**3.3 Link system design using weighting factors**

This section presents a procedure involving the use of a genetic algorithm for the optimal designs of single four-bar link systems and a double four-bar link system used in diesel engines. Studies concerning the optimal design of the double link system comprised of both an open single link system and a closed single link system which are rare, and moreover the application of the SZGA in this field is hard to find, where the shape of objective function have a broad, flat distribution 12.

During the optimal design of single four-bar link systems, one can find that for the case of equal IO angles, the initial and final configurations show certain symmetry. In the case of open single link systems, the radii of the IO links are the same and there is planar symmetry. In the case of closed single systems, the radii of the IO links are the same and there is point symmetry.

To control the Swirl Control Valve in small High Speed Direct Injection engines, there are two types of actuating systems. The first uses a single DC motor controlled by Pulse Width Modulation, while the second uses two DC motors. However, this study uses the first type of actuator for the simultaneous control of two Swirl Control Valves using a double link system. When two intake valves in a diesel engine are controlled by a single motor, they usually exhibit quite different angular responses when the design variables for the control link system are not properly selected. Therefore, in order to ensure balanced performance in diesel engines with two intake valves, an optimization problem needs to be formulated and solved to find the best set of design variables for the double four-bar link system, which in turn can be used to minimize the different responses to a single input.

Two weighting factors are introduced into the objective function to maintain balance between the multi-objective functions. The proper ratios of weighting factors between objective functions are chosen graphically. The optimal solutions provided by the SZGA and developed FORTRAN Link programs can be confirmed by monitoring the fitness. The reduction in the objective functions is listed in the tables. The responses of the output links that follow the simultaneously acting input links are verified by experiment and the Recurdyn 3-D kinematic analysis package. The experimental and analysis results show good correspondence.

14 Bio-Inspired Computational Algorithms and Their Applications

function calculations. However, when the combinational searching method was used, the number of function calculations was reduced by about 10-1~10-4 times when compared to the full-scale monitoring case, as shown in Table 4. Table 5 shows the good detection of the

Element No. 19 20 25 26 31 32

soundness factor 1 1 0.5 1 1 1

detection result 1.0 1.0 0.499999 1.0 1.0 1.0

This section presents a procedure involving the use of a genetic algorithm for the optimal designs of single four-bar link systems and a double four-bar link system used in diesel engines. Studies concerning the optimal design of the double link system comprised of both an open single link system and a closed single link system which are rare, and moreover the application of the SZGA in this field is hard to find, where the shape of objective function

During the optimal design of single four-bar link systems, one can find that for the case of equal IO angles, the initial and final configurations show certain symmetry. In the case of open single link systems, the radii of the IO links are the same and there is planar symmetry. In the case of closed single systems, the radii of the IO links are the same and there is point

To control the Swirl Control Valve in small High Speed Direct Injection engines, there are two types of actuating systems. The first uses a single DC motor controlled by Pulse Width Modulation, while the second uses two DC motors. However, this study uses the first type of actuator for the simultaneous control of two Swirl Control Valves using a double link system. When two intake valves in a diesel engine are controlled by a single motor, they usually exhibit quite different angular responses when the design variables for the control link system are not properly selected. Therefore, in order to ensure balanced performance in diesel engines with two intake valves, an optimization problem needs to be formulated and solved to find the best set of design variables for the double four-bar link system, which in

Two weighting factors are introduced into the objective function to maintain balance between the multi-objective functions. The proper ratios of weighting factors between objective functions are chosen graphically. The optimal solutions provided by the SZGA and developed FORTRAN Link programs can be confirmed by monitoring the fitness. The reduction in the objective functions is listed in the tables. The responses of the output links that follow the simultaneously acting input links are verified by experiment and the Recurdyn 3-D kinematic analysis package. The experimental and analysis results show

turn can be used to minimize the different responses to a single input.

Table 5. Result of structural damage detection using the combination method and SZGA

damage using the combination method and SZGA.

**3.3 Link system design using weighting factors** 

Actual

Damage

have a broad, flat distribution 12.

symmetry.

good correspondence.

The proposed optimal design process was successfully applied to a recently launched luxury Sports Utility Vehicle model. Table 6 shows the original response and that of the optimized model. The optimal model exhibits almost the exact left and right outputs, and the difference between the left and right responses of 0.603 is thought to be a least value for the given positions of the link centers and the double control system adopting a single input motor.


Table 6. Comparison of original and optimal models

#### **3.4 Proper band width for equality constraints**

In a problem having an equality constraint, it is not so simple for GA to satisfy the constraint while maintaining efficiency. Optimal solution lies on the line of equality constraint. It is very important to gernerate individuals on or near the equality line. However, the desirable narrow area including the equality line is very small compared with the whole area. The number of individual generated in this narrow area is much less than those in the outer area of the desirable narrow area including the equality line. Therefore, the convergence rate of GA or SZGA is significantly slow for the problems with equality constraints. The bandwidth method is proposed to overcome this kind of slow convergence rate.

For the minimization problems, we added a basic penalty function to meet the equality constraint, which will be explained soon. For this problem with the basic constraint, we can not expect a rapid convergence rate as mentioned above. Therefore, we added an additional penalty function to the region, located out of the desirable narrow area including the equality line, to make an infeasible area of a very highly increased objective function. The bandwidth denotes the half width of the narrow region with the basic penalty only.

There are three methods to handle the equality constraints using GA. One is to give both sides the penalty functions along the equality condition. The other is to give one side the monotonic function and other side the even (jump) penalty function along the equality constraint. However, the one side with the monotonic penalty should be feasible. And, the final one is to apply one side with no penalty function and the other side with the even (jump) penalty function along the equality constraint, and the one side of no penalty function should be feasible.

The penalty methods provided in Fig. 6 only with original penalty, is the basic technique for handling the equality constraint 15. With this kind of basic technique only, however, the convergence rate would be too slow to reach the optimal point. Many generated individuals are wasted because they mostly too far from the equality constraint line. Therefore we need an additional penalty function to increase the effectiveness of GA. That is an additional

The Successive Zooming Genetic Algorithm and Its Applications 17

When the band-width is bigger than about 0.3, the best fitness dropped rapidly. In other words, if we open the full range as the feasible solution range, the optimal ridge would be too narrow to be chosen by GA. In conclusion, a too narrow bandwidth may lead to a

The SZGA explained in the foregoing sections may be applied to more fields of interest, such as, the optimal design of ceramic pieces considering important factors like beauty, usage, stability, strength, lid, and exact volume. Prediction of a long -term performance of a

The most dominant characteristics of SZGA are its accuracy up to the required significant digits, and its rapid convergence rate even in the later stage. However, users have to properly select the parameters, namely, the zooming factor, number of zooms, and number of sub-domain population. A useful reference can be found in Table 1, Fig. 3, Fig. 4, and Fig. 5. The number of zooms can be determined by eq.(5) for a given upper limit of accuracy. The number of sub-domain population has been recommended as a fixed number until

[2] D.E. Goldberg, 1989, *Genetic Algorithms in Search, Optimization, and Machine Learning*,

[3] K. Krishnakumar, 1989, "Micro-genetic algorithms for stationary and non-stationary

[4] N.N. Schraudolph, R.K. Belew, 1992, "Dynamic parameter encoding for genetic

[5] D.L. Carroll, 1996, "Genetic algorithms and optimizing chemical oxygen-lodine lasers," Developments in Theoretical and Applied Mechanics, Vol. 18, pp. 411~424. [6] Y.D. Kwon, S.B. Kwon, S.B. Jin and J.Y. Kim, 2003, "Convergence enhanced genetic

[8] Y.D. Kwon, H.W. Kwon, J.Y. Kim, S.B. Jin, 2004, "Optimization and verification of

[9] Y.D. Kwon, H.W. Kwon, S.W. Cho, and S.H. Kang, 2006, "Convergence rate of the

[10] Y.D. Kwon, J.Y. Kim, Y.C. Jung, and I.S. Han, 2007, "Estimation of rubber material

problems," Computers and Structures, Vol. 81, Iss. 17, pp. 1715~1725. [7] Y.D. Kwon, S.B. Jin, J.Y. Kim, and I.H. Lee, 2004, "Local zooming genetic algorithm and

Mechanics, An International Journal, Vol. 17, No. 5, pp. 611~626.

Engineering and Technology, Vol. 18, No. 5, pp. 29~35.

Transactions on Computers, Vol. 5, Iss. 6, pp. 1200~12007.

Mechanics and Materials Engineering, Vol. 1, Iss. 6, pp. 815-826.

function optimization," SPIEP, Intelligent Control and Adaptive Systems, Vol. 1196,

algorithm with successive zooming method for solving continuous optimization

its application to radial gate support problems," Structural Engineering and

parameters used in successive zooming genetic algorithm," Journal of Ocean

successive zooming genetic algorithm using optimal control parameters," WSEAS

property by successive zooming genetic algorithm," JSME, Journal of Solid

rubber seal installed in an automotive engine is another possible application.

divergence and a too wide bandwidth may result in inefficiency.

now, however, it may be varied as a function of the zooming step.

[1] D.M. Himmelblau, 1972, *Applied Nonlinear Programming*, McGraw-Hill.

algorithms," Journal of Machine Learning, Vol. 9, pp. 9-21.

**4. Further studies and concluding remarks** 

**5. References** 

Addison-Wesley.

pp. 289~296.

penalty to the objective function if the condition is located in outer region of a certain bandwidth centered with the equality constraint.

Fig. 6. Three methods to handle the equality constraint in GA.

Using the type (c) equality constraint and additional bandwidth penalty, the design of a ceramic jar was optimized for three values of zooming factors and various bandwidths of equality constraint, as shown in Fig. 7 and Table 7. The result showed a proper range of bandwidth for the equality constraint. In Table 7, the optimal solutions were found for the jar, satisfying the equality constraint of 2 liter volume.

Fig. 7. Best fitness for band-width of an equality constraint and numbers of generation.


Table 7. Proper bandwidths and the optimal solutions for three zooming factors

This optimization problem does not converge below 0.15 of the band-width of an equality constraint, because the objective function is rather complicated and the band-width is relatively too narrow to give the most candidated optimal individual out of feasible region. When the band-width is bigger than about 0.3, the best fitness dropped rapidly. In other words, if we open the full range as the feasible solution range, the optimal ridge would be too narrow to be chosen by GA. In conclusion, a too narrow bandwidth may lead to a divergence and a too wide bandwidth may result in inefficiency.

#### **4. Further studies and concluding remarks**

The SZGA explained in the foregoing sections may be applied to more fields of interest, such as, the optimal design of ceramic pieces considering important factors like beauty, usage, stability, strength, lid, and exact volume. Prediction of a long -term performance of a rubber seal installed in an automotive engine is another possible application.

The most dominant characteristics of SZGA are its accuracy up to the required significant digits, and its rapid convergence rate even in the later stage. However, users have to properly select the parameters, namely, the zooming factor, number of zooms, and number of sub-domain population. A useful reference can be found in Table 1, Fig. 3, Fig. 4, and Fig. 5. The number of zooms can be determined by eq.(5) for a given upper limit of accuracy. The number of sub-domain population has been recommended as a fixed number until now, however, it may be varied as a function of the zooming step.

#### **5. References**

16 Bio-Inspired Computational Algorithms and Their Applications

penalty to the objective function if the condition is located in outer region of a certain

Using the type (c) equality constraint and additional bandwidth penalty, the design of a ceramic jar was optimized for three values of zooming factors and various bandwidths of equality constraint, as shown in Fig. 7 and Table 7. The result showed a proper range of bandwidth for the equality constraint. In Table 7, the optimal solutions were found for the

(Zooming factor 0.1) (Zooming factor 0.2) (Zooming factor 0.3)

Volume

(liter) Z1 Z2

Fig. 7. Best fitness for band-width of an equality constraint and numbers of generation.

0.1 0.15~0.3 0.0802 2.000 0.4790 1.000 0.2 0.15~0.3 0.0802 2.000 0.4790 1.000 0.3 0.15~0.3 0.0802 2.000 0.4790 1.000

This optimization problem does not converge below 0.15 of the band-width of an equality constraint, because the objective function is rather complicated and the band-width is relatively too narrow to give the most candidated optimal individual out of feasible region.

Weight (kg)

Table 7. Proper bandwidths and the optimal solutions for three zooming factors

(a) (b) (c)

Fig. 6. Three methods to handle the equality constraint in GA.

jar, satisfying the equality constraint of 2 liter volume.

Proper band-width

Zooming factors

bandwidth centered with the equality constraint.


**2** 

*Russia* 

**The Network Operator Method for Search** 

Askhat Diveev1 and Elena Sofronova2

*2Peoples' Friendship University of Russia* 

**of the Most Suitable Mathematical Equation** 

*1Institution of Russian Academy of Sciences Dorodnicyn Computing Centre of RAS,* 

For many applied and research problems it is necessary to find solution in the form of mathematical equation. These problems are the selection of function at approximation of experimental data, identification of control object model, control synthesis in the form of state space coordinates function, the inverse problem of kinetics and mathematical physics, etc. The main method to receive mathematical equations for solution of these problems consists in analytical transformations of initial statement formulas of the problem. A few problems have the exact analytical solution, therefore mathematicians use various assumptions, decomposition, and special characteristics of solutions. Usually mathematicians set the form of mathematical equation, and the optimal parameters are found using numerical methods and PC. Such methods as the least-square method have been applied to the problems of approximation for many years (Kahaner D. et al., 1989).

Recently the neural networks have been used to solve complex problems when the mathematical equation cannot be found analytically. The structure of any neural network is also given within the values of parameters or weight coefficients. In problems of function approximation and the neural network training the form of mathematical equation is set by the researcher, and the computer searches for optimum values of parameters in these

In 1992 a new method of genetic programming was developed. It allows to solve the problem of search of the most suitable mathematical equation. In genetic programming mathematical equations are represented in the form of symbol strings. Each symbol string corresponds to a computation graph in the form of a tree. The nodes of this graph contain operations, and the leaves contain variables or parameters ( Koza, 1992, 1994; Koza, Bennett

Genetic programming solves the problems by applying genetic algorithm. To perform the crossover it is necessary to find symbol substrings that correspond to brunches of trees. The analysis of symbol strings increases the operating time of the algorithm. If the same parameter or variable is included in the required mathematical equation several times, then to solve the problem effectively the genetic programming needs to crossover the trees so

that the leaves contain no less than the required number of parameters or variables.

**1. Introduction** 

equations (Callan, 1999; Demuth et al., 2008).

et al., 1999 & Koza, Keane et al., 2003).

