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 Email: cherk@math.utah.edu Tel : +1 801 - 581 6822 Table of contents

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:
• Homogenization of elliptic, parabolic, and hyperbolic operators. Effective properties, bounds.
• Convexity and non-linear programming. Control theory. Maximum principle.
• Control of PDE systems: optimization of coefficients (material properties) and optimization of sources (loadings).
• Examples of structural optimization, phase equilibrium, bio-materials.

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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:

• A. Cherkaev. Variational Methods for Structural Optimization. Springer, 2000.
• Notes
The reference books:

• A. Bensoussan, J.-L.Lions, G. Papanicolaou. Asymptotic Analyssis for Periodic Structures. North Holland, 1978.
• V.V.Jikov, S.M.Kozlov, O.A.Olejnik. Homogenization of differential operators and integral functionals Springer, 1991
• K. Lurie. Applied Optimal Control. Plenum 1993
• M.Bendsoe. Optimization of structural topology, shape, and materials. Springer, 1995
Papers:
• Kohn, Strang. Optimal design and relaxation of variational problems. Comm.Pure Appl.Math. 39, (1986) Parts I - III.
• Calculus of variations and homogenization by F. Murat and L. Tartar. In: Topics in mathematical modelling of composite materials Andrej Cherkaev and Robert Kohn editors, Birkhauser, 1997
• K.A. Lurie and A. V. Cherkaev. Effective characteristics of composite materials and the optimal design of structural elements. In: Topics in mathematical modelling of composite materials Andrej Cherkaev and Robert Kohn editors, Birkhauser, 1997

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Plan
 I semester Introduction: 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 Quasiconvexity. Definitions Translation method Minimizing sequences Minimal extensions. Bounds. 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