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International Journal of Impact Engineering 64 (2014) 62e74

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International Journal of Impact Engineering

journal homepage: www.elsevier .com/locate/ i j impeng

Crashing analysis and multiobjective optimization for thin-walledstructures with functionally graded thickness

Guangyong Sun a, Fengxiang Xu a, Guangyao Li a,*, Qing Li b

a State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, Hunan 410082, PR Chinab School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW 2006, Australia

a r t i c l e i n f o

Article history:Received 15 February 2013Received in revised form25 September 2013Accepted 12 October 2013Available online 19 October 2013

Keywords:Multiobjective optimizationTailor rolled blank (TRB)CrashworthinessNon-uniform thickness sheetFunctionally graded thickness

* Corresponding author. Tel.: þ86 731 8882 1717; fE-mail addresses: sgy800@126.com (G. Sun), gyli@

0734-743X/$ e see front matter Crown Copyright �http://dx.doi.org/10.1016/j.ijimpeng.2013.10.004

a b s t r a c t

Thin-walled structures have exhibited significant advantages in light weight and energy absorption andbeen widely applied in automotive, aerospace, transportation and defense industries. Unlike existingthin-walled structures with uniform thickness, this paper introduces functionally graded structures withchanging wall thickness along the longitudinal direction in a certain gradient (namely, functionallygraded thickness e FGT). Its crashing behaviors are the key topics of the present study. We examine thecrashing characteristics of functionally graded thin-walled structures and evaluate the effect of differentthickness gradient patterns on crashing behaviors. It is shown that the gradient exponent parameter nthat controls the variation of thickness has significant effect on crashworthiness. To optimize crash-worthiness of the FGT tubes, the Non-dominated Sorting Genetic Algorithm (NSGA-II) is used to seek foran optimal gradient, where a surrogate modeling method, specifically response surface method (RSM), isadopted to formulate the specific energy absorption (SEA) and peak crashing force functions. The resultsyielded from the optimization indicate that the FGT tube is superior to its uniform thickness counterpartsin overall crashing behaviors. Therefore, FGT thin-walled structures are recommended as a potentialabsorber of crashing energy.

Crown Copyright � 2013 Published by Elsevier Ltd. All rights reserved.

1. Introduction

Over the past years, research interests in crashworthiness haveresulted in a series of systematic investigations into crash re-sponses of various thin-walled structures via analytical, experi-mental and numerical approaches [1e8]. Two significant aspects,namelyweight and crashworthiness, have drawnprimary attentionin these studies. As an effective structure, thin-walled componentshave showed significant advantages over other solid elements andare capable of carrying substantial loads with desired deformation,which could be appreciably higher than the corresponding ultimateor bulking loads [9,10]. In reality, thin-walled structural membersplay a critical role on enhancing the capability of energy absorptionin impact engineering.

The automobile body in white (BIW) is mainly composed ofthin-walled structural parts, which are made by stamping orforming process of traditional metal sheets with uniform thickness[11e19]. It is of great interests in investigating the crashworthinessof thin-walled structures for improving the vehicle safety and light

ax: þ86 731 8882 2051.hnu.edu.cn (G. Li).

2013 Published by Elsevier Ltd. All

weight. In this regard, Zhang et al. [12] evaluated the energy ab-sorption characteristics of regular polygonal and rhombic columnsunder quasi-static axial compression. Song et al. [13] introducedorigami patterns into thin-walled structures and minimized theinitial peak force and subsequent fluctuations. Tang et al. [20]presented a cylindrical multi-cell column to improve energy ab-sorption. Najafi and Rais-Rohani [21] proposed a sequentiallycoupled nonlinear finite element analysis (FEA) technique toinvestigate the effects of sheet-forming process and design pa-rameters on energy absorption of thin-walled tubes made ofmagnesium alloy. Acar et al. [22] studied the crashing performancesof tapered tubes using multiobjective optimization. Although suchthin-walled structures have been extensively used as energy ab-sorbers for their high energy absorption capacity, light weight andlow cost [23], all these thin walled structures were based upon theuniform material and/or the same wall thickness. The inherentshortcoming resides on that such structures may not exert theirmaximum capacities in crashworthiness, and furthermore, a uni-formwall thickness does not necessarily make best use of materialfor meeting the requirements of vehicular light weight [24e26]. Sothere is an urgent need to develop new structural configurationwith different material and/or thickness combinations for maxi-mizing crashworthiness and material usage.

rights reserved.

G. Sun et al. / International Journal of Impact Engineering 64 (2014) 62e74 63

According to Yang et al. [27], a metal sheet with varying thick-ness could be a more desirable structure because it not only usesmaterial more efficiently, but also increases design flexibilityconsiderably. It has been demonstrated that with an optimal choiceof different materials grades (e.g. via tailored welded blanks (TWB),or hybrid blanks) and/or thicknesses (e.g. via tailor rolling blanks),crashing performance of the combined components can beimproved to a higher extent. Indeed, design of specific thin-walledcomponents with desired materials/thicknesses in a more efficientmanner could represent new potential for further reducing weightand enhancing performance of the products. Of these componentswith variable material/thickness, the TWB structures, which con-sists of laser-welded sheet metals of different thicknesses andmaterials, provide a flexible combination of component materialsand thicknesses, which has been adopted in vehicular floorcomponent [28], B-pillar [29], front-end structure [30], and doorinner panels etc. [31,32]. The main shortcoming of those blanks liesin that it consists of discrete thickness sections and may lead tostress concentration in the interfaces. To overcome such defects ofTWB, a relatively new rolling process, named tailor rolled blank(TRB), has been developed. In the newly developed TRB process, therolling gap can be varied, which leads to a continuous thicknessvariation in the workpiece. Applications of such a rolling processallow reducing more weight compared with traditional stampingor forming processes. As such, varying sheet thickness can bettermeet more and more demanding design requirements, therebyenhancing utilization of material and/or thickness comparing withtraditional stamping uniform sheets.

There have been some reports on TRB in sheet metal forming.For example, Zhang et al. [33] investigated the effects of transitionzone length, blank thickness variation, friction coefficient and dieclearance on the springback of TRB component. Meyer et al. [34]used TRB to increase the maximum drawing depth compared tothe blanks with constant thickness. Urban et al. developed a designtool by combining numerical simulation and optimization algo-rithm to improve the formability of TRB [35]. To the author’s bestknowledge, however, very limited studies on crashworthinessdesign of thin-walled TRB structures have been performed to date.

To make use of TRB thin-walled structures with functionallygraded thicknesses (FGT) for impact engineering, it is essential tounderstand the energy absorption characteristics in comparisonwith those well-studied uniform thickness (UT) thin-walledstructures. More importantly, it is of particular importance toseeking the best possible thickness gradient for crashing perfor-mance with different measures. Thus the objective of this paperresides in quantifying and improving crashing behaviors of thin-walled structures with functionally graded wall thickness. For thisreason, two critical issues need to be addressed in this paper: (1) adirect problem that quantifies the crashing characteristic of func-tionally graded thin-walled structures with variable wall thicknessand evaluates the effects of the different thickness pattern on bothspecific energy absorption (SEA) and peak impact forces; (2) aninverse problem that seeks optimal gradient for maximizing thespecific absorption energy (SAE) and minimizing peak crashingforce (Fmax).

As for the functionally graded thickness structure, the thicknessof thin-walled varies throughout the depth in an ascending ordescending gradient. It is expected that the gradient exponentialparameter (n) has a significant effect on crashworthiness. Torepresent such complex crashworthiness objective functions withrespect to gradient parameter, which has not been explored inliterature before, a surrogate model technique, namely specificallyresponse surface method (RSM), will be attempted here. To maxi-mize the energy absorption and minimize the peak crashing force,the multiobjective optimizations for the FGT structures are

formulated and the Non-dominated Sorting Genetic Algorithm(NSGA-II) is applied for its proven effectiveness in crashworthinessdesign [36,37].

2. High-strength steel column structures with functionallygraded wall thickness

2.1. Geometrical description and material properties

Dynamic axial crushing simulationwas performed in the squaretubes which were made of high-strength steel grade DP800 [38].The dynamic procedurewas conducted at velocities of 5m/s, 10m/sand 15 m/s, respectively, with an impacting mass of 600 kg in orderto assess the crash behaviors measured in the impact force andenergy absorption. Fig. 1 illustrates the geometry of thin-walledsquare structure in the dynamic tests. These specimens have anominal square core cross-section with rounded corners and theaverage dimensions of 60 mm � 60 mm. The strain-rate dependentproperties of DP800 are considered herein and the true-stressversus true-plastic strain curves at different strain rates(0.000903/s, 1.029/s, 278/s and 444/s, respectively) are plotted inFig. 2. To characterize the material behavior, an empirical consti-tutive equation for the effective yield stress as a function s of theeffective plastic strain is fit to the following formulae [39]:

sðεÞ ¼ s0 þ

X2i¼1

Qið1� expð�CiεÞÞ!�

1þ_ε

_ε0

�q

(1)

where s0 is the initial yield stress, and Qi and Ci denote strainhardening coefficients, q represents a material constant and _ε0 is auser-defined reference strain rate. The relevant material propertiesare summarized in Table 1. It is assumed that the material prop-erties remain the same regardless of variation in sheet thickness.

2.2. Structural crashworthiness criteria

The design optimization aims to generate a controllablecrashing pattern for maximizing energy absorption andminimizingthe peak forces during collapse [7]. There are several key indicatorsto evaluate crashworthiness of a structure, e.g. energy absorption(EA), specific energy absorption (SEA), average crash force (Favg),and crash force efficiency (CFE), as given in Eqs. (2)e(5) respectively(Fig. 3), are widely used in the measurement.

As a key indicator, the energy absorption (EA) of a structuremeasures the capacity of absorbing impact energy, which can bedetermined mathematically as,

EA ¼Zd0

FðdÞdd (2)

where F(d) is the instantaneous crashing force with a function ofthe displacement d.

The specific energy absorption (SEA) assesses the absorbedenergy per unit mass of a structure as,

SEA ¼ EAm

(3)

where m is the total mass of the structure. In this case, a highervalue indicates a higher energy absorption efficiency of material.

The average crashing force Favg for a given deformation alsoindicates the capacity of energy-absorption of a structure, which iscalculated as EA divided by the compressive displacement d as [35],

V0=5 m/s, 10 m/s, 15 m/s

Impactingmass=600 kgL1=310 mmL2=100 mm

R=3

a=60

b=60

t=1.2

Seam weld

Unit: mm

(a) (b)

Fig. 1. Schematic of (a) experimental set-up for dynamic tests and (b) geometry description of square tube [39].

G. Sun et al. / International Journal of Impact Engineering 64 (2014) 62e7464

Favg ¼ EAd

(4)

From Zarei and Kroger [41], the crash force efficiency (CFE) andspecific energy absorption can increase simultaneously. The CFEindicates the uniformity of forceedisplacement curve, meaningthat the higher the CFE, the more efficient the structure. Thus, CFEof a structure is formulated as other key indicator,

CFE ¼ FavgFmax

� 100% (5)

2.3. Discrete functionally graded column

It is assumed that the square column is fabricated with func-tionally graded thickness sheet whose direction of thicknessgradient is identical to the direction of the axial impact velocity.Fig. 4 illustrates the key geometrical features (in a side view) of thesquare column under consideration, in which we ignore the effectof changing inner surface and outer surfaces as this difference is notrequired in finite element modeling. The idea is similar to thedensity gradient of the functionally graded foam materials devel-oped by us previously [42]. The top wall thickness ttop is chosen atthe origin of gradient axis, whilst the bottom wall thickness tbotcorresponds to the farthest layer from the origin. In the axialgrading case, the thickness gradient function tf(x) can be defined interms of the following power law:

0.00 0.05 0.10 0.15 0.20 0.250

200

400

600

800

1000

1200

Tru

e st

ress

(M

Pa)

Plastic Strain

Strain Rate (1/s) 0.000903 1.029 278 444

Fig. 2. True-stress versus true-plastic strain curves at different strain rates for DP800high-strength steel [39].

tf ðxÞ ¼(

tmin þ ðtmax � tminÞ�xL

�n for an ascending pattern

tmax � ðtmax � tminÞ�xL

�n for a descending pattern

(6)

where x is the distance from the origin (top) of the column, and n isthe grading exponent and assumed to vary between 0 and 10, Ldenotes the total length of column. When the total layer Ns isapproaching to infinite, the wall thickness will be graded contin-uously. Thus, the wall thickness increases along the length with agradient function changing from convexity to concavity when the nvalue varies from less than 1 to greater than 1, as shown in Fig. 5a.On the contrary, the wall thickness decreases along length, anopposite tendency is observed as shown in Fig. 5b. Herein, theascending pattern is considered to perform the crashworthinessdesign with the FGT structures.

3. Numerical modeling

3.1. Finite element (FE) modeling

The FE model used to simulate the crashing process was basedupon the experimental tests conducted by Tarigopula et al. [39]. Anexplicit FE code, LS-DYNA, was used to implement the parallelcomputing. Fig. 6 shows the FE mesh using the 4-node shell ele-ments with 6 degrees of freedom at each node. In this model, thebottom end is fixed and a rigid mass-block of 600 kg was linked tothe loading end through a master node. In the crash scenario, themass block is assigned an initial velocity of 10 m/s through themaster node. The contact between the mass block surface and theFGT tubes was modeled with a friction coefficient of 0.2, while forthe self-contact of the tube, frictional effects were neglected [43].

To find the optimum mesh size for the numerical simulation, aconvergence test with five different mesh sizes was carried out. Incomparison, a UT column having the same mass as its FGT coun-terpart was also employed to perform the mesh convergence.Therefore, element size 3 mm � 3 mm is adopted in the finiteelement model. The energy absorption characteristics predicted bydifferent element sizes (exponent n ¼ 0) are summarized in Fig. 7,which shows that there is very small difference between elementsizes 2 mm � 2 mm and 2.5 mm � 2.5 mm. It can be seen that themesh size adopted for all models of 2.0 � 2.0 mm is sufficient.

As for FGT, it is assumed that thedepthof each layer is the sameanddefined as Le, an equivalent thickness of UTcolumn that has the samevolume of high-strength steel material can be calculated as follows,

tavg ¼XNS

i¼1

ðtiLeÞ=ðNSLeÞ ¼XNS

i¼1

ti=NS (7)

Table 1Material properties for DP800 high-strength steel.

E(GPa) v r (kg/m3) s0 (MPa) Q1(MPa) Q2(MPa) C1 C2 q _ε0 (1/s)

195 0.33 7850 495 200 233 76 10 0.0116 0.001

0 50 100 150 200 250 300 350 4000

100

200

300

400

500

600

700

800

900

1000

Ea

Favg

Fmax

Forc

e F

(kN

)

Displacement (mm)

Fig. 3. A typical relationship of force versus displacement of axial crashing behaviorwith progressive folding [40].

(a)

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

n=10n=5

n=2

n=1

n=0.5

n=0.2n=0.1

Thi

ckne

ss r

atio

t min/t m

ax

Normalized distance x/L

1.0

G. Sun et al. / International Journal of Impact Engineering 64 (2014) 62e74 65

where NS denotes the total number of layers of FGT thin-walledsquare column, and ti is thickness of the ith layer. As such, suchgradient exponent nwould also affect the crashing results of the UTcolumn with uniform wall thicknesses but different values.

In an ideal functionally graded continuous model, the columnwall should be divided to an infinite number of layers. In the FEframework, the minimum depth of layer would be equal to the sizeof each shell element, which could however lead to very costlycomputational time. Furthermore, increasing the number of layerscould increase the risk of numerical instability in the modelresulting from the use of smaller element sizes. In this respect,another convergence test was performed for three differentlylayered configurations of 21, 31, and 52 layers, respectively, in orderto determine the optimum number of layers. Comparison of theload-displacement response for various configurations of layers is

Loading plate (impacting mass)

Top end (ttop)

Bottom end (tbot)

Dir

ectio

nof

thic

knes

sgr

adin

g

x

Crash speed v0Crash speed v0

L

123

NS

...

456789

1011

NS-1NS-2NS-3NS-4NS-5NS-6NS-7NS-8NS-9

NS-10NS-11

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

presented in Fig. 8, in which the grading exponent (n) in Eq. (6) isdefined as 0.2 and the top (incident end) thickness ttop and thebottom (distal end) thickness tbot are set to 0.8 mm and 2.2 mm,respectively. There is no significant difference between themaximum peak loads of these three configurations and the curvesof 31 and 52 layers exhibit considerable consistency. Therefore, wechose 31 horizontal layers for modeling the thickness gradient.

3.2. Validation of the numerical models

In order to validate the developed FE models, the modeling re-sults of the UT square tubes under axial dynamic loading are firstcompared with the theoretical solutions available in the literature[44], i.e.

Favg ¼ 13:06s0b1=3w t5=3 (8)

where s0 is the characteristic stress of tube material, bw and t arethe width and wall thickness of tube, respectively. The comparisonof the crashing force versus displacement of the UT square tubes(thickness t¼ 1.2 mm) under an axial dynamic loading rate of 10m/s is shown in Fig. 9. It is seen that the present FEA results are in goodagreement with the theoretical solutions for the UT square tubes.

(b)

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

Normalized distance x/L

Thi

ckne

ss r

atio

t min/t m

ax

n=0.1

n=0.2

n=0.5

n=1

n=2

n=5

n=10

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

Fig. 6. Finite element model for dynamic impact simulation. (a) Experimental test; (b) Finite element model (2D); (c) 3D model.

G. Sun et al. / International Journal of Impact Engineering 64 (2014) 62e7466

In addition to the theoretical validation, in which the thicknessof t ¼ 1.2 mmwas adopted, the FE model is also verified against theexperimental results in literature [39]. Fig. 10 illustrates the com-parison of the force versus displacement curves between theexperiment and simulation at the velocity of 10 m/s. In terms of thepeak force (Fmax) and specific energy absorption (SEA), the exper-imental and simulation results are 135.06 kN and 130 kN, 10.6 kJ/kgand 9.7 kJ/kg, respectively. It can be seen that there is alsoreasonable agreement between the simulation and experimentalresults. The validated numerical model provides considerableconfidence for us to explore the energy absorption capacity of thin-walled sections with different gradients of wall thickness.

3.3. Numerical results of functionally graded column

In the present work, a group of graded and uniform thicknessthin-walled columns with the same weight were compared forenergy absorption and peak impact force (SAE and Fmax) respec-tively. The effects of thickness ranges and design parameter n (Eq.(6)) on the resulted crashing characteristics of the axially thickness-graded column are explored.

3.3.1. Effect of thickness rangeThickness range has a notable effect on the crashworthiness

performance. Herein, the value of ttop was varied with 0.8 mm,1.2 mm, and 1.5 mm while tbot was fixed at 2.2 mm in order toquantify the effect of selecting different thickness range (Dt) on SEAand Fmax. Variations in SEA and Fmax for various values of Dt and nare illustrated in Fig. 11. It is noted again that equivalent wallthickness was calculated as Eq. (7) for the corresponding UT tube tokeep the same weight as the functionally graded thickness (FGT)tube with different parameter n. In other words, the wall thicknessof UT tube can also be expressed in term of n hereafter. Twoimportant points can be emphasized: (1) SEA and Fmax of both GTand UT columns decrease with the increased exponent n; (2) SEA ofFGT column is generally higher than that of UT column at the sameexponent nwhilst Fmax of FGT is smaller. This indicates that the FGTcolumn is superior to the UT column. In addition, the changes inSEA and Fmax at different thickness ranges are summarized inTable 2. For 0 � n � 1, the DFmax has a maximum value at rangeDt ¼ 1.4 mm, whilst for 1 < n � 10 DSEA reaches a maximum valueat rangeDt¼ 1.4mm except for n¼ 8. Note that theDFmax and DSEAare the amplitudes of variation in the peak force and specific energyabsorption when the FGT and UT tubes are compared with each

other. Overall, FGT with larger thickness range is more effective inminimizing Fmax for 0 � n � 1; as well as in maximizing SEA for1 < n � 10.

The value of grading exponent (n) in Eq. (6) is tested in a series ofparameters of n¼ 0, 0.2, 0.4, 0.6, 0.8, 1.0, 2.0, 4.0, 6.0, 8.0, 10.0, whilethe top (incident end) thickness ttop and the bottom (distal end)thickness tbot are set to 0.8 mm and 2.2 mm (i.e. Dt ¼ 1.4 mm),respectively. As justified before, the total number of layers in theFGT thin-walled square column is prescribed to be 31, which wasconsidered sufficient for modeling axially graded FGT structuresaccurately. Note that each layer has the different color representingdifferent thickness sections for easy observation in the schematicdiagram (Fig. 12a). It is worth mentioning that the total weight ofthin-walled tube increases with the decreasing of gradientparameter n (Fig. 12b). This change would be directly related to thevariation in SEA and Fmax.

3.3.2. Effect of different deformation distancesThe relationship of SEA and Fmax versus mass of FGT and the

corresponding UT is shown in Figs. 13e15 at the timeframes of15 ms, 20 ms, and 25 ms, respectively. Importantly, it can beobserved that the SEA of FGT columns is not necessarily better atthe initial stage of crashing. In a shorter crashing instance (e.g. at15 ms in Fig. 13), the SEA of UT is even greater than that of thecorresponding FGT columns for all gradient parameters consideredhere. However, FGT becomes more and more superior to the cor-responding UT counterpart as the time increases (from Fig. 14 toFig. 15). It can be also observed that Fmax of the FGTcolumn is muchsmaller than that of the corresponding UT column no matter whatcrashing displacement is. Note that gradient exponent n controlsthe variation of thickness distributions and has a significant effecton SEA for the FGT columns. As seen in Fig. 15, SEA monotonouslydecreases with increase of n in the region of n˛ [0, 10] considered.The reason lies in that the total weight is to decrease as theexponent n increases. The similar trend can be seen for the UTcolumn.

Fig. 16 depicts the deformation modes of the FGT column andthe corresponding UT column with the same weight at differenttimeframes and different exponents n. It can be seen that thedeformation of the FGT column seems to be more stable and ad-vantageous over its uniform counterpart, especially with increasedparameter n.

To be brief, the columns with graded wall thickness have abetter crushing performance than the uniform thickness columns

Fig. 7. Internal energy versus crashing displacement (n ¼ 0).

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

Cra

shin

g fo

rce

(kN

)

Displacment (mm)

Dynamic crashing force by FE simulations Dynamic average crashing force by FE simulations Theoretical solution of F

avg by Eq. (8)

Fig. 9. Crashing force and displacement curves for square tubes under axial dynamicloading.

G. Sun et al. / International Journal of Impact Engineering 64 (2014) 62e74 67

in terms of SEA and Fmax in the later stage of impact (e.g. Fig. 15).Therefore, the FGT component has significant potential and seemsto be superior and ideal to the crashworthiness design. Whileknowing considerable effect of the exponent of thickness gradientfunction (Eq. (6)) on crashworthiness of FGT column, it remains aquestion how to seek for best possible n by optimizing the crashingcharacteristics of corresponding FGTcolumns, which forms anothergoal of this study below.

4. Multiobjective optimization for functionally gradedthickness column

4.1. Optimization methodology

4.1.1. Definition of optimization problemDesign optimization is further applied for the crashworthiness

criteria in the FGT columns. In general, the SEA and force peak(-Fmax) both are the key indicators [45,46]. An overly high Fmaxoften leads to severe injury or even death of occupant. From thevehicle safety point of view, the smaller the Fmax, the lower thedeceleration, thus the higher the safety. In this study, the optimal n

(a)

0 50 100 150 200 250 3000

20

40

60

80

100

120

140

160

180 21 layers 31 layers 52 layers

Cra

sh f

orce

F (

kN)

Displacement (mm)

O r i g i n P r o 8 E v a l u a t i o n O r i g i n P r o 8 E v a l u a t i o n

O r i g i n P r o 8 E v a l u a t i o n O r i g i n P r o 8 E v a l u a t i o n

O r i g i n P r o 8 E v a l u a t i o n O r i g i n P r o 8 E v a l u a t i o n

O r i g i n P r o 8 E v a l u a t i o n O r i g i n P r o 8 E v a l u a t i o n

O r i g i n P r o 8 E v a l u a t i o n O r i g i n P r o 8 E v a l u a t i o n

O r i g i n P r o 8 E v a l u a t i o n O r i g i n P r o 8 E v a l u a t i o n

Fig. 8. Force-displacement responses for vario

is obtained based on the same impact initial velocity to furtherperform the comparisons. However, SEA and Fmax often conflictwith each other, i.e, the increase in SEA often leads to the increasein Fmax. Therefore, to account for both different design criteria andimpose the optimum in a Pareto sense, the optimization problemcan be formulated mathematically in a multiobjective frameworkas follows:

�Max½SEAðnÞ;�FmaxðnÞ�s:t: nLower � n � nUpper

(9)

4.1.2. Surrogate model and error metricsIt is often difficult to mathematically derive the analytical

objective functions for SEA and Fmax that involve highly nonlinearcontact-impact and large deformation mechanics. As an effectivealternative, the surrogate techniques, e.g. response surface method(RSM), have proven particularly effective and been widely adoptedin crashworthiness design [45,47].

To estimate the fitting accuracy of these surrogate models, sucherror metrics as R square (R2), Relative Average Absolute Error

(b)

0 50 100 150 200 250 3000

20

40

60

80

100

120

Ave

rage

cra

sh f

orce

Fav

g (kN

)

Displacement (mm)

21 layers 31 layers 52 layers

O r i g i n P r o 8 E v a l u a t i o n O r i g i n P r o 8 E v a l u a t i o n

O r i g i n P r o 8 E v a l u a t i o n O r i g i n P r o 8 E v a l u a t i o n

O r i g i n P r o 8 E v a l u a t i o n O r i g i n P r o 8 E v a l u a t i o n

O r i g i n P r o 8 E v a l u a t i o n O r i g i n P r o 8 E v a l u a t i o n

O r i g i n P r o 8 E v a l u a t i o n O r i g i n P r o 8 E v a l u a t i o n

O r i g i n P r o 8 E v a l u a t i o n O r i g i n P r o 8 E v a l u a t i o n

us numbers of horizontal layers (n ¼ 0.2).

G. Sun et al. / International Journal of Impact Engineering 64 (2014) 62e7468

(RAAE) and Relative Maximum Absolute Error (RMAE) are adoptedin Eqs. (10)e(12), respectively [48e50].

R2 ¼ 1�Pm

j¼1

�yj � byj�2Pm

j¼1

�yj � y

�2 (10)

RAAE ¼Pm

j¼1

yj � byjPmj¼1

yj � y (11)

RMAE ¼ maxfjy1 � by1j; :::; jym � bymjgPmj¼1

yj � by=m (12)

where byj and y are the corresponding predicted (or surrogate) andmean values for the FEA value yj at each checking point j, respec-tively, m represents the number of these checking points. In gen-eral, the larger the R2 values, the more accurate the surrogatemodel. The smaller the RAAE and RMAE, the better the metamodel.

4.1.3. Multiobjective genetic algorithmGenetic algorithm (GA) is a popular global optimization tools

that was originated from mechanisms of natural evolution andgenetic principles, which is superior to many traditional optimi-zation algorithms because of capability of avoiding trapping in localoptima for searching an optimum [51]. To deal with multiobjectiveoptimization, the non-dominated sorting GA (NSGA) algorithm hasproven effective by ranking the solutions with non-dominatedsorting and assigning the fitness with the ranking. As animproved version of NSGA, NSGA-II has been demonstrated to beone of themost efficient algorithms for multiobjective optimizationin a number of benchmarking problems [52]. The NSGA-II algo-rithm was used in this study and the relevant NSGA-II parametersare listed in Table 3.

Quadratic :

*8>><>>:

SEAG�n�¼�

23:48798�4:08330nþ1:27351n2�0

21:53987�0:76198nþ0:01791n2�1

FGmax

�n�¼�286:80818�544:51939nþ429:57576n148:55867�6:74555nþ0:31197n2

�18>><>>:

SEAU�n�¼�

23:22336�9:27598nþ4:28701n2�0

19:00577�1:57404nþ0:10427n2�1

FUmax

�n�¼�328:04650�281:26548nþ159:35548n211:46400�21:87412nþ1:35758n2

�

Cubic :

*8>><>>:

SEAG�n�

¼�

23:26355� 0:51731n� 8:07786n2

19:43504þ 1:10489n� 0:38697n2

FGmax

�n�

¼�316:18392� 1011:26717nþ 1653:56469

153:91233� 11:49396nþ 1:34152n28>><>>:SEAU

�n�

¼�23:30785� 10:61836nþ 7:80724n2

20:01888� 2:47261nþ 0:29909n2

FUmax

�n�

¼�335:79308� 404:34992nþ 482:12937n

231:33800� 39:50129nþ 5:17950n2

4.1.4. Procedure of multiobjective optimization for FGT structuresAs in Fig. 17, a flowchart is provided to clarify the procedure of

multiobjective optimization for thin-walled structures withfunctionally-graded thickness (FGT). It is noted that in this case, thegradient parameter(s) (e.g. exponent n, if a power law formula isadopted herein as in Eq. (6)) should be one of the design variables inorder to reflect their significant roles in crashworthiness design.

4.2. Results and discussion

4.2.1. Comparison of different RSMAccuracies of the surrogate models can be measured in different

ways [50]. Note that it is difficult to check all metamodels fitnessaccuracies based on the existing sampling points properly. Hence, aseries of new checking points should be randomly generated tobetter verify the accuracy of the constructed metamodels. Fordesign variable n, 10 additional checking points are obtained in thedesign ranges of 0 � n � 1 and 1 < n � 10, respectively.

Based on these experimental results, the polynomial RSMmodels (Linear, Quadratic, Cubic, Quartic) of SEA and Fmax areestablished as follows. In Eqs. (13)e(16), superscripts G and Urepresent the functionally graded tubes and uniform tubes,respectively. Thus, SEAG and Fmax

G indicate the specific energy ab-sorption and maximum crashing force of functionally graded tube.Correspondingly, the SEAU and Fmax

U indicate the specific energyabsorption and maximum crashing force of uniform tube.

Linear :

*8><>:

SEAG�n�¼�

23:29696�2:80979n ð0�n�1Þ21:14585�0:56497n ð1<n�10Þ

FGmax

�n�¼�222:37182�114:94364n ð0�n�1Þ141:69533�3:31388n ð1<n�10Þ8><>:

SEAU�n�¼�

22:58031�4:98897n ð0�n�1Þ16:71194�0:42712n ð1<n�10Þ

FUmax

�n�¼�304:14318�121:91000n ð0�n�1Þ181:59733�6:94079n ð1<n�10Þ

(13)

�n�1�

<n�10�

2 �0�n�1�

<n�10�

�n�1�

<n�10�

2 �0�n�1�

1<n�10�

(14)

þ 6:23425n3�0 � n � 1

�þ 0:02453n3

�1 < n � 10

�n2 � 815:99262n3

�0 � n � 1

�� 0:06240n3

�1 < n � 10

�� 2:34682n3

�0 � n � 1

�� 0:01181n3

�1 < n � 10

�2 � 215:18260n3

�0 � n � 1

�� 0:23163n3

�1 < n � 10

�(15)

Quartic :

*8>><>>:

SEAG�n�

¼�23:06291þ 6:44927n� 42:91078n2 þ 61:96692n3 � 27:86634n4

�0 � n � 1

�22:29852� 2:56623nþ 0:95643n2 � 0:15902n3 þ 0:00834n4

�1 < n � 10

�FGmax

�n�

¼�330:47713� 1507:55942nþ 4135:02593n2 � 4786:33061n3 þ 1985:16900n4

�0 � n � 1

�199:33333� 69:72601nþ 22:64916n2 � 2:97400n3 þ 0:13235n4

�1 < n � 10

�8>><>>:SEAU

�n�

¼�23:33132� 11:43353nþ 11:88308n2 � 8:86816n3 þ 3:26067n4

�0 � n � 1

�21:74548� 4:68621nþ 1:10907n2 � 0:12249n3 þ 0:00503n4

�1 < n � 10

�FUmax

�n�

¼�335:39238� 390:43675nþ 412:56352n2 � 103:87723n3 � 55:65268n4

�0 � n � 1

�245:57500� 57:75386nþ 11:85828n2 � 1:14426n3 þ 0:04148n4

�1 < n � 10

�(16)

0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

160 Experiment (ds04) Simulation

Forc

e (k

N)

Displacement (mm)

Fig. 10. Comparison of force versus displacement curves between experiments [39]and FE simulations.

Table 2Change range of SEA and Fmax at different thickness range (Unit: %).

n Dt ¼ 1.4 mm Dt ¼ 1.0 mm Dt ¼ 0.7 mm

DSEAa DFmaxb DSEA DFmax DSEA DFmax

0.2 13.54 �45.72 �4.56 �41.14 19.10 �16.100.4 6.11 �39.36 �3.11 �30.55 10.89 �13.640.6 7.65 �38.09 12.00 �24.29 6.44 �12.630.8 13.45 �30.47 8.68 �15.58 7.84 �10.111 14.26 �23.21 13.31 �14.93 11.75 �7.982 22.44 �29.59 10.15 �15.2 1.09 �8.384 33.86 �11.64 21.07 �11.91 11.67 �7.056 19.40 �13.51 16.22 �9.52 8.67 �5.318 1.08 �14.18 14.85 �8.03 6.56 �4.6010 20.71 �8.52 16.52 �6.88 3.34 �3.90

a DSEA ¼ (SEA of FGT-SEA of UT)/SEA of UT.b DFmax ¼ (Fmax of FGT-Fmax of UT)/Fmax of UT.

(a) (b)

0.0 0.2 0.4 0.6 0.8 1.0 1.20.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

Wei

ght (

kg)

Graded exponent n (0<n<1)

0 2 4 6 8 10 12

0<n<1

1<n<10

Graded exponent n (1<n<10)

Fig. 12. (a) Schematic of FGT thin-walled square column (31 layers); (b) Mass variationwith different gradient exponent n.

G. Sun et al. / International Journal of Impact Engineering 64 (2014) 62e74 69

Tables 4 and 5 summarize the modeling accuracy, which enableus to compare different forms of RSM surrogate functions using thesame set of checking points. The higher the order, the better themodeling accuracy in such cases. In addition, they will be furthercompared through the results of multiobjective optimization later.

4.2.2. Multiobjective optimization resultsFrom NSGA-II, the Pareto fronts of the FGT column and UT col-

umn are plotted in Fig. 18 based upon different metamodels. It canbe clearly seen that the Pareto fronts of the FGT columns are muchmore predominant than those of the UT columns. Specifically, theSEA of FGT is better than that of the UT counterpart at the samelevel of Fmax. For the UT columns, the optimal SEA results are ratherclose for different orders of polynomial RSM.

Fig. 11. Variation of SEA and Fmax for various values of Dt and n.

(a) SEA (b) Fmax

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3

6

8

10

12

14

16

18

SEA

(kJ

/kg)

Mass (kg)

UT FGT

n=0

n=0.2n=0.4

n=0.6n=0.8

n=1.0n=2.0

n=4.0

n=6.0

n=8.0

n=10.0

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.350

100

150

200

250

300

350

Peak

for

ce F

max

(kN

)

Mass (kg)

UT FGT

n=0

n=0.2

n=0.4n=0.6

n=0.8n=1.0

n=2.0

n=4.0

n=6.0

n=8.0

n=10.0

Fig. 13. Comparisons of SEA and Fmax of FGT and UT columns at the timeframe of 15 ms.

(a) SEA (b) Fmax

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3

8

10

12

14

16

18

SEA

(kJ

/kg)

Mass (kg)

UT FGT

n=0

n=0.2

n=0.4

n=0.6n=0.8

n=1.0

n=2.0

n=4.0

n=8.0

n=10.0 n=6.0

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3

100

150

200

250

300

350

Peak

for

ce F

max

(kN

)

Mass (kg)

UT FGT

n=0

n=0.2

n=0.4n=0.6

n=0.8n=1.0

n=2.0

n=4.0n=8.0

n=10.0 n=6.0

Fig. 14. Comparisons of SEA and Fmax of FGT and UT columns at the timeframe of 20 ms.

G. Sun et al. / International Journal of Impact Engineering 64 (2014) 62e7470

However, the SEA of FGT may not be consistently better whenthe order of polynomial functions is higher. In Fig. 18(a), forinstance, there are higher energy absorption capacities with higherorder of the polynomial on the left of intersection A (i.e. n ¼ 0.3047,SEA ¼ 22.5568 kJ, -Fmax ¼ �136.7385 kN). This is to say that thePareto front of the 4th order polynomial function locates above theothers when the peak force level is higher than 136.7385 kN. Itindicates that the 4th order polynomial function leads to betteroptima in this range. However, this becomes different when peakforce is lower than 136.7385 kN, which shows that the 3rd orderpolynomial function may provide a better Pareto solution. Inaddition, on the right of intersection B (n ¼ 0.4983,

(a) SEA

0.4 0.6 0.8 1.0 1.2 1.410

12

14

16

18

20

22

24

SEA

(kJ

/kg)

Mass (kg)

UT FGT

n=10.0

n=8.0

n=4.0

n=2.0n=6.0

n=1.0

n=0.6n=0.8

n=0.4n=0.2

n=0

Fig. 15. Comparisons of SEA and Fmax of FGT an

SEA ¼ 21.7712 kJ, Fmax ¼ 121.8934 kN), the 2nd order polynomialfunction may yield better optimal results. It is thus concluded thatthe higher metamodeling accuracy of RSM does not necessarilyyield better final optimal results.

In order to further discuss the results of the two single objectiveoptimizations, SEA and Fmax predicted by the different RS modelsare listed in Tables 6 and 7, respectively. These single objectiveoptima given in both tables correspond to the special (or idealized)points in the Pareto space, which lie at each end of the Paretocurves as in Fig.18. From both tables, the optimum exponents of theFGT tube for the same objective function generally differ from thedifferent RS models. To minimize Fmax, for instance, the optimal

(b) Fmax

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

100

150

200

250

300

350

Peak

for

ce F

max

(kN

)

Mass (kg)

UT FGT

n=6.0

n=8.0

n=10.0

n=4.0

n=2.0

n=1.0n=0.8

n=0.6n=0.4

n=0.2

n=0

d UT columns at the timeframe of 25 ms.

Fig. 16. Deformation modes of FGT and UT columns with different graded parameters n at different time steps.

G. Sun et al. / International Journal of Impact Engineering 64 (2014) 62e74 71

Table 3Parameters for NSGA-II algorithm.

Parameters Value

Population size 20Number of generations 50Crossover probability 0.9Crossover distribution index 10.0Mutation distribution index 20.0

Define design problem: Design variables • Sample design space;Define design problem: Design variables(with gradient parameter n – Eq. (6)), objectives, constraints

Sample design space;• Create FE model with graded thickness• Conduct finite element analysis

• Construct metamodels for objectives and constraints (e.g. RSM, Eqs. (13)-(16)• Evaluate modeling error (Eqs.(10)-(12))

Satisfy?No

• Perform multiobjective optimization (Eq. (9)) (e.g. using NSGA-II)• Obtain Pareto solutions

Yes

Fig. 17. Flowchart of multiobjective optimization for thin-walled structures withfunctionally-graded thickness (FGT).

Table 4Accuracy assessment of the RSM metamodels (0 � n � 1).

Objective Order R2 Fitting indicators

RAAE RMAE

SAEG 1 0.95296 0.15160 0.750462 0.95335 0.14830 0.752923 0.95394 0.14912 0.745724 0.95496 0.14650 0.74061

FmaxG 1 0.88159 0.27700 0.66889

2 0.89639 0.23092 0.676633 0.91112 0.27446 0.533604 0.95557 0.16865 0.45332

SAEU 1 0.99374 0.06528 0.151812 0.99478 0.06732 0.116773 0.99512 0.06412 0.118134 0.99513 0.06376 0.12022

FmaxU 1 0.99473 0.06879 0.10292

2 0.99929 0.02005 0.052673 0.99976 0.01266 0.025714 0.99976 0.01285 0.02571

Table 5Accuracy assessment of the RSM metamodel (1 < n � 10).

Objective Order R2 Fitting indicators

RAAE RMAE

SAEG 1 0.98745 0.07209 0.295032 0.99113 0.06297 0.252973 0.99373 0.05478 0.186994 0.99525 0.05980 0.19098

FmaxG 1 0.98875 0.07963 0.21789

2 0.98928 0.08803 0.198393 0.98943 0.08366 0.216924 0.99017 0.07934 0.24795

SAEU 1 0.92491 0.16616 0.709092 0.92566 0.16722 0.724943 0.93214 0.20066 0.658864 0.93311 0.16312 0.66710

FmaxU 1 0.98995 0.09155 0.13862

2 0.99912 0.02413 0.066883 0.99939 0.02299 0.042594 0.99984 0.01013 0.02590

G. Sun et al. / International Journal of Impact Engineering 64 (2014) 62e7472

gradient yielded from the cubic RS model is 0.4686(Fmax ¼ 121.440 kN), whereas from the quartic model it is 0.8239(Fmax ¼ 133.173 kN).

In addition, the ideal optima of such two single objective func-tions in the FGT columns can be obtained from the different RSmodels. For example, in the region of (0� n� 1), themaximum SEA(23.488 kJ) is obtained from the quadratic RS model, and theminimum Fmax (107.428 kN) is yielded from the linear RSmodel. Onthe other hand, the maximum SEA (20.796 kJ) can be generatedfrom the quadratic RS model in the region of (1 < n � 10), and theminimum Fmax (106.6 kN) can be obtained from the quartic RSmodel. This indicates again that an optimal solution for a higherorder RS model does not necessarily yield the better solution.

5. Conclusions

This paper proposed to characterize and optimize the crashingperformance of thin-walled structures with functionally gradedthickness (FGT). The crashworthiness of the FGT structures wasexamined and the effects of the different thickness gradients onboth specific absorption energy (SEA) and peak force (Fmax) levels

-360 -320 -280 -240 -200 -160 -120 -8017

18

19

20

21

22

23

24

B

A

UT from Eq. (13) UT from Eq. (14) UT from Eq. (15) UT from Eq. (16)

FGT from Eq. (13) FGT from Eq. (14) FGT from Eq. (15) FGT from Eq. (16)

SEA

(kJ

/kg)

-Fmax

(kN)

-200 -180 -160 -140 -120 -10012

13

14

15

16

17

18

19

20

21

-Fmax

(kN)

SEA

(kJ

/kg)

FGT from Eq. (13) FGT from Eq. (14) FGT from Eq. (15) FGT from Eq. (16)

UT from Eq. (13) UT from Eq. (14) UT from Eq. (15) UT from Eq. (16)

Fig. 18. Pareto fronts of FGT thin-walled columns and corresponding UT columns.

Table 6Ideal optimums of the single objective functions for the FGT and UT columns(0 � n � 1).

Columns RS order Single objective Grade n SEA (kJ/kg) Fmax (kN)

UT Linear Ideal Max SEA 2.25e-08 22.580 304.143Ideal Min Fmax 1.0000 17.591 182.233

Quadratic Ideal Max SEA 1.92e-10 23.223 328.047Ideal Min Fmax 0.8826 18.376 203.937

Cubic Ideal Max SEA 6.34e-09 23.308 335.793Ideal Min Fmax 1.0000 18.150 198.390

Quartic Ideal Max SEA 5.19e-11 23.331 335.392Ideal Min Fmax 1.0000 18.173 197.989

FGT Linear Ideal Max SEA 7.24e-10 23.297 222.372Ideal Min Fmax 1.0000 20.487 107.428

Quadratic Ideal Max SEA 7.51e-10 23.488 286.808Ideal Min Fmax 0.6341 21.411 114.253

Cubic Ideal Max SEA 8.23e-09 23.264 316.184Ideal Min Fmax 0.4686 21.889 121.440

Quartic Ideal Max SEA 0.0924 23.339 222.855Ideal Min Fmax 0.8239 21.064 133.173

Table 7Ideal optimums of the two single objective functions for the FGT and UT columns(1 < n � 10).

Columns RS order Single objective Grade n SEA (kJ/kg) Fmax (kN)

UT Linear Ideal Max SEA 1.0000 16.285 174.657Ideal Min Fmax 10.0000 12.441 112.189

Quadratic Ideal Max SEA 1.0000 17.536 190.947Ideal Min Fmax 8.0568 13.092 123.352

Cubic Ideal Max SEA 1.0000 17.834 196.785Ideal Min Fmax 10.0000 13.394 122.643

Quartic Ideal Max SEA 1.0000 18.051 198.577Ideal Min Fmax 9.4953 13.277 123.946

FGT Linear Ideal Max SEA 1.0000 20.581 138.381Ideal Min Fmax 10.0000 15.496 108.557

Quadratic Ideal Max SEA 1.0000 20.796 142.125Ideal Min Fmax 10.0000 15.711 112.300

Cubic Ideal Max SEA 1.7073 20.316 137.889Ideal Min Fmax 10.0000 16.329 110.728

Quartic Ideal Max SEA 1.0000 20.538 149.415Ideal Min Fmax 8.9055 15.461 106.600

G. Sun et al. / International Journal of Impact Engineering 64 (2014) 62e74 73

were evaluated. As for such FGT structures, the gradient parameterin terms of power law exponent n has significant effect on crash-worthiness analysis and design. It was found that SEA of FGT col-umn is superior to that of the UT counterpart, when the crashingtime increased. In addition, Fmax of the FGT column is alwayssmaller than that of the corresponding UT column. To formulatesophisticated crashworthiness objective functions, the responsesurface method (RSM) was employed after validating the modelingaccuracy. In order to maximize the SEA and minimize the Fmax, themultiobjective optimizations were formulated and the Non-dominated Sorting Genetic Algorithm (NSGA-II) was adopted. Theoptimal results demonstrated that in general, the FGT column doesprovide better crashing performance than the UT column. Inter-estingly, a higher metamodeling accuracy of the RSM does notnecessarily imply that the final optimal results would be the better.It is concluded that FGT structure is of considerable implication andbenefits with significant potential in crashworthiness applications.

It must be noted that in this study the impact follows the axialdirection of thickness gradient. When oblique loads are involved,the results would be different and a new design optimization formultiple load cases will be needed, which is however beyond thescope of this paper. In addition, Nagel and Thambiratnam’s [53] andother’s works quantified the sensitivity of wall thickness to taperedcolumn. From functional point of view, the tapered column mighthave some similar behaviors to the FGT columns. However, spatial

and/or weight constraints sometimes may not allow to use “large”tapered angle in practice.

Acknowledgments

This work was supported from National 973 Project of China(2010CB328005), The National Natural Science Foundation of China(61232014, 11202072), The Doctoral Fund of Ministry of Educationof China (20120161120005), The Hunan Provincial Science Foun-dation of China (13JJ4036), and The Open Fund of Traction PowerState Key Laboratory of Southwest Jiaotong University (TPL1206).The authors would like to thank the Graduate Student InnovationProject of Hunan province, China (521298760).

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