boundary surface are satisfied. This paper reviews several frequency-domain and time-domain ESMs. There are several distinct advantages in these types of methods: (1) the method is a meshless approach so that it is easy and simple to implement; (2) it does not have a numerical singularity problem that occurs in the BEM; (3) the number of equivalent sources can be fewer than the number of surface collocation points so that the matrix size is reduced and a fast computation is achieved for large problems. The main issue of the ESM is that there is no rule to find out the optimal number and position of equivalent sources. In addition, the ESM suffers from the numerical instability that is associated with the ill-conditioned matrix. Some guidelines have been suggested in terms of finding the number and position of the sources, and several numerical techniques have been developed to resolve the numerical instability. This paper reviews the common theories, numerical issues and challenges of the ESM, and it summarizes recent developments and applications of the ESM to aircraft noise.
Highlights
Acoustic radiation and scattering are fundamental problems in computational acoustics
While the wave superposition method or the equivalent source method (ESM) that will be discussed have been mainly developed for acoustic or elastodynamic problems, the method of fundamental solutions (MFS) is a more general approach that was applied to various elliptic partial differential equations in two and three space variables including potential, Helmholtz, and diffusion problems
The ESM is a meshless method so that it significantly simplifies the implementation of the method and the coding effort
Summary
Acoustic radiation and scattering are fundamental problems in computational acoustics. When the boundary integral equation is solved on the surface, the self-mesh computation results in a singularity problem as the distance between the source and the surface point (collocation point) becomes zero. This singularity problem should be carefully resolved using additional numerical techniques.[3,4,5,6]. A reasonable number of equivalent sources, making the approximate solution acceptable for most engineering problems Another and a more serious issue is that the method often encounters the numerical instability problem since the linear algebraic system becomes illconditioned. The current paper covers literatures that were published under the name of the method of fundamental solutions, wave superposition method, and ESM
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