微分求积法的特性及其改进

微分求积法已在科学和工程计算中得到了广泛应用。然而,有关时域微分求积法的数值稳定性、计算精度即阶数等基本特性,仍缺乏系统性的分析结论。依据微分求积法的基本原理,推导证明了微分求积法的权系数矩阵满足V-变换这一重要特性;利用微分求积法和隐式Runge-Kutta法的等值性,证明了时域微分求积法是A-稳定、s级s阶的数值方法。在此基础上,为进一步提高传统微分求积法的计算精度,利用待定系数法和Padé逼近,推导出了一类新的s级2s阶的微分求积法。数值计算对比结果验证了所提出的新微分求积法比传统的微分求积法具有更高的计算精度。     

基于微分求积法的逐步积分法与常用时间积分法的比较

微分求积法具有数学概念简单、精度高和计算时间少等优点,是一种不断受到重视的数值方法.目前微分求积法在方法本身的研究已经相当充分和成熟,而应用方面的研究则大多集中在边值问题的求解,本文的研究集中在采用微分求积法求解动力学初值问题方面.先介绍了一种新近提出的逐步积分方法,该方法基于一种特殊的节点分布的微分求积法.然后通过理论分析与几种常用的时间积分方法进行了稳定性、精确性和计算量的比较.最后计算了一个双质点系在强迫力下的瞬态响应.比较结果表明新近提出的逐步积分方法具有无条件稳定、高精度和计算量少的优点.     

集中载荷作用下梯度复合材料梁的小波-微分求积法分析

将梯度复合材料梁作为平面应力问题处理,采用小波和微分求积混合法,对集中荷载作用下结构的响应进行了分析.考虑材料特性参数沿高度方向呈梯度分布,在该方向上采用广义微分求积法进行离散;鉴于广义微分求积法求解集中荷载问题精度不高的缺点,在梁的长度方向上引入对突变信号敏感的小波插值函数.数值计算表明,小波-微分求积混合法不仅保留了广义微分求积法高效的优点,而且能够很好地模拟结构局部化特征.     

混合微分求积法在功能梯度材料平板柱形弯曲问题中的应用

利用混合微分求积法,对任意荷载作用下不同材料梯度分布的功能梯度材料平板柱形弯曲问题进行了分析。针对广义微分求积法求解集中荷载问题精度不高的缺点,本文利用小波微分求积法进行了改进。由于小波对突变信号具有良好的自适应描述能力,因此在平板宽度方向上,利用小波微分求积法可以有效地处理集中荷载;而在材料梯度变化的板厚方向上,则利用广义微分求积法计算量小且精度高的特点进行离散计算。计算表明,混合微分求积法不仅保留了广义微分求积法高效的特点,而且能有效地求解任意荷载作用的问题。通过算例,分析了在机械荷载作用下,材料不同梯度形式、平板上下表面材料性质差异对功能梯度平板结构响应的影响。     

压电-弹性层合梁自由振动分析的新方法

将状态空间法和微分求积法相结合，分析了压电-弹性层合梁的自由振动. 通过微分求积把状态方程在每一个节点处离散，进而求得解答. 选用不同的节点数目， 分析了方法的收敛性. 计算结果与相关文献的结果能较好地符合. 该方法 对于分析压电-弹性层合梁的工程振动问题非常方便.     

基于边界值方法的微分动力系统数值计算方法

对高维非线性初值问题,微分求积法在每一步的积分过程中需要求解一个更高维的非线性方程组,因而计算量巨大。基于微分求积法与边界值方法两者之间的关系,可以将广义向后差分方法和扩展的隐式梯形积分方法看作是经典微分求积法的稀疏表达形式。将广义向后差分方法以及扩展的隐式梯形积分方法这两类边界值方法应用于微分动力系统的数值计算,提出了一类新的数值计算方法。理论分析及算例结果表明,对高维非线性微分初值问题的数值计算,本文方法相对于经典的微分求积法具有更高的计算效率。     

变截面门式刚架结构分析的微分求积单元法

本文采用一种精确、简便的数值计算方法——微分求积单元法(DQEM)对变截面门式刚架结构进行了力学分析。首先建立了一般荷载作用下变截面构件的平衡微分方程，并采用微分求积法进行离散，进而得出了较为精确的分析变截面构件的单元力学模型。该模型的刚度方程不仅反映了单元的刚度性质，而且反映了单元的实际荷载作用，可较为精确地分析变截面门式刚架结构在分布载荷作用下的受力性能。通过与有限元法计算结果的比较，表明了微分求积单元法在变截面刚架的力学分析中的正确性和优越性。微分求积单元法可用于任意形状的刚架结构的静力分析。     

新型带虚点的径向基函数微分求积法及其在薄板弯曲中的应用

本文提出了新型带虚点的径向基函数微分求积法,并将其应用于模拟薄板弯曲问题。带虚点的径向基函数微分求积法是一种基于传统径向基函数微分求积法的新型无网格方法,传统方法只将中心点放在计算域内,而该方法扩展了中心点的区域,使其既位于计算域内又位于计算域外,在不增加计算量和存储量的基础上,显著提高计算精度。本文首次尝试将此方法应用于求解薄板弯曲问题,并与解析解和传统方法进行对比,验证了此方法的优越性     

微分求积单元法在结构工程中的应用

微分求积法（Differential Quadrature Method）是求鳃偏微分方程和积分-微分方程的一种数值方法，该法具有计算简便、精度较高和易于实现等优点。微分求积单元法（Differential Quadrature Element Method）是在微分求积法的基础上结合区域分割和集成规则而形成的一种新的数值计算方法，能通过自适应地选取微分求积网点数目正确模拟构件的刚度和荷载性质，其精度可通过细分单元或增加离散点数目加以提高。微分求积单元法是一种可供选择的、性能优越的数值计算方法。本文将详细论述这一数值方法的基本原理，并通过数值算例说明该方法的应用过程及其优越性，为这一方法在结构工程中的推广应用提供参考。     

一种结构动力响应分析的快速高阶算法

为了提高基于高阶格式的结构动力响应微分求积分析方法的计算效率,发展了一种求解动力方程的快速算法．利用微分求积原理将结构动力方程转化为标准Sylvester方程的形式,通过对系数矩阵进行矩阵分解,进而将动力响应Sylvester方程化为一系列标准线性方程组,采用相关成熟算法求解这些线性方程组后即可获得结构动力时程响应的全部解答．结构动力响应微分求积分析方法为高阶数值方法,一步计算可以获得多个时点处的动力响应．基于本文快速算法,不必直接对矩阵方程进行求解．数值算例表明,本文快速算法能够准确地计算出结构动力响应,具有数值精度高、收敛性好的优点．     

轴向均布载荷下压杆稳定问题的DQ解

叙述了微分求积法(differential quadrature method)的一般方法，研究用微分求积法求解在均布轴向载荷下细长杆的稳定问题．通过Newton-Raphson法求解非线性方程组，以及对问题进行线性假设后求解广义特征值方程，得到了精度很高的后屈曲挠度数值和临界载荷数值．与解析解和其他近似解相比，微分求积法具有较高的精度和简便性．     

Mixed finite element and differential quadrature method for free and forced vibration and buckling analysis of rectangular plates

This paper presents a combined application of the finite element method (FEM) and the differential quadrature method (DQM) to vibration and buckling problems of rectangular plates. The proposed scheme combines the geometry flexibility of the FEM and the high accuracy and efficiency of the DQM. The accuracy of the present method is demonstrated by comparing the obtained results with those available in the literature. It is shown that highly accurate results can be obtained by using a small number of finite elements and DQM sample points. The proposed method is suitable for the problems considered due to its simplicity and potential for further development.     

Nonlinear dynamics of a fluid-conveying curved pipe subjected to motion-limiting constraints and a harmonic excitation

This paper investigates the dynamical behaviour of a fluid-conveying curved pipe subjected to motion-limiting constraints and a harmonic excitation. Based on a Newtonian method, the in-plane equation of motion of this curved pipe is derived. Then a set of discrete equations in spatial space obtained by the differential quadrature method (DQM) is solved numerically. Emphasis is placed on the possible dynamical behaviour of the curved pipe conveying fluid. The numerical results show that the pipe without motion-limiting constraints but with a harmonic force behaves as an ordinary linear system. If, however, the pipe is subjected to cubic motion-limiting constraints, nonlinear dynamic phenomena of the system will occur. Calculations of bifurcation diagrams, phase-plane portraits, time responses, power spectrum diagrams, and Poincaré maps of the oscillations clearly demonstrate the existence of chaotic and quasiperiodic motions. Moreover, it is shown that the route to chaos is via a sequence of period-doubling bifurcations.     

具有初始构型的MEMS驱动器的跳跃和吸合现象研究

在曲梁变形后以弧长为参数的自然坐标系中,利用曲梁大变形分析理论,建立了具有任意初始构型的微电驱动器大变形电动力学分析的数学模型,并采用微分求积法(DQM)进行空间离散,得到了一组具有强非线性的微分-代数系统方程,运用Petzold-Gear BDF方法进行时间域内的求解。研究了MEMS驱动器在电场力作用下的瞬态动力学特性,包括跳跃(snap-through)和吸合(pull-in)现象,并与已有实验结果进行了比较。     

理想直杆中轴向应力波的传播和反射导致的分叉问题

对阶跃载荷作用下弹性理想完善直杆中应力波的传播及其反射引起民的分叉问题进行了理论分析和数值计算，讨论了此类动力屈曲问题中横向惯性效应的影响；给出了一种进行后分叉分析的近似方法，得到了一些结论，在各类结构由于应力波的传播及反射引起的分叉问题具有指导意义。     

黏弹性圆柱形壳动力学高余维分岔、普适开折问题

讨论两端受到谐波激励的黏弹性圆柱形壳的非线性动力学行为,利用奇异性理论,研究了分岔方程的普适开折问题,严格证明了它是一个高余维分岔问题。余维数为5（含有一个模参数）,给出了它的所有可能的普适开折形式。在分岔参数满足某些条件时得到该分岔问题的转迁集及分岔图,展示了一些新的动力学行为,改进和完善了奇异性分析方法。     

Periodic excitation of a buckled beam using a higher order semianalytic approach

This paper considers the steady-state behavior of a transversally excited, buckled pinned–pinned beam, which is free to move axially on one side. This research focuses on higher order single-mode as well as multimode Galerkin discretizations of the beam’s partial differential equation. The convergence of the static load-paths and eigenfrequencies (of the linearized system) of the various higher-order Taylor approximations is investigated. In the steady-state analyses of the semianalytic models, amplitude–frequency plots are presented based on 7th order approximations for the strains. These plots are obtained by solving two-point boundary value problems and by applying a path-following technique. Local stability and bifurcation analysis is carried out using Floquet theory. Dynamically interesting areas (bifurcation points, routes to chaos, snapthrough regions) are analyzed using phase space plots and Poincaré plots. In addition, parameter variation studies are carried out. The accuracy of some semianalytic results is verified by Finite Element analyses. It is shown that the described semianalytic higher order approach is very useful for fast and accurate evaluation of the nonlinear dynamics of the buckled beam system.     

Three-dimensional magneto-thermo-elastic analysis of functionally graded cylindrical shell

The present paper presents the three-dimensional magneto-thermo-elastic analysis of the functionally graded cylindrical shell immersed in applied thermal and magnetic fields under non-uniform internal pressure. The inhomogeneity of the shell is assumed to vary along the radial direction according to a power law function, whereas Poisson's ratio is supposed to be constant through the thickness. The existing equations in terms of the displacement components, temperature, and magnetic parameters are derived, and then the effective differential quadrature method(DQM) is used to acquire the analytical solution. Based on the DQM, the governing heat differential equations and edge boundary conditions are transformed into algebraic equations, and discretized in the series form. The effects of the gradient index and rapid temperature on the displacement,stress components, temperature, and induced magnetic field are graphically illustrated.The fast convergence of the method is demonstrated and compared with the results obtained by the finite element method(FEM).     

Static and dynamic behaviors of belt-drive dynamic systems with a one-way clutch

This paper focuses on the nontrivial equilibrium and the steady-state periodic response of belt-drive system with a one-way clutch and belt flexural rigidity. A nonlinear piecewise discrete–continuous dynamic model is established by modeling the motions of the translating belt spans as transverse vibrations of axially moving viscoelastic beams. The rotations of the pulleys and the accessory are also considered. Furthermore, the transverse vibrations and the rotation motions are coupled by nonlinear dynamic tension. The nontrivial equilibriums of the belt-drive system are obtained by an iterative scheme via the differential and integral quadrature methods (DQM and IQM). Moreover, the periodic fluctuation of the driving pulley is modeled as the excitation of the belt-drive system. The steady-state periodic responses of the dynamic system are, respectively, studied via the high-order Galerkin truncation as well as the DQM and IQM. The time histories of the system are numerically calculated based on the 4th Runge–Kutta time discretization method. Furthermore, the frequency–response curves are presented from the numerical solutions. Based on the steady-state periodic response, the resonance areas of the dynamic system are obtained by using the frequency sweep. Moreover, the influences of the truncation terms of the Galerkin method, such as 6-term, 8-term, 10-term, 12-term, and 16-term, are investigated by comparing with the DQM and IQM. Numerical results demonstrate that the one-way clutch reduces the resonance responses of the belt-drive system via the torque-transmitting directional function. Furthermore, the comparisons in numerical examples show that the investigation on steady-state responses of the belt-drive system with a one-way clutch and belt flexural rigidity needs 16-term truncation