计算几何中几何偏微分方程的构造

平均曲率流、曲面扩散流和Willmore流等著名的几何流除了在理论方面有重要的意义之外,在计算机辅助几何设计、计算机图形学以及图像处理等领域也得到了广泛的应用．然而在解决实际问题时,人们经常要根据问题的特点构造其它具有指定性质的几何流．本文从统一的观点出发,对于参数曲面以及水平集曲面,给出了几类重要几何偏微分方程(包括L2梯度流、H-1梯度流以及H-2梯度流)的构造．这几类几何流的包容十分广泛,上述提到的几个几何流均为其特例．     

小攻角高超声速钝锥边界层中不同扰动对转捩的影响

为了研究上游不同扰动对转捩位置的影响,针对来流Ma=6,攻角1°,半锥角5°的钝锥边界层的转捩进行了数值模拟．首先研究了边界层中小扰动的演化,与流动稳定性理论进行了对比,结果表明:在所考虑的流场中,流动稳定性线性理论可以对扰动的增长率做出一个较好的预测．在此基础上,研究了不同扰动对转捩位置的影响．计算给出了在两种不同频率分布的扰动情况下,转捩位置沿周向的分布．结果表明,转捩位置沿周向分布与入口扰动的幅值和频率有关．某子午面的转捩位置由该处的最不稳定波在入口的幅值决定．根据大多数风洞中背景扰动的特性,解释了小攻角圆锥转捩实验中背风面先转捩,迎风面后转捩的现象．同时,还解释了在背风面附近转捩位置“凹陷”的现象．     

组合几何(二) 结构问题

结构和构造是整个组合几何的核心,实际上组合几何主要讨论的就是以几何元素为某元素的集合的结构及其性质,一个有关组合几何的问题大都是在解决了结构问题之后才迎刃而解的。因此,深入地探讨一下结构问题是有必要的,而且对今后很多问题的讨论     

三维可压Euler方程简单波及双重波结构

本文研究了三维可压欧拉方程简单波解和双重波解的结构.给出了简单波的流动区域是被一族相互独立的平面所覆盖,沿着每个平面u,v,w从而p,ρ,c均为常数.双重波的流动区域是被一族相互独立的直线所覆盖,沿着每个特征平面u,v,w从而p,ρ,c均为常数.     

一类通过构造三角形求解的代数问题

一类通过构造三角形求解的代数问题范文贵（辽宁省锦州师范学院数学系１２１００３）在中学数学中，代数与几何是既互相分离，又互相联系的．下面我介绍一类具有相同几何背景的代数问题．它们的数量关系具有余弦定理的结构形式，可根据这种特征构造出相应的三角形来解．例...     

简单波的解析解

Riemann于1860年给出了简单波的一般解,可是欲将其中所含的一个任意函数的具体形式按边界条件或初始条件求出来,却难以实现,因而对具体问题就难以深入探讨.本文按拟线性偏微分方程的几何理论,十分方便地给出了由边界条件或初始条件所确定的简单波解析解,并用这些解讨论了有关流动的性质,且得到了若干新颖的结果     

基于GMRES算法的弹性结构强耦合小变形流激振动分析

使用混合广义变分原理,将基于Lagrange表述的小位移变形结构振动问题与基于Euler描述的不可压缩粘性流动问题,统一在功率平衡的框架下建立流固系统的耦合控制方程．用有限元格式做空间离散后,再用广义梯形法将有限元控制方程转化为增量型的线性方程组,该方程组的系数矩阵具有非对称性,其中元素含对流效应和时间因子．将GMRES算法与振动分析的Newmark法和流动分析的Hughes预测多修正法结合,发展成一种基于GMRES-Hughes-Newmark的稳定算法,用于计算具有复杂几何边界的强耦合流激振动问题．以混流式水轮机叶道为数值算例的计算表明,模拟结果与试验实测结果吻合较好．     

论跨声速流函数计算中的人工粘性

从气体动力学基本方程组出发,根据数值计算方法的理论,分析了近年来在跨声速流动计算中广泛使用的人工密度法,指出流函数计算中采用人工密度法,理论上是有问题的,进而提出一种正确的人工粘性表达式。数值计算表明,它扩大了流函数方法可计算的Mach数范围,使激波位置接近于实验结果。     

考虑几何非线性的杆系结构弯曲变形样条有限点法

研究了杆系结构考虑几何非线性的大挠度弯曲变形问题,推算并验证了一种考虑几何非线性的杆系结构弯曲变形计算新方法.以三次B样条函数为基函数,采用广义参数法,构造出梁的样条基函数,通过最小势能原理,建立了杆系结构考虑几何非线性的刚度方程,对处于弹性范围内的杆系结构的大变形弯曲问题进行了计算,提出了考虑几何非线性时杆系结构弯曲变形计算的样条有限点法.结果表明:方法不用进行单元坐标变换、划分等分数少、收敛速度快且计算精度较高,是一种较传统有限元法更简单且可行的方法.     

地形构造中地震波传播的非对称交错网格模拟

提出了一种新的三维空间对称交错网格差分方法,模拟地形构造中弹性波传播过程．通过具有二阶时间精度和四阶空间精度的不规则网格差分算子用来近似一阶弹性波动方程,引入附加差分公式解决非均匀交错网格的不对称问题．该方法无需在精细网格和粗糙网格间进行插值,所有网格点上的计算在同一次空间迭代中完成．使用精细不规则网格处理海底粗糙界面、 断层和空间界面等复杂几何构造, 理论分析和数值算例表明, 该方法不但节省了大量内存和计算时间, 而且具有令人满意的稳定性和精度．在模拟地形构造中地震波传播时,该方法比常规方法效率更高．     

Artificial boundary conditions for two-dimensional equations of fluid dynamics. 1. Convective wave equation

The article models external flow problems on artificially bounded regions. In the linear approximation we examine the reflection of acoustic waves in a moving medium, incident at various angles on a fixed boundary. We consider the construction of various boundary conditions and estimate their reflecting properties for plane waves and waves from point sources. The plane wave approximation is justified theoretically. A method is proposed for estimating the integral reflection coefficient for waves with a whole range of incidence angles. __________ Translated from Prikladnaya Matematika i Informatika, No. 24, pp. 76–110, 2006.     

Interaction of elementary waves for equations of potential flow

Interaction of elementary waves for equations of unsteady potential flow in gas dynamics is considered. Under the assumptions on weakness of strength of the elementary waves the structure of solutions has been given in various cases of interaction of simple wave with shock, or interaction between simple waves or shocks. Hence the complete results on interaction of weak elementary waves for second-order equation of potential flow are obtained. Project supported by the National Natural Science Foundation of China and the State Education Commission of China.     

GLOBAL SIMPLE WAVE SOLUTIONS TO A KIND OF TWO DIMENSIONAL HYPERBOLIC SYSTEM OF CONSERVATION LAWS

This paper is concerned with the simple waves of a kind of two dimensional hyperbolic system of conservation laws, which can be obtained from the two dimensional relativistic membrane equation in Minkowski space. Using wave decomposition method, we get that a flow adjacent to a nonconstant state can be a global simple wave. Furthermore, the flow is covered by three families of characteristics, in which the first family of characteristics is straight and the others are curved, which is different to the almost related results.     

Exact Free Surface Flows for Shallow Water Equations I: The Incompressible Case

A comprehensive exact treatment of free surface flows governed by shallow water equations (in sigma variables) is given. Several new families of exact solutions of the governing PDEs are found and are shown to embed the well-known self-similar or traveling wave solutions which themselves are governed by reduced ODEs. The classes of solutions found here are explicit in contrast to those found earlier in an implicit form. The height of the free surface for each family of solutions is found explicitly. For the traveling or simple wave, the free surface is governed by a nonlinear wave equation, but is arbitrary otherwise. For other types of solutions, the height of the free surface is constant either on lines of constant acceleration or on lines of constant speed; in another case, the free surface is a horizontal plane while the flow underneath is a sine wave. The existence of simple waves on shear flows is analytically proved. The interaction of large amplitude progressive waves with shear flow is also studied.     

Riemann problem in non-ideal gas dynamics

In this paper we consider the Riemann problem for gas dynamic equations governing a one dimensional flow of van der Waals gases. The existence and uniqueness of shocks, contact discontinuities, simple wave solutions are discussed using R-H conditions and Lax conditions. The explicit form of solutions for shocks, contact discontinuities and simple waves are derived. The effects of van der Waals parameter on the shock and simple waves are studied. A condition is derived on the initial data for the existence of a solution to the Riemann problem. Moreover, a necessary and sufficient condition is derived on the initial data which gives the information about the existence of a shock wave or a simple wave for a 1-family and a 3-family of characteristics in the solution of the Riemann problem.     

Interaction of three and four rarefaction waves of the pressure-gradient system

We study the two-dimensional pressure-gradient system, a subsystem of the two-dimensional compressible Euler system. We consider the problem of interaction of four rarefaction waves which is one case of two-dimensional Riemann problems. It is known that, when two planar waves interact, there exists a smooth solution in the interaction region. In this paper, we establish the existence of a smooth solution in the hyperbolic domain of determinacy, in which we encounter the interaction of simple and planar waves and shock prevention in simple waves.     

桩基承台结构的波流力研究进展

我国近海风电场建设大多采用桩基承台结构．总结了不规则波浪和水流共同作用下桩基承台结构的波流力物理模型实验结果,得到了群桩效应系数及其变化规律,讨论了作用于近水面承台底部的波浪拍击力;从理论上分析了规则波作用下承台对桩基波浪力的影响;建立了规则波与桩基承台相互作用的数值模型,揭示了波浪在承台的上浪与爬高及其水动力特征．鉴于桩基承台结构包含多个斜桩和较大尺度的承台,在波浪与水流作用下该结构物附近的流场结构十分复杂,有必要针对结构附近的流动结构以及自由表面大变形开展细致的实验和数值模拟研究,以进一步揭示作用于这类结构的波流力变化规律及其机理.     

静脉血流动力学模型基本波的相互作用北大核心CSCD

静脉系统是心血管系统的重要组成部分.脉搏波在血液流动中有着突出的重要性.本文主要研究静脉血流动力学模型基本波的相互作用.血流动力学模型是2×2严格双曲型方程组,其基本波包括疏散波和激波,属于血液流动中的脉搏波.基本波相互作用后血管截面面积和血流速度发生相应的变化.     

On the acceleration of shock waves and concentration of energy

By a series of simple examples related to exact solutions of problems in gas dynamics and magnetohydrodynamics, possible mechanisms of acceleration of shock waves and concentration of energy are elucidated. The acceleration of a shock wave is investigated in the problem of motion of a plane piston at a constant velocity in the case when the initial density of the medium drops in the presence of constant counterpressure. It is shown that in this situation a “blow-up” regime is induced by a shock wave going to infinity in finite time even for limited work of the piston. A simple spherically symmetric solution with a converging shock wave is constructed and shown to lead to the concentration of energy. A general method for solving one-dimensional non-self-similar problems related to matching the equilibrium state to a motion with homogeneous deformation on a shock wave is discussed; this method leads to a solution in quadratures.