Tracing post-limit-point paths with reduced basis technique

A reduced basis technique and a problem-adaptive computational algorithm are presented for predicting the post-limit-point paths of structures. In the proposed approach the structure is discretized by using displacement finite element models. The nodal displacement vector is expressed as a linear combination of a small number of vectors and a Rayleigh-Ritz technique is used to approximate the finite element equations by a small system of nonlinear algebraic equations.To circumvent the difficulties associated with the singularity of the stiffness matrix at limit points, a constraint equation, defining a generalized arc-length in the solution space, is added to the system of nonlinear algebraic equations and the Rayleigh-Ritz approximation functions (or basis vectors) are chosen to consist of a nonlinear solution of the discretized structure and its various order derivatives with respect to the generalized arc-length. The potential of the proposed approach and its advantages over the reduced basis-load control technique are outlined. The effectiveness of the proposed approach is demonstrated by means of numerical examples of structural problems with snap-through and snap-back phenomena.     

(2+1)维KD方程的解及分岔行为

通过行波变换,将非线性偏微分方程化为常微分方程,利用辅助常微分方程的解来构造偏微分方程的精确解,获得了(2+1)维Konopelchenko-Dubrovsky方程的孤波解和周期解.然后直接研究变换以后的常微分方程,揭示该方程控制的动力系统的鞍结分岔行为,画出了系统的分岔图.     

Bifurcation and post-buckling analysis of laminated composite plates via reduced basis technique

A reduced basis technique and a problem-adaptive computational algorithm are presented for the bifurcation and post-buckling analysis of laminated anisotropic plates. The computational algorithm can be conveniently divided into three distinct stages. The first stage is that of determining the bifurcation point. The plate is discretized by using displacement finite element (or finite difference) models. The special symmetries exhibited by the response of the anisotropic plate are used to reduce the size of the analysis region. The vector of unknown nodal parameters is expressed as a linear combination of a small number of basis vectors, and a Rayleigh-Ritz technique is used to approximate the finite element equations by a small system of algebraic equations. The reduced equations are used to determine the bifurcation point and the associated eigen mode of the panel.In the second stage of the bifurcation buckling mode is used to obtain a nonlinear solution in the vicinity of the bifurcation point and new (updated) sets of basis vectors and reduced equations are generated. In the third stage the reduced equations are used to trace the post-buckling paths.The effectiveness of the proposed technique for predicting the bifurcation and post-buckling behavior of plates is demonstrated by means of numerical examples for plates loaded by means of prescribed edge displacements.     

Radial basis function neural network for approximation andestimation of nonlinear stochastic dynamic systems

This paper presents a means to approximate the dynamic and static equations of stochastic nonlinear systems and to estimate state variables based on radial basis function neural network (RBFNN). After a nonparametric approximate model of the system is constructed from a priori experiments or simulations, a suboptimal filter is designed based on the upper bound error in approximating the original unknown plant with nonlinear state and output equations. The procedures for both training and state estimation are described along with discussions on approximation error. Nonlinear systems with linear output equations are considered as a special case of the general formulation. Finally, applications of the proposed RBFNN to the state estimation of highly nonlinear systems are presented to demonstrate the performance and effectiveness of the method.     

基于吴方法的非线性分类器设计

在模式识别领域工程实际中占大多数的是非线性问题,传统的分类器设计方法是将非线性问题转化为线性问题解决,但是这样不能很好的解决日益复杂化的非线性问题.文中提出的新方法为解决非线性问题提供一个新的思路和手段.该方法立足于直接处理非线性问题的吴文俊方法,进行非线性分类器的设计.经过选择分类函数的复杂度和训练点,建立非线性方程组,使用吴方法解出这个非线性方程组,得到分类函数.使用该方法进行异或问题和鸢尾花数据的实例研究,并与神经网络方法和支持向量机方法进行比较分析.该方法是使得结构风险和经验风险总体最小化的方法,并且过程直接、简洁、运算量小.该方法尽管有不完善的部分,但是不失为有潜力发展的新方法.     

高维拟线性抛物系统能控性的近期进展

本文综述高维拟线性抛物型方程、拟线性复Ginzburg-Landau方程以及只含一个控制变量的高维耦合拟线性抛物型方程组的能控性方面的一些近期的结果.通过使用不动点技术,采用主部具有C1系数的线性抛物型方程或方程组一些新的精细的Carleman估计.这一方法的要点是在古典解的框架下考虑能控性问题,并且当给定的数据具有一定的正则性时,线性抛物型方程或方程组在H¨older空间中来选取控制函数.利用类似的方法,还建立了拟线性抛物型方程不灵敏控制的存在性,其关键是将不灵敏问题转化为由拟线性抛物型方程和线性抛物型方程构成的耦合方程组在单个控制下一个非标准的能控性问题.     

一类非线性金融系统的数值研究

利用数值模拟的方法研究了一类非线性金融系统的动力学行为.建立了由生产、资金、股份、劳动力四个部分构成的一类非线性金融系统的动力学模型.首先运用四维微分方程来描述由利率、投资需求、价格指数和平均利润率构成的四个状态变量随时间的变化,然后将金融系统简化为四维自治微分方程组.通过对四维自治微分方程组进行数值模拟发现了非线性金融系统的动力学特性,从数值模拟获得的三维相图反映了金融系统的非线性特性.从数值模拟结果发现,在特定的条件下非线性金融系统存在周期运动和混沌运动.除此之外,还观察到参数的改变对四维自治金融系统的非线性特性有着显著的影响.     

一类不可逆系统的非线性控制及预期动力学方程的选取

改进了非线性控制的逆系统方法，使之适用于一类典型的不可逆系统的控制器设计；提出了一族预期动力学方程及其参数的选取规则，简化了非线性控制系统的设计，使之满足稳定性，动态性能，静态性能和鲁棒性等方面的要求。仿真结果证实了本文结论的正确性。     

Semi‐Global Robust Output Regulation for A Class of Singular Nonlinear Systems with Unknown Algebraic Equations

This paper considers semi‐global robust output regulation problem for a class of singular nonlinear systems whose algebraic equations are not precisely known. Since the algebraic equations are not known, the output regulation problem of singular nonlinear systems cannot be solved by directly reducing the singular nonlinear system into a normal nonlinear system. Based on internal model principle, we convert the robust output regulation problem of singular nonlinear systems into a robust stabilization problem of an augmented singular nonlinear system. The augmented singular nonlinear system is also with unknown algebraic equations. However, without transforming the singular nonlinear system into a normal nonlinear system, it is shown that the augmented singular nonlinear system can be semi‐globally stabilized by a high gain output feedback control law under some reasonable assumptions. Moreover, the semi‐global stabilization control law of the augmented singular nonlinear systems also solves the semi‐global robust output regulation problem of the original singular nonlinear system.     

Numerical solution of nonlinear ordinary differential equations using flatlet oblique multiwavelets

This paper is concerned with the construction of a biorthogonal multiwavelet basis in the unit interval to form a biorthogonal flatlet multiwavelet system. The system is then used to solve nonlinear ordinary differential equations. The biorthogonality and high vanishing moment properties of this system result in efficient and accurate solutions. Finally, numerical results for some test problems with known solutions are presented and the absolute errors are compared with the errors resulting from B-spline bases.     

Nonlinear viscoelastic membranes

The subject of nonlinear viscoelastic membranes is an important class of problems within nonlinear viscoelasticity that involves interesting and important applications, computational issues and applied mathematics.The viscoelastic materials discussed in this paper are described by nonlinear single integral constitutive equations. After presenting the general constitutive framework, two fundamental membrane problems are formulated: the inflation of a circular membrane and the extension and inflation of a circular tube. Both problems involve large axially symmetric deformations and lead to a system of nonlinear partial differential–Volterra integral equations. A numerical method of solution is presented that combines methods for solving nonlinear Volterra integral equations and nonlinear ordinary differential equations. Finally, some properties of the equations are discussed that are related to the possibility that there may exist a critical time when the solution develops multiple branches.     

Solving fractional pantograph delay differential equations via fractional-order Boubaker polynomials

In the current study, we introduce fractional-order Boubaker polynomials related to the Boubaker polynomials to achieve the numerical result for pantograph differential equations of fractional order in any arbitrary interval. The features of these polynomials are exploited to construct the new fractional integration and pantograph operational matrices. Then these matrices and least square approximation method are used to reorganize the problem to a nonlinear equations system which can be resolved by means of the Newton’s iterative method. The brief discussion about errors of the used estimations is deliberated and, finally, some examples are included to demonstrate the validity and applicability of our method.

     

大系统集结比较方程的新方法

用向量Ｖ函数求解大系统稳定性要利用向量比较方程。对指数稳定型系统，本文改变了过去利用柯西不等式集结常系数线性比较方程的做法，直接集结出非线性比较方程，并利用文‘１’的结果判断稳定性，使参数稳定区域扩大。     

Numerical simulation of a coupled nonlinear model for grain coarsening and coalescence

A kinetic nonlinear model of mass transfer, grain coarsening and coalescence with potential applications in sintering processes is studied. The model involves nonlinear ordinary differential equations that determine the transport of mass between grains. The rate of mass transfer is controlled by an Arrhenius factor leading to a nonlinear model of mass transfer and grain coarsening. The resulting dynamical system of coupled nonlinear differential equations with random initial conditions (i.e., initial grain mass configuration) is solved by means of the fourth order Runge–Kutta method. We conduct an analysis of the two-grain system and identify three dynamic regimes (diffusive, growth-decay and trapping). The same regimes are shown to persist in the multigrain system. We confirm the numerical performance of the Runge–Kutta method by means of a suitable convergence measure. The influence of the activation energy parameter on the dynamic regimes is investigated. It is shown that as the parameter grows the diffusive regime is progressively restricted to smaller values of the initial grain distribution. We introduce grain coalescence in the mass transfer equations, and we show that it accelerates the growth of the larger grains. Finally, we compare the dynamic evolution of the grain size distribution with the Ostwald ripening expression.     

利用粒子滤波原理求解非线性方程组

为了提高非线性方程组的求解精度,利用粒子滤波算法对非线性方程组问题进行求解计算。系统地介绍粒子滤波算法的基本原理及其优化算法的实现过程。将非线性方程组的求解问题转化为函数优化问题,并建立基于粒子滤波算法求解非线性方程组的优化模型。通过仿真实例验证所提方法的有效性。实验结果表明该方法能够准确、有效地解决非线性方程组的求解问题,这也为非线性方程组问题的研究提供一种有效的手段。     

Another step forward in searching for cubature formulae with a minimal number of knots for the square

The knots and weights of a cubature formula are determined by a system of nonlinear equations. The number of equations and unknowns can be reduced by imposing some structure on the formula. We are concerned with the construction of cubature formulae which are invariant under rotations. Using invariant theory, we obtain a smaller system of algebraically independent equations. This is used to construct cubature formulae for the square. One of the results is a 24-point formula of degree 11.     

On Nonlinear Yaw Model for Directional Responses of Log Hauling Trucks

A nonlinear yaw model for dynamics of log hauling trucks is developed based on Kane's equations. The relationship between the articulation yaw angles, between the vehicle units, and the sliding length of the drawbar has been investigated using a new procedure and introduced into the governing equations of motion. Analytical expressions of equations of motion for a tractor/jeep/pole-trailer system are obtained. The numerical results for directional responses of the system, under some maneuvers, are checked against measured responses in field tests. Also a comparison of the results for the nonlinear and linear yaw models is carried out and the modifying effects of the nonlinear terms are noted.     

电磁与机械载荷作用下导电梁式板的超谐波共振

研究了电磁与机械载荷共同作用下梁式薄板的非线性超谐波共振问题．在给出薄板的电磁弹性运动基本方程及电磁力表达式的基础上,推得了横向稳恒磁场和机械载荷共同作用下梁式薄板的振动方程;应用伽辽金积分法,进一步导出了相应的非线性振动控制微分方程．采用多尺度法进行求解,得到了稳态运动下的幅频响应方程．最后,通过算例,给出了相应的幅频响应曲线图和时间历程图,分析了板厚、磁场及激励幅值对系统振动的影响．