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Lightweight Design of Automotive Front Side Rail Based on Robust Optimisation
ARTICLE IN PRESS
Thin-Walled Structures 45 (2007) 670–676 www.elsevier.com/locate/tws
Lightweight Design of Automotive Front Side Rail Based on Robust Optimisation Yu Zhang, Ping Zhu, Guanlong Chen School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China Received 26 March 2007; received in revised form 19 May 2007; accepted 31 May 2007 Available online 30 July 2007
Abstract Nowadays, both conventional automobiles and new energy cars require urgently the lightweight design to realise energy economy and environmental protection in a long run. The weight reduction of body structure plays a rather important role in decreasing the weight of full vehicle. In the real engineering problems, the variation in sheet gauge, geometrical size and material parameters, caused by environmental factors and other uncertainties, may affect the structural performances of body parts. Therefore, the lightweight design without considering this kind of tolerance may result in the loss of feasibility and reliability in engineering application. In this work, based on robust optimisation method, the study on the front side rail lightweight design is performed. The response surface method (RSM), coupled with design of experiment (DOE) technique, is employed to create the approximate functions of structural performances. The robust optimisation and deterministic optimisation formulations are constructed, respectively, for comparison. The solutions are obtained by using the sequential quadratic programming (SQP) algorithm. The lightweight design, considering the impact of the tolerance of sheet gauge, mechanical parameters of material and structural performances, is still guaranteed to be reliable when structural random varieties are present. The weight reduction achieved by using robust optimisation reached 29.96%. r 2007 Elsevier Ltd. All rights reserved. Keywords: Front side rail; Lightweight design; RSM; Robust optimisation; DOE; Structural crashworthiness
1. Introduction Energy economy and environmental protection are the crucial problems needed to be solved urgently facing the automotive industry all over the world in the 21st century. It is stated that oil consumption may decrease 6–8% once the lightweight effect of full vehicle reaches 10% [1]. Vehicle weight reduction is a primary and necessary way to realise energy savings and oil economy, which promotes the lightweight technology to be a hot research area. The weight reduction in body structure plays a rather important role in decreasing the weight of full vehicle, which results from the fact that body structure possesses about 30% weight of full vehicle. A perusal of available related literature, some of which are outlined here, revealed that a quantity of study in
Corresponding author. Tel.: +86 21 34206787.
E-mail address: [email protected] (P. Zhu). 0263-8231/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2007.05.007
lightweight design and application in body structures have been developed. The ultralight steel material was used in the lightweight design of automotive closures [2]. Structural foam material was applied in bumper and B pillar structures, reaching the lightweight effect of 37% with the guarantee of original structural performances [3]. Topology optimisation technique was utilised in the lightweight design of the air suspension and frame cross member [4]. The design of car doors was performed with the application of lightweight material and structures [5]. The lightweight design of body structure was developed, from the point of view of crashworthiness performance of full vehicle, with the application of the high strength steel and aluminium alloy materials, respectively [6]. The utility of aluminium alloy material and the improvement of structural design were integrated to achieve a satisfactory lightweight effect of the bonnet [7]. The response surface method (RSM) was applied to develop an optimisation formulation for the lightweight design of a front side rail structure [8].
ARTICLE IN PRESS Y. Zhang et al. / Thin-Walled Structures 45 (2007) 670–676
It is required that the structural performances should be guaranteed in the lightweight design of body structures. But in the open literature about autobody lightweight designs, the uncertainty in the gauge thickness, geometrical size and mechanical parameters of material is not considered during the design optimisation process, which may rise to the variation of the structural performance and make the optimum fail to meet the requirements of structural performances. The robust design, considering the variation of design variables, noise factors and structural performances, can avoid the above problem and make the lightweight solution still feasible even when the environmental random factors are considered. Based on the robust optimisation method, this paper addresses the lightweight design of the front side rail structure, by considering the tolerance of factors such as gauge thickness, yield limit of material and variation of structural performance.
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To guarantee the feasibility of design optimum, the variation of constraint functions must be considered. The worst-case approach is much more convenient and also suitable in the situation where the probability distribution attributes of design and uncontrollable variables are unknown. The variation of constraint function is obtained as follows: ! ! n k X X @g @g (3) Dxi þ Dzi : Dg ¼ @x @z i x;¯ i x;¯ ¯z ¯z i¼1 i¼1 In the situation where the constraint function is highly nonlinear and the fluctuation of variables is large, the second-order terms should be added to the approximate function. It can be formulated as follows: ! ! nþk nþk nþk X @g 1 X X @g Dg ¼ Dbi þ Dbi Dbj . @b 2 @b @b i x;¯ i j x;¯ ¯z ¯z i¼1 i¼1 j¼1 (4) Generally the robust optimisation problem can be formulated as follows [11]:
2. Worst-case robust design methodology Parkinson et al. [9,10] pointed out that robust design is an optimisation problem in essence. A key concept in robust design is that the variation of design variables and random parameters will be transferred to performance functions, leading to the variation of objective and constraint functions. The robustness of engineering problems can be identified in two categories: sensitivity robustness and feasibility robustness. It means that when the sensitivity of the objective function to the variables is reduced, the constraint conditions should also be satisfied when considering the variation caused by the fluctuation of variables. The conventional deterministic optimisation formulation is as follows: Min
f ðx; zÞ
Set:
gj ðx; zÞp0; x pxpx ; X ¼ ðx1 ; x2 ; ; xn ÞT ; L
U
i ¼ 1; 2; ; m .
(1)
Z ¼ ðz1 ; z2 ; ; zk ÞT ; Where x and z are vectors for design variables and uncontrollable variables, and f(x, z) and gj(x, z) are objective functions and the jth constraint function of m constraints, respectively. xL and xU are lower and upper bounds. In a general engineering problem the performance indicator is frequently a nonlinear function of both design and uncontrollable variables. The nonlinear function can be approximated with linearisation approach when the variation of design variables and noise factors is very small and it changes continuously[11]. It is reasonable to use the first-order Taylor series and the function can be formulated as follows: n k X X @f @f f f ðx; (2) Dxi þ Dzi . ¯ z¯ Þ þ @xi @zi i¼1 i¼1
Min
FðxÞ ¼ Eðf ðx; zÞÞ
Min
S:t:
gj ðx; ¯ z¯ Þ þ Dgj p0;
S:t:
or
gj ðx; ¯ z¯ Þ þ Dgj p0;
or
gj ðx; ¯ z¯ Þ þ Dgj p0; gj ðx; ¯ z¯ Þ þ ksgj p0; ðEðf ðx; zÞÞ y0 Þ2 p0;
Varðf ðx; zÞÞ p0; xL p¯x DxpxU ;
FðxÞ ¼ Varðf ðx; zÞÞ
or
xL p¯x DxpxU ;
(5) L
U
where x and x are lower and upper bounds and gj ðx; ¯ z¯ Þ and sgi are the mean and standard deviation of the jth constraint function, respectively. 3. Lightweight design of front side rail based on robust optimisation Front side rail is a key part of energy absorption in the frontal crash and its structural crashworthiness performance is commonly represented by the deformation mode and the absorbed energy. This can affect greatly the crash performance of full vehicle. Therefore the structural crashworthiness performance of the front side rail should be guaranteed primarily in the lightweight design process. The lightweight design has been treated as an optimisation problem in the previous study[8,12–14], where structure weight is the objective function subject to the structural performance constraints. But to ignore the impact of the tolerance of design variables, the random noise factor may result in the designed lightweight structure unable to meet the requirements of structural crashworthiness performances, which makes the design suffer from reliability and feasibility. In this study, the high strength steel material is used in the lightweight design of the front side rail, instead of the original mild steel. The material parameters are shown in Table 1. Based on the above determined lightweight structure with the high strength steel material, we will
ARTICLE IN PRESS Y. Zhang et al. / Thin-Walled Structures 45 (2007) 670–676
672 Table 1 Material parameters comparison
Mild steel (original) High strength steel (lightweight)
Density (Kg/m3)
Young’s modulus (GPa)
Yield limit stress (MPa)
Poisson’s ratio
7850
210
180.5
0.3
7850
210
343
0.3
focus on dealing with the problem of determining the gauge thicknesses of the front side rail by considering the variation of both gauge thicknesses and material yield limit. Therefore the thicknesses of the main body and upper cover panel of the front side rail are chosen as the design variables. The random noise factor is the yield limit stress of the high strength steel and its mean is 343 MPa. RSM and design off experiment (DOE) can be used to create the approximate function of structural performance in order to form (a) robust optimisation and (b) deterministic optimisation formulations for comparisons. One aim of robust optimisation in automotive industry would be to comply with the ever increasing design functional requirements for minimising weight and maximising structural stiffness. It is well known that analysis of a nonlinear design will result in the iterative design optimisation becoming computationally expensive. In addition, a robustness assessment will require possibly thousands of e.g. Monte Carlo simulations impractical. Therefore, combination of RSM (variable design space) and the information generated by DOE gives output variables in the design space as a function of the design variables which subsequently result in expeditious simulation runs. In this paper the RSM and DOE techniques are utilised to develop the approximation functions of crashworthiness performance indicators where the deterministic and robust optimisation formulations are created, respectively. The optimal solution is obtained by using seqalics quadric programming (SQP) algorithm.
of the front side rail in the full vehicle condition, which may reduce the extensive computation cost. The finite element model (LS-DYNA), containing 36,293 shell elements, 580 beam elements and 25,172 mass elements, consists of 30 parts and is subject to the load condition of impacting the rigid wall at a specific speed. The model is shown in Fig. 1 and the centre of gravity of the model, which plays a rather important role in the simulation result, is adjusted through assigning the additional mass (not including the front side rail structure). The magnitude of the initial velocity is 40 m/s against the rigid wall. The simulated deformation results of the front side rail using the equivalent model and full vehicle model are compared in Fig. 2. It can be seen from Fig. 2 that the axial crush mode, which determines the crashworthiness attribute, occurs in the equivalent model as well as in the full vehicle model. The deformation and crush mode of the front side rail in the equivalent model is consistent with the one of the full scale model. This indicates that the equivalent model in replacing the full vehicle model can be used in the robust lightweight design and hence achieving the intended crashworthiness performances described below.
3.2. Response surface models of structural performance indicators The DOE and the RSM are commonly used to construct approximate functions, avoiding the expensively computational cost in evaluating functions during the numerical optimisation process[17]. Using these metamodels, approximate optimisation and robust design can be performed. Among commonly used DOE techniques, D-Optimality design has been widely used in the crashworthiness related problems and possesses several advantages including no restriction in the quantity of sampling points and free addition of new sampling points in the design space [18]. Therefore, D-Optimality design is used in this study to construct an experimental design to sample the design space.
3.1. Finite element model and crash simulation In most research of the crashworthiness performance of a single closed hat-section structure, the quasi-static load condition is commonly used in a physical test or numerical simulation procedure[15,16]. But in the real-world crash, the deformation mode of a part may not be as identical to the one tested in the quasi-static load condition due to the synthetic impact of the high crash speed and the assembly interaction between parts. While the numerical crash simulation of a full vehicle entails high computational cost. To reduce the cost this study only focuses on the crashworthiness performance of the single part of the front side rail. Therefore we build up an equivalent finite element model that can be used to simulate the deformation mode
Fig. 1. Equivalent finite element model.
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Fig. 2. Comparison of deformed shape using the two finite element models: equivalent model and full vehicle model.
Table 2 D-optimality experimental design matrix and response values
Fig. 3. Front side rail structure.
The absorbed energy and peak crash force under the same deformation of the structure are crashworthiness performance indicators of the front side rail. The front side rail structure is shown in Fig. 3. Structural weight is regarded as the objective function, denoted by M(x). The performance indicators of peak crash force, absorbed energy and structural deformation are regarded as the constraint functions, denoted by F(x). E(x) S(x), respectively. The thicknesses of the main body and upper cover panel of the front side rail are design variables, denoted by x1, x2 (0.8 mmpx¯ i p2.5 mm, i ¼ 1,2). The yield limit stress of the material is the random noise factor, denoted by x3. The design variables x1 and x2 are subjected to normal probability distribution and x3 is subjected to logarithmic normal probability distribution (2di ¼ s=x¯ i ¼ 0.0083, i ¼ 1,2, x¯ 3 ¼ 343 MPa,Dx3 ¼ 20 MPa) [19]. Original weight is 0.008 t and the original thicknesses of the main body and upper cover panel are 1.5 and 1.0 mm, respectively. The MATLAB software was utilised to construct the D-Optimality design matrix given in Table 2. The regression coefficients are computed using the least square method and the response surface models are shown as follows: MðxÞ ¼ 0:543972e 7 þ 0:0045584x1 þ 0:0011927x2 ,
Number Design matrix x1 x2 x3
Response values M(x) (t) F(x) (N)
1 2 3 4 5 6 7 8 9 10 11
0.0133640 0.0143777 0.0143777 0.0066285 0.0066285 0.0046009 0.0094893 0.0123502 0.0123502 0.0046009 0.0084755
2.525 2.525 2.525 0.775 0.775 0.775 1.650 2.525 2.525 0.775 1.650
1.650 2.525 2.525 2.525 2.525 0.775 1.650 0.775 0.775 0.775 0.775
343.000 363.000 323.000 323.000 363.000 363.000 363.000 323.000 363.000 323.000 343.000
275,638.90 358,065.50 327,589.20 184,941.40 197,588.70 195,644.60 196,803.80 263,530.90 261,426.20 256,143.50 155,637.50
E(x) (1.0-3J)
S(x) (mm)
38,888,300 43,136,900 42,931,800 32,751,100 33,362,200 21,154,600 36,464,700 31,600,600 31,207,600 21,347,000 30,278,100
350.54615 312.14353 328.96742 504.50060 498.15044 552.41431 455.49456 426.36409 413.10921 558.38175 485.59813
F ðxÞ ¼ 9; 879; 710 53; 080:717x3 471; 827:32x1 307; 070:07x2 þ 73:98175x23 þ 81; 600:138x21 3569:0657x22 þ 560:46471x1 x3 þ 777:40588x2 x3 þ 39; 784:706x1 x2 , EðxÞ ¼ 198; 911; 000 þ 1; 166; 872:4x3 þ 22; 465; 992x1 þ 7; 952; 246:7x2 1712:9375x23 4; 519; 688:6x21 1; 361; 799:3x22 4460:2941x3 x1 þ 10; 305:882x3 x2 60; 761:246x1 x2 , SðxÞ ¼ 5455:82 28:392782x3 þ 44:099373x1 23:26317x2 þ 0:0413866x23 20:187467x21 þ 4:2148893x22 0:1305968x3 x1 0:0290568x3 x2 15:608535x1 x2 .
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The R square (R2), adjusted R square (R2Adj) and root mean square error (RMSE) are employed as regression fitness measures and the regression analysis results are shown in Table 3. The model fits of structural mass, absorbed energy and structural deformation regressions are very good. Although the model of the peak crash force is constructed with the adjusted R-square value of 0.94837, it is still acceptable with the RMSE% value of 1.86%. Response surfaces of peak crash force, absorbed energy and structural deformation are shown in Fig. 4. 3.3. Robust optimisation formulation In deterministic optimisation the tolerance of all variables is not considered, so the formulation can be constructed by replacing the variable x3 with the nominal value of yield limit (343 MPa) in the objective and constraint functions. The deterministic optimisation formulation of the lightweight design of the front side rail Table 3 Results of regression analysis
M(x) F(x) E(x) S(x)
R2
R2
RMSE
RMSE%
1.00000 0.99484 0.99999 0.99998
1.00000 0.94837 0.99990 0.99982
6.32e008 4.53e+003 9.56e+004 0.36
0.00065 1.86 0.29 0.08
is as follows: Min
MðxÞ
S:t:
F ðxÞX135; 993:7; EðxÞX27; 394; 800; SðxÞp535:667; 0:775pxi p2:525;
(6) i ¼ 1; 2:
But in robust optimisation all variables are considered stochastic and subjected to a probability distribution. The design variables x1 and x2 are mutually independent and subjected to normal probability distribution. Therefore the objective function M(x), which is the linear function of x1 and x2, is also subjected to normal probability distribution. The mean and variance functions are obtained as follows: EðMðxÞÞ ¼ 0:543972e 7 þ 0:0045584x1 þ 0:0011927x2 , VarðMðxÞÞ ¼ 0:1329856676e 8x21 þ 0:9104213056e 10x22 .
The statistical characteristic parameters of the design variables x1 and x2 are known. Therefore the worst-case tolerance can be obtained. The variation of the constraint responses including peak crash force, absorbed energy and structural deformation is calculated according to formulation (3) and (4). The stochastic variables x1, x2 and x3 in all functions are replaced with their means x¯ 1 ,x¯ 2 and 343, respectively. The worst-case robust optimisation of
Fig. 4. Response surfaces of structural performance indicators: force (x2 ¼ 0.8 mm), energy(x2 ¼ 2.5 mm), deformation (x2 ¼ 0.8 mm).
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lightweight design is formulated as follows: Min
EðMðxÞÞ
S:t: E 1 ðxÞp0
F 1 ðxÞp0
Table 4 Comparison of weight reduction effect of front side rail
(7)
S 1 ðxÞp0 VarðxÞp0 0:775pxi p2:525;
675
Deterministic optimisation Robust optimisation
i ¼ 1; 2;
where
Thickness of main body x1(mm)
Thickness of upper cover panel x2(mm)
Mass after reduced weight (t)
Mass reduction effect(%)
0.8
1.452
0.005378
32.78
0.8
1.640
0.005603
29.96
F 1 ðxÞ ¼ 135993:7 FðxÞ þ DFðxÞ ¼ 240; 909:276 þ 279; 868:1569x1 þ 40; 808:55614x2 81; 600:138x21 þ 3569:0657x22 39; 752:84056x1 x2 þ 0:5 102:0001725x2 4:461332125x2 þ 59; 185:4 þ 0:025x1 1
2
j279; 587:9245 þ 163; 200:276x1 þ 39; 784:706x2 j þ 0:025x2 j40; 412:8532 7138:1314x2 þ 39; 784:706x1 j þ j46; 584:73 þ 11; 209:2942x1 þ 15; 548:1176x2 j,
E 1 ðxÞ ¼ 27; 394; 800 EðxÞ þ DEðxÞ ¼ 28; 279; 125:7 20; 933; 880:97295x1 11; 482; 011:289x2 þ 4; 522; 513:405375x21 þ 1; 362; 650:4245625x22 þ 60; 799:22177876x1 x2 þ 0:025x1 j20; 936; 111:12 9; 039; 377:2x1 60; 761:246x2 j þ 0:025x2 j11; 487; 164:23 27; 23; 598:6x2 60; 761:246x1 j þ j164; 054:5 þ 89; 205:882x1 206; 117:64x2 j,
S1 ðxÞ ¼ SðxÞ þ DSðxÞ 535:667 ¼ 50:520877 0:630031x1 33:215124x2 20:187467x21 þ 4:2148893x22 15:598779666 x1 x2 þ 0:5 j0:02523433375x21 0:005268611625x22 33:1092800j þ 0:025x1 j0:6953294 40:374934x1 15:608535x2 j þ 0:025x2 j33:2296524 þ 8:4297786x2 15:608535x1 j þ j0:031488 þ 2:611936x1 þ 0:581136x2 j, VarðxÞ ¼ 0:1329856676e 8x21 þ 0:9104213056e 10x22 5:0e 9.
The SQP algorithm is employed to solve the multiconstraint optimisation problem. The results are listed as follows: x1 ¼ 0:8;
x2 ¼ 1:64;
M min ¼ 0:005603:
In order to compare the above with the results of the deterministic optimisation, we have obtained the solution to the formulation (6) using SQP algorithm. They are as follows: x1 ¼ 0:8;
x2 ¼ 1:452;
M min ¼ 0:005378:
The weight reduction effect is shown in Table 4. The thicknesses of the main body and upper cover panel of the front side rail obtained from deterministic optimisation formulation are 0.8 and 1.452 mm, respectively. The lightweight effect reaches 32.78%. While the thicknesses
of the main body and upper cover panel from robust optimisation formulation are 0.8 and 1.64 mm, respectively, and the weight reduction effect is 29.96%. As expected, the results from robust optimisation are conservative compared to the ones from strictly minimum weight formulation, which results from the fact that the variation of the design variables and the noise factor is taken into account. It means that the lightweight design optimum is still reliable when the worst-case tolerances of design variables and noise factors are considered simultaneously. 4. Conclusion The vehicle lightweight design without considering the variation of the gauge thickness and material parameter will attenuate the reliability and feasibility in engineering applications. Based on the robust optimisation method, this paper addresses the lightweight design of the front side rail in autobody structure, considering the variation of the design variable and random noise factor. The thicknesses of the main body and upper cover panel of the front side rail are design variables and the yield limit stress of the material is the random noise factor. Compared with the results of deterministic optimisation, the results from robust optimisation are more conservative. But the lightweight design utilising robust optimisation is still reliable when the worst-case tolerances of design variables and noise factors are considered simultaneously. The thicknesses of the main body and upper cover panel from the robust optimisation are 0.8 and 1.64 mm, respectively, and the lightweight effect shows a reduction of 29.96%. Acknowledgement The work presented in this paper was supported by the National High Technology Research and Development Program of China (863 Program): No.2006AA04Z126. Reference [1] Joseph C, Benedy K. Light metals in automotive applications. Light Metal Age 2000;58(10):34–5.
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[2] Koehr R. ULSAC-lightweight steel automotive closures. Transactions of SAE International, 2001-01-0076, 2001. p. 21–8. [3] Lanzerath H, Schilling R. Crash simulation of structural foam. Transactions of SAE International, 2003-01-0328, 2003. p. 73–81. [4] Vanderplaats GN. Saving energy through design optimization, Transactions of SAE International, 2003-01-1331, 2003. p. 587–93. [5] Wang HY, Xu SY. A study on lightweight design of car door. Automotive Eng 2004;3(26):349–53. [6] Ping Z, Yu Z, Guanlong C, Zhongqin L. Research on lightweight of auto-body material based on crashworthiness simulation. Chin J Mech Eng 2005;41(9):207–11. [7] Yu Z, Ping Z, Chen G-L, Lin Z-Q. Lightweight design of bonnet in auto-body based on finite element method. J Shanghai Jiao Tong Univ 2006;40(1):163–6. [8] Zhang Y, Zhu P, Chen GL, Lin ZQ. Study on structural lightweight design of automotive front side rail based on response surface method. ASME J Mech Des 2007;129:553–7. [9] Parkinson A, Sorensen C, Pourhassan N. A general approach for robust optimal design. ASME J Mech Des 1993;115(1):74–9. [10] Parkinson A. Robust mechanical design using engineering models. J Mech Des Trans ASME 1995;117:48–54. [11] Chen LZ. Robust design. Beijing: China Machine Press; 2000.
[12] Marklund PO, Nilsson L. Optimization of a car body component subjected to side impact. Struct Multidisc Optim 2001;21:383–92. [13] Sandgren E, Cameron TM. Robust design optimization of structures through consideration of variation, Comput Struct 2002;80:1605–13. [14] Youn BD, Choi KK, Yang R-J, Gu L. Reliability-based design optimization for crashworthiness of vehicle side impact. Struct Multidisc Optim 2004;26:272–83. [15] Kim HS, Wierzbicki T. ‘‘Effect of the cross-sectional shape on crash behavior of a 3-D space frame’’. Impact and Crashworthiness Laboratory Report No. 34, MIT, May 2000. [16] Wang HL. Study on optimal design of autobody structure based on crashworthiness numerical simulation [D]. Shanghai Shanghai Jiao Tong University, School of Mechanical Engineering, 2002. [17] Yang RJ, Wang N, Tho CH, et al. Metamodeling development for vehicle frontal impact simulation. ASME J Mech Des 2005;127(3): 1014–20. [18] Redhle M, Forsberg J, Jansson T, Marklund PO, Nilsson L. Using the response surface methodology and the d-optimality criterion in crashworthiness related problems. Struct Multidisc Optim 2002; 24:185–94. [19] Chen LZ. Optimization design method for engineering stochastic variables. Beijing: Science Press; 1997.
Thin-Walled Structures 45 (2007) 670–676 www.elsevier.com/locate/tws
Lightweight Design of Automotive Front Side Rail Based on Robust Optimisation Yu Zhang, Ping Zhu, Guanlong Chen School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China Received 26 March 2007; received in revised form 19 May 2007; accepted 31 May 2007 Available online 30 July 2007
Abstract Nowadays, both conventional automobiles and new energy cars require urgently the lightweight design to realise energy economy and environmental protection in a long run. The weight reduction of body structure plays a rather important role in decreasing the weight of full vehicle. In the real engineering problems, the variation in sheet gauge, geometrical size and material parameters, caused by environmental factors and other uncertainties, may affect the structural performances of body parts. Therefore, the lightweight design without considering this kind of tolerance may result in the loss of feasibility and reliability in engineering application. In this work, based on robust optimisation method, the study on the front side rail lightweight design is performed. The response surface method (RSM), coupled with design of experiment (DOE) technique, is employed to create the approximate functions of structural performances. The robust optimisation and deterministic optimisation formulations are constructed, respectively, for comparison. The solutions are obtained by using the sequential quadratic programming (SQP) algorithm. The lightweight design, considering the impact of the tolerance of sheet gauge, mechanical parameters of material and structural performances, is still guaranteed to be reliable when structural random varieties are present. The weight reduction achieved by using robust optimisation reached 29.96%. r 2007 Elsevier Ltd. All rights reserved. Keywords: Front side rail; Lightweight design; RSM; Robust optimisation; DOE; Structural crashworthiness
1. Introduction Energy economy and environmental protection are the crucial problems needed to be solved urgently facing the automotive industry all over the world in the 21st century. It is stated that oil consumption may decrease 6–8% once the lightweight effect of full vehicle reaches 10% [1]. Vehicle weight reduction is a primary and necessary way to realise energy savings and oil economy, which promotes the lightweight technology to be a hot research area. The weight reduction in body structure plays a rather important role in decreasing the weight of full vehicle, which results from the fact that body structure possesses about 30% weight of full vehicle. A perusal of available related literature, some of which are outlined here, revealed that a quantity of study in
Corresponding author. Tel.: +86 21 34206787.
E-mail address: [email protected] (P. Zhu). 0263-8231/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2007.05.007
lightweight design and application in body structures have been developed. The ultralight steel material was used in the lightweight design of automotive closures [2]. Structural foam material was applied in bumper and B pillar structures, reaching the lightweight effect of 37% with the guarantee of original structural performances [3]. Topology optimisation technique was utilised in the lightweight design of the air suspension and frame cross member [4]. The design of car doors was performed with the application of lightweight material and structures [5]. The lightweight design of body structure was developed, from the point of view of crashworthiness performance of full vehicle, with the application of the high strength steel and aluminium alloy materials, respectively [6]. The utility of aluminium alloy material and the improvement of structural design were integrated to achieve a satisfactory lightweight effect of the bonnet [7]. The response surface method (RSM) was applied to develop an optimisation formulation for the lightweight design of a front side rail structure [8].
ARTICLE IN PRESS Y. Zhang et al. / Thin-Walled Structures 45 (2007) 670–676
It is required that the structural performances should be guaranteed in the lightweight design of body structures. But in the open literature about autobody lightweight designs, the uncertainty in the gauge thickness, geometrical size and mechanical parameters of material is not considered during the design optimisation process, which may rise to the variation of the structural performance and make the optimum fail to meet the requirements of structural performances. The robust design, considering the variation of design variables, noise factors and structural performances, can avoid the above problem and make the lightweight solution still feasible even when the environmental random factors are considered. Based on the robust optimisation method, this paper addresses the lightweight design of the front side rail structure, by considering the tolerance of factors such as gauge thickness, yield limit of material and variation of structural performance.
671
To guarantee the feasibility of design optimum, the variation of constraint functions must be considered. The worst-case approach is much more convenient and also suitable in the situation where the probability distribution attributes of design and uncontrollable variables are unknown. The variation of constraint function is obtained as follows: ! ! n k X X @g @g (3) Dxi þ Dzi : Dg ¼ @x @z i x;¯ i x;¯ ¯z ¯z i¼1 i¼1 In the situation where the constraint function is highly nonlinear and the fluctuation of variables is large, the second-order terms should be added to the approximate function. It can be formulated as follows: ! ! nþk nþk nþk X @g 1 X X @g Dg ¼ Dbi þ Dbi Dbj . @b 2 @b @b i x;¯ i j x;¯ ¯z ¯z i¼1 i¼1 j¼1 (4) Generally the robust optimisation problem can be formulated as follows [11]:
2. Worst-case robust design methodology Parkinson et al. [9,10] pointed out that robust design is an optimisation problem in essence. A key concept in robust design is that the variation of design variables and random parameters will be transferred to performance functions, leading to the variation of objective and constraint functions. The robustness of engineering problems can be identified in two categories: sensitivity robustness and feasibility robustness. It means that when the sensitivity of the objective function to the variables is reduced, the constraint conditions should also be satisfied when considering the variation caused by the fluctuation of variables. The conventional deterministic optimisation formulation is as follows: Min
f ðx; zÞ
Set:
gj ðx; zÞp0; x pxpx ; X ¼ ðx1 ; x2 ; ; xn ÞT ; L
U
i ¼ 1; 2; ; m .
(1)
Z ¼ ðz1 ; z2 ; ; zk ÞT ; Where x and z are vectors for design variables and uncontrollable variables, and f(x, z) and gj(x, z) are objective functions and the jth constraint function of m constraints, respectively. xL and xU are lower and upper bounds. In a general engineering problem the performance indicator is frequently a nonlinear function of both design and uncontrollable variables. The nonlinear function can be approximated with linearisation approach when the variation of design variables and noise factors is very small and it changes continuously[11]. It is reasonable to use the first-order Taylor series and the function can be formulated as follows: n k X X @f @f f f ðx; (2) Dxi þ Dzi . ¯ z¯ Þ þ @xi @zi i¼1 i¼1
Min
FðxÞ ¼ Eðf ðx; zÞÞ
Min
S:t:
gj ðx; ¯ z¯ Þ þ Dgj p0;
S:t:
or
gj ðx; ¯ z¯ Þ þ Dgj p0;
or
gj ðx; ¯ z¯ Þ þ Dgj p0; gj ðx; ¯ z¯ Þ þ ksgj p0; ðEðf ðx; zÞÞ y0 Þ2 p0;
Varðf ðx; zÞÞ p0; xL p¯x DxpxU ;
FðxÞ ¼ Varðf ðx; zÞÞ
or
xL p¯x DxpxU ;
(5) L
U
where x and x are lower and upper bounds and gj ðx; ¯ z¯ Þ and sgi are the mean and standard deviation of the jth constraint function, respectively. 3. Lightweight design of front side rail based on robust optimisation Front side rail is a key part of energy absorption in the frontal crash and its structural crashworthiness performance is commonly represented by the deformation mode and the absorbed energy. This can affect greatly the crash performance of full vehicle. Therefore the structural crashworthiness performance of the front side rail should be guaranteed primarily in the lightweight design process. The lightweight design has been treated as an optimisation problem in the previous study[8,12–14], where structure weight is the objective function subject to the structural performance constraints. But to ignore the impact of the tolerance of design variables, the random noise factor may result in the designed lightweight structure unable to meet the requirements of structural crashworthiness performances, which makes the design suffer from reliability and feasibility. In this study, the high strength steel material is used in the lightweight design of the front side rail, instead of the original mild steel. The material parameters are shown in Table 1. Based on the above determined lightweight structure with the high strength steel material, we will
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672 Table 1 Material parameters comparison
Mild steel (original) High strength steel (lightweight)
Density (Kg/m3)
Young’s modulus (GPa)
Yield limit stress (MPa)
Poisson’s ratio
7850
210
180.5
0.3
7850
210
343
0.3
focus on dealing with the problem of determining the gauge thicknesses of the front side rail by considering the variation of both gauge thicknesses and material yield limit. Therefore the thicknesses of the main body and upper cover panel of the front side rail are chosen as the design variables. The random noise factor is the yield limit stress of the high strength steel and its mean is 343 MPa. RSM and design off experiment (DOE) can be used to create the approximate function of structural performance in order to form (a) robust optimisation and (b) deterministic optimisation formulations for comparisons. One aim of robust optimisation in automotive industry would be to comply with the ever increasing design functional requirements for minimising weight and maximising structural stiffness. It is well known that analysis of a nonlinear design will result in the iterative design optimisation becoming computationally expensive. In addition, a robustness assessment will require possibly thousands of e.g. Monte Carlo simulations impractical. Therefore, combination of RSM (variable design space) and the information generated by DOE gives output variables in the design space as a function of the design variables which subsequently result in expeditious simulation runs. In this paper the RSM and DOE techniques are utilised to develop the approximation functions of crashworthiness performance indicators where the deterministic and robust optimisation formulations are created, respectively. The optimal solution is obtained by using seqalics quadric programming (SQP) algorithm.
of the front side rail in the full vehicle condition, which may reduce the extensive computation cost. The finite element model (LS-DYNA), containing 36,293 shell elements, 580 beam elements and 25,172 mass elements, consists of 30 parts and is subject to the load condition of impacting the rigid wall at a specific speed. The model is shown in Fig. 1 and the centre of gravity of the model, which plays a rather important role in the simulation result, is adjusted through assigning the additional mass (not including the front side rail structure). The magnitude of the initial velocity is 40 m/s against the rigid wall. The simulated deformation results of the front side rail using the equivalent model and full vehicle model are compared in Fig. 2. It can be seen from Fig. 2 that the axial crush mode, which determines the crashworthiness attribute, occurs in the equivalent model as well as in the full vehicle model. The deformation and crush mode of the front side rail in the equivalent model is consistent with the one of the full scale model. This indicates that the equivalent model in replacing the full vehicle model can be used in the robust lightweight design and hence achieving the intended crashworthiness performances described below.
3.2. Response surface models of structural performance indicators The DOE and the RSM are commonly used to construct approximate functions, avoiding the expensively computational cost in evaluating functions during the numerical optimisation process[17]. Using these metamodels, approximate optimisation and robust design can be performed. Among commonly used DOE techniques, D-Optimality design has been widely used in the crashworthiness related problems and possesses several advantages including no restriction in the quantity of sampling points and free addition of new sampling points in the design space [18]. Therefore, D-Optimality design is used in this study to construct an experimental design to sample the design space.
3.1. Finite element model and crash simulation In most research of the crashworthiness performance of a single closed hat-section structure, the quasi-static load condition is commonly used in a physical test or numerical simulation procedure[15,16]. But in the real-world crash, the deformation mode of a part may not be as identical to the one tested in the quasi-static load condition due to the synthetic impact of the high crash speed and the assembly interaction between parts. While the numerical crash simulation of a full vehicle entails high computational cost. To reduce the cost this study only focuses on the crashworthiness performance of the single part of the front side rail. Therefore we build up an equivalent finite element model that can be used to simulate the deformation mode
Fig. 1. Equivalent finite element model.
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Fig. 2. Comparison of deformed shape using the two finite element models: equivalent model and full vehicle model.
Table 2 D-optimality experimental design matrix and response values
Fig. 3. Front side rail structure.
The absorbed energy and peak crash force under the same deformation of the structure are crashworthiness performance indicators of the front side rail. The front side rail structure is shown in Fig. 3. Structural weight is regarded as the objective function, denoted by M(x). The performance indicators of peak crash force, absorbed energy and structural deformation are regarded as the constraint functions, denoted by F(x). E(x) S(x), respectively. The thicknesses of the main body and upper cover panel of the front side rail are design variables, denoted by x1, x2 (0.8 mmpx¯ i p2.5 mm, i ¼ 1,2). The yield limit stress of the material is the random noise factor, denoted by x3. The design variables x1 and x2 are subjected to normal probability distribution and x3 is subjected to logarithmic normal probability distribution (2di ¼ s=x¯ i ¼ 0.0083, i ¼ 1,2, x¯ 3 ¼ 343 MPa,Dx3 ¼ 20 MPa) [19]. Original weight is 0.008 t and the original thicknesses of the main body and upper cover panel are 1.5 and 1.0 mm, respectively. The MATLAB software was utilised to construct the D-Optimality design matrix given in Table 2. The regression coefficients are computed using the least square method and the response surface models are shown as follows: MðxÞ ¼ 0:543972e 7 þ 0:0045584x1 þ 0:0011927x2 ,
Number Design matrix x1 x2 x3
Response values M(x) (t) F(x) (N)
1 2 3 4 5 6 7 8 9 10 11
0.0133640 0.0143777 0.0143777 0.0066285 0.0066285 0.0046009 0.0094893 0.0123502 0.0123502 0.0046009 0.0084755
2.525 2.525 2.525 0.775 0.775 0.775 1.650 2.525 2.525 0.775 1.650
1.650 2.525 2.525 2.525 2.525 0.775 1.650 0.775 0.775 0.775 0.775
343.000 363.000 323.000 323.000 363.000 363.000 363.000 323.000 363.000 323.000 343.000
275,638.90 358,065.50 327,589.20 184,941.40 197,588.70 195,644.60 196,803.80 263,530.90 261,426.20 256,143.50 155,637.50
E(x) (1.0-3J)
S(x) (mm)
38,888,300 43,136,900 42,931,800 32,751,100 33,362,200 21,154,600 36,464,700 31,600,600 31,207,600 21,347,000 30,278,100
350.54615 312.14353 328.96742 504.50060 498.15044 552.41431 455.49456 426.36409 413.10921 558.38175 485.59813
F ðxÞ ¼ 9; 879; 710 53; 080:717x3 471; 827:32x1 307; 070:07x2 þ 73:98175x23 þ 81; 600:138x21 3569:0657x22 þ 560:46471x1 x3 þ 777:40588x2 x3 þ 39; 784:706x1 x2 , EðxÞ ¼ 198; 911; 000 þ 1; 166; 872:4x3 þ 22; 465; 992x1 þ 7; 952; 246:7x2 1712:9375x23 4; 519; 688:6x21 1; 361; 799:3x22 4460:2941x3 x1 þ 10; 305:882x3 x2 60; 761:246x1 x2 , SðxÞ ¼ 5455:82 28:392782x3 þ 44:099373x1 23:26317x2 þ 0:0413866x23 20:187467x21 þ 4:2148893x22 0:1305968x3 x1 0:0290568x3 x2 15:608535x1 x2 .
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The R square (R2), adjusted R square (R2Adj) and root mean square error (RMSE) are employed as regression fitness measures and the regression analysis results are shown in Table 3. The model fits of structural mass, absorbed energy and structural deformation regressions are very good. Although the model of the peak crash force is constructed with the adjusted R-square value of 0.94837, it is still acceptable with the RMSE% value of 1.86%. Response surfaces of peak crash force, absorbed energy and structural deformation are shown in Fig. 4. 3.3. Robust optimisation formulation In deterministic optimisation the tolerance of all variables is not considered, so the formulation can be constructed by replacing the variable x3 with the nominal value of yield limit (343 MPa) in the objective and constraint functions. The deterministic optimisation formulation of the lightweight design of the front side rail Table 3 Results of regression analysis
M(x) F(x) E(x) S(x)
R2
R2
RMSE
RMSE%
1.00000 0.99484 0.99999 0.99998
1.00000 0.94837 0.99990 0.99982
6.32e008 4.53e+003 9.56e+004 0.36
0.00065 1.86 0.29 0.08
is as follows: Min
MðxÞ
S:t:
F ðxÞX135; 993:7; EðxÞX27; 394; 800; SðxÞp535:667; 0:775pxi p2:525;
(6) i ¼ 1; 2:
But in robust optimisation all variables are considered stochastic and subjected to a probability distribution. The design variables x1 and x2 are mutually independent and subjected to normal probability distribution. Therefore the objective function M(x), which is the linear function of x1 and x2, is also subjected to normal probability distribution. The mean and variance functions are obtained as follows: EðMðxÞÞ ¼ 0:543972e 7 þ 0:0045584x1 þ 0:0011927x2 , VarðMðxÞÞ ¼ 0:1329856676e 8x21 þ 0:9104213056e 10x22 .
The statistical characteristic parameters of the design variables x1 and x2 are known. Therefore the worst-case tolerance can be obtained. The variation of the constraint responses including peak crash force, absorbed energy and structural deformation is calculated according to formulation (3) and (4). The stochastic variables x1, x2 and x3 in all functions are replaced with their means x¯ 1 ,x¯ 2 and 343, respectively. The worst-case robust optimisation of
Fig. 4. Response surfaces of structural performance indicators: force (x2 ¼ 0.8 mm), energy(x2 ¼ 2.5 mm), deformation (x2 ¼ 0.8 mm).
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lightweight design is formulated as follows: Min
EðMðxÞÞ
S:t: E 1 ðxÞp0
F 1 ðxÞp0
Table 4 Comparison of weight reduction effect of front side rail
(7)
S 1 ðxÞp0 VarðxÞp0 0:775pxi p2:525;
675
Deterministic optimisation Robust optimisation
i ¼ 1; 2;
where
Thickness of main body x1(mm)
Thickness of upper cover panel x2(mm)
Mass after reduced weight (t)
Mass reduction effect(%)
0.8
1.452
0.005378
32.78
0.8
1.640
0.005603
29.96
F 1 ðxÞ ¼ 135993:7 FðxÞ þ DFðxÞ ¼ 240; 909:276 þ 279; 868:1569x1 þ 40; 808:55614x2 81; 600:138x21 þ 3569:0657x22 39; 752:84056x1 x2 þ 0:5 102:0001725x2 4:461332125x2 þ 59; 185:4 þ 0:025x1 1
2
j279; 587:9245 þ 163; 200:276x1 þ 39; 784:706x2 j þ 0:025x2 j40; 412:8532 7138:1314x2 þ 39; 784:706x1 j þ j46; 584:73 þ 11; 209:2942x1 þ 15; 548:1176x2 j,
E 1 ðxÞ ¼ 27; 394; 800 EðxÞ þ DEðxÞ ¼ 28; 279; 125:7 20; 933; 880:97295x1 11; 482; 011:289x2 þ 4; 522; 513:405375x21 þ 1; 362; 650:4245625x22 þ 60; 799:22177876x1 x2 þ 0:025x1 j20; 936; 111:12 9; 039; 377:2x1 60; 761:246x2 j þ 0:025x2 j11; 487; 164:23 27; 23; 598:6x2 60; 761:246x1 j þ j164; 054:5 þ 89; 205:882x1 206; 117:64x2 j,
S1 ðxÞ ¼ SðxÞ þ DSðxÞ 535:667 ¼ 50:520877 0:630031x1 33:215124x2 20:187467x21 þ 4:2148893x22 15:598779666 x1 x2 þ 0:5 j0:02523433375x21 0:005268611625x22 33:1092800j þ 0:025x1 j0:6953294 40:374934x1 15:608535x2 j þ 0:025x2 j33:2296524 þ 8:4297786x2 15:608535x1 j þ j0:031488 þ 2:611936x1 þ 0:581136x2 j, VarðxÞ ¼ 0:1329856676e 8x21 þ 0:9104213056e 10x22 5:0e 9.
The SQP algorithm is employed to solve the multiconstraint optimisation problem. The results are listed as follows: x1 ¼ 0:8;
x2 ¼ 1:64;
M min ¼ 0:005603:
In order to compare the above with the results of the deterministic optimisation, we have obtained the solution to the formulation (6) using SQP algorithm. They are as follows: x1 ¼ 0:8;
x2 ¼ 1:452;
M min ¼ 0:005378:
The weight reduction effect is shown in Table 4. The thicknesses of the main body and upper cover panel of the front side rail obtained from deterministic optimisation formulation are 0.8 and 1.452 mm, respectively. The lightweight effect reaches 32.78%. While the thicknesses
of the main body and upper cover panel from robust optimisation formulation are 0.8 and 1.64 mm, respectively, and the weight reduction effect is 29.96%. As expected, the results from robust optimisation are conservative compared to the ones from strictly minimum weight formulation, which results from the fact that the variation of the design variables and the noise factor is taken into account. It means that the lightweight design optimum is still reliable when the worst-case tolerances of design variables and noise factors are considered simultaneously. 4. Conclusion The vehicle lightweight design without considering the variation of the gauge thickness and material parameter will attenuate the reliability and feasibility in engineering applications. Based on the robust optimisation method, this paper addresses the lightweight design of the front side rail in autobody structure, considering the variation of the design variable and random noise factor. The thicknesses of the main body and upper cover panel of the front side rail are design variables and the yield limit stress of the material is the random noise factor. Compared with the results of deterministic optimisation, the results from robust optimisation are more conservative. But the lightweight design utilising robust optimisation is still reliable when the worst-case tolerances of design variables and noise factors are considered simultaneously. The thicknesses of the main body and upper cover panel from the robust optimisation are 0.8 and 1.64 mm, respectively, and the lightweight effect shows a reduction of 29.96%. Acknowledgement The work presented in this paper was supported by the National High Technology Research and Development Program of China (863 Program): No.2006AA04Z126. Reference [1] Joseph C, Benedy K. Light metals in automotive applications. Light Metal Age 2000;58(10):34–5.
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