Modular-topology optimization with Wang tilings: an application to truss structures
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RESEARCH PAPER
Modular-topology optimization with Wang tilings: an application to truss structures Marek Tyburec1
· Jan Zeman1
´ r1 · Martin Doˇskaˇ
· Martin Kruˇz´ık2
ˇ Lepˇs1 · Matej
Received: 15 April 2020 / Revised: 2 September 2020 / Accepted: 17 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Modularity is appealing for solving many problems in optimization. It brings the benefits of manufacturability and reconfigurability to structural optimization, and enables a trade-off between the computational performance of a periodic unit cell (PUC) and the efficacy of non-uniform designs in multi-scale material optimization. Here, we introduce a novel strategy for concurrent minimum-compliance design of truss modules topologies and their macroscopic assembly encoded using Wang tiling, a formalism providing independent control over the number of modules and their interfaces. We tackle the emerging bilevel optimization problem with a combination of meta-heuristics and mathematical programming. At the upper level, we employ a genetic algorithm to optimize module assemblies. For each assembly, we obtain optimal module topologies as a solution to a convex second-order conic program that exploits the underlying modularity, incorporating stress constraints, multiple load cases, and reuse of module(s) for various structures. Merits of the proposed strategy are illustrated with three representative examples, clearly demonstrating that the best designs obtained by our method exhibited decreased compliance: by 56 up to 69% compared with the PUC designs. Keywords Modular-topology optimization · Second-order cone programming · Truss microstructures · Bilevel optimization · Wang tiling
1 Introduction Modular structures, composed of repeated building blocks (modules), offer multiple appealing advantages over nonmodular designs. These include more economical mass fabrication, increased production productivity (Tugilimana et al. 2017b), and better quality control (Mikkola and Gassmann 2003). In addition, modules facilitate structural reconfigurability, conversion among designs with considerably different structural responses (Neˇzerka et al. 2018). Finally, the design of modular structures enables structural efficiency to be balanced with design complexity (Tugilimana et al. 2019), which often arises in optimal structures (Kohn and Strang 1986).
Responsible Editor: Ole Sigmund Marek Tyburec
[email protected]
Extended author information available on the last page of the article.
Our approach to designing modular structures follows recent successful applications of Wang tiles in compression and reconstruction of heterogeneous microstructures (Nov´ak et al. 2012; Doˇsk´aˇr et al. 2014; Antolin et al. 2019; Doˇsk´aˇr et al. 2020), where modularity suppresses artificial periodicity artifacts inherent to the periodic unit ˇ cell approach, e.g., Zeman and Sejnoha (2007). Here, we focus on the reverse direction: designing modular structures or materials composed of a compressed set o
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