Complementary energy based meso-level homogenization for multiscale topology optimization

Multiscale topology optimization (MTO) of structures requires the computationally efficient evaluation of the mechanical properties of each mesoscale representative volume element (RVE). This paper presents a novel method that uses a complementary energy (CE) approach to estimate the effective homogenized properties of an RVE, consisting of an embedded structure that is parametrically defined. Visualizing the meso-scale structure as a sweep cross-section makes it possible to quickly and accurately write the CE and then invoke Castigliano’s second theorem to find the force–displacement relationships. A two-dimensional (2D) running example of an anchored torus is employed to motivate the framework in which Lagrange multipliers are used to ensure the physical constraints on the mesostructure. For three-dimensional (3D) RVEs, a Frenet frame representation is used to associate the global forces and moments to a local coordinate system. Two homogenization methods are compared to ensure correctness; the first directly evaluates the constitutive matrix via the kinetic uniform boundary condition (KUBC) approach, while the second is a new finite element based optimization that directly finds the equivalent material properties. The proposed method is compared at the meso-scale against a solid finite element model and a discrete beam model to verify accuracy. Furthermore, a variety of material properties are demonstrated for a 3D example.