1 Introduction

Thin-walled tubes have been widely utilized as efficient energy absorbers in vehicles’ structural frame to reduce the harmful influences of impact loading and to protect the passengers. Peak collapsing load (PCL) and specific energy absorption (SEA) are the most important responses of energy absorbing structure for vehicle crashworthiness applications. Generally, high SEA and low PCL are preferred to ensure the lightweight design of the vehicle and the safety of the passengers. Over the past years, there was a growing interest in the crashworthiness field where the crush responses and deformation modes of various structures under quasi-static and dynamic loads were investigated using numerical, analytical, and experimental methods (Niknejad and Bonakdar 2015; Niknejad and Orojloo 2016; Zahran et al. 2018; Esa et al. 2017; Estrada et al. 2018a, 2018b, 2019; Taghipoor and Noori Mohammad 2018; Taghipoor and Damghani Nouri 2018a, 2018b; Taghipoor et al. 2020a; Tran et al. 2020; Baroutaji et al. 2021). For example, DiPaolo et al. (2004) carried out an investigation on the compression behavior of steel square tubes under quasi-static loading. Ahmad and Thambiratnam (2009) studied the energy absorption ability of foam-filled conical tubes under impact loading and considered the effects of the geometry and loading angles on the behavior of such structures. Mahdi Abedi et al. (2012) executed crush experiments on empty and foam-filled tubes to develop analytical formulas for predicting folding load. Meran (2016) examined the crashworthiness of thin-walled tubes with different geometrical shapes and revealed the influence of cross-section shape on the crashworthiness metrics. Eyvazian et al. (2020) studied the behavior of the vertical corrugated square tube under axial loading and pointed out that the height of the vertical waves strongly affects the maximum compression load of the tube. Taghipoor et al. (2020b) compared the crashworthiness characteristics of the different energy absorbers in case of low speed loading. Among the different configurations, the aluminum foam-filled square tube showed the better crashworthiness performance.

Recently, multi-cell tubes have received increased research attention as new set of structures with outstanding energy absorption capability, and their crashworthiness performance was the subject of study for many researchers. For instance, Kim (2002) introduced new multi-cell square tubes with better crashworthiness performance than the traditional single-cell configuration. Alavi and Parsapour (2014) and Vimal Kannan and Rajkumar (2020) studied the energy absorption ability of different multi-cell tubes and pointed out that the multi-cell hexagonal and octagonal tubes have better crashworthiness performance than multi-cell square tubes. Xu et al. (2018) conducted crashworthiness analysis on hierarchical hexagonal tubes and explored the main factors affecting their energy absorption responses. Generally, it was reported that a multi-cell tube has better crashworthiness performance than its single-cell counterpart because it can absorb greater SEA (Chen and Wierzbicki 2001). However, the high SEA of the multi-cell tubes is normally associated with a very high PCL which is much greater than its single-cell counterpart.

To reduce PCL of an energy absorber, crush initiators are normally implemented on the tube walls. In addition to reducing the PCL, initiators play a key role in controlling the folding process and inhibiting the undesirable global buckling deformation mode. The crush initiators come in different formats including grooves, corrugations, or holes. El-Hage et al. (2005) studied the effects of the different initiators’ mechanisms on the crushing behavior using simulation. Hosseinipour and Daneshi (2003) considered the crashworthiness performance of grooved tubes and showed that such tubes have smaller PCL and better collapsing behavior than those of the corresponding traditional tubes without grooves. Eyvazian et al. (2012, 2014, 2018) examined the effects of the corrugations on the collapsing behavior and crashworthiness indexes including SEA and PCL. The authors reported that the corrugations make the collapsing mode of the tubes more controllable. Alkhatib et al. (2017) examined the influence of corrugations on the crashworthiness behavior of corrugated tapered tubes. They have found that the corrugations with longer wavelength yield smaller PCL during the crushing. Arnold and Altenhof (2004) examined the crashworthiness of the tubes with circular discontinuities and showed that an increase in hole diameter may lead to a decrease in PCL. However, they have noticed that the holes with very big diameters, larger than 32 mm, may induce cracks on the tubes walls during the axial crushing. Bodlani et al. (2009a, 2009b) also performed similar investigation on the crashworthiness of tubes with circular holes under axial loading. Their work also proved that introducing holes to the side walls of the tube can decrease PCL. Nonetheless, PCL was not reasonably reduced when the number of the holes was greater than two. Marzbanrad et al. (2009) studied the crashworthiness performance of a circular tube with rectangular holes on its walls. They have also reported that the holes have caused a reduction in PCL during impact.

Generally, the crashworthiness performance of a structure is influenced by different factors including its geometrical parameters and material characteristics. Design of experiments, meta-modeling, and optimization tools were employed widely for various types of structures to find the optimal crashworthiness configuration. Pareto front, which is a set of optimal solutions, is normally obtained as a result of applying the optimization process. Hou et al. (2007, 2008, 2014) used the aforementioned tools to solve a multi-objective crashworthiness optimization problem of multi-cell tubes. Tran and Baroutaji (2018) performed crashworthiness optimization analysis of multi-cell triangular tubes and analyzed their energy absorption characteristics. Yang et al. (2018) carried out design optimization of multi-cell tubes with folded patterns and evaluated the impact of the design parameters on the crashworthiness behavior. Bigdeli and Nouri (2019) combined DoE and multiple objective particle swarm optimization (MOPSO) algorism to perform the optimization of multi-cell tubes and proposed novel cross-section to improve crashworthiness performance. Other studies on the crashworthiness optimization of multi-cell tubes were also executed by Shen et al. (2017), Chen et al. (2018), Zhang et al. (2018), Luo and Fan (2018), and Yin et al. (2014, 2015, 2017).

As a summary, the aforementioned studies showed that the multi-cell structures have better crashworthiness performance than the standard single-cell ones. Also, it was widely reported that using the crush initiators on tube’s wall can reduce its PCL and control its deformation behavior. However, no attempt was made in the literature to use windows crush initiators as a mean to reduce the high PCL of the multi-cell tubes. The high PCL of the multi-cell structures was considered as a limiting factor for using such structures in real-life applications (Baroutaji et al. 2017; Tran et al. 2019; Nikkhah et al. 2020). Thus, the current paper proposes new windowed hexagonal multi-cell structures and seeks their crashworthiness optimal design under axial impact loading. First, crush simulations of multi-cell hexagonal tubes without and with windows are built using LS-DYNA. Then, a two-stage optimization scheme, with SEA and PCL as objective functions, is executed to find the optimal crashworthiness design of the standard and windowed tubes. In the first step of the optimization procedure, the optimal configurations of the standard multi-cell hexagonal tubes are determined. In the second step, windows were implemented on the optimal configuration of the standard tubes, and their sizes were optimized to obtain the best windowed tubes. DoE, response surface method (RSM), and MOPSO are combined to solve the crashworthiness multi-objective optimization problem. Also, the influence of the geometrical parameters of the tube and windows on the crashworthiness behavior and collapsing modes is presented and discussed.

2 Numerical modeling

2.1 Geometry and material

Figure 1 shows the different standard and windowed multi-cell hexagonal tubes, abbreviated as STs and WTs, respectively, investigated in this paper. The circumcircle diameter, the wall thickness, and the length of the hexagonal tubes are denoted as a, t, and L0, respectively. For WTs, rectangular windows with depth c and width d were applied to the sidewalls of the hexagonal tubes, and they were distributed equally along the tube’s length. A length of 250 mm was used for all STs and WTs. Three different multi-cell configurations, as shown in Fig. 1, were created. The distinguishing feature among the three configurations is the way in which the corners and walls of the hexagonal tube are connected together to form the multi-cell geometry. Wall-to-wall (WTW) connecting webs were used in type I, while corner-to-corner (CTC) webs were employed in types II and III. The three multi-cell types have the same values of d, L0, and t, but they are different in weight. Tube III is the heaviest one, while tube I is the lightest. All the tubes were subjected to axial impact loading. The crush test set-up is shown in Fig. 2.

Fig. 1
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Cross-sectional geometry and outlines of tubes

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Fig. 2
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Dynamic model setup: a STs and b WTs

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The tubes were made of AA6060-T4 aluminum alloy with Young’s modulus of E = 68,200 MPa, initial yield stress of σy = 80 MPa, ultimate stress of σu = 173 MPa, Poisson’s ration of υ = 0.3, and power law exponent of n = 0.23 for representing the stress-strain response in the plastic region (Santosa et al. 2000).

2.2 Finite element modeling

The finite element (FE) models of all tubes investigated in this paper are constructed using nonlinear explicit finite element code LS-DYNA. Hypermesh was used in the preprocessor stage to define the material model, meshing, loading, and boundary conditions. The model consists of a stationary rigid wall and a thin-walled tube traveling at 10 m/s. An additional mass of 500 kg was attached to the moving object to provide sufficient deformation for the tube upon impacting on the rigid wall. The thin-walled tubes were meshed using the four-node quadrilateral shell elements with one integration point in the element plane and three other integration points through the thickness. Such shell element was extensively used in the literature for modeling the crush behavior of thin-walled structures as they can accurately capture the deformation modes under the impact loading. To decrease the impact of mesh size on the accuracy of the model’s results, a mesh convergence test is performed. The mesh convergence results are presented in Fig. 3a which shows the variation of peak collapse load (PCL) of the tube with seven different mesh sizes. As it can be seen, the difference in load’s value becomes very insignificant when the mesh size is less than 2.5 mm. Thus, it can be stated that an element size of 2.5 mm is sufficient to provide accurate numerical results within reasonable solution time and for this it was selected for the simulation. The attached mass and the rigid wall were set as rigid bodies (*MAT_20). The piecewise linear plastic material model (*MAT_24) was adopted to represent the mechanical behavior of AA6060-T4 aluminum alloy. It is evident from previous researches that AA6060-T4 is insensitive to strain rate and for this the strain rate effects were not considered in the current FE model (Santosa et al. 2000). Surface-to-surface contact with static and dynamic friction coefficients of 0.3 is employed to model the interactions between the tube and rigid walls. The self-contact experienced by the tube during the deformation was modeled using the single_surface_contact.

Fig. 3
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a Convergence of load versus mesh density for tube, b comparison of experimental and numerical crush responses of standard square tube, and c comparison of experimental and numerical crush responses of windowed square tube

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2.3 FE model validation

The developed FE model was first validated using a closed-form formula that was extracted by Yin et al. (2020) based on the super folding element method, and it describes the MCL of ST-III multi-cell hexagonal tube as function of tube’s thickness (t) and circumcircle diameter (a) as given in (1).

$$ MCL=\gamma \frac{6}{2k}{\sigma}_o{t}^{1.5}{\left(a/2\right)}^{0.5}\sqrt{2\pi \left(1+2\cos \left(\frac{\pi }{2}-\frac{\pi }{6}\right)\right)\left(1+2\tan \left(\frac{\pi }{4}-\frac{\pi }{12}\right)+\tan \left(\frac{\pi }{2}-\frac{\pi }{6}\right)\right)} $$
(1)

where σo is the material’s flow stress, γ is the dynamic enhancement factor and was set as 1.1, and k is the crushing distance coefficient and was set as 0.75.

Based on the above equation, the MCL for ST-III was calculated as 58.3 kN when D and t were selected as 80 mm and 2 mm, respectively. The average force obtained using the FE simulation was 62 kN. By comparing the analytical and the simulated MCL, it can be seen that the FE model can predict the MCL with only 6.5% relative error.

To further validate the material model used in the finite element model developed in this paper, experimental test data reported by Nikkhah et al. (2019) on the crushing of simple and windowed square tubes made of AA6060-T4 was adopted. Crush simulations of the square tubes were also executed in LS-DYNA with the same boundary conditions as those used in the experiments. A comparison of the numerical and experimental results is presented in Fig. 3b and Fig. 3c. As it can be seen from the crushing response, the numerical crushing forces are slightly lower than the experimental ones, particularly for the windowed tubes, which might be due to the inability of the FE model to accurately capture the inhomogeneous axial stiffness of the tube caused by the material cutaway in the windows’ zone. However, the experimental and simulation results in terms of the collapsed configuration and load-displacement responses are still in a satisfactory agreement. Thus, it can be considered that the developed FE model is sufficiently accurate and can be used to simulate the responses of STs and WTs tubes under axial impact.

3 Crashworthiness optimization designs

3.1 Crashworthiness metrics

The crashworthiness performance of STs and WTs is appraised via different crashworthiness indices, such as energy absorption capacity (EA), PCL, SEA, and mean collapsing load (MCL), which are deliberated previously in the literature and extensively utilized for evaluating the crush and energy absorption behaviors of thin-walled tubes (Sun et al. 2011, 2014, 2017).

The energy absorption capacity or the energy absorbed by tube (EA) is the strain energy absorbed during the collapsing process, and it is normally given as shown in (2)

$$ {E}_A=\underset{0}{\overset{\delta }{\int }}P(u) du $$
(2)

where P(u) is the instantaneous collapsing load as a function of collapsing distance and δ is the total collapsing distance.

Specific energy absorption (SEA) is a key index representing the amount of energy that can be absorbed per unit mass of a structure; SEA can be calculated as demonstrated in (3)

$$ SEA=\frac{E_A}{m} $$
(3)

where m is the mass of the tube.

PCL is the maximum value of P(u) which corresponds to peak deceleration. Considering the vehicle impact, a high PCL often leads to a large damage and severe injury or death of the vehicle’s occupants. The PCL of an absorber should therefore be reduced and constrained to the survivable levels.

Collapsing load efficiency (CLE) is another valuable dimensionless crashworthiness indicator which can be defined as the ratio between the mean collapse load (MCL) and the PCL, as shown in (4)

$$ \mathrm{CLE}=\frac{\mathrm{MCL}}{\mathrm{PCL}} $$
(4)

MCL can be obtained using (5)

$$ MCL=\frac{E_A}{\delta } $$
(5)

3.2 Design of experiment

The first step of establishing the meta-model of the crashworthiness metrics is to choose a sampling technique to generate the design points. Sampling strategy is very crucial for balancing the accuracy of the meta-models and the number of runs required therefore an effective sampling technique should be used. Design of experiment (DoE) approach is usually applied to build an effective mesh of training points within the design space (Hou et al. 2011, 2012, 2013; Baroutaji et al. 2014, 2015; Acar et al. 2011). The main sampling techniques offered by DoE are Latin hypercube sampling, orthogonal design, and full factorial design (FFD). Taking the benefits of furnishing the design region with uniformly distributed sampling points, FFD is adopted in this study to create the design points. The geometrical parameters of the tubes and the windows, namely, a, t, c, and d, as presented in Section 2.1, were selected as design variables. The upper and lower levels of these design variables are presented in Table 1. Applying the FFD on the four design variables has resulted in total 16 design points within the design space.

Table 1 Upper and lower levels of the design variables
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3.3 Response surface method

After generating the design points, axial crush simulations of the different STs and WTs were conducted at the design points using LS-DYNA, and the SEA and PCL values were calculated. The design variables, i.e., a, t, c, and d, and the design responses, i.e., SEA and PCL, as obtained from the numerical simulations are presented in Tables 2 and 3 for STs and WTs, respectively. Following obtaining the SEA and PCL values at the training points, response surface method (RSM) was then adopted to construct the meta-models of SEA and PCL responses required for crashworthiness optimization and parametric analysis. The aforementioned crashworthiness metrics, i.e., SEA and PCL, were calculated up to a crushing distance of 140 mm in order to avoid any undesirable crush behavior at high displacement, thus providing more realistic crashworthiness analysis within the practical crush zone. The RSM demonstrated good approximation accuracy for highly nonlinear crashworthiness optimization problem, and it was adopted extensively in previous researchers (Hou et al. 2007, 2008, 2009; Nguyen et al. 2016, 2017).

Table 2 Design matrix for STs
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Table 3 Design matrix for WTs
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After building the meta-models, they should be tested for accuracy to ensure their validity for the analysis. Hence, the analysis of variance (ANOVA), which provides detailed information about the model statistical significance measures, such as coefficient of determination (R2), adjusted coefficient of determination (Adj-R2), and Adeq precision, was used to check the adequacy of the developed meta-models for SEA and PCL. Table 4 shows a summary of ANOVA analysis for STs and WTs, for which the statistical measures are listed. As it can be seen, all meta-models exhibit high F values and small p values. The F values, ranging from 12.76 to 3840.96, point out that these models are significant. The small p values confirm that there is very small opportunity that the F value could occur due to noise. Additionally, the R2, of all meta-models, ranges from 0.86 to 0.99 indicating that the meta-models provide an excellent fitting of the data points. The values of Pre-R2, ranging from 0.7 to 0.99, are in very good agreement with the Adj-R2 values. The Adeq precision, which measures the signal to noise ratio, is above 4 for all meta-models indicating excellent adequacy for them.

Table 4 Summary of ANOVA analysis for STs and WTs
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3.4 Formulation of optimization problem

One of the problematics in design optimization for vehicle crashworthiness is the selection of appropriate objective functions. Different combinations of the various crush responses such as EA, peak acceleration (PA), PCL, and SEA tend to be employed as objective functions in the crashworthiness optimization problems. When maximizing SEA is considered to be one objective, it should be accompanied by restraints on PCL to ensure the safety performance of the vehicle. Contrary to the single-objective optimization which offers only one optimal solution, the multi-objective optimization provides a set of optimal solutions, which permits larger design range for designers. For the current study, PCL and SEA are selected as the objective functions to be minimized and maximized, respectively. The geometrical parameters of tubes and windows including a, t, c, and d were set to vary within their limits. The optimization problem of STs and WTs can be formulated as shown in (6)

$$ \left|\begin{array}{l}{1}^{\mathrm{st}}\;\mathrm{process}(STs):\operatorname{}\left\{\begin{array}{l}\left[\begin{array}{l}\operatorname{Maximize}\operatorname{} SEA\\ {}\operatorname{Minimize}\operatorname{} PCL\end{array}\right.\\ {}s.t\operatorname{}80 mm\le a\le 100 mm\\ {}\operatorname{}1 mm\le t\le 2 mm\end{array}\right.\\ {}{2}^{\mathrm{nd}}\;\mathrm{process}(WTs):\operatorname{}\left\{\begin{array}{l}\left[\begin{array}{l}\operatorname{Maximize}\operatorname{} SEA\\ {}\operatorname{Minimize}\operatorname{} PCL\end{array}\right.\\ {}s.t\operatorname{}4 mm\le c\le 16 mm\\ {}\operatorname{}8 mm\le d\le 32 mm\end{array}\right.\end{array}\right. $$
(6)

3.5 Multi-objective optimization procedure

Optimization algorithms are required to solve the multi-objective optimization problem presented in (6) and to perform the trade-off between the two conflicting design objectives and to generate a series of optimal solutions known as the Pareto front. Among the most common optimization algorithms such as Multi-Island Genetic Algorithm (MIGA), Non-dominated Sorting Genetic Algorithm (NSGA), and Multi-Objective Particle Swarm Optimization (MOPSO), MOPSO is characterized by the fast convergence and well-distributed Pareto front, and it was selected for finding the Pareto fronts of the different STs and WTs. A flowchart illustrating the multi-objective optimization process of STs and WTs is introduced in Fig. 4.

Fig. 4
figure 4

Flowchart of multi-objective optimization

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The optimization procedure contains two processes with identical steps. Each optimization process has three main steps. The first step is to define the crashworthiness optimization problem for STs and to select the design variables and design space. In the second step, crush simulations using FE models are carried out to get the crashworthiness indicators at the sampling points and then construct the meta-model of the crush responses using RSM. In the last step, the MOPSO solves the crashworthiness optimization problem, and the Pareto fronts, i.e., a series of optimal solutions, of the STs are obtained. A favorite optimal solution for each type of STs is chosen from its corresponding Pareto front based on specific design requirements using the minimum distance selection method. Following that, the obtained optimal design of each ST is combined with windows to build WTs, and then the second optimization process is executed to get the Pareto fronts for each WT.

4 Results and discussion

4.1 Deformation modes and crush responses of STs and WTs

In order to analyze the crushing behavior of STs, their deformation processes of STs with a = 93.3333 mm and t = 1.3333 mm are laid out in Fig. 5. The crashworthiness metrics and the crush responses were calculated up to a displacement of 140 mm in order to avoid the extreme increase in the crush force at the end of the deformation process caused by the crumpling of the tube’s material. The selected crush distance, i.e., 140 mm, allows for more sensible estimation of the crashworthiness behavior of the investigated tubes. Although all STs deform in a progressive folding way, they reveal different folding aspects. The CTC and WTW connection webs play an important role in determining the deformation mode of the standard multi-cell tubes. The deformation mode of ST-I with WTW connection webs is more regular with shorter wavelength and greater number of folds than those observed for ST-II and ST-III. This could be due to the fact that the WTW webs may have improved the lateral stiffness of the sidewalls of ST-I leading to more stable and better deformation mechanism. The CTC webs in ST-II and ST-III do not seem to enhance the lateral stiffness of the tubes, and therefore, their deformation mechanisms involve lower number of folds with greater wavelength than those of ST-I. So in summary, it can be reported that ST-I has more favorable progressive collapsing mode than the other standard tubes. Such folding mode is desirable as it can improve the energy absorption capacity of the structure.

Fig. 5
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Deformation history of STs

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To analyze the effects of implementing windows on the sidewalls of the STs, the final deformation patterns of WTs with various windows’ sizes are illustrated in Fig. 6. It is clear that all WTs deform in a progressive folding fashion. The wrinkling of the WT-I and WT-II occur at their middle regions while WT-III wrinkles at its upper part. Generally, as it has been previously reported in the literature, by introducing windows into a tube, its folding mode can be efficiently controlled, and the global buckling can be prevented, and this appears to be the case for WT-II and WT-III. However, the WT-I with larger windows’ size seems to undergo some sort of global bending which negatively affects its crashworthiness characteristics. Generally, the plastic deformation is initiated in the zones possessing lower stiffness due to its relatively low ability to support load, and then it continuously progresses to zones with higher stiffness during the crushing process. Based on this feature, the desired collapse mode of the WTs can be achieved by controlling the locations of the windows. Comparing the failure modes of all WTs, it seems that the WT-II has the most regular and stable failure mode with the higher number of folds. When increasing the window’s size, the global buckling phenomenon appears in WT-I, while the deformations of WT-II and WT-III remain progressive and stable. For WT-I, the larger the windows are, the more unstable the deformation mode is.

Fig. 6
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Deformation history of WTs: a) WT-I, b) WT-II, and c) WT-III

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The load-displacement curves of STs and WTs are plotted in Fig. 7. The load histories of tubes during crushing are valuable data to estimate the overall collapsing performance. It is clear that the different tubes offer similar force-displacement responses. After the first loading peak, the collapsing load fluctuates several times to form wrinkles during the crushing process. For STs, the level of collapsing load of ST-I is greater than those of ST-II and ST-III indicating a higher MCL index. However, for WTs, the levels of the crushing load of WT-II and WT-III are greater than that of WT-I. Therefore, MCLs of WT-II and WT-III are improved compared to WT-I. Figure 7 also shows that the load fluctuation of WTs with windows’ sizes of 4 × 32 mm is smaller than those of WTs with windows’ size of 8 × 8 mm. This can be linked with the fact that the WTs with bigger windows’ size require less force to generate the successive folds during the crushing.

Fig. 7
figure 7

Force-displacement responses of a STs with a = 93.3333 mm and t = 1.3333 mm, b WTs with windows’ size of 8 × 8 mm, and c WTs with windows’ size of 4 × 32 mm

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4.2 Parametric analysis

The geometrical shape of the energy absorption structure dominates its responses to the dynamic loading. Thus, in order to fully understand and evaluate the performance and safety of STs and WTs, the influence of their main geometrical parameters, such as a, t, c, and d, on their crashworthiness indexes is presented in this section.

4.2.1 Influence of a and t on the crashworthiness metrics of STs

Figure 8 shows surface plots depicting the variation of SEA and PCF with the geometrical parameters for all STs. It can be seen that ST-I has a slightly lower PCL and greater SEA than the other STs for all values of t and a. ST-III offers slightly higher PCL and SEA than those of ST-II. Such observations confirm that WTW connection implemented in ST-I is more effective for the standard tubes than the CTC connection type used in ST-II and ST-III.

Fig. 8
figure 8

Variation of a PCL and b SEA with geometrical parameters of STs

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Figure 9 illustrates the changes of PCL and SEA with a when t = 1 mm. It is clear that crashworthiness indexes of STs are considerably influenced by a. From the figure, it can be seen that by increasing a, PCL increases and SEA decreases. Under the same values of t and a, ST-I shows smaller PCL and greater SEA than those recorded in ST-II and ST-III. When a reaches to 80 mm, the SEA of ST-II falls more rapidly compared to those of ST-I and ST-III.

Fig. 9
figure 9

Effects of a on a) PCL and b) SEA of STs

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Figure 10 depicts the variation of SEA and PCL with t when a equals to 80 mm. It is clear that both PCL and SEA steadily rise with the increase of t. Also, it can be seen that ST-I exhibits greater SEA and smaller PCL than ST-II and ST-III for all values of t. PCLs and SEAs of ST-II and ST-III are approximately equal. When t is above 1.3333 mm, the rate of increase of SEA with t of ST-I is smaller than those of ST-II and ST-III. Although increasing t can improve the energy absorption, it also causes PCF to be substantially increased.

Fig. 10
figure 10

Effects of t on a) PCL and b) SEA of STs

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The values of CLE index for all STs at the 16 design points are exhibited in Fig. 11. This index is also a key crashworthiness response which gives a good idea about the collapse character and the bearing capability of the tubes. As shown in Fig. 11, it is clear that ST-I is the best performing design in terms of CLE. Figure 11 can also be used to extract the influence of t and a on CLE. It can be seen that when t increases, CLE of all STs increase.

Fig. 11
figure 11

CLE response of all STs at the design points

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4.2.2 Influence of c and d on the crashworthiness metrics of WTs

The variations of PCL and SEA with c and d are presented in Fig. 12 for all WTs. Comparing with STs, PCL and SEA are reduced substantially in WTs and continuously reduced with increasing windows’ dimensions. PCL is directly related to the stiffness of the weakest region of the tube, and therefore, the larger windows’ size means a less stiffness of the tube and thus smaller PCL is required. Contrary to the observations on STs, PCLs and SEAs of WT-II and WT-III are considerably larger than those of WT-I. WT-I exhibits slightly lower PCL than both WT-II and WT-III which can be considered an advantage for this tube. However, WT-I does not seem to perform well in terms of SEA as it absorbs less energy than WT-II and WT-III, respectively. This might be due to the undesirable interactions between the windows located on the sidewalls of WT-I and the WTW webs which are connecting its walls together. The PCL of WT-III is close to that of WT-II, but it offers higher SEA than WT-II.

Fig. 12
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Variation of a PCL and b SEA with the geometrical parameters of WTs

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The variations of PCLs of all WTs with windows’ size are illustrated in Fig. 13, which shows that the PCLs of WT-II and WT-III are greater than that of WT-I for all windows’ sizes. The general trend is that WTs with larger windows size yield smaller PCL. The changes in PCL are quite limited and ignorable for all WTs with small windows’ sizes. The PCL changes become more observable for WTs with larger windows sizes. When d = 32 mm, PCLs of all WTs decrease significantly by increasing c. Interestingly, PCL shows a negligible variation with c when d = 8 mm. For WT-II with d of 16 mm, PCL variation with c seems to be unstable as it decreases first, then rising, before declining again.

Fig. 13
figure 13

Effects of c and d on PCLs of all WTs

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Figure 14 shows the variations of SEAs of all WTs with the windows’ parameters including c and d. Irrespective of windows’ size, WT-I performs less than the other WTs in terms of SEA. Interestingly, WT-II and WT-III show very comparable SEA values for the different windows’ parameters indicating that the configuration of the CTC connection webs has limited effects on SEA. SEAs of WT-I with d of 8 mm, 16 mm, and 32 mm are monotonically decreasing with increasing of c. However, for all WT-II and WT-III, the SEA curves do not show any regular pattern with respect to their changes with the windows’ sizes.

Fig. 14
figure 14

Effects of c and d on SEAs of all WTs

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The values of CLE for all WTs at the design points are introduced in Fig. 15. It can be seen that WT-III shows the highest CLE among all WTs where the WT-III with the smallest windows’ size exhibits the maximum CLE response.

Fig. 15
figure 15

CLE response of all WTs at the design points

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4.3 Optimization results

As described by the optimization flowchart in Fig. 4, the generated meta-models of SEA and PCL were used in MOPSO algorithm to solve the optimization problem of STs and WTs as presented in Eq. (6). Three Pareto fronts of PCL against SEA for STs and WTs are plotted respectively in Figs. 16 and 17, which show that PCL and SEA indexes conflict with each other and any reduction in PCL is associated with a decrease in SEA. The most satisfactory optimal design of each of STs or of WTs is then determined from the Pareto fronts using the minimum distance selection method (Branke et al. 2004; Sun et al. 2011; Tran et al. 2014) and shown in Table 5. For STs Pareto front as shown in Fig. 16, the Pareto fronts of ST-II and ST-III are located on the left side of the graph, while Pareto front of ST-I is on the right side, and this reveals that ST-I delivers higher SEA than ST-II and ST-III for the same value of PCL. This gives strong evidence that ST-I is preferable to ST-II and ST-III in terms of crashworthiness performance. Concerning the WTs and as shown in Fig. 17, the Pareto fronts of WT-II and WT-III are above and to the right of WT-I, which points out that the crashworthiness characteristics of the WT-I are inferior to those of WT-II and WT-III. STs exhibit greater SEA than WTs which is a desirable feature for real-world crashworthiness applications. Such behavior is due to the fact that the STs have more material than WTs to deform plastically and dissipate the kinetic energy. Numerical crush simulations were undertaken using the FE model to verify the chosen favorite solution. As it can be seen from Table 6, the percentage errors between the optimization and simulation values of SEA and PCF for all STs and WTs are small, and this confirms the validity of the obtained optimization results.

Fig. 16
figure 16

Pareto fronts for STs

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Fig. 17
figure 17

Pareto fronts for WTs

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Table 5 Selected optimal solution of (a) STs and (b) WTs
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Table 6 Verification of the chosen optimal solutions for STs and WTs
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It is worth noting that the geometrical dimensions of the optimized WTs, i.e., a and t, were determined based on the results of the first optimization process conducted on STs as detailed in the multi-objective optimization scheme shown in Fig. 4. Thus, there might be better optimized configurations for the WTs if a different optimization scheme that simultaneously considers all the geometrical and windows parameters was adopted.

5 Conclusion

In this study, the crashworthiness optimization analysis was conducted on six different multi-cell standard and windowed hexagonal tubes. Wall-to-wall (WTW) and corner-to-corner (CTC) webs were used to create their different multi-cell configurations. A two-stage sequential optimization procedure was used to find the most suitable design for the standard configuration and then to optimize the windows’ size that should be implemented on it. Validated FE model was established and used to run the crush simulations required for the crashworthiness analysis of the different designs. The influence of geometrical parameters on the crashworthiness responses of the standard and windowed tubes was determined via conducting comprehensive parametric analysis. Meta-modeling, i.e., RSM, and multiple objective optimization, i.e., MOPSO, tools were combined to find the optimal design of all examined structures. It was found that the standard design with WTW connection web performs better than all other designs and absorbs the greatest energy per unit mass. However, for the windowed tubes, the WTW webs were not effective, and the tube with this type of webs performed less than the other windowed tubes with CTC webs. Generally, all the windowed tubes absorbed less energy than their standard counterparts, and also they did not provide sufficient reduction for the peak collapsing load.