Comparison between the radial point interpolation and the Kriging interpolation used in meshfree methods K. Y. Dai, G. R. Liu, K. M. Lim, Y. T. Gu Abstract The radial point interpolation method (RPIM) based on local supported radial basis function (RBF) and the Galerkin weak form has been developed and success- fully applied to many engineering problems. Recently, a new meshfree method was proposed based on the uni- versal moving Kriging interpolation. This paper studies the difference between the meshfree shape functions created based on the point interpolation and the Kriging interpolation. It is found that both the two methods yield the same shape function as long as the same radial basis function or semivariogram is adopted for interpolation. Although the two methods lead to the same shape func- tion, the theorem in Kriging formulation may provide an alternative theoretical support for the RPIM. Some com- mon semivariograms used in Kriging may also be incor- porated in the RPIM. In addition, in order to satisfy the conformability requirements, a penalty technique is introduced in this paper to form a conforming Kriging, which can pass the standard patch test exactly. Keywords Point interpolation method, Kriging method, Radial basis functions, Meshfree method 1 Introduction In recent years, the element-free, or meshfree method has been developed and achieved remarkable progress in computational mechanics and related fields due to its flexibility and the good convergence rate. Different ver- sions of meshfree methods have been proposed and they have been successfully applied to the analyses of solids and structures as well as fluid flow and heat transfer problems (Liu 2002). The element free Galerkin (EFG) method (Belytschko et al. 1994; Belytschko et al. 1996) is one of the most viable methods in which the moving least square (MLS) approach is used to construct the meshfree shape functions. Although the EFG method has been applied to many problems, there still exists some inconvenience or disadvantages when using EFG. Because the MLS shape functions lack the Kronecker delta function property, it is not easy to accurately impose essential boundary condi- tions in the EFG method. The Point interpolation method (PIM) has been proposed by Liu and his coworkers (Liu 2002). It uses the nodal values in the local support domain to construct the shape functions and the shape functions so constructed possess the Kronecker delta function property. According to the basis function employed, it can be mainly classified into two types, i.e., polynomial PIM (Liu and Gu 2001a, c) and radial PIM (or RPIM) (Wang and Liu 2002a, b). For the former, the moment matrix cannot always be inversed and special techniques are needed to overcome the possible singularity problem, which inevitably increases the computational cost (Liu and Gu 2001b). For the RPIM, the moment matrix is nonsin- gular for arbitrary nodal distributions. If the shape parameters in radial basis functions (RBF) are properly selected (Wang and Liu 2002b), the RPIM works well for practical problems. The RPIM has found applications in many engineering areas (Wang et al. 2001; Liu et al. 2003). Although great efforts have already been spent in this area, one of the key issues in the development of meshfree methods is still to find an efficient, stable and preferably conformable method to construct the meshfree shape functions. Recently Gu (2002) found some good properties using the Kriging interpolation method and reported them in a conference abstract. Kriging is a form of generalized linear regression of an optimal estimator in a minimum mean square error sense, named after a South African mining engineer D. G. Krige. It was originally developed for mapping in the fields of geology and geophysics, mining, and photogrammetry. Kriging method has become a fundamental tool in the field of geostatistics (Olea 1999; Wackernagel 1995; Cressie 1993). It has also found applications in a variety of fields so far, including environmental monitoring and assessment. Based on the structure and characteristics of spatially located data, Kriging method takes into account the interdependency of samples that are close to each other while allowing for a certain independence of the sample points. Apart from being an exact interpolator, it has many advantages, such as minimum mean square error, zero estimation variance, estimation interval not restricted to the data interval and so on. It also possesses good char- acteristics when employed to a meshless method. First the shape functions constructed by Kriging have the delta function property; hence the imposition of boundary Computational Mechanics 32 (2003) 60–70 Ó Springer-Verlag 2003 DOI 10.1007/s00466-003-0462-z 60 Received: 27 August 2002 / Accepted: 26 May 2003 K. Y. Dai (&), K. M. Lim, Y. T. Gu Center for Advanced Computations in Engineering Science, Department of Mechanical Engineering, The National University of Singapore, 10 Kent Ridge Crescent, Singapore, 119260 e-mail: mpeliugr@nus.edu.sg; http://www.nus.edu.sg/ACES G. R. Liu SMA Fellow, Singapore-MIT Alliance