Numerical Analysis For Elasticity Problem And Its Inverse Problem In Boundary Condition

This dissertation is concerned with the problem of elastic mechanics. First-ly, we introduce the contact problem which take the displacement. field as the research object, and illustrate the hypothetical condition for the existence and uniqueness of solution of the contact problem with friction and the elastic prob-lem with damage, respectively. Then we take the strain tensor as the research object for the contact problem, that is the dual problem of the original prob-lem. It is proved that the intercommunity for solutions between the primal and dual problems. Moreover, under the extra assumption condition, we proof the existence and uniqueness of the solution of the dual problem.Secondly, by the viscoelastic contact model, we study the case of the con-tact boundary is inclusion, in other words, the contact conditions controlled by the non-monotone operator, and the operator is a single or multi-valued op-erator with Clarke subdifferential. From frictionless viscoelastic contact model, we simplify the formulation and derive a parabolic variational-Hemivariational inequality with pseudo-monotone elliptic operator. We analysis the semi-discrete scheme of the inequality on the time, that is to say Rothe problem. We prove that the existence of solution of the Rothe problem, and obtain the convergence and regularity results. At last, we show the two-dimensional numerical simulation.Finally, the focus of this dissertation is a class of inverse elastic problem, the goal of the study is to reconstruct traction which acts on boundary by observed data. We use optimal control problem to describe this target, and then the inverse problem has at least one solution. Then the objective function is introduced into the Tikhonov regularization to prove the uniqueness of solution of the inverse problem, and as the regularization parameter goes to 0, the solution converges to the solution which is the smallest in the solution set of the original inverse problem under the L1 norm. Furthermore, we derive error estimate of the numerical solution which depends on the regularization parameter. The most important point is that we are derived adjoint problems of the truncation elastic problem with damage and contact problems with frictionless, respectively. Then by the adjoint problem, the constraints inequality of solution of the inverse problem is obtained, and we construct the iterative algorithm for numerical simulation.