INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2008; 74:1928–1954 Published online 2 November 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2237 Higher-order natural element methods: Towards an isogeometric meshless method David Gonz´ alez, El´ ıas Cueto ∗, † and Manuel Doblar´ e Group of Structural Mechanics and Material Modelling, Arag´ on Institute of Engineering Research (I3A), University of Zaragoza, Mar´ ıa de Luna, 5, Campus Rio Ebro, E-50018 Zaragoza, Spain SUMMARY 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 2007 John Wiley & Sons, Ltd. Received 9 May 2007; Revised 26 September 2007; Accepted 1 October 2007 KEY WORDS: meshless; natural element method; Sibson interpolation; Laplace interpolation; isogeometric analysis; de Boor’s algorithm; B-splines ∗ Correspondence to: El´ ıas Cueto, Mechanical Engineering Department, Edificio Betancourt, University of Zaragoza, Mar´ ıa de Luna, 5, E-50018 Zaragoza, Spain. † E-mail: ecueto@unizar.es Contract/grant sponsor: Spanish Ministry of Education and Science; contract/grant number: CICYT-DPI2005-08727- C02-01 Contract/grant sponsor: European Union, sixth Framework Program; contract/grant number: DeSSOS project, STREP project number 027252 Copyright 2007 John Wiley & Sons, Ltd.