Explicit form and efficient computation of MLS shape functions and their derivatives

This work presents a general and e$cient way of computing both di!use and full derivatives of shape functions for meshless methods based on moving least-squares approximation (MLS) and interpolation. It is an extension of the recently introduced consistency approach based on Lagrange multipliers which provides a general framework for constrained MLS along with robust algorithms for the computation of shape functions and their di!use derivatives. The particularity of the proposed algorithms is that they do not involve matrix inversion or linear system solving. The previous approach is limited to di!use derivatives of the shape functions and not their full derivatives which are usually much more expensive to obtain. In the present paper we propose to e$ciently compute the full derivatives by a new algorithm based on the formal di!erentiation of the previous one. In this way, we obtain a uni"ed low-cost consistent methodology for evaluating the shape functions and both their di!use and full derivatives. In the second part of the paper we introduce explicit forms of MLS shape functions in 1D, 2D and 3D for an arbitrary number of nodes. These forms are especially useful for comparing "nite element and MLS approximations. Finally we present a general architecture of an MLS program