# Routing Design for Pipe System Considering Natural Frequency
## 7 Conclusions  
In this article, a novel and effective pipe path planning method considering natural frequency has been developed. This article presents an adaptive decomposition method for modeling the narrow routing space in actual aero-engine applications. The adaptive decomposition method contributes to the accurate modeling of the narrow routing space, enabling more effective and efficient pipe design in aero-engine applications. Then, the proposed method addresses the issues of illegal initial paths by improving the initial population using RRT. Additionally, several novel operators are introduced to enhance the performance of the GA. The evaluation function considers the trade-off between natural frequency and length by applying appropriate weights. Moreover, a new co-evaluation algorithm is proposed for multi-pipe path planning, which initializes the chromosomes of each pipe using the connection points generated around the routed pipe and evolves them through the iterative process. Simulations demonstrate that the proposed algorithm effectively generates optimal layouts for different piping cases. Several conclusions are listed as follows:  
1. This article proposes a new algorithm for the spatial modeling of aero engines and ships. This algorithm accelerates the recognition efficiency of obstacles in pipeline layout space by adaptively adjusting the grid size, merging and decomposing existing grids. Thus, existing algorithms can be applied to the spatial extraction of real engines and ships. Compared with traditional cell decomposition, the efficiency of obstacle identification has been improved by more than 50%.
2. This article integrates an RRT algorithm, a genetic algorithm, and other optimization algorithms to introduce novel selection, mutation, crossover, and other operators, leading to the development of a new multi-pipe routing algorithm. By formulating appropriate fitness functions, the natural frequency of the longest segment in the pipeline route is considered, ensuring that the laid pipeline meets practical requirements, such as avoiding resonance, and can be directly applied to real engineering environments. This comprehensive approach addresses the challenges of pipeline layout and optimization, enhancing the efficiency and effectiveness of the entire process.
3. The proposed algorithms in this article demonstrate their ability to generate connected paths between the pipe nozzles while satisfying various constraints. These constraints include maintaining the direction constraints on the nozzles, ensuring collision avoidance between pipes and the layout environment. The algorithms consider the complex routing space, obstacles, and diversity of constraints to optimize the pipe paths effectively. By incorporating techniques such as genetic algorithms, RRT, and co-evolutionary methods, the algorithm in this article can consider the maximum interval in the pipeline route, thereby reducing the maximum interval length and reducing its vibration response. This comprehensive approach contributes to the generation of optimal and feasible pipe arrangements in complex engineering scenarios, such as aero engines and other applications where precise and efficient pipe routing is crucial.  
Conceptualization, J.F.; methodology, writing, H.X.; validation, Q.M.; formal analysis, Q.M.; investigation, Y.S. All authors have read and agreed to the published version of the manuscript. This study was supported by the National Science and Technology Major Project of China (J2019-I-0008-0008 and J2019-IV-0002-0069).  
Not applicable.  
Data Availability Statement: The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors declare no conflicts of interest.