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
Notes will be distributed

Please contact me if you have question.

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.

  • A. Cherkaev. Variational Methods for Structural Optimization. Springer, 2000
  • Notes
    The topics covered in the first semester

    I semester

    Introduction: One-dimensional systems

      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.
    Announcement and plan for the entire course

    Optimization and Homogenization are placed on

    Fishes (by Esher)

    "Homogenized" Fishes
    To Andrej Cherkaev homepage