A Cartesian-grid Collocation Technique with Integrated Radial Basis Functions for mixed boundary value problems

In this paper, high order systems are reformulated as first order systems which are then numerically solved by a collocation method. The collocation method is based on Cartesian discretisation with 1D-integrated radial basis function networks (1D-IRBFN) [1]. The present method is enhanced by a new boundary interpolation technique based on 1D-IRBFN which is introduced to obtain variable approximation at irregular points in irregular domains. The proposed method is well suited to problems with mixed boundary conditions on both regular and irregular domains. The main results obtained are (a) the boundary conditions for the reformulated problem are of Dirichlet type only; (b) the integrated RBFN approximation avoids the well known reduction of convergence rate associated with differential formulations; (c) the primary variable (e.g. displacement, temperature) and the dual variable (e.g. stress, temperature gradient) have similar convergence order; (d) the volumetric locking effects associated with incompressible materials in solid mechanics are alleviated. Numerical experiments show that the proposed method achieves very good accuracy and high convergence rates.