COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China
©2006 Tsinghua University Press & Springer
Optimization Studies for Crashworthiness Design using Response Surface
Method
Xingtao Liao 1*, Qing Li 1, 2, Weigang Zhang 1
1
2
State Key Laboratory of Advanced Design and Manufacture for Vehicle Body, Hunan University, Changsha, 410082
China
School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW 2006
Australia
Email: [email protected], [email protected]
Abstract In the automotive industry, structural optimization related to crashworthiness and energy absorption
capability is of great importance, which involves highly nonlinear computational analysis and design with many
material and structure parameters. Unfortunately, conventional design analysis techniques can only improve the
structural crashworthiness to a limited extent. This paper developed an innovative Response Surface Method to tackle
the crashworthiness design problems, where the numbers of FE analyses are significantly reduced. This paper also
combined Response Surface Method with variable screening technique, the sequential approximation optimization
method and nonlinear finite element code to improve the performance of Response Surface Method. The optimization
method is applied to solve crashworthiness design problems, which include one thin plate with a square hole and one
simplified front rail structure of vehicle. The results demonstrate that the new computational design method is efficient
and effective in solving crashworthiness design optimization problems.
Key words: Response Surface Method, Optimization, Crashworthiness, Nonlinear finite element analysis
INTRODUCTION
During recent years, the use of Finite Element Method (FEM) has been an important means to vehicle structural design
related to crashworthiness, mainly due to rapidly-increasing computer power and improved algorithms. The objective
and constraints of vehicle crashworthiness involves many parameters of material and structure. Unfortunately,
conventional design method requires trial-and-error. The crash simulation and experiment can only improve the
structural crashworthiness to a certain extent, but usually can not achieve a global optimization performance. On the
other hand, conventional optimization method would largely suffer from the difficulties when it applies to the
crashworthiness optimization. First, conventional optimization method necessitates sensitivity analysis, which usually
there are significant difficulties for a non-linear sensitivity analysis as a result of unilateral contact and large
deformation. Second, a ‘noisy’ response may also result in sub-optimal solutions caused by the multitude of local
minima. Third, crashworthiness simulations require extensive computational resources and time. As a result, the
crashworthiness design should be performed using a few crashworthiness analyses as possible.
For this reason, this paper develops an alternative optimization method: Response Surface Method [1, 2], which is a
kind of global optimization method and has capability to tackle the ‘noisy’ response problems. Response Surface
Method constructs smooth approximations of computationally expensive and non-smooth objective and constraint
functions using limited number of sampling crashworthiness simulations. Successively using these smooth
approximations in the optimization sub-problems can result in Successive Response Surface Method [3].
In general, the Successive Response Method needs a series of iterations and is time-consuming for the crashworthiness
analyses. Hence, this paper utilizes variable screening method and linear response surface model to improve the
computing efficiency of the Successive Response Method and remarkably reduce the computing time.
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RESPONSE SURFACE METHOD
1. Response Surface Method Response Surface Method is a method that uses basis functions to construct global
approximations of the objective and constraint functions in the design space. The selection of basis functions is
essential. These functions can be polynomials with any order or the sum of different basis functions, e.g. sine and
cosine functions. Response function can be thus defined as:
y ( x ) = ∑ j =1 a jϕ j ( x )
N
(1)
Where N represents the numbers of basis function ϕ j ( x ) , a j represents the tuning parameters for the basis function
ϕ j ( x ) , which represent design variables [3] .When a linear model is employed, the forms of ϕ j ( x ) are taken as
1, x1 , x2 , L , xn
(2)
While in a full quadratic model, the forms are
1, x1 , x2 , L , xn , x12 , x1 x2 , L , x1 xn , L , xn2
If the finite element simulation results f = ( f
(3)
(1)
f (2) L f ( M ) )T at some carefully selected M design sampling points
( M > N ) are obtained, the unknown tuning parameters a = ( a1 a2 L a N ) can be numerically determined by means
of the least-squares method. At the i-th design point xi , the error between the finite element analysis and response
surface representation can be expressed as
ε i = f ( i ) − y ( i ) = f ( i ) − ∑ j =1 a jϕ j ( x ( i ) )
N
(4)
And the sum of the square of the errors is given as
E ( a) = ∑ i =1 ε i2 = ∑ i =1 ⎡ y ( i ) − ∑ j =1 a jϕ j ( x ( i ) ) ⎤
⎣
⎦
M
M
N
2
(5)
In order to minimize the error ε i , the least square method is used to find the estimations of a j , which lead to,
a = ( X T X ) −1 ( X T y )
(6)
where X is the matrix consisting of basis function results evaluated at the design sample points and is given as
X = [ Xui ] = [φ i ( xu )]
(7)
It should be pointed out that samples of designs need to be well selected. Randomly selected designs may cause an
inaccurate surface being constructed or even lead to wrong response. For this reason, the theory of experimental design
(DOE) is needed.
2. Experimental design (DOE) This paper adopts the D-optimal design [4] [5], which takes a subset of all the possible
design points for a basis to determine X T X⏐. The subset is usually selected from factorial design and X T X⏐ is
inversely proportional to the volume of the confidence region of the regression coefficients. Hence, a better
approximation is reached if X T X⏐ is maximized.
3. Successive approximate optimization Mathematical and engineering optimization literature usually presents the
problem in a standard form as
min （or max）f0(x)
(8)
s.t. f j ( x ) ≤ 0 ；j=1, 2, …, m
xil ≤ xi ≤ xiu ； i=1, 2, …, n
where f 0 (x), f j are functions of independent variables x1, x2, x3,…,xn. The function f0(x), referred to as the cost or
objective function, identifies the quantity to be minimized or maximized. The functions f j (x) denote constraints which
represent the design restrictions. The variables collectively described by the vector x are often referred to as design
variables. In a vehicle crash problem, the function f 0 (x) and f j (x) usually relate to the occupant’s safety and the
vehicle structure integrity. When the highly non-linear problems such as vehicle crashworthiness are involved, the
⎯ 1026 ⎯
optimization may converge to local minima, instead of a true global optimum solution. The paper presents a more
effective strategy: Successive approximate optimization [3], which represents the optimization problem in (8) in a series
of simpler approximate sub-problems. In this approach, the solutions to the sub-problems are expected to yield the
optimum of the original optimization problem. As such, the k-th sub-problem in the method is defined as:
(k ) ( x )
min （or max） y0
(9)
(k ) ( x ) ≤ 0 ；j=1,2,…,m
s.t. y j
xil(k) ≤ xi ≤ xiu(k) ； i=1,2,…,n
(k )
(k)
(k)
(k)
where x(k)
il ≥ xil , xiu ≤ xiu , xil and xiu define the bounds of the k-th sub-region. y j ( x ) （ j=0,…,m ） are
approximations of f j (x) constructed by the Response Surface Method in the k-th sub-region. The sub-region problem
can be easily solved by a conventional optimization.
∗
In the sequential optimization approach, the optimal point of the k-th sub-problem x ( k ) becomes the reference point
for the (k+1)-th sub-problem. The process of creating new sub-problems with new approximations is repeated until the
convergence criterion is reached. Convergence criterion used in the paper is the relative change in the approximate
objective values in these last two iterations and its value is set to 1%. Construction of objective and constraints by the
Response Surface Method, experimental design and the updating method of sub-region are the key steps of the
successive approximate optimization. The details of the updating method of sub-region are given below.
(k )
(k )
4. Updating of successive sub-regions The sizes of successive sub-regions ( xiu − xil ) are highly influential on
the accuracy of the approximations to be constructed. Generally speaking, the smaller the size of the sub-region, the
greater the accuracy of the approximation. Accordingly, in the process of the optimization, whenever approximation is
not accurate enough, the size of sub-regions should be reduced or otherwise kept the same in order to ensure the fast
convergence to a true optimum. In this study, the scheme in [3] is followed. In the light of the scheme, the starting
design of the k+1-th sub-region (Figure 1) is a fraction λi of the corresponding range in the k-th region and is centered
around the k-th optimum, x (k )* .
Figure 1: Size of successive sub-regions for two design variable cases
The fraction parameter, λi, for the i-th design variable is calculated as follows:
λi( k +1)
*
x ( k ) +x ( k )
⏐xi( k ) − iu il ⏐
2
= η + (γ − η )
xiu( k ) -xil( k )
2
(10)
In this paper, η ＝0.5 and γ ＝0.8 are adopted for linear response model. The maximum value of λi( k +1)
( λi( k +1) = max λi( k +1) (i = 1,..., n) ) is chosen as the fraction value to be applied to all design variables so as to maintain
the aspect ratio of the design regions during the updating process. The upper and lower bounds of the i-th design
variable for k+1-th sub-region can be determined by:
⎯ 1027 ⎯
*
1
xil( k +1) = xi( k ) − λ ( k +1) ( xiu( k ) -xil( k ) )
2
(11)
*
1
xiu( k +1) = xi( k ) + λ ( k +1) ( xiu( k ) -xil( k ) )
2
VARIABLE SCREENING
Since the number of regression coefficients determines the number of simulation runs, it is important to remove those
variables which have little contribution to the design model. This can be done by performing a preliminary study
involving a design of experiments and regression analysis. The statistical results are used in an analysis of variable
(ANOVA) [1] to rank the variables for screening purposes. By doing so, we can assess the relative importance between
the variables such as the shape, the size and material parameters.
OPTIMIZATION EXAMPLES
1. Optimization of a thin plate with a square hole impacting onto a rigid wall The numerical model is depicted in
Figure 2 with the initial velocity of 20m/s and additional mass of 264kg shown. The maximum energy absorbed by the
plate U is defined as the objective function which will be maximized for a purpose of improving the structural
crashworthiness. The problem, although simple, presents some challenging to design analysis such as nonlinearity,
buckling, and contact.
Figure 2: Plate impacting a rigid wall
The FE model is parameterized in terms of two design variables: the location x and the width a. The area of the hole A
is supposed to be constant. Thus, the optimization problem is formulated as:
max U
(12)
s.t. x ≥ 0.2m
x + a ≤ 0.8m
A
≤ 0.8m
a
According to the theory of DOE, enough design points(x, a) are chosen to construct the Response Surface. A
commercial FEM analysis code LS-DYNA [6] is used to find a numerical solution of the internal energy absorbed by
the structure.
At first, the quadratic response surface was constructed from the results of the design points. However it is found that
the response surface model was not accurate enough to predict the true performance. Then a cubic response surface
was constructed as follows:
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U = −17.197 − 41.194 x + 543.14a + 402.01x 2 − 421.26 xa − 1211.7 a 2 − 402.9 x 3
(13)
− 95.733 x 2 a + 617.09 xa 2 + 848.64a 3
According to the result of model adequacy checking, the cubic function was capable of predicting the true response.
After optimizing the response function, the optimum design point (0.2m, 0.6m) is achieved and the relative difference
between the optimum internal energy U (51.9813KJ) by the response and the internal energy U (51.904kJ) is only
0.15%. Figure 3 exhibits the deformations with the optimal design. Figure 4 shows the energy absorbed with respect to
time for the initial design (0.2m, 0.2m) and optimum design (0.2m, 0.6m), where a significant improvement can be
clearly observed.
Figure 3: The deformation of the plate after impact
60
Initial
Optimum
Internal Energy[KJ]
50
40
30
20
10
0
0
5
10
15
20
25
Time[ms]
30
35
40
Figure 4: The energy absorbed for the initial and optimum design
2. Optimization of frontal structural crashworthiness of vehicle
(1) Problem description In this optimization problem, a simplified vehicle frontal structure with initial velocity of
13.6m/s and attached mass of 275kg crashing into a rigid wall is considered. In order to obtain safe and light-weight
vehicle structure, the optimization of this problem is to minimize the mass of the frontal energy absorbing structure,
whereas the constraints are the maximum rigid wall force and the structure intrusion. The thicknesses and the yield
stress are chosen to be the design variables.
The FE model of the frontal structure, as illustrated in Figure 5, was established by Altair Hypermesh, and solved by
LS-DYNA. The structure was composed of the upper cross member, the middle connect structure, the lower cross
member, the upper plate of the side rail and the lower plate of the side rail. The initial design variables are the
thicknesses and yield stress of these five parts. As mentioned before, if all the ten design variables are considered
⎯ 1029 ⎯
simultaneously in the problem, it takes at least 17 FE runs to construct a linear response surface and the iteration
process of the successive approximate optimization will require a large number of FE simulations. In the study, the
variable screening method is employed. By means of ANOVA and an initial linear response surface, the thickness of
the upper cross member t1, the thickness of the upper plate of the side rail t2, the thickness of the lower plate of the side
rail t3, the yield stress of the middle connect structure σ 1 , the yield stress of the lower cross member σ 2 , the yield
stress of the lower plate of the side rail σ 3 are chosen as the final design variables. Thus, the optimization problem is
formulated as:
min Mass
(14)
s.t. F max ≤ 600 KN
D max ≤ 130mm
1mm ≤ t1, t 2, t 3 ≤ 3mm
0.3GPa ≤ σ 1, σ 2, σ 3 ≤ 0.38GPa
Figure 5: The simplified frontal structure
(2) The optimization process and the results In the baseline design as shown in Figure 5, the initial thicknesses are
2.4mm and the yield stresses are 0.36GPa. The initial sub-region for the thickness is 1.2mm and the initial sub-region
for the material parameter is 0.04GPa. By using the LS-DYNA, the result of the baseline shows that the rigid wall force
transmitted is 738.539KN, which is far higher than the constraint value of 600KN. The Successive Response Surface
Method sequentially updates the thicknesses and the yielding stresses, till the design doesn’t violate the constraints.
After 4 iterations of the linear response surface, the optimum design point is achieved as [2.999, 1.102, 2.624, 0.327,
0.380, 0.371]. The mass of the frontal structure is 22.976kg, the maximum force that the rigid wall transmits is
598.18KN and the maximum intrusion of the frontal structure is 129.95mm. The optimization process is summarized
in Table 1 and Figure 7 and the deformation of the optimum design is shown in Fig. 6.
Table 1 Optimization results of the frontal structure impacting into a rigid wall
Initial sub-region size for t 1, t 2, t 3 are 1.2mm, Initial sub-region size for σ 1, σ 2, σ 3 are 0.04GPa
Mass
Iteration
Numbers
t1
Initial
2.4
2.4
1
2.904
1.8
2
3
t2
t3
σ1
σ2
σ3
Approx.
2.4
0.36
0.36
0.36
2.173 0.38
0.38
0.34
1.526 2.417 0.38 0.364 0.324
Fmax
Dmax
FE
FE
FE
Approx.
Approx.
analysis
analysis
analysis
22.233
738.539
130.006
22.62
22.626
599.9
605.252
135.02
134.512
23
23
600
592.374
131.7
131.557
3
2.999 1.343 2.434 0.38 0.377
0.34
22.91
22.913
600
601.264
129.61
130.166
-
2.999 1.102 2.624 0.38 0.371 0.327
22.97
22.976
599.51
598.18
130
129.950
⎯ 1030 ⎯
Figure 6: The optimum deformation result
606
23
Response Surface Approx.
FEM analysis
22.95
22.9
Rigid wall Force[kN]
602
Mass[KG]
22.85
22.8
22.75
600
598
596
22.7
594
22.65
22.6
Response Surface Approx.
FEM analysis
604
1
2
3
4
592
1
2
3
4
Iteration Numbers
Iteration Numbers
136
Response Surface Approx.
FEM analysis
135
Intrusion[mm]
134
133
132
131
130
129
1
2
3
4
Iteration Numbers
Figure 7: Values in the process of optimization
In the light of the relative change between the approximate values and the FE analysis values from Fig. 7, there is
relatively big noise in the response of the rigid wall force. The reason might be that the rigid wall force hasn’t been
filtered. In this study, the accuracy of the linear response surface is in an acceptable level for the problem and
optimization process is completed successively.
⎯ 1031 ⎯
CONCLUSION
The investigation in this paper reveals that the Response Surface Method coupled with variable screening technique,
sequential approximation optimization method and nonlinear finite element code leads to an efficient and effective
optimization approach to crashworthiness design applications. This is proved by two illustrative examples; one thin
plate with a square hole with a cubic response surface and one simplified front rail structure of vehicle with successive
linear response surface. The both optimization processes converge to the optimal designs compared to the initial
design and demonstrate that the Response Surface Method is a useful tool in structural optimization design involved
complex nonlinear dynamic behavior.
Acknowledgements
The support of National Natural Science Foundation of China (10372029) is gratefully acknowledged.
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