Dual reciprocity hybrid boundary node method for nonlinear problems

In this paper, a boundary type meshless method of dual reciprocity hybrid boundary node method (DHBNM) is proposed to solve complicate Poisson type linear and nonlinear problems. Firstly, the solutions are divided into the complementary solutions related to homogeneous equation and the particular solutions solved by nonhomogeneous terms, for the latter, they are approximated by the radial basis function interpolation based on dual reciprocity method, and the complementary solutions are obtained based on simple Poisson's equation by hybrid boundary node method, by which a simple fundamental solution of the Laplacian operator is employed instead of some other complicated ones; then a function of field functions and their derivatives on any point can be easily obtained, employing the concept of the analog equation of Katsikadelis, the field functions and their derivatives can be expressed as the function of unknown series of coefficients, and a series of nonlinear equivalent equations can be established by collocating the original governing equation at discrete points in the interior and on boundary of the domain. As a result, a new meshless method of dual reciprocity hybrid boundary node method is proposed to solve nonlinear Poisson type problems, because of the usage of those techniques, the boundary type meshless properties can be kept for any type of nonlinear equations. Different types of classical nonlinear problems are presented to validate the effectiveness and the accuracy of the present method.