Studies On Mathematical Programming Approaches For Nonsmooth Mechanics Problems

The field of nonsmooth mechanics is a newly developed research area in mechanics, which represents a natural framework for the studies of non-differentiable, discontinuous and nonconvex problems in mechanics. The appearance of non-smoothness makes the mathematical programming methods and non-smooth analysis be adopted as the necessary mathematical tools for research of non-smooth mechanics problems. This thesis concerns the application of some modern mathematical programming approaches for solving certain problems of non-smooth mechanics. Attention is given to develop the more accuracy and efficiency numerical methods for some advanced non-smooth mechanics problems such as limit analysis, dynamic problems and structural optimal design problems.The first three Chapters of this dissertation are devoted to studying the general methodology for nonsmooth mechanics, while the remainders are focused on studying some new numerical methods for three special nonsmooth mechanics problems.In Chapter 1, the background and motivation of this dissertation are introduced together with a review of some basic concepts and existing methods for nonsmooth mechanics. Finally, main research work of this dissertation is discussed.In Chapter 2, some necessary mathematical tools required in this dissertation are presented. Especially, some new numerical methods for complementarity problems (CP) and variational inequalities (VI) are introduced. Then, attention is given to the construction of a smoothing GAP function for VI. An integral global optimality condition is employed to provide a smooth approximation for the classical GAP function, through which the variational inequality problem may be reformulated as a new differentiable optimization problem.Chapter 3 is devoted to developing a regularization solving methodology for nonsmooth mechanics problems. The idea of entropic regularization from the field of mathematical programming is employed to derive a smooth approximation that uniformly approaches the nonsmooth models of nonsmooth mechanics problems. The proposed regularization methodology greatly improve the practicability of the existing theories of non-smooth mechanicsIn Chapter 4, nonlinear programming methods are utilized to study the problem of plastic limit analysis of continuum. Firstly, the Lagrange duality theory of convex programs is employed to discover the duality relationship in the limit analysis theory, especially, we...