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.     

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.     

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.     

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.     

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.     

Design of experiments and energy dissipation analysis for a contact mechanics 3D model of frictional bolted lap joints

Bolted lap joints allow structural assemblies to be made. The answer to requirements, both static and dynamic, depends on the joint behaviour. Bolted joints are a primary source of energy dissipation in dynamic built-up and space structures among others. This paper presents an analysis of a bolted lap joint, subjected to a relative displacement after applying a pre-stress on the bolt in order to characterise the joint behaviour. For this purpose a 3D modelling is made by means of finite elements, using design techniques of experiments (DOE) to fit constitutive contact parameters. The theoretical results relative to elasto-plastic hysteresis cycles of the joint are experimentally validated. Finally, the preload effect and the magnitude of the displacement on the non-linear joint behaviour are analysed to determine equivalent stiffness and dissipated energy in the hysterical loops of the joint.     

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.     

Preconditioned Krylov subspace methods for solving radiative transfer problems with scattering and reflection

Two Krylov subspace methods, the GMRES and the BiCGSTAB, are analyzed for solving the linear systems arising from the mixed finite element discretization of the discrete ordinates radiative transfer equation. To increase their convergence rate and stability, the Jacobi and block Jacobi methods are used as preconditioners for both Krylov subspace methods. Numerical experiments, designed to test the effectiveness of the (preconditioned) GMRES and the BiCGSTAB, are performed on various radiative transfer problems: (i) transparent, (ii) absorption dominant, (iii) scattering dominant, and (iv) with specular reflection. It is observed that the BiCGSTAB is superior to the GMRES, with lower iteration counts, solving times, and memory consumption. In particular, the BiCGSTAB preconditioned by the block Jacobi method performed best amongst the set of other solvers. To better understand the discrete systems for radiative problems (i) to (iv), an eigenvalue spectrum analysis has also been performed. It revealed that the linear system conditioning deteriorates for scattering media problems in comparison to absorbing or transparent media problems. This conditioning further deteriorates when reflection is involved.     

An augmented method for free boundary problems with moving contact lines

An augmented immersed interface method (IIM) is proposed for simulating one-phase moving contact line problems in which a liquid drop spreads or recoils on a solid substrate. While the present two-dimensional mathematical model is a free boundary problem, in our new numerical method, the fluid domain enclosed by the free boundary is embedded into a rectangular one so that the problem can be solved by a regular Cartesian grid method. We introduce an augmented variable along the free boundary so that the stress balancing boundary condition is satisfied. A hybrid time discretization is used in the projection method for better stability. The resultant Helmholtz/Poisson equations with interfaces then are solved by the IIM in an efficient way. Several numerical tests including an accuracy check, and the spreading and recoiling processes of a liquid drop are presented in detail.     

Evolution strategies for solving discrete optimization problems

A method to solve discrete optimization problems using evolution strategies (ESs) is described. The ESs imitate biological evolution in nature and have two characteristics that differ from other conventional optimization algorithms: (a) ESs use randomized operators instead of the usual deterministic ones; (b) instead of a single design point, the ESs work simultaneously with a population of design points in the space of variables. The important operators of ESs are mutation, selection and recombination. The ESs are commonly applied for continuous optimization problems. For the application to discrete problems, several modifications on the operators mutation and recombination are suggested here. Several examples from the literature are solved with this modified ES and the results compared. The examples show that the modified ES is robust and suitable for discrete optimization problems.