Mathematics 7710. II Semester. Optimization of Structures

Spring semester, 1999 ( 3 hours)
Time and Place: JWB 208, T,Th, 12:25-1:45

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

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

Plan Spring semester
Quasiconvexity, Bounds
  • Definitions of quasiconvexity, Null-Lagrangians.
  • Minimizing sequences. Laminates
  • Translation method
  • Weierstrass test and Minimal extensions. Fields in optimal structures.

  • The technique for bounds of G-closures
  • Bounds on conducting constants. Polycrystals
  • Bounds on complex properties
  • Several materials
  • Bounds on elastic moduli. Elastic polycrystals.
Variational problems for elastic structures.
  • Optimization of stiffness
  • Optimization of the mean stiffness
  • Optimization of eigenvalues
Various problems of structural optimization.
  • Optimization of single-loaded system by arbitrary criteria
  • Min-max problems of optimization: load versus structure
  • Optimization and bio-materials. What does Nature want?

Course objectives

The course discusses structural optimization. The main problems are to find "the best" geometrical composition of the structure and to determine the distribution of the optimal structures in the design.

Examples of optimal design include:

Mathematically, we are dealing with variational problems with non-convex Lagrangians. We develop the technique that enables to correctly formulate and solve these variational problems with non-stable solutions. We discuss special methods based on the quasiconvex envelopes. Applying the technique to the mechanical and transport problems, we are able to find optimal structures for the above mentioned problems.

First semester: M-7710 Homogenization (Fall Semester}

Announcement and the preliminary plan for the entire course

Optimization and Homogenization are placed on

Fishes (by Esher)

"Homogenized" Fishes
To Andrej Cherkaev homepage