Optimal design for the heat equation
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Transcript of Optimal design for the heat equation
Optimal design for the heat equation
Francisco Periago
Polythecnic University of Cartagena, Spain
joint work with
Arnaud MünchUniversité de Franche-Comte, Besançon,
France and
Pablo PedregalUniversity of Castilla-La Mancha, Spain
PICOF’08 Marrakesh, April 16-18, 2008
Outline of the talk
• The time-independent design case
• The time-dependent design case
1. Problem formulation
1. Problem formulation
2. Relaxation.
2. Relaxation.
The homogenization method.
A Young measure approach.
3. Numerical resolution of the relaxed problem: numerical experiments
3. Numerical resolution of the relaxed problem: numerical experiments
• Open problems
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
?
Mathematical Model
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
Relaxation. The homogenization method
G-closure problem
A Relaxation Theorem
Numerical resolution of (RP) in 2D
A numerical experiment
The time-dependent design case
A Young measure approach
Structure of the Young measure
Importance of the Young measure
What is the role of this Young measure in our optimal design problem ?
A Young measure approach
Variational reformulation
relaxation
constrained quasi-convexification
Computation of the quasi-convexification
first-order div-curl laminate
A Relaxation Theorem
Numerical resolution of (RPt)
A final conjecture
Numerical experiments 1-D
Numerical experiments 2-D
time-dependent design
time-independent design
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.)