# Routing Design for Pipe System Considering Natural Frequency
## 6 Simulations and Results
### Application  
After validating the proposed method in 2D plane cases, it is demonstrated that the algorithm can effectively consider the natural frequency of the longest segment, ensuring that the routing results can withstand the actual engineering loads. In case study 3, the algorithm is applied to a real engine model. The model used in this article is the Garrett GTCp85-98D gas turbine engine, which was obtained through an open-source model acquisition method. The model is a scaled model. The routing space in this model is filled with numerous obstacles and accessories, posing challenges for pipe arrangement (Figure 20). Figure 19: Fitness value in case study 2. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Case** & **Pipe** & **Length (mm)** & **Frequency (Hz)** &
\begin{tabular}{c} **Minimum** \\ **Segment (mm)** \\ \end{tabular} & **Fitness Value** & **Generations** \\ \hline \multirow{3}{*}{2} & 1 & 650.2 & 2686.6 & 79.7 & 370.4 & 2520 \\  & 2 & 544.6 & 20,862.5 & 46.7 & 2.50362 & 3510 \\  & 3 & 77.8 & 46,053.8 & 43.4 & 402.752 & 2200 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Information of the paths in case study 2. Figure 18: Simulation result in case study 2. The different stars indicate the different start and end points of the pipe. Using the adaptive decomposition method in this article to model the routing space. The final grid size achieved by the adaptive decomposition and the entire process takes a total of 1461 s in a 2.9 GHz CPU computer. In case study 3, the number of pipes is 12, and the diameters are 5 mm, 10 mm, and 20 mm. The material is 0 cr18 ni9 which is commonly used in aero engines. In the surrogate model of natural frequency for the longest segment, the parameters are shown in the Table 2.  
Pretreatment on the Direction Constraint of the Pipe Nozzles  
The initial pipe nozzles may be covered by the obstacle extension; to eliminate the impact and meet the direction constraints on pipe nozzles, the locations of the connection points will be extended outward appropriately in advance. The grid paths between the initial nozzles and the new nozzles are straight, which should be kept and marked as obstacles. When finishing the routes between the new pipe nozzles, both kinds of paths will be connected together to obtain a complete layout. Note that the extension distance must be greater than the maximum distance of obstacle extension. In general, the same extension distance will be specified for all pipe nozzles at first; to obtain better results, the extension distances of some nozzles could be adjusted according to the feedback of the initial layouts, in this case, when the extension distance of each nozzle is set to 8 mm (Table 6). After these pretreatments, the pipes can be arranged by following steps: (a) extend the pipe nozzles outward by 8 mm; (b) search the path of the current pipe with the proposed algorithms; (c) smooth the route to allow for actual processing. In Example 3 of this article, the fillet radius of all the pipelines is 3 mm. Figure 21a shows the full view of the engine routing space. Figure 21b shows the pipe view after hiding the engine parts. The obtained layout is shown as Figure 21. As can be seen, to avoid collisions between pipes, many bending parts are involved. In Figure 21, all turning points are the fixed positions of the clamps, thereby dividing the pipeline into multiple segments and increasing the natural frequency of the pipeline. From the values of the evaluation function in Table 7, it can be found that the fitness function designed in this article effectively normalizes the magnitude mismatch between pipe length and natural frequency, achieving the goal of considering natural frequency during the routing process. When the fitness value is 1, this