An advanced meshless LBIE/RBF method for solving two-dimensional incompressible fluid flows

Abstract

The present work presents a meshless local boundary integral equation (LBIE) method for the solution of two-dimensional incompressible fluid flow problems governed by the Navier–Stokes equations. The method uses, for its meshless implementation, nodal points spread over the analyzed domain and employs in an efficient way the radial basis functions (RBF) for the interpolation of the interior and boundary variables. The inverse matrix of the RBF is computed only once for every nodal point and the interpolation functions are evaluated by the inner product of the inverse matrix with the weight vector associated to the integration point. This technique leads to a fast and efficient meshless approach, the locality of the method is maintained and the system matrices are banded with small bandwidth. The velocity–vorticity approach of the Navier–Stokes equations is adopted and the LBIEs are derived for the velocity and the vorticity field, resulting in a very stable and accurate implementation. The evaluation of the volume integrals is accomplished via a very efficient and accurate technique by triangularizing the local area of the nodal point to the minimum number of well formed triangles. Numerical examples illustrate the proposed methodology and demonstrate its accuracy.