On choosing interpolation domains and functions in topology optimization of multi-scale structures: a comparative study

Abstract

In this work, we evaluate the behavior of several material interpolation schemes in conducting topology optimization of composite structures made of 28 orthotropic materials with similar costs and densities. The materials are categorized into three families according to their geometrical features and arranged in a collection of interpolation domains, including 2D and 3D domains, as well as orthogonal and non-orthogonal ones. The concept of Coons patches is used to extend three interpolation functions, namely polynomial fitting, ordered Solid Isotropic Material with Penalization (ordered SIMP), and smooth single-variable-based interpolation, from 1D to 2D and 3D. Several material interpolation schemes are built using different combinations of interpolation domains and interpolation functions for a representative comparison. Numerical experiments are conducted using a two-scale topology optimization procedure to solve a benchmark problem given by the Messerschmitt-Bölkow-Blohm (MBB) beam problem. The obtained results show that, when taking the simple case of compliance minimization, as well as candidate lattices of same densities, the selection of the interpolation domain and of the interpolation function does not have a significant effect on the optimization process. Accordingly, we conclude that taking the simplest combination of a 2D square domain together with a piecewise linear function is sufficient to get the optimized design, if the candidate lattices are well categorized and sorted.