A STUDY ON THE WEIGHT FUNCTION OF THE MOVING LEAST SQUARE APPROXIMATION IN THE LOCAL BOUNDARY INTEGRAL EQUATION METHOD

The meshless method is a new numerical technique presented in recent years.It uses the moving least square(MLS)approximation as a shape function.The smoothness of the MLS approximation is determined by that of the basic function and of the weight function,and is mainly determined by that of the weight function.Therefore,the weight function greatly affects the accuracy of results obtained.Different kinds of weight functions,such as the spline function, the Gauss function and so on,are proposed recently by many researchers.In the present work,the features of various weight functions are illustrated through solving elasto-static problems using the local boundary integral equation method.The effect of various weight functions on the accuracy, convergence and stability of results obtained is also discussed.Examples show that the weight function proposed by Zhou Weiyuan and Gauss and the quartic spline weight function are better than the others if parameters c and α in Gauss and exponential weight functions are in the range of reasonable values,respectively,and the higher the smoothness of the weight function,the better the features of the solutions.

STUDY ON PARAMETERS FOR TOPOLOGICAL VARIABLES FIELD INTERPOLATED BY MOVING LEAST SQUARE APPROXIMATION

This paper presents a new approach to the structural topology optimization of continuum structures. Material-point independent variables are presented to illustrate the existence condition,or inexistence of the material points and their vicinity instead of elements or nodes in popular topology optimization methods. Topological variables field is constructed by moving least square approximation which is used as a shape function in the meshless method. Combined with finite element analyses,not only checkerboard patterns and mesh-dependence phenomena are overcome by this continuous and smooth topological variables field,but also the locations and numbers of topological variables can be arbitrary. Parameters including the number of quadrature points,scaling parameter,weight function and so on upon optimum topological configurations are discussed. Two classic topology optimization problems are solved successfully by the proposed method. The method is found robust and no numerical instabilities are found with proper parameters.

Assessing dynamic response of multispan viscoelastic thin beams under a moving mass via generalized moving least square method

Dynamic response of multispan viscoelastic thin beams subjected to a moving mass is studied by an efficient numerical method in some detail. To this end, the unknown parameters of the problem are discretized in spatial domain using generalized moving least square method （GMLSM） and then, discrete equations of motion based on Lagrange＇s equation are obtained. Maximum deflection and bending moments are considered as the important design parameters. The design parameter spectra in terms of mass weight and velocity of the moving mass are presented for multispan viscoelastic beams as well as various values of relaxation rate and beam span number. A reasonable good agreement is achieved between the results of the proposed solution and those obtained by other researchers. The results indicate that, although the load inertia effects in beams with higher span number would be intensified for higher levels of moving mass velocity, the maximum values of design parameters would increase either. Moreover, the possibility of mass separation is shown to be more critical as the span number of the beam increases. This fact also violates the linear relation between the mass weight of the moving load and the associated design parameters, especially for high moving mass velocities. However, as the relaxation rate of the beam material increases, the load inertia effects as well as the possibility of moving mass separation reduces.

ELASTO-PLASTICITY ANALYSIS BASED ON COLLOCATION WITH THE MOVING LEAST SQUARE METHOD

A meshless approach based on the moving least square method is developed for elasto-plasticity analysis,in which the incremental formulation is used.In this approach,the dis- placement shape functions are constructed by using the moving least square approximation,and the discrete governing equations for elasto-plastic material are constructed with the direct collo- cation method.The boundary conditions are also imposed by collocation.The method established is a truly meshless one,as it does not need any mesh,either for the purpose of interpolation of the solution variables,or for the purpose of construction of the discrete equations.It is simply formu- lated and very efficient,and no post-processing procedure is required to compute the derivatives of the unknown variables,since the solution from this method based on the moving least square approximation is already smooth enough.Numerical examples are given to verify the accuracy of the meshless method proposed for elasto-plasticity analysis.

Local geoid height approximation and interpolation using moving least squares approach

In this paper an introduction of the moving least squares approach is presented in the context of data approximation and interpolation problems in Geodesy.An application of this method is presented for geoid height approximation and interpolation using different polynomial basis functions for the approximant and interpolant,respectively,in a regular grid of geoid height data in the region 16.0417°≤φ≤47.9583°and 36.0417°≤λ≤69.9582°,with increment 0.0833°in both latitudal and longitudal directions.The results of approximation and interpolation are then compared with the geoid height data from GPS-Levelling approach.Using the standard deviation of the difference of the results,it is shown that the planar interpolant,with reciprocal of distance as weight function,is the best choice in this local approximation and interpolation problem.

A meshless algorithm with moving least square approximations for elliptic Signorini problems

Based on the moving least square （MLS） approximations and the boundary integral equations （BIEs）, a meshless algorithm is presented in this paper for elliptic Signorini problems. In the algorithm, a projection operator is used to tackle the nonlinear boundary inequality conditions. The Signorini problem is then reformulated as BIEs and the unknown boundary variables are approximated by the MLS approximations. Accordingly, only a nodal data structure on the boundary of a domain is required. The convergence of the algorithm is proven. Numerical examples are given to show the high convergence rate and high computational efficiency of the presented algorithm.

Calculation of stratum surface principal curvature based on a moving least square method

With the east section of the Changji sag Zhunger Basin as a case study, both a principal curvature method and a moving least square method are elaborated. The moving least square method is introduced, for the first time, to fit a stratum surface. The results show that, using the same-degree base function, compared with a traditional least square method, the moving least square method can produce lower fitting errors, the fitting surface can describe the morphological characteristics of stratum surfaces more accurately and the principal curvature values vary within a wide range and may be more suitable for the prediction of the distribution of structural fractures. The moving least square method could be useful in curved surface fitting and stratum curvature analysis.

Global interpolating meshless shape function based on generalized moving least-square for structural dynamic analysis

A global interpolating meshless shape function based on the generalized moving least-square （GMLS） is formulated by the transformation technique. Both the shape function and its derivatives meet the Kronecker delta function property. With the interpolating GMLS （IGMLS） shape function, an improved element-free Galerkin （EFG） method is proposed for the structural dynamic analysis. Compared with the conven- tional EFG method, the obvious advantage of the proposed method is that the essential boundary conditions including both displacements and derivatives can be imposed by the straightforward way. Meanwhile, it can greatly improve the ill-condition feature of the standard GMLS approximation, and provide good accuracy at low cost. The dynamic analyses of the Euler beam and Kirchhoff plate are performed to demonstrate the feasi- bility and effectiveness of the improved method. The comparison between the numerical results of the conventional method and the improved method shows that the proposed method has better stability, higher accuracy, and less time consumption.

Moving least squares-based multi-functional sensing technique for estimating viscosity and density of ternary solution

In the osmotic dehydration process of food,on-line estimation of concentrations of two components in ternary solution with NaCl and sucrose was performed based on multi-functional sensing technique.Moving Least Squares were adopted in approximation procedure to estimate the viscosity of such interested ternary solution with the given data set.As a result,in one mode of using total experimental data as calibration data and validation data,the relative deviations of estimated viscosities are less than ±1.24%.In the other mode,by taking total experimental data except the ones for estimation as calibration data,the relative deviations are less than ±3.47%.In the same way,the density of ternary solution can be also estimated with deviations less than ± 0.11% and ± 0.30% respectively in these two models.The satisfactory and accurate results show the extraordinary efficiency of Moving Least Squares behaved in signal approximation for multi-functional sensors.

A meshless algorithm with the improved moving least square approximation for nonlinear improved Boussinesq equation

An improved moving least square meshless method is developed for the numerical solution of the nonlinear improved Boussinesq equation. After the approximation of temporal derivatives, nonlinear systems of discrete algebraic equations are established and are solved by an iterative algorithm. Convergence of the iterative algorithm is discussed. Shifted and scaled basis functions are incorporated into the method to guarantee convergence and stability of numerical results. Numerical examples are presented to demonstrate the high convergence rate and high computational accuracy of the method.

Matrices Associated with Moving Least-Squares Approximation and Corresponding Inequalities

In this article, some properties of matrices of moving least-squares approximation have been proven. The used technique is based on known inequalities for singular-values of matrices. Some inequalities for the norm of coefficients-vector of the linear approximation have been proven.

基于MLS拟合带容差的测量数据加工刀位面生成算法

传统的刀位面生成算法无法直接生成存在缺陷的离散数据的刀位面,因此,针对球头铣刀提出了一种基于MLS拟合带容差的离散数据加工刀位面生成算法。该方法首先依据球头铣刀构造离散数据的点膨胀球模型,然后根据凸包原理获取该模型的最外层包络面,最终通过MLS投影拟合方法剔除奇异点、修补裂缝,生成刀位面网格模型。实例表明,该算法具有较高的稳定性,能够有效解决存在缺陷的离散数据刀位面生成问题,适用于实际工程应用加工。

二维MLSPH无网格方法

给出MLSPH无网格方法的控制方程、移动最小二乘近似、MLSPH守恒格式等,重点研究MLSPH守恒格式中MLS面积向量的数值求解方法,以提高求解的精度.对激波管问题和平面Noh问题进行模拟,得到较好的结果,确定用三次样条积分公式计算面积向量.

新型无网格法(IMLS方形域法)的研究及其应用

根据最新无网格法的前沿研究,提出了一种新型无网格法-IMLS方形域法。该法在场函数近似、权函数选取以及方形域数值积分等方面给出了全新的格式。方法中未知变量的近似采用IMLS技术,局部影响域形状采用方形几何形态。这些技术的具体实施展现了节点布置和数值积分的无网格特点,并自然满足Dirichlet边界条件。该方法可以容易求解偏微分方程初值或边值问题上实现,此外,计算了一维固结问题、二维弹性力学平面问题的算例,所得计算结果证明:该方法是一种具有收敛快、精度高、简便有效的通用方法,在工程中具有广阔的应用前景。

基于MLS改进算法的曲线拟合技术研究

针对散乱点拟合曲线的算法问题,分析了移动最小二乘逼近法(MLS)的特点,提出了用MLS改进算法对散乱点进行曲线拟合,并结合实例进行了验证.结果表明,MLS改进算法在拟合曲线时,选取不同阶的基函数和不同的权函数,在保证精度和光滑度的基础上,解决了传统曲线拟合方法在拟合曲线时出现的运算不稳定现象,显示了MLS改进算法在曲线拟合中的优越性.

骨架驱动的MLS卡通角色变形

传统的基于移动最小二乘法(MLS)的图像变形方法中,由于没有考虑图像变形中的角色形状的拓扑关系,使得变形结果产生局部扭曲的现象.为此提出一种骨架驱动的MLS卡通角色变形方法.首先抽取图像中的角色,并对抽取出的角色模型进行三角剖分得到其网格模型;然后根据网格模型的拓扑关系手工输入骨架,并将模型的骨架自动绑定到该角色模型上;最后用户可交互地修改模型骨架,而骨架通过MLS将形变传递到整个角色模型上去.由于该方法是局部相关的,将每一个网格顶点绑定在相应骨架上,限定了全局骨架变形的影响,从而克服了变形失真现象.对比实验结果表明,文中方法可以使卡通角色的变形效果更加自然,具有广泛的应用前景.

图像形变中基于MLS的光滑纹理映射技术

提出了一种基于移动最小二乘的二次拟合多项式纹理映射模型。该模型主要应用于使用控制网格的图像形变技术。图像中的物体被嵌入一个控制网格,并且对控制网格进行形变操作。在控制网格上应用纹理映射,以绘制形变后的物体。在纹理映射时通过将控制网格顶点在形变前后的位移向量作为拟合函数值,采用移动最小二乘方法进行拟合得到映射所使用的二次函数。该方法能够明显改善图像形变后的光滑性,实现逼真、自然的图像形变。

基于MLS的FLAC3D复杂三维地质建模

针对岩土力学分析软件FLAC3D建模效率较低的不足之处,在分析FLAC3D建模命令以及网格模型*.FLAC3D文件格式特点的基础上,介绍了空间坐标变换、MLS插值的方法与意义,并提出了新的建模方法,开发了相应的计算机辅助建模程序。该程序直接读取原始数据点,无需借助第三方软件,通过坐标变换、MLS插值,可快速生成具有复杂边界的地质模型,且便于材料参数和应力边界条件参数的赋值,可有效地提高工作效率。

IMLS方形影响域法

展示了一些最新无网格法的研究进展,给出了一种新型无网格法-IMLS方形域无网格法。该法中未知变量的近似采用IMLS技术,局部影响域形状采用方形几何形态。这些技术的具体实施展现了节点布置和数值积分的无网格特点,并自然满足Dirichlet边界条件。该方法可以容易推广到求解非线性问题以及非均匀介质的力学问题。此外,还计算了两个弹性力学平面问题的例子,所得计算结果证明:该方法是一种具有收敛快、精度高、简便有效的通用方法,在工程中具有广阔的应用前景。

基于MLS-SVM的结构整体可靠度与全局灵敏度分析

作为一种有效的代理模型,支持向量机(SVM)以统计学习中的结构风险最小化原则为基本原理,在具有隐式极限状态函数的结构可靠度分析中得到了广泛的应用。然而,传统的支持向量机在核函数的选择、全局基本变量空间建模、计算效率等方面还存在许多不足。针对这些不足,该文提出一种新的基于移动最小二乘(MLS)技术的支持向量机模型(MLS-SVM),可以在全局基本变量空间中具备自适应能力。该文将MLSSVM应用于复杂结构的整体可靠度和全局灵敏度分析,并将该模型与基于再生核函数的支持向量机(RPKSVM)及基于最小二乘的支持向量机(LS-SVM)进行比较分析,结果表明:该文提出的模型相较其他两种模型具有更高的精度和计算效率。