An augmented Lagrangian optimization method for contact analysis problems, 2: numerical evaluation

The ALM2 solution procedure is evaluated by solving two simple contact analysis problems for different friction conditions. These example problems are devised to have closed form solutions. This way there is no uncertainty about the target solution for evaluation of the proposed algorithm as well as existing algorithms. The numerical results with ALM2 are compared with the analytical solutions as well as with the penalty, Lagrange multiplier and existing augmented Lagrangian methods. All the algorithms are analysed for stick and slip friction conditions. The example problems are used to show clearly the dependence of the existing solution methods on the number of load steps and penalty values. It is concluded that convergence of incremental solution schemes employed in these methods does not guarantee accuracy of the contact solution even with the use of solution enhancement schemes such as automatic load stepping and contact load prediction. The example problems are also used to demonstrate solution independence of the proposed ALM2 procedure from penalty values, and from the number of load steps. The proposed formulation for calculation of frictional forces and the ALM solution algorithm have worked quite well for the example problems. However, the algorithm needs to be developed and evaluated for more complex contact analysis problems.     

An augmented Lagrangian optimization method for contact analysis problems, 1: formulation and algorithm

A review of existing augmented Lagrangian methods (ALM) for contact analysis problems reveals that they have not been implemented with automatic penalty updates as intended in their original development. Therefore, although the methods are an improvement over the penalty methods, solution with them still depends on the user-specified penalty values for the contact constraints. To overcome this drawback, an ALM is developed and discussed for contact analysis problems that automatically update the user-specified penalty values to obtain the final appropriate values. Further, to solve the frictional contact analysis problem accurately, a two-phase formulation is proposed. Solution of the Phase 1 problem removes penetration of the contacting nodes and brings them exactly to their initial contact points. In addition, a new contact constraint is introduced which allows determination of the precise friction force at the contacting nodes. Phase 2 of the formulation checks the friction conditions and solves the friction problem to bring the structure to an equilibrium state. Phases 1 and 2 are then combined to provide a general algorithm for multi-node frictional contact problems. The two-phase procedure also removes dependence of the contact solution on the number of load steps for the elastostatic problem. Numerical evaluation of the formulation and the algorithm is presented in Part 2 of the paper.     

Study of variational inequality and equality formulations for elastostatic frictional contact problems

Summary An overview ofvariational inequality andvariational equality formulations for frictionless contact and frictional contact problems is provided. The aim is to discuss the state-of-the-art in these two formulations and clearly point out their advantages and disadvantages in terms of mathematical completeness and practicality. Various terms required to describe the contact configuration are defined.Unilateral contact law and classical Coulomb’s friction law are given.Elastostatic frictional contact boundary value problem is defined. General two-dimensional frictionless and frictional contact formulations for elastostatic problems are investigated. An example problem of a two bar truss-rigid wall frictionless contact system is formulated as an optimization problem based on the variational inequality approach. The problem is solved in a closed form using the Karush-Kuhn-Tucker (KKT) optimality conditions. The example problem is also formulated as a frictional contact system. It is solved in the closed form using a new two-phase analytical procedure. The procedure avoids use of the incremental/iterative techniques and user defined parameters required in a typical implementation based on the variational equality formulation. Numerical solutions for the frictionless and frictional contact problems are compared with the results obtained by using a general-purpose finite element program ANSYS (that uses variational equality formulation). ANSYS results match reasonably well with the solutions of KKT optimality conditions for the frictionless contact problem and the two-phase procedure for the frictional contact problem. The validity of the analytical formulation for frictional contact problems (with one contacting node) is verified. Thevariational equality formulation for frictionless and frictional, contact problems is also studied in detail. The incremental/iterative Newton-Raphson scheme incorporating the penalty approach is utilized. Studies are conducted to provide insights for the numerical solution techniques. Based on the present study it is concluded that alternate formulations and computational procedures need to be developed for analysis of frictional contact problems.     

Boundary element analysis of elastic contact problems using gap finite elements

Gap finite elements provide a practical approach for dealing with elastic contact problems, and it is not possible to derive something similar with boundary elements. This work introduces a simple technique for the analysis of elastic contact problems by coupling a gap finite element subregion with boundary element subregions. The developed algorithm proves to be accurate and reliable, combining the advantages within contact problems of both the gap finite element and the boundary element method.     

车轮滑行轮轨热接触耦合有限元分析

为研究轮轨接触温升和热应力规律,用有限元法分别建立锥型踏面车轮和磨耗型踏面车轮在60kg／m钢轨上滑行的三维热接触耦合模型．考虑温度场与结构场相互影响、相关材料参数随温度变化以及轮轨接触问题,对在一定速度下抱死滑行时轮轨温度场和应力场的热一结构直接耦合进行分析．结果表明磨耗型踏面车轮的接触斑面积大于锥型踏面车轮的接触斑面积,且前者接触斑趋近于圆形,后者接触斑为细长椭圆形;材料参数随温度的变化对轮轨温度场和应力场影响很大,不可忽略;温度场对应力场的影响很大,温度升高的趋势与应力升高的趋势相同;磨耗型踏面对轮轨的热损伤比锥型踏面小很多．     

摩擦式离合器滑摩过程及温度场分析

为更真实地反映摩擦式离合器滑摩过程中的温度场,提出1种利用动力学软件与有限元软件相互辅助分析的方法.在试验测得的负载扭矩的基础上,建立摩擦片和内鼓轮及气胎的模型;利用MSC Adams对摩擦式离合器进行动力学仿真,通过对摩擦片加压计算出滑摩时间和主、从动端的相对转速;将该相对转速用于有限元热机耦合场分析,通过摩擦片和内鼓轮的相对转动形成滑摩、生成热量,得离合器各时刻的温度场.结果表明：离合器在接触后3.064 s时主、从动端接合完毕,且摩擦片瞬态最高温度只有65.83℃.该方法对摩擦式离合器滑摩过程及温度场分析具有一定的实际意义.     

An iterative procedure for finite-element stress analysis of frictional contact problems

A simple, efficient, versatile and easily adaptable, iterative finite-element technique is described for solving frictional contact problems. The method is based on logical steps to establish the contact geometry and regions of slip and nonslip. Unlike previous techniques, the approach can be extended readily to multiple contact surfaces. The scheme is demonstrated by analyzing a mechanical joint in orthotropic wood. In this case, mixed coordinate systems are used to enhance accuracy of the stresses near the pin contact region. The numerically computed values agree with those reported elsewhere.     

Review of formulations for structural and mechanical system optimization

Alternative formulations for optimization and simulation of structural and mechanical systems and other related fields are reviewed. The material is divided roughly into two parts. Part 1 focuses on the developments in structural and mechanical systems, including configuration and topology optimization. Here the formulations are classified into three broad categories: (i) the conventional formulation where only the structural design variables are treated as optimization variables, (ii) simultaneous analysis and design (SAND) formulations where design and some of the state variables are treated as optimization variables, and (iii) a displacement-based two-phase approach where the displacements are treated as unknowns in the outer loop and the design variables as the unknowns in the inner loop. Part 2 covers more general formulations that are applicable to diverse fields, such as economics, optimal control, multidisciplinary problems and other engineering disciplines. In these fields, SAND-type formulations have been called mathematical programs with equilibrium constraints (MPEC), and partial differential equations (PDE)-constrained optimization problems. These formulations are viewed as generalizations of the SAND formulations developed in the structural optimization field. Based on the review, it is concluded that the basic ideas of the formulations presented in diverse fields can be integrated to conduct further research and develop alternative formulations and solution procedures for practical engineering applications. The paper lists 187 references on the subject.     

Piecewise constant level set method for topology optimization of unilateral contact problems

The paper deals with the structural optimization of the elastic body in unilateral contact with a rigid foundation using the level set approach. A piecewise constant level set method is used to represent the evolution of interfaces rather than the standard method. The piecewise constant level set function takes distinct constant values in each subdomain of a whole design domain. Using a two-phase approximation the original structural optimization problem is reformulated as an equivalent constrained optimization problem in terms of the piecewise constant level set function. Necessary optimality condition is formulated. Finite difference and finite element methods are applied as the approximation methods. Numerical examples are provided and discussed.     

An analysis of dual formulations for the finite element solution of two-body contact problems

This paper examines the convergence properties of dual finite element formulations of the two-dimensional frictionless two-body contact problem under the assumption of infinitesimal kinematics. The centerpiece of the proposed analysis is the well-known Babuška–Brezzi condition, suitably adapted to the present problem. It is demonstrated for certain canonical geometries that several widely used methods that employ pressure or force interpolations derived from the discretizations of both surfaces violate the Babuška–Brezzi condition, thus producing increasingly oscillatory solutions under mesh refinement. Alternative algorithms are proposed that circumvent this difficulty and are shown to yield convergent solutions.     

Optimization of a contact surface

The paper introduces ideas from shape optimization to multibody system dynamics. A disk rolling down a given slope is taken as a simple example, for which it is the goal of the optimization to shape the rolling contour of the disk such that it takes a minimum time to cover a certain distance. The shape of the contour is described by its radius of curvature. The governing equations of motion result from the kinematics of relative motion and the Newton–Euler formalism. Three different kinds of spirals are defined and optimized.     

基于边权约束法实现接触问题多约束图剖分

为提高接触问题并行计算的效率,分析内力计算和接触计算过程的并行性,提出基于边权约束法构造接触多约束图的方法,对比和分析多约束图剖分算法和双重区域剖分算法的负载平衡和通信性能.数值实验表明,在典型二维模型中多约束图剖分算法的负载平衡性能略低于双重区域剖分算法,但仍可将负载不平衡度控制在较好的范围内,简化并行计算的通信过程,减少总通信量并降低动态通信量比例.     

A Petrov Galerkin finite-element method for interface problems arising in sensitivity computations

Continuous sensitivity equation methods have been applied to a variety of applications ranging from optimal design, to fast algorithms in computational fluid dynamics to the quantification of uncertainty. In order to make use of these methods for interface problems, one needs fast and accurate numerical methods for computing sensitivities for problems defined by partial differential equations with solutions that have spatial discontinuities such as shocks and interfaces. In this paper we develop a discontinuous Petrov Galerkin finite-element scheme for solving the sensitivity equation resulting from a 1D interface problem. The 1D example is sufficient to motivate the theoretical and computational issues that arise when one derives the corresponding boundary value problem for the sensitivities. In particular, the sensitivity boundary value problem must be formulated in a very weak sense, and the resulting variational problem provides a natural framework for developing and analyzing numerical schemes. Numerical examples are presented to illustrate the benefits of this approach.     

Frictional properties of VA‐mode alignment‐film surfaces studied by frictional force microscopy

Abstract— The surface‐friction characteristics of the post‐bake temperature were compared to that of the rubbing condition on VA‐mode alignment films (AFs) using frictional force microscopy (FFM). The surface roughness and the surface frictional force increased when the temperature was elevated. However, the frictional asymmetry could not be detected by the frictional loop due to the instability of the side‐chain molecular arrangement. For the inter‐influences of the total friction and surface roughness, the density of the side chain was changed and it was intentionally rubbed for evaluation. The results showed that the total friction was higher when the rubbing was stronger and side‐chain density lower, and the surface roughness also increased in the same manner. This can be explained by the potential interaction due to the curved structure of the side‐chain surface that establishes the probe that is scanned and the top surface layer. In order to confirm the influence of the pre‐tilt angle on the tilt of the side chain, a test cell was produced under the same conditions and evaluated. The results showed that the pre‐tilt‐angle decreased according to the difference in density of the side chain and strength of rubbing after injecting liquid‐crystal (LC).     

人工髋骨接触状态仿真

利用有限元法对髋臼内衬在一个步态周期载荷下的应力状况进行仿真,探讨髋臼内衬在不同股骨头-髋臼材料组合和不同接触位置下的接触状态.结果表明:髋臼内衬的最大接触压力集中在髋臼内衬后1/4部分,其大小与髋骨所受合力一致.髋臼内衬应力受其接触位置影响很大.在某些状态下,不同的股骨头-髋臼材料组合往往会对其接触状态产生根本性影响.     

Scalable Total BETI based algorithm for 3D coercive contact problems of linear elastostatics

A Total BETI (TBETI) based domain decomposition algorithm with the preconditioning by a natural coarse grid of the rigid body motions is adapted for the solution of contact problems of linear elastostatics and proved to be scalable for the coercive problems, i.e., the cost of the solution is asymptotically proportional to the number of variables. The analysis is based on the original results by Langer and Steinbach on the scalability of BETI for linear problems and our development of optimal quadratic programming algorithms for bound and equality constrained problems. Both theoretical results and numerical experiments indicate a high efficiency of the algorithms presented.      

Two guaranteed equilibrated error estimators for Harmonic formulations in eddy current problems

In this paper a guaranteed equilibrated error estimator is developed for the 3D harmonic magnetodynamic problem of Maxwell’s system. This system is recasted in the classical $\mathbf{A}?\phi$ potential formulation and solved by the Finite Element method. The error estimator is built starting from the $\mathbf{A}?\phi$ numerical solution by a local flux reconstruction technique. Its equivalence with the error in the energy norm is established. A comparison of this estimator with an equilibrated error estimator already developed through a complementary problem points out the advantages and drawbacks of these two estimators. In particular, an analytical benchmark test illustrates the obtained theoretical results and a physical benchmark test shows the efficiency of these two estimators.     

某清扫车车架结构分析与拓扑优化设计

为了提高某自卸清扫车车架的力学性能并减轻重量,建立了车架的有限元模型,对车架进行了强度和刚度校核以及模态分析。并借助LS-DYNA分析了车架中间矩形框在外力激励输入下的位移时程曲线,得到了车架的变形及变化趋势,针对车架的动态变形分析结果,对车架中间矩形框的前后横梁加强区域进行了局部拓扑优化,以减少材料使用提高车架刚度,改善了车架的动态特性。分析结果表明:车架静态强度满足工作要求,工作时路面激励及怠速不会引起车架共振。     

A research on the contact stress of roller bearing based on crowning analysis

According to elastic contact theory,contact model between roller and race is established.Compared with the Hertz results,the results are proved,based on which contact stress distribution of different crowning and initial contact length is given,then the appropriate value is derived.On the basis,inertia force and different radial force is given into consideration.Via analysis,it concludes that under balanced pure radial load condition the largest contact stress between roller and race increases along with crowning value increasing.With the same crowning value,the largest contact stress between roller and race decreases in the first and increases at the end along with initial contact length increasing.Contact stress between roller and outer race increases along with revolution speed increasing.