  This work is concerned with the micro-architecture of multi-layer material
that globally exhibits desired mechanical properties, for instance a negative
apparent Poisson ratio. We use inverse homogenization, the level set method,
and the shape derivative in the sense of Hadamard to identify material regions
and track boundary changes within the context of the smoothed interface. The
level set method and the shape derivative obtained in the smoothed interface
context allows to capture, within the unit cell, the optimal micro-geometry. We
test the algorithm by computing several multi-layer auxetic micro-structures.
The multi-layer approach has the added benefit that contact during movement of
adjacent "branches" of the micro-structure can be avoided in order to increase
its capacity to withstand larger stresses.
