Mathematics 7710. Optimization and homogenization

Fall and spring semesters, 1998-1999
JWB 333, T/TH, 12:25-1:45.

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

Course description

The course discusses homogenization and structural optimization. These topics are closely connected: both are dealing with PDE with fast variable coefficients. The focus of the course is the optimization of properties of inhomogeneous bodies by varying their structures.

I. We start with the theory of homogenization which will be taught from the book by Bensoussan, Lions, and Papanicolaou, plus from recent research papers.

Fishes (by Esher)

Homogenized Fishes

The homogenization is the natural procedure to describe processes in complicated structures with known microstructures. It allows to replace a highly inhomogeneous medium with an equivalent homogeneous material, to estimate the norm of fluctuations of fields, etc.

We also formulate the central problem of structural optimization about "the best" geometrical composition of the structure.

Homogenization reduces the original problem to a problem which is doable, and this simplified problem reflects most of important features of the original one.

The coefficients of homogenized equations significantly depend on the structure. If the structure is unknown , we can only determine the bounds of the coefficients, that are independent of the structure.

II. The optimization of structures naturally follows the previous topic. We give an introduction to the optimization theory, optimal control, necessary conditions, minimizing sequences.
Then several structural optimization problems are discussed, such as the maximization of the stiffness of a structure, a game between the load and the structure, structures of optimal composites . Finally, we will discuss "suboptimal" projects that are much simpler than the truly optimal ones, and possess almost the same cost.

Applications include: structural optimization of electro-conducting and mechanical constructions, phase equilibrium and phase transition, the biological systems, which are both structured and rational.

During the class, we cover several hot topics and techniques of the modern applied math. Specifically:
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The course is addressed to graduate students in math, science, and engineering.

Several topics for course projects will be suggested. Most welcome are the students own projects.

Text for I semester:

The reference books:

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I semester


      Homogenization in studying and control of complex systems.

    Part 1. Problems with one independent variable.

    One dimensional homogenization.

    • Canonical form, averaging.
    • Examples. effective conductivity, speed of sound, etc.

    Introduction to optimal control

    • Control theory: variables, controls, functionals. Examples.
    • Canonical form and Pontriagin's maximum principle.
    • Chattering control and averaging.
    • Dynamical programming.

Part 2. Homogenization and control in PDE.

Homogenization of elliptic PDE

  • Equation of second order.
  • Homogenization by multi-scale expansions. Effective properties.
  • Bounds for effective properties.
  • Elasticity equations. Homogenization.
  • Non-linear problems.

Control of systems described by elliptic PDE.

  • Examples of optimal control: variable domain, variable load, variable properties.
  • Necessary conditions of Weierstrass type.
  • Chattering regimes and homogenization.

Homogenization of parabolic and hyperbolic equations.

  • Problems
  • Stochastic averaging
  • Waves and dissipation

II semester

    Part 3. Quasiconvexity, Bounds, G-closure


    • Definitions
    • Translation method
    • Minimizing sequences
    • Minimal extensions.


    • Bounds on conducting constants
    • Bounds on elastic constants
    • Some other bounds

Part 4. Structural Optimization .

Variational problems

  • Optimization of stiffness (conductivity)
  • Optimization of eigenfrequencies

Various optimization problems.

  • Optimization of single-loaded system by arbitrary criterium
  • Min-max problems of optimization: load versus structure
  • Optimization and bio-materials. What the nature wants?

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Notes (will be posted)

To Andrej Cherkaev homepage