Mathematical Analysis and Numerical Resolution of Contact Problems: Combined Penalization, Semi-Smooth Newton and Boundary Element Methods

Résumé

In this thesis, we develop and analyze fast procedure for numerical resolution of the two dimensional Signorini contact problems in linear elasticity, including the frictional conditions. For this purpose, the static linear elasticity model, the statics frictionless and frictional contact models, and quasi-static frictional model are successively considered. Since the contact can be occurred at the boundary, the boundary integral operators are then used to develop the boundary variational formulation of our models. For the linear elasticity model with mixed boundary conditions, the approximate solution is determined by computing the boundary Cauchy data. An efficient computational of the discrete boundary integral operators matrices is presented and the convergence results of the solution are given. For the frictionless and frictional contact models, since the boundary variational formulation leads to a constrained minimization problem and to a constrained non differentiable minimization problem respectively, the Fenchel duality theory is used in the both cases to present a dual formulation which can be written as an inequality-constrained maximization of smooth function. After proving the well posedness of the regularized dual problem and convergence to continuous contact problem, the generalized Newton method based on active-set iterative method is constructed and local as well as global convergence are established. In particular case of contact problem with friction, Coulomb’s law is considered. This problem is then approximated by the sequence of contact problems with given friction known as Tresca problems. The Coulomb problem is then finally analyzed by using a particular fixed point method. In the last model, where we have considered the quasi-static Signorini problem with Coulomb friction, we remark that this model leads at each time step to a system of static contact problem. The time discretization of the model and mixed duality-fixed point formulation combined with the augmented lagrangian approach are then presented. Finally, for all these models, the spatial discretization is carried out by using the Galerkin boundary element method with the help of B-spline functions. The resulting linear system is solved by using preconditioned Conjugate Gradient iterative solver via Fast Fourier Transform method. An error estimate for the discretization is computed and some numerical examples are presented to show the effectiveness of the theoretical results obtained.

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