Research on Ratio-Extremum Algorithm for Topology Optimization of Continuum Structures Including Active Constraint Identification

Ou Disheng; Zheng Xuefen; Zhou Xiongxin

doi:doi:10.11648/j.ijmea.20221001.11

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Research on Ratio-Extremum Algorithm for Topology Optimization of Continuum Structures Including Active Constraint Identification

Ou Disheng, Zheng Xuefen, Zhou Xiongxin

Published in International Journal of Mechanical Engineering and Applications (Volume 10, Issue 1)

Received: 16 January 2022 Accepted: 29 January 2022 Published: 16 February 2022

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Abstract

To obtain the topology optimization algorithm of continuum structure which can effectively identify the effective constraints and quickly converge, based on the original Ratio-Extremum algorithm theory based on truss structure optimization, the emitter algorithm theory is introduced into the topology optimization of continuum structure. Firstly, taking pseudo density as design variables, mathematical model of the minimization mass with constraints of nodal displacements and element stresses is constructed. Secondly, according to essential extremum conditions of Dual objective function, iterative optimization direction and analytical step-size of constraint multipliers are derived. And, according to essential extremum conditions of Generalized Lagrange function, iterative optimization direction and analytical step-size of pseudo densities are derived. Analytical step-sizes are used to avoid one-dimensional optimization and then the calculation quantity of iterative optimization can be decreased. Thirdly, first-order partial derivatives of nodal displacement and element equivalent stress constraints with respect to pseudo densities are given. After that, by using self-compiled MATLAB program for continuum structure analysis, partial derivative calculation and optimization iteration, 4 optimization examples of different beam structures are used to show the changes of active nodal displacement and element equivalent stress constraints, and structural mass in the optimization iteration process, and to show the effectiveness of Ratio-Extremum algorithm in topology optimization of continuum structures.

Published in	International Journal of Mechanical Engineering and Applications (Volume 10, Issue 1)
DOI	10.11648/j.ijmea.20221001.11
Page(s)	1-6
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Continuum Structure, Topology Optimization, Ratio-Extremum Algorithm, Optimization Direction, Step-size, Active Constraint Identification

References

[1]	ZHOU Kemin, LI Junfeng, LI Xia. A review on topology optimization of structures [J]. Advances in Mechanics, 2005 (01): 69-76.
[2]	Sigmund O, Maute K. Topology optimization approaches: A comparative review [J]. Structural and Multidisciplinary Optimization, 2013, 48 (6): 1031-1055.
[3]	DING Mao, GENG Da, ZHOU Mingdong, et al. Topology optimization strategy of structural strength based on variable density method [J]. Journal of Shanghai Jiao Tong University, 2021, 55 (06): 764-773.
[4]	Bendsøe MP. Optimal shape design as a material distribution problem [J]. Structural Optimization, 1989, 1 (4): 193-202.
[5]	Bendsøe MP, Sigmund O. Material interpolations schemes in topology optimization [J]. Archive of Applied Mechanics, 1999, 69 (9): 635-654.
[6]	Sigmund O. A 99 line topology optimization code written in Matlab [J]. Structural and Multidisciplinary Optimization, 2001, 21 (2): 120-127.
[7]	ZHAO Longbiao, GAO Liang, CHEN Zhimin, et al. A proportional and differential optimality criterion method for topology optimization [J]. China Mechanical Engineering, 2011, 22 (03): 345-350.
[8]	ZHANG Richeng, ZHAO Jiong, WU Qinglong, et al. Variable density topology optimization method considering structural stability [J]. Chinese Journal of Engineering Design, 2018, 25 (04): 441-449.
[9]	YU Chengtian. Topology optimization method of MMC based on smoothing technology [D]. Dalian University of Technology, 2021.
[10]	LI Hao. A variable density topology optimization method for continuum structure with smooth boundary and its application in bridge selection [D]. South China University of Technology, 2020.
[11]	LIU Yunlong, MAMTIMIN Geni. Smoothing method for edge chessboard format in topological structure [J]. Machinery Design &Manufacture, 2021 (08): 171-175.
[12]	XU Xiaokui, GUO Baofeng, JIN Miao. Structural topology optimization based on density-volume interpolation scheme [J]. China Mechanical Engineering, 2017, 28 (11): 1269-1273.
[13]	DU Yixian, ZHANG Yan, LI Hanzhao. Topology optimization of iterative algorithm with the approximation of 0/1 discrete properties [J]. Machine Design & Research, 2017, 33 (01): 58-62.
[14]	YAN Xiaolei, XIE Lu, CHEN Jiawen, et al. A density-constrained topological optimization method [J]. Mechanical Science and Technology for Aerospace Engineering, 2021, 40 (03): 350-355.
[15]	LONG Kail, FU Xiaojin. Sensitivity filtering method considering density gradient [J]. Journal of Computer- Aided Design & Computer Graphics, 2014, 26 (04): 669-674.
[16]	LI Qihong, LI Haiyan. Continuum structure topology optimization method based on improved SIMP method [J]. Journal of Mechanical & Electrical Engineering, 2021, 38 (04): 428-433.
[17]	ZHANG Guofeng, XU Lei, LI Dashuang, et al. Research on sensitivity filtering of continuum topology optimization [J]. Modular Machine Tool & Automatic Manufacturing Technique, 2021 (06): 29-32.
[18]	GAO Xiang, WANG Linjun, DU Yixian, et al. Gray element filtering technology of topology structure based on improved guide-weight method [J/OL]. Journal of Beijing University of Aeronautics and Astronautics: 1-12 [2021-4-12].
[19]	ZHANG Zhifei, XU Wei, XU Zhongming, et al. Double-SIMP method for gray-scale elements suppression in topology optimization [J]. Transactions of the Chinese Society for Agricultural Machinery, 2015, 46 (11): 405-410.
[20]	DU Yixian, ZHANG Yan, LI Hanzhao, et al. Topology optimization of node density interpolation method with polarization properties [J]. Journal of Machine Design, 2018, 35 (03): 80-85.
[21]	CHEN Zhimin, ZHAO Longbiao, QIU Haobo, et al. A topology optimization method based on memetic algorithm [J]. China Mechanical Engineering, 2010, 21 (24): 2983-2988.
[22]	LUO Zhen, CHEN Liping, HUANG Yuying, et al. Topological optimization using RAMP interpolation scheme [J]. Chinese Journal of Computational Mechanics, 2005 (05): 585-591.
[23]	Namhee Ryu, Minsik Seo, Seungjae Min. Multi-objective topology optimization incorporating an adaptive weighed- sum method and a configuration-based clustering scheme [J]. Computer Methods in Applied Mechanics and Engineering, 2021, 385.
[24]	YI Jijun, RONG Jianhua, ZENG Tao. A new structural topology optimization method subject to compliance [J]. Journal of Central South University (Science and Technology), 2011, 42 (07): 1953-1959.
[25]	ZHANG Qiang. Study on topological optimization method and its application of continuum structure based on design space adjustments [D]. Changsha University of Science & Technology, 2009.
[26]	LI Yue, LI Xuan, LI Minchen, et al. Lagrangian-Eulerian multi-density topology optimization with the material point method [J]. International Journal for Numerical Methods in Engineering, 2021, 122 (14): 3400-3424.
[27]	LIAO Zhongyuan, WANG Yingjun, WANG Shuting. Graded-density lattice structure optimization design based on topology optimization [J]. Chinese Journal of Engineering Design, 2019, 55 (08): 65-72.
[28]	OU Disheng, ZHOU Xiongxin. Work-ratio-extremum method for topology optimization of trusses subjected to loads including self-weight [J]. Engineering Mechanics, 2011, 28 (07): 136-142.
[29]	ZHOU Xiongxin, OU Disheng. Application of optimal step-size factor methods in truss sizing optimization [J]. Chinese Journal of Applied Mechanic, 2014, 31 (4): 551-555.
[30]	OU Disheng, ZHOU Xiongxin, Lin Mao-hua, et al. Singular solutions of truss size optimization for considering fundamental frequency constraints [J]. Archive of Applied Mechanics, 2019, 89 (4): 649-658.

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APA Style

Ou Disheng, Zheng Xuefen, Zhou Xiongxin. (2022). Research on Ratio-Extremum Algorithm for Topology Optimization of Continuum Structures Including Active Constraint Identification. International Journal of Mechanical Engineering and Applications, 10(1), 1-6. https://doi.org/10.11648/j.ijmea.20221001.11

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ACS Style

Ou Disheng; Zheng Xuefen; Zhou Xiongxin. Research on Ratio-Extremum Algorithm for Topology Optimization of Continuum Structures Including Active Constraint Identification. Int. J. Mech. Eng. Appl. 2022, 10(1), 1-6. doi: 10.11648/j.ijmea.20221001.11

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AMA Style

Ou Disheng, Zheng Xuefen, Zhou Xiongxin. Research on Ratio-Extremum Algorithm for Topology Optimization of Continuum Structures Including Active Constraint Identification. Int J Mech Eng Appl. 2022;10(1):1-6. doi: 10.11648/j.ijmea.20221001.11

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@article{10.11648/j.ijmea.20221001.11,
  author = {Ou Disheng and Zheng Xuefen and Zhou Xiongxin},
  title = {Research on Ratio-Extremum Algorithm for Topology Optimization of Continuum Structures Including Active Constraint Identification},
  journal = {International Journal of Mechanical Engineering and Applications},
  volume = {10},
  number = {1},
  pages = {1-6},
  doi = {10.11648/j.ijmea.20221001.11},
  url = {https://doi.org/10.11648/j.ijmea.20221001.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijmea.20221001.11},
  abstract = {To obtain the topology optimization algorithm of continuum structure which can effectively identify the effective constraints and quickly converge, based on the original Ratio-Extremum algorithm theory based on truss structure optimization, the emitter algorithm theory is introduced into the topology optimization of continuum structure. Firstly, taking pseudo density as design variables, mathematical model of the minimization mass with constraints of nodal displacements and element stresses is constructed. Secondly, according to essential extremum conditions of Dual objective function, iterative optimization direction and analytical step-size of constraint multipliers are derived. And, according to essential extremum conditions of Generalized Lagrange function, iterative optimization direction and analytical step-size of pseudo densities are derived. Analytical step-sizes are used to avoid one-dimensional optimization and then the calculation quantity of iterative optimization can be decreased. Thirdly, first-order partial derivatives of nodal displacement and element equivalent stress constraints with respect to pseudo densities are given. After that, by using self-compiled MATLAB program for continuum structure analysis, partial derivative calculation and optimization iteration, 4 optimization examples of different beam structures are used to show the changes of active nodal displacement and element equivalent stress constraints, and structural mass in the optimization iteration process, and to show the effectiveness of Ratio-Extremum algorithm in topology optimization of continuum structures.},
 year = {2022}
}

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TY  - JOUR
T1  - Research on Ratio-Extremum Algorithm for Topology Optimization of Continuum Structures Including Active Constraint Identification
AU  - Ou Disheng
AU  - Zheng Xuefen
AU  - Zhou Xiongxin
Y1  - 2022/02/16
PY  - 2022
N1  - https://doi.org/10.11648/j.ijmea.20221001.11
DO  - 10.11648/j.ijmea.20221001.11
T2  - International Journal of Mechanical Engineering and Applications
JF  - International Journal of Mechanical Engineering and Applications
JO  - International Journal of Mechanical Engineering and Applications
SP  - 1
EP  - 6
PB  - Science Publishing Group
SN  - 2330-0248
UR  - https://doi.org/10.11648/j.ijmea.20221001.11
AB  - To obtain the topology optimization algorithm of continuum structure which can effectively identify the effective constraints and quickly converge, based on the original Ratio-Extremum algorithm theory based on truss structure optimization, the emitter algorithm theory is introduced into the topology optimization of continuum structure. Firstly, taking pseudo density as design variables, mathematical model of the minimization mass with constraints of nodal displacements and element stresses is constructed. Secondly, according to essential extremum conditions of Dual objective function, iterative optimization direction and analytical step-size of constraint multipliers are derived. And, according to essential extremum conditions of Generalized Lagrange function, iterative optimization direction and analytical step-size of pseudo densities are derived. Analytical step-sizes are used to avoid one-dimensional optimization and then the calculation quantity of iterative optimization can be decreased. Thirdly, first-order partial derivatives of nodal displacement and element equivalent stress constraints with respect to pseudo densities are given. After that, by using self-compiled MATLAB program for continuum structure analysis, partial derivative calculation and optimization iteration, 4 optimization examples of different beam structures are used to show the changes of active nodal displacement and element equivalent stress constraints, and structural mass in the optimization iteration process, and to show the effectiveness of Ratio-Extremum algorithm in topology optimization of continuum structures.
VL  - 10
IS  - 1
ER  -

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Author Information

Ou Disheng

School of Innovation and Entrepreneurship, Center of Engineering Practice and Innovation Education, Guangxi University of Science and Technology, Liuzhou, China
Zheng Xuefen

School of Mechanical and Automotive Engineering, Guangxi University of Science and Technology, Liuzhou, China
Zhou Xiongxin

School of Innovation and Entrepreneurship, Center of Engineering Practice and Innovation Education, Guangxi University of Science and Technology, Liuzhou, China

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Plain Text BibTeX RIS

APA Style

Ou Disheng, Zheng Xuefen, Zhou Xiongxin. (2022). Research on Ratio-Extremum Algorithm for Topology Optimization of Continuum Structures Including Active Constraint Identification. International Journal of Mechanical Engineering and Applications, 10(1), 1-6. https://doi.org/10.11648/j.ijmea.20221001.11

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ACS Style

Ou Disheng; Zheng Xuefen; Zhou Xiongxin. Research on Ratio-Extremum Algorithm for Topology Optimization of Continuum Structures Including Active Constraint Identification. Int. J. Mech. Eng. Appl. 2022, 10(1), 1-6. doi: 10.11648/j.ijmea.20221001.11

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AMA Style

Ou Disheng, Zheng Xuefen, Zhou Xiongxin. Research on Ratio-Extremum Algorithm for Topology Optimization of Continuum Structures Including Active Constraint Identification. Int J Mech Eng Appl. 2022;10(1):1-6. doi: 10.11648/j.ijmea.20221001.11

Copy | Download

@article{10.11648/j.ijmea.20221001.11,
  author = {Ou Disheng and Zheng Xuefen and Zhou Xiongxin},
  title = {Research on Ratio-Extremum Algorithm for Topology Optimization of Continuum Structures Including Active Constraint Identification},
  journal = {International Journal of Mechanical Engineering and Applications},
  volume = {10},
  number = {1},
  pages = {1-6},
  doi = {10.11648/j.ijmea.20221001.11},
  url = {https://doi.org/10.11648/j.ijmea.20221001.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijmea.20221001.11},
  abstract = {To obtain the topology optimization algorithm of continuum structure which can effectively identify the effective constraints and quickly converge, based on the original Ratio-Extremum algorithm theory based on truss structure optimization, the emitter algorithm theory is introduced into the topology optimization of continuum structure. Firstly, taking pseudo density as design variables, mathematical model of the minimization mass with constraints of nodal displacements and element stresses is constructed. Secondly, according to essential extremum conditions of Dual objective function, iterative optimization direction and analytical step-size of constraint multipliers are derived. And, according to essential extremum conditions of Generalized Lagrange function, iterative optimization direction and analytical step-size of pseudo densities are derived. Analytical step-sizes are used to avoid one-dimensional optimization and then the calculation quantity of iterative optimization can be decreased. Thirdly, first-order partial derivatives of nodal displacement and element equivalent stress constraints with respect to pseudo densities are given. After that, by using self-compiled MATLAB program for continuum structure analysis, partial derivative calculation and optimization iteration, 4 optimization examples of different beam structures are used to show the changes of active nodal displacement and element equivalent stress constraints, and structural mass in the optimization iteration process, and to show the effectiveness of Ratio-Extremum algorithm in topology optimization of continuum structures.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - Research on Ratio-Extremum Algorithm for Topology Optimization of Continuum Structures Including Active Constraint Identification
AU  - Ou Disheng
AU  - Zheng Xuefen
AU  - Zhou Xiongxin
Y1  - 2022/02/16
PY  - 2022
N1  - https://doi.org/10.11648/j.ijmea.20221001.11
DO  - 10.11648/j.ijmea.20221001.11
T2  - International Journal of Mechanical Engineering and Applications
JF  - International Journal of Mechanical Engineering and Applications
JO  - International Journal of Mechanical Engineering and Applications
SP  - 1
EP  - 6
PB  - Science Publishing Group
SN  - 2330-0248
UR  - https://doi.org/10.11648/j.ijmea.20221001.11
AB  - To obtain the topology optimization algorithm of continuum structure which can effectively identify the effective constraints and quickly converge, based on the original Ratio-Extremum algorithm theory based on truss structure optimization, the emitter algorithm theory is introduced into the topology optimization of continuum structure. Firstly, taking pseudo density as design variables, mathematical model of the minimization mass with constraints of nodal displacements and element stresses is constructed. Secondly, according to essential extremum conditions of Dual objective function, iterative optimization direction and analytical step-size of constraint multipliers are derived. And, according to essential extremum conditions of Generalized Lagrange function, iterative optimization direction and analytical step-size of pseudo densities are derived. Analytical step-sizes are used to avoid one-dimensional optimization and then the calculation quantity of iterative optimization can be decreased. Thirdly, first-order partial derivatives of nodal displacement and element equivalent stress constraints with respect to pseudo densities are given. After that, by using self-compiled MATLAB program for continuum structure analysis, partial derivative calculation and optimization iteration, 4 optimization examples of different beam structures are used to show the changes of active nodal displacement and element equivalent stress constraints, and structural mass in the optimization iteration process, and to show the effectiveness of Ratio-Extremum algorithm in topology optimization of continuum structures.
VL  - 10
IS  - 1
ER  -

Copy | Download