Optimal design of periodic linear elastic microstructures

This paper presents two computational models to design the periodic microstructure of cellular materials for optimal elastic properties. The material equivalent mechanical properties are obtained through a homogenization model. The two formulations address the problem of finding the optimal representative microstructural element for periodic media that maximizes either the weighted sum of the equivalent strain energy density for specified multiple macroscopic strain fields, or a linear combination of the equivalent mechanical properties. Constraints on material volume fraction and material symmetries are considered. The computational models are established using finite elements and mathematical programming techniques and tested in several numerical examples.