Design of a New Energy Absorption Box with Honeycomb Structure for Vehicles and Research on Vehicle Crashworthiness

Design of a New Energy Absorption Box with Honeycomb Structure for Vehicles and Research on Vehicle Crashworthiness | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Design of a New Energy Absorption Box with Honeycomb Structure for Vehicles and Research on Vehicle Crashworthiness Honglin Wang, Zhanyu Wang, Xuejing Du, Ning Wang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6258409/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 21 May, 2025 Read the published version in Iranian Journal of Science and Technology, Transactions of Mechanical Engineering → Version 1 posted 11 You are reading this latest preprint version Abstract The energy absorbing box plays the role of buffering energy absorption in the process of frontal collision of automobiles, but the current structure of the energy absorbing box is single, and the energy absorbing effect is limited. In this study, the honeycomb structure was filled into a thin-walled square tube to design a new type of automotive energy absorbing box. Firstly, a frontal collision finite element model is established with a vehicle model as the research object, and the crashworthiness defects of the vehicle model under the frontal collision condition are pointed out. Next, two filling forms of new energy absorbing boxes were designed, and the crashworthiness was compared through impact simulation, followed by multi-objective optimization to further enhance their performance. Finally, the original thin-walled square tube energy absorbing box was replaced with the honeycomb-filled design. A simplified model was used to compare the crashworthiness of the vehicle before and after the replacement. The results showed that, after the replacement, the vehicle's acceleration decreased by 11.49%, occupant compartment intrusion was reduced by 17.48%, and the energy absorption ratio of the energy absorbing box increased by 12.51%. Crashworthiness and occupant safety have been improved. Honeycomb structure Energy absorbing box Crashworthiness Multi-objective optimization Replacement design Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 Figure 16 Figure 17 Figure 18 Figure 19 Figure 20 Figure 21 1 Introduction With the popularization of automobiles, the issue of crash safety has garnered significant attention. Among various traffic accidents, frontal collisions are the most severe in terms of causing occupant injuries(Gulshan et al. 2022 ). In the past, automotive energy absorbing boxes primarily featured thin-walled tube structures, which allowed for lightweight design while providing a certain level of buffering and energy absorption(Min et al. 2017). However, as vehicle speeds continue to increase, the impact forces have grown larger, and the traditional thin-walled tube structures of energy absorbing boxes can no longer meet the energy absorption requirements. In recent years, with the continuous advancement of manufacturing capabilities, various complex structures and new materials have gradually been applied in the automotive industry. For instance, honeycomb structures, as a type of biomimetic porous design, exhibit excellent energy absorption characteristics and load-bearing capacity under impact loads(Xiang et al. 2017). Biomimetic honeycombs with functionally graded designs show significant improvements in energy absorption and crashworthiness compared to traditional honeycombs(Pattaramon et al. 2024 ; Mohammad et al. 2024; Wu et al. 2021 ; Deng et al. 2024 ). Additionally, studies have shown that the energy absorption characteristics of honeycomb structures are greatly influenced by their structural parameters and geometric configurations. Sun et al. ( 2021 ) conducted impact tests on honeycomb structures with different parameters under four energy levels, demonstrating that the height of the honeycomb has no effect on energy absorption, while increasing the wall thickness significantly enhances energy absorption performance. Building on this, some scholars have designed new biomimetic honeycomb structures inspired by natural biological features, such as elliptical arc concave honeycombs(Tang et al. 2025 ), 3D-printed spherical honeycombs(Deng et al. 2024 ), origami-inspired negative Poisson's ratio honeycombs(Zhou et al. 2024 ), and biomimetic glass sponge-like honeycombs(Wang et al. 2024 ). Compared to traditional hexagonal honeycomb structures, these designs exhibit notable improvements in energy absorption and impact resistance. The material properties of honeycomb structures play a crucial role in their energy absorption and crashworthiness. As a result, researchers have conducted extensive studies on composite honeycomb structures in recent years, further enhancing their dynamic performance(Guo et al. 2024 ; Zhou et al. 2018 ; Hussain et al. 2025 ; Hua et al. 2025 ; Liu et al. 2025) . In the field of automotive energy absorption box research, Nia et al. (2014) investigated energy absorbing boxes with hexagonal and octagonal cross-sections. They found that both exhibited similar energy absorption values, but the hexagonal cross-section had a lower peak collision force, making it more suitable for designing new energy absorbing boxes. Building on this, researchers have developed novel energy absorbing boxes by filling thin-walled tubes with materials such as aluminum foam, porous structures, and negative Poisson's ratio structures, thereby improving crashworthiness(Zhao et al. 2024 ; Xie et al. 2023 ; Sorrawit et al. 2025 ). Some scholars have explored the energy absorption capabilities of energy absorbing boxes by selecting different material types, discovering that composite materials outperform traditional materials in terms of energy absorption and crashworthiness(Yoo et al. 2010 ; Fatima et al. 2024 ; Xing et al. 2025 ; Francesco et al. 2024 ). Multi-objective optimization methods are widely used in engineering design optimization problems, enabling the identification of optimal design parameters that balance conflicting objectives to achieve the best overall performance. For instance, Zhang et al. ( 2024 ) established a response surface model and applied a multi-objective optimization algorithm to enhance the mechanical performance of torsional plate energy absorbing components. Yang et al. ( 2025 ) used a three-level Box-Behnken method to perform multi-objective optimization on a biomimetic diaphragm-reinforced bamboo system, validating its performance advantages over typical energy absorbing boxes. Wang et al. ( 2022 ) proposed a novel hexagonal pyramid-shaped energy absorbing box composed of an internal negative Poisson's ratio structure and an outer shell. They optimized the box using response surface methodology and multi-objective optimization to improve its crashworthiness. The optimization results demonstrated that the new energy absorbing box achieves both crashworthiness and lightweight design. In summary, significant progress has been made in enhancing the energy absorption characteristics of honeycomb structures and designing automotive energy absorbing boxes using porous materials. However, biomimetic honeycomb structures with relatively low densities have not been widely applied in automotive energy absorbing boxes. Moreover, existing research has primarily focused on the crashworthiness of individual energy absorbing boxes without integrating them into full-vehicle crash scenarios. During a frontal collision, the stress conditions of the energy absorbing box are highly complex. Only by incorporating the new energy absorbing box into a full-vehicle model for crash analysis can its effectiveness in improving the vehicle's crashworthiness be convincingly demonstrated. Therefore, this study proposes a novel automotive energy absorbing box by filling thin-walled square tubes with honeycomb structures. This design will be installed in the front bumper beam of a vehicle for full-vehicle frontal crash analysis. The aim is to leverage the honeycomb structure to enhance the crashworthiness of the energy absorbing box during frontal collisions, thereby improving the vehicle's overall crashworthiness and occupant safety. 2 Establishment of the Frontal Collision Finite Element Model and Analysis of Simulation Results 2.1 Basic Parameters of the Vehicle Model The subject of this study is a compact car with a length of 4635 mm, a width of 1780 mm, and a height of 1455 mm. The wheelbase is 2700 mm, and the curb weight is 1320 kg. The front bumper beam consists of a crossbeam, an energy absorbing box, and longitudinal beams, as shown in Fig. 1 . The vehicle employs the simplest form of thin-walled square tube energy absorbing boxes. It is required that, in the event of a frontal collision, the energy absorbing box can quickly absorb and dissipate collision energy, thereby enhancing the vehicle's safety performance. The key component parameters that play a decisive role in the crashworthiness of the vehicle during frontal collisions are listed in Table 1 . Table 1 Parameters of some components Parts Length (mm) Width (mm) Height (mm) Thickness (mm) material Mass (kg) Crossbeam 1000 35 110 1.8 Carbon 13.3 Energy absorbing box 270 200 105 1.5 Aluminum alloy 0.4 Stringers 2500 100 105 1.8 Carbon 21.5 2.2 Full-Vehicle Finite Element Modeling First, the three-dimensional model of the vehicle was imported into Hypermesh software for geometric cleanup. Components that had no impact on the collision were removed, and geometric defects and missing edges caused by software incompatibility were repaired. The full vehicle was then meshed, with the crossbeam, energy absorption box, and longitudinal beams using 8 mm quadrilateral shell elements, while other parts were meshed with a size of 15 mm. The occupant compartment was meshed using solid elements. The full-vehicle model consisted of 19,032 elements, including 15,631 shell elements, 2,312 solid elements, and 1,089 triangular elements, accounting for 5.72% of the total, which is less than 10%. To ensure the mesh quality met the requirements, the entire model underwent a mesh quality check, with the aspect ratio, Jacobian, maximum internal angle, and warping factor all satisfying the criteria. After meshing, material and property definitions were assigned to the components. Rigid materials were assigned to the rigid wall and non-deformable parts at the rear of the vehicle to save computation time, while elastic-plastic materials were assigned to the crossbeam, energy absorbing box, and longitudinal beams, which undergo deformation during collisions. The material parameters for some components of the full-vehicle model are listed in Table 2 . RBE2 rigid connections were used to connect adjacent components in the model, and spot welding was applied between the crossbeam, energy absorbing box, and longitudinal beams. In the full-vehicle model, the vehicle's self-contact was set as single-sided automatic contact, while surface-to-surface contact was defined between the bumper beam and the rigid wall, as well as between the crossbeam, energy absorbong box, and longitudinal beams, with both static and dynamic friction coefficients set to 0.2. The simulation termination time was set to 0.15 s, with a time step of 0.0015 s. The global hourglass control method was used for computation, with a global stiffness coefficient of 0.1. To obtain the required simulation results, output data such as node deformation, velocity, acceleration, kinetic energy, internal energy, hourglass energy, and slip energy were configured. The established finite element model for the full-vehicle frontal collision is shown in Fig. 2 . 2.3 Model Reliability Validation To ensure the accuracy of the simulation results, the model must undergo reliability validation. A commonly used method is to verify whether the model satisfies the law of energy conservation during the collision process and to ensure that the hourglass energy does not exceed 5% of the total energy. Otherwise, the model's accuracy would be considered low, and the simulation results would be unreliable for this study. Figure 3 shows the energy variation curves during the simulation. The total energy remains stable at approximately 208 kJ, with kinetic energy decreasing and internal energy increasing in a symmetrical manner. The hourglass energy is 4.99 kJ, accounting for 2.39% of the total energy, which is below the 5% threshold. Table 2 Material properties of some components in the full-vehicle finite element model Parts Density (g·cm − 3 ) Yield strength (MPa) Modulus of elasticity (GPa) Poisson's ratio Crossbeam 7.82 260 210 0.27 Energy absorbing box 2.75 240 68.3 0.30 Stringers 7.81 235 190 0.27 In this study, to further validate the reliability of the model, the deformation of the simulation model was compared with that of a real vehicle test conducted under C-NCAP for the same vehicle type. As shown in Figs. 4 and 5 , which compare the vehicle deformation and acceleration between the real vehicle test and the simulation model, the two exhibit a high degree of similarity. Based on the above analysis, it can be concluded that the full-vehicle frontal collision model established in this study is highly reliable. The simulation results align well with the actual collision outcomes, making this model suitable for subsequent research. 2.4 Analysis of Simulation Results Figure 6 shows the deformation diagram of the vehicle after the frontal collision. Significant deformation occurs in the front part of the vehicle. The crossbeam deforms first, then compresses the front fender outward and the energy absorbing box inward. The deformation of the wheel axle causes the wheels to cave inward, affecting the force transmission path of the front bumper beam. During the collision, the engine hood bends. All these deformed components can impact the occupant compartment, thereby endangering the safety of the occupants. Figure 7 illustrates the deformation of the crossbeam, energy absorbing box, and longitudinal beams. It can be observed that the crossbeam undergoes severe deformation and compresses the energy absorbing box, while the right longitudinal beam also deforms. This indicates that the stiffness of the crossbeam is insufficient to effectively resist the intrusion of the rigid wall. Figure 8 shows the energy absorption curve of the energy absorbing box, which accounts for 7.69% of the total energy. This demonstrates that the energy absorption capacity of this ordinary thin-walled tube energy absorbing box is severely inadequate. It cannot effectively absorb collision energy or reduce the impact force transmitted to the occupants. 3 Design and Multi-objective Optimization of the New Energy Absorbing Box 3.1 Structural Design of the New Energy Absorbing Box Based on the analysis above, the traditional thin-walled square tube energy absorbing box no longer meets the crashworthiness requirements. Therefore, this study proposes the use of a biomimetic honeycomb structure with high energy absorption characteristics to fill the thin-walled square tube, designing a new automotive energy absorbing box to improve the vehicle's crashworthiness. 3.1.1 In-Plane Honeycomb-filled Energy Absorbing Box Design In reference (Shi et al. 2024 ), quasi-static compression tests were conducted on traditional hexagonal honeycomb structures with different relative densities. It was found that when the pore size ρ is 2.4 mm, the energy absorption characteristics of the honeycomb structure are optimal. The initial wall thickness t 1 of the honeycomb structure is set to 0.06 mm. In Fig. 9 (a) , during the design process of the in-plane honeycomb-filled energy absorbing box, to ensure compatibility between the honeycomb structure and the thin-walled square tube (length L = 270 mm, width W = 200 mm, height H = 105 mm, thickness t 2 = 1.5 mm), the width W 2 of the honeycomb structure is required to be 197 mm ( W 2 = W − 2 T ), the height H 2 to be 270 mm ( H 2 = L ), and the length L 2 to be 102 mm ( L 2 = H − 2 T ). The thin-walled square tube and the filled honeycomb structure are assembled in Solidworks to form the new in-plane honeycomb-filled automotive energy absorbing box. To more clearly observe the internal honeycomb structure, the external thin-walled square tube of the new energy absorbing box is treated with a frame removal and hidden line retention process. 3.1.2 Out-Plane Honeycomb-filled Energy Absorbing Box Design For the design of the out-plane honeycomb-filled energy absorbing box, the dimensions of the thin-walled square tube remain unchanged. The difference lies in the dimensions of the honeycomb structure: the height H 1 is 102 mm ( H 1 = H − 2 T ), the width W 1 is 197 mm ( W 1 = W − 2 T ), and the length L 1 is 270 mm ( L 1 = L ). The thin-walled square tube and the filled honeycomb structure are assembled to form the new out-plane honeycomb-filled automotive energy absorbing box, as shown in Fig. 9 (b) . 3.2 Crashworthiness Analysis of the Energy Absorbing Box 3.2.1 Establishment of the Impact Finite Element Model The geometric model of the automotive energy absorbing box is imported into Abaqus in IGS file format. The Explicit dynamic analysis module is used to simulate the impact process of the in-plane/out-plane filled new automotive energy absorbing box and the traditional thin-walled square tube energy absorbing box. The material of the energy absorbing box is simulated using aluminum alloy 6061, with a density of 2.75 g·cm⁻³, an elastic modulus of 68,300 MPa, a yield strength of 250 MPa, and a Poisson's ratio of 0.30. A material section is created and assigned to the energy absorbing box, completing the material property assignment. Two rigid plates are created at the top and bottom of the energy absorbing box and assembled with it. The dynamic explicit analysis step is set to a duration of 0.10 s. General contact is selected as the interaction property between the upper and lower rigid plates and the automotive energy absorbing box, with both static and dynamic friction coefficients set to 0.2. The lower plate is fixed, while the upper plate is constrained with a velocity of 5 m/s and assigned a mass of 500 kg. A global mesh seed size of 0.05 mm is used to generate quadrilateral meshes, balancing computation time and convergence criteria. Outputs include the energy absorption value and displacement of the energy absorbing box. The impact finite element model of the automotive energy absorbing box is shown in Fig. 10 . The impact finite element models for the traditional thin-walled tube energy absorbing box and the new honeycomb-filled automotive energy absorbing box are identical in terms of materials, dimensions, impact speed, and other parameters, except for the presence or absence of the filling structure. 3.2.2 Model Reliability Verification To verify the reliability of the model, referring to Lee's study(Lee et al. 2008 ), thin-walled square tubes of the same size and material were designed, and impact simulations were conducted under identical loading conditions. Figure 11 shows a comparison of the deformation of the thin-walled square tubes between the simulation and the experiment. The deformation modes of the two are consistent. In the simulation, the initial deformation of the thin-walled square tube starts from the triggered side and reaches the maximum load at the beginning of the subsequent collapse. Therefore, it can be concluded that the finite element model of the thin-walled square tube designed for comparison with the experimental deformation results has high reliability. The energy absorbing box finite element model established in this study and the thin-walled square tube finite element model are both explicit analysis models under the same Abaqus/Explicit setup. Moreover, the material properties and mesh size, which have the greatest impact on the simulation results, are the same. Only the dimensions and shape of the impacted structure, which have a smaller impact on the result errors, were modified. Thus, it can be considered that the energy absorbing box finite element model and the thin-walled square tube finite element model have similar accuracy in terms of results. Based on the above analysis, the impact finite element model of the energy absorbing box established in this study is reliable and can be used for subsequent research. 3.2.3 Crashworthiness Analysis Figure 12 shows the deformation results of the energy absorbing box under impact loading. It can be observed that the traditional thin-walled square tube energy absorbing box, without internal honeycomb structure support, undergoes buckling deformation of the thin wall under impact loading. When the thin-walled tube is filled with a honeycomb structure in-plane, the honeycomb structure experiences bending deformation of adjacent cell walls and plastic deformation at the joints of adjacent cell walls. When the thin-walled tube is filled with a honeycomb structure out-plane, the honeycomb structure undergoes buckling and deformation of the cell walls, which is similar to the traditional thin-walled tube energy absorbing box. From the perspective of deformation mode, the energy absorbing box with out-plane filled honeycomb structure exhibits buckling of the honeycomb cell walls, which squeeze against each other, forming a layered structure that continues to compress other honeycomb cells inward. Under the same energy absorption conditions, this deformation model can reduce the amount of deformation. Typically, the crashworthiness evaluation indicators for energy absorbing boxes include total energy absorption ( E ), specific energy absorption ( E a ), initial peak impact force ( F p ), average impact force ( F m ), and maximum deformation ( D ). Among these, the initial peak load represents the maximum load value first reached by the structure during the impact loading process. The other indicators are calculated using Equations ( 1 ) to ( 3 ), as follows: $$\:\text{}\text{E}={\int\:}_{\text{0}}^{\text{D}}\text{F}\text{d}\text{y}$$ 1 $$\:{\text{E}}_{\text{a}}=\frac{\text{E}}{\text{m}}$$ 2 $$\:{\text{F}}_{\text{m}}=\frac{\text{E}}{\text{D}}$$ 3 Here, F represents the instantaneous impact force, m is the structural mass, and y is the instantaneous impact displacement. An ideal energy absorbing box with optimal crashworthiness should satisfy a lower F p and higher E , E a , and F m . Since the mass and velocity of the impact wall are the same, the impact kinetic energy is 6.25 kJ. Therefore, the maximum energy absorption for the three energy absorbing boxes is 6.25 kJ. Under the same impact energy, the relationship between the impact force F and time t is shown in Fig. 13 . Based on Equations ( 1 ) to ( 3 ), the crashworthiness evaluation indicators of the energy absorbing boxes can be calculated, as listed in Table 3 . Through comparison, under the same impact energy, the three types of energy absorbing boxes exhibit the same maximum energy absorption. However, the traditional thin-walled square tube energy absorbing box shows the largest deformation, approaching complete collapse, indicating that it has reached its energy absorption limit. In contrast, the in-plane and out-plane filled energy absorbing boxes exhibit smaller deformations. If the impact energy is appropriately increased, their maximum energy absorption will also increase, demonstrating greater energy absorption potential. The out-plane filled honeycomb energy absorbing box has the smallest initial peak impact force F p , the largest average impact force F m , and the smallest deformation D . Therefore, it can be concluded that the out-plane filled honeycomb energy absorbing box has the best crashworthiness performance. 3.3 Multi-objective Optimization of Energy Absorbing Box 3.3.1 Establishment of Multi-objective Optimization Model Based on the above analysis, the new automotive energy absorbing box with out-plane filled honeycomb structure exhibits the best crashworthiness. In this section, a multi-objective optimization of the energy absorbing box will be conducted. First, a mathematical model for the multi-objective optimization needs to be established. Since the new energy absorbing box is intended to protect occupant safety in vehicles, the factors that most significantly affect occupant safety are the deformation and specific energy absorption of the energy absorbing box. According to Eq. ( 3 ), the larger the average impact force F m , the more energy the energy absorbing box absorbs per unit of collision displacement. Under the premise of a fixed total collision energy, this can reduce the collision displacement, thereby decreasing the deformation of the energy absorbing box and minimizing the intrusion into the occupant compartment. The specific energy absorption E a , which represents the energy absorbed per unit mass, reflects both the crashworthiness and lightweight characteristics of the energy absorbing box. Therefore, the optimization objectives can be defined as maximizing the average impact force F m and maximizing the specific energy absorption E a . The optimization variables are the honeycomb aperture ρ , the wall thickness of the honeycomb t 1 , and the wall thickness of the thin-walled square tube t 2 . Typically, the range of optimization variables is set within ± 20% of their initial values. The multi-objective optimization model is shown in Eq. ( 4 ). $$\:\left\{\begin{array}{c}\text{m}\text{i}\text{n}\text{(}-{\text{F}}_{\text{m}}\text{,}-{\text{E}}_{\text{a}}\text{)}\\\:\text{x}=\left(\text{ρ}\text{,}{\text{t}}_{\text{1}}\text{,}{\text{t}}_{\text{2}}\right)\\\:\text{s}\text{.}\text{t}\text{.}\left\{\begin{array}{c}\text{1.920}\text{m}\text{m}\le\:\text{ρ}\le\:\text{2.880}\\\:\text{0.048}\text{m}\text{m}\le\:{\text{t}}_{\text{1}}\le\:\text{0.072}\text{m}\text{m}\\\:\text{1.200}\text{m}\text{m}\le\:{\text{t}}_{\text{2}}\le\:\text{1.800}\text{m}\text{m}\end{array}\right.\end{array}\text{mm}\right.$$ 4 3.3.2 Latin Experimental Design In the multi-objective optimization model, there are three optimization variables, which require at least 10 sample points. According to the Latin hypercube sampling design rules, 16 sample points are randomly generated within the design space. The finite element simulation model is then used to obtain the average impact force F m and specific energy absorption E a for each sample point. The data for each sample are presented in Table 4 . Table 3 Evaluation index of crashworthiness of energy absorbing box Structure type F p (kN) F m (kN) D (mm) E (kJ) No filled 174.84 24.67 253.35 6.25 In-place filled 137.36 37.73 165.63 6.25 Out-place filled 111.45 42.35 147.59 6.25 3.3.3 Establishment of Response Surface Model The Latin hypercube sample data from Table 4 are imported into the Design-Expert software. Using the Response Surface module's Box-Behnken design, a response surface model is established. A second-order polynomial fitting is performed for the optimization variables and output response values. The requirement is that the goodness of fit (R2) must be greater than 0.90; otherwise, the constructed response surface model may have significant errors. The second-order response surface equations for the optimization objectives and optimization variables are shown in Equations ( 5 ) and ( 6 ). $$\:{\text{F}}_{\text{m}}=\text{26.26942}+\text{3.99078}\text{ρ}+\text{29.38653}{\text{t}}_{\text{1}}+\text{3.69007}{\text{t}}_{\text{2}}-\text{0.85423}\text{ρ}{\text{t}}_{\text{1}}-\text{2.9875}\text{ρ}{\text{t}}_{\text{2}}+\text{1.89511}{\text{t}}_{\text{1}}{\text{t}}_{\text{2}}+\text{2.77143}{\text{ρ}}^{\text{2}}+\text{1.11942}{{\text{t}}_{\text{1}}}^{\text{2}}-\text{1.86430}{{\text{t}}_{\text{2}}}^{\text{2}}$$ 5 $$\:{\text{E}}_{\text{a}}=\text{19.40808}-\text{4.94493}\text{ρ}+\text{6.85921}{\text{t}}_{\text{1}}-\text{5.47}{\text{174}\text{t}}_{\text{2}}-\text{2.88109}\text{ρ}{\text{t}}_{\text{1}}+\text{0.921529}\text{ρ}{\text{t}}_{\text{2}}-\text{2.64498}{\text{t}}_{\text{1}}{\text{t}}_{\text{2}}+\text{0.435542}{\text{ρ}}^{\text{2}}+\text{23.79775}{{\text{t}}_{\text{1}}}^{\text{2}}+\text{0.589691}{{\text{t}}_{\text{2}}}^{\text{2}}$$ 6 Table 4 Latin square experiment sample point data Numbering ρ (mm) t 1 (mm) t 2 (mm) F m (kN) E a D (mm) m (kg) 1 1.927 0.051 1.206 40.581 7.881 154.014 0.793 2 1.952 0.049 1.571 41.246 7.015 151.531 0.891 3 2.056 0.055 1.378 41.119 7.184 151.997 0.870 4 2.177 0.069 1.712 42.261 6.339 147.891 0.986 5 2.512 0.050 1.301 41.965 6.490 148.933 0.963 6 2.471 0.048 1.479 42.881 6.269 145.752 0.997 7 2.294 0.063 1.736 44.508 6.127 140.423 1.020 8 2.384 0.065 1.589 43.603 6.213 143.337 1.006 9 2.400 0.060 1.500 42.347 6.332 147.590 0.987 10 2.099 0.053 1.289 40.884 7.284 152.871 0.858 11 2.871 0.071 1.755 45.486 5.337 137.404 1.171 12 2.710 0.058 1.401 45.034 6.015 138.782 1.039 13 2.657 0.056 1.299 43.277 6.250 144.420 1.000 14 2.539 0.067 1.605 45.276 5.958 138.042 1.049 15 2.444 0.070 1.666 45.109 5.998 138.553 1.042 16 2.330 0.061 1.353 41.448 6.699 150.793 0.933 For Eq. ( 5 ), the goodness of fit (R 2 ) is 0.92, and the adjusted goodness of fit is 0.91. For Eq. ( 6 ), the goodness of fit (R 2 ) is 0.99, and the adjusted goodness of fit is 0.99. To verify the accuracy of the response surface model, three additional sample points were generated using the optimal Latin hypercube sampling method. The finite element model was used to calculate the simulation values corresponding to these sample points, and the results were compared with the response values obtained from the aforementioned response surface model. The results are shown in Table 5 . The average error between the response values of the response surface model and the true values of the simulation results for the average impact force F m and specific energy absorption E a is 2.022%, which is less than 5%. The maximum error is 7.083%, which is less than 10%. Therefore, it can be concluded that the response surface model has high accuracy and can replace the simulation model for subsequent optimization design. 3.3.4 Multi-objective Optimization Solution In Matlab, NSGA-Ⅱ algorithm is used to solve Equations ( 4 ). By defining the upper and lower bound constraint objective function of design variables, Gamultiobj function is selected as the solution function, population size is set to 200, crossover probability is set to 0.9, mutation probability is set to 0.2, and Pareto ratio is set to 0.35. After 300 iterations, the Pareto frontier optimal solution set shown in Fig. 14 is obtained. As shown in Fig. 14 , the average impact force F m is negatively correlated with specific energy absorption E a , that is, the solution with the maximum specific energy absorption when the average impact force is maximum cannot be found. The actual situation should be considered when selecting the final optimal solution. For the crashability of the energy absorbing box, the deformation and specific energy absorption can reflect the resistance to deformation and energy absorption. It determines the invasion amount of the crew cabin and the safety of the crew after the collision. The optimal solution in Fig. 14 can ensure that the average impact force is maximum when the specific absorption energy meets the requirements. According to Equations ( 3 ), when the impact energy is constant, the greater the average impact force is, the smaller the deformation is. The optimal solution corresponds to the honeycomb aperture ρ of the energy absorbing box, the honeycomb wall thickness t 1 is 0.071 mm, and the thin-walled square tube wall thickness t 2 is 1.315 mm. The average impact force F m is 43.281 kN, the deformation D is 144.405 mm, and the specific absorption energy is 6.981. Table 5 Comparison between the response surface model and the simulation model Num-bering ρ (mm) t 1 (mm) t 2 (mm) F m (kN) E a Emu-lation Mo-del Error Emu-lation Mo-del Error 1 2.105 0.055 1.575 41.387 39.918 3.549% 6.713 6.711 0.030% 2 2.432 0.052 1.388 42.158 45.372 7.083% 6.474 6.472 0.031% 3 2.340 0.062 1.608 43.098 42.546 1.281% 6.256 6.246 0.160% 3.4 Comparative Analysis Before and After Optimization Figure 15 shows a comparison of the impact force versus time curves for the out-plane filled honeycomb energy absorbing box before and after multi-objective optimization. The peak impact force F p after optimization is smaller than that before optimization. The crashworthiness evaluation indicators of the energy absorbing box calculated using Equations ( 1 ) to ( 3 ) are listed in Table 6 . After optimization, the specific energy absorption E a of the out-plane filled honeycomb energy absorbing box is improved, and the deformation D is reduced. Overall, the energy absorbing box obtained through the NSGA-II algorithm for multi-objective optimization shows significant improvements in both comprehensive crashworthiness and lightweight performance. 4 Crashworthiness Analysis of the Whole Vehicle After Replacing the Design 4.1 Simplification of the Finite Element Model This study primarily investigates the impact of replacing a new energy absorbing box on the crashworthiness of the entire vehicle. To save computational time, it is necessary to simplify the full-vehicle frontal crash simulation model. Under the premise of maintaining the material properties and connection methods of each component, the crash wall, crossbeam, energy absorbing box, longitudinal beam, and other connecting parts are directly extracted from the full-vehicle model. To ensure that the motion behavior of the simplified model remains consistent with that of the full-vehicle model, motion constraints are applied to the contact areas between the simplified model and the rest of the vehicle. The displacement-time curves of the crossbeam constraint nodes in the x, y, and z axes, which are representative of the full-vehicle crash process, are extracted. The boundary conditions of the simplified model are defined, and displacement sensors are added at the ends of the longitudinal beams to evaluate the crash outcomes for the passenger compartment. The simplified model is shown in Fig. 16 . 4.2 Simplify Model Reliability Verification Figure 17 shows a comparison of the energy absorption curves of the energy absorbing box before and after simplifying the full-vehicle frontal crash model. The two curves exhibit a high degree of agreement. Figure 18 presents a comparison of the displacement contour maps of the crossbeam before and after model simplification, revealing minimal errors. The simplified model demonstrates strong consistency with the original model, confirming its suitability for use in subsequent research. Table 6 Comparison of crashworthiness evaluation indexes of energy absorbing boxes before and after optimization Evaluation indicators F p (kN) F m (kN) E a D (mm) m (kg) Before optimization 111.453 42.347 6.332 147.590 0.987 After optimization 86.918 43.281 6.981 144.405 0.895 The amount of change 22.014% 2.158% 9.297% 2.158% 9.321% 4.3 Comparison of the Crashworthiness of the Whole Vehicle Before and After the Replacement Design The out-plane filled honeycomb energy absorbing box, optimized for crashworthiness in Chap. 3, was installed into the simplified model for frontal crash simulation. The crash speed and other constraint conditions remained unchanged. Figure 19 shows a comparison of the energy absorption curves of the energy absorbing box before and after the replacement design. It can be observed that the energy absorption value of the new energy absorbing box accounts for 8.79% of the total crash energy, representing a 12.51% increase in energy absorption proportion compared to the traditional thin-walled square tube energy absorbing box used before the replacement. Excessive crash acceleration can lead to occupant injuries such as fractures and internal organ damage. Reducing the overall crash acceleration of the vehicle can mitigate the severity of occupant injuries after a collision. Figure 20 compares the vehicle crash acceleration before and after the replacement design. The results show that the acceleration decreased by 11.49%, and the acceleration curve became smoother with reduced oscillations, improving the secondary impact forces experienced by the occupants. Based on the data from the displacement sensors placed at the ends of the longitudinal beams in the simplified model, the intrusion curve of the passenger compartment over time was obtained, as shown in Fig. 21 . The intrusion of the passenger compartment was reduced by 17.48%. Based on the above analysis, the proposed out-plane filled honeycomb energy absorbing box for vehicles can enhance the crashworthiness of the vehicle, reduce occupant injuries, and improve occupant safety. 5 Conclusion This study demonstrates that replacing the energy absorbing box with a new filled honeycomb structure enhances the vehicle's crashworthiness and occupant safety. The specific conclusions are as follows: Taking a certain sedan as the research subject, a simulation environment for side pole collision was established based on the C-NCAP testing requirements. The reliability of the model was verified by comparing it with relevant tests. It was identified that the vehicle's poor crashworthiness under frontal collision conditions is due to insufficient stiffness of the crossbeam and inadequate energy absorption by the energy absorbing box. Two new types of energy-absorbing boxes with different filling configurations were designed. Impact simulations demonstrated that the out-plane filled honeycomb energy absorbing box exhibits the best crashworthiness. Multi-objective optimization was performed on the energy absorbing box, achieving lightweight design while improving crash performance. The new energy absorbing box was installed in a simplified model for frontal collision simulation. By comparing the vehicle crash acceleration, energy absorption ratio of the energy absorbing box, and passenger compartment deformation before and after the replacement design, it was verified that the proposed new energy absorbing box reduces the vehicle acceleration by 11.49%, decreases passenger compartment intrusion by 17.48%, and increases the energy absorption ratio by 12.51%. This addresses the shortcomings of the original vehicle's crashworthiness and enhances occupant safety. Declarations Disclosure statement No potential conflict of interest was reported by the author(s). Funding This study was supported by NSFC 52472357, partially supported by Heilongjiang Province Key R&D Program Guidance Category Project GZ20220027 and Heilongjiang Provincial Transportation Investment Group Co., LTD Project JT-100000-ZC-FW-2021-0119. Author Contribution W. H.L. wrote the manuscript text,W. Z.Y.provided guidance on ideas,D.X.J. provided financial support,W. N. drew all the charts. 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Acta Mechanica Solida Sinica 21(04):383-388. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 21 May, 2025 Read the published version in Iranian Journal of Science and Technology, Transactions of Mechanical Engineering → Version 1 posted Editorial decision: Revision requested 04 Apr, 2025 Reviews received at journal 30 Mar, 2025 Reviews received at journal 30 Mar, 2025 Reviewers agreed at journal 23 Mar, 2025 Reviewers agreed at journal 21 Mar, 2025 Reviews received at journal 21 Mar, 2025 Reviewers agreed at journal 20 Mar, 2025 Reviewers invited by journal 20 Mar, 2025 Editor assigned by journal 20 Mar, 2025 Submission checks completed at journal 20 Mar, 2025 First submitted to journal 19 Mar, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-6258409","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":432126217,"identity":"ab46d93b-9fe6-46a5-8153-416c90b1801f","order_by":0,"name":"Honglin Wang","email":"","orcid":"","institution":"Northeast Forestry University","correspondingAuthor":false,"prefix":"","firstName":"Honglin","middleName":"","lastName":"Wang","suffix":""},{"id":432126218,"identity":"9cce74bf-3d01-4ce1-8a92-cb708a352412","order_by":1,"name":"Zhanyu 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1","display":"","copyAsset":false,"role":"figure","size":203251,"visible":true,"origin":"","legend":"\u003cp\u003eFront bumper beam of the car\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-6258409/v1/2bd02c5923a41e5c277c5252.png"},{"id":79087278,"identity":"9a10c998-0648-466a-8830-b30bf9cbae46","added_by":"auto","created_at":"2025-03-24 09:24:10","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":106754,"visible":true,"origin":"","legend":"\u003cp\u003eFinite element model of vehicle frontal collision\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-6258409/v1/05e122851d753d77ff08960e.png"},{"id":79087261,"identity":"f1809ff9-cddc-40f8-9f1c-cebc4567a9eb","added_by":"auto","created_at":"2025-03-24 09:24:09","extension":"png","order_by":3,"title":"Figure 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5","display":"","copyAsset":false,"role":"figure","size":45158,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of vehicle acceleration between the real vehicle experiment and the simulation model\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-6258409/v1/13d55943213f8fdca996959e.png"},{"id":79088978,"identity":"d97e9891-6a77-42f3-81e2-2aba5be99197","added_by":"auto","created_at":"2025-03-24 09:40:12","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":433113,"visible":true,"origin":"","legend":"\u003cp\u003eVehicle frontal collision deformation diagram\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-6258409/v1/3cf9f79fa7a37df7f993f6dd.png"},{"id":79087138,"identity":"66e6d14a-ef51-4dbd-bf01-5d60d8253f0a","added_by":"auto","created_at":"2025-03-24 09:24:07","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":121373,"visible":true,"origin":"","legend":"\u003cp\u003eDeformation diagram of the front collision beam of the vehicle\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-6258409/v1/0554229c79b24ab4e6d68a8e.png"},{"id":79087961,"identity":"5ff90fb8-7ef3-4bfa-a746-4f7fe55a53c8","added_by":"auto","created_at":"2025-03-24 09:32:09","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":32304,"visible":true,"origin":"","legend":"\u003cp\u003eEnergy absorption curves of the energy-absorbing box\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-6258409/v1/ca1a526cfab5c530a19141e5.png"},{"id":79087077,"identity":"0e6cb889-6674-4a4e-8782-49dbba314184","added_by":"auto","created_at":"2025-03-24 09:24:03","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":316507,"visible":true,"origin":"","legend":"\u003cp\u003eA new type of automotive energy absorbing box filled with a honeycomb structure\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-6258409/v1/84f0b3e3eb9e8966960177de.png"},{"id":79087299,"identity":"ec0fec54-337f-45a7-a41a-05ff4720462d","added_by":"auto","created_at":"2025-03-24 09:24:12","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":21528,"visible":true,"origin":"","legend":"\u003cp\u003eFinite element model of energy absorbing box impact\u003c/p\u003e","description":"","filename":"10.png","url":"https://assets-eu.researchsquare.com/files/rs-6258409/v1/6dcb1811d214b00f5bca651a.png"},{"id":79087983,"identity":"508ef5b4-71e1-4cdd-8ed6-65427e440b67","added_by":"auto","created_at":"2025-03-24 09:32:12","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":120942,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of deformation between simulation and experimental results\u003c/p\u003e","description":"","filename":"11.png","url":"https://assets-eu.researchsquare.com/files/rs-6258409/v1/c2d24d84790c7623405a5ba6.png"},{"id":79087955,"identity":"3a3319b2-ae7c-401d-906b-dcd176516f3d","added_by":"auto","created_at":"2025-03-24 09:32:08","extension":"png","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":247458,"visible":true,"origin":"","legend":"\u003cp\u003eDeformation results of the energy absorbing box\u003c/p\u003e","description":"","filename":"12.png","url":"https://assets-eu.researchsquare.com/files/rs-6258409/v1/3a4cc9ea78bceebfcd2bd81e.png"},{"id":79087919,"identity":"f1669567-c2b0-4596-96d4-f61af9f83edc","added_by":"auto","created_at":"2025-03-24 09:32:07","extension":"png","order_by":13,"title":"Figure 13","display":"","copyAsset":false,"role":"figure","size":87409,"visible":true,"origin":"","legend":"\u003cp\u003eCollision force and time curve of the energy absorbing box\u003c/p\u003e","description":"","filename":"13.png","url":"https://assets-eu.researchsquare.com/files/rs-6258409/v1/db8c8fea621c7fdffb7b48cb.png"},{"id":79087132,"identity":"5e47964c-f29e-4847-bc29-7b571ebb22ae","added_by":"auto","created_at":"2025-03-24 09:24:06","extension":"png","order_by":14,"title":"Figure 14","display":"","copyAsset":false,"role":"figure","size":42234,"visible":true,"origin":"","legend":"\u003cp\u003eOptimal solution set of Pareto frontier after multi-objective optimization\u003c/p\u003e","description":"","filename":"14.png","url":"https://assets-eu.researchsquare.com/files/rs-6258409/v1/984043dd8380b2628e6574e3.png"},{"id":79087269,"identity":"fa3180b6-4d94-43c0-8879-578a3723ef85","added_by":"auto","created_at":"2025-03-24 09:24:09","extension":"png","order_by":15,"title":"Figure 15","display":"","copyAsset":false,"role":"figure","size":80863,"visible":true,"origin":"","legend":"\u003cp\u003eshows the comparison of the collision force and time curves of the energy absorbing box before and after optimization\u003c/p\u003e","description":"","filename":"15.png","url":"https://assets-eu.researchsquare.com/files/rs-6258409/v1/ce475120330598b809f63831.png"},{"id":79087963,"identity":"9a4db239-3422-4714-906c-0a0aaecb3fc9","added_by":"auto","created_at":"2025-03-24 09:32:09","extension":"png","order_by":16,"title":"Figure 16","display":"","copyAsset":false,"role":"figure","size":49350,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic diagram of the simplified model\u003c/p\u003e","description":"","filename":"16.png","url":"https://assets-eu.researchsquare.com/files/rs-6258409/v1/a8ba0a330922989066ab6d40.png"},{"id":79087256,"identity":"fb904376-f1c9-44e3-b500-97b2d2aa7527","added_by":"auto","created_at":"2025-03-24 09:24:09","extension":"png","order_by":17,"title":"Figure 17","display":"","copyAsset":false,"role":"figure","size":47758,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of energy absorption curves of energy absorption boxes before and after simplification\u003c/p\u003e","description":"","filename":"17.png","url":"https://assets-eu.researchsquare.com/files/rs-6258409/v1/ce6caf9ef881fe92c32aea75.png"},{"id":79087959,"identity":"74ebc2cd-626a-4f08-96f1-40bc72d28ec8","added_by":"auto","created_at":"2025-03-24 09:32:09","extension":"png","order_by":18,"title":"Figure 18","display":"","copyAsset":false,"role":"figure","size":102295,"visible":true,"origin":"","legend":"\u003cp\u003eDisplacement contour of the front and rear beams simplified by the model\u003c/p\u003e","description":"","filename":"18.png","url":"https://assets-eu.researchsquare.com/files/rs-6258409/v1/d9c935198e4db9476b42f144.png"},{"id":79087290,"identity":"46933b27-c207-4017-9027-85776de4d698","added_by":"auto","created_at":"2025-03-24 09:24:11","extension":"png","order_by":19,"title":"Figure 19","display":"","copyAsset":false,"role":"figure","size":54610,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of energy absorption curves of energy absorbing boxes before and after the replacement design\u003c/p\u003e","description":"","filename":"19.png","url":"https://assets-eu.researchsquare.com/files/rs-6258409/v1/19ad93744642b848f5628474.png"},{"id":79087323,"identity":"ae87a563-61fd-4817-98b6-b0802feea379","added_by":"auto","created_at":"2025-03-24 09:24:14","extension":"png","order_by":20,"title":"Figure 20","display":"","copyAsset":false,"role":"figure","size":55687,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of acceleration curves before and after replacement design\u003c/p\u003e","description":"","filename":"20.png","url":"https://assets-eu.researchsquare.com/files/rs-6258409/v1/fc2e7577f7f030b14fd75376.png"},{"id":79087131,"identity":"b8cb57e8-1da6-4357-9b5a-cc7703165fc9","added_by":"auto","created_at":"2025-03-24 09:24:06","extension":"png","order_by":21,"title":"Figure 21","display":"","copyAsset":false,"role":"figure","size":59964,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of the intrusion of the acceleration cabin before and after the replacement design\u003c/p\u003e","description":"","filename":"21.png","url":"https://assets-eu.researchsquare.com/files/rs-6258409/v1/023a1ebc78995f23aa06c12a.png"},{"id":83460214,"identity":"77f4abd2-66ca-4ea0-8fac-e59c26be436d","added_by":"auto","created_at":"2025-05-26 16:12:08","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3850895,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6258409/v1/23300f4c-a489-4147-a92e-62313410fc76.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Design of a New Energy Absorption Box with Honeycomb Structure for Vehicles and Research on Vehicle Crashworthiness","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eWith the popularization of automobiles, the issue of crash safety has garnered significant attention. Among various traffic accidents, frontal collisions are the most severe in terms of causing occupant injuries(Gulshan et al. \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). In the past, automotive energy absorbing boxes primarily featured thin-walled tube structures, which allowed for lightweight design while providing a certain level of buffering and energy absorption(Min et al. 2017). However, as vehicle speeds continue to increase, the impact forces have grown larger, and the traditional thin-walled tube structures of energy absorbing boxes can no longer meet the energy absorption requirements.\u003c/p\u003e \u003cp\u003eIn recent years, with the continuous advancement of manufacturing capabilities, various complex structures and new materials have gradually been applied in the automotive industry. For instance, honeycomb structures, as a type of biomimetic porous design, exhibit excellent energy absorption characteristics and load-bearing capacity under impact loads(Xiang et al. 2017). Biomimetic honeycombs with functionally graded designs show significant improvements in energy absorption and crashworthiness compared to traditional honeycombs(Pattaramon et al. \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2024\u003c/span\u003e; Mohammad et al. 2024; Wu et al. \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2021\u003c/span\u003e; Deng et al. \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2024\u003c/span\u003e). Additionally, studies have shown that the energy absorption characteristics of honeycomb structures are greatly influenced by their structural parameters and geometric configurations. Sun et al. (\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2021\u003c/span\u003e) conducted impact tests on honeycomb structures with different parameters under four energy levels, demonstrating that the height of the honeycomb has no effect on energy absorption, while increasing the wall thickness significantly enhances energy absorption performance. Building on this, some scholars have designed new biomimetic honeycomb structures inspired by natural biological features, such as elliptical arc concave honeycombs(Tang et al. \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2025\u003c/span\u003e), 3D-printed spherical honeycombs(Deng et al. \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2024\u003c/span\u003e), origami-inspired negative Poisson's ratio honeycombs(Zhou et al. \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2024\u003c/span\u003e), and biomimetic glass sponge-like honeycombs(Wang et al. \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2024\u003c/span\u003e). Compared to traditional hexagonal honeycomb structures, these designs exhibit notable improvements in energy absorption and impact resistance. The material properties of honeycomb structures play a crucial role in their energy absorption and crashworthiness. As a result, researchers have conducted extensive studies on composite honeycomb structures in recent years, further enhancing their dynamic performance(Guo et al. \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2024\u003c/span\u003e; Zhou et al. \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2018\u003c/span\u003e; Hussain et al. \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2025\u003c/span\u003e; Hua et al. \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2025\u003c/span\u003e; Liu et al. 2025) .\u003c/p\u003e \u003cp\u003eIn the field of automotive energy absorption box research, Nia et al. (2014) investigated energy absorbing boxes with hexagonal and octagonal cross-sections. They found that both exhibited similar energy absorption values, but the hexagonal cross-section had a lower peak collision force, making it more suitable for designing new energy absorbing boxes. Building on this, researchers have developed novel energy absorbing boxes by filling thin-walled tubes with materials such as aluminum foam, porous structures, and negative Poisson's ratio structures, thereby improving crashworthiness(Zhao et al. \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2024\u003c/span\u003e; Xie et al. \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2023\u003c/span\u003e; Sorrawit et al. \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2025\u003c/span\u003e). Some scholars have explored the energy absorption capabilities of energy absorbing boxes by selecting different material types, discovering that composite materials outperform traditional materials in terms of energy absorption and crashworthiness(Yoo et al. \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2010\u003c/span\u003e; Fatima et al. \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2024\u003c/span\u003e; Xing et al. \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2025\u003c/span\u003e; Francesco et al. \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e2024\u003c/span\u003e). Multi-objective optimization methods are widely used in engineering design optimization problems, enabling the identification of optimal design parameters that balance conflicting objectives to achieve the best overall performance. For instance, Zhang et al. (\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e2024\u003c/span\u003e) established a response surface model and applied a multi-objective optimization algorithm to enhance the mechanical performance of torsional plate energy absorbing components. Yang et al. (\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e2025\u003c/span\u003e) used a three-level Box-Behnken method to perform multi-objective optimization on a biomimetic diaphragm-reinforced bamboo system, validating its performance advantages over typical energy absorbing boxes. Wang et al. (\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e2022\u003c/span\u003e) proposed a novel hexagonal pyramid-shaped energy absorbing box composed of an internal negative Poisson's ratio structure and an outer shell. They optimized the box using response surface methodology and multi-objective optimization to improve its crashworthiness. The optimization results demonstrated that the new energy absorbing box achieves both crashworthiness and lightweight design.\u003c/p\u003e \u003cp\u003eIn summary, significant progress has been made in enhancing the energy absorption characteristics of honeycomb structures and designing automotive energy absorbing boxes using porous materials. However, biomimetic honeycomb structures with relatively low densities have not been widely applied in automotive energy absorbing boxes. Moreover, existing research has primarily focused on the crashworthiness of individual energy absorbing boxes without integrating them into full-vehicle crash scenarios. During a frontal collision, the stress conditions of the energy absorbing box are highly complex. Only by incorporating the new energy absorbing box into a full-vehicle model for crash analysis can its effectiveness in improving the vehicle's crashworthiness be convincingly demonstrated. Therefore, this study proposes a novel automotive energy absorbing box by filling thin-walled square tubes with honeycomb structures. This design will be installed in the front bumper beam of a vehicle for full-vehicle frontal crash analysis. The aim is to leverage the honeycomb structure to enhance the crashworthiness of the energy absorbing box during frontal collisions, thereby improving the vehicle's overall crashworthiness and occupant safety.\u003c/p\u003e"},{"header":"2 Establishment of the Frontal Collision Finite Element Model and Analysis of Simulation Results","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 Basic Parameters of the Vehicle Model\u003c/h2\u003e \u003cp\u003eThe subject of this study is a compact car with a length of 4635 mm, a width of 1780 mm, and a height of 1455 mm. The wheelbase is 2700 mm, and the curb weight is 1320 kg. The front bumper beam consists of a crossbeam, an energy absorbing box, and longitudinal beams, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. The vehicle employs the simplest form of thin-walled square tube energy absorbing boxes. It is required that, in the event of a frontal collision, the energy absorbing box can quickly absorb and dissipate collision energy, thereby enhancing the vehicle's safety performance. The key component parameters that play a decisive role in the crashworthiness of the vehicle during frontal collisions are listed in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eParameters of some components\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"7\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eParts\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eLength\u003c/p\u003e \u003cp\u003e(mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eWidth\u003c/p\u003e \u003cp\u003e(mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eHeight\u003c/p\u003e \u003cp\u003e(mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eThickness\u003c/p\u003e \u003cp\u003e(mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003ematerial\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eMass\u003c/p\u003e \u003cp\u003e(kg)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCrossbeam\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e35\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e110\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eCarbon\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e13.3\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eEnergy absorbing box\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e270\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e200\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e105\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eAluminum alloy\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.4\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eStringers\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2500\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e105\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eCarbon\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e21.5\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 Full-Vehicle Finite Element Modeling\u003c/h2\u003e \u003cp\u003eFirst, the three-dimensional model of the vehicle was imported into Hypermesh software for geometric cleanup. Components that had no impact on the collision were removed, and geometric defects and missing edges caused by software incompatibility were repaired. The full vehicle was then meshed, with the crossbeam, energy absorption box, and longitudinal beams using 8 mm quadrilateral shell elements, while other parts were meshed with a size of 15 mm. The occupant compartment was meshed using solid elements. The full-vehicle model consisted of 19,032 elements, including 15,631 shell elements, 2,312 solid elements, and 1,089 triangular elements, accounting for 5.72% of the total, which is less than 10%. To ensure the mesh quality met the requirements, the entire model underwent a mesh quality check, with the aspect ratio, Jacobian, maximum internal angle, and warping factor all satisfying the criteria. After meshing, material and property definitions were assigned to the components. Rigid materials were assigned to the rigid wall and non-deformable parts at the rear of the vehicle to save computation time, while elastic-plastic materials were assigned to the crossbeam, energy absorbing box, and longitudinal beams, which undergo deformation during collisions. The material parameters for some components of the full-vehicle model are listed in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. RBE2 rigid connections were used to connect adjacent components in the model, and spot welding was applied between the crossbeam, energy absorbing box, and longitudinal beams. In the full-vehicle model, the vehicle's self-contact was set as single-sided automatic contact, while surface-to-surface contact was defined between the bumper beam and the rigid wall, as well as between the crossbeam, energy absorbong box, and longitudinal beams, with both static and dynamic friction coefficients set to 0.2. The simulation termination time was set to 0.15 s, with a time step of 0.0015 s. The global hourglass control method was used for computation, with a global stiffness coefficient of 0.1. To obtain the required simulation results, output data such as node deformation, velocity, acceleration, kinetic energy, internal energy, hourglass energy, and slip energy were configured. The established finite element model for the full-vehicle frontal collision is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e2.3 Model Reliability Validation\u003c/h2\u003e \u003cp\u003eTo ensure the accuracy of the simulation results, the model must undergo reliability validation. A commonly used method is to verify whether the model satisfies the law of energy conservation during the collision process and to ensure that the hourglass energy does not exceed 5% of the total energy. Otherwise, the model's accuracy would be considered low, and the simulation results would be unreliable for this study. Figure\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e shows the energy variation curves during the simulation. The total energy remains stable at approximately 208 kJ, with kinetic energy decreasing and internal energy increasing in a symmetrical manner. The hourglass energy is 4.99 kJ, accounting for 2.39% of the total energy, which is below the 5% threshold.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eMaterial properties of some components in the full-vehicle finite element model\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eParts\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDensity\u003c/p\u003e \u003cp\u003e(g\u0026middot;cm\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003e)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eYield strength\u003c/p\u003e \u003cp\u003e(MPa)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eModulus of elasticity\u003c/p\u003e \u003cp\u003e(GPa)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003ePoisson's ratio\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCrossbeam\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e7.82\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e260\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e210\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.27\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eEnergy absorbing box\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.75\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e240\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e68.3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.30\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eStringers\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e7.81\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e235\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e190\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.27\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eIn this study, to further validate the reliability of the model, the deformation of the simulation model was compared with that of a real vehicle test conducted under C-NCAP for the same vehicle type. As shown in Figs.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e and \u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e, which compare the vehicle deformation and acceleration between the real vehicle test and the simulation model, the two exhibit a high degree of similarity. Based on the above analysis, it can be concluded that the full-vehicle frontal collision model established in this study is highly reliable. The simulation results align well with the actual collision outcomes, making this model suitable for subsequent research.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e2.4 Analysis of Simulation Results\u003c/h2\u003e \u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e shows the deformation diagram of the vehicle after the frontal collision. Significant deformation occurs in the front part of the vehicle. The crossbeam deforms first, then compresses the front fender outward and the energy absorbing box inward. The deformation of the wheel axle causes the wheels to cave inward, affecting the force transmission path of the front bumper beam. During the collision, the engine hood bends. All these deformed components can impact the occupant compartment, thereby endangering the safety of the occupants.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e illustrates the deformation of the crossbeam, energy absorbing box, and longitudinal beams. It can be observed that the crossbeam undergoes severe deformation and compresses the energy absorbing box, while the right longitudinal beam also deforms. This indicates that the stiffness of the crossbeam is insufficient to effectively resist the intrusion of the rigid wall. Figure\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e shows the energy absorption curve of the energy absorbing box, which accounts for 7.69% of the total energy. This demonstrates that the energy absorption capacity of this ordinary thin-walled tube energy absorbing box is severely inadequate. It cannot effectively absorb collision energy or reduce the impact force transmitted to the occupants.\u003c/p\u003e \u003c/div\u003e"},{"header":"3 Design and Multi-objective Optimization of the New Energy Absorbing Box","content":"\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Structural Design of the New Energy Absorbing Box\u003c/h2\u003e \u003cp\u003eBased on the analysis above, the traditional thin-walled square tube energy absorbing box no longer meets the crashworthiness requirements. Therefore, this study proposes the use of a biomimetic honeycomb structure with high energy absorption characteristics to fill the thin-walled square tube, designing a new automotive energy absorbing box to improve the vehicle's crashworthiness.\u003c/p\u003e \u003cdiv id=\"Sec9\" class=\"Section3\"\u003e \u003ch2\u003e3.1.1 In-Plane Honeycomb-filled Energy Absorbing Box Design\u003c/h2\u003e \u003cp\u003eIn reference (Shi et al. \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e2024\u003c/span\u003e), quasi-static compression tests were conducted on traditional hexagonal honeycomb structures with different relative densities. It was found that when the pore size \u003cem\u003eρ\u003c/em\u003e is 2.4 mm, the energy absorption characteristics of the honeycomb structure are optimal. The initial wall thickness \u003cem\u003et\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e of the honeycomb structure is set to 0.06 mm. In Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e\u003cb\u003e(a)\u003c/b\u003e, during the design process of the in-plane honeycomb-filled energy absorbing box, to ensure compatibility between the honeycomb structure and the thin-walled square tube (length \u003cem\u003eL\u003c/em\u003e\u0026thinsp;=\u0026thinsp;270 mm, width \u003cem\u003eW\u003c/em\u003e\u0026thinsp;=\u0026thinsp;200 mm, height \u003cem\u003eH\u003c/em\u003e\u0026thinsp;=\u0026thinsp;105 mm, thickness \u003cem\u003et\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;1.5 mm), the width \u003cem\u003eW\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e of the honeycomb structure is required to be 197 mm (\u003cem\u003eW\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eW\u003c/em\u003e \u0026minus;\u0026thinsp;2\u003cem\u003eT\u003c/em\u003e), the height \u003cem\u003eH\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e to be 270 mm (\u003cem\u003eH\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eL\u003c/em\u003e), and the length \u003cem\u003eL\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e to be 102 mm (\u003cem\u003eL\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eH\u003c/em\u003e \u0026minus;\u0026thinsp;2\u003cem\u003eT\u003c/em\u003e). The thin-walled square tube and the filled honeycomb structure are assembled in Solidworks to form the new in-plane honeycomb-filled automotive energy absorbing box. To more clearly observe the internal honeycomb structure, the external thin-walled square tube of the new energy absorbing box is treated with a frame removal and hidden line retention process.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec10\" class=\"Section3\"\u003e \u003ch2\u003e3.1.2 Out-Plane Honeycomb-filled Energy Absorbing Box Design\u003c/h2\u003e \u003cp\u003eFor the design of the out-plane honeycomb-filled energy absorbing box, the dimensions of the thin-walled square tube remain unchanged. The difference lies in the dimensions of the honeycomb structure: the height \u003cem\u003eH\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e is 102 mm (\u003cem\u003eH\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eH\u003c/em\u003e \u0026minus;\u0026thinsp;2\u003cem\u003eT\u003c/em\u003e), the width \u003cem\u003eW\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e is 197 mm (\u003cem\u003eW\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eW\u003c/em\u003e \u0026minus;\u0026thinsp;2\u003cem\u003eT\u003c/em\u003e), and the length \u003cem\u003eL\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e is 270 mm (\u003cem\u003eL\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eL\u003c/em\u003e). The thin-walled square tube and the filled honeycomb structure are assembled to form the new out-plane honeycomb-filled automotive energy absorbing box, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e\u003cb\u003e(b)\u003c/b\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Crashworthiness Analysis of the Energy Absorbing Box\u003c/h2\u003e \u003cdiv id=\"Sec12\" class=\"Section3\"\u003e \u003ch2\u003e3.2.1 Establishment of the Impact Finite Element Model\u003c/h2\u003e \u003cp\u003eThe geometric model of the automotive energy absorbing box is imported into Abaqus in IGS file format. The Explicit dynamic analysis module is used to simulate the impact process of the in-plane/out-plane filled new automotive energy absorbing box and the traditional thin-walled square tube energy absorbing box. The material of the energy absorbing box is simulated using aluminum alloy 6061, with a density of 2.75 g\u0026middot;cm⁻\u0026sup3;, an elastic modulus of 68,300 MPa, a yield strength of 250 MPa, and a Poisson's ratio of 0.30. A material section is created and assigned to the energy absorbing box, completing the material property assignment. Two rigid plates are created at the top and bottom of the energy absorbing box and assembled with it. The dynamic explicit analysis step is set to a duration of 0.10 s. General contact is selected as the interaction property between the upper and lower rigid plates and the automotive energy absorbing box, with both static and dynamic friction coefficients set to 0.2. The lower plate is fixed, while the upper plate is constrained with a velocity of 5 m/s and assigned a mass of 500 kg. A global mesh seed size of 0.05 mm is used to generate quadrilateral meshes, balancing computation time and convergence criteria. Outputs include the energy absorption value and displacement of the energy absorbing box. The impact finite element model of the automotive energy absorbing box is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e. The impact finite element models for the traditional thin-walled tube energy absorbing box and the new honeycomb-filled automotive energy absorbing box are identical in terms of materials, dimensions, impact speed, and other parameters, except for the presence or absence of the filling structure.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec13\" class=\"Section3\"\u003e \u003ch2\u003e3.2.2 Model Reliability Verification\u003c/h2\u003e \u003cp\u003eTo verify the reliability of the model, referring to Lee's study(Lee et al. \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e2008\u003c/span\u003e), thin-walled square tubes of the same size and material were designed, and impact simulations were conducted under identical loading conditions. Figure\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e11\u003c/span\u003e shows a comparison of the deformation of the thin-walled square tubes between the simulation and the experiment. The deformation modes of the two are consistent. In the simulation, the initial deformation of the thin-walled square tube starts from the triggered side and reaches the maximum load at the beginning of the subsequent collapse. Therefore, it can be concluded that the finite element model of the thin-walled square tube designed for comparison with the experimental deformation results has high reliability. The energy absorbing box finite element model established in this study and the thin-walled square tube finite element model are both explicit analysis models under the same Abaqus/Explicit setup. Moreover, the material properties and mesh size, which have the greatest impact on the simulation results, are the same. Only the dimensions and shape of the impacted structure, which have a smaller impact on the result errors, were modified. Thus, it can be considered that the energy absorbing box finite element model and the thin-walled square tube finite element model have similar accuracy in terms of results. Based on the above analysis, the impact finite element model of the energy absorbing box established in this study is reliable and can be used for subsequent research.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec14\" class=\"Section3\"\u003e \u003ch2\u003e3.2.3 Crashworthiness Analysis\u003c/h2\u003e \u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e12\u003c/span\u003e shows the deformation results of the energy absorbing box under impact loading. It can be observed that the traditional thin-walled square tube energy absorbing box, without internal honeycomb structure support, undergoes buckling deformation of the thin wall under impact loading. When the thin-walled tube is filled with a honeycomb structure in-plane, the honeycomb structure experiences bending deformation of adjacent cell walls and plastic deformation at the joints of adjacent cell walls. When the thin-walled tube is filled with a honeycomb structure out-plane, the honeycomb structure undergoes buckling and deformation of the cell walls, which is similar to the traditional thin-walled tube energy absorbing box. From the perspective of deformation mode, the energy absorbing box with out-plane filled honeycomb structure exhibits buckling of the honeycomb cell walls, which squeeze against each other, forming a layered structure that continues to compress other honeycomb cells inward. Under the same energy absorption conditions, this deformation model can reduce the amount of deformation.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eTypically, the crashworthiness evaluation indicators for energy absorbing boxes include total energy absorption (\u003cem\u003eE\u003c/em\u003e), specific energy absorption (\u003cem\u003eE\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e), initial peak impact force (\u003cem\u003eF\u003c/em\u003e\u003csub\u003ep\u003c/sub\u003e), average impact force (\u003cem\u003eF\u003c/em\u003e\u003csub\u003em\u003c/sub\u003e), and maximum deformation (\u003cem\u003eD\u003c/em\u003e). Among these, the initial peak load represents the maximum load value first reached by the structure during the impact loading process. The other indicators are calculated using Equations (\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) to (\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e), as follows:\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:\\text{}\\text{E}={\\int\\:}_{\\text{0}}^{\\text{D}}\\text{F}\\text{d}\\text{y}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\:{\\text{E}}_{\\text{a}}=\\frac{\\text{E}}{\\text{m}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\:{\\text{F}}_{\\text{m}}=\\frac{\\text{E}}{\\text{D}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eHere, \u003cem\u003eF\u003c/em\u003e represents the instantaneous impact force, \u003cem\u003em\u003c/em\u003e is the structural mass, and \u003cem\u003ey\u003c/em\u003e is the instantaneous impact displacement. An ideal energy absorbing box with optimal crashworthiness should satisfy a lower \u003cem\u003eF\u003c/em\u003e\u003csub\u003ep\u003c/sub\u003e and higher \u003cem\u003eE\u003c/em\u003e, \u003cem\u003eE\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e, and \u003cem\u003eF\u003c/em\u003e\u003csub\u003em\u003c/sub\u003e. Since the mass and velocity of the impact wall are the same, the impact kinetic energy is 6.25 kJ. Therefore, the maximum energy absorption for the three energy absorbing boxes is 6.25 kJ. Under the same impact energy, the relationship between the impact force \u003cem\u003eF\u003c/em\u003e and time \u003cem\u003et\u003c/em\u003e is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e13\u003c/span\u003e. Based on Equations (\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) to (\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e), the crashworthiness evaluation indicators of the energy absorbing boxes can be calculated, as listed in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eThrough comparison, under the same impact energy, the three types of energy absorbing boxes exhibit the same maximum energy absorption. However, the traditional thin-walled square tube energy absorbing box shows the largest deformation, approaching complete collapse, indicating that it has reached its energy absorption limit. In contrast, the in-plane and out-plane filled energy absorbing boxes exhibit smaller deformations. If the impact energy is appropriately increased, their maximum energy absorption will also increase, demonstrating greater energy absorption potential. The out-plane filled honeycomb energy absorbing box has the smallest initial peak impact force \u003cem\u003eF\u003c/em\u003e\u003csub\u003ep\u003c/sub\u003e, the largest average impact force \u003cem\u003eF\u003c/em\u003e\u003csub\u003em\u003c/sub\u003e, and the smallest deformation \u003cem\u003eD\u003c/em\u003e. Therefore, it can be concluded that the out-plane filled honeycomb energy absorbing box has the best crashworthiness performance.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Sec15\" class=\"Section2\"\u003e \u003ch2\u003e3.3 Multi-objective Optimization of Energy Absorbing Box\u003c/h2\u003e \u003cdiv id=\"Sec16\" class=\"Section3\"\u003e \u003ch2\u003e3.3.1 Establishment of Multi-objective Optimization Model\u003c/h2\u003e \u003cp\u003eBased on the above analysis, the new automotive energy absorbing box with out-plane filled honeycomb structure exhibits the best crashworthiness. In this section, a multi-objective optimization of the energy absorbing box will be conducted. First, a mathematical model for the multi-objective optimization needs to be established. Since the new energy absorbing box is intended to protect occupant safety in vehicles, the factors that most significantly affect occupant safety are the deformation and specific energy absorption of the energy absorbing box. According to Eq.\u0026nbsp;(\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e), the larger the average impact force \u003cem\u003eF\u003c/em\u003e\u003csub\u003em\u003c/sub\u003e, the more energy the energy absorbing box absorbs per unit of collision displacement. Under the premise of a fixed total collision energy, this can reduce the collision displacement, thereby decreasing the deformation of the energy absorbing box and minimizing the intrusion into the occupant compartment. The specific energy absorption \u003cem\u003eE\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e, which represents the energy absorbed per unit mass, reflects both the crashworthiness and lightweight characteristics of the energy absorbing box. Therefore, the optimization objectives can be defined as maximizing the average impact force \u003cem\u003eF\u003c/em\u003e\u003csub\u003em\u003c/sub\u003e and maximizing the specific energy absorption \u003cem\u003eE\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e. The optimization variables are the honeycomb aperture \u003cem\u003eρ\u003c/em\u003e, the wall thickness of the honeycomb \u003cem\u003et\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, and the wall thickness of the thin-walled square tube \u003cem\u003et\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e. Typically, the range of optimization variables is set within \u0026plusmn;\u0026thinsp;20% of their initial values. The multi-objective optimization model is shown in Eq.\u0026nbsp;(\u003cspan refid=\"Equ4\" class=\"InternalRef\"\u003e4\u003c/span\u003e).\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$\\:\\left\\{\\begin{array}{c}\\text{m}\\text{i}\\text{n}\\text{(}-{\\text{F}}_{\\text{m}}\\text{,}-{\\text{E}}_{\\text{a}}\\text{)}\\\\\\:\\text{x}=\\left(\\text{\u0026rho;}\\text{,}{\\text{t}}_{\\text{1}}\\text{,}{\\text{t}}_{\\text{2}}\\right)\\\\\\:\\text{s}\\text{.}\\text{t}\\text{.}\\left\\{\\begin{array}{c}\\text{1.920}\\text{m}\\text{m}\\le\\:\\text{\u0026rho;}\\le\\:\\text{2.880}\\\\\\:\\text{0.048}\\text{m}\\text{m}\\le\\:{\\text{t}}_{\\text{1}}\\le\\:\\text{0.072}\\text{m}\\text{m}\\\\\\:\\text{1.200}\\text{m}\\text{m}\\le\\:{\\text{t}}_{\\text{2}}\\le\\:\\text{1.800}\\text{m}\\text{m}\\end{array}\\right.\\end{array}\\text{mm}\\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec17\" class=\"Section3\"\u003e \u003ch2\u003e3.3.2 Latin Experimental Design\u003c/h2\u003e \u003cp\u003eIn the multi-objective optimization model, there are three optimization variables, which require at least 10 sample points. According to the Latin hypercube sampling design rules, 16 sample points are randomly generated within the design space. The finite element simulation model is then used to obtain the average impact force \u003cem\u003eF\u003c/em\u003e\u003csub\u003em\u003c/sub\u003e and specific energy absorption \u003cem\u003eE\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e for each sample point. The data for each sample are presented in Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eEvaluation index of crashworthiness of energy absorbing box\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eStructure type\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003eF\u003c/em\u003e\u003csub\u003ep\u003c/sub\u003e(kN)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cem\u003eF\u003c/em\u003e\u003csub\u003em\u003c/sub\u003e(kN)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cem\u003eD\u003c/em\u003e(mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u003cem\u003eE\u003c/em\u003e(kJ)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNo filled\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e174.84\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e24.67\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e253.35\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e6.25\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eIn-place filled\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e137.36\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e37.73\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e165.63\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e6.25\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eOut-place filled\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e111.45\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e42.35\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e147.59\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e6.25\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec18\" class=\"Section3\"\u003e \u003ch2\u003e3.3.3 Establishment of Response Surface Model\u003c/h2\u003e \u003cp\u003eThe Latin hypercube sample data from Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e are imported into the Design-Expert software. Using the Response Surface module's Box-Behnken design, a response surface model is established. A second-order polynomial fitting is performed for the optimization variables and output response values. The requirement is that the goodness of fit (R2) must be greater than 0.90; otherwise, the constructed response surface model may have significant errors. The second-order response surface equations for the optimization objectives and optimization variables are shown in Equations (\u003cspan refid=\"Equ5\" class=\"InternalRef\"\u003e5\u003c/span\u003e) and (\u003cspan refid=\"Equ6\" class=\"InternalRef\"\u003e6\u003c/span\u003e).\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$\\:{\\text{F}}_{\\text{m}}=\\text{26.26942}+\\text{3.99078}\\text{\u0026rho;}+\\text{29.38653}{\\text{t}}_{\\text{1}}+\\text{3.69007}{\\text{t}}_{\\text{2}}-\\text{0.85423}\\text{\u0026rho;}{\\text{t}}_{\\text{1}}-\\text{2.9875}\\text{\u0026rho;}{\\text{t}}_{\\text{2}}+\\text{1.89511}{\\text{t}}_{\\text{1}}{\\text{t}}_{\\text{2}}+\\text{2.77143}{\\text{\u0026rho;}}^{\\text{2}}+\\text{1.11942}{{\\text{t}}_{\\text{1}}}^{\\text{2}}-\\text{1.86430}{{\\text{t}}_{\\text{2}}}^{\\text{2}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$$\\:{\\text{E}}_{\\text{a}}=\\text{19.40808}-\\text{4.94493}\\text{\u0026rho;}+\\text{6.85921}{\\text{t}}_{\\text{1}}-\\text{5.47}{\\text{174}\\text{t}}_{\\text{2}}-\\text{2.88109}\\text{\u0026rho;}{\\text{t}}_{\\text{1}}+\\text{0.921529}\\text{\u0026rho;}{\\text{t}}_{\\text{2}}-\\text{2.64498}{\\text{t}}_{\\text{1}}{\\text{t}}_{\\text{2}}+\\text{0.435542}{\\text{\u0026rho;}}^{\\text{2}}+\\text{23.79775}{{\\text{t}}_{\\text{1}}}^{\\text{2}}+\\text{0.589691}{{\\text{t}}_{\\text{2}}}^{\\text{2}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eLatin square experiment sample point data\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"8\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNumbering\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003eρ\u003c/em\u003e(mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cem\u003et\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e(mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cem\u003et\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e(mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u003cem\u003eF\u003c/em\u003e\u003csub\u003em\u003c/sub\u003e(kN)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u003cem\u003eE\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003e\u003cem\u003eD\u003c/em\u003e(mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c8\"\u003e \u003cp\u003e\u003cem\u003em\u003c/em\u003e(kg)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.927\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.051\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.206\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e40.581\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e7.881\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e154.014\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0.793\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.952\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.049\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.571\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e41.246\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e7.015\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e151.531\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0.891\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.056\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.055\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.378\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e41.119\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e7.184\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e151.997\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0.870\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.177\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.069\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.712\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e42.261\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e6.339\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e147.891\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0.986\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.512\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.050\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.301\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e41.965\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e6.490\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e148.933\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0.963\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.471\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.048\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.479\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e42.881\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e6.269\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e145.752\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0.997\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.294\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.063\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.736\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e44.508\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e6.127\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e140.423\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e1.020\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.384\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.065\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.589\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e43.603\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e6.213\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e143.337\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e1.006\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.400\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.060\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.500\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e42.347\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e6.332\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e147.590\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0.987\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.099\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.053\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.289\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e40.884\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e7.284\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e152.871\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0.858\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.871\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.071\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.755\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e45.486\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e5.337\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e137.404\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e1.171\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.710\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.058\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.401\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e45.034\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e6.015\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e138.782\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e1.039\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e13\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.657\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.056\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.299\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e43.277\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e6.250\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e144.420\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e1.000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.539\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.067\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.605\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e45.276\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e5.958\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e138.042\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e1.049\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.444\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.070\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.666\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e45.109\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e5.998\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e138.553\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e1.042\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.330\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.061\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.353\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e41.448\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e6.699\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e150.793\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0.933\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eFor Eq.\u0026nbsp;(\u003cspan refid=\"Equ5\" class=\"InternalRef\"\u003e5\u003c/span\u003e), the goodness of fit (R\u003csup\u003e2\u003c/sup\u003e) is 0.92, and the adjusted goodness of fit is 0.91. For Eq.\u0026nbsp;(\u003cspan refid=\"Equ6\" class=\"InternalRef\"\u003e6\u003c/span\u003e), the goodness of fit (R\u003csup\u003e2\u003c/sup\u003e) is 0.99, and the adjusted goodness of fit is 0.99. To verify the accuracy of the response surface model, three additional sample points were generated using the optimal Latin hypercube sampling method. The finite element model was used to calculate the simulation values corresponding to these sample points, and the results were compared with the response values obtained from the aforementioned response surface model. The results are shown in Table\u0026nbsp;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e5\u003c/span\u003e. The average error between the response values of the response surface model and the true values of the simulation results for the average impact force \u003cem\u003eF\u003c/em\u003e\u003csub\u003em\u003c/sub\u003e and specific energy absorption \u003cem\u003eE\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e is 2.022%, which is less than 5%. The maximum error is 7.083%, which is less than 10%. Therefore, it can be concluded that the response surface model has high accuracy and can replace the simulation model for subsequent optimization design.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec19\" class=\"Section3\"\u003e \u003ch2\u003e3.3.4 Multi-objective Optimization Solution\u003c/h2\u003e \u003cp\u003eIn Matlab, NSGA-Ⅱ algorithm is used to solve Equations (\u003cspan refid=\"Equ4\" class=\"InternalRef\"\u003e4\u003c/span\u003e). By defining the upper and lower bound constraint objective function of design variables, Gamultiobj function is selected as the solution function, population size is set to 200, crossover probability is set to 0.9, mutation probability is set to 0.2, and Pareto ratio is set to 0.35. After 300 iterations, the Pareto frontier optimal solution set shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig14\" class=\"InternalRef\"\u003e14\u003c/span\u003e is obtained. As shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig14\" class=\"InternalRef\"\u003e14\u003c/span\u003e, the average impact force \u003cem\u003eF\u003c/em\u003e\u003csub\u003em\u003c/sub\u003e is negatively correlated with specific energy absorption \u003cem\u003eE\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e, that is, the solution with the maximum specific energy absorption when the average impact force is maximum cannot be found. The actual situation should be considered when selecting the final optimal solution. For the crashability of the energy absorbing box, the deformation and specific energy absorption can reflect the resistance to deformation and energy absorption. It determines the invasion amount of the crew cabin and the safety of the crew after the collision. The optimal solution in Fig.\u0026nbsp;\u003cspan refid=\"Fig14\" class=\"InternalRef\"\u003e14\u003c/span\u003e can ensure that the average impact force is maximum when the specific absorption energy meets the requirements. According to Equations (\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e), when the impact energy is constant, the greater the average impact force is, the smaller the deformation is. The optimal solution corresponds to the honeycomb aperture ρ of the energy absorbing box, the honeycomb wall thickness \u003cem\u003et\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e is 0.071 mm, and the thin-walled square tube wall thickness \u003cem\u003et\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e is 1.315 mm. The average impact force \u003cem\u003eF\u003c/em\u003e\u003csub\u003em\u003c/sub\u003e is 43.281 kN, the deformation \u003cem\u003eD\u003c/em\u003e is 144.405 mm, and the specific absorption energy is 6.981.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab5\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 5\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eComparison between the response surface model and the simulation model\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"10\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c9\" colnum=\"9\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c10\" colnum=\"10\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eNum-bering\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e\u003cem\u003eρ\u003c/em\u003e\u003c/p\u003e \u003cp\u003e(mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e\u003cem\u003et\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e\u003c/p\u003e \u003cp\u003e(mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e\u003cem\u003et\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e\u003c/p\u003e \u003cp\u003e(mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"3\" nameend=\"c7\" namest=\"c5\"\u003e \u003cp\u003e\u003cem\u003eF\u003c/em\u003e\u003csub\u003em\u003c/sub\u003e(kN)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"3\" nameend=\"c10\" namest=\"c8\"\u003e \u003cp\u003e\u003cem\u003eE\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eEmu-lation\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eMo-del\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eError\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c8\"\u003e \u003cp\u003eEmu-lation\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c9\"\u003e \u003cp\u003eMo-del\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c10\"\u003e \u003cp\u003eError\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.105\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.055\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.575\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e41.387\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e39.918\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e3.549%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e6.713\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e6.711\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0.030%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.432\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.052\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.388\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e42.158\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e45.372\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e7.083%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e6.474\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e6.472\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0.031%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.340\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.062\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.608\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e43.098\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e42.546\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1.281%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e6.256\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e6.246\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0.160%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Sec20\" class=\"Section2\"\u003e \u003ch2\u003e3.4 Comparative Analysis Before and After Optimization\u003c/h2\u003e \u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig15\" class=\"InternalRef\"\u003e15\u003c/span\u003e shows a comparison of the impact force versus time curves for the out-plane filled honeycomb energy absorbing box before and after multi-objective optimization. The peak impact force \u003cem\u003eF\u003c/em\u003e\u003csub\u003ep\u003c/sub\u003e after optimization is smaller than that before optimization. The crashworthiness evaluation indicators of the energy absorbing box calculated using Equations (\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) to (\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e) are listed in Table\u0026nbsp;\u003cspan refid=\"Tab6\" class=\"InternalRef\"\u003e6\u003c/span\u003e. After optimization, the specific energy absorption \u003cem\u003eE\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e of the out-plane filled honeycomb energy absorbing box is improved, and the deformation \u003cem\u003eD\u003c/em\u003e is reduced. Overall, the energy absorbing box obtained through the NSGA-II algorithm for multi-objective optimization shows significant improvements in both comprehensive crashworthiness and lightweight performance.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"4 Crashworthiness Analysis of the Whole Vehicle After Replacing the Design","content":"\u003cdiv id=\"Sec22\" class=\"Section2\"\u003e \u003ch2\u003e4.1 Simplification of the Finite Element Model\u003c/h2\u003e \u003cp\u003eThis study primarily investigates the impact of replacing a new energy absorbing box on the crashworthiness of the entire vehicle. To save computational time, it is necessary to simplify the full-vehicle frontal crash simulation model. Under the premise of maintaining the material properties and connection methods of each component, the crash wall, crossbeam, energy absorbing box, longitudinal beam, and other connecting parts are directly extracted from the full-vehicle model. To ensure that the motion behavior of the simplified model remains consistent with that of the full-vehicle model, motion constraints are applied to the contact areas between the simplified model and the rest of the vehicle. The displacement-time curves of the crossbeam constraint nodes in the x, y, and z axes, which are representative of the full-vehicle crash process, are extracted. The boundary conditions of the simplified model are defined, and displacement sensors are added at the ends of the longitudinal beams to evaluate the crash outcomes for the passenger compartment. The simplified model is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig16\" class=\"InternalRef\"\u003e16\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec23\" class=\"Section2\"\u003e \u003ch2\u003e4.2 Simplify Model Reliability Verification\u003c/h2\u003e \u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig17\" class=\"InternalRef\"\u003e17\u003c/span\u003e shows a comparison of the energy absorption curves of the energy absorbing box before and after simplifying the full-vehicle frontal crash model. The two curves exhibit a high degree of agreement. Figure\u0026nbsp;\u003cspan refid=\"Fig18\" class=\"InternalRef\"\u003e18\u003c/span\u003e presents a comparison of the displacement contour maps of the crossbeam before and after model simplification, revealing minimal errors. The simplified model demonstrates strong consistency with the original model, confirming its suitability for use in subsequent research.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab6\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 6\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eComparison of crashworthiness evaluation indexes of energy absorbing boxes before and after optimization\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eEvaluation indicators\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003eF\u003c/em\u003e\u003csub\u003ep\u003c/sub\u003e(kN)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cem\u003eF\u003c/em\u003e\u003csub\u003em\u003c/sub\u003e(kN)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cem\u003eE\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u003cem\u003eD\u003c/em\u003e(mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u003cem\u003em\u003c/em\u003e(kg)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBefore optimization\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e111.453\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e42.347\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e6.332\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e147.590\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.987\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAfter optimization\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e86.918\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e43.281\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e6.981\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e144.405\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.895\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eThe amount of change\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e22.014%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2.158%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e9.297%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e2.158%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e9.321%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec24\" class=\"Section2\"\u003e \u003ch2\u003e4.3 Comparison of the Crashworthiness of the Whole Vehicle Before and After the Replacement Design\u003c/h2\u003e \u003cp\u003eThe out-plane filled honeycomb energy absorbing box, optimized for crashworthiness in Chap.\u0026nbsp;3, was installed into the simplified model for frontal crash simulation. The crash speed and other constraint conditions remained unchanged. Figure\u0026nbsp;\u003cspan refid=\"Fig19\" class=\"InternalRef\"\u003e19\u003c/span\u003e shows a comparison of the energy absorption curves of the energy absorbing box before and after the replacement design. It can be observed that the energy absorption value of the new energy absorbing box accounts for 8.79% of the total crash energy, representing a 12.51% increase in energy absorption proportion compared to the traditional thin-walled square tube energy absorbing box used before the replacement.\u003c/p\u003e \u003cp\u003eExcessive crash acceleration can lead to occupant injuries such as fractures and internal organ damage. Reducing the overall crash acceleration of the vehicle can mitigate the severity of occupant injuries after a collision. Figure\u0026nbsp;\u003cspan refid=\"Fig20\" class=\"InternalRef\"\u003e20\u003c/span\u003e compares the vehicle crash acceleration before and after the replacement design. The results show that the acceleration decreased by 11.49%, and the acceleration curve became smoother with reduced oscillations, improving the secondary impact forces experienced by the occupants.\u003c/p\u003e \u003cp\u003eBased on the data from the displacement sensors placed at the ends of the longitudinal beams in the simplified model, the intrusion curve of the passenger compartment over time was obtained, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig21\" class=\"InternalRef\"\u003e21\u003c/span\u003e. The intrusion of the passenger compartment was reduced by 17.48%. Based on the above analysis, the proposed out-plane filled honeycomb energy absorbing box for vehicles can enhance the crashworthiness of the vehicle, reduce occupant injuries, and improve occupant safety.\u003c/p\u003e \u003c/div\u003e"},{"header":"5 Conclusion","content":"\u003cp\u003eThis study demonstrates that replacing the energy absorbing box with a new filled honeycomb structure enhances the vehicle's crashworthiness and occupant safety. The specific conclusions are as follows:\u003c/p\u003e \u003cp\u003e \u003col\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eTaking a certain sedan as the research subject, a simulation environment for side pole collision was established based on the C-NCAP testing requirements. The reliability of the model was verified by comparing it with relevant tests. It was identified that the vehicle's poor crashworthiness under frontal collision conditions is due to insufficient stiffness of the crossbeam and inadequate energy absorption by the energy absorbing box.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eTwo new types of energy-absorbing boxes with different filling configurations were designed. Impact simulations demonstrated that the out-plane filled honeycomb energy absorbing box exhibits the best crashworthiness. Multi-objective optimization was performed on the energy absorbing box, achieving lightweight design while improving crash performance.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eThe new energy absorbing box was installed in a simplified model for frontal collision simulation. By comparing the vehicle crash acceleration, energy absorption ratio of the energy absorbing box, and passenger compartment deformation before and after the replacement design, it was verified that the proposed new energy absorbing box reduces the vehicle acceleration by 11.49%, decreases passenger compartment intrusion by 17.48%, and increases the energy absorption ratio by 12.51%. This addresses the shortcomings of the original vehicle's crashworthiness and enhances occupant safety.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003c/ol\u003e \u003c/p\u003e "},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eDisclosure statement\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNo potential conflict of interest was reported by the author(s).\u003c/p\u003e\u003ch2\u003eFunding\u003c/h2\u003e \u003cp\u003eThis study was supported by NSFC 52472357, partially supported by Heilongjiang Province Key R\u0026amp;D Program Guidance Category Project GZ20220027 and Heilongjiang Provincial Transportation Investment Group Co., LTD Project JT-100000-ZC-FW-2021-0119.\u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eW. H.L. wrote the manuscript text,W. Z.Y.provided guidance on ideas,D.X.J. provided financial support,W. N. drew all the charts.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eGulshan N, Svitlana R, Kjell G, et al. (2022) Vehicle crashworthiness performance in frontal impact: mathematical model using elastic pendulum. Mechanics Research Communications 124:103954.\u003c/li\u003e\n\u003cli\u003eMin B, Cho J (2017) Impact characteristic according to the structure of crash box at the vehicle. Archives of Metallurgy and Materials 62(02):1047-1050.\u003c/li\u003e\n\u003cli\u003eXiang J, Du J (2017) Energy absorption characteristics of bio-inspired honeycomb structure under axial impact loading. Materials Science \u0026amp; Engineering 696:283-289.\u003c/li\u003e\n\u003cli\u003ePattaramon J, Sittha T, Suphanut K, et al. (2024) Optimizing functionally graded hexagonal crash boxes with honeycomb filler for enhanced crashworthiness. 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Acta Mechanica Solida Sinica 21(04):383-388.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"iranian-journal-of-science-and-technology-transactions-of-mechanical-engineering","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"istm","sideBox":"Learn more about [Iranian Journal of Science and Technology, Transactions of Mechanical Engineering](https://link.springer.com/journal/40997)","snPcode":"40997","submissionUrl":"https://submission.springernature.com/new-submission/40997/3","title":"Iranian Journal of Science and Technology, Transactions of Mechanical Engineering","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Honeycomb structure, Energy absorbing box, Crashworthiness, Multi-objective optimization, Replacement design","lastPublishedDoi":"10.21203/rs.3.rs-6258409/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6258409/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe energy absorbing box plays the role of buffering energy absorption in the process of frontal collision of automobiles, but the current structure of the energy absorbing box is single, and the energy absorbing effect is limited. In this study, the honeycomb structure was filled into a thin-walled square tube to design a new type of automotive energy absorbing box. Firstly, a frontal collision finite element model is established with a vehicle model as the research object, and the crashworthiness defects of the vehicle model under the frontal collision condition are pointed out. Next, two filling forms of new energy absorbing boxes were designed, and the crashworthiness was compared through impact simulation, followed by multi-objective optimization to further enhance their performance. Finally, the original thin-walled square tube energy absorbing box was replaced with the honeycomb-filled design. A simplified model was used to compare the crashworthiness of the vehicle before and after the replacement. The results showed that, after the replacement, the vehicle's acceleration decreased by 11.49%, occupant compartment intrusion was reduced by 17.48%, and the energy absorption ratio of the energy absorbing box increased by 12.51%. Crashworthiness and occupant safety have been improved.\u003c/p\u003e","manuscriptTitle":"Design of a New Energy Absorption Box with Honeycomb Structure for Vehicles and Research on Vehicle Crashworthiness","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-03-24 09:23:44","doi":"10.21203/rs.3.rs-6258409/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-04-05T03:02:46+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-03-30T09:41:16+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-03-30T04:24:12+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"335262459950872258558078956476573355586","date":"2025-03-24T00:55:50+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"96617101922948538604842962272942388577","date":"2025-03-21T13:23:27+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-03-21T12:25:41+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"310843962129186511292123864304337982186","date":"2025-03-21T03:41:00+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-03-21T03:07:38+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-03-20T07:49:33+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-03-20T07:48:57+00:00","index":"","fulltext":""},{"type":"submitted","content":"Iranian Journal of Science and Technology, Transactions of Mechanical Engineering","date":"2025-03-19T06:10:05+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"iranian-journal-of-science-and-technology-transactions-of-mechanical-engineering","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"istm","sideBox":"Learn more about [Iranian Journal of Science and Technology, Transactions of Mechanical Engineering](https://link.springer.com/journal/40997)","snPcode":"40997","submissionUrl":"https://submission.springernature.com/new-submission/40997/3","title":"Iranian Journal of Science and Technology, Transactions of Mechanical Engineering","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"338a1f5f-af43-4dc4-ab94-ede42628146b","owner":[],"postedDate":"March 24th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2025-05-26T16:06:28+00:00","versionOfRecord":{"articleIdentity":"rs-6258409","link":"https://doi.org/10.1007/s40997-025-00877-z","journal":{"identity":"iranian-journal-of-science-and-technology-transactions-of-mechanical-engineering","isVorOnly":false,"title":"Iranian Journal of Science and Technology, Transactions of Mechanical Engineering"},"publishedOn":"2025-05-21 15:57:52","publishedOnDateReadable":"May 21st, 2025"},"versionCreatedAt":"2025-03-24 09:23:44","video":"","vorDoi":"10.1007/s40997-025-00877-z","vorDoiUrl":"https://doi.org/10.1007/s40997-025-00877-z","workflowStages":[]},"version":"v1","identity":"rs-6258409","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6258409","identity":"rs-6258409","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}