非完整约束动力系统的离散积分方法

采用了微分-代数方程求解的雷同策略,只在离散区段内满足无约束的积分,而在离散节点处则用约束冲量(Lagrange乘子向量)让轨道发生转折,给出了非完整约束动力系统的离散积分格式.数值例题的效果表明,本文给出的积分算法严格满足节点的非完整约束,具有良好的长时间积分性能.     

非线性动力学系统最优控制问题的保辛求解方法

基于对偶变量变分原理提出了求解非线性动力学系统最优控制问题的一种保辛数值方法.以时间区段一端状态和另一端协态作为混合独立变量,在时间区段内采用拉格朗日插值近似状态变量与协态变量,然后利用对偶变量变分原理并将非线性最优控制问题转化为非线性方程组的求解,最终得到求解非线性动力学系统最优控制问题的保辛数值方法.数值实验验证了本文算法在求解精度与求解效率上的有效性.     

具有不等式路径约束的微分代数方程系统的动态优化

针对具有不等式路径约束的微分代数方程（Differential-algebraic equations，DAE）系统的动态优化问题，通常将DAE中的等式路径约束进行微分处理，或者将其转化为点约束或不等式约束进行求解.前者需要考虑初值条件的相容性或增加约束，在变量间耦合度较高的情况下这种转化求解方法是不可行的；后者将等式约束转化为其他类型的约束会增加约束条件，增加了求解难度.为了克服该缺点，本文提出了结合后向差分法对DAE直接处理来求解上述动态优化问题的方法.首先利用控制向量参数化方法将无限维的最优控制问题转化为有限维的最优控制问题，再利用分点离散法用有限个内点约束去代替原不等式路径约束，最后用序列二次规划（Sequential quadratic programming，SQP）法使得在有限步数的迭代下，得到满足用户指定的路径约束违反容忍度下的KKT（Karush Kuhn Tucker）最优点.理论上证明了该算法在有限步内收敛.最后将所提出的方法应用在具有不等式路径约束的微分代数方程系统中进行仿真，结果验证了该方法的有效性.     

空间太阳能电站太阳能接收器二维展开过程的保结构分析

针对传统数值方法求解微分-代数方程过程中经常遇到的违约问题,本文以空间太阳能电站太阳能接收器的简化二维模型为例,采用辛算法模拟了简化模型的展开过程,研究了辛算法在求解过程中约束违约问题.首先,基于Hamilton变分原理,将描述简化二维模型展开过程的Euler-Lagrange方程导入Hamilton体系,建立其Hamilton正则方程;随后,采用s级PRK离散方法离散正则方程,得到其辛格式;最后,采用辛PRK格式模拟太阳能接收器的二维展开过程.模拟结果显示:本文构造的辛PRK格式能够很好地满足系统的位移约束.     

PDE与DAE耦合系统求解方法

针对目前Modelica语言只能解决由微分代数方程(DAE)描述的问题,而不能解决由偏微分方程(PDE)表达的问题,提出一种求解PDE与DAE耦合系统的方法.首先采用径向基函数构造近似函数,将未知量场函数的时空变量分开;然后运用配点法对空间变量进行离散,从而将PDE问题转化为DAE问题;最后采用成熟的DAE求解器进行求解,得到场函数在任意时空点的函数值.实例结果表明,该方法在不改变Modelica语法的前提下,能较好地实现PDE与DAE耦合系统的一致求解,且求解精度高、稳定性好、边界条件处理简单.     

多体系统动力学Lie 群微分⁃代数方程约束稳定方法

针对多体系统动力学微分-代数方程求解问题,研究基于Lie群表达的约束稳定方法.首先引入新的Lagrange乘子,结合位移约束、速度级约束和加速度级约束方程,构造了新的Lie群微分-代数方程.然后使用向后差商隐式方法和CG(Crouch-Grossman)方法,对微分–代数方程进行离散求解,得到精确度较高的动力学仿真结果.该方法在精确保持各级约束方程的同时,保持旋转矩阵的正交性,并且使系统总能量误差较小.     

LQ控制区段混合能矩阵的微分方程及其应用

本文根据计算结构力学与线性二次控制的对应关系,提出了连续时间有限区段的混合能 分块子矩阵Q2,G2及Φ2.推导出适用于LQ控制非定常课题的二区段连接的凝聚消元公式及 这些子矩阵的微分方程,可用级数展开求解这些方程.当△t很小时,这些分块子矩阵的高次 近似可以大大加速里卡提代数方程算法的收敛性.     

萤火虫算法求解多体系统动力学微分-代数方程

针对多体系统动力学微分-代数方程求解问题,研究基于萤火虫算法的求解方法.首先将广义坐标和广义速度进行Lagrange插值,结合Gauss数值积分方法,将微分-代数方程求解问题转化成求解最优化问题.然后用萤火虫算法对问题进行优化求解.最后,通过对平面双连杆机械臂的多体系统仿真实验,验证了萤火虫算法在求解动力学方程中既保持了约束又较好地保证了能量精度.结果表明智能优化算法在求解多体动力学问题上具有较好的应用前景.     

分析结构力学与有限元

分析力学历来是在动力学范围内论述的,结构力学与最优控制模拟关系的共同基础就是分析力学．这表明在结构力学与最优控制理论的架构内也应有分析力学的整套理论．本文就结构力学讲述分析力学,称分析结构力学．保守体系可用Hamilton体系的方法描述,其特点是保辛．保辛给出保守体系结构最重要的特性．有限元法是从结构力学发展的,有限元的单元刚度阵应保持对称性,其实这就是保辛．根据区段单元变形能只与其两端位移有关,就可通过数学分析得到Lagrange括号与Poisson括号,展示了其辛对偶体系、正则方程、正则变换等的内容．     

基于响应面方法的波导介质层PBG结构滤波器的优化设计

在辛体系下利用精细积分对矩形波导纵向排列介质层PGB结构进行分析的基础之上,用响应面方法对滤波器进行了优化设计.采用棱单元对波导的横截面进行离散,然后导向哈密顿体系,运用基于黎卡提微分方程的精细积分求出一段介质层和一段空气层的出口刚度阵,再将两区段合并得到一个周期段的出口刚度阵,从而可对所有周期进行合并以对问题求解.在分析的基础上建立了滤波器的优化设计模型,利用响应面方法将目标函数和约束函数近似显式化,运用二次规划法对优化模型进行求解,得到了滤波性能最优的设计参数.算例表明本文方法是可行有效的.     

基于祖冲之类方法的多体动力学方程保能量保约束积分

针对一类多体动力学问题导出的微分-代数方程,提出一种保能量、保约束的算法.该算法基于祖冲之类方法和欧拉中点保辛差分,利用祖冲之类方法保证在时间格点上精确满足约束方程,避免约束违约问题;并进一步证明该算法在时间格点上可以精确保能量.数值算例进一步验证该算法的可靠性.     

Null Space Integration Method for Constrained Multibody Systems with No Constraint Violation

A method for integrating equations of motion of constrained multibodysystems with no constraint violation is presented. A mathematical model,shaped as a differential-algebraic system of index 1, is transformedinto a system of ordinary differential equations using the null-spaceprojection method. Equations of motion are set in a non-minimal form.During integration, violations of constraints are corrected by solvingconstraint equations at the position and velocity level, utilising themetric of the system's configuration space, and projective criterion to thecoordinate partitioning method. The method is applied to dynamicsimulation of 3D constrained biomechanical system. The simulation resultsare evaluated by comparing them to the values of characteristicparameters obtained by kinematic analysis of analyzed motion based onmeasured kinematic data.     

Singularity-free time integration of rotational quaternions using non-redundant ordinary differential equations

A novel ODE time stepping scheme for solving rotational kinematics in terms of unit quaternions is presented in the paper. This scheme inherently respects the unit-length condition without including it explicitly as a constraint equation, as it is common practice. In the standard algorithms, the unit-length condition is included as an additional equation leading to kinematical equations in the form of a system of differential-algebraic equations (DAEs). On the contrary, the proposed method is based on numerical integration of the kinematic relations in terms of the instantaneous rotation vector that form a system of ordinary differential equations (ODEs) on the Lie algebra \(\mathit{so}(3)\) of the rotation group \(\mathit{SO}(3)\). This rotation vector defines an incremental rotation (and thus the associated incremental unit quaternion), and the rotation update is determined by the exponential mapping on the quaternion group. Since the kinematic ODE on \(\mathit{so}(3)\) can be solved by using any standard (possibly higher-order) ODE integration scheme, the proposed method yields a non-redundant integration algorithm for the rotational kinematics in terms of unit quaternions, avoiding integration of DAE equations. Besides being ‘more elegant’—in the opinion of the authors—this integration procedure also exhibits numerical advantages in terms of better accuracy when longer integration steps are applied during simulation. As presented in the paper, the numerical integration of three non-linear ODEs in terms of the rotation vector as canonical coordinates achieves a higher accuracy compared to integrating the four (linear in ODE part) standard-quaternion DAE system. In summary, this paper solves the long-standing problem of the necessity of imposing the unit-length constraint equation during integration of quaternions, i.e. the need to deal with DAE’s in the context of such kinematical model, which has been a major drawback of using quaternions, and a numerical scheme is presented that also allows for longer integration steps during kinematic reconstruction of large three-dimensional rotations.     

Solvability and asymptotic behavior of generalized Riccatiequations arising in indefinite stochastic LQ controls

The optimal control problem in a finite time horizon with an indefinite quadratic cost function for a linear system subject to multiplicative noise on both the state and control can be solved via a constrained matrix differential Riccati equation. In this paper, we provide general necessary and sufficient conditions for the solvability of this generalized differential Riccati equation. Furthermore, its asymptotic behavior is investigated along with its connection to the generalized algebraic Riccati equation associated with the linear quadratic control problem in finite time horizon. Examples are presented to illustrate the results established     

基于祖冲之类方法的多体动力学方程保能量保约束积分

针对一类多体动力学问题导出的微分 代数方程,提出一种保能量、保约束的算法.该算法基于祖冲之类方法和欧拉中点保辛差分,利用祖冲之类方法保证在时间格点上精确满足约束方程,避免约束违约问题;并进一步证明该算法在时间格点上可以精确保能量.数值算例进一步验证该算法的可靠性.     

H∞滤波问题数值求解的精细积分算法

有限时间H∞滤波的Riccati方程和滤波方程分别为非线性矩阵微分方程和线性变系 数微分方程,而且Riccati微分方程解的存在性还依赖于参数 γ-2,因此求这些方程的数值解一 般比较困难.按照结构力学与最优控制的模拟关系,Riccati方程解存在的临界参数 γ-2cr对应于 广义Rayleigh商的一阶本征值.因此可以用精细积分法结合扩展的Wittrick-Williams(W-W) 算法计算 γ-2cr .并求解Ricclati方程,滤波微分方程的解也可以由精细积分法计算.     

Optimum design of beams surrounded by a Winkler medium

In the present work the analytical approach is used to study the optimization of a beam loaded laterally at the top, not constrained at its ends and surrounded by a Winkler medium. In the literature there are a few examples of studies regarding the optimization of structures constrained both by local constraints and by contact with a medium; in this case the Winkler medium around the beam is the only constraint. The nonlinear system of differential equations governing the problem is solved with an adequate iterative finite difference algorithm. The optimum design of such a beam distributes the great part of the mass near the top; it is seen that the length of beams that are too long is also optimized limiting, in this way, the driving. In the particular case where the moment of inertia is proportional to the area of the cross-section, an analytical solution is also obtained.     

时间-空间混和有限元

分析动力学与分析结构力学在数学理论上是一致的.振动与结构力学问题,其实只是一个符号之差.分析力学方法对两方面可通用.双曲型偏微分方程与椭圆型偏微分方程也是差一个符号.虽然性质不同,但分析上有共同之处.本文提出在有限元分析方面,不用对时间、空间分别离散而是组成混和的时空混和有限元网格.数值结果表明,时空混和有限元是有前途的.     

NICE h : a higher-order explicit numerical scheme for integration of constitutive models in plasticity

The article introduces, as a result of further development of the first-order scheme NICE, a simple and efficient higher-order explicit numerical scheme for the integration of a system of ordinary differential equations which is constrained by an algebraic condition (DAE). The scheme is based on the truncated Taylor expansion of the constraint equation with order h of the scheme being determined by the highest exponent in the truncated Taylor series. The integration scheme thus conceived will be named NICE ^h , considering both principal premises of its construction. In conjunction with a direct solution technique used to solve the boundary value problem, the NICE ^h scheme is very convenient for integrating constitutive models in plasticity. The plasticity models are defined mostly by a system of algebraic and differential equations in which the yield criterion represents the constraint condition. To study the properties of the new integration scheme, which, like the forward-Euler scheme, is characterised by its implementation simplicity due to the explicitness of its formulations, a damage constitutive model (Gurson–Tvergaard–Needleman model) is considered. The general opinion that the implicit backward-Euler scheme is much more accurate than the thus-far known explicit schemes is challenged by the introduction of the NICE ^h scheme. The accuracy of the higher-order explicit scheme in the studied cases is significantly higher than the accuracy of the classical backward-Euler scheme, if we compare them under the condition of a similar CPU time consumption.