# A Novel Optimization Layout Method for Clamps in a Pipeline System
## 3 Results and Discussion
### Optimization by Genetic Algorithm  
In order to implement the generic algorithm, 6 main factors (parameter coding, initial population setting, fitness function design, control parameter setting and constraint setting) need to be taken into consideration. Commonly used coding methods include binary coding and real-number encoding. The real-number encoding method is more suitable in this case and the efficiency is high. The selection of some important parameters (mainly include population n, mutation rate Pm, crossover rate Pc, and maximum number of iterations maxgen) directly affects the performance of the genetic algorithm. Currently there is no accepted method for determining the value of parameters in genetic algorithm. It is normally taken that n = 20-100, the mutation rate Pm = 0.0001-0.1, and the crossover rate Pc = 0.4-0.99. In this paper, the sets of data n = 50, the maximum number of iterations maxgen = 100, the crossover probability Pc = 0.7 and the mutation probability Pm = 0.05 are determined as parameters in genetic algorithm due to limited computing resources. The implementation of constraints in this paper is by constructing the penalty function. The selection of the adaptability function directly affects the convergence speed and the accuracy of the optimal solution of the genetic algorithm. Genetic algorithms basically do not need other information in evolutionary search, except the adaptability function based on the search of individuals. Therefore, the design of the adaptability function should be simple, efficient, and distinguished. There are two cases to optimize the clamp-pipeline system. One is to maximize the frequency difference and the other is to minimize the displacement or stress. When maximizing the frequency difference, the fitness function should be:  
\[F(x)\ =\ \left\{ \begin{array}{cc}f(x)-C_{\min},&\text{Meet all constraints}\\ 0,&\text{Any one of the constraints is not met}\end{array} \right.. \tag{8}\]  
where \(f(x)\) is the absolute value of the difference between the first-order frequency and the excitation frequency. \(C_{\min}\) is the minimum estimate value of \(f(x)\). When minimizing the displacement or stress, the fitness function should be:  
\[F(x)\ =\ \left\{ \begin{array}{cc}C_{\max}-f(x),&\text{Meet all constraints}\\ 0,&\text{Any one of the constraints is not met}\end{array} \right.. \tag{9}\]  
Figure 10: Errors of neural network prediction. where \(f(x)\) is the displacement or stress under certain excitation. \(C_{\text{max}}\) is the maximum estimate value of \(f(x)\). The results predicted by the neural network surrogate model shows that the possible maximum displacement is \(6\times 10^{-5}\) m and the maximum stress value is 50 MPa. The peak frequency of the vibration in a certain aircraft under working conditions consist of the frequency of the high and low-pressure rotors and their multiplications. The low-pressure rotor frequency is 159 Hz and the high-pressure rotor frequency is 206 Hz. The first-order frequency in the trial data is generally smaller than the low-pressure rotor frequency. When the first-order frequency difference is maximized, the value of the clamp position L1-L3 is shown in Figure 11. The frequency difference convergence curve is shown in Figure 12.  
Figures 11 and 12 shows that after serval iterations, L1, L2, L3 converge  
Figures 13 and 14 shows that after serval iterations, L1, L2, L3 converge at 111.3 mm, 100.8 mm, 110.7 mm respectively. The maximum displacement converges at 0.0338 mm at 77 th generation. When minimizing the stress under random loads, the extraction process of clamp positions (L1-L3) is shown in Figure 15. The maximum stress convergence curve is shown in Figure 16.  
Figures 15 and 16 shows that after serval iterations, L1, L2, and L3 converge at 128.3 mm, 85.7 mm, 122.7 mm respectively. The maximum displacement converges at 18.22 MPa at 11 th generation.