作平面运动的二维平面板的热耦合动力学问题

研究受到热冲击的作平面运动的二维平面板的温度场和动力学响应.考虑温度场和应变场的耦合,通过计算熵密度,建立了平面板的热传导变分方程,此外,在本构关系式中考虑热应变,基于平面应力假设建立了作平面运动的二维平面板的动力学变分方程.用有限元方法分别对温度场和变形场进行离散.最后,计算了作直线平动的平面板的温度场和变形场,揭示了温度和变形的相互耦合特征,体现了板运动对温度和变形的影响.     

Forward dynamical analysis of flexible multibody systems using system-level model reduction

Fast simulation (e.g., real-time) of flexible multibody systems is typically restricted by the presence of both differential and algebraic equations in the model equations, and the number of degrees of freedom required to accurately model flexibility. Model reduction techniques can alleviate the problem, although the classically used body-level model reduction and general-purpose system-level techniques do not eliminate the algebraic equations and do not necessarily result in optimal dimension reduction. In this research, Global Modal Parametrization, a model reduction technique for flexible multibody systems is further developed to speed up simulation of flexible multibody systems. The reduction of the model is achieved by projection on a curvilinear subspace instead of the classically used fixed vector space, requiring significantly less degrees of freedom to represent the system dynamics with the same level of accuracy. The numerical experiment in this paper illustrates previously unexposed sources of approximation error: (1) the rigid body motion is computed in a forward dynamical analysis resulting in a small divergence of the rigid body motion, and (2) the errors resulting from the transformation from the modal degrees of freedom of the reduced model back to the original degrees of freedom. The effect of the configuration space discretization coarseness on the different approximation error sources is investigated. The trade-offs to be defined by the user to control these approximation errors are explained.     

悬臂梁大变形的向量式有限元分析

为分析悬臂梁的几何非线性行为,用向量式有限元法将结构离散成质点系以及质点间的连接单元.根据牛顿第二定律得到每个质点在内力和外载荷作用下的运动方程以及悬臂梁在每个时刻的变形用该时刻质点系的运动表示.结合刚架元的节点内力和等效质量得出质点位移的迭代计算公式,采用FORTRAN编制计算程序,对悬臂梁分别承受集中载荷和弯矩下的大变形进行算例分析.计算结果与理论解吻合较好,表明该方法能很好地模拟分析悬臂梁的大变形.     

Finite element analysis of viscous,incompressible fluid flow: Part 1 : Basic methodology

A numerical procedure is developed for the analysis of general two-dimensional flows of viscous, incompressible fluids using the finite element method. The partial differential equations describing the continuum motion of the fluid are discretized by using an integral energy balance approach in conjunction with the finite element approximation. The nonlinear algebraic equations resulting from the discretization process are solved using a Picard iteration technique.A number of computational procedures are developed that allow significant reductions to be made in the computational effort required for the analysis of many flow problems. These techniques include a coarse-to-fine-mesh rezone procedure for the detailed study of regions of particular interest in a flow field and a special finite element to model far-field regions in external flow problems.     

A distributed parameter model for the dynamics of flexible-link robots

In this paper a model is developed for kinematic and dynamic analysis of flexible robots undergoing general three-dimensional motion. For modeling robotic links, distributed mass and flexibility are considered without discretization. Some modeling issues are discussed, and parameters characterizing the real design of a robot are introduced into the analysis. The concept of a fictitious rigid link is presented to consider the rigid body motion of a link separately, and to account for possibly complex link shapes. Based on Jourdain's principle, an alternative formulation is proposed to derive the dynamic equations of flexible robots. The equations of motion are developed and analyzed in detail. The vibrations of links are described by linear, inhomogeneous partial differential equations, with homogeneous, nonlinear, time-dependent boundary conditions. © 1998 John Wiley & Sons, Inc.     

Rigid body dynamics with a scalable body,quaternions and perfect constraints

In this paper, we present a formulation of the quaternion constraint for rigid body rotations in the form of a standard perfect bilateral mechanical constraint, for which the associated Lagrangian multiplier has the meaning of a constraint force. First, the equations of motion of a scalable body are derived. A scalable body has three translational, three rotational, and one uniform scaling degree of freedom. As generalized coordinates, an unconstrained quaternion and a displacement vector are used. To the scalable body, a perfect bilateral constraint is added, restricting the quaternion to unit length and making the body rigid. This way a quaternion based differential algebraic equation (DAE) formulation for the dynamics of a rigid body is obtained, where the 7×7 mass matrix is regular and the unit length restriction of the quaternion is enforced by a mechanical constraint. Finally, the equations of motion in the form of a DAE are linked to the Newton–Euler equations of motion of a rigid body. The rigid body DAE formulation is useful for the construction of (energy) consistent integrators.     

3-D beam element of composite cross section including warping and shear deformation effects

In this paper a boundary element method is developed for the construction of the 14 × 14 stiffness matrix and the nodal load vector of a member of arbitrary homogeneous or composite cross section taking into account both warping and shear deformation effects. The composite member consists of materials in contact each of which can surround a finite number of inclusions. To account for shear deformations, the concept of shear deformation coefficients is used. In this investigation the definition of these factors is accomplished using a strain energy approach. Seven boundary value problems are formulated and solved employing a pure BEM approach, that is only boundary discretization is used. The evaluation of the shear deformation coefficients is accomplished from stress functions using only boundary integration. Numerical results are presented to illustrate the method and demonstrate its efficiency and accuracy. The influence of the warping effect especially in composite members of open form cross section is analyzed through examples demonstrating the importance of the inclusion of the warping degrees of freedom in the analysis of a space frame. Moreover, the discrepancy of both the deflections and the internal forces of a member of a spatial structure arising from the ignorance of the shear deformation effect necessitates the inclusion of this additional effect, especially in thick walled cross section members.     

The elastic problem for fractal media: basic theory and finite element formulation

In a previous paper [Comput. Methods Appl. Mech. Eng. 190 (2001) 6053], the framework for the mechanics of solids, deformable over fractal subsets, was outlined. Anomalous mechanical quantities with fractal dimensions were introduced, i.e., the fractal stress [σ∗], the fractal strain [ε∗] and the fractal work of deformation W∗. By means of the local fractional operators, the static and kinematic equations were obtained, and the Principle of Virtual Work for fractal media was demonstrated. In this paper, the constitutive equations of fractal elasticity are put forward. From the definition of the fractal elastic potential φ∗, the linear elastic constitutive relation is derived. The physical dimensions of the second derivatives of the elastic potential depend on the fractal dimensions of both stress and strain. Thereby, the elastic constants undergo positive or negative scaling, depending on the topological character of deformation patterns and stress flux. The direct formulation of elastic equilibrium is derived in terms of the fractional Lamé operators and of the equivalence equations at the boundary. The variational form of the elastic problem is also obtained, through minimization of the total potential energy. Finally, discretization of the fractal medium is proposed, in the spirit of the Ritz-Galerkin approach, and a finite element formulation is obtained by means of devil’s staircase interpolating splines.     

Relative Finite Element Coordinates in Multibody System Simulation

In the paper a numerical approach for deriving the nonlinear explicitform dynamic equations of rigid and flexible multibody systems ispresented. The dynamic equations are obtained as Ordinary DifferentialEquations for generalized coordinates and without algebraic constraints.The Finite Element Theory is applied for discretization of flexiblebodies. The minimal set of the generalized coordinates includesindependent joint motions, as well as independent small flexibledeflections of finite element nodes. The node deflections and stiffnessmatrices are calculated with respect to the moving relative coordinatesystems of the flexible bodies. The positions and orientations ofelement and substructure coordinate systems are updated according to thenode deflections. A major step of the numerical process is the kinematicanalysis and calculation of matrices of partial derivatives of thequasi-coordinates (dependent joint motions and coordinates of points andnodes) with respect to the generalized coordinates. The inertia terms inthe dynamic equations are obtained multiplying the matrices of thepartial derivatives by the mass matrices of the rigid and flexiblebodies. Stiffness properties of flexible bodies are presented in thedynamic equations by stiff forces that depend on the generalizedrelative flexible deflections only. Several examples of large motion ofbeam structures show the effectiveness of the algorithm.     

Study of the ligament tension and cross-section deformation using nonlinear finite element/multibody system algorithms

In this investigation, the effects of the knee-joint movements on the ligament tension and cross-section deformation are examined using large displacement nonlinear finite element/multibody system formulations. Two knee-joint models that employ different constitutive equations and significantly different deformation kinematics are developed and implemented to analyze the ligament dynamics in a computational solution procedure that integrates large displacement finite element and multibody system algorithms. The first model employs a lower fidelity large displacement cable element that does not capture the cross-section deformations and allows for using only nonlinear classical beam theory with a linear Hookean material law instead of a general continuum mechanics approach. In the second model, a higher fidelity large displacement beam model that captures more coupled deformation modes including Poisson modes as well the cross-section deformation is used. This higher fidelity model also allows for a straight forward implementation of general nonlinear constitutive models, such as Neo Hookean material laws, based on a general continuum mechanics approach. Cauchy stress tensor and Nanson’s formula are used to obtain an accurate expression for the ligament tension forces, which as shown in this investigation depend on the ligament cross section deformation. The two models are implemented in a general multibody system algorithm that allows introducing general constraint and force functions. The finite element/multibody system computational algorithm used in this investigation is based on an optimum sparse matrix structure and ensures that the kinematic constraint equations are satisfied at the position, velocity, and acceleration levels. The results obtained in this investigation show that models that ignore coupled deformation modes including some Poisson modes and the cross-section deformations can lead to inaccurate prediction of the ligament forces. These simpler models, as demonstrated in this investigation, can be used to obtain only simplified expressions for the ligament tensions. A three-dimensional knee-joint model that consists of five bodies including two flexible bodies that represent the medial collateral ligament (MCL) and lateral collateral ligament (LCL) is used in the numerical comparative study presented in this paper. The large displacement procedure presented in this investigation can be applied to other types of Ligaments, Muscles, and Soft Tissues (LMST) in biomechanics applications.     

Discrete hill's equations

In the work, based on the formulas of discrete mechanics in a rotating frame, a discretization of classical Hill's equations of the moon motion is worked out. It is proved that the proposed discretization conserves Jacobi's constant of motion. For computational purposes an algorithm to the solution of obtained discrete Hill's equations is given.     

Analysis of Large Flexible Body Deformation in Multibody Systems Using Absolute Coordinates

To consider large deformation problems in multibody system simulations afinite element approach, called absolute nodal coordinate.formulation,has been proposed. In this formulation absolute nodal coordinates andtheir material derivatives are applied to represent both deformation andrigid body motion. The choice of nodal variables allows a fullynonlinear representation of rigid body motion and can provide the exactrigid body inertia in the case of large rotations. The methodology isespecially suited for but not limited to modeling of beams, cables andshells in multibody dynamics.This paper summarizes the absolute nodal coordinate formulation for a 3D Euler–Bernoulli beam model, in particular the definition of nodal variables, corresponding generalized elastic and inertia forces and equations of motion. The element stiffness matrix is a nonlinear function of the nodal variables even in the case of linearized strain/displacement relations. Nonlinear strain/displacement relations can be calculated from the global displacements using quadrature formulae.Computational examples are given which demonstrate the capabilities of the applied methodology. Consequences of the choice of shape.functions on the representation of internal forces are discussed. Linearized strain/displacement modeling is compared to the nonlinear approach and significant advantages of the latter, when using the absolute nodal coordinate formulation, are outlined.     

Flexible multibody modeling of reeving systems including transverse vibrations

This paper presents a discretization procedure for the flexible multibody modeling of reeving systems. Reeving systems are assumed to include a set of rigid bodies connected by wire ropes using a set of sheaves and reels. The method is capable to model the deformation of the varying-length wire-rope spans. Wire ropes are assumed to deform axially, transversally and in torsion. This paper shows the capability of the presented method to model transverse vibrations. The discretization procedure uses a combination of absolute position coordinates, relative-transverse deformation coordinates and longitudinal material coordinates. Each wire-rope span is modeled using a single two-noded element under an arbitrary Lagrangian–Eulerian approach. The discretization method is validated using analytical and numerical reference solutions found in the literature that describe the dynamics of varying-length strings. In addition, the dynamics of a three-dimensional tower crane is simulated.     

A network model for rigid-body motion

In the formulation of equations of motion of three-dimensional mechanical systems, the techniques utilized and developed to analyze the electrical networks based on linear graph theory can conveniently be used. The success of this approach, however, relies on the availability of a complete and adequate mathematical model of the rigid body valid in the three-dimensional motion. This article is devoted to the derivation of such a mathematical model for the rigid body as a (k + 1)-port component. In this derivation, the dynamic properties of the rigid body are automatically included as a consequence of the analytical procedures used in the article. In this model, a general form of the terminal equations is given. In many applications, however, its special form, also given in this article, is used.     

弹性细杆弯扭度有突变时的Lagrange方程

根据弹性细杆静力学的Kirchhoff动力学比拟方法,将弹性细杆截面的弯扭度和形心应变矢有突变的弹性变形比拟为动力学中的打击运动现象.分别从精确Cosserat弹性细杆和Kirchhoff弹性细杆静力学的Lagrange方程出发,导出了弯扭度和形心应变矢有突变时的Lagrange方程,其形式与打击运动的Lagrange方程形式相同.分析了弯扭度和形心应变矢的突变对挠曲线光滑性的影响.为弹性细杆弯扭度有突变时的平衡分析提供分析力学方法.     

Explicit approximate inverse preconditioning techniques

Summary  The numerical treatment and the production of related software for solving large sparse linear systems of algebraic equations, derived mainly from the discretization of partial differential equation, by preconditioning techniques has attracted the attention of many researchers. In this paper we give an overview of explicit approximate inverse matrix techniques for computing explicitly various families of approximate inverses based on Choleski and LU—type approximate factorization procedures for solving sparse linear systems, which are derived from the finite difference, finite element and the domain decomposition discretization of elliptic and parabolic partial differential equations. Composite iterative schemes, using inner-outer schemes in conjunction with Picard and Newton method, based on approximate inverse matrix techniques for solving non-linear boundary value problems, are presented. Additionally, isomorphic iterative methods are introduced for the efficient solution of non-linear systems. Explicit preconditioned conjugate gradient—type schemes in conjunction with approximate inverse matrix techniques are presented for the efficient solution of linear and non-linear system of algebraic equations. Theoretical estimates on the rate of convergence and computational complexity of the explicit preconditioned conjugate gradient method are also presented. Applications of the proposed methods on characteristic linear and non-linear problems are discussed and numerical results are given.     

Rigid-flexible coupling dynamics of three-dimensional hub-beams system

In the previous research of the coupling dynamics of a hub-beam system, coupling between the rotational motion of hub and the torsion deformation of beam is not taken into account since the system undergoes planar motion. Due to the small longitudinal deformation, coupling between the rotational motion of hub and the longitudinal deformation of beam is also neglected. In this paper, rigid-flexible coupling dynamics is extended to a hub-beams system with three-dimensional large overall motion. Not only coupling between the large overall motion and the bending deformation, but also coupling between the large overall motion and the torsional deformation are taken into account. In case of temperature increase, the longitudinal deformation caused by the thermal expansion is significant, such that coupling between the large overall motion and the longitudinal deformation is also investigated. Combining the characteristics of the hybrid coordinate formulation and the absolute nodal coordinate formulation, the system generalized coordinates include the relative nodal displacement and the slope of each beam element with respect to the body-fixed frame of the hub, and the variables related to the spatial large overall motion of the hub and beams. Based on precise strain-displacement relation, the geometric stiffening effect is taken into account, and the rigid-flexible coupling dynamic equations are derived using velocity variational principle. Finite element method is employed for discretization. Simulation of a hub-beams system is used to show the coupling effect between the large overall motion and the torsional deformation as well as the longitudinal deformation. Furthermore, conservation of energy in case of free motion is shown to verify the formulation.     

A Continuum Mechanical Approach to Geodesics in Shape Space

In this paper concepts from continuum mechanics are used to define geodesic paths in the space of shapes, where shapes are implicitly described as boundary contours of objects. The proposed shape metric is derived from a continuum mechanical notion of viscous dissipation. A geodesic path is defined as the family of shapes such that the total amount of viscous dissipation caused by an optimal material transport along the path is minimized. The approach can easily be generalized to shapes given as segment contours of multi-labeled images and to geodesic paths between partially occluded objects. The proposed computational framework for finding such a minimizer is based on the time discretization of a geodesic path as a sequence of pairwise matching problems, which is strictly invariant with respect to rigid body motions and ensures a 1–1 correspondence along the induced flow in shape space. When decreasing the time step size, the proposed model leads to the minimization of the actual geodesic length, where the Hessian of the pairwise matching energy reflects the chosen Riemannian metric on the underlying shape space. If the constraint of pairwise shape correspondence is replaced by the volume of the shape mismatch as a penalty functional, one obtains for decreasing time step size an optical flow term controlling the transport of the shape by the underlying motion field. The method is implemented via a level set representation of shapes, and a finite element approximation is employed as spatial discretization both for the pairwise matching deformations and for the level set representations. The numerical relaxation of the energy is performed via an efficient multi-scale procedure in space and time. Various examples for 2D and 3D shapes underline the effectiveness and robustness of the proposed approach.     

BDDC preconditioning for high-order Galerkin Least-Squares methods using inexact solvers

A high-order Galerkin Least-Squares (GLS) finite element discretization is combined with a Balancing Domain Decomposition by Constraints (BDDC) preconditioner and inexact local solvers to provide an efficient solution technique for large-scale, convection-dominated problems. The algorithm is applied to the linear system arising from the discretization of the two-dimensional advection–diffusion equation and Euler equations for compressible, inviscid flow. A Robin–Robin interface condition is extended to the Euler equations using entropy-symmetrized variables. The BDDC method maintains scalability for the high-order discretization of the diffusion-dominated flows, and achieves low iteration count in the advection-dominated regime. The BDDC method based on inexact local solvers with incomplete factorization and p = 1 coarse correction maintains the performance of the exact counterpart for the wide range of the Peclet numbers considered while at significantly reduced memory and computational costs.