An iterative defect-correction type meshless method for acoustics

Accurate numerical simulation of acoustic wave propagation is still an open problem, particularly for medium frequencies. We have thus formulated a new numerical method better suited to the acoustical problem: the element-free Galerkin method (EFGM) improved by appropriate basis functions computed by a defect correction approach. One of the EFGM advantages is that the shape functions are customizable. Indeed, we can construct the basis of the approximation with terms that are suited to the problem which has to be solved. Acoustical problems, in cavities 9 with boundary F, are governed by the Helmholtz equation completed with appropriate boundary conditions. As the pressure p(x, v) is a complex variable, it can always be expressed as a function of cos theta(x, y) and sin theta(x, y) where theta(x, y) is the phase of the wave in each point (x, y). If the exact distribution theta(x, y) of the phase is known and if a nieshIcss basis {1, cos theta(x, y), sin theta(x, y)} is used, then the exact solution of the acoustic problem can be obtained. Obviously, in real-life cases, the distribution of the phase is unknown. The aim of our work is to resolve, as a first step, the acoustic problem by using a polynomial basis to obtain a first approximation of the pressure field p(I)(h)(x, y). As a second step, from p(I)(h)(x, y) we compute the distribution of the phase theta(I)(h)(x, y) and we introduce it in the meshless basis in order to compute a second approximated pressure field p(II)(h)(x, y). From p(II)(h)(x, y), a new distribution of the phase is computed in order to obtain a third approximated pressure field and so on until a convergence criterion, concerning the pressure or the phase, is obtained. So, an iterative defect-correction type meshless method has been developed to compute the pressure field in Omega. This work will show the efficiency of this meshless method in terms of accuracy and in terms of computational time. We will also compare the performance of this method with the classical finite element method. Copyright (C) 2003 John Wiley Sons, Ltd.