A quasi-consistent integration method for efficient meshfree analysis of Helmholtz problems with plane wave basis functions

A quasi-consistent integration method is presented for the efficient meshfree analysis of Helmholtz problems. The plane wave basis functions are employed for the reproducing kernel meshfree approximation to accurately represent the acoustic field resulting from Helmholtz problems. In order to improve the computational efficiency of Galerkin meshfree analysis of Helmholtz problems, a reproducing kernel gradient smoothing approach is introduced into the meshfree formulation with plane wave basis functions. In the proposed method, the smoothed gradients of meshfree shape functions with plane wave basis functions are built upon a reproducing kernel gradient representation and the integration consistency of Galerkin meshfree formulation is implicitly ensured. Furthermore, a quasi-consistent integration scheme is proposed to compute the smoothed gradients, which aims to balance the efficiency and accuracy for meshfree analysis of Helmholtz problems. The proposed integration method leads to fully consistent integration when one wave direction is considered, and nearly consistent integration if two wave directions are taken into account, where the boundary sample points of integration cells are particularly preferred since they are simultaneously used by neighboring integration cells with efficiency gain. Numerical results demonstrate that the proposed methodology is much more efficient and accurate for Galerkin meshfree analysis of Helmholtz problems, in comparison with the standard meshfree formulations using high order Gauss quadrature rules.