Multi-objective crashworthiness optimization of lattice structure filled thin-walled tubes

Thin–Walled Structures 149 (2020) 106630

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Multi-objective crashworthiness optimization of lattice structure filled thin-walled tubes �lu a, Cengiz Baykasog �lu b, *, Erhan Cetin b Adil Baykasog a b

Dokuz Eylul University, Faculty of Engineering, Department of Industrial Engineering, Izmir, Turkey Hitit University, Faculty of Engineering, Department of Mechanical Engineering, Çorum, Turkey

A R T I C L E I N F O

A B S T R A C T

Keywords: Thin-walled tubes Lattice structures Hybrid structures Multi-objective crashworthiness optimization Axial impact loading

Thin-walled tubes have been mostly used in passive vehicle safety systems as crash energy absorber. With the use of additive manufacturing technology, it is possible to produce novel filler materials to further enhance the crashworthiness performance of thin-walled tubes. In this study, optimal designs of novel lattice structure filled square thin-walled tubes are investigated under axial impact loading by using a compromise programming based multi-objective crashworthiness optimization procedure. Types of filler lattice structures (i.e., body-centered cubic, BCC and body-centered cubic with vertical strut, BCC-Z), diameter of lattice member, number of lattice unit cells and tube thickness are considered as design parameters, and the optimum values of these design pa­ rameters are sought for minimizing the peak crash force (PCF) and maximizing the specific energy absorption (SEA) values. The validated finite element models are utilized in order to construct the sample design space and carrying out results verification; an artificial neural network is employed for predicting values of the objective functions; the weighted superposition attraction algorithm is used to generate design alternatives and searching for their optimal combination. The compromise programming approach is used to combine multi-objectives and to produce various optimal design alternatives. The optimization results showed that the proposed approach is able to provide good solutions with high accuracy and proper selection of design parameters can effectively enhance the crashworthiness performance of the lattice structure filled thin-walled tubes. The optimum results revealed that BCC hybrid designs have generally superior crashworthiness performance compared to that of their BCC-Z counterparts for the same compromise solutions. In particular, the PCF value of the optimized BCC-Z hybrid structures is up to 44% higher than that of BCC hybrid structures while these structures have similar energy absorption performances. The compromise solutions also show that the SEA of BCC and BCC-Z hybrid structures increases respectively by 29% and 51% depending on the selected weight factors for the design objectives.

1. Introduction Thin-walled tubes as crash energy absorbers have been widely used in the passive vehicle safety systems due to their superior crashworthi­ ness performances [1–3]. In recent decades, extensive research has been conducted to improve the crashworthiness performance of thin-walled tubes and numerous novel designs have been proposed in the related literature. At this point, several materials (e.g., steel, aluminum and composite), cross-sections (e.g., triangular, hexagonal and octagonal), patterns (e.g., corrugated, windowed, origami and functionally graded) etc. are considered in order to enhance the crashworthiness

characteristics of thin-walled tubes [4–7]. Thin-walled tubes are expected to have high crashworthiness per­ formances with the least possible mass. In this respect, researchers have tried several methods in order to improve the crashworthiness perfor­ mance of thin-walled tubes [1–7]. At this point, the energy absorption capacity of thin-walled tubes can be improved by increasing the tube thickness, however, thickening tubes also causes an increase in the peak crush force, and this makes the structures inefficient by considering crashworthiness indices. At this point, one of the most effective methods for enhancing the crashworthiness of thin-walled tubes without increasing weight and volume too much, is filling or covering these

* Corresponding author. Hitit University, Faculty of Engineering, Department of Mechanical Engineering, 19030, Çorum, Turkey. E-mail addresses: [email protected] (A. Baykaso� glu), [email protected], [email protected] (C. Baykaso� glu), [email protected] (E. Cetin). https://doi.org/10.1016/j.tws.2020.106630 Received 20 October 2019; Received in revised form 21 January 2020; Accepted 21 January 2020 Available online 31 January 2020 0263-8231/© 2020 Elsevier Ltd. All rights reserved.

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Fig. 1. Schematic view of the BCC and BCC-Z structures.

experimental investigations on the low-velocity impact behavior of sandwich panels with BCC and BCC-Z micro-lattice structures. Gümrük and Mines [27] focused on the compression behavior of lattice structures by using theoretical and numerical approaches as well as experimental studies. In another study, Gümrük et al. [28] investigated the static behaviors of lattice structures for both different cell topologies (e.g. BCC and BCC-Z) and loading scenarios (e.g. compression and tension). McKown et al. [29] carried out quasi-static and blast tests for BCC and BCC-Z lattice structures. Smith et al. [30] developed numerical models to predict the compressive response of BCC and BCC-Z lattice structures. Besides, Merkt et al. [31] focused on the mechanical behavior of BCC and BCC-Z lattice structures under quasi-static and dynamic compres­ sion tests. Furthermore, Maskery et al. [32] examined the effect of cell size and number of cells on the mechanical behavior of BCC structures under tensile loading. In our recent work [33], the BCC and BCC-Z lattice structures are also proposed as filler materials for thin-walled tubes for the first time in the literature, and the results show the promising energy absorption per­ formances of these hybrid structures. The limited number of design parameters (e.g., the number of lattice unit cells and the diameter of lattice members) are considered in detail in our previous work [33] and the effectiveness of these structures are discussed for certain hybrid designs. In the present work, the number of design parameters of the previous study are extended by including the effect of tube thickness, and the crashworthiness performance of square tubes filled with BCC and BCC-Z lattice structures under axial impact loading which are optimized by a multi-objective crashworthiness optimization framework for the first time in literature. In the optimization scheme, tube thick­ ness, number of lattice unit cells and the diameter of lattice members are

structures with lightweight materials or structures such as honeycombs [8,9], aluminum foams [10,11], polymeric foams [12,13], carbon nano polyurethane foams [14], and composites [15–19]. It has been shown that filling or covering thin-walled tubes with lightweight materials could significantly increase the total amount of energy absorption and improve deformation stability under severe crushing with relatively small increase in total mass [7,20–22]. In recent decades, Additive Manufacturing (AM) technologies have been attracted the attention of researchers since they allow easy pro­ duction of complex parts. With this feature, the AM technique enables the researchers to create novel complex shaped filler materials for impact energy absorber applications. At this point, the filling of thinwalled tubes with low density lattice structures could improve the en­ ergy absorption capability of these structures without significant increasing in the peak crash force in a certain volume. Note that although the deformation mechanisms of lattice structures are different from metal thin-walled tubes, recent studies have shown that these structures have also promising energy absorption capacity [23–32]. Moreover, the changing of the tube thickness can alter the deformation modes of tubes depending also on the aspect ratio of structures, and these modes could be efficiently controlled by hybrid interactions be­ tween the tube wall and lattice structures under both axial and oblique loading conditions. At this point, different kinds of low-density lattice structures [23] have been already proposed in the related literature. Among them, the body-centered cubic (BCC) and the body-centered cubic with vertical strut (BCC-Z) lattice structures have been exten­ sively studied in the literature due to their promising energy absorption performances. The related studies are summarized as follows. Mines et al. [24], Shen et al. [25] and Turner et al. [26] carried out 2

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Fig. 2. The FE model and boundary conditions of the hybrid structures under axial impact.

considered as the design parameters and minimizing the peak crushing force (PCF) and maximizing the specific energy absorption (SEA) are considered as the design objectives. Different optimization schemes have been proposed in crashworthiness studies of thin-walled structures in recent decades [34–40] and various optimization methods such as multi-objective particle swarm optimization (MOPSO), response surface methods (RSM), Kriging model (KM), genetic algorithm (GA) and arti­ ficial neural networks (ANN) have been suggested in the related litera­ ture [41]. In the present work, a recent metaheuristic optimizer that is known as Weighted Superposition Attraction (WSA) and Artificial Neural Networks (ANN) are used within a compromise programming logic to search for optimal design of thin-walled tubular structures for the first time in the literature. WSA has proven its effectiveness and superiority on several other well-known metaheuristic optimization al­ gorithms on solving constraint design optimization problems with discrete and continuous design variables [42,43]. Moreover, WSA re­ quires very few algorithm specific parameters to set during its search in comparison to other alternative optimization procedures. The validated finite element models are used beforehand to construct the sample design space and to verify optimal solutions at the end for the optimi­ zation procedure. Computational results confirmed that the proposed optimization procedure is able to provide optimal solutions with high accuracy and proper selection of design parameters can effectively enhance the crashworthiness performance of the tubes filled with lattice structures under axial impact loading condition. The numerical results revealed that the optimized hybrid structures show promising crash­ worthiness performances, and these structures are recommended as passive protection elements used in a broad range of road vehicles, rail vehicles and aerospace applications and packaging protection elements of heavy equipment.

Fig. 3. The true stress-strain curves of a) Al6063-T5 and b) AlSi10Mg.

are assumed to be perfectly straight. As shown in Fig. 1, different aspect ratios (i.e., L/a) are considered depending on the number of unit cells, where L and a are the initial length and width of the structure, respec­ tively. The initial length of all structures is chosen as 120 mm, and the width of the corresponding lattice structures having 3, 4, 5, 6 and 7 lattice unit cells are 40, 30, 24, 20 and 17.14 mm, respectively. 2.2. Numerical modeling and validation The geometric lattice models are created in SolidWorks platforms, and numerical simulations are performed in the Abaqus/Explicit finite element (FE) software. In the typical scenario, the hybrid structures are situated between a fixed rigid plate and a moving rigid plate with initial velocity (v0) of 10 m/s and impacting mass of 250 kg (Fig. 2). Note that the moving rigid plate is permitted to move only in the axial direction. General contact property is defined to all interactions between the rigid plates and hybrid structures, and penalty formulation is used for the contact behavior in the tangential direction [47]. A friction coefficient of 0.25 is adopted for all of the contacts similar to the Refs. [38,48], and the contact behavior in normal direction is introduced as ‘hard contact’. A reference point is defined for both rigid plates, and the reaction force history is recorded from the fixed rigid plate while the displacement history is recorded from the moving rigid plate. The eigenvalue buckling simulations are performed in order to account for the initial geometric imperfections, where the initial imperfections corresponding to the first elastic buckling modes of shell are introduced with amplitude of 0.02t [33] by using a trial-and-error method. The buckling mode is introduced in the FE mode by using ‘imperfection’ option in Abaqus. The lattice structures, thin-walled tubes and rigid plates are modeled using four-node linear tetrahedron C3D4 elements, four-node S4R shell ele­ ments and four-node linear quadrilateral R3D4 elements, respectively. It should be noted that the thin-walled tubes are designed by considering mid-surface shell element formulation and overclosure is prevented in designs. It should be also noted that, the beam elements can be used instead of solid elements for simulation of lattice structures in order to reduce computational cost. However, due to the limitation of the beam element formulation and lack of stiffness at the junction regions, the plateau and densification regions cannot be predicted accurately by using these elements [27,30,33]. Therefore, the beam elements are not considered for lattice structure modeling in FE simulations. In order to

2. Problem description 2.1. Description of geometric features The square thin-walled tubes are filled with the BCC and BCC-Z lattice structures in this study. At this point, two types of hybrid struc­ tures with different lattice member diameters (1 � d � 5 mm), different number of lattice unit cells (3 � n � 7) and different tube thickness (0.5 � t � 1.5 mm) are considered in designs. It should be mentioned that the limits of design parameters are selected in the range where the efficient hybrid structure designs can be obtained by unit lattice structure-tube interactions [33]. The schematic view of the BCC and BCC-Z struc­ tures are presented in Fig. 1. The BCC and BCC-Z lattice structures are designed to fit exactly in­ side the square thin-walled tubes, therefore the lattice unit cells are formed similar to the Refs. [44–46]. The lattice unit cells are replicated to the vertical direction depending on the desired number of unit cells. It should be mentioned that the lattice structures are designed as an idealized geometry, in which the members have constant geometry and 3

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Fig. 4. The comparison of the experimental and FE simulation results.

obtain acceptable results, at least three elements are used through the diameter of the lattice structures [49] and the global element size of 1 and 2 mm is chosen for the tubes and the lattice structures, respectively [33]. The square tubes are made of the aluminum alloy Al6063-T5 with mechanical properties of Young’s modulus E ¼ 68.2 GPa, yield strength σ y ¼ 187 MPa, mass density ρ ¼ 2700 kg/m3 and Poisson’s ratio ν ¼ 0.33 [50,51]. On the other hand, the BCC and BCC-Z lattice structures are made of AlSi10Mg with mechanical properties of Young’s modulus E ¼ 69.3 GPa, yield strength σy ¼ 160 MPa, mass density ρ ¼ 2670 kg/m3 and Poisson’s ratio ν ¼ 0.3 [33,52]. True stress and true strain graphs of Al 6063 T5 and AlSi10Mg are shown in Fig. 3. To model the plasticity of the materials, the von Mises type yield criterion (J2-plasticity) in conjunction with isotropic hardening is used in the FE model. In addi­ tion, available strain rate dependency models and parameters for Al6063-T5 and AlSi10Mg from the literature are used in order to introduce the strain rate behavior of the materials. The

Cowper-Symonds overstress power law R ¼ 1þ(εpD)1n , where εp is the equivalent plastic strain rate, R is the ratio of dynamic stress to the static flow stress, and n and D are material constants, is included in the FE model to define the strain-rate-sensitive behavior of Al6063-T5. The D and n strain rate parameters in the Cowper-Symonds overstress power law are chosen to be 128,800 s 1 and 4, respectively [48,50]. On the other hand, Johnson-Cook plasticity model, R ¼ 1þCln(εpε0), where, C and ε0 are material parameters and the reference strain rate is used to capture the strain-rate-sensitive behavior of AlSi10Mg in the FE simu­ lation. The C and ε0 strain rate parameters in the Johnson-Cook formula are selected as 0.02 and 0.001s 1 respectively [52]. The crashworthiness performance of the proposed structures may be significantly influenced by material failures such as crack formation and propagation. Therefore, Johnson-Cook and ductile damage models, which are based on the value of von Mises equivalent plastic strain at element integration points, are used along with plasticity models, and a cumulative failure model controlled by a damage parameter is employed in the FE models. More 4

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detailed information on the damage models and their parameters for Al6063-T5 and AlSi10Mg can be found in Refs. [33,52,53]. In order to validate the proposed FE model, the lattice structures are first manufactured using direct metal laser sintering (DMLS) based on AM process via EOS M 290 equipment; a AlSi10Mg powder material with size of 30 μm is used to manufacture lattice structures. Then, the produced lattice structures are assembled in the Al6063-T5 square tube to build hybrid structures. Since the inertia effect is secondary, defor­ mation modes are similar both in dynamic and quasi-static compression for low-velocity impact [4,54,55] and 6 series aluminum alloys are low strain rate sensitive materials [56,57], the FE models are validated under quasi-static loading conditions similar to the approach in Refs. [18,58–60]. The experimental axial crushing tests are carried out using a Shimadzu Autograph AG-IS universal testing machine having capacity of 100 kN. The hybrid structures are situated between the two flat platens and compressed with a crushing velocity of 2 mm/min. The quasi-static loading conditions in the experimental tests are also intro­ duced into the FE models. The very small time increments generally have to be used in explicit FE time integration methods, thus use of too slow experimental test velocity in FE model is quite costly in terms of computational time. Hence, an artificial velocity profile is described similar to Refs. [18,27,61,62] for increasing the solution efficiency. At this point, different velocity profiles are considered to investigate effects of velocity profiles on the ratio of the total kinetic energy to the total internal energy (does not exceed 5%) and the force-displacement response which should be independent from the applied velocity. Finally, the artificial velocity is ramped during the first 15 ms to 2 m/s, then this velocity is held constant for minimizing the secondary inertia effects and still achieving quasi-static loading conditions. The compar­ ison of the experimental and FE results (Fig. 4) showed that the elastic, elastic-plastic, plateau, densification regions and deformation modes of the FE model results are in considerably good agreement with the experimental results. Namely, by comparing the FE and the experi­ mental results for BCC and BCC-Z hybrid structures, the maximal dif­ ferences for the peak crushing force and for the total energy absorption values are found to be lower than 5.5%. It should be also mentioned that the experimental test results are also compared with the results of low-velocity FE impact models with and without strain rate effects. The results show that the ratio of mean dynamic force to mean static force are found to be around 12–15% in FE benchmark simulations for all cases similar to those in Ref. [48]. A more detailed information on the material models, manufacturing of specimens and the experimental test procedure can be found in Ref. [33].

Fig. 5. Typical deformation mode shapes and corresponding forcedisplacement curves of selected hybrid tubes having tube thickness of 0.5 mm (black solid line) and 1.5 mm (red dotted line) obtained by FE results. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

structures have a sharp rise in the elastic region followed by a stress plateau. Next, the densification region begins, in which the entire structure is deformed plastically and collapsed on itself with a consid­ erable rise in hybrid structure stiffness. At this point, the onset of the densification region is considered as the beginning of the sudden rise in stress value (e.g., Fig. 5) similar to Refs. [23,27,29]. One energy-based metric (i.e., SEA) and one injury-based metric (i.e., PCF) are selected as the design objectives in this study. It should be noted that these objectives are generally in conflict with each other (e.g. Fig. 6). Namely, increase in SEA usually leads to increase in PCF. Hence, the problem should be considered as a multi-objective optimization (MOO) problem [41] and a compromise solution should be found. Crashworthiness performance of the hybrid structures mainly de­ pends on their deformation modes. At this point, the deformation modes are directly affected by the aspect ratio (i.e., L/a) of the hybrid struc­ tures. When hybrid structures with high aspect ratio (i.e., hybrid structures with 7 unit lattice cells) are subjected to axial loading, they may fail due to global bending with local buckling (Fig. 5), and this mode reduces the energy absorption performance of the structures. Some representative results of the BCC and BCC-Z hybrid structures are discussed in the following lines. The deformation modes, SEA and PCF values of BCC and BCC-Z hybrid structures having different selected design parameters are depicted in Fig. 6. It is clearly shown that the SEA and PCF values for both BCC and BCC-Z hybrid structures increase by increasing the tube thickness, and the deformation modes of the hybrid structures are considerably enhanced when the tube thickness is increased from 0.5 to 1.5 mm. For instance, the BCC-Z hybrid structure having the tube thickness of 1.5 mm shows progressive buckling behavior, while the BCC-Z hybrid structure with a tube thickness of 0.5 deforms with global bending with local buckling. Figure also shows that the increment in the diameter of the lattice member affects the SEA and PCF values of the hybrid structures. The crushing behavior of the BCC and BCC-Z hybrid structures are mainly controlled by the tubes for the member diameter range of 1–3 mm, on the other hand the lattice structures have a significant effect on the SEA and PCF values of the hybrid for the member diameter of 4–5 mm. Furthermore, it is clearly seen from Fig. 6 that the BCC-Z hybrid structures with tube thickness of 0.5 mm have global bending mode for all cases. On the other hand, while the PCF values of both the BCC and BCC-Z hybrid structures decrease as the number of unit cells increases, the SEA values generally increase except for high aspect ratio values of BCC-Z hybrid structures due to the tendency to global bending of these structures (Fig. 6 (c)). Hence, the proper selection of the design parameters is crucial to optimize the crashworthiness performance of hybrid structures and constitutes the main motivation of this study. All data set (sample design space, n, d, t) and the corresponding fitness space, which is obtained from FE

2.3. Crashworthiness parameters and fitness space The design objectives in the multi-objective crashworthiness opti­ mization studies include the following crashworthiness parameters: The energy absorption (EA) is defined as the total absorbed energy (including elastic energy) during crushing distance (δ) and it is obtained from the force-displacement curve as: Z δmax EA ¼ FðδÞdδ (1) 0

where δmax denotes the maximum crush displacement. The specific en­ ergy absorption (SEA) is the EA per total mass (m) of the structure. A higher SEA value means that an efficient crushing occurs [63]. The SEA is calculated as: SEA ¼ EA=m

(2)

The peak crush force (PCF) is defined as the maximum force value, and excessively high PCF values cause undesirable consequences for safety of passengers due to a large deceleration [64]. The crush effi­ ciency (CE) indicates the maximum crushing distance per initial struc­ ture length. Note that the stress-strain curves of crushing hybrid 5

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Fig. 6. The effect of a) tube thickness, b) diameter of lattice member and c) number of unit cell on the SEA and PCF values for some selected BCC and BCC-Z hybrid structures.

simulations (SEA, PCF) for BCC and BCC-Z lattice structures are depicted in the experimental design scheme in Fig. 7.

present work is depicted in Fig. 8. Artificial neural network is utilized in order to predict objective function values (peak crash force, PCF and specific energy absorption, SEA) for alternative designs, which are generated by the WSA algorithm. WSA algorithm makes use of the compromise programming approach in order to merge objective func­ tions in searching for a compromise design. Alternative compromise solutions can also be generated by varying user-defined weights in the compromise programming model. Otherwise stated, ANN acts as objective function evaluator. The whole multi-objective optimization system is programmed and realized in MATLAB 2019a environment. The details of the systems’ components are defined in the following sections.

3. Multi-objective optimization process 3.1. Optimization approach The purpose of the current work is multi-objective crashworthiness optimization of thin-walled tubes filled with BCC and BCC-Z lattice structures under axial impact loading condition. In order to achieve this purpose we need to perform extensive numerical simulations for con­ structing the design space. Nevertheless, performing all these simula­ tions by means of finite element modeling is unaffordable, as it necessitates extensive computational work and time [65]. It is a very well-known fact that envisaging behavior of mechanical structures by finite element modeling is a very difficult task [66]. Consequently, we employed artificial neural networks in order to reproduce the crash­ worthiness behavior of thin-walled tubes filled with BCC and BCC-Z lattice structures under axial crushing in the current study. The gen­ eral multi-objective design optimization process that is employed in the

3.2. Model building and analysis via ANN ANN is frequently used in industrial applications including crash­ worthiness studies due to its high prediction capabilities [64]. A typical ANN operates like a black box to convert inputs into outputs by employing various learning procedures [67]. ANN mimics biological neurons by having many non-linear computing agents working in par­ allel. All these computing agents are tightly connected via weights, 6

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Fig. 7. (a) Sample design space and corresponding fitness space for (b) BCC hybrid and (c) BCC-Z hybrid structures.

which are carefully modified with learning algorithms in order to improve their prediction performance. In operating ANN, input data is fed into input-nodes of ANN. The input data propagates along the con­ nections to the other portions of ANN. As they propagate, they are amalgamated at common joints and altered based on the predefined computational procedures. Outcomes from the output nodes of ANN institute the numeric results that are sought for the specific application. It is possible to develop ANN systems by making use of any usual computer programming languages, but this may be a challenging task for non-professional programmers. On the other hand, there are several good quality ANN development platforms that can be utilized for practical and professional applications. Some of the prominent software platforms are MATLAB ANN Tool Box, NeuroSolutions and Mathematica [68]. In the present study, the MATLAB’s Neural Network Tool Box is utilized for developing a suitable Generalized Feedforward ANN for predicting PCF and SEA. The datasets, which are generated through finite element simulations [33] as described in Section 2, are used to model PCF and SEA. Much more details about the datasets and the related finite element simulations can be found in our recent work [33]. The main purpose is to discover the best attainable weights (i.e. trained ANN) of the ANN to connect input variables, namely, number of lattice unit cells (n), lattice member diameter (d), and tube thickness (t) to the output variables, PCF, SEA. An appropriate ANN structure should be

determined keeping in mind the training performance (accuracy). Since the number of input/output variables institutes the neurons along with the transfer functions for input/output layers, it is necessary to designate a suitable organization for the hidden layers. The common practice to achieve a suitable ANN structure is the trial-and-error approach as there is not an agreed procedure for this task. In this study, 12 combinations of the ANN structure for BCC and BCC-Z hybrid structures are evaluated with the purpose of having a firm idea on the capability of the ANN in predicting PCF and SEA. ANN test parameters are presented in Table 1. It should be also mentioned that the hybrid structures having global bending modes are not taken into account in training since the desired energy absorption performance could not be achieved via this defor­ mation modes. The Levenberg-Marquardt algorithm that is considered as a very effective approach in ANN training is selected to train all ANN configurations. Data is divided as 70% for training 15% for validation and 15% for testing. The average results that are obtained after 15 in­ dependent runs from these tests are depicted in Table 2. The best ANN on test has R ¼ 0.96847 (Pearson correlation coefficient) for BCC lattice structures and R ¼ 0.95728 for BCC-Z lattice structures (see Table 2). Results indicate that there is considerable high correlation between the actual and predicted outputs. The ANN structure with one hidden-layer and 8 neurons in the hidden-layer produced the best results for both cases. Computational results for the most suitable ANN structure are shown in Figs. 9–14. 7

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Fig. 8. Compromise programming based multi-objective optimization process. Table 1 Parameters for ANN. Number of Hidden Layers : Number of Neurons in the Hidden Layersa : Structure of the Neural Network :

Table 2 Simulation results for alternative ANNs. 1, 2, 3 2, 4, 6, 8 Generalized Feed-forward

Extra values are estimated by subtracting 2 from the average value and incre­ menting by 2 from the mean. a Average number of neurons in hidden-layers can be set by RoundUp (2*SquareRoot(Number of Inputsþ1)).

3.3. Weighted superposition attraction (WSA) algorithm WSA is developed and improved by the first author for solving difficult engineering optimization problems [42,43,69]. WSA is a swarm intelligence based algorithm that imitates the superposition principle 8

Test

Number of hiddenlayers

Number of neurons in the hidden-layers

R value on test for BCC lattice structures

R value on test for BCC-Z lattice structures

1 2 3 4 5 6 7 8 9 10 11 12

1 1 1 1 2 2 2 2 3 3 3 3

2 4 6 8 2 4 6 8 2 4 4 8

0.89254 0.91232 0.93337 0.96847 0.95622 0.92511 0.94251 0.95311 0.94511 0.89219 0.88365 0.90011

0.87521 0.90521 0.93254 0.95728 0.93687 0.92478 0.92631 0.92657 0.94368 0.88365 0.87659 0.88254

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Fig. 9. Correlation coefficients for the most suitable ANN for BCC hybrid structures.

Fig. 10. Test results of the best ANN for BCC hybrid structures for PCF.

Fig. 11. Test results of the best ANN for BCC hybrid structures for SEA.

from physics. Two superpositions (attractive superposition & repulsive superposition) are determined in each stage of a WSA. The original WSA was designed with only attractive superposition; later repulsive super­ position is also considered [72]. Superpositions are essentially weighted composition of solution vectors where good quality solutions get higher weights in composing the “attractive superposition” and poor quality solutions get higher weights in composing the “repulsive superposition”.

Subsequently, the fitness of superpositions are evaluated/compared. If “repulsive superposition” comes with a better fitness value its label is changed to “attractive superposition”, and vice versa (this is an exceptional case nonetheless may occur in complex search spaces with many local op­ tima). Solution vectors may move randomly during WSA search if their fitness is better than the “attractive superposition”, otherwise they move 9

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Fig. 12. Correlation coefficients for the most suitable ANN for BCC-Z hybrid structures.

Fig. 13. Test results of the best ANN for BCC-Z hybrid structures for PCF. Fig. 14. Test results of the best ANN for BCC-Z hybrid structures for SEA.

in the direction of the “attractive superposition” and farther away from the “repulsive superposition”. In other words, “superpositions determination” and “attracted movement of solutions agents” are two main instruments 10

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Fig. 15. Main steps of the WSA algorithm [69].

of search in WSA. The basic steps of the WSA algorithm are outlined in Fig. 15. Main components of the WSA are explained in the following subsections in relation to the present application; more details about WSA can be obtained from Refs. [42,43,69].

under axial impact loading. The position vector of the ith solution vector is symbolized by SVi ¼ ½ni ; di ; ti � .Solution vectors are randomly initial­ ized within the range of variables boundaries as defined in Equation (3). � � (3) SVij ¼ SV Lj þ rand½0; 1� SV Uj SV Lj

3.3.1. Initialization of solution vectors The number of lattice unit cells (n), lattice member diameter (d), and tube thickness (t) are the decision variables and they are encoded as the elements of the solution vectors in WSA based crashworthiness optimi­ zation of thin-walled tubes filled with BCC and BCC-Z lattice structures

where i ¼ 1; 2; …; nPop, j ¼ 1; 2; …; D, rand[0,1], Aij, AjL and AjU characterize a uniformly distributed random number 2[0,1], the value of the ith solution vector at the jth dimension, lower and upper bounds for the jth dimension, correspondingly. The variable boundaries are set

Table 3 Biased randomized sampling procedure for superposition vector determination.

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empty vector is generated first. Afterwards, a random number 0, 1 is assigned to each cell of this empty vector. These random numbers are compared with the rank based weights (weight columns in Table 3, where τ is set to 0.8) of every solution vector. The computation pro­ cedure of rank based weights is as follows: weight(i) ¼ i-τ where i is the rank of the solution vector (solution vector with the best fitness and the worst fitness value gets the first rank in attractive and repulsive superposition determination). τ is the main parameter of the WSA. If τ is set to larger values, low ranking solutions will have a small chance to occupy in superpositions, resulting in a greedy search behavior; otherwise a more randomized search behavior is likely. If a solution vector’s rank based weight is greater than the preassigned random numbers the corre­ sponding element of the related solution becomes a candidate for the unfilled position of superpositions. After determining the candidate set, one of the candidates is selected as an element of the superpositions with the roulette wheel procedure. This procedure is repeated iteratively until complete superposition vectors are determined. A numerical example for the process is shown in Table 3. Notice that solution vectors are ranked from “best to worst” and from “worst to best”. τ is set to 0.80 (τ is commonly set to 0.80 in WSA applications as this value provides a good balance between diversification and intensification) and random numbers for each dimension are [0.350, 0.298, 0.741], [0.809, 0.658, 0.345] for attractive and repulsive superpositions. Consistent with these threshold values, the solution vectors that will enter roulette wheel se­ lection are highlighted with gray color in Table 3. For example, let us consider the first dimension, where the first three solution vectors enter roulette wheel selection. Suppose that the first solution vector wins. Consequently, the first dimension of the attractive superposition vector becomes 3, which is also presented with a star symbol in Table 3. The third dimension of the repulsive superposition becomes 0.69 by applying the same procedure. All other elements of the superposition vectors are determined in the same way, iteratively. Subsequent to determination of the superpositions, their fitness values are computed and their labels are changed if repulsive super­ position gets better fitness. Superpositions are determined after each iteration (i.e. after moving all solution vectors) during search.

Table 4 Compromise solutions from WSA. Solution number

wSEA

wPCF

Compromise Solution for BCC lattice structure (n, d, t – SEA, PCF )

Compromise Solution for BCC-Z lattice structure (n, d, t – SEA, PCF )

1

0.05

0.95

2

0.10

0.90

3

0.15

0.85

4

0.20

0.80

5

0.25

0.75

6

0.30

0.70

7

0.35

0.65

8

0.40

0.60

9

0.45

0.55

10

0.50

0.50

11

0.55

0.45

12

0.60

0.40

13

0.65

0.35

14

0.70

0.30

15

0.75

0.25

16

0.80

0.20

17

0.85

0.15

18

0.90

0.10

19

0.95

0.05

7, 1.0000, 1.0573–25.5741, 17.2344 7, 1.6544, 1.2113–28.1859, 18.9331 7, 1.6950, 1.2471–28.6799, 19.3458 7, 1.7241, 1.2792–29.1239, 19.8408 7, 1.7461, 1.3089–29.5435, 20.4126 7, 1.7629, 1.3368–29.9484, 21.0522 7, 1.7756, 1.3630–30.3426, 21.7492 7, 1.7852, 1.3877–30.7280, 22.4935 7, 1.7921, 1.4112–31.1057, 23.2766 7, 1.7969, 1.4336–31.4766, 24.0919 7, 1.7999, 1.4551–31.8418, 24.9346 7, 1.8014, 1.4759–32.2023, 25.8018 7, 1.8016, 1.4961–32.5591, 26.6918 7, 1.7518, 1.5000–32.7018, 27.0690 7, 1.6930, 1.5000–32.7820, 27.3299 7, 1.6365, 1.5000–32.8511, 27.6106 7, 1.5804, 1.5000–32.9095, 27.9218 7, 1.5220, 1.5000–32.9570, 28.2838 7, 1.4559, 1.5000–32.9921, 28.7432

7, 3.6498, 0.8381–21.8027, 13.4498 7, 3.7726, 1.0300–23.8812, 17.1684 7, 3.8064, 1.1130–24.9979, 19.4836 7, 3.7752, 1.1758–26.4193, 22.6204 7, 2.2834, 1.5000–28.1738, 26.5420 7, 3.7690, 1.2238–28.3853, 26.8799 7, 3.7722, 1.2437–29.1154, 28.4657 7, 3.7780, 1.2649–29.7326, 29.8139 7, 3.7857, 1.2903–30.2632, 30.9814 7, 3.7922, 1.3266–30.7236, 32.0036 7, 3.7939, 1.3818–31.0511, 32.7443 7, 3.8231, 1.4069–31.1973, 33.1083 7, 3.8677, 1.4182–31.3341, 33.5011 7, 3.9223, 1.4266–31.4896, 34.0141 7, 3.9880, 1.4343–31.6753, 34.7084 7, 4.0682, 1.4422–31.8945, 35.6599 7, 4.1690, 1.4510–32.1614, 36.9664 7, 4.3032, 1.4617–32.4822, 38.7585 7, 4.5075, 1.4775–32.8571, 41.2499

3.3.3. Moving of solution vectors The distinguishing swarming feature of WSA is to direct solution vectors towards attractive superposition with regard to their fitness. Nonetheless, solution vectors are also able to decide whether to move towards attractive superposition or not. A solution vector compares its fitness with the attractive superposition’s fitness to determine its move direction. If attractive superposition has a better fitness, the solution vector moves towards it by utilizing a specially devised move function (Equation (4)) [69], which also aids the solution vector to move away from the repulsive superposition. Otherwise, the solution vector com­ pares a random number ½0; 1� by the value computed as eðfðiÞ fðasÞÞ , where fðiÞ is the fitness score of the solution vector i and fðasÞ is the fitness score of the attractive superposition vector. If the random number is lower than the obtained value, the related solution vector moves towards the attractive superposition. Otherwise, it moves randomly by utilizing a specially devised move function (Equation (5)) [69]. Moving towards target superposition vector: A solution vector i updates its existing position on dimension jð1 � j � DÞ at iteration t by using Equation (4) which is similar to Jaya algorithm’s move equation [69,70] if its fitness is worse than the attractive superposition’s fitness or the condition random½0; 1� > eðfðiÞ fðasÞÞ is true.

as follows in this research: 3�n�7 ; 1 �d�5 mm; 0.5�t�1.5 mm nPop expresses the number of solution vectors and D denotes the number of variables. The first element of SV, n is rounded to nearest integer value within its range. Other algorithm specific parameters are discussed in the relevant sections alongside with their descriptions, which are also initialized at the beginning of WSA.

SVij ðt þ 1Þ ¼ SVij ðtÞ þ rand½0; 1� * SVas ðtÞ

3.3.2. Determination of superposition vectors A biased randomized sampling procedure that was developed by Baykasoglu [69] is utilized to determine attractive and repulsive su­ perpositions in WSA. The details of the superposition determination mechanism that is central to WSA can be summarized as follows: an

� �� �SVij ðtÞ�

rand½0; 1� * SVrs ðtÞ

� ��� �SVij ðtÞ� (4)

Where SVij ðtÞ is the value of the position of solution vector i on dimen­ sion j at iteration t ; SVas ðtÞ and SVrs ðtÞ are the attractive and repulsive superpositions at iteration t. j:j entitles absolute value. 12

A. Baykaso�glu et al.

Thin-Walled Structures 149 (2020) 106630

Random move of solution vectors: Solution vectors perform random search if they do not move towards the target superposition vector. In this case, Equation (5) [69] that includes a random walk step sizing function which is proposed by Baykasoglu [69] is utilized to realize random walking of each decision variable. � � SVij ðt þ 1Þ ¼ SVij ðtÞ þ ssðtÞ * unifrndð 1; 1Þ*�SVðr1Þj ðtÞ SVðr2Þj ðtÞ� (5)

" G X L ðSVÞ ¼ p

g ¼ 1; …::G are objective functions and f *g designates the ideal solution

vector (the best design vectors which also indicate the ideal points in the dataset as found by the finite element simulation). wg indicates the weight assigned by decision maker to the objective function g. The value of parameter p states a particular distance metric. For example, when p ¼ 1 , it corresponds to Pythagorean distance. When p ¼ 2; it corresponds to Euclidean distance. In fact, p ¼ 1 and p ¼ ∞ provide the longest and shortest distances. The most widely used compromise solutions are for p ¼ 1; 2; ∞ values. p ¼ 1; ∞ can be converted into linear forms by ab­ solute value and min-max linearization respectively if analytical forms of the objective functions are provided and they are linear [73]. Conversely, other p values 1 < p < ∞ result in non-linear models. For instance, the compromise program will be in quadratic form ifp ¼ 2. In this case, it is also not necessary to employ absolute value function in Equation (8). In the present crashworthiness optimization problem the analytical form of the objective functions, namely, PCF and SEA are not available, ANN predicts their values. Therefore, we actually have a non-linear model. Consequently, p ¼ 2can be considered a suitable se­ lection for the present problem. Based on this information, the compromise programming model for the present multi-objective crashworthiness optimization can be stated as follows:

Design optimization problems usually require simultaneous optimi­ zation of several and conflicting objectives. The present crashworthiness optimization of thin-walled tubes filled with BCC and BCC-Z lattice structures under axial impact loading involves two conflicting objec­ tives, minimizing the peak crash force, PCF and maximizing the specific

��2

(7)

g¼1

Where, Lp ðSVÞ denotes distance metric for alternative solution vector (SV), which is aimed to be minimized in an optimization setting. fg ðSVÞ;

3.4. Multi-objective optimization with compromise programming

SEAmin

#1=p

p

Where ϕstep is an user-defined parameter. The step sizing function has a decreasing tendency as iterations carry on; nevertheless, step size in­ creases for some random incidents during the search. Owing to this distinguishing feature, WSA is able to explore the search space more intensively.

� SEAðn; d; tÞ Þ SEAmax

�p � fg ðSVÞ�

Equation (7) needs to be normalized in order to be used in the pre­ sent multi-objective design optimization problem because objective functions have different (PCF (kN), SEA (kJ/kg)) units. Normalizing between the range [0, 1], Equation (7) becomes; " � * �p #1=p G X �f g fg ðSVÞ� � L ðSVÞ ¼ wg �� (8) Mg mg � g¼1

Where SVðr1Þj ðtÞ and SVðr2Þj characterize the randomly selected solution vectors from the swarm at iteration t. r1 and r2 are random indexes and r1 6¼ r2 6¼ i. Determination of an appropriate step size, ss can help to attain a high level of intensification during the search process. WSA implements a random walk variable step sizing tactic for solution vectors that choose to make random move. Random walk step sizing function starts with an initial step size (ssð0Þ) for each type of variable, n, d, t and regulates the step size as the search progresses by using a proportional rule [69,71]. The proportional rule requires a randomly generated number and a user-defined parameter, λ. Equation (6) presents the step size updating function [69,71]. � ssðtÞ e t=ðtþ1Þ � ϕstep � ssðtÞ; if randð0; 1Þ � λ ssðt þ 1Þ ¼ (6) ssðtÞ þ e t=ðtþ1Þ � ϕstep � ssðtÞ; if randð0; 1Þ > λ

minimize wSEA *ððSEAmax

� � wg �f *g

þ wPCF * ðPCFðn; d; tÞ

energy absorption, SEA. This is essentially due to the fact that PCF and SEA are typically in disagreement with each other, i.e. increase in SEA usually leads to increase in PCF. In mathematical programming, these problems are recognized as MOO problems. There are numerous ap­ proaches to resolve MOO problems. In most of these approaches, it is a common practice to search for a compromise solution, which is also a Pareto efficient solution. Compromise programming (CP) technique that was introduced by Zeleny [72] is a frequently utilized and mathemati­ cally proven MOO technique. CP does not require any subjective infor­ mation from the decision maker. Specifically, the decision maker does not need to state preferences and/or weights about the objectives. However, decision maker can also assign weights to the objectives in order to generate alternative solutions. Furthermore, a relatively low computational effort is needed in CP, since it can select a unique optimal solution from a set of Pareto efficient solutions. Based on these facts, CP can be considered as an appropriate multi-objective optimization tech­ nique to find a compromise solution for the present crashworthiness optimization problem. CP mainly relies on the selection of a unique solution that is the closest one to the ideal solution. Closeness to the ideal solution vector is determined by employing the following distance function (Lp-metrics) in CP [72]:

PCF min

�.

PCF max

PCF min

��2

(9)

Subject to; nlower � n � nupper

(10)

dlower � d � dupper

(11)

tlower � t � tupper

(12)

n; d; t � 0 & n is integer

(13)

In order to generate compromise solutions under different weights (wSEA , wPCF ) the compromise programming model (equations 9–13) is solved by the WSA algorithm that is explained in Section 3.3. The ideal points of SEAmin ; SEAmax ; PCFmin ; PCFmax for BCC structures are respec­ tively 7.115154, 35.06419214, 7.11378 and 98.417, on the other hand, these values are respectively 8.837287356, 33.52967213, 5.16003 and 119.258 for BCC-Z lattice structures. Upper and lower bounds for the decision variables (nlower ; nupper ; dlower ; dupper ; t lower ; tupper ) are 3, 7, 1, 5, 0.5, 1.5 respectively. While generating alternative solutions via equations (4) and (5), the WSA algorithm controls the variable boundaries and if a violation occurs, the corresponding variable is equalized to its boundary 13

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Thin-Walled Structures 149 (2020) 106630

Fig. 16. Frontiers of compromise solutions for (a) BCC and (b) BCC-Z hybrid structure.

Fig. 17. Deformation modes of BCC and BCC-Z hybrid structures for some selected compromise solutions.

value. As revealed in the previous sections, the objective functions values of PCF and SEA are determined by the trained ANN and the compromise solutions are determined by WSA as depicted in Fig. 8. The key parameter of the WSA algorithm is τ, which controls the composition of superpositions. In the current research and our former WSA practices [42,43,69,71–77] we recognized that a value around 0.8 for τ provides a good balance between intensification and diversification ability of WSA. Consequently τ is set to 0.8 in the present work. The other parameters, which are used in random walk step sizing function, are ϕ and λ. Numerous combinations of these parameters are examined and no sig­ nificant difference on the WSA performance is detected, values around 0.003 for ϕ and 0.7 for λ are found acceptable. Consequently, values for ϕ and λ are set to 0.003 and 0.7 correspondingly. Moreover, a small

population size around 10 is found sufficient to obtain competitive re­ sults. Thus, population size is set to 10 in all tests. WSA algorithm is run for 1000 iterations in order to generate alternative compromise solu­ tions for BCC and BCC-Z lattice structures. All results are presented in Table 4 and Fig. 16. Designer maker can select one particular solution for implementation based on the weights assigned to each solution. The energy absorption performance of BCC and BCC-Z hybrid structures is expected to increase with the increase in the number of unit lattice cell in the structures since these structures may collapse with more folds along the length of the body. On the other hand, the aspect ratio of the hybrid structures also increases by increasing the number of unit lattice cells; the desired energy absorption performance could not be reached due to global bending of hybrid structures (i.e. Fig. 6). The compromise solutions presented in Table 4 revealed that the optimum 14

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Thin-Walled Structures 149 (2020) 106630

almost all cases. Namely, the lattice member diameter values in opti­ mum designs of BCC-Z hybrid structures are up to 3.65 times larger than that of BCC hybrid designs. Thus, the PCF values of BCC hybrid designs are considerably lower than that of BCC-Z hybrid designs for the most of the cases since higher lattice member diameters are required to achieve the optimum energy absorption performance for BCC-Z hybrid designs, and this leads to a significant increase in the PCF values of these struc­ tures. By considering the compromise solutions having high PCF weight factors, the SEA values of BCC hybrid designs are slightly larger than that of BCC-Z ones while both structure have similar PCF. On the other hand, by considering the SEA dominant compromise solutions, although the hybrid structures have the similar SEA values, the PCF values of BCCZ hybrid designs are up to 44% higher than those of BCC hybrid designs. Overall, the compromise results show that the optimum BCC hybrid designs have generally superior crashworthiness performance compared to that of their BCC-Z counterparts in terms of the proposed design objectives. Fig. 17 shows the deformation modes of BCC and BCC-Z hybrid structures for some selected compromise results. It is seen from Fig. 17 that the BCC and BCC-Z hybrid structures have similar deformation modes and collapse with folds along the body, and global bending is not observed for these structures. These results also showed that the crashworthiness performance of BCC and BCC-Z hybrid structures can be enhanced by selecting the optimum structural parameters. These results revealed that the proposed hybrid designs have promising crashwor­ thiness performance and are suggested in crashworthiness applications as potential candidates. Verification of the solutions are also carried out by making use of two well-known error measurement statistics that are known as MAPE (Mean Absolute Percent Error) and MSE (Mean Squared Error). MAPE and MSE equations are given by equations (14) and (15). Smaller values of MAPE and MSE mean better accuracy. As it can be seen from Table 5 and Fig. 18, results are very close to the actual results (verified by the finite element model). � X � 1 n jActual Predictionj MAPE ¼ *100 (14) n i¼1 jActualj

Table 5 Verification of WSA results. Solution number

BCC hybrid structure

BCC-Z hybrid structure

SEA (kJ/kg) (from finite element model)

PCF (kN) (from finite element model)

SEA (kJ/kg) (from finite element model)

PCF (kN) (from finite element model)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

25.4072 28.0020 28.5703 29.2010 29.6918 29.8017 30.3622 30.6108 31.0571 31.5178 31.8542 32.4017 33.2166 33.4011 33.4542 33.6081 33.6220 33.7540 33.8487

15.7814 17.3369 18.9113 20.1404 20.4703 20.5333 21.3005 22.2380 21.8554 23.9015 25.6598 25.7656 26.4502 27.0404 27.3628 27.1553 27.5626 27.7169 28.1671

21.7789 22.1167 24.8781 26.4755 28.2539 28.6271 29.1267 29.5393 29.5809 30.5728 31.0210 31.2262 31.4251 31.4759 31.7702 31.9901 32.2578 32.5717 32.9893

15.8792 19.1546 20.6920 23.6009 24.7083 26.3588 28.3668 29.9404 31.2826 31.7746 31.8420 32.0009 33.0344 34.2014 34.4317 35.3756 36.6717 38.4495 40.9210

balance between two design objectives can be established by appro­ priate selection of tube thickness and lattice member diameter values in BCC and BCC-Z hybrid designs having 7 unit lattice cells. It is clearly observed from Fig. 6 that the tendency of global bending also increases especially in BCC and BCC-Z hybrid structures having low tube thick­ ness. It is seen from Table 4 and Fig. 17 that the increasing tube thickness improves the energy absorption capacity of the hybrid structures and contribute the global bending resistance of the hybrid structures. When the compromise solutions are examined, the optimum results are ob­ tained in a similar tube thickness range for BCC and BCC-Z hybrid de­ signs (i.e, about 1.06–1.50 mm for BCC and 0.84–1.47 mm for BCC-Z designs). However, the lattice member diameter values for compro­ mise solutions for BCC and BCC-Z hybrid designs are quite different (i.e, about 1.00–1.80 mm for BCC and 2.28–4.50 mm for BCC-Z designs) for

Fig. 18. Comparison of results obtained from WSA based multi-objective optimization system and FE simulation for a) BCC hybrid and b) BCC-Z hybrid. 15

A. Baykaso�glu et al. n �X

MSE ¼

ðActual

Thin-Walled Structures 149 (2020) 106630

PredictionÞ2

�. n

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(15)

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i¼1

Error measurement statistics for PCF and SEA: MAPEPCF ¼ 2.4600%, MSEPCF ¼ 0.4795, MAPESEA ¼ 1.0362 %, MSESEA ¼ 0.2107 for BCC hybrid; MAPEPCF ¼ 3.1468%, MSEPCF ¼ 0.9901, MAPESEA ¼ 0.8156%, MSESEA ¼ 0.1992 for BCC-Z hybrid structures. These results showed that the proposed optimization procedure has high prediction accuracy. 4. Conclusions Crashworthiness performances of square thin-walled tubes filled with BCC and BCC-Z lattice structures are investigated under axial impact loading by using multi-objective crashworthiness optimization procedure. Number of lattice cells, diameter of lattice members and tube thickness are considered as design parameters, and the optimum values of these design parameters are sought for maximizing the SEA and minimizing the PCF values. The proposed compromise programming for multi-objective optimization approach is based on finite element simu­ lation results in order to determine the sample design space and verifi­ cation. An artificial neural network is utilized in order to predict objective function values for alternative designs that are generated by a novel swarm intelligence based optimizer, which is known as weighted superposition attraction algorithm. Compromise programming approach is employed to combine multiple objectives and to produce various optimal design alternatives. The result revealed that the opti­ mum design objectives can be obtained by suitable selection of diameter and lattice member and tube thickness and number of lattice unit cells for BCC and BCC-Z hybrid structures. The results showed that the opti­ mized BCC hybrid designs have generally superior crashworthiness performance than that of their BCC-Z counterparts for providing the same design objectives. Namely, the PCF values of BCC hybrid designs are considerably lower than those of BCC-Z hybrid designs for most of the cases since higher lattice member diameters are required for BCC-Z hybrid designs in order to achieve the similar energy absorption per­ formances with BCC hybrid designs. Particularly, the BCC-Z hybrid structures have up to 50% higher PCF values than the BCC hybrid structures while these structures have similar energy absorption per­ formances. The compromise solutions also showed that the SEA of BCC and BCC-Z hybrid structures increases respectively 29% and 51% by considering the selected weight factors for the design objectives. The simulation results also showed that the proposed optimization proced­ ure has high prediction accuracy/search capability. In future works, the crashworthiness performance of lattice structure filled thin-walled tubes will be investigated and optimized under oblique and lateral impact loading conditions. In addition, further studies will also focus on accu­ rate beam element models of lattice structures to obtain fast and precise solutions. CRediT authorship contribution statement �lu: Methodology, Software, Investigation, Formal Adil Baykasog �lu: analysis, Visualization, Writing - review & editing. Cengiz Baykasog Conceptualization, Methodology, Software, Validation, Investigation, Writing - review & editing, Supervision. Erhan Cetin: Methodology, Software, Validation, Investigation, Visualization, Writing - original draft. Acknowledgments This work is supported by the Scientific Research Projects Governing Unit of Hitit University, Çorum, Turkey, project No: MUH19004.18.001.

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