Dispersion analysis for acoustic problems using the point interpolation method

A typical meshfree point interpolation method (PIM) is presented to investigate the dispersion error in the numerical solutions of acoustic problems which is governed by the Helmholtz equation. It is well-known that those results from several numerical approaches, such as the finite element method (FEM) and several meshfree techniques, will suffer from the pollution effect, leading to the incorrect acoustic wave propagation for high wave numbers. The reason for this phenomenon is that the numerical solutions of wave number do not accord with the exact wave number, which is the so-called dispersion issue. In addition, to overcome the possible singularity issue in constructing the shape functions for the PIM with the polynomial basis functions (PBFs), the Gauss-Jordan elimination (GJE) technique is employed here. Several numerical examples concerning dispersion analysis and acoustic wave propagation are performed to verify the accuracy of results from the PIM. It is found that the PIM can reduce the dispersion error effectively and hence generate more accurate results than the FEM with the same set of nodes.