Local Heaviside‐weighted LRPIM meshless method and its application to two‐dimensional potential flows

In this paper, the local radial point interpolation meshless method (LRPIM) is used for the analysis of two‐dimensional potential flows, based on a local‐weighted residual method with the Heaviside step function as the weighting function over a local subdomain. Trial functions are constructed using radial basis functions. The present method is a truly meshless method based only on a number of randomly located nodes. Integration over the subdomains requires only a simple integration cell to obtain the solution. No element matrix assembly is required and no special treatment is needed to impose the essential boundary conditions. The novelty of the paper is the use of a local Heaviside weight function in the LRPIM, which does not need local domain integration and integrations only on the boundary of the local domains are needed. Effects of the sizes of local subdomain and interpolation domain on the performance of the present method are investigated. The behavior of shape parameters of multiquadrics has been systematically studied. Two numerical tests in groundwater and fluid flows are presented and compared with closed‐form solutions and finite element method. The results show that the use of a local Heaviside weight function in the LRPIM is highly accurate and possesses no numerical difficulties. Copyright © 2008 John Wiley & Sons, Ltd.     

基于局部Petrov-Galerkin离散方案的无网格法

基于局部Petrov-Galerkin离散方案,选用自然邻近插值构造试函数,用Shepard函数作为权函数,提出了一种无网格方法(MNNPG),这种方法充分发挥了局部Petrov-Galerkin法的优势,并且结合了自然邻近插值的特点,方便引入边界条件,由于以Shepard函数的圆形支集作为积分子域,用分片中点插值来完成区域积分,无需额外背景网格,是一种真正的无网格法。本文将该无网格方法用于求解二维弹性力学边值问题,算例结果很好地吻合了精确解,表明该方法具有良好的数值精度和稳定性。     

用无网格局部径向点插值法分析非均质中厚板的弯曲问题

用无网格局部径向点插值法分析了非均质中厚板的弯曲问题.利用虚位移原理推导了中厚板的离散系统方程.采用径向基函数耦合多项式基函数来近似试函数,用四次样条函数作为加权残值公式中的权函数.所构造成的形函数具有Kronecker delta性质,可以很方便地施加本质边界条件.此方法不需要任何形式的网格划分,所有的积分都在规则形状的子域及其边界上进行,是一种真正的无网格方法.在计算过程中,取积分中的高斯点的材料参数来模拟问题域材料特性的变化.算例结果表明这种无网格方法具有效率高、精度高和易于实现等优点.     

大变形问题分析的局部Petrov-Galerkin法

在微机电系统(MEMS)的建模和模拟研究中,大变形或大移动要充分予以考虑.用有限元法分析这类问题,由于难以避免的网格畸变,使模拟效率精度降低甚至失效,无网格方法(Meshless Method)则能在分析这类问题时显示出明显的优势,无网格局部Petrov-Galerkin(MLPG)法被誉为是一种有发展前景的真正无网格法.本文进一步发展了MLPG法,通过对任意的离散分布节点采用局部径向基函数构造插值形函数和Heaviside权函数,分析方程采用局部加权弱形式离散,建立了变量仅依赖于初始构型的完全Lagrange分析格式,最后用Newton-Raphson法迭代求解.文中分析了悬臂梁典型算例和微机电开关非线性大变形问题,通过与有限元结果的比较,表明本文提出的大变形问题无网格局部Petrov-Galerkin法具有稳定性好及收敛性快等优点.     

非均质中厚板的无网格LRPIM动力学分析

用局部加权残值法建立了非均质中厚板的局部径向点插值离散系统方程,采用无网格局部径向点插值法分析了非均质中厚板的自由振动和强迫振动问题。用径向基函数耦合多项式基函数来近似试函数,用四次样条函数做为加权残值法中的权函数。所构造的形函数具有Kronecker delta性质,可以很方便地施加本质边界条件。该方法不需要任何形式的网格划分,所有的积分都在规则形状的子域及其边界上进行。在计算过程中,取积分中的高斯点的材料参数来模拟问题域材料特性的变化。计算结果表明,利用该方法计算非均质中厚板的自由振动和强迫振动问题可以得到具有较高精度的解。     

ELASTIC DYNAMIC ANALYSIS OF MODERATELY THICK PLATE USING MESHLESS LRPIM

A meshless local radial point interpolation method （LRPIM） for solving elastic dy-namic problems of moderately thick plates is presented in this paper. The discretized system equation of the plate is obtained using a locally weighted residual method. It uses a radial basis function （RBF） coupled with a polynomial basis function as a trial function,and uses the quartic spline function as a test function of the weighted residual method. The shape function has the properties of the Kronecker delta function,and no additional treatment is done to impose essen-tial boundary conditions. The Newmark method for solving the dynamic problem is adopted in computation. Effects of sizes of the quadrature sub-domain and influence domain on the dynamic properties are investigated. The numerical results show that the presented method can give quite accurate results for the elastic dynamic problem of the moderately thick plate.     

一种高效的局部径向基点插值无网格方法

提出了一种弹性动力分析的高效局部径向基点插值无网格方法(MLRPI).该方法采用径向基点插值形函数近似解变量,运用局部Petrov-Galerkin法推导出了相应的离散方程,并根据波动模拟的精度要求,得到某一结点的动力方程.然后采用Newmark常平均加速度法和中心差分法相结合的显式积分格式进行时域积分,得到每个自由度的一种解耦递推格式.最后,对一平面应变问题进行了求解,比较了该文提出的解耦MI.RPI方法、常规MLRPI方法和ANSYS有限元方法的精度和计算时间,结果表明解耦MLRPI方法与常规MLRPI方法的精度相当,但计算效率大大提高.     

FREE VIBRATION ANALYSIS OF ARBITRARY SHAPED PLATES BY BOUNDARY KNOT METHOD

The boundary knot method （BKM） is a truly meshless boundary-type radial basis function （RBF） collocation scheme, where the general solution is employed instead of the fundamental solution to avoid the fictitious outside boundary of the physical domain of interest. In this study, the BKM is first used to calculate the free vibration of free and simply-upported thin plates. Compared with the analytical solution and ANSYS （a commercial FEM code） results, the present BKM is highly accurate and fast convergent.     

无网格局部Petrov-Galerkin方法在弹塑性断裂力学问题中的应用

采用无网格局部Petroy-Galerkin方法来分析弹塑性断裂力学问题.这种无网格方法采用移动最小二乘法(MLS)来构造近似试函数和采用Heaviside函数作为加权残值法中的权函数,由于近似函数不满足KroneckerDelta条件,因此采用直接插值法来施加本质边界条件.如果不考虑体力,所形成的整体刚度矩阵只包含局部边界积分,而不包含局部域积分和奇异积分.采用增量Newton-Raphson迭代法来求解弹塑性增量形式的局部Petrov-Galerkin方程.数值算例结果表明,该文方法对于弹塑性断裂力学问题的求解是可行的和有效的,并且所得到的结果具有较好的精度.     

h-ADAPTIVE ANALYSIS BASED ON MESHLESS LOCAL PETROV-G ALERKIN METHOD WITH B SPLINE WAVELET FOR PLATES AND SHELLS

Using the two-scale decomposition technique, the h-adaptive meshless local Petrov- Galerkin method for solving Mindlin plate and shell problems is presented. The scaling functions of B spline wavelet are employed as the basis of the moving least square method to construct the meshless interpolation function. Multi-resolution analysis is used to decompose the field variables into high and low scales and the high scale component can commonly represent the gradient of the solution according to inherent characteristics of wavelets. The high scale component in the present method can directly detect high gradient regions of the field variables. The developed adaptive refinement scheme has been applied to simulate actual examples, and the effectiveness of the present adaptive refinement scheme has been verified.     

中厚板的耦合多项式基径向点插值无网格法

采用Mindlin平板理论,通过最小位能原理建立了各向同性中厚板的伽辽金整体弱式方 程,形函数采用耦合多项式基的径向点插值法构造,可以直接施加本质边界条件. 算例表明, 用耦合多项式基的径向点插值无网格法分析中厚板问题,具有效率高、精度高和易于实现等 优点,可以避免薄板弯曲时的剪切自锁现象.     

FREE VIBRATION ANALYSIS OF CARBON NANOTUBES BASED ON THE MESHLESS METHOD1)

采用无网格方法分析了碳纳米管的自由振动问题。通过几个数值算例来证明本方法的收敛性。将不同长径比的碳纳米管的固有频率计算结果与公开文献的结果进行对比。对比结果表明：本文方法具有较高的计算精度,基于薄板样条径向基函数的无网格方法可以成功地分析碳纳米管的自由振动问题。     

ADAPTIVE MESHLESS METHOD BASED ON LOCAL FIT TECHNOLOGY

An h-adaptive meshless method is proposed in this paper. The error estimation is based on local fit technology, usually confined to Voronoi Cells. The error is achieved by comparison of the computational results with smoothed ones, which are projected with Taylor series. Voronoi Cells are introduced not only for integration of potential energy but also for guidance of refinement.New nodes are placed within those cells with high estimated error. At the end of the paper, two numerical examples with severe stress gradient are analyzed. Through adaptive analysis accurate results are obtained at critical subdomains, which validates the efficiency of the method.     

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。     

Winkler地基上厚板分析的自然单元法

以自然单元法中自然邻接点的Laplace插值形函数为基础,基于Mindlin厚板理论,建立了Winkler地基上厚板弯曲挠度的自然单元法求解控制方程,并进行了相应的程序实现,最后通过算例分析,表明了该文方法的可行性和有效性.     

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 a 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.     

基于局部移动Kriging无网格方法的层合板自由振动分析

利用基于局部移动Kriging插值无网格法对层合板自由振动进行了数值分析,基于一阶剪切层合理论导出了层合板振动的控制方程和边界条件,进一步得到了自由振动的离散化特征方程。由于Kriging插值函数具有Kronecker delta函数性质,可以直接施加本质边界条件。通过本文给出的方法,对不同边界条件、不同跨厚比、不同材料参数和铺设角度的层合板的振动频率进行了计算,均得到满意结果。最后用该方法对层合板的铺设角度进行优化设计,得到了与已有文献完全一致的优化结果。数值结果充分表明了无网格Kriging方法分析层合板自由振动问题的有效性和高精确度。     

形状设计灵敏度分析的改进的再生核质点法

基于物质导数概念和直接微分法,将再生核质点法应用于形状设计灵敏度分析（DSA）中。导出了基于无网格近似的灵敏度方程,特别强调了在考虑形状函数关于设计变量的物质导数时无网格方法与有限元法的不同。通过对RKPM形状函数及其物质导数进行矩式显式表述,提高了无网格方法的计算效率。对两个二维线弹性问题进行了位移灵敏度和应力灵敏度分析,计算结果与解析解吻合的很好;同时通过对通常的RKPM和改进的RKPM计算耗时的比较,显示了该方法不仅有效,而且可以显著地提高计算效率。     

Development of a local MQ‐DQ‐based stencil adaptive method and its application to solve incompressible Navier–Stokes equations

In this paper, a local stencil adaptive method is presented, which is designed for solving computational fluid dynamics (CFD) problems with curved boundaries accurately. A local multiquadric‐differential quadrature (MQ‐DQ) method is used to discretize the governing equations, taking advantage of its meshless nature. The present method bears the properties of both local MQ‐DQ method and local stencil adaptive method and is thus named the local MQ‐DQ‐based stencil adaptive method. Two test problems with curved boundaries are solved to investigate the performance of this solution‐adaptive method. The numerical results indicate that the proposed method is effective and efficient by combining the advantages of meshless property for complex geometries and local adaptation for accuracy improvement. Copyright © 2007 John Wiley & Sons, Ltd.     

复合材料层合板自由振动分析的无网格自然邻接点Petrov-Galerkin法

基于一阶剪切变形理论,提出了复合材料层合板自由振动分析的无网格自然邻接点Petrov-Galerkin法。计算时在复合材料层合板中面上仅需要布置一系列的离散节点,并利用这些节点构建插值函数。在板中面上的局部多边形子域上,采用加权余量法建立复合材料层合板自由振动分析的离散化控制方程,并且这些子域可由Delaunay三角形方便创建。自然邻接点插值形函数具有Kronecker delta函数性质,因而无需经过特别处理就能准确地施加本质边界条件。对不同边界条件、不同跨厚比、不同材料参数和不同铺设角度的复合材料层合板,由本文提出的无网格自然邻接点Petrov-Galerkin法进行自由振动分析时均可得到满意的结果。数值算例结果表明,本文方法求解复合材料层合板的自由振动问题是行之有效的。