Ito型耦合KdV方程的高阶保能量方法

构造具有能量守恒特性的Ito型耦合KdV方程的高阶保能量格式在模拟方程的运动中有重要的意义.本文利用四阶平均向量场方法和拟谱方法得到了Ito型耦合KdV方程的高阶保能量格式,并利用高阶保能量格式数值模拟方程孤立波的演化行为.数值结果表明新的高阶保能量格式能很好地模拟Ito型耦合KdV方程孤立波的行为,且精确地保持了方程的离散能量守恒.     

一类非局部粘性水波模型的数值格式

本文主要讨论带非局部粘性项水波方程的数值方法.我们建立了一种求解这类粘性水波方程的数值方案.该方案有效解决了非局部粘性项与非线性项的离散问题.所提的格式包括对α阶分数阶项的2-α阶格式和对非线性项的线性化处理的混合格式.我们证明了这种格式是无条件稳定的,并得出线性Crank-Nicolson加2-α格式的收敛阶是O(?t32+N1-m)的结论.一系列的数值例子验证了理论证明的正确性.最后,我们用所提数值格式研究了粘性水波方程的渐近衰减率,并讨论了各种参数项对衰减率的影响.     

偏心冲击下阻尼薄圆板动力学响应的广义多辛分析

对称破缺是复杂动力学系统的本质属性,决定着系统诸多非线性性质。关注荷载不对称性和结构耗散这两类引起对称破缺的因素,采用广义多辛分析方法研究了偏心冲击荷载作用下阻尼薄圆板振动问题。在哈密顿体系下,建立偏心冲击荷载作用下阻尼薄圆板振动问题的动力学控制方程,构造其广义多辛近似对称形式,研究由于以上因素引起的守恒律误差。随后采用显式中点差分离散方法构造其广义多辛格式,并用于研究圆板阻尼和冲击荷载偏心对振动过程的影响。研究思路为进一步探索动力学系统的对称性与耗散效应之间的内在联系奠定基础。     

线性阻尼杆振动问题基于变量变换的多辛算法

提出线性阻尼杆振动问题基于变量变换的多辛离散算法。首先通过变量变换把耗散系统化为保守系统。其次以变换变量组成状态向量并采用中点离散方法构造中点Box多辛离散格式,然后,基于空间层的状态变量建立矩阵形式的递推关系,最后结合空间边界条件和初始条件建立线性方程组求解。研究结果表明,构造的多辛算法不仅能够保持系统守恒型几何性质,通过状态变量合理表示了边界条件,而且能够较准确地体现系统的耗散效应。     

基于谱元方法的三维矢量波动方程的辛离散格式

本文给出了三维矢量波动方程的无穷维Hamilton系统形式并提出了一个新的数值逼近格式.基于Gauss-Lobatto-Legendre多项式,建立了该无穷维系统的矢量谱元方法空间离散格式,并得到一个有限维Hamilton系统.进而,利用辛差分方法对该有限维系统进行全离散,以期保持系统的结构和能量.最后,借助于对角化技...     

长短波方程的两个守恒型紧致有限差分格式

本文对一类耦合非线性长短波方程组进行了数值研究，提出了两个四阶紧致有限差分格式，并证明新格式在离散意义下保持原问题的两个守恒性质，即总质量守恒和总能量守恒．数值实验表明本文格式在时间和空间方向分别具有二阶和四阶精度，具有良好的稳定性且在离散意义下很好地保持总质量和总能量守恒．     

弹性力学极坐标辛体系Hamilton函数的守恒律

用弹性力学直角坐标辛体系中类似的形式,定义了极坐标问题径向和环向辛体系的Hamilton函数,对其守恒性进行了研究,由Hamilton对偶方程推出了Hamilton函数的守恒律,同时给出了守恒条件,指出两种极坐标辛体系中Hamilton函数是否守恒均取决于两侧边的荷载和位移情况。在径向和环向辛体系中都给出了算例,验证了Hamilton函数的守恒律。这一守恒律丰富了弹性力学辛体系的理论内容,不仅对于弹性力学极坐标问题的理论分析有所帮助,也为极坐标问题的数值计算分析提供了一个判断依据。     

缓增分数阶扩散方程的高阶时间离散LDG方法

研究了缓增分数阶扩散方程的高阶时间离散局部间断Galerkin (Local Discontinuous Galerkin, LDG)方法，不是直接求解缓增分数阶扩散方程，而是首先通过变换将其转化成Caputo型时间分数阶扩散方程。接着，采用L1-2差分逼近离散Caputo型分数阶导数，间断有限元离散空间变量，构造求解模型的全离散LDG格式。证明了所建立的全离散格式为无条件稳定的且具有最优误差阶，两个数值算了验证了所建立数值格式的精度和鲁棒性。数值实验结果表明所建立格式在时间和空间方向均具有高精度。     

脉冲微分方程的一个修正block-by-block数值格式

本文利用修正的block-by-block方法针对脉冲微分方程构造了高阶数值格式.修正的block-by-block方法是传统的block-by-block方法的改进,其优点是除第一块外其余每块都能够解耦求解积分方程的高阶数值方法.首先,把脉冲微分方程等价转化为脉冲型积分方程,并利用修正的block-by-block方法进行离散,得到在两个相邻脉冲点中除第一块外其余每块都解耦的高阶数值格式.其次,利用离散的Grownwall不等式证明了数值解逼近精度为四阶.最后,一系列的数值算例验证了理论分析的正确性.     

KdV方程的一个紧致差分格式

本文基于经典的有限差分方法,讨论了满足周期边界条件的KdV方程的高精度差分格式的构造问题.通过引入中间函数及紧致方法对空间区域进行离散,提出了KdV方程的一个两层隐式紧致差分格式.利用泰勒展开法得出,该格式在时间方向具有二阶精度,但在空间方向可达到六阶精度.采用线性稳定性分析法证明了该格式是稳定的.数值结果表明:本文所提出的紧致差分格式是有效的,在空间方向拥有较高的精度,还能够很好地保持离散动量和能量守恒性质.     

弹性杆纵波弥散效应数值分析

弹性杆中的纵波弥散效应一直是力学界研究的热点问题之一。本文基于Hamiltonian体系变分原     

Numerical Solution and Experimental Test for Corona Discharge Between Blade and Plate

This paper presents a numerical technique to solve the problem of space charge distribution in a blade-plate electrodes system. The coupled equations are: Poisson equation solved by finite element method (FEM) to determine the distributions of potential, and charge conservation equation solved by the method of characteristics (MOC) to obtain the charge density between the two electrodes. The structured mesh is redefined at each step of the iterative scheme. A simplified injection law at the blade is retained; it allows us to obtain solutions which compare very favorably with experimental results concerning the current density distribution at the plate     

An energy-based discontinuous Galerkin method for coupled elasto-acoustic wave equations in second-order form

We consider wave propagation in a coupled fluid-solid region separated by a static but possibly curved interface. The wave propagation is modeled by the acoustic wave equation in terms of a velocity potential in the fluid, and the elastic wave equation for the displacement in the solid. At the fluid solid interface, we impose suitable interface conditions to couple the two equations. We use a recently developed energy-based discontinuous Galerkin method to discretize the governing equations in space. Both energy conserving and upwind numerical fluxes are derived to impose the interface conditions. The highlights of the developed scheme include provable energy stability and high order accuracy. We present numerical experiments to illustrate the accuracy property and robustness of the developed scheme.     

拆除爆破中触地诱发震动的震动模型

在高层建筑物爆破拆除中,塌落体触地冲击地面引起的震动往往成为拆除爆破负面效应控制的关键因素之一.从动力平衡方程出发,运用波动理论和动量守恒定理,建立了计算触地震动的等效集总单自由度震动模型.该模型从理论上揭示了触地诱发震动的产生机理,反映了触地震动的衰减规律,与数值模拟和工程实践吻合.     

Hybrid finite element/volume method for shallow water equations

A hybrid numerical scheme based on finite element and finite volume methods is developed to solve shallow water equations. In the recent past, we introduced a series of hybrid methods to solve incompressible low and high Reynolds number flows for single and two‐fluid flow problems. The present work extends the application of hybrid method to shallow water equations. In our hybrid shallow water flow solver, we write the governing equations in non‐conservation form and solve the non‐linear wave equation using finite element method with linear interpolation functions in space. On the other hand, the momentum equation is solved with highly accurate cell‐center finite volume method. Our hybrid numerical scheme is truly a segregated method with primitive variables stored and solved at both node and element centers. To enhance the stability of the hybrid method around discontinuities, we introduce a new shock capturing which will act only around sharp interfaces without sacrificing the accuracy elsewhere. Matrix‐free GMRES iterative solvers are used to solve both the wave and momentum equations in finite element and finite volume schemes. Several test problems are presented to demonstrate the robustness and applicability of the numerical method. Copyright © 2010 John Wiley & Sons, Ltd.     

A Petrov–Galerkin finite element scheme for the regularized long wave equation

A Petrov-Galerkin finite element method (FEM) for the regularized long wave (RLW) equation is proposed. Finite elements are used in both the space and the time domains. Dispersion correction and a highly selective dissipation mechanism are introduced through additional streamline upwind terms in the weight functions. An implicit, conditionally stable, one-step predictor–corrector time integration scheme results. The accuracy and stability are investigated by means of local expansion by Taylor series and the resulting equivalent differential equation. An analysis based on a linear Fourier series solution and the Von Neumanns stability criterion is also performed. Based on the order of the analytical approximations and of the domain discretization it is concluded that the scheme is of third order in the nonlinear version and of fourth order in the linear version. Three numerical experiments of wave propagation are presented and their results compared with similar ones in the literature: solitary wave propagation, undular bore propagation, and cnoidal wave propagation. It is concluded that the present scheme possesses superior conservation and accuracy properties.This work has been partially supported by the Fundação para a Ciência e Tecnologia, under project POCTI/ECM/41800/2001.     

Free‐surface tracking in 2D with the harmonic polynomial cell method: Two alternative strategies

Several cases of nonlinear wave propagation are studied numerically in two dimensions within the framework of potential flow. The Laplace equation is solved with the harmonic polynomial cell (HPC) method, which is a field method with high‐order accuracy. In the HPC method, the computational domain is divided into overlapping cells. Within each cell, the velocity potential is represented by a sum of harmonic polynomials. Two different methods denoted as immersed boundary (IB) and multigrid (MG) are used to track the free surface. The former treats the free surface as an IB in a fixed Cartesian background grid, while the latter uses a free‐surface fitted grid that overlaps with a Cartesian background grid. The simulated cases include several nonlinear wave mechanisms, such as high steepness and shallow‐water effects. For one of the cases, a numerical scheme to suppress local wave breaking is introduced. Such scheme can serve as a practical mean to ensure numerical stability in simulations where local breaking is not significant for the result. For all the considered cases, both the IB and MG method generally give satisfactory agreement with known reference results. Although the two free‐surface tracking methods mostly have similar performance, some differences between them are pointed out. These include aspects related to modeling of particular physical problems as well as their computational efficiency when combined with the HPC method.     

Multigrid and Runge-Kutta time stepping applied to the uniformly non-oscillatory scheme for conservation laws

It is known that Harten's uniformly non-oscillatory scheme is a second-order accurate scheme for discretizing coservation laws. In this paper a multigrid technique and Runge-Kutta time stepping with frozen dissipation is applied to Harten's scheme in order to obtain the steady-state solution. It is shown that these techniques applied to Harten's scheme lead to a better convergence to the steady-state solution of a first-order conservation law than applied to Jameson's scheme.     

Application of finite element methods to thermohydrodynamic lubrication

A finite element formulation is presented for the equations governing the steady thermohydrodynamic behaviour of liquid lubricated bearings. This formulation permits application of the iterative solution scheme to bearings of arbitrary geometry. A generalized Reynolds equation resulting from the combination of the mass and momentum conservation equations is cast into variational form and used to derive general finite element equations. The method of weighted residuals with Galerkin's criterion is used to generate finite element matrix equations for the thermal energy equation. In addition to the finite element formulation, a discussion of appropriate finite difference techniques is also given for problems without complex geometry. As an example, the formulations are applied to obtain numerical solutions for a three-dimensional sector thrust bearing operating in the thermohydrodynamic regime. Pressure, velocity and temperature distributions are give, and the thermohydrodynamic solutions are compared with the results of classical isothermal theory.     

An analysis of the convection-diffusion problems using meshless and meshbased methods

The numerical solution of the convection-diffusion equation represents a very important issue in many numerical methods that need some artificial methods to obtain stable and accurate solutions. In this article, a meshless method based on the local Petrov-Galerkin method is applied to solve this equation. The essential boundary condition is enforced by the transformation method, and the MLS method is used for the interpolation schemes. The streamline upwind Petrov-Galerkin (SUPG) scheme is developed to employ on the present meshless method to overcome the influence of false diffusion. In order to validate the stability and accuracy of the present method, the model is used to solve two different cases and the results of the present method are compared with the results of the upwind scheme of the MLPG method and the high order upwind scheme (QUICK) of the finite volume method. The computational results show that fairly accurate solutions can be obtained for high Peclet number and the SUPG scheme can very well eliminate the influence of false diffusion.