A-new-multi-objective-approach-to-finite-element-model-updating

A new multi-objective approach to finite element

model updating

Seung-Seop Jin a,Soojin Cho b,n,Hyung-Jo Jung a,Jong-Jae Lee c,

Chung-Bang Yun b

a Department of Civil and Environmental Engineering,KAIST(Korea Advanced Institute of Science and Technology),Daejeon305-701,

Republic of Korea

b School of Urban and Environmental Engineering,UNIST(Ulsan National Institute of Science and Technology),Ulsan689-798,

Republic of Korea

c Department of Civil an

d Environmental Engineering,Sejong University,Seoul143-747,Republic of Korea

a r t i c l e i n f o

Article history:

Received22May2013

Received in revised form

19December2013

Accepted22January2014

Handling Editor:M.P.Cartmell

Available online16February2014

a b s t r a c t

The single objective function(SOF)has been employed for the optimization process in the

conventional finite element(FE)model updating.The SOF balances the residual of multiple

properties(e.g.,modal properties)using weighting factors,but the weighting factors are hard to

determine before the run of model updating.Therefore,the trial-and-error strategy is taken to

find the most preferred model among alternative updated models resulted from varying

weighting factors.In this study,a new approach to the FE model updating using the multi-

objective function(MOF)is proposed to get the most preferred model in a single run of

updating without trial-and-error.For the optimization using the MOF,non-dominated sorting

genetic algorithm-II(NSGA-II)is employed to find the Pareto optimal front.The bend angle

related to the trade-off relationship of objective functions is used to select the most preferred

model among the solutions on the Pareto optimal front.To validate the proposed approach,

a highway bridge is selected as a test-bed and the modal properties of the bridge are obtained

from the ambient vibration test.The initial FE model of the bridge is built using SAP2000.The

model is updated using the identified modal properties by the SOF approach with varying the

weighting factors and the proposed MOF approach.The most preferred model is selected using

the bend angle of the Pareto optimal front,and compared with the results from the SOF

approach using varying the weighting factors.The comparison shows that the proposed MOF

approach is superior to the SOF approach using varying the weighting factors in getting smaller

objective function values,estimating better updated parameters,and taking less computational

time.

&2014Elsevier Ltd.All rights reserved.

1.Introduction

Finite element(FE)model updating is the process of modifying the initial FE model to represent the dynamic behavior of current structures for the condition assessment of existing structures or to realize the required dynamic behavior for the structural modification.The FE model updating mostly employs an optimization technique that minimizes the given objective function.The objective function is formulated in terms of the discrepancy between the FE model and the reference

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journal homepage:https://www.wendangku.net/doc/213241773.html,/locate/jsvi

Journal of Sound and Vibration

https://www.wendangku.net/doc/213241773.html,/10.1016/j.jsv.2014.01.015

0022-460X&2014Elsevier Ltd.All rights

reserved.

n Corresponding author.Tel.:t82522172831.

E-mail addresses:seungsab@kaist.ac.kr(S.-S.Jin),soojin@unist.ac.kr(S.Cho),hjung@kaist.ac.kr(H.-J.Jung),jongjae@sejong.ac.kr(J.-J.Lee),

cbyun@unist.ac.kr(C.-B.Yun).

Journal of Sound and Vibration333(2014)2323–2338

properties obtained from the structure[1].In many cases,modal properties,such as natural frequencies and mode shapes, are used as the reference properties to compose the objective function.

The conventional FE model updating uses an optimization technique to minimize the single-objective function(SOF). Though many problems in a variety of engineering applications are characterized as the qualification of multiple objective functions or criteria,they are generally treated as the problem with the SOF that combines all criteria in a single function. Several researchers,however,have reported that the FE model updating using the SOF can result in various alternative updated models.Berman[2]demonstrated from theoretical and numerical examples that a large number of alternative updated models may exist in the SOF-based model updating.Robert-Nicoud et al.[3]showed that various combinations of the dynamic characteristics may represent almost identical dynamic responses when limited information is available.Zarate and Caicedo[4]showed that the large complex structures with a limited number of sensors are expected to have multiple updated FE models with similar objective function values through numerical verification and experimental implementation of the FE model updating on the Bill Emerson Memorial Bridge.Caicedo and Yun[5]identified multiple updated models using the cosine similarity in the parameter space for the ASCE benchmark problem.They found that multiple local minima in the objective function space can yield alternative updated models including the correct one,and different weighting factors given to the multiple criteria to construct SOF lead to different updated models due to limited reference properties.

Though some researchers have studied the effect of the weighting factors on finding the most preferred model,little is known about how to choose the best weighting factors[6,7].If the user is experienced,weighting factors may be determined substantially. However,the choice of weighting factors is not an easy task.The even distribution of weighting factors does not correspond to the even distribution of feasible solutions on the Pareto optimal front[6],and thus,slightly misaligned weighting factors may lead to the large change of the objective function[8].Therefore,the trial-and-error approach with varying the weighting factors is often taken to confirm the chosen weighting factors or to validate the updated model.The trial-and-error approach,however,requires long computational time to construct the alternative updated models among which the most preferred solution will be chosen.

Multi-objective function(MOF)approaches are the promising alternatives to overcome the limitation of the SOF approach.Kim and Park[9]introduced multi-objective optimization for FE model updating of a hard disk to avoid the difficulty in choosing weighting factors.They obtained the updated FE model based on the preferred criterion of each modal property,which should be set based on expertise and experience prior to the updating.Christodoulou et al.[10]updated the FE model of a lab-scaled shear building using the Pareto optimal front based on the normal-boundary intersection method.The FE model of a three-story shear building structure was updated with3updating parameters,and by comparing the Pareto optimal fronts from low(3-DOF)and high fidelity(546-DOF)FE models,it was figured out that the higher fidelity model tends to reduce the variability among the feasible solutions on the Pareto optimal front in the predictions.Perera and Ruiz[11]proposed a multi-stage damage identification method using the multi-objective FE model updating.Two independent objective functions related to the modal flexibility and the combination of frequency and mode shapes have been formulated based on a piecewise linear damage function.These two objective function are complementary to each other.One objective function for modal flexibility is highly sensitive to damage,while the other objective function is defined to complement the main deficiency of the modal flexibility.The multi-objective optimization was used to separate the two conflicting objective functions,because it was not easy to find the most suitable objective function. Jaishi and Ren[12]showed the FE model updating of a bridge using the multi-objective optimization technique.They used two objective functions composed of the residuals of the natural frequencies and the modal strain energy,and updated the FE model of Hongtang Bridge successfully.To solve the multi-objective problem,they employed the Goal Attainment method[13]that finds the optimal model satisfying a set of design goals determined by the user's expertise.Their researches showed the potential of the MOF approaches to handle the difficulties in constructing a proper SOF without trial-and-error efforts or significant expertise.

This paper presents a new approach to the FE model updating using the MOF to select the most preferred model among a set of alternative updated models.To select the most preferred model among the alternative models on the Pareto optimal front, a decision making(DM)strategy is taken using the bend angle[14]which is closely related to the trade-off relationship of the MOFs.To validate the proposed approach,a highway bridge was selected as a test-bed,and the modal properties of the bridge are obtained from the ambient vibration test.The initial FE model of the bridge is built using SAP2000and updated using the identified modal properties by the SOF approach with varying the weighting factors and the proposed MOF approach.To minimize the dominance of the initial point,whose selection is critical in finding the optimum for the local optimization techniques,such as simplex method,the genetic algorithm(GA)and the non-dominated sorting genetic algorithm(NSGA-II)are used for the SOF and MOF approaches,respectively.The most preferred model is selected using the bend angle of the Pareto optimal front,and compared with the results from the SOF approach using varying weighting factors.Finally,the most preferred models obtained from both approaches are compared each other in regard of final objective function values,updated parameters,and computational time.

2.Theoretical background

2.1.FE model updating using single-objective function(SOF approach)

The conventional FE model updating employs an optimization technique with the single-objective function(SOF). General formulation of the SOF approach can be stated as follows:

min∑

n

i?1ωi F ieXT?ωT FeXT

S.-S.Jin et al./Journal of Sound and Vibration333(2014)2323–2338 2324

subject to g jeXTo0;j?1;2;:::

h keXT?0;k?1;2;:::

X L r X r X U;∑ωi?1;ωi Z0(1) where X is an updating parameter vector;F ieXTis the sub-objective function that returns a scalar value;n is the number of the sub-objective functions;ωi is the corresponding weighting factor to each sub-objective function;g ieXTand h ieXTare inequality and equality constraints,respectively;and X L and X U are lower and upper constraint vectors of the updating parameter vector,respectively.The weighting factors play an important role in balancing the significance of the sub-objective functions in the SOF,whereas they are generally selected by a user based on his/her experience and expertise.

In the FE model updating,the objective function is formulated in terms of the discrepancy between finite element and reference properties[1].When experimental modal properties(i.e.,natural frequencies and mode shapes)are used as reference properties,the SOF can be formulated as the weighted sum of two sub-objective functions–natural frequency residual and mode shape residual.Moller and Friberg[15]tried several expressions of the weighted sum and suggested the SOF that is well-balanced between the residuals of natural frequency and mode shape as

F?∑

N m

i ωf

i

f FEM

i

àf EXP

i

f EXP

i

!2

t∑N m

i

ω?

i

e1à

????????????

MAC i

p

T2

MAC i

!

subject to∑

i

eωf

i

tω?

i

T?1;

ωf

i

;ω?

i

Z0(2)

where N m is total number of the used modes;f FEM

i and f EXP

i

denote the i-th natural frequencies from the FE model and

experiment,respectively;MAC i is the modal assurance criterion(MAC)value between the i-th mode shapes from the FE

model and experiment;andωf

i andω?

i

are the weighting factors for the residuals of natural frequencies and mode shapes,

respectively,to make both residuals condensed to a scalar value F.In this study,Eq.(2)is selected as the representative SOF, and genetic algorithm(GA)is used for the single-objective optimization in this study.

The FE model updating aims to find an updated model corresponding to global minimum in the objective function space. However,the objective function space in the SOF approach varies with selection of the weighting factors.When more than one type of experimental properties are to be included in the SOF,the weighting factors must be determined based on the relative contribution of each property to the objective function,which is very hard to estimate.Mostly,the weighting factors are determined reasonably considering the uncertainty of experimental properties.

2.2.FE model updating using multi-objective function(MOF approach)

Multi-objective function(MOF)approach is to explore all objective function spaces independently without assigning weighting factors.M-objective optimization problem can be expressed as

mineF1eXT;F2eXT;…;F MeXTT

subject to g jeXTo0;j?1;2;:::

h keXT?0;k?1;2;:::

X L r X r X U(3) The MOF approach aims to find a set of preferred solutions called the Pareto optimal front.Fig.1shows an example of the Pareto optimal front with two objective functions.The markers in Fig.1represent feasible solutions found during search of solutions,and the shadowed area is considered as the feasible solution domain.In the feasible solution domain,non-dominated solutions,at which an objective function value cannot be improved without degradation of the other objective function values,constitute the Pareto optimal front.In the case of Solution C,for example,F1can be improved without degradation of F2by taking Solution A,or F2can be improved without degradation of F1by taking Solution B.Therefore, Solution C is dominated by both points A and B.Solutions A and B do not have any dominated solution and thus they are on the Pareto optimal front.

For the multi-objective optimization,non-dominated sorting genetic algorithm-II(NSGA-II)is implemented as follows. NSGA-II creates a new offspring population(O i)by the reproduction process of the GA.After the reproduction,a combined population(R i?P i[O i)is generated from parent population(P i)and offspring population(O i).Then fast non-dominated sorting is carried out on the combined population(R i)to obtain the candidate solutions on the Pareto optimal front.Then, crowding distance assignment is carried out based on the objective function values to maintain the diversity of new solutions(i.e.,updated parent population)by distributing the solutions evenly on the Pareto optimal front.The process is iteratively carried out every time until a given stopping criteria is met.The flowchart of NSGA-II is described in Fig.2.The details of NSGA-II may be referred to[16,17].

S.-S.Jin et al./Journal of Sound and Vibration333(2014)2323–23382325

For the FE model updating using MOF,the SOF shown in Eq.(2)is divided into two independent objective functions –natural frequency residual and mode shape residual –as

F 1?∑N m i f FEM i àf EXP i f i !2;F 2?∑N m i e1à????????????MAC i p T2i (4)

The two objective functions will help to compare the results by SOF approach and MOF approach.Since the MOF

approach does not require weighting factors as shown in Eq.(4),the feasible solutions are obtained with preserving information of each objective function.

2.3.Selection of the most preferred FE model in MOF approach

After finding the Pareto optimal front,substantial selection of the most preferred FE model among the solutions on the Pareto optimal front is the core process of the MOF approach.Since the solutions on the Pareto optimal front are non-dominated (i.e.,having trade-off relationship),the substantial selection requires reasonable criteria to avoid investigation of the whole Pareto optimal solutions.

In this study,decision making (DM)strategy based on the feature of the Pareto optimal front is employed to the end [14].In the case of two objective functions,their trade-off relationship can be used as the major feature in bi-criteria problems.

1

F 2Ideal Solution

Front

Fig.1.Pareto optimal front in two-objective optimization problem.

Fig.2.Flowchart of NSGA-II.

S.-S.Jin et al./Journal of Sound and Vibration 333(2014)2323–2338

2326

For a solution on the front x ,the gain from x to the other solution is the decrease of one objective function,and the sacrifice is the increase of the other objective function.Then,the most preferred solution can be chosen at the point where a large sacrifice is required to make a small gain to move to any other solution in the Pareto optimal front.The point is called edge-knee point at which the maximum bend angle of the front is observed.

The bend angle (θ)of a solution x on the front can be obtained using the slopes of two lines connecting x with its adjacent neighbors x L and x R as shown in Fig.3.The bend angle can be formulated as

θ?θL àθR

(5)

and the slopes can be formulated as

tan θL ?

F 2ex L TàF 2ex TF 1ex TàF 1ex L T?

S L

G L (6)tan θR ?

F 2ex TàF 2ex R T1ex TàF 1ex T?

G R

S (7)

where G L and S L denote the gain and sacrifice of x when x moves to x L ,and G R and S R denote the gain and sacrifice of x when

x moves to x R .Eqs.(6)and (7)show that θL is the sacrifice-to-gain ratio when x moves to x L ,while θR is the gain-to-sacrifice ratio (i.e.,inverse of the sacrifice-to-gain ratio)when x moves to x R .Therefore,the bend angle in Eq.(5)quantifies the sacrifice against the gain when moving a solution x to its adjacent solutions x L and x R ,and the solution with the maximum bend angle is where the maximum sacrifice to the gain is required to move to the adjacent solutions.The calculation of the bend angles facilitates a user to find the most preferred model without investigation of all updated models on the Pareto optimal front.

https://www.wendangku.net/doc/213241773.html,parison between SOF and MOF approaches

Fig.4shows the comparison of the SOF and MOF approaches.In SOF approach,the preferred model is chosen from alternative FE models obtained by varying the weighting factors.The SOF approach depends on the weighting factors given by the subjective preference,experience or expertise,it may require multiple runs of optimization to validate the updated model by cross-checking of several candidates.Meanwhile,the MOF approach searches all alternative updated models in a single run,and the most preferred FE model can be selected with a single run of the optimization with the help of taking the DM strategy stated in the previous section.Especially,the users who lack experience in the FE model updating can significantly save the time for the selection of the most preferred FE model by taking the MOF approach.3.Ambient vibration test on a multispan highway bridge 3.1.Test bridge and ambient vibration test

To investigate the performance of the MOF approach by comparison of the SOF approach,a bridge was used as a test-bed of the FE model updating by the support of Korea Expressway Corporation (KEX).The test bridge is a curved steel-box girder bridge and has 5continuous spans whose total length is 230m as depicted in Fig.5.

Ambient vibration test was performed to identify the modal properties of the bridge that will be used as reference properties for the FE model updating.A total of 14seismic accelerometers (PCB 393B12)were installed in the box girder at the locations shown in Fig.5.Six sensors (sensors 5–10marked in Fig.5)were densely installed in the middle span,since sinuous mode shapes are expected to appear in the middle span first at the multi-span continuous bridge when the mode order is increasing [18,19].The data acquisition system was composed of a laptop computer,a signal conditioner (PCB Model 481A03)equipped with high-performance anti-aliasing filter,a terminal block (NI BNC-2090),and 16-bit analog-to-digital converter (NI DAQCard-6036E).The ambient vibration data excited by the ordinary vehicle traffic on the bridge

were

Fig.3.Trade-off relationship of a solution on Pareto optimal front.

S.-S.Jin et al./Journal of Sound and Vibration 333(2014)2323–23382327

measured.The data was collected for 50min at the sampling rate of 100Hz with low pass filtering at 45Hz.Fig.6shows an example of the measured ambient acceleration from the sensor 11.It shows that the bridge was excited sufficiently by plenty of traffics passing on the bridge.3.2.Output-only modal identification

Two output-only modal identification methods –frequency domain decomposition (FDD)and stochastic subspace identification (SSI)–were used together to identify the modal properties,such as natural frequencies and mode

shapes,

https://www.wendangku.net/doc/213241773.html,parison of FE model updating in SOF and MOF

approaches.

Fig.5.Plan view of test bridge with sensor configuration.

-0.2

-0.1

0.1

0.2

Time (sec)

A c c e l e r a t i o n (g )

Fig.6.Measured ambient acceleration from sensor 11.

S.-S.Jin et al./Journal of Sound and Vibration 333(2014)2323–2338

2328

from the ambient vibration data [20,21].Fig.7is the dual-axis plot of the first singular values (SVs)obtained from FDD and the stabilization chart obtained from SSI.In this study,the modal properties were identified at the frequencies where the peaks of the first SVs and the stable poles of the stabilization chart are collocated.Six bending modes (marked as B)and one torsional mode (marked as T)were identified as shown in Fig.7.The corresponding mode shapes are depicted in Fig.8.3.3.Initial finite element model

A structural analysis software,SAP2000,was used to construct an initial FE model of the test bridge.Brownjohn et al.[22]have reported that an initial FE model should be constructed to represent relatively changeless geometric and structural details as much as possible.The FE model was built with 281elements prudently considering geometric and structural details of the test bridge.The main box girder was modeled by 93frame elements including prismatic and non-prismatic sections,and the slab was modeled by 188shell elements.Considering support bearings of the test bridge,two abutments are modeled as hinges while four piers are as rollers.The total number of degree-of-freedom (DOF)of the model is 1701.Fig.9shows a segment of the initial FE model.The properties of materials used in the FE model are tabulated in Table 1.

The modal properties from the initial FE model are compared with the modal properties identified from ambient vibration test (using SSI)in Table 2.The errors of natural frequencies range from 4to 16percent,and the MAC values range from 0.497to 0.971.The 4th mode of the initial FE model (at 4.01Hz)is correlated to both of B4and B5from the experiment.The MAC values of 4th and 5th bending modes and 1st torsional mode are not close to unity,which means that the initial FE model does not reproduce the actual dynamic response of the test bridge.Inferring from the revealed discrepancies of the natural frequencies and the mode shapes,the initial FE model requires the model updating to represent the existing bridge precisely.

4.Finite element model updating

4.1.GUI-based finite element model updating software

To facilitate the FE model updating with variable optimization options,a GUI-based FE model updating software was developed using MATLAB as shown in Fig.10.The software was designed to handle an FE model built using commercial structural analysis package,SAP2000,to use existing FE models of bridges or to facilitate the initial FE model design.The analysis engine of SAP2000is called to execute the iterative modal analyses of the FE model in SAP2000for the updating.The software can select either the SOF or MOF approach.The software was designed to use the GA for the SOF approach,and NSGA-II for the MOF approach.4.2.Updating parameters

To avoid making optimization problem complex due to large number of updating parameters,it is preferred to define as small updating parameters as one can [23].In the case of the test bridge,the properties of the steel box girders have less uncertainty than those of the concrete slabs,since the steel girders were manufactured using a deterministic material

(i.e.,

Fig.7.Dual-axis plot of the first singular values (FDD)and stabilization chart (SSI).

S.-S.Jin et al./Journal of Sound and Vibration 333(2014)2323–23382329

steel)made with relatively sophisticated process in the factory.After the sensitivity analysis,the updating parameters were selected as Table 3.The properties related to the aging or damage (i.e.,2nd moment of inertia for strong axis (I 33)and torsional constant (T ))were chosen for the steel box girder,while the ones related to the uncertainty in the material (i.e.,mass density (M )and elastic modulus (E ))were chosen for the concrete slab.Grouping of I 33was based on the proximity of the elements according to the geometric representation (prismatic and non-prismatic sections).Considering smaller number (one)of the identified torsional mode than that (six)of the bending modes,J was grouped for each span to

M o d e V e c t o r

M o d e V e c t o r

M o d e V e c t o r

M o d e V e c t o r

M o d e V e c t o r

M o d e V e c t o r

M o d e V e c t o r

Fig.8.Identified mode shapes (Solid:Mode shape,dashed:undeformed shape):(a)Bending mode #1,(b)Bending mode #2,(c)Bending mode #3,(d)Bending mode #4,(e)Bending mode #5,(f)Torsional mode #1,and (g)Bending mode #6.

S.-S.Jin et al./Journal of Sound and Vibration 333(2014)2323–2338

2330

optimally minimize the number of updating parameters.The grouping of the updating parameters is graphically shown in Fig.11.

To get the physically meaningful values,the upper and lower constraints were imposed on the updating parameters for both of the SOF and MOF approaches.The constraints were set around the initial values according to the engineering judgment.In the studies of Zhang et al.[24,25]and Brownjohn et al.[22],the constraints range from 720to 740percent.The constraints are also tabulated in Table 3

.

Fig.9.Segment of initial FE model (SAP2000).

Table 1

Material properties used in the initial FE model.Type

Material properties Mass density (N m à3)

Elasticity modulus (GPa)Poisson's ratio Concrete slab (Shell element)2400250.2Steel-box girder (frame element)

7830

200

0.3

Table 2

Validation of initial FE model.Mode f

FEM

(Hz)f

EXP

(Hz)

Error (%)MAC B1 2.153 1.92911.630.971B2 2.527 2.372 6.530.905B3 3.153 2.990 5.440.923B4 4.010 3.7157.940.497B5 4.010 3.844 4.310.637T1 5.180 6.159à15.90.721B6

5.882

6.409

à8.22

0.903

Fig.10.GUI-based FE model updating software.

S.-S.Jin et al./Journal of Sound and Vibration 333(2014)2323–23382331

4.3.SOF approach for test bridge

The initial FE model of the test bridge was updated by the SOF approach.Eq.(2)was used with six different weighting factors as shown in Table 4.When determining weighting factors,larger values should be given to the stable properties with less error or significance.Since the natural frequencies can be estimated more accurately than the mode shapes,larger or equal weighting factors were assigned to the residual of natural frequencies (i.e.,the first term of Eq.(2))than the residual of MAC values (i.e.,the second term of Eq.(2))[24,26].The same values were assigned regardless of the mode order.The number of generation and the population size for the GA were set as 200,respectively,which were figured out to be sufficient to converge in the SOF approach with avoiding local minima based on the preliminary trial-and-errors.The initial

Table 3

Selected updating parameters and their constraints.Elements

Updating parameters

Constraints (?Initial value)Lower constraints

Upper constraints Concrete slab (shell element)E (5sets)0.8 1.2M (5sets)0.8 1.2Steel box girder (frame element)J (5sets)0.7 1.3I 33(19sets)0.7 1.3Total

34sets

–

–

Span A

Span B

Span C

Span D

Span E

#11

#12

#13

#14

#15

Span A

Span B Span C

Span D Span E #16

#17#18#19#20

#21#22#23#24

#25#26

#27

#28

#29

#30

#31

#32#33

#34

Span A

Span B

Span C

Span D

Span E

#6

#7

#8

#9

#10

Span A

Span B

Span C

Span D

Span E

#1

#2

#3

#4

#5

Fig.11.Graphical presentation of updating parameters:(a)E (Elastic Modulus):5sets,(b)M (Mass density):5sets,(c)J (Torsional stiffness):5sets,and (d)

I 33(Moment of inertial):19sets.

Table 4

Different combinations of weighting factors.Case ωf ω?Case ωf ω?F1M10.50.5F1M0.50.670.33F1M0.050.950.05F1M0.010.990.01F1M0.005

0.995

0.005

F1M0.001

0.999

0.001

S.-S.Jin et al./Journal of Sound and Vibration 333(2014)2323–2338

2332

population was created randomly with uniform distribution within the bounds given in Table 3.The other details of GA are tabulated in Table 5.

For each case,the updating was carried out six times under the identical condition to yield the best result considering the stochastic characteristics of GA.After 36times of FE model updating,21alternative updated models were found to be the duplication of the others.Finally,15alternative updated models were obtained resulting in different objective function values and updating parameters.The most preferred model was selected for each case based on the minimum objective function values.

Fig.12shows the improvement (i.e.,reduction of the natural frequency residual and increase of the MAC)of each mode made by the six updating cases.It shows larger weighting to the residual of natural frequency generally leads to worse agreement in the residual of MAC value.Among the six cases,F1M0.01has provided the best updating result with significant improvement on the mode shape of T1while keeping moderate improvement on the other modes.Considering that the weighting factors with the best result is hard to assign ahead,the repeated trials of updating are inevitable to get the most preferred model in the SOF approach.4.4.MOF approach for test bridge

The initial FE model of the test bridge was updated by the MOF approach using NSGA-II algorithm.The number of generation was set as 1400that is seven-fold of those used in the SOF approach while the population size kept same (i.e.,200),since the calculation of the crowding distance at each generation is the key operation to update the Pareto optimal front.The initial population was created randomly with bounds as it was created in the SOF approach.Although the initial populations of both approaches were not identical,their dominance to the initial population was not so critical in finding the optimum since the generation and population sizes are sufficient in this study.The other details of NSGA-II used for MOF approaches are tabulated in Table 6.

The MOF approach produced a total of 70alternative updated models consisting of the Pareto-optimal front shown in Fig.13.Several discontinuities are found on the front,indicating the infeasible space by the constraints and finite number (seventy)of preserving the elitism which depends on the Pareto front population fraction [27].

Fig.14shows the improvement of each mode by the MOF approach.Most of the updated models were significantly improved in both natural frequencies and mode shapes from the initial FE model.Especially,the MAC values of B4and B5were significantly improved from 0.497and 0.637,respectively,to the values larger than 0.93.In the case of the torsional mode (T1),the MAC values have large variation ranging from 0.67to 0.95,because the number of updating parameters for the torsional mode is relatively smaller than those for the bending mode.

For the 70alternative updated models (feasible solutions on the Pareto optimal front),the bend angles were calculated using Eq.(5)as Fig.15.Among the 70models,six updated models with the largest bend angles (#33,#36,#43,#46,#48,

Table 5

Details of GA in SOF approach.Parameters

Values and operators Parameters Values and operators Population size //Type 200//Double

Tolerance 10à6

Selection

Tournament (size ?4)Crossover Intermediate (Ratio ?1)No.of generation

200

Mutation

Adaptive feasible

R e l a t i v e E r r o r (%)

B1

B2

B3

B4

B5

T1

B6

-20

-15-10-50510

15FEM0F1M1F1M0.5F1M0.05F1M0.01F1M0.005F1M0.001M A C

B1B2B3B4B5T1B6

0.4

0.50.60.70.8

0.9

1Fig.12.Relative errors and MAC values of the initial model and updated models obtained by SOF approach (FEM0:the initial FE model):(a)Relative errors of natural frequencies,and (b)MAC values.

S.-S.Jin et al./Journal of Sound and Vibration 333(2014)2323–23382333

#51marked with circles in Fig.15)were selected to investigate the improvement by the model updating.Fig.16shows that the six models were improved significantly.As the number of model is increasing,the improvement of MAC values gets larger than that of natural frequencies.

Among the six models,the alternative updated model #43was chosen as the most preferred model due to its largest bend angle.Fig.17shows the comparison of the relative errors of the natural frequencies and the MAC values of the most preferred model with those of the initial FE model.The model has the relative error significantly decreased up to less than 5percent,and the MAC values for all modes dramatically increased close to https://www.wendangku.net/doc/213241773.html,parison of updated FE models by SOF and MOF approaches

To compare the updated FE models by the SOF and MOF approaches directly,the two sub-objective functions (F 1and F 2)are recalculated using Eq.(4)for the 15updated models obtained by the SOF approach.The calculated sub-objective

Table 6

Details of NSGA-II in MOF approach.Parameters

Values and operators Parameters

Values and operators Population size //Type 200//Double

Tolerance 10à6

Selection

Tournament (size ?4)Crossover Intermediate (Ratio ?1)No.of generation 1400

Mutation

Adaptive feasible Distance measure

Distance-crowding

Pareto front Population fraction

0.35

Fig.13.Alternative updated models on Pareto optimal front.

246

8101214

16

R e l a t i v e E r r o r (%

)

1M A C

Fig.14.Relative errors and MAC values of alternative updated models obtained by MOF approach (FEM0:initial FE model):(a)Relative errors of natural frequencies (b)MAC values.

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function values are plotted in the objective function space along

approach in Fig.18.The figure shows that all updated models by the (i.e.,far from the origin).This implies that the updated FE models are several (i.e.six)repetition under identical conditions.Though the approach (it is also shown in Fig.18),it is still a little behind the front and quite far from the most preferred model (#43)obtained by the MOF approach.

Alternative updated models

B e n d a n g l e (D e g r e e )

Fig.15.Bend angles of alternative updated models on Pareto optimal front.

R e l a t i v e E r r o r (%)

B1

B2

B3

B4

B5

T1

B6

-8

-6-4-202468

10#33#36#43#46#48#51

M A C

B1B2B3B4B5T1B6

0.4

0.5

0.6

0.7

0.8

0.91

Fig.16.Relative errors and MAC values of six updated models with largest bend angles:(a)Relative errors of natural frequencies,and (b)MAC values.

M A C

R e l a t i v e E r r o r (%)

0.4

0.50.60.70.8

0.91Fig.17.Relative errors and MAC values of the initial and the most preferred FE S.-S.Jin et al./Journal of Sound and Vibration 333(2014)2323–2338

2335

Fig.18provides another important insight on the SOF approach.In the SOF approach,the weighting factors are expected to balance the sub-objective functions.Some cases,however,deviate from the expectation.For example,the updated model of F1M0.5has better (i.e.,smaller)values of F 1than that of F1M0.05,and worse values of F 1than that of F1M0.05.It may result from the worse capability of finding global or near-global optimum.On the other hand,the MOF approach provides alternative updated models which have obvious trade-off relationship between the natural frequency and MAC value.Therefore,the MOF approach can facilitate a user to select the most preferred FE model substantially based on the DM strategy (i.e.,using bend angles)or even based on user's preference.

The updated parameters of F1M0.01(i.e.,the most preferred model)in the SOF approach and the most preferred model (#43)are compared in Fig.19.The updated parameters that have the final values very near (i.e.,within 0.03)the lower and upper constraints are marked with diamond (?),since they are finalized by the constraints so that they may not contain the physical meaning.Though the #43of the MOF approach has better objective function values as shown in Fig.18,six

updated

Fig.18.Updated models of SOF and MOF approaches in objective function space.

1

U p d a t e d V a l u e / I n i t i a l V a l u e

0.8

11.2Updating Parameters

U p d a t e d V a l u e / I n i t i a l V a l u e

https://www.wendangku.net/doc/213241773.html,parison of updated parameters of F1M0.01of SOF approach and #43of MOF approach (?:near the constraints).

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S.-S.Jin et al./Journal of Sound and Vibration333(2014)2323–23382337

Table7

Comparison of computation required for SOF and MOF approaches.

Approach Pop.size Generation No.of runs No.of alternative updated models No.of evaluation Total computational time(s)

SOF20020036151,440,0004,342,145

MOF2001400170280,000844,976

parameters are found near the constraints while seventeen updated parameters(50percent of34total updating parameters)of F1M0.01are near the constraints.

Computational time is one of the performance indicators for the optimization process,since more computation tends to find the global optimum better.While performing the model updating,the computational time was calculated.An identical computer(Intel s Core(TM)i7-950processor running with3.06GHz,3GB RAM,and Windows7OS)was used for both of the SOF and MOF approaches.The turnaround of a generation can be categorized as

Step1:Generation of the initial population.

Step2:Evaluation of the objective functions for the populations in a generation.

Step3:GA operation(selection,crossover and mutation to generate the offspring).

Step4:Update of the Pareto optimal front(non-dominated sorting and crowding distance).

Note that the SOF approach iterates Steps(1)–(3),while the MOF approach iterates Steps(1)–(4).In a single generation, Step2requires a series of evaluations of the objective functions followed by modal analyses of the FE model composed of the updating parameters renewed by the GA operation.The computational time for a single evaluation of the objective function(including the modal analysis using SAP2000)was3.015s,and Step2requires200evaluations(same as the number of populations).Meanwhile,Steps1,3,4took0.007,0.076,and0.479s,respectively.Therefore,the number of evaluations can be used as a measure of computational time.The numbers of evaluations and total computational time are tabulated in Table7.In the SOF approach,36updates were carried out and15alternative updated models were found with 1,440,000evaluations,while a single update was carried out in the MOF approach and70alternative updated models were identified with280,000evaluations.The total execution time was4,342,145s for the SOF approach,whereas it was 844,976s for the MOF approach.This clearly shows that the MOF approach is more efficient than the SOF approach to get the most preferred model with less computational effort.

Though Table7shows the computational effectiveness of the proposed MOF approach,the computational time of 844,976s(almost10days)can be still a big burden to the users in the aspect of practicality.The large computational time may be a significant limitation of stochastic optimization algorithms,such as the GA.Instead,the GA has strength in searching the objective function space with high complexity due to its heuristic characteristics,and thus,the large computational time may be swallowed to get the most preferred model with high accuracy.In the case the model should be updated repeatedly in short intervals,the present GA-based approach can be used for the first update and local optimization techniques can be used to track the changes quickly in the short intervals.Hybrid optimization technique[28]or fitness approximation technique[29,30]can be combined with the present approach to shorten the computational time.

5.Conclusion

In this study,a new multi-objective function(MOF)approach using evolutionary algorithm(NSGA-II)and a decision making strategy using bend angles is presented to find the most preferred model in the single run of FE model updating without trial-and-errors.To validate the proposed approach,a highway bridge was selected as a test-bed and the modal properties of the bridge were obtained from the ambient vibration test.The initial FE model of the bridge built using SAP2000was updated using the identified modal properties by the SOF approach with varying the weighting factors and the proposed MOF approach.The most preferred model was selected using the bend angles of the Pareto optimal front,and compared with the results from the SOF approach using varying the weighting factors.The conclusion of this study can be summarized as:

1.From36FE model updates in the SOF approach,15updated models were identified with different improvement

according to varying the weighting factors.The case F1M0.01provided the most preferred model whose relative errors of natural frequencies were significantly improved up to less than4percent and MAC values were dramatically increased above0.85from the initial FE model.

2.The MOF approach produced70alternative updated models in a single run.Alternative updated models were found to be

significantly improved in the residual of natural frequencies and the MAC values.The calculation of bend angle provided six preferred models,and the FE model with largest bend angle was selected as the most preferred https://www.wendangku.net/doc/213241773.html,paring the most preferred model with the initial FE model,the relative errors of natural frequencies were significantly improved up to less than5percent and MAC values were dramatically increased above0.9from the initial FE model.

3.In the comparison of results from both the SOF and MOF approaches,the MOF approach in a single run provided better

alternative updated models than the SOF approach in the improvement of the natural frequency and MAC values.

4.The updated parameters from the MOF approach seemed to contain the physical meaning with less objective function values (i.e.,better fitness),while the SOF approach resulted in about 50percent of updated parameters near the constraints.

5.In terms of computational time,the MOF approach required 844,976s while the SOF approach required 4,342,145s which is 5times longer than the MOF approach.

6.From 3to 5,it can be concluded that the proposed MOF approach is superior to the SOF approach using varying the weighting factors in getting smaller objective function values,estimating better updated parameters,and taking less computational time.

Acknowlegements

This work was partially supported by the expressway and transportation research institute of Korea Expressway Corporation and financially supported by the Korea Minister of Ministry of Land,Transport and Maritime Affairs (MLTM)as The U-City Master and Doctor Course Grant Program.References

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