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.     

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.     

A novel hybrid finite element analysis of inplane singular elastic field around inclusion corners in elastic media

This paper deals with the inplane singular elastic field problems of inclusion corners in elastic media by an ad hoc hybrid-stress finite element method. A one-dimensional finite element method-based eigenanalysis is first applied to determine the order of singularity and the angular dependence of the stress and displacement field, which reflects elastic behavior around an inclusion corner. These numerical eigensolutions are subsequently used to develop a super element that simulates the elastic behavior around the inclusion corner. The super element is finally incorporated with standard four-node hybrid-stress elements to constitute an ad hoc hybrid-stress finite element method for the analysis of local singular stress fields arising from inclusion corners. The singular stress field is expressed by generalized stress intensity factors defined at the inclusion corner. The ad hoc finite element method is used to investigate the problem of a single rectangular or diamond inclusion in isotropic materials under longitudinal tension. Comparison with available numerical results shows the present method is an efficient mesh reducer and yields accurate stress distribution in the near-field region. As applications, the present ad hoc finite element method is extended to discuss the inplane singular elastic field problems of a single rectangular or diamond inclusion in anisotropic materials and of two interacting rectangular inclusions in isotropic materials. In the numerical analysis, the generalized stress intensity factors at the inclusion corner are systematically calculated for various material type, stiffness ratio, shape and spacing position of one or two inclusions in a plate subjected to tension and shear loadings.     

A pseudo-time integral method for non-isothermal viscoelastic flows and its application to extrusion simulation

A pseudo-time integral scheme based on a finite streamline element method is developed to combine variable temperature with viscoelasticity. A specific KBKZ integral model for isothermal flow is transformed to its non-isothermal version by introducing a pseudo-time and applying the Morland-Lee hypothesis. The coupling between momentum and energy equations is through the time-temperature shifting factor by which the pseudo-time is defined. The observer time and the pseudo-time are simultaneously calculated when tracing the strain history for the stress calculation in a non-homogeneous temperature field. Using this scheme, a full non-isothermal numerical simulation of some IUPAC extrusion experiments is carried out. Results show that while the temperature distribution near the die exit plane is an important factor controlling extrudate swell, either self-heating inside the die tube or external cooling on the free surface dominantly determines the temperature distribution near the die exit when the wall temperature is kept constant, depending on whether the Péclet number is large or small. The hot layer effect predicted by the inelastic swell mechanism is confirmed and well illustrated by the computation. Calculations with reasonable thermal boundary conditions further convince us that the isothermal assumption in our earlier numerical simulation is a good approximation in this particular case.     

Analytical removal of singularity and direct evaluation of singular integrals in 3-D elastoplastic finite deformation analysis by BEM

As a further development of the present authors' research work [1,2], in this paper a method of the so-called quadratic pentahedron polar co-ordinate transformation and analytical removal of singularity of Cauchy principal value singular integrals is proposed to evaluate the strongly singular integrals in the sense of Cauchy principal values and the weakly singular integrals over quadratic internal cells in 3-D elastoplastic finite deformation analysis by BEM. First, a quadratic pentahedron polar co-ordinate transformation technique is used to reduce the order of singularity of the singular integrals. Then, a form of Gauss' theorem is introduced to remove the singularity in the Cauchy principal value singular integrals analytically. Therefore, the evaluation of all those strongly and weakly singular integrals can be carried out by standard Gaussian quadrature accurately and efficiently. Numerical examples of the 3-D elastoplastic problem and 3-D finite deformation problem are given to demonstrate that the method possesses good accuracy and numerical stability, and is convenient to implement. The method in this paper can be applied extensively to evaluating the singular integrals over cubic and higher order elements.     

Numerical analysis of singular solutions of two-dimensional problems of asymmetric elasticity

An algorithm for the numerical analysis of singular solutions of two-dimensional problems of asymmetric elasticity is considered. The algorithm is based on separation of a power-law dependence from the finite-element solution in a neighborhood of singular points in the domain under study, where singular solutions are possible. The obtained power-law dependencies allow one to conclude whether the stresses have singularities and what the character of these singularities is. The algorithm was tested for problems of classical elasticity by comparing the stress singularity exponents obtained by the proposed method and from known analytic solutions. Problems with various cases of singular points, namely, body surface points at which either the smoothness of the surface is violated, or the type of boundary conditions is changed, or distinct materials are in contact, are considered as applications. The stress singularity exponents obtained by using the models of classical and asymmetric elasticity are compared. It is shown that, in the case of cracks, the stress singularity exponents are the same for the elasticity models under study, but for other cases of singular points, the stress singularity exponents obtained on the basis of asymmetric elasticity have insignificant quantitative distinctions from the solutions of the classical elasticity.     

一种确定结合材料界面端应力强度因子的数值方法

基于弹性力平面问题的基本方程，给出了结合材料界面端的应力奇异性特征方程以及位移场和奇异应力场。提出了一种确定结合材料界面端应力强度因子的数值外插方法。对界面端区域进行了有限元网格单元划分。经过具体实例检验进一步确定了求解应力强度因子的最佳方向，该数值外插法的计算结果精度符合工程应用的要求，为工程材料强度的评价提供了有效的计算途径。     

Analysis of bonded anisotropic wedges with interface crack under anti-plane shear loading

The antiplane stress analysis of two anisotropic finite wedges with arbitrary radii and apex angles that are bonded together along a common edge is investigated. The wedge radial boundaries can be subjected to displacement-displacement boundary condi- tions, and the circular boundary of the wedge is free from any traction. The new finite complex transforms are employed to solve the problem. These finite complex transforms have complex analogies to both kinds of standard finite Mellin transforms. The traction free condition on the crack faces is expressed as a singular integral equation by using the exact analytical method. The explicit terms for the strength of singularity are extracted, showing the dependence of the order of the stress singularity on the wedge angle, material constants, and boundary conditions. A numerical method is used for solving the resul- tant singular integral equations. The displacement boundary condition may be a general term of the Taylor series expansion for the displacement prescribed on the radial edge of the wedge. Thus, the analysis of every kind of displacement boundary conditions can be obtained by the achieved results from the foregoing general displacement boundary condition. The obtained stress intensity factors （SIFs） at the crack tips are plotted and compared with those obtained by the finite element analysis （FEA）.     

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。     

Flow of a viscous fluid past a flexible membrane at low Reynolds numbers

The numerical calculation of a steady two-dimensional viscous flow past a flexible membrane is treated. Both edges of the membrane are fixed in the flow and its chord is set normal to the flow. The Navier-Stokes equation in terms of the stream function and the vorticity is transformed to the body fitted coordinate system. The numerical calculations, based on a finite difference method and relaxation method, are carried out for several values of the membranes tension for cases when the Reynolds numbers are 5, 10 and 20. It is found that two different shapes of the membranes are possible at a given value of tension and Reynolds number: one with a small deformation, and the other with a large deformation. Two vortices appear in the concave region of the membrane if its deformation increases beyond a certain extent.     

三维弹塑性有限变形问题边界元法中奇异积分的直接分析去奇解法

作为本文作者研究工作的继续,本文提出了处理三维弹塑性有限变形问题边界元法中二次元区域弱奇及Cauchy 主值奇异积分的二次极坐标变换—分析去奇法.该方法先通过适当的二次极坐标变换降低奇异积分的奇异性,然后利用Causs 散度定理去除Cauchy主值积分的奇异性.通过三维弹塑性及三维有限变形问题数值算例说明该方法具有良好的精度及数值稳定性,并且实施较方便.本文方法可直接推广应用于二阶以上高阶元离散模型奇异积分处理.     

PLANAR LIQUID SHEETS AT LOW REYNOLDS NUMBERS

Asymptotic methods are employed to derive the leading-order equations which govern the fluid dynamics of time-dependent, incompressible, planar liquid sheets at low Reynolds numbers using as small parameter the slenderness ratio. Analytical and numerical solutions of relevance to both steady film casting processes and plane stagnation flows are obtained with the leading-order equations. It is shown that for steady film casting processes the model which accounts for both gravity and low-Reynolds-number effects predicts thicker and slower planar liquid sheets than those which neglect a surface curvature term or assume that Reynolds number is zero, because the neglect of the curvature term and the assumption of zero Reynolds number are not justified at high take-up velocities owing to the large velocity gradients that occur at the take-up point. It is also shown that for Reynolds number/Froude number ratios larger than one, models which neglect the surface curvature or assume a zero Reynolds number predict velocity profiles which are either concave or exhibit an inflection point, whereas the model which accounts for both curvature and low-Reynolds-number effects predicts convex velocity profiles. For plane stagnation flows it is shown that models which account for both low-Reynolds-number and curvature effects predict nearly identical results to those of models which assume zero Reynolds number. These two models also predict a faster thickening of the planar liquid sheet than models which account for low- Reynolds-number effects but neglect the surface curvature. This curvature term is very large near the stagnation point and cannot be neglected there. It is also shown that the thickening of the sheet occurs closer to the stagnation point as the Reynolds number/Froude number ratio is increased, i.e. as the magnitude of the gravitational acceleration is increased. In addition it is shown that large surface tension introduces a third-order spatial derivative in the axial momentum equation at leading order.     

与界面相交的裂纹尖端的应力奇异性分析

为了确定与结合材料的界面相交的裂纹尖端附近的应力奇异性次数，提出了一种基于最小势能原理的一维特殊有限元法，以奇异点为原点半径r0的扇形奇异区域，可以简化为一维线性领域，即一条以代表结合材料的两个自由表面为端点的线段。对该一维线性领域作网格划分，采用三节点一维等参数二次单元。数值计算结果与已有理论解的比较表明，该方法具有很高的精度和效率，最后，利用文中给出的方法，得到了各向异性结合材料中与界面以任意角相交的裂纹尖端的奇异性次数随裂纹的变化规律。     

Collapsed 27 node Lagrangian element for three-dimensional stress intensity computations

Considered are the details of a collapsed 27 node (singular) Lagrangian element for use in three-dimensional stress intensity factor calculations. The element is shown to capture both the singularity near the front as required for linear elastic fracture mechanics and a 1/r singularity under a relaxed constraint for plastic fracture. It is shown that the radial and circumferential strains exhibit the singularity, but that direct strain along the front is zero, thus providing plane strain conditions. Of equal relevance is the quality of mesh at the front. For this we adopt a standard suite of elements comprising a number of singular ‘wedge-shaped' elements around the crack tip in the form of a tube, with the major axis placed along the front. A number of three-dimensional fracture problems are analysed to demonstrate the accuracy and efficiency of the collapsed element, even when used with moderate mesh sizes.     

积分型Maxwell流体挤出胀大的数值模拟

应用Luo提出的基于常规有限单元的应力计算方法和范毓润回避奇点的方法,模拟了积分型Maxwell黏弹流体的挤出胀大流动,计算得到了Weissenberg数达1．0下的合理结果。     

三维双材料结构的应力奇异性分析

应用有限单元法子模型技术，对具有不同界面角的三维双材料结构的应力奇异性进行了分析。结果表明，应用子模型技术估算三维双材料结构的应力奇异性指数是有效的。然后分析了界面端线和界面端点处附近奇异性指数，得到了一些重要而有趣的结果。最后对消除三维双材料结构应力奇异性的几何条件进行了讨论。     

The steady annular extrusion of a Newtonian liquid under gravity and surface tension

The steady extrusion of a Newtonian liquid through an annular die and its development outside and away from the die are studied under the influence of gravitational and surface tension forces. The finite element method (FEM) is used for the simulations. The positions of the inner and outer free surface profiles are calculated simultaneously with the other unknown fields, i.e. using the Newton–Raphson iterative scheme. The effects of three relevant parameters, i.e. the Reynolds, the Stokes and the capillary numbers, on the shape of the annular film are studied for two values of the inner to the outer diameter ratio, corresponding to a thick and a thin annular film respectively. A one‐dimensional model for the extrudate region, valid for thin annular films, is also presented, and its predictions are compared with the two‐dimensional finite element calculations. Despite the fact that it is valid away from the die exit, the one‐dimensional model predicts satisfactorily the effects of the Stokes and capillary numbers. Copyright © 2000 John Wiley & Sons, Ltd.     

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.      

Stress intensification in cracked shank of tightened bolt

Defects or cracks in the shank of bolts can degrade their load carrying capacity. The ways with which loading and residual stress intensify the crack border stress field can be reflected through the stress intensity factor quantity as defined in the linear elastic fracture mechanics theory. Use is made of the stiffness derivative method where quarter-point singular finite elements are used in the numerical calculation. Improved accuracy is achieved by considering the displacements not only of the main nodes but also of those quarter-point nodes in plane normal and adjacent to the crack.Numerical results are obtained for a semi-elliptical shaped crack in the bolt shank owing to tension, bending, residual stress and stress caused by tightening of the bolt. Maximum value of the Mode I stress intensity factor K_a due to tension or bending could prevail either at the deepest point on the crack border or at the root of the shank where the crack border terminates depending on the aspect ratio of the ellipse. In general, K₁ at the deepest point of crack penetration is larger than that at the free surface for tension and bending for a fixed crack depth with reference to the bolt diameter. Tightening of the bolt tends to increase K₁ at the free surface if the crack depth is small. The opposite is obtained for deeper cracks. Assumed residual stress effect obtained from experimental data is found to have negligible influence on the stress intensity factor when compared with that arising from tensile load.     

一类新的具有无外力圆柱表面的杂交应力元

根据Hellinger-Reissner原理,建立了进一步改进的具有一个无外力圆柱表面三维杂交应力元.元内假定应力场满足以柱坐标表示的平衡方程,及圆柱面上的无外力边界条件.当退化为二维时,也满足协调方程.数值算例表明,当分析带圆弧的槽孔板、块时,在稀疏的有限元网格下,这类单元即可提供较以前各类特殊元、一般假定位移元及一般假定应力元远为准确的三维及二维应力分布.