×
Zhou, Pingzhang; Du, Jianbin; Lü, Zhenhua

A generalized DCT compression based density method for topology optimization of 2D and 3D continua. (English) Zbl 1440.74456

Comput. Methods Appl. Mech. Eng. 334, 1-21 (2018).
Summary: In this paper, a novel topology optimization method based on discrete cosine transform (DCT) and density interpolation is proposed for layout designs of 2D and 3D continua. As one of the most frequently used transforms in digital image compression, the DCT may significantly reduce the number of design variables in density-based topology optimization, and can hereby improve the efficiency of solving the topology optimization problems to a great extent. This way the DCT compression based density method (DCDM) could be quite attractive in the topology optimization of large-scale engineering structures where a huge number of design variables may present. Effectiveness and efficiency of the proposed method is demonstrated with several 2D and 3D examples including both mechanical and heat conduction problems. Through these examples, some interesting features of DCDM are revealed and discussed. Since high frequency components are inherently filtered in DCDM, there is no need to introduce additional density filter or sensitivity filter in the present model. It is shown by numerical examples that there is no sharp corners present in the final optimized layout obtained by DCDM, which is beneficial when considering the stress of structures.

Cited in 4 Documents

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74P15 Topological methods for optimization problems in solid mechanics

Keywords:

topology optimization; discrete cosine transform (DCT); digital image compression; density method

Software:

top.m
Cite Review PDF
Full Text: DOI

References:

[1] Bendsoe, M.; Kikuchi, N., Generating optimal topologies in structural design using a homogenization method, Comput. Methods Appl. Mech. Engrg., 71, 197-224 (1988) · Zbl 0671.73065
[2] Bendsoe, M., Optimal shape design as a material distribution problem, Struct. Optim., 1, 193-202 (1989)
[3] Rozvany, G.; Zhou, M.; Birker, T., Generalized shape optimization without homogenization, Struct. Optim., 4, 250-254 (1992)
[4] Sigmund, O., A 99 line topology optimization code written in matlab, Struct. Multidiscip. Optim. (2001)
[5] Huang, X.; Xie, M., Evolutionary Topology Optimization of Continuum Structures: Methods and Applications (2010), John Wiley & Sons · Zbl 1279.90001
[6] Xie, Y. M.; Steven, G., A simple evolutionary procedure for structural optimization, Comput. & Structures, 49, 5, 885-896 (1993)
[7] Wang, M. Y.; Wang, X.; Guo, D., A level set method for structural topology optimization, Comput. Methods Appl. Mech. Engrg., 192, 1, 227-246 (2003) · Zbl 1083.74573
[8] Guo, X.; Zhang, W.; Zhong, W., Doing topology optimization explicitly and geometrically: A new moving morphable components based framework, J. Appl. Mech. (2014)
[9] Haftka, R. T.; Gürdal, Z., Elements of Structural Optimization, Vol. 11 (2012), Springer Science & Business Media
[10] Park, J.; Sutradhar, A., A multi-resolution method for 3d multi-material topology optimization, Comput. Methods Appl. Mech. Engrg., 285, 571-586 (2015) · Zbl 1423.74757
[11] Zhang, W.; Yuan, J.; Zhang, J.; Guo, X., A new topology optimization approach based on moving morphable components (mmc) and the ersatz material model, Struct. Multidiscip. Optim., 53, 6, 1243-1260 (2016)
[12] Guo, X.; Zhang, W.; Zhang, J.; Yuan, J., Explicit structural topology optimization based on moving morphable components (mmc) with curved skeletons, Comput. Methods Appl. Mech. Engrg., 310, 711-748 (2016) · Zbl 1439.74272
[13] Zhang, W.; Li, D.; Zhang, J.; Guo, X., Minimum length scale control in structural topology optimization based on the moving morphable components (mmc) approach, Comput. Methods Appl. Mech. Engrg., 311, 327-355 (2016) · Zbl 1439.74312
[14] Gonzalez, R. G.; Woods, R. E., Digital Image Processing (2007), Publishing House of Electronics Industry
[15] Rao, K. R.; Yip, P., Discrete Cosine Transform: Algorithms, Advantages and Applications (1990), Academic Press · Zbl 0726.65162
[16] Zhang, W.; Chen, J.; Zhu, X.; Zhou, J.; Xue, D.; Lei, X.; Guo, X., Explicit three dimensional topology optimization via moving morphable void (mmv) approach, Comput. Methods Appl. Mech. Engrg., 322, 590-614 (2017) · Zbl 1439.74311
[17] Zhang, W.; Song, J.; Zhou, J.; Du, Z.; Zhu, Y.; Sun, Z.; Guo, X., Topology optimization with multiple materials via moving morphable component (mmc) method, Internat. J. Numer. Methods Engrg. (2017)
[18] Gottlieb, D.; Shu, C.-W., On the gibbs phenomenon and its resolution, SIAM Rev., 39, 4, 644-668 (1997) · Zbl 0885.42003
[19] Bourdin, B., Filters in topology optimization, Internat. J. Numer. Methods Engrg., 50, 9, 2143-2158 (2001) · Zbl 0971.74062
[20] Sigmund, O., Morphology-based black and white filters for topology optimization, Struct. Multidiscip. Optim., 33, 4-5, 401-424 (2007)
[21] Kolda, T. G.; Bader, B. W., Tensor decompositions and applications, SIAM Rev., 51, 3, 455-500 (2009) · Zbl 1173.65029
[22] Zhou, P.; Du, J.; Lü, Z., Highly efficient density-based topology optimization using dct-based digital image compression, Struct. Multidiscip. Optim., 57, 1, 463-467 (2018)
[23] Zhou, M.; Shyy, Y.; Thomas, H., Checkerboard and minimum member size control in topology optimization, Struct. Multidiscip. Optim., 21, 2, 152-158 (2001)
[24] Sigmund, O.; Petersson, J., Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima, Struct. Optim., 16, 1, 68-75 (1998)
[25] Wang, F.; Lazarov, B. S.; Sigmund, O., On projection methods, convergence and robust formulations in topology optimization, Struct. Multidiscip. Optim., 43, 6, 767-784 (2011) · Zbl 1274.74409
[26] The Numerical Algorithms Group, The nag library, 2017. URL https://www.nag.com; The Numerical Algorithms Group, The nag library, 2017. URL https://www.nag.com
[27] Aage, N.; Lazarov, B. S., Parallel framework for topology optimization using the method of moving asymptotes, Struct. Multidiscip. Optim., 47, 4, 493-505 (2013) · Zbl 1274.74302
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.