Optimization of three-dimensional truss-like periodic materials considering isotropy constraints

In recent years, much attention has been directed to the study of ultralight periodic cellular materials, such as the so called Periodic Truss Materials (PTMs), which are made up of truss-like unit cells. Application of these materials has its best potential in structures subjected to multifunctional, and sometimes conflicting, engineering requirements. Hence, optimization techniques can be employed to help finding the shape of the optimal unit cell for a given multifunctional application. Although the result can be geometrically complex, this difficulty can be minored in view of modern additive manufacturing technologies. This work presents a parameter/topology optimization procedure to design the particular unit cell geometry (that is, finding the cross sections of the bars) that results in a macroscopic material with optimum homogenized elastic or/and thermal constitutive properties. Emphasis is devoted to analyze the effect of enforcing independently elastic or thermal isotropy in the macroscopic material behavior. Although isotropic behavior can be imposed through adequate cell symmetries, an equivalent effect can be achieved by satisfying equality constraints relating constitutive coefficients. A sequential quadratic programming algorithm (SQP) is adopted, thus enforcing equality constraints gradually and within a tolerance range. This results in an enlarged search space at intermediate stages, rendering an effective strategy to solve the optimization problem. Different 3D cases with engineering appeal are solved and the results discussed.