Optimum design of lattice structures based on continuum expression using micropolar continuum theory

Abstract

We construct a structural optimization method targeting lattice structures composed of a sufficiently large array of unit cells where each unit cell consists of several beam elements, with the aim of maximizing the stiffness of the entire structure under a volume constraint. In this method, micropolar continuum theory is introduced for continuum modeling of the lattice structures because micropolar continua and beam elements both have rotational degrees of freedom in addition to translational degrees of freedom. That is, the material behavior of a micropolar continuum that is equivalent to the lattice structure is obtained so that the strain energy density of the micropolar continuum is equal to the volume average of the strain energy stored in a unit cell of the lattice structure, which is derived from the framework of classical beam theory. The conventional finite element method is expanded for numerical analyses of the micropolar continuum to take the rotational degrees of freedom into account. The optimization algorithm has the width of each frame element set as a design variable and represented as a continuous distribution in the modeling of the micropolar continuum. Finally, two design examples are provided to confirm the validity and effectiveness of the proposed method.