基于镜面反射算法的可靠性稳健优化设计研究

Qi Qisong; Wang Jun; Xu Gening; Fan Xiaoning

doi:10.16383/j.aas.2017.e150129

基于镜面反射算法的可靠性稳健优化设计研究

doi: 10.16383/j.aas.2017.e150129

戚其松^1, ,,
王君^1,,
徐格宁^1,,
范小宁^1,

1.
太原科技大学机械工程学院太原 030024

基金项目:

the National Natural Science Foundation of China 51275329

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- 被引次数: 0
出版历程
- 收稿日期: 2015-08-04
- 录用日期: 2016-02-01
- 刊出日期: 2017-08-20

Reliability-based Robust Optimization Design Based on Specular Reflection Algorithm

Qi Qisong^{1
, ,},
Wang Jun^1
,,
Xu Gening^1
,,
Fan Xiaoning^1
,

1.
Taiyuan University of Science and Technology, Taiyuan 030024, China

Funds:

the National Natural Science Foundation of China 51275329

More Information

Author Bio:
Jun Wang is a Ph.D.candidate at Taiyuan University of Science and Technology, China.Her research interests include intelligent optimization algorithm and hydraulic servo control system.E-mail:best_wangj@126.com

Gening Xu received the Ph.D.degree from Tongji University, China.He is currently a Professor of Taiyuan University of Science and Technology.His research interests include theory and method of structure design and analysis and structure reliability design.E-mail:xugening@sina.com

Xiaoning Fan received the Ph.D.degree from Dalian University of Technology, China.She is currently a Professor of Taiyuan University of Science and Technology.Her research interests include computational intelligence, optimization design and structure fatigue life evaluation.E-mail:fannyfxn@163.com

Corresponding author: Qisong Qi is a Ph.D.candidate at Taiyuan University of Science and Technology, China.His research interests include intelligent optimization algorithm, structure reliability design and structure design of Crane.Corresponding author of this paper.E-mail:qiqisong_19870328@126.com

摘要

摘要: 本文通过模拟镜子反射光线的能力，提出了一种全新的全局优化方法-镜面反射算法（Specular Reflection Algorithm）.镜面反射算法不仅具有全局收敛、计算效率高，适合求解高维、多峰的复杂函数等优点，而且算法的自适应能力强，能够根据不同的优化对象，自适应调整参数，以达到最好的优化效果.本文利用4个典型的优化问题验证该算法的计算能力，并同经典的粒子群算法、果蝇算法和模拟退火算法比较.最后，以桥式起重机桥架结构为研究对象，将镜面反射算法和稳健可靠性设计方法相结合对其进行优化设计，证明了镜面反射算法具有很高的工程研究价值，而且可以预见算法能够广泛的同生产实际结合，创造更大的价值.
- 工程 /
- 镜子 /
- 可靠性稳健优化 /
- 镜面反射算法
Abstract: In this paper, a novel global optimization method-specular reflection algorithm (SRA) is proposed, which simulates the unique optical property of mirror-reflection function. Combining the computing features of the SRA with traditional mathematical theories, the global convergence ability of the SRA is verified. The reasonable value of the SRA's control parameter is analysed, so that the best control parameter which is suitable for current optimization problems can be acquired. Four numerical examples are researched using the SRA and other 4 classical intelligent optimization methods, such as particle swarm optimization, Kalman swarm optimization, etc. Simulation results of numerical examples demonstrated the effectiveness and superiority of the SRA, especially its suitability for solving high dimensional, multi-peak complex functions. Finally the structure of general bridge crane is investigated and designed by SRA for robust reliability optimization design. The results illustrate that the SRA is reasonable, accurate and can be treated as an effective analysis technique in reliability-based robust optimization design. It can be predicted that the SRA can be widely used in engineering for creating more value.
- Engineering /
- mirror /
- robust reliability optimization /
- specular reflection algorithm (SRA)
Recommended by Associate Editor Qinglai Wei
注释:

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Fig. 1 Coordinate update of the specular reflection system.

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Fig. 2 Optimization flow chart of the SRA.

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Fig. 3 Three-dimensional surface of test function.

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Fig. 4 Iteration curve of sphere.

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Fig. 5 Iteration curve of griewank.

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Fig. 6 Iteration curve of rosenbrock.

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Fig. 7 Iteration curve of restrigin.

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Fig. 8 Mechanical model diagram and sectional dimension.

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Table Ⅰ JUDGEMENT OF $\xi$

Value of $\xi$	N =2		N =10		N=20		N=50		N = 100
Value of $\xi$	Optimal solution (10^-6)	Iteration times	Optimal solution (10^-6)	Iteration times	Optimal solution (10^-6)	Optimal solution (10³)	Optimal solution (10^-6)	Optimal solution (10³)	Optimal solution (10^-6)	Optimal solution (10⁴)
0.4	4.7776	402.70	7.1895	1103	7.2883	2.0369	8.9324	6.1015	9.5844	1.4945
0.5	3.3845	341.04	6.4267	940.12	7.3111	1.8001	8.4771	5.3149	9.6383	1.2586
0.6	3.9884	737.76	5.4844	936.46	7.2327	1.6802	9.0155	4.8292	9.3691	1.1344
0.7	3.5625	515.18	6.9587	810.24	7.2858	1.5971	8.5419	4.4544	9.5971	1.0679
0.8	4.2770	509.46	6.7379	747.90	7.5046	1.4992	8.8811	4.2697	9.3384	1.0741
0.9	4.0589	259.08	6.3850	732.90	7.4304	1.4562	8.3421	4.2036	9.4009	1.0976
1.0	4.9287	193.26	5.9257	694.18	6.8977	1.3677	8.3414	4.3603	9.5947	1.1404
1.1	4.6702	142.60	6.1496	674.28	8.0852	1.2946	9.4538	4.2854	9.4944	1.1889
1.2	4.6250	142.42	5.8875	626.54	7.7654	1.3608	8.6969	4.4775	9.6771	1.2434
1.3	5.1501	139.08	6.5208	654.72	7.2172	1.4050	8.9588	4.5342	9.5792	1.3215
1.4	5.4409	131.02	5.6072	695.40	6.9556	1.4699	8.9053	4.6930	9.6898	1.3675
1.5	4.7099	103.72	5.7050	675.02	7.6612	1.4740	9.0472	4.8329	9.5134	1.4173
1.6	4.7625	93.82	5.8038	713.20	6.4546	1470	8.9756	4.8634	9.7748	1.4768
1.7	4.9327	91.94	4.9871	783.90	5.8034	1.6036	9.1825	5.0851	9.6612	1.4985
1.8	5.9076	87.32	5.4104	856.30	7.1143	1.6917	8.7372	5.3202	9.4446	1.5536
1.9	4.9402	82.44	5.5724	832.12	6.4092	1.8641	8.7754	5.5962	9.6617	1.6423
2.0	4.7168	89.08	4.8307	998.300	5.7508	2.0544	8.1780	6.3700	9.4975	2.5117

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Table Ⅱ NUMERICAL CALCULATION FUNCTION

Name	Expression	Interval of convergence	Global extreme	Dimension
Sphere	$f_1 = \sum\limits_{i = 1}^n {x_i^2 }$	$x_i\in [-50,50]$	0 (0, 0, $\ldots$ , 0)	$n$ = 30 100
Griewank	$f_2 = 1 + \sum\limits_{i = 1}^n {\left( {\frac{x_i^2 }{4000}} \right) -\prod\limits_{i = 1}^n {\cos \left( {\frac{x_i }{\sqrt i }} \right)} }$	$x_i\in [-600,600]$	0 (0, 0, $\ldots$ , 0)	$n$ = 30 100
Rosenbrock	$f_3 = \sum\limits_{i = 1}^{n - 1} {[100(x_{i + 1}-x_i^2 )^2 + (x_i-1)^2]}$	$x_i\in [-100,100]$	0 (1, 1, $\ldots$ , 1)	$n$ = 30 100
Restrigin	$f_4 = \sum\limits_{i = 1}^n {[10 + x_i^2-10\cos (2\pi x_i )]}$	$x_i\in [-5.0, 5.0]$	0 (0, 0, $\ldots$ , 0)	$n$ = 30 100

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Table Ⅲ CALCULATION RESULTS OF TEST FUNCTION

Name	PSO (n = 30) [10]	Kalman swarm (n = 30) [10]	Chaos ant colony optimization (n = 30)[10]	Chaos PSO (n = 30) [10]	New chaos PSO (n = 30) [10]	SRA (n = 30)	SRA (n = 100)
Sphere	3.7004×10²	4.723	3.815×10^-1	2.4736×10^-3	2.0729×10^-9	1.1080×10^-24	2.3160×10^-12
Griewank	2.61×10⁷	3.28×10³	23.414	6.8481×10^-2	9.9051×10^-11	4.6629×10^-15	2.7978×10^-14
Rosenbrock	13.865	9.96×10^-1	4.669×10^-1	1.0404×10^-2	2.9068×10^-4	9.8730×10^-7	6.1173×10^-5
Restrigin	1.0655×10²	53.293	22.6361	9.5258×10^-1	4.3741×10^-4	3.9373×10^-21	8.7727×10^-7

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Table Ⅳ CALCULATION RESULTS

Design method		Design variables (mm)					Objective function (mm²)	Reliability	Sensitivity of reliability/(10^-3)
		x₁	x₂	x₃	x₄	x₅	A	R_v	$\frac{\partial R_v}{\partial S}$	$\frac{\partial R_v}{\partial F}$	$\frac{\partial R_v}{\partial E}$	$\frac{\partial R_v}{\partial \rho}$
SRA	Optimization	6	6	205	635	257	10 704	0.5071	14.3985	0.0017	9.15×10^-9	0.0011
	Reliability Optimization	6	6	258	632	310	11 304	0.9968	13.0816	0.0015	8.77×10^-9	0.0010
	Robust Reliability Optimization	6	6	324	595	376	11 652	0.9813	12.6270	0.0015	9.23×10^-9	0.0010
PSO	Optimization	10	6	185	567	619	11 544	0.5314	13.0714	0.0015	9.8×10^-9	0.0010
	Reliability Optimization	7	7	222	605	276	12 334	0.9806	12.9119	0.0015	9.73×10^-9	0.0011
	Robust Reliability Optimization	9	6	302	534	354	12 780	0.9810	11.5262	0.0013	1.01×10^-8	0.0010
FOA	Optimization	9	6	190	581	633	11 328	0.5132	13.3500	0.0016	9.64×10^-9	0.0010
	Reliability Optimization	6	6	491	532	543	13 068	0.9802	11.3697	0.0013	1.01×10^-8	9.95×10^-4
	Robust Reliability Optimization	8	11	237	536	299	16 576	1.0	11.6479	0.0013	1.27×10^-9	0.0013
Note: The index of reliability R₀ = 0.98 is deflned.

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参考文献(21)

[1]	W. T. Pan, "A new fruit fly optimization algorithm: Taking the financial distress model as an example, " Knowl. Based Syst. , vol. 26, pp. 69-74, Feb. 2012. http://www.sciencedirect.com/science/article/pii/S0950705111001365
[2]	H. Cheng and C. Z. Liu, "Mixed fruit fly optimization algorithm based on chaotic mapping, " Comput. Eng. , vol. 39, no. 5, pp. 218-221, May 2013. http://en.cnki.com.cn/Article_en/CJFDTOTAL-JSJC201305050.htm
[3]	T. Li, C. F. Wang, W. B. Wang, and W. L. Su, "A global optimization bionics algorithm for solving integer programming-Plant growth simulation algorithm, " Syst. Eng. Theory Pract. , vol. 25, no. 1, pp. 76-85, Jan. 2005. http://en.cnki.com.cn/Article_en/CJFDTotal-XTLL200501012.htm
[4]	D. Karaboga and B. Basturk, "On the performance of artificial bee colony (ABC) algorithm, " Appl. Soft Comput. , vol. 8, no. 1, pp. 687-697, Jan. 2008. http://www.sciencedirect.com/science/article/pii/S1568494607000531
[5]	X. J. Bi and Y. J. Wang, "A modified artificial bee colony algorithm and its application, " J. Harbin Eng. Univ. , vol. 33, no. 1, pp. 117-123, Jan. 2012. http://en.cnki.com.cn/Article_en/CJFDTOTAL-HEBG201201023.htm
[6]	J. H. Holland, Adaptation in Natural and Artificial Systems. Ann Arbor, MI:University of Michigan Press, 1975.
[7]	M. Franchini, "Use of a genetic algorithm combined with a local search method for the automatic calibration of conceptual rainfall-runoff models, " Hydrol. Sci. J. , vol. 41, no. 1, pp. 21-39, Feb. 1996.
[8]	C. Y. Liu, C. Q. Yan, J. J. Wang, and Z. H. Liu, "Particle swarm genetic algorithm and its application, " Nucl. Power Eng. , vol. 33, no. 4, pp. 29-33, Aug. 2012. http://en.cnki.com.cn/Article_en/CJFDTOTAL-HDLG201204009.htm
[9]	L. B. Zhang, C. G. Zhou, M. Ma, and X. H. Liu, "Solutions of multi-objective optimization problems based on particle swarm optimization, " J. Comput. Res. Dev. , vol. 41, no. 7, pp. 1286-1291, Jul. 2004. http://en.cnki.com.cn/Article_en/CJFDTotal-JFYZ200407034.htm
[10]	X. B. Xu, K. F. Zheng, D. Li, M. Ma, and Y. X. Yang, "New chaos-particle swarm optimization algorithm, " J. Commun. , vol. 33, no. 1, pp. 24-30, 37, Jan. 2012. http://en.cnki.com.cn/Article_en/CJFDTOTAL-TXXB201201005.htm
[11]	H. Modares, A. Alfi, and M. B. Naghibi Sistani, "Parameter estimation of bilinear systems based on an adaptive particle swarm optimization, " Eng. Appl. Artif. Intell. , vol. 23, no. 7, pp. 1105-1111, Oct. 2010. http://www.sciencedirect.com/science/article/pii/S0952197610001089
[12]	S. Kirkpatrick, C. D. Gelatt Jr, and M. P. Vecchi, "Optimization by simulated annealing, " Science, vol. 220, no. 4598, 671-680, May 1983.
[13]	K. Abdi, M. Fathian, and E. Safari, "A novel algorithm based on hybridization of artificial immune system and simulated annealing for clustering problem, " Int. J. Adv. Manuf. Technol. , vol. 60, no. 5-8, pp. 723-732, May 2012. doi: 10.1007/s00170-011-3632-8
[14]	R. Storn and K. Price, "Differential evolution-A simple and efficient heuristic for global optimization over continuous spaces, " J. Global Optim. , vol. 11, no. 4, pp. 341-359, Dec. 1997.
[15]	B. V. Babu and M. M. L. Jehan, "Differential evolution for multi-objective optimization, " in Proc. The 2003 Congress on Evolutionary Computation, Canberra, ACT, Australia, vol. 4, pp. 2696-2703, Dec. 2003. http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1299429
[16]	Z. X. Liu and H. Liang, "Parameter setting and experimental analysis of the random number in particle swarm optimization algorithm, " Control Theory Appl. , vol. 27, no. 11, pp. 1489-1496, Nov. 2010. http://en.cnki.com.cn/Article_en/CJFDTOTAL-KZLY201011009.htm
[17]	C. K. Dimou and V. K. Koumousis, "Reliability-based optimal design of truss structures using particle swarm optimization, " J. Comput. Civil Eng. , vol. 23, no. 2, pp. 100-109, Mar. 2009.
[18]	D. A. McAdams and W. Li, "A novel method to design and optimize flat-foldable origami structures through a genetic algorithm, " J. Comput. Inform. Sci. Eng. , vol. 14, no. 3, Article ID: 031008, Jun. 2014.
[19]	Z. Z. Lv, S. F. Song, H. S. Li, and X. K. Yuan, Structure and Mechanism Reliability Analysis and Reliability Sensitivity Analysis. Beijing, China:Science Press, 2009.
[20]	H. X. Guo, "Current status of research on robust design and development of research on fuzzy robust design, " J. Mach. Design, vol. 22, no. 2, pp. 1-5, Feb. 2005. http://en.cnki.com.cn/Article_en/CJFDTOTAL-JXSJ200502000.htm
[21]	Y. M. Zhang, R. Y. Liu, and F. H. Yu, "Multi-objective reliability based on robust optimization design for vehicle front-axles, " Mach. Design and Res. , vol. 22, no. 4, pp. 82-84, 106, Aug. 2006. http://en.cnki.com.cn/article_en/cjfdtotal-jsyy200604021.htm