Optimization techniques in landslides modelling
Riad Hassani, Ioan R. Ionescu, Thomas Lachand-Robert · 2005
Abstract. The stationary anti-plane problem for a two dimensional Bingham fluid is consid-ered. 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 is equivalent to a shape optimization problem. We describe a numerical method to compute the safety factor through this equiva-lence. For the stress optimization problem we give a stream function formulation in order to deduce a minimum problem in W 1,∞(Ω) and we prove the existence of an minimizer. The Lp(Ω) approximation technique is used to get a sequence of minimum problems for smooth functionals. The finite element approach and a Newton method is used to obtain a numerical scheme for the safety factor. 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 con-vergence 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.