Road to chaos for a soft spring system under weak periodic disturbance

The Mel’nikov-Holmes method is used to discuss the road to chaos for an elastic system with soft spring constant and under weak periodic disturbance. It is found that according to how the disturbance is applied, forced vibration or parametric excitation, the response may pass to chaos through different sequences of subharmonic bifurcations, and that the orders in the sequences may be different for different disturbance frequencies.     

Optimal control of parabolic variational inequalities with state constraint

1　IntroductionandProblemWeshallstudytheoptimalcontrolproblemsgovernedbynonlinearparabolicvariationalinequalitiesoftheformy′+Ay +β(y) ∈Bu+f(a.e .(x,t)∈Q =Ω× [0 ,t]) ,y(0 ) =y0 , ( )withthestateconstraintF(y) S ,andthecostfunctionalI(y,u) .Whereβisadiscontinuous,nonlinearandnonmonotonemulti_valuedmapping .Theoptimalcontrolproblemsofthedifferentialsystemshavebeenstudiedforalongtime.Manyscholars,suchasJ.L .Lions ,V .Barbu ,D .Tiba,andF .Mignotetal.,haveresearchedtheoptimalcontrolpr…     

瞬态热传导问题的加权最小二乘无网格法

加权最小二乘无网格法是一种基于节点信息的纯无网格法,该方法使用最小二乘法建立系统的变分原理,通过移动最小二乘法构造近似函数,控制方程在节点处的残量使用最小二乘法予以消除,边界条件通过罚函数法引入。本文推导了瞬态热传导问题的加权最小二乘无网格计算格式,编制了相应的计算程序,算例结果表明,该方法具有精度高、前后处理简单的优点,是一种高效的的新型无网格法。     

New smooth gap function for box constrained variational inequalities

A new smooth gap function for the box constrained variational inequality problem(VIP) is proposed based on an integral global optimality condition.The smooth gap function is simple and has some good differentiable properties.The box constrained VIP can be reformulated as a differentiable optimization problem by the proposed smooth gap function.The conditions,under which any stationary point of the optimization problem is the solution to the box constrained VIP,are discussed.A simple frictional contact problem is analyzed to show the applications of the smooth gap function.Finally,the numerical experiments confirm the good theoretical properties of the method.     

Parametric variational solution of linear-quadratic optimal control problems with control inequality constraints

A parametric variational principle and the corresponding numerical algo- rithm are proposed to solve a linear-quadratic （LQ） optimal control problem with control inequality constraints. Based on the parametric variational principle, this control prob- lem is transformed into a set of Hamiltonian canonical equations coupled with the linear complementarity equations, which are solved by a linear complementarity solver in the discrete-time domain. The costate variable information is also evaluated by the proposed method. The parametric variational algorithm proposed in this paper is suitable for both time-invariant and time-varying systems. Two numerical examples are used to test the validity of the proposed method. The proposed algorithm is used to astrodynamics to solve a practical optimal control problem for rendezvousing spacecrafts with a finite low thrust. The numerical simulations show that the parametric variational algorithm is ef- fective for LQ optimal control problems with control inequality constraints.     

Projected Runge-Kutta methods for constrained Hamiltonian systems

Projected Runge-Kutta (R-K) methods for constrained Hamiltonian systems are proposed. Dynamic equations of the systems, which are index-3 differential-algebraic equations (DAEs) in the Heisenberg form, are established under the framework of Lagrangian multipliers. R-K methods combined with the technique of projections are then used to solve the DAEs. The basic idea of projections is to eliminate the constraint violations at the position, velocity, and acceleration levels, and to preserve the total energy of constrained Hamiltonian systems by correcting variables of the position, velocity, acceleration, and energy. Numerical results confirm the validity and show the high precision of the proposed method in preserving three levels of constraints and total energy compared with results reported in the literature.     

Solution of certain variational problems of thermal resilience for thin shells considering the selection of optimum conditions for localized heat treatment

One of the possible ways of stating and solving the selection problem for optimum temperature fields for localized axisymmetric heating of shells is investigated. The minimum of shell elastic energy is taken as the optimization criterion. An infinite cylindrical shell was considered in a similar formulation in [1], The corresponding variational problem is formulated for the functional of elastic energy with additional limitations imposed on the function of twist angle at specified shell sections. The variational problem is reduced to an isoperimetric by the use of singular functionals of the -function kind. The related Euler equation is obtained, and this together with the problem resolvent equation constitute a complete set of equations for determining the extremum temperature field with related stress-strain state of the shell. Cylindrical, conical, and spherical shells are considered separately. A numerical analysis is made for the simplest conditions of localized heating of cylindrical and conical shells.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 9, No. 4, pp. 47–54, July–August, 1968.     

A duality recipe for non-convex variational problems

The aim of this paper is to present a general convexification recipe that can be useful for studying non-convex variational problems. In particular, this allows us to treat such problems by using a powerful primal–dual scheme. Possible further developments and open issues are given.     

The application of the variational principle in the constrained control system

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基于对偶变量变分原理的完整约束系统保辛算法

基于对偶变量变分原理,选择积分区间两端位移为独立变量,构造了求解完整约束哈密顿动力系统的高阶保辛算法。首先,利用拉格朗日多项式对作用量中的位移、动量及拉格朗日乘子进行近似;然后,对作用量中不包含约束的积分项采用Gauss积分近似,对作用量中包含约束的积分项采用Lobatto积分近似,从而得到近似作用量;最后,在此近似作用量的基础上,利用对偶变量变分原理,将求解完整约束哈密顿动力系统问题转化为一组非线性方程组的求解。算法具有保辛性和高阶收敛性,能够在位移的插值点处高精度地满足完整约束。算法的收敛阶数及数值性质通过数值算例验证。     

TIME PRECISE INTEGRATION METHOD FOR CONSTRAINED NONLINEAR CONTROL SYSTEM

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Region-wise variational principles and generalized variational principles on large strain for consolidation theory

Introduction Thedeformationofsaturatedsoftclayisoneofimportantquestionsingeotechnical engineering.Thenotablecharacteristicisthatthedegreeofitsstrainisgenerallylargerthan10percent.Oneofthecauses[1],Whichleadtoaquantitativedifferencebetweenthe numericalsimu…     

约束多体系统独立广义坐标的数值选取

多体系统的完整约束定义了一个嵌入到欧氏空间的微分流型,多体系统独立的广义坐标选取问题等价于该流型的坐标选取问题。据此,本文提出了一种新的独立广义坐标数值选取理论,可将多体系统的微分-代数混合型运动方程转化为较易求解的纯微分方程,并且不以求解非线性方程组作为必须的手段。     

加权最小二乘法求解Rayleigh阻尼系数的讨论

合理确定Rayleigh阻尼矩阵比例系数对于准确计算场地地震响应有重要影响。本文提出以H2iH′2iA2i为权重系数,采用加权最小二乘法求解Rayleigh阻尼矩阵比例系数,以提高土层地震反应时程分析的计算精度。以上海某处地质剖面建立深覆盖土层有限元模型,计算其在28条各具代表性的基岩场地地震波激励下的动力反应,与其他研究成果对比分析,验证本文方法的精度及适用性。     

求解变分型积分方程的一种新型数值方法——有限变分法

将作者提出的多虚拟裂纹扩展法(MVCE法)拓展为求解变分型积分方程问题的一种新型数值方法——有限变分法(FVM)。它的基本思想是,给定有限个(N个)局部变分模式,将所求解的未知量用适当的方法离散化,针对这N个局部变分模式列出N个方程,求解N个未知系数,从而求得未知量。单一未知变量FVM的最终方程组的系数矩阵通常是一个对称的窄带矩阵,对角元是大数,有很好的数值计算性能。用FVM求解了三维I型裂纹前缘的应力强度因子(SIF)分布。利用基于FVM的通用权函数法计算程序,可以高精度和高效率地求解表面力、体积力和温度载荷共同作用情况下三维裂纹前缘SIF的分布及其时间历程。FVM可以被推广到更广泛的领域,是一个求解变分型积分方程问题的普遍适用的新型数值方法。     

Classification of parametrically constrained bifurcations

If the constraint boundary relates to a bifurcation parameter, a bifurcation is said to be parametrically constrained. Relying upon some substitution, a parametrically constrained bifurcation is transformed to an unconstrained bifurcation about new variables. A general form of transition sets of the parametrically constrained bifurcation is derived. The result indicates that only the constrained bifurcation set is influenced by parametric constraints, while other transition sets are the same as those of the corresponding nonparametrically constrained bifurcation. Taking parametrically constrained pitchfork bifurcation problems as examples, effects of parametric constraints on bifurcation classification are discussed.     

Hydrodynamic coefficients for vertical plane wave motion problems

In this paper a numerical method for calculating the hydro-dynamic coefficients for vertical plane wave motion problems in deep water is described.This procedure is developed by using the wave source method based on Green‘s theorem.The applications of the method to the cases of semi-circular and rectangular section bodies subjected to linear waves are pre-sented here,and the computed results are compared with the earlier experimental data of Vugts.     

Well-posedness of the fundamental boundary value problems for constrained anisotropic elastic materials

We consider the equations of linear homogeneous anisotropic elasticity admitting the possibility that the material is internally constrained, and formulate a simple necessary and sufficient condition for the fundamental boundary value problems to be well-posed. For materials fulfilling the condition, we establish continuous dependence of the displacement and stress on the elastic moduli and ellipticity of the elasticity system. As an application we determine the orthotropic materials for which the fundamental problems are well-posed in terms of their Young's moduli, shear moduli, and Poisson ratios. Finally, we derive a reformulation of the elasticity system that is valid for both constrained and unconstrained materials and involves only one scalar unknown in addition to the displacements. For a two-dimensional constrained material a further reduction to a single scalar equation is outlined.This paper is dedicated to Professor Joachim Nitsche on the occasion of his sixtieth birthday