Séance Séminaire

Séminaire ACSIOM

Optimisation des formes et minimisation supremale dans l'analyse de la charge limite (Shape optimization and supremal minimization approaches in limit analysis of collapse states)

The anti-plane (unidirectional, 2D) flow for a Bingham (rigid-visco-plastic) fluid is considered in landslides modeling. The blocking property, associated to the rigid-plastic model, is analyzed through the critical load of the collapse state. The limit load analysis is connected to two optimization problems in terms of velocities and stresses. Concerning the velocity analysis we prove that the minimum problem in $BV(\Omega)$ is equivalent to a shape optimization problem. The optimal set is the part of the structure which slides whenever the loading parameter becomes greater than the critical load. This is proved in the one dimensional case and conjectured for the two dimensional flow. For the stress optimization problem a stream function formulation is given in order to deduce a minimum problem in $W^{1,\infty}(\O)$ and to prove the existence of a minimizer. The $L^p(\O)$ approximation technique is used to get a sequence of minimum problems for smooth functionals. Two numerical approaches, following the two approaches presented before, are proposed. The finite element and a Newton method are used to obtain a numerical scheme for the stress formulation. Some numerical results are given in order to compare the two approaches. Finally the above technics are adapted for the in-plane (vectorial, 2D) problem and some numerical results are presented.