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Front. Struct. Civ. Eng.    2020, Vol. 14 Issue (4) : 907-929    https://doi.org/10.1007/s11709-020-0628-1
RESEARCH ARTICLE
An efficient two-stage approach for structural damage detection using meta-heuristic algorithms and group method of data handling surrogate model
Hamed FATHNEJAT, Behrouz AHMADI-NEDUSHAN()
Department of Civil Engineering, Yazd University, Yazd 89195-741, Iran
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Abstract

In this study, the performance of an efficient two-stage methodology which is applied in a damage detection system using a surrogate model of the structure has been investigated. In the first stage, in order to locate the damage accurately, the performance of the modal strain energy based index for using different numbers of natural mode shapes has been evaluated using the confusion matrix. In the second stage, to estimate the damage extent, the sensitivity of most used modal properties due to damage, such as natural frequency and flexibility matrix is compared with the mean normalized modal strain energy (MNMSE) of suspected damaged elements. Moreover, a modal property change vector is evaluated using the group method of data handling (GMDH) network as a surrogate model during damage extent estimation by optimization algorithm; in this part of methodology, the performance of the three popular optimization algorithms including particle swarm optimization (PSO), bat algorithm (BA), and colliding bodies optimization (CBO) is examined and in this regard, root mean square deviation (RMSD) based on the modal property change vector has been proposed as an objective function. Furthermore, the effect of noise in the measurement of structural responses by the sensors has also been studied. Finally, in order to achieve the most generalized neural network as a surrogate model, GMDH performance is compared with a properly trained cascade feed-forward neural network (CFNN) with log-sigmoid hidden layer transfer function. The results indicate that the accuracy of damage extent estimation is acceptable in the case of integration of PSO and MNMSE. Moreover, the GMDH model is also more efficient and mimics the behavior of the structure slightly better than CFNN model.
Keywords two-stage method      modal strain energy      surrogate model      GMDH      optimization damage detection     
Corresponding Author(s): Behrouz AHMADI-NEDUSHAN   
Just Accepted Date: 25 May 2020   Online First Date: 14 July 2020    Issue Date: 27 August 2020
 Cite this article:   
Hamed FATHNEJAT,Behrouz AHMADI-NEDUSHAN. An efficient two-stage approach for structural damage detection using meta-heuristic algorithms and group method of data handling surrogate model[J]. Front. Struct. Civ. Eng., 2020, 14(4): 907-929.
 URL:  
https://academic.hep.com.cn/fsce/EN/10.1007/s11709-020-0628-1
https://academic.hep.com.cn/fsce/EN/Y2020/V14/I4/907
Fig.1  Flowchart of damage localization.
Fig.2  Flowchart of damage extent estimation procedure.
Fig.3  A 52-bar space truss: (a) elevation view; (b) plan view.
property unit value
E (modulus of elasticity) N·m−2 2.1×101
ρ (material density) kg·m−3 7800
added mass kg 50
cross section m2 2×10−4
Tab.1  Material and section properties for the 52-bar space truss
scenario 1 scenario 2 scenario 3
element number damage extent element number damage extent element number damage extent
13 0.10 12 0.25 7 0.15
21 0.35 28 0.10 17 0.35
39 0.15 43 0.30 26 0.10
? ? 49 0.15 39 0.25
? ? ? ? 51 0.20
Tab.2  Three different damage scenarios induced in 52-bar space truss
item 1 mode 3 modes 7 modes
no. of predicted healthy elements no. of predicted damaged elements no. of predicted healthy elements no. of predicted damaged elements no. of predicted healthy elements no. of predicted damaged elements
no. of actual healthy elements (49) TN=28 FP=21 TN=44 FP=5 TN=49 FP=0
no. of actual damaged elements (3) FN=0 TP=3 FN=0 TP=3 FN=0 TP=3
28 24 44 8 49 3
TP rate 1 1 1
FP rate 0.4286 0.1020 0
predicted damaged elements 5,6,9,12,13,15,21,22,23,… 5,6,13,21,28,29,30,39 13,21,39
Tab.3  Confusion matrix using first 7, 3 and 1st mode shapes to compute MSEBI for 52-bar space truss (scenario 1)
Fig.4  MSEBI identification performance up to first 10 mode shapes of 52-bar space truss (scenario 1).
item 1 mode 3 modes 7 modes
no. of predicted healthy elements no. of predicted damaged elements no. of predicted healthy elements no. of predicted damaged elements no. of predicted healthy elements no. of predicted damaged elements
no. of actual healthy elements (48) TN=33 FP=15 TN=46 FP=2 TN=47 FP=1
no. of actual damaged elements (4) FN=0 TP=4 FN=0 TP=4 FN=0 TP=4
33 19 46 6 47 5
TP rate 1 1 1
FP rate 0.3125 0.0417 0.0208
predicted damaged elements 9,12,13,15,21,22,28,29,30,… 12,28,41,43,49,51 12,28,43,49,51
Tab.4  Confusion matrix using first 7, 3 and 1st mode shapes to compute MSEBI for 52-bar space truss (scenario 2)
Fig.5  MSEBI identification performance up to first 10 mode shapes of 52-bar space truss (Scenario 2).
item 1 mode 3 modes 7 modes
no. of predicted healthy elements no. of predicted damaged elements no. of predicted healthy elements no. of predicted damaged elements no. of predicted healthy elements no. of predicted damaged elements
no. of actual healthy elements (47) TN=34 FP=13 TN=47 FP=0 TN=47 FP=0
no. of actual damaged elements (5) FN=2 TP=3 FN=0 TP=5 FN=0 TP=5
36 16 47 5 47 5
TP rate 1 1 1
FP rate 0.2766 0 0
predicted damaged elements 7,12,15,17,39,40,41,42,43,… 7,17,26,39,51 7,17,26,39,51
Tab.5  Confusion matrix using first 7, 3 and 1st mode shapes to compute MSEBI for 52-bar space truss (scenario 3)
Fig.6  MSEBI identification performance up to first 10 mode shapes of 52-bar space truss (Scenario 3).
parameter PSO BA CBO
population size (NP) 30 (scenario 1)
40 (scenarios 2 & 3)		 30 (scenario 1)
40 (scenarios 2 & 3)		 30 (scenario 1)
40 (scenarios 2 & 3)
the maximum number of iterations 30 (scenario 1)
40 (scenarios 2 & 3)		 30 (scenario 1)
40 (scenarios 2 & 3)		 30 (scenario 1)
40 (scenarios 2 & 3)
cognitive parameter (C1) 2.0 – –
social parameter (C2) 2.0 – –
minimum of inertia weight ( ωmin) 0.4 – –
maximum of inertia weight ( ωmax) 0.9 – –
minimum frequency (fmin) – 0 –
maximum frequency (fmax) – 1 –
initial loudness (lmin) – 1 –
loudness adaption parameter (α) – 0.95 –
initial pulse rate (rmin) – 0.5 –
pulse rate adaption parameter (γ) – 0.98 –
Tab.6  The specifications of the PSO, BA and CBO algorithms
metaheuristic method RMSE
first 3 natural frequencies first 3 diagonal elements of flexibility matrix first 2 mode shapes to compute MNMSE
PSO 4.90E–05 2.57E–04 3.06E–05
BA 8.24E–04 1.87E–03 7.16E–04
CBO 2.34E–04 7.62E–04 1.92E–04
suspected damaged elements 13,21,39
Tab.7  Performance evaluation of damage extent estimation using RMSD objective function for 52-bar truss, the best results of 10 independent runs (scenario 1)
run no. PSO BA CBO
estimated extent of suspected damaged elements RMSE estimated extent of suspected damaged elements RMSE estimated extent of suspected damaged elements RMSE
13 21 39 13 21 39 13 21 39
1 0.1019 0.3503 0.1501 1.12E–03 0.0960 0.3506 0.1516 2.51E–03 0.1003 0.3500 0.1500 1.92E–04
2 0.1005 0.3501 0.1501 2.96E–04 0.1127 0.3507 0.1518 7.40E–03 0.0996 0.3500 0.1500 2.42E–04
3 0.0994 0.3500 0.1500 3.63E–04 0.1114 0.3498 0.1498 6.57E–03 0.1046 0.3502 0.1503 2.67E–03
4 0.1013 0.3502 0.1501 7.38E–04 0.1423 0.3487 0.1494 2.45E–02 0.0960 0.3498 0.1497 2.30E–03
5 0.0997 0.3499 0.1499 1.63E–04 0.1049 0.3498 0.1506 2.83E–03 0.1066 0.3502 0.1505 3.80E–03
6 0.1007 0.3500 0.1501 4.25E–04 0.0990 0.3502 0.1506 7.16E–04 0.1090 0.3498 0.1506 5.22E–03
7 0.1000 0.3500 0.1500 3.06E–05 0.1025 0.3498 0.1492 1.54E–03 0.1802 0.3511 0.1559 4.65E–02
8 0.1006 0.3499 0.1499 3.76E–04 0.0998 0.3522 0.1539 2.58E–03 0.0955 0.3500 0.1497 2.58E–03
9 0.0993 0.3500 0.1500 4.00E–04 0.1097 0.3492 0.1490 5.65E–03 0.0992 0.3500 0.1499 4.91E–04
10 0.1003 0.3500 0.1500 1.91E–04 0.0956 0.3480 0.1470 3.26E–03 0.0996 0.3500 0.1500 2.11E–04
average 0.1004 0.3500 0.1500 4.10E–04 0.1074 0.3499 0.1503 5.75E–03 0.1091 0.3501 0.1507 6.42E–03
standard deviation 3.12E–04 6.90E–03 1.42E–02
actual 0.10 0.35 0.15 0.10 0.35 0.15 0.10 0.35 0.15
Tab.8  Estimated extent of suspected damaged elements by PSO, BA, and CBO with MNMSE for 52-bar truss (scenario 1)
metaheuristic method RMSE
first 3 natural frequencies first 3 diagonal elements of flexibility matrix first 2 mode shapes to compute MNMSE
PSO 4.63E–02 2.29E–02 1.23E–03
BA 4.99E–02 3.21E–02 8.92E–03
CBO 7.23E–02 3.11E–02 1.28E–02
suspected damaged elements 12,28,43,49,51
Tab.9  Performance evaluation of damage extent estimation using RMSD objective function for 52-bar truss; the best results of 10 independent runs (Scenario 2)
item PSO BA CBO
average 1.57E–02 6.58E–02 1.07E–01
standard deviation 1.33E–02 7.87E–02 1.24E–01
Tab.10  RMSE of estimated extent of suspected damaged elements by PSO, BA, and CBO with MNMSE for 52-bar truss over 10 independent runs (Scenario 2)
metaheuristic method RMSE
first 3 natural frequencies first 3 diagonal elements of flexibility matrix first 2 mode shapes to compute MNMSE
PSO 6.88E–03 2.39E–02 1.06E–04
BA 1.36E–02 4.84E–02 1.99E–03
CBO 1.50E–02 4.26E–02 2.37E–04
suspected damaged elements 7,17,26,39,51
Tab.11  Performance evaluation of damage extent estimation using RMSD objective function for 52-bar truss; the best results of 10 independent runs (Scenario 3)
item PSO BA CBO
average 1.73E–04 3.39E–03 1.58E–02
standard deviation 6.49E–05 9.82E–04 4.04E–02
Tab.12  RMSE of estimated extent of suspected damaged elements by PSO, BA and, CBO with MNMSE for 52-bar truss over 10 independent runs (Scenario 3)
Fig.7  Convergence history of optimization algorithms for damage extent estimation with MNMSE for 52-bar truss. (a) Scenario 1; (b) scenario 2; (c) scenario 3.
item 1 mode 4 modes 8 modes
no. of predicted healthy elements no. of predicted damaged elements no. of predicted healthy elements no. of predicted damaged elements no. of predicted healthy elements no. of predicted damaged elements
no. of actual healthy elements (193) TN=162 FP=31 TN=174 FP=19 TN=190 FP=3
no. of actual damaged elements (7) FN=1 TP=6 FN=1 TP=6 FN=0 TP=7
163 37 175 25 190 10
TP rate 0.8571 0.8571 1
FP rate 0.1606 0.0984 0.0155
predicted damaged elements 4,9,14,25,26,27,35,36,40,… 9,26,27,56,73,78,… 9,27,30,55,78,80
100,150,152,195
Tab.13  Two different damage scenarios induced in 200-bar double layer grid
Fig.8  200-bar double layer grid.
item 1 mode 4 modes 8 modes
no. of predicted healthy elements no. of predicted damaged elements no. of predicted healthy elements no. of predicted damaged elements no. of predicted healthy elements no. of predicted damaged elements
no. of actual healthy elements (193) TN=162 FP=31 TN=174 FP=19 TN=190 FP=3
no. of actual damaged elements (7) FN=1 TP=6 FN=1 TP=6 FN=0 TP=7
163 37 175 25 190 10
TP rate 0.8571 0.8571 1
FP rate 0.1606 0.0984 0.0155
predicted damaged elements 4,9,14,25,26,27,35,36,40,… 9,26,27,56,73,78,… 9,27,30,55,78,80
100,150,152,195
Tab.14  Confusion matrix using first 8, 4 and 1st mode shapes to compute MSEBI for 200-bar double layer grid (scenario 1)
Fig.9  MSEBI identification performance up to first 10 mode shapes of 200-bar double layer grid (Scenario 1).
item 1 mode 4 modes 8 modes
no. of predicted healthy elements no. of predicted damaged elements no. of predicted healthy elements no. of predicted damaged elements no. of predicted healthy elements no. of predicted damaged elements
no. of actual healthy elements (193) TN=162 EP=31 TN=174 EP=19 TN=189 EP=4
no. of actual damaged elements (7) FN=1 TP=6 FN=1 TP=6 FN=0 TP=7
163 37 175 25 189 11
TP rate 0.8571 0.8571 1
FP rate 0.1606 0.0984 0.0207
predicted damaged elements 1,4,9,20,25,27,31,35,… 9,26,27,56,73,78,… 9,26,27,30,55,78,80,
100,150,152,195
Tab.15  Confusion matrix using first 8, 4 and 1st mode shapes to compute MSEBI for 200-bar double layer grid with considering measurement noise (scenario 1)
Fig.10  MSEBI identification performance up to first 10 mode shapes of 200-bar double layer grid with considering measurement noise (Scenario 1).
item 1 mode 4 modes 7 modes
no. of predicted healthy elements no. of predicted damaged elements no. of predicted healthy elements no. of predicted damaged elements no. of predicted healthy elements no. of predicted damaged elements
no. of actual healthy elements (190) TN=174 FP=16 TN=186 FP=4 TN=189 FP=1
no. of actual damaged elements (10) FN=0 TP=10 FN=0 TP=10 FN=0 TP=10
174 26 186 14 189 11
TP rate 1 1 1
FP rate 0.0842 0.0211 0.0053
predicted damaged elements 10,11,15,20,25,33,40,45,55,… 11,25,33,45,56,93,… 11,25,33,45,98,100,112
136,152,176,191
Tab.16  Confusion matrix using first 8, 4 and 1st mode shapes to compute MSEBI for 200-bar double layer grid (scenario 2)
Fig.11  MSEBI identification performance up to first 10 mode shapes of 200-bar double layer grid (Scenario 2).
method RMSE suspected damaged elements
first 7 natural frequencies first 7 diagonal elements of flexibility matrix first 2 mode shapes to compute mnmse
metaheuristic method (without noise effect)
PSO 1.02E–01 7.17E–02 5.09E–05 9,27,30,55,78,80
100,150,152,195
BA 1.25E–01 8.77E–02 3.55E–02
CBO 1.13E–01 1.09E–01 3.02E–03
metaheuristic method (with noise effect ( ηf=1% & ηϕ=3%))
PSO 1.27E–01 1.50E–01 3.52E–02 9,26,27,30,55,78,80,
100,150,152,195
BA 2.40E–01 2.96E–01 5.72E–02
CBO 1.96E–01 3.00E–01 3.91E–02
Tab.17  Performance evaluation of damage extent estimation for 200-bar double layer grid for both cases of with and without measurement noise effect; the best results of 10 independent runs (scenario 1)
item metaheuristic method metaheuristic method (with noise effect ηφ =3%)
PSO BA CBO PSO BA CBO
average 1.25E–02 9.09E–02 9.30E–02 7.71E–02 1.04E–01 1.13E–01
standard deviation 3.76E–02 3.01E–02 9.04E–02 3.58E–02 3.48E–02 6.12E–02
Tab.18  RMSE of estimated extent of suspected damaged elements by PSO, BA, and CBO with MNMSE for 200-bar double layer grid for both cases of with and without measurement noise effect over 10 independent runs (scenario 1)
Fig.12  Average RMSE of estimated extent of suspected damaged elements by PSO with MNMSE for 200-bar double layer grid over 10 independent runs by considering different measurement noise percentages (Scenario 1).
Fig.13  Convergence history of optimization algorithms for damage extent estimation with MNMSE for 200-bar double layer grid (Scenario 1). (a) Without noise effect; (b) with noise effect.
Fig.14  RMSE of GMDH test data sets as a function of the number of training and test data sets.
suspected damaged element ID damage extent estimation using GMDH damage extent estimation using CFNN damage extent estimation using direct FE model exact damage extent
9 0.3061 0.3022 0.3000 0.30
27 0.2583 0.2513 0.2500 0.25
30 0.0001 0.0064 0.0000 0.00
55 0.1069 0.0953 0.1000 0.10
78 0.0022 0.0005 0.0000 0.00
80 0.3547 0.3517 0.3500 0.35
100 0.1984 0.2144 0.2001 0.20
150 0.3442 0.3536 0.3500 0.35
152 0.0015 0.0015 0.0000 0.00
195 0.1419 0.1504 0.1499 0.15
Tab.19  Estimated extent of suspected damaged elements using two different surrogate models ; the best results over 10 independent runs (Scenario 1)
item damage extent estimation using GMDH damage extent estimation using CFNN damage extent estimation using direct FE model
damage extent estimation process time (s) 95 186 245
the best RMSE of estimated
damage extent		 0.005336 0.005435 5.09E–05
number of iterations 80 80 80
number of FE analyses 400 400 4800
Tab.20  Comparison of results between using two different surrogate models in terms of computational time and accuracy (scenario 1)
suspected damaged element ID damage extent estimation using GMDH damage extent estimation using CFNN damage extent estimation using direct FE model exact damage extent
9 0.3044 0.3019 0.2997 0.30
26 0.0702 0.0418 0.0434 0.00
27 0.2497 0.2498 0.2496 0.25
30 0.0864 0.1247 0.0715 0.00
55 0.1194 0.0963 0.0989 0.10
78 0.0094 0.0010 0.0016 0.00
80 0.3418 0.3513 0.3504 0.35
100 0.2179 0.1690 0.1906 0.20
150 0.3404 0.3475 0.3458 0.35
152 0.0830 0.0978 0.0805 0.00
195 0.1549 0.1416 0.1430 0.15
Tab.21  Estimated extent of suspected damaged elements using two different surrogate models with considering measurement noise ; the best results over 10 independent runs (scenario 1)
item damage extent estimation using GMDH damage extent estimation using CFNN damage extent estimation using direct FE model
damage extent estimation process time (sec) 98 189 248
the best RMSE of estimated
damage extent		 0.0447 0.0504 3.52E–02
number of iterations 80 80 80
number of FE analyses 400 400 4800
Tab.22  Comparison of results between using two different surrogate models in terms of computational time and accuracy with considering measurement noise (scenario 1)
suspected damaged element ID damage extent estimation using GMDH damage extent estimation using CFNN damage extent estimation using direct FE model exact damage extent
11 0.0998 0.1008 0.0999 0.10
25 0.3542 0.3492 0.3500 0.35
33 0.1607 0.1501 0.1503 0.15
45 0.3966 0.3506 0.3993 0.40
98 0.4547 0.4518 0.4500 0.45
100 0.0285 0.0000 0.0000 0.00
112 0.3451 0.3482 0.3498 0.35
136 0.1607 0.1459 0.1471 0.15
152 0.1815 0.2038 0.2003 0.20
176 0.2189 0.1989 0.2009 0.20
191 0.1002 0.0968 0.1005 0.10
Tab.23  Estimated extent of suspected damaged elements using two different surrogate models; the best results over 10 independent runs (scenario 2)
item damage extent estimation using GMDH damage extent estimation using CFNN damage extent estimation using direct FE model
damage extent estimation process time (sec) 109 198 305
the best RMSE of estimated
damage extent		 0.0128 0.0150 9.53E-04
number of iterations 100 100 100
number of FE analyses 400 400 6000
Tab.24  Comparison of results between using two different surrogate models in terms of computational time and accuracy (Scenario 2)
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