In this study, a thin-walled S-shaped absorber having its geometry described by a 3D model was used. Four geometric parameters such as cross-sectional width (C), wall thickness (t), radius of curvature (R), and curvature angle were selected as the main design variables. The total length of the absorber (L=1 m was fixed for all absorbers. Geometric description of the absorber and the geometric parameters used in this study are

The application of the axial loading was achieved by placing the absorber between two rigid plates. The upper rigid plate was taken as the fixed support, while the lower one played the role of an impactor. The general structure of the impact test system along with the S-shaped absorber is shown in Fig. 2.

It was assumed that the energy absorber was made of steel. The elastic material characteristics, such as Young's modulus (E = 206 GPa), Poisson's ratio ν = 0.33, and density ρ = 7800)kg/m³, were introduced in the numerical model. The nonlinear plastic behavior of the material was described with the help of the Johnson-Cook constitutive model, which is used extensively for characterization of the material under large plastic deformations and dynamic loads.

σ = A + Bεn p, (1)

where σ is the flow stress, εp is the equivalent plastic strain, and (A), (B), and (n) are the material-model parameters. The plastic stress-strain data that were adopted in the numerical analyses were based on the material data presented in [39].

In order to model the crushing process, both the contact behavior between the absorber and the rigid plates as well as self-contact between the various sections of the absorber during the crushing process were considered. The contact behavior between the absorber and the rigid plates was modeled through the surface-to-surface contact approach. Moreover, the self-contact option was taken into account to prevent any interpenetration among different parts of the structure.

The coefficient of friction between the absorber and the rigid plates was assumed to be 0.3 as suggested in Ref. [17]. The frictional behavior was represented through the penalty approach.

A concentrated mass of 500 kg was used to model the mass of the impactor. Then, an impactor velocity of 10 m/s was considered along the longitudinal axis of the absorber, where an initial velocity was applied to the lower plate. The upper rigid plate was restrained completely, while the lower plate was restrained from moving along the principal axis of the structure. Hence, the initial kinetic energy of the impactor was converted into plastic deformation and crushing of the S-shaped absorber.

In order to reproduce the local deformation mode due to the thin-walled structure, the absorber was modeled using four-node S4R shell elements. This is because reduced-integration shell elements are appropriate for analyzing thin-walled structures under large deformation and plastic collapse. The mesh size was selected through a mesh independence study, where no substantial changes were obtained in the main responses of the structure (absorbed energy, maximum crushing force. The resulting finite element mesh used in the present simulations is shown in Fig. 3.

After establishing the geometric parameters, material properties, interaction between elements, boundary conditions, and meshing, an explicit dynamic simulation was conducted in ABAQUS/Explicit software. In the initial state of the simulation, the absorber was in its undeformed state, whereas the lower rigid plate was moving at a speed of 10 m/s. With the motion of the lower plate in the upward direction, the absorber experiences axial compression leading to plastic deformation.

The deformation consists of buckling initiation, folding, and crushing processes. The final deformed configuration of the absorber under axial loading is shown in Fig. 4. It is clear that the deformation behavior shows the ability of the developed numerical model to replicate the main crushing process of the S-shaped energy absorber.

In order to study the influence of the geometric parameters on the performance of the absorber and to create the dataset for surrogate modeling using neural networks, 58 different designs of the S-shaped energy absorber were numerically simulated. In the numerical simulations, four geometric parameters (C), (θ), (R), and (t) varied in the investigated ranges, while the other geometrical parameters and material properties remained unchanged.

The absorbed energy and peak crushing force of each configuration have been determined from the numerical data and used as output variables of the surrogate models and objective functions in the optimization process.

In order to verify the correctness of the created numerical model, its results have been compared with the results reported by Xu et al. [40]. For this purpose, 58 numerical models with different combinations of geometric parameters were created. Some of the obtained results were used for the training of the neural network, while the other part formed the independent testing dataset.

There was reasonable agreement between the results of the current finite element modeling and the reference values. The maximum difference between the two for the independent test data was 2.86% in terms of absorbed energy and 0.75% for the peak crushing force. Such reasonable differences suggest that the current finite element model offers adequate accuracy in the prediction of the crushing behavior of the S-shaped energy absorber .

For the purpose of comparing the current results with the reference values, Fig. 5 shows the results of the absorbed energy and peak crushing force from the current modeling and the reference values. The closeness of the data points to the (y=x) line suggests reasonable agreement between the two sets of results.

Since the developed FEM model has proved to have satisfactory levels of accuracy after being validated, it is possible to use it as the main computational model for generating the required data. Such data was later used for developing the nonlinear relationship between the design parameters and performance characteristics of the absorber based on the RBF and Group Method of Data Handling (GMDH) models.

The GMDH network was created first to predict the absorbed energy of the S-shaped energy absorber. The optimized architecture of the network is made up of a number of second-order polynomial functions that are used to capture the nonlinear interactions between the geometric input parameters and the absorbed energy. These equations corresponding to the network neurons were derived as follows:

In addition, the outputs of the intermediary neurons were then combined as follows:

0.00017893 + 0.618805y1 + 0.254887y2 + y5 = ( 2 2 )y6 0.00014485y1 + 0.00014470y2 − 0.00027354y1y2 0.000003568 + 0.038247y3 + 0.732736y4 +

= ( 2 2 ). (3)

0.00050152y3 + 0.00051146y4 − 0.00097897y3y4

The absorbed energy was finally predicted using the following equation:

−0.00001972 + 0.601432y5 − 0.093318y6

Eabs = ( 2 2 ) . (4)

−0.00003085y5 + 0.000008186y6 + 0.00012231y5y 6

The performance of the model can be estimated by comparing the absorbed energy values predicted using the GMDH network and those obtained from the finite element analysis. Figure 6 shows the scatter diagrams of the absorbed energy values.

testing data, and c. overall dataset.

As shown in Fig. 6a, the data points corresponding to the training dataset are closely distributed around the ideal line, yielding a coefficient of determination of R2 = 0.9868. The high R2 value demonstrates that the GMDH network has a very good capability to learn the nonlinear relationship between the geometric parameters of the energy absorber and the absorbed energy.

The generalization capability of the developed model was evaluated using an independent testing dataset. As shown in Fig. 6b, the coefficient of determination for the testing dataset was (R2 = 0.9940). The close agreement between the predicted data points and the fitting line, together with the high (R2) value, indicates that the model maintains excellent predictive performance even for data that were not used during the training process.

Finally, the simultaneous comparison of all training and testing data in Fig. 6c resulted in an overall coefficient of determination of (R2 = 0.9882). Therefore, the small discrepancies between the FEM and GMDH results demonstrate that the developed surrogate model can accurately reproduce the absorbed-energy behavior of the S-shaped energy absorber.

In the next step, the GMDH model was developed to predict the maximum crushing force of the S-shaped energy absorber. This parameter is of particular importance in the design of energy absorbers because an excessively high crushing force can result in severe load transfer to the protected structure or occupant. Therefore, accurate prediction of (Fmax) is essential for the design and optimization of the energy absorber. The optimized GMDH network structure for predicting the maximum crushing force was composed of the following quadratic polynomial relationships:

5.750596 − 0.145030x2 − 0.000449x3 + y1 = 10000 ( 2 −6 2 ) 0.001025x2 − 2.205 × 10 x3 + 0.000241x2x3 −6.866123 + 0.039342x3 + 4.338069x4 − y2 = 1000 ( 2 2 ) 0.00002950x3 + 3.260853x4 + 0.010015x3x4 −0.0002125 + 1.044552y1 − 0.022386x2 − y3 = ( 2 2 ) 0.000001674y1 − 1.719680x2 + 0.002496y1x2

−0.00009523 − 0.164273y2 − 0.006509x1 − y4 = ( 2 2 ).

0.000000488y2 − 0.446169x1 + 0.020414y2x1 Finally, the maximum crushing force was calculated from the following expression:

0.00002157 + 0.432650y3 + 0.6549997y4 −

F ( 2 2 ) . (6)

0.000011582y3 − 0.000001645y4 + 0.000011499y3y4 max

The performance of the developed GMDH model in predicting the maximum crushing force is presented in

As can be observed in Fig. 7a, the predicted data points for the training dataset are very close to the trend line of FEM results, with a coefficient of determination of (R2=0.9712). The relatively high value of (R2) represents the strong relationship between the results obtained by the GMDH model and the numerical results, which indicates the effectiveness of the established model in capturing the nonlinear relationship between geometric parameters and maximum crushing force.

For the testing dataset that is independent of the training dataset, as illustrated in Fig. 7b, the coefficient of determination is increased to (R2=0.9830).

The coefficient of determination (R2=0.9725) for the entire dataset, consisting of both training and testing samples, is R2=0.9725 and is illustrated in Fig. 7c. Even though there is some scatter in the data points, which is more significant compared to the absorbed energy model, the GMDH model captures the general trend of the finite element method solution. The relatively higher scatter in the data points is because the maximum crushing force is a rather complex parameter depending on several factors, including the point of crushing initiation, development of plastic folds, and geometrical changes occurring during the progressive collapse of the S-shaped structure .

Overall, it is confirmed that the developed GMDH-based surrogate model is an effective computational tool for predicting the maximum crushing force of the S-shaped thin-walled energy absorbers. The high correlation between the data points in both training and testing datasets further confirms the applicability of the model for multiobjective optimization .

For the purpose of comparing the performance of both models, the values of the coefficients of determination (R2) calculated from training, testing, and total datasets are tabulated below.

The data clearly show that both models possess high accuracy in prediction. Nevertheless, the GMDH model proves more effective in predicting the absorbed energy compared to the maximum crushing force. The maximum value of (R2) is calculated for the testing dataset of the absorbed-energy model as (R2 = 0.9940). However, the value of (R2=0.9830) achieved for the test dataset of the maximum crushing force also indicates the high prediction accuracy of the proposed neural network model. The reason for the discrepancy in the performance of the two models is due to the difference in the nature of these two response variables. The absorbed energy is a cumulative quantity based on the overall deformation of the structure and, therefore, behaves in a relatively smooth way. On the other hand, the maximum crushing force is a local response that depends heavily on the first fold formation and local buckling.

For the optimal design of the S-shaped energy absorbers, a bi-objective optimization problem was formulated considering the aforementioned design parameters. The mathematical polynomial models obtained from the GMDH and RBF modeling presented in the previous section were employed as surrogate models for the multiobjective optimization process. Based on these mathematical relationships, the optimization procedure was performed without the need to conduct computationally expensive finite element simulations repeatedly. One of the most important requirements in the optimal design of energy absorbers is to increase the energy absorption capability while simultaneously reducing the maximum crushing force. An energy absorber can be considered to have a desirable design when it is capable of absorbing a large amount of energy while maintaining a relatively low maximum crushing force. Conversely, a design that absorbs a large amount of energy but generates an excessively high crushing force is not considered optimal. Similarly, an energy absorber with a low crushing force but insufficient energy absorption capacity cannot be regarded as a desirable design.

Accordingly, the absorbed energy (Eabs) and the maximum crushing force (Fmax) were considered as two conflicting objective functions in the present optimization problem. The objective was therefore to maximize the absorbed energy while simultaneously minimizing the maximum crushing force. The multiobjective optimization problem can consequently be formulated as follows:

Maximize f1 = E(θ, R, C, t) (Absorbed Energy) (8) Minimize f2 = F(θ, )( R, C, t Peak Crushing Force)

R ≤ , 2(1 − Cosθ) where (C), (θ), (R), and (t) represent the geometric design variables of the S-shaped energy absorber. The resulting set of non-dominated solutions represents the Pareto-optimal front, from which an appropriate design can be selected according to the desired balance between energy absorption capacity and crushing-force level.

The DE approach was used to solve the above-stated optimization problem. The DE approach starts with the generation of an initial population of solutions within the design space and develops this population through successive generations. In each generation, trial vectors are created through the application of the mutation and crossover processes and then compared to the existing solutions.

As far as the multiobjective problem formulation is concerned, the solution selection is done according to the principle of non-dominance. Thus, the solutions that are not dominated by other solutions concerning the two objective functions are selected. Moreover, the crowding distance approach is used to avoid the overcrowding of certain parts of the Pareto front with solutions and maintain diversity among the non-dominated solutions.

In the current case, the initial population size was equal to 100, the maximum number of generations was 500, the mutation factor equaled 0.1, and the crossover rate was 0.8. The process of evolution was carried out until the predefined number of generations was achieved. Finally, the set of non-dominated solutions was obtained as the solution to the multiobjective optimization problem.

Eight non-dominated optimal solutions were obtained in the process of optimization, as illustrated in the relevant figure. All of them could be regarded as alternative optimal designs, and the final choice depends on particular design requirements.

One can see that improving the value of one objective function leads to the degradation of the value of another objective function. It means that there exists some trade-off between the two objective functions. From among the Pareto-optimal solutions, Point A is characterized by minimum absorbed energy and minimum maximum crushing force. At the same time, Point B corresponds to maximum absorbed energy but higher maximum crushing force. Thus, neither of them could be considered the most appropriate solution.

Taking into account that the primary design aims to find the right balance between energy absorption and limitation of the crushing force, Point C was chosen as the most preferable compromise solution. The value of absorbed energy at Point C is approximately 33,400 J, and the maximum crushing force is about 8,713 N. The geometric design parameters are as follows:

[x1 = 50, x2 = 45, x3 = 1284, x4 = 2.59]. (9)

Fig. 8 illustrates the Pareto front achieved using the multiobjective optimization procedure. As seen in the figure, the movement along the Pareto front towards higher absorbed energy is accompanied by an increase in the maximum crushing force. Such a tendency demonstrates the existence of an intrinsic trade-off between the two objective functions; that is, the increase in the capacity for energy absorption is usually associated with an increased crushing force.

Of all the Pareto solutions, Point A corresponds to the lower performance area of the Pareto front in terms of both absorbed energy and crushing force. At the same time, Point B lies in the area of maximal capacity of energy absorption with the increase in the maximum crushing force. Consequently, none of these points can be considered the best design option.

In accordance with the primary purpose of obtaining a proper balance between energy absorption and crushing-force minimization, Point C was chosen as the best compromise solution. It offers an absorbed energy value of approximately 33,400 J and the maximum crushing force of approximately 8,713 N. The geometric parameters of the chosen optimal design are:

C = 50; mm, θ = 45°, R = 1284;mm, t = 2.59; mm. (10)

The above results clearly show that the joint use of the GMDH surrogate model and DE algorithm allows for an effective exploration of the design space of the S-shaped energy absorber by using significantly fewer computations than a large number of finite element simulations. In addition, the proposed methodology allows for obtaining a set of non-dominated solutions that enables the designer to choose a design depending on the desirable compromise between energy absorption and crushing force.

To assess the effectiveness of the obtained designs, the Pareto-optimal solutions were compared with the data for the reference specimens. The results of the comparison are shown in Fig. 9.

As can be seen from Fig. 7, the designs obtained through the optimization are located in a performance region providing a good trade-off between absorbed energy and maximum crushing force. Thus, it can be said that the proposed methodology is able to find the promising design regions and to generate competitive designs compared with the reference specimens.

Last but not least, since it is a multiobjective problem, there is no absolute best solution. Hence, the optimal choice can only be made based on the design criteria set. Point C, in this case, is chosen as the optimal design for the reason that it makes a perfect balance between maximizing energy absorption and minimizing the peak crushing load.