MORTAR型旋转Q_1元的多重网格方法

本文讨论了mortar型旋转Q_1元的多重网格方法.证明了W循环的多重网格法是最优的,即收敛率与网格尺寸及层数无关.同时给出了一种可变的V循环多重网格算法,得到了一个条件数一致有界的预条件子.最后,数值试验验证了我们的理论结果.     

等几何分析的多重网格共轭梯度法

提高NURBS基函数阶数可以提高等几何分析的精度,同时也会降低多重网格迭代收敛速度.将共轭梯度法与多重网格方法相结合,提出了一种提高收敛速度的方法,该方法用共轭梯度法作为基础迭代算法,用多重网格进行预处理.对Poisson(泊松)方程分别用多重网格方法和多重网格共轭梯度法进行了求解,计算结果表明:等几何分析中采用高阶NURBS基函数处理三维问题时,多重网格共轭梯度法比多重网格法的收敛速度更快.     

无限元多重网格算法

在求偏微分方程数值解时,往往需要解一个规模很大的代数方程组,而多重网格是一种十分有效的迭代方法.大量数值试验证明,它具有很高的收敛速度.理论分析表明,这种迭代法的收敛速度并不随网格的加密而降低,这一突出优点是其它迭代方法望尘莫及的. 在使用有限元多重网格算法时,如果区域边界的角点使解具有奇性,理论分析会遇到     

非匹配网格上Stokes-Darcy模型的非协调元方法及其预条件技术

本文讨论了非匹配网格上Stokes-Darcy模型的两种低阶非协调元方法,证明了离散问题的适定性并得到了最优的误差估计.对离散出来的非对称不定线性方程组,我们提出了几种有效的预条件子,证明了预条件子的最优性.最后,数值试验验证了我们的理论结果.     

任意区域的网格自动生成和任意高阶有限元方法

本文首先简要介绍非拟合网格有限元方法求解复杂区域上椭圆问题的发展现状.然后结合最近本文作者发展的非拟合网格有限元方法,针对二阶椭圆方程提出一种任意光滑区域上的任意高阶协调有限元方法.本文在带悬点的Cartesian网格上自动生成诱导网格,在诱导网格上构造协调的高阶有限元空间,采用Nitsche技术处理Dirichlet边界条件,并证明方法的适定性和hp先验误差估计.数值算例验证了本文的理论结果.     

一种新的并行代数多重网格粗化算法

近年来,受实际应用领域中大规模科学计算问题的驱动,在大规模并行机上实现代数多重网格（AMG）算法成为数值计算领域的研究热点。本文针对经典AMG方法,提出一种新的并行网格粗化算法一多阶段并行RS算法（MPRS）。我们将新算法集成到了高性能预条件子软件包Hypre中。大量数值实验结果显示,新算法适合更广泛的问题,相对其他并行粗化算法,明显地改善了AMG并行计算的可扩展性。对三维27点格式有限差分离散的Poisson方程,在64个处理机上并行AMG求解,含8百万个未知量,新算法比RS3算法减少了近60的三维Poisson方程,近32万个未知量,在16个处理机上并行AMG—GMRES求解,新算法所需的迭代步数大约为其他粗化算法的一半,显示了很好的算法可扩展性。     

有限元网格的超收敛测度及其应用

给出线性有限元求解二阶椭圆问题的有限元网格超收敛测度及其应用.有限元超收敛经常是在具有一定结构的特殊网格条件下讨论的,而本文从一般网格出发,导出一种网格的范数用来描述超收敛所需要的网格条件以及超收敛的程度.并且通过对这种网格范数性质的考察,可以证明对于通常考虑的一些特殊网格的超收敛的存在性.更进一步,我们可以通过正则细分的方式在一般区域上也可以自动获得超收敛网格.最后给出相关的数值结果来验证本文的理论分析.     

三维泊松方程的高精度多重网格解法

利用对称网格点泰勒展开式中各阶导数项明显的对称性,得到了数值求解三维泊松方程的四阶和六阶精度的紧致差分格式,其推导过程简便直接.为了克服传统迭代法在求解高维问题时计算量大、收敛速度慢的缺陷,采用了多重网格加速技术,设计了相应的多重网格算法,求解了三维泊松方程的Dirichlet边值问题.数值实验结果表明,本文所提出的高精度紧致格式达到了期望的精度并且多重网格方法的加速效果是非常显著的.     

求解Oseen流的交替线松弛多重网格方法

利用Riemann解的通量差分分裂法——Godunov方法对Oseen流控制方程进行离散,得到了基于一阶上迎风格式的离散方程,并给出了使用多重网格方法求解该离散方程的V-循环算法和W-循环算法的收敛性分析.通过局部Fourier分析方法,对获得的离散方程的聚对称交替线GaussSeidel松弛的光滑性质进行了研究.结果表明:使用多重网格的两层网格及三层网格算法求解具有不同Reynolds数的Oseen流,即便是在高Reynolds数情况下,聚对称交替线Gauss-Seidel松弛具有很好的光滑性质,多重网格W-循环算法收敛性比V-循环算法好.     

非嵌套有限元的W循环多重网格方法

本文讨论了下述情形：１非嵌套网格；２曲边有限元；３非协调元；４拟协调元；５有限元的型函数有特殊性质，都能导致非嵌套的有限元空间．对一个包括上述情形的问题给出了非嵌套有限元的Ｗ循环多重网格方法，并证明了它的收敛性。     

曲边有限元和自然边界元求解外边值问题的耦合法

1　引　　言边界元与有限元耦合法在科学和工程计算中有着独特的作用 .由于区域的无限性给人们常用的有限元方法带来困难 ,边界元方法又难以独立处理非线性和非均质的问题以及具有不规则边界的区域上的问题 ,而两者相结合却可以克服各自的缺点 ,故边界元与有限元耦合法在处理一般区域问题特别是无界区域问题时便得到科学与工程界的青睐 ,获得了比较广泛的应用 .自然边界元方法并不引入新的变量 ,属于直接边界元方法[2 ] [8] .它保持能量不变和原边值问题的许多有用性质 ,例如双线性型的对称性和强制性 ,从而自然积分方程的解的存在唯一性及…     

NONMONOTONE CURVED SEARCH METHODS FOR UNCONSTRAINED OPTIMIZATION

In this paper two nonmonolone curved search (NCS) algorithms fur unconstrained optimization are presented. The NCS algorithms possess both a global convergence properly and a quadratic rale of convergence. Some numerical results are also reported which show that the NCS algorithn is superior to the usual curved search (UCS)aIgorithm for typical lest problems.     

非协调曲边四边形十二自由度平板弯曲单元*

本文利用精确元法[1],给出一个十二自由度曲边四边形板弯曲单元.该方法不需要变分原理,适用于任意正定和非正定偏微分方程.利用这个方法,单元之间的协调条件很容易满足,仅须位移和内力在单元节点上连续,即可保证所得到的解收敛于精确解.利用本文方法所获得的解,无论是位移还是内力可同时有二阶收敛精度.文末给出数值算例.表明了本文所得到的单元有非常好的精度.     

A second order isoparametric finite element method for elliptic interface problems

A second order isoparametric finite element method (IPFEM) is proposed for elliptic interface problems. It yields better accuracy than some existing second-order methods, when the coefficients or the flux across the immersed curved interface is discontinuous. Based on an initial Cartesian mesh, a mesh optimization strategy is presented by employing curved boundary elements at the interface, and an incomplete quadratic finite element space is constructed on the optimized mesh. It turns out that the number of curved boundary elements is far less than that of the straight one, and the total degree of freedom is almost the same as the uniform Cartesian mesh. Numerical examples with simple and complicated geometrical interfaces demonstrate the efficiency of the proposed method.     

On the Convergence of Quasi-Conforming Elements for Linear Elasticity Problems

Continuing the work in [1,2], we discuss the convergence conditions and the error estimates of quasi-conforming elements for linear elasticity problems. Some results about curved elements for second-order boundary value problems are also given.     

Derivation of the model of elastic curved rods from three-dimensional micropolar elasticity

In this paper we derive a model of curved elastic rods from the threedimensional linearized micropolar elasticity. Derivation is based on the asymptotic expansion method with respect to the thickness of the rod. The method is used without any a priori assumption on the scaling of the unknowns. The leading term, displacement and microrotation, is identified as the unique solution of a certain one-dimensional problem. Appropriate convergence results are proved.      

Asymptotic analysis of elastic curved rods

We consider a sequence of curved rods which consist of isotropic material and which are clamped on the lower base or on both bases. We study the asymptotic behaviour of the stress tensor and displacement under the assumptions of linearized elasticity when the cross‐sectional diameter of the rods tends to zero and the body force is given in the particular form. The analysis covers the case of a non‐smooth limit line of centroids. We show how the body force and the choice of the approximating curved rods can affect the strong convergence and the limit form of the stress tensor for the curved rods clamped on both bases. Copyright © 2006 John Wiley & Sons, Ltd.     

Consensus of second-order and high-order discrete-time multi-agent systems with random networks

This paper studies the convergence and convergence speed for the second-order and the high-order discrete-time multi-agent systems with random networks and arbitrary weights. Random networks mean that the existence of any edge is probabilistic and independent of any other edge. By introducing the agreement set, velocity control gain and high-order state control gain, some consensus protocols are provided for the discrete-time random networks. Moreover, the per-step and asymptotic convergence factors are proposed to measure the convergence and convergence speed. Some examples and simulation results are given to illustrate the effectiveness of the obtained theoretical results.     

Higher-order conformal decomposition finite element method

Numerical simulations of structures using higher-order finite elements is still a challenging task, in particular for domains with curved boundaries. A new higher-order accurate approach is proposed, combining the advantages of the classical p-FEM with embedded domain methods. Boundaries and/or interfaces are described implicitly using the level set method. In the elements cut by the zero level set, an automatic decomposition into interface aligned, i. e. conforming sub-elements is realized. Transfinite mappings are utilized to construct higher-order sub-elements by mappings of reference elements to the two sides of the boundary or interface. It is shown that although the resulting sub-elements are not always well-shaped, optimal convergence rates are possible. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

旋转曲面的面积及围成立体体积的求法

首先给出空间简单光滑曲线Γ绕空间直线l旋转所得到的旋转曲面面积以及围成立体的体积求法,作为特例又给出了空间曲线Γ绕坐标轴旋转所得到的旋转曲面面积及围成立体的体积求法,同时也得到了平面曲线Γ绕直线l及坐标轴旋转分别所得到的旋转曲面面积和围成立体的体积求法.