Mathematics 7710. II Semester. Optimization of
Spring semester, 1999 ( 3 hours)
Time and Place: JWB 208, T,Th, 12:25-1:45
Plan Spring semester
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.
- Elastic structures of maximal stiffness.
- Conducting structures, able to concentrate current.
- A game between the load and the structure.
- Composite with extremal properties.
A. Cherkaev. Variational Methods for Structural Optimization. Springer, 2000
The topics covered in the first semester
Announcement and plan for the entire course
Introduction: One-dimensional systems
- Inhomogeneous conducting medium.
- Elasticity equations.
Effective properties and microstructures.
- Asymptotic expansion: Effective coefficients.
- Examples: laminates.
- Homogenization and the boundary conditions.
- Homogenization and Gamma- and G-convergence.
- G-closures. Topological properties.
- Checker board structure, 2d polycrystal.
- Laminates of high rank.
- Algebra of laminates from contrast materials.
Optimization of conducting structures
Homogenization are placed on
To Andrej Cherkaev
Fishes (by Esher)