移动最小二乘法研究进展与述评

为使移动最小二乘法能更好地应用到无网格方法中,详细阐述移动最小二乘逼近法、移动最小二乘插值法、MUKHERJEE改进的移动最小二乘法以及程玉民等提出的改进的移动最小二乘法和复变量移动最小二乘法等的研究进展,述评各种移动最小二乘法的优缺点,并概述各种移动最小二乘法形成的无网格方法的研究进展.     

改进的无单元Galerkin方法在机场复合道面中的应用

运用改进的无单元Galerkin（Improved Element Free Galerkin,IEFG）方法计算机场复合道面的位移和应力,分析不同的节点数量和影响域比例参数对计算结果的影响.结果表明：节点数在4 000以上、影响域比例参数取2.5~3.5时计算结果较好;IEFG方法比无单元Galerkin(Element Free Galerkin,EFG)方法的精度更高,计算速度更快.计算结果可为机场道面设计提供参考.     

热传导方程的无网格Galerkin方法数值模拟研究

为了更好地数值模拟热传导方程,将无网格Galerkin( EFG)方法引入热传导问题的求解中,时间导数采用θ加权方法离散,同时与有限元(FE)方法的数值结果进行了比较,并研究了EFG方法中若干参数的选取对数值结果的影响.计算结果表明:相对于有限元方法,EFG方法能更好地吻合微分方程的解析解,EFG方法在节点布置较稀疏时,也可以获得很高的计算精度;θ≥1/2,EFG方法无条件稳定,且θ=1时数值解精度最高;算例中影响半径取为1.2h≤r＜2.8h,EFG方法可获得较为理想的计算结果.     

基于泰勒基函数的移动最小二乘法及误差分析

移动最小二乘法通常选用不超过m次单项式生成基函数空间,本文选用了以计算点(?)为平移点的泰勒基函数生成基空间.理论和数值试验发现:选用此种基函数后会降低形函数及导数计算的复杂性,并且有效减小广义逆矩阵的条件数,提高了计算效率同时增加了计算稳定性,并用该方法推导出移动最小二乘近似的收敛阶及误差主部.     

移动最小二乘法导数近似讨论

在移动最小二乘法（moving least squares method, MLS）构造无网格形函数的数值方法中,通常采用无单元伽辽金法（element-free Galerkin method, EFG）的建议,将系数向量a参与导数运算。为探讨这种导数近似算法在更一般无网格法中的适用性和合理性,针对系数向量a是否应参与运算的问题进行讨论和数值检验。结果表明：单纯从近似意义上讲,这种将系数向量代入导数运算的算法并不具有优势;从数值方法的应用意义上讲,这种导数近似算法对数值求解,特别是强式无网格法,会带来一系列潜在不稳定的问题。建议在MLS导数近似中,系数向量a不应当参与导数运算,并提出采用一种由核基函数代替普通基函数的核近似法。     

三维断裂弹性动力学的改进无单元Galerkin法

为提高断裂弹性动力学问题数值计算的精度,避免出现病态或奇异方程组,基于改进的移动最小二乘法建立三维弹性动力学问题的积分弱形式,采用罚函数法施加位移边界条件,引入隐式时间积分并且结合三维断裂力学的形函数考虑裂纹尖端的奇异性,探究将改进的无单元Galerkin(improved element-free Galerkin,IEFG)法用于断裂弹性动力学问题的数值计算.通过悬臂梁、柱和矩形板等3个算例,讨论节点分布、影响域比例参数、罚因子和时间步长等参数对计算精度的影响,证明IEFG法用于求解三维断裂弹性动力学问题的正确性和有效性.     

无网格伽辽金法节点分布求解精度研究

摘　要 基于移动最小二乘法的无网格伽辽金法（EFG）的计算精度的影响因素很多。为了分析无网格伽辽金法对节点布置的影响,文中通过分析经典的悬臂梁问题,定义了一个能量范数作为误差指标,用Matlab程序开发了无网格法计算程序。研究了离散节点的布置方法及节点个数对无网格法伽辽金法求解精度的影响。考察了均匀布点法及随机布点法情况下的无网格法的计算精度,最后提出了两种布点方法结合起来应用的混合布点方法,得出了一些有益的结论.。     

基于平均值插值求解势问题的宏单元法

在有限元法中采用宏单元(大单元)可以减少单元数量,进而降低网格生成的工作量,提高计算效率.本文利用多边形上的平均值插值构造宏单元的形函数,采用标准的Galerkin法推导出求解二维势问题的宏单元法.给出了宏单元形函数偏导数计算的方法.运用提出的宏单元法,分别采用多边形宏单元和含有任意数量边节点的宏单元,对一些二维势问题进行了分析计算.数值算例表明,本文提出的宏单元法具有较好的计算精度.     

基于移动最小二乘法的气动力数据建模方法

针对一体化飞行器高度耦合的非线性气动问题,提出了一种基于移动最小二乘法的气动力数据建模方法;首先,对影响模型精度的因素进行了分析;接着,在构建移动最小二乘模型时采用遗传算法获取最佳支撑域半径以及最佳影响因子β,提高近似精度从而达到减少样本点的目的;得到泛化能力较强的气动力模型,并与偏最小二乘方法的建模结果进行对比;实验结果表明:移动最小二乘法的建模效果优于偏最小二乘方法,预测误差较小,证明了将该方法应用于气动数据建模是可行的。     

人工智能技术在电力系统状态估计的应用研究

为了有效地提高状态估计的计算精度和鲁棒性,将人工智能技术与电网数据相结合,提出了基于偏最小二乘(PLS)和极限学习(ELM)的电力系统状态估计方法。针对量测量之间的强相关性问题,采用偏最小二乘(PLS)对各量测量进行重要信息提取和变量选择,将得到的最优变量输入ELM模型,从而建立了状态量的PLS-ELM模型,然后,采用IEEE14节点系统数据样本和实际电网历史数据对所提方法进行了验证,并将该方法与其他方法进行对比。结果表明,所提状态估计方法降低了模型的复杂程度,能够有效地抵抗量测量中的不良数据,具有较高的估计精度和较强的鲁棒性。     

The element-free Galerkin method for the nonlinear p-Laplacian equation

The element-free Galerkin (EFG) method is developed in this paper for solving the nonlinear p-Laplacian equation. The moving least squares approximation is used to generate meshless shape functions, the penalty approach is adopted to enforce the Dirichlet boundary condition, the Galerkin weak form is employed to obtain the system of discrete equations, and two iterative procedures are developed to deal with the strong nonlinearity. Then, the computational formulas of the EFG method for the p-Laplacian equation are established. Numerical results are finally given to verify the convergence and high computational precision of the method.     

A generalized element-free Galerkin method for Stokes problem

A generalized element-free Galerkin (GEFG) method is developed in this paper for solving Stokes problem in primitive variable form. To obtain stable numerical results for both velocity and pressure, extended terms are only introduced into the approximate space of velocity in a special way as that in the generalized finite element method. Theoretical analysis shows that the GEFG method implies a stabilized formulation similar to that in the variational multiscale element-free Galerkin (VMEFG) method. Numerical results show the efficiency of the present method and reveal that both computational errors and CPU times of the present method are less than those of the VMEFG and the finite element methods.     

A paralleled element-free Galerkin analysis for structures with cyclic symmetry

This paper presents a paralleled element-free Galerkin (EFG) algorithm to analyze the structure with cyclic symmetry. The global EFG equation system is proved to be block-circulant under a corresponding symmetry-adapted coordinate system, and is partitioned into a number of smaller independent sub-problems, which are solved simultaneously on a multiprocessor parallel computer in parallel virtual machine. Numerical examples are given to illustrate the correctness, speedup, and efficiency of the proposed algorithm.     

接触力学自适应无网格计算系统设计

在求解弹塑性接触问题时,为了保证计算精度同时减少计算耗费,提出一种接触力学自适应无网格计算系统.利用无网格法结点排布灵活便于删减等优点,对采用基于应变能梯度的自适应无网格伽辽金一有限元方法求解接触问题的计算系统的原理、算法流程进行阐述,并考虑到自适应影响域半径、弹塑性等关键问题.采用模块化思想进行程序设计.通过圆柱体与弹塑性平面接触的算例验证了计算系统的正确性,并将自适应密化解与整体密化解比较,显现出良好的计算精度和效率.为接触力学自适应无网格分析软件商业化提供了一种理论和程序设计基础.     

E(FG)<Superscript>2</Superscript>: a new fixed-grid shape optimization method based on the element-free galerkin mesh-free analysis

We propose a shape optimization method over a fixed grid. Nodes at the intersection with the fixed grid lines track the domain’s boundary. These “floating” boundary nodes are the only ones that can move/appear/disappear in the optimization process. The element-free Galerkin (EFG) method, used for the analysis problem, provides a simple way to create these nodes. The fixed grid (FG) defines integration cells for EFG method. We project the physical domain onto the FG and numerical integration is performed over partially cut cells. The integration procedure converges quadratically. The performance of the method is shown with examples from shape optimization of thermal systems involving large shape changes between iterations. The method is applicable, without change, to shape optimization problems in elasticity, etc. and appears to eliminate non-differentiability of the objective noticed in finite element method (FEM)-based fictitious domain shape optimization methods. We give arguments to support this statement. A mathematical proof is needed.     

Numerical modeling of air mass motion by the mesh-free method

The Navier-Stokes equation, applied to the calculation of wind velocity without accounting for the turbulent motion of the atmosphere, is considered in this work. The main flow characteristics were computed with the use of the Lagrange discrete vortex method for finding the solution of the Poisson equation under the Dirichlet and Neumann boundary conditions. To do this, two mesh-free methods: the element-free Galerkin (EFG) and the Finite Pointset (FP) methods, as well as the modification of the latter, have been analyzed. It is shown that the computation speed of the EFG method is higher than of the FP-method. It is determined that a serious disadvantage of the FP-method is its low rate convergence, while the computational complexity of each iteration is reasonable. The use of the modified FP-method has shown its computational speed to be comparable with that of the EFG method, although the advantage of the FP-method is not obvious when the size of the problem increases.     

Analyzing three-dimensional wave propagation with the hybrid reproducing kernel particle method based on the dimension splitting method

By introducing the dimension splitting method into the reproducing kernel particle method (RKPM), a hybrid reproducing kernel particle method (HRKPM) for solving three-dimensional (3D) wave propagation problems is presented in this paper. Compared with the RKPM of 3D problems, the HRKPM needs only solving a set of two-dimensional (2D) problems in some subdomains, rather than solving a 3D problem in the 3D problem domain. The shape functions of 2D problems are much simpler than those of 3D problems, which results in that the HRKPM can save the CPU time greatly. Four numerical examples are selected to verify the validity and advantages of the proposed method. In addition, the error analysis and convergence of the proposed method are investigated. From the numerical results we can know that the HRKPM has higher computational efficiency than the RKPM and the element-free Galerkin method.

     

Topology optimization of free vibrating continuum structures based on the element free Galerkin method

In this paper, the topology optimization design of the free vibrating continuum structures is formulated based on the element free Galerkin (EFG) method. Considering the relative density of nodes as design variable, and the maximization of the fundamental eigenvalue as an objective function, the mathematical formulation of the topology optimization model is developed using the solid isotropic microstructures with penalization (SIMP) interpolation scheme. The topology optimization problem is solved by the optimality criteria method. Finally, the feasibility and efficiency of the proposed method are illustrated with several 2D examples that are widely used in the topology optimization design.