The partition of unity quadrature in meshless methods

Abstract

In dealing with mesh-free formulations a major problem is connected to the computation of the quadratures appearing in the variational principle related to the differential boundary value problem. These integrals require, in the standard approach, the introduction of background quadrature subcells which somehow make these methods not 'truly meshless'. In this paper a new general method for computing definite integrals over arbitrary bounded domains is proposed, and it is applied in particular to the evaluation of the discrete weak form of the equilibrium equations in the framework of an augmented Lagrangian element-free formulation. The approach is based on splitting the integrals over the entire domain into the sum of integrals over weight function supports without modifying in any way the variational principle or requiring background quadrature cells. The accuracy and computational cost of the technique compared to standard Gauss subcells quadrature are discussed. Copyright © 2002 John Wiley & Sons, Ltd.