Research On Variational-Hemivariational Inequalities And Related Problems With Applications

Variational-hemivariational inequality is a generalization of hemivariational inequality,which involves both convex and non-convex potential functions in the inequality.In recent years,in view of its important applications in mechanics,engineering,control and other fields,variational-hemivariational inequalities have attached much more attention,and a large number of excellent research works have emerged.However,in light of the short history of variational-hemivariational inequality,there are still many problems to be solved and its applicationss in complex realistic problems arising from mechanics and engineering deserves further exploration,therefore,it is of great significance to study the theory and applications of variational-hemivariational inequalities.This dissertation aims to delve into the issues relating to variational-hemivariational inequalities,including penalty method,regularization method,optimal control problems and numerical method and explore the applications of theoretical results to contact mechanics.Firstly,a generalized penalty method is discussed for elliptic variational-hemivariational inequalities,which is an approximating method for solving variational-hemivariational inequalities.Initially,based on Tikhonov well-posedness and traditional penalty method,the generalized penalty method for a single-valued elliptic variational-hemivariational inequality is studied.It is proved that the solution of the generalized penalty problem converges strongly to the unique solution of the original problem.Then,a class of set-valued elliptic variational-hemivariational inequalities are concerned.Since the involved convex function is an extended real-valued function,traditional penalty method is not available.We construct a generaized penalty problem for the set-valued variational-hemivariational inequality based on traditional penalty method and Moreau-Yosida approximating technique,and by applying smoothness and differentiability of Moreau-Yosida approximation function,the convergence of solution sequence to the generalized penalty problem is proved.Finally,variational formulas for two kinds of elastic contact problems with unilateral constraint are established and we prove the existence and the approximating results of weak solutions to the contact problems by means of the obtained theoretical results for the considered elliptic variational-hemivariational inequalities.Then,we focuses on the regularization method,which is another approximating method for solving variational-hemivariational inequalities.We consider the regularization method for an elliptic quasi-variational hemivariational inequality.Firstly,the solvability of the elliptic quasi-variational hemivariational inequality is obtained by employing fixed point theorem and variational selection,which generalizes the existing solvability results obtained in the literature.Then we study the solvability of regularization problem and the convergence of regularized solutions.Specifically,the weak convergence of regularized solutions is proved under mild assumptions,and furthermore the strong convergence result is obtained by enhancing some conditions.Finally,an elastic frictional contact problem with deformation constraint is considered and the existence of weak solution to the contact problem and the stability with respect to perturbation of data are discussed.Subsequently,the solvability and optimal control problems of a class of parabolic quasi-variational hemivariational inequalities are considered.First,it is proved that the parabolic quasi-variational hemivariational inequality is equivalent to its integral version,and the solvability of the parabolic quasi-variational hemivariational inequality is obtained by employing fixed point theorem and surjective theorem.This technique avoids the requirement of uniqueness of the solution.Moreover,the continuous dependence of solution on initial value and vector f is investigated by applying Mosco continuity of the constraint set,and based on the continuous dependence result,we prove the solvability of associated optimal control problem.Finally,the theoretical results are applied to a class of viscoelastic frictional contact problems with deformation constraint and the existence of weak solution for the contact problem and the solvability of associated optimal control problem are delivered.Finally,we devote to studying numerical method for a class of first-order evolutionary variational-hemivariational inequalities arising from a dynamic contact problem with multi-contact zones and unilateral constraint.First,we construct a fully discrete finite element scheme and derive its well-posedness and stability,and the optimal order error is analyzed under appropriate assumptions on the regularity of exact solution.Moreover,in order to simultaneously deal with nonmonotone conditions acting on two contact zones,a projection-iterative algorithm based on Lagrange multiplier method is provided for computation,whose convergence is proved theoretically under some assumptions.Finally,two numerical examples are carried out to confirm the theoretical results and investigate the applicability of the projection-iterative algorithm.