Optimization of Structures Mathematics 7710-1. Optimization of Structures (3 hours)

Instructor: Andrej Cherkaev, Time and Place: JWB 208, T,Th, 12:25-1:45

Department of Mathematics, JWB Office 225, email: cherk@math.utah.edu Telephone: +1 801 - 581 6822

Focus

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:
  • Elastic structures of maximal stiffness.
  • Conducting structures, able to concentrate current.
  • A game between the load and the structure.
  • Composite with extremal properties.
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.

Plan

Quasiconvexity and Nonconvex Variational Problems
  • Definitions
  • Translation method
  • Minimizing sequences
  • Minimal extensions.

Structural Optimization (Examples of Nonconvex Problems)

  • Optimization of stiffness and of eigenfrequencies
  • G-closures
  • Optimization of loaded mechanical constructions
  • Min-max problems of optimization: load versus structure
  • Suboptimal projects
  • Phase transition and bio-materials. What does Nature want?

The course is addressed to graduate students in math, science, and engineering.

Lecture notes will be distributed.