Mathematics 7710-1.

Part 1. Homogenization

Instructor: Andrej Cherkaev
Department of Mathematics JWB Office 225
University of Utah
Email: Tel : +1 801 - 581 6822

Fishes (by Esher)

"Homogenized" Fishes
To Andrej Cherkaev homepage

    Andrej Cherkaev. Variational methods for structural optimization. Springer, 2000.
The covered topics

I semester


    One dimensional homogenization.
    • Canonical form, averaging.
    • Examples. effective conductivity, speed of sound, etc.
    • Vibrology: constructions under vibrations. Averaged equations. Examples. Vibro-viscosity, granular media.

    Introduction to optimal control

    • Control theory: variables, controls, functionals. Examples.
    • Canonical form and Pontriagin's maximum principle.
    • Chattering control and averaging in the optimal systems.
    • Inhomogeneous conducting medium.
    • Elasticity equations.
    Homogenization technique
    • Asymptotic expansion: Effective coefficients.
    • Correctors.
    • Examples: laminates.
    • Homogenization and the boundary conditions.
    Effective properties and microstructures.
    • Homogenization and Gamma- and G-convergence.
    • G-closures. Topological properties.
    Exact solutions
    • Checker board structure, 2d polycrystal.
    • Laminates of high rank.
    • Algebra of laminates from contrast materials.
Optimization of conducting structures
    Optimization of the conducting structures
    • Wiener bounds or effective properties.
    • Minimal energy. Relaxation and homogenization
    • Necessary conditions of Weierstrass type.
    • Example: an annulus of optimal conductivity.
    • Multi-material design.
    • Lower weakly semi-continuous functionals. Structure of solutions.
    • Examples. How to turn the current away from the field? The thermal lense.

    Optimal caverns in an elastic plane

    • Optimal cavern in hydrostatic field. Periodic array.
    • Optimal cavern in the shear field field. Collective effect.

Text for I semester:

The reference books:

Announcement and the preliminary plan for the entire course

Optimization and Homogenization are placed on