基于Foliation条件的离散动力系统二维流形计算

研究离散动力系统双曲不动点的二维流形计算,利用不变流形轨道上Jacobian矩阵能够传递导数这一特殊性质,提出一种新的一维流形计算方法,通过预测-校正两个步骤迅速确定流形上新网格点,避免重复计算,并简化精度控制条件.在此基础上,将基于流形面Foliation条件进行推广,推广后的Foliation条件能够控制二维流形上的一维子流形的增长速度,从而实现二维流形在各个方向上的均匀增长.此外,算法可以同时用于二维稳定和不稳定流形的计算.以超混沌三维Hénon映射和具有蝶形吸引子的Lorenz系统为例验证了算法的有效性.     

基于自适应因子轨道延拓法的不变流形计算

提出自适应因子和轨道延拓相结合的二维流形计算方法,利用以平衡点为中心的椭圆对局域流形的近似,通过轨道的等距延拓和椭圆初始点的自适应调节,在精度要求下自适应的添加轨道,完成二维双曲不变流形的计算.此方法比"轨道弧长法"精度高,包含更多细节信息;同时要比"盒子细分法"更能反映流形的延拓趋势.     

求解二维Euler方程有限单元边插值的降维重构算法

数值求解二维Euler方程的有限体积法（如k-exact，WENO重构、紧致重构等），无一例外地要进行耗时的网格单元上的二维重构.然而这些二维重构最后仅用于确定网格单元边界上高斯积分点处的解值，单元上二维重构似乎并非必需的.因此，文章提出用网格边上的一维重构来取代有限体积法中网格单元上的二维重构，分别在一致矩形网格和非结构三角形网格上发展了基于网格边重构的求解二维Euler方程的新方法，称为降维重构算法.数值算例表明该算法可以计算有强激波的无黏流动问题，且有较高的计算效率.      

空中点爆炸问题的数值解-细网格校正技术

本文对空中点爆炸问题的数值计算提出了一种自适应网格算法。即在某一时间层上局部截断误差较大的区域内细分网格,重新计算,并用所得的结果修正粗网格上的解。这种自這应网格算法大大提高了计算的效率。     

求解参与性介质内辐射传递问题的无网格有限差分法

本文应用无网格有限差分法求解参与性介质内辐射传递问题。无网格有限差分法源于广义有限差分法,利用局域泰勒多项式展开和具有插值特性的移动最小二乘,直接构造任意计算点待求函数的近似值及其各阶导数的差分格式,计算效率高,满足插值特性。本文检验了该算法求解辐射传输方程的稳定性;分别求解一维和二维参与性介质内辐射换热问题,算例结果与已有文献对比,表明了无网格有限差分法求解辐射传递问题的有效性和高精度。     

带源参数的热传导反问题的无网格方法

利用无网格有限点法求带有源参数的一维热传导反问题，推导了相应的离散方程.与其他基于网格的方法相比，有限点法采用移动最小二乘法构造形函数，只需要节点信息，不需要划分网格，用配点法离散求解方程，可以直接施加边界条件，不需要在区域内部求积分，减小了计算量.用有限点法求解热传导反问题具有数值实现简单、计算量小、可以任意布置节点等优点.最后通过算例验证了该方法的有效性.     

弹性力学的复变量无网格方法

在移动最小二乘法的基础上，提出了复变量移动最小二乘法.复变量移动最小二乘法的优点是采用一维基函数建立二维问题的逼近函数，所形成的无网格方法计算量小.然后，将复变量移动最小二乘法应用于弹性力学的无网格方法，提出了复变量无网格方法，推导了复变量无网格方法的公式.与传统的无网格方法相比，复变量无网格方法具有计算量小、精度高的优点.最后给出了数值算例. 关键词： 移动最小二乘法 复变量移动最小二乘法 无网格方法 弹性力学 复变量无网格方法     

运动介质中电磁场计算的一种迎风有限元方法

本文提出了运动介质中正弦稳态电磁场问题的一种迎风有限元解法。用伽僚金法求解这类问题,当离散网格的Peclet数大于1时,计算结果会出现伪振荡。为了抑制这种振荡,引入了采用在迎风面与背风面具有不同迎风参数的权函数的迎风有限元法。该方法对一维问题,在均匀网格下能在节点上给出问题的精确解,在一维结果的基础上,提出了相应的二维解法,并用一个二维模型进行了验证。     

基于非结构变形网格的间断装配法原理

在激波捕捉法计算得到的流场基础上采用辨识算法得到初始间断位置, 从ALE方程出发, 考虑离散几何守恒律, 采用变形网格和网格重构技术解决计算过程中间断运动和变形, 新旧网格之间流场采用高精度信息传递方法保持时间精度, 建立了基于非结构动网格技术的间断装配方法.通过激波管问题的二维模拟, 模拟了初始间断分解为激波和接触间断激波遇到固壁反射后与接触间断相交的非定常流动过程, 对这种新方法的基本原理进行了介绍.      

连续时间系统二维不稳定流形的异构算法

非线性系统的二维流形通常具有复杂几何结构和丰富动力学信息,因此在流形计算与可视化时存在大量的不可避免的数值计算.因此,如何高效地完成这些计算就成了关键问题.鉴于当今计算机的异构发展趋势(包含多核CPU和通用GPU),本文在兼顾精度和通用性的基础上,提出了适用于新一代计算平台的快速流形计算方法.本算法将计算任务分为轨道延伸和三角形生成两部分,前者运算量大而单一适合GPU完成,后者运算量小而复杂适合CPU执行.通过对Lorenz系统原点稳定流形的计算,表明本算法能充分发挥异构平台的综合性能,可大幅度提高计算速 关键词： 不稳定流形 流形计算 异构计算 Lorenz系统     

An adaptive moving mesh method for two-dimensional ideal magnetohydrodynamics

This paper presents an adaptive moving mesh algorithm for two-dimensional (2D) ideal magnetohydrodynamics (MHD) that utilizes a staggered constrained transport technique to keep the magnetic field divergence-free. The algorithm consists of two independent parts: MHD evolution and mesh-redistribution. The first part is a high-resolution, divergence-free, shock-capturing scheme on a fixed quadrangular mesh, while the second part is an iterative procedure. In each iteration, mesh points are first redistributed, and then a conservative-interpolation formula is used to calculate the remapped cell-averages of the mass, momentum, and total energy on the resulting new mesh; the magnetic potential is remapped to the new mesh in a non-conservative way and is reconstructed to give a divergence-free magnetic field on the new mesh. Several numerical examples are given to demonstrate that the proposed method can achieve high numerical accuracy, track and resolve strong shock waves in ideal MHD problems, and preserve divergence-free property of the magnetic field. Numerical examples include the smooth Alfvén wave problem, 2D and 2.5D shock tube problems, two rotor problems, the stringent blast problem, and the cloud–shock interaction problem.     

Strange attractors are classified by bounding Tori

There is at present a doubly discrete classification for strange attractors of low dimension, d(L)<3. A branched manifold describes the stretching and squeezing processes that generate the strange attractor, and a basis set of orbits describes the complete set of unstable periodic orbits in the attractor. To this we add a third discrete classification level. Strange attractors are organized by the boundary of an open set surrounding their branched manifold. The boundary is a torus with g holes that is dressed by a surface flow with 2(g-1) singular points. All known strange attractors in R3 are classified by genus, g, and flow type.     

Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics

In this paper we apply dynamical systems techniques to the problem of heteroclinic connections and resonance transitions in the planar circular restricted three-body problem. These related phenomena have been of concern for some time in topics such as the capture of comets and asteroids and with the design of trajectories for space missions such as the Genesis Discovery Mission. The main new technical result in this paper is the numerical demonstration of the existence of a heteroclinic connection between pairs of periodic orbits: one around the libration point L(1) and the other around L(2), with the two periodic orbits having the same energy. This result is applied to the resonance transition problem and to the explicit numerical construction of interesting orbits with prescribed itineraries. The point of view developed in this paper is that the invariant manifold structures associated to L(1) and L(2) as well as the aforementioned heteroclinic connection are fundamental tools that can aid in understanding dynamical channels throughout the solar system as well as transport between the "interior" and "exterior" Hill's regions and other resonant phenomena. (c) 2000 American Institute of Physics.     

On a simple recursive control algorithm automated and applied to an electrochemical experiment

We review a simple recursive proportional feedback (RPF) control strategy for stabilizing unstable periodic orbits found in chaotic attractors. The method is generally applicable to high-dimensional systems and stabilizes periodic orbits even if they are completely unstable, i.e., have no stable manifolds. The goal of the control scheme is the fixed point itself rather than a stable manifold and the controlled system reaches the fixed point in d+1 steps, where d is the dimension of the state space of the Poincare map. We provide a geometrical interpretation of the control method based on an extended phase space. Controllability conditions or special symmetries that limit the possibility of using a single control parameter to control multiply unstable periodic orbits are discussed. An automated adaptive learning algorithm is described for the application of the control method to an experimental system with no previous knowledge about its dynamics. The automated control system is used to stabilize a period-one orbit in an experimental system involving electrodissolution of copper. (c) 1997 American Institute of Physics.     

An Eulerian approach for computing the finite time Lyapunov exponent

We propose efficient Eulerian methods for approximating the finite-time Lyapunov exponent (FTLE). The idea is to compute the related flow map using the Level Set Method and the Liouville equation. There are several advantages of the proposed approach. Unlike the usual Lagrangian-type computations, the resulting method requires the velocity field defined only at discrete locations. No interpolation of the velocity field is needed. Also, the method automatically stops a particle trajectory in the case when the ray hits the boundary of the computational domain. The computational complexity of the algorithm is O(Δx^?(d+1)) with d the dimension of the physical space. Since there are the same number of mesh points in the x–t space, the computational complexity of the proposed Eulerian approach is optimal in the sense that each grid point is visited for only O(1) time. We also extend the algorithm to compute the FTLE on a co-dimension one manifold. The resulting algorithm does not require computation on any local coordinate system and is simple to implement even for an evolving manifold.     

Three-dimensional matched interface and boundary (MIB) method for treating geometric singularities

This paper reports the three-dimensional (3D) generalization of our previous 2D higher-order matched interface and boundary (MIB) method for solving elliptic equations with discontinuous coefficients and non-smooth interfaces. New MIB algorithms that make use of two sets of interface jump conditions are proposed to remove the critical acute angle constraint of our earlier MIB scheme for treating interfaces with sharp geometric singularities, such as sharp edges, sharp wedges and sharp tips. The resulting 3D MIB schemes are of second-order accuracy for arbitrarily complex interfaces with sharp geometric singularities, of fourth-order accuracy for complex interfaces with moderate geometric singularities, and of sixth-order accuracy for curved smooth interfaces. A systematical procedure is introduced to make the MIB matrix optimally symmetric and banded by appropriately choosing auxiliary grid points. Consequently, the new MIB linear algebraic equations can be solved with fewer number of iterations. The proposed MIB method makes use of Cartesian grids, standard finite difference schemes, lowest order interface jump conditions and fictitious values. The interface jump conditions are enforced at each intersecting point of the interface and mesh lines to overcome the staircase phenomena in finite difference approximation. While a pair of fictitious values are determined along a mesh at a time, an iterative procedure is proposed to determine all the required fictitious values for higher-order schemes by repeatedly using the lowest order jump conditions. A variety of MIB techniques are developed to overcome geometric constraints. The essential strategy of the MIB method is to locally reduce a 2D or a 3D interface problem into 1D-like ones. The proposed MIB method is extensively validated in terms of the order of accuracy, the speed of convergence, the number of iterations and CPU time. Numerical experiments are carried out to complex interfaces, including the molecular surfaces of a protein, a missile interface, and van der Waals surfaces of intersecting spheres.     

CCD camera automatic calibration technology and ellipse recognition algorithm

A novel two-dimensional (2D) pattern used in camera calibration is presented. With one feature circle located at the center, an array of circles is photo-etched on this pattern. An ellipse recognition algorithm is proposed to implement the acquisition of interest calibration points without human intervention. According to the circle arrangement of the pattern, the relation between three-dimensional (3D) and 2D coordinates of these points can be established automatically and accurately. These calibration points are computed for intrinsic parameters calibration of charge-coupled device (CCD) camera with Tsai method. A series of experiments have shown that the algorithm is robust and reliable with the calibration error less than 0.4 pixel. This new calibration pattern and ellipse recognition algorithm can be widely used in computer vision.     

Orbit Sum Rules for the Quantum Wave Functions of the Strongly Chaotic Hadamard Billiard in Arbitrary Dimensions

Sum rules are derived for the quantum wave functions of the Hadamard billiard in arbitrary dimensions. This billiard is a strongly chaotic (Anosov) system which consists of a point particle moving freely on a D-dimensional compact manifold (orbifold) of constant negative curvature. The sum rules express a general (two-point)correlation function of the quantum mechanical wave functions in terms of a sum over the orbits of the corresponding classical system. By taking the trace of the orbit sum rule or pre-trace formula, one obtains the Selberg trace formula. The sum rules are applied in two dimensions to a compact Riemann surface of genus two, and in three dimensions to the only non-arithmetic tetrahedron existing in hyperbolic 3-space. It is shown that the quantum wave functions can be computed from classical orbits. Conversely, we demonstrate that the structure of classical orbits can be extracted from the quantum mechanical energy levels and wave functions (inverse quantum chaology).     

基于Steiner点扰动和矢量边界推进技术的三维约束非结构网格生成方法

提出一种结合Steiner点扰动和矢量边界推进三角化技术的三维约束非结构四面体网格生成方法.在由物面节点生成的Delaunay网格基础上,利用Conforming方法恢复计算域形状,然后利用点扰动技术迫使所有Steiner点全部从约束边、面上转移,并利用矢量边界推进三角化方法重构约束面三角剖分,进而得到完全恢复所有约束条件的Constrained网格,并证明方法的收敛性和稳定性.