Multimaterial topology optimization with multiple volume constraints: Combining the ZPR update with a ground‐structure algorithm to select a single material per overlapping set

Abstract

Multimaterial topology optimization often leads to members containing composite materials. However, in some instances, designers might be interested in using only one material for each member. Therefore, we propose an algorithm that selects a single preferred material from multiple materials per overlapping set. We develop the algorithm, based on the evaluation of both the strain energy and the cross-sectional area of each member, to control the material profile (ie, the number of materials) in each subdomain of the final design. This algorithm actively and iteratively selects materials to ensure that a single material is used for each member. In this work, we adopt a multimaterial formulation that handles an arbitrary number of volume constraints and candidate materials. To efficiently handle such volume constraints, we employ the ZPR (Zhang-Paulino-Ramos) design variable update scheme for multimaterial optimization, which is based upon the separability of the dual objective function of the convex subproblem with respect to Lagrange multipliers. We provide an alternative derivation of this update scheme based on the Karush-Kuhn-Tucker conditions. Through numerical examples, we demonstrate that the proposed material selection algorithm, which can be readily implemented in multimaterial optimization, along with the ZPR update scheme, is robust and effective for selecting a single preferred material among multiple materials. KEYWORDSground structure method, material nonlinearity, material selection algorithm, multimaterial topology optimization, multiple volume constraints, ZPR update algorithm

Open-Source Codes of Topology Optimization: A Summary for Beginners to Start Their Research

“…Without introducing extra variables, Zuo et al [41] presented an ordered multi-material SIMP interpolation scheme for the TO optimization issue involving multiple materials (multitop199). Zhang et al [42] proposed the so-called ZPR optimizer for the multimaterial TOs with considerable volume limitations.…”

A comprehensive review of educational articles on structural and multidisciplinary optimization

“…Thus, we report this paper in the density-based method section. Based on the multi-material formulation and the efficient ZPR (Zhang-Paulino-Ramos) update scheme proposed by Zhang et al (2018), Sanders et al (2018) developed an educational MATLAB code PolyMat built upon PolyTop, to solve multi-material design problems on 2D polygonal discretization with many volume constraints.…”

Topology optimization of periodic beam lattices using Cosserat elasticity

“…However, the optimization process considers the microstructure effect exclusively by its effective (homogenized) behavior. Various previous studies have investigated the potential macro-micro design optimization [37,31,28,71,65,26,55,70,63,42]], and most of the corresponding papers are based on scale separation assumptions and periodic homogenization theory.…”