双参数地基上Kirchhoff板计算的无网格自然单元法

自然单元法是一种基于Voronoi图及Delaunay三角形剖分图,以自然邻接点插值函数为试函数的无网格数值方法。以目前该方法中自然邻接点的Laplace插值形函数为基础,求出了其一阶及二阶导函数,建立了双参数地基上Kirchhoff板弯曲挠度的自然单元法求解控制方程,并编制了相应的计算程序。通过算例分析表明了该文方法的可行性和有效性。     

弹性地基圆形加肋板静力弯曲及弯曲自由振动分析的无网格法EI北大核心CSCD

基于一阶剪切理论和移动最小二乘法,提出分析弹性地基圆形加肋板线性弯曲的静力和自由振动问题的无网格方法。对弹性地基采用Winkler地基模型,对圆形加肋板则将平板和肋条分开考虑,通过位移协调条件建立两者的参数转换方程。平板和肋条均采用一系列点来离散,用移动最小二乘法建立的形函数来分别描述两者的位移场,再分别通过最小势能原理和Hamilton原理导出弹性地基的静力弯曲和弯曲自由振动控制方程,最后采用完全转换法处理边界条件。以一系列不同基床系数、荷载、边界、肋条布置方式的弹性地基圆形加肋板为例,研究所提出方法的收敛性和稳定性,并将计算结果与有限元、文献结果对比。研究表明该方法可以有效分析弹性地基圆形加肋板线性弯曲的静力和自由振动问题,且肋条位置改变时,可避免网格重构。     

基于双参数地基系统的改进傅里叶级数振动方法研究

双参数地基系统中,在含自由边界或弹性边界处,其上板结构弯曲位移不为零,将导致基底压力向地基板外域传递和扩散,从而使板外域地基产生沉降。然而,现有的研究因边界条件的局限性忽略了板外域地基沉降,使得该类边界的计算结果产生较大的误差。针对双参数地基上矩形薄板的弯曲振动问题,基于改进Fourier级数法和Vlazov假定,提出了一种能在任意边界条件下考虑板外域地基沉降的振动分析方法。通过参数化分析,并与文献和有限元计算结果进行对比,验证了该方法具有较快的收敛速度和较高的计算精度;在此基础上,研究了不同自由边界数组合和弹性边界下地基沉降对振动特性的影响。     

弹性地基板广义边值问题的边界元法

本文利用Ｈａｎｋｅｌ变换导出了弹性地基板弯曲问题的基本解，该基本解对于Ｗｉｎｋｌｅｒ地基、Ｐａｓｔｅｒｎａｋ地基和弹性半空间地基模型具有统一的表达形式。在此基础上，建立了适用于弹性地基板广义边值问题的边界积分方程组，最后文中给出了若干数值算例。     

考虑地基耦合效应的矩形中厚板的非线性静力分析

基于能量变分原理,考虑板的横向剪切和地基耦合效应,建立了双参数地基上弹性中厚板的非线性控制方程。应用伽辽金法对双参数地基上四边自由矩形中厚板的非线性弯曲问题进行了探讨。数值计算中考虑了系统各种参数变化对弯曲问题的影响。     

双参数地基板的精度分析

为了板与地基相互作用理论的严密性和计算的可靠性,该文先纠正了双参数地基自由矩形板研究中长期存在的板外地基沉降衰减指数取值不当的问题;然后基于弹性半空间地基上四边自由矩形板的弯曲解析解,数值计算验证了实际地基反力与挠度不具有Vlazov双参数地基模型所导出的关系式;最后,通过对算例的数值计算及结果的对比分析,从板的弯曲变形角度,说明该文给出的衰减指数值才是适当的;同时,也分析给出了Vlazov双参数地基自由矩形板各种近似边界条件的计算精度。     

弹性地基圆形加肋板静力弯曲及弯曲自由振动分析的无网格法

基于一阶剪切理论和移动最小二乘法,提出分析弹性地基圆形加肋板线性弯曲的静力和自由振动问题的无网格方法.对弹性地基采用Winkler地基模型,对圆形加肋板则将平板和肋条分开考虑,通过位移协调条件建立两者的参数转换方程.平板和肋条均采用一系列点来离散,用移动最小二乘法建立的形函数来分别描述两者的位移场,再分别通过最小势能原...     

弹性地基中厚板的拟薄板域外奇点法分析

本文从简化的Reissner 理论出发，将中厚板问题模拟成薄板问题，导出类似于求解弹性地基上薄板问题的边界积分方程。本文利用域外奇点法提出的方法适用于弹性地基上的任意边界，任意荷载的薄板，中厚板的弯曲问题。该方法简单，易于编程序，能方便的应用于工程计算中。     

等代数结构面网格剖分下三维弹性问题的代数多重网格法

在一种等代数结构面网格剖分下,建立了求解三维弹性问题有限元方程的代数多重网格法及相应的预处理共轭梯度法,详细描述了代数多重网格方法中网格粗化技术与插值算子的构造,并将所构造的代数多重网格法应用于某些实际问题如非均匀介质、高应力梯度问题的数值求解。结果表明,建立的代数多重网格法对求解三维弹性问题是十分有效的,具有很好的鲁棒性,较直接解法和其它常用迭代方法具有明显的优越性。     

层合板分析的无网格局部Petrov-Galerkin方法

基于Kirchhoff均匀各向异性板控制方程的等效积分弱形式和对挠度函数采用移动最小二乘近似函数进行插值, 进一步研究无网格局部Petrov-Galerkin方法在纤维增强对称层合板弯曲问题中的应用。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,其问题的本质边界条件采用罚因子法来施加。通过数值算例和与其他方法的结果比较, 表明无网格局部Petrov-Galerkin法求解层合薄板弯曲问题具有解的精度高、收敛性好等一系列优点。      

Dual reciprocity hybrid boundary node method for acoustic eigenvalue problems

The hybrid boundary node method (HBNM) is a truly meshless method, and elements are not required for either interpolation or integration. The method, however, can only be used for solving homogeneous problems. For the inhomogeneous problem, the domain integration is inevitable. This paper applied the dual reciprocity hybrid boundary node method (DRHBNM), which is composed by the HBNM and the dual reciprocity method (DRM) for solving acoustic eigenvalue problems. In this method, the solution is composed of two parts, i.e. the complementary solution and the particular solution. The complementary solution is solved by HBNM and the particular one is obtained by DRM. The modified variational formulation is applied to form the discrete equations of HBNM. The moving least squares (MLS) is employed to approximate the boundary variables, while the domain variables are interpolated by the fundamental solutions. The domain integration is interpolated by radial basis function (RBF). The Q–R algorithm and Householder algorithm are applied for solving the eigenvalues of the transformed matrix. The parameters that influence the performance of DRHBNM are studied through numerical examples. Numerical results show that high convergence rates and high accuracy are achievable.     

A hybrid radial boundary node method based on radial basis point interpolation

The hybrid boundary node method (HBNM) retains the meshless attribute of the moving least squares (MLS) approximation and the reduced dimensionality advantages of the boundary element method. However, the HBNM inherits the deficiency of the MLS approximation, in which shape functions lack the delta function property. Thus in the HBNM, boundary conditions are implemented after they are transformed into their approximations on the boundary nodes with the MLS scheme.This paper combines the hybrid displacement variational formulation and the radial basis point interpolation to develop a direct boundary-type meshless method, the hybrid radial boundary node method (HRBNM) for two-dimensional potential problems. The HRBNM is truly meshless, i.e. absolutely no elements are required either for interpolation or for integration. The radial basis point interpolation is used to construct shape functions with delta function property. So unlike the HBNM, the HRBNM is a direct numerical method in which the basic unknown quantity is the real solution of nodal variables, and boundary conditions can be applied directly and easily, which leads to greater computational precision. Some selected numerical tests illustrate the efficiency of the method proposed.     

Development of hybrid boundary node method in two-dimensional elasticity

As a truly meshless method, the Hybrid Boundary Node Method (HBNM) does not require a ‘boundary element mesh’, either for the purpose of interpolation of the solution variables or for the integration of ‘energy’. It has been applied to solve the potential problems. This paper presents a further development of the HBNM to the 2D elastic problems.In this paper, the hybrid displacement variational formulations have been coupled with the Moving Least Squares (MLS) approximation. The rigid body movement method is employed to solve the hyper-singular integrations. The ‘boundary layer effect’, which is the main drawback of the original HBNM, has been circumvented by an adaptive integration scheme.In the present method, the source points of the fundamental solution are arranged directly on the boundary. Thus, the uncertain scale factor taken in the Regular Hybrid Boundary Node Method (RHBNM) can be avoided. The parameters that influence the performance of this method are studied through several numerical examples and the known analytical solutions. The treatment of singularity and further integration has been given by a series of effective approaches. The computation results obtained by the present method are shown that good convergence and high accuracy with a small node number are achievable.     

Multiple reciprocity hybrid boundary node method for potential problems

Meshless methods have some obvious advantages such as they do not require meshes in the domain and on the boundary, only some nodes are needed in the computation. Furthermore, for the boundary-type meshless methods, the nodes are even not needed in the domain and only distributed on the boundary. Practice shows that boundary-type meshless methods are effective for homogeneous problems. But for inhomogeneous problems, the application of these boundary-type meshless methods has some difficulties and need to be studied further.The hybrid boundary node method (HBNM) is a boundary-only meshless method, which is based on the moving least squares (MLS) approximation and the hybrid displacement variational principle. No cell is required either for the interpolation of solution variables or for numerical integration. It has a drawback of ‘boundary layer effect’, so a new regular hybrid boundary node method (RHBNM) has been proposed to avoid this pitfall, in which the source points of the fundamental solutions are located outside the domain. These two methods, however, can only be used for solving homogeneous problems. Combining the dual reciprocity method (DRM) and the HBNM, the dual reciprocity hybrid boundary node method (DRHBNM) has been proposed for the inhomogeneous terms. The DRHBNM requires a substantial number of internal points to interpolate the particular solution by the radial basis function, where approximation based only on boundary nodes may not guarantee sufficient accuracy.Now a further improvement to the RHBNM, i.e., a combination of the RHBNM and the multiple reciprocity method (MRM), is presented and called the multiple reciprocity hybrid boundary node method (MRHBNM). The solution comprises two parts, i.e., the complementary and particular solutions. The complementary solution is solved by the RHBNM. The particular solution is solved by the MRM, i.e., a sum of high-order homogeneous solutions, which can be approximated by the same-order fundamental solutions. Compared with the DRHBNM, the MRHBNM does not require internal points to obtain the particular solution for inhomogeneous problems. Therefore, the present method is a real boundary-only meshless method, and can be used to deal with inhomogeneous problems conveniently. The validity and efficiency of the present method are demonstrated by a series of numerical examples of inhomogeneous potential problems.     

Analysis of functionally graded plates by meshless method: A purely analytical formulation

Functionally graded plates under static and dynamic loads are investigated by the local integral equation method (LIEM) in this paper. Plate bending problem is described by the Reissner moderate thick plate theory. The governing equations for the functionally graded material with respect to the neutral plane are presented in the Laplace transform domain and therefore the in-plane and bending problems are uncoupled. Both isotropic and orthotropic material properties are considered. The local integral equation method is developed with the locally supported radial basis function (RBF) interpolation. As the closed forms of the local boundary integrals are obtained, there are no domain or boundary integrals to be calculated numerically in this approach. The solutions of the nodal values for the entire plate are obtained by solving a set of linear algebraic equation system with certain boundary conditions. Details of numerical procedures are presented and the accuracy and convergence characteristics of the method are examined. Several examples are presented for the functionally graded plates under static and dynamic loads and the accuracy for proposed method has been observed compared with 3D analytical solutions.     

A new hybrid boundary node method based on Taylor expansion and the Shepard interpolation method

A novel meshless method based on the Shepard and Taylor interpolation method (STIM) and the hybrid boundary node method (HBNM) is proposed. Based on the Shepard interpolation method and Taylor expansion, the STIM is developed to construct the shape function of the HBNM. In the STIM, the Shepard shape function is used as the basic function, which is the zero‐level shape function, and the high‐power basic functions are constructed through Taylor expansion. Four advantages of the STIM are the interpolation property, the arbitrarily high‐order consistency, the absence of inversion for the whole process of shape function construction, and the low computational expense. These properties are desirable in the implementation of meshless methods. By combining the STIM and the HBNM, a much more effective meshless method is proposed to solve the elasticity problems. Compared with the traditional HBNM, the STIM can improve accuracy because of the use of high‐power basic functions and can also improve the computational efficiency because there is no inversion for the shape function construction process. Numerical examples are given to demonstrate the accuracy and efficiency of the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.     

Local RBF-based differential quadrature collocation method for the boundary layer problems

In this article, the local RBF-based differential quadrature (LRBFDQ) collocation method is presented for the boundary layer problems, i.e., the singularly perturbed two-point boundary value problems. This novel method has an advantage over the globally supported RBF collocation method because it approximates the derivatives by RBF interpolation using a small set of nodes in the neighborhood of any collocation node. So it needs much less computational work than the globally supported RBF collocation method. It also could easily use the nodes in local support domain on the upwind side to obtain the non-oscillatory solution of boundary layer problems. Numerical examples are made by the multiquadric (MQ) RBF. Compared with the globally supported RBF collocation method and the finite difference method, numerical results demonstrate the accuracy and easy implementation of the LRBFDQ collocation method, even for the extremely thin layers in the boundary layer problems.     

Development of a meshless hybrid boundary node method for Stokes flows

The meshless hybrid boundary node method (HBNM) is a promising method for solving boundary value problems, and is further developed and numerically implemented for incompressible 2D and 3D Stokes flows in this paper. In this approach, a new modified variational formulation using a hybrid functional is presented. The formulation is expressed in terms of domain and boundary variables. The moving least-squares (MLS) method is employed to approximate the boundary variables whereas the domain variables are interpolated by the fundamental solutions of Stokes equation, i.e. Stokeslets. The present method only requires scatter nodes on the surface, and is a truly boundary type meshless method as it does not require the ‘boundary element mesh’, either for the purpose of interpolation of the variables or the integration of ‘energy’. Moreover, since the primitive variables, i.e., velocity vector and pressure, are employed in this approach, the problem of finding the velocity is separated from that of finding pressure. Numerical examples are given to illustrate the implementation and performance of the present method. It is shown that the high convergence rates and accuracy can be achieved with a small number of nodes.     

A fast multipole boundary element method for solving the thin plate bending problem

A fast multipole boundary element method (BEM) for solving large-scale thin plate bending problems is presented in this paper. The method is based on the Kirchhoff thin plate bending theory and the biharmonic equation governing the deflection of the plate. First, the direct boundary integral equations and the conventional BEM for thin plate bending problems are reviewed. Second, the complex notation of the kernel functions, expansions and translations in the fast multipole BEM are presented. Finally, a few numerical examples are presented to show the accuracy and efficiency of the fast multipole BEM in solving thin plate bending problems. The bending rigidity of a perforated plate is evaluated using the developed code. It is shown that the fast multipole BEM can be applied to solve plate bending problems with good accuracy. Possible improvements in the efficiency of the method are discussed.     

薄板屈曲分析的局部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,研究了无网格局部Petrov-Galerkin方法在薄板屈曲问题中的应用。它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,并用罚因子法施加本质边界条件。数值算例表明,无网格局部Petrov-Galerkin法不但能够求解弹性静力学问题,而且在求解弹性稳定性问题时仍具有收敛快,稳定性好,精度高的特点。