Study On Nonsmooth Equations Method For Three Dimensional Contact Problems With Friction

The contact problem with friction is an important topic in solids mechanics and often encountered in the engineering practice. It takes much difficulty to the research of the methods because of the strong nonlinearity of the contact problems and it often appears as nonsmooth status. The available methods mostly include two types: the trial and error iterative method and the mathematical programming method. The former is based on the mechanical intuitions and is widely used in the engineering, but it has rarely strict mathematical foundation and difficult to ensure convergence. The latter include the quardratic programming for the frictionless case and the linear and nonlinear complementary models for the frictional case, they have the strict mathematical foundation and can ensure convergence. The nonsmooth analysis is an area on the nonsmooth (non-differentiable) function and its corresponding algorithms which developed rapidly in the recent years. As the development of the nonsmooth theory and many efficient algorithms were presented, a lot of problems which are difficult to analyze using the classic smooth function theory can be solved well now. In this thesis, we do some work on the contact problems with friction based on the nonsmooth analysis theory. Two nonsmooth methods for three dimensional frictional contact problems are presented and we try to find the theoretical basis for the iterative method of two and three dimensional contact problems using the nonsmooth analysis, and improve the iterative method to make it more convenient for application. The outline of works in this thesis is as follows: A nonsmooth method of the nonsmooth equations model (NEQ3D-l) from the nonlinear complementary description for the three dimensional contact problem with friction is presented based on the nonsmooth theory. The nonsmooth damped Newton method is given based on the definitions of the generalized derivative. No smoothing procedure is needed and then this method simplifies the computing. The numerical results suggest that this method is asefficient as the smoothing one.APPlying the theory of the nonsmooth analysis the contact conditions aretranslated to the nonsmooth equaions. And then a new nonsmooth eqUaionsmodel(NEQ3D-2) fOr the three dimensional conod problems with ffiction ispresented and a corresPOnding damped NeWton method is given. No extemalvedables are introduced and less storage and computing are requlred. msmodel is formulated with the relative displacement and the contact press direCtiyin the orthogonal coordinate syStem, without introducing the slip angle asvariable and transforming the variab1e to the polar coordinate system. Theformulation of thes model is simpler and the genetalized derivative is easier tocompute, so the method is easier tO imPlement. Based on the nonsmooth thcorywe give itS convergence analysis.The itendive method for the tWo dimensional contact problems edfuction is widely used in the engineering, Which is based on the mechanicalinndhons and need not introduce any extemal variables or comPute thederivative. The iterative methOd is a heuristic method and there are differentsortS of itCfative forms, bot the clue and the Procedure are moedy same. Theform described in thes thesis is a tyPical and efficient one of them. Thrughanalysis of the procedures of the nonsmooth eqUatin methodopQ2D) of thetWo dfornsionaI contat Problems with ffichon and the itefative method, wePrOve the equlvalence of these two methods. It can be regarded as thefoWon for the conVe4ence of the iterative method. Then variable sorts ofiterative methods can be modified to uns iterative form to improve theirconvergence, this Inak it more conv...