Meshless analysis and applications of a symmetric improved Galerkin boundary node method using the improved moving least-square approximation

This paper combines variational formulations of boundary integral equations (BIEs) and the improved moving least-square (IMLS) approximation to develop a symmetric meshless method, the improved Galerkin boundary node method (IGBNM) for boundary-only analysis of boundary value problems in potential theory and viscous fluid flow. The IMLS approximation is used in the IGBNM to construct meshless shape functions. In the IMLS approximation, the system of algebra equations can be solved without the inverse matrix. Compared with other MLS-based meshless methods, the IGBNM is a direct numerical method in which the basic unknown quantities are the actual nodal values. Besides, boundary conditions in the IGBNM are implemented directly and easily, and the resulting 'stiffness' matrices conserve the symmetry and positive definiteness of the variational formulations. Thus, it gives a higher computational efficiency. Total details of numerical implementation and error analysis of the IGBNM are first given for general BIEs. Then, taking potential problems and viscous fluid flow problems as examples, we set up a framework for numerical implementation and asymptotic error estimates of the IGBNM. Finally, some numerical tests are presented to demonstrate the efficiency of the method.