Materials design using inverse homogenization

We have applied our method for multi-phase optimization in the design of materials using inverse homogenization. The distribution of one or multiple phases in the periodic unit cell is optimized, so that the homogenized coefficients of the microstructure approach some target values (see Chapter 6 in my Ph.D Thesis: "Manufacturing Constraints and Multi-Phase Shape and Topology Optimization via a Level-Set Method").

Negative Poisson ratio using one phase and void.
Initialization (unit cell). Optimized shape (unit cell). Microstructure (16 unit cells).

Unit Poisson ratio using one phase and void.
Initialization (unit cell). Optimized shape (unit cell). Microstructure (16 unit cells).

Zero Poisson ratio using one phase and void.
Initialization (unit cell). Optimized shape (unit cell). Microstructure (16 unit cells).

Low thermal expansion coefficients using two phases and void.
Initialization (unit cell). Optimized shape (unit cell). Microstructure (16 unit cells).

Opposite thermal expansion coefficients using two phases and void.
Initialization (unit cell). Optimized shape (unit cell). Microstructure (16 unit cells).