Overview and construction of meshfree basis functions: from moving least squares to entropy approximants

Abstract

SUMMARYIn this paper, an overview of the construction of meshfree basis functions is presented, with particular emphasis on moving least-squares approximants, natural neighbour-based polygonal interpolants, and entropy approximants. The use of information-theoretic variational principles to derive approximation schemes is a recent development. In this setting, data approximation is viewed as an inductive inference problem, with the basis functions being synonymous with a discrete probability distribution and the polynomial reproducing conditions acting as the linear constraints. The maximization (minimization) of the Shannon-Jaynes entropy functional (relative entropy functional) is used to unify the construction of globally and locally supported convex approximation schemes. A JAVA applet is used to visualize the meshfree basis functions, and comparisons and links between different meshfree approximation schemes are presented.

Smooth, second order, non‐negative meshfree approximants selected by maximum entropy

“…See also [10,11] for the relation of the present formulation with the relative entropy maximization approach. Here, for the sake of completeness, we review some of the topics of particular relevance to the present work.…”

Maximum-entropy meshfree method for compressible and near-incompressible elasticity

“…Arroyo and Ortiz [13] realized a meshfree approximation using a modified entropy functional-with emphasis on establishing a smooth transition between finite element and meshfree methods. Sukumar and Wright [50] generalized the construction of max-ent meshfree basis functions by using the relative (Shannon-Jaynes) entropy functional with a prior [54]. On using compactly-supported prior weight functions that are at least C 0 , compactly-supported max-ent basis functions are realized.…”

“…[50] to present expressions for max-ent basis functions and their derivatives. To this end, let the prior weight function be denoted by w a (x).…”

“…Applying the procedure of Lagrange multipliers, the following expression for max-ent basis functions is obtained [50]:…”

“…Since numerical errors are prevalent when standard tensorproduct Gauss quadrature is used in meshfree methods, we appeal to assumed strain methods [49] and nodal integration techniques [20,21,22] to define a modified strain tensor. Maximum-entropy basis functions [13,50] are used to define the modified strain matrix, and Gauss quadrature is adopted in the numerical integration. The procedure so devised alleviates numerical integration errors in meshfree methods using minimal number of integration points and ensures patch test satisfaction to within machine precision.…”