Relaxation through homogenization for optimal design problems with gradient constraints

Journal of Optimization Theory and Applications

Volumn 114, Issue 1, 2002, Pages 27-53

  Lipton, R ^a  

^a LOUISIANA STATE UNIVERSITY   (United States)

Author keywords

gradient constraints; optimal design; relaxation

Indexed keywords

COMPOSITES STRUCTURES; DESIGN PROBLEMS; FINITE NUMBER; GRADIENT CONSTRAINT; INFINITE NUMBERS; MINIMIZING SEQUENCE; MULTIPHASE COMPOSITES; OPTIMAL DESIGN; RELAXATION; STATE-VARIABLES;

OPTIMAL SYSTEMS;

EID: 0036265665     PISSN: 00223239     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1015408020092     Document Type: Article

References (27)

1. NELSON, J. K., Some Steps Toward the Automation of the Design of Composite Dielectric Structures, IEEE Transactions on Dielectrics and Electrical Insulation, Vol. 1, pp. 663-671, 1994.
2. LI, Y. S., and DUXBURY, P. M., From Moduli Scaling to Breakdown Scaling: A Moment Spectrum Analysis, Physical Review, Vol. 40B, pp. 4889-4897, 1989.
3. CABIB, E., and DAL MASO, G., On a Class of Optimum Problems in Structural Design, Journal of Optimization Theory and Applications, Vol. 56, pp. 39-65, 1988.
4. LURIE, K. A., On the Optimal Distribution of the Resistivity Tensor of the Working Substance in a Magnetohydrodynamic Channel, Journal of Applied Mathematics and Mechanics, Vol. 34, pp. 255-274, 1970.
5. MURAT, F., Un Contre-Example pour le Problème du Contrôle dans les Coefficients, Comptes Rendus de l'Académie des Sciences, Paris, Vol. 273A, pp. 708-711, 1971.
6. ROZVANY, I. N., OLHOFF, N., CHENG, K. T., and TAYLOR, J. E., On the Solid Plate Paradox in Structural Optimization, Journal of Structural Mechanics, Vol. 10, pp. 1-32, 1982.
7. LURIE, K. A., and CHERKAEV, A. V., Effective Characteristics of Composite Materials and the Optimal Design of Structural Elements, Uspekhi Mekhaniki (Advances in Mechanics), Vol. 9, pp. 3-81, 1986.
8. MURAT, F., and TARTAR, L., Calcul des Variations et Homogé néisation, Les Méthodes de l'Homogénéisation: Théorie et Applications en Physique, Edited by D. J. Bregman et al., Collection de la Direction des Études et Recherches d'Electricité de France, Eyrolles, Paris, France, Vol. 57 pp. 319-369, 1985.
9. SPAGNOLO, S., Convergence in Energy for Elliptic Operators, Proceedings of the 3rd Symposium on the Numerical Solutions of Partial Differential Equations, College Park, Maryland, 1975; Edited by B. Hubbard, Academic Press, New York, NY, pp. 469-498, 1976.
10. MURAT, F., and TARTAR, L., H-Convergence, Topics in the Mathematical Modelling of Composite Materials, Edited by A. Cherkaev and R. Kohn, Birkhäuser, Boston, Massachusetts, pp. 21-43, 1997.
11. DAL MASO, G., An Introduction to Γ-Convergence, Birkhäuser, Boston, Massachusetts, 1993.
12. BENDSOE, M. P., Optimization of Structural Topology, Shape, and Material, Springer, Berlin, Germany, 1995.
13. TARTAR, L., An Introduction to the Homogenization Method in Optimal Design, Lecture Notes in Mathematics, Springer, Berlin, Germany, Vol. 1740, pp. 45-156, 2000.
14. BERGMAN, D. J., The Dielectric Constant of a Composite Material: A Problem in Classical Physics, Physics Reports, Vol. 43C, pp. 377-407, 1978.
15. LIPTON, R., Optimal Bounds on Electric Field Fluctuations for Random Composites, Journal of Applied Physics, Vol. 88, pp. 4287-4296, 2000.
16. LIPTON, R., Optimal Bounds on Field Fluctuations for Random Composites: Three-Dimensional Problems, Journal of Applied Physics, Vol. 89, pp. 1371-1376, 2001.
17. LIPTON, R., Homogenization of the Product of Weakly Converging Sequences of Gradients, Technical Report MS-10-00-13, Mathematical Sciences, Worcester Polytechnic Institute, 2000.
18. EKELAND, I., and TEMAM, R., Convex Analysis and Variational Problems, North-Holland, Amsterdam, Holland, 1976.
19. KOHN, R. V., and STRANG, G., Optimal Design and Relaxation of Variational Problems, Parts 1-3, Communications on Pure and Applied Mathematics, Vol. 34, pp. 113-137, 1986; Vol. 34, pp. 139-182, 1986; Vol. 34, pp. 357-377, 1986.
20. KOHN, R. V., and STRANG, G., Optimal Design and Relaxation of Variational Problems, Parts 1-3, Communications on Pure and Applied Mathematics, Vol. 34, pp. 113-137, 1986; Vol. 34, pp. 139-182, 1986; Vol. 34, pp. 357-377, 1986.
21. KOHN, R. V., and STRANG, G., Optimal Design and Relaxation of Variational Problems, Parts 1-3, Communications on Pure and Applied Mathematics, Vol. 34, pp. 113-137, 1986; Vol. 34, pp. 139-182, 1986; Vol. 34, pp. 357-377, 1986.
22. LIPTON, R., and VELO, A. P., Nonlinear Partial Differential Equations and Their Applications: Seminaire du College de France J. L. Lions. Series in Applied Mathematics, Elsevier-Gauthier-Villars (to appear).
23. GRABOVSKY, Y., Optimal Design Problems for Two-Phase Conducting Composites with Weakly Discontinuous Objective Functionals, Advances in Applied Mathematics, Vol. 27, pp. 683-704, 2001.
24. PEDREGAL, P., Constrained Quasiconvexity and Structural Optimization, Archive for Rational Mechanics and Analysis, Vol. 154, pp. 325-342, 2000.
25. PEDREGAL, P., Fully Explicit Quasiconvexification of the Mean-Square Deviation of the Gradient of the State in Optimal Design, Electronic Research Announcements of the American Mathematical Society, Vol. 7, pp. 72-78, 2001.
26. TARTAR, L., Remarks on Optimal Design Problems, Calculus of Variations, Homogenization, and Continuum Mechanics, Edited by G. Buttazzo, G. Bouchitte, and P. Suquet, World Scientific, Singapore, pp. 279-296, 1994.
27. RAITUMS, U., Optimal Control Problems for Elliptic Equations: Mathematical Questions, Zinatne, Riga, Latvia, 1989.