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.

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.

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.

Re-parameterization of Bézier Curves by Square Approximation with Endpoint Constraints

In order to get an approximation with better effect of pararneterization of Bezier curves, we proposed a method for arc-length parameterization and the corresponding algorithms by square approximation for the discrete even de-parameterization of the curves. This method is simple and easy to implement, and the property of the approximation has no change compared with the original curve. A quantitative criterion for estimating the effect of parameterization is also built to quantitatively characterize the parameterization effect of the algorithms. As a result, the nearly arc-length parameterized curve has a smaller relative deviation using either the algorithm with point constraint at endpoints or the algorithm with point constraint plus the first derivative constraint at endpoints. Experiments show that after re-parameterization with our algorithms, the relative deviation will have at least a 20% reduction.

Application of nonlinear partial least square in catalyst modeling

The neural network partial least square (NNPLS) method was used to establish a robust reaction model for a multi-component catalyst of methane oxidative coupling. The details, including the learning algorithm, the number of hidden units of the inner network, activation function, initialization of the network weights and the principal components, are discussed. The results show that the structural organizations of inner neural network are 1-10-5-1, 1-8-4-1, 1-8-5-1, 1-7-4-1, 1-8-4-1, 1-8-6-1, respectively. The Levenberg-Marquardt method was used in the learning algorithm, and the central sigmoidal function is the activation function. Calculation results show that four principal components are convenient in the use of the multi-component catalyst modeling of methane oxidative coupling. Therefore a robust reaction model expressed by NNPLS succeeds in correlating the relations between elements in catalyst and catalytic reaction results. Compared with the direct network modeling, NNPLS model can be adjusted by experimental data and the calculation of the model is simpler and faster than that of the direct network model.

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.

Neural Network inverse Adaptive Controller Based on Davidon Least Square

General neural network inverse adaptive controller has two flaws: the first is the slow convergence speed; the second is the invalidation to the non-minimum phase system. These defects limit the scope in which the neural network inverse adaptive controller is used. We employ Davidon least squares in training the multi-layer feedforward neural network used in approximating the inverse model of plant to expedite the convergence, and then through constructing the pseudo-plant, a neural network inverse adaptive controller is put forward which is still effective to the nonlinear non-minimum phase system. The simulation results show the validity of this scheme.

Modelling of the Relaxation Least Squares-Based Neural Networks and Its Application

A relaxation least squares-based learning algorithm for neual networks is proposed. Not only does it have a fast convergence rate, but it involves less computation quantity. Therefore, it is suitable to deal with the case when a network has a large scale but the number of training data is very limited. It has been used in converting furnace process modelling, and impressive result has been obtained.

Robust On-Line Fault Diagnosis for Nonlinear Difference-Algebraic Systems Using Least Squares Estimate

A robust on-line fault diagnosis methor based on least squares estimate for nonlinear difference-algebraic systems (DAS) with uncertainties is proposed. Based on the known nominal model of the DAS, this method firstly constructs an auxiliary system consisting of a difference equation and an algebraic equation, then, based on the relationship between the state deviation and the faults in the difference equation and the relationship between the algebraic variable deviation and the faults in algebraic equation, it identifies the faults on-line through least squares estimate. This method can not only detect, isolate and identify faults for DAS, but also give the upper bound of the error of fault identification. The simulation results indicate that it can give satisfactory diagnostic results for both abrupt and incipient faults.

Weight-based shape modification of NURBS curves by constrained optimization

A new method for shape modification of non-uniform rational B-splines (NURBS) curves was presented, which was based on constrained optimization by means of altering the corresponding weights of their control points. Using this method, the original NURBS curve was modified to satisfy the specified geometric constraints, including single point and multi-point constraints. With the introduction of free parameters, the shapes of modified NURBS curves could be further controlled by users without destroying geometric constraints and seem more naturally. Since explicit formulae were derived to compute new weights of the modified curve, the method was simple and easy to program. Practical examples showed that the method was applicable for computer aided design (CAD) system.

Study on the tradeoff between interpretability and precision in fuzzy modeling

An approach to identifying fuzzy models considering both interpretability and precision was proposed. Firstly, interpretability issues about fuzzy models were analyzed. Then, a heuristic strategy was used to select input variables by increasing the number of input variables, and the Gustafson-Kessel fuzzy clustering algorithm, combined with the least square method, was used to identify the fuzzy model. Subsequently, an interpretability measure was described by the product of the number of input variables and the number of rules, while precision was weighted by root mean square error, and the selection objective function concerning interpretability and precision was defined. Given the maximum and minimum number of input variables and rules, a set of fuzzy models was constructed. Finally, the optimal fuzzy model was selected by the objective function, and was optimized by a genetic algorithm to achieve a good tradeoff between interpretability and precision. The performance of the proposed method was illustrated by the well-known Box-Jenkins gas furnace benchmark; the results demonstrate its validity.

A MESHLESS LOCAL PETROV-GALERKIN METHOD FOR GEOMETRICALLY NONLINEAR PROBLEMS

Nonlinear formulations of the meshless local Petrov-Galerkin （MLPG） method are presented for geometrically nonlinear problems. The method requires no mesh in computation and therefore avoids mesh distortion difficulties in the large deformation analysis. The essential boundary conditions in the present formulation axe imposed by a penalty method. An incremental and iterative solution procedure is used to solve geometrically nonlinear problems. Several examples are presented to demonstrate the effectiveness of the method in geometrically nonlinear problems analysis. Numerical results show that the MLPG method is an effective one and that the values of the unknown variable are quite accurate.

LOCAL PETROV-GALERKIN METHOD FOR A THIN PLATE

The meshless local Petrov_Galerkin (MLPG) method for solving the bending problem of the thin plate were presented and discussed. The method used the moving least_squares approximation to interpolate the solution variables, and employed a local symmetric weak form. The present method was a truly meshless one as it did not need a finite element or boundary element mesh, either for purpose of interpolation of the solution, or for the integration of the energy. All integrals could be easily evaluated over regularly shaped domains (in general, spheres in three_dimensional problems) and their boundaries. The essential boundary conditions were enforced by the penalty method. Several numerical examples were presented to illustrate the implementation and performance of the present method. The numerical examples presented show that high accuracy can be achieved for arbitrary grid geometries for clamped and simply_supported edge conditions. No post processing procedure is required to computer the strain and stress, since the original solution from the present method, using the moving least squares approximation, is already smooth enough.

Improved non-singular local boundary integral equation method

When the source nodes are on the global boundary in the implementation of local boundary integral equation method （LBIEM）, singularities in the local boundary integrals need to be treated specially. In the current paper, local integral equations are adopted for the nodes inside the domain and moving least square approximation （MLSA） for the nodes on the global boundary, thus singularities will not occur in the new al- gorithm. At the same time, approximation errors of boundary integrals are reduced significantly. As applications and numerical tests, Laplace equation and Helmholtz equation problems are considered and excellent numerical results are obtained. Furthermore, when solving the Helmholtz problems, the modified basis functions with wave solutions are adapted to replace the usually-used monomial basis functions. Numerical results show that this treatment is simple and effective and its application is promising in solutions for the wave propagation problem with high wave number.

A SPACE THEORY BASIS FOR CURVE FITTING PROBLEMS

itherto, a precision Concept for curve fitting problems has not been set. By using the theory of functional analysis, the author of this paper established a space theory basis for curve fitting problems. Also given in the paper is the precision concept of the curve fitting problems and the method for constructing the fitting of a curve satisfying given precision requirements.

Extension of Direct Citation Model Using In-Text Citations

Citations based relevant research paper recommendations can be generated primarily with the assistance of three citation models:(1)Bibliographic Coupling,(2)Co-Citation,and(3)Direct Citations.Millions of new scholarly articles are published every year.This flux of scientific information has made it a challenging task to devise techniques that could help researchers to find the most relevant research papers for the paper at hand.In this study,we have deployed an in-text citation analysis that extends the Direct Citation Model to discover the nature of the relationship degree-ofrelevancy among scientific papers.For this purpose,the relationship between citing and cited articles is categorized into three categories:weak,medium,and strong.As an experiment,around 5,000 research papers were crawled from the CiteSeerX.These research papers were parsed for the identification of in-text citation frequencies.Subsequently,0.1 million references of those articles were extracted,and their in-text citation frequencies were computed.A comprehensive benchmark dataset was established based on the user study.Afterwards,the results were validated with the help of Least Square Approximation by Quadratic Polynomial method.It was found that degreeof-relevancy between scientific papers is a quadratic increasing/decreasing polynomial with respect to-increase/decrease in the in-text citation frequencies of a cited article.Furthermore,the results of the proposed model were compared with state-of-the-art techniques by utilizing a well-known measure,known as the normalized Discount Cumulative Gain(nDCG).The proposed method received an nDCG score of 0.89,whereas the state-of-the-art models such as the Content,Bibliographic-coupling,and Metadata-based Models were able to acquire the nDCG values of 0.65,0.54,and 0.51 respectively.These results indicate that the proposed mechanism may be applied in future information retrieval systems for better results.

A 3D shell-like approach using element-free Galerkin method for analysis of thin and thick plate structures

A new efficient meshless method based on the element-free Galerkin method is proposed to analyze the static deformation of thin and thick plate structures in this paper. Using the new 3D shell-like kinematics in analogy to the solid-shell concept of the finite element method, discretization is carried out by the nodes located on the upper and lower surfaces of the structures. The approximation of all unknown field variables is carried out by using the moving least squares （MLS） approximation scheme in the in-plane directions, while the linear interpolation is applied through the thickness direction. Thus, different boundary conditions are defined only using displacements and penalty method is used to enforce the essential boundary conditions. The constrained Galerkin weak form, which incorporates only dis- placement degrees of freedom （d.o.f.s）, is derived. A modified 3D constitutive relationship is adopted in order to avoid or eliminate some self-locking effects. The numeric efficiency of the proposed meshless formulation is illustrated by the numeric examples.

ANALYSIS OF 2-D THIN STRUCTURES BY THE MESHLESS REGULAR HYBRID BOUNDARY NODE METHOD

Thin structures are generally solved by the Finite Element Method(FEM), using plate or shell finite elements which have manylimitations in applications, such as numerical locking, edge effects,length scaling and the cnvergence problem. Recently, by proposing anew approach to tranting the nearly- singular integrals, Liu et al.developed a BEM to successfully solve thin structures with thethickness-to- length ratios in the micro-or nano-scales. On the otherhand, the meshless Regular Hybrid Boundary Node Method (RHBNM), whichis proposed by the current authors and based on a modified functionaland the Moving Least-Square (MLS) approximation, has very promisingapplications for engineering problems owing To its meshless natureand dimension-reduction advantage, and not involving any singular ornearly-singular Integrals. Test examples show that the RHBNM can alsobe applied readily to thin structures with high accu- Racy withoutany modification.

Quaternion based joint DOA and polarization parameters estimation with stretched three-component electromagnetic vector sensor array

The three-component electromagnetic vector sensor (EMVS) consisting of co-centered orthogonally oriented x-dipole, z-dipole and z-loop is considered. In order to make full use of the spatial aperture of each component, the original uniform linear three-component EMVS array (ULTEA) is stretched into one half-wavelength spaced uniform linear loop subarray (ULLSA) along the z axis, and one sparse uniform linear co-centered orthogonally oriented dual-dipole (CODD) subarray (SULCSA) along the x axis. Then, a generalized rotation invariance based quaternion multiple signal classification (GRIQ-MUSIC) algorithm is presented for direction of arrival (DOA) and polarization parameters estimation. According to the proposed algorithm, the elevation angles are firstly estimated based on the half-wavelength spaced ULLSA. Then the polarization phase differences and azimuth angles are obtained based on the coupling relationship between the angle domain and polarization domain, but the azimuth angles are in coarse-resolution since the array aperture is not utilized. Next, the SULCSA is used to re-estimate the azimuth angles in fine-resolution, and the ambiguity problem can be resolved by the least square method. Finally, based on the estimated elevation angles, azimuth angles and polarization phase differences, the corresponding auxiliary polarization angles can be estimated by N times one-dimensional parameter search, where N is the sources number, and the parameters are matched automatically. Based on the GRIQ-MUSIC algorithm, the high dimensional parameters search problem of the conventional Q-MUSIC algorithm is simplified to a one-dimensional parameter search problem, thus the proposed algorithm not only reduces the computation complexity considerably, but also avoids the performance degradation caused by the failure in parameters pairing. The simulation examples demonstrate the effectiveness and feasibility of the proposed algorithm. © 1990-2011 Beijing Institute of Aerospace Information.