Two-Scale Concurrent Topology Optimization with Multiple Micro Materials Based on Principal Stress Direction

Abstract

This paper studies two-scale concurrent topology optimization with multiple micro heterogeneous materials subject to volume constraints. Unlike the existing work on concurrent two-scale optimization where only one material with optimal microstructure is used or that with multiple micro material where each material is distributed in a number of prescribed geometrical domains, selection of micro heterogeneous materials in this work is based on direction of principal stresses in macro structure. For a structure composed of m micro materials, the macro elements are classified into m categories according to their principal stress direction and each category is assigned with a uniform micro material. The interpolation scheme for macro elements is based on Discrete Material Optimization (DMO), where each element is assigned with m macro design variables. The categorization process of the macro structure is achieved by proper modification of volume constraints, where the macro design variables are multiplied by penalty functions. The penalty functions make it uneconomical for the usage of micro materials, which do not correspond to the principal stress direction of their macro element. The macro structure and micro material are connected through effective property, which is calculated through novel numerical implementation of asymptotic homogenization method (NIAH). Both macro structure and micro materials are optimized concurrently and analytical sensitivities are calculated with adjoint method. One minimum compliance numerical example of an L-bracket subject to volume constraints, where one micro material correspond to macro domain with principal stress angle near 0 or 90 degrees and another correspond to that with principal stress angle near 45 or 135 degrees, is presented to show the potential of the proposed method.