2012 International Conference on Software and Computer Applications (ICSCA 2012)
IPCSIT vol. 41 (2012) © (2012) IACSIT Press, Singapore
Topology Optimization for Dynamic Problems Using Modified
Artificial Bee Colony Algorithm
Dae-Ho Chang1, Ji-Yong Park1, Jae-Yong Park1 and Seog-Young Han2+
1
Department of Mechanical Engineering, Graduate School, Hanyang University, 17 Haengdang-Dong,
Seongdong-Gu, Seoul, Korea
2
School of Mechanical Engineering, Hanyang University, 17 Haengdang-Dong, Seongdong-Gu, Seoul,
Korea
Abstract. The artificial bee colony algorithm (ABCA) is suggested for dynamic problems of topology
optimization. The objective is to obtain a structure with the highest fundamental natural frequency in a
certain amount of material, based on the contributed structural sensitivity of each element calculated by
design variables and eigenvalue. Examples are provided to examine the applicability and effectiveness of the
ABCA comparing to bi-directional evolutionary structural optimization (BESO).
Keywords: Artificial bee colony algorithm, Topology optimization, Natural frequency, Stochastic search
method, Finite element method.
1. Introduction
The goal of structural optimization is to achieve the best performance from a structure while satisfying
various constraints, such as a certain amount of material and specific mechanical conditions. For the past
several decades, topology optimization techniques such as the homogenization method [1], the solid isotropic
material with penalization (SIMP) method [2], the evolutionary structural optimization (ESO/BESO) method
[3], and the level set technique [4] have been developed. These methods have been successfully adopted in
topology optimization for static, dynamic, and practical engineering problems.
Recently, swarm-intelligence based algorithms, modeled after an artificial bee colony, have been applied
in engineering design optimization processes. Karaboga et al. [5-7] published a survey of the artificial bee
colony algorithm (ABCA) and its applications.
The purpose of this study is to suggest a method of applying the ABCA in the frequency topology
optimization of structural models using finite element analysis. Since the original ABCA was used to find the
optimum solution of continuous functions between upper and lower bounds, considerable modifications are
made to the algorithm so that it may be applied for the frequency optimization of structural models.
Examples are provided to examine the effectiveness and applicability of modified ABCA, and then
compared to the soft-kill BESO. Form the results, the algorithm with the proposed methods is very
applicable and effective in topology optimization for obtaining a stable and robust optimal layout.
2. Formulation of Natural Frequency Optimization
Excessive vibration due to resonance occurs when the frequency of a dynamic excitation is close to one
of the natural frequencies of a structure. Therefore, it is necessary to restrict the fundamental or higher
natural frequencies of a structure to a prescribed range in order to avoid severe vibration [8].
+
Corresponding author. Tel.: +82-2-2220-0456; fax: +82-2-2220-2299.
E-mail address: syhan@hanyang.ac.kr.
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In the finite element method, dynamic behavior of a structure is represented by the following general
eigenvalue problem;
([ K ] − ω j2 [ M ]){u j } = {0}
(1)
where, [K ] is the global stiffness matrix, [M ] is the global mass matrix, ω j is the j-th natural frequency,
and {u j } is the eigenvector corresponding to ω j .
Here we consider topology optimization problems for maximizing the j-th natural frequency of vibrating
continuum structures. For a solid-void design, the optimization problem can be stated as Eq. (2);
Minimize : ω j
⎧1
χi = ⎨
N
Subjected to : χ − ∑ χ i = 0
*
⎩ χ min
i =1
if element is solid
if element is void
(2)
*
where, χ and χi denote a prescribed volume constraint and the presence/absence of the i-th element,
respectively. χ i = χ min and χ i = 1 denote void and solid elements, respectively. N is the total number of
elements in a structure. χ min (e.g., 0.001) is a small value used for denoting a void element to avoid highly
localized modes.
3. Modification of Topology Optimization using ABCA
3.1. Waggle index update rule
A waggle index update rule is developed as a simple algebraic equation based on the amount of
information shared on each element from the previous iteration and the presence/absence of the employed
bees in the present iteration, which is updated by Eq. (5);
Ii (k ), update = δ × Ii (k −1) + (1 − δ ) × ei (k )
(3)
where, k is the number of k-th iteration, Ii is the waggle of information shared on the i-th element, ei is a
employed bee of presence (1) or absence ( χ min ), δ is the waggle index update coefficient. δ is empirically
appropriate to be 0.8 for natural frequency optimization problems. The waggle index represents the amount
of information that should be transferred to bee colony in the next iteration.
3.2. Changing filtering size scheme
A changing filtering size scheme is suggested in order to suppress the cleft or biased layout as follows
[3];
M
∑ w(r
ik ) f k
fi =
k =1
node
M
∑ w(r
ik )
⎛ dω j ⎞
Vi ⎜
⎟
dI i ⎠
⎝
node
i =1
where, f k =
m
Vi
m
∑
, w( rik ) = rmin − rik ( k = 1, 2,..., M )
∑
k =1
rmin = element size ×
element
(4)
i =1
2
(1 ≤ iteration < limit value_step(1))
rmin = element size × 2
(limit value_step(1) ≤ iteration < limit value_step(2))
rmin = element size × 2 2
(limit value_step(2) ≤ iteration < limit value_step(3))
Where, fi denote the elemental sensitivity number of the i-th element, M is the total number of nodes
included in the circular sub-domain, w(rik ) is the weight factor for the distance from the center of the i-th
element to the k-th node and m is the total number of elements connected to the k-th node. Vi is the volume
of an individual element.
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3.3. Evaluating method of fitness values
In order to give clear differences of fitness that calculated by the original ABCA, an evaluation method
of fitness values, denoted by fiti as Eq. (3) suggests. This makes the distribution linear from 0 to 1 by
setting the median value of the distribution to be 0.5, is suggested as follows;
fiti =
fiti =
0.5 × {temp _ fiti − min(temp _ fiti )}
[med (temp _ fiti ) − min(temp _ fiti )]
( min(temp _ fiti ) ≤ temp _ fiti < med (temp _ fiti ) )
0.5 × {temp _ fiti − med (temp _ fiti )}
[max(temp _ fiti ) − med (temp _ fiti )]
(5)
+0.5
( med (temp _ fiti ) ≤ temp _ fiti < max(temp _ fiti ) )
where, min : min ium vaule, max : max imum value, med : median value
4. Numerical Examples
4.1. A short beam with the first natural frequency
The beam with dimensions 5 m ｘ 1 m is clamped on both sides as shown in Fig. 1. The material is
assumed with Young’s modulus E = 10 MPa, Poisson’s ratio ν = 0.3 and mass density ρ = 1 kg/m3 . The
objective is to obtain a structure having the highest first natural frequency with volume constraint 90% of the
original volume. The filter radius rmin = 1.5 and penalty factor p = 3 is used in calculating sensitivity numbers.
Fig. 2. shows optimal topologies corresponding to some iterations as well as evolution history of the first
natural frequency of the short beam in the soft-kill BESO and the modified ABCA algorithms. Fig. 3. shows
the optimal topologies obtained from the two algorithms. The optimal topologies are very close each other.
The first natural frequency of the initial design domain was 0.8339 Hz. The first natural frequency of the
optimal topology obtained from the ABCA algorithm is obtained as 0.9121 Hz. Whereas, the first natural
frequency of the optimal topology obtained from the soft-kill BESO algorithm is obtained as 0.9156 Hz.
That is, the ratios of the first natural frequency increment for the ABCA and soft-kill BESO algorithms are
9.37% and 9.79%, respectively.
Fig. 1: A short beam under plane stress conditions
(a) Soft-kill BESO
(b) Modified ABCA
Fig. 2: Evolution history of the first natural frequency of the short beam
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(a) Soft-kill BESO
(b) Modified ABCA
Fig. 3: Optimal topologies for maximizing the first natural frequency.
4.2. A rectangular plate
Fig. 4. shows an aluminum plate of dimensions 0.15 m ｘ 0.1 m. The plate is fixed at two corners on its
diagonal. Only in-plane vibration is considered here. The material is assumed with Young’s modulus E = 10
MPa, Poisson’s ratioν = 0.3 , mass density ρ = 1 kg/m3 . The objective is to obtain a structure having the
highest natural frequency with volume constraint 50% of the original volume.
Fig. 5. shows optimal topologies corresponding to some iterations as well as evolution history of the first
natural frequency of the rectangular plate in the soft-kill BESO and the ABCA algorithms. Fig. 6. shows the
optimal topologies obtained from the two algorithms. The optimal topologies are obtained very similarly.
The first natural frequency of the initial design domain is 0.7560 Hz. The first natural frequency of the
optimal topology obtained from the ABCA algorithm is obtained as 1.0480 Hz. Whereas, the first natural
frequency of the optimal topology obtained from the soft-kill BESO algorithm is obtained as 1.0491 Hz.
That is, the ratios of the first natural frequency increment for the ABCA and soft-kill BESO algorithms are
38.62% and 38.76%, respectively.
Fig. 4: A rectangular plate under plane stress conditions
(a) Soft-kill BESO
(b) Modified ABCA
Fig. 5: Evolution history of the first natural frequency of the rectangular plate
(a) Soft-kill BESO
(b) Modified ABCA
Fig. 6: Optimal topologies for maximizing the first natural frequency
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5. Conclusions
As a stochastic method, the modified ABCA with three improved or developed methods is applied in a
topology optimization procedure for dynamic problems. From the results of the presented examples, the
followings can be concluded:
(1) The ABCA is very effective and applicable in topology optimization for dynamic problems
compared to the soft-kill BESO algorithm.
(2) The convergence rate of the ABCA algorithm is very fast in the case of a small volume fraction
constraint.
(3) It is verified that the three suggested algorithms based on the ABCA provide both stable and robust
optimum layouts.
(4) The optimal topology of the ABCA algorithm is almost obtained in the initial or early stage of
topology optimization because of the characteristics of the ABCA algorithm.
6. References
[1] H. Lazarus, T. Hagiwara, and I. Hagiwara. Static and vibrational shape and topology optimization using
homogenization and mathematical programming. Comput. Meth. Appl. Mech. Eng. 1993, 109 (1-2): 143-154.
[2] M. P. Bendsøe, and O. Sigmund. Topology Optimization: Theory, Methods and Applications. Springer, 2003.
[3] X. Huang, and Y. M. Xie. Evolutionary topology optimization of continuum structures. Wiley, 2009.
[4] S. Yamasaki, S. Nishiwaki, T. Yamada, K. Izui, and M. Yoshimura. A structural optimization method based on the
level set method using a new geometry re-initialization scheme. Int. J. Numer. Meth. Eng. 2010, 83 (12): 15801624.
[5] D. Karaboga. An idea based on honey bee swarm for numerical optimization. Technical Report. Erciyes University.
Engineering Faculty Computer Engineering Department. Turkey, 2005.
[6] D. Karaboga, and B. Basturk. A powerful and efficient algorithm for numerical function optimization: artificial bee
colony (ABC) algorithm. J. Glob. Opt. 2007, 39 (3): 459-471.
[7] B. Akay, and D. Karaboga. A modified Artificial Bee Colony algorithm for real-parameter optimization. Inf. Sci.
2010, article in press.
[8] X. Huang, Z. H. Zuo, and Y. M. Xie. Evolutionary topological optimization of vibrating continuum structures for
natural frequencies. Comput. Struct. 2010, 88 (5-6): 357-364.
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