  Micro-structured materials consisting of an array of microstructures are
engineered to provide the specific material properties. This present work
investigates the design of cellular materials under the framework of level set,
so as to optimize the topologies and shapes of these porous material
microstructures. Firstly, the energy-based homogenization method (EBHM) is
applied to evaluate the material effective properties based on the topology of
the material cell, where the effective elasticity property is evaluated by the
average stress and strain theorems. Secondly, a parametric level set method
(PLSM) is employed to optimize the microstructural topology until the specific
mechanical properties can be achieved, including the maximum bulk modulus, the
maximum shear modulus and their combinations, as well as the negative Poisson's
ratio (NPR). The complicated topological shape optimization of the material
microstructure has been equivalent to evolve the sizes of the expansion
coefficients in the interpolation of the level set function. Finally, several
numerical examples are fully discussed to demonstrate the effectiveness of the
developed method. A series of new and interesting material cells with the
specific mechanical properties can be found.
