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.

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 moving Kriging interpolation-based boundary node method for two-dimensional potential problems

In this paper, a meshfree boundary integral equation （BIE） method, called the moving Kriging interpolation- based boundary node method （MKIBNM）, is developed for solving two-dimensional potential problems. This study combines the DIE method with the moving Kriging interpolation to present a boundary-type meshfree method, and the corresponding formulae of the MKIBNM are derived. In the present method, the moving Kriging interpolation is applied instead of the traditional moving least-square approximation to overcome Kronecker＇s delta property, then the boundary conditions can be imposed directly and easily. To verify the accuracy and stability of the present formulation, three selected numerical examples are presented to demonstrate the efficiency of MKIBNM numerically.

Elastoplastic Large Deformation Using Meshless Integral Method

In this paper, the meshless integral method based on the regularized boundary integral equation [1] has been extended to analyze the large deformation of elastoplastic materials. The updated Lagrangian governing integral equation is obtained from the weak form of elastoplasticity based on Green-Naghdi’s theory over a local sub-domain, and the moving least-squares approximation is used for meshless function approximation. Green-Naghdi’s theory starts with the additive decomposition of the Green-Lagrange strain into elastic and plastic parts and considers aJ2elastoplastic constitutive law that relates the Green-Lagrange strain to the second Piola-Kirchhoff stress. A simple, generalized collocation method is proposed to enforce essential boundary conditions straightforwardly and accurately, while natural boundary conditions are incorporated in the system governing equations and require no special handling. The solution algorithm for large deformation analysis is discussed in detail. Numerical examples show that meshless integral method with large deformation is accurate and robust.

CLASSIFICATION OF POSITIVE SOLUTIONS FOR NONLINEAR DIFFERENTIAL AND INTEGRAL SYSTEMS WITH CRITICAL EXPONENTS

We classify all positive solutions for the following integral system：{ui（x）=∫Rn1/│x-y│^n-α fi（u（y））dy,x∈R^n,i=1,…,m,0〈α〈n,and u（x）=（u1（x）,u2（x）…,um（x））.Here fi（u）, 1 ≤ i ≤m, monotone nondecreasing are real-valued functions of homogeneous degree n＋α/n-α and are monotone nondecreasing with respect to all the independent variables U1, u2, ..., urn.In the special case n ≥ 3 and α = 2. we show that the above system is equivalent to thefollowing elliptic PDE system：This system is closely related to the stationary SchrSdinger system with critical exponents for Bose-Einstein condensate

AN IMPROVED HYBRID BOUNDARY NODE METHOD IN TWO-DIMENSIONAL SOLIDS

The hybrid boundary node method （HBNM） is a promising method for solving boundary value problems with the hybrid displacement variational formulation and shape functions from the moving least squares（MLS） approximation. The main idea is to reduce the dimensionality of the former and keep the meshless advantage of the latter. Following its application in solving potential problems, it is further developed and numerically implemented for 2D solids in this paper. The rigid movement method is employed to solve the hyper-singular integrations. Numerical examples for some 2D solids have been given to show the characteristics. The computation results obtained by the present method are in excellent agreement with the analytical solution. The parameters that influence the performance of this method are studied through numerical examples.

Diffraction of plane P waves around an alluvial valley in poroelastic half-space

Diffraction of plane P waves around an alluvial valley of arbitrary shape in poroelastic half-space is investigated by using an indirect boundary integral equation method. Based on the Green＇s fimctions of line source in poroelastic half-space, the scattered waves are constructed using the fictitious wave sources close to the interface of the valley and the density of ficti- tious wave sources are determined by boundary conditions. The precision of the method is verified by the satisfaction extent of boundary conditions, and the comparison between the degenerated solutions and available results in single-phase case. Finally, the nature of diffraction of plane P waves around an alluvial valley in poroelastic half-space is investigated in detail through nu- merical examples.

CLASSIFICATION OF SOLUTIONS FOR A CLASS OF SINGULAR INTEGRAL SYSTEM

In this paper, we consider the following integral system： u（x） = R n v q （y） ｜ x y ｜ nα dy, v（x） = R n u p （y） ｜ x y ｜ nμ dy, （0.1） where 0 〈 α, μ 〈 n; p, q ≥ 1. Using the method of moving planes in an integral form which was recently introduced by Chen, Li, and Ou in [2, 4, 8], we show that all positive solutions of （0.1） are radially symmetric and decreasing with respect to some point under some general conditions of integrability. The results essentially improve and extend previously known results [4, 8].

ITERATIVE ALGORITHM FOR AXIALLY ACCELERATING STRINGS WITH INTEGRAL CONSTITUTIVE LAW

A numerical method is proposed to simulate the transverse vibrations of a viscoelastic moving string constituted by an integral law. In the numerical computation, the Galerkin method based on the Hermite functions is applied to discretize the state variables, and the Runge- Kutta method is applied to solve the resulting differential-integral equation system. A linear iterative process is designed to compute the integral terms at each time step, which makes the numerical method more efficient and accurate. As examples, nonlinear parametric vibrations of an axially moving viscoelastic string are analyzed.

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.

Liouville-Type Theorems for Some Integral Systems

In this paper, Liouville-type theorems of nonnegative solutions for some elliptic integral systems are considered. We use a new type of moving plane method introduced by Chen-Li-Ou. Our new ingredient is the use of Stein-Weiss inequality instead of Maximum Principle.

The Method of Moving Planes for Integral Equation in an Extremal Case

In this paper, we study the symmetry results and monotonicity of solutions for an integral equation u（x）=-cn∫RNeu（y）log｜x-y｜dy in an external case.

NONEXISTENCE OF POSITIVE SOLUTIONS FOR A SEMI-LINEAR EQUATION INVOLVING THE FRACTIONAL LAPLACIAN IN R^N

In this paper, we consider the semilinear equation involving the fractional Laplacian in the Euclidian space R^n：（-△）^α/2u（x） ： f（xn）u^p（x）, x ∈R^n（0.1）in the subcritical case with 1〈 p〈n＋a/n-a.Instead of carrying out direct investigations on pseudo-differential equation （0.1）, we first seek its equivalent form in an integral equation as below：u（x）=∫R^nG∞（x, y） f（yn） u^p（y）dy,where G∞（x, y） is the Green＇s function associated with the fractional Laplacian in R^n. Employing the method of moving planes in integral forms, we are able to derive the nonexistence of positive solutions for （0.2） in the subcritical case. Thanks to the equivalence, same con- clusion is true for （0.1）.

A MESHLESS METHOD OF A THIN PLATE

A meshless approach to analysis of arbitrary Kirechhoff plates bythe local boundary integral equation (LBIF) method is presented. Themethod combines the advantageous features of all the three meth- ods:the Galerkin finite element method (GFEM), the boundary elementmethod (BEM) and the element- free Galerkin method (EFGM). It is atruly meshless method, which means that the discretization is inde-pendent of geometric subdivision into elements or cells, but is onlyboundary integration, however over a local boundary cen- tered) overa domain in question.

A Direct Method of Moving Planes to Fractional Power Sub Laplace Equations on the Heisenberg Group

We give the direct method of moving planes for solutions to the conformally invariant fractional power sub Laplace equation on the Heisenberg group.The method is based on four maximum principles derived here.Then symmetry and nonexistence of positive cylindrical solutions are proved.

薄板的局部Petrov-Galerkin方法

利用薄板控制微分方程的等效积分对称弱形式和对变量(挠度)采用移动最小二乘近似函数进行插值,研究了薄板弯曲问题的无网格局部Petrov＿Galerkin方法· 这是一种真正的无网格方法,它不需要任何有限元或边界元网格,不管这种网格是用于能量积分还是进行插值的目的· 所有的积分都在规则形状的子域及其边界上进行,并用罚因子法施加本质边界条件· 数值例子表明,无网格局部Petrov＿Galerkin法不但能够求解二阶微分方程的边值问题,而且求解四阶微分方程的边值问题也很有效,也具有收敛快、稳定性好。

壳结构的无网格局部Petrov-Galerkin方法

无网格近似函数具有高度光滑性,能够很好的逼近曲壳表面及其位移场。无网格局部Petrov-Galerkin方法不论插值还是离散都不需要单元,是一种真正的无网格方法。本文基于无网格局部Petrov-Galerkin方法的基本原理,采用移动最小二乘插值,利用控制微分方程弱形式,建立了Mindlin壳结构的无网格局部Petrov-Galerkin分析方法,用屋顶壳、受夹圆柱壳、几何非线性圆柱壳作为计算实例分析了求解精度、收敛性和稳定性,并与精确解和有限元计算结果进行了对比,表明该方法计算精度高及收敛性好。

混合局部-非局部椭圆方程奇异解的单调性和对称性

该文运用移动平面法研究了一类混合局部-非局部半线性椭圆方程奇异解的单调性和对称性.

一类分数阶Laplace方程解的径向对称性研究

通过直接移动平面法研究了一类具有以下形式的分数阶Laplace方程,(-Δ)^(α/2) u(x)=|x|^(β)u^(p)(x)+|x|^(γ)u^(q)(x),x∈R^(n),0<α<2,n≥3。得到了其在全空间及不同约束条件下方程正解的径向对称性。

SYMMETRY OF POSITIVE SOLUTIONS FOR THE FRACTIONAL HARTREE EQUATION

In this paper, by using the method of moving planes, we are concerned with the symmetry and monotonicity of positive solutions for the fractional Hartree equation.