基于隐式微分/代数方程的多体系统动力学设计灵敏度分析方法

基于一般性的积分型目标函数、隐式相容初始条件及终止时刻表达式,系统建立了含设计参数的用隐式微分／代数方程表达的多体系统动力学设计灵敏度分析的直接微分方法和伴随变量方法．为降低目标函数及其对设计变量导数的计算复杂性。将其积分形式的计算转化为微分形式．所得到的结果可方便地应用于高效的间接最优化设计方法．最后通过采用绝对坐标建模的平面两连杆机械臂模型对该方法进行了验证．     

灵敏度在车辆动力学分析中的应用

针对系统时变与非时变参数对系统变量的影响问题,建立了四分之一车辆系统动力学模型,借助Matlab交互式高级编程语言编写函数文件实现算法,建立了路面激励仿真模型与悬架仿真模型;并以此为基础,利用灵敏度分析方法,采用一阶标准灵敏度函数形式.通过四分之一车体模型实例计算,分析在路面激励下,车身速度、车身位移与车轮位移的标准灵敏度响应,分析系统结构参数的改变对结构动态特性变化的敏感程度,提高了设计效率并减少了设计成本.     

分维自适应稀疏网格积分非线性滤波器

针对含加性高斯噪声的非线性离散系统,提出了可分别根据各维状态及量测方程的非线性函数特性来确定采样点及其权重的积分滤波器.设计了基于嵌入式高斯采样积分和稀疏网格法则的自适应多变量采样积分方法,可在匹配函数高阶泰勒展开项时,利用低阶采样点,提出了高效的数据结构和遍历算法,便于采用该积分方法分别估计系统状态/量测的预测均值和协方差矩阵.该滤波器既能根据各维非线性函数的特性确定采样点,又实现了对采样值和权重的完全复用,保证了算法效率.理论分析和仿真表明,该滤波算法中自适应调整的运算量小于计算非线性函数采样值.该滤波器与无迹卡尔曼滤波相比,提高了滤波精度,与固定形式的稀疏网格滤波器相比,提高了采样效率,且该方法为两者的广义形式.仿真实验也验证了状态估计的精确性和函数采样的高效性.     

复杂电路分析的最优化程序方法与应用

应用最优化基本原理，提出了以支路电流和节点位为设计变量，以电路平衡条件为目标函数的任意复杂电路分析的框架式目标函数化程序方法，该方法所构造的框架式目标函数，将根据具体电路输入参数，合理地设计变量分配与排序和完成目标函数的累积运算，可快速实现任意组合电路的分析计算。电路问题算例验证了这种程序方法的可行性和通用性。     

变拓扑柔性多体系统接触碰撞动力学研究

在实际工程领域中存在着大量接触碰撞等非连续动力学问题,现有的解决柔性多体系统连续动力学过程的建模理论与方法,已经无法解决或无法很好解决这些问题.本文基于变拓扑思想,提出了附加接触约束的柔性多体系统碰撞动力学建模理论；通过设计柔性圆柱杆接触碰撞实验,验证了所提出附加约束接触碰撞模型的有效性；针对柔性多体系统全局动力学仿真面临时间和空间的多尺度问题,提出多变量的离散方法,从而提高了柔性多体系统非连续动力学的仿真效率.     

最小二乘法在高斯拘束函数求解中的应用

高斯最小拘束原理是一种典型的微分变分原理,以加速度为变量,通过寻求拘束函数极值的变分方法直接得出系统的运动规律.目前,国内常用的求解高斯拘束函数的方法为拉格朗日乘子法,通过引入拉格朗日乘子将高斯拘束函数的条件极值问题转化为带有拉格朗日乘子的无条件极值问题,这种求解方法会增加未知变量的个数.为减少变量个数,进一步提高运算效率,文章首先对高斯拘束函数进行简单变形,引入加速度形式的约束方程将高斯拘束函数化为最小二乘形式,直接运用最小二乘法导出使高斯拘束函数取极小值时系统真实加速度的表达式.最后通过对曲柄滑块机构正动力学问题的分析和计算,验证了该方法的有效性.     

二阶非线性多智能体系统有限时间分布式优化

本文研究一类具有未知常参数的二阶非线性多智能体系统的有限时间自适应分布式优化.首先,通过给定各个智能体的二次目标函数,并结合多智能体系统达到一致性的条件,构造含有惩罚因子的惩罚函数,提出加速智能体状态收敛至目标函数最优解的控制策略.其次,在给定惩罚因子下,基于幂积分方法和有限时间稳定理论,设计有限时间分布式自适应控制协议,使得惩罚函数的梯度在有限时间内收敛到零的邻域内.再次,通过增大惩罚因子,保证多智能体系统的状态最终达到一致,并收敛到总体目标函数的最优解.最后,仿真算例验证了结果的可行性和有效性.     

势问题的复变量无单元Galerkin方法

为提高无单元Galerkin(Element-Free Galerkin, EFG)方法的计算效率,将复变量移动最小二乘法与EFG方法结合,利用控制方程的积分弱形式并采用Lagrange乘子法引入边界条件,提出势问题的复变量无单元Galerkin(Complex Variable EFG,CVEFG)方法,并推导相关公式.与传统的EFG方法相比,该方法采用复变量移动最小二乘法可以减少试函数中的待定系数,从而减少计算量、提高计算效率. 最后,给出数值算例验证该方法的有效性.     

具有双面约束单点摩擦多体系统的数值计算方法

研究了一类具有双面约束单点摩擦的单自由度多体系统动力学方程的算法问题.首先给出了系统的动力学方程,该方程具有很强的非光滑性,不能应用已有的一些光滑系统的数值方法研究系统的动力学特性.因此,本文利用方程的特点和所求变量的物理含义,给出了一种简便的数值计算方法.该方法的计算效率和精度与迭代法相比均较高.     

运动滑道含摩擦多体系统的建模和数值方法

研究了运动约束面含摩擦多体系统动力学方程的建立和算法问题．首先利用第一类Lagrange方程给出了系统的动力学方程,并以矩阵形式给出了这类系统摩擦力的广义力的一般表达式．为便于摩擦力和铰链约束力的分析与计算,采用笛卡尔坐标和约束方程的局部方法,使得系统的约束力与Lagrange乘子一一对应．应用增广法将微分一代数方程组转化为常微分方程组并用分块矩阵的形式给出,以便于方程的编程与计算,提高计算效率．最后用一个算例验证了该方法的有效性．     

Second order adjoint sensitivity analysis of multibody systems described by differential–algebraic equations

The optimization strategies employing second order sensitivity information has higher accuracy, but its computation is complex. In this paper, an adjoint variable method applied for the second order design sensitivity analysis of multibody design problems is developed. Based on Lagrange equations of multibody system dynamics, a general objective function, constraint conditions, initial and end conditions, the adjoint variable equations for first order sensitivity analysis and design sensitivity formulations are derived firstly. Then, second order sensitivity analysis formulations, as well as the detailed computation steps, are given based on the previous results. For simplification, the second derivative of the objective function with respect to design variables is translated into an initial value problem of an ordinary differential equation with one variable. Finally, a numerical example of slider–crank mechanism validates the accuracy and efficiency of the method for second order sensitivity analysis.     

Design sensitivity analysis and optimization of dynamic response

The paper presents methods of design sensitivity analysis and optimization of dynamic response of mechanical and structural systems. A key feature of the paper is the development of procedures to handle point-wise state variable constraints. Difficulties with a previous treatment where such constraints were transformed to equivalent integral constraints are noted and explained from theoretical as well as engineering standpoints. An alternate treatment of such constraints is proposed, developed and evaluated. In this treatment each point-wise state variable constraint is replaced by several constraints that are imposed at all the local max-points for the original constraint function. The differential equations of motion are formulated in the first-order form so as to handle more general problems. The direct differentiation and adjoint variable methods of design sensitivity analysis to deal with the point-wise constraints are presented. With the adjoint variable methods, there are two ways of calculating design sensitivity coefficients. The first approach uses an impulse load and the second approach uses a step load for the corresponding adjoint equation. Since the adjoint variable methods are better for a large class of problems, an efficient computational algorithm with these methods is presented in detail. Optimum results for several problems are obtained and compared with those available in the literature. The new formulation works extremely well as precise optimum designs are obtained.     

First order sensitivity analysis of flexible multibody systems using absolute nodal coordinate formulation

Design sensitivity analysis of flexible multibody systems is important in optimizing the performance of mechanical systems. The choice of coordinates to describe the motion of multibody systems has a great influence on the efficiency and accuracy of both the dynamic and sensitivity analysis. In the flexible multibody system dynamics, both the floating frame of reference formulation (FFRF) and absolute nodal coordinate formulation (ANCF) are frequently utilized to describe flexibility, however, only the former has been used in design sensitivity analysis. In this article, ANCF, which has been recently developed and focuses on modeling of beams and plates in large deformation problems, is extended into design sensitivity analysis of flexible multibody systems. The Motion equations of a constrained flexible multibody system are expressed as a set of index-3 differential algebraic equations (DAEs), in which the element elastic forces are defined using nonlinear strain-displacement relations. Both the direct differentiation method and adjoint variable method are performed to do sensitivity analysis and the related dynamic and sensitivity equations are integrated with HHT-I3 algorithm. In this paper, a new method to deduce system sensitivity equations is proposed. With this approach, the system sensitivity equations are constructed by assembling the element sensitivity equations with the help of invariant matrices, which results in the advantage that the complex symbolic differentiation of the dynamic equations is avoided when the flexible multibody system model is changed. Besides that, the dynamic and sensitivity equations formed with the proposed method can be efficiently integrated using HHT-I3 method, which makes the efficiency of the direct differentiation method comparable to that of the adjoint variable method when the number of design variables is not extremely large. All these improvements greatly enhance the application value of the direct differentiation method in the engineering optimization of the ANCF-based flexible multibody systems.     

A contact force model considering constant external forces for impact analysis in multibody dynamics

The adjoint method is an elegant approach for the computation of the gradient of a cost function to identify a set of parameters. An additional set of differential equations has to be solved to compute the adjoint variables, which are further used for the gradient computation. However, the accuracy of the numerical solution of the adjoint differential equation has a great impact on the gradient. Hence, an alternative approach is the discrete adjoint method, where the adjoint differential equations are replaced by algebraic equations. Therefore, a finite difference scheme is constructed for the adjoint system directly from the numerical time integration method. The method provides the exact gradient of the discretized cost function subjected to the discretized equations of motion.     

Design sensitivity analysis for transient response of non-viscously damped dynamic systems

A design sensitivity analysis for the transient response of the non-viscously damped dynamic systems is presented. The non-viscously (viscoelastically) damped system is widely used in structural vibration control. The damping forces in the system depend on the past history of motion via convolution integrals. The non-viscos damping is modeled by the generalized Maxwell model. The transient response is calculated with the implicit Newmark time integration scheme. The design sensitivity analysis method of the history dependent system is developed using the adjoint variable method. The discretize-then-differentiate approach is adopted for deriving discrete adjoint equations. The accuracy and the consistency of the proposed method are demonstrated through a single dof system. The proposed method is also applied to a multi-dof system. The validity and accuracy of the sensitivities from the proposed method are confirmed by finite difference results.     

Singular value decomposition for dynamic system design sensitivity analysis

A computer-based method for automatic generation and efficient numerical solution of mixed differential-algebraic equations for dynamic and design sensitivity analysis of dynamic systems is developed. The equations are written in terms of a maximal set of Cartesian coordinates to facilitate general formulation of kinematic and design constraints and forcing functions. Singular value decomposition of the system Jacobian matrix generates a set of composite generalized coordinates that are best suited to represent the system. The coordinates naturally partition into optimal independent and dependent sets, and integration of only the independent coordinates generates all of the system information. An adjoint variable method is used to compute design sensitivities of dynamic performance measures of the system. A general-purpose computer program incorporating these capabilities has been developed. A numerical example is presented to illustrate accuracy and properties of the method.     

Shape design sensitivity analysis of nonlinear structural systems

A unified approach is presented for shape design sensitivity analysis of nonlinear structural systems that include trusses and beams. Both geometric and material nonlinearities are considered. Design variables that specify the shape of components of built-up structures are treated, using the continuum equilibrium equations and the material derivative concept. To best utilize the basic character of the finite element method, shape design sensitivity information is expressed as domain integrals. For numerical evaluation of shape design sensitivity expressions, two alternative methods are presented: the adjoint variable and direct differentiation methods. Advantages and disadvantages of each method are discussed. Using the domain formulation of shape design sensitivity analysis, and the adjoint variable and direct differentiation methods, design sensitivity expressions are derived in the continuous setting in terms of shape design variations. A numerical method to implement the shape design sensitivity analysis, using established finite element codes, is discussed. Unlike conventional methods, the current approach does not require differentiation of finite element stiffness and mass matrices.     

CAD-based shape optimisation using adjoint sensitivities

Existing approaches to CAD-based design optimisation using adjoint sensitivities are reviewed and their shortcomings are recalled. An alternative approach is presented which uses the control points of the boundary representation (BRep) as design parameters. The sensitivity of the objective function with respect to the design variables is calculated using automatic differentiation (AD). Results for a 2-D aerofoil are presented.     

A study of solution algorithms for shape design sensitivity analysis on a supermini computer with an attached array processor

This paper presents a study and comparison of shape design sensitivity analysis algorithms that are based on the continuum adjoint variable method, the continuum direct differentiation method, and the finite difference method, implemented on a supermini computer with an attached array processor. The basic algorithms and their differences in evaluating shape design sensitivity coefficients are outlined. A solution method for solving a system of equations, using a general sparse storage technique, is used for numerical implementation of shape design sensitivity analysis. It is found that computing shape design sensitivity coefficients using the direct differentiation method is significantly more efficient than using the adjoint variable method or the finite difference method. A detailed performance evaluation of the methods, using an attached array processor, is presented. The performance of the attached array processor, compared to a supermini computer is shown to depend strongly on the type of computations to be carried out. When only parts of a program are running on an attached array processor, the CPU time distribution among the different subroutines of the program can change significantly, compared to using the host processor only.     

On the numerical implementation of continuous adjoint sensitivity for transient heat conduction problems using an isogeometric approach

A crucial problem of continuous adjoint shape sensitivity analysis is the numerical implementation of its lengthy formulations. In this paper, the numerical implementation of continuous adjoint shape sensitivity analysis is presented for transient heat conduction problems using isogeometric analysis, which can serve as a tutorial guide for beginners. Using the adjoint boundary and loading conditions derived from the design objective and the primary state variable fields, the numerical analysis procedure of the adjoint problem, which is solved backward in time, is demonstrated. Following that, the numerical integration algorithm of the shape sensitivity using a boundary approach is provided. Adjoint shape sensitivity is studied with detailed explanations for two transient heat conduction problems to illustrate the numerical implementation aspects of the continuous adjoint method. These two problems can be used as benchmark problems for future studies.