Enhancing material efficiency of energy absorbers through graded thickness structures

Thin-Walled Structures 97 (2015) 250–265

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Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws

Enhancing material efficiency of energy absorbers through graded thickness structures Fengxiang Xu a,b,n a b

Hubei Key Laboratory of Advanced Technology of Automotive Components, Wuhan University of Technology, Wuhan 430070, China Hubei Collaborative Innovation Center for Automotive Components Technology, Wuhan University of Technology, Wuhan 430070, China

art ic l e i nf o

a b s t r a c t

Article history: Received 14 May 2015 Received in revised form 16 September 2015 Accepted 22 September 2015 Available online 6 October 2015

At present, there has been constant aspiration of advanced thin-walled structures in vehicular industries for more efficient usage of materials to achieve much lighter weight and even higher energy absorption. In this paper, functionally graded thickness (FGT) tubes with a varying wall thickness are introduced and their energy-absorbing efficiency is enhanced. Apart from the geometrical parameters such as diameter and length, the gradient exponent that controls the variation of thickness distributions has also a significant effect on the increase in absorbed-energy. Numerical model is validated by performed crashing experiments of FGT tube. Parametric analysis demonstrates that the FGT column is superior to the uniform thickness (UT) column. Further, the multiobjective optimization (MOO) of FGT tubes is conducted for axial impacting by considering specific energy absorption (SEA) and crashing force efficiency (CFE) as objectives, and the diameter, initial length and gradient exponent of thickness variation as the design variables. The multiobjective particle swarm optimization algorithm (MOPSO) is applied to obtain the Pareto optimal solutions. In addition, a comparative study on different surrogate models, such as response surface method (RSM), Kriging method (KRM), and radial basis function (RBF), is also carried out to gain insights into their relative performance and features in computational modeling and design optimization. It is indicated that the performance of FGT tubes can be significantly improved by optimizing the geometrical parameters and gradient exponent. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Optimization design Energy absorption Thin-walled structures Graded thickness Tailor rolling blank (TRB)

1. Introduction Lured by significant advantages of efficient energy absorption and low cost of manufacturing, a range of thin-walled metal tubes have been widely utilized in transportation and defense industries as one of the most important energy absorbers in different ways [1]. Over the past decades, two critical characteristics of lightweight and crashworthiness have drawn increasing attentions through analytical, experimental and numerical approaches to understanding of collapse mode, peak force, mean force and the energy-absorption [2–11]. The axial crashing behavior of thinwalled structures has been a key topic with great interest for many researchers. The automobile body structures are largely composed of thinwalled structural parts, which are typically made by forming process of traditional metal sheets with uniform thickness. Crashworthiness of such sheet metal formed thin-walled structures has been of great n Correspondence address: Hubei Key Laboratory of Advanced Technology of Automotive Components, Wuhan University of Technology, Wuhan 430070, China. E-mail address: [email protected]

http://dx.doi.org/10.1016/j.tws.2015.09.020 0263-8231/& 2015 Elsevier Ltd. All rights reserved.

interests for improving the material utilization and vehicle safety. Exhaustive studies were performed for developing different hollow or foam-filled structural configurations [12]. For example, Tang et al. [13] proposed a cylindrical multi-cell column to improve energy absorption. Ghamarian et al. [14] compared the crashworthiness of end-capped cylindrical and conical tubes using experimental and numerical approaches to searching more efficient and lighter energy absorbers. Marzbanrad et al. [15] systematically studied square, circular, and elliptic tubes of steel and aluminum tubes with different geometric dimensions by using finite element simulation. Acar et al. [16] investigated the crashing behaviors of tapered tubes using multiobjective optimization. Zhang et al. [17] evaluated the energy absorption characteristics of regular polygonal and rhombi columns under quasi-static axial compression. Despite its significance, all those abovementioned thin-walled structures were made of uniform materials and/or the same wall thickness. The main drawbacks lie in that such structures may not exert their maximum capacities of crashworthiness. In other words, the tubes with uniform wall thickness may not necessarily make best use of materials for meeting the requirements of vehicular lightweight [18–20]. Therefore, there is an urgent demand

F. Xu / Thin-Walled Structures 97 (2015) 250–265

Rigid Wall

V=V0

t

O

D O

L

251

t

D

L

O

Added Mass

Fig. 1. Geometrical details of the circular column with uniform thickness used in axial crashing.

225 (Unit=mm)

50

200

75 200

Engineering stress (MPa)

20

12.5

R20

175 150 125 100 75 50 25

Tensile specimen-1 Tensile specimen-2

0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 Engineering strain Fig. 2. (a) Standard specimens specified in the ASTM; (b) Tensile coupon test specimen; (c) Engineering stress and engineering strain relationship of metal aluminum material.

55 50 Fmax

45

Crash force (kN)

40 35 30

Favg

25 20 15

EA

10 5 0

0

25

50

75 100 125 150 Displacement (mm)

175

200

Fig. 3. A typical relationship between force versus displacement of axial crashing behavior of circular tubes [35].

to develop new structural configurations with different material and/or thickness combinations for maximizing crashing capacities and material utilization. It has been proven that crashworthiness of non-uniform components could be improved by an optimal design of different materials and/or thicknesses [21]. Of various technologies to combine different materials or thickness, tailor-welded blank (TWB) structures have drawn major attentions over the recent years. They are fabricated by welding metal sheets with different

thicknesses and/or materials first and then formed to be the desired thin-walled structures. Thus, it provides a more flexible combination of different sheets and allows better utilizing materials for improving the crashworthiness characteristics. For this reason, such structures have been extensively adopted in vehicular floor component, B-pillar, front-end structure, and door inner panels etc. [22–26]. However, the main shortcoming of those welded blanks lies in that they need to combine the sheets with different thickness/materials, and the material properties in the welding zone can be rather different from those in the base materials, potentially causing stress concentration in the interfaces. To overcome such defects of TWB, a new rolling technology, namely tailor rolling blank (TRB), which leads to a continuous thickness variation in the workpiece, has been developed in recent years [27]. From the study by Yang et al. [27,28], a metal sheet with variations of thickness would be a more desirable because the technology not only uses material more efficiently, but also makes design more flexible. As such, thin-walled structures with varying sheet thicknesses can better meet different design requirements, thereby providing enhanced material/thickness utilization comparing with those made of traditional uniform sheets [29–31]. Besides that, such special structures enable to absorb energy in a more controlled manner during a crashing situation. To date, there have been some published reports available concerning TRB technology. For example, Urban et al. [32] developed a design tool by combining numerical simulation and optimization algorithm to improve the formability of TRB. Meyer et al. [33] used TRB to increase the maximum drawing depth compared with constant thickness. Zhang et al. [34] investigated the effects of the transition zone length, the blank thickness variation, friction coefficient and die clearance on the springback of TRB component. Nevertheless, those published works focused mainly on the design

252

F. Xu / Thin-Walled Structures 97 (2015) 250–265

Loading plate (impacting mass) Crash speed V0 Top end (ttop)

Crash speed V0

tmin

Direction of thickness grading ascending

x

(Top end)

L

Bottom end (tbot)

tmax

(Bottom end)

Fig. 4. Schematic showing thickness grading patterns in the axial direction.

1.0

1.0

0.8

0.8

n=0.1 n=0.2

0.6

Thickness ratio tmin/tmax

Thickness ratio t min/tmax

n=10

n=0.5 n=1

0.4

n=2 n=5

0.2

n=10

0.0 0.0

0.2

0.4 0.6 0.8 Normalized distance x/L

1.0

n=5 n=2

0.6 n=1 0.4 0.2

n=0.5 n=0.2 n=0.1

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Normalized distance x/L

Fig. 5. Variation of thickness vs. normalized distance: (a) ascending gradient pattern, and (b) descending gradient pattern.

of metal sheet forming processes. To the author's best knowledge, there have been rare reports available on crashworthiness design of thin-walled TRB structures to date. In order to make full use of TRB structures with functionally graded sheet thicknesses for automotive body shell components, it is essential to quantify their energy absorption characteristics in comparison with those well-studied uniform thickness tubes. In this paper, functionally graded thickness (FGT) circular columns with a defined thickness distribution along longitudinal direction are considered. It is of particular significance to seek the best possible thickness distribution for crashworthiness with different measures. Experiments from FGT circular tube are first performed to validate the finite element model (FEM). A comparative study on the different surrogate models, such as response surface method (RSM), Kriging method (KRM), and radial basis function (RBF), which have been widely used for a variety of crashworthiness designs of thin-walled structures with uniform wall thickness, is presented to gain insights into their relative performance and features for designing FGT structures. The multiobjective

optimization (MOO) of FGT circular tubes is carried out herein by considering specific energy absorption (SEA) and crashing force efficiency (CFE) as objectives and the tubal diameter D, initial length L and gradient exponent parameter n of wall thickness as the design variables. The multiobjective particle swarm optimization algorithm (MOPSO) is applied to obtain the Pareto solutions. The results yielded from the multiobjective optimization indicate that the FGT tube is superior to its uniform thickness counterparts in overall crashworthiness.

2. Material design and descriptions 2.1. Geometrical description and material property Axisymmetric and circular shapes provide the widest range of all choices for use as absorbing elements because of their favorable plastic behavior under axial forces, as well as their common occurrence as structural elements. Fig. 1 illustrates the schematic

F. Xu / Thin-Walled Structures 97 (2015) 250–265

253

Loading end Block (Master node: 140kg)

Velocity direction V0

Vel

oci

ty d

irec

ti o n

V0

Mass block (140kg)

Fix end

Loading end Fix end Fig. 6. Finite element model for dynamic impact simulation of FGT circular tubes: (a) FE model (2D); (b) 3D model (isometric view).

35

Dynamic crashing force of FGT tube (Experiment) Dynamic crashing force of FGT tube (Simulation) Mean crashing force of FGT tube (Experiment) Mean crashing force of FGT tube (Simulation)

30

Force (kN)

25 20 15 10 5 0

0

5

10 15 20 25 30 35 40 45 50 55 60 Displacement (mm)

Fig. 7. Comparison of numerical and experimental results from FGT thin-walled circular columns.

geometry of thin-walled circular tubes with uniform thickness. These specimens have a nominal circular cross-section with diameter D, total initial lengths L and wall thickness t. In this work, mechanical properties of the aluminum alloy (AA6061-T5) tubes are considered and the elastic modulus of this material is E¼ 68.2 GPa, density ρ ¼ 2700 kg/m3, Poisson ratio v ¼0.33. The material model is considered insensitive to strain rate and defined as non-linear isotropic work hardening in the plastic region. The material characteristics of the aluminum alloy material were obtained from quasi-static tensile tests (2 mm/min) on standard specimens, whose dimensions are specified in the ASTM (American Society for Testing and Materials) standard E8M-04 (Fig. 2(a) and (b)). The obtained stress–strain curve is plotted in Fig. 2(c).

2.2. Structural crashworthiness criteria The relationship of axial crashing force and displacement for a typical crash model is illustrated in Fig. 3. when the tubal structures are crashed at a constant velocity. It can be seen that the first peak force typically corresponds to the initial impact when the cylindrical tube behaves mainly elastically, follows by a rapid decrease in force magnitude as plastic folding takes a shape, and then fluctuates around an average value of the axial crashing force. There are several key indicators to evaluate crashworthiness of a structure [35–39]. Typically, energy absorption (EA), specific energy absorption (SEA), average crash force (Favg), and crash force efficiency (CFE), as given in Eqs. (1)–(4),respectively, are commonly adopted.

254

F. Xu / Thin-Walled Structures 97 (2015) 250–265

80

60 50

70 Peak crashing force (kN)

70

Crashing force (kN)

80

D/L=0.33 D/L=0.50 D/L=0.66 D/L=0.83 D/L=0.99

40 30 20 10 0

0

20

40

60

80

100

60 50 40 30 20

120

0.3

0.4

0.5

S p e c if ic e n e r g y a b s o r p tio n ( k J /k g )

3.00

Energy absorption (kJ)

0.7

0.8

0.9

1.0

8

3.25

2.75 2.50 2.25 2.00 1.75 1.50

0.6

Ratio D/L

Displacement (mm)

7 6 5 4 3

0.3

D/L=0.33

0.4

0.5

0.6 0.7 Ratio D/L

0.8

D/L=0.50

0.9

1.0

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Ratio D/L

D/L=0.66

D/L=0.83

D/L=0.99

Fig. 8. Comparisons of crashworthiness performance of UT circular tubes with different diameters D: (a) force versus displacement; (b) peak force; (c) energy absorption; (d) specific energy absorption and (e) final deformation modes.

EA =

∫0

xmax

F ( xL ) dxL

(1)

SEA = EA/mass

(2)

Favg = EA/xmax

(3)

CFE = EA/( Fmax xmax ) × 100% = Favg/Fmax × 100%

(4)

2.3. Functionally graded thickness (FGT) column In this paper, the tubes to be considered have varying thicknesses but the same nominal cross-sectional profile and fixed lengths. It is assumed that the circular tube is fabricated with variable thickness configuration related to the direction of axial impact velocity [40]. The key geometrical features of the column under consideration are shown in Fig. 4.

The idea of thickness distribution is originated from the density gradient of the functionally graded foam materials [41] and keeps the same as other definitions from us [42–47]. The top wall thickness ttop is chosen at the origin of grading axis, whilst the bottom wall thickness tbot corresponds to the farthest layer from the origin. Depending on the grading direction, i.e., ascending or descending, ttop/tbot is assumed to be the values of tmin/tmax or tmax/tmin, respectively. In the axial grading case, the thickness gradient function tf(x, n) can be defined through the following power law equation: n ⎧ for an ascending pattern ⎪ tmin + ( tmax − tmin ) (x / L ) t f (x, n) = ⎨ n ⎪ ⎩ tmax − ( tmax − tmin ) (x/L ) for a descending pattern (5)

where x is the distance from the top of the column, n is the grading exponent that governs the variation of wall thickness and assumed to vary between 0 and 10, and L represents the total length of the tube. Thus, the wall thickness increases along the length with a

F. Xu / Thin-Walled Structures 97 (2015) 250–265

be calculated as follows,

2.50

E q u iv a le n t w a ll th ic k n e s s o f U T tu b e s

255

Δt =1.0 mm

2.25

Δt =1.5 mm Δt =2.0 mm

2.00

NS

tavg =

NS

∑ ( ti L e )/( NS L e ) = ∑ ti/NS i=1

i=1

(6)

1.75

where NS denotes the total number of layers of FGT thin-walled circular tube, and ti is the thickness of the ith layer.

1.50

3.2. Validation of the numerical model

1.25

Before numerical analysis is carried out to investigate the energy absorption of FGT circular tube, relevant experiments must be conducted to validate the modeling accuracy in terms of force– displacement curves and collapse modes. Due to limitation of manufacturing technology, those tubes are fabricated by a cutting operation from the counterpart with uniform thickness. The thickness of the crashing (top) end has minimum value tmin and that of the fixed (bottom) end is defined as maximum value tmax. In the experiment, the FGT tube are set as tmin ¼1.2 mm, tmax ¼1.5 mm, D¼ 45 mm, L ¼120 mm and n¼ 1.0 which indicates a linear thickness distribution over the length. The crashing force vs. displacement relationship and the crashing modes obtained from numerical simulation and experimental tests are compared in Fig. 7. Good agreement between numerical analysis and experimental results can be observed, which provides us with confidence to predict the collapse shape and behavior of tubes with sufficient accuracy. For example, the peak crashing forces from simulation and experimental results are 31.5 kN and 30.1 kN, respectively. The relative error is kept at about 5%, which is acceptable in actual engineering. In sum, the established numerical model could capture some important crashing information and is enough to perform the further parametric analysis and optimal design with different size parameters and graded thickness variations. Generally, there are three typical crashing deformation patterns, i.e., axisymmetric mode, non-axisymmetric mode and column buckling [49,50]. Such pattern difference depends mainly on the parametrical changes (specifically diameter D, initial length L and wall thickness t). It is well-known that the slenderness ratios (D/t and D/L) are of primary effect on the buckling pattern of tubes [51,52]. The linear elastic bifurcation buckling capacity of thin cylindrical shells subjected to uniform axial pressure can be predicted by the classical theory of elasticity as [53]

1.00 0.75 0.50

0

0.2

0.4

0.6

0.8

1.0

2.0

4.0

6.0

8.0 10.0

Exponent n

Fig. 9. Relationship between equivalent thickness of UT tube and exponent n of FGT tube.

gradient function changing from convexity to concavity correspondingly when the n value varies from less than 1 to greater than 1, as shown in Fig. 5(a). On the contrary, the wall thickness decreases along the length direction and an opposite tendency can be observed as shown in Fig. 5(b). Generally, the peak crashing force depends on how easy the plastic folding takes place. To ease the formation of such an ideal deformation mode, we consider only the ascending pattern here, which places less material and lower structural stiffness in the incidental end of the tube, to perform the crashworthiness design for the FGT columns.

3. Numerical analysis 3.1. Finite element modeling A circular tube with an added mass colliding with a rigid wall with the initial velocity V0 is studied here by using the finite element (FE) method. An explicit FE code, LS-DYNA 3D, is used to carry out the numerical simulations of axial crashing of cylindrical tubes under impact loading. The FE model used to simulate the crashing process was based upon the experimental validation. Fig. 6 shows the FE mesh using the 4-node shell elements suitable for large deformation analysis with 6 degrees of freedom at each node [39,48]. Five integration points are used across the shell thickness to model bending. In this model, the bottom end is fixed whilst a rigid mass-block of 95 kg was connected to the free loading end through a master node. In the crash scenario, the mass block is assigned an initial velocity V0 of 4.4 m/s through the master node. The surface-to-surface contact between the mass block surface and the circular tubes was modeled with a friction coefficient of 0.2, while self-contact with a friction coefficient equal to 0.2 is defined for the inner and the outer surfaces of circular tubular. To simplify the modeling, we adopted discrete layered thickness configuration with the same depth Le as shown in Fig. 6. Herein, the mesh size of 2.0  2.0 mm2 is enough and suitable for simulating the graded tubular structures with multi-layer thickness, which has been performed by author [42]. For the layered FGT structures, the depth of each graded layer is assumed to be the same and defined as Le. The equivalent thickness of uniform thickness (UT) column that has the same mass as the FGT tube can

Ncr =

⎛ t2 ⎞ ⎜ ⎟ ⎝ D⎠

2E

(

)

3 1 − v2

(7)

where E is the Young's modulus, v the Poisson's ratio, t the wall thickness, and D the diameter of the shell. For a cylindrical shell uniformly compressed in the axial direction, buckling symmetrically with respect to the axis of the shell may occur at a certain value of the crashing load. Thus, the length of half-waves into which the cylinder buckles is assumed to express as follows,

L D2 t 2 = π4 i 48 1 − v2

(

)

(8)

where L is the length of the circular tube and i the integer. Therefore, the geometric parameters of the circular shell are of decisive influence in the dynamic response of the cylindrical tubes. For this reason, those dimensional parameters will be defined as the design variables in the following optimization. 3.3. Parametric analyses Depending on geometrical parameters such as the ratios of D/t

256

F. Xu / Thin-Walled Structures 97 (2015) 250–265

Table 1 Characteristics for the UT and FGT columns at different thickness intervals and gradient exponents. n

Mass (kg)

Fmax (kN)a

UT/FGT

UT

FGT

IPb (%)

UT

FGT

IP (%)

UT

54.29

20.63

 62.00

3.11

3.09

 0.64

6.51

EA (kJ)

SEA (kJ/kg) FGT

Thickness interval Δ t¼ 1.0 0.2 0.478 0.4

0.455

0.6

0.440

0.8

0.438

1.0

0.424

2.0

0.413

4.0

0.380

6.0

0.353

8.0

0.342

10.0

0.336

6.46

 60.79

18.83

 1.30

3.03

 1.33

6.75

3.03

3.04

0.33

6.89

0.29 6.89

 59.73

19.12

3.01

3.02

0.33

6.87

0.29 7.10

45.43

 55.38

20.27

3.08

 2.27

3.01

 2.20

7.26 7.26

44.27

 38.99

27.01

3.09

 2.91

3.00

 2.94

7.48 7.82

40.31

 19.82

32.32

2.92

2.97

1.71

7.68

1.82 8.41

37.19

 12.64

32.49

2.82

2.97

5.32

7.99

5.26 8.65

35.75

 8.95

32.55

2.70

2.96

9.63

7.89

9.63 8.81

34.93

b

3.07

6.91

47.48

a

 61.83

19.39

48.02

Thickness interval Δ t¼ 2.0 0.2 0.440 48.02 0.4 0.394 42.01 0.6 0.359 37.92 0.8 0.332 34.46 1.0 0.310 31.58 2.0 0.243 21.64 4.0 0.190 13.50 6.0 0.168 9.36 8.0 0.155 8.06 10.0 0.147 8.49

 0.77 6.66

50.80

Thickness interval Δ t¼ 1.5 0.2 0.459 51.39 0.4 0.425 45.50 0.6 0.398 42.55 0.8 0.378 40.11 1.0 0.361 38.17 2.0 0.312 31.82 4.0 0.272 26.15 6.0 0.255 23.52 8.0 0.245 21.99 10.0 0.239 20.96

IP (%)

 6.84

32.54

2.67

2.96

10.86

7.95

10.82

0.90 14.15

 72.47

3.06

3.05

 0.33

6.67

6.73

12.30

 72.97

3.08

2.96

 3.90

7.25

7.13

10.96

 74.24

3.01

3.03

0.66

7.56

7.64

10.56

 73.67

2.93

3.03

3.41

7.75

7.99

10.52

 72.44

2.91

3.02

3.78

8.06

8.34

10.48

 67.06

2.56

2.95

15.23

8.21

9.62

12.50

 52.20

2.75

3.02

9.82

10.11

10.92

14.66

 37.67

2.81

2.99

6.41

11.02

11.65

15.63

 28.92

2.81

2.98

6.05

11.47

12.08

15.79

 24.67

2.84

2.98

4.93

11.88

12.38

 1.66 1.06 3.10 3.47 17.17 8.01 5.72 5.32 4.21

4.08

 91.50

3.03

3.03

0

6.89

6.89

0

4.00

 90.48

3.06

3.04

 0.65

7.77

7.72

 0.64

3.41

 91.01

2.87

3.01

4.88

7.99

8.38

4.88

2.77

 91.96

2.61

3.02

15.71

7.86

9.10

15.78

2.55

 91.93

2.52

2.96

17.46

8.13

9.54

17.34

2.68

 87.61

2.84

2.92

2.82

11.69

12.02

2.82

2.92

 78.37

2.78

2.96

6.47

14.63

15.58

6.49

2.68

 71.37

2.86

3.01

5.24

17.02

17.92

5.29

2.60

 67.74

2.89

2.98

3.11

18.65

19.23

3.11

2.66

 68.67

2.99

2.81

 6.02

20.34

19.12

 6.00

Fmax herein indicates the initial peak crashing force; IP indicates the improvement proportion.

F. Xu / Thin-Walled Structures 97 (2015) 250–265

n=0.4

n=4.0

specific energy absorption, are presented in Fig. 8(a)–(d). From those plots, the ratio of diameter to initial length has a fairly significant effect on the crashing performance. With the increase of the ratio, the peak crashing force increases and specific energy absorption decreases, monotonically. Fig. 8(e) summarizes the deformation modes of such five tubes with different ratios D/L. Obviously, the deformation appears to be more typical folding pattern with an increase in diameter, especially when the diameter is closer to the initial length.

Δ t=1.0

Δ t=1.5

Δ t=2.0 Fig. 10. Final deformation modes of UT and FGT tubes (Note that the comparative figures of exponent n¼ 0.4 are presented at the time of 40 ms and those of n¼ 4.0 are at 25 ms).

40 36

Maximum Crashing force

32

A

Crashing force (kN)

28

Initial maximum force

24 B

20 16 12 8 4 0 0

20

40

60

80

257

100

120

Displacement (mm) Fig. 11. Force versus displacement of FGT tubes with exponent n¼ 0.4.

(diameter/thickness) and D/L (diameter/length), there are a variety of possible collapse modes of circular tubes. For the FGT tubes, the thickness can distribute non-uniformly along the length direction. Thus, it is difficult to directly analyze the effect of ratio of D/t on the collapse modes. When the gradient parameter n is given, the thickness distribution of the special thin-walled structure is determined by the maximum (bottom) and minimum (top) thicknesses in Fig. 4. Therefore, effect factors of crashworthiness of the graded thickness tubes include gradient exponent n and thickness interval ( Δt¼tbottom ttop) herein. 3.3.1. Effect of different slenderness D/L of the UT circular tube To better investigate and compare the influences of different diameter size, five different slenderness ratios D/L of 0.33, 0.50, 0.66, 0.83 and 0.99 are considered herein. Their effects on dynamic crashing characteristics, including initial peak crashing force and

3.3.2. Effect of wall thickness distributions of the FGT circular tube For the FGT tube, the thickness distribution is defined according to a power law formula (Eq. (5)). Only the minimum and maximum thicknesses are used to determine the thickness distribution of the whole structure for a given gradient parameter n. In this section, different gradient intervals of wall thickness ( Δ t¼tbottom ttop) are explored for a given slenderness ratio D/L¼ 0.5. The value of ttop is varied with 0.5 mm, 1.0 mm, and 1.5 mm in order to quantify the effect of different thickness intervals on peak crashing force and specific energy absorption. The gradient exponent n is also varied from 0 to 10, in which tbot was fixed at 2.5 mm. It must be noted that the corresponding wall thickness of the UT tube has the same weight as the graded thickness tube in different parameter n according to Eq. (6). In other words, the wall thickness of the UT tubes can be expressed as functions of n hereafter to make them comparable. The relationship between the equivalent thickness of UT tube and exponent n is illustrated in Fig. 9. Note that the equivalent thickness of UT tubes follows a declining trend with the increase in exponent n. From Fig. 9, it is seen that for a larger thickness interval (e.g., Δt¼ 2.0 mm, the equivalent thickness is much smaller for the same exponent n). Variations in peak crashing force (Fmax) and specific energy absorption (SEA) for different Δt and n are summarized in Table. 1. It is noted that Fmax and energy absorption (EA) of UT and FGT columns decrease whilst SEA increases with increase in exponent n. It is also noted that SEA of FGT column is generally higher than that of UT column at the same exponent n. This feature demonstrates that the FGT column could be superior to the UT column. In addition, FGT with a larger thickness range is more effective in minimizing Fmax for 0 ≤ n ≤ 1; as well as in maximizing SEA for 1 on ≤ 10. Fig. 10 depicts the deformation modes of the FGT column and the corresponding UT column with the same weight at different thickness intervals and exponents n. It can be seen that the deformations of the FGT columns seem to be more stable and advantageous over their uniform counterparts especially when the thickness interval is much larger.

4. Optimization design 4.1. Optimization techniques 4.1.1. Definition of optimization problem Design optimization is further applied for the crashworthiness criteria in the FGT columns in this section. Let us first look at the historic plot of force versus displacement of the FGT tubes again before defining the optimization problem. From Fig. 11, the maximum crashing force (point A) over the whole process and initial maximum crashing force (point B) could have a big difference. At the final crashing stage, more reduction in the maximum crashing force does not necessarily lead to more reduction in the energy absorption. But an overly high maximum crashing force could often associate with severe injury or even death of occupant. To balance them, the crashing force efficiency (CFE) (see Eq. (4)) and

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Table 2 Design matrix and responses for exponent 0 ≤ n ≤ 1. No.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Design variables

Indicators of dynamic response

D (mm)

L (mm)

n

EA (kJ)

SEA (kJ/kg)

Fmax (kN)

Favg (kN)

CFE (%)

60.34 96.55 189.66 65.52 143.10 127.59 148.28 55.17 117.24 70.69 200.00 153.45 122.41 168.97 194.83 86.21 81.03 179.31 158.62 50.00 112.07 75.86 132.76 184.48 91.38 163.79 137.93 174.14 101.72 106.90

131.034 389.655 100.000 327.586 400.000 110.345 265.517 193.103 306.897 348.276 141.379 296.552 317.241 213.793 120.69 224.138 337.931 162.069 286.207 358.621 244.828 203.448 368.966 275.862 379.310 234.483 151.724 172.414 182.759 255.172

0.00 0.03 0.06 0.10 0.13 0.17 0.20 0.24 0.27 0.31 0.34 0.37 0.41 0.44 0.48 0.51 0.55 0.58 0.62 0.65 0.69 0.72 0.75 0.79 0.82 0.86 0.89 0.93 0.96 1.00

2.99 3.18 3.32 3.09 3.17 3.17 3.03 3.00 3.03 2.94 3.20 3.00 3.07 3.12 3.16 3.03 2.94 3.04 3.03 2.64 3.03 3.05 3.00 3.06 2.88 3.02 3.04 2.99 3.03 3.02

13.907 3.136 6.561 5.450 2.116 8.685 2.991 11.029 3.326 4.750 4.539 2.662 3.211 3.533 5.524 6.488 4.468 4.380 2.808 6.226 4.690 8.425 2.627 2.591 3.596 3.416 6.320 4.359 7.163 4.887

93.88 57.90 80.02 42.12 73.84 62.75 51.01 29.43 38.32 31.83 63.48 39.35 44.29 47.54 57.89 29.91 28.53 47.16 43.29 19.66 34.41 34.51 36.62 41.85 30.26 37.89 46.51 49.41 40.18 31.45

24.94 26.32 42.69 19.85 28.90 32.68 28.95 19.16 24.11 18.36 37.08 27.42 23.55 30.09 37.22 21.34 18.31 30.58 25.88 14.16 22.20 20.01 20.71 26.18 17.19 26.09 26.32 27.98 22.63 20.40

26.57 45.45 53.35 47.11 39.14 52.08 56.75 65.11 62.90 57.68 58.41 69.69 53.17 63.29 64.31 71.33 64.20 64.84 59.78 72.01 64.51 58.00 56.56 62.55 56.80 68.85 56.58 56.63 56.33 64.87

Table 3 Design matrix and responses for exponent 1 on ≤ 10. No.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Design variables

Indicators of dynamic response

D (mm)

L (mm)

n

EA (kJ)

SEA (kJ/kg)

Fmax (kN)

Favg (kN)

CFE (%)

50.00 194.83 65.52 189.66 122.41 91.38 179.31 200.00 70.69 96.55 81.03 148.28 75.86 184.48 153.45 106.90 143.10 60.34 137.93 86.21 163.79 101.72 168.97 127.59 55.17 112.07 117.24 158.62 132.76 174.14

265.517 172.414 348.276 389.655 203.448 131.034 337.931 162.069 379.31 213.793 244.828 193.103 255.172 368.966 286.207 234.483 141.379 400.000 327.586 110.345 120.690 296.552 100.000 317.241 275.862 358.621 224.138 182.759 151.724 306.897

1.00 1.31 1.62 1.93 2.24 2.55 2.86 3.17 3.48 3.79 4.10 4.41 4.72 5.03 5.34 5.66 5.97 6.28 6.59 6.90 7.21 7.52 7.83 8.14 8.45 8.76 9.07 9.38 9.69 10.0

2.68 3.02 2.54 3.03 3.03 3.03 3.01 3.03 2.34 3.03 2.83 3.02 2.72 2.94 2.93 2.96 3.05 2.10 2.84 2.84 3.02 2.68 3.06 2.76 2.30 2.58 3.00 3.07 3.03 2.96

8.904 4.081 5.163 1.939 5.838 12.267 2.437 4.633 4.358 7.372 7.219 5.364 7.177 2.216 3.435 6.103 7.821 4.526 3.279 15.604 8.011 4.669 9.533 3.598 7.986 3.399 6.048 5.623 7.995 2.948

21.45 41.88 27.52 68.23 32.24 117.51 55.94 51.47 30.56 41.16 31.52 39.66 30.53 63.70 53.78 38.08 80.00 25.95 52.87 455.50 103.79 37.29 135.69 50.72 23.99 46.47 39.79 44.46 80.06 59.93

14.47 27.70 13.16 20.99 21.70 25.09 20.04 25.87 11.96 18.72 15.74 21.90 14.66 18.69 18.27 17.60 24.83 10.22 16.54 27.27 28.18 14.74 32.63 15.76 11.30 14.17 18.25 21.97 22.95 19.03

67.47 66.16 47.81 30.76 67.31 21.35 35.82 50.26 39.14 45.47 49.94 55.21 48.02 29.34 33.97 46.23 31.03 39.39 31.29 5.99 27.15 39.52 24.05 31.08 47.10 30.50 45.85 49.42 28.66 31.75

F. Xu / Thin-Walled Structures 97 (2015) 250–265

259

Sampling points Validation points

Sampling points Validation points 10

0.8

Thickness exponent n

Thickness exponent n

1

0.6 0.4 0.2

8 6 4 2

0 400

400

Init 300 ial le n gth L

200 200 (m m)

100

50

Ini 300 tial le n gth L

150 ) 100 D (mm r te e Diam

200 150 200 (m m)

100 100

50

eter D Diam

(mm)

Fig. 12. Sampling points (30) and validation points (20): (a) 0 ≤ n ≤ 1; (b) 1 on ≤ 10.

Table 4 Functions of surrogate models and error metric for model accuracy. Descriptions of surrogate models Model RSM KRM RBF

Indicators Metrics MSE RMSE

Approximate function y˜RSM (x) = α1ϕ1 (x) + α 2 ϕ2 (x) + ... + αn ϕn (x) y˜ (x) = ^ β + rT (x) R−1(Y − f^ β)

Correlation function None

y˜RBF (x) = ∑iN= 1 wi ϕ (‖x − xi ‖) + ∑M j = 1 Cj pj (x)

ϕ (r ) = a 2 + r 2 r = ‖x − xi ‖, a = 0.5

KRM

Formulations

⌢2 ∑m i = 1 (yi − yi ) / m ⌢2 ∑m i = 1 (yi − yi ) / m

R2

1−

RAAE

∑m i=1

RMAE

−θk x i − x j k k

2

β = (FTR−1F)−1FTR−1Y ; R (xi , x j ) = ∏km= 1 e

, where θk > 0

Features [54] Mean square error (MSE): measure the average of the error between the predicted and true values. Root mean square error (RMSE): a better measure than MSE.

¯ 2 −⌢ yi )2/∑m i = 1 (yi − y )

Usually correlated with MSE. A global error measurement.

¯ yi − ⌢ yi /∑m i = 1 yi − y ¯ max { y1 − ⌢ y1 , ⋯ , ym − ⌢ ym }/∑m i = 1 yi − y / m

Usually correlated with MSE. A global error measurement.

∑m i = 1 (yi

A relative error measurement in a local region.

specific energy absorption (SEA) are considered as design objectives. The diameter D, initial length L and thickness gradient n are taken as the design variables.

⎧ Max [SEA (D, L, n), CFE (D, L, n)] ⎪ s . t . 50mm ≤ D ≤ 200mm ⎪ ⎪ ⎨ 100mm ≤ L ≤ 400mm ⎪ ⎪ { case1: 0 ≤ n ≤ 1 ⎪ ⎩ case2: 1 < n ≤ 10

(9)

4.1.2. Surrogate models To formulate the objective functions, some sample points are needed to explore the design space. In this study, 30 sample points are generated in a random but homogeneous fashion inside the design domain by using Latin hypercube sampling (LHS). The range of wall thickness is assumed as 1.6 mm (i.e. ttop ¼0.6 mm; tbottom ¼2.2 mm) to formulate the surrogate models. In the ascending thickness distributions seen in Fig. 5(a), the wall thickness increases along the length with a gradient function changing from convexity to concavity when the n value varies from less than 1 to greater than 1. It is fairly difficult to construct a unified surrogate model for the whole domain of 0≤ n≤ 10 for capturing the detailed

nonlinear responses in such a non-uniform range. Thus, we formulate the surrogate models in two cases: 0 ≤ n ≤ 1 and 1 on ≤ 10 for multiobjective optimization. The design matrix and responses at those sampling points with different exponents in 0 ≤ n ≤ 1 and 1 on ≤ 10 are summarized in Tables 2 and 3 respectively. To validate the surrogate models, 20 extra validation points are selected uniformly inside the design domain (Fig. 12). Without knowing much of a design problem which involved a new design variable of thickness gradient exponent, it would be interesting to test one or more surrogate models before deciding which performs best. Several most typical functions of surrogate models, include response surface method (RSM), Kriging model (KRM), and radial basis function (RBF) are considered herein. As summarized in Table 4, three commonly used fitting indicators are used, in which ⌢ yi and y¯ are the corresponding predicted (or surrogate) and mean values for the exact function value yi at each checking point i, m represents the number of those checking points. 4.2. Results and discussion For its simplicity in implementation and high computational

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F. Xu / Thin-Walled Structures 97 (2015) 250–265

efficiency, particle swarm optimization has been used in a range of applications [11,55–59]. It is a population-based search procedure where individuals (called particles) continuously change position (called state) within the search area. In other words, these particles go around in the design space to look for the best position. The best position encountered by a particle and its neighbors along with the current velocity and inertia are used to decide the next better position of the particle. This algorithm is chosen to optimize

geometric parameters of circular columns in this paper. 4.2.1. Error analysis of different surrogate models The 3D contours of SEA and CFE are plotted in Fig. 13 to gain more insights into those three different surrogate models constructed. In each figure, the exponent n is set as 0.4 and 4.0, respectively. It can be observed that the contours with RSM, KRM and RBF models are similar and change almost monotonically

70 60 50

15

CFE (%)

SEA (kJ/kg)

20

10

40 30 20

5

10

400

0 400

Ini 300 tIa l le ngt hL

50 200 (m m)

Ini 300 tial len gth L

100 150 100 200

) r D (mm Diamete

25

150 100 200

) r D (mm Diamete

60 40

15

CFE (%)

SEA (kJ/kg)

200 (m m)

80

20

10

20 0 -20

5

-40 -60 400

0 400

Ini tial 300 len gth L

50 200 (m m)

Ini tial 300 len gth L

100 150 100 200

m)

r D (m Diamete

25

80

20

60

15

CFE (%)

SEA (kJ/kg)

50 100

10

-40 400

200 (m m)

150 100 200

) r D (mm Diamete

) r D (mm Diamete

0

0 400 50

150 100 200

20

-20

100

200 (m m)

40

5

Ini tial 300 len gth L

50 100

Ini tial 300 len gth L

50 200 (m m)

100 150 100 200

) r D (mm Diamete

Fig. 13. Surface plots of SEA and CFE responses for the three different surrogate models: (a) RSM model; (b) KRM model; (c) RBF model.

F. Xu / Thin-Walled Structures 97 (2015) 250–265

especially for SEA. However, some difference is seen in the CFE objective at the exponent n ¼0.4. It implies that optimal results may not be a unique when the different surrogate models are used. The comparisons of modeling errors are given in Fig. 14. It can be clearly seen that in general these three models approximate the two different responses (SEA and CFE) rather well and each is considered to have sufficient accuracy (e.g., all the R2 values are very close to 1.0). Specifically, the RSM models perform the best for both SEA and CFE with exponent 0 ≤ n ≤ 1, while the RBF models perform the best in 1 on≤ 10. In fact, each surrogate model has its cons and pros in practical applications [11,60]. Therefore, a satisfactory surface contour or error estimate may not necessarily indicate a best surrogate for practical applications because optimization process typically depends on the gradient information when searching for an optimum solution. Thus, all of these models are used for the optimization as defined in Eq. (9), and then are compared to provide the best solutions. 4.2.2. Optimization results The optimization problem involves two different design objectives and it is very difficult to achieve them simultaneously. In other words, some trade-off must be made in a form of Pareto set. The MOPSO Pareto optima of the FGT columns with the 10, 30, 50, 80, 100 and 200 generations are respectively plotted in Fig. 15

261

based on different surrogate models, which indicate that the 100 generations converged fairly stable and is considered adequate.

4.2.2.1. Case 1: Single objective optimization. The optimization determines the optimal parameter D, L and gradient exponent n to either maximize SEA, or maximize CFE of the FGT column. In order to identify the ideal optima, two single objective optimizations are also conducted as summarized in Table 5. It is noted that those single idealized optima well correspond to the special points which lie at each end of the Pareto fronts as in Fig. 15. For example, to maximize SEA in the region of 0 ≤ n ≤ 1, almost the same optimal parameters (D ¼50.00 mm, L ¼100.00 mm, n ¼0) can be obtained from the different surrogate models, which shows that lower limit of design variables is the best. Such optimal dimensions can be also generated from the models in the region of 1 on ≤ 10. For maximizing CFE, nevertheless, the optimum exponent n of the FGT tube in 1 on≤ 10 are set as 1.0 for the different surrogate models, which indicates that the thickness with linear distribution along the axial coordinate (Eq. (5)) is the best graded design formation. It is found that the most accurate surrogate models generally yield the optimum for the objective response for the single objective optimization. In addition, the optimum for exponent 1 ≤ n ≤ 10 seems to be better than that for exponent 0 ≤ n ≤ 1 of the objective either SEA or CFE.

Fig. 14. Error comparison of the different surrogate models: (a) Errors for SEA of FGT tubes with exponent 0 ≤ n ≤ 1; (b) Errors for CFE of FGT tubes with exponent 0 ≤ n ≤ 1; (c) Errors for SEA of FGT tubes with exponent 1o n≤ 10; (d) Errors for CFE of FGT tubes with exponent 1o n≤ 10. (Note: SEA and CFE values at the validation points have been ranked according to the ascending sequence).

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F. Xu / Thin-Walled Structures 97 (2015) 250–265

4.2.2.2. Case 2: Multiobjective optimization. While the single objective optimization can generate some meaningful results, it is insufficient to simultaneously optimize multiobjective and explore their interactions. The corresponding design variables of optimal solutions using different surrogate models in the multiobjective optimization are plotted in Fig. 16. It appears different surrogate models generate different optimal parameters. Nevertheless, some close solutions from these different models can be also observed (e.g., in the circled small region in Fig. 16). For example, it is

evident that for n ∈ [0, 1], the optima concentrate in a vertical line (i.e., D¼50 mm, L¼100 mm) in Fig. 16(a). Similarity, the solutions focus on the zones enveloped by D ¼50 mm and L ¼ 240 for n ∈ [1,10], as in Fig. 16(b). Some chosen optimal solutions from the Pareto fronts which meet some constraints and corresponding design variables are listed in Tables 6 and 7. From both tables, designers may choose the optimum solutions according to different emphases. Take the exponent 0 ≤ n ≤ 1 an example (Table 6), if the designer wishes to

Fig. 15. Pareto fronts of the FGT tubes with different surrogate models and generations: (a)–(c) 0 ≤ n ≤ 1; (d)–(f) 1o n ≤ 10.

F. Xu / Thin-Walled Structures 97 (2015) 250–265

Table 5 Ideal optimums of the two single objective functions for FGT tube with different surrogate models. Model

Single objective

Optimal parameters

Objectives

D (mm)

L (mm)

n

SEA (kJ/kg)

CFE (%)

0≤n≤1 RSM SEA (D, L, n) CFE (D, L, n) KRM SEA (D, L, n) CFE (D, L, n) RBF SEA (D, L, n) CFE (D, L, n)

50.00 50.00 50.00 50.00 51.76 81.68

100.00 299.77 117.47 100.00 100.00 228.84

0 0.7343 0 0.5120 0 0.5159

16.86 7.38 14.00 13.11 13.98 6.74

25.29 70.41 37.61 68.68 29.58 70.39

1 ≤ n ≤ 10 RSM SEA (D, L, n) CFE (D, L, n) KRM SEA (D, L, n) CFE (D, L, n) RBF SEA (D, L, n) CFE (D, L, n)

50.00 195.29 77.53 135.84 50.00 124.97

100.00 203.27 100.00 211.15 100.00 255.02

3.056 1.000 5.662 1.000 10.00 1.000

19.44 2.85 15.45 5.07 19.49 4.93

9.27 71.79 6.77 76.72 7.06 77.14

This paper provided a parametric study and multiobjective optimization of the circular columns with functionally-graded thickness (FGT). The gradient exponent n controls the variation pattern of thickness distributions and was found to have significant effect on crashworthiness. Some important conclusions are summarized as follows: (1) From parametric analysis, it was observed that the deformation modes were more desirable with an increase in diameter especially when the diameter and initial length are nearly equal. The work highlighted the fact that knowing the influence of gradient exponent on the crashing performance is of considerable importance. The comparison results exhibited that the FGT column is superior to the UT column. (2) A comparative study on three surrogate models such as RSM, KRM, and RBF was also conducted. For three design variables of the FGT tubes, it can be seen that the smallest diameter are more appropriate for crashworthiness; the linear thickness distribution performed best; and the optimal initial length is remarkably different when different surrogate models were adopted. (3) For the CFE and SEA objectives, optimal results or the improved rates were different for those three different surrogate models. This comparative study showed that the direct modeling accuracy of different surrogate models may not necessarily guarantee the best optimum and some comparison may be needed. The obtained results suggest that the thin-walled structures with variable thickness may become appropriate structures for innovative applications that require lightweight and high energy absorption. In addition, more experimental tests of FGT tubes with other thickness distributions should be further performed to compare the deformation patterns and confirm the feasible of the advanced energy-absorbed structures.

RSM KRM RBF

RSM KRM RBF

10 8 Exponent n

0.8

Exponent n

may not necessarily generate the optimum solutions for a specific objective function with the multiobjective optimization design. Based on this reason, simultaneous tests of multiple surrogate models are important to find the real optimum design, especially for computationally expensive simulation such as the design of FGT thin-walled structures.

5. Remarkable conclusions

emphasize the SEA, Solution no. 4, 7 and 5 for three surrogate models, respectively, can be a choice. If the designer would like to pay more attention on the CFE, Solution no. 5, 1, and 2 may be better, respectively. In addition, it can be seen from Table 6 that the FGT tubes with diameter D¼ 50 mm that reaches the smallest are more appropriate regardless of thickness distributions. From Table 7, it seems that the linear thickness distribution in the axial coordinate is best. Interestingly, the initial length L remarkably differs using different surrogate models. For the CFE objective with 0 ≤ n ≤ 1 (Table 6), optimal results are slightly smaller than the initial solutions using the RSM and RBF models, however, the KRM model improves by 15.6% (solution no. 1 as bolded character listed in the eleventh row). For 1 ≤ n ≤ 10 (Table 7), the improved rate becomes 59.1%, 66.5%, and 75.8% respectively for these three models (Solutions no. 2, 3, and 9 as bolded characters listed in the sixth, twelfth and sixteenth row). As for the SEA objective, the optimal results are significantly better than the initial values for whichever surrogate model, especially the SEA is maximized to 14.28 kJ/kg over 100% based on the RSM model. According to the precious discussions, i.e., the RSM models perform the best for both SEA and CFE with exponent 0 ≤ n ≤ 1, while the RBF models perform the best in 1 ≤ n ≤ 10. It demonstrates that the most accurate surrogate model

1

263

0.6 0.4

6 4

0.2

2 0 400 300 Init ial l eng th L 200 (mm )

400 200 150 100

50

) 100 (mm eter D Diam

Init 300 ial l eng th L

200 200 (mm )

100

50

150 100 (mm) D r ete Diam

Fig. 16. The corresponding design variables of optimal solutions generated from the different surrogate models in the multiobjective optimization.

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F. Xu / Thin-Walled Structures 97 (2015) 250–265

Table 6 Optimal solutions (chosen from Pareto set) and corresponding design variables (Fig. 16(a)) for multiobjective optimization (0 ≤ n ≤ 1). No.

Parameters D (mm)

Objectives

No.

L (mm)

n

SEA (kJ/kg)

CFE (%)

147.37

0.211

4.38

59.42

RSM: SEA 412 kJ/kg and CFE 450% 1 50.00 146.49 2 50.00 162.52 3 50.77 154.60 4 50.00 128.75 5 50.00 166.86

0.78 0.50 0.52 0.73 0.63

13.32 12.69 12.99 14.28 12.38

55.25 56.94 55.91 51.86 59.10

KRM: SEA4 12 kJ/kg and CFE 450% 1 60.29 100.00 2 50.00 100.00 3 50.00 113.04 4 50.00 124.13 5 50.00 114.14 6 50.00 124.02

0.51 0.37 0.54 0.41 0.24 0.44

12.95 13.42 13.12 13.41 13.75 13.34

RBF: SEA4 12 kJ/kg and CFE 450% 1 50.00 124.74 2 50.00 115.08 3 50.00 147.05 4 50.00 139.86 5 53.40 151.28

0.32 0.31 0.24 0.28 0.25

12.27 12.46 12.38 12.26 12.06

Initial value 192.11

Parameters

Objectives

D (mm)

L (mm)

n

SEA (kJ/kg)

CFE (%)

6 7 8 9 10

50.00 50.00 50.00 50.00 50.00

160.45 158.21 143.39 130.22 145.20

0.86 0.54 0.70 0.67 0.54

12.59 12.87 13.52 14.25 13.54

56.99 56.80 54.97 52.24 54.33

68.67 65.51 68.60 67.10 57.13 67.91

7 8 9 10 11

50.00 50.00 50.00 50.00 51.45

100.00 100.00 100.00 100.00 100.00

0.26 0.32 0.30 0.46 0.47

13.60 13.50 13.53 13.23 13.19

58.78 63.18 61.88 68.28 68.44

52.14 50.23 51.95 52.97 54.35

6 7 8 9

50.00 50.00 51.01 50.00

148.39 158.11 141.75 129.87

0.25 0.22 0.24 0.34

12.24 12.22 12.40 12.10

53.40 53.45 51.44 53.59

Table 7 Optimal solutions (chosen from Pareto set) and corresponding design variables (Fig. 16(b)) for multiobjective optimization (1 o n ≤ 10). No.

Parameters D (mm)

Objectives

No.

L (mm)

n

SEA (kJ/kg)

CFE (%)

278.68

4.386

3.52

36.72

RSM: SEA 410 kJ/kg and CFE 450% 1 50.00 212.76 2 50.00 233.49 3 50.00 208.82 4 50.00 208.72

1.00 1.00 1.00 1.00

11.43 10.29 11.65 11.66

54.05 58.42 53.09 53.06

KRM: SEA4 9 kJ/kg and CFE 4 50% 1 50.00 230.45 2 50.00 228.26 3 50.00 233.71

1.07 1.00 1.00

9.31 9.32 9.28

RBF: SEA4 10 kJ/kg and CFE 450% 1 50.00 206.63 2 50.00 210.02 3 50.00 211.01 4 50.00 208.56 5 50.00 221.09 6 50.00 224.56

1.00 1.00 1.00 1.00 1.09 1.00

11.74 11.55 11.50 11.63 10.96 10.78

Initial value 152.61

Parameters

Objectives

D (mm)

L (mm)

n

SEA (kJ/kg)

CFE (%)

5 6 7

50.00 52.13 50.00

233.09 216.99 213.31

1.00 1.00 1.00

10.31 10.97 11.40

58.35 55.22 54.18

59.76 59.30 61.13

4 5

50.00 50.00

227.95 223.11

1.74 1.00

9.39 9.36

55.66 57.40

58.27 59.27 59.55 58.85 61.62 62.98

7 8 9 10 11 12

50.00 50.00 50.00 50.00 50.00 50.00

214.36 183.11 232.34 199.82 210.12 206.74

1.00 1.00 1.00 1.00 1.00 1.00

11.32 13.12 10.39 12.12 11.55 11.73

60.48 50.19 64.56 56.13 59.30 58.31

Acknowledgments The support of this work by the Program for Innovative Research Team in University (IRT13087) is greatly appreciated.

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