三维数值流形方法的点-面接触模型

建立了三维流形元的点-面接触模型,给出了嵌入准则并应用罚方法强迫接触界面满足不嵌入和无拉伸条件约束求解,详细推导了法向弹簧、切向弹簧和摩擦力子矩阵并给出了接触状态(张开、滑动和固定)的模式变化准则,算例表明该模型的有效性。     

A combined programming and iteration algorithm for finite element analysis of three-dimensional contact problems

Comparing with two-dimensional contact problems, three-dimensional frictional contact problems are more difficult to deal with, because of the unknown slip direction of the tangential force and enormous computing time. In order to overcome these difficulties, a combined PQP (Parametric Quadratic Programming) and iteration method is derived in this paper. The iteration algorithm, which alleviates the difficulty of unknown slip direction, is used along with the PQP method to cut down computing costs. Numerical example is given to demonstrate the validity of the present algorithm. The project supported by the Machinary and Electronics Ministry of China     

Numerical simulation of contact problems in vane machinery by a parametric quadratic programming method

Interference fits are widely used for connecting impeller and shaft assembly that are forced together slowly by pressing. The interference fit design ensures stable balance behavior and allows for positive contact between the impeller and shaft assembly throughout the range of operating speeds. In addition to maintaining radial contact, sufficient net radial interface pressure must remain in order to transmit torque when the rotational speed is very high. Therefore, the interference fit between the impeller and the shaft assembly is one of the most important factors influencing the performance of the turbo unit in the design of turbocharger compressor. A suitable fit tolerance needs to be considered in the structural design. A locomotive-type turbocharger compressor with 24 blades under combined centrifugal and interference fit loading is used for the analysis. The finite-element (FE) parametric quadratic programming (PQP) method developed based on the parametric variational principle (PVP) is used for the analysis of the stress distribution in the three-dimensional (3D) contact problem of impeller. The advantages of the parametric programming method compared with conventional approaches are that the penalty factors can be canceled and that solutions can be obtained directly without tedious iterative procedures such as the general incremental iterative method. To save time in the computation, a~multi-substructure technique is adopted for structural modeling. This not only simplifies the calculation, but also provides a convenient service for process computer-aided design (CAD) by means of FE simulation. The effects of the fit tolerance, coefficient of friction and rotational speed (centrifugal force), wall thickness of the shaft sleeve and the contact stress on the interference-fitting surfaces are studied in detail in the numerical computation. It is found that a nonuniform initial amount of interference in the structural design avoids the relative displacement generated and ensures uniformity of the contact stress. To assure quality of press-fitting, the amount of interference between the shaft sleeve and shaft should be strictly controlled to avoid the rapid increase of the contact stress. The numerical results demonstrate the high accuracy and good convergence of the algorithm presented here, which provides an effective approach that achieves more-reliable interference-fitted connections and more-precise assembly accuracy with lower manufacturing cost in the structural design.     

Two kinds of contact problems in three-dimensional icosahedral quasicrystals

Two kinds of contact problems, i.e., the frictional contact problem and the adhesive contact problem, in three-dimensional(3D) icosahedral quasicrystals are discussed by a complex variable function method. For the frictional contact problem, the contact stress exhibits power singularities at the edge of the contact zone. For the adhesive contact problem, the contact stress exhibits oscillatory singularities at the edge of the contact zone. The numerical examples show that for the two kinds of contact problems, the contact stress exhibits singularities, and reaches the maximum value at the edge of the contact zone. The phonon-phason coupling constant has a significant effect on the contact stress intensity, while has little impact on the contact stress distribution regulation. The results are consistent with those of the classical elastic materials when the phonon-phason coupling constant is 0. For the adhesive contact problem, the indentation force has positive correlation with the contact displacement, but the phonon-phason coupling constant impact is barely perceptible. The validity of the conclusions is verified.     

Effective numerical methods for elasto-plastic contact problems with friction

Several effective numerical methods for solving the elasto-plastic contact problems with friction are presented. First, a direct substitution method is employed to impose the contact constraint conditions on condensed finite element equations, thus resulting in a reduction by half in the dimension of final governing equations. Second, an algorithm composed of contact condition probes and elasto-plastic iterations is utilized to solve the governing equation, which distinguishes two kinds of nonlinearities, and makes the solution unique. In addition, Positive-Negative Sequence Modification Method is used to condense the finite element equations of each substructure and an analytical integration is introduced to determine the elasto-plastic status after each time step or each iteration, hence the computational efficiency is enhanced to a great extent. Finally, several test and practical examples are presented showing the validity and versatility of these methods and algorithms. The Project Supported by National Natural Science Foundation of China.     

EXACT AUGMENTED LAGRANGIAN FUNCTION FOR NONLINEAR PROGRAMMING PROBLEMS WITH INEQUALITY CONSTRAINTS

An exact augmented Lagrangian function for the nonlinear nonconvex programming problems with inequality constraints was discussed. Under suitable hypotheses, the relationship was established between the local unconstrained minimizers of the augmented Lagrangian function on the space of problem variables and the local minimizers of the original constrained problem. Furthermore, under some assumptions, the relationship was also established between the global solutions of the augmented Lagrangian function on some compact subset of the space of problem variables and the global solutions of the constrained problem. Therefore, from the theoretical point of view, a solution of the inequality constrained problem and the corresponding values of the Lagrange multipliers can be found by the well-known method of multipliers which resort to the unconstrained minimization of the augmented Lagrangian function presented.     

The boundary element method applied to elastic contact problems with friction

In the current paper the boundary integral equations (BIE) for elastic contact problems with friction are derived from the incremental virtual work principle. After introducing contact conditions of adhesion and slip into BIE all variants on boundary are made to discretize by quadratic isoparametric boundary element. In the current paper not only an auto-increment loading law is presented but also the iterative calculation laws for open, slip and adhesion condition are given. The results of numerical examples are satisfactory.     

A unified treatment of axisymmetric adhesive contact problems using the harmonic potential function method

A unified treatment of axisymmetric adhesive contact problems is provided using the harmonic potential function method for axisymmetric elasticity problems advanced by Green, Keer, Barber and others. The harmonic function adopted in the current analysis is the one that was introduced by Jin et al. (2008) to solve an external crack problem. It is demonstrated that the harmonic potential function method offers a simpler and more consistent way to treat non-adhesive and adhesive contact problems. By using this method and the principle of superposition, a general solution is derived for the adhesive contact problem involving an axisymmetric rigid punch of arbitrary shape and an adhesive interaction force distribution of any profile. This solution provides analytical expressions for all non-zero displacement and stress components on the contact surface, unlike existing ones. In addition, the newly derived solution is able to link existing solutions/models for axisymmetric non-adhesive and adhesive contact problems and to reveal the connections and differences among these solutions/models individually obtained using different methods at various times. Specifically, it is shown that Sneddon’s solution for the axisymmetric punch problem, Boussinesq’s solution for the flat-ended cylindrical punch problem, the Hertz solution for the spherical punch problem, the JKR model, the DMT model, the M-D model, and the M-D-n model can all be explicitly recovered by the current general solution.     

空间接触问题的参变量方法

本文借助于参变量方法给出了空间接触问题的解法.空间接触单元刚度矩阵是通过接触压力和摩擦力的惩罚函数表示式而引入的.两种空间接触本构关系(圆锥型和棱锥型)可以类似于弹塑性问题的模型得到,一定的数学技巧可使惩罚因子最后得以消去,而获得参变量线性互补问题.两个实例证明了这个方法的有效性.     

12结点三维等参奇异单元的构造和应用

通过改变三维8结点六面体等参单元的结点位置、结点数目和形函数,构造了一种12结点三维等参奇异单元,该单元的应力场具有1/(√r)奇异性,可以模拟裂缝前沿的奇异应力场;该单元的位移模式在其中两个坐标方向是线性变化的,因此,该单元与线性单元连接时不需要过渡单元,仍能保证交界面位移协调,克服了20结点三维等参奇异单元不能与线性单元协调连接的缺陷;文章最后将该奇异单元布置在裂缝前沿,应用有限元法计算了三点弯曲梁预制裂缝前沿的应力强度因子,该结果与规范公式计算值基本一致.     

任意多孔多裂纹板的裂纹闭合接触分析

基于摩擦接触问题的数学规划解法，采用各向异性体平面弹性理论中的复势方法，建立了含多椭圆孔及裂纹群有限大各向异性板，在任意载荷作用下裂纹闭合或局部闭合问题的有效分析方法。通过在可能闭合的裂纹边界引入互补变量函数并将其展成Fourier级数形式，以Faber级数为工具，应用保角映射技术和最小二乘边界配点法，导出无卸载情况下裂纹面摩擦接触的线性互补模型，并通过算例验证了方法的有效性。数值结果表明，由于采用级数解描述板应力场和位移场，该方法具有较高的计算精度和效率，便于研究裂纹闭合对应力强度因子等断裂参数的影响。     

Thermoelastic instability in contact problems for rotating solids with heat generation

In this paper we discuss certain cases of thermal instability in axisymmetric contact problems with heat generation and heat exchange. Two solids of revolution pressed against each other are considered. One of the solids rotates about its symmetry axis. A characteristic feature of the problem is that the distance between the centers of gravity of the solids cannot be greater than a certain value Published in Prikladnaya Mekhanika, Vol. 43, No. 1, pp. 134–144, January 2007.     

A necklace model for vesicles simulations in 2D

The aim of this paper is to propose a new numerical model to simulate 2D vesicles interacting with a Newtonian fluid. The inextensible membrane is modeled by a chain of circular rigid particles, which are maintained in cohesion by using two different types of forces. First, a spring force is imposed between neighboring particles in the chain. Second, in order to model the bending of the membrane, each triplet of successive particles is submitted to an angular force. Numerical simulations of vesicles in shear flow have been run using FEM and the FreeFem++ software. Exploring different ratios of inner and outer viscosities, we recover the well‐known ‘tank‐treading’ and ‘tumbling’ motions predicted by a theory and experiments. Moreover, for the first time, 2D simulations of the ‘vacillating‐breathing’ regime without special ingredient such as thermal fluctuations are recovered. Copyright © 2014 John Wiley & Sons, Ltd.     

On interface crack models with contact zones situated in an anisotropic bimaterial

Summary A boundary value problem for two semi-infinite anisotropic spaces with mixed boundary conditions at the interface is considered. Assuming that the displacements are independent of the coordinate x ₃, stresses and derivatives of displacement jumps are expressed via a sectionally holomorphic vector function. By means of these relations the problem for an interface crack with an artificial contact zone in an orthotropic bimaterial is reduced to a combined Dirichlet-Riemann problem which is solved analytically. As a particular case of this solution, the contact zone model (in Comninou's sense) is derived. A simple transcendental equation and an asymptotic formula for the determination of the real contact zone length are obtained. The classical interface crack model with oscillating singularities at the crack tips is derived from the obtained solution as well. Analytical relations between fracture mechanical parameters of different models are found, and recommendations concerning their implementation are given. The dependencies of the contact zone lengths on material properties and external load coefficients are illustrated in graphical form. The practical applicability of the obtained results is demonstrated by means of a FEM analysis of a finite-sized orthotropic bimaterial with an interface crack. Received 19 October 1998; accepted for publication 13 November 1998     

自由面势流问题边界元法的数值色散误差

以连续及离散Fourier分析研究自由面势流问题边界元法的数值色散误差,并从理论上探讨有关计算中数值色散误差的改善问题.研究表明:对于该问题的数值色散误差而言,重要的在于以问题相应的离散算子考察计及各种数值手段后的总体色散误差,而非仅考虑该数值手段自身的数值色散误差大小.高阶面元、自由面域外奇点或适当的耦合方法是降低有关问题算子总体色散误差的较好选择.     

Transient three-dimensional flow in an enclosure with a hot spot on a vertical wall

A numerical method for the solution of the vector potential/vorticity vector formulation of the transient, fully three-dimensional Navier-Stokes energy and continuity equations has been applied to simulate the development of natural convective flow within a ‘box’ after a sudden temperature change on a vertical portion of the wall. Only one cavity size has been considered, this having a vertical height of three times its width and a horizontal length of six times its width. A single heated rectangular hot spot or ‘element’ on an otherwise adiabatic wall is centred between the vertical end walls. The opposite vertical wall is held at the intial fluid temperature, and all other walls are assumed to be adiabatic. Fluid properties have been assumed constant except for the density change with temperature that gives rise to the buoyancy force. The numerical method is an underrelaxation Gauss-Seidel method using finite differencing at each time step. Solutions have been obtained for a Prandtl number of 0.71, for Rayleigh numbers, based on the width, of between 0 and 100000 and for a number of heated element locations and sizes.     

Critical equilibrium state of a bushing in a contact pair with cracks possessing plastic tip zones

The problem of mechanics of contact fracture is considered for a bushing in a friction pair. It is assumed that multiple reciprocating motion of the plunger leads to fracture of the bushing material owing to friction caused by contact interaction and accompanied by the joint effect of loading and temperature. It is assumed that there are several arbitrarily located straight-line cracks with tip zones near the contact surface of the bushing. The stress state of the bushing is examined in the presence of regions where the crack faces (or some part of them) come into contact. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 145–156, September–October, 2006.     

The three-dimensional fundamental solution to Stokes flow in the oblate spheroidal coordinates with applications to multiples spheroid problems

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