The point collocation method with a local maximum entropy approach

Meshless methods have long been a topic of interest in computational modelling in solid mechanics and are broadly divided into weak and strong form-based approaches. The need for numerical integration in the former remains a challenge often met by using a background mesh or complex stabilised nodal approaches. It is only strong form-based point collocation methods (PCMs) which dispense with meshing and integration entirely, and for this reason PCMs remain of interest. In this paper, a new point collocation method is developed which is based on maximum entropy basis functions which bring benefits in terms of accuracy and efficiency. These basis functions possess non-negativity and a weak Kronecker delta property which decreases the errors on boundaries to improve overall accuracy of solutions. After a discussion of implementation issues in the new method, numerical examples are presented, including 1D and 2D problems with linear elasticity and Poisson PDEs, on both convex and non-convex domains to show the performance. Comparisons of convergence properties with respect to accuracy and computational cost (both CPU time and floating point operations) are made with an existing method, the reproducing kernel collocation method (RKCM), to show the effectiveness of the proposed method. In all examples, higher order convergence rates are obtained using the developed method with increasingly reduced computational effort for higher levels of accuracy due to the fundamental advantages.