离散时间系统Ляпунов函数的构造与稳定性判定

本文旨在以矩阵的相似变换理论为基础,以电子计算机为工具,提供一种无需求解离散?矩阵方程,而直接由系统矩阵来构造离散时间系统?函数和判定系统稳定性的方法.     

A novel approach with smallest transition matrix for milling stability prediction

The introduction of chatter limits the processing efficiency of milling. Chatter prediction is an off-line strategy to select chatter-free cutting parameters. This paper presents a novel method for prediction of milling chatter which aims to reduce the dimension of transition matrix of the discrete map, where the eigenvalues of the transition matrix determine the system stability by using Floquet theory. A linear weight function is introduced when the weighted residual method is applied to the delay differential equation on discrete time intervals. Thus the displacement item can be removed from the state vectors. When the number of discrete intervals is fixed, it is concluded that the transition matrix obtained by the proposed method is the smallest among the time-domain methods. Meanwhile, the acceleration continuity condition is naturally satisfied on discrete time nodes which endows the method with competitive accuracy.     

多尺度嵌入式离散裂缝模型模拟方法

天然裂缝性油藏和人工压裂油藏内裂缝形态多样,分布复杂,传统的离散裂缝模型将裂缝作为基岩网格的边界,采用非结构化网格进行网格划分,其划分过程复杂,计算量大。嵌入式离散裂缝模型划分网格时不需要考虑油藏内的裂缝形态,只需对基岩系统进行简单的网格剖分,可以大大降低网格划分的复杂度,从而提高计算效率。然而,在油藏级别的数值模拟和人工压裂裂缝下的产能分析中,仍然存在计算量巨大、模拟时间过长的问题。本文提出嵌入式离散裂缝模型的多尺度数值计算格式,使用多尺度模拟有限差分法研究嵌入式离散裂缝模型渗流问题。通过在粗网格上求解局部流动问题计算多尺度基函数,多尺度基函数可以捕捉裂缝与基岩间的相互关系,反映单元内的非均质性,因此该方法既有传统尺度升级法的计算效率,又可以保证计算精度,数值结果表明这是一种有效的裂缝性油藏数值模拟方法。     

多体系统传递矩阵法研究进展

作为一种多体系统动力学新方法, 多体系统传递矩阵法由于其无需系统总体动力学方程和快速计算的特点, 已被广泛用于各种多管火箭、自行火炮、舰炮等复杂大型机械系统动力学分析与设计. 本文介绍了该方法的研究进展, 包括: 线性多体系统传递矩阵法、多体系统离散时间传递矩阵法、二维系统传递矩阵法、受控多体系统传递矩阵法、多体系统传递矩阵法和通常动力学方法的混合方法等, 给出了该方法解决自行火炮、多管火箭武器多体系统动力学的重大工程应用实例.      

Efficient approximation of Sparse Jacobians for time‐implicit reduced order models

This paper introduces a sparse matrix discrete interpolation method to effectively compute matrix approximations in the reduced order modeling framework. The sparse algorithm developed herein relies on the discrete empirical interpolation method and uses only samples of the nonzero entries of the matrix series. The proposed approach can approximate very large matrices, unlike the current matrix discrete empirical interpolation method, which is limited by its large computational memory requirements. The empirical interpolation indices obtained by the sparse algorithm slightly differ from the ones computed by the matrix discrete empirical interpolation method as a consequence of the singular vectors round‐off errors introduced by the economy or full singular value decomposition (SVD) algorithms when applied to the full matrix snapshots. When appropriately padded with zeros, the economy SVD factorization of the nonzero elements of the snapshots matrix is a valid economy SVD for the full snapshots matrix. Numerical experiments are performed with the 1D Burgers and 2D shallow water equations test problems where the quadratic reduced nonlinearities are computed via tensorial calculus. The sparse matrix approximation strategy is compared against five existing methods for computing reduced Jacobians: (i) matrix discrete empirical interpolation method, (ii) discrete empirical interpolation method, (iii) tensorial calculus, (iv) full Jacobian projection onto the reduced basis subspace, and (v) directional derivatives of the model along the reduced basis functions. The sparse matrix method outperforms all other algorithms. The use of traditional matrix discrete empirical interpolation method is not possible for very large dimensions because of its excessive memory requirements. Copyright © 2016 John Wiley & Sons, Ltd.     

DIFFERENCE DISCRETE SYSTEM OF EULER-BEAM WITH ARBITRARY SUPPORTS AND SIGN-OSCILLATORY PROPERTY OF STIFFNESS MATRICES

The difference discrete system of Euler-beam with arbitrary supports was constructed by using the two order central difference formulas. This system is equivalent to the spring-mass-rigidrod model. By using the theory of oscillatory matrix, the sign-oscillatory property of stiffness matrices of this system was proved, and the necessary and sufficient condition for the system to be positive was obtained completely.     

充液弯管连续/离散模型振动的动刚度解法

由充液弯管三维振动模型切入,应用动刚度法构建了弯管及直管单元的振动求解方法,进而用于组装求解充液管系的振动,可同时适用于含弯管单元的连续模型或只含直管单元的离散模型;通过算例对比,证明动刚度法比传递矩阵法和有限元法在计算效率和精度上有所提升;与充液L型管道振动实验测得的加速度频响曲线对比,验证了本文对于管道组装的计算方法的有效性,此外还分析了连续模型和离散模型的区别及适用范围。     

滚动体几何相似条件在轴承边界元法中的应用

滚动轴承作为多物体接触系统,具有高度的非线性。为了求解滚动轴承的三维载荷分布,考虑所有滚动体的形状和边界条件相同,利用几何相似条件通过初始位置滚动体的离散模型,得到其他滚动体的离散数据。将所有滚动体看作是一个物体,轴承系统可以用3个物体无摩擦接触边界元法进行模拟计算。考虑全部滚动体作为一个整体的特殊性,叙述了边界积分方程和耦合的矩阵方程的建立,编制了基于滚动体几何相似条件的轴承边界元法Fortran源程序。利用该程序对轧机四列圆锥滚子轴承进行数值模拟,得到轴承的接触压力和载荷,滚动体接触宽度和接触滚动体个数,并说明了方法的有效性。     

完整系统的多刚体系统离散时间传递矩阵法研究

基于多刚体系统离散时间传递矩阵法，采用提高计算精度的方法，研究具有大运动、非线性特征的完整系统在平面、空间中的动力学响应。提出了对部分变量重新赋值的违约修正方法，计算机仿真表明了其有效性。多刚体系统离散时间传递矩阵法不须进行违约修正，体现了该方法建模灵活性较强、程式化程度较高的优点。     

Development of a family of explicit algorithms for structural dynamics with unconditional stability

A new family of explicit integration algorithms is developed based on discrete control theory for solving the dynamic equations of motion. The proposed algorithms are explicit for both displacement and velocity and require no factorisation of the damping matrix and the stiffness matrix. Therefore, for a system with nonlinear damping and stiffness, the proposed algorithms are more efficient than the common explicit algorithms that provide only explicit displacement. Accuracy and stability properties of the proposed algorithms are analysed theoretically and verified numerically. Certain subfamilies are found to be unconditionally stable for any system state (linear elastic, stiffness softening or stiffness hardening) that may occur in earthquake engineering of a practical structure. With dual explicit expression and excellent stability property, the proposed family of algorithms can potentially solve complicated nonlinear dynamic problems.     

Calculation of the added mass matrix of water impact of elastic wedges by the discrete vortex method

The discrete vortex method is used to obtain the added mass matrix of water entry of elastic structures. Based on the Wagner theory, the hydrodynamic loads are evaluated by the “flat-disc” approximation. The structural deformation is described by modal decomposition and the velocity potential of the flow can be decomposed by the principal coordinates. The impacts of a rigid wedge and an elastic plate are first investigated to analytically validate our method in obtaining the added mass matrix. Then the impact of an elastic wedge made up of two beams symmetrically onto the water surface is analyzed. It is shown that the discrete vortex method can be used to obtain the added mass matrix of elastic wedge impact accurately and efficiently.     

Development of the direct stress solution technique for three-dimensional hydrodynamic models using finite elements

Velocity varies rapidly near sheared boundaries. Therefore in many practical fluid problems it can be inefficient to solve discrete equations with velocity as the dependent variable. Conversely, shear stress varies slowly near sheared boundaries, suggesting that it may be well suited for use as the dependent variable in discrete equations. This paper describes a formulation of the internal mode equations for a three-dimensional hydrodynamic model using shear stress as the dependent variable. The resulting direct stress solution (DSS), coupled with a spatial discretization using linear finite elements, yields a system matrix that can be set up and solved with the efficiency of a banded matrix with bandwidth 8. If the eddy viscosity distribution is assumed to be piecewise linear over the depth (with an arbitrary number of time-varying segments), the recovery of velocity from stress can be easily accomplished in closed form, thereby avoiding any difficulty resulting from the logarithmic singularity in the velocity profile that occurs at a boundary. Results from tidal and wind-driven test cases with realistic boundary layers are used to demonstrate the accuracy and computational advantages of a DSS formulation versus a standard velocity-based formulation.     

Precise integration method for solving singular perturbation problems

This paper presents a precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end. The method divides the interval evenly and gives a set of algebraic equations in a matrix form by the precise integration relationship of each segment. Substituting the boundary conditions into the algebraic equations, the coefficient matrix can be transformed to the block tridiagonal matrix. Considering the nature of the problem, an efficient reduction method is given for solving singular perturbation problems. Since the precise integration relationship introduces no discrete error in the discrete process, the present method has high precision. Numerical examples show the validity of the present method.     

Estimates of accuracy of approximate solutions of some combined modular elastic continuum systems with discrete interaction

At present, it happens increasingly often that one needs to study complicated modular systems with discrete interaction between the units. If a system contains a distributed unit, then it is said to be combined. One method for studying such systems is to analyze their frequency models; in this case, it is sometimes required to calculate the Green functions of the distributed unit and study the block structures generated by a model of a system with superelements. A technique for studying such systems was developed in [1] and is called the factorized perturbation method; in foreign literature, this technique is called the Green function method (e.g., see [2]).The main idea of such methods is first to construct an equivalent system in the characteristic space that arises in the discrete macrostructure of the original system as a result of the discrete interaction between the units of the latter and then to reduce the characteristic solution to the original solution by using a simple transfer relation. A typical characteristic of structural methods is their algorithmic universality, which is independent of the special form of the discrete microstructure and requires solving a characteristic matrix equation. To obtain approximate solutions by these methods, one should approximately calculate the Green functions of the distributed units and explicitly indicate their operating frequency ranges ( see [3, 4]). The concept of structural methods is to obtain a system solution by using a priori studied elements.There are a variety of methods for studying combined modular systems. In view of this, the accuracy of the approximate solutions, which are also obtained by numerous methods, should be estimated from the algorithmic standpoint. At present, this is done at the level of practical-empirical considerations like doubling the number of grid points for the analysis of the distributed unit, taking into account additional vibration modes in the Green functions, etc.In the present paper, we describe a structural method for solving elastic one-dimensional distributed systems with discrete interaction and present an efficient scalar a priori estimate and a universal a posteriori estimate for the accuracy of an approximate solution, which can be used in many approximate methods, e.g., FEM [5].     

自由-自由运行体系动态载荷反演问题研究

对自由-自由运行体系动态载荷识别问题进行深入研究,将自由-自由运行体系振动积分方程时域离散化,建立了体系动态载荷反演的力学模型。在动态载荷反演模型求解过程中,由于结构矩阵的病态特性以及测量噪声的影响,常规最小二乘法往往失效;通过对载荷反演模型进行奇异值分解,指出该病态问题的本质,并提出相应的正则化求解方法。数值仿真表明了本文方法能够得到满足工程要求的稳定近似解。本文研究对于实际飞行器运行过程所受动载荷评估具有一定的意义。     

Unified proof to oscillation property of discrete beam

The oscillation property （OP） is a fundamental and important qualitative property for the vibrations of single span one-dimensional continuums such as strings, bars, torsion bars, and Euler beams. Any properly discretized continuum model should keep the OP. In literatures, the OP of discrete beam models is discussed essentially by means of matrix factorization. The discussion is model-specific and boundary-condition- specific. Besides, matrix factorization is difficult in handling finite element （FE） models of beams. In this paper, according to a sufficient condition for the OP, a new approach to discuss the property is proposed. The local criteria on discrete displacements rather than global matrix factorizations are given to verify the OP. Based on the proposed approach, known results such as the OP for the 2-node FE beams via the Heilinger- Reissener principle （HR-FE beams） as well as the 5-point finite difference （FD） beams are verified. New results on the OP for the 2-node PE-FE beams and the FE Timoshenko beams with small slenderness are given. Through a simple manipulation, the qualitative property of discrete multibearing beams can also be discussed by the proposed approach.     

提高多体系统离散时间传递矩阵法计算精度的研究

深入探讨多体系统离散时间传递矩阵法对平面、空间刚体-光滑铰多体系统运动响应的研究。提出提高该方法计算精度和计算稳定性的方法，导出相应的多端刚体传递矩阵。设计了角坐标的迭代循环，该变量不必采用近似形式，因而提高了计算精度和计算稳定性。其它方法有：1)增加泰勒展开式的高阶项；2)合理选择将速度、加速度表示为位移的线性函数的方法(本文简称线性化方法)；3)合理确定迭代循环的初值等。对4种多体系统进行了计算机仿真研究，表明本文提出的方法有效。     

Discrete time transfer matrix method for dynamics of multibody system with flexible beams moving in space

In this paper,by defining new state vectors and developing new transfer matrices of various elements moving in space,the discrete time transfer matrix method of multi-rigid-flexible-body system is expanded to study the dynamics of multibody system with flexible beams moving in space.Formulations and numerical example of a rigidflexible-body three pendulums system moving in space are given to validate the method.Using the new method to study the dynamics of multi-rigid-flexible-body system moving in space,the global dynamics equations of system are not needed,the orders of involved matrices of the system are very low and the computational speed is high,irrespective of the size of the system.The new method is simple,straightforward,practical,and provides a powerful tool for multi-rigid-flexible-body system dynamics.     

First-order perturbation solution to the complex eigenvalues

The matrix perturbation method is extended to discrete linear nonconservative system with unsymmetrical matrices in this article. By introducing the concept of the adjoint complex eigenvector and by making use of the orthogonality relationship in the complex mode theory, the first-order perturbation solution to the complex eigenvalues is derived. Numerical example shows that this method is efficient and practicable.     

On the application of mixed method to solve solids of revolution with discrete fixed supports

In this paper, the force method of statically indeterminate structure mechanics is used to treat the solids of revolution with discrete fixed supports. The reactionary forces of discrete fixed supports are considered as statically indeterminate unknown variables. The force-method canonical equations, in which the coefficient matrix and the right-hand vector are computed by semi-analytical finite element method, are solved. Then the finite element solution of solid of revolution with discrete fixed supports is calculated with the external loads superposed from the assigned external loads and the reactionary forces of discrete supports.