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Optimal design for the heat equation

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.

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Optimal design for the heat equation

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

  3. Time-independent design • differential: evolutionary heat equation • volume : amount of the black material to be used ? black material : whitematerial : 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

  4. Mathematical Model

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

  6. Relaxation.The homogenization method G-closure problem

  7. ARelaxation Theorem

  8. Numerical resolution of (RP)in2D A numerical experiment

  9. The time-dependent design case

  10. AYoung measure approach

  11. Structure of theYoung measure

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

  13. AYoung measure approach constrained quasi-convexification Variational reformulation relaxation

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

  15. ARelaxation Theorem

  16. Numerical resolution of (RPt) Afinalconjecture

  17. Numerical experiments 1-D

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

  19. Some related openproblems 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.)

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