Theoretical analysis and crashworthiness optimization of hybrid multi-cell structures

Thin-Walled Structures 142 (2019) 116–131

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Theoretical analysis and crashworthiness optimization of hybrid multi-cell structures

T

Tengteng Chena, Yong Zhanga,b,∗, Jiming Lina, Yong Lua a b

College of Mechanical Engineering and Automation, Huaqiao University, Xiamen, China School of Mechanical and Manufacturing Engineering, University of New South Wales, Sydney, Australia

ARTICLE INFO

ABSTRACT

Keywords: Hybrid design Energy absorption Crashworthiness optimization Theoretical prediction Multiple surrogate models

Thin-walled structure is widely used in automobile, aviation and other industrial fields due to its lightweight, high energy absorption efficiency and low cost. Four thin-walled hybrid multi-cell structures with circular and square sections are proposed in this paper. The energy absorption characteristics and crushing deformation of the hybrid structures are investigated by experimental testing and numerical analysis. Meanwhile, the theoretical models of mean crushing force and specific energy absorption of the hybrid structures are developed by simplified super folding element (SSFE) theory. It is found from the sensitivity analysis of parameters that the dimension of the external tube (D) and the wall thickness (T) of the structure have significant effects on energy absorption. Furthermore, the multi-objective optimization including multiple surrogate models is performed to obtain optimal crashworthiness of the hybrid multi-cell structures. The results show that multiple surrogate models are more favorable and accurate for the crashworthiness design, and the hybrid multi-cell structure with the outer circle and inner square section (CS2) has the best crashworthiness performance. Finally, the multiobjective optimization solutions are analyzed and chosen by the normal boundary intersection (NBI) method to carry out the crashworthiness comparison with the typical multi-cell structures of the same mass, and the hybrid structure (CS2) outperforms multi-cell tubes with the single circular or square section.

1. Introduction Thin-walled structure, as an efficient energy-absorbing structure, has been widely used in various engineering fields due to their advantages of lightweight. A large number of studies on the deformation mode [1–3] and energy absorption [4–7] of thin-walled structures have been performed. Recent years, the focus of the research has shifted from thin-walled single tube to multi-cell tube owing to increasing the requirements of collision resistance of thin-walled structures [8,9]. Therefore, some new design methods, such as hierarchical design [10–13] (Fig. 1(a) and (b)), bionic design [14,15] (Fig. 1(c) and (d)) and hybrid design [16] were used to design multi-cell structure with better crashworthiness performance. For example, Zhang et al. [10] proposed a new self-similar hierarchical fractal method to design thinwalled tubes, and the results showed that this method could effectively improve the impact resistance of thin-walled structures. Zhang et al. [15] proposed a group of bionic multi-cell tubes (BMCTs) inspired by the microstructure of beetle forewings and found BMCTs showed superior crashworthiness than traditional multi-cell structures. In term of the hybrid design, Vinayagar et al. [16] experimentally studied hybrid

∗

ribless structures with multi-section (Fig. 1(g)) and found that the hybrid of different sections had a significant impact on the mechanical impact characteristics. The energy absorption characteristics of metal/ composites hybrid structures (Fig. 1(e)) under axial and oblique impact were investigated by Sun et al. [17,18] through experimental test and simulation analysis. In addition, they also studied the metal-foamcomposite hybrid structures [19] (Fig. 1(f)), and found that these multimaterial hybrid structures could promote energy absorption of single material structure. These studies have found that hybrid design was one of the effective methods to improve the crash-resistance of thin-walled structures. However, to the best of the authors’ knowledge, these studies mainly focused on the experiment and simulation of the multimaterial structures and internal unsupported structures. Therefore, the crashworthiness study of multi-cell hybrid structures was valuable based on experimental testing and theoretical analysis. In addition, surrogate modeling technology has also been applied more and more widely in the optimization design of the thin-walled structure to find the optimal parameter of the structure and further improve the anti-collision performance [20–23]. For example, Qiu et al. [22] established the Kriging model to optimize the section parameters

Corresponding author. College of Mechanical Engineering and Automation, Huaqiao University, Xiamen, China. E-mail address: [email protected] (Y. Zhang).

https://doi.org/10.1016/j.tws.2019.05.002 Received 23 October 2018; Received in revised form 25 February 2019; Accepted 2 May 2019 Available online 14 May 2019 0263-8231/ © 2019 Elsevier Ltd. All rights reserved.

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Fig. 1. Design methods of thin-wall structures: (a) [10]; (b) [11]; (c) [15]; (d) [14]; (e) [18]; (f) [19]; (g) [16].

of multi-cell thin-walled structures under multiple loads. Ying et al. [23] performed multi-objective optimization for the thin-walled structure with functional gradient combined with Radial Basis Function neural networks and Non-Dominated Sorting Genetic Algorithm II algorithm to minimize peak impact force and maximize energy absorption. Although approximate model techniques have achieved good applications in engineering optimization problems, it was unclear which surrogate model was most suitable for a particular engineering problem. Hence comprehensive research on diversiform surrogate models for accuracy became essential. Driven by these promising findings, four hybrid multi-cell structures with square and circular sections are proposed and studied by experimental testing, simulation analysis, theoretical prediction and multiobjective optimization. The configuration of the paper is as follows. The geometric configuration is given in Section 2. In Section 3, the quasistatic crushing behaviors of hybrid tubes are studied by experimental and numerical simulation. Section 4 derives theoretical expressions for the mean crushing force of hybrid tubes. Section 5 introduces the multiobjective optimization of multiple surrogate models, and Section 6 is conclusions.

collision environment [24]; the crushing distance keeps 140 mm. The bottom of the thin-walled structure is constrained by a rigid wall. Details are described in the following sections. 3. Experimental tests and numerical modeling 3.1. Specimens and experimental method The tensile specimens of 6061-O aluminum alloy were first prepared, and the engineering stress-strain curve of aluminum A6061-O was obtained by standard uniaxial tensile test on the specimens, as shown in Fig. 3. The main material properties are listed in Table 1. Secondly, the test specimens shown in Fig. 4 of the four hybrid tubes were fabricated by the wire cut electrical discharge machining (WEDM) technique. The wall thicknesses (T), D and d for the four hybrid multicell tubes was 0.8 mm, 80 mm and 40 mm, respectively. Since the maximum length limit of the specimen machined by WEDM machine, the length of the four specimens in this paper was 120 mm. The quasi-static crushing test of the four hybrid structures was completed by a Universal Material Testing Machine (Fig. 5(a)) in order to analyze the crashworthiness. The specimens were placed on the bottom platen (Fig. 5(b)), which was moved upward at a constant speed of 5 mm/min, while the upper platform was fixed. The load was terminated when the crushing displacement of the specimen reached 70 mm. SANS data acquisition system collected the force and displacement data of the crushing process of specimens.

2. Structural description The geometric sections of the hybrid multi-cell structures are shown in Fig. 2. According to the position of the ribs, there are four hybrid multi-cell tubes: a) the ribs connecting the external circular tube and the vertices of the inner square tube (CS1); b) the ribs connecting the external circular tube and the middle wall of the inner square tube (CS2); c) the ribs connecting the vertices of external square tube and the inner circular tube (SC1); d) the ribs connecting the middle wall of external square tube and the inner circular tube (SC2), as shown in Fig. 2(a). The diameter D of the outer circular tube is the same as the side length of the outer square tube; the length d of the diagonal of the inner square tube is the same as the diameter d of the inner circular tube, and the wall thickness of the hybrid structures is T. The parameter D, d and T will be varied, their ranges are 80 mm ≤ D ≤ 120 mm, 20 mm ≤ d ≤ 60 mm and 0.8 mm ≤ T ≤ 2.0 mm, respectively. According to the energy absorber shown in Fig. 2(c) at the front of the vehicle, a brief diagram of the axial load of the thin-walled structure is formed in Fig. 2(b). The height of the hybrid tube is 200 mm. The mass of the upper impactor is 500 kg, and the thin-walled tube is compressed downward at the speed of 50 km/h to simulate the typical

3.2. Experimental evaluation and results For evaluating the crashworthiness of different hybrid structures, several evaluation indicators (Total Energy Absorption (EA), Specific Energy Absorption (SEA), Peak Crushing Force (PCF)) are introduced. EA is the total strain energy absorbed during plastic deformation, and its equation is as follows: d

EA =

F (x ) dx 0

(1)

where F(x) is the instantaneous collision force and d is the compression displacement. SEA is the energy absorption per unit mass of a thin-walled tube, which can be calculated from EA as: 117

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Fig. 2. Schematic diagram: (a) the cross-section of the hybrid tubes; (b) schematic diagram of the crushing process; (c) the circular energy absorber in the body of an Audi car [25].

SEA =

Table 1 Mechanical properties of aluminum A6061-O. Symbol

Value

Young's modulus Density Poison's ratio Initial yield stress Ultimate stress

E ρ μ

68.0 GPa 2.7 × 103 kg/m3 0.33 71 MPa 130.7 MPa

y

u

(2)

where m is the total mass of the thin-walled structure. PCF is the maximum instantaneous collision force closely related to speed reduction, and high PCF is very unfavorable to the safety of passengers. Therefore, PCF should be reduced as much as possible, or it should be limited to the safety range. Figs. 6(a) and 7(a) firstly presented collapse shapes of CS1 and CS2 at different crushing displacement, it was seen that CS1 and CS2 were crushed layer-by-layer in the process of collapse. CS1 and CS2 gradually collapse at the bottom, followed by regular folds stacked on top of the first folds, and the collapsed shape was considered as an effective energy absorption mode [26]. Figs. 6(b) and 7(b) was the internal profile of CS1 and CS2, which was clearly seen the phenomenon of internal layer folding. Moreover, there was shorter folded wavelength inside CS2, which was more folds than CS1, and short folded wavelength was favorable for energy absorption. Fig. 8 showed the collapse processes of SC1 and SC2 with external square cross sections, it was seen that their collapse modes were quite different, in which SC2 folded regularly layer by layer, and the length of the local buckling wave for each fold remained constant during the formation of buckling, however SC1 had very large folding wavelength, which was unfavorable for energy absorption. The force-displacement curves of the experimental test shown in Fig. 9 can be used to calculated SEA and PCF of the hybrid structures (CS1, CS2, SC1, SC2), which is listed in Table 2. It was found that the CS2 structure showed the highest SEA compared with the other three structures, while SC1 structure had the worst crashworthiness performance due to the largest PCF and the smallest SEA.

Fig. 3. The engineering stress-strain curve of aluminum alloy 6061-O.

Property

EA m

3.3. Numerical modeling and validation According to the geometric models of Fig. 2, the numerical models of the hybrid structures were established in LS-DYNA (Fig. 10). The 118

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Fig. 4. Experimental specimens of four hybrid multi-cell tubes.

four-node Belytschko-Tsay shell elements with five integration points through thickness was used to discretize the hybrid structure [27]. Under the premise of ensuring that the simulation calculation met the accuracy, the mesh size of the finite element model in the paper was set to 1.0 mm × 1.0 mm to save the computational cost. The elastoplastic material model MAT_24 and the rigid material model MAT_20 in LSDYNA were used to simulate thin-walled tubes and upper-end compression rigid plates, respectively. The bottom end of the tube was supported by a rigid wall, and the top end of the tube was compressed by an impactor. The crash velocity of the impactor in FE models was 0.5 m/s to simulate the quasi-static crushing process. Node-to-Surface was established between the hybrid tubes and the impactor and Automatic single-Surface contact was applied to hybrid tubes to prevent penetrating during the crushing process. The static friction coefficient and dynamic friction coefficient were set to 0.3 and 0.2 respectively in the numerical model [25]. The effect of strain rate was ignored in the FE models because aluminum was not sensitive to strain rate [10,28]. Comparisons of the force-displacement curves between the

experimental test and numerical analysis for the four hybrid tubes were presented in Fig. 11. It was found that the fluctuation of the crush load of numerical analysis was in good agreement with the experimental results during the collapse process. Moreover, Fig. 12 further gave the comparisons of deformation shapes of CS1, CS2, SC1, and SC2 at 70 mm crushing displacement. It was found that the predicted collapse shapes showed stable and almost identical deformation collapses with experimental results. In addition, the results of numerical predictions were shown in Table 2. It was seen that the error of simulation value and the experimental value was less than 5% except for SEA value of the SC1 structure, which was because of the crack in the SC1 structure during the test crushing process (as indicated in Fig. 8(a)). Hence, it can be judged that the computational model can well predict the crash behavior of the hybrid tubes. The finite element model can effectively and accurately reflect the experimental data and can be used for further study.

Fig. 5. Quasi-static axial compression test of hybrid structures: (a) the universal material testing machine; (b) Partial enlargement of the compression test.

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Fig. 6. Collapse test of CS1: (a) Detailed of collapse modes at different distances; (b) internal profile.

4. Theoretical prediction of the hybrid multi-cell structures

According to the mechanical properties of aluminum A6061-O in Table 1, 0 can be calculated as 100.85 MPa. The flanges are completely flattened according to the SSFE theory in the axial deformation in the 2H wavelength, and the angles of rotation of the three hinge lines are π/ 2, π, and π/2, respectively. So:

Although numerical simulation technology is widely used to simulate the collision behavior of thin-walled structures, theoretical prediction is still a key method, which can directly analyze the structure's crashworthiness without test and numerical simulation. Therefore, this section analyzes the axial crushing of hybrid tubes by the Simplified Super Folding Element (SSFE) theory [29–33]. Suppose that each folded wave has a wavelength of 2H and is consistent and the wall thickness of the structure is uniform in the SSFE theoretical model. The mean crushing force can be calculated on the basis of a basic folded component, including three stationary hinge lines and extensional and compressional elements (as shown in Fig. 13(a) and (b)). Therefore, during the whole collapse process of a folded layer (2H), the energy balance equation of the system is:

4.2. Membrane deformation dissipated energy Fig. 14 shows the plastic strain diagrams of the four hybrid tubes. It is clearly seen that the energy dissipation in the compressive process is mainly determined through the yield strain of the different corner elements (square wireframe). Furthermore, both external and internal arcs of the circular cross-section have the obvious strain in the compressive process (elliptical frame), so it is necessary to consider the membrane deformation of the circular element [10,34]. Therefore, dividing four hybrid structures into seven basic elements to analyze the membrane energy dissipation under axial crushing: namely Circular element, 2-panel element, 3-panel element І, 3-panel element II, Convex T-shape element, Concave T-shape element, and Tshape element, as shown in Fig. 15. The membrane energy is generated by compressing or expanding the shell element for the Circular element, which can be calculated during a whole fold formation as [34]:

(3)

Pm 2H k = Eb + Em

where Pm represents the mean crushing force; Eb and Em are energy consumed by bending and membrane deformation, respectively. Fig. 13(c) shows the ideal status of folding element for the flange. Regarding as the practical status, effective crashing coefficient k which is the ratio of effective crashing distance S (see Fig. 13(d)) and the wavelength 2H is introduced, k is taken as 0.7 here [29]. 4.1. Bending dissipated energy Eb can be calculated by integrating the energy dissipation at three stationary hinge lines:

EmCircular = 8

3

Eb =

M0 µ Lc 1 1

0

=

y

2

µ

M0 H 2 T

(7)

It is seen from Fig. 15 that all the included angle of the 2-panel elements contained in the four hybrid structures is 90°. According to the previous studies [28], the membrane energy of the 2-panel element with central angle θ:

(4)

where, M0 = 4 0 T 2 is the completely plastic bending moment of the flange; µ is the rotation angle of each static hinge line, and LC is the total length of all flanges. 0 denotes the flow stress of the material, which can be calculated by the following formula [1]:

+

(6)

Eb = 2 M0 Lc

Em2

(5)

is:

panel

=

tan( /2) 4M0 H 2 T (tan( /2) + 0.05/tan( /2))/1.1

So, the membrane energy dissipation of 2-panel element (θ = 90°)

Fig. 7. Collapse test of CS2: (a) Detailed of collapse modes at different deformation; (b) internal profile.

120

(8)

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Fig. 8. Collapse test at different deformations: (a) SC1; (b) SC2.

Fig. 10. FE models of different hybrid tubes.

panel element tend to deform in Pattern-1; when α > 120°, the 3-panel element tend to deform in Pattern-2; when 90°≤α ≤ 120°, both types of deformation are likely to occur. When the 3-panel element experiences Pattern-1 deformation, its membrane energy is expressed as [15,27]:

Em3

Fig. 9. Force-Displacement curves of experimental tests of hybrid tubes.

CS1 CS2 SC1 SC2

Em2

panel

PCF (kN) FEA

Error (%)

Test

FEA

Error (%)

29.65 29.58 33.12 30.94

30.14 29.39 33.89 31.70

1.65 −0.64 2.32 2.46

9.74 9.55 6.13 9.04

9.29 9.35 7.06 8.97

−4.62 −2.10 15.17 −0.77

( = 900) =

4.19M0 H 2 T

tan( ) 4M0 H 2 + 2 tan T (tan( ) + 0.05/tan( ))/1.1 2

When the 3-panel element undergoes Pattern-2 deformation, its membrane energy is expressed as [27,35]:

SEA (kJ/kg)

Test

( )=

(10)

Table 2 Experimental and numerical results of hybrid tubes under quasi-static compression. structures

panel element

Em3

panel element

( )=

2M0 H 2 4 tan + 2 sin + 3 sin T 4 2

(11)

where β = 2(π-α), α and β are the included angle of the two panels in the 3-panel element, their position relationships are shown in Fig. 16. The value of α、β for the 3-panel element I and 3-panel element II referred to herein are 45° and 90°, respectively, which can be seen from Fig. 15. Therefore, their membrane energy dissipation is: (9)

The deformation modes of the 3-panel elements are divided into two types: Pattern-1 and Pattern-2 (shown in Fig. 16(a) and (b)), in which the broken lines indicate the deformation directions of the panel elements. According to SSFE theory: when included angle α < 90°, the 3-

Em3

panel element

Em3

panel element

( = 450) =

7.50M0 H 2 T

(12)

( = 900) =

12.14M0 H 2 T

(13)

For Convex and Concave T-shape elements, since the membrane

121

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Fig. 11. Comparison of force-displacement curves obtained by experiment and numerical simulation: (a) CS1; (b) CS2; (c) SC1; (c) SC2.

Fig. 12. Comparison of the deformation modes between experiment and simulation for four hybrid structures: (a) CS1, (b) CS2, (c) SC1, (d) SC2.

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Fig. 13. Energy dissipation in bending deformation: (a) extensional elements; (b) stationary hinge lines; (c) ideal completely folding; (d) actual folding.

Fig. 14. Plastic strain of hybrid structures.

energy is mainly dissipated in the intersection region [32], a simplified method by replaced arc-panel to the tangent plane which is formed from the intersection point of the panels is taken to calculate its membrane energy, as shown in Fig. 16(c); therefore, the Convex Tshape element and Concave T-shape element are simplified as T-shape element. Then

EmConvex

T shape

= EmConcave

T shape

= ET

shape

EmTotal = EmCircular + N1 E2 + N3 E3

ET

shape

=

( =

900)

12.3M0 H 2 = T

panel element

+ N2 E3

+ NT ET

panel element

(16)

shape

where N1, N2, N3, and NT represent the number of the 2-panel element, 3-panel element I、3-panel element II and T-shape element, respectively. Taking CS1 structures as an example, according to Fig. 15(a), CS1 is divided into one circular element, four T-shape elements and four 3of CS1 is expanel elements I. Therefore, the membrane energy EmTotal cs1 pressed as:

(14)

T-shape element is a special case for = 90 0 of 3-panel element I. Therefore, the membrane energy of T-shape element is expressed as:

Em3 panel element

panel element

EmTotal = EmCircular + N2 E3 cs1

(15)

E3 =8

Therefore, the total membrane deformation consumption EmTotal of a hybrid multi-cell structure is:

panel element M0 H2 T

panel element

+ 4ET

+4×

+ NT ET

shape

= EmCircular + 4

shape

12.14M0 H2 T

+4×

12.3M0 H2 T

(17)

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Fig. 15. Basic structural forms.

Fig. 16. 3-panel element and Simplification: (a) Pattern-1; (b) Pattern-2; (c) Simplification of convex and concave T-shape elements.

The membrane energy of the other three hybrid tubes (CS2, SC1, SC2) is also obtained in the same way.

It is assumed that the folding unit is deformed in the most ideal crushing manner during the folding and crushing process, that is, with a minimum average crushing [29]. So:

4.3. The mean crushing force

Pm _ CS1 =0 H

Substituting Eqs (6) and (16) into (3) to calculate the mean crushing force Pm _ CS of the hybrid multi-cell structures under axial compression. Then:

Thus, H can be derived and calculated as:

H=

(18)

Pm _ CS 2H k = 2 M0 Lc + EmTotal Taking an example of CS1, Substitute Eqs. (17) into (18):

Pm _ CS1 2H k = 2 M0 Lc + 8

H2

M0 T

+4×

12.14M0 T

H2

+4×

12.3M0 T

(20)

LC T 61.44

(21)

Substituting the value of H into Eq. (19) can be obtained:

H2

Pm _ CS1 = 6.94

(19)

0

k

( D + 2 2 d + 2(D

d))0.5T1.5

(22)

The mean crushing force of the other three hybrid tubes can be obtained by the same method and are listed in Table 3. The influence of structural dynamic effects on the dynamic load is

where Pm _ CS1 represents the mean crushing force of CS1 under axial quasi-static compression. Table 3 Mean crushing force for four hybrid structures. Structures

Circular element

CS1

1

CS2

1

SC1

1

SC2

1

2-panel element

T-shape element

3-panel element-I

4 4 4

4

8 4

3-panel element-II

Pm _ CS 6.94( 0/k )( D + 2 2 d + 2(D 7.42( 0/k )( D + 2 2 d + 2(D

4

6.40( 0/k )(4D + d + 2( 2 D

8

7.42( 0/k )(4D + d + 2(D

124

0.5

d)) T1.5 0.5

2 d/2)) T1.5 0.5

d)) T1.5

d )) 0.5T1.5

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Table 4 The numerical simulation results of EC.

Pmd _ SC2 = 1.11

No.

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

hybrid structures

d

CS1

CS2

SC1

SC2

D

T

Dynamic d Pm _ CS

Quasi-static Pm _ CS

(mm)

(mm)

(mm)

(kN)

(kN)

25 55 52 31 20 33 44 23 39 36 41 49 57 47 60 28

96 117 107 80 109 104 101 88 93 120 85 83 99 115 91 112

1.52 1.36 0.88 1.68 1.20 0.80 1.44 0.96 2.00 1.28 1.04 1.60 1.84 1.92 1.12 1.76

46.14 43.89 21.11 52.41 34.39 19.70 48.86 23.44 74.04 36.40 26.90 50.89 77.25 82.36 34.38 65.88

42.87 38.70 18.53 50.03 31.00 17.16 45.65 22.19 68.65 32.87 23.40 44.45 70.69 74.31 30.97 56.42

EC

( D + 2 2 d + 2(D

Pmd _ CS2

100.85 = 1.11 0.7

7.36

5. Multi-objective optimization 5.1. Optimization process The multi-objective optimization flowchart of the hybrid structure is presented in Fig. 17. Firstly, the structural parameters are analyzed to determine design variables, and the optimized mathematical model is established; then some sample points are selected by the optimal Latin hypercube experimental design method (OLHS) [36]. Secondly, numerical models are established and computed in the sample points. Four approximate methods are used to select a high accuracy metamodels for the four hybrid multi-cell tubes. Then, the Archive-Based Micro Genetic Algorithm (AMGA) algorithm is employed to seek the optimal solutions, and the multi-objective solutions are analyzed and chosen by the normal boundary intersection (NBI) method.

(

2 2

d

))

0.5

T1.5 = 1.187

1000 0.5

T1.5

100.85 6.34(4D + d + 2( 2 D = 1.11 0.7 1000 (4D + d + 2( 2 D

The sensitive analysis of parameters effectively determines the most sensitive factors affecting the energy absorption and considered as design variables of optimization. Based on Eqs. (24)–(27), the theoretical expressions on the SEA of the four hybrid structures are obtained as follows, respectively.

(24)

2d + 2 D 2 d 2

5.2. Sensitive analysis of parameters

= 1.110

d )) 0.5T1.5

( D+2

D + 2 2d + 2 D

Pmd _ SC1

d ))0.5T1.5

SEACS1 =

(25)

d ))0.5T1.5

Pmd _ CS1 MCS1 140 =

= 1.023

d )) 0.5T1.5

(27)

The 32 dynamic numerical simulations of the four hybrid tubes with different geometrical parameters listed in Table 5 are performed to verify the accuracy of the theoretical expressions of the four structures. The simulation values of the mean crushing force of the four structures are taken at 70% of the height of the structure, and the theoretical prediction of the four structures are obtained by Eqs. (24)–(27). It is seen that the value of the maximum error is 12.4%, which is within an acceptable reasonable range. Therefore, it is believed that the theoretical prediction models of the hybrid tubes have sufficient accuracy to be applied in subsequent studies.

1.08 1.13 1.14 1.05 1.11 1.15 1.07 1.06 1.08 1.11 1.15 1.14 1.09 1.11 1.11 1.17

not considered in the theoretical derivation of the above section. The dynamic effects of the structure mainly consider the inertial effect. A series of random dynamic numerical simulations with an impact velocity of 50 km/h and quasi-static numerical simulations with an impact velocity of 5 mm/min was performed to estimate the dynamic enhancement factor EC. The numerical results are shown in Table 4. The EC values of different structures are close to each other, and the average value is about 1.11. Based on Eq. (23) and Table 3, the prediction models of mean crushing force for CS1, CS2, SC1, SC2 structures under dynamic loading are, respectively:

100.85 6.91( D + 2 2 d + 2(D 0.7 1000

= 1.187

4.4. Verification of dynamic expressions Pmd _ CS

Thus, theoretical expression Pmd _ CS of mean crushing force under dynamic impact can be expressed as.

Pmd _ CS1 = 1.11

d ))0.5T1.5

d )) 0.5T1.5

(4D + d + 2(D

(23)

Pmd _ CS = EC Pm _ CS = 1.11 Pm _ CS

100.85 7.36(4D + d + 2(D 0.7 1000

h=

10

6

1.11 ( D + 2 2 d + 2(D d )) 0.5T1.5 2.7 × 103 200( D + 2 2 d + 2(D d )) T

287.78T 0.5 ( D + 2 2 d + 2(D

d)) 0.5

(28)

(26)

Table 5 Theoretical value (Theo.) and simulation value (FEA) of MCF. No.

1 2 3 4 5 6 7 8

Variables(mm)

d Pm _ CS1 (kN)

d Pm _ CS2 (kN)

d Pm _ SC1 (kN)

d Pm _ SC2 (kN)

D

d

T

Theo.

FEA

Error (%)

Theo.

FEA

Error (%)

Theo.

FEA

Error (%)

Theo.

FEA

Error (%)

80 86 91 97 103 109 114 120

31 43 54 60 26 49 37 20

1.49 1.83 1.66 1.31 0.8 2.0 1.14 0.97

42.60 44.19 60.50 54.36 39.35 77.49 33.90 27.61

42.81 44.06 61.37 54.81 38.35 76.86 31.88 26.00

−0.50 0.30 −1.42 −0.82 2.61 0.82 6.34 6.17

46.50 47.87 66.38 59.90 43.41 84.81 36.88 30.28

48.21 45.58 69.39 61.68 43.64 86.77 34.66 28.80

−3.54 5.05 −4.33 −2.89 −0.54 −2.25 6.42 5.15

45.30 46.97 64.35 57.83 41.87 82.41 36.04 29.37

43.66 42.33 64.52 55.38 41.51 81.62 33.04 26.13

3.75 10.97 −0.26 4.44 0.87 0.97 9.08 12.40

49.48 51.22 70.37 63.30 45.83 90.07 39.34 32.11

49.63 47.61 71.00 63.19 44.46 88.51 35.40 28.87

−0.30 7.59 −0.90 0.17 3.08 1.76 11.11 11.20

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Fig. 17. Flowchart of the optimization.

Fig. 18. SEA and PCF with different T (D = 100 mm, d = 40 mm): (a) SEA; (b) PCF.

SEACS2 =

Pmd _ CS2

SEASC1 =

MCS2 1.187

(

2.7 ×

103

h= 10 140 =

6

(

(

2 2

D + 2 2d + 2 D 200

( D+2

(

(

))

2d + 2 D

305.41T 0.5 D + 2 2d + 2 D

d

2 2

d

))

0.5

d

MSC1 140 =

T1.5

2 2

Pmd _ SC1

)) T

SEASC2 =

0.5

Pmd _ SC2 MSC2 140 =

(29)

0.5

h=

1.023 (4D + d + 2( 2 D d)) T1.5 10 2.7 × 103 200(4D + d + 2( 2 D d)) T 6

265.22T 0.5 (4D + d + 2( 2 D

h=

10

6

d ))

0.5

(30)

1.187 (4D + d + 2(D d))0.5T1.5 2.7 × 103 200(4D + d + 2(D d )) T

305.41T 0.5 (4D + d + 2(D d))0.5

(31)

where M (MCSi or MSCi,i = 1,2) is the mass of hybrid structures, it is calculated as: 126

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Fig. 19. SEA for four tubes with different D and d (T = 1.4 mm): (a) CS1; (b) CS2; (c) SC1; (d) SC2.

M = 10

6

×

L LC T

Network (ANN) [21]. However, these different surrogate techniques typically provide different modeling accuracy [39], gradient information and design outcomes [40]. It is not clear which kind of surrogate model is best for a particular problem. Therefore, four different surrogate models, quadratic response surface RSM2, KRG, RBF, and ANN are all considered for crashworthiness applications in this study, they are summarized in Table 6. Two error indicators (R-square (R2) and the root-mean-square error (RMSE)) are introduced to evaluate the accuracy of these alternative models. R2 ranges from 0 to 1 and is typically greater than 0.9 as standard acceptability. R2 can be obtained by the following formula:

(32)

where ρ and h are the density and the collision distance of the hybrid tubes, and they are 2.7 × 103 kg/m3 and 140 mm, respectively.

According to the theoretical expression of the SEA, the variables of the hybrid multi-cell tubes mainly include D, d and thickness T. The parameter sensitivity of the thickness T when D and d is 100 mm and 40 mm and different D and d when T is 1.4 mm were analysed, respectively. The results of design responses (SEA and PCF) are shown in Figs. 18–20. It's obvious from Fig. 18 that SEA and PCF of the four tubes gradually increase with the increase of T. Furthermore, SEA and PCF shown in Figs. 19 and 20 for each structure has strong sensitivity on D. However, SEA only shows a slight variation when the d increases from 20 mm to 60 mm. Meanwhile, the curves of PCF are close to horizontal; therefore, the d is not sensitive on the SEA or PCF. Overall, D and T are considered as design variables owing to the strong sensitivity on the SEA and PCF for the following optimization study. As an energy absorber, the goal of structural optimization is to find the appropriate structural parameter to ensure that the absorber can absorb as much energy as possible under the mass of the unit and to reduce the PCF of the structure. Therefore, the mathematic optimization problems of the hybrid multi-cell structures are:

Min { SEA(D , T ), PCF(D , T ) s. t. 80 mm D 120 mm 0.8 mm T 2.0 mm

R2 =

N i=1

f¯ )2

(fi

N i=1

(fi

N (f i=1 i ¯ 2

f)

fi ) 2 (34)

fi’

where, fi and is the value of numerical simulation and the approximation model output of the ith sample point, respectively; f¯ represents the average numerical simulation value of all sampling points; N is the number of the sample points. The values of RMSE can be obtained by normalizing the mean square error of the true value and the predicted value, and as a standard of acceptability when it is less than 0.2, which is calculated by the following formula:

(33)

RMSE =

N i=1

(fi N

fi ) 2

(35)

To accurately build surrogate models, 15 sample points are generated using OLHS for each hybrid structures with 2 design variables. Based on the simulation results, four hybrid multi-tube surrogate models (RSM2, KRG, RBF, and ANN) of SEA and PCF are established. In addition, 10 evaluation points are generated to evaluate the accuracy of the RSM2, KRG, RBF, and ANN models. The results of R2, RMSE are listed in Table 7. It is seen that the ANN model has the best accuracy for SEA of four structures, but accuracy of ANN for PCF is lower than RBF

5.3. Accuracy analysis of multiple surrogate models The surrogate models are widely used in the optimization process to describe the relationship between design variables and responses to reduce computational costs. The main approximate modeling techniques used in crashworthiness designs include response surface method (RSM) [35], Kriging approximation model (KRG) method [37,38], Radial Basis Function (RBF) neural networks [27] and Artificial Neural 127

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Fig. 20. PCF for four tubes with different D and d (T = 0.8 mm): (a) CS1; (b) CS2; (c) SC1; (d) SC2.

and RSM2. It is concluded from Table 7 that the RSM2 is the most accurate model of the CS1, CS2, and SC2 structures, and the RBF is the most accurate approximation model for SC1. Therefore, RSM2 is used to model CS1, CS2 and SC2, and RBF to SC1.

tubes are distributed in SEA and PCF spaces in a good manner. Obviously, the Pareto front of CS2 is in the lower right position, which indicates that CS2 has the highest SEA in the same PCF or the lowest PCF in the same SEA. In contrast, the Pareto front of SC1 is in the upper left position, which is the worst performer. The optimal SEA of the four hybrid structures can be obtained from Fig. 21 to quantitatively compare the crashworthiness of the four structures when the constraint PCF≤95 kN. Compared with CS1, SC1, and SC2 structures, the SEA of the CS2 structure is higher by 7.86%, 54.57%, and 17.94%, respectively.

5.4. Pareto fronts of multi-objective optimization The Archive-Based Micro Genetic Algorithm (AMGA) [42] is used to seek the multi-objective optimal solutions for the hybrid structures due to relatively less computational cost and faster convergence in solving crashworthiness problems. Fig. 21 presents the Pareto fronts obtained by the AMGA algorithm. The optimized Pareto fronts of the four hybrid Table 6 Functions of surrogate models [41]. Surrogate models

Approximation function yˆ (x )

RSM2

b0 + b1 x1 + b2 x2 + …+ bi xi+ bM + 1 x12 + bM + 2 x 22 + …+ b2M x 22M +

KRG

ˆ + r T (x ) R

1 (Y

Supplementary instruction i j bij xi yi

Sample Size

f ˆ)

ˆ = (FT R 1F ) FT R 1Y R (x i , x j ) =

RBF ANN

N i = 1 wi

(x

xi) +

(M + 1)(M + 2) 2

M j = 1 cj pj (x )

(r ) =

None Fitting indicators

m k=1 e

j i k x k xk

a2 + r 2 ; r = x

2

, where

Three-level feed forward network

128

k

xi ; a = 0.5

>0

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Table 7 Error analysis results of four approximate models. Structures

Response

CS1

SEA PCF SEA PCF SEA PCF SEA PCF

CS2 SC1 SC2

KRG

RBF 2

RSM2 2

RMSE

R

RMSE

R

0.041 0.051 0.048 0.036 0.048 0.051 0.042 0.043

0.987 0.971 0.977 0.990 0.986 0.978 0.980 0.980

0.015 0.015 0.058 0.024 0.036 0.022 0.040 0.033

0.997 0.998 0.966 0.993 0.986 0.996 0.983 0.990

ANN 2

RMSE

R

0.011 0.012 0.045 0.016 0.042 0.052 0.039 0.028

0.999 0.998 0.979 0.998 0.977 0.973 0.987 0.993

RMSE

R2

0.007 0.014 0.027 0.068 0.028 0.060 0.008 0.056

0.999 0.997 0.991 0.976 0.993 0.972 0.998 0.973

Table 8 The standardized results of NBI method for partial Pareto solutions. No.

D (mm)

T (mm)

SEA (kJ/kg)

PCF (kN)

NSEA

NPCF

R2

1 2 3 . . . 84 85 86 . . . 156 157 158

80.067 80.085 80.478 . . . 80.054 80.007 80.016 . . . 80.067 80.076 80.007

1.294 1.716 0.838 . . . 1.922 1.406 1.979 . . . 1.423 0.956 0.806

16.912 19.979 13.191 . . . 21.360 17.766 21.740 . . . 17.888 14.224 12.971

65.237 90.914 40.353 . . . 104.252 71.779 108.080 . . . 72.855 46.446 38.564

0.6420 0.3658 0.9771 . . . 0.2414 0.5651 0.2072 . . . 0.5541 0.8841 0.9969

0.5115 0.7975 0.2343 . . . 0.9461 0.5844 0.9887 . . . 0.5963 0.3022 0.2144

0.8680 0.8736 0.9008 . . . 0.8873 0.8657 0.8921 . . . 0.8663 0.8867 0.9008

( = 0.2 + 0.8 (

NSEA = 0.2 + 0.8 Fig. 21. Pareto fronts of four structures using AMGA.

NPCF

5.5. Selection of Pareto solution sets

Ni = µa + (µb

µa )

µa )

xi _ max xi x i _ max x i _ min x i x i _ min x i _ max x i _ min

PCFi_max PCFi PCFi_max PCFi_min

) )

(38)

The second step is to calculate the search radius. The search radius used in this article is R2, which is calculated by the search domain formula.

According to Fig. 21, the CS2 has the best crashworthiness for the four hybrid multi-cell structures. However, it is more difficult to select the most suitable design point from the optimized Pareto solution sets. Therefore, this section uses the normal boundary intersection (NBI) method to find the optimal solution in the obtained Pareto front. The first step is to standardize the Pareto solution. For the problem of finding the maximum or minimum value of the objective function, the dimensionless normalized values are calculated by the following formulas separately:

Ni = µa + (µb

SEAi_max SEAi SEAi_max SEAi_min

0.5

n

R2 =

(Ni

0.2)

(39)

i=1

where n is the number of variables in the objective function in multidimensional space; R2 is the Euclidean distance from the lower limit. The smallest solution of R2 is the optimal solution, which can be expressed as: 0.5

n

min R2 =

(36)

(Ni i=1

0.2)

(40)

The results of the calculations are shown in Table 8, and it can be seen that the 85th Pareto solution has the smallest R2; therefore, the 85th Pareto solution (shown in red in Table 8) is considered as the optimum design to coordinate these two goals, and the optimal parameter configuration of CS2 is D = 80 mm, d = 40 mm, and T = 1.406 mm. The comparative analysis of CS2 and several popular single crosssection multi-cell structures of the same mass are performed to evaluate the relative merits of the hybrid multi-cell design on the basis of the optimal design parameters of the CS2 structure. The histograms of SEA and PCF for numerical simulation of these structures are shown in Fig. 22. Firstly, the values of SEA and PCF in the numerical analysis of CS2 structure were 17.56 kJ/kg and 70.82 kN, and the errors of the optimization values were 1.24% and 1.35%, respectively, which proved the reliability of the optimization results. Secondly, it is found that the

(37)

where Ni is the ith objective function value after normalization; ua and ub are the lower and upper limit of the boundary of the coordinate axis, respectively. xi is the objective function value obtained by optimization; x i _ min, x i _ max are the minimum and maximum values of the ith objective function in the Pareto solution set, respectively. The standardized Pareto solution is distributed between ua and ub after normalization, and the standardized upper limit ub is set to 1.0 and the lower bound ua is set to 0.2 in order to make the solution point spread in a more concentrated manner. Then, the standardized objective functions can be calculated as follows, respectively.

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Fig. 22. Comparison of SEA and PCF values of different sections: (a) SEA; (b) PCF.

hybrid multi-cell design proposed in this study shows the higher energy absorption than typical square and circle multi-cell structures, and maximum improvement ratio of SEA reaches 70.13%. In addition, the SEA of CS2 also increases by 5.35% than the circle multi-cell structure. On the other hand, the PCF values for all multi-cell tubes of the same mass are very close. Therefore, it draws a conclusion that the hybrid multi-cell design absorbs energy more than wide circular or square multi-cell structures. In other words, the hybrid multi-cell structure design has great potential for use as an energy absorber.

Acknowledgments This work is supported by The National Natural Science Foundation of China (51675190), Program for New Century Excellent Talents in Fujian Province University, Promotion Program for Young and Middleaged Teacher in Science and Technology Research of Huaqiao University (ZQN-PY202). References [1] X. Zhang, H. Zhang, Energy absorption of multi-cell stub columns under axial compression, Thin-Walled Struct. 68 (2013) 156–163. [2] S. Chen, H. Yu, J. Fang, A novel multi-cell tubal structure with circular corners for crashworthiness, Thin-Walled Struct. 122 (2018) 329–343. [3] X. Deng, W. Liu, L. Jin, On the crashworthiness analysis and design of a lateral corrugated tube with a sinusoidal cross-section, Int. J. Mech. Sci. 141 (2018) 330–340. [4] M.S. Zahran, P. Xue, M.S. Esa, M.M. Abdelwahab, A novel tailor-made technique for enhancing the crashworthiness by multi-stage tubular square tubes, Thin-Walled Struct. 122 (2018) 64–82. [5] B.F. Xing, D.Y. Hu, Y.X. Sun, J.L. Yang, T.X. Yu, Effects of hinges and deployment angle on the energy absorption characteristics of a single cell in a deployable energy absorber, Thin-Walled Struct. 94 (2015) 107–119. [6] Z. Tang, S. Liu, Z. Zhang, Analysis of energy absorption characteristics of cylindrical multi-cell columns, Thin-Walled Struct. 62 (2013) 75–84. [7] Y. Zhang, X. Xu, G. Sun, X. Lai, Q. Li, Nondeterministic optimization of tapered sandwich column for crashworthiness, Thin-Walled Struct. 122 (2018) 193–207. [8] A.A. Nia, M. Parsapour, Comparative analysis of energy absorption capacity of simple and multi-cell thin-walled tubes with triangular, square, hexagonal and octagonal sections, Thin-Walled Struct. 74 (2014) 155–165. [9] X. Zhang, H. Zhang, Energy absorption of multi-cell stub columns under axial compression, Thin-Walled Struct. 68 (2013) 156–163. [10] Y. Zhang, X. Xu, J. Wang, T. Chen, C.H. Wang, Crushing analysis for novel bioinspired hierarchical circular structures subjected to axial load, Int. J. Mech. Sci. 140 (2018). [11] W. Li, Y. Luo, M. Li, F. Sun, H. Fan, A more weight-efficient hierarchical hexagonal multi-cell tubular absorber, Int. J. Mech. Sci. 140 (2018) 241–249. [12] Y. Luo, H. Fan, Investigation of lateral crushing behaviors of hierarchical quadrangular thin-walled tubular structures, Thin-Walled Struct. 125 (2018) 100–106. [13] X. Xu, Y. Zhang, J. Wang, F. Jiang, C.H. Wang, Crashworthiness design of novel hierarchical hexagonal columns, Compos. Struct. 194 (2018) 36–48. [14] Y. Zhang, J. Wang, C. Wang, Y. Zeng, T. Chen, Crashworthiness of bionic fractal hierarchical structures, Mater. Des. 158 (2018) 147–159. [15] L. Zhang, Z. Bai, F. Bai, Crashworthiness design for bio-inspired multi-cell tubes with quadrilateral, hexagonal and octagonal sections, Thin-Walled Struct. 122 (2018) 42–51. [16] K. Vinayagar, A.S. Kumar, Multi-response optimization of crashworthiness parameters of bi-tubular structures, Steel Compos. Struct. 23 (2017) 31–40. [17] G. Zhu, G. Sun, H. Yu, S. Li, Q. Li, Energy absorption of metal, composite and metal/ composite hybrid structures under oblique crushing loading, Int. J. Mech. Sci. 135 (2018) 458–483. [18] G. Zhu, G. Sun, Q. Liu, G. Li, Q. Li, On crushing characteristics of different

6. Conclusions Four hybrid multi-cell structures are proposed and analyzed in point of collision performance based on the experimental testing, theoretical prediction, and deterministic multi-objective design optimization, the following conclusions are drawn: 1) The hybrid structures with external circle sections (CS1, CS2) have better deformation modes than that of inner circle sections (SC1, SC2) by the experimental testing. The crush loads of finite element models of the hybrid structures are good agreement with the experimental results during the collapse process. 2) The theoretical models of MCF and SEA of the hybrid structures are established by using SSFE, which can well predict the crushing force and energy absorption of the hybrid tubes. 3) The RSM2 is most suitable approximate models for the CS1, CS2, and SC2, and RBF outperforms the other approximate technologies for SC1. CS2 structure has the best crashworthiness than other three hybrid structures according to deterministic multi-objective Pareto solutions. 4) The optimal solution of CS2 is obtained by the normal boundary intersection method, and the CS2 structure with optimal parameter offers greater energy absorption than typical multi-cell structures with the single cross-section of the same mass, indicating the excellent potential of hybrid design as an energy absorber design. Conflicts of interest The authors declare that we have no conflict of interest.

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