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Frequency response function-based model updating using Kriging model
Mechanical Systems and Signal Processing 87 (2017) 218–228
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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Frequency response function-based model updating using Kriging model
cross
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J.T. Wanga, , C.J. Wanga,b, J.P. Zhaoa a b
School of Mechanical Engineering and Automation, Beihang University, Beijing, China State Key Laboratory of Virtual Reality Technology and Systems, Beihang University, Beijing, China
A R T I C L E I N F O
ABSTRACT
Keywords: Model updating Metamodel Frequency response function Optimization
An acceleration frequency response function (FRF) based model updating method is presented in this paper, which introduces Kriging model as metamodel into the optimization process instead of iterating the finite element analysis directly. The Kriging model is taken as a fast running model that can reduce solving time and facilitate the application of intelligent algorithms in model updating. The training samples for Kriging model are generated by the design of experiment (DOE), whose response corresponds to the difference between experimental acceleration FRFs and its counterpart of finite element model (FEM) at selected frequency points. The boundary condition is taken into account, and a two-step DOE method is proposed for reducing the number of training samples. The first step is to select the design variables from the boundary condition, and the selected variables will be passed to the second step for generating the training samples. The optimization results of the design variables are taken as the updated values of the design variables to calibrate the FEM, and then the analytical FRFs tend to coincide with the experimental FRFs. The proposed method is performed successfully on a composite structure of honeycomb sandwich beam, after model updating, the analytical acceleration FRFs have a significant improvement to match the experimental data especially when the damping ratios are adjusted.
1. Introduction Finite element model updating method can be employed as an important tool to calibrate the finite element model (FEM) for mitigating modeling error. It can make the updated FEM have the same behavior to the corresponding real structure as much as possible. There are two main dynamic model updating method: modal parameters based method and frequency response function (FRF) based method. The former has been thoroughly studied, but the FRF based method [1] introduced by Hemez and Brown [2] and other researchers [3,4] has received considerable attention and applications [5–9] in recent years due to its advantage: Firstly, the measured FRF data can be utilized directly without data transformation. In some special software, the calculation of modal parameters is based on the measured FRF data. And the modal analysis is more complex, the error might arise from the modal identification. The identification error might greater than the modeling error. Secondly, the FRF can be measured in more locations of the structure and taken as an objective so it can provide more data. Model updating process is intrinsically an inverse problem [10] and usually formulated as optimization problem, which aims to minimize the differences between the FEM behavior and the corresponding experimental behavior. However, the conventional
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Corresponding author. E-mail address: [email protected] (J.T. Wang).
http://dx.doi.org/10.1016/j.ymssp.2016.10.023 Received 26 April 2016; Received in revised form 23 August 2016; Accepted 22 October 2016 Available online 03 November 2016 0888-3270/ © 2016 Elsevier Ltd. All rights reserved.
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sensitivity based optimization algorithms have disadvantages, such as: (1) the sensitivity analysis may be computation expensive, and the sensitivity results might not be obtained easily; (2) huge computational cost may be required to meet the convergence point for optimization algorithm. Although the modern intelligent algorithms (Genetic Algorithm, Particle Swarm Optimization and Simulated Annealing etc.) can avoid calculating the sensitivity. But to the structural dynamic analysis of FEM, its large number function calls of finite element analysis (FEA) is time consuming, this even cause the intelligent algorithms is impractical applied to the complex structure. Fortunately, the barrier between the practical applications and the intelligent algorithm can be overcome via the metamodel technique which is known as approximate model or surrogate model. This technique considers the relationship between the input and output as a black-box system, and other information of the system such as internal process of dynamic analysis are not required. It can create a fast running surrogate model to replace the exact FEA, and then the solving time of optimization will be reduced significantly. In addition, in-house FEA codes and existing commercial software can be integrated directly, and it is suitable for parallel computing. Thereby the potential of metamodel techniques is indisputable in model updating field. There are several types of construction method of metamodel are commonly used: response surface methodology (RSM), radial basis function (RBF), Kriging method, polynomial correlated function expansion [11], Polynomial Chaos expansion [12] and Support Vector Regression. Most of them are constructed based on training samples which include input and output information of the interested system. The metamodel technique has been applied in fields such as: structural reliability analysis [13], structural static model updating [14], structural dynamic model updating based on modal parameters [15], and structural damage identification [16,17]. The comparison [18,19] and recommendation [20] of the main metamodels also have been studied. As one of the main method, Kriging model is constructed based on the correlation function theory. Particularly, it is an exact interpolation of given data and goes through all the sampling points. So the Kriging model usually has a higher approximation accuracy than traditional RSM. Khodaparast et al. [21] solved the problem interval model updating by using the Kriging method, and the good accuracy of Kriging method was illustrated by beam experiment. Liu et al. [22] calibrated the FEM based on the modal parameters of a complex structure, the Kriging model was taken as a surrogate model. But there is few application of Kriging model in FRF based model updating in nowadays. In this article, the Kriging model is applied to the model updating based on FRF data. The training samples for the Kriging model are obtained via the design of experiment (DOE), whose factors and response are corresponding to design variables and FRF based data respectively. The case study is followed by an impact hammer test of cantilever beam of honeycomb sandwich structure. The acceleration FRF (AFRF) data are measured and used for DOE and optimization. A two-step DOE method is proposed to reduce the design variables and training samples. The model updating result shows the updated damped FEM has significant improvement to match the experimental AFRF data, and demonstrate the effectiveness of the proposed approach. Moreover, the proposed method can be extended to stochastic model updating [23–25] where usually require a huge number function calls of FEA.
2. AFRF based model updating and objective function Model updating method aims to reduce the modeling error of FEM making the FEM agree better with the real experimental situation. In AFRF based model updating, both FEM data and experimental data corresponding to AFRF amplitude curves, are simulated and measured at the interested degree of freedoms (DOFs) of the structure, respectively. Because of the modeling error, the mass matrix, damping matrix and stiffness matrix of FEM have deviation from the experimental situation, therefore, the two analogous amplitude curves cannot overlap. Then minimizing the difference between the coupled curves is considered as an optimization problem. In which, the design variables appointed by the user are taken as input variables. Optimal results (updated input variables) are obtained via minimizing the objective function. The results can in turn change the matrixes for FEA, and finally lead the AFRF curve of FEM to coincide with the experimental AFRF curve as much as possible. The optimization problem can be formulated in the following form:
Minimize F, s. t .
xiL ≤ xi ≤ xiU ,
i = 1, 2, …, n
(1)
where F is the objective function, xi is the design variables of the structure, xiL and xiU are the lower and upper bound of the input variables, respectively. In this study, the objective function is established based on the differences between the corresponding AFRF curves at each selected frequency point, it can be formulated as follows: np
F=
nf
⎛ Atn (ωi ) − Aan (ωi ) ⎞2 ⎟ Atn (ωi ) ⎝ ⎠
∑ ∑ wi⎜ n =1 i =1
(2)
where wi is the weights, np is the number of measured DOFs, nf is the number of selected frequency points of AFRF, ωi is the selected frequency points, Atn (ωi ) is the experimental acceleration amplitude at ωi, Aan (ωi ) is the corresponding amplitude of the FEM. As mentioned in the previous section, during the optimization process, the FEA process can be replaced by Kriging model, which does not require the internal information of the matrix operation in FEA. The construction method of Kriging model will be introduced in the following section. 219
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3. Construction of Kriging model The Kriging method is a linear unbiased estimation which consists of linear part and nonparametric part. And the latter one can be considered as the realization of a stochastic process [26]. The Kriging model can be described as the form:
y(x) = F (β, x) + z(x) = f T (x)β + z(x)
(3)
where z(x) is the realization of the stochastic process, f(x) is a polynomial vector of training sample x, and β is the coefficient of the linear regression. The f(x) term approximates the drift of the Kriging model, and z(x) approximates the local deviation of the Kriging model. z(x) follows normal distribution N(0, σz2 ), and has nonzero covariance which estimated by:
Cov[z(xi ), z(xj )] = σz2 R
(4)
where xi and xj are two training samples, σz is the variance of z(x), R is a symmetry matrix that composed by Rij(xi, xj). Rij(xi, xj) is the correlation function specified by the user which can characterize the correlation between any two training points. Different Rij(xi, xj) may induce different approximation accuracy of the constructed Kriging model. The function Rij(xi, xj) is given by: Nv
Rij (xi , xj ) =
∏ Rk (θk , dk ),
dk = xik − xjk
(5)
k =1
where Nv denotes the number of design variables; θk is the unknown coefficient of correlation; and are the kth components of xi and xj, respectively; dk is the distance between the different values of selected design variable. Because of its smoothness characteristic, the Gaussian correlation function which has widely application [27] and is employed in this study. It has the following form:
xik
Rk (θk , dk ) = exp( − θkdk2 )
xjk
(6)
The detailed estimation of θk and σz can be found in the reference [28], σz is the function of θk. In short, the θk andσz can be evaluated by solving the maximum likelihood estimated problem which has the form as:
⎛ N ln(σ 2 ) + ln R ⎞ z ⎟⎟ max − ⎜⎜ s θk >0 2 ⎝ ⎠
(7)
where Ns is the number of training samples, |R| is the determinant of R which is a function of θk. f(x) in Eq. (3) is enough to describe the trends of the output response, in most cases, f(x) can be simply taken to be constant vector [26]. Thus any output response y(x) can be dealt with stochastic process with normal distribution during the construction of Kriging model. When the Kriging model is constructed with the obtained θk, it can be used to predict the output response at untried location x0 with unbiased estimation. The predicted response yˆ(x 0 ) is given by:
yˆ(x 0) = β* + r T (x 0)R−1(Y − fβ*),
β* = (f T R−1f )−1f T R−1Y
(8)
where Y is a column vector composed by the output of training samples; f is a column vector filled with ones when f(x) is constant vector; r(x0) is a relevant vector composed between training samples and predict point x0.
r(x 0 ) = [R01(x 0 , x1), R02(x 0 , x 2 ), …, R0Ns(x 0 , xNs )]T
(9)
Briefly, the Kriging model will be determined once the values of θk are given. In this study, the training samples come from DOE results of FEM. The input variables of Kriging model are directly taken as the input variables of the formulated optimization problem, and the input variables come from the appointed design variables of FEM. Meanwhile, the optimization objective corresponds to the output response of the DOE. 4. Model updating procedure In order to take advantage the fast running characteristic of metamodel, the Kriging model is applied to the AFRF data based model updating. During the optimization process, the constructed Kriging model is taken as a surrogate of the dynamic analysis of FEM. Model updating result will be achieved based on the optimization results. The training samples are obtained by DOE, whose factors are selected by users based on design variables of FEM. The design variables are considered have modeling error. The Kriging model and DOE are set to have the same output response. The design matrix of DOE is generated by the method of Optimal Latin Hypercube, which is a more evenly sampling method than conventional Latin Hypercube method. DOE results are taken as training samples for solving the basic parameters θk and then constructing Kriging model. Nevertheless, too many factors (design variables) need more DOE samplings which are also a time consuming process. In particular, a considerable number of parameters may involve into factors when the boundary conditions are taken into account. Therefore, a two-step DOE procedure is proposed to reduce the training samples for saving time. The first step is to select the 220
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sensitive variables from the parameters of boundary conditions based on the DOE results, and the selected variables will be taken as a part of factors in the second step. The second step will generate training samples for constructing Kriging model based on all selected variables. The accuracy of the constructed Kriging model will be checked before the optimization. The optimization results correspond to the updated values of the design variables. And then the updated values are taken to calibrate the FEM as model updating result until the updated error is acceptable. The proposed procedure for model updating can be summarized in the flow chart shown in Fig. 1.
5. Case study: experimental honeycomb beam Because of the characteristic of lightweight, high intensity and low density, honeycomb sandwich structure has been widely applied in many industries, especially in the aerospace industry. The structure consists of 5 parts: one thick lightweight honeycomb core, two very thin face sheets, and two thin adhesive layers. Fig. 2 shows its components. In this section, to demonstrate the effectiveness of the proposed method, a cantilever beam test of honeycomb sandwich structure is performed. It has aluminum face sheets (0.5 mm respectively) and aluminum honeycomb core (20 mm). The test structure has the dimension of 500×40×21.6 mm shown in Fig. 3, the adhesive layers have a nominal thickness of 0.3 mm. Fig. 0.3 also shows the dimension of honeycomb cell of honeycomb core. The embedded part is utilized for clamping which is aluminum. The measurement set-up of FRF test is shown in Fig. 4, the impact point is located at 70 mm from the edge of clamping zone. The impact location is close to the clamping zone and can obtain low noise contaminated experimental data [29]. The accelerometer is the type of Integrate Circuit Piezoelectric.
Fig. 1. Flow chart of the proposed model updating procedure.
Fig. 2. Honeycomb sandwich structure.
Fig. 3. Honeycomb sandwich beam test structure.
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Fig. 4. Measurement set-up of FRF test.
Fig. 5. The FEM of honeycomb sandwich beam.
Table 1 Thickness and angles of each layer of the composite beam. Layer
Component
Thickness (mm)
Angle
1 2 3 4 5
Aluminum face Adhesive Honeycomb core Adhesive Aluminum face
0.5 0.3 20 0.3 0.5
0° 0° 0° 0° 0°
5.1. Description of the FEM The FEM is modeled by 188 shell elements of laminated composite in the commercial software Patran. Fig. 5 shows the FEM without showing the thickness. The details of thickness and angles of each layer are given in Table 1. The embedded part of the structure is used for clamping the beam, so the exact cantilever length in the FEM is 470 mm excluding the embedded part. Moreover, the mass of accelerometer is taken into account for the dynamic response. The accelerometer is simplified as a mass point with 10 g of its real mass. As can be seen in Fig. 5, the boundary condition (point A) of the clumped embedded part is equivalent to a 6 springs system: 1 vertical spring (Kz), 2 horizontal springs (Kx and Ky), and 3 directional torsional springs (Ktx, Kty and Ktz). Each of the spring stiffness is given as 106 N/mm. Fig. 5 also shows the load case: point B is the excitation DOF, and point C is the observation DOF of AFRF and associates the mass point. Under 3% modal damping, the structure is excited by a 1N harmonic force (in Z direction) at DOF B. The analytical AFRF data in Z direction can be obtained after FEA which corresponds to the AFRF of the impact hammer test. 5.2. First step DOE for boundary condition In order to select the parameters from the boundary conditions for updating, the insensitivity parameter will be excluded from the design variables. The sensitivity is assessed based on the DOE results. The 6 spring stiffness parameters are taken as the factors of the first step DOE, and the output response of the DOE corresponds to the value of objective function Eq. (2), whose weights wi=1. Particularly, the AFRF amplitudes close to the resonances are sensitive to noise and damping, so the output response should be evaluated in the ranges excluding the vicinity of the resonances [30]. Thereby the frequency points for the DOE response are come from the ranges at every 1.25 Hz step: 10–60 Hz, 90–390 Hz, 460–800 Hz. And then 21 sampling points are generated. F-test method [31] is applied to evaluate the factor significance to the response based on the sampling points, the quantification of the significance can be formulated as: 222
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P{FA ≥ F1− p(fA , fE )} = p ,
FA =
SA / fA SE / fE
∼ F (fA , fE )
(10)
where SA is the square of deviation caused by the DOE factors; SE is the square of experimental deviation; fA and fE are the DOFs of SA and SE, respectively. When the p-value is lower than the threshold set by user, the corresponding factor is considered sensitive to the response and should be selected as the design variable for updating. The p-value of all the 6 factors are shown in Fig. 6, the threshold is set by 0.1, and then the Kty is selected as a factor of the second step DOE. 5.3. Design variables choice and the second step DOE Honeycomb sandwich structure usually has a honeycomb core with large numbers of honeycomb cells. An appropriate method is the honeycomb core equivalent to orthotropic material, three kinds of equivalent theories [32,33] are commonly used: sandwich plate theory, honeycomb plate theory and equivalent plate theory. The sandwich plate theory is a classic method which is adopted in this study. The equivalent properties are given in Table 2 which also lists the properties of the other two main material of the FEM. The design variables for the model updating should be the FEM parameters which prone to existing modeling error. Firstly, the material properties of the honeycomb core should be taken as design variables since it is an equivalent component. Secondly, the
Fig. 6. The p-value of the 6 spring stiffness.
Table 2 Main material Properties of the FEM. Material property Elastic modulus Shear modulus
Poisson ratio Density (kg/m3)
Ex(Pa) Ey(Pa) Gxy(Pa) Gxz(Pa) Gyz(Pa)
Equivalent core
Adhesive
Aluminum face
1.6×105 1.6×105 6×104 2.3×108 1.6×108
7×109
7.1×1010
0.35 1500
0.3 2700
31
Fig. 7. Boundaries between honeycomb cell wall and face sheet.
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adhesive is smeared on the face sheet to combine the honeycomb core, so the elastic modulus and Poisson ratio of the aluminum maybe affected and then taken to be the design variables. Thirdly, the material properties of the adhesive are taken to be design variables, since its status cannot be measured easily. Finally, the boundaries are fuzzy between the cell wall and the adhesive which are illustrate in Fig. 7, therefore the thicknesses of both honeycomb core and adhesive are added to the design variables. Consequently, the 14 design variables and the previously selected stiffness Kty constitute all the design variables, which are summarized in the second column of Table 3. Then the selected 15 design variables are taken as the factors of the second step DOE. The lower bound and upper bound of every factor are listed in Table 3. The design matrix of the DOE is generated based on the bounds by using Optimal Latin Hypercube method. Based on the same frequency points as the first step DOE, 120 sampling points are obtained after the DOE. The response is also evaluated by Eq. (2). The number of sampling points is given by the following equation:
Ns =
Nv(Nv + 1) 2
(11)
5.4. Construction and accuracy assessment of Kriging model The 120 sampling points generated by the second step DOE are taken as the training samples for constructing Kriging model, whose input variables adopt the 15 factors of the DOE. The output responses correspond to the DOE response. Then the parameters of the Kriging model can be obtained based on the training samples. The basic parameters θk in Eq. (7) are found by using Genetic Algorithm: θ1=0.15850, θ2=0.024570, θ3=0.059123, θ4=0.016713, θ5=0.0094339, θ6=0.17948, θ7=0.053560, θ8=0.25231,
Table 3 Selected design variables of FEM. Material name Spring stiffness Honeycomb core
Aluminum face Adhesive
Design variable (properties) Kty(N/mm) Ex(Pa) Ey(Pa) Gxy(Pa) Gyz(Pa) Gxz(Pa) DH (kg/m3) TH(mm) EAL PAL(Poisson ratio) EA(Pa) PA(Poisson ratio) DA (kg/m3) TUP(mm) TDOWN(mm)
Initial value 6
1×10 1.6×105 1.6×105 6×104 2.3×108 1.6×108 31 20 7.1×1010 0.3 7×109 0.35 1500 0.3 0.3
Lower bound 5
1×10 −50% −50% −50% −50% −50% −50% 12 −50% 0.2 −50% 0.2 −50% 0.2 0.2
Upper bound 1×10 100% 100% 100% 100% 100% 100% 20 100% 0.4 100% 0.48 100% 3 3
Fig. 8. The response values of checking points.
224
8
Updated value 7
6.558×10 1.236×105 2.089×105 7.804×104 2.131×108 2.674×108 50.53 16.034 9.657×1010 0.275 9.119×109 0.222 2462.6 2.272 1.107
Percent change +6558% −22.8% +30.6% +30.7% −7.35% +67.1% +63.0% −19.8% +36.0% −8.33% +30.3% −36.6% +64.2% +354% +121%
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Fig. 9. Overlay of initial AFRF, experimental AFRF and updated AFRF.
Table 4 Updated results of damping ratios. Damping ratio
0–300 Hz
300–500 Hz
500–800 Hz
800 Hz and beyond
Initial value Updated value
3% 2.8%
3% 3.0%
3% 4.2%
3% 19%
θ9=0.43850, θ10=0.025487, θ11=0.056517, θ12=0.010368, θ13=1.0688, θ14=0.025359, θ15=0.0099887. The θk values correspond to the design variables listed in Table 3 from top to bottom. The Kriging model is achieved based on these 15 basic parameters, and then the accuracy of the constructed Kriging metamodel will be assessed. A classical accuracy measurement is to assess the capability of the metamodel to reproduce the training samples, but Kriging model is an interpolating model which goes through all the training samples. So the root mean square error (RMSE) is an appropriate method to evaluate the accuracy of the Kriging model. The RMSE can reflect the response error between the Kriging model and FEM at each checking point. The RMSE is defined as follows:
RMSE =
1 ky
k
∑ (yi − yˆ)i 2
(12)
i =1
where k is the number of checking points, yi is the FEM response at checking points, y is the mean value of yi, yˆi is the Kriging response at the corresponding checking points. 12 checking points are generated and used to assess the accuracy. The 12 points are different from the training samples. Fig. 8 shows the response values at each checking point between the Kriging model and the FEM. The RMSE is evaluated as about 3.70% which quantify the error between the Kriging model and the FEM, and the error is acceptable. 5.5. Model updating results Model updating result is achieved based on the optimization results. During the optimization process, the input variables scope is taken as the same of the DOE factor bounds. The Multi-Island Genetic Algorithm (MIGA) [34] is used to solve the minimization of the optimization problem. The optimal results of the input variables shown in Table 3 are obtained after 1000 iterations, the percent changes of the input variables are also given in Table 3. The variable Kty has the biggest changes followed by the variable TUP and TDOWN. The initial FEM is updated based on the updated values in Table 3. For comparison, Fig. 9 plots the overlay of the three AFRF curves at the interested DOF C in Z direction. The three curves are updated AFRF, experimental AFRF and initial FEM AFRF, respectively. Because of the modeling error, a huge error exists between the initial FEM AFRF and the experimental AFRF. After model updating, a significant improvement is achieved that the updated AFRF tends to coincide with the experimental AFRF. Fig. 9 also proves the choice of the design variables is efficient. But the resonance peaks have a residual error between the updated AFRF and the experimental AFRF, the damping ratio is crucial to the resonance peak and can be modified independently. García-Palencia AJ et al. [35] proposed a two-step methodology for updating of stiffness, mass, and damping matrices based on the FRF, the damping matrices is updated in the second step. So in order to update the resonance peak for further model updating, the damping ratio of FEM can be modified via a manual adjustment, since it only has four independent damping ratios. The damping ratios only affect the vicinity of the resonance peak respectively and can easily be adjusted to match the resonance peaks [36]. The modified results of damping ratios are listed in Table 4 after manually adjusting. Fig. 10 shows the AFRF overlay before and after adjusting the damping ratios. As can be seen, the new updated AFRF has 225
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a better match with the experimental AFRF. It also can demonstrate that the constructed Kriging model has the capability to replace the FEA to participate in the model updating process, and the MIGA is an effective algorithm to solve the optimization problem. As can be seen in Fig. 10, the deviation between FEM and experiment in the higher frequency range 500–800 Hz is larger than in lower range, this may be influenced by the mass of the accelerometer [37] which may not be given ideally equivalent mass. And the actual location of the accelerometer is on the surface of the beam which destructs the symmetry of the structure slightly, but the simplified mass point associating with the shell element is in the middle of FEM which may cause the difference of AFRF data. Besides, the measurement error of the accelerometer itself may become larger when measuring data in the high frequency range. Consequently, to mitigate the mass influence of the accelerometer in model updating, using a lighter accelerometer is an appropriate way. 5.6. Model updating results based on RSM for comparison In order to further demonstrate the efficiency of Kriging model, the widely used RSM is applied into the model updating procedure instead of the Kriging model for comparison. But the RSM of the complete 2-order polynomial of 15 factors needs 136 sampling points at least, 20 additional sampling points are generated and added into the existing 120 sampling points. Then the RSM is constructed based on the 140 sampling points. The MIGA is used to solve the optimization problem, and the updated values of the 15 design variables are obtained after 1000 iterations. Fig. 11 plots the comparison of the updated AFRF between the RSM and Kriging model. Though the amplitude of the resonance peak can be improved by adjusting the damping ratios, the important values of the resonance frequency still have a large deviation from the experimental value. And the error nearby the 400 Hz is not acceptable after several re-optimization. Consequently, the Kriging model is a more effective metamodel in this case. 5.7. Model updating results based on different numbers of spring variables In order to investigate the influence of different numbers of spring variables on model updating results, the parameter Ktx is added into the design variables for model updating. In other words, the threshold in Fig. 6 is moved from 0.1 to 0.2. So the second step DOE will have 16 factors including 2 spring variables, and 136 sampling points are generated. The model updating result can be
Fig. 10. Comparison of model updating results for updated damping ratios.
Fig. 11. Comparison of model updating results between different metamodel.
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Fig. 12. Comparison of model updating results among different numbers of spring variables.
achieved after optimization based on the constructed Kriging model. In the same way for another model updating that includes 3 spring variables. The threshold continues to widen that Kx is added into the design variables, and then the DOE has 17 factors including the 3 spring variables. The Kriging model can be constructed based on the generated 153 sampling points of DOE results. Similarly, the model updating results is achieved after optimization. The two new model updating results are achieved without adjusting their initial damping ratios (3%), Fig. 12 plots the AFRF overlay among different numbers of spring variables, and the curve of ‘1 spring variable’ corresponds to the ‘updated’ curve in Fig. 10. As can be seen, they have almost the same frequency in the first peak, and have a close trend at the second peak. On the other hand, the peak of the three curves can be modified to coincide with the experimental peak via adjusting their damping ratios respectively. Consequently, the two added variables have a slight influence on the model updating results. 6. Conclusions Metamodel techniques can simplify the FEA to a surrogate model as a fast running model which can facilitate the application of the intelligent algorithms in model updating. The Kriging model is introduced to replace the computational expensive FEA during the optimization process of model updating. This method is applied to the model updating of a composite sandwich beam. In addition to the material properties and geometry parameters, the boundary condition is also taken into account via an equivalent spring system. A two-step DOE method is proposed to the selection of design variables for reducing the number of training samples. The Kriging model is constructed based on the training samples coming from the results of the second step DOE, whose factors include: selected spring stiffness in the first step DOE, material properties and geometry parameters. Based on the experimental AFRF data, the objective function is calculated at the selected frequency ranges, which exclude the vicinity of resonance peaks. The MIGA is applied to solve the optimization problem based on the constructed Kriging model. A significant improvement is achieved after model updating, especially when the damping ratios are adjusted. So the proposed method is working successfully for the model updating of the honeycomb sandwich beam. In addition, a model updating result comparison between different metamodels demonstrates the Kriging model is a more effective metamodel. The influence of different boundary condition on the model updating result is also investigated. Though another two variables are successively added into the design variables for the second step DOE, there is no evident improvement of the model updating result. Therefore, the effectiveness of variable selection for the boundary condition is proved. The next step of this research is to study the dynamic model updating of honeycomb structure with carbon fiber face sheet. References [1] M. Imregun, W.J. Visser, D.J. Ewins, Finite element model updating using frequency response function data: I. Theory and initial investigation, Mecha. Syst. Signal Process. 9 (2) (1995) 187–202. http://dx.doi.org/10.1006/mssp.1995.0015. [2] F.M. Hemez, G.W. Brown, Improving structural dynamics models by simulated to measured frequency response functions, AIAA-98-1789, 39thAIAA/ASME/ ASCE/AHS/ASC structural, Struct. Dyn. Mater. Conf. (1998) 772–782. http://dx.doi.org/10.2514/6.1998-1789. [3] R.M. Lin, J. Zhu, Model updating of damped structures using FRF data, Mecha. Syst. Signal Process. 20 (8) (2006) 2200–2218. http://dx.doi.org/10.1016/ j.ymssp.2006.05.008. [4] M.W. Li, J.Z. Hong, Research on model updating method based on frequency response functions, J. Shanghai Jiaotong Univ. 45 (10) (2011) 1455–1459 (in Chinese) 〈http://en.cnki.com.cn/Article_en/CJFDTotal-SHJT201110008.html〉. [5] A.J. García-Palencia, E. Santini-Bel, Structural model updating using dynamic data, J. Civ. Struct. Health Monit. 5 (2) (2014) 115–127. http://dx.doi.org/ 10.1007/s13349-014-0073-8. [6] V. Arora, S.P. Singh, T.K. Kundra, On the use of damped updated FE model for dynamic design, Mecha. Syst. Signal Process. 23 (3) (2009) 580–587. http:// dx.doi.org/10.1016/j.ymssp.2008.07.010. [7] P. Moyo, R. Tait, Structural performance assessment and fatigue analysis of a railway bridge, Struct. Infrastruct. Eng. 6 (5) (2010) 647–660. http://dx.doi.org/ 10.1080/15732470903068912. [8] R.M. Lin, Identification of modal parameters of unmeasured modes using multiple FRF modal analysis method, Mecha. Syst. Signal Process. 25 (1) (2011) 151–162. http://dx.doi.org/10.1016/j.ymssp.2010.03.002. [9] F. Shadan, F. Khoshnoudian, A. Esfandiari, A frequency response-based structural damage identification using model updating method, Struct. Control Health Monit. (2015). http://dx.doi.org/10.1002/stc.1768.
227
Mechanical Systems and Signal Processing 87 (2017) 218–228
J.T. Wang et al.
[10] S. Pradhan, S.V. Modak, Normal response function method for mass and stiffness matrix updating using complex FRFs, Mech. Syst. Signal Process. 32 (2012) 232–250. http://dx.doi.org/10.1016/j.ymssp.2012.04.019. [11] S. Chakraborty, R. Chowdhury, Polynomial correlated function expansion for nonlinear stochastic dynamic analysis, J. Eng. Mech. 141 (3) (2014) 04014132. http://dx.doi.org/10.1061/(ASCE)EM.1943-7889.0000855. [12] G. Blatman, B. Sudret, Adaptive sparse polynomial chaos expansion based on least angle regression, J. Comput. Phys. 230 (6) (2011) 2345–2367. http:// dx.doi.org/10.1016/j.jcp.2010.12.021. [13] D.L. Allaix, V.I. Carbone, An improvement of the response surface method, Struct. Saf. 33 (2) (2011) 165–172. http://dx.doi.org/10.1016/j.strusafe.2011.02.001. [14] S. Chakraborty, A. Sen, Adaptive response surface based efficient finite element model updating, Finite Elem. Anal. Des. 80 (3) (2014) 33–40. http://dx.doi.org/ 10.1016/j.finel.2013.11.002. [15] H.P. Wan, W.X. Ren, A residual-based Gaussian process model framework for finite element model updating, Comput. Struct. 156 (2015) 149–159. http:// dx.doi.org/10.1016/j.compstruc.2015.05.003. [16] S.E. Fang, R. Perera, Damage identification by response surface based model updating using D-optimal design, Mech. Syst. Signal Process. 25 (2) (2011) 717–733. http://dx.doi.org/10.1016/j.ymssp.2010.07.007. [17] P. Dey, S. Talukdar, D.J. Bordoloi, Multiple-crack identification in a channel section steel beam using a combined response surface methodology and genetic algorithm, Struct. Control Health Monit. (2015). http://dx.doi.org/10.1002/stc.1818. [18] A.G.Anthony, T.W.Layne, A Comparison of Approximation Modeling Techniques: Polynomial Versus Interpolating Models, Aiaa/usaf/nasa/issmo Symposium on Multidisciplinary Analysis & Optimization, 1999. http://dx.doi.org/10.2514/6.1998-4758. [19] L.R. Zhou, G.R. Yan, J.P. Ou, Response surface method based on radial basis functions for modeling large-scale structures in model updating, Comput.-Aided Civ. Infrastruct. Eng. 28 (3) (2013) 210–226. http://dx.doi.org/10.1111/j.1467-8667.2012.00803.x. [20] T.W. Simpson, J.D. Poplinski, P.N. Koch, et al., Metamodels for computer-based engineering design: survey and recommendations, Eng. Comput. 17 (2) (2001) 129–150. http://dx.doi.org/10.1007/PL00007198. [21] H.H. Khodaparast, J.E. Mottershead, K.J. Badcock, Interval model updating with irreducible uncertainty using the Kriging predictor, Mech. Syst. Signal Process. 25 (4) (2011) 1204–1226. http://dx.doi.org/10.1016/j.ymssp.2010.10.009. [22] Y. Liu, Y. Li, D. Wang, S. Zhang, Model updating of complex structures using the combination of component mode synthesis and Kriging predictor, Sci. World J. (2014). http://dx.doi.org/10.1155/2014/476219 13 pp., (Article ID 476219). [23] I. Behmanesh, B. Moaveni, G. Lombaert, et al., Hierarchical Bayesian model updating for structural identification, Mech. Syst. Signal Process. 64 (2015) 360–376. http://dx.doi.org/10.1016/j.ymssp.2015.03.026. [24] Y. Govers, H.H. Khodaparast, M. Link, et al., A comparison of two stochastic model updating methods using the DLR AIRMOD test structure, Mech. Syst. Signal Process. 52 (2015) 105–114. http://dx.doi.org/10.1016/j.ymssp.2014.06.003. [25] H.G. Pasha, K. Kohli, R.J. Allemang, et al., Structural Dynamics Model Calibration and Validation of a Rectangular Steel Plate Structure[M]//Model Validation and Uncertainty Quantification 3, Springer International Publishing, 2015, pp. 351–362. http://dx.doi.org/10.1007/978-3-319-15224-0_37. [26] T.W. Simpson, M. Mauery, J.J. Korte, F. Mistree, Kriging models for global approximation in simulation- based multidisciplinary design optimization, AIAA J. 39 (12) (2001) 2233–2241. http://dx.doi.org/10.2514/3.15017. [27] S.E. Gano, J.E. Renaud, J.D. Martin, T.W. Simpson, Update strategies for Kriging models for use in variable fidelity optimization, Struct. Multidiscip. Optim. 32 (4) (2005) 287–298. http://dx.doi.org/10.1007/s00158-006-0025-y. [28] J.D.Martin, T.W.Simpson, On the use of Kriging models to approximate deterministic computer models, in: Proceedings of the ASME 2004 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, American Society of Mechanical Engineers, 2004 pp. 481–492. http://dx.doi.org/10.2514/1.8650. [29] E.R. Fotsing, M. Sola, A. Ross, Edu Ruiz, Lightweight damping of composite sandwich beams: experimental analysis, J. Compos. Mater. 47 (12) (2013) 1501–1511. http://dx.doi.org/10.1177/0021998312449027. [30] K.S. Kwon, R.M. Lin, Frequency selection method for FRF-based model updating, J. Sound Vib. 278 (1) (2004) 285–306. http://dx.doi.org/10.1016/ j.jsv.2003.10.003. [31] W.X. Ren, H.B. Chen, Finite element model updating in structural dynamics by using the response surface method, Eng. Struct. 32 (8) (2010) 2455–2465. http://dx.doi.org/10.1016/j.engstruct.2010.04.019. [32] A. Boudjemai, R. Amri, A. Mankour, et al., Modal analysis and testing of hexagonal honeycomb plates used for satellite structural design, Mater. Des. 35 (2012) 266–275. http://dx.doi.org/10.1016/j.matdes.2011.09.012. [33] X.Q. Zhao, G. Wang, D.L. Yu, Experiment verification of equivalent model for vibration analysis of honeycomb sandwich, Struct., Appl. Mech. Mater. 624 (2014) 280–284. http://dx.doi.org/10.4028/www.scientific.net/AMM.624.280. [34] M. Miki, T. Hiroyasu, M. Kaneko, K. Hatanaka, A parallel genetic algorithm with distributed environment scheme, IEEE Int. Conf. Syst., Man, Cybern. 1 (1999) 695–700. http://dx.doi.org/10.1109/ICSMC.1999.814176. [35] A.J. García-Palencia, E. Santini-Bel, A. Two-Step, Model Updating algorithm for parameter identification of linear elastic damped structures, Comput.-Aided Civ. Infrastruct. Eng. 28 (7) (2013) 509–521. http://dx.doi.org/10.1111/mice.12012. [36] J.D. Sipple, M. Sanayei, Finite element model updating of the UCF grid benchmark using measured frequency response functions, Mech. Syst. Signal Process. 46 (1) (2014) 179–190. http://dx.doi.org/10.1016/j.ymssp.2014.01.008. [37] J.D. Sipple, M. Sanayei, Finite element model updating using frequency response functions and numerical sensitivities, Struct. Control Health Monit. 21 (5) (2014) 784–802. http://dx.doi.org/10.1002/stc.1601.
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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Frequency response function-based model updating using Kriging model
cross
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J.T. Wanga, , C.J. Wanga,b, J.P. Zhaoa a b
School of Mechanical Engineering and Automation, Beihang University, Beijing, China State Key Laboratory of Virtual Reality Technology and Systems, Beihang University, Beijing, China
A R T I C L E I N F O
ABSTRACT
Keywords: Model updating Metamodel Frequency response function Optimization
An acceleration frequency response function (FRF) based model updating method is presented in this paper, which introduces Kriging model as metamodel into the optimization process instead of iterating the finite element analysis directly. The Kriging model is taken as a fast running model that can reduce solving time and facilitate the application of intelligent algorithms in model updating. The training samples for Kriging model are generated by the design of experiment (DOE), whose response corresponds to the difference between experimental acceleration FRFs and its counterpart of finite element model (FEM) at selected frequency points. The boundary condition is taken into account, and a two-step DOE method is proposed for reducing the number of training samples. The first step is to select the design variables from the boundary condition, and the selected variables will be passed to the second step for generating the training samples. The optimization results of the design variables are taken as the updated values of the design variables to calibrate the FEM, and then the analytical FRFs tend to coincide with the experimental FRFs. The proposed method is performed successfully on a composite structure of honeycomb sandwich beam, after model updating, the analytical acceleration FRFs have a significant improvement to match the experimental data especially when the damping ratios are adjusted.
1. Introduction Finite element model updating method can be employed as an important tool to calibrate the finite element model (FEM) for mitigating modeling error. It can make the updated FEM have the same behavior to the corresponding real structure as much as possible. There are two main dynamic model updating method: modal parameters based method and frequency response function (FRF) based method. The former has been thoroughly studied, but the FRF based method [1] introduced by Hemez and Brown [2] and other researchers [3,4] has received considerable attention and applications [5–9] in recent years due to its advantage: Firstly, the measured FRF data can be utilized directly without data transformation. In some special software, the calculation of modal parameters is based on the measured FRF data. And the modal analysis is more complex, the error might arise from the modal identification. The identification error might greater than the modeling error. Secondly, the FRF can be measured in more locations of the structure and taken as an objective so it can provide more data. Model updating process is intrinsically an inverse problem [10] and usually formulated as optimization problem, which aims to minimize the differences between the FEM behavior and the corresponding experimental behavior. However, the conventional
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Corresponding author. E-mail address: [email protected] (J.T. Wang).
http://dx.doi.org/10.1016/j.ymssp.2016.10.023 Received 26 April 2016; Received in revised form 23 August 2016; Accepted 22 October 2016 Available online 03 November 2016 0888-3270/ © 2016 Elsevier Ltd. All rights reserved.
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sensitivity based optimization algorithms have disadvantages, such as: (1) the sensitivity analysis may be computation expensive, and the sensitivity results might not be obtained easily; (2) huge computational cost may be required to meet the convergence point for optimization algorithm. Although the modern intelligent algorithms (Genetic Algorithm, Particle Swarm Optimization and Simulated Annealing etc.) can avoid calculating the sensitivity. But to the structural dynamic analysis of FEM, its large number function calls of finite element analysis (FEA) is time consuming, this even cause the intelligent algorithms is impractical applied to the complex structure. Fortunately, the barrier between the practical applications and the intelligent algorithm can be overcome via the metamodel technique which is known as approximate model or surrogate model. This technique considers the relationship between the input and output as a black-box system, and other information of the system such as internal process of dynamic analysis are not required. It can create a fast running surrogate model to replace the exact FEA, and then the solving time of optimization will be reduced significantly. In addition, in-house FEA codes and existing commercial software can be integrated directly, and it is suitable for parallel computing. Thereby the potential of metamodel techniques is indisputable in model updating field. There are several types of construction method of metamodel are commonly used: response surface methodology (RSM), radial basis function (RBF), Kriging method, polynomial correlated function expansion [11], Polynomial Chaos expansion [12] and Support Vector Regression. Most of them are constructed based on training samples which include input and output information of the interested system. The metamodel technique has been applied in fields such as: structural reliability analysis [13], structural static model updating [14], structural dynamic model updating based on modal parameters [15], and structural damage identification [16,17]. The comparison [18,19] and recommendation [20] of the main metamodels also have been studied. As one of the main method, Kriging model is constructed based on the correlation function theory. Particularly, it is an exact interpolation of given data and goes through all the sampling points. So the Kriging model usually has a higher approximation accuracy than traditional RSM. Khodaparast et al. [21] solved the problem interval model updating by using the Kriging method, and the good accuracy of Kriging method was illustrated by beam experiment. Liu et al. [22] calibrated the FEM based on the modal parameters of a complex structure, the Kriging model was taken as a surrogate model. But there is few application of Kriging model in FRF based model updating in nowadays. In this article, the Kriging model is applied to the model updating based on FRF data. The training samples for the Kriging model are obtained via the design of experiment (DOE), whose factors and response are corresponding to design variables and FRF based data respectively. The case study is followed by an impact hammer test of cantilever beam of honeycomb sandwich structure. The acceleration FRF (AFRF) data are measured and used for DOE and optimization. A two-step DOE method is proposed to reduce the design variables and training samples. The model updating result shows the updated damped FEM has significant improvement to match the experimental AFRF data, and demonstrate the effectiveness of the proposed approach. Moreover, the proposed method can be extended to stochastic model updating [23–25] where usually require a huge number function calls of FEA.
2. AFRF based model updating and objective function Model updating method aims to reduce the modeling error of FEM making the FEM agree better with the real experimental situation. In AFRF based model updating, both FEM data and experimental data corresponding to AFRF amplitude curves, are simulated and measured at the interested degree of freedoms (DOFs) of the structure, respectively. Because of the modeling error, the mass matrix, damping matrix and stiffness matrix of FEM have deviation from the experimental situation, therefore, the two analogous amplitude curves cannot overlap. Then minimizing the difference between the coupled curves is considered as an optimization problem. In which, the design variables appointed by the user are taken as input variables. Optimal results (updated input variables) are obtained via minimizing the objective function. The results can in turn change the matrixes for FEA, and finally lead the AFRF curve of FEM to coincide with the experimental AFRF curve as much as possible. The optimization problem can be formulated in the following form:
Minimize F, s. t .
xiL ≤ xi ≤ xiU ,
i = 1, 2, …, n
(1)
where F is the objective function, xi is the design variables of the structure, xiL and xiU are the lower and upper bound of the input variables, respectively. In this study, the objective function is established based on the differences between the corresponding AFRF curves at each selected frequency point, it can be formulated as follows: np
F=
nf
⎛ Atn (ωi ) − Aan (ωi ) ⎞2 ⎟ Atn (ωi ) ⎝ ⎠
∑ ∑ wi⎜ n =1 i =1
(2)
where wi is the weights, np is the number of measured DOFs, nf is the number of selected frequency points of AFRF, ωi is the selected frequency points, Atn (ωi ) is the experimental acceleration amplitude at ωi, Aan (ωi ) is the corresponding amplitude of the FEM. As mentioned in the previous section, during the optimization process, the FEA process can be replaced by Kriging model, which does not require the internal information of the matrix operation in FEA. The construction method of Kriging model will be introduced in the following section. 219
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3. Construction of Kriging model The Kriging method is a linear unbiased estimation which consists of linear part and nonparametric part. And the latter one can be considered as the realization of a stochastic process [26]. The Kriging model can be described as the form:
y(x) = F (β, x) + z(x) = f T (x)β + z(x)
(3)
where z(x) is the realization of the stochastic process, f(x) is a polynomial vector of training sample x, and β is the coefficient of the linear regression. The f(x) term approximates the drift of the Kriging model, and z(x) approximates the local deviation of the Kriging model. z(x) follows normal distribution N(0, σz2 ), and has nonzero covariance which estimated by:
Cov[z(xi ), z(xj )] = σz2 R
(4)
where xi and xj are two training samples, σz is the variance of z(x), R is a symmetry matrix that composed by Rij(xi, xj). Rij(xi, xj) is the correlation function specified by the user which can characterize the correlation between any two training points. Different Rij(xi, xj) may induce different approximation accuracy of the constructed Kriging model. The function Rij(xi, xj) is given by: Nv
Rij (xi , xj ) =
∏ Rk (θk , dk ),
dk = xik − xjk
(5)
k =1
where Nv denotes the number of design variables; θk is the unknown coefficient of correlation; and are the kth components of xi and xj, respectively; dk is the distance between the different values of selected design variable. Because of its smoothness characteristic, the Gaussian correlation function which has widely application [27] and is employed in this study. It has the following form:
xik
Rk (θk , dk ) = exp( − θkdk2 )
xjk
(6)
The detailed estimation of θk and σz can be found in the reference [28], σz is the function of θk. In short, the θk andσz can be evaluated by solving the maximum likelihood estimated problem which has the form as:
⎛ N ln(σ 2 ) + ln R ⎞ z ⎟⎟ max − ⎜⎜ s θk >0 2 ⎝ ⎠
(7)
where Ns is the number of training samples, |R| is the determinant of R which is a function of θk. f(x) in Eq. (3) is enough to describe the trends of the output response, in most cases, f(x) can be simply taken to be constant vector [26]. Thus any output response y(x) can be dealt with stochastic process with normal distribution during the construction of Kriging model. When the Kriging model is constructed with the obtained θk, it can be used to predict the output response at untried location x0 with unbiased estimation. The predicted response yˆ(x 0 ) is given by:
yˆ(x 0) = β* + r T (x 0)R−1(Y − fβ*),
β* = (f T R−1f )−1f T R−1Y
(8)
where Y is a column vector composed by the output of training samples; f is a column vector filled with ones when f(x) is constant vector; r(x0) is a relevant vector composed between training samples and predict point x0.
r(x 0 ) = [R01(x 0 , x1), R02(x 0 , x 2 ), …, R0Ns(x 0 , xNs )]T
(9)
Briefly, the Kriging model will be determined once the values of θk are given. In this study, the training samples come from DOE results of FEM. The input variables of Kriging model are directly taken as the input variables of the formulated optimization problem, and the input variables come from the appointed design variables of FEM. Meanwhile, the optimization objective corresponds to the output response of the DOE. 4. Model updating procedure In order to take advantage the fast running characteristic of metamodel, the Kriging model is applied to the AFRF data based model updating. During the optimization process, the constructed Kriging model is taken as a surrogate of the dynamic analysis of FEM. Model updating result will be achieved based on the optimization results. The training samples are obtained by DOE, whose factors are selected by users based on design variables of FEM. The design variables are considered have modeling error. The Kriging model and DOE are set to have the same output response. The design matrix of DOE is generated by the method of Optimal Latin Hypercube, which is a more evenly sampling method than conventional Latin Hypercube method. DOE results are taken as training samples for solving the basic parameters θk and then constructing Kriging model. Nevertheless, too many factors (design variables) need more DOE samplings which are also a time consuming process. In particular, a considerable number of parameters may involve into factors when the boundary conditions are taken into account. Therefore, a two-step DOE procedure is proposed to reduce the training samples for saving time. The first step is to select the 220
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sensitive variables from the parameters of boundary conditions based on the DOE results, and the selected variables will be taken as a part of factors in the second step. The second step will generate training samples for constructing Kriging model based on all selected variables. The accuracy of the constructed Kriging model will be checked before the optimization. The optimization results correspond to the updated values of the design variables. And then the updated values are taken to calibrate the FEM as model updating result until the updated error is acceptable. The proposed procedure for model updating can be summarized in the flow chart shown in Fig. 1.
5. Case study: experimental honeycomb beam Because of the characteristic of lightweight, high intensity and low density, honeycomb sandwich structure has been widely applied in many industries, especially in the aerospace industry. The structure consists of 5 parts: one thick lightweight honeycomb core, two very thin face sheets, and two thin adhesive layers. Fig. 2 shows its components. In this section, to demonstrate the effectiveness of the proposed method, a cantilever beam test of honeycomb sandwich structure is performed. It has aluminum face sheets (0.5 mm respectively) and aluminum honeycomb core (20 mm). The test structure has the dimension of 500×40×21.6 mm shown in Fig. 3, the adhesive layers have a nominal thickness of 0.3 mm. Fig. 0.3 also shows the dimension of honeycomb cell of honeycomb core. The embedded part is utilized for clamping which is aluminum. The measurement set-up of FRF test is shown in Fig. 4, the impact point is located at 70 mm from the edge of clamping zone. The impact location is close to the clamping zone and can obtain low noise contaminated experimental data [29]. The accelerometer is the type of Integrate Circuit Piezoelectric.
Fig. 1. Flow chart of the proposed model updating procedure.
Fig. 2. Honeycomb sandwich structure.
Fig. 3. Honeycomb sandwich beam test structure.
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Fig. 4. Measurement set-up of FRF test.
Fig. 5. The FEM of honeycomb sandwich beam.
Table 1 Thickness and angles of each layer of the composite beam. Layer
Component
Thickness (mm)
Angle
1 2 3 4 5
Aluminum face Adhesive Honeycomb core Adhesive Aluminum face
0.5 0.3 20 0.3 0.5
0° 0° 0° 0° 0°
5.1. Description of the FEM The FEM is modeled by 188 shell elements of laminated composite in the commercial software Patran. Fig. 5 shows the FEM without showing the thickness. The details of thickness and angles of each layer are given in Table 1. The embedded part of the structure is used for clamping the beam, so the exact cantilever length in the FEM is 470 mm excluding the embedded part. Moreover, the mass of accelerometer is taken into account for the dynamic response. The accelerometer is simplified as a mass point with 10 g of its real mass. As can be seen in Fig. 5, the boundary condition (point A) of the clumped embedded part is equivalent to a 6 springs system: 1 vertical spring (Kz), 2 horizontal springs (Kx and Ky), and 3 directional torsional springs (Ktx, Kty and Ktz). Each of the spring stiffness is given as 106 N/mm. Fig. 5 also shows the load case: point B is the excitation DOF, and point C is the observation DOF of AFRF and associates the mass point. Under 3% modal damping, the structure is excited by a 1N harmonic force (in Z direction) at DOF B. The analytical AFRF data in Z direction can be obtained after FEA which corresponds to the AFRF of the impact hammer test. 5.2. First step DOE for boundary condition In order to select the parameters from the boundary conditions for updating, the insensitivity parameter will be excluded from the design variables. The sensitivity is assessed based on the DOE results. The 6 spring stiffness parameters are taken as the factors of the first step DOE, and the output response of the DOE corresponds to the value of objective function Eq. (2), whose weights wi=1. Particularly, the AFRF amplitudes close to the resonances are sensitive to noise and damping, so the output response should be evaluated in the ranges excluding the vicinity of the resonances [30]. Thereby the frequency points for the DOE response are come from the ranges at every 1.25 Hz step: 10–60 Hz, 90–390 Hz, 460–800 Hz. And then 21 sampling points are generated. F-test method [31] is applied to evaluate the factor significance to the response based on the sampling points, the quantification of the significance can be formulated as: 222
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P{FA ≥ F1− p(fA , fE )} = p ,
FA =
SA / fA SE / fE
∼ F (fA , fE )
(10)
where SA is the square of deviation caused by the DOE factors; SE is the square of experimental deviation; fA and fE are the DOFs of SA and SE, respectively. When the p-value is lower than the threshold set by user, the corresponding factor is considered sensitive to the response and should be selected as the design variable for updating. The p-value of all the 6 factors are shown in Fig. 6, the threshold is set by 0.1, and then the Kty is selected as a factor of the second step DOE. 5.3. Design variables choice and the second step DOE Honeycomb sandwich structure usually has a honeycomb core with large numbers of honeycomb cells. An appropriate method is the honeycomb core equivalent to orthotropic material, three kinds of equivalent theories [32,33] are commonly used: sandwich plate theory, honeycomb plate theory and equivalent plate theory. The sandwich plate theory is a classic method which is adopted in this study. The equivalent properties are given in Table 2 which also lists the properties of the other two main material of the FEM. The design variables for the model updating should be the FEM parameters which prone to existing modeling error. Firstly, the material properties of the honeycomb core should be taken as design variables since it is an equivalent component. Secondly, the
Fig. 6. The p-value of the 6 spring stiffness.
Table 2 Main material Properties of the FEM. Material property Elastic modulus Shear modulus
Poisson ratio Density (kg/m3)
Ex(Pa) Ey(Pa) Gxy(Pa) Gxz(Pa) Gyz(Pa)
Equivalent core
Adhesive
Aluminum face
1.6×105 1.6×105 6×104 2.3×108 1.6×108
7×109
7.1×1010
0.35 1500
0.3 2700
31
Fig. 7. Boundaries between honeycomb cell wall and face sheet.
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adhesive is smeared on the face sheet to combine the honeycomb core, so the elastic modulus and Poisson ratio of the aluminum maybe affected and then taken to be the design variables. Thirdly, the material properties of the adhesive are taken to be design variables, since its status cannot be measured easily. Finally, the boundaries are fuzzy between the cell wall and the adhesive which are illustrate in Fig. 7, therefore the thicknesses of both honeycomb core and adhesive are added to the design variables. Consequently, the 14 design variables and the previously selected stiffness Kty constitute all the design variables, which are summarized in the second column of Table 3. Then the selected 15 design variables are taken as the factors of the second step DOE. The lower bound and upper bound of every factor are listed in Table 3. The design matrix of the DOE is generated based on the bounds by using Optimal Latin Hypercube method. Based on the same frequency points as the first step DOE, 120 sampling points are obtained after the DOE. The response is also evaluated by Eq. (2). The number of sampling points is given by the following equation:
Ns =
Nv(Nv + 1) 2
(11)
5.4. Construction and accuracy assessment of Kriging model The 120 sampling points generated by the second step DOE are taken as the training samples for constructing Kriging model, whose input variables adopt the 15 factors of the DOE. The output responses correspond to the DOE response. Then the parameters of the Kriging model can be obtained based on the training samples. The basic parameters θk in Eq. (7) are found by using Genetic Algorithm: θ1=0.15850, θ2=0.024570, θ3=0.059123, θ4=0.016713, θ5=0.0094339, θ6=0.17948, θ7=0.053560, θ8=0.25231,
Table 3 Selected design variables of FEM. Material name Spring stiffness Honeycomb core
Aluminum face Adhesive
Design variable (properties) Kty(N/mm) Ex(Pa) Ey(Pa) Gxy(Pa) Gyz(Pa) Gxz(Pa) DH (kg/m3) TH(mm) EAL PAL(Poisson ratio) EA(Pa) PA(Poisson ratio) DA (kg/m3) TUP(mm) TDOWN(mm)
Initial value 6
1×10 1.6×105 1.6×105 6×104 2.3×108 1.6×108 31 20 7.1×1010 0.3 7×109 0.35 1500 0.3 0.3
Lower bound 5
1×10 −50% −50% −50% −50% −50% −50% 12 −50% 0.2 −50% 0.2 −50% 0.2 0.2
Upper bound 1×10 100% 100% 100% 100% 100% 100% 20 100% 0.4 100% 0.48 100% 3 3
Fig. 8. The response values of checking points.
224
8
Updated value 7
6.558×10 1.236×105 2.089×105 7.804×104 2.131×108 2.674×108 50.53 16.034 9.657×1010 0.275 9.119×109 0.222 2462.6 2.272 1.107
Percent change +6558% −22.8% +30.6% +30.7% −7.35% +67.1% +63.0% −19.8% +36.0% −8.33% +30.3% −36.6% +64.2% +354% +121%
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Fig. 9. Overlay of initial AFRF, experimental AFRF and updated AFRF.
Table 4 Updated results of damping ratios. Damping ratio
0–300 Hz
300–500 Hz
500–800 Hz
800 Hz and beyond
Initial value Updated value
3% 2.8%
3% 3.0%
3% 4.2%
3% 19%
θ9=0.43850, θ10=0.025487, θ11=0.056517, θ12=0.010368, θ13=1.0688, θ14=0.025359, θ15=0.0099887. The θk values correspond to the design variables listed in Table 3 from top to bottom. The Kriging model is achieved based on these 15 basic parameters, and then the accuracy of the constructed Kriging metamodel will be assessed. A classical accuracy measurement is to assess the capability of the metamodel to reproduce the training samples, but Kriging model is an interpolating model which goes through all the training samples. So the root mean square error (RMSE) is an appropriate method to evaluate the accuracy of the Kriging model. The RMSE can reflect the response error between the Kriging model and FEM at each checking point. The RMSE is defined as follows:
RMSE =
1 ky
k
∑ (yi − yˆ)i 2
(12)
i =1
where k is the number of checking points, yi is the FEM response at checking points, y is the mean value of yi, yˆi is the Kriging response at the corresponding checking points. 12 checking points are generated and used to assess the accuracy. The 12 points are different from the training samples. Fig. 8 shows the response values at each checking point between the Kriging model and the FEM. The RMSE is evaluated as about 3.70% which quantify the error between the Kriging model and the FEM, and the error is acceptable. 5.5. Model updating results Model updating result is achieved based on the optimization results. During the optimization process, the input variables scope is taken as the same of the DOE factor bounds. The Multi-Island Genetic Algorithm (MIGA) [34] is used to solve the minimization of the optimization problem. The optimal results of the input variables shown in Table 3 are obtained after 1000 iterations, the percent changes of the input variables are also given in Table 3. The variable Kty has the biggest changes followed by the variable TUP and TDOWN. The initial FEM is updated based on the updated values in Table 3. For comparison, Fig. 9 plots the overlay of the three AFRF curves at the interested DOF C in Z direction. The three curves are updated AFRF, experimental AFRF and initial FEM AFRF, respectively. Because of the modeling error, a huge error exists between the initial FEM AFRF and the experimental AFRF. After model updating, a significant improvement is achieved that the updated AFRF tends to coincide with the experimental AFRF. Fig. 9 also proves the choice of the design variables is efficient. But the resonance peaks have a residual error between the updated AFRF and the experimental AFRF, the damping ratio is crucial to the resonance peak and can be modified independently. García-Palencia AJ et al. [35] proposed a two-step methodology for updating of stiffness, mass, and damping matrices based on the FRF, the damping matrices is updated in the second step. So in order to update the resonance peak for further model updating, the damping ratio of FEM can be modified via a manual adjustment, since it only has four independent damping ratios. The damping ratios only affect the vicinity of the resonance peak respectively and can easily be adjusted to match the resonance peaks [36]. The modified results of damping ratios are listed in Table 4 after manually adjusting. Fig. 10 shows the AFRF overlay before and after adjusting the damping ratios. As can be seen, the new updated AFRF has 225
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a better match with the experimental AFRF. It also can demonstrate that the constructed Kriging model has the capability to replace the FEA to participate in the model updating process, and the MIGA is an effective algorithm to solve the optimization problem. As can be seen in Fig. 10, the deviation between FEM and experiment in the higher frequency range 500–800 Hz is larger than in lower range, this may be influenced by the mass of the accelerometer [37] which may not be given ideally equivalent mass. And the actual location of the accelerometer is on the surface of the beam which destructs the symmetry of the structure slightly, but the simplified mass point associating with the shell element is in the middle of FEM which may cause the difference of AFRF data. Besides, the measurement error of the accelerometer itself may become larger when measuring data in the high frequency range. Consequently, to mitigate the mass influence of the accelerometer in model updating, using a lighter accelerometer is an appropriate way. 5.6. Model updating results based on RSM for comparison In order to further demonstrate the efficiency of Kriging model, the widely used RSM is applied into the model updating procedure instead of the Kriging model for comparison. But the RSM of the complete 2-order polynomial of 15 factors needs 136 sampling points at least, 20 additional sampling points are generated and added into the existing 120 sampling points. Then the RSM is constructed based on the 140 sampling points. The MIGA is used to solve the optimization problem, and the updated values of the 15 design variables are obtained after 1000 iterations. Fig. 11 plots the comparison of the updated AFRF between the RSM and Kriging model. Though the amplitude of the resonance peak can be improved by adjusting the damping ratios, the important values of the resonance frequency still have a large deviation from the experimental value. And the error nearby the 400 Hz is not acceptable after several re-optimization. Consequently, the Kriging model is a more effective metamodel in this case. 5.7. Model updating results based on different numbers of spring variables In order to investigate the influence of different numbers of spring variables on model updating results, the parameter Ktx is added into the design variables for model updating. In other words, the threshold in Fig. 6 is moved from 0.1 to 0.2. So the second step DOE will have 16 factors including 2 spring variables, and 136 sampling points are generated. The model updating result can be
Fig. 10. Comparison of model updating results for updated damping ratios.
Fig. 11. Comparison of model updating results between different metamodel.
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Fig. 12. Comparison of model updating results among different numbers of spring variables.
achieved after optimization based on the constructed Kriging model. In the same way for another model updating that includes 3 spring variables. The threshold continues to widen that Kx is added into the design variables, and then the DOE has 17 factors including the 3 spring variables. The Kriging model can be constructed based on the generated 153 sampling points of DOE results. Similarly, the model updating results is achieved after optimization. The two new model updating results are achieved without adjusting their initial damping ratios (3%), Fig. 12 plots the AFRF overlay among different numbers of spring variables, and the curve of ‘1 spring variable’ corresponds to the ‘updated’ curve in Fig. 10. As can be seen, they have almost the same frequency in the first peak, and have a close trend at the second peak. On the other hand, the peak of the three curves can be modified to coincide with the experimental peak via adjusting their damping ratios respectively. Consequently, the two added variables have a slight influence on the model updating results. 6. Conclusions Metamodel techniques can simplify the FEA to a surrogate model as a fast running model which can facilitate the application of the intelligent algorithms in model updating. The Kriging model is introduced to replace the computational expensive FEA during the optimization process of model updating. This method is applied to the model updating of a composite sandwich beam. In addition to the material properties and geometry parameters, the boundary condition is also taken into account via an equivalent spring system. A two-step DOE method is proposed to the selection of design variables for reducing the number of training samples. The Kriging model is constructed based on the training samples coming from the results of the second step DOE, whose factors include: selected spring stiffness in the first step DOE, material properties and geometry parameters. Based on the experimental AFRF data, the objective function is calculated at the selected frequency ranges, which exclude the vicinity of resonance peaks. The MIGA is applied to solve the optimization problem based on the constructed Kriging model. A significant improvement is achieved after model updating, especially when the damping ratios are adjusted. So the proposed method is working successfully for the model updating of the honeycomb sandwich beam. In addition, a model updating result comparison between different metamodels demonstrates the Kriging model is a more effective metamodel. The influence of different boundary condition on the model updating result is also investigated. Though another two variables are successively added into the design variables for the second step DOE, there is no evident improvement of the model updating result. Therefore, the effectiveness of variable selection for the boundary condition is proved. The next step of this research is to study the dynamic model updating of honeycomb structure with carbon fiber face sheet. References [1] M. Imregun, W.J. Visser, D.J. Ewins, Finite element model updating using frequency response function data: I. Theory and initial investigation, Mecha. Syst. Signal Process. 9 (2) (1995) 187–202. http://dx.doi.org/10.1006/mssp.1995.0015. [2] F.M. Hemez, G.W. Brown, Improving structural dynamics models by simulated to measured frequency response functions, AIAA-98-1789, 39thAIAA/ASME/ ASCE/AHS/ASC structural, Struct. Dyn. Mater. Conf. (1998) 772–782. http://dx.doi.org/10.2514/6.1998-1789. [3] R.M. Lin, J. Zhu, Model updating of damped structures using FRF data, Mecha. Syst. Signal Process. 20 (8) (2006) 2200–2218. http://dx.doi.org/10.1016/ j.ymssp.2006.05.008. [4] M.W. Li, J.Z. Hong, Research on model updating method based on frequency response functions, J. Shanghai Jiaotong Univ. 45 (10) (2011) 1455–1459 (in Chinese) 〈http://en.cnki.com.cn/Article_en/CJFDTotal-SHJT201110008.html〉. [5] A.J. García-Palencia, E. Santini-Bel, Structural model updating using dynamic data, J. Civ. Struct. Health Monit. 5 (2) (2014) 115–127. http://dx.doi.org/ 10.1007/s13349-014-0073-8. [6] V. Arora, S.P. Singh, T.K. Kundra, On the use of damped updated FE model for dynamic design, Mecha. Syst. Signal Process. 23 (3) (2009) 580–587. http:// dx.doi.org/10.1016/j.ymssp.2008.07.010. [7] P. Moyo, R. Tait, Structural performance assessment and fatigue analysis of a railway bridge, Struct. Infrastruct. Eng. 6 (5) (2010) 647–660. http://dx.doi.org/ 10.1080/15732470903068912. [8] R.M. Lin, Identification of modal parameters of unmeasured modes using multiple FRF modal analysis method, Mecha. Syst. Signal Process. 25 (1) (2011) 151–162. http://dx.doi.org/10.1016/j.ymssp.2010.03.002. [9] F. Shadan, F. Khoshnoudian, A. Esfandiari, A frequency response-based structural damage identification using model updating method, Struct. Control Health Monit. (2015). http://dx.doi.org/10.1002/stc.1768.
227
Mechanical Systems and Signal Processing 87 (2017) 218–228
J.T. Wang et al.
[10] S. Pradhan, S.V. Modak, Normal response function method for mass and stiffness matrix updating using complex FRFs, Mech. Syst. Signal Process. 32 (2012) 232–250. http://dx.doi.org/10.1016/j.ymssp.2012.04.019. [11] S. Chakraborty, R. Chowdhury, Polynomial correlated function expansion for nonlinear stochastic dynamic analysis, J. Eng. Mech. 141 (3) (2014) 04014132. http://dx.doi.org/10.1061/(ASCE)EM.1943-7889.0000855. [12] G. Blatman, B. Sudret, Adaptive sparse polynomial chaos expansion based on least angle regression, J. Comput. Phys. 230 (6) (2011) 2345–2367. http:// dx.doi.org/10.1016/j.jcp.2010.12.021. [13] D.L. Allaix, V.I. Carbone, An improvement of the response surface method, Struct. Saf. 33 (2) (2011) 165–172. http://dx.doi.org/10.1016/j.strusafe.2011.02.001. [14] S. Chakraborty, A. Sen, Adaptive response surface based efficient finite element model updating, Finite Elem. Anal. Des. 80 (3) (2014) 33–40. http://dx.doi.org/ 10.1016/j.finel.2013.11.002. [15] H.P. Wan, W.X. Ren, A residual-based Gaussian process model framework for finite element model updating, Comput. Struct. 156 (2015) 149–159. http:// dx.doi.org/10.1016/j.compstruc.2015.05.003. [16] S.E. Fang, R. Perera, Damage identification by response surface based model updating using D-optimal design, Mech. Syst. Signal Process. 25 (2) (2011) 717–733. http://dx.doi.org/10.1016/j.ymssp.2010.07.007. [17] P. Dey, S. Talukdar, D.J. Bordoloi, Multiple-crack identification in a channel section steel beam using a combined response surface methodology and genetic algorithm, Struct. Control Health Monit. (2015). http://dx.doi.org/10.1002/stc.1818. [18] A.G.Anthony, T.W.Layne, A Comparison of Approximation Modeling Techniques: Polynomial Versus Interpolating Models, Aiaa/usaf/nasa/issmo Symposium on Multidisciplinary Analysis & Optimization, 1999. http://dx.doi.org/10.2514/6.1998-4758. [19] L.R. Zhou, G.R. Yan, J.P. Ou, Response surface method based on radial basis functions for modeling large-scale structures in model updating, Comput.-Aided Civ. Infrastruct. Eng. 28 (3) (2013) 210–226. http://dx.doi.org/10.1111/j.1467-8667.2012.00803.x. [20] T.W. Simpson, J.D. Poplinski, P.N. Koch, et al., Metamodels for computer-based engineering design: survey and recommendations, Eng. Comput. 17 (2) (2001) 129–150. http://dx.doi.org/10.1007/PL00007198. [21] H.H. Khodaparast, J.E. Mottershead, K.J. Badcock, Interval model updating with irreducible uncertainty using the Kriging predictor, Mech. Syst. Signal Process. 25 (4) (2011) 1204–1226. http://dx.doi.org/10.1016/j.ymssp.2010.10.009. [22] Y. Liu, Y. Li, D. Wang, S. Zhang, Model updating of complex structures using the combination of component mode synthesis and Kriging predictor, Sci. World J. (2014). http://dx.doi.org/10.1155/2014/476219 13 pp., (Article ID 476219). [23] I. Behmanesh, B. Moaveni, G. Lombaert, et al., Hierarchical Bayesian model updating for structural identification, Mech. Syst. Signal Process. 64 (2015) 360–376. http://dx.doi.org/10.1016/j.ymssp.2015.03.026. [24] Y. Govers, H.H. Khodaparast, M. Link, et al., A comparison of two stochastic model updating methods using the DLR AIRMOD test structure, Mech. Syst. Signal Process. 52 (2015) 105–114. http://dx.doi.org/10.1016/j.ymssp.2014.06.003. [25] H.G. Pasha, K. Kohli, R.J. Allemang, et al., Structural Dynamics Model Calibration and Validation of a Rectangular Steel Plate Structure[M]//Model Validation and Uncertainty Quantification 3, Springer International Publishing, 2015, pp. 351–362. http://dx.doi.org/10.1007/978-3-319-15224-0_37. [26] T.W. Simpson, M. Mauery, J.J. Korte, F. Mistree, Kriging models for global approximation in simulation- based multidisciplinary design optimization, AIAA J. 39 (12) (2001) 2233–2241. http://dx.doi.org/10.2514/3.15017. [27] S.E. Gano, J.E. Renaud, J.D. Martin, T.W. Simpson, Update strategies for Kriging models for use in variable fidelity optimization, Struct. Multidiscip. Optim. 32 (4) (2005) 287–298. http://dx.doi.org/10.1007/s00158-006-0025-y. [28] J.D.Martin, T.W.Simpson, On the use of Kriging models to approximate deterministic computer models, in: Proceedings of the ASME 2004 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, American Society of Mechanical Engineers, 2004 pp. 481–492. http://dx.doi.org/10.2514/1.8650. [29] E.R. Fotsing, M. Sola, A. Ross, Edu Ruiz, Lightweight damping of composite sandwich beams: experimental analysis, J. Compos. Mater. 47 (12) (2013) 1501–1511. http://dx.doi.org/10.1177/0021998312449027. [30] K.S. Kwon, R.M. Lin, Frequency selection method for FRF-based model updating, J. Sound Vib. 278 (1) (2004) 285–306. http://dx.doi.org/10.1016/ j.jsv.2003.10.003. [31] W.X. Ren, H.B. Chen, Finite element model updating in structural dynamics by using the response surface method, Eng. Struct. 32 (8) (2010) 2455–2465. http://dx.doi.org/10.1016/j.engstruct.2010.04.019. [32] A. Boudjemai, R. Amri, A. Mankour, et al., Modal analysis and testing of hexagonal honeycomb plates used for satellite structural design, Mater. Des. 35 (2012) 266–275. http://dx.doi.org/10.1016/j.matdes.2011.09.012. [33] X.Q. Zhao, G. Wang, D.L. Yu, Experiment verification of equivalent model for vibration analysis of honeycomb sandwich, Struct., Appl. Mech. Mater. 624 (2014) 280–284. http://dx.doi.org/10.4028/www.scientific.net/AMM.624.280. [34] M. Miki, T. Hiroyasu, M. Kaneko, K. Hatanaka, A parallel genetic algorithm with distributed environment scheme, IEEE Int. Conf. Syst., Man, Cybern. 1 (1999) 695–700. http://dx.doi.org/10.1109/ICSMC.1999.814176. [35] A.J. García-Palencia, E. Santini-Bel, A. Two-Step, Model Updating algorithm for parameter identification of linear elastic damped structures, Comput.-Aided Civ. Infrastruct. Eng. 28 (7) (2013) 509–521. http://dx.doi.org/10.1111/mice.12012. [36] J.D. Sipple, M. Sanayei, Finite element model updating of the UCF grid benchmark using measured frequency response functions, Mech. Syst. Signal Process. 46 (1) (2014) 179–190. http://dx.doi.org/10.1016/j.ymssp.2014.01.008. [37] J.D. Sipple, M. Sanayei, Finite element model updating using frequency response functions and numerical sensitivities, Struct. Control Health Monit. 21 (5) (2014) 784–802. http://dx.doi.org/10.1002/stc.1601.
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