Shape Optimization and Supremal Minimization Approaches in Landslides Modeling
Résumé
The steady-state unidirectional (anti-plane) flow for a Bingham fluid is considered. We take into account the inhomogeneous yield limit of the fluid, which is well adjusted to the description of landslides. The blocking property is analyzed and we introduce the safety factor which is connected to two optimization problems in terms of velocities and stresses. Concerning the velocity analysis the minimum problem in BV(Ω) is equivalent to a shape-optimization problem. The optimal set is the part of the land which slides whenever the loading parameter becomes greater than the safety factor. This is proved in the one-dimensional case and conjectured for the two-dimensional flow. For the stress-optimization problem we give a stream function formulation in order to deduce a minimum problem in W1,∞(Ω) and we prove the existence of a minimizer. The Lp(Ω) approximation technique is used to get a sequence of minimum problems for smooth functionals. We propose two numerical approaches following the two analysis presented before. First, we describe a numerical method to compute the safety factor through equivalence with the shape-optimization problem. Then the finite-element approach and a Newton method is used to obtain a numerical scheme for the stress formulation. Some numerical results are given in order to compare the two methods. The shape-optimization method is sharp in detecting the sliding zones but the convergence is very sensitive to the choice of the parameters. The stress-optimization method is more robust, gives precise safety factors but the results cannot be easily compiled to obtain the sliding zone.