Projection methods in flexible multibody dynamics. Part II: Dynamics and recursive projection methods

In Part I of this paper the kinematic relationships between the absolute, elastic and joint accelerations are developed. In this paper, these kinematic equations are used with the generalized Newton-Euler equations and the relationship between the actual and generalized reaction forces to develop a recursive projection algorithm for the dynamic analysis of open-loop mechanical systems consisting of a set of interconnected rigid and deformable bodies. Optimal matrix permutation, partitioning and projection methods are used to eliminate the elastic accelerations while maintaining the inertia coupling between the rigid body motion and the elastic deformation. Recursive projection methods are then applied in order to project the inertia of the leaf bodies onto their parent bodies. This leads to an optimal symbolic factorization which recursively yields the absolute and joint accelerations, and the joint reaction forces. The method presented in this paper avoids the use of Newton-Raphson algorithms in the numerical solution of the constrained dynamic equations of open-loop kinematic chains since the joint accelerations are readily available from the solution of the resulting reduced system of equations. Furthermore, the method requires only the inversion or decomposition of relatively small matrices and the numerical integration of a minimum number of co-ordinates. Open-loop multibody robotic manipulator systems are used to compare the results and efficiency of the recursive methods with that of the augmented formulations that employ Newton-Raphson algorithms.     

A simple and efficient preconditioning scheme for heaviside enriched XFEM

The extended finite element method (XFEM) is an approach for solving problems with non-smooth solutions, which arise from geometric features such as cracks, holes, and material inclusions. In the XFEM, the approximate solution is locally enriched to capture the discontinuities without requiring a mesh which conforms to the geometric features. One drawback of the XFEM is that an ill-conditioned system of equations results when the ratio of volumes on either side of the interface in an element is small. Such interface configurations are often unavoidable, in particular for moving interface problems on fixed meshes. In general, the ill-conditioning reduces the performance of iterative linear solvers and impedes the convergence of solvers for nonlinear problems. This paper studies the XFEM with a Heaviside enrichment strategy for solving problems with stationary and moving material interfaces. A generalized formulation of the XFEM is combined with the level set method to implicitly define the embedded interface geometry. In order to avoid the ill-conditioning, a simple and efficient scheme based on a geometric preconditioner and constraining degrees of freedom to zero for small intersections is proposed. The geometric preconditioner is computed from the nodal basis functions, and therefore may be constructed prior to building the system of equations. This feature and the low-cost of constructing the preconditioning matrix makes it well suited for nonlinear problems with fixed and moving interfaces. It is shown by numerical examples that the proposed preconditioning scheme performs well for discontinuous problems and \(C^0\) -continuous problems with both the stabilized Lagrange and Nitsche methods for enforcing the continuity constraint at the interface. Numerical examples are presented which compare the condition number and solution error with and without the proposed preconditioning scheme. The results suggest that the proposed preconditioning scheme leads to condition numbers similar to that of a body-fitted mesh using the traditional finite element method without loss of solution accuracy.     

An arbitrary Lagrangian‐Eulerian framework with exact mass conservation for the numerical simulation of 2D rising bubble problem

An arbitrary Lagrangian‐Eulerian framework, which combines the advantages of both Lagrangian and Eulerian methods, is presented to solve incompressible multiphase flow problems. The incompressible Navier‐Stokes equations are discretized using the side‐centered unstructured finite volume method, where the velocity vector components are defined at the midpoint of each cell face, while the pressure term is defined at element centroids. The pressure field is treated to be discontinuous across the interface with the discontinuous treatment of density and viscosity. The surface tension term at the interface is treated as a force tangent to the interface and computed with several different approaches including the use of Legendre polynomials. In addition, the several different discretizations of interface kinematic boundary conditions are investigated. For the application of the interface kinematic boundary condition, a special attention is given to satisfy both local and global discrete geometric conservation law to conserve the total mass of both species at machine precision. The mesh vertices are deformed by solving the linear elasticity equations due to the normal displacement of interface. The resulting algebraic equations are solved in a fully coupled manner, and a one‐level restricted additive Schwarz preconditioner with a block‐incomplete factorization within each partitioned subdomain is used for the resulting fully coupled system. The method is validated by simulating the classical benchmark problem of a single rising bubble in a viscous fluid due to buoyancy. The results of numerical simulations are found out to be in an excellent agreement with the earlier results in the literature. The mass of the bubble is conserved, and discontinuous pressure field is obtained to avoid errors due to the incompressibility condition in the vicinity of the interface, where the density and viscosity jumps occur.     

星轮偏心误差对浮动式星型齿轮传动动态特性的影响

以太阳轮浮动式星型齿轮传动系统为研究对象,基于集中参数理论,建立了星型传动广义动力学模型,建模中考虑了齿轮制造偏心误差、时变啮合刚度以及间隙浮动机构等因素。采用数值解法对系统的动力学微分方程进行求解,获得了系统的受迫振动响应,利用时间历程、相平面、Poincare截面图及Fourier频谱分析了系统的动态特性。着重研究各星轮偏心误差及间隙浮动机构对星型轮系动态特性的影响规律。结果表明:星轮偏心误差增强了系统振动;不同位置、不同数量的星轮偏心误差作用,对应的系统动态响应不同;间隙浮动结构影响了系统的稳定性,不利于振动噪声的控制。     

Optimal design of dynamically loaded continuous structures

A computational algorithm is developed and applied for optimization of beam and plate structures, subject to constraints on transient dynamic response. A continuous design formulation is retained, with dynamic response governed by partial differential operator equations. Adjoint equations are employed for sensitivity analysis and a function space gradient projection optimization approach is presented. Finite element analysis methods are applied for solution of the system dynamic and adjoint differential equations. Displacement constrained beam and plate minimum weight examples are solved, with a variety of boundary conditions.     

Solution of frictional contact problems using ILU and coarse/fine preconditioners

In this paper we discuss the efficiency of two preconditioners (incomplete factorization and coarse/fine factorization) to solve non symmetric and ill-conditioned linear systems arising from the resolution of frictional contact problems. Mixed formulation is introduced and generalized Newton method is reviewed. We describe a multilevel interpolation adapted to the coarse/fine preconditioner. Numerical examples are proposed to show strong and weak points associated to both methods and optimal approximations are given to improve their efficiency.     

Geometrical numerical algorithms for a plasticity model with Armstrong–Frederick kinematic hardening rule under strain and stress controls

In this paper, we approach the numerical integration problem of a plasticity model with the Armstrong–Frederick kinematic hardening rule on back stress through a combination of the techniques of integral representation and geometrical integrator. First, the internal symmetry group of the constitutive model is investigated. Then, we develop two geometrical integrators for strain control and stress control, respectively. These integrators are obtained by a discretization of the integral representation of the constitutive equations and an exponential approximation of the quasilinear differential equations system for the relative stress, which guarantee to retain the consistency condition exactly without the need for any iterations. Some numerical examples are used to assess the performance of the new algorithms. The measures in terms of stress relative errors and also isoerror maps confirm that our schemes are superior to the classical radial return methods. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical solution of a parabolic system with coupled nonlinear boundary conditions

A numerical technique has been developed to solve a system that consists of m linear parabolic differential equations with coupled nonlinear boundary conditions. Such a system may represent chemical reactions, chemical lasers and diffusion problems. An implicit finite difference scheme is adopted to discretize the problem, and the resulting system of equations is solved by a novel technique that is a modification of the cyclic odd–even reduction and factorization (CORF) algorithm. At each time level, the system of equations is first reduced to m nonlinear algebraic equations that involve only the m unknown grid points on the nonlinear boundary. Newton's method is used to determine these m unknowns, and the corresponding Jacobian matrix can be computed and updated easily. After convergence is achieved, the remaining unknowns are solved directly. The efficiency of this technique is illustrated by the numerical computations of two examples previously solved by the cubic spline Galerkin method.     

脉冲风洞测力系统建模与载荷辨识方法研究

将载荷辨识技术应用于脉冲燃烧风洞模型测力。用子结构综合法建立了测力试验系统的动力学模型,在时域内将动力学方程进行离散,建立起天平测量信号与模型气动载荷历程之间的线性关系,作为载荷辨识的模型。采用Tikhonov正则化和子空间投影法相结合的混合正则化方法,将高维的、不适定的载荷辨识问题转化为低维的适定问题,以利于快速求解。提出了一种新方法来确定合适的投影子空间维数,然后应用L曲线准则来寻找低维正则化问题的最优正则化参数。最后通过算例验证了系统建模方法的精度和载荷辨识算法的有效性与稳定性。     

The method of fundamental solutions for fracture mechanics—Reissner's plate application

The application of the method of fundamental solutions (MFS), a mesh-free technique, to solve cracked Reissner's plates is discussed in this work. Here, the numerical Green's function (NGF) previously developed by the authors is used as the fundamental solution required by the method. Stress intensity factors or the related force intensity factors are obtained using the generalized crack openings at a single point near the tip, computed through a summation of the fundamental generalized openings at that point weighted by their influence factors. Despite the ill-conditioning of the equations system, which may require appropriate handling to solve (such as the singular value decomposition method), examples show good results for problems with embedded cracks. The method can be a good option to evaluate stress intensity factors of given problems due to its simple and intuitive implementation.     

基于建模误差位置识别的有限元模型修正方法

提出了一种修正有限元模型的方法。将模型修正分为两步，首先用矩阵型方法求出系统质量、刚度矩阵的误差矩阵，通过定义的误差百分比确定建模误差位置；然后针对建模误差比较大的单元（子结构）依据特征方程和固有振型对质量矩阵的加权正交条件建立参数修正方程组，方程求解采用奇异值分解法。算例表明它能有效地避免先前方法中参数修正的盲目性，修改结果物理意义明确，精度较好。     

周边固支的矩形板的动力耦合热弹性问题分析

本文研究的是周边固支的矩形板上表面受均匀分布热流冲击的动力耦合热弹性问题，在热冲击过程中，板的周边温度场保持稳定，下表面绝热。采用算子法将热传导方程由三维降为二维问题，以和二维的热弹性动力方程相协调，二方程相互耦合，故用带补充项的双重富里叶级数消去对坐标的微分项，再用拉普拉斯变换消去方程中对时间的微分项，然后尝试用加权最佳逼近法作数值拉普拉斯逆变换，给出问题在原象空间的解。为防止误差的扩大，本文利用正交齐异分解法求解方程，并对计算结果中动力项和耦合项的影响作出分析。     

Parallel block-preconditioned monolithic solvers for fluid-structure interaction problems

In this work, we consider the solution of fluid-structure interaction (FSI) problems using a monolithic approach for the coupling between fluid and solid subproblems. The coupling of both equations is realized by means of the arbitrary Lagrangian-Eulerian framework and a nonlinear harmonic mesh motion model. Monolithic approaches require the solution of large ill-conditioned linear systems of algebraic equations at every Newton step. Direct solvers tend to use too much memory even for a relatively small number of degrees of freedom and, in addition, exhibit superlinear growth in arithmetic complexity. Thus, iterative solvers are the only viable option. To ensure convergence of iterative methods within a reasonable amount of iterations, good and, at the same time, cheap preconditioners have to be developed. We study physics-based block preconditioners, which are derived from the block-LDU factorization of the FSI Jacobian, and their performance on distributed memory parallel computers in terms of two- and three-dimensional test cases permitting large deformations.     

A compliant track link model for high‐speed,high‐mobility tracked vehicles

Several modelling methods have recently been developed for the dynamic analysis of low‐speed tracked vehicles. These methods were used to demonstrate the significant effect of the force of the interaction between the track links and vehicle components, even when low speeds are considered. It is the objective of this investigation to develop compliant track link models and investigate the use of these models in the dynamic analysis of high‐speed, high‐mobility tracked vehicles. There are two major difficulties encountered in developing the compliant track models discussed in this paper. The first is due to the fact that the integration step size must be kept small in order to maintain the numerical stability of the solution. This solution includes high oscillatory signals resulting from the impulsive contact forces and the use of stiff compliant elements to represent the joints between the track links. The characteristics of the compliant elements used in this investigation to describe the track joints are measured experimentally. A numerical integration method having a relatively large stability region is employed in order to maintain the solution accuracy, and a variable step size integration algorithm is used in order to improve the efficiency. The second difficulty encountered in this investigation is due to the large number of the system equations of motion of the three‐dimensional multibody tracked vehicle model. The dimensionality problem is solved by decoupling the equations of motion of the chassis subsystem and the track subsystems. Recursive methods are used to obtain a minimum set of equations for the chassis subsystem. Several simulations scenarios including an accelerated motion, high‐speed motion, braking, and turning motion of the high‐mobility vehicle are tested in order to demonstrate the effectiveness and validity of the methods proposed in this investigation. Copyright © 2000 John Wiley & Sons, Ltd.     

Frequency‐domain analysis of time‐integration methods for semidiscrete finite element equations—part I: Parabolic problems

Time‐integration methods for semidiscrete equations emanating from parabolic differential equations are analysed in the frequency domain. The discrete‐time transfer functions of three popular methods are derived, and subsequently the forced response characteristics of single modes are studied in the frequency domain. To enable consistent comparison of the frequency responses of different algorithms, three characteristic numbers are identified. Frequency responses and L₂‐norms of the phase and magnitude errors are compared for the three time‐integration algorithms. The examples demonstrate that frequency‐domain analysis provides substantial insight into the time‐domain properties of time‐integration algorithms. Copyright © 2001 John Wiley & Sons, Ltd.     

中厚椭球壳自由振动动力刚度法分析

介绍精确动力刚度法分析中厚椭球壳自由振动具体实施方法,据环向波数不同将中厚椭球壳自由振动分解为一系列确定环向波数的一维振动;利用控制方程Hamilton形式建立动力刚度关系,用常微分方程求解器COLSYS求解控制方程获得单元动力刚度,用Wittrick-Williams算法求得该环向波数下椭球壳自振频率。数值算例给出中厚圆球壳及椭球壳不同边界条件的自振频率,验证动力刚度法高效、可靠、精确。     

Projection methods in flexible multibody dynamics. Part I: Kinematics

In this paper a recursive projection method for the dynamic analysis of open-loop mechanical systems that consist of a set of interconnected deformable bodies is presented. The configuration of each body in the system is identified using a coupled set of reference and elastic co-ordinates. The absolute velocities and accelerations of leaf or child bodies in the open-loop system are expressed in terms of the absolute velocities and accelerations of the parent bodies and the time derivatives of the relative co-ordinates of the joints between the bodies. The dynamic differential equations of motion are developed for each link using the generalized Newton-Euler equations. The relationship between the actual joint reactions and the generalized forces combined with the kinematic relationships and the generalized Newton-Euler equations are used to develop a system of loosely coupled equations which has a sparse matrix structure. Using matrix partitioning and recursive projection techniques based on optimal block factorization an efficient solution for the system accelerations and joint reaction forces is obtained. This solution technique yields a much smaller operations count and can more effectively exploit vectorization and parallel processing. It also allows a systematic procedure for decoupling the joint and elastic accelerations.     

The solution of ill-conditioned linear systems arising from Fredholm equations of the first kind by steepest descents and conjugate gradients

It is shown that the methods of steepest descents and conjugate gradients give smooth solutions of the ill-conditioned systems of linear equations obtained by the discretization of Fredholm equations of the first kind without the use of any regularization techniques. A modification of the usual method of steepest descents is described which is equivalent to applying two cycles of the usual method with very little additional work. A number of examples are worked including one with a degenerate kernel in which the algebraic system is singular.     

Probabilistic approach to manipulator kinematics and dynamics

A high performance, high speed robotic arm must be able to manipulate objects with a high degree of accuracy and repeatability. As with any other physical system, there are a number of factors causing uncertainties in the behavior of a robotic manipulator. These factors include manufacturing and assembling tolerances, and errors in the joint actuators and controllers. In order to study the effect of these uncertainties on the robotic end-effector and to obtain a better insight into the manipulator behavior, the manipulator kinematics and dynamics are modeled using a probabilistic approach. Based on the probabilistic model, kinematic and dynamic performance criteria are defined to provide measures of the behavior of the robotic end-effector. Techniques are presented to compute the kinematic and dynamic reliabilities of the manipulator. The effects of tolerances associated with the various manipulator parameters on the reliabilities are studied. Numerical examples are presented to illustrate the procedures.     

由路面不平顺引起的车桥响应频谱分析

本文用频谱分析的方法研究了车桥系统由路面不平顺引起的动力响应.为定性地分析车桥系统提供了一个新的方法,文中视车桥为两个子系统,用选代法解出它们的频谱响应,避免了以往所使用的数值积分分析法。从分析车桥系统传递函数的频谱入手,研究了车桥系统的基频与车速的关系。并且分析了车桥的质量比,固有频率比,车速以及阻尼对系统动力稳定性的影响,从而导出较有利的行车速度。