International Journal of Computational Methods Vol. 4, No. 3 (2007) 521–541 c  World Scientific Publishing Company THE UPPER BOUND PROPERTY FOR SOLID MECHANICS OF THE LINEARLY CONFORMING RADIAL POINT INTERPOLATION METHOD (LC-RPIM) G. Y. ZHANG *,†,‡ , G. R. LIU †,‡ , T. T. NGUYEN † and C. X. SONG † † Centre for Advanced Computations in Engineering Science (ACES) Department of Mechanical Engineering, National University of Singapore 9 Engineering Drive 1, 117576, Singapore ‡ The Singapore — MIT Alliance (SMA), E4-04-10 4 Engineering Drive 3, 117576, Singapore * smazg@nus.edu.sg X. HAN, Z. H. ZHONG and G. Y. LI State Key Laboratory of Advanced Technology for Vehicle Body Design & Manufacture Hunan University, Changsha, 410082, P. R. China Received 2 July 2007 Accepted 6 September 2007 It has been proven by the authors that both the upper and lower bounds in energy norm of the exact solution to elasticity problems can now be obtained by using the fully compatible finite element method (FEM) and linearly conforming point interpola- tion method (LC-PIM). This paper examines the upper bound property of the linearly conforming radial point interpolation method (LC-RPIM), where the Radial Basis Func- tions (RBFs) are used to construct shape functions and node-based smoothed strains are used to formulate the discrete system equations. It is found that the LC-RPIM also provides the upper bound of the exact solution in energy norm to elasticity problems, and it is much sharper than that of LC-PIM due to the decrease of stiffening effect. An effective procedure is also proposed to determine both upper and lower bounds for the exact solution without knowing it in advance: using the LC-RPIM to compute the upper bound, using the standard fully compatible FEM to compute the lower bound based on the same mesh for the problem domain. Numerical examples of 1D, 2D and 3D problems are presented to demonstrate these important properties of LC-RPIM. Keywords : Meshfree methods; point interpolation method; radial basis functions; strain smoothing; error bound; elasticity. 1. Introduction The finite element method (FEM) has been well developed and is now widely and routinely used to provide a numerical solution to engineering problems. However, to certify the solution or to provide an error bounds to the numerical solution of FEM 521