Optimal design of double-layer barrel vaults using genetic and pattern search algorithms and optimized neural network as surrogate model

doi:10.1007/s11709-022-0899-9

Frontiers of Structural and Civil Engineering

2023, Vol. 17

Issue (3): 378-395 https://doi.org/10.1007/s11709-022-0899-9

本期目录

Optimal design of double-layer barrel vaults using genetic and pattern search algorithms and optimized neural network as surrogate model

Reza JAVANMARDI, Behrouz AHMADI-NEDUSHAN(

)

Department of Civil Engineering, Yazd University, Yazd 89195, Iran

全文: PDF(9838 KB) HTML

Abstract：

This paper presents a combined method based on optimized neural networks and optimization algorithms to solve structural optimization problems. The main idea is to utilize an optimized artificial neural network (OANN) as a surrogate model to reduce the number of computations for structural analysis. First, the OANN is trained appropriately. Subsequently, the main optimization problem is solved using the OANN and a population-based algorithm. The algorithms considered in this step are the arithmetic optimization algorithm (AOA) and genetic algorithm (GA). Finally, the abovementioned problem is solved using the optimal point obtained from the previous step and the pattern search (PS) algorithm. To evaluate the performance of the proposed method, two numerical examples are considered. In the first example, the performance of two algorithms, OANN + AOA + PS and OANN + GA + PS, is investigated. Using the GA reduces the elapsed time by approximately 50% compared with using the AOA. Results show that both the OANN + GA + PS and OANN + AOA + PS algorithms perform well in solving structural optimization problems and achieve the same optimal design. However, the OANN + GA + PS algorithm requires significantly fewer function evaluations to achieve the same accuracy as the OANN + AOA + PS algorithm.

Key words： optimization surrogate models artificial neural network SAP2000 genetic algorithm

收稿日期: 2022-05-26 出版日期: 2023-05-24

Corresponding Author(s): Behrouz AHMADI-NEDUSHAN

引用本文:

. [J]. Frontiers of Structural and Civil Engineering, 2023, 17(3): 378-395.
Reza JAVANMARDI, Behrouz AHMADI-NEDUSHAN. Optimal design of double-layer barrel vaults using genetic and pattern search algorithms and optimized neural network as surrogate model. Front. Struct. Civ. Eng., 2023, 17(3): 378-395.

链接本文:

https://academic.hep.com.cn/fsce/CN/10.1007/s11709-022-0899-9
https://academic.hep.com.cn/fsce/CN/Y2023/V17/I3/378

Fig.1

Fig.2

function	figure
$1 : l o g s i g (n) = 1 1 + e − n$
$2 : t a n s i g (n) = 2 1 + e − 2 n − 1$
$3 : r a d b a s (n) = e − n 2$

Tab.1

Fig.3

Fig.4

Fig.5

items	remarks	type	values
E	modulus of elasticity of steel	constant	210 × 10³ N/mm²
$F y$	steel yield stress	constant	355 N/mm²
$γ$	specific gravity of steel	constant	7850 kg·f/m³
$F 1,DL$	dead load applied to the structure	constant	$18 kN$
$F 1,LL$	live load applied to the structure	constant	$10 kN$
$F 2,DL$	dead load applied to the structure	constant	$18 kN$
$F 2,LL$	live load applied to the structure	constant	$10 kN$
$F 3,DL$	dead load applied to the structure	constant	$18 kN$
$F 3,LL$	live load applied to the structure	constant	$10 kN$
$S$	dimensional parameter	constant	$3000 mm$
$H$	dimensional parameter	constant	$3000 mm$
$d 1 (D G 1)$	circular cross-section diameter, group 1	design variable	$20 : 0.1 : 150 mm$
$d 2 (t G 1)$	circular cross-section thickness, group 1	design variable	$2 : 0.1 : 10 mm$
$d 3 (D G 2)$	circular cross-section diameter, group 2	design variable	$20 : 0.1 : 150 mm$
$d 4 (t G 2)$	circular cross-section thickness, group 2	design variable	$2 : 0.1 : 10 mm$

Tab.2

Tab.3

Fig.6

Fig.7

Fig.8

Fig.9

Fig.10

Fig.11

item	remark	value
$N st$	number of records required to create dataset	300
$X OANN$	optimal variables associated with the neural network	[1,4,1]
$C$	quality index of surrogate model	0.176

Tab.4

Tab.5

Fig.12

Fig.13

item	remark	value
$d 0$	starting point	[83,2,64,2]
$d opt$	final point (optimal design)	[86,2,63,2]
$W t,opt (N)$	optimal weight of the structure	$1.26 × 103 (N)$

Tab.6

Tab.7

Fig.14

Fig.15

item	remark	value
$d 0$	search starting point	[85,2,74,2]
$d opt$	the optimal	[86,2,63,2]
$W t,opt (N)$	optimal weight of the structure	$1.26 × 103 (N)$

Tab.8

Tab.9

Tab.10

Fig.16

Fig.17

Fig.18

Fig.19

items	remarks	type	values
E	modulus of elasticity of steel	constant	210 × 10³ N/mm²
$F y$	steel yield stress	constant	335 N/mm²
$γ$	specific gravity of steel	constant	7850 kg·f/m³
$S L$	amount of snow load	constant	1000 N/mm²
$D L$	amount of dead load	constant	2500 N/mm²
$W X$	area dimension in the direction of the x-axis	constant	40000 mm
$W Y$	area dimension in the direction of the y-axis	constant	60000 mm
$N D X$	number of panels in x-direction	constant	7
$N D Y$	number of panels in y-direction, in one span	constant	4
$N B Y$	number of spans in y-direction	constant	3
$K R$	effective buckling length coefficient in the elements used in the roof of the structure	constant	1
$d 1 (H Roof)$	roof thickness	design variable	$1000 : 50 : 4000 mm$
$d 2 (f)$	height of the highest point of the parabolic roof	design variable	$3000 : 50 : 6000 mm$
$d 3 (D UL1)$	diameter of the sections used in the upper layer of the roof, group 1	design variable	$50 : 5 : 250 mm$
$d 4 (t UL1)$	thickness of the sections used in the upper layer of the roof, group 1	design variable	$3 : 1 : 5 mm$
$d 5 (D UL2)$	diameter of the sections used in the upper layer of the roof, group 2	design variable	$50 : 5 : 250 mm$
$d 6 (t UL2)$	thickness of the sections used in the upper layer of the roof, group 2	design variable	$3 : 1 : 5 mm$
$d 7 (D BR1)$	diameter of the sections used in the braces, group 1	design variable	$50 : 5 : 250 mm$
$d 8 (t BR1)$	thickness of the sections used in the braces, group 1	design variable	$3 : 1 : 5 mm$
$d 9 (D BR2)$	diameter of the sections used in the braces, group 2	design variable	$50 : 5 : 250 mm$
$d 10 (t BR2)$	thickness of the sections used in the braces, group 2	design variable	$3 : 1 : 5 mm$
$d 11 (D LL1)$	diameter of the sections used in the lower layer of the roof, group 1	design variable	$50 : 5 : 250 mm$
$d 12 (t LL1)$	thickness of the sections used in the lower layer of the roof, group 1	design variable	$3 : 1 : 5 mm$
$d 13 (D LL2)$	diameter of the sections used in the lower layer of the roof, group 2	design variable	$50 : 5 : 250 mm$
$d 14 (t LL2)$	thickness of the sections used in the lower layer of the roof, group 2	design variable	$3 : 1 : 5 mm$

Tab.11

Fig.20

Fig.21

Fig.22

item	remarks	value
$N st$	number of records required to create data	1850
$X OANN$	optimal variables related to the neural network	[3,11,1]
$C$	surrogate model quality index	0.394

Tab.12

Fig.23

Fig.24

Fig.25

Fig.26

Fig.27

Fig.28

Fig.29

Fig.30

item	remark	value
$d 0$	search starting point	$[2350, 5900, 220, 4, 245, …,$ $4, 220, 4, 115, 3, 235, 5, 150, 4]$
$d o p t$	the optimal	$[2050, 5500, 205, 5, 230, …,$ $4, 225, 4, 60, 5, 235, 5, 145, 4]$
$W t, o p t (N)$	optimal weight of the structure	$8.5602 × 105$

Tab.13

1	R T HaftkaZ Gürdal. Elements of Structural Optimization. 3rd ed. Berlin: Springer Science & Business Media, 1992
2	V Shobeiri, B Ahmadi-Nedushan. Bi-directional evolutionary structural optimization for strut-and-tie modelling of three-dimensional structural concrete. Engineering Optimization, 2017, 49(12): 2055–2078 https://doi.org/10.1080/0305215X.2017.1292382
3	H Fathnejat, B Ahmadi-Nedushan. An efficient two-stage approach for structural damage detection using meta-heuristic algorithms and group method of data handling surrogate model. Frontiers of Structural and Civil Engineering, 2020, 14(4): 907–929 https://doi.org/10.1007/s11709-020-0628-1
4	G Bekdaş, M Yücel, S M Nigdeli. Estimation of optimum design of structural systems via machine learning. Frontiers of Structural and Civil Engineering, 2021, 15(6): 1441–1452
5	T Nguyen-Thoi, A Tran-Viet, N Nguyen-Minh, T Vo-Duy, V Ho-Huu. A combination of damage locating vector method (DLV) and differential evolution algorithm (DE) for structural damage assessment. Frontiers of Structural and Civil Engineering, 2018, 12(1): 92–108 https://doi.org/10.1007/s11709-016-0379-1
6	MS Es-HaghiA ShishegaranT Rabczuk. Evaluation of a novel Asymmetric Genetic Algorithm to optimize the structural design of 3D regular and irregular steel frames. Frontiers of Structural and Civil Engineering, 2020, 14(5):1110–1130
7	H Ghasemi, H S Park, T Rabczuk. A multi-material level set-based topology optimization of flexoelectric composites. Computer Methods in Applied Mechanics and Engineering, 2018, 332: 47–62 https://doi.org/10.1016/j.cma.2017.12.005
8	B Ahmadi-NedushanH Varaee. Optimal design of reinforced concrete retaining walls using a swarm intelligence technique. In: The First International Conference on Soft Computing Technology in Civil, Structural and Environmental Engineering. Funchal: Civil-Comp Press, 2009
9	M ShakibaB Ahmadi-Nedushan. Engineering optimization using opposition based differential evolution. In: The First International Conference on Soft Computing Technology in Civil, Structural and Environmental Engineering. Funchal: Civil-Comp Press, 2009
10	B Ahmadi-Nedushan, H Varaee. Minimum cost design of concrete slabs using particle swarm optimization with time varying acceleration coefficients. World Applied Sciences Journal, 2011, 13(12): 2484–2494
11	B Ahmadi-Nedushan, H Fathnejat. A modified teaching–learning optimization algorithm for structural damage detection using a novel damage index based on modal flexibility and strain energy under environmental variations. Engineering with Computers, 2022, 38: 847–874 https://doi.org/10.1007/s00366-021-01577-3
12	L Abualigah, A Diabat, S Mirjalili, M Abd Elaziz, A H Gandomi. The arithmetic optimization algorithm. Computer Methods in Applied Mechanics and Engineering, 2021, 376: 113609 https://doi.org/10.1016/j.cma.2020.113609
13	D E Goldberg. Genetic Algorithms in Search and Optimization. Optimization. 13th ed. Boston: Addison-Wesley Professional, 1989
14	A J Booker, J E Dennis, P D Frank, D B Serafini, V Torczon, M W Trosset. A rigorous framework for optimization of expensive functions by surrogates. Structural Optimization, 1999, 17(1): 1–13 https://doi.org/10.1007/BF01197708
15	M A Latif, M P Saka. Optimum design of tied-arch bridges under code requirements using enhanced artificial bee colony algorithm. Advances in Engineering Software, 2019, 135: 102685 https://doi.org/10.1016/j.advengsoft.2019.102685
16	T H NguyenA T Vu. Using neural networks as surrogate models in differential evolution optimization of truss structures. In: International Conference on Computational Collective Intelligence. Da Nang: Springer, 2020
17	A Vellido, P J G Lisboa, J Vaughan. Neural networks in business: A survey of applications (1992−1998). Expert Systems with Applications, 1999, 17(1): 51–70 https://doi.org/10.1016/s0957-4174(99)00016-0
18	M Papadrakakis, N D Lagaros, Y Tsompanakis. Optimization of large-scale 3-D trusses using evolution strategies and neural networks. International Journal of Space Structures, 1999, 14(3): 211–223 https://doi.org/10.1260/0266351991494830
19	A Kaveh, Y Gholipour, H Rahami. Optimal design of transmission towers using genetic algorithm and neural networks. International Journal of Space Structures, 2008, 23(1): 1–19 https://doi.org/10.1260/026635108785342073
20	E Krempser, H S Bernardino, H J C Barbosa, A C C Lemonge. Differential evolution assisted by surrogate models for structural optimization problems. Civil-Comp Proc., 2012, 100 https://doi.org/10.4203/ccp.100.49
21	V Penadés-Plà, T García-Segura, V Yepes. Accelerated optimization method for low-embodied energy concrete box-girder bridge design. Engineering Structures, 2019, 179: 556–565 https://doi.org/10.1016/j.engstruct.2018.11.015
22	J H Wang, J H Jiang, R Q Yu. Robust back propagation algorithm as a chemometric tool to prevent the overfitting to outliers. Chemometrics and Intelligent Laboratory Systems, 1996, 34(1): 109–115 https://doi.org/10.1016/0169-7439(96)00005-6
23	G Panchal, A Ganatra, P Shah, D Panchal. Determination of over-learning and over-fitting problem in back propagation neural network. International Journal of Soft Computing, 2011, 2(2): 40–51 https://doi.org/10.5121/ijsc.2011.2204
24	T Dietterich. Overfitting and undercomputing in machine learning. ACM Computing Surveys, 1995, 27(3): 326–327 https://doi.org/10.1145/212094.212114
25	S LawrenceC L GilesA C Tsoi. Lessons in neural network training: overfitting may be harder than expected. In: Proceedings of the 14th National Conference on Artificial Intelligence. Menlo Park: AAAI, 1997
26	W S Sarle. Stopped Training and Other Remedies for Overfitting. In: Proceedings of the 27th Symposium on the Interface of Computing Science and Statistics. Pittsburgh: Interface Foundation of North America, 1995
27	MathWorks. MATLAB Documantation. Natick: MathWorks, 2019
28	R R BartonM Meckesheimer. Handbooks in Operations Research and Management Science 13. 2006
29	T W Simpson, J D Peplinski, P N Koch, J K Allen. Metamodels for computer-based engineering design: Survey and recommendations. Engineering with Computers, 2001, 17(2): 129–150 https://doi.org/10.1007/PL00007198
30	R Jin, W Chen, T W Simpson. Comparative studies of metamodelling techniques under multiple modelling criteria. Structural and Multidisciplinary Optimization, 2001, 23(1): 1–13 https://doi.org/10.1007/s00158-001-0160-4
31	Y F Li, S H Ng, M Xie, T N Goh. A systematic comparison of metamodeling techniques for simulation optimization in Decision Support Systems. Applied Soft Computing, 2010, 10(4): 1257–1273 https://doi.org/10.1016/j.asoc.2009.11.034
32	B S Kim, Y Bin Lee, D H Choi. Comparison study on the accuracy of metamodeling technique for non-convex functions. Journal of Mechanical Science and Technology, 2009, 23(4): 1175–1181 https://doi.org/10.1007/s12206-008-1201-3
33	M MoustaphaJ M BourinetB GuillaumeB Sudret. Comparative study of Kriging and support vector regression for structural engineering applications. ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems. Part A: Civil Engineering, 2018, 4(2): 0000950
34	C Jiang, X Cai, H Qiu, L Gao, P Li. A two-stage support vector regression assisted sequential sampling approach for global metamodeling. Structural and Multidisciplinary Optimization, 2018, 58(4): 1657–1672 https://doi.org/10.1007/s00158-018-1992-5
35	B Ahmadi NedushanL E Chouinard. Statistical analysis in real time of monitoring data for Idukki Arch Dam. In: Annual Conference of the Canadian Society for Civil Engineering. Moncton: Canadian Society for Civil Engineering, 2003
36	R GhanemP D Spanos. Stochastic Finite Elements: A Spectral Approach. New York: Springer, 2003
37	L Y Deng. The design and analysis of computer experiments. Technometrics, 2004, 46(4): 488–489 https://doi.org/10.1198/tech.2004.s230
38	A J Smola, B Schölkopf. A tutorial on support vector regression. Statistics and Computing, 2004, 14(3): 199–222 https://doi.org/10.1023/B:STCO.0000035301.49549.88
39	B Ahmadi NedushanL E Chouinard. Use of artificial neural networks for real time analysis of dam monitoring data. In: Annual Conference of the Canadian Society for Civil Engineering. Moncton: Canadian Society for Civil Engineering, 2003
40	J Deng, D Gu, X Li, Z Q Yue. Structural reliability analysis for implicit performance functions using artificial neural network. Structural Safety, 2005, 27(1): 25–48 https://doi.org/10.1016/j.strusafe.2004.03.004
41	A Mukhopadhyay, A Iqbal. Prediction of mechanical properties of hot rolled, low-carbon steel strips using artificial neural network. Materials and Manufacturing Processes, 2005, 20(5): 793–812 https://doi.org/10.1081/AMP-200055140
42	A Mukhopadhyay, A Iqbal. Comparison of ANN and MARS in prediction of property of steel strips. Applied Soft Computing Technologies, 2006, 34: 329–341 https://doi.org/10.1007/3-540-31662-0_26
43	A R Shahani, S Setayeshi, S A Nodamaie, M A Asadi, S Rezaie. Prediction of influence parameters on the hot rolling process using finite element method and neural network. Journal of Materials Processing Technology, 2009, 209(4): 1920–1935 https://doi.org/10.1016/j.jmatprotec.2008.04.055
44	CSI. SAP2000 Integrated Software for Structural Analysis. Berkeley, CA: Computers & Structures Inc., 2011
45	R Javanmardi, B Ahmadi-Nedushan. Cost optimization of steel-concrete composite I- girder bridges with skew angle and longitudinal slope, using the Sm toolbox and the parallel pattern search algorithm. International Journal of Optimization in Civil Engineering, 2021, 11(3): 357–382
46	J S ShahBahrami. Parallel Processing and Programming by GPU (In Persian). Tehran: Nas, 2016
47	S Mirjalili. Genetic algorithm. Nature Computational Science, 2015, 28: 21–42 https://doi.org/10.1007/978-3-662-43631-8_3
48	J. Shapiro. Genetic algorithms in machine learning. In Advanced Course on Artificial Intelligence, pp. 146-168. Springer, Berlin, Heidelberg, 1999
49	J H Holland. Genetic algorithms. Scientific American, 1992, 267(1): 66–72 https://doi.org/10.1038/scientificamerican0792-66
50	W A Green. Vibrations of space structures. In: Hodnett, F. In: Hodnett, F. Verlag: European Consortium for Mathematics in Industry, 1991
51	M Shimoda, Y Liu, M Wakasa. Free-form optimization method for frame structures aiming at controlling time-dependent responses. Structural and Multidisciplinary Optimization, 2021, 63(1): 479–497 https://doi.org/10.1007/s00158-020-02708-y
52	Z S Makowski. Book review: analysis, design and construction of steel space frames. International Journal of Space Structures, 2002, 17(2–3): 243–243
53	A Kaveh, M Moradveisi. Size and geometry optimization of double-layer grids using CBO and ECBO algorithms. Iranian Journal of Science and Technology. Transaction of Civil Engineering, 2017, 41(2): 101–112 https://doi.org/10.1007/s40996-016-0043-y
54	Z S Makowski. Analysis, Design and Construction of Braced Barrel Vaults. Boca Raton: CRC Press, 2016
55	A Kaveh, A Eskandari, A Eskandary. Analysis of double-layer barrel vaults using different neural networks; a comparative study. International Journal of Optimization in Civil Engineering, 2021, 11(1): 113–141
56	A S NowakK R Collins. Reliability of Structures. 2nd ed. Boca Raton: CRC Press, 2012
57	CSI. Open Application Programming Interface (OAPI). Berkeley: Computers and Structures Inc., California, 2011
58	Z HuX Du. A random field approach to reliability analysis with random and interval variables. ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part B: Mechanical Engineering, 2015, 1(4): 041005
59	C JiangW X LiX HanL X LiuP H Le. Structural reliability analysis based on random distributions with interval parameters. Computers & Structures, 2011, 89(23−24): 2292−2302
60	L Wang, X Wang, R Wang, X Chen. Reliability-based design optimization under mixture of random, interval and convex uncertainties. Archive of Applied Mechanics, 2016, 86(7): 1341–1367 https://doi.org/10.1007/s00419-016-1121-0

Viewed

Full text

Abstract

Cited

Shared

Discussed