Interval Arithmetic and Static Interval Finite Element Method

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基于单元的静力区间有限元法

在许多工程问题中 ,结构参数和荷载具有某种程度的误差或不确定性。若不将它们定量化或模型化加以考虑 ,就不能作出合理的分析和设计。考虑到有限元法在科学界和工程界的广泛应用 ,本文以连续梁结构为例 ,建立了基于单元的静力区间有限元法。为了说明本方法的有效性 ,本文给出了一个数值例子 ,并把所得结果与文献 [1 2 ]进行了比较。     

线性区间有限元静力控制方程的组合解法

区间有限元的静力控制方程常被归结为区间方程组来求解。但实际上两者并不等价。本文根据不确定结构有限元分析的力学背景，直接从问题的基本参量的不确定性出发，将基本区间参量的边界组合与求解区间方程组的有关解法相结合，提出了线性区间有限元静力控制方程的两种组合解法－参量边界全组合法和组合迭代法。可以以较小的计算量获得或逼近位移和应力区间的准确界限。且不受基本参量变化范围的限制。算例分析表明文中方法是实用和可行的。     

基于材料相变的声子晶体带隙可调结构设计与性能分析

提出一种基于材料相变的穿孔型带隙可调声子晶体结构.其结构形式为含缝隙的形状记忆合金和环氧树脂的组合体,通过温度变化诱发相变引起的形状记忆合金材料性质的变化,实现声子晶体的带隙变化;通过合理布置缝隙与形状记忆合金相变材料的位置,实现声子晶体带隙性质的可调设计.基于有限元方法,建立了可调声子晶体的分析模型,分析了形状记忆合金的填充分数以及相变等对带隙性能的影响规律.分析结果表明,通过合理设计微结构形式,材料相变可实现带隙位置和宽度的调节,同时可实现特定频段内带隙的有无.     

基于单元的子区间摄动有限元方法研究

针对区间相关性导致区间扩张的问题,探讨了区间数之间的相关性并给出了降低区间扩张的子区间摄动方法。文中给出了基于单元的子区间摄动有限元计算公式和子区间划分数目的近似计算公式,同时文中讨论了区间有限元计算精度问题,给出了可提高计算效率的一些措施。对桁架结构和平面应力问题梁结构算例分析结果表明文中方法可以达到一定的计算精度,并且是合理可行的。     

FUZZY ARITHMETIC AND SOLVING OF THE STATIC GOVERNING EQUATIONS OF FUZZY FINITE ELEMENT METHOD

The key component of finite element analysis of structures with fuzzy parameters, which is associated with handling of some fuzzy information and arithmetic relation of fuzzy variables , was the solving of the governing equations of fuzzy finite element method. Based on a given interval representation of fuzzy numbers, some arithmetic rules of fuzzy numbers and fuzzy variables were developed in terms of the properties of interval arithmetic. According to the rules and by the theory of interval finite element method, procedures for solving the static governing equations of fuzzy finite element method of structures were presented. By the proposed procedure, the possibility distributions of responses of fuzzy structures can be generated in terms of the membership functions of the input fuzzy numbers. It is shown by a numerical example that the computational burden of the presented procedures is low and easy to implement. The effectiveness and usefulness of the presented procedures are also illustrated.     

AN INTERVAL FINITE ELEMENT METHOD BASED ON THE NEUMANN SERIES EXPANSION 1)

实际工程问题中通常存在大量的不确定参数, 区间有限元方法是一种结合有限元数值计算工具对结构进行不确定性分析的区间方法. 区间有限元的目的是获得在含有区间不确定性参数条件下的结构响应上下边界, 其关键问题在于区间平衡方程组的求解, 而这属于一类往往很难求解的NP-hard问题. 本文归纳了一类工程实际中常见的结构不确定性问题, 即可线性分解式区间有限元问题, 并针对此提出一种基于Neumann级数的区间有限元方法. 在区间有限元分析中, 当区间不确定参数表示为一组独立区间变量线性叠加时, 若结构的刚度矩阵也可表示为这些独立区间变量的线性叠加形式, 则称此类区间有限元问题为可线性分解式区间有限元问题. 对于此类问题, 采用Neumann级数对其刚度矩阵的逆矩阵进行表示, 可获得结构响应关于区间变量的显式表达式, 从而可高效求解结构响应的上下边界. 最后通过两个算例验证了本文所提方法的有效性.     

一维声子晶体中声波能带的解析研究

为了对一维声子晶体的能带进行解析研究,采用一维声子晶体的色散法,推导出一维声子晶体禁带频率和通带频率的解析公式;并利用此式对一维声子晶体的能带结构进行了解析研究,得到了禁带和导带的频率中心、频率宽度分别随介质厚度和声阻抗两者的变化规律,并与转移矩阵方法的计算结果进行了比较.研究结果表明:两种方法得到的结论相同,但解析法能更准确、更深刻地反映出一维声子晶体的能带随介质厚度和声阻抗等因素的变化规律;并且解析法克服了转移矩阵法等其他方法不能对能带结构进行解析研究的不足,是一种研究一维声子晶体能带更有效的方法.     

薄壁箱梁广义坐标法刚度矩阵的推导及应用

在符拉索夫广义坐标法初参数方程的基础上,推导出可用于均布扭转荷载作用下薄壁箱梁翘曲分析的刚度矩阵,该刚度矩阵具有较高的单元精度,可用于由较多薄壁箱梁组成的复杂结构的整体有限元分析。通过对广义坐标法刚度矩阵和乌曼斯基理论、修正乌曼斯基理论求得薄壁箱梁的位移和应力进行分析比较,为各方法在实际工程中的应用提供一定的参考。     

Interval finite element method and its application on anti-slide stability analysis

The problem of interval correlation results in interval extension is discussed by the relationship of interval-valued functions and real-valued functions. The methods of reducing interval extension are given. Based on the ideas of the paper, the formulas of sub-interval perturbed finite element method based on the elements are given. The sub-interval amount is discussed and the approximate computation formula is given. At the same time, the computational precision is discussed and some measures of improving computational efficiency are given. Finally, based on sub-interval perturbed finite element method and anti-slide stability analysis method, the formula for computing the bounds of stability factor is given. It provides a basis for estimating and evaluating reasonably anti-slide stability of structures.     

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.     

基于小波有限元的车辆-轨道-桥梁系统竖向振动分析

基于小波有限元建立了车辆-轨道-桥梁系统竖向运动方程。将车辆、轨道和桥梁作为一个整体系统,钢轨和桥梁采用区间B样条小波单元离散,钢轨与桥梁之间的钢轨基础采用均布的弹簧和阻尼模拟,采用虚功原理建立了基于小波有限元的四轴车辆-轨道-桥梁竖向振动分析模型。结果表明,采用区间B样条小波单元可较大程度上减小系统的自由度数,缩减计算量,节省计算时间。     

铁路道床有砟-无砟过渡段动力特性的DEM-FEM耦合分析

针对铁路道床有砟-无砟过渡段的结构特点,采用离散元-有限元耦合模型分析散体道砟和无砟道床间过渡段的动力特性。散体道砟道床和无砟道床分别采用离散元方法 DEM和有限元方法 FEM模拟,而在过渡段将道砟颗粒嵌入无砟道床以增加道砟颗粒与无砟道床间的咬合力,并在离散元和有限元耦合区域实现了力学参数的传递。采用以上DEM-FEM耦合方法对有砟-无砟道床及其过渡段在列车荷载作用下的沉降过程进行了数值分析。计算结果表明,离散元方法中道砟颗粒间的力链呈现非对称梯形分布,其与有限元方法中的应力分布趋势一致;采用嵌入式道砟颗粒的方法可以增加有砟-无砟过渡段道砟间的咬合力,有效约束道砟颗粒的位移,减少有砟-无砟道床间的沉降差异。本文计算模型可以合理地分析有砟道床的力链分布以及无砟道床的应力分布,确定列车荷载下道床有砟-无砟过渡段的动力学行为。     

配点型区间有限元法

在分析Taylor展开``点逼近'区间有限元法不足的基础上, 提出了基于Chebyshev第一类正交多项式全局逼近目标函数的配点型区间有限元法. 该方法不需要计算目标函数对不确定性变量的灵敏度, 不要求不确定性变量的变化范围为小区间, 并适合求解目标函数为不确定变量非线性函数的情形. 目标函数正交展开式的系数采用Gauss-Chebyshev求积公式得到,故需要在不确定性变量所在区间内配置高斯积分点. 计算目标函数在高斯点的取值是该方法的主要工作量, 当不确定性变量数为m, 并选用高斯十点法进行积分时, 需要对系统进行12$m$次分析. 算例表明, 在其他区间有限元法失效的情况下, 配点型区间有限元法依然能够得到几乎精确的区间界限.      

Optimum design of phononic crystal perforated plate structures for widest bandgap of fundamental guided wave modes and maximized in-plane stiffness

This paper presents a topology optimization of single material phononic crystal plate (PhP) to be produced by perforation of a uniform background plate. The primary objective of this optimization study is to explore widest exclusive bandgaps of fundamental (first order) symmetric or asymmetric guided wave modes as well as widest complete bandgap of mixed wave modes (symmetric and asymmetric). However, in the case of single material porous phononic crystals the bandgap width essentially depends on the resultant structural integration introduced by achieved unitcell topology. Thinner connections of scattering segments (i.e. lower effective stiffness) generally lead to (i) wider bandgap due to enhanced interfacial reflections, and (ii) lower bandgap frequency range due to lower wave speed. In other words higher relative bandgap width (RBW) is produced by topology with lower effective stiffness. Hence in order to study the bandgap efficiency of PhP unitcell with respect to its structural worthiness, the in-plane stiffness is incorporated in optimization algorithm as an opposing objective to be maximized. Thick and relatively thin Polysilicon PhP unitcells with square symmetry are studied. Non-dominated sorting genetic algorithm NSGA-II is employed for this multi-objective optimization problem and modal band analysis of individual topologies is performed through finite element method. Specialized topology initiation, evaluation and filtering are applied to achieve refined feasible topologies without penalizing the randomness of genetic algorithm (GA) and diversity of search space. Selected Pareto topologies are presented and gradient of RBW and elastic properties in between the two Pareto front extremes are investigated. Chosen intermediate Pareto topology, even not extreme topology with widest bandgap, show superior bandgap efficiency compared with the results reported in other works on widest bandgap topology of asymmetric guided waves, available in the literature. Finally, steady state and transient frequency response of finite thin PhP structures of selected Pareto topologies are studied and validity of obtained bandgaps is confirmed.     

Dispersion error reduction for acoustic problems using the smoothed finite element method (SFEM)

The smoothed finite element method (SFEM), which was recently introduced for solving the mechanics and acoustic problems, uses the gradient smoothing technique to operate over the cell‐based smoothing domains. On the basis of the previous work, this paper reports a detailed analysis on the numerical dispersion error in solving two‐dimensional acoustic problems governed by the Helmholtz equation using the SFEM, in comparison with the standard finite element method. Owing to the proper softening effects provided naturally by the cell‐based gradient smoothing operations, the SFEM model behaves much softer than the standard finite element method model. Therefore, the SFEM can significantly reduce the dispersion error in the numerical solution. Results of both theoretical and numerical experiments will support these important findings. It is shown clearly that the SFEM suits ideally well for solving acoustic problems, because of the crucial effectiveness in reducing the dispersion error. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

平面广义四节点等参元GQ4及其性能探讨

广义节点有限元是将传统有限元方法中的节点广义化,在不增加节点个数的前提下,仅通过提高广义节点的插值函数的阶次,从而达到提高有限元解精度的目的.与现有的p型和hp型有限元不同,在这种新的有限元中,节点自由度全部定义在节点处,在理论与程序实现上与传统有限元方法具有很好的相容性,传统有限元方法是这种新方法的广义节点退化为0阶时的特殊情形.文中主要讨论了这一新方法的四节点等参元（记为GQ4）的形式.对GQ4进行的各种数值试验表明,所发展的广义四节点等参单元具有精度高且无剪切自锁与体积自锁等的特点.     

点式压电智能结构的模型修正与振动主动控制

提出了在点式压电智能结构中应用摄动有限元方法对结构的有限元模型进行修正 ,从而达到提高建模精度 ,改善实际结构振动主动控制效果的目的。通过对一悬臂梁在模型修正前后进行振动主动控制的不同的控制效果验证了该方法的有效性     

扩展有限元法(XFEM)及其应用

扩展有限元法(extended finite element method,XFEM)是1999年提出的一种求解不连续力学问题的数值方法, 它继承了常规有限元法(CFEM)的所有优点, 在模拟界面、裂纹生长、复杂流体等不连续问题时特别有效, 短短几年间得到了快速发展与应用. XFEM与CFEM的最根本区别在于, 它所使用的网格与结构内部的几何或物理界面无关, 从而克服了在诸如裂纹尖端等高应力和变形集中区进行高密度网格剖分所带来的困难, 模拟裂纹生长时也无需对网格进行重新剖分.重点介绍XFEM的基本原理、实施步骤及应用实例等, 并进行必要的评述. 单位分解概念保证了XFEM的收敛, 基于此, XFEM通过改进单元的形状函数使之包含问题不连续性的基本成分, 从而放松对网格密度的过分要求. 水平集法是XFEM中常用的确定内部界面位置和跟踪其生长的数值技术, 任何内部界面可用它的零水平集函数表示. 第2和第3节分别简要介绍单位分解法和水平集法;第4节和第5节介绍XFEM的基本思想、详细实施步骤和若干应用实例, 同时修正了以往文献中的一些不妥之处; 最后, 初步展望了该领域尚需进一步研究的课题.