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A symbolic transformation language and its application to a multiscale method. (English) Zbl 1310.68260

Summary: The context of this work is the design of a software, called MEMSALab, dedicated to the automatic derivation of multiscale models of arrays of micro- and nanosystems. In this domain a model is a partial differential equation. Multiscale methods approximate it by another partial differential equation which can be numerically simulated in a reasonable time. The challenge consists in taking into account a wide range of geometries combining thin and periodic structures with the possibility of multiple nested scales. { } In this paper we present a transformation language that will make the development of MEMSALab more feasible. It is proposed as a Maple\(^{\text{TM}}\) package for rule-based programming, rewriting strategies and their combination with standard Maple\(^{\text{TM}}\) code. We illustrate the practical interest of this language by using it to encode two examples of multiscale derivations, namely the two-scale limit of the derivative operator and the two-scale model of the stationary heat equation.

MSC:

68W30 Symbolic computation and algebraic computation
35K05 Heat equation
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
68Q42 Grammars and rewriting systems

Software:

MEMSALab; Maple; Tom
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References:

[1] Balland, E.; Brauner, P.; Kopetz, R.; Moreau, P.-E.; Reilles, A., Tom: Piggybacking rewriting on Java, (The Proceedings of the 18th International Conference on Rewriting Techniques and Applications RTA 07 (2007)), 36-47
[2] Belkhir, W.; Giorgetti, A., Lazy AC-pattern matching for rewriting, (Escobar, S., Proceedings 10th International Workshop on Reduction Strategies in Rewriting and Programming, WRS 2011. Proceedings 10th International Workshop on Reduction Strategies in Rewriting and Programming, WRS 2011, Novi Sad, Serbia. Proceedings 10th International Workshop on Reduction Strategies in Rewriting and Programming, WRS 2011. Proceedings 10th International Workshop on Reduction Strategies in Rewriting and Programming, WRS 2011, Novi Sad, Serbia, Electr. Proc. Theor. Comput. Sci., vol. 82 (29 May, 2011)), 37-51
[3] Bensoussan, A.; Lions, J.-L.; Papanicolaou, G., Asymptotic Analysis for Periodic Structures, Stud. Math. Appl., vol. 5 (1978), North-Holland Publishing Co.: North-Holland Publishing Co. Amsterdam · Zbl 0411.60078
[4] Borovansky, P.; Kirchner, C.; Kirchner, H.; Ringeissen, C., Rewriting with strategies in ELAN: a functional semantics, Int. J. Found. Comput. Sci., 12, 01, 69-95 (2001) · Zbl 1319.68125
[5] Bündgen, R., Combining computer algebra and rule based reasoning, (Proceedings of the International Conference on Integrating Symbolic Mathematical Computation and Artificial Intelligence. Proceedings of the International Conference on Integrating Symbolic Mathematical Computation and Artificial Intelligence, Lect. Notes Comput. Sci., vol. 958 (1995), Springer), 209-223
[6] Casado-Díaz, J., Two-scale convergence for nonlinear Dirichlet problems in perforated domains, Proc. R. Soc. Edinb. A, 130, 2, 249-276 (2000) · Zbl 0977.35019
[7] Ciarlet, P. G., Mathematical Elasticity, vol. I, Stud. Math. Appl., vol. 20 (1988), North-Holland Publishing Co.: North-Holland Publishing Co. Amsterdam
[8] Cioranescu, D.; Damlamian, A.; Griso, G., Periodic unfolding and homogenization, C. R. Math. Acad. Sci. Paris, 335, 1, 99-104 (2002) · Zbl 1001.49016
[9] Cioranescu, D.; Damlamian, A.; Griso, G., The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40, 4, 1585-1620 (2008) · Zbl 1167.49013
[10] Cirstea, H.; Kirchner, C., The rewriting calculus - parts I and II, Log. J. IGPL, 9, 3, 427-498 (May 2001)
[11] Cirstea, H.; Kirchner, C.; Liquori, L., The rho cube, (Honsell, F.; Miculan, M., Foundations of Software Science and Computation Structures. Foundations of Software Science and Computation Structures, Lect. Notes Comput. Sci., vol. 2030 (2001), Springer), 168-183 · Zbl 0978.68072
[12] Cirstea, H.; Kirchner, C.; Liquori, L.; Wack, B., Rewrite strategies in the rewriting calculus, (Gramlich, B.; Lucas, S., 3rd International Workshop on Reduction Strategies in Rewriting and Programming. 3rd International Workshop on Reduction Strategies in Rewriting and Programming, Electron. Notes Theor. Comput. Sci., vol. 86(4) (2003), Elsevier: Elsevier Valencia, Spain), 18-34
[13] (Clavel, M.; Durán, F.; Eker, S.; Lincoln, P.; Martí-Oliet, N.; Meseguer, J.; Talcott, C. L., All About Maude - A High-Performance Logical Framework, How to Specify, Program and Verify Systems in Rewriting Logic. All About Maude - A High-Performance Logical Framework, How to Specify, Program and Verify Systems in Rewriting Logic, Lect. Notes Comput. Sci., vol. 4350 (2007), Springer) · Zbl 1115.68046
[14] Dautray, R.; Lions, J.-L., Mathematical Analysis and Numerical Methods for Science and Technology, vol. 3 (1990), Springer: Springer Berlin
[15] Fèvre, S.; Wang, D., Combining algebraic computing and term-rewriting for geometry theorem proving, (Calmet, J.; Plaza, J., Artificial Intelligence and Symbolic Computation. Artificial Intelligence and Symbolic Computation, Lect. Notes Comput. Sci., vol. 1476 (1998), Springer), 145-156 · Zbl 0914.03014
[16] Geuvers, H., Introduction to type theory, (Bove, A.; Barbosa, L.; Pardo, A.; Pinto, J., Language Engineering and Rigorous Software Development. Language Engineering and Rigorous Software Development, Lect. Notes Comput. Sci., vol. 5520 (2009), Springer), 1-56 · Zbl 1250.68088
[17] Lenczner, M., Homogénéisation d’un circuit électrique, C. R. Acad. Sci. Paris Sér. II b, 324, 9, 537-542 (1997) · Zbl 0887.35016
[18] Lenczner, M.; Senouci-Bereksi, G., Homogenization of electrical networks including voltage-to-voltage amplifiers, Math. Models Methods Appl. Sci., 9, 6, 899-932 (1999) · Zbl 0963.35014
[19] Lenczner, M.; Smith, R. C., A two-scale model for an array of AFM’s cantilever in the static case, Math. Comput. Model., 46, 5-6, 776-805 (2007) · Zbl 1230.74145
[20] Marin, M.; Piroi, F., Rule-based programming with Mathematica, (Sixth Mathematica Symposium (IMS 2004) (2004)), 1-6
[21] Yang, B.; Belkhir, W.; Dhara, R. N.; Lenczner, M.; Giorgetti, A., Computer-aided multiscale model derivation for MEMS arrays, (EuroSimE 2011, 13-th Int. Conf. on Thermal, Mechanical and Multi-Physics Simulation and Experiments in Microelectronics and Microsystems (April 2011), IEEE Computer Society: IEEE Computer Society Linz, Austria), 6 pp., Electronic proceedings
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