Éditeur(s) : HAL CCSD American Mathematical Society
Résumé : International audience
We define an abstract notion of subdifferential operator and an associated notion of smoothness of a norm covering all the standard situations. In particular, a norm is smooth for the Gâteaux (Fréchet, Hadamard, Lipschitz-smooth) subdifferential if it is Gâteaux (Fréchet, Hadamard, Lipschitz) smooth in the classical sense, while on the other hand any norm is smooth for the Clarke-Rockafellar subdifferential. We then show that lower semicontinuous functions on a Banach space satisfy an Approximate Mean Value Inequality with respect to any subdifferential for which the norm is smooth, thus providing a new insight on the connection between the smoothness of norms and the subdifferentiability properties of functions. The proof relies on an adaptation of the ''smooth'' variational principle of Borwein-Preiss. Along the same vein, we derive subdifferential criteria for coercivity, Lipschitz behavior, cone-monotonicity, quasiconvexity, and convexity of lower semicontinuous functions which clarify, unify and extend many existing results for specific subdifferentials.
ISSN: 0002-9947

hal-00699220
https://hal.archives-ouvertes.fr/hal-00699220

Éditeur(s) : HAL CCSD
Résumé : International audience
This paper deals with variational inclusions of the form 0 2 f(x) + g(x) K where f is smooth function from a re exive Banach space X into a Banach space Y , g is a Lipschitz function from X into Y and K is a nonempty closed convex cone in the space Y . We show that the previous problem can be solved by an extension of the Zincenko's method which can be seen as a perturbed Newton's method. Numerical results are given at the end of the paper.
Revista Investigacion Operacional

hal-00924143
https://hal.archives-ouvertes.fr/hal-00924143

Éditeur(s) : HAL CCSD
Résumé : This work is devoted to the theoritical research and to the numerical calculus of weak solutions (in the sens of generalized functions) for the non linear transport equation((∂u)/(∂t))(x,y,t)+a(x,y,t)((∂f(u))/(∂x))(x,y,t)+b(x,y,t)((∂g(u))/(∂y))(x,y,t)=0 pour t>0with the initial condition u(x,y,0)=u₀(x,y) where the functions {(x,y,t)→a(x,y,t)} and {(x,y,t)→b(x,y,t)} belong to L^{∞}(R²×R⁺) (but can be discontinuous), the functions f and g are smooth and monotonous, the function {(x,y)→u₀(x,y)} belongs to L^{∞}(R²). We recall the necessary notions on nonlinear generalized functions for introducing their tensorial product. The main results (to determine the weak solutions) are sufficient conditions so that, when a sum of généralized functions (like Heaviside or Dirac products) is associated with zero, each terms of the sum is equal to zero. Thanks to these theoretical results, we can solve the Riemann problem with the help of a solver written like tensorial product of Heaviside functions (or like a sum of tensorial product of Heaviside functions) in order to obtain the weak solutions. These weak solutions allow to develop two dimensional numerical Godunov type schemes. Then, numerical tests are performed which give a comparison between the results obtained by these 2D schemes and the ones of the splitting method. These tests prove that the 2D numerical schemes are as reliable as the ones obtained by splitting. They are also more simple in their expression. Moreover, a more detailed comparative study of the two types of numerical schemes show that the 2D schemes are far less expensive in the linear case as well as in the non linear case. They are stable for the L^{∞} norm, unlike the splitting schemes. At least some perspectives as well as an existence theorem of a strong solution "almost eveywhere" for linear hyberbolic systems are studied.
Nous nous intéressons à la recherche et au calcul numérique de solutions faibles (au sens des fonctions généralisées) de l'équation de transport non linéaire((∂u)/(∂t))(x,y,t)+a(x,y,t)((∂f(u))/(∂x))(x,y,t)+b(x,y,t)((∂g(u))/(∂y))(x,y,t)=0 pour t>0complétée de la condition initiale : u(x,y,0)=u₀(x,y) où les fonctions {(x,y,t)→a(x,y,t)} et {(x,y,t)→b(x,y,t)} appartiennent à L^{∞}(R²×R⁺) (mais peuvent être discontinues), les fonctions f et g sont lisses et monotones, la fonction {(x,y)→u₀(x,y)} appartient à L^{∞}(R²). Des rappels sur les fonctions généralisées nous permettent d'introduire leur produit tensoriel. Un des résultats clés (pour déterminer ultérieurement les solutions faibles cherchées) donne des conditions suffisantes pour que, lorsqu'une somme de fonctions généralisées (de type produit d'Heaviside ou de Dirac) est associée à 0, chacun des termes de la somme est nul. Grâce à ces résultats théoriques, on résout le problème de Riemann 2D à l'aide d'un solveur s'écrivant comme produit tensoriel de fonctions type Heaviside (ou comme somme de produit tensoriel de fonctions type Heaviside) afin d'obtenir les solutions faibles. Ces solutions faibles permettent la construction des schémas numériques de type Godunov 2D. Nous les validons par des test numériques comparant les résultats obtenus par ces schémas 2D et ceux de la méthode du splitting. Ces tests montrent que les schémas numériques 2D sont aussi fiables que ceux par splitting, alors qu'ils sont plus simples dans leur écriture. Une étude comparative plus complète entre les deux types de schémas numériques montre de plus que les schémas 2D sont nettement moins coûteux à la fois dans les cas linéaire et non linéaire et qu'ils sont stables pour la norme L^{∞}, contrairement aux schémas par splitting. Enfin des perspectives ainsi qu'un théoréme d'existence de solutions fortes "presque partout" pour des systèmes hyperboliques linéaires sont abordés.
https://hal.archives-ouvertes.fr/tel-01487366

Droits : info:eu-repo/semantics/OpenAccess
tel-01487366
https://hal.archives-ouvertes.fr/tel-01487366
https://hal.archives-ouvertes.fr/tel-01487366/document
https://hal.archives-ouvertes.fr/tel-01487366/file/th%C3%A8se_olivier_jouannelle_HAL.pdf

Éditeur(s) : HAL CCSD Elsevier Masson
Résumé : Extension fractures of two types defined by the mean stress σ were generated in a synthetic rock analogue material. When σ is very small, the fractures are mode I cracks with smooth surfaces. At higher σ, these surfaces have plumose topography, with the amplitude increasing with σ. Both SEM observations and mechanical measurements show that fractures/discontinuities in the latter case are initiated as dilatancy localization bands. They form under tensile or slightly compressive normal stress and can be seen as running constitutive instabilities. The similarity between the plumose fractography of the experimental fractures and natural joints suggests similarity in the formation mechanism.
ISSN: 1631-0721

hal-00585886
https://hal.archives-ouvertes.fr/hal-00585886
DOI : 10.1016/j.crme.2011.01.002

Éditeur(s) : HAL CCSD
Résumé : International audience
This paper deals with variational inclusions of the form 0 ∈ K − f(x) − g(x) where f is a smooth function from a reflexive Banach space X into a Banach space Y , g is a function from X into Y admitting divided differences and K is a nonempty closed convex cone in the space Y . We show that the previous problem can be solved by a combination of two methods: the Newton and the Secant methods. We show that the order of the semilocal method obtained is equal to (1 + √5)/2. Numerical results are also given to illustrate the convergence at the end of the paper.
Serdica Math. J.

hal-00924150
https://hal.archives-ouvertes.fr/hal-00924150

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
ISSN: 0362-546X

hal-00699203
https://hal.archives-ouvertes.fr/hal-00699203

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
We conducted a laser-equipped analog experiment aimed at quasi-continuously monitoring the growth of a dense population of normal faults in homogeneous conditions. To further understand the way geological faults progressively gain in slip and length as they accumulate more strain, we measured with great precision the incremental slip and length changes that the analog faults sustain as they grow. These measurements show that the analog faults share common features with the natural ones. In particular, during their growth, the faults develop and maintain cumulative slip profiles that are generally triangular and asymmetric. The growth takes place through two distinct phases: an initial, short period of rapid lateral lengthening, followed by a longer phase of slip accumulation with little or no lengthening. The incremental slip is found to be highly variable in both space (along the faults) and time, resulting in variable slip rates. In particular, ‘short- and long-term' slip rates are markedly different. We also find that slip measurements at local points on fault traces do not contain clear information on the slip increment repeat mode. Finally, while the fault growth process is highly heterogeneous when considered at the scale of a few slip events, it appears homogeneous and self-similar at longer time scales which integrate many slip increments. This is likely to be the result of a feedback between stress heterogeneities and slip development. The long-term scale homogeneity also implies that the long-term faulting process is primarily insensitive to the short-term heterogeneities that are rapidly smoothed or redistributed. We propose a new conceptual scenario of fault growth that integrates the above observations and we suggest that faults grow in a bimodal way as a result of a self-driven and self-sustaining process.
ISSN: 0012-821X

Droits : info:eu-repo/semantics/OpenAccess
insu-00351998
https://hal-insu.archives-ouvertes.fr/insu-00351998
https://hal-insu.archives-ouvertes.fr/insu-00351998/document
https://hal-insu.archives-ouvertes.fr/insu-00351998/file/SchlagenhaufEtAl2008_EPSL_author.pdf
DOI : 10.1016/j.epsl.2008.06.042