Analysis and control of a nonlinear boundary value problem

We consider a nonlinear two-dimensional boundary value problem which models the frictional contact of a bar with a rigid obstacle. The weak formulation of the problem is in the form of an elliptic variational inequality of the second kind. We establish the existence of a unique weak solution to the problem, then we introduce a regularized version of the variational inequality for which we prove existence, uniqueness and convergence results. We proceed with an optimal control problem for which we prove the existence of an optimal pair. Finally, we consider the corresponding optimal control problem associated to the regularized variational inequality and prove a convergence result

Time-dependent variational inequalities for viscoelastic contact problems

AbstractWe consider a class of abstract evolutionary variational inequalities arising in the study of contact problems for viscoelastic materials. We prove an existence and uniqueness result, using standard arguments of time-dependent elliptic variational inequalities and Banach's fixed point theorem. We then consider numerical approximations of the problem. We use the finite element method to discretize the spatial domain and we introduce spatially semi-discrete and fully discrete schemes. For both schemes, we show the existence of a unique solution, and derive error estimates. Finally, we apply the abstract results to the analysis and numerical approximations of a viscoelastic contact problem with normal compliance and friction

On the Tykhonov Well-posedness of an Antiplane Shear Problem

We consider a boundary value problem which describes the frictional antiplane shear of an elastic body. The process is static and friction is modeled with a slip-dependent version of Coulomb's law of dry friction. The weak formulation of the problem is in the form of a quasivariational inequality for the displacement field, denoted by \cP. We associated to problem \cP a boundary optimal control problem, denoted by \cQ. For Problem \cP we introduce the concept of well-posedness and for Problem \cQ we introduce the concept of weakly and weakly generalized well-posedness, both associated to appropriate Tykhonov triples. Our main result are Theorems \ref{t1} and \ref{t2}. Theorem \ref{t1} provides the well-posedness of Problem \cP and, as a consequence, the continuous dependence of the solution with respect to the data. Theorem \ref{t2} provides the weakly generalized well-posedness of Problem \cQ and, under additional hypothesis, its weakly well posedness. The proofs of these theorems are based on arguments of compactness, lower semicontinuity, monotonicity and various estimates. Moreover, we provide the mechanical interpretation of our well-posedness results.Comment: 21 page

Modélisation mathématique en Mécanique du Contact

International audienceDans ce travail nous présentons quelques considérations sur la modélisation et l'analyse variationnelle des modèles mathématiques décrivant le contact entre un corps déformable et un obstacle. Nous exemplifions ces propos a travers l'étude d'un problème élasto-visco-plastique avec compliance normale et frottement de Coulomb

Advances in variational and hemivariational inequalities : theory, numerical analysis, and applications

Highlighting recent advances in variational and hemivariational inequalities with an emphasis on theory, numerical analysis and applications, this volume serves as an indispensable resource to graduate students and researchers interested in the latest results from recognized scholars in this relatively young and rapidly-growing field. Particularly, readers will find that the volume’s results and analysis present valuable insights into the fields of pure and applied mathematics, as well as civil, aeronautical, and mechanical engineering. Researchers and students will find new results on well posedness to stationary and evolutionary inequalities and their rigorous proofs. In addition to results on modeling and abstract problems, the book contains new results on the numerical methods for variational and hemivariational inequalities. Finally, the applications presented illustrate the use of these results in the study of miscellaneous mathematical models which describe the contact between deformable bodies and a foundation. This includes the modelling, the variational and the numerical analysis of the corresponding contact processes. Furthermore, it can be used as supplementary reading material for advanced specialized courses in mathematical modeling for students with a strong background knowledge on nonlinear analysis, numerical analysis, partial differential equations, and mechanics of continua