1 Introduction
The aeroengine piping system is mainly used to transport various media, such as fuel, lubrication oil, and air. Most pipes between engine casing and accessories are interconnected through pipe joints and clamps, forming a complex piping system. The routing process must consider various factors, leading to potential continuous adjustments to the pipe-routing. Therefore, the constraints are gradually determined. Despite the support of modern computer-aided design systems for the digital design of pipes, the process of mechanical routing design still takes approximately 2–3 months or more. Therefore, automated routing has significant engineering value.
In recent years, the issue of automatic pipe-routing has drawn significant scholarly attention, leading to the proposal of various assortment route optimization algorithms, which are mainly based on A* algorithm [
1,
2], visible graph [
3], escape algorithm [
4–
6], genetic algorithm (GA) [
7–
10], particle swarm optimization (PSO) [
10–
12], ant colony optimization (ACO) algorithm [
13,
14], and artificial fish swarm [
15].
Zhang and Bai [
16] developed a single-pipe routing technique based on an improved artificial fish swarm algorithm and ensured that the generated pipes adhered to the manufacturing process. Wang and Liu [
17] introduced a heuristic pipe-routing technique based on geodesic lines and the degree of interference, which is proved beneficial for routing pipes in complex curved shapes of aeroengine casing. Liu and Jiao [
18] proposed a pipe-routing method considering vibration for aeroengines using the Kriging model and Nondominated Sorting Genetic Algorithm II. Qu et al. [
19] conceived an adaptive octree modeling method anchored on routing space characteristics. They also introduced an enhanced Modified Max-Min Ant System optimization algorithm for pipe-routing, which adopts a layered point selection and an active update mechanism to optimize routing efficiently. Liu et al. [
20] introduced an enhanced multi-objective ant lion optimizer algorithm and accounted for clamp effects on pipe-routing. Yuan et al. [
21] presented an approach for the parallel routing of multiple grouped pipes for aircraft engines, effectively addressing the challenges in the installation of parallel, grouped pipes. Neumaier et al. [
22] conducted a design practice implementing a graph-based design language for automated pipe-routing within the Airbus A320 aircraft’s landing gear compartment.
Wang et al. [
23] proposed a method to enhance the efficiency and effectiveness of the ship’s branch pipe-routing design process. The method is based on an improved particle swarm GA and supported by collaborative co-evolution. Chen et al. [
24] introduced a method for optimizing the spatial path and adjusting multicorner pipes. This method considers assembly requirements and is built upon the coding structure of pipe pose parameters. Dong et al. [
25] demonstrated the application of a heightened multi-objective ACO algorithm in a ship pipe-routing design. Kim et al. [
26] introduced a process to identify and prioritize key search points during ship pipe-routing, thereby enhancing the speed of pipe-routing by reducing the search points needed for automatic ship pipe-routing realization. Lin et al. [
27] proposed a discrete hybrid algorithm, a blend of differential evolution (DE) and cuckoo search for optimizing ship pipe-routing. Kim et al. [
28] initiated a reinforcement learning method rooted in course learning for the dynamic adjustment of ship pipes. This method swiftly generates new solutions upon modification of accessories. Blanco et al. [
29] formulated a minimum-cost multicommodity network flow model to address all feasible technical requirements for pipe routing, and this method was applied to a real naval case. Lin et al. [
30] developed a versatile multi-objective cooperative PSO methodology for ship pipe route design, featuring a multiparticle dimension strategy and a cooperative update mechanism.
Current research has limitations mainly in three aspects.
I. Adequate consideration of route constraints is lacking. Most studies do not allow designers to guide the solution of pipe-routing through route constraints, such as clamps, control points, waypoints, or parallel pipe segments.
II. The fixed coding length in the optimization algorithm limits the number of pipe segments in advance. Most of the time, the optimal number of pipe segments is unknown before routing.
III. The acquired solutions are too similar in pipe shape to provide designers with enough alternatives.
Designers need routing tools that can generate as many differentiated solutions with routing constraints as possible. This study proposes an multi-solution pipe-routing (MSPR) algorithm to aid designers obtain multiple pipe-routing solutions for decision-making. The main contributions are as follows:
(1) The pipe routing of the aeroengine poses a multi-solution, multi-objective optimization problem with constraints during the design phase. During this phase, the MSPR algorithm can generate multiple path solutions for a pipe while responding to progressively determined route constraints. Furthermore, the generated multiple solutions are remarkably differentiated.
(2) A hybrid encoding mechanism is introduced. In this mechanism, routing constraints are encoded as fixed-length encoding, and free-exploration points are encoded as variable-length encoding. The fixed-length encoding describes route constraints, such as control points, clamps, parallel pipe segments, and waypoints.
(3) The variable-length encoding iteration uses a modified genetic iteration mechanism. The mechanism includes selection, crossover, mutation, and adding point and removing point mechanisms. It also allows an appropriate number of control points to be obtained through iteration.
(4) A niche method based on K-means clustering is proposed to classify pipe shapes and guide the pipe shape differentiation of routing results.
The rest of this paper is organized as follows. In Section 2, the mathematical model of the multi-objective constrained optimization for the issue of multi-solution pipe-routing is established. Section 3 presents the algorithm of multi-solution pipe-routing. In Section 4, routing simulation experiments are constructed and compared with the DE algorithm and the multi-objective PSO (MOPSO) algorithm to demonstrate the effectiveness of the MSPR algorithm. Finally, Section 5 concludes this paper.
2 Mathematical modeling of pipe-routing problem
The issue of multi-solution pipe-routing is a multi-objective constrained optimization problem. The aim is to obtain all optimal solutions within a constrained decision space with multiple conflicting optimization objectives.
2.1 Objectives and constraints
As the core part of the MSPR algorithm, the routing objectives and the constraints mainly considered are as follows:
Objective 1: Minimize the total length of the pipe.
Objective 2: Maximize the alignment of the pipe with the axial and circumferential directions of the aeroengine.
Objective 3: Minimize the distance between the pipe and the casing surface.
Objective 4: Minimize the number of control points.
Constraint 1: Avoid obstacles such as accessories, port, and pipe segments.
Constraint 2: Avoid pipes embedding into the aeroengine casing or out-of-bounds.
Constraint 3: Satisfy the requirements for minimum length of pipe segment and maintain pipe uniformity.
Constraint 4: Ensure proper sequencing of control points and avoid the wrong points in segments formed by clamp or waypoint constraints, as shown in Fig.1.
Fig.1 Wrong point in segments formed by the clamp. |
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Constraint 5: Adhere to the appointed control points.
Constraint 6: Pass through the appointed clamps and waypoints.
Constraint 7: Maintain parallel with the appointed pipe segments.
2.2 Multi-objective fitness design
The following fitness design, which aims at the above optimization objectives, is proposed.
First, the total pipe length can be expressed as follows:
where n is the number of pipe segments, li is the length of the ith pipe segment, and ω1 is the scaling factor for f1.
Second, the axial or circumferential routing is expressed as follows:
where is the length of the ith pipe segment arranged along the axial and circumferential directions, lmin is the minimum length of pipe segments required by the process, ω2 is the correction factor per unit pipe length, and b is a parameter that prevents the fitness from being too small when the pipe segment is too long and is set to −3.
The proximity to the engine casing is expressed as follows:
where ρi is the radius of the ith pipe control point, and ρc is the minimum radius of the engine casing. The scaling factor ω3 is used to adjust the value due to the large casing radius.
Finally, the number of control points is expressed as follows:
Based on the constraint objective, the following penalty term design is proposed:
I. The interference penalty is expressed as follows:
where nobs is the number of discrete points that interfere between the pipe segments and obstacles and between the pipe segments.
II. The out-of-bounds penalty is expressed as follows:
where nbounds is the number of out-of-bounds discrete points in the pipe segment.
III. The length requirement penalty is expressed as follows:
where lsuitmin and lsuitmax are the minimum and maximum suitable lengths for the pipe segment, respectively, and lsuit is the suitable length for the pipe segment. The penalty is applied when the length of the pipe segment is outside the appropriate interval. In particular, when the length of the pipe segment is less than the minimum length of the straight segment, the pipe cannot be processed normally. This penalty is rounded up in the calculation.
IV. The penalty for improper control point position is expressed as follows:
where npos is the number of wrong points in the pipe segments formed by waypoint or clamp constraints and improperly ordered points.
The route constraints 5–7, which are incorporated into the design of the encoding, are detailed in Section 3.2.
2.3 Mathematical model of multi-solution pipe-routing
According to the above optimization objectives and constraints, as well as the fitness functions and penalty terms, the multi-solution, multi-objective constrained optimization problem for a pipe can be expressed as follows:
where Oi is the obtained feasible solution, and D is the problem dimension. Here, h5–h7 denote the constraints posed by Constraints 5–7, respectively.
3 MSPR algorithm design
The pipe-routing algorithm of aeroengines must first establish the map of the slewing routing space and then obtain the information about the routing, such as relationships of logical connection and route constraints. After all the information is acquired, the MSPR algorithm can be executed to generate multiple feasible solutions.
3.1 Grid construction and interference judgment
Before the automated routing for aeroengine pipes, the environmental information of the routing task must be extracted. We use a fast and accurate method of grid generation to obtain the information in routing space. First, the routing space is determined according to the routing task. Then, the axis-aligned bounding boxes (AABBs) of all accessories are obtained to determine quickly whose AABB is in the routing space. Second, for AABBs that belong to the routing space, the oriented bounding box (OBB) is further discretized according to the grid resolution. Each grid point is traversed in the OBB to determine whether an entity exists. The value of grid cells is set to 1 if an entity exists and 0 if otherwise. Finally, the grid points around the outermost grid point of the entity are also assigned the value 1 to ensure a safe distance between the pipe and the obstacle.
The grid is constructed based on the ρ-, θ-, and Z-axes in the cylindrical coordinate system. However, the generatrix Rc(z) of the casing is first extracted because of the mixed cylindrical-conical shape of the engine casing. Subsequently, the edges are affixed to the height of the casing generatrix during grid extraction. The grid is partitioned along each axis to obtain a matrix M. The coordinate transformation of the corresponding points to the grid matrix M is expressed by Eq. (10).
where is the matrix index value corresponding to the control point of a pipe, is the resolution of the matrix, θmin is the circumferential minimum boundary value, and zmin is the minimum boundary value in the axial direction.
As shown in Fig.2, the assessment of the pipe segment interference in the grid is accomplished through an interference determination based on resolution sampling. The pipe segments are discretized and dilated according to the grid resolution. Subsequently, the discrete points of the pipe segments are traversed, and the corresponding grid values are examined. The interference is determined when the grid values are equal to one. The pseudocode for the interference judgment based on resolution is shown in Tab.1.
Tab.1 Pseudocode for interference decision based on resolution |
| Algorithm 1 Interference decision based on resolution |
| Input: set of pipe control points , grid matrix . |
| Output: interference result. |
| 1 begin |
| 2 Initialize the judgment value |
| 3 for do |
| 4 |
| 5 |
| 6 Generate the set to be discriminated Pinterference∪ PD∪Pdil |
| 7 end for |
| 8 for do |
| 9 |
| 10 If then |
| 11 Break |
| 12 end if |
| 13 end for |
| return interference result |
Fig.2 Interference judgment based on grid and resolution. |
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3.2 Encoding design
In the design stage of an aeroengine pipe-routing, many route constraints are gradually determined. These route constraints must be considered to help in the convergence to the ideal pipe shape. The constraints 5–7 in Section 2.1 are designed into fixed-length encoding. Thus, the length of this encoding segment remains constant during iteration. The route constraints include the extension distance of the starting and the ending pipe segments, the appointed control points, the waypoints and clamps by which the pipe should pass, and the existing pipe segments with which the pipe should parallel. Moreover, the variable length segment of encoding denotes the free control points. The encoding style is shown in Fig.3.
Fig.3 Cylindrical coordinate-based encoding mode for the pipe with route constraints. |
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(1) The pipe segments that originate from the starting point and the ending point must initially extend a certain distance along the nominal directions of the two ports. Therefore, their encoding is set as .
(2) The appointed control points are added directly at the time of decoding and are not involved in encoding.
(3) The waypoint is a point that a pipe segment must pass through. The coordinates of the two endpoints of the pipe segment can be represented by the direction vector that passes through the waypoint and the extension distances on both sides. Therefore, the encoding is designed as .
(4) The center point position and center line orientation of the selected clamps can be obtained. Therefore, the pipe segment can be extended at certain distances along the center line to both sides. The encoding is set as .
(5) Parallel pipes are fixed using duplex clamps to determine the center points and direction vectors of the parallel pipes. The spacing between the two pipes is specified in the design manual. Therefore, the constraint of the parallel pipe segments is analogous to the constraint of the clamp. The encoding is designated as .
(6) Free control point uses its coordinates for the encoding. In this section, the number of free points has no limit. However, it is determined by the iteration results.
3.3 MSPR algorithm process
The MSPR algorithm is based on the MOPSO framework with various improvements, and the whole process can be summarized in the following five steps:
Step 1. Initialize parameters and populations.
Step 2. Decode and control point sorting. Then, perform fitness calculation.
Step 3. Evaluate the shape similarity of the pipe and cluster to obtain a new elite solution set.
Step 4. Conduct split encoding. Then, perform iteration separately for different encodings. Finally, merge the encodings.
Step 5. Repeat steps 2–4 until the maximum number of iterations is reached and generate the solutions.
The details are expressed in the following. The process of the MSPR algorithm is shown in Fig.4.
Fig.4 Flowchart of the MSPR algorithm. |
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3.4 Initialization of population
The population is initialized through a random procedure. For the fixed-length encoding, numerical values are generated randomly within the specified range for each position in the encoding. For the variable-length encoding, a constant number of points are randomly selected in the routing space to generate the encoding. The number of generated free points is determined by the empirical Eq. (11).
where nv1 is the number of control points for initializing variable-length encoding, Dm is the distance between the two endpoints of the routing pipe segment, nw is the number of waypoint constraints, nc is the number of clamp constraints, and np is the number of parallel pipe constraints.
3.5 Control points sorting and fitness calculation
A greedy sorting method is introduced when decoding to obtain the shortest path solution because of the inherent absence of sequencing in the encoding design. This solution serves as the pipe solution represented by the encoding. The initial pipe before sorting is shown in Fig.5(a). In the greedy sorting procedure, the first control point is the initial extension point. The subsequent control point is the one nearest to the previous control point, and it is incrementally selected until a complete connection is made. This process helps to derive an accurate sequence of pipe control points. The result of sorting is shown in Fig.5(b). In addition, the sorting of pipe control points helps the algorithm converge quickly. After a reasonable sequence of the control points for the pipe is obtained, the fitness of the pipe is calculated according to Eqs. (1)–(8).
Fig.5 Sorting method for obtaining the correct pipe: (a) unordered pipeline before sorting; (b) correct pipe after sorting. |
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3.6 Pipe shape similarity evaluation
Given that the length of the particle encoding is variable, the similarity evaluation of the shapes between pipes must be considered. The projection of pipes on the casing surface generally represents the approximate pipe shape. Within the projection plane, the angle each pipe segment makes with the Z-axis can characterize the geometric direction of the pipe. First, the particle encoding is decoded. Then, the angle of each pipe segment to the Z-axis is calculated. Finally, the ratio of the length of the pipe segment to the minimum length of pipe segments is calculated, and the angle can represent the shape of the pipe. This representation is denoted as the pipe shape vector , as shown in Fig.6. Subsequently, a piecewise quadratic interpolation method is employed to align the lengths of the short vectors to a 64-dimension vector . Thus, the alignment for each individual is completed. Then, all aligned vectors are compiled into a pipe shape set , which contains individuals of equal length and represents the shape of the pipes.
Fig.6 The obtained pipe shape vector. |
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3.7 Niche approach based on pipe shape clustering
The niche method [
31] is an assistance algorithm employed to seek multiple global or local optimal solutions. This approach divides the entire population into different niches, thereby allowing each niche to explore different areas of the value at the same time. Clustering, a common task in unsupervised learning, serves as a frequently used tool in niche methods [
32,
33]. In the problem of routing pipes with multiple solutions, the generated solutions must show diversity, which can be reflected intuitively through variations in geometric shapes. Here, the K-means clustering algorithm [
34] is used to partition individuals during the iterative process and to group individuals with similar geometric shapes into the same cluster. It involves partitioning a given sample set
to minimize the square error
E by using Eq. (12) and obtaining a cluster division
, where
μi represents the mean vector of the cluster
Ci, as shown in Eq. (13).
A small E value indicates high similarity among individuals within a cluster. The number of clusters k is a hyperparameter that requires adjustment and setting.
After individuals are clustered based on pipe shape, a nondominated solution set is selected within each cluster. Subsequently, the crowding distance of individuals for each nondominated solution set is computed based on their fitness. The larger the crowding distance is, the higher the probability of being selected is. Then, all selected elite individuals are merged to form a new elite solution set. This new elite solution set serves as guidance for individual iterations and produces global optimal individuals. Individuals are guided by elite individuals within their respective clusters.
3.8 Iteration mechanisms
The uncertainty in the length of individual encodings poses a challenge to the iteration of PSO [
35,
36]. In response to this problem, we introduce an iteration mechanism for variable-length encoding, which includes selection, crossover, mutation, and adding and removing mechanisms. Therefore, the fixed-length encoding and the variable-length encoding are treated separately during the iteration phase. The iteration mechanisms of the fixed-length encoding are based on PSO, as shown in Eq. (14).
where ω4 is the inertia factor, ω5 is the self-learning factor, ω6 is the best learning factor, Xi,t is the fixed-length encoding of the ith individual in generation t, Xp,t is the individual optimal fixed-length encoding of the ith individual, Xe,t is the fixed-length encoding of the corresponding elite individual, Vi,t is the current velocity of the ith individual in generation t, Vi,t+1 is the updated velocity of the ith individual in generation t, and Xi,t+1 is the fixed-length encoding after flight.
The selection mechanism for variable-length segments involves directly preserving individuals from the elite solution set for the subsequent generation.
where Yi,t is the variable-length encoding of the ith individual in generation t.
As shown in Fig.7, the crossover mechanism adopts three consecutive codes of control point as the minimum unit of crossover, which have a geometric meaning. When the crossover individual and the corresponding elite solution individual cross, each point position has a half probability of retaining the original point or selecting the point of the elite solution. In Fig.7, is the random number used to determine whether to leave.
Fig.7 Uniform crossover mechanism. |
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As shown in Fig.8, the mechanisms of mutation and adding points involve moving or generating points of practical significance. Two options exist for mutation or adding points. One is the diagonal point formed by two adjacent points, and the other is the midpoint of the line between two adjacent points. Mutation and adding point mechanisms make a rapid design convergence to the required piping route, which is along the axial or circumferential direction of the aeroengine as far as possible.
Fig.8 Mutation and adding point mechanisms. |
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The mechanism of removing points also has practical implications, as shown in Fig.9. Two consecutive control points that are too close to each other do not meet the design requirements for the minimum length of pipe segments. Moreover, small-angle pipes are not permitted in manufacturing. Therefore, if the spatial distance between two points is too close, one of the points is randomly removed. The control point that makes the adjacent segments form an acute angle should also be removed.
Fig.9 Removing point mechanism. |
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The updated formula for the variable length encoding is as follows:
where Ni represents mutation, crossover, adding point, and removing point mechanisms. ci is the probability of the corresponding operation occurring. The probability is expressed as in Eq. (17), where T is the total number of iterations, and t is the current iteration number. The probability of adding or removing is kept low to prevent drastic changes in the number of control points. Furthermore, the probability of removing points is slightly higher than the probability of adding points to minimize the number of control points.
3.9 Termination criterion evaluation
The algorithm is terminated when a specified number of iterations of the algorithm have been reached. The elite solution set for each generation is output. All the elite solution sets are reevaluated. Starting from the last number of iterations, the evaluation is carried out progressively forward, thereby eliminating similar results and those that do not meet the manufacturing requirements. The final result is output once 10 satisfactory solutions have been accumulated or the entire solution set has been thoroughly screened.
3.10 Postprocessing of the pipe
Maintaining uniform lengths for each pipe segment is beneficial for fluid transportation. Pipes generated through iterations may not be inherently uniform. Postprocessing is applied to ensure uniformity by homogenizing the pipe segments (Fig.10). First, the pipe segments in the same direction are merged. Second, a multiple number of shorter pipe segments along the same direction are used instead of long pipe segments. The control points of discrete pipe segments are given the same radial height as long pipe segments. Such postprocessing aligns with manufacturing process requirements and design standards while preserving the overall shape of the pipes.
Fig.10 Pipe homogenization and discretization. |
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4 Simulation experiments
Currently, our team has conducted research and development on a pipe-routing system for aeroengines called “AeroPiping” on SIEMENS NX. This system integrates automated routing of external pipes, automatic assembly, inspection of routing rules, and rapid adjustment of the associated components. Building upon the foundation of “AeroPiping,” three types of experiments were designed to investigate the effectiveness and quality of algorithms for the multi-solution routing of external pipes on the surface of the aeroengine casing. These experiments were performed on a desktop computer equipped with an Intel Core i5-13600K 3.5 GHz CPU and 32 GB RAM.
4.1 Experiment comparison without route constraint
The parameters of the MSPR algorithm and the comparison algorithm are shown in Tab.2. The scaling factors ω1 and ω3 appropriately scale down the fitness values to enhance the effectiveness of the penalty term when the penalty values are fixed. The scaling factor ω2 is determined from empirical data in design practice. The factors ω4, ω5, and ω6 are hyperparameters, determined according to the convergence rate of the algorithm.
Tab.2 Algorithm parameter settings for simulation experiments |
| Parameter | Population, m | Iterations, T | Differential scaling factor, CD | Crossover probability, Cp | Preset bending count, Pb | Number of clusters, k | Scaling factor, ω1 | Scaling factor, ω2 | Scaling factor, ω3 | Inertia factor, ω4 | Self-learning factor, ω5 | Best learning factor, ω6 |
| MSPR | 50 | 300 | – | – | – | 4 | 0.05 | 3 | 0.01 | 0.8 | 0.1 | 0.1 |
| DE | 50 | 300 | 0.5 | 0.5 | 6 | – | 0.05 | 3 | 0.01 | – | – | – |
| MOPSO | 50 | 300 | – | – | 8 | – | 0.05 | 3 | 0.01 | 0.8 | 0.1 | 0.1 |
The same pipe-routing problem is performed using the DE algorithm and the MOPSO algorithm to make a further comparison. The DE algorithm, which has been proposed in recent years, is designed according to previous research [
21]. This algorithm has the same application scene and application object as the method presented in this study. It is also an exemplary algorithm for conducting weighted processing of multiple optimization objectives. The MSPR algorithm is based on the MOPSO framework with various improvements. Consequently, the MOPSO algorithm is also subjected to comparison. The MOPSO algorithm adopts the standard algorithm, with an encoding design based on commonly employed schemes in previous research [
21]. By contrast, the MOPSO algorithm lacks the niche method based on clustering and the iterative mechanism proposed in this study. In addition, the fitness function is calculated using Eqs. (1)–(8) in the MOPSO algorithm.
In Fig.11, the DE-based single-objective optimization algorithm, represented by DE, can only yield a solution, denoted as Pipe A. The generated solution, under the weighted fitness, represents a global or local optimum with the shortest total pipe length of 1608.62 mm. However, such solutions often face the challenge of meeting the practical design requirements.
Fig.11 Solution from the DE algorithm for pipes. |
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Fig.12 shows that MOPSO produces two distinct nondominated solutions, namely, Pipe B and Pipe C, under the same fitness conditions. Compared with these two solutions, other solutions display slight variations. However, the difference is not significant. The shape of Pipe C is irregular, possibly because of the competitive selection mechanism across the entire nondominated solution set. Multi-objective optimization algorithms usually aim to achieve a uniform distribution of solutions along the Pareto front. However, aligning these solutions with the design preferences of human experts can be challenging.
Fig.12 Typical results from the MOPSO algorithm for pipes without constraints. |
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Fig.13 illustrates that the MSPR algorithm endeavors to obtain pipes with pronounced geometric structural differences, such as Pipes D–G. These pipes exhibit clear circumferential and axial structural characteristics, providing engineers with a highly comprehensive set of reference options. In routing practice, this type of structure facilitates the installation of clamps and the fixation of pipes. Moreover, the engineer prefers Pipe F when Pipes B and C are compared with Pipe F. Although the MSPR algorithm attempts to obtain additional results with significant differences, it still cannot obtain all the results. Furthermore, the MSPR and MOPSO algorithms are highly sensitive to the initial population in the test, thereby affecting the results obtained.
Fig.13 Typical results from the MSPR algorithm for pipes without constraints. |
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The comparative results of the experiments are shown in Tab.3. In Tab.4, the fitness values of the typical solutions obtained by MOPSO and MSPR algorithms are presented. Penalty terms are imposed on each fitness. The solutions obtained by the MSPR algorithm exhibit mutual dominance rather than forming a Pareto solution set. This phenomenon is due to the use of niche methods and the reevaluation of solutions. Furthermore, the use of niche methods in the MSPR algorithm leads to a lower exploration of individuals for each cluster than that in the MOPSO algorithm. This scenario can affect the optimality and stability of solutions obtained under similar conditions. However, it is the case that many mutually distinct solutions are obtained.
Tab.3 Result comparison of each algorithm |
| Item compared | Multiple solutions | Differential solution ratio | Axial and circumferential routing |
| DE | No | – | – |
| MOPSO | Yes | 2/10 | Existence |
| MSPR | Yes | 4/10 | Significant |
Tab.4 Typical result fitness of the MOPSO and MSPR algorithms |
| Algorithm | Pipe | f1 | f2 | f3 | f4 |
| MOPSO | B | 40.39 | −14 | 67.93 | 13 |
| C | 35.10 | 1 | 62.00 | 12 |
| MSPR | D | 53.72 | −17 | 81.45 | 23 |
| E | 58.30 | 0 | 74.58 | 26 |
| F | 59.51 | −5 | 75.81 | 27 |
| G | 60.46 | 2 | 76.84 | 28 |
4.2 Experiment comparison with route constraints
This section evaluates the effectiveness of various route constraints in the encoding design of the study. Experiments 1–4 evaluated the implementation performance of the algorithm when each route constraint was introduced individually, as shown in Tab.5. Experiment 5 assessed the implementation performance when all four constraints were introduced simultaneously. The experimental result consists of the refined set of solution sets obtained after the elimination of approximate solutions.
Tab.5 Constraint settings for the experiment |
| Experiment | Route constraint |
| 1 | Appointed control point |
| 2 | Clamp |
| 3 | Waypoint |
| 4 | Parallel pipe |
| 5 | All constraints |
As shown in Fig.14–Fig.18, the MSPR algorithm can obtain diverse typical pipe-routing solutions under the conditions of four distinct route constraints. The solutions for Pipes P and Q in Fig.17 are quite similar because of the reduced decision space compared with other cases.
Fig.14 Multi-solution routing for an appointed control point constraint. |
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Fig.15 Multi-solution routing for a clamp constraint. |
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Fig.16 Multi-solution routing for a waypoint constraint. |
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Fig.17 Multi-solution routing for a parallel constraint. |
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Fig.18 Routing result for all four constraints. |
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In Fig.18, the typical solution routed under the presence of four different route constraints is presented, and other generated results exhibit only minor variations compared with the results obtained with this solution. These results demonstrate the effectiveness of the designed route constraints. The MSPR algorithm can obtain multi-solution routing under various constraint cases.
4.3 Routing in a relatively complex scenario
Multiple pipe routes were generated in a relatively complex scenario, as shown in Fig.19. The example of 16 pipes routing on an aeroengine with a diameter of more than 1000 mm and a length of 2570 mm was carried out. The entire routing work took approximately 2 h to complete by using MSPR and route constraints. This routing work includes the addition of branch ports and clamp fixing. The generation time of the grid is approximately 35 s, and the generation time of the single pipeline is approximately 32–70 s.
Fig.19 Routing result in a complex scenario. |
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5 Conclusions
In this study, the MSPR algorithm is proposed to address the issue of multi-solution in pipe-routing with route constraints. Unlike the conventional aeroengine pipe-routing algorithms, the MSPR algorithm enables the direct generation of multiple solutions exhibiting distinct geometric variations while satisfying four types of route constraints.
(1) The routing objectives and constraints are established based on the engineering requirements of pipeline laying to generate multiple solutions for a pipe with four types of routing constraints and an uncertain number of control points. Subsequently, fitness and penalty functions are formulated accordingly. Additionally, hybrid encoding is proposed, wherein fixed-length encoding represents route constraints, whereas variable-length encoding represents free control points. Greedy sorting is used to determine the correct order of the control points to ensure accurate decoding.
(2) The iteration of variable-length coding is enhanced by proposing improved iterative mechanisms for selection, crossover, mutation, adding point, and removing point. These enhancements facilitate the determination of an optimal number of control points for the pipe and enable the generation of a pipe with axial and circumferential shape.
(3) A way of calculating the similarity of the pipe shape to different numbers of control points is proposed to increase the difference between the obtained solutions. Then, the niche method based on the clustering of pipe shapes is introduced to divide the populations and guide their evolution.
The feasibility of the proposed method is demonstrated through simulation results and experimental cases conducted on “AeroPiping.” Future research should consider multi-solution routing for managing multiple pipes and accounting for the impact of vibration characteristics on the routing solution.
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Fig.1 Wrong point in segments formed by the clamp.
Tab.1 Pseudocode for interference decision based on resolution
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