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 with boundary , are governed by the Helmholtz equation completed with appropriate boundary conditions. As the pressure p(x; y) is a complex variable, it can always be expressed as a function of cos (x; y) and sin (x; y) where (x; y) is the phase of the wave in each point (x; y). If the exact distribution (x; y) of the phase is known and if a meshless basis {1; cos (x; y); sin (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 rst step, the acoustic problem by using a polynomial basis to obtain a rst approximation of the pressure eld ph I (x; y). As a second step, from ph I (x; y) we compute the distribution of the phase h I (x; y) and we introduce it in the meshless basis in order to compute a second approximated pressure eld ph II(x; y). From ph II(x; y), a new distribution of the phase is computed in order to obtain a third approximated pressure eld 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 eld in . This work will show the e ciency 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 nite element method