Reproducing kernel particle boundary element-free method for potential problems

Reproducing kernel particle method (RKPM) is one of theimportant methods to obtain the shape functions in meshless (or mesh-free)methods. In this paper, combining the reproducing kernel particle method andboundary integral equations for two-dimensional potential problems, thereproducing kernel particle boundary element-free (RKP-BEF) method fortwo-dimensional potential problems is presented. The formulae of the RKP-BEFmethod for two-dimensional potential problems based on Poisson's equationare obtained. The discrete boundary integral equations of the RKP-BEF methodare formed, and the corresponding numerical integral methods are discussed.The boundary integral equations of the RKP-BEF method for the potentials atinterior points are obtained. The smoothness of the shape function of theRKPM is the same as that of the reproducing kernel function, and the valuesof polynomials at interpolating points can be exactly reconstructed, thenthe RKP-BEF method has higher precision. In comparison with other existingmeshless boundary integral equation methods, such as boundary node method(BNM) and local boundary integral equation (LBIE) method, the RKP-BEF methodis a direct numerical method, in which the basic unknown quantity is thereal solution of the nodal variables, of meshless boundary integral equationmethods. And the boundary conditions can be implemented directly. Thenumerical examples of 2-D potential problems are given for verifying theeffectiveness and correctness of the method in this paper.