DESIGN OPTIMIZATION 1

54 (2012) 3 © Carl Hanser Verlag, München Materials Testing

Over the past few years, the studies on evolutionary algorithms have shown that these methods can be efficiently used to eliminate most of the difficulties of classi-cal methods. Evolutionary algorithms are widely used to solve engineering optimiza-tion problems with complex nature. Vari-ous research works are carried out to en-hance the performance of evolutionary al-gorithms [1 – 23].

For instance, in our previous work [2], Yildiz and Saito developed a novel approach for multi-component topology optimization of continuum structures using a multi-objec-tive genetic algorithm to obtain Pareto opti-mal solutions that exhibits trade-offs among stiffness, weight, manufacturability, and as-semble ability. The developed approach is applied to multi-component topology opti-mization of a vehicle floor frame.

The Cuckoo search (CS) algorithm is in-troduced by Yang and Deb [20]. The CS al-gorithm has been used in many areas of optimization studies. The use of the Cuckoo search algorithm in the optimum solution of problems resulted better solutions com-pared to classical methods [21-24].

Cuckoo Search Algorithm for Structural Optimization

In this paper, the CS algorithm is used to solve structural design optimization problems.

innovative approaches, such as tabu search, genetic algorithm, simulated an-nealing, particle swarm optimization algo-rithm, ant colony algorithm, and immune algorithm have been developed and widely applied in various fields of science [1-13].

There is an increasing interest to apply the new approaches and to further improve the performance of optimization techniques for the solution of structural design optimi-zation problems. Although some improve-ments regarding structural design optimi-zation issues are achieved, the complexity of design problems presents shortcomings. The main goal of present research is to solve real world design optimization prob-lems using Cuckoo search algorithm (CS).

The CS algorithm is applied to a vehicle part design optimization problem taken from automotive industry to demonstrate the application of the present approach to real world design problems. The results of the CS approach show that the proposed optimization method converges rapidly to the global optimum solution and provides reliable and accurate solutions.

Literature Review

Recently, new approaches in the area of op-timization research are presented to fur-ther improve the solution of optimization problems with complex nature.

Designing new products possessing desired properties is important in the design indus-try. In the past few decades, computer-aided product design has been proved to be an al-ternative to the traditional trial-and-error method. With the advent of ever faster com-puting platforms, computer aided-design and optimization tools are becoming more attractive due to its great contribution to cost, material and time savings in the proce-dures of the engineering design. The applica-tion of these tools allows a more rapid design process and more detailed design studies.

The optimal design of structures includes sizing, shape, and topology optimization. The purpose of design optimization is to deter-mine the optimal shape of a continuum me-dium to maximize or minimize a given crite-rion (often called an objective function), such as minimize the weight of the body, maxi-mize the stiffness of the structure or remove the stress concentrations, subjected to the stress or displacement constraint conditions.

Numerous optimization techniques have been developed to solve structural design optimization problems in the last four dec-ades. The early works on the topic mostly use various mathematical techniques. These methods are not only time consum-ing in solving complex nature problems but also they may not be used efficiently in finding global or near global optimum solu-tions. In the past few decades, a number of

In order to meet today’s vehicle design requirements and to improve the cost and fuel efficiency, there is an increasing interest to design light-weight and cost-effective vehicle components. In this research, a new optimization algorithm, called the Cuckoo Search Algorithm (CS) algo-rithm, is introduced for solving structural design optimization problems. This research is the first application of the CS to the shape design optimi-zation problems in the literature. The CS algorithm is applied to the struc-tural design optimization of a vehicle component to illustrate how the present approach can be applied for solving structural design problems. Results show the ability of the CS to find better optimal structural design.

İsmail Durgun and Ali R. YildizBursa, Turkey

Structural Design Optimization of Vehicle Components Using Cuckoo Search Algorithm

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The Cuckoo Search algorithm (CS) is in-spired by some species of a bird family called Cuckoo because of their special life-style and aggressive reproduction strategy. These species lay their eggs in the nests of other host birds (almost other species) with amazing abilities such as selecting the re-cently spawned nests and removing exist-ing eggs that increase hatching probability of their eggs. On the other hand, some of host birds are able to combat this parasite behaviour of Cuckoos and throw out the dis-covered alien eggs or build their new nests in new locations. This algorithm contains a population of nests or eggs. For simplicity, the following representations are used, where each egg in a nest represents a solu-tion and a Cuckoo egg represents a new one. If the Cuckoo egg is very similar to the host’s, then this Cuckoo egg is less likely to be discovered; thus, the fitness should be related to the difference in solutions. The

aim is to employ the new and potentially better solutions (Cuckoos) to replace a not-so-good solution in the nests [20, 24].

For simplicity in describing the CS, the fol-lowing three idealized rules are utilized [20]: a)  each Cuckoo lays one egg at a time and

dumps it in a randomly chosen nestb)  the best nests with high quality of eggs

are carried over to the next generationsc)  the number of available host nests is con-

stant, and the egg, which is laid by a Cuckoo, is discovered by the host bird with a probability of pa in the range of [0, 1].

The later assumption can be approximated by the fraction pa of the n nests are re-placed by new ones (with new random so-lutions). With these three rules, the basic steps of the CS can be summarized as the pseudocode shown in Figure 1.

In the first step according to the pseudo code, one of the randomly selected nests (except the best one) is replaced by a new

solution, which is produced by random walk with a Lévy flight around the so far best nest, considering the quality. But in the new version, all of the nests except the best one are replaced in one step by new solutions. To generate new solutions xi

(t+1) for the ith Cuckoo, a Lévy flight is per-formed using the following equation:

xi(t+1) = xi

(t) + α · S (1)

where α > 0 is the step size parameter and should be chosen considering the scale of the problem, is set to unity in the CS [20] and decreases function as the number of generations increases in the modified CS [22, 23] . It should be noted that in this new version, the current positions of the solu-tions are used instead of the best solution so far as the origin of the Lévy flight is con-cerned. The step size is considered as 0.1 in this work, because it results in an effi-cient performance of algorithm in our ex-ample. The parameter S is the length of a random walk with Lévy flights according to Mantegna’s algorithm as described in Equation (2).

In the second step, the pa fraction of the worst nests is discovered and replaced by new ones. However, in the new version, the parameter pa is considered as the probabil-ity of a solution’s component to be discov-ered. Therefore, a probability matrix is pro-duced as

Pif rand paif rand paij

= <≥

⎧⎨⎪

⎩⎪10      

(2)

where rand is a random number in [0, 1] interval and Pi,j is the discovering probabil-ity for the jth variable of the ith nest. Then, all of the nests are replaced by new ones produced by random walks (point-wise multiplication of random step sizes with probability matrix) from their current posi-tions according to quality.

In this paper, the CS algorithm is used for optimal design of vehicle components. As a supplement to help readers to implement the CS correctly, a demo version is provided in the paper by Yang and Deb [20].

Structural Design Optimi- zation Using ImprovedCuckoo Search Algorithm

The CS algorithm is applied to the struc-tural design optimization of an automobile bracket part problem taken from automo-tive industry for the optimal design of a vehicle component in this section.

Objective function ! ! , ! = !!, !!,… . , !! ; Generate initial population of ! host nests !!   ! = 1,2,… ,! ; while (stop criterion) Get a Cuckoo randomly by Lévy flights; Evaluate its quality/fitness !! ; Choose a nest among !  (say j) randomly; if !! ≥ !! end Abandon a fraction (!") of worse nests [and build new ones at new locations via Lévy flights] Keep the best solutions (or nests with quality solutions); Rank the solutions and find the current best; end while Post process results and visualization;

Figure 1. Pseudo code of Cuckoo Search

Figure 2. Initial design domain and boundary conditions of an automobile bracket part

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Minimization of volume is chosen as ob-jective function. Maximum stress is chosen as constraint function in this problem.

Initial design domain and boundary con-dition of example application part is shown in Figure 2. Compliance minimization is chosen as objective function and volume re-duction with % 80 is chosen as constraint function. Material distribution that is shown in Figure 3 is obtained. In this study, ANSYS 12 is used for topology optimization.

The darker density colours represent the material, which should be removed, and the density lighter colours represent the mate-rial, which should be kept as shown in Figure 3. According to the results of the topology optimisation, the structure is redefined as be-ing based on material distribution in Figure 4. This is the initial optimal topology of exam-ple part which is used for shape optimisation.

In this research, then structural optimi-zation is performed using the Cuckoo search algorithm approach. The four de-sign variables x1, x2, x3, and x4 are se-lected as shown in Figure 5. The range of

Figure 3. Material distribution after topology optimization

Figure 4. Design domain after topology optimization

Figure 5. Design variables for shape optimization

Figure 6. The optimal structural layout as well as stress and displacement distributions

X1(mm)

X2 (mm)

X3(mm)

X4 (mm)

Volume(cm3)

Stress(MPa)

Initial design 18 24 11 32 89669 237

CAD optimum design 114 133 61 22 66431 277

PSO 116 136 65 23 54120 290

CS 128 142 69 24 50855 295

Table 1. Comparison of the optimization results for the automobile bracket design

design variables in shape optimization is used as 80 < X1 < 128, 115 < X2 < 142, 30 < X3 < 69, 22 < X4 < 28.

The results of the CS are given in Table 1. It can be seen that a volume of 50885 mm3 with 295 MPa is obtained.

It is clearly seen that the structural de-sign optimization performance is improved compared to traditional CAD and particle swarm optimization algorithm solutions. ANSYS is used for the CAD optimization process. The structural layout results of the CS algorithm for the vehicle part is given in Figure 6.

Conclusions

Recently, computer aided design and anal-ysis scenarios (design-build-test) are widely employed in the automotive indus-try, and savings in development time and cost reduction are obtained. From this study, it can be seen that there is a crucial need to consider structural optimization techniques to support the innovative de-sign and further to reduce development time and cost. Therefore, the optimal struc-tural design of components is of great im-portance in the area of automotive indus-

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22 E. Valian, S. Mohanna, S. Tavakoli: Improved Cuckoo search algorithm for feedforward neu-ral network training, International Journal of Artificial Intelligence and Applications 2 (2011), No. 3, pp.36-43

23 S. Walton, O. Hassan, K. Morgan, M. R. Brown: Modified Cuckoo search: A new gradient free optimization algorithm, Chaos, Solitons and Fractals 44 (2011), pp. 710-718

24 A. Kaveh, T. Bakhshpoori: Optimum design of steel frames using Cuckoo Search algorithm with Lévy flights, Struct. Design Tall Spec. Build. (2011)

The Authors of This Contribution

Dr. Ismail Durgun received B.Sc. degree in Me-chanical Engineering from Uludag University in 1988. He worked as a research assistant for Ul-udag University from 1989 to 1993. He received M.Sc. degree in Mechanical Engineering from Is-tanbul Technical University in 1991 with his the-sis on “Transient Heat Transfer and Cooling Load in Building” and started Ph.D. education in Me-chanical Engineering in 1992. He has been work-ing for TOFAS since 1993 and he is the adminis-trator of Prototype Production Department. Dr. Ali Riza Yildiz is an Associate professor at the Department of Mechanical Engineering, Bursa Technical University (BTU). Dr. Yildiz is a Vice Dean of Natural Science & Engineering Faculty of Bursa Technical University. He is also director of Multidisciplinary Product Design and Optimiza-tion Laboratory (MPDOL) at BTU. His research in-terests are vehicle design, vehicle crashworthi-ness, vehicle and pedestrian safety, crush box de-sign and optimization, shape and topology optimization of vehicle components, advanced op-timization techniques, sheet metal forming.

particle swarm based approach, International Journal of Advance Manufacturing Technology, in press, DOI: 10.1007/s00170-011-3496-y

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15 A. R. Yildiz, N. Ozturk, N. Kaya, F. Ozturk: Hybrid multi-objective shape design optimization using Taguchi’s method and genetic algo-rithm, Structural and Multidisciplinary Optimization 34 (2007), pp. 277-365

16 A. R. Yildiz: An effective hybrid immune-hill climbing optimization approach for solving de-sign and manufacturing optimization prob-lems in industry, Journal of Materials Process-ing Technology 209 (2009), pp. 2773-2780

17 A. R. Yildiz, F. Ozturk: Hybrid enhanced genetic algorithm to select optimal machining parameters in turning operation, Proceedings of the Institution of Mechanical Engineers Part B, Journal of Engineering Manufacture 220 (2006), pp. 2041-2053

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try. This research describes an optimiza-tion strategy based on the Cuckoo search algorithm for solving structural design problems. The Cuckoo search algorithm is applied to a vehicle component taken from automotive industry. It is seen that better results can be achieved with the CS. There-fore, the CS is a suitable optimization tech-nique for the solution of structural design optimization problems.

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Abstract

Struturdesignoptimierung von Fahrzeugkomponenten mittels des Cuckoo-Suchalgorithmusʼ. Um heutige Anforderungen an das Fahrzeug-design zu berücksichtigen und um die Kosten- und Kraftstoffeffektivität zu erhöhen, nimmt das Interesse am Design leichter und kosteneffektiver Fahrzeugkomponenten weiterhin zu. In der diesem Beitrag zugrunde lie-genden Studie wurde ein neuer Optimierungsalgorithmus angewendet, der so genannte Cuckoo Suchalgorithmus (CS). Es handelt sich um die erste CS-Applikation für das Formdesign in der Literatur. Der CS-Algorith-mus wird hierbei zur Strukturdesignoptimierung einer Fahrzeugkompo-nente angewendet, um zu zeigen, wie er bei der Lösung von Strukturdesi-gnaufgaben angewendet werden kann. Die Ergebnisse zeigen, wie damit ein verbessertes Design erreicht werden kann.

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