Modeling shallow avalanche onset over complex basal topography

Abstract

This paper deals with modeling of the onset of a shallow avalanche (soils, snow or other geomaterials) over various bottom topologies (mountains, valleys, ...). In order to do that, we use the shallow visco-plastic model with topography, developed in Ionescu (J. Non-Newtonian Fluid Mech. 193:116–128 2013), and we introduce a simple criterion to distinguish if an avalanche occurs or not. This criterion, relating the yield limit (material resistance) to the distribution of the external forces, is deduced from an optimization problem, called limit load analysis. The plastic dissipation functional which is involved in the limit load problem is non smooth and non coercive in the classical Sobolev spaces. To prove the existence of an onset velocity field (collapse flow) the appropriate functional space consists of bounded tangential deformation functions. We propose therefore a numerical strategy to solve the limit load problem and to get the onset flow field. A mesh free method, called the discontinuous velocity domain splitting (DVDS), is adapted here. The limit load problem is thus reduced to the minimization of a shape dependent functional. The discontinuous collapse flow velocity field is associated to a sub-domain and a rigid flow. With a level set of a Fourier function we give a description of the sub-domains and then we use genetic algorithms to solve the resulted non convex and non smooth global optimization problem. Finally, we illustrate the proposed numerical approach by solving several safety factor problems.