# Topology Optimisation in Structural Steel Design for Additive Manufacturing
## 1 Introduction
### Seminal Works and Systematic Reviews  
Most of the former developments are well documented, explained, and exemplified in extensive and, mostly, user-friendly books. That is the case of [PERSON] and [PERSON]'s Elements of Structural Optimisation [112] with several editions, the profusely cited [PERSON]  
Figure 1: (a) Original [PERSON]’s minimum frame [9], (b) structural design by [PERSON] and [PERSON] [105], and (c) CITIC financial centre in Shenzhen by SOM [105]. Figure 2: (a) From cellular to topology optimised beam [106] (reproduced under the Creative Commons Attribution 4.0 International License, [[http://creativecommons.org/licenses/by/4.0/](http://creativecommons.org/licenses/by/4.0/)]([http://creativecommons.org/licenses/by/4.0/](http://creativecommons.org/licenses/by/4.0/)) accessed on 25 February 2021) and (b) bridge topology optimisation from problem statement to optimal solution [104]. and [PERSON]'s \"Topology Optimisation-Theory, Methods, and Applications\" [113] and, more recently, with [PERSON], [PERSON], and [PERSON] books [114; 115] and [PERSON] et al. 's book [105]. More specific works addressing computational methods [116] or buckling [117] in TO have also recently been made available. [PERSON] and [PERSON] [1] presented an extensive systematic review, encompassing the TO evolution from early structural optimisation to the fundamentals of TO, formulated in Michell structures [9], until the last century's late developments. There, microstructure and macrostructure approaches have been described within a product design context, paving the way for the concepts and approaches developed in the new millennium. The latter include numerical methods such as SIMP, enclosed into the density approaches, and ESO, whose retrospective analysis can be found in another useful Review Article by [PERSON] [35]. Moreover, such article is equally crucial for understanding the controversy around ESO (and BESO, its bi-directional development), with several authors referring to it as Sequential Element Rejections and Admissions (SERA) for not employing evolutionary processes nor yielding necessarily an optimal solution. Before that, [PERSON] and [PERSON] [118; 119] had summarised the basic concepts and criteria for the emerging TO. Yet, the history of TO Review Articles is mostly made of sectorial perspectives. Among the first, one can find [PERSON]'s [120] revision to TO in the automotive industry, focusing on the applications rather than on approaches or methods. Contrariwise, methods-driven reviews have been more frequent, and include assessments about ESO discrete approach developments [121] on Level-Set Methods (LSM or LS) [18]. Broader critical investigations of the former, added to the remaining density approaches, such as Rational Approximation of Material Properties (RAMP), Topological Derivatives (or the Bubble-Method), and Phase-Field Approach, as well as Lagrangian approaches [122; 123] are also available. Optimisation methods in AM have been a theme for a profusion of Review Articles along the last decade. While some assess manufacturing technologies and how optimisation methods can be accounted for in the process [124; 125; 126; 127; 128], others analysed the suitability of TO approaches for given AM endeavours, such as construction in [PERSON] and [PERSON] [129], aerospace structures in [PERSON] and [PERSON] [130] or the automotive industry in [PERSON] et al. [131]. Moreover, optimisation approaches impact the AM customised healthcare, environmental impact, supply chain efficiency, life-cycle analysis, hazards and energy consumption, which is part of [PERSON] et al. 's review [132]. The same decade also brought landmark review articles on particular topics, including TO applications in vibration problems [133], fluid problems [134], and materials design [135]. However, recent reviews on TO-assisted structural engineering are the most meaningful for this article's scope. Those include [PERSON] et al.