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International Journal for Multiscale Computational Engineering
IF: 1.016 5-Year IF: 1.194 SJR: 0.554 SNIP: 0.68 CiteScore™: 1.18

ISSN Print: 1543-1649
ISSN Online: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2014006595
pages 91-114

COMPUTER-AIDED DERIVATION OF MULTISCALE MODELS: A REWRITING FRAMEWORK

Bin YANG
Department of Applied Mathematics, Northwestern Polytechnical University, 710129 Xi'an Shaanxi, China, and University of Franche-Comte, 26 Chemin de l'Epitaphe, 25030 Besancon Cedex, France
Walid Belkhir
LIFC, University of Franche-Comte, 16 route de Gray, 25030, Besancon Cedex, France, and INRIA Nancy - Grand Est, CASSIS project, 54600 Villers-les-Nancy, France
Michel Lenczner
FEMTO-ST, Departement Temps-Frequence, University of Franche-Comte, 26 Chemin de l'Epitaphe, 25030 Besancon Cedex, France

ABSTRACT

We introduce the first part of a framework for computer-aided derivation of multiscale models. It relies on a combination of an asymptotic method used in the field of partial differential equations with term-rewriting techniques coming from computer science. In our approach, a multiscale model derivation is characterized by the features taken into account in the asymptotic analysis. Its formulation consists in a derivation of a reference model associated to an elementary nominal model, and in a set of transformations to apply to this proof until it takes into account the wanted features. In addition to the reference model proof, the framework includes first-order rewriting principles designed for asymptotic model derivations, second-order rewriting principles dedicated to elementary extensions of model derivations, and their combinations. The latter point is only briefly sketched and will be detailed in another work. We report implementation results regarding three simple extensions of the reference proof.

REFERENCES

  1. Arbogast, T., Douglas, Jr., J., and Hornung, U., Derivation of the double porosity model of single phase flow via homogenization theory. DOI: 10.1137/0521046

  2. Baader, F. and Nipkow, T., Term Rewriting and All That.

  3. Balland, E., Brauner, P., Kopetz, R., Moreau, P.-E., and Reilles, A., Tom: Piggybacking rewriting on Java. DOI: 10.1007/978-3-540-73449-9_5

  4. Belkhir, W., Giorgetti, A., and Lenczner, M., A symbolic transformation language and its application to a multiscale method.

  5. Bensoussan, A., Lions, J., and Papanicolaou, G., Asymptotic Methods for Periodic Structures.

  6. Bertot, Y. and Castéran, P., Interactive Theorem Proving and Program Development. Coq‘Art: The Calculus of Inductive Constructions.

  7. Borovansky, P., Kirchner, C., Kirchner, H., and Ringeissen, C., Rewriting with strategies in ELAN: A functional semantics. DOI: 10.1142/S0129054101000412

  8. Bouchitte, G. and Bellieud, M., Homogenization of a soft elastic material reinforced by fibers.

  9. Bourgeat, A., Luckhaus, S., and Mikelic, A., Convergence of the homogenization process for a double-porosity model of immiscible two-phase flow. DOI: 10.1137/S0036141094276457

  10. Casado-Díaz, J., Two-scale convergence for nonlinear Dirichlet problems in perforated domains. DOI: 10.1017/S0308210500000147

  11. Cioranescu, D. and Donato, P., An Introduction to Homogenization.

  12. Cioranescu, D., Damlamian, A., and Griso, G., Periodic unfolding and homogenization. DOI: 10.1016/S1631-073X(02)02429-9

  13. Cioranescu, D., Damlamian, A., and Griso, G., The periodic unfolding method in homogenization. DOI: 10.1137/080713148

  14. Cirstea, H. and Kirchner, C., The rewriting calculus—Parts I and II.

  15. Cirstea, H., Kirchner, C., Liquori, L., and Wack, B., Rewrite strategies in the rewriting calculus, Gramlich, B. and Lucas, S. (Eds.).

  16. Cirstea, H., Faure, G., and Kirchner, C., A ρ-calculus of explicit constraint application. DOI: 10.1016/j.entcs.2004.06.029

  17. Gascón, A., Godoy, G., and Jacquemard, F., Closure of tree automata languages under innermost rewriting. DOI: 10.1016/j.entcs.2009.03.033

  18. Hui, H., Yakoubi, Y., Lenczner, M., Cogan, S., Meister, A., Favre, M., Couturier, R., and Domas, S., Modeling, filtering and optimization for AFM arrays. DOI: 10.1109/ESIME.2011.5765783

  19. Hui, H., Lenczner, M., Pillet, E., and Cogan, S., A two-scale model for one-dimensional arrays of cantilevers and its verification. DOI: 10.1016/j.mechatronics.2011.09.010

  20. Jikov, V., Zhikov, V., Kozlov, M., and Oleinik, O., Homogenization of Differential Operators and Integral Functionals. DOI: 10.1007/978-3-642-84659-5

  21. Lenczner, M., Homogénéisation d‘un circuit électrique. DOI: 10.1016/S1251-8069(97)83186-7

  22. Lenczner, M., Homogenization of linear spatially periodic electronic circuits. DOI: 10.3934/nhm.2006.1.467

  23. Lenczner, M., A multiscale model for atomic force microscope array mechanical behavior. DOI: 10.1063/1.2710001

  24. Lenczner, M. and Smith, R. C., A two-scale model for an array of AFM‘s cantilever in the static case. DOI: 10.1016/j.mcm.2006.12.028

  25. Marino, D. and Millstein, T., A generic type-and-effect system. DOI: 10.1145/1481861.1481868

  26. Slonneger, K. and Kurtz, B. L., Formal Syntax and Semantics of Programming Languages: A Laboratory Based Approach.

  27. Tarski, A., A Lattice-theoretical fixpoint theorem and its applications.

  28. Terese, Term Rewriting Systems.

  29. Wong, W., A proof checker for HOL.


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