COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Evolutionary Topological Design of Frame for Impact Loads
Xianyan Chen 1*, Qing Li 1, 2, Shuyao Long 1, Xujing Yang 3
1
2
3
Department of Engineering Mechanics, Hunan University, Changsha, Hunan, 410082 China
School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW 2006
Australia
Key Laboratory of Advanced Technology for Vehicle Body Design & Manufacture of Ministry of Education, Hunan
University, Changsha, 410082 China
Email: [email protected] , [email protected], [email protected]
Abstract The evolutionary structural optimization algorithm is proposed in this paper to solve the topological design
problems with crashworthiness criterion. In this method, the ratio of elemental strain energy to the highest strain
energy is adopted as a factor to determine the relative efficiency of material usage, and the ratio of total stain energy to
total structural weight is established in order to decide whether an optimum has been reached. The non-linear explicit
finite element code LS-DYNA is employed to simulate the deformation and stain energy of the structure under impact
load. An example of topology design is shown in this study to demonstrate the capabilities of the present method.
Key words: crashworthiness, evolutionary structural optimization, energy absorption, LS-DYNA
INTRODUCTION
Over the last few decades, substantial attention of research has been paid to the subject of behavior of impact and
energy absorption [1]. The developments of various numerical techniques for crashworthiness and impact analysis
have been an important topic in the field. Nowadays, many commercial packages, like DYNA3D and PAM-CRASH,
are extensively available to solve for a range of direct problems with accepted accuracy. These numerical tools provide
design engineers with an effective means to the development of industrial products.
However, topology and shape optimizations for crashworthiness, as one of the most typical inverse problems, have
achieved far less popularity. The difficulty is primarily raised from two folders on (1) the nonlinear sensitivity analysis
and computational cost [2], which become particularly challenging in various topological designs; (2) dynamic
multi-modals and non-convex design space, which do not lend the crash problems themselves well to classical gradient
techniques.
In recent years, several different optimization schemes combined with finite element modeling have been used to
improve the capability of energy absorption of structures which are particularly designed for crashworthiness criteria.
Marklund and Nilsson use the shape optimization to enhance the crashworthiness of a car body subjected to side
impact, the first order sensitivities are obtained by using an computational expensive finite difference approach [3];
C.B.W. Pedersen carried out the crashworthiness design of frame structures by using topology optimization, a
quasi-static non-linear finite element solution is attained with an implicit backward Euler algorithm, and the analytical
sensitivities are computed by the direct differentiation method [4-6]. In these pieces of work, the nonlinear sensitivity
analysis has been considered. To avoid the complexity of the sensitivity analysis, Shen-yeh Chen introduced a more
practical approach using the robust genetic algorithm for impact structure and crashworthiness optimization [7];
K.Yamazaki and J. Han used the response surface approximation technique to maximize the absorbing energy of the
cylindrical shell subjected to an axial impact force [8]; J. Forsberg and L. Nilsson investigated two different
methodologies⎯classical response surface methodology and Kriging⎯to construct an intermediate approximation to
the optimization problem [9].
In this paper, Evolutionary Structural Optimization (ESO) algorithm is employed to optimize the topology of frame
structures subjected to impact loads. This technique was demonstrated the successful applications for various
structural problems with linear static loading. The most significant advantages of this method are its simplicity in
⎯ 1064 ⎯
physical concept and easiness in computer implementation [10-13]. In the present algorithm, the non-linear explicit
finite element code DYNA3D [14] is adopted as an analysis tool to simulate the complicated crushing behavior of
frame structures.
EVOLUTIONARY OPTIMIZATION PROCEDURES
In a crash simulation, it is frequently found that plastic deformation in some locations may be much higher than that in
others. This implies that the material of the crashing elements may make different contributions to the crashworthiness
goal. To represent the relative performance of element’s material, a dimensionless factor is formulated by dividing the
crash energy absorption by each element to the highest one as,
α i = U i U max
(1)
where U i and U max denote the strain energy absorbed by the candidate element (ith) and the maximum of the strain
energies of all beam elements. It is worth pointing out that the total internal energy should contain the elastic and
plastic components, i.e. U = U e + U p . The factor α i can be used to identify the relative performance and
contributions of elemental material. Clearly, the efficiency factor satisfies
0 ≤αi ≤1
(2)
Ideally, the energy absorption levels in all location can be nearly identical, so it is logical that the less efficiently
utilized material is gradually removed from the structure. In the ESO method, the criterion of element removal is
determined by a threshold efficiency level as
α i ≤ RRSS
(3)
where, RRSS is called Rejection Ratio or threshold efficiency level. If the efficiency factor of an element α i is less
than a threshold level, this element is considered to be relatively structurally inefficient and its material will be
partially removed from the structure. The process of the elemental material removal is repeated using the same value of
RRSS until an ESO Steady State ( SS ) is reached, which means that there are no more elements whose material can be
removed at the current iteration. At this stage an evolutionary Rate ( ER ) is introduced so that
RRSS +1 = RR SS + ER
(4)
With the increased threshold efficiency or rejection ratio, the iterations take place again until a new steady state is
attained. The evolution rate is set to increase the threshold efficiency to a higher level, whereby the usage efficiency of
the remaining material becomes more uniform than before. For clarification, the evolutionary iteration procedure is
given as follows.
(1) Step 1: Setup a cross-section area set A = (A1 , A2 , L A j , A j +1 ,L , Am ) ( A j > A j +1 , j = 1,2,L , m ); define an initial
rejection ratio RR0 and an evolutionary rate ER ; and set SS = 0 .
(2) Step 2: Carry out an impact FEA, find the strain energies of each beam element ( U i ) and the maximum strain
energy among all beams ( U max ), compute the relative efficiency factor α i of each beam element.
(3) Step 3: If condition of α i ≤ RRSS holds, reduce the cross-sectional area of the ith beam from current value Aj to
next lower value Aj+1.
(4) Step 4: If there is no more beam element satisfying the condition α i ≤ RRSS , increase RRSS by ER and set
SS = SS + 1 , repeat Step 3; otherwise, repeat Step 2 to 3 until an optimum is attained.
During the evolving optimization process, the ratio of total strain energy to the total structural weight (REW) is
examined to see if the efficiency of material usage is progressively improved. The optimization procedure can be
terminated at a point where some desirable level of the REW is attained or the REW value can not improved any more.
EXAMPLE
The following example is used to demonstrate the capability of the proposed ESO method to solve topology
optimizations of frame structure for a crashworthiness criterion. In the optimization process, an initial rejection ration
⎯ 1065 ⎯
of RR0 = 0 and an evolution rate of ER = 0.1% are set up. The frame structure in this paper is the same as the first
ground structure in reference [5].
The ground structure is shown in Fig. 1(a). The lumped mass of M 1 of 1500 kg is attached in the center of the free
end, the initial velocity v0 is 10m/s. The FE model for the ground structure is shown in Fig. 1(b). The rectangle
cross-section area of the beams is set as A = (10, 8, 6, 4, 2) mm2, the initial design domain is prescribed as all beams
having the identical area of 10mm2. The beam will be removed if its area is required to be reduced less than the
minimum area. The material adopted in Ref. [5] is assumed to be mild steel, and the material’s strain-hardening is
assumed to be an elastic, linear hardening model. The material properties are taken as follows: Young's modulus of
elasticity E=210GPa, Poisson's ratio v=0.3, yield stress σ0=510MPa，density ρ=7800kg/m3, the hardening modulus
Ep=10.5GPa. The mass in the ground structure is simplified as the lumped mass element in the FE model. The crushing
time T is 0.04 s.
v 0 = 10 m /s
M1 = 1500kg
2m
3m
(a)
(b)
Figure 1:
(a) The ground structure; (b) FE model for the ground structure
To observe the evolution process of beam material removal, Figs. 2(a-d) display several ESO steady states. As more
and more beam elements which absorb far less stain energy are removed from the structure, the material usage of the
remaining structure becomes higher and higher. As a result, the energy absorption levels in all location become more
identical.
(a)
(b)
(c)
(d)
Figure 2: Evolution process at the time of 0.04s: (a) REW = 306.396 , V / Vo = 80.2% , SS = 6 , RR = 0.1% ; (b)
REW = 367.415 , V / Vo = 66.3% , SS = 21 , RR = 0.8% ; (c) REW = 404.743 , V / Vo = 64.1% , SS = 24 ,
RR = 0.8% ; (d) REW = 522.004 , V / Vo = 52.1% , SS = 36 , RR = 1.7% .
Fig. 3 shows the evolution histories of (a) the ratio of total strain energy to the total structural weight (REW) and (b) the
⎯ 1066 ⎯
total stain energy of the structure. It is clear that the REW becomes higher and higher until a constant situation is reached
at iteration 46. But the total strain energy increases until the iteration 46, and then decreases. This indicates that the total
strain energy that the whole structure absorbs may reach the maximum at iteration 46. However, this does not necessarily
mean that this design stage of structure is of the highest efficiency of material usage. Although the total strain energy
decreases in the following iteration, the material reduction catches the loss of strain energy absorption, which may have
the same or even higher level of material usage efficiency. From this angle, it seems that REW is a better performance
indicator for topological crashworthiness design.
7.0x10
4
6.8x10
4
6.6x10
4
6.4x10
4
6.2x10
4
6.0x10
4
550
500
Strain Energy (J)
REW (J/kg)
450
400
350
300
250
0
10
20
30
40
50
0
10
20
30
40
50
Iteration Number
Iteration Number
(a)
(b)
Figure 3: The evolution histories of the ratio of total strain energy to the total structural weight (REW)
and the total stain energy of the structure: (a) REW, (b) Total strain energy.
CONCLUDING REMARKS
In this paper, the evolutionary structural optimization algorithm is extended to solve the topological design problems
with crashworthiness criterion. A factor of elemental strain energy to the highest strain energy is used to determine the
relative efficiency of material usage. To monitor the overall structural performance, the ratio of total stain energy to
total structural weight is computed so as to decide whether an optimum has been reached. By gradually removing the
inefficient material of the beam elements from the structure, the remaining structure evolves toward an optimum. The
non-linear explicit finite element code LS-DYNA is employed to simulate the deformation and stain energy of the
structure under impact load. The ESO method is implemented in an environment of LS-DYNA.
REFERENCES
1. Lu Guoxing, Yu Tongxi. Energy absorption of structures and materials. CRC, New York, USA, 2003.
2. Bendsøe MP, Sigmund O. Topology Optimization—Theory, Methods and Applications. Springer, New York,
USA, 2003.
3. Marklund PO, Nilsson L. Optimization of a car body component subjected to side impact. Structural and
Multidisciplinary Optimization, 2001; 21: 383-392.
4. Pedersen CBW. Topology optimization design of crushed 2D-frames for desired energy absorption history.
Structural and Multidisciplinary Optimization, 2003; 25: 368-382.
5. Pedersen CBW. Topology optimization for crashworthiness of frame structures. International Journal of
Crashworthiness, 2003; 8(1): 29-39.
6. Pedersen CBW. Crashworthiness design of transient frame structures using topology optimization. Computer
methods in applied mechanics and engineering, 2004; 193: 653-678.
7. Chen Shen-Yeh. An approach for impact structure optimization using the robust genetic algorithm. Finite
Elements in Analysis and design, 2001; 37: 431-446.
8. Yamazaki K, Han J. Maximization of the crushing energy absorption of cylindrical shells. Advances in
Engineering Software, 2000; 31: 425-434.
⎯ 1067 ⎯
9. Forsberg J, Nilsson L. Evaluation of response surface methodologies used in crashworthiness optimization.
International Journal of Impact Engineering, 2006; 32: 759-777.
10. Xie YM, Steven GP. Evolutionary Structural Optimization. Springer, Berlin, Germany, 1997.
11. Steven GP, Li Q, Xie YM. Evolutionary topology and shape design for general physical field problems.
Computational Mechanics, 2000; 26: 129-139.
12. Li Q, Steven GP, Querin OM, Xie YM. Stress based optimization of torsional shafts using an evolutionary
procedure. International Journal of Solids and Structures, 2001; 38: 5661-5677.
13. Li Q, Steven GP, Xie YM. Thermoelastic topology optimization for problems with varying temperature fields.
Journal of Thermal Stresses, 2001; 24: 347-366.
14. LS-DYNA3D Theoretical Manual. Livermore Software Technology Corporation.
⎯ 1068 ⎯