Size optimization of micro-frame structures for designing multiscale structures

Abstract

In this paper, we propose a multi-scale optimal design method for determining the member diameters of minute micro-frame structures in a macro-structure. The microstructures and the macrostructure are bridged by the homogenization method. In order to efficiently calculate the homogenized elastic tensors of complicated micro-frame structures consisting of slender members, the NIAH (New implementation of asymptotic homogenization) method is employed instead of the general method with applying the initial strains to the unit cells consisted by SOLID elements. In the NIAH method, the initial displacements equivalent to the initial unit strains are applied to the unit cells consisting of BEAM elements. The objective function is the squared displacements of the loading points. The volume of the macrostructure including the volume of the microstructures and the state equation of the macrostructure are used as the constraints. The micro-frame structures are assigned in the macro structure in advance, and the member diameters of all micro-frame structures are optimized. This multi-scale size optimization problem is formulated as a distributed-parameter optimization problem, and the sensitivity function is derived. The sensitivity function thus obtained is applied to the H¹ gradient method for scalar variables to obtain the diameter distribution of all members. The several design problems of multi-scale structures are solved with the proposed method, and the effectiveness and practicality of the method are confirmed.