B-样条函数极小曲面造型

极小曲面在建筑、航空、轮船制造等领域有着重要应用,但由于极小曲面表示复杂,给实际应用带来了很大的困难.研究了具有给定边界的极小曲面的B-样条函数曲面逼近.基于非线性约束优化方法和有限单元方法,求极小曲面方程的近似解.在算法中使用数值延拓方法,使非线性问题的初值选择问题自动化,同时,使用一个简单的线性化策略对非线性问题进行线性化.给出了几个数值结果.     

推广Dirichlet 方法用于B 样条极小曲面设计

为弥补当前NURBS系统无法有效设计工程所急需的B样条极小曲面的缺陷,将构造Bézier极小曲面的Dirichlet方法成功地推广到了B样条极小曲面设计.提出了插值控制网格边界的B样条曲面模型,运用B样条基函数的求导公式及求值割角算法,将计算极小曲面内部控制顶点的问题转化为一个线性方程组的求解,从而避免了强非线性问题所导致的困惑,极大地提高了运算效率.最后,用大量实例对理论和算法进行了验证.     

插值边界曲线的NURBS 近似极小曲面设计

探索性地设计了一个插值给定边界曲线的NURBS 近似极小曲面算法,弥补了当前NURBS 系统无法有效地设计工程所急需的一般NURBS 极小曲面的缺陷.运用NURBS 曲面的节点插入、Hybrid 多项式逼近等多种技术,将NURBS 曲面转化为相对简单的分片Bézier 曲面求解,并运用各子曲面片的控制顶点优化、整体曲面不断更新的迭代方法,成功地得到高精度的近似分片Bézier 极小曲面.最后,可以按用户的各种要求选择运用相应不同的迭代逼近算法,求取插值给定边界曲线的近似NURBS 极小曲面.     

有理曲线曲面的降阶逼近

基于齐次坐标空间，提出了一种NURBS曲线曲面和有理Bezier曲线曲面降阶的简便方法。在齐次坐标空间中，使降阶后的曲线曲面与原曲线曲面的差的L2范数达到极小，将有理曲线曲面降多阶问题转化为二次规划问题求解，并给出了误差估计。实验结果表明，该方法计算速度快，降阶逼近效果好。     

基于偏微分方程的隐式曲面光顺方法

提出隐式曲面的光顺问题．针对该问题，给出刻画隐式曲面光顺程度的能量模型，并将能量解释为关于隐函数的泛函．基于变分原理，构造出隐函数关于时间的偏微分方程。通过求解该方程得到隐函数序列，使得光顺能量逐渐变小，从而达到光顺隐式曲面的目的．另外．针对光顺问题提出的其它约束条件，如尽可能保持面积不变，保持原有的形状特征等，对模型进行修正．最后，给出方程的实用解法及实验结果。并作简单讨论．实验结果表明该方法通用、灵活、有效，而且程序易于实现．     

旋转极小曲面中微分方程通解的解法

微分方程解析解(即通解)的求解方法十分复杂,数学领域对微分方程的研究着重在几个不同的方面,但大多数都是关心微分方程的解.只有少数简单的微分方程可以求得解析解.不过即使没有找到其解析解,仍然可以确认其解的部份性质.本文用加减消元法、微分方程解析解的求法及一些数学技巧给出了旋转极小曲面中微分方程的通解.和其他文献中该方程的解法进行比较,本文的方法更加简单易懂。     

一类时间最优控制问题中开关曲线的求法

对目标集是光滑超曲面的一类线性定常系统的时间最优控制问题,本文提出了一种求开关曲线参数方程的方法,计算表明,该方法简单实用。文末,对一般线性定常系统的时间最优控制问题进行了分类,并讨论了求解方法。     

填充函数算法研究综述

该文主要介绍填充函数方法求解全局优化问题。利用填充函数方法可以有效的求解大规模的全局优化问题。填充函数方法的思想就是该算法的思想是在求得总体优化问题的一个局部极小点后,构造填充函数,通过极小化该填充函数找到比当前局部极小值更好的解。     

三角域上的Plateau-Bézier问题求解新方法

从极小曲面上平均曲率处处为零出发求解三角域上的Plateau-Bézier问题.首先提出了一种新的线性能量函数,称之为平均曲率平方能量.基于该能量函数的极小化,推导出了内部控制顶点应满足的充要条件.通过造型实例,与基于Dirichlet能量极小化的求解方法进行了比较,发现两者各有千秋.特别地,若给定的边界曲线恰巧为三角...     

由离散点和特征线构建DEM的优化控制方法

基于拉普拉斯方程和优化控制(OC)理论,提出一种构建地形的OC方法.以极小化拉普拉斯方程左端项平方和为目标函数、采样离散点作为等式约束条件、河流线作为不等式约束条件、格网点附近的离散点集的高程范围为上下界约束条件,形成一个标准的OC问题,求解该问题即可获得融合原始数据中隐含信息的数字高程模型(DEM).通过调节OC的参数值,可以获得不同的DEM.实际案例表明,OC方法既能保证地形曲面的整体光滑性,又能保证DEM结果对于原始数据的忠实性,其模拟结果优于约束不规则三角网方法.     

An energy-preserving algorithm for nonlinear Hamiltonian wave equations with Neumann boundary conditions

In this paper, a novel energy-preserving numerical scheme for nonlinear Hamiltonian wave equations with Neumann boundary conditions is proposed and analyzed based on the blend of spatial discretization by finite element method (FEM) and time discretization by Average Vector Field (AVF) approach. We first use the finite element discretization in space, which leads to a system of Hamiltonian ODEs whose Hamiltonian can be thought of as the semi-discrete energy of the original continuous system. The stability of the semi-discrete finite element scheme is analyzed. We then apply the AVF approach to the Hamiltonian ODEs to yield a new and efficient fully discrete scheme, which can preserve exactly (machine precision) the semi-discrete energy. The blend of FEM and AVF approach derives a new and efficient numerical scheme for nonlinear Hamiltonian wave equations. The numerical results on a single-soliton problem and a sine-Gordon equation are presented to demonstrate the remarkable energy-preserving property of the proposed numerical scheme.     

Multigrid Methods for a Mixed Finite Element Method of the Darcy–Forchheimer Model

An efficient nonlinear multigrid method for a mixed finite element method of the Darcy–Forchheimer model is constructed in this paper. A Peaceman–Rachford type iteration is used as a smoother to decouple the nonlinearity from the divergence constraint. The nonlinear equation can be solved element-wise with a closed formulae. The linear saddle point system for the constraint is reduced into a symmetric positive definite system of Poisson type. Furthermore an empirical choice of the parameter used in the splitting is proposed and the resulting multigrid method is robust to the so-called Forchheimer number which controls the strength of the nonlinearity. By comparing the number of iterations and CPU time of different solvers in several numerical experiments, our multigrid method is shown to convergent with a rate independent of the mesh size and the Forchheimer number and with a nearly linear computational cost.     

Numerical Solution of the Kohn-Sham Equation by Finite Element Methods with an Adaptive Mesh Redistribution Technique

A mesh redistribution method is introduced to solve the Kohn-Sham equation. The standard linear finite element space is employed for the spatial discretization, and the self-consistent field iteration scheme is adopted for the derived nonlinear generalized eigenvalue problem. A mesh redistribution technique is used to optimize the distribution of the mesh grids according to wavefunctions obtained from the self-consistent iterations. After the mesh redistribution, important regions in the domain such as the vicinity of the nucleus, as well as the bonding between the atoms, may be resolved more effectively. Consequently, more accurate numerical results are obtained without increasing the number of mesh grids. Numerical experiments confirm the effectiveness and reliability of our method for a wide range of problems. The accuracy and efficiency of the method are also illustrated through examples.     

Numerical solution of nonlinear elliptic partial differential equations by a generalized conjugate gradient method

We have studied previously a generalized conjugate gradient method for solving sparse positive-definite systems of linear equations arising from the discretization of elliptic partial-differential boundary-value problems. Here, extensions to the nonlinear case are considered. We split the original discretized operator into the sum of two operators, one of which corresponds to a more easily solvable system of equations, and accelerate the associated iteration based on this splitting by (nonlinear) conjugate gradients. The behavior of the method is illustrated for the minimal surface equation with splittings corresponding to nonlinear SSOR, to approximate factorization of the Jacobian matrix, and to elliptic operators suitable for use with fast direct methods. The results of numerical experiments are given as well for a mildy nonlinear example, for which, in the corresponding linear case, the finite termination property of the conjugate gradient algorithm is crucial.     

Markov 控制过程在紧致行动集上的迭代优化算法

研究一类连续时间Markov控制过程(CTMCP)在紧致行动集上关于平均代价性能准则的优化算法。根据CTMCP的性能势公式和平均代价最优性方程，导出了求解最优或次最优平稳控制策略的策略迭代算法和数值迭代算法，在无需假设迭代算子是sp—压缩的条件下，给出了这两种算法的收敛性证明。最后通过分析一个受控排队网络的例子说明了这种方法的优越性。     

Analysis of unsteady compressible boundary layer flow via finite elements

This paper is concerned with the discrete formulation and numerical solution of unsteady compressible boundary layer flows using the Galerkin-finite element method. Linear interpolation functions for the velocity, density, temperature and pressure are used in the momentum equation and equations of continuity, energy and state. The coupled nonlinear finite element equations are approximated by a third order Taylor series expansion as temporal operator to integrate in time with Newton-Raphson type iterations performed until convergence within each time step. As an example, a boundary layer problem of a perfect gas behind a normal shock wave is solved. A comparison of the results with those by other method indicates a favorable agreement.     

A faster modified newton-raphson iteration

The paper describes an accelerated modified Newton-Raphson iteration in which the iterative deflection change is a scalar times the previous iterative change plus a further scalar times the usual unaccelerated change. These scalars are automatically recalculated at each iteration. They are related to inner products involving the iterative deflections and the present and past out-of-balance force vectors. The extra computation for each iteration is negligble, the only penalty being the storage of two extra vectors. The method is based on a secant approach and leads to a significant reduction in the required number of iterations. Examples are presented in which the method is applied to the nonlinear analysis of thin plates and shells. The technique is used in conjunction with the finite element method.     

Axisymmetric bending of circular and annular sandwich plates with nonlinear elastic core material

This paper compares the analytical model of the axisymmetric bending of a circular sandwich plate with the finite element method (FEM) based numerical model. The differential equations of the bending of circular symmetrical sandwich plates with isotropic face sheets and a nonlinear elastic core material are obtained. The perturbation method of a small parameter is used to represent the nonlinear differential equations as a sequence of linear equations specifying each other. The linear differential equations are solved by reducing them to the Bessel equation. The results of the calculations with the use of the analytical and FEM models are compared with the results obtained by other authors by the example of the following problems: (1) axisymmetric transverse bending of a circular sandwich plate; (2) axisymmetric transverse bending of an annular sandwich plate. The effect of the nonlinear elasticity of the core material on the strained state of the sandwich plate is described.     

Explicit monotone iterations providing upper and lower bounds for finite element solution with nonlinear radiation boundary conditions

In this paper we consider explicit monotone iterations of the finite element solutions for the radiation cooling problem with the nonlinear boundary conditions. These iterations provide upper and lower bounds, and convergence proofs are given. Finally, we give some numerical examples to demonstrate the effectiveness.