位势问题改进的无网格局部Petrov-Galerkin法

将滑动Kriging插值法与无网格局部Petrov-Galerkin法相结合,采用Heaviside分段函数作为局部弱形式的权函数,提出改进的无网格局部Petrov-Galerkin法,进一步将这种无网格法应用于位势问题,并推导相应的离散方程.因为滑动Kriging插值法构造的形函数满足Kronecker函数性质,所以本文建立的改进的无网格局部Petrov-Galerkin法可以像有限元法一样直接施加边界条件;由于采用Heaviside分段函数作为局部弱形式的权函数,因此在计算刚度矩阵时只涉及边界积分,而没有区域积分.此外,还对本方法中一些重要参数的选取进行了研究.数值算例表明,本文建立的改进的无网格局部Petrov-Galerkin法具有数值实现简单、计算量小以及方便施加边界条件等优点.     

搅拌器内部流动的无网格法三维数值模拟

本文将移动粒子半隐式法(MPS)的基本算法由二维扩展至三维。将圆柱坐标系引入到初场粒子的布置中,避免了在笛卡儿坐标系下处理不规则形状(如斜边或曲边)问题时粒子初场布置困难和精确度较低的问题,改善了对计算边界条件表达的精确性。引入移动边界模型,对直叶片搅拌器的内部流动进行了三维数值模拟。还提出了一种新的初始粒子布置简易方法,明显简化粒子初始布置时的复杂程度,提高了对三维复杂几何形状问题的可操作性。     

管道内壁侵蚀形状识别的无网格法研究

本文使用直接配点无网格法结合共轭梯度法对管道内壁面侵蚀状况进行了反演识别。直接配点无网格法使用节点离散求解区域,采用移动最小二乘近似构造试函数,直接配点法构造线性方程组进行导热正问题求解;反演过程采用共轭梯度法使目标函数最小化,Akima三次样条插值将连续的几何边界反演问题转化为离散点几何位置的反演,并最终将这些离散点拟合成为光滑曲线。文中选择两个典型算例对数值方法进行验证,模拟结果表明使用直接配点无网格法结合共轭梯度法进行管道内壁几何边界识别具有较高精度。     

基于局部加密纯无网格法非线性Cahn-Hilliard方程的模拟

为数值求解描述不同物质间相位分离现象的高阶非线性Cahn-Hilliard(C-H)方程,发展了一种基于局部加密纯无网格有限点集法(local refinement finite pointset method,LR-FPM).其构造过程为:1)将C-H方程中四阶导数降阶为两个二阶导数,连续应用基于Taylor展开和加权最小二乘法的FPM离散空间导数;2)对区域进行局部加密和采用五次样条核函数以提高数值精度;3)局部线性方程组求解中准确施加含高阶导数Neumann边值条件.随后,运用LR-FPM求解有解析解的一维/二维C-H方程,分析粒子均匀分布/非均匀分布以及局部粒子加密情况的误差和收敛阶,展示了LR-FPM较网格类算法在非均匀布点情况下的优点.最后,采用LR-FPM对无解析解的一维/二维C-H方程进行了数值预测,并与有限差分结果相比较.数值结果表明,LR-FPM方法具有较高的数值精度和收敛阶,比有限差分法更易数值实现,能够准确展现不同类型材料间相位分离非线性扩散现象随时间的演化过程.     

基于无网格点插值法的旋转悬臂梁的动力学分析

将基于多项式点插值的无网格方法用于旋转悬臂梁的动力学分析. 利用无网格点插值方法对柔性梁的变形场进行离散, 考虑梁的纵向拉伸变形和横向弯曲变形, 并计入横向弯曲变形引起的纵向缩短, 即非线性耦合项, 运用第二类Lagrange方程推导得到系统刚柔耦合动力学方程. 与有限元法相比, 该方法只需节点信息, 无需定义单元, 具有前处理简单的优势; 构造的形函数采用更多的节点插值, 具有高阶连续性. 将无网格点插值方法的仿真结果与有限元和假设模态法进行比较分析, 验证了该方法的正确性, 并表明其作为一种柔性体离散方法在刚柔耦合多体系统动力学的研究中具有可推广性.     

基于无网格法的水中气泡上升运动数值模拟

本文对无网格法的一种-移动粒子半隐式法(MPS)进行了较深入的研究,介绍了该方法的基本原理和算法,推导并建立了表面张力和气液两相流等若干粒子作用模型,编程实现了二维情况下水中气泡自由上升运动的数值模拟;对计算结果进行分析并与相关实验结果进行了定性比较,模拟结果显示了移动粒子半隐式法在模拟自由表面和气液两相流问题的优越性,也为研究涉及到大变形的相关问题提供了很好的解决思路.     

无网格法剖析及基扩充EFG法在电磁计算中的应用

主要论述基扩充的无网格法(MLM)用于2D电磁问题计算时的具体算法及编程问题。以独特的分步骤操作方法,介绍了无网格方法;从数值拟合的角度,对无网格伽辽金法(EFG)的核心技术——移动最小二乘法进行了深入剖析;严格按照加权余量法原理,利用偏微分方程的余量加权在节点支持域上的积分,导出了基扩充的EFG离散格式;应用FEM和基扩充的EFG两种方法对一些实例进行了计算验证。     

三维无网格法中权函数影响半径的优化

从函数逼近的角度建立三维无网格法中权函数影响半径的优化计算模型,针对一次基、二次基情形求解该模型.分析不同影响半径下函数的逼近误差,以及计算量、条件数等因素对无网格法计算性能的影响,结合数值试验,确认给出的权函数影响半径是有效、可靠、综合最优的.     

基于DEM的伊犁河谷气温空间插值研究

为了研究伊犁河谷地区气温空间变异性,提出基于DEM修正的空间插值方法,以1961-2008年伊犁河谷地区及周边19个气象站点月平均气温数据为基础,分析多年平均气温与海拔的相关关系,并与反距离权重法(IDW)、克里格插值法(Kriging)等传统方法计算结果进行对比.结果表明:该研究区的气温直减率为0.564℃·100 ...     

FGMs稳态热传导分析的重心Lagrange插值配点法

提出一种求解二维功能梯度材料（FGMs）稳态热传导问题的重心Lagrange插值配点法.基于Chebyshev节点构造二维重心Lagrange插值函数及其偏导数,然后基于配点法将其直接代入FGMs热传导问题的控制方程和边界条件,得到系统离散方程.重心Lagrange插值配点法是一种真正的无网格方法,很好地融合了重心Lagrange插值和配点格式的优势,具有高效、稳定、高精度和易于数值实现的优点.采用重心Lagrange插值配点法分别对指数型、二次型和三角型FGMs热传导问题进行数值模拟.结果表明：该方法具有较高的计算效率和计算精度,对材料梯度参数的变化不敏感.可以进一步拓展到FGMs瞬态问题和FGMs的热力耦合分析.     

A meshless method based on moving Kriging interpolation for a two-dimensional time-fractional diffusion equation

Fractional diffusion equations have been the focus of modeling problems in hydrology, biology, viscoelasticity, physics, engineering, and other areas of applications. In this paper, a meshfree method based on the moving Kriging inter- polation is developed for a two-dimensional time-fractional diffusion equation. The shape function and its derivatives are obtained by the moving Kriging interpolation technique. For possessing the Kronecker delta property, this technique is very efficient in imposing the essential boundary conditions. The governing time-fractional diffusion equations are transformed into a standard weak formulation by the Galerkin method. It is then discretized into a meshfree system of time-dependent equations, which are solved by the standard central difference method. Numerical examples illustrating the applicability and effectiveness of the proposed method are presented and discussed in detail.     

薄板弯曲问题的变分-差分方法

从最小势能原理出发,使用变分-差分方法构造带有弯曲边梁的薄板的小挠度弯曲问题的差分格式,所得格式仅依赖板面网格结点,从而避免了由于引入虚拟网格结点而带来的问题;编制求解差分方程组的MATLAB程序,给出数值模拟结果.     

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 interpolationbased boundary node method(MKIBNM),is developed for solving two-dimensional potential problems.This study combines the BIE 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.     

弹性力学的复变量无网格局部 Petrov-Galerkin 法

复变量移动最小二乘法构造形函数, 其优点是采用一维基函数建立二维问题的试函数, 使得试函数中所含的待定系数减少, 从而有效提高计算效率. 文章基于复变量移动最小二乘法和局部Petrov-Galerkin弱形式, 采用罚函数法施加边界条件, 推导相应的离散方程, 提出弹性力学的复变量无网格局部Petrov-Galerkin法. 数值算例验证了该方法的有效性.     

A scaled boundary node method applied to two-dimensional crack problems

A boundary-type meshless method called the scaled boundary node method(SBNM) is developed to directly evaluate mixed mode stress intensity factors(SIFs) without extra post-processing.The SBNM combines the scaled boundary equations with the moving Kriging(MK) interpolation to retain the dimensionality advantage of the former and the meshless attribute of the latter.As a result,the SBNM requires only a set of scattered nodes on the boundary,and the displacement field is approximated by using the MK interpolation technique,which possesses the δ function property.This makes the developed method efficient and straightforward in imposing the essential boundary conditions,and no special treatment techniques are required.Besides,the SBNM works by weakening the governing differential equations in the circumferential direction and then solving the weakened equations analytically in the radial direction.Therefore,the SBNM permits an accurate representation of the singularities in the radial direction when the scaling center is located at the crack tip.Numerical examples using the SBNM for computing the SIFs are presented.Good agreements with available results in the literature are obtained.     

On Feasibility of Variable Separation Method Based on Hamiltonian System for a Class of Plate Bending Equations

The eigenfunction system of infinite-dimensional Hamiltonian operators appearing in the bending problem of rectangular plate with two opposites simply supported is studied. At first, the completeness of the extended eigenfunction system in the sense of Cauchy＇s principal value is proved. Then the incompleteness of the extended eigenfunction system in general sense is proved. So the completeness of the symplectic orthogonal system of the infinite-dimensional Hamiltonian operator of this kind of plate bending equation is proved. At last the general solution of the infinite dimensional Hamiltonian system is equivalent to the solution function system series expansion, so it gives to theoretical basis of the methods of separation of variables based on Hamiltonian system for this kind of equations.     

Meshless analysis of three-dimensional steady-state heat conduction problems

Steady-state heat conduction problems arisen in connection with various physical and engineering problems where the functions satisfy a given partial differential equation and particular boundary conditions, have attracted much attention and research recently. These problems are independent of time and involve only space coordinates, as in Poisson’s equation or the Laplace equation with Dirichlet, Neuman, or mixed conditions. When the problems are too complex, it is difficult to find an analytical solution, the only choice left is an approximate numerical solution. This paper deals with the numerical solution of three-dimensional steady-state heat conduction problems using the meshless reproducing kernel particle method (RKPM). A variational method is used to obtain the discrete equations. The essential boundary conditions are enforced by the penalty method. The effectiveness of RKPM for three-dimensional steady-state heat conduction problems is investigated by two numerical examples.     

Meshless analysis of an improved element-free Galerkin method for linear and nonlinear elliptic problems

We first give a stabilized improved moving least squares(IMLS) approximation, which has better computational stability and precision than the IMLS approximation. Then, analysis of the improved element-free Galerkin method is provided theoretically for both linear and nonlinear elliptic boundary value problems. Finally, numerical examples are given to verify the theoretical analysis.