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.

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.

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.

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.

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.

The Approximation Error of Ordinary Differential Equations Based on the Moved Node Method

One of the central issues in solving differential equations by numerical methods is the issue of approximation. The standard way of approximating differential equations by numerical methods (particularly difference methods) is to question the degree of approximation in the form O(hp). Here h is the grid step. In this case we have an implicit approximation. Based on the difference equation approximating the differential equation, the order of approximation is obtained using the Taylor series. However, it is possible to calculate the approximation error at nodal points based on the method of moving nodes. The method of moving nodes allows obtaining an approximate analytical expression. On the basis of the approximate form, it is possible to calculate the approximation error. The analytical form of the approximation makes it possible to efficiently calculate this error. On the other hand, the property of this error allows the construction of new improved circuits. In addition, based on these types of errors, you can create a differential analog of the difference equation that gives an exact approximation.

ICM Method Combined with Meshfree Approximation for Continuum Structure

The independent continuous mapping（ICM） method is integrated into element free Galerkin method and a new implementation of topology optimization for continuum structure is presented.To facilitate the enforcement of the essential boundary condition and derivative of various sensitivities,a singular weight function in element free Galerkin method is introduced.Material point variable is defined to illustrate the condition of material point and its vicinity instead of element or node.The topological variables field is constructed by moving least square approximation which inherits the continuity and smoothness of the weight function.Due to reciprocal relationships between the topological variables and design variables,various structural responses sensitivities are derived according to the method for calculating the partial derivatives of compound functions.Numerical examples indicate that checkerboard pattern and mesh-dependence phenomena are overcome without additional restriction methods.

Convergence Properties of Piecewise Power Approximations

We address the problem of convergence of approximations obtained from two versions of the piecewise power-law representations arisen in Systems Biology. The most important cases of mean-square and uniform convergence are studied in detail. Advantages and drawbacks of the representations as well as properties of both kinds of convergence are discussed. Numerical approximation algorithms related to piecewise power-law representations are described in Appendix.

浸入运动边界-格子Boltzmann方法4种固含率计算方法对比研究

为了达到流固耦合,格子Boltzmann方法(LBM)可采用浸入运动边界法(IMB)实现移动颗粒边界上的无滑移条件.该耦合方式(IMB-LBM)中固含率计算方法对流固耦合计算精度和效率有影响.对常用的固含率4种计算方法,即蒙特卡洛法(MCM)、单元分解法(UDM)、近似多边形法(APM)和闭合边界法(CBM),分别阐述其具体算法,对比了它们的计算精度和计算效率;最后通过圆盘颗粒非连续变形分析方法(DDDA)与IMB-LBM耦合模型下的一个多颗粒沉降流固耦合算例,对比分析了它们在流固耦合计算过程中的耗时.结果表明:1)CBM无误差,MCM和UDM在随机点数取1000,子单元数取100时误差稳定在1%以下,APM在颗粒直径大于格子长度10倍时,误差小于0.44%;2)MCM和UDM的计算精度及耗时分别与随机点数和子单元数相关,它们的计算耗时大于APM和CBM;3)计算效率上,APM>CBM>UDM>MCM,其中CBM计算耗时略微大于APM,APM和UDM计算耗时分别比MCM少2个和1个数量级.该结果可为IMB-LBM耦合模型中固含率计算方法优选提供借鉴.

基于改进Reddy型三阶剪切变形理论的弹性地基上FG-CNTRC板屈曲无网格分析

针对含碳纳米管转向的Pasternak地基上功能梯度碳纳米管增强复合材料FG-CNTRC(functionally graded carbon nanotube-reinforced composite)板的屈曲问题,提出了一种基于改进Reddy型三阶剪切变形理论TSDT(third-order shear deformation theory)和移动最小二乘近似MLS(moving-least square)的无网格分析模型。该模型避免了无网格法第二类边界条件的施加困难问题,且能够满足中厚/厚板的自由表面条件,无需额外引入剪切修正因子。基于最小势能原理推导了弹性地基上FG-CNTRC板的无网格屈曲控制方程,采用完全转换法处理本质边界条件。通过基准算例验证了本文方法的收敛性及有效性,讨论了碳纳米管的转向角、体积分数、分布形式、地基系数、宽厚比和边界条件等对FG-CNTRC板临界屈曲荷载的影响。

改进Reddy型三阶剪切变形理论下FG-GRC板弯曲和模态分析的无网格法

基于含7个自由度变量的改进Reddy型三阶剪切变形理论(third-order deformation theory,TSDT)假设,采用稳定移动最小二乘近似(stabilized moving least-square approximation,SMLS)的无网格法研究了功能梯度石墨烯增强复合材料(functionally graded graphene-reinforced composite,FG-GRC)板结构的静态线性弯曲和自振模态。通过Halpin-Tsai模型来估算材料的有效弹性模量,有效质量密度和泊松比由混合定律确定。利用最小势能原理和Hamition原理分别推导了FG-GRC板的线性弯曲和自振频率无网格控制方程。由于基于SMLS构造的形函数不满足克罗内克条件,故采用完全转换法处理本质边界条件。该研究首先介绍了基于TSDT下FG-GRC板的SMLS离散模型;随后通过与已有成果进行比较,检验了本文方法的收敛性及准确性;最后数值分析了石墨烯片(graphene platelets,GPLs)分布模式,质量分数、几何参数、总层数及边界条件等对FG-GRC板结构弯曲和模态的影响。

Least-Squares Solutions of Matrix Equations(AX = B,XC = D) for Hermitian Reflexive (Anti-Hermitian Reflexive) Matrices and Its Approximation

In this paper,the Hermitian reflexive（Anti-Hermitian reflexive）least-squares so-lutions of matrix equations（AX = B,XC = D）are considered.With special properties of partitioned matrices and Hermitian reflexive（Anti-Hermitian reflexive）matrices,the general expression of the solution is obtained.Moreover,the related optimal approximation problem to a given matrix over the solution set is considered.

MOVING BOUNDARY PROBLEM FOR DIFFUSION RELEASE OF DRUG FROM A CYLINDER POLYMERIC MATRIX

An approximate analytical solution of moving boundary problem for diffusion release of drug from a cylinder polymeric matrix was obtained by use of refined integral method. The release kinetics has been analyzed for non-erodible matrices with perfect sink condition. The formulas of the moving boundary and the fractional drug release were given. The moving boundary and the fractional drug release have been calculated at various drug loading levels, mid the calculated results were in good agreement with those of experiments. The comparison of the moving boundary in spherical, cylinder, planar matrices has been completed. An approximate formula for estimating the available release time was presented. These results are useful for the clinic experiments. This investigation provides a new theoretical tool for studying the diffusion release of drug from a cylinder polymeric matrix and designing the controlled released drug.

Anomalous diffusion in a variable area whose boundary moves with a constant speed

In this paper, we study a space-fractional anomalous diffusion in a variable area. The moving boundary is assumed moving with constant speed. The numerical scheme was present by changing the moving boundary to a fixed one. The steady-state approximation was also given to show the properties of the diffusion process.

High-Dimensional Regression on Sparse Grids Applied to Pricing Moving Window Asian Options

The pricing of moving window Asian option with an early exercise feature is considered a challenging problem in option pricing. The computational challenge lies in the unknown optimal exercise strategy and in the high dimensionality required for approximating the early exercise boundary. We use sparse grid basis functions in the Least Squares Monte Carlo approach to solve this “curse of dimensionality” problem. The resulting algorithm provides a general and convergent method for pricing moving window Asian options. The sparse grid technique presented in this paper can be generalized to pricing other high-dimensional, early-exercisable derivatives.

基于逆散射成像条件的转换波弹性波动方程自动偏移速度分析

在地震偏移成像过程中,成像精度依赖于速度模型,因此偏移速度分析对高精度叠前偏移成像至关重要。基于高频近似下的逆散射成像条件,提出了基于剩余曲率分析的转换波弹性波动方程偏移速度分析方法。首先基于高频近似下的逆散射成像条件,采用逆时偏移近似估计转换S波散射位,然后结合入射P波和转换S波散射位,重构得到上行转换S波,最后利用伴随状态法导出剩余时差关于波速的梯度。基于逆散射成像条件,有效保证了弹性波动方程偏移速度分析中梯度的稳定求解。数值实验表明该方法导出了高质量的梯度。初步测试表明反演的速度模型有效地校平了共成像点道集,改善了成像质量。此外,剩余时差曲率智能识别等技术简化了中间处理流程,构建的智能化速度分析框架有望提升计算效率。

Moving Multiple Curves/Surfaces Approximation of Mixed Point Clouds

We propose a local model called moving multiple curves/surfaces approximation to separate mixed scanning points received from a thin-wall object,where data from two sides of the object may be mixed due to measurement error.The cases of two curves(including plane curves and space curves)and two surfaces in one model are mainly elaborated,and a lot of examples are tested.

A STOCHASTIC MOVING BALLS APPROXIMATION METHOD OVER A SMOOTH INEQUALITY CONSTRAINT

We consider the problem of minimizing the average of a large number of smooth component functions over one smooth inequality constraint.We propose and analyze a stochastic Moving Balls Approximation(SMBA)method.Like stochastic gradient(SG)met hods,the SMBA method's iteration cost is independent of the number of component functions and by exploiting the smoothness of the constraint function,our method can be easily implemented.Theoretical and computational properties of SMBA are studied,and convergence results are established.Numerical experiments indicate that our algorithm dramatically outperforms the existing Moving Balls Approximation algorithm(MBA)for the structure of our problem.