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Conférence externe
Shape gradients for three-dimensional contact problems with Tresca friction
B. Chaudet
Département de mathématiques et de statistique, Université Laval

Part of an ongoing research on shape optimization for hyperelastic contact problems, this presentation deals with three dimensional elastic bodies in contact with given friction (Tresca model). The proposed method is based on shape gradients combined with a level set representation of the shape. Due to the irregularity (non-linearity, non-differentiability) of the boundary conditions, the computation of shape gradients requires a specific treatment. Indeed, since the solution of the contact problem is non-differentiable with respect to the shape, we introduce a regularized Lagrange multiplier approach. The main advantages of this approach are its variational formulation, which takes the form of a non-linear variational equality, and the shape-differentiability of the regularized solution obtained.

After proving the convergence of the regularized solutions to the original one, we express the shape gradient of a general functional for the regularized problem, and try to establish sufficient conditions for those regularized shape gradients to converge. Numerical experiments, based on the finite element method for the augmented Lagrangian formulation and finite differences for the advection of the level set, will be presented. The method benefits from an original mesh cutting algorithm allowing sharp representation of the boundary at each iteration of the optimization process.

Date: 2018-07-24 à 11:30
Endroit: IFIP TC7 Conference on System Modelling and Optimization, Universität Duisburg-Essen, Essen (Allemagne)