INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING Int. J. Numer. Meth. Biomed. Engng. 2010; 26:252–261 Published online 22 July 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/cnm.1149 COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING A local multivariate Lagrange interpolation method for constructing shape functions Yunhua Luo ∗, † Department of Mechanical and Manufacturing Engineering, University of Manitoba, Winnipeg, MB, Canada R3T 5V6 SUMMARY In this paper, the conventional global univariate Lagrange interpolation method is transformed into a local multivariate interpolation method. The method has the following attractive features: it can be used to interpolate irregularly distributed data points; it does not need to solve local problems; if used for constructing shape functions, the obtained shape functions satisfy the Kronecker delta condition and they have the reproducing properties. The performance of the local multivariate Lagrange interpolation method was examined by applying it to function approximation. Copyright 2008 John Wiley & Sons, Ltd. Received 5 September 2007; Revised 26 April 2008; Accepted 20 May 2008 KEY WORDS: local multivariate interpolation; patch; data point; evaluation point 1. INTRODUCTION Polynomial interpolation method, especially multivariate polynomial interpolation, is a major building block of many numerical methods [1–9]. Among the various applications of polynomial interpolation is the construction of shape functions. The quality of shape functions has a significant effect on the performance of a numerical method. This is especially true for a group of numerical methods collectively called meshless or meshfree methods, e.g. [1–3, 10–12] among many others. In spite of starting from the same variational formulation, meshless methods may have very different performance if shape functions are constructed by different methods [2, 13]. Univariate interpolation methods have a long history that can even be traced back to ancient Babylon and Greece around 300 BC. Therefore, the theory of univariate polynomial interpolation methods is well developed. Nevertheless, the theory of multivariate polynomial interpolation is still underdeveloped [14–38]. Multivariate polynomial interpolation methods are widely adopted in meshless methods for constructing shape functions. Two common deficiencies of shape function construction methods used in the existing meshless methods are the obtained shape functions do not satisfy the Kronecker delta condition, which brings about inconvenience of applying essential boundary conditions; they need to solve local problems in the construction, which leads to the increase in computational time; furthermore, if a set of selected data points are at inappropriate locations, the local problem may be ill-conditioned. The inherent relationship between interpolation functions and generalized finite difference schemes was revealed in [39]: once an interpolation function is obtained, one can immediately obtain a finite difference scheme simply by differentiating the interpolation function ∗ Correspondence to: Yunhua Luo, Department of Mechanical and Manufacturing Engineering, University of Manitoba, Winnipeg, MB, Canada R3T 5V6. † E-mail: luoy@cc.umanitoba.ca Copyright 2008 John Wiley & Sons, Ltd.