A geometric projection method for designing three‐dimensional open lattices with inverse homogenization

Abstract

Summary Topology optimization is a methodology for assigning material or void to each point in a design domain in a way that extremizes some objective function, such as the compliance of a structure under given loads, subject to various imposed constraints, such as an upper bound on the mass of the structure. Geometry projection is a means to parameterize the topology optimization problem, by describing the design in a way that is independent of the mesh used for analysis of the design's performance; it results in many fewer design parameters, necessarily resolves the ill‐posed nature of the topology optimization problem, and provides sharp descriptions of the material interfaces. We extend previous geometric projection work to 3 dimensions and design unit cells for lattice materials using inverse homogenization. We perform a sensitivity analysis of the geometric projection and show it has smooth derivatives, making it suitable for use with gradient‐based optimization algorithms. The technique is demonstrated by designing unit cells comprised of a single constituent material plus void space to obtain light, stiff materials with cubic and isotropic material symmetry. We also design a single‐constituent isotropic material with negative Poisson's ratio and a light, stiff material comprised of 2 constituent solids plus void space.

A Heaviside function‐based density representation algorithm for truss‐like buckling‐induced mechanism design

“…Thus, the composite density is defined as follows: Note that, if p tends to +∞, the value in p‐norm formulation earlier approximates the maximum of density ρ ij , whereas, for finite p value, p‐norm function always exceeds the maximum density. As mentioned by Watts and Tortorelli, composite density may exceed 1. However, for two‐dimensional design, it is necessary to restrict composite density between 0 and 1.…”

“…Considering the requirement of practical manufacturing, optimal design with controllable structural complexity is preferred such as bars or beams. Actually, several methods were reported for controlling the complexity in topology optimization design such as in the works of Norato et al, Zhang et al, and Tortorelli et al However, there still exist some weaknesses such as generating reasonable initial configuration and avoiding exceeding material 0‐1 bounds at local regions. To address the aforementioned challenges, an alternative geometric mapping method is proposed in this article as follows.…”

“… ρ min is a small nonnegative value. Obviously, discrete geometric component is successfully represented by projection based on Heaviside function, which is a more direct approach to define geometry projection compared to the method proposed by Watts and Tortorelli …”

“…To overcome this numerical issue, a special density projection (SDP) function is introduced as follows: The curve property of the aforementioned projection function is shown in Figure . Compared to the work of Watts and Tortorelli, differentiability during optimization is guaranteed using SDP function instead of applying discontinuous minimum function to avoid artificially high stiffness at local regions. Using p‐norm function combined with SDP can effectively constrain the composite density between 0 and 1, the similar numerical approach applying lower bound KS (LKS) function can be found in the work of Zhang et al…”

“…Furthermore, Zhang et al developed this method to resolve stress constraint problem, where optimal design is made by assemble of discrete geometric components like bars or plates. Lately, Watts and Tortorelli extended geometric projection work to three dimensions and design unit cell for lattice materials based on inverse homogenization, where a negative Poisson's ratio lattice material is achieved. Inspired by discrete geometry projection method, we further propose an alternative geometry representation approach based on Heaviside function to design damping metamaterial in the context of density‐based TO.…”