1. Optimal design for the heat equation Francisco Periago Polythecnic University of Cartagena, Spain joint work with Arnaud Münch Université de Franche-Comte, Besançon, France and Pablo Pedregal University of Castilla-La Mancha, Spain PICOF’08 Marrakesh, April 16-18, 2008

2. Outline of the talk The time-independent design case The time-dependent design case 1. Problem formulation 2. Relaxation. The homogenization method. A Young measure approach. 3. Numerical resolution of the relaxed problem: numerical experiments Open problems

3. Time-independent design black material : white material : Goal : to find the best distribution of the two materials in order to optimize some physical quantity associated with the resultant material design variable (independent of time !) Optimality criterium (to be precised later on) Constraints differential: evolutionary heat equation volume : amount of the black material to be used ?

4. Mathematical Model

5. Ill-posedness: towards relaxation This type of problems is ussually ill-posed Not optimal Optimal We need to enlarge the space of designs in order to have an optimal solution Relaxed problem ?? Original (classical) problem Relaxation

6. Relaxation. The homogenization method G-closure problem

7. A Relaxation Theorem

8. Numerical resolution of (RP) in 2D A numerical experiment

9. The time-dependent design case

10. A Young measure approach

11. Structure of the Young measure

12. Importance of the Young measure What is the role of this Young measure in our optimal design problem ?

13. A Young measure approach Variational reformulation relaxation constrained quasi- convexification

14. Computation of the quasi-convexification first-order div-curl laminate

15. A Relaxation Theorem

16. Numerical resolution of (RPt) A final conjecture

17. Numerical experiments 1-D

18. Numerical experiments 2-D time-dependent design time-independent design

19. Some related open problems 1. Prove or disprove the conjecture on the harmonic mean. 2. Consider more general cost functions. 3. Analyze the time-dependent case with the homogenization approach. For the 1D-wave equation: K. A. Lurie (1999-2003.)