  In this work, we present an efficiently computational approach for designing
material micro-structures by means of topology optimization. The central idea
relies on using the isogeometric analysis integrated with the parameterized
level set function for numerical homogenization, sensitivity calculation and
optimization of the effective elastic properties. Design variables, which are
level set values associated with control points, are updated from the optimizer
and represent the geometry of the unit cell. We further improve the
computational efficiency in each iteration by employing reduced order modeling
when solving linear systems of the equilibrium equations. We construct a
reduced basis by reusing computed solutions from previous optimization steps,
and a much smaller linear system of equations is solved on the reduced basis.
Two- and three-dimensional numerical results show the effectiveness of the
topology optimization algorithm coupled with the reduced basis approach in
designing metamaterials.
