Long time behavior of a two-phase optimal design for the heat equation

SIAM Journal on Control and Optimization

Volumn 48, Issue 8, 2010, Pages 5333-5356

  Allaire, Grégoire ^a   Münch, Arnaud ^b   Periago, Francisco ^c  

^a CENTRE DE MATHÉMATIQUES APPLIQUÉES   (France)

^b UNIVERSITÉ BLAISE PASCAL   (France)

^c TECHNICAL UNIVERSITY OF CARTAGENA   (Spain)

Author keywords

Asymptotic behavior; Heat equation; Numerical simulation; Optimal design; Optimality conditions; Relaxation by the homogenization method

Indexed keywords

ASYMPTOTIC ANALYSIS; COMPUTER SIMULATION; HEAT TRANSFER; HOMOGENIZATION METHOD; NUMERICAL METHODS; OPTIMAL SYSTEMS; OPTIMIZATION;

ASYMPTOTIC BEHAVIORS; DISSIPATED THERMAL ENERGIES; HEAT EQUATION; NECESSARY OPTIMALITY CONDITION; OPTIMAL DESIGN; OPTIMALITY CONDITIONS; OPTIMIZATION PROBLEMS; TWO-PHASE OPTIMIZATIONS;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 79251469736     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/090780481     Document Type: Article

References (23)

1. G. ALLAIRE, Shape Optimization by the Homogenization Method, Appl. Math. Sci. 146, Springer, New York, 2002.
2. N. ANTONIĆ AND M. VRDOLJAK, Gradient methods for multiple state optimal design problems, Ann. Univ. Ferrara Sez. VII Sci. Mat., 53 (2007), pp. 177-187.
3. A. CHERKAEV, Variational Methods for Structural Optimization, Springer, New York, 2000.
4. R. GLOWINSKI, Numerical simulation for some problems originating from continuum mechanics, in Trends in Applications of Pure Mathematics to Mechanics, Lecture Notes in Phys. 195, Springer, Berlin, 1984, pp. 96-145.
5. J. S. JENSEN, Space-time topology optimization for one-dimensional wave propagation, Comput. Methods Appl. Mech. Engrg., 198 (2009), pp. 705-715.
6. R. V. KOHN AND G. STRANG, Optimal design and relaxat ion of variational problems, Comm. Pure Appl. Math., 39 (1986), pp. 113-182.
7. J. L. LIONS and E. MAGENES, Non-Homogeneous Boundary Value Problems and Applications, Vol. I, Springer, New York, 1972.
8. K. A. LURIE, Control in the coefficients of linear hyperbolic equations via spatio-temporal components. Homogenization, in Ser. Adv. Math. Appl. Sci. 50, World Sci. Publ., Hackensack, NJ, 1999, pp. 285-315.
9. K. A. LURIE, Some new advances in the theory of dynamic materials, J. Elasticity, 72 (2003), pp. 229-239.
10. F. MAESTRE, A. MÜNOH, AND P. PEDREGAL, A spatio-temporal design problem for a damped wave equation, SIAM J. Appl. Math., 68 (2007), pp. 109-132.
11. A. MÜNOH, P. PEDREGAL, AND F. PERIAGO, Relaxation of an optimal design problem for the heat equation, J. Math. Pures Appl. (9), 89 (2008), pp. 225-247.
12. F. MURAT, Contre-exemples pour divers problèmes où le contrôle intervient dans les coefficients, Ann. Mat. Pura Appl. (4), 112 (1977), pp. 49- 68.
13. F. MURAT AND L. TARTAR, Calcul des Variations et Homogéné isation, Les Méthodes d'Homogénéisation Théories et Applications en Physique, Coll. Dir. Etudes et Recherches EDF 57, Eyrolles, Paris, 1985, pp. 319-369 (in French);
14. Topics in the mathematical modelling of composite materials, in Progr. Nonlinear Differential Equations Appl. 31, Birkhäuuser, Boston, 1997, pp. 319-369 (in English).
15. O. PANTZ AND K. TRABELSI, A post-treatment of the homogenization method for shape optimization, SIAM J. Control Optim., 47 (2008), pp. 1380-1398.
16. P. PEDREGAL, Vector variational problems and applications to optimal design, ESAIM Control Optim. Calc. Var., 11 (2005), pp. 357-381.
17. P. PEDREGAL, Optimal design in two-dimensional conductivity for a general cost depending on the field, Arch. Ration. Mech. Anal., 182 (2006), pp. 367-385.
18. U. RAITUMS, The extension of extremal problems connected with a linear elliptic equation, Soviet Math. Lecture Notes in Math., 19 (1978), pp. 1342-1345.
19. L. TARTAR, An Introduction to the Homogenization Method in Optimal Design, Lecture Notes in Math. 1740, Springer, New York, 1998, pp. 47-156.
20. S. TURTELTAUB, Optimal material properties for transient problems, Struct. Multidiscip. Optim., 22 (2001), pp. 157-166.
21. S. TURTELTAUB, Functionally graded materials for prescribed field evolution, Comput. Methods Appl. Mech. Engrg., 192 (2002), pp. 2283-2296.
22. M. VRDOLJAK, Necessary conditions for optimal design in hyperbolic problems, in Proceedings of the 5th Conference on Applied Mathematics and Scientific Computing, Tambaca, Josip, ed., University of Zagreb, Zagreb, Croatia, 2007.
23. M. VRDOLJAK, Multiple state optimal design problems and Hashin-Shtikman bounds, Int. J. Pure Appl. Math., 42 (2008), pp. 319-324.