Numerical study of the axially symmetric motion of an incompressible viscous fluid in an annulus between two concentric rotating spheres

The research reported herein involved the study of the transient motion of a system consisting of an incompressible Newtonian fluid in an annulus between two concentric, rotating, rigid spheres. The primary purpose of the research was to study the use of a numerical method for analysing the transient motion that results from the interaction between the fluid in the annulus and the spheres which are started suddenly by the action of prescribed torques. The problems considered in this research included cases where: (a) one or both spheres rotate with prescribed constant angular velocities and (b) one sphere rotates due to the action of an applied constant or impulsive t?orque. In this research the coupled solid and fluid equations were solved numerically by employing the finite difference technique. With the approach adopted in this research, only the derivatives with respect to spatial variables were approximated with the use of the finite difference formulae. The steady state problem was also solved as a separate problem (for verification purposes), and the results were compared with those obtained from the solution of the transient problem. Newton's algorithm was employed to solve the algebraic equations which resulted from the steady state problem, and the Adams fourth-order predictor–corrector method was employed to solve the ordinary differential equations for the transient problem. Results were obtained for the streamfunction, circumferential function, angular velocity of the spheres and viscous torques acting on the spheres as a function of time for various values of the system dimensionless parameters.     

带有障碍物溃坝流动问题的有限元数值模拟研究

针对下游带有障碍物的溃坝流动问题,本文基于两相流动模型,在有限元算法框架下对其进行数值模拟研究。依据水平集(Level Set)方法追踪运动界面,并引入了一个简单的修正技术,保证较好的质量守恒性。为了精确表示运动界面,采用稳定和有效的间断有限元方法求解双曲型Level Set及其重新初始化方程。对于两相统一Navier-Stokes方程,首先利用分裂格式对其解耦,然后通过SUPG(Streamline Upwind Petrov Galerkin)方法进行数值求解。模拟研究了下游带有障碍物的牛顿流体溃坝流动问题,得到的数值结果与文献已有模拟结果及实验结果均吻合较好。此外,还考虑了幂律型非牛顿流体,并分析了不同特性非牛顿流体对于溃坝流动过程和界面形态等的影响。     

Analysis of impact interactions in dynamics of many-body mechanical systems

Nowadays, methods for representing the dynamic equations of coupled systems of bodies in a form suitable for numerical solution are used widely. On the basis of such approaches, universal software aimed at solving certain classes of problems is developed [1–3].In design diagrams of mechanical systems, it is necessary to take into account various nonlinearities, including those of impact type. The representation of impact interactions with the use of nonlinear characteristics of positional forces results in increased stiffness of the system of differential equations and increased time expenditures for the solution, because it is necessary to decrease the integration step or use implicit integration methods.In this connection, it is expedient to consider approaches based on assumptions of the impact theory, i.e., the assumptions that the time of impact interaction is negligibly small and the impact is absolutely inelastic or partially elastic.In the general statement, an impact in a system with arbitrarily many stationary and impact constraints was studied in [4], where a special term—system impact—was introduced to denote this problem, showing that the constant and discontinuous constraints are multidimensional.In the present paper, we give a method for the numerical implementation of the methods proposed in [4] with the use of the FRUND software designed for modeling the dynamics of systems of rigid and elastic bodies [2]. We analyze the efficiency of their application.According to [4], the system impact problem is stated as follows. For a many-body mechanical system with arbitrary kinematic constraints, it is required to determine the impact discontinuity of velocities as the number of constraints varies instantaneously.     

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.     

冲击载荷作用下矩形薄板的弹性动力屈曲

为了研究冲击载荷作用下考虑应力波效应弹性矩形薄板的动力屈曲,根据动力屈曲发生瞬间的能量转换和守恒准则,导出板的屈曲控制方程和波阵面上的补充约束条件,真实的屈曲位移应同时满足控制方程和波阵面上的附加约束条件。满足上述条件,建立了该问题的完整数值解法,对屈曲过程中冲击载荷、屈曲模态和临界屈曲长度之间的关系进行研究,定量计算了横向惯性效应对提高薄板动力屈曲临界应力的贡献。研究表明:板的厚宽比一定时,临界屈曲长度随冲击载荷的增大而减小;由于屈曲时的横向惯性效应,应力波作用下薄板一阶临界力参数是相应边界板的静力失稳临界力参数的1.5倍;随着边界约束逐渐减弱,板临界力参数逐渐减小,动力特征参数逐渐增大。     

基于Hamilton体系的弹性力学问题的比例边界有限元方法

比例边界有限元方法是求解偏微分方程的一种半解析半数值解法。对于弹性力学问题,可采用基于力学相似性、基于比例坐标相似变换的加权余量法和虚功原理得到以位移为未知量的系统控制方程,属于Lagrange体系。但在求解时,又引入了表面力为未知量,控制方程属于Hamilton体系。因而,本文提出在比例边界有限元离散方法的基础上,利...     

平面柔性多体系统正碰撞动力学建模理论研究

针对目前柔性多体系统碰撞动力学建模方法存在的不足,对影响碰撞动力学仿真的主要因素如柔性体建模和碰撞初始条件进行分析,建立起基于变约束的柔性体碰撞动力学方程。首先,为了解决子结构法在处理碰撞界面搜索时面临的难题,引入多体系统柔性体有限元描述方法,推导出凸形柔性体接触点间法向位移约束的二阶导数形式。其次,从碰撞引起的接触界面速度不连续机理出发,结合连续介质力学间断面理论,给出碰撞瞬时由物体本身物理性质决定的接触位置处速度跳跃公式。最后对两弹性圆盘低速碰撞问题进行数值仿真。结果表明本文提出的改进方法符合力学基本原理,仿真结果满足收敛性要求。     

AN EXACT ELEMENT METHOD FOR THE BENDING OF NONHOMOGENEOUS REISSNER’S PLATE

In this paper,based on the step reduction method and exact analytic method,a new method,the exact element method for constructing finite element,is presented.Since the new method doesn’t need variational principle,it can be applied to solve non-positive and positive definite partial differential equations with arbitrary variable coefficients.By this method,a triangle noncompatible element with15 degrees of freedom is derived to solve the bending of nonhomogenous Reissner’s plate.Because the displacement parameters at the nodal point only contain deflection and rotation angle.it is convenient to deal with arbitrary boundary conditions.In this paper,the convergence of displacement and stress resultants is proved.The element obtained by the present method can be used for thin and thick plates as well,Four numerical examples are given at the end of this paper,which indicates that we can obtain satisfactory results and have higher numerical precision.     

冲击动力系统的鲁棒稳定性分析

考虑冲击动力系统的ｋ－ｐ周期运动的鲁棒稳定性问题。首先，根据微分方程的解、冲击条件和衔接条件，应用迭代法给出了系统存在ｋ－ｐ周期运动的充分必要条件，并利用稳定性的等价原理，通过周期运动的扰动差分方程导出其稳定条件；然后，着重对含有不确定参数的冲击动力系统的ｋ－ｐ周期运动的稳定性进行了分析，得出了鲁棒稳定的充分条件，文末给出了用于阐明理论结果的算例。     

Solve a contact problem by optimization method—to calculate the stresses for the softwheel in a harmonic gear

It is difficult to solve the contact problem by usual finite element program. In this paper, we express the contact problem as an optimization problem. In this form we do not need to know all boundary condition in advance. We only need to know the constraint conditions. This method is especially good for solving contact problem. Using this method, we calculate the stresses of the softwheel in the harmonic gear given by Shanghai Jiaotong University, and the results are in good agreement with the experimental results.     

The BEM analysis on bending fracture problem of thin plate

In this paper, based on Betti's reciprocal theorem, a set of boundary integral equations of thin plate with a crack is introduced. These boundary integral equations can be used to solve the bending fracture problem of thin plate. In the analysis process a higher-order singular integral equation will be induced. Using the concept of the Hadamard's principal value the higher-order singular integral can be solved conveniently. By this method we have found the analytical solution of a thin plate with a straight crack under pure bending load and indicated how to use the BEM to analyze the bending and fracture problem for thin plate with a straight crack. At the end of the paper some numerical examples are given. Numerical results show the accuracy and efficiency of the algorithms.     

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.     

Determination of a system consisting of 2<Emphasis Type="Italic">n</Emphasis> shock pairs realizing the total pressure maximum

The problem of a symmetric system consisting of 2n pairs of intersecting shock waves in a plane breaking duct which realizes the maximum total pressure is solved for given Mach numbers upstream of the leading shocks and downstream of the closing shocks provided that in each pair consisting of impinging and reflected waves the flow turning angles are equal in absolute values and have opposite directions. The corresponding necessary conditions of optimality of this shock-wave system, which constitutes a system of nonlinear algebraic equations, are obtained. An efficient iteration method of solving this system of equations, which makes it possible to solve the above-mentioned problem with high accuracy, is developed. An approximate analytic solution is obtained for large n. The results of solving the problem make it possible to select the optimum configuration of the plane internal-compression duct.     

Transient analysis of Cosserat rod with inextensibility and unshearability constraints using the least-squares finite element model

In this study, time-dependent fully discretized least-squares finite element model is developed for the transient response of Cosserat rod having inextensibility and unshearability constraints to simulate a surgical thread in space. Starting from the kinematics of the rod for large deformation, the linear and angular momentum equations along with constraint conditions for the sake of completeness are derived. Then, the α-family of time derivarive approximation is used to reduce the governing equations of motion to obtain a semi-discretized system of equations, which are then fully discretized using the least-squares approach to obtain the non-linear finite element equations. Newton׳s method is utilized to solve the non-linear finite element equations. Dynamic response due to impulse force and time-dependent follower force at the free end of the rod is presented as numerical examples.     

Dynamics of a novel nonlinear SIR model with double epidemic hypothesis and impulsive effects

In this paper, the propagation of a nonlinear delay SIR epidemic using the double epidemic hypothesis is modeled. In the model, a system of impulsive functional differential equations is studied and the sufficient conditions for the global attractivity of the semi-trivial periodic solution are drawn. By use of new computational techniques for impulsive differential equations with delay, we prove that the system is permanent under appropriate conditions. The results show that time delay, pulse vaccination, and nonlinear incidence have significant effects on the dynamics behaviors of the model. The conditions for the control of the infection caused by viruses A and B are given.     

Representation of Normal Cone Inclusion Problems in Dynamics Via Non-linear Equations

Analyzing non-smooth mechanical systems requires often the solution of inclusion problems of normal cone type. These problems arise for example in the event-driven or time-stepping simulation approaches. Such inclusion problems can be written as non-linear equations, which can be solved iteratively. In this paper we discuss three different methods to derive the non-linear equations representing the inclusions arising in the event-driven simulation approach. First, we formulate inclusions describing the individual non-smooth constraints and solve them successively. Secondly, we interpret the non-linear equations as the conditions for the saddle point of the augmented Lagrangian function. As a third possibility we discuss the exact regularization of set-valued force laws. All three methods lead to the same numerical scheme, but give different insight into the problem. Especially the factor r occurring in the non-linear equations is discussed. Two iterative methods for solving the non-linear equations are presented together with some remarks on convergence.     

Similarity in stationary motions of gas and two-phase medium with incompressible component

In this paper we suggest the transformation between the equations for a perfect gas and the equations describing in one-velocity approach the two-phase medium with any volume occupied by the incompressible phase. It is proved that the motion of a two-phase medium in the transformed coordinate system is similar with certain accuracy to that of a perfect gas. It means that the solutions obtained for perfect gas can be used to solve wave problems for media with incompressible component. There is no necessity directly to solve the problem for medium with incompressible component, and it is only sufficient to transform the known solution of the similar problem for a homogeneous medium. Thus, the solutions of many hydrodynamic problems for multi-component media with incompressible phase can be obtained without solving the original set of equations. The scope for the suggested transformation is demonstrated by reference to the strong explosion in a two-phase medium.     

The equations of motion for a nonholonomic system under the action of impulsive forces

The equations of impact for a nonholonomic system described with generalized coordinated have been discussed in detail in the general references of classical dynamics. But these equations contain undetermined multipliers which made the problem complicated. Through the appropriate treatment of mathematics, using the -function and expression of matrix in this paper, the author derived equations of impact for a nonholonomic system without undetermined multipliers. Therefore, the problem can be solved more simply.     

二维弹性接触问题中随机边界元法与可靠性分析

发展了二维弹性接触问题中的随机边界元法，推导并建立了相应的随机边界元基本方程，并将所发展的方法用于静强度的可靠性分析，讨论了其数值解技术。通过算例分析表明，本文发展的方法是可行的。     

The Kinematics and the Full Minimal Dynamic Model of a 6-DOF Parallel Robot Manipulator

In this paper we present a particular architecture of parallel robots which has six-degrees-of-freedom (6-DOF) with only three limbs. The particular properties of the geometric and kinematic models with respect to that of a classical parallel robot are presented. We show that inverse problems have an analytical solution. However, to solve the direct problems, an efficient numerical procedure which needs to inverse only a 3 × 3 passive Jacobian matrix is proposed. In a second step, dynamic equations are derived using the Lagrangian formalism where the joint variables are passive and active joint coordinates. Based on the geometrical properties of the robot, the equations of motion are derived in terms of only nine coordinates related by three kinematic constraints instead of 18 joint coordinates. The computational cost of the dynamic model obtained is reduced by using a minimum set of base inertial parameters.