A truly meshfree method for solving acoustic problems using local weak form and radial basis functions

Abstract

It has been known that numerical solutions to Helmholtz problems obtained using several numerical methodologies (e.g. finite element approach) are always plagued by the pollution error effect particularly for high wave numbers, giving rise to erroneous results of numerical acoustic wave. To mitigate this effect, a truly meshfree technique using a local weak form and radial basis functions is employed to analyze acoustic problems. On the basis of the local Petrov–Galerkin weak form, numerical integration is performed over the quadrature domains related to all field nodes, instead of the mesh grid required in the global Galerkin weak form like finite element approach and element-free Galerkin method. Hence, the present methodology is totally independent of the mesh grid either in forming shape functions, or in the integration procedure for system matrices. Also, owing to the use of radial point interpolation by passing all the relevant nodes, the constructed shape functions hold the practical feature of the Kronecker delta function, resulting in easy enforcement of the essential boundary condition as in finite element approach. The results of several numerical examples have shown that with same group of field nodes the present methodology can lead to much more accurate solutions than finite element approach, in particular for relatively high frequencies, and can also generate comparable solutions in comparison to other global Galerkin meshfree techniques.