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- W3118242102 abstract "This thesis is motivated by a research program between the LAMA (Mathematics, Univ. GustaveEiffel) and the Institut de Physique du Globe of Paris (Earth Sciences) on granular media and theirmathematical description.We consider here a continuous description: the material is described as a fluid with viscoplasticrheology, that allows us to model the transition between static (solid) states and mobile (liquid)states. Incompressible models have been used since the introduction of the so called µ(I) rheology(Jop et al. 2006). However such models do not represent accurately real flows, even in laboratoryexperiments. Recent studies indicate that volume variations, even if not significantly large, play akey role in the dynamics. Therefore compressible models have been recently considered (Barker etal. 2017). Although particular rheologies such as Bingham or Herschel-Bulkley models have beenoften considered in mathematical studies such as Malek et al. 2010, not much can be found ongeneral nonlinearities in terms of the trace and the norm of the strain rate tensor. We considerhere compressible models with general nonlinearities σ ∈ ∂F(D) where σ is the stress, D is thestrain rate and F is a convex viscoplastic potential. Under technical assumptions on F suchas subquadratic growth and superlinearity we prove the existence of solutions to the associatedvariational problem. This is obtained in the viscous as well as in the inviscid cases. We establishEuler-Lagrange characterizations of these solutions. No regularity is assumed on F, thus yieldstress rheologies are included. Numerical methods for viscoplastic laws have been classically used:augmented Lagrangian or regularization methods. However these methods were designed merelyfor Bingham or Herschel-Bulkley fluids, and moreover their cost is still too high for applicationsto real configurations. Here we consider an iterative but explicit method in the sense that thereis no linear system to solve, inherited from the minimizing of total variation functionals used inimaging (Chambolle, Pock 2011). It is applicable to any kind of nonlinearity, and includes a kindof projection on some convex sets. We prove the convergence of the method discretized in spacewith finite elements. Numerical tests confirm the theoretical results." @default.
- W3118242102 created "2021-01-18" @default.
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- W3118242102 date "2020-12-14" @default.
- W3118242102 modified "2023-09-23" @default.
- W3118242102 title "Analysis and approximation of compressible viscoplastic models with general nonlinearity for granular flows" @default.
- W3118242102 hasPublicationYear "2020" @default.
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