Two examples of optimization of periodic microstructures by varying the vectors defining the periodicity cell. The optimization algorithm alternates shape optimization with periodicity optimization.

In the first example, we start with a 60° periodicity cell (hexagonal periodicity) and maximize the shear modulus of the homogenized material. The optimization algorithm changes the periodicity cell to a 45° parallelogram.

In the second example, we start with square periodicity and maximize the Young modulus of the homogenized material along a direction with slope 30°. The optimization algorithm changes the periodicity cell to a rectangle with slope 30°.