Optimization for multi-cell thin-walled tubes under quasi-static three-point bending

Approaches such as changing the cell number, changing the rib direction, and adding internal structure are utilized to acquire a multi-cell thin-walled structure, and these approaches have meaningful effects on the crashworthiness performance of multi-cell thin-walled tubes. In this study, a comprehensive review is done by using and comparing these approaches together under quasi-static three-point bending conditions. A different crashworthiness indicator is better for each of the produced multi-cell thin-walled structures. The overall best tubes are determined by the complex proportion assessment (COPRAS), a multi-criteria decision-making technique. The weights used in the COPRAS technique are calculated by the entropy method. Thus, two different tubes are chosen as the best ones. Then, multi-objective optimization is performed on these tubes with the multi-objective genetic algorithm (MOGA). The surrogate models of PCF and SEA, which are defined as the objectives in multi-objective optimization studies, are obtained by the (radial basis functions) RBF. Multi-objective optimized multi-cell thin-walled W1L1 and W1L1S1 tubes achieved the same SEA values as the W0L0 square tube at 13.1% and 15.4% lower PCF values, respectively.


Introduction
It is important to absorb the impact energy generated to ensure the safety of the passengers and driver during a traffic accident. The B pillar, side door beam and bumper are important components used to absorb this impact energy. Recently, multi-cell thin-walled tubes have drawn increasing attention because of their great energy absorption characteristic and lightweight [1][2][3]. Multi-cell thin-walled structures are investigated under various loading conditions such as axial [4][5][6], oblique [7][8][9] and lateral [10,11] loading cases.
In recent years, different approaches have been tried by researchers to increase the crashworthiness capacity of multi-cell thin-walled structures [12][13][14][15]. Wang et al. [16] studied the performance of the multi-cell thin-walled square tubes, with the cell amounts ranging from 1 × 1 to 15 × 15 under axial loading cases. Their study indicated that for a multi-cell thin-walled tube, the most critical element in enhancing crashworthiness performance is optimizing the half-wavelength. Wang et al. [17] studied the bending performance of multi-cell square structures under three-point bending. They presented that the number of cells has a meaningful effect on the bending strength of the tube and that high cell number does not always lead to high energy absorption efficiency. Zhang et al. [18] studied bending collapse responses of embedded multi-cell thin-walled tubes under three-point bending. They revealed that the bending capacity of embedded multi-cell thin-walled structures is met to account for about 65-72% of corresponding conventional multi-cell structures.
Multi-cell thin-walled tubes can also be created by joining inner ribs into simple conventional structures and multi-cell tubes or inspired by nature [19][20][21][22]. Albak [23] investigated the crashworthiness capacity of multi-cell square thin-walled tubes under axial loading cases by adding square, hexagonal, octagonal and circular structures to the intersections of the walls. The study revealed that the structures added to the wall intersections improve the crashworthiness performance.
Hu et al. [24] comprehensively explored energy absorption performance of a bionic honeycomb tubular nested structure inspired by bamboo under axial impact conditions. In their studies, they added circular structures to the wall intersection areas and analyzed the effects of the diameter of the circular structure, length of the junction plate and wall thickness of the circular structure parametrically. The study showed that the influence of the diverse mean diameter of the circular structure and length of the junction plate on the energy absorption performance. Du et al. [25] investigated the bending collapse performance and energy absorption characteristics of thin-walled tubes with the mono-box multi-cell cross section under three-point bending condition. The energy absorption performance of thin-walled tubes with mono-box multi-cell cross sections is high to thin-walled tubes with mono-box mono-cell sections under the equal mass, and its energy absorption performance will enhance gradually with the number of web regions increased. Huang et al. [26] studied the lateral indentation responses of different multi-cell thin-walled tubes. For thin-walled tubes of equal mass and uniform wall thickness, two-cell and four-cell thin-walled tubes show better energy absorption performance than a mono-cell tube. The specific energy absorption of two-cell and four-cell tubes is 10% and 26% higher, respectively, than that of single-cell at quasi-static loading. Zheng et al. [27] investigated the energy absorption characteristics of fully clamped multi-cell square structures under transverse loading conditions. They investigated the influence of rib direction and rib number on energy absorption characteristics. Their study showed that the multi-cell thin-walled structures with rib parallel to the load direction have higher energy absorption characteristics than that with rib perpendicular to the load direction. They also revealed that the effect of increasing the rib number is different according to the rib direction. Wang et al. [28] studied the collision performance of circular multi-cell thin-walled tubes inspired by the lotus leaf vein branched structure under three-point bending. Their studies revealed that the number of cells, the ratio of the diameters of the cells, and the wall thicknesses have an effect on the crashworthiness performance.
When the above studies are examined, changing the cell number, changing the rib orientation and adding internal ribs are used to create a multi-cell thin-walled tube, and these approaches have significant effects on crashworthiness performance. In studies in the literature, these approaches are generally examined in separate studies. The innovative aspect of this study is that these approaches are examined comparatively in a single study and their advantages and disadvantages are revealed relative to each other. In the study, new multi-cell thin-walled structures are obtained by changing the rib direction vertically, horizontally and diagonally, creating different cell numbers and adding inner ribs. Three-point bending analyses of fifteen different multi-cell thin-walled tubes created with different approaches are examined, and their crashworthiness performance is compared. Then, the best tubes are determined by the complex proportion assessment (COPRAS) method, and these tubes are optimized. The weights required for the COPRAS method are found using the entropy approach. Finally, multiobjective optimization is applied with the MOGA for the selected best tubes.

Geometrical description
In this study, the crash performance of fifteen different multi-cell thin-walled structures under quasi-static threepoint bending is investigated. The tubes have a square outer wall with a side length of 30 mm. In order to obtain multicell structures, different numbers of vertical, horizontal and diagonal walls are added to the square wall. Also, multi-cell tubes are created by adding the square and circular internal structures. The diameter of the circles and the diagonal length of the squares used in the interior are 12.5 mm. While addressing the names of the tubes, if there are vertical, horizontal and diagonal structures, their first letters are used and their numbers are indicated. In addition, the first letters of the square and circular cylindrical internal structures are used. All tubes used in this paper and their nomenclature are given in Fig. 1.
In this study, fifteen basic cross sections that can be obtained by the above-mentioned approaches are examined in order to prevent the increase in the complexity of the multi-cell thin-walled tubes. Similar multi-cell thin-walled structures can also be obtained using different approaches Fig. 1 The cross sections of multi-cell thin-walled tubes and different geometric cross sections. Although simpler structures are preferred as energy absorbers, multi-cell thin-walled tubes can be preferred in conditions where more performance is desired. In addition, with the development of advanced production methods such as 3D printers, the production of multi-cell thin-walled tubes will be easier and cheaper.

Crashworthiness indicators
To measure the crashworthiness performance of the multicell thin-walled structures, it is critical to state crashworthiness indicators. The most used crash indicators by researchers are the peak crushing force (PCF), specific energy absorption (SEA), energy absorption (EA), mean crushing force (MCF) and crushing force efficiency (CFE) [29,30]. PCF is defined as the maximum crushing force during the whole impact process. SEA is the energy absorption of the tube per mass. The SEA is described as: where M is the overall mass of the tube, and EA is the total energy absorption of the thin-walled tube. The EA can be given as: where F(x) is the instantaneous crushing force and S is the total crushing displacement of the indenter.
CFE is calculated as the ratio of the mean impact force to the highest impact force: where MCF is the average crushing force and can be described as:

Simulation model
The researchers have examined the three-point bending test of energy absorbers under dynamic load [28] or quasi-static conditions [31][32][33]. In order to evaluate the crashing performance of multi-cell thin-walled structures, the finite element (FE) model of the tubes under the quasi-static three-point bending is established using explicit non-linear finite element code RADIOSS [34]. The schematic and finite element model of the multi-cell thin-walled tubes is given in Fig. 2. The finite element model consists of a specimen, an impactor and two supports. As shown in Fig. 2, the specimen is located on two cylindrical supports and crushed by a cylindrical impactor in the center. All components are created using qeph shell formulation four-node element using five through-thickness integration points. Contact definitions are made between specimen and impactor and supports. Also, the contact is carried out to simulate self-contact of the multi-cell thin-walled tubes to prevent mutual interpenetration. The friction coefficients for all contacts are described as 0.3 [35].
The mesh convergence analysis is performed as presented in Fig. 3. The analysis is executed on an Intel ® Core ® i5-8265U CPU running at 1.8 GHz with 8 GB of memory. MCF values less than 1.25 mm are very close to each other. And, computational time increased after 1 mm. So, the average element size is determined as 1 mm, considering accuracy and computational time.
The material AA6063-O is modeled by isotropic elastoplastic material model Mat36 in RADIOSS. Mat36 material definition simulates an isotropic elasto-plastic material with user-defined functions for the work-hardening portion of the stress-strain curve. The mechanical characteristics of AA6063-O are presented in Table 1. The

Validation of the simulation model
Literature has been utilized to validate the finite element model to be employed in the paper. Huang and Zhang [35] have conducted quasi-static three-point bending experiments for thin-walled square section tubes. In order to validate the accuracy of the finite element model, the same structural parameters, section properties and experimental requirements as in Ref. [35] are applied. Experimental and finite element analysis deformation modes are given in Fig. 5. The deformation patterns of finite element analysis have good agreement with the experiment. In Fig. 6, force-displacement curves of the experiment and finite element analysis are given. As with the deformation modes of the finite element model, the force-displacement curves have a parallel trend with the experimental test data. The comparison of the initial peak force (IPCF) and MCF values received from the curve of the experimental test and the values obtained as a result of the finite elements is given in Table 2. The maximum relative error for these two indicators is below 4%. Overall, the finite element model developed in this paper has been adequately validated and can be used to explore the crashworthiness characteristics of the multi-cell thinwalled tubes.

Finite element analysis results
In this section, the crashworthiness performance of the multi-cell thin-walled tubes is discussed. All multi-cell thinwalled tube variants have different wall thicknesses with the same weight. Thus, all comparisons are made with the same weight. The deformation patterns, impact force-displacement curves and crash criteria of multi-cell thin-walled tubes are given in Figs. 7, 8 and 9, respectively. When the deformation modes of W1L0, W2L0, W1L0D2, W0L1D2 and W1L1D2 tubes are studied, it is seen that local bending collapse occurs. When the force-displacement  In summary, the best alternatives are W1L1 and W2L2 multi-cell thin-walled tubes with both vertical and horizontal walls, and W1L1C, W1L1S1 and W1L1S2 multi-cell thinwalled structures with additional structures at wall intersections. However, the best multi-cell thin-walled structure cannot be determined directly. It should be determined which of the PCF, CFE and SEA values is more important or at what values to choose for the best multi-cell thin-walled tube. Therefore, in the next sections, selection with the complex proportional assessment (COPRAS) will be explained to determine the best alternatives.

Selection of the best multi-cell thin-walled tubes
In this study, fifteen different multi-cell thin-walled structures are investigated using three different crashworthiness criteria, PCF, CFE, and SEA. Therefore, it is difficult to choose the best among the considered multi-cell thin-walled structures. So the multi-criteria decision-making study has been utilized for this complex best multi-cell thin-walled structure selection problem. In this work, the complex proportion assessment (COPRAS), which is one of the multicriteria decision-making approaches, is utilized. COPRAS is commonly used in multi-criteria decision-making cases because of its simplicity [36,37].

The complex proportional assessment methodology (COPRAS)
The preference ranking technique of complex proportional assessment (COPRAS) is presented by Zavadskas et al. [38]. The specific steps of the evaluation approach are as follows: Step 1 Developing the initial decision-making matrix (X) Force-displacement curves of the experiment [35] and finite element analysis where x ij is the quality score of ith alternative with respect to jth indicator, m is the number of alternatives compared, and n is the number of the indicator.
Step 2 Acquiring the normalized decision matrix (R): Step 3 Calculating the weighted normalized decision matrix (D): where d ij is the weighted normalized score of the ith alternative with the jth indicator.
Step 4 Summing the weighted normalized values for beneficial and non-beneficial attributes: where d +ij and d −ij represent the normalized weighted values of beneficial and non-beneficial attributes, respectively.
Step 5 Determining the relative importance (Q i ) of each alternative: Step 6 Determining the quantitative utility (U i ) for the ith alternative, which presents the rank of the ith alternative.
Q max is the highest Q i rate among the Q i rates. The alternative with the maximum U i value (100%) is selected as the ideal alternative.

Entropy method
Weighting is an important element that determines the best alternative for multi-criteria decision-making approaches. Determining the weight depending on subjective preferences may cause the best alternative to be different. Therefore, in this study, the weights are specified by the entropy technique. The entropy technique is frequently used in thinwalled tube selection problems with the ideal crashworthiness characteristics [39,40]. The entropy technique can be implemented by the following several steps: Step 1 Developing the initial decision-making matrix (X): Step 2 The decision-making matrix should be normalized as p ij with Eq. (13): Step 3 Obtain the entropy of the jth indicator, E j : Step 4 Calculate the distance, d j : Step 5 Indicator weights, w j , can be given as:

Finding the best alternatives with COPRAS and entropy
In this section, multi-cell thin-walled tubes will be ranked in terms of their crashworthiness performance with COPRAS and entropy methods. PCF, CFE and SEA indicators are taken into account when ranking the crashworthiness of the tubes. Effective energy absorbers should have high SEA and CFE and low PCF during crashworthiness. Therefore, SEA and CFE are chosen as beneficial indicators and high values are desired, PCF is chosen as the non-beneficial indicator and low value is desired. In the COPRAS method, firstly, the initial decision-making matrix is created with 15 alternative multi-cell thin-walled tubes examined over 3 crashworthiness criteria. The initial decision-making matrix is given in Table 3. The initial decision-making matrix is normalized using Eq. (7) and presented in Table 4. The weight values of the crashworthiness criteria are calculated by the entropy method and are given in Table 5. After obtaining the weighted normalized matrix, beneficial values and nonbeneficial values are obtained. Finally, these values and the relative importance, the quantitative utility and ranking of  the alternatives obtained from these values are given in Table 6. The COPRAS method presented the best alternative as W1L1. W1L1S1 is seen as the second-best alternative.

Crashworthiness optimization
In this section, the multi-objective optimization of W1L1 and W1L1S1 tubes, which are determined as the best multi-cell thin-walled tubes by the COPRAS method, will be explained. W1L1 tube is chosen as the best tube in the COPRAS method, while the W1L1S1 is chosen because it is the best tube with an inner structure.

Description of the optimization problem
It is generally expected that energy absorbers can absorb as much impact energy as possible per unit mass and have a smaller peak crushing force when it is used as a vehicle safety component [28]. For this reason, in this study, multiobjective optimization is carried out with the PCF and SEA indicators as the objectives. Therefore, the optimization case for W1L1 can be expressed as follows: where three design variables, namely the outer wall thickness t 1 , vertical wall thickness t 2 and lateral wall thickness t 3 , which range from 0.5 to 2.0 mm, are csidered. The optimization problem for W1L1S1 can be formulated as follows: where four design variables, namely the outer wall thickness t 1 , vertical wall thickness t 2 , lateral wall thickness t 3 and inner structure wall thickness t 4 which range from 0.5 to 2.0 mm, are considered. The design variables of W1L1 and W1L1S1 are given in Fig. 10.

Surrogate model
After establishing the optimization case, it is very critical to create the functional relationship between optimization objectives and design variables. It is hard to derive analytically objective functions for PCF and SEA. Because they involve highly nonlinear contact-impact and large deformation mechanics. For this reason, surrogate modeling is generally preferred to obtain objective functions in crashworthiness cases [41,42]. Compared to the other approaches, the (radial basis functions) RBF technique has ensured higher accuracy in crashworthiness optimization cases [43]. Therefore, in this study, the RBF method is chosen to obtain surrogate models of objective functions. The first step of the surrogate modeling is to generate the design area using (design of experiment) DOE techniques. In this study, 30 sample points for W1L1 and 40 sample points for W1L1S1 are created using the Latin hypercube method. The accuracy of the optimization results depends on the accuracy of the created surrogate models. Therefore, the created surrogate models need to be validated. To measure the accuracy of the created surrogate models, five checking designs are created using the Latin hypercube approach. The relative error (RE) between the response value [f RBF (x)] and the finite element simulation [f fea (x)] can be expressed as: The RE values of the RBF models for W1L1 and W1L1S1 are presented in Fig. 11. All RE values are less than 7%, indicating that the generated surrogate models have acceptable accuracy and can therefore be applied to multiobjective crashworthiness optimization.

Multi-objective optimization results
In this study, the multi-objective genetic algorithm (MOGA) is used to seek the optimal solutions for W1L1 and W1L1S1. MOGA is one of the multi-objective optimization methods preferred in crashworthiness problems [44,45]. Figure 12 presents the Pareto frontiers obtained by the MOGA algorithm. As shown in Fig. 12, the Pareto frontiers are declared by two conflicting crashworthiness criteria PCF and SEA. That is, as one increases, the other decreases and vice versa. Truly, any design in the Pareto front could be ideal. Which design should be chosen is fully determined by the engineering design case. As an example, the force value (1.943 kN) of the W0L0 tube, which consists of only the outer wall, is considered and is indicated by the green line in Fig. 12. In the Pareto frontiers obtained for W1L1 and W1L1S1, the designs corresponding to this force value are considered. Finite element analyses of the optimum models obtained at these design points are performed, and their accuracy is examined. In Table 7

Conclusions
In this study, the crashworthiness performance of fifteen different multi-cell thin-walled tubes under three-point bending is extensively investigated by finite element analysis. The finite element model has been validated with literature test data. The crashworthiness performance of each tube is different, so the best alternatives are determined by the COPRAS method. Since weighting is an important factor in the COPRAS method, the weighting is calculated with the entropy method. W1L1 and W1L1S1 tubes, which are determined as the best alternatives, are optimized with the MOGA method. The surrogate models of PCF and SEA, which are selected as the objectives in multi-objective optimization studies, are obtained by the RBF method. Finally, the optimum designs corresponding to the force value of the W0L0 tube are chosen from the Pareto frontiers to compare the results of the optimizations. Within its limitation of the study, the major conclusions are summarized as follows: • W1L0 and W2L0 tubes obtained with vertical walls have high PCF values but low CFE and SEA values. This shows that not only vertical walls improve crashworthiness performance.