The effective numerical model of stitched sandwich composite plays crucial role in dynamics analysis and structural design. A sensitivitybased multistage model updating method is proposed for modeling of the stitched sandwich panel using the experimental modal frequencies. Based on applying the periodic boundary condition, the initial equivalent finite element model of a stitched sandwich composite is constructed; the measured frequencies are obtained from the modal test. According to relative sensitivity analysis of modal frequencies with respect to updating parameters, the different wellconditioned groups of parameters are selected to be updated. This method is applied to a stitched sandwich panel with established configuration of the stitches. Results show that the proposed method with the smallest condition number of relative sensitivity matrix has a better performance than the multistage method using typebased parameter group selection and the traditional model updating approach.
The stitched sandwich composites have widely been applied in engineering because of the brilliant property in throughthethickness direction [
The effective material parameters of the stitched sandwich composite can be determined from the experimental, theoretical, and numerical methods. Singh and Saponara [
In general, model updating is adopted to reduce the error in the parameter estimations of composites [
Without loss of generality, the parameter identification and model updating algorithms for other composites can also be applied to the stitched sandwich composites. In order to identify the material properties of the individual layer of laminate plates, Lauwagie et al. [
Mottershead et al. [
A relative sensitivitybased multistage model updating method is proposed to update the equivalent properties of the stitched sandwich panel. In Section
The stitched sandwich specimen investigated in this study is shown in Figure
The stitched composite sandwich panel.
Due to the diversity in the consisted components, a refined numerical model of the stitched sandwich composite decreases the computation efficiency. Based on the previous study, the homogeneous equivalent modelling method [
The unit cell of the stitched sandwich specimen consisted of multicomponents as shown in Figure
Figure
Estimation procedure of the equivalent elastic properties of the stitched sandwich composite changing the different forced boundary conditions in every loop.
Equivalent material properties of skin and core of the stitched sandwich panel.
Type 











Skin  5230  5230  5425  0.31  0.30  0.29  5630  2815  2815  1.63 × 10^{3} 
Core  192.3  181.8  277.8  0.06  0.02  0.03  104  29  29  3.37 × 10^{3} 
The refined and equivalent FEM of the stitched sandwich panel are shown in Figure
Refined (a) and equivalent (b) FEM of the stitched sandwich panel.
According to the equivalent properties determined using the homogenized parameter estimation method, the orthotropic material properties of the shell and solid elements in Figure
Material properties of shell element in the equivalent finite element model.
Type 








Skin  5230  5230  0.31  5630  2815  2815  1.63 × 10^{3} 
Material properties of solid elements in the equivalent finite element model.
Type 











Core  193.1  11.1  6.1  182.5  5.9  278.2  104.0  29.0  29.0  3.37 × 10^{3} 
Experimental modal analysis (EMA) performs well on determining the natural frequencies and mode shapes of structures. The impact hammer testing has become widespread as an efficient and economical mean of finding the modal parameters in EMA [
Procedure of the typical modal test with impact hammer.
Figure
Modal test setup for the stitched sandwich panel.
Figure
Measured summation frequency response function estimated from all measured data and three measured data at three points.
Comparison of modal frequencies between refined and equivalent models.
Mode order  Experimental (Hz)  Numerical  

Refined model (Hz)  Error (%)  Equivalent model (Hz)  Error 
Diff. (%)  
1  189.90  213.17  12.25  213.32  12.33  −0.02 
2  204.59  125.05  −38.88  125.37  −38.72  0.26 
3  387.64  414.38  6.90  415.23  7.12  0.21 
4  439.56  322.22  −26.69  324.71  −26.13  0.77 
5  583.81  609.20  4.35  610.97  4.65  0.29 
6  672.74  578.64  −13.99  561.10  −16.59  −3.03 
Error = (
First six mode shapes obtained from experiment and FE analysis.
The purpose of model updating is to modify the property parameters of the numerical model and improve the accuracy of the analysis results. The model updating method is essentially an optimization which minimizes the discrepancies between the experiment and analytical outputs [
Resorting to the Taylor expansion, the optimization problem in equation (
The outputs (
This iteration procedure continues until consecutive estimates
To deal with the problem that the inaccurate inverse solution caused by large condition number of the sensitivity matrix and enhance the accuracy of the equivalent model updating, the method multistage parameter selection for model updating is developed in this section. In the proposed multistage model updating method, updating parameters are divided into several groups and only one group of parameter is adopted to be updating in one stage procedure. The modified procedure will be executed several times until every group of parameters is modified and the residual error between numerical and experimental data sufficiently converged to be minimum. Compared with the traditional method, the multistage method could decrease the condition number of the sensitivity matrix. The major step of the multistage updating procedure is parameter group selection, which is introduced as follows.
In the equivalent model of the stitched sandwich panel, updating parameters should be determined from the modal testing data, and the parameter groups
Two methods of parameter group selection are introduced in this paper. A concise method is to divide parameters according to the type of parameters. In the case of the sandwich panel, parameters related to the tensile modulus can be put in the same one group, so are others related to shear modules. The advantage of the typebased method is easy and visualized to operate. In order to be quantitative, the second relative sensitivitybased grouping method is an alternative approach to multistage parameter grouping. The parameter sensitivity matrix in equation (
The implementation process of the sensitivitybased multistage updating method, as shown in Figure
Initial finite element model of the stitched sandwich panel is firstly established using the homogeneous equivalent method and estimate the initial equivalent parameters.
Relative sensitivity analysis is used for the multistage parameters selection. The updating parameter vector
Begining of multistage finite element model updating. Choose the
When all the parameter groups are renewed and the accuracy is satisfied, go to step (5) and the updated finite element model is obtained; Otherwise, set
The updated finite element model of stitched sandwich is obtained.
Flow chart of the multistage model updating procedure.
In brief, parameters in one group are modified at the same time in multistage model updating method, while parameters in the next group adopt the terminal value of the identification before and execute the modification in sequence. The multistage model updating method ensures that the experimental data are abundant at every step of model updating and the stability of every optimization iteration is enhanced. With the iteration times increasing of multistage updating, the parameter will converge at the true value.
To verify the accuracy of the proposed method for the equivalent model updating, an established stitched sandwich panel as shown in Figure
Table
Relative sensitivity matrix of first four modes with respect to updating parameters.
Mode 









1  1.05  16.96  0.02  37.93  2.14  158.56  15.51  0.28 
2  0.62  12.98  2.54  46.54  40.75  121.22  12.19  43.96 
3  18.68  0.79  0.13  2.41  87.07  −11.05  194.40  2.35 
4  4.38  0.25  7.32  61.88  63.72  −1.62  43.20  126.43 
The elastic parameters listed in Table
Figure
Convergence of selected parameters with different weighting matrices: (a)
Comparison of updated modal frequencies between experiment and analysis with different weighting matrices.
Mode order  Experiment (Hz)  Case I: 
Case II: 


Numerical (Hz)  Error (%)  Numerical (Hz)  Error (%)  
1  189.90  190.66  0.40  190.78  0.46 
2  204.59  201.94  −1.29  199.57  −2.46 
3  387.64  383.47  −1.08  384.03  −0.93 
4  439.56  450.75  2.55  449.97  2.37 
5  583.81  593.42  1.65  595.61  2.02 
6  672.74  664.87  −1.17  661.87  −1.62 
7  751.18  769.54  2.44  770.28  2.54 
8  790.16  817.25  3.43  822.75  4.12 
9  878.64  909.26  3.48  913.50  3.97 
10  916.01  970.10  5.91  979.15  6.89 
Mean error  2.34  2.74 
Mean error means absolute average error, mean error =
The results in Table
As shown in Figure
To compare the effect of different parameter group methods, two cases as shown in Section
Two cases of parameter groups.
Case number 



1 


2 


Figures
Convergence procedure of different parameter groups in each stage of case 1.
Convergence of groups of parameters in each stage in case 2.
Comparison of modal frequencies between test and numerical models of cases 1 and 2.
Mode order  Test (Hz)  Numerical mode  

Initial (Hz)  Error (%)  Case 1  Case 2  
Updated (Hz)  Error (%)  Updated (Hz)  Error (%)  
1  189.90  213.32  12.33  187.29  −1.37  189.16  −0.39 
2  204.59  125.37  −38.72  201.34  −1.59  201.17  −1.67 
3  387.64  415.25  7.12  380.19  −1.92  378.37  −2.39 
4  439.56  324.71  −26.13  441.70  0.49  447.81  1.88 
5  583.81  610.97  4.65  587.73  0.67  584.30  0.08 
6  672.74  561.10  −16.59  655.02  −2.63  670.85  −0.28 
7  751.18  768.72  2.33  767.95  2.23  763.80  1.68 
8  790.16  817.67  3.48  803.54  1.69  805.00  1.88 
9  878.64  875.59  −0.35  912.01  3.80  893.22  1.66 
10  916.01  994.60  8.58  937.48  2.34  961.01  4.91 
Mean error  12.03  1.87  1.68 
Condition number of the relative sensitivity matrix of two stages.
Iteration stage  Case 1  Case 2 

One  12429  49.26 
Two  8141  8.20 
The final updated elastic parameters of the equivalent finite element model of the stitched sandwich panel are listed in Table
Updated elastic parameters of equivalent finite element of stitched sandwich composite from case 2.
Foam core properties  Skin properties  









193.31  177.14  103.97  15.64  19.42  22500.41  17169.98  4572.30 
Modal assurance criterion (MAC) between experimental and analytical mode shapes.
A sensitivitybased multistage model updating method is proposed on a stitched sandwich panel using the experimental modal frequencies. In this method, the multistage parameter selections are based on the relative parameter sensitivity analysis, which avoids the large condition number of the sensitivity matrix in the updating procedure. The proposed method is used for an equivalent model of a stitched sandwich panel with established configuration of the stitches. The comparison between the proposed method and the traditional method utilizing two different weighting matrices is also conducted. The two types of weighting matrix have small impact on the decreasing condition number of the sensitivity matrix. Results show that the computational efficiency for modal analysis is improved using the equivalent model, and the equivalent model of the stitched sandwich panel can successfully be updated by the proposed method.
Homogenized stress and strain vector
Stiffness matrix of composites
Compliance matrix of composites
Young’s modulus
Shear modulus
Poisson’s ratio
Vector of updating parameters
Diagonal weighting matrix
Experimental and analytical modal data
Weighed sensitivity matrix and relative sensitivity matrix
Maximum elements of an array
Diagonal elements of matrix
Round to nearest integer.
The data of this study are included within the article.
The authors declare that they have no conflicts of interest.
The authors are grateful for the support from the National Natural Science Foundation of China (nos. 11602112 and 11572086) and the Distinguished Youth Scholar Foundation of Jiangsu Province (nos. BK20170022 and BK20180062).