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Evolutionary topology optimization of continuum structures with smooth boundary representation

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Abstract

This paper develops an extended bi-directional evolutionary structural optimization (BESO) method for topology optimization of continuum structures with smoothed boundary representation. In contrast to conventional zigzag BESO designs and removal/addition of elements, the newly proposed evolutionary topology optimization (ETO) method, determines implicitly the smooth structural topology by a level-set function (LSF) constructed by nodal sensitivity numbers. The projection relationship between the design model and the finite element analysis (FEA) model is established. The analysis of the design model is replaced by the FEA model with various elemental volume fractions, which are determined by the auxiliary LSF. The introduction of sensitivity LSF results in intermediate volume elements along the solid-void interface of the FEA model, thus contributing to the better convergence of the optimized topology for the design model. The effectiveness and robustness of the proposed method are verified by a series of 2D and 3D topology optimization design problems including compliance minimization and natural frequency maximization. It has been shown that the developed ETO method is capable of generating a clear and smooth boundary representation; meanwhile the resultant designs are less dependent on the initial guess design and the finite element mesh resolution.

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Acknowledgements

This work was supported by State Key Program of National Natural Science of China (61232014) and Australian Research Council (FT130101094). The first author is partially supported by the scholarship (201606130105) provided by China Scholarship Council (CSC).

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Authors and Affiliations

  1. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, People’s Republic of China

    Daicong Da, Guangyao Li & Xiaodong Huang

  2. Collaborative Innovation Center of Intelligent New Energy Vehicle, 200092, Shanghai, People’s Republic of China

    Daicong Da & Guangyao Li

  3. State Key laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan, People’s Republic of China

    Liang Xia

  4. Faculty of Science, Engineering and Technology, Swinburne University of Technology, Hawthorn, Melbourne, VIC, 3122, Australia

    Xiaodong Huang

Authors
  1. Daicong Da
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  2. Liang Xia
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  3. Guangyao Li
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  4. Xiaodong Huang
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Corresponding authors

Correspondence to Guangyao Li or Xiaodong Huang.

Appendix

Appendix

This appendix contains an ETO Matlab code for benchmark designs of structures from full material of the design domain. The code is developed on top of the 88-line code (Andreassen et al. 2011) with the implementation of the ETO method. The design domain is assumed rectangular and discretized into square plane stress elements. The main program is called form the Matlab prompt by the commands

$$ \mathrm{eto}\left(\mathrm{nelx},\mathrm{nely},\mathrm{volfrac},\mathrm{er},\mathrm{rmin},\mathrm{ctp}\right) $$

where nelx and nely denote the total number of elements in the horizontal and vertical directions respectively, volfrac is the prescribed volume fraction, er is the evolutionary rate, rmin is the filter radius, and ctp specifies the case type of benchmark design. The ctp takes values 1, 2, and 3 denoting three benchmark design cases of stiffness maximization design subject to volume fraction constraint: half-MBB beam design, clamped cantilever design, and roller-supported half-wheel design.

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Da, D., Xia, L., Li, G. et al. Evolutionary topology optimization of continuum structures with smooth boundary representation. Struct Multidisc Optim 57, 2143–2159 (2018). https://doi.org/10.1007/s00158-017-1846-6

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