Projet «Jeunes Chercheurs» N° JCJC06_139561 Financé par l'ANR de 2006 à 2009

  In many physical situations, one has to consider objects whose geometry involves different scales. Typically, to the macroscopic description should be added a microscopic level of details : this is the case of gas bubble inside a melted material, granulates inside a concrete, or bumps on a shell. The questions we are interested in deal in particular with mechanical or electromagnetic properties of such materials. The mathematical modelling of these situations consists usually of a system of partial differential equations in a (2D or 3D) domain standing for the real geometry. While the theoretical aspects are usually unaffected by microscopic inhomogeneities, this is not the case for numerics. Indeed, taking into account two different scales in a finite element code for instance requires an adapted mesh refinement in the vicinity of the defects. Resulting computations can become prohibitively costly. Hence, usually, only the macroscopic description is preserved. The influence of local inhomogeneities on the global behavior of the material is then ignored. We aim at designing a numerical method involving the two geometric scales, with a reasonable computation cost.

  Our approach is based on a precise asymptotic analysis of the state equation with respect to the characteristic size ε of the microdefects. The limit solution when ε tends to 0 corresponds to the solution in the unperturbed domain, a coarse mesh is enough to compute it satisfactorily. In the framework of our project, the perturbation resulting from the microdefects essentially oncentrates near the defects. Precisely, in a model case, it has been shown that the first corrector consists of a profile, i.e. a function defined on an infinite dimensionless domain, arising in the rapid variable x/e, that is to the scale of the erturbation. This structure suggests a numerical method based on a superposition of the unperturbed solution and the profile. The analysis and implementation of this method is a major objective: the main difficulties rely on the practical computation of the profile and the study of the performance of the algorithm. To determine the profile, we have to solve an elliptic equation posed in an unbounded domain with infinite boundary. Three directions seem promising: the use of a finite element method with truncation, the introduction of infinite elements, and a boundary element method. The application of the last method is known to present some difficulties, specific to the dimension 2, due to the logarithmic potential. Once the profile is computed, it has to be added to the unperturbed solution on a patch of elements, in the neighborhood of the microdefects. The implementation of such a superposition method requires accurate error and computation cost evaluations. We started this work thanks to a collaboration with the mechanical engineering community. We aimed at incorporating surfacic defects which generate cracks in concrete. The multiscale superposition method presented above is a preliminary step for detecting cracks. This is followed by a model of damage and propagation, previously developed by a member of the project.

  Concerning the mathematical analysis, we focus on the asymptotic analysis of the perturbed problem. We aim at fully justifying the expansions founding the superposition numerical method. We partially dealt with the model case of the Laplace operator with various boundary conditions and we plan to extend the results to more general framework (geometry or physical model). To answer the practical needs of engineers, we want to delimitate the validity of our method. Furthermore, the software modules implemented during the project will improve the finite element library Mélina developed at IRMAR and POEMS team at ENSTA Paris. Besides applications of these asymptotic expansions allow to describe the behavior of shape functionals in cases untreated by boundary variations or topological gradients. Moreover one of the major concerns is to deal with the case of several microdefects close one another. This introduces a new scale corresponding to the distance between two asperities. The coexistence of three scales makes the study of this question much more involved ; technically we have to understand the interaction between different profiles.

1. V. Bonnaillie-Noël, D. Brancherie, M. Dambrine, F. Hérau, S. Tordeux and G. Vial, Multiscale expansion and numerical approximation for surface defects, Submitted (2010).
2. V. Bonnaillie Noël, D.Brancherie, M. Dambrine, S. Tordeux and G. Vial, Effect of micro-defects on structure failure : coupling asymptotic analysis and strong discontinuity approach, Eur. Journal Comput. Mech., 1-2-3/2010, 19 pp. 165-175 (2010).
3. V. Bonnaillie-Noël, M. Dambrine, F. Hérau and G. Vial, On generalized Ventcel's type boundary conditions for Laplace operator in a bounded domain, SIAM J. Math. Anal., 42, 2 pp. 931-945 (2010). PDF
4. V. Bonnaillie-Noël, M. Dambrine, S. Tordeux and G. Vial, Interactions between moderately close inclusions for the Laplace equation, M3 AS: Mathematical Models and Methods in Applied Sciences, 10, 19 pp. 1853-1882 (2009). PDF
5. D. Brancherie, M. Dambrine, G. Vial and P. Villon, Effect of surface defects on structure failure: a two-scale approach, Eur. Journal Comput. Mech., 17, 5-7 pp. 613-624 (2008). PDF
6. V. Bonnaillie Noël, M. Dambrine, S. Tordeux and G. Vial, On moderately close inclusions for the Laplacian equations, C. R. Acad. Sci. Paris, 345, 11 pp. 609-614 (2007). PDF
7. V. Bonnaillie-Noël and S. Fournais, Superconductivity in domains with corners, Rev. Math. Phys., 19, 6 pp.607-637 (2007). PDF
8. M. Dambrine and G. Vial, A multiscale correction method for local singular perturbations of the boundary, ESAIM : M2AN, 41 pp.111-128 (2007). PDF

Congrès

1. V. Bonnaillie-Noël, D. Brancherie, M. Dambrine, F. Hérau, S. Tordeux, G. Vial, Asymptotique multi-échelle et approximation numérique pour des défauts surfaciques, 10e forum Jeunes Mathématiciennes, CIRM, Marseille (Novembre 2010).
2. V. Bonnaillie-Noël, D. Brancherie, M. Dambrine, F. Hérau, S. Tordeux, G. Vial, Asymptotique multi-échelle et approximation numérique pour des défauts surfaciques, CANUM 2010, Carcans Maubuisson (Juin 2010).
3. V. Bonnaillie-Noël, D. Brancherie, M. Dambrine, F. Hérau, S. Tordeux, G. Vial, Interactions between moderately close inclusions for the Laplace equation and applications in mechanics, Workshop 6th Singular Days on Asymptotic Methods for PDEs, Berlin (April 2010).
4. V. Bonnaillie-Noël, D. Brancherie, M. Dambrine, F. Hérau, S. Tordeux, G. Vial, Interactions between moderately close inclusions for the Laplace equation and applications in mechanics, Workshop Some mathematical problems of material science: effects of multiple scales and extreme aspect ratios, Banff Canada (February 2010).
5. V. Bonnaillie-Noël, D. Brancherie, M. Dambrine, S. Tordeux, G. Vial, Effect of "small" heterogeneities on structure failure, European Congress on Computational Mechanics, Paris (May 2010). PDF
6. V. Bonnaillie-Noël, D. Brancherie, M. Dambrine, F. Hérau, S. Tordeux, G. Vial, Interactions entre inclusions relativement proches pour l’équation de Laplace, Journée thématique "Méthodes numériques et applications", Marseille (November 2009). PDF
7. V. Bonnaillie-Noël, D. Brancherie, M. Dambrine, F. Hérau, S. Tordeux, G. Vial, Some mathematical questions studied by Macadam, Workshop Asymptotic methods, mechanics and other applications, Rennes (August 2009). PDF
8. V. Bonnaillie-Noël, D. Brancherie, M. Dambrine, S. Tordeux, G. Vial, From micro-defects to rupture: coupling asymptotic analysis and strong discontinuity approach, Workshop Asymptotic methods, mechanics and other applications, Rennes (August 2009). PDF
9. V. Bonnaillie-Noël, D. Brancherie, M. Dambrine, S. Tordeux, G. Vial, Couplage analyse asymptotique/méthode à discontinuité forte pour l'étude à rupture en présence de perturbations géométriques, 19^e congrès français de mécanique, Marseille (August 2009). PDF
10. V. Bonnaillie-Noël, D. Brancherie, M. Dambrine, S. Tordeux, G. Vial, Effet de micro-défauts sur la rupture des structures : couplage d’une analyse asymptotique et des méthodes à discontinuité forte, 7^e congrès national de calcul des structures, Giens (May 2009). PDF
11. F. Hérau, V. Bonnallie-Noël, M. Dambrine, G. Vial, Conditions au bord de type Wentzell généralisées pour l’opérateur de Laplace sur des domaines bornés, Journée Amiéno-Rémoise, Reims (14 avril 2009). PDF
12. V. Bonnaillie-Noël, D. Brancherie, M. Dambrine, S. Tordeux, G. Vial, Interactions entre inclusions relativement proches pour l'équation de Laplace, Workshop en l'honneur du soixantième anniversaire de Jacques Laminie, Orsay (December 2008). PDF
13. D. Brancherie, M. Dambrine, G. Vial, P. Villon, Effect of Micro Defects on Structure Failure: A Two Scale Approach, World Congress on Computational Mechanics VIII, Venise (Juy 2008).
14. V. Bonnaillie-Noël, D. Brancherie, M. Dambrine, S. Tordeux, G. Vial, Multiscale Asymptotics and Computational Approximation for surface Defects and Applications in Mechanics, Colloque de suivi à mi-parcours 2006, Giens (May 2008). PDF
15. M. Dauge, S. Tordeux, G. Vial, Développements asymptotiques raccordés et développement multi-échelle : quelle différence ? 20^e aniversaire du CERFACS, Toulouse (October 2007). PDF
16. D. Brancherie, M. Dambrine, G. Vial, P. Villon, Approche à deux échelles pour la prise en compte de défauts surfaciques dans l'analyse à rupture des structures, 8^e congrès national en calcul des structures, Giens (May 2007).
17. V. Bonnaillie-Noël, M. Dauge, S. Fournais, D. Martin and G. Vial, Superconductivity in domains with corners, International Conference on Mthematical Theory of Superconductivity and Liquid Crystals, Shanghai (May 2007). PDF
18. V. Bonnaillie-Noël, M. Dambrine and G. Vial, Interactions between close inclusions for the Laplace equation, 5^es journées singulières, Marseille (April 2007). PDF
19. D. Brancherie, M. Dambrine, G. Vial, P. Villon, Ultimate load : effect of surfacic defects, World Congress on Computational Mechanics VII, Los Angeles (July 2006).

Séminaires

1. G. Vial, Méthode multi-échelle pour la prise en compte de micro-défauts, Lyon (Janvier 2011).
2. V. Bonnaillie-Noël, Asymptotique multi-échelle et approximation numérique pour des défauts surfaciques, Villetaneuse (Décembre 2010).
3. M. Dambrine, Perturbations singulières du bord d'un domaine : application en rupture de structures mécaniques, Clermont-Ferrand (Février 2009).
4. V. Bonnaillie-Noël, Interactions entre inclusions relativement proches pour l'équation de Laplace, IHP Paris (Janvier 2009).
5. M. Dambrine, Perturbations singulières du bord d'un domaine : application en rupture de structures mécaniques, Chambéry (Avril 2008).
6. G. Vial, Développements asymptotiques raccordés et développement multi-échelle : quelles différences ? Clermont-Ferrand (Mars 2008). PDF
7. M. Dambrine, Perturbations singulières d'un domaine : analyse asymptotique de la solution d'une EDP elliptique et application en rupture de structures mécaniques, Pau (Mars 2008). PDF
8. M. Dambrine, Perturbations singulières du bord d'un domaine : application en rupture de structures mécaniques, Tours (Janvier 2008).
9. G. Vial, Développements asymptotiques raccordés et développement multi-échelle : quelles différences ? Nantes (Janvier 2008).
10. M. Dambrine, Une approche multi-échelle pour étudier l'influence des défauts de surface sur la rupture des structures, Reims (Janvier 2008).
11. G. Vial, Développements asymptotiques raccordés et développement multi-échelle : comparaison sur un problème-modèle, Grenoble (Avril 2007).