Finite element model updating of a satellite honeycomb sandwich plate in structural dynamics

Abstract

The honeycomb sandwich structures have a crucial participation in aerospace industry, especially in the design of satellite structures due to their exceptional mechanical properties. The equivalent finite element modeling of such structures is initially presented through the implementation of modal analysis via the three-layered sandwich theory. Subsequently, the computational results are validated by carrying out an experimental modal testing. In addition, sensitivity analysis based upon design of experiments and parameters correlation, is executed for the sake of selecting the most appropriate design parameters for the optimization problem. Finally, finite element model updating of a honeycomb sandwich plate is thoroughly introduced using three optimization algorithms including genetic algorithms, adaptive-multiple optimization, and response surface method. A good agreement between the previously-mentioned optimization algorithms is obtained. Meanwhile, response surface method and its related design of experiments tool succeed in avoiding such time-consuming process and reduce the involved computational expense with an acceptable accuracy.

Keywords

Sandwich structures sandwich theory design of experiments response surface method finite element model updating

Introduction

Employing honeycomb sandwich structures in building up the primary and secondary structures of satellites has become of a special concern to many researchers.¹ The usage of such structures has figured out a considerable number of obstacles related to strength, stiffness, mass, and dimensional stability issues. A significant amount of literature on the dynamic properties of these structures is available. Dynamic properties were originally estimated using the first order shear deformation theory (FOSDT),² finite-strip method,³ and high order shear deformation theory.⁴ Subsequently, the finite element (FE) method was commonly used in calculating the honeycomb sandwich structures modal properties.^5,6 However, the detailed finite element modeling (FEM) of such structures involves a high computational expense. Thus, developing an equivalent model either for the core or for the whole structure is regarded as the most efficient way for modeling these structures but such model should be validated using experimental work.⁷

Xia et al.⁸ introduced the use of three different theories in modeling honeycomb sandwich structures represented in the sandwich theory, the honeycomb plate theory, and the equivalent plate theory. The mentioned theories were reliable and robust in the FE analysis of honeycomb plate. Hao et al.⁹ compared between the introduced three equivalent theories. Based upon the comparison, the satellite structure was modeled using the honeycomb-plate theory. They performed modal and harmonic response analyses and results were considered as a foundation for the satellite optimal structural design. Boudjemai et al.¹⁰ carried out a modal analysis on a honeycomb sandwich beam using both detailed and equivalent FEM. The error between experimental and numerical results, concerning the first three natural frequencies, did not exceed 10%. Jiang et al.¹¹ utilized the three-layered sandwich theory in the FEM of a honeycomb plate. They calculated the four natural frequencies and their mode shapes, and then compared the numerical results with experimental modal testing results. They obtained an acceptable mean error of 6.17%. Recently, Fu et al.¹² estimated the dynamic properties of a satellite solar array fabricated of a honeycomb sandwich structure numerically and experimentally. They calculated the equivalent elastic properties of the core. A good matching between both computational and experimental results was obtained.

The aforementioned survey showed the consistent existence of a deviation between experimental and numerical results. By other words, these FEM’s do not certainly reflect the measured data accurately. Thus, such models need to be updated to predict the measured data in a better way.¹³ The main domains for finite element model updating are the natural frequencies, mode shapes and the frequency response function (FRF). Thus, it can be concluded that FEM updating is an optimization problem with the objective to minimize the difference between the experimental and numerical results, and the design parameters are the uncertain material properties. Jaishi and Ren¹⁴ utilized a single-objective optimization technique in a successful FE model updating of real bridge structures using experimental results. In 2006, they performed a sensitivity-based FEM updating with the sake of damage detection.¹⁵ They identified the damage pattern in an excellent way. In 2007, they used multi-objective optimization algorithm in the FEM updating of a simply supported beam and real bridge.¹⁶ Both case studies depict that the proposed algorithm is accurate and robust. In 2008, Huang and Zhu¹⁷ introduced the FEM updating of a bridge using the sensitivity analysis and optimization techniques. The updated model predicted the bridge dynamic properties more accurately and can be utilized as a basis for damage detection. In 2009, Liu et al.¹⁸ utilized the FRF shapes for locating the damage using FEM updating techniques. They used the fuzzy theory and they successfully applied their model on a real concrete bridge. In order to circumnavigate the high computational expense related to large finite element models and the involved huge number of optimization computations, approximate meta-data models were introduced by Ren and Chen.¹⁹ They presented the FEM updating of an actual bridge using response surface method. They demonstrated that the usage of the response surface model results in an efficient and convergent model updating process. Latter, Guo et al.^20,21 provided a precise dynamic structural finite element model based on strain mode shapes and natural frequencies instead of the displacement modes. Since the strain mode shape is sensitive to local changes in structures, it enhanced the model updating. They adopted the hybrid genetic/pattern-search optimization technique to perform the FEM updating.

In 2014, Jiang et al.¹¹ investigated the FEM updating of a three-layered honeycomb sandwich plate. They updated the out-of-plane shear moduli and minimized the mean error from 6.17% to 1.7%. In 2017, Sun and Cheng⁷ discussed the FEM updating of two equivalent FEM’s of a honeycomb sandwich plate. They introduced two optimization methods based upon response surface method in addition to particle swarm optimization. They assessed that updated FEM’s were efficient enough to analyze the dynamic characteristics of the honeycomb sandwich plate. Recently, Luo et al.²² introduced the FEM updating of a satellite honeycomb sandwich sailboard. They selected appropriate design variables based upon sensitivity analysis results. They reduced the mean error in the natural frequency of the first seven modes from 8.94% to 3.05%.

It is evident from the aforementioned survey that the equivalent FEM of sandwich structures in structural dynamics is inevitable. However, it should be accompanied with an experimental modal testing for validation. In addition, the process of FEM updating has a crucial influence on minimizing the difference between numerical and experimental data and thus, enhancing the behavior of the numerical finite element model. In this article, the dynamic properties of a honeycomb sandwich plate are estimated numerically using sandwich theory. The numerical results are validated with the aid of an experimental modal testing. Subsequently, A FEM updating is carried out to reduce the difference between experimental and numerical results. The FEM updating is preceded by a sensitivity analysis in order to properly select the most effective design parameters for the optimization problem. Finally, different optimization techniques are employed seeking for the best one. The utilized algorithms include genetic algorithms (GA), adaptive-multiple optimization (AMO), and response surface method (RSM).

Theoretical approaches and governing equations

Sandwich theory

The sandwich theory represents the honeycomb sandwich structure as its real nature; three different layers where the core only is treated as a homogeneous continuum with orthotropic properties. An important requirement here is estimating the equivalent properties of the homogenized core. Such properties rely on the single cell geometry in addition to the material properties of the honeycomb core material. For a double wall thickness core, the core properties are shown in Figure 1.

Figure 1.

Honeycomb core cell.

For a regular hexagon core where l = h and $θ$ = 30°, the in-plane and out-of-plane properties are computed as follows:

In-plane equivalent properties

The in-plane properties include four elastic properties; shear modulus ( $G_{x z}$ ), two moduli of elasticity ( $E_{x}$ , $E_{z}$ ), and Poisson’s ratio (υ_xz). They are estimated according to the following formulas^23,24:

E_{x} = E_{z} = \frac{4}{\sqrt{3}} {(\frac{t}{h})}^{3} E_{c}

(1)

G_{x z} = \frac{4}{5 \sqrt{3}} {(\frac{t}{h})}^{3} E_{c}

(2)

υ_{x z} = υ_{z x} = 1

(3)

where $t$ is the cell wall thickness, $h$ is the cell wall length, and $E_{c}$ is the core material modulus of elasticity.

Out-of-plane equivalent properties

The out-of-plane equivalent elastic properties incorporate five elastic moduli; the modulus of elasticity ( $E_{y}$ ), two shear moduli ( $G_{x y}, G_{z y})$ , and two Poisson’s ratios ( $υ_{x y}$ , $υ_{z y}$ ). The modulus of elasticity ( $E_{y}$ ) is calculated from static analysis²³ as follows:

E_{y} = \frac{8}{3 \sqrt{3}} (\frac{t}{h}) E_{c}

(4)

The two Poisson’s ratios ( $υ_{x y}$ ), ( $υ_{z y}$ ) are calculated as follows:

υ_{x y} = \frac{E_{x}}{E_{y}} υ_{c} \approx 0, υ_{z y} = \frac{E_{z}}{E_{y}} υ_{c} \approx 0

(5)

The upper and lower bound for the two shear moduli are formulated according to the published work²⁵ by computing the strain energy related to both strain distribution and stress distribution. If they coincide, then this is the exact solution (as in the case of $G_{x y}$ ), if not, then the exact solution is located in between (as in the case of $G_{z y}$ ) as follows:

G_{x y} = \frac{1}{\sqrt{3}} (\frac{t}{h}) G_{c}

(6)

\frac{\sqrt{3}}{2} (\frac{t}{h}) G_{c} \leq G_{z y} \leq \frac{5}{3 \sqrt{3}} (\frac{t}{h}) G_{c}

(7)

An approximate formula for calculating $G_{z y}$ based upon the ( $\frac{c}{h}$ )²⁶ is obtained as follows:

G_{z y} = G_{z y l o w e r} + \frac{0.787}{(\frac{c}{h})} (G_{z y u p p e r} - G_{z y l o w e r})

(8)

where $υ_{c}$ is the core material Poisson’s ratio, $G_{c}$ is the core material shear modulus, and $c$ is the core height.

Response surface method (RSM)

Finite element models of complex structures are quite large and require a high computational time. Optimizing such structures is extremely difficult due to the enormous number of calculation cycles and the related computational expense.²⁷ Such time-consuming calculations can be avoided using RSM. The term “response surface” is analogous to “meta-model” or “surrogate model,” which refer to a simple mathematical relationship between input parameters and output response that is based on a limited data derived from design of experiments (DOE). In order to produce an accurate response surface of a given model, it should be preceded by an efficient DOE.

Design of experiments

DOE is a technique used to locate the sampling points such that the space of the design parameters is explored in an efficient way. Thus, DOE is a statistical tool that allows obtaining the required data with minimum number of sampling points. Efficient locations of sample points not only reduce their required number but also increase the accuracy of the response surface derived from them.²⁸

Wide range of DOE algorithms are found, Central Composite Design (CCD) algorithm and optimal space-filling (OSF) algorithm are used in this research. CCD is an orthogonal method, where most of samples are located at the boundaries of the space and few in the center. CCD algorithms are widely used in the sensitivity analysis to identify the most effective design parameters.²⁹ Such algorithm includes one central point, points along the boundaries of the input parameters, and the points determined by a fractional factorial design as follows:

N u m b e r o f s a m p l i n g p o i n t s = 1 + (2 * N) + (2^{N - ϵ})

(9)

where N is the number of design parameters and $ϵ$ is the factorial number. On the other hand, OSF algorithm is capable of distributing sampling points equally throughout the design space with the goal of having the maximum accuracy with the fewest number of points. Figure 2 shows an example for distributing six points using OSF.²⁸

Figure 2.

OSF pattern for six sampling points.

RSM models

Different approximate RSM models are generally utilized including polynomial models, Kriging, and neural network. In this article, both full second order polynomial and Kriging are utilized to implement the regression analysis.

Full second order polynomial model

Such model is employed according to the following form¹⁹:

f (x) = β_{0} + \sum_{i = 1}^{n} β_{i} x_{i} + \sum_{i = 1}^{n} β_{i i} x_{i}^{2} + \sum_{i} \sum_{j} β_{i j} x_{i} x_{j}

(10)

where $f (x)$ is the output response, x are the input design parameters, and $(β_{i}, β_{i i}, β_{i j})$ are the regression parameters estimated using the least square method.

Kriging

Kriging is a meta-modeling algorithm that is capable of supporting an improved response quality. It is a combination of the aforementioned polynomial model $f (x)$ plus stochastically component $k (x)$ which adds local deviations as follows:

y = f (x) + k (x)

(11)

The effectiveness of such algorithm is based on the ability of its internal error estimator to improve the response surface quality by generating refinement points and adding them to the areas of the response surface most in need of improvement.²⁸

In order to check the accuracy of the built response surface, Goodness of Fit is estimated for each output parameter. It includes the following parameters:

Coefficient of determination (R²)^19,28: it is the percentage of variation of the output parameter, the best value is 1.

R^{2} = 1 - \frac{\sum_{i = 1}^{n} {(y_{i} - y_{r s})}^{2}}{\sum_{i = 1}^{n} {(y_{i} - \bar{y})}^{2}}

(12)

Relative root mean square error (RRMSE)^19,28: it is the square root of the average square error divided by the true values of the output parameter, best value is 0%.

R R M S E = \sqrt{\frac{1}{n} \sum_{i = 1}^{n} {(\frac{y_{i} - y_{r s}}{y_{i}})}^{2}}

(13)

where $y_{i}$ is the true value of the output parameter, $y_{r s}$ is the response surface value of the output parameter, $\bar{y}$ is the mean of all real values, and $n$ is the number of sampling points.

Modal analysis and testing of honeycomb sandwich structure

In this section, the equivalent FEM of a honeycomb sandwich plate is presented while implementing modal analysis via the sandwich theory. In addition, an experimental modal testing is executed to validate numerical results.³⁰

The square honeycomb sandwich plate dimensions are as follows: length = 400 mm, total thickness = 10 mm, core thickness = 8 mm, and each facing sheet = 1 mm. Core specifications are listed in Table 1.

Table 1.

Honeycomb core specifications.

Material	Wall length (l = h)	Wall thickness (t)	Density ( $ρ_{eq .}$ )	$E_{c}$	$ρ_{c}$	$υ_{c}$
AL-5052	1.83 mm	0.05 mm	130 kg/m³	70 GPa	2700 kg/m³	0.33

Modal analysis of honeycomb sandwich plate

The honeycomb sandwich plate is modeled using the sandwich theory according to the shell-volume-shell (SVS) approach. The facing sheets are meshed via shell element “shell 181,” while the core is meshed via solid element “solid 186.”^11,7,31 Mesh sensitivity analysis is carried out, where meshing details are shown in Table 2.

Table 2.

Number of nodes and elements used to mesh the honeycomb plate.

Modeling approach	Element size, mm	Number of elements	Number of nodes
SVS	8	7500	23,205

The honeycomb core equivalent elastic properties are estimated according to equations (1)–(8) as depicted in Table 3. The facing sheets are fabricated of Aluminum alloy 2024-T3 with isotropic material properties shown in Table 4.

Table 3.

Core equivalent properties.

$E_{x}$	3.29 MPa	$E_{y}$	2944.58 MPa
$E_{z}$	3.29 MPa	$G_{x y}$	425.9 MPa
$G_{x z}$	1.99 MPa	$G_{z y}$	651.67 MPa
$υ_{x z}$	1	$υ_{x y} = υ_{z y}$	0

Table 4.

Facing sheets material properties.

$E$	$ρ_{f}$	$υ_{f}$
70 GPa	2800 kg/m³	0.33

The free-free boundary conditions are utilized while calculating the first four modal frequencies and the corresponding mode shapes.

Experimental modal testing

The dynamic properties of the honeycomb plate are measured via an impact modal testing. The boundary conditions are simulated by suspending the plate using soft bands as shown in Figure 3.³⁰ The plate is excited by an impact hammer equipped with a force transducer. Two piezoelectric accelerometers are secured to the plate to detect its response. The whole process is controlled via LabVIEW software installed on a laptop that is connected to an analyzing system.

Figure 3.

Experimental modal testing setup.

Results comparison and discussion

The results of the four natural frequencies concerning the experimental testing and numerical analysis are listed in Table 5.

Table 5.

Experimental and results comparison.

Mode order	Experimental, Hz	Numerical, Hz	Error, %
First natural frequency	259	272.7	5.29
Second natural frequency	391	415.7	6.32
Third natural frequency	494	519.9	5.24
Fourth natural frequency	664	694.1	4.53
Mean error, %			5.35

The table shows a good agreement between both results; mean deviation does not exceed 6%. However, FEM updating is capable of minimizing this mean deviation and provides precise values of the design parameters (core equivalent elastic properties).

FEM updating

In this section, the finite element model (FEM) updating of honeycomb sandwich structures is thoroughly discussed. The process of FEM updating is carried out in order to obtain the most accurate equivalent elastic properties and thus, reduce the mean deviation between the experimental modal testing results and the related computational results. The FEM updating domain is the first four natural frequencies of the sandwich plate utilized during the aforementioned experimental modal testing.

The FEM updating process can be identified as an optimization problem with the objective of minimizing the mean deviation between numerical and experimental natural frequencies. The design variables are the honeycomb six equivalent elastic properties ( $E_{x}$ , $E_{z}$ , $E_{y}$ , $G_{x z}$ , $G_{x y}$ , $G_{z y}$ ) and the equivalent density $ρ_{e q}$ . Nevertheless, the number of design variables is too large to be optimized. Thus, selecting the appropriate design parameters for the optimization problem is a vital initial step.

Sensitivity analysis

The goal of such analysis is to accurately select the suitable design parameters for the optimization problem and thus, reduce the inherent computational expense. In this research, the sensitivity analysis is performed twice using both the design of experiments (DOE) module and the parameter correlation module. Both modules are part of “Designxplorer” module in ANSYS workbench software.

Sensitivity analysis for parameters identification using DOE

Central Composite Design (CCD) algorithm is used in this analysis. The total number of sampling points is calculated according to equation (9) to be 45 points. This number of iterations is implemented using DOE module in ANSYS workbench software where each design parameter is allowed to vary according to the range mentioned in Table 6. The results are exported to statistical analysis software named as “Minitab” so as to be analyzed and interpreted.

Table 6.

Design parameters range.

Design parameter	From	To
$E_{x}$ , $E_{z}$ (MPa)	2.35	4.25
$E_{y}$ (MPa)	2088	3781.8
$G_{x z}$ (MPa)	1.4	2.55
$G_{x y}$ (MPa)	303	550
$G_{z y}$ (MPa)	463.5	840.44
$ρ_{e q}$ (kg/m³)	113	130

An important analytical tool for analyzing such results is the main effects plot. A main effect plot is a plot of the mean value of an output parameter at different levels of a selected input parameter.³² This plot can be used to compare the between the relative strength of the effects of different input design parameters. The main effects plots of the first four natural frequency of the honeycomb sandwich plate are shown in Figures 4 to 7.

Figure 4.

Main effects plot of the plate first natural frequency.

Figure 5.

Main effects plot of the plate second natural frequency.

Figure 6.

Main effects plot of the plate third natural frequency.

Figure 7.

Main effects plot of the plate fourth natural frequency.

The main effects plots of the first four natural frequencies show the following conclusions:

( $E_{x}$ , $E_{z}$ , $E_{y}$ , $G_{x z}$ ) have no clear effect on the honeycomb plate natural frequency.

( $G_{x y}$ , $G_{z y}$ ) has a positive effect on the natural frequency, this means that increasing these parameters leads to significant increment in the modal frequencies.

( $ρ_{e q}$ ) has a negative effect on the natural frequency, this means that increasing the density leads to obvious decrease in the modal frequencies.

Parameters correlation

Correlation analysis is implemented with the same goal as the aforementioned DOE to locate the design parameters that have a clear influence on the honeycomb sandwich plate natural frequencies. In this research, Pearson correlation method is used, such method utilizes real data to estimate the correlation coefficients and sensitivities chart. The correlation coefficient always lies between −1 and +1, when it is close to +1 or −1, this means that this parameter has a strong effect on the natural frequency and it is either directly or inversely promotional to it, respectively. Whereas, a value close to 0 indicates that they are independent. Table 7 shows the correlation coefficients of the input parameters for each natural frequency.

Table 7.

Correlation coefficients of the input parameters for each natural frequency.

Input parameters	First natural frequency	Second natural frequency	Third natural frequency	Fourth natural frequency	Average
$E_{x}$ , $E_{z}$	0.02375	0.0163	0.0255	0.0207	0.0216
$E_{y}$	−0.017	−0.014	−0.0009	−0.038	−0.017
$G_{x z}$	0.0122	0.01168	0.0126	0.011	0.01187
$G_{x y}$	0.697	0.657	0.663	0.778	0.6987
$G_{z y}$	0.548	0.518	0.578	0.515	0.5397
$ρ_{e q}$	−0.4562	−0.549	−0.481	−0.303	−0.447

It is obvious that the highlighted parameters ( $G_{x y}, G_{z y}, ρ_{e q}$ ) has a clear effect on the first four natural frequencies, while the effect of the other parameters is almost undetectable. It can be noted also that the density correlation coefficients are of –ve values as it is inversely proportional to natural frequencies.

Finally, sensitivity chart shows the global sensitivity of the natural frequencies with respect to the input parameters. Figure 8 shows the average sensitivities of the four natural frequencies. A high sensitivity indicates that the output parameter is more sensitive to the input parameter, and vice-versa.

Figure 8.

Average sensitivities of natural frequencies.

Based upon the aforementioned sensitivity analyses, and considering that the density accurate value is the measured one (130 kg/m³), ( $G_{x y}$ , $G_{z y}$ ) are selected as the parameters for FEM updating of the honeycomb sandwich plate.

FE model optimization

As previously mentioned, three different optimization techniques are utilized for the sake of comparing results to elect the best technique. The optimization problem can be formulated as minimizing the mean error between the experimental and numerical modal frequencies as follows:

\min f u n . = \frac{1}{4} \sum_{i = 1}^{4} | \frac{f_{e x p} - f_{n u m}}{f_{e x p}} | \times 100 %

(14)

Subject to:

\begin{array}{l} 100 M P a \leq G_{x y} \leq 500 M P a \\ 100 M P a \leq G_{z y} \leq 700 M P a \\ G_{x y} < G_{z y} \end{array}

(15)

where f is the natural modal frequencies, i is the order of the natural frequency.

Multi-objective genetic algorithm (MOGA)

Genetic algorithms are search algorithms based on the principle of evolutionary computation and survival of the fittest.³³ Genetic Algorithms differ from more traditional optimization techniques; in that they involve a search from a “population” of solutions, not from a single point. The general scheme of such algorithm is shown in Figure 9.

Figure 9.

Genetic algorithm scheme.

MOGA is a direct optimization algorithm which utilizes real evaluations within ANSYS workbench software. In this article, the initial population is selected to be 100 samples, the number of population to be iterated and updated is 50 samples for maximum number of iteration to be 20 iterations, and finally the convergence criteria is whether the maximum allowable Pareto percentage is set at 70% or the convergence stability percentage to be set at 2%. After 334 evaluations, the MOGA converges and a candidate point is obtained.

Adaptive multiple-objective algorithm (AMO)

Adaptive multiple-objective (AMO) is a hybrid algorithm that utilizes a combination of both genetic algorithms and Kriging within “Direct Optimization” module in ANSYS workbench software. It uses the same procedure as MOGA, but a Kriging response surface is employed; part of the population is “simulated” by evaluations of the Kriging and the Kriging error predictor minimizes the number of evaluations used in finding the global optimum. The AMO scheme is shown in Figure 10.

Figure 10.

AMO scheme.

The settings used in AMO is the same as MOGA concerning the initial population, the number of population to be iterated and updated, maximum number of iteration, the maximum allowable Pareto percentage, and the convergence stability percentage. The results indicate that the candidate point obtained using MOGA is re-obtained exactly using AMO but with reduced number of evaluations; 297 evaluations.

Response surface optimization (RSO)

Response surface optimization is an indirect optimization technique that relies on building meta-model using approximation techniques. Such technique is composed of three modules in ANSYS workbench: design of experiments (DOE), response surface method (RSM), and finally applying the optimization technique.

Firstly, DOE is implemented using Optimal Space-Filling (OSF) algorithm in order to determine accurately the location of the sampling points. The number of sampling point is calculated according to equation (9) to be nine sampling points, where $ϵ$ = 0 for two design parameters. The results of DOE are shown in Table 8.

Table 8.

DOE results.

No.	$G_{x y}$	$G_{z y}$	f₁	f₂	f₃	f₄	Mean error, %
1	388.89	200	265.26	407.2	506.26	660.67	0.502
2	477.78	333.33	269.9	412.55	514.86	682.39	2.77
3	211.11	533.33	267.14	409.19	510.17	668.12	0.62
4	344.44	466.67	269.99	412.63	514.99	682.73	2.82
5	166.67	133.33	256.12	396.63	486.64	621.96	6.33
6	255.56	266.67	264.93	407.08	505.25	661.53	0.37
7	300	666.67	270.65	413.34	516.5	684.35	3.06
8	433.33	600	272.41	415.35	519.39	693.19	4.4
9	122.22	400	260.53	400.58	498.18	636.48	4.15

The subsequent step is building the response surface within the design space utilizing both second order polynomial model and Kriging according to equations (10) and (11) respectively. Two refinement points are created in order to improve the response surface quality of the polynomial model, while Kriginig model is automatically refined. Four supplementary verification points are generated to assess the quality of the response surface and validate its accuracy. Goodness of fit information is estimated for each output parameter in order to check the accuracy of the built response surface as shown in Table 9.

Table 9.

Goodness of fit information.

Output parameters	Full polynomial model		Kriging
Output parameters	Coefficient of determination (R²)	Relative root mean square error (RRMSE)	Coefficient of determination (R²)	Relative root mean square error (RRMSE)
First natural frequency	0.999	0.0559	1	0
Second natural frequency	0.9991	0.0434	1	0
Third natural frequency	0.9996	0.0355	1	0
Fourth natural frequency	0.9995	0.0677	1	0

The above-mentioned Goodness of Fit information ensures the efficient accuracy of the both generated response surfaces. Finally, GA is employed to find the optimum candidate point with the same aforesaid settings.

Results comparison and discussion

A comparison between the obtained candidate points using MOGA, AMO, and RSM optimization is listed in Table 10.

Table 10.

Comparison between different optimization techniques.

Input and output parameters	Numerical data prior to updating	Experimental data	Numerical data post updating
Input and output parameters	Numerical data prior to updating	Experimental data	MOGA	AMO	Polynomial model	Kriging
$G_{x y}$ , MPa	425.9	–	148.68	148.68	155.15	150.67
$G_{z y},$ MPa	651.67	–	232.45	232.45	247.16	251.97
First natural frequency, Hz	272.7	259	259.62	259.62	259.38	259.17
Second natural frequency, Hz	415.7	391	400.7	400.7	400.02	399.66
Third natural frequency, Hz	519.9	494	494.6	494.6	494.995	494.94
Fourth natural frequency, Hz	694.1	664	636.08	636.08	636.93	632.34
Mean error, % (objective function)	5.35	–	1.75	1.75	1.67	1.77
Number of evaluations	–	–	334	297	9 + 2 + 4 = 15	9 + 4 = 13

It can be denoted from the previous results that the three optimization techniques employed in the honeycomb sandwich plate FEM updating, succeed in capturing a candidate point with minimum deviation in results. MOGA converges and offers a candidate optimum point after 334 evaluations. Exactly, the same candidate point is obtained using AMO but with reduced number of evaluations; 297 evaluations. Thus, using the AMO leads to a reasonable reduction in the computational expense while maintaining the same accuracy. However, 297 evaluations are still considered of high computational expense. The usage of an approximate technique as RSM, both in the polynomial form and Kriging, helps to circumnavigate such time-consuming process and reduces the involved computational expense with an acceptable accuracy. The mean deviation of both polynomial model and kriging results when compared to AMO results concerning input parameters ( $G_{x y}$ , $G_{z y}$ ) is 5.34% and 4.86% respectively. Meanwhile, the deviation concerning the objective function is 4.57% and 1.14%. Therefore, kriging model shows better accuracy than the polynomial model due to its internal error estimator and the inherent refinement points. Finally, it can be concluded that the FEM updating techniques succeed in minimizing the deviation between experimental and numerical results from 5.35% up to about 1.67%∼1.77%.

Conclusions

The dynamic properties of a honeycomb sandwich plate are estimated numerically using sandwich theory and its related shell-volume-shell (SVS) approach. The numerical results are validated with the aid of an experimental modal testing. A good agreement between both results, concerning the first four natural frequencies, is obtained; mean error does not exceed 6%. Subsequently, A FEM updating is carried out to reduce the mean error between experimental and numerical results. The FEM updating is preceded by a sensitivity analysis which selects both $G_{x y}$ and $G_{z y}$ as the most effective design parameters for the optimization problem. Finally, different optimization techniques, including MOGA, AMO, and RSM, are employed and the obtained results are compared. The three optimization techniques succeed in capturing a candidate point with no clear difference in results accuracy. However, MOGA and AMO evaluations are still of a high computational expense. That’s why, the usage of an approximate technique as RSM is inevitable, as it exhibits an acceptable accuracy with an outstanding reduced computational expense. The proposed FEM updating techniques succeed to reduce the mean error from 6% to 1.67%~1.77% which shows their ability to be effectively involved in the FEM updating process of the satellite honeycomb sandwich structures.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Ali Aborehab

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