Construction of n-sided polygonal spline element using area coordinates and B-net method

In general, triangular and quadrilateral elements are commonly applied in two-dimensional finite element methods. If they are used to compute polycrystalline materials, the cost of computation can be quite significant. Polygonal elements can do well in simulation of the materials behavior and provide greater flexibility for the meshing of complex geometries. Hence, the study on the polygonal element is a very useful and necessary part in the finite element method. In this paper, an n-sided polygonal element based on quadratic spline interpolant, denoted by PS2 element, is presented using the triangular area coordinates and the B-net method. The PS2 element is conforming and can exactly model the quadratic field. It is valid for both convex and non-convex polygonal element, and insensitive to mesh distortions. In addition, no mapping or coordinate transformation is required and thus no Jacobian matrix and its inverse are evaluated. Some appropriate examples are employed to evaluate the performance of the proposed element.     

Local and parallel finite element algorithms based on two-grid discretization for steady Navier-Stokes equations

Local and parallel finite element algorithms based on two-grid discretization for Navier-Stokes equations in two dimension are presented. Its basis is a coarse finite element space on the global domain and a fine finite element space on the subdomain. The local algorithm consists of finding a solution for a given nonlinear problem in the coarse finite element space and a solution for a linear problem in the fine finite element space, then droping the coarse solution of the region near the boundary. By overlapping domain decomposition, the parallel algorithms are obtained. This paper analyzes the error of these algorithms and gets some error estimates which are better than those of the standard finite element method. The numerical experiments are given too. By analyzing and comparing these results, it is shown that these algorithms are correct and high efficient.     

框架结构屈曲的精确有限元求解

基于屈曲微分控制方程的一般解,构造了Euler梁在轴力作用下的精确形函数,建立了用于框架结构屈曲分析的精确有限单元,得到了单元刚度矩阵和几何刚度矩阵的显式表达,并提出了基于常规特征值计算的迭代算法以确定屈曲载荷及相应失稳模态的精确解. 研究表明, 对于线性稳定性分析而言,常规框架有限单元可视为精确有限单元的一种近似. 若采用精确单元,无需进行网格细分就可以获得精确的屈曲载荷和失稳模态. 数值算例证明了新单元和算法的效率和精度.      

Corotational nonlinear analyses of laminated shell structures using a 4-node quadrilateral flat shell element with drilling stiffness

A new 4-node quadrilateral flat shell element is developed for geometrically nonlinear analyses of thin and moderately thick laminated shell structures. The fiat shell element is constructed by combining a quadrilateral area co- ordinate method （QAC） based membrane element AGQ6- II, and a Timoshenko beam function （TBF） method based shear deformable plate bending element ARS-Q12. In order to model folded plates and connect with beam elements, the drilling stiffness is added to the element stiffness matrix based on the mixed variational principle. The transverse shear rigidity matrix, based on the first-order shear deformation theory （FSDT）, for the laminated composite plate is evaluated using the transverse equilibrium conditions, while the shear correction factors are not needed. The conventional TBF methods are also modified to efficiently calculate the element stiffness for laminate. The new shell element is extended to large deflection and post-buckling analyses of isotropic and laminated composite shells based on the element independent corotational formulation. Numerical re- sults show that the present shell element has an excellent numerical performance for the test examples, and is applicable to stiffened plates.     

弹性地基上Timoshenko梁的精确数值解

研究了弹性地基上Timoshenko梁的高精度有限元分析方法,利用控制微分方程的基本解建立了单元形函数,提出了弹性地基上Timoshenko梁分析的Trefftz单元。通过对引入的非节点自由度进行静力凝聚,得到的精确单元与常规单元具有相同的节点自由度。文中还讨论了有效降低计算过程中舍入误差的方法。算例结果表明,采用提出...     

一种基于自适应单元删除率的BESO方法

在ESO中采用动态删除率能有效地提高优化效率和稳定性,但现有的动态删除率策略都含有经验参数,确定删除率的过程较为复杂。本文提出了一种用于BESO的无经验参数自适应单元删除率确定方法。通过分析单元删除率对优化稳定性的影响,得到了结构优化过程中单元删除率的理想变化规律和单元灵敏度均匀化信息对删除率的影响情况,并据此分析了经验参数引入的原因,从而构造了评价单迭代步的单元灵敏度均匀化程度指标。然后,基于单迭代步的单元灵敏度均匀化程度指标,构造了全部迭代步信息下的单元灵敏度均匀化程度相对指标,结合单元删除率的推荐范围值,给出了一种自适应于结构优化进程的单元删除率自适应函数。最后,给出了基于自适应单元删除率的BESO方法实现流程。经典算例的结果对比说明,本文方法在保证优化质量相近的情况下,具有更好的优化效率和稳定性。     

基于扩展有限元的应力强度因子的位移外推法

针对平面裂纹问题,阐述了扩展有限元法的单元位移模式、推导了扩展有限元法的控制方程、介绍了特殊单元的数值积分技术.基于最小二乘法,建立了应力强度因子位移外推法的计算公式.利用MATLAB编写计算程序,对平面裂纹问题用扩展有限元法进行了计算.基于扩展有限元法的计算结果,分别利用位移外推法和相互作用积分法,对平面裂纹的应力强度因子进行了计算.计算结果表明,位移外推法比相互作用积分法能更方便和准确地计算平面裂纹的应力强度因子.     

Dual variational formulation for Trefftz finite element method of elastic materials

A dual variational principle is presented for Trefftz finite element analysis. The proof of the stationary conditions of the variational functional and the theorem on the existence of extremum are provided in this paper. They are boundary displacement condition, surface traction condition and interelement continuity condition. Based on the assumed intraelement and frame fields, element stiffness matrix equation is obtained which can easily be implemented into computer programs for numerical analysis with Trefftz finite element method. Two numerical examples are considered to illustrate the effectiveness and applicability of the proposed element model.     

The application of 2D base force element method (BFEM) to geometrically non-linear analysis

Based on the concept of the base forces by Gao, a new finite element method – the base force element method (BFEM) on complementary energy principle for two-dimensional geometrically non-linear problems is presented. A 4-mid-node plane element model of the BFEM for geometrically non-linear problem is derived by assuming that the stress is uniformly distributed on each sides of a plane element. The explicit formulations of the control equations for the BFEM are derived using the modified complementary energy principle. The BFEM is naturally universal for small displacement and large displacement problems. A number of example problems are solved using the BFEM and the results are compared with corresponding analytical solutions and those obtained from the standard displacement finite element method. A good agreement of the results, and better performance of the BFEM, compared to the displacement model, in the large displacement and large rotation calculations, is observed.     

自然单元法研究进展

自然单元法是一种基于Voronoi图和Delaunay三角化几何结构,以自然邻点插值为试函数的一种新型数值方法.其既具有无网格方法和经典有限元方法的优点,又克服了两者的一些缺陷,是一种发展前景广阔的求解微分方程的数值方法.自然单元法的形函数满足插值性质,可以像有限元法一样直接施加本质边界条件,不存在基于移动最小二乘拟合的无网格方法不能直接施加本质边界条件的难题.由于自然单元法是无网格方法,可以方便处理有限元方法较难处理的一些问题,例如移动边界和大变形等问题.自然单元法与其他数值方法的最根本区别于其插值格式的不同.将自然邻点插值用于Galerkin过程,就得到基于Voronoi结构的自然单元Galerkin法.自然邻点插值有自然邻点Sibson插值和Laplace插值(非Sibson插值)两种.Laplace插值比Sibson插值在计算上要简单的多,并且不论对凸的或非凸的区域都能精确施加本质边界条件.以Laplace插值为试函数的自然单元法在数值实施上比以Sibson插值为试函数的自然单元法简单.本文对基于Voronoi结构的自然邻点插值和自然单元法的基本思想作了介绍,综述了国内外关于自然单元法的研究成果,总结了自然单元法的优点和尚需解决的问题.      

Modes computation and analysis for long multi-span bundle conductors of power transmission lines with galloping control devices

A new type of element which is suitable for solving the modes of the galloping long multi-span bundle conductor structures is presented. The element is composed of all sub-conductor segments between two spacers. Based on the linearized governing differential equations of the conductors, the mass matrix and stiffness matrix of the element in consideration of the constrained relations imposed on the conductors by spacers are derived. The dynamic characteristics of the galloping control devices can be directly added to the element. The modes for an actual power line structure are computed by using the element formula and FEM procedures, where seven cases of different galloping control device allocations are considered. Compared with the measured data, the method is shown to be reliable and effective. Analysis and discussions of the computational results are given. Some hints that are helpful to further investigation of galloping are also obtained.     

AN INTEGRATED COUPLING ELEMENT FOR VEHICLE-RAIL-BRIDGE INTERACTION SYSTEM WITH A NON-UNIFORM CONTINUOUS BRIDGE

An integrated coupling element considering wheel-rail interface for analyzing the dynamic responses of vehicle-rail-bridge interaction system with a non-uniform continuous bridge is presented. The governing equations of the interaction system are established first, and the solution procedure and assembly method of the coupling element are demonstrated. Finally, the accuracy, efficiency and function of the integrated coupling element are tested using two numerical examples. The influences of different combinations of rail and bridge element length in the coupling element on the solution are investigated, and the effects of different rail irregularities on the dynamic responses are discussed.     

三维时间-空间混和有限元

基于分析动力学与分析结构力学在数学理论上的一致性,在有限元分析方面,同时对时间、空间坐标离散组成混和的时空有限元网格.然后利用Hamilton变分原理,取一次变分为零,导出三维混和元列式;混和元列式矩阵的对称性,保证了混和有限元保辛的性质.数值例题表明,时-空混和有限元能灵活地处理多尺度波动问题和变动边界问题.     

基于8节点平面单元的RC梁柱节点单元

为了分析RC框架结构的非线性滞回性能,基于平面8节点单元,本文提出了一个新的针对受循环荷载作用钢筋混凝土的梁柱节点单元．单元中梁与节点交界面和柱与节点交界面被划分成“节点截面”和“梁柱截面”,节点核心区的力学性能由8节点单元描述,而梁柱受力钢筋与节点核心区的粘结滑移由存在于“节点截面”和“梁柱截面”之间的8根弹簧控制,梁柱与节点之间的剪切由4根剪切弹簧表示．单元具有4个外节点和8个内节点,每个内节点具有2个自由度,每个外节点具有3个自由度,该3个自由度与普通梁单元一致,从而确保本单元能够同普通一维梁柱单元一起进行钢筋混凝土结构平面非线性分析．通过将内节点上的自由度依附到外节点上,单元在数值表现上具有4个节点和28个自由度．通过对比试验和模拟分析结果,验证了本模型适合于循环荷载作用下平面框架结构的非线性响应分析．     

非连续问题中单元分割的模板方法

扩展有限元法 (extended finite element method, XFEM) 因具有裂纹几何独立于模拟网格、裂纹扩展时无需网格重分重映、计算精度高等优点,成为裂纹分析的主流数值方法之一. 但该方法在工程实践中存在单元被裂纹分割的几何困难 —— 现有精确几何分割方法实现复杂、计算量大、鲁棒性差. 为克服这一困难, 本文提出一种基于单元水平集的模板分割方法, 用于非连续单元子剖分和数值积分. 首先, 遍历单元水平集值所有形态并建立标准单元分割模板库; 然后, 根据单元水平集值, 对非标准单元进行形态查询和模板插值; 最后, 套用标准单元分割模板实现单元高效分割和子剖分. 将该方法与常规XFEM、改进型XFEM进行结合,从而应用于孔洞、夹杂、裂纹等非连续问题分析中. 算例分析表明, 本文提出的模板分割方法具有较高计算精度. 由于不引入复杂几何操作, 该模板分割方法同时具有较高计算效率和鲁棒性, 故可为XFEM类方法在实际工程应用中提供有效支撑.      

动态断裂力学的无限相似边界元法

对弹性动力学的相似边界元法进行了进一步研究,推导了相应的计算公式,并在此基础上提出了动态断裂力学的无限相似边界元法.与传统的边界元法相比,相似边界元法由于只需在少数单元上进行数值积分,大大减少了计算量.对动态断裂力学问题,无限相似边界元法由于在裂纹尖端的边界上设置了逼近于裂纹尖端的无限个相似边界单元,可直接得到裂纹尖端具有奇异性的应力,而不需要设置奇异单元,从而突破了奇异单元对应力奇异性阶次的局限.另外,还讨论了无限相似边界元法得到的无限阶的线性代数方程组的求解方法.     

Dealing with extremely large deformation by Nearest-Nodes FEM with algorithm for updating element connectivity

In the recently developed Nearest-Nodes Finite Element Method (NN-FEM), elements are mainly used for numerical integration; while shape functions are constructed in a similar way as in meshless methods. Based on this strategy, NN-FEM inherits major merits from both the classical Finite Element Method and meshless methods. One of them is that NN-FEM is nearly not affected by element distortion. So NN-FEM is more efficient than the classical FEM on dealing with large deformation problems. Nevertheless, NN-FEM still has a requirement on finite element meshes, that is, elements in a mesh are required not to overlap or penetrate to each other, to avoid difficulty in numerical integration. To eliminate overlapped elements, NN-FEM is supplemented with an algorithm for updating element connectivity. With this supplement, NN-FEM is able to deal with extremely large deformation. In updating element connectivity, element nodes are kept not changed and all information associated with nodes are not touched. Therefore, there is no need to transfer solution data, and error introduced by solution transfer is avoided.     

Formulation and implementation of a singular anisotropic finite element

The formulation and implementation of a singular finite element for analyzing homogeneous anistropic materials is presented in this paper. Lekhnitskii's stress function method is used to formulate the boundary value problem with the stress function expressed as a Laurent series. The development of the element stiffness matrix and the method of integrating the element to conventional displacement based finite element programs is shown. The stiffness matrix generation is based on a least squates collocation technique to satisfy displacement continuity boundary conditions at the element interface. Implementation of the element is demonstrated for cracked anisotropic materials subjected to inplane loading. Center cracked, on and off-axis coupons under tensile loading are analyzed using the element. It is shown that the stress distributions and intensity factors compare well with those obtained using other methods.     

连续胶体材料化学-力学耦合有限元理论及其在ABAQUS中UEL程序实现

基于哈密顿原理,得到水凝胶的化学-力学耦合控制方程的等效积分形式和有限元形式。在整体坐标系下推导出用形函数表示的化学-力学耦合应变矩阵和单元刚度矩阵,并且得到在局部坐标系下的离散化形式。结合ABAQUS软件,编制了用户单元子程序UEL,通过数值算例验证了所开发单元的正确性,为在ABAQUS软件中实现各种耦合问题的有限元UEL编程提供了参考依据。     

改进型扩展比例边界有限元法

结合了扩展有限元法(extended finite elementmethods,XFEM)和比例边界有限元法(scaled boundary finite elementmethods,SBFEM)的主要优点,提出了一种改进型扩展比例边界有限元法(improvedextended scaled boundary finite elementmethods,$i$XSBFEM),为断裂问题模拟提供了一条新的途径.类似XFEM,采用两个正交的水平集函数表征材料内部裂纹面,并基于水平集函数判断单元切割类型;将被裂纹切割的单元作为SBFE的子域处理,采用SBFEM求解单元刚度矩阵,从而避免了XFEM中求解不连续单元刚度矩阵需要进一步进行单元子划分的缺陷;同时,借助XFEM的主要思想,将裂纹与单元边界交点的真实位移作为单元结点的附加自由度考虑,赋予了单元结点附加自由度明确的物理意义,可以直接根据位移求解结果得出裂纹与单元边界交点的位移;对于含有裂尖的单元,选取围绕裂尖单元一圈的若干层单元作为超级单元,并将此超级单元作为SBFE的一个子域求解刚度矩阵,超级单元内部的结点位移可通过SBFE的位移模式求解得到,应力强度因子可基于裂尖处的奇异位移(应力)直接获得,无需借助其他的数值方法.最后,通过若干数值算例验证了建议的$i$XSBFEM的有效性,相比于常规XFEM,$i$XSBFEM的基于位移范数的相对误差收敛性较好;采用$i$XSBFEM通过应力法和位移法直接计算得到的裂尖应力强度因子均与解析解吻合\较好.