多体系统动力学的常用积分器算法

本文将以单步法中的广义 α族积分器和多步法中的BDF族积分器为主要讨论对象,详细介绍大型多体系统动力学软件中常见类型的积分器的算法细节.每族积分器都给出了不止一套计算公式,而且其对应求解微分代数方程组(DAE)的index可以为1、2或者3.除此以外,本文还着重介绍了微分代数方程组的误差估计、变阶变步长策略等关键技术;...     

变尺度法与Huang族公式

变尺度算法是求解无约束优化问题的有效而著名的算法,算法易于实现,计算量较小,并形成了完整的算法体系,对工程应用有重要的影响。文中简述了变尺度算法的基本思想,介绍了Huang算法族和Broyden族及它们的基本性质,分析比较了近年来对变尺度算法不同角度的研究。通过数值实验证明了变尺度算法的有效性。     

变尺度法与Huang族公式

变尺度算法是求解无约束优化问题的有效而著名的算法，算法易于实现，计算量较小，并形成了完整的算法体系，对工程应用有重要的影响。文中简述了变尺度算法的基本思想，介绍了Huang算法族和Broyden族及它们的基本性质，分析比较了近年来对变尺度算法不同角度的研究。通过数值实验证明了变尺度算法的有效性。     

Modelica软件中微分代数方程指标约简算法分析

Modelica 建模软件会产生高指标的微分代数方程(DAE),因为现有数值求解器一般不能直接对高指标DAE进行求解,所以Modelica建模软件必须对高指标DAE进行指标约简.为提高Modelica建模软件性能,针对现有指标约简算法,展开了理论分析和实验比较,着重对负权二部图算法进行了详细分析,并给出了负权二部图算法的时间复杂度.理论分析和实验结果表明,负权二部图算法相对Pantelides算法有较好的性能,因此可以考虑在Modelica建模软件中使用负权二部图算法来提高性能.     

基于队列长度和速率的拥塞控制神经网络方法

文中研究了网络拥塞控制问题。PID控制器是实现网络拥塞控制非常有效的方法,能够实现对网络的主动队列管理。文中根据队列长度和变化速率,利用神经网络实现传统的比例微分积分器（PID）功能,从而提出了基于队列长度和速率的拥塞控制神经网络方法（RSPID）。该方法利用神经网络的加权动量梯度学习算法,自动调节控制参数,克服了传统PID控制方法由于控制器参数固定带来的适应性和稳定性问题。仿真结果表明,RSPID算法的鲁棒性和队列长度性能要优于PID算法。     

一种智能积分器

本文提出一种新型智能积分器,它融汇了控制专家的控的制经验.文中给出它的数学模型,并用谐波线性化原理讨论它的频率特性:幅频特性起积分作用,而相位滞后仅27.6度;高次谐波分量小;抗干扰能力强.最后应用这种积分器设计一个双目标优化的二次优化系统.结果表明它的控制性能好,具有广泛的应用前景.     

基于多个自主水下航行器的分布式协同流场估计

本文考虑利用多个自主式水下航行器(AUV)实现流场估计, 提出了一种基于树型网络的分布式方法来估计 水下流场. 在本文中, 借助绝对运动积分误差和相对运动积分误差, 流场估计问题被描述为求解一个以未知流场为 变元的非线性方程组. 继而本文在多AUV系统内建立一个低通讯成本的树型网络, 并在该网络上运行一种分布式 算法以求解与流场估计相关的非线性方程组. 在该算法中, 每个AUV将当前的流场估计值连续地投影到自身拥有 的约束方程的解集中, 并通过扩散和池化两个步骤在树型网络间传递流场估计值. 本文证明了上述算法的收敛性, 并通过仿真实验验证了所述分布式协同流场估计方法的有效性.     

基于PXI的数字式长时间积分器

传统模拟积分器存在着零点漂移、非线性误差和电容泄漏等问题,通过对传统模拟积分器及数字式积分器的深入调研,以及PXI系统的学习,设计了基于PXI的数字式长时间积分器,该积分器减小了非线性误差和泄漏现象。主要思想是首先对信号进行数字积分,然后通过某种算法扣除数字积分的零漂,来修正积分结果。测试结果证明基于PXI的数字式长时间积分器有效地降低了零点漂移,比传统模拟积分器更加灵活的功能。     

自动补偿低零漂积分器设计

传统模拟积分器存在着零点漂移、非线性误差和泄漏等问题,通过对其进行的深入调研,给出了实际积分电路的误差分析;提出一种自动补偿零漂的方法,采用CPLD控制时序,给出了两种实现方案,并对这两种方案做了比较;设计的积分器在100 s内漂移很低,在此基础上,提出了一种改进型积分器。比单个积分器效果更好,能够满足现有HT-7超导托卡马克实验的需要。     

基于FPGA的低零漂数字积分器

针对模拟积分器存在的零点漂移、非线性误差和电容泄漏等问题,通过对积分器实现的形式及误差的分析.设计了基于现场可编程门阵列(Field Programmable Gate Array)技术的低零漂数字积分器;主要思想是通过模拟开关切换将输入信号交替反向,再用FPGA将其还原累加,进而达到扣去噪声,减小零漂的目的;实验结果表明基于FPGA的数字积分器具有极低的零漂和较好的线性,适合于托卡马克装置放电实验中诊断信号的测量.     

Comparison of several formulations and integration methods for the resolution of DAEs formulations in event-driven simulation of nonsmooth frictionless multibody dynamics

This article is devoted to the comparison of numerical integration methods for nonsmooth multibody dynamics with joints, unilateral contacts and impacts in an industrial context. With an event-driven strategy, the smooth dynamics, which is integrated between two events, can be equivalently formulated as a Differential Algebraic Equation (DAE) of index 1, 2 or 3. It is well known that these reformulations are no longer equivalent when a numerical time-integration technique is used. The drift-off effect and the stability of the numerical scheme strongly depend on the index of the formulation. But, besides the standard properties of accuracy and stability of the DAE solvers, the event-driven context imposes some further requirements that are crucial for a robust and efficient event-driven strategy. In this article, several state-of-the-art numerical time integration methods for each formulation are compared: the generalized-\(\alpha\) scheme for index-3 formulation and stabilized index-2 formulation, (Partitioned) Runge–Kutta Half-Explicit Method of order 5 (HEM5 and PHEM56) for index-2 DAEs with projection techniques, and Runge–Kutta explicit scheme of order 5, the Dormand–Prince scheme (DOPRI5), for index-1 DAEs with projection techniques (MDOP5). We compare these schemes in terms of efficiency, violation of the constraints and the way they handle stiff dynamics on numerous industrial benchmarks, where a CAD software is in this loop. One of the major conclusions is that the index-2 DAEs solvers prove to be better than other schemes to maintain low violations at position and acceleration levels. The best compromise allows us to design efficient event-driven solvers. When the dynamics is stiff, implicit schemes outperform explicit and half-explicit methods which are sometimes unable to compute the dynamics when the system’s frequency range is wide. Furthermore, in industrial context, some solvers fail to reproduce the properties that they enjoy in theory. This is particularly true for half-explicit schemes when the Jacobian of the constraints has not full rank.     

基于ADAMS动力学仿真参数设置的研究

在ADAMS动力学仿真过程中,针对同样的模型没置不同的仿真参数,其仿真结果都不相同,甚至出现仿真失败的现象。该文根据动力学数值计算流程,对于不同的积分器以及积分格式作了比较研究,分析了不同积分器和积分格式的优缺点和适用场合,并对动力学仿真中其它参数设置作了介绍,针对动力学仿真过程中几种常见的仿真失败进行了原因分析并提出解决措施,并通过具体的实例对．卜述分析进行说明,最后提出了基于ADAMS动力学仿真参数设置一般步骤和原则,为获得最佳仿真效果奠定了基础,具有一定的理论价值。     

Symbolic preprocessing for simulation of PDE models of chemical processes

First principle modeling of chemical processes very often leads to a mixed system of partial differential equations (PDEs) and differential algebraic equations (DAEs) which must be preprocessed for use in standard DAE numerical simulation or optimization tools. This contribution presents the symbolic preprocessing tool SyPProT developed for the simulation environment Diva in order to apply DAE numerics also to PDEs. The method-of-lines (MOL) approach for the required PDE discretization is implemented in SyPProT by configurable finite-difference and finite-volume schemes. The model as well as the MOL parameters are represented in a tailor-made Mathematica data structure (MDS). The preprocessing of a PDE model is illustrated by the example of a circulation-loop-reactor (CLR).     

Regularity of distributional differential algebraic equations

Time-varying differential algebraic equations (DAEs) of the form E[(x)\dot]=Ax+f{E\dot{x}=Ax+f} are considered. The solutions x and the inhomogeneities f are assumed to be distributions (generalized functions). As a new approach, distributional entries in the time-varying coefficient matrices E and A are allowed as well. Since a multiplication for general distributions is not possible, the smaller space of piecewise-smooth distributions is introduced. This space consists of distributions which could be written as the sum of a piecewise-smooth function and locally finite Dirac impulses and derivatives of Dirac impulses. A restriction can be defined for the space of piecewise-smooth distributions, this restriction is used to study DAEs with inconsistent initial values; basically, it is assumed that some past trajectory for x is given and the DAE is activated at some initial time. If this initial trajectory problem has a unique solution for all initial trajectories and all inhomogeneities, then the DAE is called regular. This generalizes the regularity for classical DAEs (i.e. a DAE with constant coefficients). Sufficient and necessary conditions for the regularity of distributional DAEs are given.     

Starting algorithms for a class of RK methods for index-2 DAEs

When semiexplicit differential-algebraic equations are solved with implicit Runge-Kutta methods (RK), the computational effort is dominated by the cost of solving the nonlinear systems, and therefore it is important to have good starting values to begin the iterations. For semiexplicit index-2 DAEs, starting algorithms without additional cost for RK methods with regular matrix coefficient were studied in a previous paper. However, the regularity condition on the matrix coefficient excludes some interesting methods like Lobatto IIIa and ESDIRK methods. In this paper, we study starting algorithms, without additional computational cost, for a class of Runge-Kutta methods in the case of index-2 DAEs.     

A distributed memory parallel algorithm for the efficient computation of sensitivities of differential-algebraic systems

An efficient algorithm for the evaluation of the parametric sensitivities for mixed systems of differential and algebraic equations (DAEs) on computers involving multiple processors operating in parallel is presented. The algorithm derives its efficiency by decoupling the integration of the sensitivity equations from that of the original DAE system, and by allowing tasks associated with the evaluation of sensitivities at multiple time points to overlap instead of being carried out in sequence. Numerical experiments demonstrating the efficiency of the proposed algorithm with systems of more than 850 DAEs and 45 parameters are presented. In all the cases studied, the simultaneous integration of the original DAEs and their sensitivity equations is carried out in less than 10% more time than that of the original DAEs alone.     

Automatic h-Scaling for the Efficient Time Integration of Stiff Mechanical Systems

When solving stiff mechanical systems, implicit time integrators overestimate the error and tend to use small stepsizes due to the order reduction phenomenon. This article introduces an algorithm for the detection of stiff components that allows automatic scaling of the stepsize h in the error estimation. It is based on an investigation of the local error and applies existing embedded formulas of implicit Runge–Kutta methods. Thus, implementation and calculation effort are low. Three examples are integrated with the code RADAU5, a linear DAE in two different formulations and a slider crank mechanism.     

A Hopf bifurcation theorem for singular differential–algebraic equations

We prove a Hopf bifurcation result for singular differential–algebraic equations (DAE) under the assumption that a trivial locus of equilibria is situated on the singularity as the bifurcation occurs. The structure that we need to obtain this result is that the linearisation of the DAE has a particular index-2 Kronecker normal form, which is said to be simple index-2. This is so-named because the nilpotent mapping used to define the Kronecker index of the pencil has the smallest possible non-trivial rank, namely one. This allows us to recast the equation in terms of a singular normal form to which a local centre-manifold reduction and, subsequently, the Hopf bifurcation theorem applies.     

On the optimal scaling of index three DAEs in multibody dynamics

We propose a preconditioning strategy for the governing equations of multibody systems in index-3 differential-algebraic form. The method eliminates the amplification of errors and the ill-conditioning which affect numerical solutions of high index differential algebraic equations for small time steps. We develop a new theoretical analysis of the perturbation problem and we apply it to the derivation of preconditioners for the Newmark family of integration schemes. The theoretical results are confirmed by numerical experiments.