Homogenization

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Homogenization deals with determining effective material properties of media with a microstructure. For example, one might be interested in mechanical properties of a composite material or electromagnetic properties of a laminated transformer core. Such problems are faced often in engineering.

Numerical simulation of media with a microstructure requires special methods. For example, direct application of standrad finite elements to model laminated transformer cores leads to unrealistically fine meshes. When the microstructure is taken into account in the design of the numerical method, much coarser grids can be used and the computational time is reduced considerably.

This lecture provides an introduction to the theory of homogenization and to numerical methods used to simulate materials with a microstructure. A lecture note on the content of the course will be provided.

Literature

  • D. Cioranescu, P. Donato, An Introduction to Homogenization
  • A. Bensoussan, J. L. Lions, G. Papanicolaou, Asymptotic analysis of periodic structures


Lectures, Thu 13-14:30

Lectures are given by Dr. Antti Hannukainen. The current version of Lecture notes

Lectures

  1. 25.10. Introduction, Convergence in Banach spaces.
  2. 31.10 Convergence in Banach spaces, periodic boundary conditions
  3. 8.11. Periodic boundary conditions, asymptotic expansion
  4. 15.11. Weak convergence via Tartar's method of oscillating test functions
  5. 22.11. Correctors, preparing for the error estimate
  6. 29.11. Error estimate for corrector

Exercises, Thu 14:45 - 15:30

  1. 8.11. Exercise sheet
  2. 15.11. Exercise sheet
  3. 4.12. Exercise sheet