On the optimum support size in meshfree methods: a variational adaptivity approach with local maximum-entropy approximants

ADRIAN ROSOLEN; DANIEL MILLÁN; MARINO ARROYO

Paris

Congreso; IV European Conference on Computational Mechanics (ECCM 2010); 2010

European Community in Computational Methods in Applied Sciences

A notable advantage of meshfree methods over conventional mesh-based methods is the flexibility they offer to scatter nodes in the domain at will and to select the support size (or locality) of the shape functions. As it is known that the accuracy of the solutions depends strongly on those choices, several adaptive strategies to arrange optimally the nodes have been proposed in the literature. However, and in spite of more than thirty years of meshfree developments, the systematic optimization of the shape functions support size has never been tackled. In this work, we explore the automatic adaption of the support size of the shape functions in the context of meshfree local maximum-entropy approximants and partial differential equations stemming from a minimum principle. Local maximum-entropy approximants are non-negative convex approximation schemes completely defined by the node set and a set of locality parameters (associated to the support size of the shape functions). These approximants exhibit smooth basis functions which fulfil zeroth and first order consistency conditions, and also satisfy ab initio a weak Kronecker-delta property at the boundary of the convex hull of the nodes. The proposed method is based on a variational approach, and the central idea is that the variational principle selects both the discretized physical fields and the discretization parameters, here those defining the support size of each basis function. A key technical fact behind this method is that the local maximum-entropy approximants allow for an easy calculation of the sensitivities with respect to the locality parameter. We illustrate by Poisson, linear and nonlinear elasticity problems the effectivity of the method, which produces very accurate solutions with very coarse discretizations, and finds unexpected patterns of the support size of the shape functions.