Moving Least-squares Method Research Articles

Development of a CAE system based on the node-by-node meshless method

The authors propose a node-by-node meshless method (NBNM), which discretizes the weak-formed governing equations of continuum mechanics using only distributed nodal data. This method uses the following three core methodologies: (i) interpolation using the moving least squares method (MLSM) (ii) estimation of stiffness by nodal integration with stabilization terms, and (iii) a node-by-node iterative solver. This paper discusses the effect of the stabilization term introduced to the NBNM and examines the convergence of the NBNM approximation. An error indicator for NBNM analysis is proposed, and the development of a prototype CAE system based on this method is outlined. Moreover, results of two-dimensional plane stress analysis with adaptation are shown. In conclusion, this paper will show that the NBNM is capable of utilizing CAD data easily and performing effective adaptation analyses, and therefore may be used as a convenient CAE tool.

Node-by-node meshless approach and its applications to structural analyses

The meshless method is expected to become an effective procedure for realizing a CAD/CAE seamless system for analyses ranging from modelling to computation, because time-consuming mesh generation processes are not required. In the present study, a new meshless approach, referred to as the Node-By-Node Meshless method is proposed, in which only nodal data is utilized to discretize the governing equations, which are derived using either the principle of virtual work or the Galerkin method. In this method, three key methodologies are utilized: (i) nodal integration using stabilization terms, (ii) interpolation by the Moving Least-Squares Method, and (iii) a node-by-node iterative solver. This paper presents the formulation of the proposed method along with numerical results obtained for two-dimensional elastostatic and eigenvalue problems. Copyright © 1999 John Wiley & Sons, Ltd.

Formulation of Geometrically Nonlinear Element Free Galerkin Method and its Application to Analysis of Membrane Structures.

The element free Galerkin method (EFGM) is a meshless method presented by Belytschko, Lu and Gu in 1994. A feature of the method is that only nodes are required in modeling of objects. Another feature is that distributions of stresses become continuous because the interpolation function used in EFGM is based on the moving least-squares method (MLSM). In this paper a formulation for geometrically nonlinear EFGM is presented, and it is applied to the analysis of membrane structures that are generally supported by the geometrical stiffness. As an illustrative example, a membrane structure whose shape is represented by bi-quadratic function is analyzed. It is also analyzed by finite element method, and results are compared.

Reproducing kernel particle methods

AbstractA new continuous reproducing kernel interpolation function which explores the attractive features of the flexible time‐frequency and space‐wave number localization of a window function is developed. This method is motivated by the theory of wavelets and also has the desirable attributes of the recently proposed smooth particle hydrodynamics (SPH) methods, moving least squares methods (MLSM), diffuse element methods (DEM) and element‐free Galerkin methods (EFGM). The proposed method maintains the advantages of the free Lagrange or SPH methods; however, because of the addition of a correction function, it gives much more accurate results. Therefore it is called the reproducing kernel particle method (RKPM). In computer implementation RKPM is shown to be more efficient than DEM and EFGM. Moreover, if the window function is C∞, the solution and its derivatives are also C∞ in the entire domain. Theoretical analysis and numerical experiments on the 1D diffusion equation reveal the stability conditions and the effect of the dilation parameter on the unusually high convergence rates of the proposed method. Two‐dimensional examples of advection‐diffusion equations and compressible Euler equations are also presented together with 2D multiple‐scale decompositions.

Moving least squares interpolation with thin-plate splines and radial basis functions

Moving least-squares methods for interpolation or approximation of scattered data are well known, and can suffer from defects, such as flat spots in the Shepard method, and edge effects inherited from a polynomial basis in the higher degree cases. We investigate methods based on thin-plate splines and on other radial basis functions. It turns out that a small support of the weight function leads to a small support for the “spline basis” and associated efficiency in the evaluation of the approximant. The edge effects seem minimal and good interpolants of scattered data can be obtained.