A 101-line MATLAB code for topology optimization using binary variables and integer programming
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EDUCATIONAL PAPER
A 101-line MATLAB code for topology optimization using binary variables and integer programming Renato Picelli1
· Raghavendra Sivapuram2 · Yi Min Xie3
Received: 19 May 2020 / Revised: 5 August 2020 / Accepted: 10 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper presents a MATLAB code with the implementation of the Topology Optimization of Binary Structures (TOBS) method first published by Sivapuram and Picelli (Finite Elem Anal Des 139: pp. 49–61, 2018). The TOBS is a gradientbased topology optimization method that employs binary design variables and formal mathematical programming. Besides its educational purposes, the 101-line code is provided to show that topology optimization with integer linear programming can be efficiently carried out, contrary to the previous reports in the literature. Compliance minimization subject to a volume constraint is first solved to highlight the main features of the TOBS method. The optimization parameters are discussed. Then, volume minimization subject to a compliance constraint is solved to illustrate that the method can efficiently deal with different types of constraints. Finally, simultaneous volume and displacement constraints are investigated in order to expose the capabilities of the optimizer and to serve as a tutorial of multiple constraints. The 101-line MATLAB code and some simple enhancements are elucidated, keeping only the integer programming solver unmodified so that it can be tested and extended to other numerical examples of interest. Keywords Topology optimization · Binary variables · Integer linear programming · Educational code
1 Introduction Topology optimization (TO) has exceeded the level of maturity to start pushing the engineering design beyond its previous capabilities. A considerable contribution in the dissemination of the TO methods can be granted to educational papers and codes, such as the well-known 99line work by Sigmund (2001a). Educational papers help
Responsible Editor: Ming Zhou Electronic supplementary material The online version of this article (https://doi.org/10.1007/s00158-020-02719-9) contains supplementary material, which is available to authorized users. Renato Picelli
[email protected] 1
Department of Mining and Petroleum Engineering, University of S˜ao Paulo, Prac¸a Narciso de Andrade s/n, Vila Mathias, Santos - SP, 11013-560, Brazil
2
Structural Engineering Department, University of California San Diego, 9500 Gilman Drive, La Jolla, CA, 92093, USA
3
Center for Innovative Structures and Materials, School of Engineering, RMIT University, Melbourne, 3001, Australia
to ease the learning curve and attract new practitioners to the field. Other methods are also documented in the form of educational papers or published compact codes (Allaire 2009; Challis 2010; Zuo and Xie 2015; Zhang et al. 2016; Ansola et al. 2018; Xia et al. 2018; Wei et al. 2018; Liang and Cheng 2020), shedding light on TO knowledge. This paper aims to advocate the use of TO using binary design
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