A Kriging‐based error‐reproducing and interpolating kernel method for improved mesh‐free approximations

Abstract

SUMMARYAn error-reproducing and interpolating kernel method (ERIKM), which is a novel and improved form of the error-reproducing kernel method (ERKM) with the nodal interpolation property, is proposed. The ERKM is a non-uniform rational B-splines (NURBS)-based mesh-free approximation scheme recently proposed by Shaw and Roy (Comput. Mech. 2007; 40(1):127-148). The ERKM is based on an initial approximation of the target function and its derivatives by NURBS basis functions. The errors in the NURBS approximation and its derivatives are then reproduced via a family of non-NURBS basis functions. The non-NURBS basis functions are constructed using a polynomial reproduction condition and added to the NURBS approximation obtained in the first step. In the ERKM, the interpolating property at the boundary is achieved by repeating the knot (open knot vector). However, for most problems of practical interest, employing NURBS with open knots is not possible because of the complex geometry of the domain, and consequently ERKM shape functions turn out to be non-interpolating. In ERIKM, the error functions are obtained through localized Kriging based on a minimization of the squared variance of the estimate with the reproduction property as a constraint. Interpolating error functions so obtained are then added to the NURBS approximant. While enriching the ERKM with the interpolation property, the ERIKM naturally possesses all the desirable features of the ERKM, such as insensitivity to the support size and ability to reproduce sharp layers. The proposed ERIKM is finally applied to obtain strong and weak solutions for a class of linear and non-linear boundary value problems of engineering interest. These illustrations help to bring out the relative numerical advantages and accuracy of the new method to some extent.

Moving Kriging reconstruction for high-order finite volume computation of compressible flows

“…Recently, Bui and Nguyen [43] used a MK-based meshfree method for the eigenvalue analysis of Kirchhoff thin plate structures with complex geometric shapes. Improved mesh-free approximations were developed by Shaw et al [44] using a Kriging based errorreproducing and interpolating kernel method (ERIKM). In order to prevent numerical instabilities, the derivatives of the basis functions are not computed directly but using a polynomial reproducing condition.…”