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.)