Least‐squares collocation meshless method

Abstract

SUMMARYA ÿnite point method, least-squares collocation meshless method, is proposed. Except for the collocation points which are used to construct the trial functions, a number of auxiliary points are also adopted. Unlike the direct collocation method, the equilibrium conditions are satisÿed not only at the collocation points but also at the auxiliary points in a least-squares sense. The moving least-squares interpolant is used to construct the trial functions. The computational e ort required for the present method is in the same order as that required for the direct collocation, while the present method improves the accuracy of solution signiÿcantly. The proposed method does not require any mesh so that it is a truly meshless method. Three numerical examples are studied in detail, which show that the proposed method possesses high accuracy with low computational e ort.

A Cartesian‐grid collocation technique with integrated radial basis functions for mixed boundary value problems

“…At the finest grid, the relative error L u 2 and L σ 2 are 1.147×10 −3 and 8.086×10 −4 , respectively. Unlike standard collocation method, which is very unstable for elasticity problems with traction boundary conditions [13,17], the present method shows a superior accuracy and stable convergence.…”

“…Recently, Zhang et al [13] suggested Least-squares collocation meshless method which can improve the accuracy of the solution in comparison with the one of standard collocation method. Onate et al [14] introduced a stabilization technique by adding artificial terms in both governing equations and Neumann boundary conditions, however, these terms only serve the stabilization purpose and their suitability is restricted to some special problems.…”

A meshless method based on Taylor series

“…A least square method combined with collocation method will be used. This technique has been suggested by Zhang et al [7]. One chooses a set of nodes x j on the boundary of the domain and one minimizes the error between the approximate value and the given value of u in these points.…”

“…First we discuss the way to account for the boundary conditions. It is established that a pure collocation generally diverges and that better results are obtained by applying a least square method as proposed by Zhang et al [7]. Next a computational technique is sketched that permits to apply Taylor series to PDE's, what permits to compute explicitly the shape functions.…”

Collocation Discrete Least Square (CDLS) Method for Elasticity Problems and Grid Irregularity Effect Assessment

“…The other group of meshless methods is built on collocation schemes. The SPH, FPM, least-square collocation meshless method [16] , and point weighted least-square (PWLS) method [17] all belong to this group. These methods are very efficient and easy to program, but they usually suffer from poor stability, and the accuracy often goes down near the boundary.…”