非线性有限元的若干基本问题

本文介绍了非线性有限元中的若干基本问题。其中包括有关应变、应力和非线性平衡方程的一些基本概念,基于不同非线性广义变分原理的位移模式、杂交模式和拟协调模式几何非线性有限元及其在壳体屈曲问题中的应用等。      

以互信息为基础的广义相关系数及其应用

以往分析变量间相关性时，大多局限于仅反映线性相关程度的互相关函数或自相关函数上，但我们所研究的问题，却常常是非线性，因而对这种非线性结构上的广义相关程度进行度量显得更为重要。本文着重讨论了描述变量间这种广义相关程度的一种以互信息为基础的广义相关系数的概念、算法及操作中的技术问题，并实例其应用。     

夹层板的屈曲状态

1 引言关于夹层结构的非线性弯曲和屈曲问题的研究已有一些工作,本文根据Reissn-er 假设,首先建立了夹层板的广义变分原理,然后用有限元方法并结合延续算法获得了受压方形夹层板在大范围载荷下的屈曲状态.文中还给出了一种选取初值问题的有效方     

等直杆单元切线刚度矩阵的精确分析方法

为了有效完成大型铰接单层网壳结构的后屈曲分析,本文采用对杆单元杆端力函数求导的方法推导出了等直杆单元切线刚度矩阵的精确形式。该切线刚度矩阵不受结构小变形限制,适用于结构产生任意大结点位移情况。以六角星桁架、平面圆拱桁架和大跨K8单层网壳结构为算例,采用广义位移控制法进行非线性后屈曲分析,其中预测子采用本文杆单元切线刚度矩阵。算例分析结果表明,本文杆单元切线刚度矩阵在大型铰接单层网壳结构的非线性后屈曲分析中有很强的预测能力。     

等直杆单元切线刚度矩阵的精确分析方法

为了有效完成大型铰接单层网壳结构的后屈曲分析,本文采用对杆单元杆端力函数求导的方法推导出了等直杆单元切线刚度矩阵的精确形式。该切线刚度矩阵不受结构小变形限制,适用于结构产生任意大结点位移情况。以六角星桁架、平面圆拱桁架和大跨K8单层网壳结构为算例,采用广义位移控制法进行非线性后屈曲分析,其中预测子采用本文杆单元切线刚度矩阵。算例分析结果表明,本文杆单元切线刚度矩阵在大型铰接单层网壳结构的非线性后屈曲分析中有很强的预测能力。     

具有非线性homologous变形约束的桁架结构形态分析

研究了具有非线性homologous变形约束条件的桁架结构形态分析问题。在已有的线性homologous变形约束桁架形态分析的基础上,将结构的节点分成三类:homologous变形约束节点,形状可变节点和边界点。运用Moore-Penrose广义逆矩阵性质,将基础方程组解的存在条件表示为包含形状可变节点未知坐标的非线性方程组,为采用Newton-Raphson方法求解非线性方程组,对AA (A为任意矩阵,A 为A的Moore-Penrose广义逆矩阵)求偏导数,找到了满足保型要求的形态,给出的桁架算例说明了本文方法的有效性。     

广义van der Pol非线性振子的多吸引子共存和跳跃现象

高余维 经分岔和跳威风昌非线性动力学中的极为重要的研究内容。在本文里我们研究了广义ｖａｎ ｄｅｒ Ｐｏｌ非线性振子中的余维３经分岔、多吸引子共存和极限环振动的跳跃问题，用数值模拟方法研究了这个系统，数值结果证明了理论结果的正确性。     

广义非线性强度理论在岩石材料中的应用

在已提出的广义非线性强度理论的基础上，结合岩石材料的力学特性，建立了岩石广 义非线性强度理论，该理论在$\pi$平面上的破坏函数为介于SMP准则和Mises准则 之间的光滑曲线，在子午面上的破坏函数为幂函数曲线. 通过已有不同岩石的真三 轴试验数据对岩石广义非线性强度理论的验证表明，岩石广义非线性强度理论可以 广泛地适用于各类岩石，描述其$\pi$平面上及子午面上的非线性强度特性；并利 用5种不同类型岩石的真三轴试验结果对岩石广义非线性强度理论和Hoek-Brown准 则进行比较,反映了所提岩石广义非线性强度理论的优越性.     

层合析参固支边界条件下的不稳定后屈曲分析

为了深入地研究复合材料层板所独具的后屈曲特性,利用能量变分原理和非线性几何方程建立了具有弹性约束的复合材料层板在面内载荷作用下的非线性稳定性控制方程组,并运用广义傅立叶级数法对其进行求解。重点分析了非对称层板在固支边界条件下的稳定性问题,发现层板在此条件下有可能存在非对称的失稳临界点和不稳定的后屈曲路径,进而构造了简化的物理模型进行解释,指出后屈曲的非对称性是由于结构关于Z轴不对称,而不稳定性是由于固支边界条件阻碍了前屈曲的发生。     

非线性广义杂交元

选择新的三类变量建立了放松单元间连续性条件的用于非线性有限元分析的泛函，并由此建立了基于不协调模式的非线性广义杂交元方法。     

Postbuckling Analysis of Nonconservative Elastic Systems

Abstract

Previous work on the postbuckling and imperfection-sensitivity of elastic structures has concentrated on conservative systems. The results of Koiterand others have led to a general theory of nonlinear stability behavior for these systems. The theory must be modified when nonconservative forces are present, and this is the aim of the present paper.

Discrete, nonconservative, elastic systems which exhibit static (divergence) instability are considered. The nonlinear behavior in the neighborhood of a critical point is analyzed by means of a perturbation procedure. When the critical point is simple, the results are similar to those for conservative systems. When a coincident critical point exists, however, different types of behavior occur. In many cases there is no bifurcation at all, with only the fundamental (trivial) equilibrium path passing through the critical point. Imperfection-sensitivity is more severe than for the typical bifurcation points and can even occur when the perfect system has no bifurcation. The results are illustrated with the use of a nonlinear double pendulum model subjected to a partial follower load.     

Computing the structural buckling limit load by using dynamic relaxation method

The numerical structural analysis schemes are extensively developed by progress of modern computer processing power. One of these approximate approaches is called "dynamic relaxation (DR) method." This technique explicitly solves the simultaneous system of equations. For analyzing the static structures, the DR strategy transfers the governing equations to the dynamic space. By adding the fictitious damping and mass to the static equilibrium equations, the corresponding artificial dynamic system is achieved. The static equilibrium path is required in order to investigate the structural stability behavior. This path shows the relationship between the loads and the displacements. In this way, the critical points and buckling loads of the non-linear structures can be obtained. The corresponding load to the first limit point is known as buckling limit load. For estimating the buckling load, the variable load factor is used in the DR process. A new procedure for finding the load factor is presented by imposing the work increment of the external forces to zero. The proposed formula only requires the fictitious parameters of the DR scheme. To prove the efficiency and robustness of the suggested algorithm, various geometric non-linear analyses are performed. The obtained results demonstrate that the new method can successfully estimate the buckling limit load of structures.     

Nonlinear axisymmetric deformation of anisotropic spherical shells

A method of solving problems of nonlinear deformation of anisotropic spherical shells with consideration of critical points and postcritical behavior is outlined. The method employs the method of incremental loading in which the load increment is specified with an unknown coefficient determined as an unknown function equivalent to the other ones. The algorithm is based on the numerical discrete-orthogonalization method, which allows analyzing the deformation path for a number of shells with different anisotropy parameters     

On the equilibrium configurations of an elastically constrained rotating disk: An analytical approach

According to the linear theory of vibration for spinning disks, the backward traveling waves of some of the modes may have zero natural frequency at what are called the critical speeds. At these speeds, the linear equations of motion cannot properly predict the amplitude response of the spinning disk, and nonlinear equations of motion must be used. In this paper, geometrical nonlinear equations of motion based on Von Karman plate theory are employed to study the dynamics of an elastically constrained disk near its critical speeds. A one-mode approximation is used to examine the effect of elastic constraint on the amplitude response. Presenting the equations in a space-fixed coordinate system, this study aims to find closed-form solutions for some of the equilibrium configurations of an elastically constrained spinning disk. Also, the stability of these configurations is studied using analytical techniques. It is shown that below the critical speed, one neutrally stable equilibrium solution exists, while above it, a bifurcation occurs. In this situation, two more branches of equilibrium configurations emerge, one of which is neutrally stable and the other unstable. Closed-form expressions for the bifurcation points are obtained. Due to the effect of an elastic constraint, a bifurcation occurs and the previously neutrally stable equilibrium configuration turns unstable. Also at this bifurcation point, two more branches of equilibrium solutions emerge.     

基于Wilson-θ和Newmark-β法的非线性动力学方程改进算法

Wilson-θ法和Newmark-β法是非线性动力学方程求解的常用方法。它们的一个基本步骤是,将方程改写为增量平衡的形式,在每一个积分步长内用状态参量修正平衡方程的系数矩阵,其本质是在单个步长内对系统的非线性环节进行了线性化处理。本文基于增量思想分别改进了Wilson-θ法和Newmark-β法,根据即时解给出下一步的猜测解,然后对猜测解进行迭代校正,最终得到收敛的近似解。算例表明,改进算法的精度更高,且收敛准则简单。更为重要的是,本文方法无须对非线性项进行线性化处理,因而计算效率更高,适应范围更广。     

New Principles of Power and Energy Rate of Incremental Rate Type for Generalized Continuum Field Theories

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Improvements on the arc-length-type method

Arc-length-type and energy-type methods are two main strategies used in structural nonlinear tracing analysis, but the former is widely used due to the explicitness and clarity in conception, as well as the convenience and reliability in calculation. It is very important to trace the complete load-deflection path in order to know comprehensively the characteristics of structures subjected to loads. Unfortunately, the nonlinear analysis techniques are only workable for tracing the limit-point-type equilibrium path. For the bifurcation-point-type path, most of them cannot secure a satisfactory result. In this paper, main arc-length-type methods are reviewed and compared, and the possible reasons of failures in tracing analysis are briefly discussed. Some improvements are proposed, a displacement perturbation method and a force perturbation method are presented for tracing the bifurcation-point-type paths. Finally, two examples are analyzed to verify the ideas, and some conclusions are drawn with respect to the arc-length-type methods. The project supported by the Special Research Fund for Doctor Program of Universities (9424702)     

极限下限分析的正交基无单元Galerkin法

基于极限分析的下限定理，建立了用正交基无单元Galerkin法进行理想弹塑性结构极 限分析的整套求解算法．下限分析所需的虚拟弹性应力场可由正交基无单元Galerkin法直接 得到，所需的自平衡应力场由一组带有待定系数的自平衡应力场基矢量的线性组合进行模 拟．这些自平衡应力场基矢量可由弹塑性增量分析中的平衡迭代得到．通过对自平衡应力场 子空间的不断修正，整个问题的求解将化为一系列非线性数学规划子问题，并通过复合形法 进行求解．算例表明该方法有效地克服了维数障碍问题，使计算效率得到了充分的提高，是 切实可行的．     

非完善板屈曲路径的有限元增量摄动法

本文以Thompson一般稳定性理论为基础,提出非完善结构(有初始几何缺陷)屈曲平衡路径分析的有限元增量摄动法,它克服了增量迭代法及摄动法各自的缺点,利用增量摄动法,还建立了非完善板屈曲平衡路径的数学分析模型,对非完善矩形板在各种边界条件下的屈曲路径实现了数值程序分析,得到了符合实验的结果。     

求解预定位移水平的改进弧长法

为了在使用弧长法进行结构非线性分析时能够收敛到指定的位移,基于广义弧长的概念,对传统弧长法进行改进。改进后的方法不仅能够自动跟踪结构非线性平衡路径,同时能求得位于结构平衡路径的任意区段的任意预定位移时的受力状态。数值算例表明本文计算方法的精确度、效率以及可靠性较好。