Study on the Cell Magniﬁcation Equivalent Method in Out-of-Plane Compression Simulations of Aluminum Honeycomb

: The large scale and long calculation times are unavoidable problems in modeling honeycomb structures with large sizes and dense cells. The cell magniﬁcation equivalent is the main method to solve those problems. This study ﬁnds that honeycomb structures with the same thickness-to-length ratios have the same mechanical properties and energy absorption characteristics. The improved equivalent ﬁnite element models of honeycomb structures with the same thickness-to-length ratios were established and validated by experiments. Based on the validated ﬁnite element model of the equivalent honeycomb structures, the out-of-plane compression behaviors of honeycomb structures were analyzed by LS-DYNA software. The results show that the performance of honeycomb structures is not equivalent before and after cell magniﬁcation. Thus, the cell magniﬁcation results were further subjected to CORA (correlation analysis) to determine the magniﬁcation time and prove the accuracy of the cell magniﬁcation time through drop-weight impact tests. In addition, a ﬁrst-order decay exponential function (ExpDec1) for predicting cell magniﬁcation time was obtained by analyzing the relationship between the cell wall length and the cell magniﬁcation time.


Introduction
Aluminum honeycomb structures are widely used in automobile, aviation, and rail transit, and other fields, due to their higher rigidity, strength, and energy absorption capacity [1][2][3][4]. A large number of scholars have carried out extensive research on out-ofplane compression in theory and with experimentation and simulation [5][6][7][8]. Selecting a reasonable equivalent method for finite element analysis of honeycomb structures has great significance due to the complexity of honeycomb structures. Allen et al. [9] were the first to study the calculation method of the in-plane equivalent modulus of honeycombs. The method proposed by Allen has been widely used. The model is easy to calculate, but the simplistic simplification method causes a large deviation in the structural analysis results. Subsequently, a large number of scholars have conducted equivalent research on irregular, negative Poisson's ratio and other honeycombs. Mukhopadhyay [10] and Chen [11] et al. studied the equivalent mechanical properties of irregular honeycomb structures and derived the expression of the equivalent elastic modulus. The results showed that the established formula can be used to predict the equivalent in-plane elastic Sustainability 2023, 15, 1882 2 of 15 modulus of irregular honeycomb structures with spatially varying material properties and cell wall thicknesses. Reis et al. [12] deduced the equivalent elastic modulus of negative Poisson's ratio structures using the discrete asymptotic homogenization method. Gibson et al. [13] studied the equivalent parameter theory of honeycomb structures and deduced the expressions of the shear stiffness and shear modulus of the honeycomb. Pan et al. [14] proposed a modified Gibson formula, according to the honeycomb shear test. Wu et al. [15] deduced equivalent theories about the elastic modulus and Poisson's ratios considering bending, tensile, hinged, and shear deformations. The results showed that the obtained equivalent theory is effective and applicable to honeycomb structure. The above studies calculated the equivalent mechanical properties of different honeycomb structures by theoretical methods. However, the equivalent theoretical calculation shows the mechanical properties but cannot show the load-bearing and failure process of the structure. Therefore, the finite element model, based on the cell magnification equivalent method, has received extensive attention. Yasuki and Kojima [16,17] successfully introduced the finite element method of aluminum honeycomb structures based on shell elements. The cell size of the honeycomb was enlarged to increase the mesh size, and the thickness of the shell element controlled the strength. Asadi M et al. [18,19] created the new ODB FE model using LS-DYNA software. To reduce the analysis time, the cell sizes were increased to 52 mm in the FE model while the cell size in the physical barrier was 19.1 mm. Similarly, Li et al. [20] and Tryland [21] enlarged the cell wall length of the honeycomb to improve the computational efficiency in the simulation analysis of automobile barriers. Wang [22] established eight kinds of aluminum honeycomb simulation models for cell magnification based on the explicit finite element method. The results verified the high efficiency and feasibility of the cell magnification equivalent method. Although the above studies have improved the computational efficiency of the honeycomb model through the cell magnification method, there is not much information about how to select the cell magnification times and the mechanical properties of the honeycombs for which the theoretically calculated cell magnification times are different from those of the original honeycombs. And less attention has been paid to further optimizing the cell magnification times for theoretical calculations to improve the model accuracy. Therefore, it is necessary to study the cell magnification equivalent method of the honeycomb. This paper focuses on the out-of-plane compression behavior of the honeycombs with large sizes and dense cells, and we comprehensively analyze the select method of cell magnification equivalent time. Based on the verified finite element model, equivalent models with the same thickness-to-length ratios are established to analyze the compression performance of the honeycomb before and after the cell magnification. Subsequently, the CORA correlation analysis of the simulation results is carried out to determine the cell magnification times. Furthermore, a first-order decay exponential function is established to predict the cell magnification time. When establishing the simulation model of the barrier based on shells in the future, the optimal magnification times of honeycomb cells can be calculated by using the first-order decay exponential function obtained by the fitting. The mechanical properties of the honeycomb structure before and after the cell expansion are equivalent, and the calculation time is effectively reduced, which is of great significance to the economy of barrier simulation calculation.

Basic Properties of Honeycombs and Evaluation Indicators of Honeycombs
The geometry of honeycomb structures has three overall dimensions: length, width, and thickness, which are represented by L, W, and T, respectively. Their periodical representative cell has 4 main parameters. These are the foil thickness t, cell wall length l, width h, and the elevation angle θ. All honeycomb specimens used here were fabricated from laminated sheets with lateral magnification sections bonded to each other. Thus, one third of the sides of the honeycomb structure were double the thickness. Figure 1 shows the structure. This paper only discusses the situations where l = h and θ = 30 • . h, and the elevation angle θ. All honeycomb specimens used here were fabricated from laminated sheets with lateral magnification sections bonded to each other. Thus, one third of the sides of the honeycomb structure were double the thickness. Figure 1 shows the structure. This paper only discusses the situations where l = h and θ = 30°. As shown schematically in Figure 2, for honeycombs, the compression includes four stages: AB is the linear elastic stage, BC is the collapse stage, CD is the flat plateau stress stage, and DE is the densification stage [23]. The commonly used indicators of energy absorption characteristics of honeycomb structures are divided into two categories: mechanical indicators and energy indicators [24]. The main evaluation indicators are as follows: (1) Peak force ( ) refers to the load required for the initial failure of the honeycomb, generally the initial peak value in the load time history curve.
(2) Plateau force (F) refers to the average load in the plateau area. The plateau force is the mean force in the calculation interval, and it takes 16.7%~66.7% of the axial length as the calculation interval [25].
(3) Energy absorption (EA) refers to the capacity of absorbing impact energy, which can be calculated by integrating the area under the force-compression displacement curve [26]. As shown schematically in Figure 2, for honeycombs, the compression includes four stages: AB is the linear elastic stage, BC is the collapse stage, CD is the flat plateau stress stage, and DE is the densification stage [23]. laminated sheets with lateral magnification sections bonded to each other. Thus, one t of the sides of the honeycomb structure were double the thickness. Figure 1 shows structure. This paper only discusses the situations where l = h and θ = 30°. As shown schematically in Figure 2, for honeycombs, the compression includes stages: AB is the linear elastic stage, BC is the collapse stage, CD is the flat plateau st stage, and DE is the densification stage [23]. The commonly used indicators of energy absorption characteristics of honeyco structures are divided into two categories: mechanical indicators and energy indica [24]. The main evaluation indicators are as follows:

Finite Element Model and Test Scheme
(1) Peak force ( ) refers to the load required for the initial failure of the ho comb, generally the initial peak value in the load time history curve.
(2) Plateau force (F) refers to the average load in the plateau area. The plateau f is the mean force in the calculation interval, and it takes 16.7%~66.7% of the axial len as the calculation interval [25].
(3) Energy absorption (EA) refers to the capacity of absorbing impact energy, w can be calculated by integrating the area under the force-compression displacement cu [26]. The commonly used indicators of energy absorption characteristics of honeycomb structures are divided into two categories: mechanical indicators and energy indicators [24]. The main evaluation indicators are as follows:

Finite Element Model and Test Scheme
(1) Peak force (F peak ) refers to the load required for the initial failure of the honeycomb, generally the initial peak value in the load time history curve.
(2) Plateau force (F) refers to the average load in the plateau area. The plateau force is the mean force in the calculation interval, and it takes 16.7%~66.7% of the axial length as the calculation interval [25].
(3) Energy absorption (EA) refers to the capacity of absorbing impact energy, which can be calculated by integrating the area under the force-compression displacement curve [26]. In this study, LS-DYNA was used to establish a finite element model, due to its high reliability and wide applicability for nonlinear dynamic problems [2]. The detailed geometry of the honeycomb core was meshed using the mixed modeling approach [27]. Using a Belytschko-Tsay shell element to construct a cell wall discrete model, the whole mesh control of the structure adopted the same scale to ensure that every buckling behavior could be captured. The shell element can effectively simulate the complex deformation of a honeycomb. In the FE model, at least 3 elements were divided on the edge of the cell [25]. The geometric size of the finite element model was 100 × 100 × 100 mm (in W × L × T direction), and the cell wall lengths of the honeycomb cells were 6 mm and 10 mm. The honeycomb model (MAT_24) in LS-DYNA was selected to define the equivalent materials for honeycombs [20,21]. The mechanical properties of the aluminum material were as follows: density ρ = 2.70 g/cm 3 , Young's module E = 70 GPa, Poisson ratio ν = 0.33, and yield stress = 105 MPa. When the honeycomb structure was subjected to low-speed impact, the adhesive surface was not torn. Therefore, the adhesive could not be considered in the model [28]. The honeycomb block was sandwiched between two rigid plates and the initial velocity of the moving rigid wall was 5 m/s. The mass of the rigid wall was set at 16.22 kg according to the experimental design. The total compression time was set at 18 ms. To avoid penetration between the cell walls, an automated single face-to-face contact algorithm was employed. In this study, 0.20 was used as the friction factor of the metal material [22]. The numerical model and boundary condition is shown in Figure 3. In this study, LS-DYNA was used to establish a finite element model, due to its high reliability and wide applicability for nonlinear dynamic problems [2]. The detailed geometry of the honeycomb core was meshed using the mixed modeling approach [27]. Using a Belytschko-Tsay shell element to construct a cell wall discrete model, the whole mesh control of the structure adopted the same scale to ensure that every buckling behavior could be captured. The shell element can effectively simulate the complex deformation of a honeycomb. In the FE model, at least 3 elements were divided on the edge of the cell [25]. The geometric size of the finite element model was 100 × 100 × 100 mm (in W × L × T direction), and the cell wall lengths of the honeycomb cells were 6 mm and 10 mm. The honeycomb model (MAT_24) in LS-DYNA was selected to define the equivalent materials for honeycombs [20,21]. The mechanical properties of the aluminum material were as follows: density = 2.70 g/cm 3 , Young's module E = 70 GPa, Poisson ratio = 0.33, and yield stress = 105 MPa. When the honeycomb structure was subjected to low-speed impact, the adhesive surface was not torn. Therefore, the adhesive could not be considered in the model [28]. The honeycomb block was sandwiched between two rigid plates and the initial velocity of the moving rigid wall was 5 m/s. The mass of the rigid wall was set at 16.22 kg according to the experimental design. The total compression time was set at 18 ms. To avoid penetration between the cell walls, an automated single face-to-face contact algorithm was employed. In this study, 0.20 was used as the friction factor of the metal material [22]. The numerical model and boundary condition is shown in Figure 3.

Specimens and Testing Procedure
In the dynamic drop-weight impact test, the kinetic energy generated by the free fall of the hammer tup was used to load. When the hammer tup contacted the specimen, the switch signal sensor at the hammer tup recorded the force between the hammer tup and the specimen for hammer tup displacement data. A high-speed camera was employed to record the impact process. The test setup is shown in Figure 4a. In this study, the mass of the hammer tup was 16.22 kg, and the maximum initial impact velocity was 8 m/s. The impact velocity was determined according to the energy expected to be absorbed by the specimen. The impact speed was 5 m/s. Under axial impact, the test specimens had a crosssection size of 100 × 100 mm 2 ; one cell wall length was 6 mm, the wall thickness was 0.03 mm, the other cell wall length was 10 mm, and the wall thickness was 0.05 mm ( Figure  4b).

Specimens and Testing Procedure
In the dynamic drop-weight impact test, the kinetic energy generated by the free fall of the hammer tup was used to load. When the hammer tup contacted the specimen, the switch signal sensor at the hammer tup recorded the force between the hammer tup and the specimen for hammer tup displacement data. A high-speed camera was employed to record the impact process. The test setup is shown in Figure 4a. In this study, the mass of the hammer tup was 16.22 kg, and the maximum initial impact velocity was 8 m/s. The impact velocity was determined according to the energy expected to be absorbed by the specimen. The impact speed was 5 m/s. Under axial impact, the test specimens had a cross-section size of 100 × 100 mm 2 ; one cell wall length was 6 mm, the wall thickness was 0.03 mm, the other cell wall length was 10 mm, and the wall thickness was 0.05 mm ( Figure 4b).   Table 1 and Figure 5 compare the deformation pattern and force-compression displacement curves between the experimental and the simulation results. The deformation mode of a honeycomb with a cell wall length of 6 mm had a good agreement in the test and simulation results. It had the characteristics of different stages, with initial elastic deformation, followed by progressive cell wall bending and folding, and then the ultimate stacking. Under the condition that the impact speed was 5 m/s and the mass of the hammer tup was 16.22 kg, the honeycomb block did not completely enter the densification stage, which was the same as the test and simulation.

Validation of the Finite Element Models
The plateau stage is the main energy absorption stage of the honeycomb compression, and the plateau force and energy absorption of a honeycomb are the focus of this study [29]. It can be seen from Figure 5 that the simulation plateau force of a honeycomb block with a cell wall length of 6 mm was 3.72 kN, and the test plateau force of a honeycomb block with a cell wall length of 6 mm was 3.63 kN. The error of the plateau force between the simulation and the test was 2.48%. Similarly, the simulation plateau force of a honeycomb block with a cell wall length of 10 mm was 3.41 kN, and the test plateau force of a honeycomb block with a cell wall length of 10 mm was 3.58 kN. The error of the plateau force between the simulation and the test was 4.75%. The errors of the plateau force of the simulation and test were both within 5%. Thus, the developed finite element models are reliable and can be used for subsequent simulation to obtain the cell magnification times of different size honeycombs, so as to further fit the equation that can predict the cell magnification times.  Table 1 and Figure 5 compare the deformation pattern and force-compression displacement curves between the experimental and the simulation results. The deformation mode of a honeycomb with a cell wall length of 6 mm had a good agreement in the test and simulation results. It had the characteristics of different stages, with initial elastic deformation, followed by progressive cell wall bending and folding, and then the ultimate stacking. Under the condition that the impact speed was 5 m/s and the mass of the hammer tup was 16.22 kg, the honeycomb block did not completely enter the densification stage, which was the same as the test and simulation.

Validation of the Finite Element Models
The plateau stage is the main energy absorption stage of the honeycomb compression, and the plateau force and energy absorption of a honeycomb are the focus of this study [29]. It can be seen from Figure 5 that the simulation plateau force of a honeycomb block with a cell wall length of 6 mm was 3.72 kN, and the test plateau force of a honeycomb block with a cell wall length of 6 mm was 3.63 kN. The error of the plateau force between the simulation and the test was 2.48%. Similarly, the simulation plateau force of a honeycomb block with a cell wall length of 10 mm was 3.41 kN, and the test plateau force of a honeycomb block with a cell wall length of 10 mm was 3.58 kN. The error of the plateau force between the simulation and the test was 4.75%. The errors of the plateau force of the simulation and test were both within 5%. Thus, the developed finite element models are reliable and can be used for subsequent simulation to obtain the cell magnification times of different size honeycombs, so as to further fit the equation that can predict the cell magnification times.

Expanded Cell Equivalence Method
According to the honeycomb density expression given by Gibson [30], it is known that the equivalent apparent densities need to satisfy the following: where is the aluminum foil density and is the thickness-to-length ratio of the honeycomb. Therefore, we have the following:

Expanded Cell Equivalence Method
According to the honeycomb density expression given by Gibson [30], it is known that the equivalent apparent densities need to satisfy the following: where is the aluminum foil density and is the thickness-to-length ratio of the honeycomb. Therefore, we have the following:

Expanded Cell Equivalence Method
According to the honeycomb density expression given by Gibson [30], it is known that the equivalent apparent densities need to satisfy the following: where is the aluminum foil density and is the thickness-to-length ratio of the honeycomb. Therefore, we have the following:

Expanded Cell Equivalence Method
According to the honeycomb density expression given by Gibson [30], it is known that the equivalent apparent densities need to satisfy the following: where is the aluminum foil density and is the thickness-to-length ratio of the honeycomb. Therefore, we have the following:

Expanded Cell Equivalence Method
According to the honeycomb density expression given by Gibson [30], it is known that the equivalent apparent densities need to satisfy the following: where is the aluminum foil density and is the thickness-to-length ratio of the honeycomb. Therefore, we have the following:   Figure 5. Force-compression displacement curves of honeycomb structures with cell wall lengths of (a) 6 mm and (b) 10 mm.

Expanded Cell Equivalence Method
According to the honeycomb density expression given by Gibson [30], it is known that the equivalent apparent densities need to satisfy the following: where is the aluminum foil density and is the thickness-to-length ratio of the honeycomb. Therefore, we have the following:   Figure 5. Force-compression displacement curves of honeycomb structures with cell wall lengths of (a) 6 mm and (b) 10 mm.

Expanded Cell Equivalence Method
According to the honeycomb density expression given by Gibson [30], it is known that the equivalent apparent densities need to satisfy the following: where is the aluminum foil density and is the thickness-to-length ratio of the honeycomb. Therefore, we have the following:   Figure 5. Force-compression displacement curves of honeycomb structures with cell wall lengths of (a) 6 mm and (b) 10 mm.

Expanded Cell Equivalence Method
According to the honeycomb density expression given by Gibson [30], it is known that the equivalent apparent densities need to satisfy the following: where is the aluminum foil density and is the thickness-to-length ratio of the honeycomb. Therefore, we have the following:

Expanded Cell Equivalence Method
According to the honeycomb density expression given by Gibson [30], it is known that the equivalent apparent densities need to satisfy the following: where is the aluminum foil density and is the thickness-to-length ratio of the honeycomb. Therefore, we have the following: where ( ) is the equivalent density before magnification, ( ) is the equivalent density after magnification, is the thickness-to-length ratio before magnification, and is the thickness-to-length ratio after magnification. The mass m can be calculated as:

Expanded Cell Equivalence Method
According to the honeycomb density expression given by Gibson [30], it is known that the equivalent apparent densities ρ c need to satisfy the following: where ρ 0 is the aluminum foil density and µ is the thickness-to-length ratio of the honeycomb. Therefore, we have the following: where (ρ c ) 1 is the equivalent density before magnification, (ρ c ) 2 is the equivalent density after magnification, µ 1 is the thickness-to-length ratio before magnification, and µ 2 is the thickness-to-length ratio after magnification. The mass m can be calculated as: where A is the bearing surface area and α T is the thickness of the honeycomb in the T direction.
The bearing surface areas of the honeycomb core blocks before and after the equivalent were equal. When the honeycomb bearing surface was large, the error of the bearing surface area before and after cell magnification was very small, and the honeycomb core thickness α T was equal. Therefore, we have the following: Similarly, the out-of-plane modulus of the honeycomb is: where E s is the elastic modulus of the honeycomb foil base material. From this, the relationship between the out-of-plane moduli (E * 3 ) 1 and (E * 3 ) 2 of the honeycomb structure before and after the equivalent can be deduced as: It is known that the relationship between the plateau strengths (σ m ) 1 and (σ m ) 2 of the equivalent honeycomb before and after is obtained by the empirical formula [30] According to Equations (2), (4), (6), and (7), the mechanical properties of a honeycomb are related to the cell thickness-to length ratio. In theory, when the honeycomb structure has the same thickness-to-length ratio, it has the same mechanical properties. Properties and energy absorption characteristics can be directly considered equivalent.

Equivalent Finite Element Model
The equivalent finite element models with consistent thickness-to-length ratios were established using the validated finite element method above. The geometrical size of the finite element model was 300 × 300 × 100 mm (W × L × T). The honeycomb included at least 13 cells [2], so the lengths of the honeycomb cells were 4 mm, 5 mm, 6 mm, 8 mm, 10 mm, 12 mm, 16 mm, 18 mm, 20 mm, and 24 mm [25,31,32]. The cell wall thickness was obtained by the thickness-to-length ratio µ. Here, the thickness-to-length ratio (µ) of the honeycombs was 0.005. As described in the expanded cell equivalence method, the ten groups of honeycomb models satisfied the expanded cell equivalence relation. Figure 6 shows the cell magnification relationship of the honeycomb models with a 4 mm cell wall length to explain the expanded cell equivalence method. The mesh, loads, boundary conditions, solutions, and post-processing of the finite element model were the same as those mentioned above. The mass of the moving rigid wall was set at a ton to make the models fully compacted.  Figure 7 shows that the force-compression displacement curves of the honeycomb under different cell magnification times have four stages. The curve of the honeycomb with a cell wall length of 4 mm is in the middle, and the honeycombs of other sizes are distributed symmetrically on both sides of the curve of the honeycomb with a cell wall length of 4 mm. The fluctuation of the curve with a small cell wall length is relatively stable. As the cell wall length increases, the plateau force fluctuation gradually increases. The main reason for this phenomenon is that the cell wall is compressed according to the wavelength λ asymptotically folded. This wavelength is generally equal to the cell wall length l. Therefore, in the process of equivalent treatment, the increase of l will inevitably lead to the fluctuation of the buckling wavelength and load [33].  Table 2 shows the specific results of peak force, plateau force, energy absorption, and calculation time with the honeycomb models. The calculation time refers to the time used by the computer for the out-of-plane compression simulation process, and the calculation was completed on a workstation with multiple CPUs with the same configurations and the same operating systems. Table 2 shows that the peak force before and after the cell magnification was not very different, but some aluminum honeycombs before and after cell magnifications had a large difference in plateau force and energy absorption. For example, the difference between the plateau force of a honeycomb with a cell wall length of 8 mm and a honeycomb with a cell wall length of 16 mm was 15.72 kN, and the difference in energy absorption was 1030 J. The thickness-to-length ratio was the same, and the energy absorption characteristics of the honeycomb with a cell wall length of 8 mm after double magnification were not equivalent to that before the cell magnification. In addition, it can be seen from Table 2 that the larger the cell wall length of the honeycomb, the  The main reason for this phenomenon is that the cell wall is compressed according to the wavelength λ asymptotically folded. This wavelength is generally equal to the cell wall length l. Therefore, in the process of equivalent treatment, the increase of l will inevitably lead to the fluctuation of the buckling wavelength and load [33].  Figure 7 shows that the force-compression displacement curves of the honeycomb under different cell magnification times have four stages. The curve of the honeycomb with a cell wall length of 4 mm is in the middle, and the honeycombs of other sizes are distributed symmetrically on both sides of the curve of the honeycomb with a cell wall length of 4 mm. The fluctuation of the curve with a small cell wall length is relatively stable. As the cell wall length increases, the plateau force fluctuation gradually increases. The main reason for this phenomenon is that the cell wall is compressed according to the wavelength λ asymptotically folded. This wavelength is generally equal to the cell wall length l. Therefore, in the process of equivalent treatment, the increase of l will inevitably lead to the fluctuation of the buckling wavelength and load [33].  Table 2 shows the specific results of peak force, plateau force, energy absorption, and calculation time with the honeycomb models. The calculation time refers to the time used by the computer for the out-of-plane compression simulation process, and the calculation was completed on a workstation with multiple CPUs with the same configurations and the same operating systems. Table 2 shows that the peak force before and after the cell magnification was not very different, but some aluminum honeycombs before and after cell magnifications had a large difference in plateau force and energy absorption. For example, the difference between the plateau force of a honeycomb with a cell wall length of 8 mm and a honeycomb with a cell wall length of 16 mm was 15.72 kN, and the difference in energy absorption was 1030 J. The thickness-to-length ratio was the same, and the energy absorption characteristics of the honeycomb with a cell wall length of 8 mm after double magnification were not equivalent to that before the cell magnification. In addition, it can be seen from Table 2 that the larger the cell wall length of the honeycomb, the  Table 2 shows the specific results of peak force, plateau force, energy absorption, and calculation time with the honeycomb models. The calculation time refers to the time used by the computer for the out-of-plane compression simulation process, and the calculation was completed on a workstation with multiple CPUs with the same configurations and the same operating systems. Table 2 shows that the peak force before and after the cell magnification was not very different, but some aluminum honeycombs before and after cell magnifications had a large difference in plateau force and energy absorption. For example, the difference between the plateau force of a honeycomb with a cell wall length of 8 mm and a honeycomb with a cell wall length of 16 mm was 15.72 kN, and the difference in energy absorption was 1030 J. The thickness-to-length ratio was the same, and the energy absorption characteristics of the honeycomb with a cell wall length of 8 mm after double magnification were not equivalent to that before the cell magnification. In addition, it can be seen from Table 2 that the larger the cell wall length of the honeycomb, the shorter the calculation time and the higher the calculation efficiency. For instance, the calculation time of the honeycomb with a cell wall length of 24 mm was reduced by 26.39 times compared to the original honeycomb. This proves that the time and resource consumption can be greatly reduced when applying the cell magnification equivalent method to simulate the out-of-plane compression of dense cellular honeycombs, which is consistent with the results of previous studies [22].

Equivalent Model Results
From the above results, the mechanical properties and energy absorption characteristics of some honeycombs before and after cell magnification is not completely equivalent. These results indicate the necessity of exploring the cell magnification times of honeycombs.

Determination of Cell Magnification Times
The correlation and analysis (CORA) rating method is a commonly used curve correlation evaluation method. It is calculated by using two different metrics: the corridor metric and the cross-correlation metric. The cross-correlation metric contains three closely related sub-ratings: the progression rating, phase shift rating, and size rating [34,35]. The results of the two metrics are summed up for the total CORA rating by using individual weighting factors for each metric. The corridor rating calculates the deviation of two curves utilizing corridor fitting. The cross-correlation rating calculates the phase shift, size, and progression of the two curves. The degree of correlation between the two curves is evaluated with a rating result from 0 to 100%. The closer the rating result is to 100%, the higher the correlation between the two curves. Figure 8 shows the correlation degree rating results of the force-compression displacement curves of the aluminum honeycomb before and after cell magnification. The specific rating results are given in Appendix A. The CORA ratings obtained for the curves correlation before and after cell magnification (Appendix A) varied from poor (0 to 64%) to good (65% to 85%) to excellent (86% to 100%). Table 3 shows the aluminum honeycomb cell sizes and the corresponding cell magnification times with the rating results that were above 90%. Table 3. Cell magnification times of honeycomb cells. The CORA ratings obtained for the curves correlation before and after cell magnification (Appendix A) varied from poor (0 to 64%) to good (65% to 85%) to excellent (86% to 100%). Table 3 shows the aluminum honeycomb cell sizes and the corresponding cell magnification times with the rating results that were above 90%.

Validation of Cell Magnification Times
The theoretical derivation of the cell magnification equivalent method above shows that the equivalence of mechanical properties and energy absorption is only related to the honeycomb thickness-to-length ratio. Thus, the cell magnification relationship is also applicable to honeycombs of different sizes. From the above study, the honeycomb with a cell wall length of 6 mm can have equivalent properties and energy absorption characteristics, after expanding 1.67 times, to those of the honeycomb with a cell wall length of 10 mm.
The drop-weight impact tests of honeycombs with cell wall lengths of 6 mm and 10 mm are used to verify the accuracy of cell magnification times in this section, and the sizes of the honeycombs were 100 × 100 × 100 mm and 150 × 150 × 100 mm (W × L × T). Figures 9 and 10 show the experimental force-compression displacement curves of the honeycomb structures that were 100 × 100 × 100 mm and 150 × 150 × 100 mm (W × L × T), respectively.  Figure 9 shows the plateau force and energy absorption results. The test plateau force of a honeycomb block with a cell wall length of 6 mm was 3.63 kN, and the energy absorption was 159.41 J. The test plateau force of a honeycomb block with a cell wall length of 10 mm was 3.58 kN, and the energy absorption was 151.53 J. The plateau force error of the honeycombs with wall lengths of 6 mm and 10 mm was 1.4%, and the energy absorption error of the honeycombs with wall lengths of 6 mm and 10 mm was 4.9%. The error was within 5%. Similarly, Figure 10 shows that the plateau force with a cell wall length of 6 mm was 7.31 kN, and the energy absorption was 322.64 J; the plateau force with a cell wall length of 10 mm was 7.28 kN, and the energy absorption was 319.76 J. The plateau force error of the honeycombs with wall lengths of 6 mm and 10 mm was 0.4%, and the energy absorption error of the honeycombs with wall lengths of 6 mm and 10 mm was  Figure 9 shows the plateau force and energy absorption results. The test plateau force of a honeycomb block with a cell wall length of 6 mm was 3.63 kN, and the energy absorption was 159.41 J. The test plateau force of a honeycomb block with a cell wall length of 10 mm was 3.58 kN, and the energy absorption was 151.53 J. The plateau force error of the honeycombs with wall lengths of 6 mm and 10 mm was 1.4%, and the energy absorption error of the honeycombs with wall lengths of 6 mm and 10 mm was 4.9%. The error was within 5%. Similarly, Figure 10 shows that the plateau force with a cell wall length of 6 mm was 7.31 kN, and the energy absorption was 322.64 J; the plateau force with a cell wall length of 10 mm was 7.28 kN, and the energy absorption was 319.76 J. The plateau force error of the honeycombs with wall lengths of 6 mm and 10 mm was 0.4%, and the energy absorption error of the honeycombs with wall lengths of 6 mm and 10 mm was 0.9%. The error was within 5%.
The plateau force and energy absorption measures of the honeycombs with cell wall lengths of 6 mm and 10 mm were very close. The rating results (force-compression displacement curves) of the honeycombs with cell wall lengths of 6 mm and 10 mm are shown in Table 4; good correlation was achieved between the honeycombs with a cell wall length of 6 mm and the honeycombs with a cell wall length of 10 mm. It can be seen from Table 4 that the size of the honeycomb block is 100 × 100 × 100 mm or 150 × 150 × 100 mm, and the CORA rating results of the force-compression displacement curves of the cell wall lengths of 6 mm and 10 mm were all above 90%.
The comprehensive plateau force, energy absorption error, and rating results, and the equivalent relationship of 1.67 times magnification of the cells with different sizes of honeycombs with a cell wall length of 6 mm, is accurate and reliable. This test proves that the cell magnification of 1.67 times is effective, which shows that the cell magnification times of different sizes of honeycomb obtained by simulation is applicable, and the results obtained in the previous article have credibility.

Function Prediction
The predicted function of the equivalent fold of magnification cells is studied in this section. The cell magnification times for different cell sizes are listed in Table 3. The honeycombs with cell wall lengths of 4, 5, 6, and 8 mm have two or more cell magnification times; therefore, the first step in fitting a function is to determine the optimal cell magnification times. The results in Table 2 show that the larger the cell wall length, the shorter the calculation time. Therefore, the optimal cell magnification times of honeycombs with cell wall lengths of 4 mm, 5 mm, 6 mm, and 8 mm were selected as the maximum values, and the optimal cell magnification times of the honeycombs with cell wall lengths of 4, 5, 6, and 8 mm were 6 times, 3.60 times, 3 times and 2.25 times, respectively. Figure 11 shows the calculated cell magnification times and fitting function. The honeycomb includes at least 13 cells [2]; therefore, the cell magnification cannot be infinitely expanded. With the increase in cell wall lengths, the cell magnification times are close to 1.

. Function Prediction
The predicted function of the equivalent fold of magnification cells is studied in this section. The cell magnification times for different cell sizes are listed in Table 3. The honeycombs with cell wall lengths of 4, 5, 6, and 8 mm have two or more cell magnification times; therefore, the first step in fitting a function is to determine the optimal cell magnification times. The results in Table 2 show that the larger the cell wall length, the shorter the calculation time. Therefore, the optimal cell magnification times of honeycombs with cell wall lengths of 4 mm, 5 mm, 6 mm, and 8 mm were selected as the maximum values, and the optimal cell magnification times of the honeycombs with cell wall lengths of 4, 5, 6, and 8 mm were 6 times, 3.60 times, 3 times and 2.25 times, respectively. Figure 11 shows the calculated cell magnification times and fitting function. The honeycomb includes at least 13 cells [2]; therefore, the cell magnification cannot be infinitely expanded. With the increase in cell wall lengths, the cell magnification times are close to 1.
A first-order decay exponential function (ExpDec1) that fits the trend of the curve was selected for fitting. A characteristic of the first-order decay exponential function is that the y value changes exponentially in the initial stage, and then it gradually slows down the change speed as x increases. When x increases to a certain extent, the y value gradually converges. The ExpDec1 function expression is as follows: After 11 iterations, the obtained ExpDec1 function parameter value A 1 was 37.64, t 1 was 1.85, and y 0 was 1.48. Substituting these values into Equation (8) shows the following relationship between the optimal cell magnification time and the honeycomb cell wall length: y = 1.48 + 37.64/exp(x/1.85) (9) where x is the cell wall length and y is the corresponding optimal cell magnification time.
The fitting curve is shown in Figure 11.

Predictive Effect Evaluation
The function that can predict the optimal cell magnification time is above. To evaluate the fit of the model, the goodness of fit is often used as a measure. Its coefficient of determination is calculated as follows: where SSE is the residual sum of squares and SST is the total deviation sum of squares. The calculation formulas of SSE and SST are as follows: = ( ) (11) = ( ) (12) where is the true value, is the fitted value, and is the mean of . The variance analysis results are shown in Table 5. The closer the value is to 1, the better the fit. According to Table 5 and Equation (10), is 0.96. According to the principle that 0.95 is a good fitting effect [36], the ExpDec1 function can predict the relationship between the cell wall length of the honeycomb cell and the optimal cell magnification Figure 11. Prediction curve of optimal cell magnification times.

Predictive Effect Evaluation
The function that can predict the optimal cell magnification time is above. To evaluate the fit of the model, the goodness of fit is often used as a measure. Its coefficient of determination is calculated as follows: where SSE is the residual sum of squares and SST is the total deviation sum of squares. The calculation formulas of SSE and SST are as follows: where y i is the true value,ŷ i is the fitted value, and y i is the mean of y i . The variance analysis results are shown in Table 5. The closer the R 2 value is to 1, the better the fit. According to Table 5 and Equation (10), R 2 is 0.96. According to the principle that > 0.95 is a good fitting effect [36], the ExpDec1 function can predict the relationship between the cell wall length of the honeycomb cell and the optimal cell magnification time.

Conclusions
In this paper, the improved equivalent method of exchanging small cells for large cells of honeycombs with the same thickness-to-length ratios is presented. The equivalent finite element models of honeycomb structures are established and validated by experiments. The prediction function of the optimal cell magnification times is also derived. Based on the analyses mentioned above, major conclusions can be drawn as follows: The honeycomb structures with the same thickness-to-length ratios have the same mechanical properties and energy absorption characteristics, according to the theoretical calculation. Therefore, a series of finite element models with thickness-to-length ratios of 0.005 were established to study the cell magnification equivalence relationship. Numerical results showed that the mechanical properties and energy absorption characteristics of some honeycombs before and after cell magnification could not be completely equivalent according to the traditional equivalent method. The main reason for this phenomenon is that the bearing area becomes smaller after the cell expansion, which causes errors in the mechanical properties before and after the cell expansion. However, after reasonably and accurately matching the bearing area before and after the equivalent method, the equivalent accuracy can be effectively improved. Thus, the adjusted cell magnification times of different sizes of honeycombs are obtained by the CORA rating of the honeycomb simulation compression curves, which proves that the results of the cell magnification time relationships are credible by the verified drop-weight impact test, and the rating results are above 90%. The out-of-plane compression simulation results show that the larger the honeycomb cell wall length, the shorter the computation time. The optimal cell magnification times for different cell wall lengths were determined according to CORA rating results and calculation times.
In addition, the ExpDec1 function satisfying the relationship between the optimal cell magnification time and the cell wall length was obtained according to the results of experiments and simulations. This function is y = 1.48 + 37.64/ exp x 1.85 . A cell magnification time for the honeycomb structures that have directly equivalent energy absorption characteristics and mechanical properties can be obtained under the same thickness-to-length ratio. The sparseness of the honeycomb structures with dense cells is realized, and the cell scale of the honeycomb numerical simulation is effectively reduced.