**3. Example of the SZGA**

The value of the zooming factor α, an optimal parameter was obtained in reference [8], and was found to show good match with the empirical one. Using this zooming factor in SZGA, the displacement of a truss structure was derived by minimizing the total potential energy of the system. The capacity of the servomotor, which operates the wicket gate mounted in a Kaplan type turbine of the electric power generator, was optimized using SZGA with the value of zooming factor 8.

This is just one parameter among the full optimal parameters discussed in sec.2.1 9. Therefore, the analysis done with this factor 8 is a simplified analysis. As commented in section 2.1, the values of the parameters of a well-behaved test model suggested in the Table 1 can be used for an optimization, or the values of the parameters obtained in another way as discussed in the next section can be used.

Several additional examples of SZGA optimization are presented in the following sections to provide more insight on SZGA and to find another way of choosing the values of the SZGA parameters. The first example finds the Moony-Rivlin coefficients of a rubber material to compare with those from the least square method. The second example is a damage detection problem in which the difference between the measured natural frequencies and those of the assumed damage in the structure is minimized. The third example finds the

The Successive Zooming Genetic Algorithm and Its Applications 11

Errors to be minimized Haines & Wilson Least Square SZGA

Simple extension 0.757932 0.709209 0.921277

Pure shear 0.702015 0.620089 0.370579 Equi-biaxial 13.2580 0.242475 0.139983 Total error 14.7180 1.57177 1.43184

Table 3. Comparisons of errors among the different methods for obtaining Mooney-Rivlin 6

Structures can sometimes experience failures far earlier than expected, due to fabrication errors, material imperfections, fatigue, or design mistakes, of which fatigue failure is perhaps the most common . Therefore, to protect a structure from any catastrophic failure, regular inspections that include knocking, visual searches, and other nondestructive testing are conducted. However, these methods are all localized and depend strongly on the skill and experience of the inspector. Consequently, smart and global ways of searching for damages have recently been investigated by using rational algorithms, powerful computers,

 The objective function of the difference between the measured data and the computed data is minimized according to an assumed structural damage to find the locations and intensities of possible damages in a structure. The measured data can be the displacement of certain points or the natural frequencies of the structure, while the computed data are obtained by FEM using an assumed structural damage, whose severity is graded between 0 and 1. For example, Chou et al. used static displacements at a few locations in a discrete structure composed of truss members, and adopted a kind of mixed string scheme as an implicit redundant representation. Meanwhile, Rao adopted a residual force method, where the fitness is the inverse of an objective function, which is the vector sum of the residual forces, and Koh adopted a stacked mode shape correlation that could locate multiple

Yet, a typical structure can be sub-divided into many finite elements and has many degrees of freedom. Thus, FEM for a static analysis, as well as for a frequency analysis, takes a long time. For a GA, the analysis time is related to the number of functions used for evaluating fitness. This number can become uncontrollable when monitoring a full structure, and as a result, the RAM or memory space required becomes too large and the access rate too slow

Accordingly, the proposed SZGA is very effective in this case, as it does not require so many chromosomes, even as few as 4, thereby overcoming the slow-down of the convergence rate of the conventional GA, which need many chromosomes in determining the extent of a damage. Furthermore, the issue of many degrees of freedom can also be solved by subdividing the monitoring problem into smaller sub-problems because the number of damages will likely be between 1~4, as long as the structure was designed properly. Moreover, the fact that cracks usually initiate at the outer and tensile stressed locations of a

coefficients

and FEM.

**3.2 Damage detection of structures** 

damages without incorporating sensitivity information 11.

when handling so much data.

optimal link specification (lengths and initial angular positions of members) to control the double link system with one motor in an automotive diesel engine. The fourth and last example finds an optimal specification (parametric sizes at specified positions) of a ceramic jar that satisfies the required holding capacity.

### **3.1 Determination of Mooney-Rivlin coefficients**

The rubber is a very important mechanical material in everyday life, used widely in mechanical engineering and automotive engineering. Rubber has low production cost and many advantages such as its characteristic softness, processability, and hyper-elasticity. The development of the rubber parts including most process of the shape design, product process, test evaluation, ingredient blending for the required property has used the empirical methods. CAE based on advances in computer-aided structural analysis software is applied to many products. FEM method is applied on various models of rubber parts to evaluate the non-linearity property and the theoretical hyper-elastic behavior of rubber, and to develop analysis codes for large, non-linear deformation.

The structure of rubber-like materials are difficult to analyze because of their material nonlinearity and geometric non-linearity as well as their incompressibility. Furthermore, unlike other linear materials, rubber materials have hyper-elasticity, which is expressed by the strain energy function. The representative strain energy functions in the finite element analysis of rubber are the extension ratio invariant function (Mooney-Rivlin model) and the principal extension ratio function (Ogden model). This case uses the Mooney-Rivlin model to investigate the behavior of a rubber material.

The value of the zooming factor changes according to the number of variables and the population number of a generation. If the population number is large, more exact solution can be obtained than the approach with smaller one. For a large population number, which is inevitable in the case of many design variables, longer computation time is needed. In this case, because six design valuables are used to solve the six material properties, nine hundred population units per one generation are used. At this time, whenever zooming is needed, the function is calculated 90,000 times, where, 900 is the population number per one generation and 100 is generation number per one zooming because zooming is implemented after 100 generations . So the point number searched per one valuable is 6 units (=90,0001/6). To search the optimum point, the zooming factor must be not less than 1/6. Therefore, the zooming factor of 0.2 is used.

The maximum generation number must be decided after the zooming factor is chosen. If the zooming factor is large, the exact solution can be solved as increasing zooming step. Generation numbers have to be decided by the user because they affect the amount of calculation like the population numbers do. For example, when zooming factor of 0.3 is chosen and Maxgen (maximum allowed generation number) is decided as 1000 (NZOOM = 10), the accuracy of the final searching range becomes ZRANGE = α(Nzoom-1) = 0.3(10-1) = 1.97E-05, and if Maxgen is decided by 1500 (NZOOM = 15) the final searching range becomes ZRANGE = α(Nzoom-1) = 0.3(15-1) = 4.78E-08, where ZRANGE is the value related with the resolution of solution and is the searching range after N steps of zooming. The smaller this value is, the more exact the solution becomes. In this case, Maxgen=900 is adopted. SZGA minimized the total error better than the other two methods.


Table 3. Comparisons of errors among the different methods for obtaining Mooney-Rivlin 6 coefficients

#### **3.2 Damage detection of structures**

10 Bio-Inspired Computational Algorithms and Their Applications

optimal link specification (lengths and initial angular positions of members) to control the double link system with one motor in an automotive diesel engine. The fourth and last example finds an optimal specification (parametric sizes at specified positions) of a ceramic

The rubber is a very important mechanical material in everyday life, used widely in mechanical engineering and automotive engineering. Rubber has low production cost and many advantages such as its characteristic softness, processability, and hyper-elasticity. The development of the rubber parts including most process of the shape design, product process, test evaluation, ingredient blending for the required property has used the empirical methods. CAE based on advances in computer-aided structural analysis software is applied to many products. FEM method is applied on various models of rubber parts to evaluate the non-linearity property and the theoretical hyper-elastic behavior of rubber, and

The structure of rubber-like materials are difficult to analyze because of their material nonlinearity and geometric non-linearity as well as their incompressibility. Furthermore, unlike other linear materials, rubber materials have hyper-elasticity, which is expressed by the strain energy function. The representative strain energy functions in the finite element analysis of rubber are the extension ratio invariant function (Mooney-Rivlin model) and the principal extension ratio function (Ogden model). This case uses the Mooney-Rivlin model

The value of the zooming factor changes according to the number of variables and the population number of a generation. If the population number is large, more exact solution can be obtained than the approach with smaller one. For a large population number, which is inevitable in the case of many design variables, longer computation time is needed. In this case, because six design valuables are used to solve the six material properties, nine hundred population units per one generation are used. At this time, whenever zooming is needed, the function is calculated 90,000 times, where, 900 is the population number per one generation and 100 is generation number per one zooming because zooming is implemented after 100 generations . So the point number searched per one valuable is 6 units (=90,0001/6). To search the optimum point, the zooming factor must be not less than 1/6. Therefore, the

The maximum generation number must be decided after the zooming factor is chosen. If the zooming factor is large, the exact solution can be solved as increasing zooming step. Generation numbers have to be decided by the user because they affect the amount of calculation like the population numbers do. For example, when zooming factor of 0.3 is chosen and Maxgen (maximum allowed generation number) is decided as 1000 (NZOOM = 10), the accuracy of the final searching range becomes ZRANGE = α(Nzoom-1) = 0.3(10-1) = 1.97E-05, and if Maxgen is decided by 1500 (NZOOM = 15) the final searching range becomes ZRANGE = α(Nzoom-1) = 0.3(15-1) = 4.78E-08, where ZRANGE is the value related with the resolution of solution and is the searching range after N steps of zooming. The smaller this value is, the more exact the solution becomes. In this case, Maxgen=900 is adopted. SZGA

jar that satisfies the required holding capacity.

**3.1 Determination of Mooney-Rivlin coefficients** 

to develop analysis codes for large, non-linear deformation.

minimized the total error better than the other two methods.

to investigate the behavior of a rubber material.

zooming factor of 0.2 is used.

Structures can sometimes experience failures far earlier than expected, due to fabrication errors, material imperfections, fatigue, or design mistakes, of which fatigue failure is perhaps the most common . Therefore, to protect a structure from any catastrophic failure, regular inspections that include knocking, visual searches, and other nondestructive testing are conducted. However, these methods are all localized and depend strongly on the skill and experience of the inspector. Consequently, smart and global ways of searching for damages have recently been investigated by using rational algorithms, powerful computers, and FEM.

 The objective function of the difference between the measured data and the computed data is minimized according to an assumed structural damage to find the locations and intensities of possible damages in a structure. The measured data can be the displacement of certain points or the natural frequencies of the structure, while the computed data are obtained by FEM using an assumed structural damage, whose severity is graded between 0 and 1. For example, Chou et al. used static displacements at a few locations in a discrete structure composed of truss members, and adopted a kind of mixed string scheme as an implicit redundant representation. Meanwhile, Rao adopted a residual force method, where the fitness is the inverse of an objective function, which is the vector sum of the residual forces, and Koh adopted a stacked mode shape correlation that could locate multiple damages without incorporating sensitivity information 11.

Yet, a typical structure can be sub-divided into many finite elements and has many degrees of freedom. Thus, FEM for a static analysis, as well as for a frequency analysis, takes a long time. For a GA, the analysis time is related to the number of functions used for evaluating fitness. This number can become uncontrollable when monitoring a full structure, and as a result, the RAM or memory space required becomes too large and the access rate too slow when handling so much data.

Accordingly, the proposed SZGA is very effective in this case, as it does not require so many chromosomes, even as few as 4, thereby overcoming the slow-down of the convergence rate of the conventional GA, which need many chromosomes in determining the extent of a damage. Furthermore, the issue of many degrees of freedom can also be solved by subdividing the monitoring problem into smaller sub-problems because the number of damages will likely be between 1~4, as long as the structure was designed properly. Moreover, the fact that cracks usually initiate at the outer and tensile stressed locations of a

The Successive Zooming Genetic Algorithm and Its Applications 13

Fig. 5. Number of function calculations with respect to the number of variables

*n k <sup>n</sup> <sup>C</sup> kn k* !

guaranteed by the successively zoomed infinitesimal range.

No. of cracks nCk

The SZGA can pinpoint an optimal solution by searching a successively zoomed domain. Yet, in addition to its fine-tuning capability, the SZGA only requires several chromosomes for each zoomed domain, which is a very useful characteristic for structural damage detection of a large structure that has a great number of solution variables. In the present study, just four or six digits of chromosomes were used. The accuracy of optimal solution is

Most structures have few cracks, which may exist at different locations. Therefore, a combinational search method is suggested to search for separate cracks by choosing probable damage site as nCk. n denotes the number of total elements and k denotes the number of possible crack sites (1~4). Thus, up to four cracks (k) were considered in a continuum structure modelled with n ( = 20) elements, and the number of function calculations between the combinational search and the full scale search was compared.

1 20 0.580671×105 0.578096×109 0.100445×10-3 2 190 0.950000×106 0.578096×109 0.164332×10-2 3 1140 0.990843×107 0.578096×109 0.171398×10-1 4 4845 0.740788×108 0.578096×109 0.128143

Table 4. Result of combinational searching method to reduce amount of calculation in SZGA

When monitoring the entire structure, the number of function calculations became about six hundred million based on the relation between the number of variables and the number of

!( )! <sup>=</sup> <sup>−</sup> (6)

No. of function calculation Ratio (Combinational/Full) Combinational search Full scale search

structure is also an advantage. As a result, the number of sub-problems becomes manageable, and the required time is much reasonable.

Several tests were performed first to determine the effectiveness of the SZGA for structure monitoring, where regional zooming is not necessary. Next, the procedure used to subdivide the monitoring problem is presented, along with a comparison of the amount of computation required between a full-scale monitoring analysis and a sub--divide monitoring analysis according to the number of probable damage sites. The optimization problem for various cases of structural damage detection was solved by using three or six variables, zooming factor of 0.2 or 0.3, and total number of function evaluations of 100,000 or 150,000, which is NZOOM × sub-iteration population number. The sub-iteration population number means the total population number in a sub-generation of one zooming.

Fig. 3. Zooming factor with respect to the number of variables

Fig. 4. Number of sub-iteration population with respect to the number of variables

Fig. 3, Fig. 4 and Fig. 5 are the fitting curves of 'NVAR α ', 'NVAR - NSP' and 'NVAR - Number of function calculation' relationship data, respectively, based on Table 1. These figures are prepared for the data point not shown in Table 1 for interpolation purpose.

12 Bio-Inspired Computational Algorithms and Their Applications

structure is also an advantage. As a result, the number of sub-problems becomes

Several tests were performed first to determine the effectiveness of the SZGA for structure monitoring, where regional zooming is not necessary. Next, the procedure used to subdivide the monitoring problem is presented, along with a comparison of the amount of computation required between a full-scale monitoring analysis and a sub--divide monitoring analysis according to the number of probable damage sites. The optimization problem for various cases of structural damage detection was solved by using three or six variables, zooming factor of 0.2 or 0.3, and total number of function evaluations of 100,000 or 150,000, which is NZOOM × sub-iteration population number. The sub-iteration population number means the total population number in a sub-generation of one

manageable, and the required time is much reasonable.

Fig. 3. Zooming factor with respect to the number of variables

Fig. 3, Fig. 4 and Fig. 5 are the fitting curves of 'NVAR -

Fig. 4. Number of sub-iteration population with respect to the number of variables

prepared for the data point not shown in Table 1 for interpolation purpose.

of function calculation' relationship data, respectively, based on Table 1. These figures are

α

', 'NVAR - NSP' and 'NVAR - Number

zooming.

Fig. 5. Number of function calculations with respect to the number of variables

The SZGA can pinpoint an optimal solution by searching a successively zoomed domain. Yet, in addition to its fine-tuning capability, the SZGA only requires several chromosomes for each zoomed domain, which is a very useful characteristic for structural damage detection of a large structure that has a great number of solution variables. In the present study, just four or six digits of chromosomes were used. The accuracy of optimal solution is guaranteed by the successively zoomed infinitesimal range.

Most structures have few cracks, which may exist at different locations. Therefore, a combinational search method is suggested to search for separate cracks by choosing probable damage site as nCk. n denotes the number of total elements and k denotes the number of possible crack sites (1~4). Thus, up to four cracks (k) were considered in a continuum structure modelled with n ( = 20) elements, and the number of function calculations between the combinational search and the full scale search was compared.

$$\mathbf{C}\_{n}\mathbf{C}\_{k} = \frac{n!}{k!(n-k)!} \tag{6}$$


Table 4. Result of combinational searching method to reduce amount of calculation in SZGA

When monitoring the entire structure, the number of function calculations became about six hundred million based on the relation between the number of variables and the number of

The Successive Zooming Genetic Algorithm and Its Applications 15

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

Original 0-90 0-89.144 0-91.958 2.044

Optimal 0-90 0-89.999 0-89.999 0.603

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

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

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

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

bandwidth denotes the half width of the narrow region with the basic penalty only.

Output( degree )

Left Right Max.

Difference

motor.

Model Input

Table 6. Comparison of original and optimal models

**3.4 Proper band width for equality constraints** 

function should be feasible.

( degree )

method is proposed to overcome this kind of slow convergence rate.

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 damage using the combination method and SZGA.


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