离散元法及其耦合算法的研究综述

介绍了离散单元法的基本理论及其研究现状,以及离散单元法与有限单元法、边界单元法、界面单元法等数值计算方法耦合的研究现状和最新进展,并讨论了离散单元法今后的发展趋势及亟待解决的问题.     

DEM与FEM动态耦合算法研究

离散单元法作为一种有效的数值分析方法,能够模拟脆性材料的裂纹扩展及碎片飞散等破坏特性,但是无法从根本上克服计算效率低下的诟病;传统有限单元法具有计算高效稳定的优点,却难以描述脆性材料冲击破坏过程中连续体向非连续体的转化。本文首先提出一种基于罚函数法的改进型离散单元和有限单元耦合方法,以提高耦合分析精度。在此基础上提出了动态耦合算法:即在初始阶段,模型全部为有限单元,当局部即将发生破坏时,仅使即将发生破坏的有限单元及相邻单元自动转化为离散单元,在离散单元区域研究破坏问题。这种算法充分利用有限单元法计算高效的优点,同时最大限度克服了离散单元法计算效率的不足。最后,通过两个简单算例验证了改进型耦合算法和动态耦合算法的有效性。     

圆筒内旋转细长管与流体界面耦合的数值方法

圆筒内旋转细长管是石油钻采工程中特有结构,细长管不仅与圆筒发生碰撞接触,还与管内流体和管外环空流体耦合,是一个复杂的非线性流固耦合系统。细长管固体域离散成梁单元,采用非线性碰撞接触动力学方程描述;管内外流体离散成六面体单元,采用计算流体动力学方程描述,在耦合界面处用任意拉格朗日欧拉法动网格来处理运动界面。根据梁单元位移...     

针对有砟道床动力特性分析的嵌入式离散元-有限元耦合方法

在离散元-有限元耦合方法中,离散元和有限元交界面处的耦合方式对整体有砟道床的力学行为影响显著.采用基于球形单元的镶嵌单元或粘结单元模拟有砟道床时,由于球形单元和有限单元表面的自锁能力较差,使道砟层在列车载荷作用下容易产生侧向滑移,导致数值模型不稳定.此外,在实际铁路道床中,底部道砟均不同程度地嵌入路堤.为此,发展了一种嵌入式离散元-有限元耦合方法,通过设置一层嵌入地基有限元模型中的球形颗粒传递离散元域和有限元域间的力学参数,实现离散元和有限元方法的耦合.数值结果表明,嵌入式离散元-有限元耦合模型能够有效降低有砟道床的侧向位移,数值结果更加稳定,在处理与有砟道床类似的连续介质与散体介质的耦合问题时推荐采用嵌入式耦合算法.     

基础-饱和土地基耦合系统分析的微分求积单元法

研究了基础-饱和土地基耦合系统的动力学特性.首先根据多孔介质理论,在小变形的假设下,分别建立了耦合系统中不可压流体饱和土地基和弹性基础的运动微分方程以及相应的边界条件,连接条件和初始条件;然后在空间域采用微分求积单元法对基础-饱和土地基的控制方程进行离散,并提供了正确处理耦合系统界面之间连接条件和间断性条件的方法,从而得到时间域内的一组代数-微分方程;接着运用隐式二阶向后差分格式处理了代数-微分方程组;最后利用牛顿迭代法数值求解了该方程组,得到了耦合系统的数值解,考察了所布节点数和参数对数值结果的影响.     

基于离散单元法和人工神经网络的近壁颗粒动力学特征研究

颗粒与壁面的相互作用往往对颗粒流动具有显著影响. 为研究颗粒与壁面作用机理, 对滚筒内颗粒流动过程进行离散单元法(DEM)数值模拟. 基于模拟结果统计分析靠近壁面处颗粒的运动特征, 结果表明, 小摩擦系数时颗粒平动和旋转速度均近似满足正态分布, 但由于壁面影响, 摩擦系数增大时颗粒沿滚筒轴向的旋转速度偏离正态分布, 颗粒动力学理论推导壁面边界条件时应考虑速度正态分布的修正及速度脉动的各向异性. 采用人工神经网络(ANN)构建了颗粒无因次旋转温度、滑移速度和平动温度之间的函数模型, 进而可以在常规双流模型壁面边界条件中考虑颗粒旋转的影响. 基于DEM模拟及结果分析可以为壁面边界条件的理论构造和半经验修正提供基础数据和封闭模型.      

考虑局部切向约束接触问题的直接刚度法

针对工程中大量存在的切向滑移受到约束的接触问题,提出了基于Lagrange乘子的点-面及点-点接触直接刚度法,该方法生成的接触协调条件可以直接组装到结构刚度阵中,从而可直接用于考虑存在结构几何非线性及材料非线性的接触问题分析中。采用这一算法进行了考虑拱坝横缝张合效应的地震响应分析,结果表明考虑切向滑移约束对接触中的张开度有明显影响。     

离散颗粒集合体-Cosserat连续体模型的连接尺度方法

基于针对分子动力学-Cauchy连续体模型提出的连接尺度方法(BSM)[1,2],发展了耦合细尺度上基于离散颗粒集合体模型的离散单元法(DEM)和粗尺度上基于Cosserat连续体模型的有限元法(FEM)的BSM。仅在有限局部区域内采用DEM以从细观层次模拟非连续破坏现象,而在全域则采用花费计算时间和存储空间较少的FEM。通过连接尺度位移(包括平移和转动)分解,和基于作用于Cosserat连续体有限元节点和颗粒集合体颗粒形心的离散系统虚功原理,得到了具有解耦特征的粗细尺度耦合系统运动方程。讨论和提出了在准静态载荷条件下粗细尺度域的界面条件,以及动态载荷条件下可以有效消除粗细尺度域界面上虚假反射波的非反射界面条件(NRBC)。本文二维数值算例结果说明了所提出的颗粒材料BSM的可应用性和优越性,及所实施界面条件对模拟颗粒材料动力学响应的有效性。     

弹性力学的重构核粒子边界无单元法与有限元的耦合法

将重构核粒子边界无单元法(RKP-BEFM)与有限元法(FEM)耦合,形成求解具有区域特征的弹性力学问题的重构核粒子边界无单元与有限元的耦合方法RKP-BEF/FE.推导了重构核粒子边界无单元与有限元耦合方法的离散化公式,建立了节点未知量的耦合方程.重构核粒子边界无单元法和有限单元法的较高精度保证了这一直接耦合方法的成功实现与求解精度.最后给出了平面问题的数值算例,验证了提出的耦合方法RKP-BEF/FE的有效性.     

框架结构P-△效应分析的微分求积单元法

采用一种新的数值方法——微分求积单元法分析框架结构的P-△效应。微分求积单元法采用微分求积法直接求解微分方程的技术，并结合有限分割技术而形成。首先建立考虑剪切变形和轴力二阶效应的框架结构单元平衡微分方程，通过微分求积离散而得到梁单元的一般弹性刚度方程；同时考虑变形后节点的平衡条件和变形协调条件，导出框架结构整体二阶分析的微分求积单元法力学模型。由于该分析模型中包括了单元及结构的所有离散形式的控制方程，因此采用该模型进行结构分析可得出较为精确的解。数值算例的分析比较，表明了该法用于框架结构P-△效应分析的正确性和有效性。本文导出的框架结构二阶分析的微分求积单元法力学模型可用于框架结构剪切变形与几何非线性的耦合效应分析。     

Nonlinear coupling modeling and dynamics analysis of rotating flexible beams with stretching deformation effect

Dynamic coupling modeling and analysis of rotating beams based on the nonlinear Green-Lagrangian strain are introduced in this work. With the reservation of the axial nonlinear strain, there are more coupling terms for axial and transverse deformations. The discretized dynamic governing equations are obtained by using the finite element method and Lagrange’s equations of the second kind. Time responses are conducted to compare the proposed model with other previous models. The stretching deforma...     

A fictitious domain approach for the Stokes problem based on the extended finite element method

In the present work, we propose to extend to the Stokes problem a fictitious domain approach inspired by extended finite element method and studied for the Poisson problem in a paper of Renard and Haslinger of 2009. The method allows computations in domains whose boundaries do not match. A mixed FEM is used for the fluid flow. The interface between the fluid and the structure is localized by a level‐set function. Dirichlet boundary conditions are taken into account using Lagrange multiplier. A stabilization term is introduced to improve the approximation of the normal trace of the Cauchy stress tensor at the interface and avoid the inf‐sup condition between the spaces for the velocity and the Lagrange multiplier. Convergence analysis is given, and several numerical tests are performed to illustrate the capabilities of the method. Copyright © 2013 John Wiley & Sons, Ltd.     

CRACK FACE CONTACT PROBLEM ANALYSIS USING THE SCALED BOUNDARY FINITE ELEMENT METHOD

比例边界有限元侧面上有任意荷载时,将侧面载荷分解成关于径向方向局部坐标的多项式函数的和,推导给出了考虑侧面载荷存在的新型形函数,并基于该形函数推导了刚度矩阵和等效节点载荷列阵.首次对比例边界有限元法求解裂纹面接触问题进行了研究,运用Lagrange乘子引入接触界面约束条件,推导给出了比例边界有限元求解裂纹面接触问题的控制方程.将裂纹面单元分为非裂尖单元和含有侧面的裂尖单元.在非裂尖单元中的裂纹面,裂纹面作为多边形单元的边界,边界上的接触力可等效到节点上,通过在节点上构造Lagrange乘子,采用点对点接触约束进行处理.对于含有侧面的裂尖单元,在整个侧面上构造Lagrange乘子的插值场,采用边对边接触约束进行处理.对三个不同的接触约束状态下的算例进行了数值计算,通过与解析解及有限元软件ABAQUS计算结果的对比,验证了本文提出的比例边界有限元点对点和边对边接触求解裂纹面接触问题的精确性与有效性.     

浅曲梁大挠度分析的余能有限元方法

以高玉臣提出的弹性大变形余能原理为基础,利用Lagrange乘子,放松平衡方程和力边界条件对余能泛函的约束,推导出广义的余能原理.根据极分解定理,将变形分为刚性转动和纯变形两部分,则余能也包含相应的两部分,一部分与刚性转动有关,而另一部分与纯变形有关.使用线弹性本构关系,建立了可用于几何非线性计算的有限元模型.应用更新的Lagrange列式法,给出了增量形式的有限元公式.数值计算结果表明,该方法可用于浅曲粱的几何大变形计算.     

A coupled interface-multilayer approach for mixed mode delamination and contact analysis in laminated composites

An accurate laminate model developed by using multilayer shear deformable plate modeling and interface elements, based on fracture mechanics and contact mechanics, is proposed to analyze mixed mode delamination in composite laminates. Perfect adhesion along the undelaminated portion of the delamination plane is simulated by treating interface stiffnesses as penalty parameters, whereas to enforce interface displacement continuity between plate elements constituting each sub-laminate above or below the delamination plane, the Lagrange multiplier method is used. The governing differential equations are derived through a variational procedure by using a modified total potential energy functional. Results are obtained by numerical integration of the non-linear three-point boundary value problem modeling mixed-mode delamination of the laminate plate subjected to end loading, which accounts also for the frictionless contact condition.The coupling of a penalty procedure with the Lagrange multiplier method, results in an accurate and direct energy release rate evaluation. Comparisons with results available from the literature obtained with a local continuum approach, show that mode partition may be performed to the desired accuracy by refining multilayer plate models for each sub-laminate. In addition, original analytical formulas for mode partition are obtained by coupling the interface approach and fracture mechanics concepts, evidencing the effectiveness of the proposed approach and gaining a better insight into the influence of shear effects on mode decomposition. Numerical computations for practical problems, evidence both the relative simplicity and the efficiency of the proposed model to represent mixed mode interlaminar fracturing as well as crack–face interaction.     

A new Finite Element Method for magneto-dynamical problems: two-dimensional results

A mixed Lagrange finite element technique is used to solve the Maxwell equations in the magneto-hydrodynamic (MHD) limit in an hybrid domain composed of vacuum and conducting regions. The originality of the approach is that no artificial boundary condition is enforced at the interface between the conducting and the insulating regions and the non-conducting medium is not approximated by a weakly conducting medium as is frequently done in the literature. As a first evaluation of the performance of the method, we study two-dimensional (2D) configurations, where the flow streamlines of the conducting fluid are planar, i.e., invariant in one direction, and either the magnetic field (“magnetic scalar” case) or the electric field (“electric scalar” case) is parallel to the invariant direction. Induction heating, eddy current generation, and magnetic field stretching are investigated showing the usefulness of finite element methods to solve magneto-dynamical problems with complex insulating boundaries.     

模拟相变问题的精细积分边界元法

将精细积分边界元法和界面追踪法相结合求解相变问题。因为边界元法只需要将待求解空间域的边界离散,方便连续追踪移动界面位置和重构网格,所以边界元法适合应用于移动边界问题的模拟。首先,利用精细积分边界元法在固相区域和液相区域分别求解相应的瞬态热传导控制方程,从而求得温度场和边界热流密度。然后,根据固-液相变界面上的能量平衡方程,利用热流密度求得相变界面的移动速度,再采用界面追踪法预测移动相变界面的位置变化。最后,给出了几个数值算例,并通过与参考解的对比验证本文方法的准确性。     

基于非重叠Mortar方法的比例边界等几何分析

在比例边界等几何分析的建模剖分过程中会出现交界面网格非匹配现象,给计算分析带来困难。为了能够处理此类问题,提出了基于非重叠Mortar方法的比例边界等几何分析。该方法能在将全域分解为若干子域时,针对每个子域分别建模和剖分网格,交界面网格无需逐点匹配;采用交叉点修正的非均匀有理B-样条基函数构造Lagrange乘子空间,子域交界面连续条件可通过Mortar条件满足;并根据交界面连续条件进行自由度凝聚,得到对称正定的系数矩阵,可直接求解。分片试验、U形结构和半无限空间上的柔性基础等数值算例验证了本文方法的有效性,计算精度满足要求。     

A parallel finite element sliding mesh technique for the simulation of viscous flows in agitated tanks

A parallel sliding mesh algorithm for the finite element simulation of viscous fluid flows in agitated tanks is presented. Lagrange multipliers are used at the sliding interfaces to enforce the continuity between the fixed and moving subdomains. The novelty of the method consists of the coupled solution of the resulting velocity–pressure‐Lagrange multipliers system of equations by an ILU(0)‐QMR solver. A penalty parameter is introduced for both the interface and the incompressibility constraints to avoid pivoting problems in the ILU(0) algorithm. To handle the convective term, both the Newton–Raphson scheme and the semi‐implicit linearization are tested. A penalty parameter is introduced for both the interface and the incompressibility constraints to avoid the failure of the ILU(0) algorithm due to the lack of pivoting. Furthermore, this approach is versatile enough so that it allows partitioning of sliding and fixed subdomains if parallelization is required. Although the sliding mesh technique is fairly common in CFD, the main advantage of the proposed approach is its low computational cost due to the inexpensive and parallelizable calculations that involve preconditioned sparse iterative solvers. The method is validated for Couette and coaxial stirred tanks. Copyright © 2011 John Wiley & Sons, Ltd.     

自由单元法及其在结构分析中的应用

通过吸收有限元与无网格法的优点,提出了一种新的数值方法------自由单元法.此方法在离散方面,采用有限元法中的等参单元,表征几何形状和进行物理量的插值;在算法方面,采用单元配点技术,逐点产生系统方程.主要特点是,在每个配置点只需要一个和周围自由选择的节点而形成的一个独立的等参单元,因而不需要考虑物理量在单元之间的相互连接关系与导数连续性问题. 本文介绍强形式与弱形式两种自由单元法,前者直接由控制方程和边界条件直接产生系统方程,后者通过在自由单元上建立控制方程的加权余量式产生弱形式积分式,并通过像传统有限元法中的积分过程建立系统方程组.本文提出的方法是一种单元配点法,对于域内点为了获得较高的导数精度,需要采用至少具有一个内部点的等参单元,为此除了可使用各阶次的拉格朗日四边形单元外, 还 给出了七节点三角形等参单元,用于模拟较为复杂的几何形状问题.