A precise singular finite element for crack analysis

The multi-variable finite element algorithm based on the generalized Galerkin’smethod is more flexible to establish a finite element model in the continuum mechanies.Byusing this algorithm and numerical tests a new singular finite element for elasto-plasticfracture analysis has been formulated.The results of numerical tests show that the newelement possesses high accuracy and good performance.Some rules for formulating amulti-variable singular finite element are also discussed in this paper.     

孔洞对平面裂纹应力强度因子影响分析的广义参数Williams单元

本文采用圆形奇异区广义参数Williams单元（W单元）建立了中心裂纹与圆孔共存的平面应力模型,奇异区外围利用ABAQUS有限元软件自动网格离散技术与FORTRAN95编程前处理相结合,克服了自主编程中网格离散的局限性．算例分析了圆孔位置和几何参数对I-II混合型裂纹尖端应力强度因子(SIFs)的影响,并与扩展有限元法(XFEM)计算结果进行比较．结果表明：靠近圆孔一侧的裂尖SIFs大于远离圆孔一侧的裂尖SIFs;控制圆孔左边缘到裂纹中心的距离,则两侧裂尖SIFs随圆孔半径的增大而增大;圆孔中心与裂纹中心水平距离越远,圆孔对裂纹扩展的影响越小．同时,基于圆形奇异区的W单元直接计算得到的裂尖SIFs与扩展有限元法得到的解吻合较好,证明了W单元对奇异区离散形状不敏感,且具有高效率和高精度．     

多参量断裂力学广义杂交/混合裂纹有限元法

通过研究广为人知的断裂力学单变量八节点位移裂纹QPE元和Akin族奇异单元法，本文运用经典局部裂纹解析解，与非协调假设应力杂交-混合元列式方法相结合，提出用于分层各向异性材料的多变量半解析假设应力奇异广义杂交／混合裂纹有限元法，能克服现有位移裂纹元法的域应力分布精度低和高次单元所需计算容量大的局限性，互为补充，更有利于结构裂纹扩展分析和应用研究。文中设计了一个半解析奇异裂纹平面单元，各向同性材料板算例验证了退化二次八节点协调位移裂纹元及六节点非协调奇异应力裂纹元，说明采用稀疏及加密单元网格，两类裂纹单元分别从上下逼近收敛于实验和理论参考解，可得到吻合程度较好的1／√r奇异应变和应力分量以及应力强度因子值，表明了本文奇异裂纹单元理论的优越性。     

局部有限元模型边界条件的直接传递方法

提出一种将整体分析得到的节点力或节点位移直接传递到精细化局部有限元模型的方法,即部分混合单元法。沿精细化局部有限元模型周边建立一组过渡单元,该组过渡单元采用与整体模型一致的单元类型和模拟方式,其外侧边界上的节点与整体模型节点的相对坐标对应,内侧边界与精细化局部有限元模型采用基于面约束的方式连接。在外侧边界上根据节点坐标对应施加整体分析获得的节点力或节点位移,过渡单元就可直接将边界条件传递到精细化局部有限元模型。通过贵州红水河特大桥钢-混结合段的精细化有限元分析,验证了本文方法的实用性和有效性。     

Singular finite elements for the sudden-expansion and the die-swell problems

The singular finite element method is used to solve the sudden-expansion and the die-swell problems in order to improve the accuracy of the solution in the vicinity of the singularity and to speed up the convergence. The method requires minor modifications to standard finite element schemes, and even coarse meshes give more accurate results than refined ordinary finite element meshes. Improved normal stress results for the sudden-expansion problem have been obtained for various Reynolds numbers up to 100 using the singular elements constructed for the creeping flow problem. In addition, the normal stresses at the walls appear to be insensitive to the singularity powers used in the construction of the singular basis functions. The die-swell problem is solved using the singular elements constructed for the stick–slip problem. The singular elements accelerate the convergence of the free surface dramatically.     

基于新型裂尖杂交元的压电材料断裂力学研究

提出了一种裂尖邻域杂交元模型，将其与标准杂交应力元结合来求解压电材料裂纹尖 端的奇性电弹场和断裂参数的数值解．裂纹尖端杂交元的建立步骤为：1) 利用高次内插有限元特征法求解特征问题，得到反映裂尖奇异性电弹场状况的特 征值和特征角分布函数；2) 利用广义Hellinger-Reissner变分泛函以及特征问题的解来建立裂尖邻域杂交元模型．该 方法求解电弹场时，摒弃了传统有限元方法中裂尖奇异性场需要借助解析解的做法，也避免 了单纯有限元方法中需要在裂尖端部进行高密度单元划分．采用PZT5板中心裂纹问题 作为考核例，数值结果显示了良好的精确性．作为进一步应用，求解了含中心界面裂纹 的PZT4-PZT5两相压电材料的应力强度因子和电位移强度因子．所有的算例都考虑 了3种裂纹面电边界条件．     

Converged solutions of the Newtonian extrudate‐swell problem

Both the axisymmetric and the planar Newtonian extrudate‐swell problems are solved using the standard and the singular finite element methods. In the latter method, special elements that incorporate the radial form of the stress singularity are used around the exit of the die. The convergence of each of the two methods with mesh refinement is studied for various values of the Reynolds and the capillary numbers. The numerical results show that the singular finite elements perform well if coarse or moderately refined meshes are used, and appear to be superior to the standard finite elements only when the Reynolds number is low and the surface tension is not large. The standard finite elements perform better as the surface tension or the Reynolds number are increased. This implies that the effect of the stress singularity on the accuracy of the numerical solution in the neighborhood of the die exit becomes less significant when the Reynolds number is high or the surface tension is large. Copyright © 1999 John Wiley & Sons, Ltd.     

A high‐order triangular discontinuous Galerkin oceanic shallow water model

A high‐order triangular discontinuous Galerkin (DG) method is applied to the two‐dimensional oceanic shallow water equations. The DG method can be characterized as the fusion of finite elements with finite volumes. This DG formulation uses high‐order Lagrange polynomials on the triangle using nodal sets up to 15th order. Both the area and boundary integrals are evaluated using order 2N Gauss cubature rules. The use of exact integration for the area integrals leads naturally to a full mass matrix; however, by using straight‐edged triangles we eliminate the mass matrix completely from the discrete equations. Besides obviating the need for a mass matrix, triangular elements offer other obvious advantages in the construction of oceanic shallow water models, specifically the ability to use unstructured grids in order to better represent the continental coastlines for use in tsunami modeling. In this paper, we focus primarily on testing the discrete spatial operators by using six test cases—three of which have analytic solutions. The three tests having analytic solutions show that the high‐order triangular DG method exhibits exponential convergence. Furthermore, comparisons with a spectral element model show that the DG model is superior for all polynomial orders and test cases considered. Copyright © 2007 John Wiley & Sons, Ltd.     

三维电磁场辛有限元列式及电磁共振腔的计算

针对三维共振腔的电磁场分析,利用Maxwell方程的对偶方程体系形式,从其相应的对偶变量变分原理出发,导出了三维电磁场辛有限单元的详细列式。为了有限元列式的保辛,变分原理被积函数可导向对于对偶变量为对称的形式。变分原理的边界积分项对于相邻单元相互抵消。由于采用了对偶变量的插值函数,使得电磁场单元构造可以在层面上进行,从而避免了所谓的连续性问题。无物理意义的零本征解可采用奇异值分解加以排除。文末分别对矩形及圆柱形的共振腔做了数值计算并与解析解和棱边元计算结果进行对比,算例表明了列式及算法的有效性。     

尖锐夹杂角端部局部应力场杂交有限元分析

本文首先利用作者曾提出的一维有限元特征分析方法计算所得到的尖锐夹杂角端部应力奇异指数和奇异应力场、位移场角分布函数,并依据Hellinger-Reissner原理,开发出了一个特殊的、能够反映夹杂角端部局部弹性现象的n结点多边形超级角端部单元,然后将该超级单元与标准的4结点杂交应力单元耦合在一起构建了一种分析异形夹杂角端部奇异弹性场的新型特殊杂交应力有限元方法.文中给出了两个应用算例,算例结果表明:本文方法不仅使用单元少、计算结果精度高,而且适用范围广,可拓展应用于分析复合材料微结构组织与力学行为关系.     

一种新的计算各向异性材料裂纹尖端应力强度因子杂交元法

利用有限元特征分析法研究了平面各向异性材料裂纹端部的奇性应力指数以及应力场和位移场的角分布函数,以此构造了一个新的裂纹尖端单元。文中利用该单元建立了研究裂纹尖端奇性场的杂交应力模型,并结合Hellinger-Reissner变分原理导出应力杂交元方程,建立了求解平面各向异性材料裂纹尖端问题的杂交元计算模型。与四节点单元相结合,由此提出了一种新的求解应力强度因子的杂交元法。最后给出了在平面应力和平面应变下求解裂纹尖端奇性场的算例。算例表明,本文所述方法不仅精度高,而且适应性强。     

大跨度斜拉桥动力特性分析

本文提出一种计算大距度钢桁梁斜拉桥动力特性的方法。文中分别采用桁段有限单元、空间梁元、空间杆元计算斜拉桥中桁架，桥塔、拉索的刚度矩阵与质量矩阵，采用子空间迭代法求解特征方程，所得结果可供设计参考。     

基于非协调元特征法的奇异性问题分析

提出了一个基于位移的、分析平面尖劈尖端奇性应力场和位移场问题的非协调FE特征分析法．该方法与过去原有求解裂纹尖端近似场的有限元特征分析方法导出公式的出发点不同，并且采用的单元形式为非协调元，尖劈尖端邻域内的位移场假定没有采用奇异变换技术，运用该方法处理了若干尖劈和接头的算例，所有的计算结果表明，该方法较原有方法使用的单元少而且精度高，具有应用广泛性。     

复合材料尖劈和接头端部奇性场的反平面问题研究

提出了一个基于位移的分析尖劈端部奇性位移场和应力场反平面问题的非协调元特征法．该方法与过去原有求解裂纹尖端近似场的有限元特征法有几点不同：(1)导出虚功原理的出发点为二维扇区的散度原理;(2)有限元的单元形式为非协调元;(3)尖劈端部邻域内的位移场假定没有采用奇异变换技术．运用该方法给出了求解正交各向异性复合材料尖劈端部附近奇性应力指数、奇性位移和应力角分布函数的算例．计算结果表明,该方法较原来的有限元特征法所用的单元少而且精度高．     

A singular finite element for Stokes flow: The stick–slip problem

Abrupt changes in boundary conditions in viscous flow problems give rise to stress singularities. Ordinary finite element methods account effectively for the global solution but perform poorly near the singularity. In this paper we develop singular finite elements, similar in principle to the crack tip elements used in fracture mechanics, to improve the solution accuracy in the vicinity of the singular point and to speed up the rate of convergence. These special elements surround the singular point, and the corresponding field shape functions embody the form of the singularity. Because the pressure is singular, there is no pressure node at the singular point. The method performs well when applied to the stick–slip problem and gives more accurate results than those from refined ordinary finite element meshes.     

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。      

A non-linear finite element model of human L4-L5 lumbar spinal segment with three-dimensional solid element ligaments

This paper establishes a non-linear finite element model (NFEM) of L4-L5 lumbar spinal segment with accurate three-dimensional solid ligaments and intervertebral disc. For the purpose, the intervertebral disc and surrounding ligaments are modeled with four-nodal three-dimensional tetrahedral elements with hyper-elastic material properties. Pure moment of 10 N·m without preload is applied to the upper vertebral body under the loading conditions of lateral bending, backward extension, torsion, and forward flexion, respectively. The simulate relationship curves between generalized forces and generalized displacement of the NFEM are compared with the in vitro experimental result curves to verify NFEM. The verified results show that: (1) The range of simulated motion is a good agreement with the in vitro experimental data; (2) The NFEM can more effectively reflect the actual mechanical properties than the FE model using cable and spring elements ligaments; (3) The NFEM can be used as the basis for further research on lumbar degenerative diseases.     

含界面裂纹Reissner板弯曲问题分析的奇异单元

首先,采用特征函数渐近展开法,推导了Reissner板弯曲界面裂纹尖端附近位移场渐近展开的前两阶显式表达式,并利用所获得的位移场渐近表达式构造了一种可用于Reissner板弯曲界面裂纹分析的奇异单元。然后,将该奇异单元与外部的常规有限单元相结合,开展了含界面裂纹Reissner板弯曲断裂问题的数值分析。奇异单元可以较好地描述裂纹尖端附近的内力场与位移场,其优势是它与常规单元进行连接时不需要使用过渡单元,并且可以直接给出应力强度因子等断裂参数的高精度数值结果。最后,通过两个数值算例验证了本文方法的有效性。     

A novel singular finite element of mixed-mode crack problems with arbitrary crack tractions

A novel singular finite element is presented to study cracked plates with arbitrary traction acting on crack surfaces. Firstly, the analytical solution around crack tips is determined using the symplectic dual approach. Subsequently, the solution is used to develop a novel singular finite element, which depicts accurately the characteristic of singular stresses field near crack tips. And the novel element can be applied to solve cracked plates, and both Mode I and Mode II stress intensity factors can be determined directly and accurately. Lastly, two numerical examples are given to illustrate the present method.     

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.