Higher-order natural element methods:: Towards an isogeometric meshless method

The problem of generalizing the natural element method (NEM) in terms of higher-order consistency and continuity is addressed here. It is done by means of the de Boor algorithm, the same employed to obtain B-splines by linear combinations of linear interpolants in one dimension. By noting that any form of natural neighbour interpolation can be considered as a suitable generalization of linear interpolation to two and higher dimensions, the de Boor algorithm is extended to two or higher dimensions, thus obtaining a new form of interpolation that can be used in a Galerkin framework to develop a new class of meshless methods. This new class of meshless methods closely resembles the isogeometric analysis developed by Hughes et al. (Comput. Methods Appl. Mech. Eng. 2005; 194:4135-4195). However, unlike B-splines, the new class of interpolants does not rely on an underlying tensor-product quadrilateral mesh. It is based on the Delaunay triangulation of the cloud of knots and does not require any regularity on the connectivity. In addition, the new method conserves many of the attractive features of the NEM, such as strict interpolation on the boundary, and thus directs imposition of essential boundary conditions. After a theoretical description of the proposed method, some numerical examples are shown to test its performance in the context of linear elastostatics. Copyright (C) 2007 John Wiley & Sons, Ltd.