An 89-line code for geometrically nonlinear topology optimization written in FreeFEM
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EDUCATIONAL PAPER
An 89-line code for geometrically nonlinear topology optimization written in FreeFEM Benliang Zhu1,2 · Xianmin Zhang1 · Hai Li1 · Junwen Liang1 · Rixin Wang1 · Hao Li2 · Shinji Nishiwaki2 Received: 19 June 2020 / Revised: 24 August 2020 / Accepted: 28 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Topology optimization has emerged as a powerful tool for structural configuration design. To further promote the development of topology optimization, many computer programs have been published for educational purposes over the past decades. However, most of the computer programs are constructed based on a linear assumption. This paper presents an 89line code for nonlinear topology optimization written in FreeFEM based on the popular SIMP (solid isotropic material with penalization) method. Excluding thirteen lines which are used for explanation, only 76 lines are needed for the initialization of the design parameters, nonlinear finite element analysis, sensitivity calculation, and updated design variables. Different design problems can be solved by modifying several lines in the proposed program. The complete program is given in the Appendix and is intended for educational purposes only. Keywords Topology optimization · Nonlinearity · SIMP · FreeFEM · Mean compliance
1 Introduction The structural topology is the material layout. Topology optimization is a design method that can determine the best material layout within a given design domain that can satisfy a set of constraints while maximizing a certain kind of performance. Since 1988, when Bendsøe and Kikuchi (Bendsøe and Kikuchi 1988) introduced the homogenization-based topology optimization method, various techniques have been developed for achieving the design goal of finding the optimal structural topology, such as density-based methods (Lazarov et al. 2016; Dilgen et al. 2018; Rozvany et al. 1992; Bendsøe and Sigmund Responsible Editor: Emilio Carlos Nelli Silva Benliang Zhu
[email protected] Xianmin Zhang [email protected] 1
Guangdong Key Lab. of Precision Equipment and Manufacturing Technology, South China University of Technology, Guangzhou, 510640, China
2
Department of Mechanical Engineering and Science, Graduate School of Engineering, Kyoto University, Kyoto, 606-8501, Japan
1999; Zhu et al. 2014), level set methods (LSM) (Allaire et al. 2005; Wang et al. 2003; Yamada et al. 2010; Zhu et al. 2015), evolutional structural optimization (ESO) methods (Huang and Xie 2009; Xie and Steven 1997), and moving morphable components (MMC) methods (Guo et al. 2016; Wang et al. 2019). For a comprehensive review of established topology optimization methods and their applications, the readers may refer to (Rozvany 2009; van Dijk et al. 2013; Sigmund and Maute 2013; Zhu et al. 2020). At the same time, computational programs concerning structural topology optimization have been gradually published in the literature. These programs, which are constructed for educational purposes, have paved the way for the flou
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