École Nationale Supérieure de
Techniques Avancées

Abstract

We are interested in the mathematical and numerical modeling of wave propagation in underground media. We present two propagation model: (i) a generalization of Zener's model for viscoelastic media, (ii) Biot's model for poroelastic media. For each model we achieve a mathematical analysis, In particular, an existence and uniqueness of solution and an energy decay result. For the numerical resolution we construct a method specific to each model, based on a variational approach, a mixed finite elements approximation in space and a finite difference in time. We prove for each scheme obtained, a result of discrete energy decay which provides a sufficient stability condition. To simulate the waves propagation in unbounded domains, we adapt the perfectly matched layers techniques to viscoelastic and poroelastic waves. Finally, we present various numerical validations of the developed methods.