Kinematic limit analysis of frictional materials using nonlinear programming

In this paper, a nonlinear numerical technique is developed to calculate the plastic limit loads and failure modes of frictional materials by means of mathematical programming, limit analysis and the conventional displacement-based finite element method. The analysis is based on a general yield function which can take the form of the Mohr–Coulomb or Drucker–Prager criterion. By using an associated flow rule, a general nonlinear yield criterion can be directly introduced into the kinematic theorem of limit analysis without linearization. The plastic dissipation power can then be expressed in terms of kinematically admissible velocity fields and a nonlinear optimization formulation is obtained. The nonlinear formulation only has one constraint and requires considerably less computational effort than a linear programming formulation. The calculation is based entirely on kinematically admissible velocities without calculation of the stress field. The finite element formulation of kinematic limit analysis is developed and solved as a nonlinear mathematical programming problem subject to a single equality constraint. The objective function corresponds to the plastic dissipation power which is then minimized to give an upper bound to the true limit load. An effective, direct iterative algorithm for kinematic limit analysis is proposed in this paper to solve the resulting nonlinear mathematical programming problem. The effectiveness and efficiency of the proposed method have been illustrated through a number of numerical examples.     

基于能力评估的空间翻滚目标抓捕策略优化

由于目标的翻滚运动, 空间双臂机器人对动态目标的抓捕相比于静态目标更具有挑战性. 对抓捕策略进行优化可以提高空间双臂机器人对翻滚目标的操作能力以保证任务的成功. 本文提出了一种基于能力评估的抓捕策略优选方法. 空间双臂机器人捕获目标时, 双臂末端执行器与目标同时接触形成闭链系统, 闭链约束的引入使操作能力的评估更加复杂. 在对双臂空间机器人协调操作翻滚目标的运动学与动力学分析基础上, 建立了考虑闭链约束的协调工作空间, 并分析了基于任务兼容度的消旋能力评估指标. 建立的协调工作空间同时包含位置和姿态信息, 可以用于灵巧度的计算. 接着, 基于协调工作空间的全局灵巧度指标确定机械臂末端执行器对目标的最优抓捕点, 以及考虑相机视角约束和末端执行器对目标速度跟踪约束下的力任务兼容度指标确定空间双臂机器人捕获翻滚目标时的最优抓捕构型. 利用能力评估确定抓捕策略可以充分利用双臂的协调性以增加对动态目标的操作能力, 通过仿真验证了所提抓捕策略的可行性和有效性.      

基于改进的双向渐进结构优化法的应力约束拓扑优化

工程结构设计时经常需要限制最大名义应力,以避免发生断裂或疲劳破坏,一个有效的策略是采用拓扑优化方法. 常规的双向渐进结构优化法(bi-evolutionary structural optimization, BESO)不能有效求解应力约束拓扑优化问题,为此本文提出一种改进的双向渐进结构优化方法,处理体积和应力约束下的最小柔顺性问题. 引入基于K-S函数的全局应力度量,以减小大量局部应力约束引起的计算代价. 采用拉格朗日乘子法将应力约束函数引入到目标函数,然后由二分法确定合适的拉格朗日乘子的值使得应力约束得到满足. 而且,详细推导了基于BESO方法的应力约束拓扑优化模型及其灵敏度列式,最后通过三个典型拓扑优化算例验证改进方法的有效性. 为展示考虑应力约束的优点,将应力约束设计与传统的基于刚度的设计进行了比较. 结果表明, 改进的BESO方法优化迭代过程稳健,获得了边界灰度单元很少的清晰的拓扑构型,并实现了有效降低应力集中效应的设计.      

基于非概率集合可靠性的结构优化设计

在结构非概率集合可靠性模型的基础上,考虑结构系统中的参数不确定性,提出了基于非概率可靠性的结构优化方法。该方法将不确定量看作是区间数,通过区间运算得到结构的非概率可靠性,并以结构的非概率可靠性小于指定可靠性指标为约束条件,利用乘子法对结构的优化问题进行求解。最后应用本文方法对一桁架结构进行总质量优化,优化结果验证了本文...     

On the reduction of the normality conditions in equality-constrained optimization problems in mechanics

A large class of problems in mechanics leads to the minimization of an objective function under equality constraints. In fact, inequality constraints can always be transformed into equality constraints by means of slack variables. The classical approach to solve equality-constrained problems relies on Lagrange multipliers, whose first-order normality conditions (FONC) lead to a system of nonlinear algebraic equations. This system of equations involves as many equations as unknowns, composed of the design variables and Lagrange multipliers, and hence, is amenable to a host of solution methods. In this paper, two methods to eliminate the Lagrange multipliers are reported, by which a reduced system of normality conditions is obtained. Reduction is conducted here either symbolically or numerically using an isotropic orthogonal complement L of the Jacobian matrix of the equality constraints. The relations thus resulting are cast into what is termed the dual form of the FONC. When the problem allows for symbolic calculations, a semi-graphical approach is applied, which leads to the global optimum of the problem at hand. However, the main novelty of the paper lies in an algorithm that returns the stationary points of a constrained optimization problem without requiring the closed-form expressions of the dual form of the FONC. Moreover, numerically efficient and stable procedures are given for the intermediate computational steps. The application of this algorithm is demonstrated with three examples from mechanics.     

A REDUCED QUADRATIC PROGRAMMING METHOD FOR ELASTIC CONTACT PROBLEMS

基于势能原理以节点位移为设计变量、以接触条件为约束方程构建了无摩擦弹性接触问题的二次规划数学模型,在此基础之上利用力平衡线性约束方程的特解和由基础解向量构成的奇异模态矩阵,提出一种新的基于奇异坐标变换的自由度缩减方法,大大降低了二次规划的规模,并使得二次规划模型不再含显性等式约束;根据弹性接触力学体系的特点,通过人为假定接触自由度位移模式,提出了一种简单高效的奇异模态矩阵的计算方法。通过两圆柱接触、轴孔间隙配合接触两个数值算例的对比分析,验证了对于弹性接触问题的求解,缩减二次规划方法有效克服了传统方法计算量大、对求解参数设置敏感、收敛困难的问题。     

解决约束违背问题的一种自适应调整方法

带约束的优化问题的目的是要找到满足等式或者不等式约束的最优点。在某些情况下,优化求解得到的"最优点"可能会使得某个或某几个约束条件超出目标约束限,或者在所有约束条件中的最大值远远小于目标约束限。针对这一类问题,本文提出一种在寻优过程的每一次迭代中自适应调整约束限的方法,通过动态调整迭代过程中迭代模型约束限的值,将约束条件中最大值的约束条件变为等式约束,使得迭代解始终在可行域范围内,且收敛后的最优解不违背任何约束条件。本文将该方法成功应用于位移约束下结构重量最小化拓扑优化模型,原来不满足约束条件的情况在使用该方法后都能使约束得到满足,解决了约束条件被违背的问题。     

多体系统接触碰撞问题的牛顿-欧拉线性互补方法

基于非光滑动力学方法的多体系统接触碰撞分析是目前多体系统动力学的研究热点.本文采用牛顿-欧拉方法建立多体系统接触、碰撞问题的动力学模型,给出一种牛顿-欧拉型线性互补公式.该建模方法与目前一般采用的拉格朗日建模方法的不同之处是约束条件中除了库仑摩擦、单边约束之外还含有光滑等式约束.在建立系统动力学模型时,首先解除摩擦约束和单边约束得到原系统对应的基本系统.牛顿-欧拉方法采用最大数目坐标建立基本系统的动力学方程,由于坐标不相互独立,因此基本系统中带有等式约束,其数学模型为一组微分代数方程.借助约束雅可比矩阵,在基本系统微分代数方程中添加摩擦接触和单边约束对应的拉氏乘子,就可以得到系统全局运动的具有变拓扑结构特征的动力学方程,再结合非光滑约束互补条件便可构成完备的系统动力学模型.完备的动力学模型由动力学微分方程以及等式约束和不等式约束组成.线性互补公式采用分块矩阵形式进行推导,简化了推导过程.数值计算采用基于线性互补的时间步进算法.时间步进算法是目前流行的非光滑数值算法,其突出特点是可以免去数值积分中繁琐的事件检测过程,而数值积分过程中通过对线性互补问题的求解可以确定系统的触-离状态.通过对典型的曲柄滑块间隙机构进行数值分析,验证本文方法的有效性.     

On some optimization procedures for shock localization

We consider a problem on shock wave localization in the numerical solution of one-dimensional unsteady problems of gas dynamics in Eulerian variables obtained on the basis of finite difference shock-capturing schemes. An optimization method for strong discontinuity localization proposed previously by Miranker and Pironneau is investigated by means of methods of classical variational calculus. This method may be difficult to implement when the entropy condition is included in the formulation of Miranker and Pironneau's optimization problem as an active constraint. In this connection we suggest an alternative optimization problem using artificial viscosity in the variational principle. It is shown theoretically that the application of such a variational principle yields a trajectory which coincides with the true discontinuity trajectory in the case of a shock wave moving at a constant speed. On the basis of this modification one more algorithm is proposed which reduces the shock localization problem to a problem of minimization of a univariate function. Numerical tests corroborate completely the theoretical conclusions. In particular, a higher shock localization accuracy is obtained on the basis of the proposed algorithms as compared to the original Miranker-Pironneau method.     

On single and bimodal optimum buckling loads of clamped columns

We study the problem of determining the optimum shape of a thin, elastic, clamped column of given length and volume, such that the fundamental buckling load is a maximum. The column cross-sections are assumed to be geometrically similar, and a minimum allowable value is specified for the cross-sectional area.Investigating the optimization problem parametrically in terms of this minimum constraint, we reveal a significant feature. There exists a threshold value for the constraint, beyond which the optimum columns are all associated with single mode optimum buckling loads, whereas, for any value of the constraint less than the threshold value, the optimum columns are associated with bimodal fundamental buckling loads.This bimodal behaviour necessitates an extension and a mathematical reformulation of the current optimization problem, which is outlined and solved in the paper. In particular, we revise the result hitherto considered to be the optimum solution for an unconstrained column with clamped ends.     

A new method for predicting the maximum vibration amplitude of periodic solution of non-linear system

An original method based on the proposed framework for calculating the maximum vibration amplitude of periodic solution of non-linear system is presented. The problem of determining the worst maximum vibration is transformed into a non-linear optimization problem. The harmonic balance method and the Hill method are selected to construct the general non-linear equality and inequality constraints. The resulting constrained maximization problem is then solved by using the MultiStart algorithm. Finally, the effectiveness of the proposed approach is illustrated through two numerical examples. Numerical examples show that the proposed method can, at much lower cost, give results with higher accuracy as compared with numerical results obtained by a parameter continuation method.     

Controllable velocity projection for constraint stabilization in multibody dynamics

The direct numerical solution of the index-3 algebraic-differential equation system (DAE) associated with the constrained dynamics of a multibody system poses several computational difficulties mainly related to stability. Specially in long simulations, the instability is related to the drift of the solution from the velocity constraint manifold. One of the techniques proposed in the literature to overcome this problem is the velocity projection, which is a post-stabilization method that brings the solution back to the invariant manifold. This projection introduce a nonnegative artificial dissipation of energy when performed with the system mass matrix, which can affect the long term quality of the solution. This paper proposes a novel controllable velocity projection procedure capable of meeting certain requirements in the projected solution; namely, a maximum constraint value or a maximum dissipated energy. Several issues related to the efficiency of the implementation are discussed and two numerical simulations are presented. These simulations illustrate the performance of the proposed methodology and provide interesting insights about the relevance of the accuracy of the projection in the stabilization effect.     

Two novel nonlinear multivariate grey models with kernel learning for small-sample time series prediction

For many applications, small-sample time series prediction based on grey forecasting models has become indispensable. Many algorithms have been developed recently to make them effective. Each of these methods has a specialized application depending on the properties of the time series that need to be inferred. In order to develop a generalized nonlinear multivariable grey model with higher compatibility and generalization performance, we realize the nonlinearization of traditional GM(1,N), and we call it NGM(1,N). The unidentified nonlinear function that maps the data into a better representational space is present in both the NGM(1,N) and its response function. The original optimization problem with linear equality constraints is established in terms of parameter estimation for the NGM(1,N), and two different approaches are taken to solve it. The former is the Lagrange multiplier method, which converts the optimization problem into a linear system to be solved; and the latter is the standard dualization method utilizing Lagrange multipliers, that uses a flexible estimation equation for the development coefficient. As the size of the training data increases, the estimation results of the potential development coefficient get richer and the final estimation results using the average value are more reliable. The kernel function expresses the dot product of two unidentified nonlinear functions during the solving process, greatly lowering the computational complexity of nonlinear functions. Three numerical examples show that the LDNGM(1,N) outperforms the other multivariate grey models compared in terms of generalization performance. The duality theory and framework with kernel learning are instructive for further research around multivariate grey models to follow.

     

Forward and inverse dynamics of nonholonomic mechanical systems

This paper deals with the forward and the inverse dynamic problems of mechanical systems subjected to nonholonomic constraints. The intrinsically dual nature of these two problems is identified and utilised to develop a systematic approach to formulate and solve them according to an unified framework. The proposed methodology is based on the fundamental equations of constrained motion which derive from Gauss’s principle of least constraint. The main advantage arising from using the fundamental equations of constrained motion is that they represent an effective method capable to derive the generalised acceleration of a mechanical system, constrained in general by a set of nonholonomic constraints, together with the generalized constraint forces (forward dynamics). When the constraint equations are used to represent the desired behaviour of the mechanical system under study, the generalised constraint forces deriving from the fundamental equations of constrained motion provide the control actions which reproduce the specified motion for the system (inverse dynamics). This approach is systematically extended to underactuated mechanical systems introducing a new method named underactuation equivalence principle. The underactuation equivalence principle is founded on the key idea that the underactuation property of a mechanical system can be mathematically represented using a particular set of nonholonomic constraint equations. Two simple case-studies are reported to exemplify the proposed methodology. In the first case-study the computation of the generalised constraint forces relative to the revolute joint constraints of a physical pendulum is illustrated. In the second case-study the calculation of the control action which solves the swing-up problem for an inverted pendulum is described.     

人体肌骨的多柔体系统动力学研究进展

人体肌肉骨骼系统简称肌骨系统, 包括骨骼、骨骼肌与关节连接, 其力学模型是典型的多柔体系统. 从多体动力学角度研究肌骨系统, 主要关注其在运动过程中的肌肉内力、关节力矩及产生的动力学影响, 属于动力学与生物力学的交叉融合. 肌骨系统的多体动力学模型已被广泛地应用于临床医学、竞技体育、军事训练、人机工程等诸多领域, 其仿真结果可为提高人体运动能力、降低关节载荷与能耗、避免运动损伤、加快康复进程等提供重要计算参考数据. 与此同时, 上述研究亦对肌骨动力学研究提出了许多新挑战. 本文综述了人体肌骨多柔体系统动力学相关研究进展, 包括骨骼肌功能解剖与生物力学建模、神经与肌肉控制理论、肌骨系统动力学问题与求解方法, 以及近年来肌骨多体动力学在步态分析、飞行员抗荷动作、口颌手术规划等领域的典型应用. 与工程领域的机械多体系统相比, 人体肌骨多体系统具有肌肉内力主动性与肌肉控制冗余性两大特征. 现有骨骼肌模型难以同时考虑肌肉的解剖结构、三维几何与肌力产生的生物化学机制. 已有大多数肌骨模型采用静态优化假设消除肌肉冗余性, 忽略了肌肉与肌腱内力平衡及兴奋收缩耦联机制. 此外, 目前仍缺乏实现肌骨模型个性化的无创在体测试手段. 未来, 人体肌骨多体动力学研究将会向更精确、智能、个性化的方向发展, 成为动力学与生物力学交叉的热点研究领域.      

两类变量综合处理的结构形状优化设计方法

本文针对截面变量为离散变量为连续变量的结构优化问题提出了一种优化设计的方法,首先将单元内力作一阶近似,利用凝聚函数多约束问题转化了单约束问题。在解过程中,把定义在连续区间上的形状变量看成是在一些离散以值的离工用变量,然后将两类变量统一考虑并利用相对差商法求解。将该算法应用于几个经典的结构优化算例,运算结果显示了该方法是可行的,优化结果也比较满意。     

基于高斯原理的非理想系统动力学建模

高斯原理给出了通过求函数极值、从可能运动中鉴别出真实运动的规则, 它可以使得多体系统动力学问题不需通过求解微分(代数)方程, 而是采用求解最小值的优化方法来解决, 从而提供了一种适用于优化算法的建模思路, 因此, 如何定义恰当的高斯拘束函数是动力学优化方法得以实现的前提. 对于理想系统而言, 约束对系统的作用可以通过约束方程来体现, 故高斯拘束可表达为系统质点加速度的函数, 系统的动力学问题因此可以描述为目标函数为高斯拘束函数、优化变量为质点加速度的约束最优化问题; 当系统中需要考虑干摩擦等非理想因素时, 部分相互作用不能被所定义的约束方程所涵盖而需要采用额外的物理规律来描述, 这种相互作用破坏了原有的针对理想系统的高斯拘束函数的极值特性. 基于变分类的高斯原理, 推导并证明了目标函数以理想约束力所表达的非理想系统的极值原理, 针对目前文献中用于非理想系统的高斯原理进行了讨论, 指出其实际为文中的极值原理在非理想约束力与理想约束力无明显关联时的一种特殊表达形式, 当非理想约束力与理想约束力有明显的函数关系(如库仑摩擦定律中滑动摩擦力与法向约束力间的线性关系)时, 该形式失效; 同时根据文中的极值原理, 得到了考虑库仑摩擦时非理想的多体系统动力学问题的优化模型. 例子中分析了优化模型及相应的线性互补性模型的关系, 分析发现在满足刚体滑动问题的唯一性条件下二者互为充分必要条件, 从而证明了文中优化模型的可靠性; 并采用优化计算方法进行了动力学模拟, 模拟结果显示了将高斯原理与优化算法相结合的可行性及有效性.      

基于非线性优化的单目视觉/惯性组合导航算法

针对自主移动机器人在GPS拒止环境下准确实时定位的问题,提出一种单目视觉/惯性组合导航算法。为解决视觉/惯导工作频率不一致问题,利用预积分技术预先处理惯性测量值;引入了一种快速高精度线性初始化方法,分步估计初始系统尺度、重力方向、速度及零偏;结合全局地图三维点约束削弱累积误差,基于是否更新地图点,构建了两种基于非线性优化的单目视觉/惯性紧耦合模型,以保证导航的局部精度及全局一致性。实验结果表明:初始化方法可快速(15 s内)实现高精度状态初始化;与单目视觉导航算法相比,提出的算法不仅可获取绝对尺度信息,且导航精度更高;与传统滑动窗口非线性优化方法相比,提出的算法在窗口优化过程中加入全局地图三维点约束,可有效削弱累积误差,验证了算法的正确性和可行性。     

基于阈值因子的结构可靠性解耦优化方法

基于结构可靠性理论,引入阈值因子的概念,其物理意义为当前最优解满足可靠性约束需将阈值增减的幅度。随着迭代的进行,对于有效约束,阈值因子收敛于1,能够使得最可能失效点快速向满足可靠性约束的极限状态曲面靠拢,优化效率得以提升。解耦模型中,优化变量可为随机变量也可为非随机变量,当优化变量为随机变量时,采取优化变量拆解方式进行计算。数值算例表明,本文方法对优化变量拆解方式不敏感,对有效约束和非有效约束均能够获得满意的优化结果,且计算效率明显高于经典方法。     

DYNAMIC MODELING OF NONIDEAL SYSTEM BASED ON GAUSS'S PRINCIPLE$^{\bf 1)}$

高斯原理给出了通过求函数极值、从可能运动中鉴别出真实运动的规则, 它可以使得多体系统动力学问题不需通过求解微分(代数)方程, 而是采用求解最小值的优化方法来解决, 从而提供了一种适用于优化算法的建模思路, 因此, 如何定义恰当的高斯拘束函数是动力学优化方法得以实现的前提. 对于理想系统而言, 约束对系统的作用可以通过约束方程来体现, 故高斯拘束可表达为系统质点加速度的函数, 系统的动力学问题因此可以描述为目标函数为高斯拘束函数、优化变量为质点加速度的约束最优化问题; 当系统中需要考虑干摩擦等非理想因素时, 部分相互作用不能被所定义的约束方程所涵盖而需要采用额外的物理规律来描述, 这种相互作用破坏了原有的针对理想系统的高斯拘束函数的极值特性. 基于变分类的高斯原理, 推导并证明了目标函数以理想约束力所表达的非理想系统的极值原理, 针对目前文献中用于非理想系统的高斯原理进行了讨论, 指出其实际为文中的极值原理在非理想约束力与理想约束力无明显关联时的一种特殊表达形式, 当非理想约束力与理想约束力有明显的函数关系(如库仑摩擦定律中滑动摩擦力与法向约束力间的线性关系)时, 该形式失效; 同时根据文中的极值原理, 得到了考虑库仑摩擦时非理想的多体系统动力学问题的优化模型. 例子中分析了优化模型及相应的线性互补性模型的关系, 分析发现在满足刚体滑动问题的唯一性条件下二者互为充分必要条件, 从而证明了文中优化模型的可靠性; 并采用优化计算方法进行了动力学模拟, 模拟结果显示了将高斯原理与优化算法相结合的可行性及有效性.