# The homogenization method for topology optimization of structures: old and new

## TL;DR

This paper reviews the homogenization method for topology optimization, focusing on mathematical tools, applications to lattice structures, and practical exercises using FreeFem++, highlighting both classical and recent developments.

## Contribution

It provides a comprehensive overview of homogenization techniques in topology optimization, including new theoretical insights and practical implementation methods.

## Key findings

- Effective homogenization techniques for lattice structure optimization
- Numerical methods demonstrated with FreeFem++ software
- Enhanced understanding of mathematical tools for topology optimization

## Abstract

These are the lecture notes of a short course on the homogenization method for topology optimization of structures, given by Gr\'egoire Allaire, during the "GSIS International Summer School 2018" at Tohoku University (Sendai, Japan). The goal of this course is to review the necessary mathematical tools of homogenization theory and apply them to topology optimization of mechanical structures. The ultimate application, targeted in this course, is the topology optimization of structures built with lattice materials. Practical and numerical exercises are given, based on the finite element free software FreeFem++.