Gradient Enhanced Localized Radial Basis Collocation Method for Inverse Analysis of Cauchy Problems

Abstract

This work proposes a gradient enhanced localized radial basis collocation method (GL-RBCM) for solving boundary value problems. In particular, the attention is paid to the solution of inverse Cauchy problems. It is known that the approximation by radial basis functions often leads to ill-conditioned systems due to the global nature. To this end, the reproducing kernel shape function and gradient reproducing kernel shape function are proposed to localize the radial basis function while the gradient approximation is aimed at reducing the computational intensity of carrying out the second derivatives of reproducing kernel shape function. In the proposed weighted collocation method, the weights on Neumann and Dirichlet boundary conditions are determined for both direct problems and inverse problems. From stability analysis, it is shown that the GL-RBCM can maintain high accuracy of approximating the first derivatives even under irregular perturbation added to boundary conditions. By comparing with the localized RBCM, the CPU saving of the GL-RBCM is manifested. The efficacy of the proposed method is therefore demonstrated.