Multi-pipeline collaborative topology layout optimization method based on braid group theory

Abstract

The application discloses a kind of based on braid group theory's multi-pipeline collaborative topology layout optimization method, including establishing three-dimensional placeable space model;According to pipeline endpoint and constraint condition generates initial layout;Pipeline relative order and cross relationship are encoded as braid group word;According to braid group relationship formula generates topological transformation;With path length cost, collision cost, topological change cost, structure stress cost as target carries out iterative optimization;Finally output meets the three-dimensional pipeline center line and topological layout result of engineering constraint;Achieve in the scene of simultaneous design of multiple pipelines avoid global conflict caused by local greedy strategy, reduce invalid intersection and rework, improve the purpose of layout quality, construction efficiency and maintainability.

Description

Technical Field

[0001] This invention relates to the field of intelligent design and spatial layout optimization technology for complex equipment, and in particular to a multi-pipeline collaborative topology layout optimization method and system based on algebraic topology theory, Lie group optimization theory and intelligent computing methods. Background Technology

[0002] As modern equipment evolves towards miniaturization and integration, the spatial layout of flexible components such as pipes and cables faces extremely stringent geometric constraints. Existing layout optimization methods are mainly based on traditional path planning or intelligent algorithms, with a core focus on minimizing geometric indices (such as total length and number of bends). However, these methods have a fundamental limitation: they cannot effectively describe and control the topology of pipe bundles (i.e., the sequential relationship of intertwining among the pipes). Even with optimal geometric indices, a chaotic topology increases assembly difficulty and maintenance costs. Summary of the Invention

[0003] The purpose of this invention is to solve the problem that the existing technology cannot effectively describe and control the topology of the pipeline bundle. Even if the geometric indicators are optimal, if the topology is chaotic, it will increase the assembly difficulty and maintenance cost.

[0004] To address the above problems, this invention provides a multi-pipeline cooperative topology layout optimization method based on braid group theory, comprising: Step 1: Construct a parametric representation of a multi-pipeline spatial curve; No. The spatial curve of the pipeline is represented by the formula:

[0005] in, For the first The spatial curve of the pipeline, This represents the actual physical distance along the pipeline from the starting point to the current point. Let i be the three-dimensional coordinates of the pipeline; i ∈ n, where n is the total number of pipelines. The set of all pipeline space curves The formula is:

[0006] Step 2: Construct a three-dimensional, arrangeable spatial model of multiple pipelines; Based on the three-dimensional spatial model of the pipeline to be laid out, the pipeline start and end points, and the pipeline engineering constraint parameters, a three-dimensional arrangeable spatial model is established. Define the pipeline laying channel as the available space Ω R³, the three-dimensional arrangeable space model Ω is represented as: Ω={(x,y,z)∈R³∣d(x,y,z)>δ} Where R³ is the three-dimensional Euclidean space, d(x,y,z) is the distance from the three-dimensional coordinates of the pipeline to the surface of the nearest obstacle, and δ is the preset safety distance threshold. Step 3: Construct the braid group topological coding model; The interpenetration relationships between multiple pipes in space are mapped to a braid group, as shown in the formula:

[0007] in, A braided group consisting of n pipes, The positional relationship between the i-th pipeline and the (i+1)-th pipeline is the braid group generator; Concatenate all braid group generators in the order of their intersection to obtain braid group terms representing arbitrary pipeline topology states. The formula is:

[0008] Step 4: Establish a constraint model based on topological invariants; The topology change cost function is defined by comparing the topology invariants before and after optimization. The formula is as follows:

[0009] Where F is the topology change cost function, The term is the topological state braid group term after optimization, and ||.|| is the modulo operation on the vector; F equals 0, indicating that the topology is equivalent before and after optimization; F not equal to 0 indicates that the topology has changed, the optimization scheme is rejected and a penalty is imposed; Step 5: Establish a joint optimization formula: U = Bn × SE (3) Among them, SE(3) is a Lie group for rigid body motion in three-dimensional space, and the state of the Lie group describes the geometric changes in space; Step 6: Establish a multi-objective collaborative optimization model, the formula is:

[0010] Where L is the path length cost, C is the collision cost, T is the topology change cost, and S is the structural stress cost. These are the weighting coefficients; Step 7: Braided group genetic search optimization; Using a genetic algorithm for optimization search, the formula for generating chromosome X, i.e., individuals in the population, is:

[0011] Iterative optimization of chromosome code X yields X ; Step 8: Output the optimal piping layout; Based on the optimized X from step 7 The optimal solution is obtained using the following formula: X =( P1 P2 ,…,Pn )

[0012] Among them, Pi Let Qi,j be the optimal pipeline space curve for the i-th pipeline. Let be the optimal spatial coordinates of the j-th control point of the i-th pipeline, where j ∈ m and m is the total number of control points; The final optimized piping layout is obtained using the following formula:

[0013] Among them, P This is the set of all optimized pipeline space curves.

[0014] In the preferred embodiment, the three-dimensional spatial model in step S2 adopts a CAD solid model, specifically including: exporting solid model files in STEP and IGES formats from CAD software such as SolidWorks, CATIA, UG, and NX, and obtaining the outer contour surface, internal cavity, and obstacle boundary information of the equipment by parsing the model file.

[0015] In a preferred embodiment, the topological invariants in step S4 include at least one of the following: braided index, Burau matrix representation, or Alexander polynomial.

[0016] In the preferred approach, step S5, Lie group space optimization, employs transformation matrix operations, with the following formula:

[0017] in, Let R be the transformation matrix, R be a 3×3 orthogonal matrix with a determinant of 1, and t be the translation vector. Continuous optimization is achieved through exponential mapping.

[0018] in, , Let ξ be the spatial pose transformation matrix for the (k+1)th and kth iterations of optimization, where ξ is an element of the Lie algebra containing rotation and translation vector components.

[0019] In the preferred method, the collision cost C is determined based on the available space Ω. When the control point of the pipeline is within the available space Ω, C is 0; when the control point of the pipeline is not within the available space Ω, C is infinity.

[0020] In the preferred approach, the optimization process in step 7 includes: The initial population is generated in the following way: Step a: For each pipeline, randomly generate intermediate control points between the start and end points. These control points are located within the available space Ω. Calculate the set of spatial curves representing the pipeline based on these intermediate control points. ; Step b: Calculate the corresponding topological state braid group term β' based on the spatial intersection relationship between the pipelines; Based on step a 1. Obtain β' from step b, generate individual X', perform feasibility verification on individual X', and remove individuals that collide with obstacles; Step c: Crossover operator calculation: A topological crossover operator based on braid group generators is adopted. For two parent individuals, a crossover position k is randomly selected, and the first k braid group generators of the topological state braid group word βa in the first parent individual are concatenated with the braid group generators of the latter half of the topological state braid group word βb in the second parent individual to obtain the offspring braid group βc. Correspondingly, for the pipeline space curves Pa and Pb of the two parent individuals, when the generator fragments of the topological state braid group word βa are exchanged at position k, the pipeline also exchanges the control point fragments at the same position. A crossover position l is randomly selected, and the first l control points of Pa are concatenated with the latter half of Pb to obtain the offspring pipeline space curve Pc. The offspring inherits the topological category of one of the parents through topological invariant detection. Step d: Mutation operator calculation; Using the topology-preserving mutation operator, randomly select the braid group generator σi and execute:

[0021] in, The transformed braided words are obtained by performing conjugate transformations on the braided words to maintain homotopy equivalence. Apply random perturbations to the coordinates of the control points of the pipeline, using the following formula: Qi,j'=Qi,j+Δ Where Qi,j is the initial spatial coordinate of the j-th control point of the i-th pipeline, Qi,j' is the mutated spatial coordinate of the j-th control point of the i-th pipeline, and Δ is a small-amplitude random vector that satisfies a Gaussian distribution; In addition, a spatial constraint correction operator is used: the mutated control points are checked, and if there is a collision with an obstacle, that is, the mutated control points are not within Ω, the control points are pushed away from the obstacle surface along the collision normal direction by the local gradient descent method until the safety distance requirement is met.

[0022] The optimization proceeds iteratively until the termination condition is met, i.e., the maximum number of iterations is reached or the fitness change is less than a threshold, thus obtaining the optimal solution.

[0023] In the preferred approach, the group generator generation methods include: Choose the main extension direction of the pipeline as the projection direction, and project the pipeline spatial curve. Projecting the image onto a two-dimensional plane perpendicular to the projection direction yields the projection curve; Detect all intersections between pipes on the projection plane and sort the intersections according to the spatial order along the projection direction; For the intersection of the i-th pipeline and its adjacent (i+1)-th pipeline, the braid group generator of the two adjacent pipelines is determined based on their actual spatial elevation relationship along the projection direction at the intersection. Specific methods include: The i-th pipeline is located above and in front of the (i+1)-th pipeline. These two adjacent pipelines are defined as having a positive intersection, with the intersection direction being... =+1, the generator of the braid group is denoted as the positive crossover. ; The i-th pipeline is located below and behind the (i+1)-th pipeline. These two adjacent pipelines are defined as crossing in opposite directions. =-1, the generator of the braid group is denoted as the reverse crossover. .

[0024] The beneficial effects of this invention: The multi-pipeline collaborative topology layout optimization method disclosed in this application innovatively utilizes the mathematical structure of braid groups to describe the topological relationship of spatial pipelines. Compared with the prior art, this invention has the following advantages: it breaks through the problem that traditional graphical models cannot describe three-dimensional interpenetration relationships, achieves topology preservation optimization, ensures the stability of topology categories during the optimization process, improves the assembly efficiency of complex pipelines, enhances maintenance convenience, and is applicable to solid pipelines and flexible cables. Attached Figure Description

[0025] Figure 1 This is a schematic diagram of the overall process of the present invention. Detailed Implementation

[0026] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0027] Example 1: A multi-pipeline cooperative topology layout optimization method based on braid group theory, combined with Figure 1 As shown, the process includes: S1: acquiring and establishing a three-dimensional spatial model; S2: parametric representation of multi-pipeline spatial curves; S3: constructing a braided group topology coding model; S4: establishing a constraint model based on topological invariants; S5: establishing a joint optimization model; S6: establishing a multi-objective collaborative optimization objective function; S7: optimization based on braided group genetic search; S8: outputting the optimal pipeline layout.

[0028] The first step is to obtain and build a three-dimensional spatial model; Specifically, the three-dimensional spatial model of the equipment to be laid out, the start and end points of the pipelines, and the pipeline engineering constraint parameters are obtained to establish a three-dimensional arrangeable spatial model.

[0029] The three-dimensional spatial model can be a CAD solid model. Specifically, a solid model file in STEP or IGES format is exported from CAD software (such as SolidWorks, CATIA, UG / NX). The outer contour surface, internal cavity, and obstacle boundary information of the equipment are obtained by parsing the model file. The pipeline laying channel is defined as the arrangeable space Ω. R³, the available space is represented as: Ω={(x,y,z)∈R 3 |d(x,y,z)>δ} Where d(x,y,z) is the distance from the point to the nearest obstacle surface, and δ is the preset safe distance threshold.

[0030] The second step is to parametrically represent the spatial curves of multiple pipelines. The multiple pipelines to be laid out are parametrically represented by spatial curves to obtain the geometric set of the multiple pipelines:

[0031] Each pipeline is represented by a three-dimensional continuous curve, and the pipeline is parametrically represented by a cubic B-spline curve. The start and end points of the pipeline are fixed by engineering constraints, and the intermediate control points are variables to be optimized.

[0032] The third step is to construct a braid group topological coding model; Based on the spatial intersection, crossing, and relative exchange relationships among multiple pipelines, a topological representation model of a multi-pipeline braid group is established. The mutual crossing relationships between n pipelines in space are mapped to the braid group Bn:

[0033] Where σi represents a single exchange operation between the i-th pipeline and the (i+1)-th pipeline. The topology of any pipeline is represented as:

[0034] in, ∈{+1, -1} indicates a change of direction. +1 indicates that the i-th pipe crosses from the left front to the right rear (i.e., the i-th pipe crosses the i+1-th pipe), and -1 indicates a cross in the opposite direction.

[0035] In practice, the construction of braided words is achieved through the following steps: (1) Obtain the spatial coordinate sequence of the centerline of each pipeline; (2) Project the pipeline onto a two-dimensional plane along a certain projection direction (such as the z-axis direction); (3) Detect the intersections between pipelines on the projection plane and sort them according to the projection positions of the intersections; (4) Whenever an intersection between the i-th and (i+1)-th pipelines is detected, record the corresponding generator. ,in The sign is determined by the vertical spatial relationship between the two pipelines at the intersection. (5) Connect the generators in the order of their crossover to form braided word β.

[0036] The fourth step is to establish a constraint model based on topological invariants; A constraint model based on topological invariants is established to constrain the braid group state during the optimization process. The topological invariants include at least one of the following: braid exponent, Burau matrix representation, or Alexander polynomial. During the optimization process, the cost of topological change is defined by comparing the topological invariants before and after optimization.

[0037] in, This represents the initial topology state. The optimized topology state; Determine whether the optimization result retains the original pipeline topology category. If the result is positive, it means that the topology is equivalent before and after optimization; otherwise, it means that the topology has changed, and the optimization scheme should be rejected or a penalty should be imposed.

[0038] Fifth step: Establish a joint optimization model; Establish a joint optimization model that includes the braid group topological state space and the three-dimensional motion state space: X = Bn × SE (3) Where SE(3) represents the Lie group of rigid body motion in three-dimensional space; braid group states are used to describe topological changes, and Lie group states are used to describe spatial geometric changes.

[0039] The Lie group space optimization adopts the following approach:

[0040] This represents the spatial attitude change of the pipeline, where R∈SO(3) is the rotation matrix, t∈R³ is the translation vector, and it is mapped through an exponential method:

[0041] Complete continuous optimization.

[0042] In practice, each control point of each pipeline can be assigned an SE(3) transformation to adjust the spatial position and orientation of the control point during the optimization process. Since SE(3) is a Lie group, its optimization is performed on the manifold, avoiding the problem of difficulty in maintaining the orthogonality constraint of the rotation matrix in Euclidean space.

[0043] Step 6: Establish a multi-objective collaborative optimization objective function; The overall optimization objective function is:

[0044] Where: L - path length cost, C - collision cost, T - topology change cost, S - structural stress cost. The weighting coefficients satisfy Σwi=1.

[0045] Step 7: Optimize based on braid group genetic search; A genetic algorithm is used for optimization search, and the chromosome encoding is defined as:

[0046] Where β∈Bn is the braided group topology code, and P is the control point matrix of the i-th pipeline.

[0047] The initial population is generated in the following way: (1) For each pipeline, randomly generate intermediate control points that satisfy obstacle constraints between the start and end points; (2) Calculate the corresponding braided group term β based on the spatial intersection relationship between pipelines; (3) To verify the feasibility of the generated individuals, individuals that collide with obstacles are removed.

[0048] Crossover operator: A topological crossover operator based on braid group generators is adopted. For two parent individuals, a crossover position k is randomly selected; the first k generators of βa are concatenated with the latter half of βb to obtain the offspring braid group βc; correspondingly, the parent control points are fragment-exchanged to obtain the offspring control points; topological invariant detection ensures that the offspring inherits the topological category of one of the parents.

[0049] Mutation operator: A topology-preserving mutation operator is used. A generator σi is randomly selected, and the following is executed:

[0050] This involves performing conjugate transformations on braided groups so that the transformed braided groups maintain homotopy equivalence.

[0051] In addition, a spatial constraint correction operator is used: the mutated control points are checked, and if there is a collision with an obstacle, the control points are pushed away from the obstacle surface along the collision normal direction by the local gradient descent method until the safety distance requirement is met.

[0052] Step 8: Output the optimal piping layout; Iterative optimization continues until the termination condition is met (reaching the maximum number of iterations or the fitness change is less than a threshold), yielding the optimal solution. Output includes: a sequence of 3D pipeline centerline coordinates, pipeline connection relationships, and braided group topology encoding β. Manufacturing constraint parameters (minimum bending radius, pipe spacing, etc.).

[0053] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the present invention. Various changes and modifications can be made to the present invention without departing from its spirit and scope. All such changes and modifications fall within the scope of the present invention as claimed, which is defined by the appended claims and their equivalents.

Claims

1. A multi-pipeline cooperative topology layout optimization method based on braid group theory, characterized in that, include: Step 1: Construct a parametric representation of a multi-pipeline spatial curve; No. The spatial curve of the pipeline is represented by the formula: in, For the first The spatial curve of the pipeline, This represents the actual physical distance along the pipeline from the starting point to the current point. Let i be the three-dimensional coordinates of the pipeline; i ∈ n, where n is the total number of pipelines. The set of all pipeline space curves The formula is: Step 2: Construct a three-dimensional, arrangeable spatial model of multiple pipelines; Based on the three-dimensional spatial model of the pipeline to be laid out, the pipeline start and end points, and the pipeline engineering constraint parameters, a three-dimensional arrangeable spatial model is established. Define the pipeline laying channel as the available space Ω R³, the three-dimensional arrangeable space model Ω is represented as: Ω={(x,y,z)∈R³∣d(x,y,z)>δ} Where R³ is the three-dimensional Euclidean space, d(x,y,z) is the distance from the three-dimensional coordinates of the pipeline to the surface of the nearest obstacle, and δ is the preset safety distance threshold. Step 3: Construct the braid group topological coding model; The interpenetration relationships between multiple pipes in space are mapped to a braid group, as shown in the formula: in, A braided group consisting of n pipes, The positional relationship between the i-th pipeline and the (i+1)-th pipeline is the braid group generator; Concatenate all braid group generators in the order of their intersection to obtain braid group terms representing arbitrary pipeline topology states. The formula is: Step 4: Establish a constraint model based on topological invariants; The topology change cost function is defined by comparing the topology invariants before and after optimization. The formula is as follows: Where F is the topology change cost function. The term is the topological state braid group term after optimization, and ||.|| is the modulo operation on the vector; F equals 0, indicating that the topology is equivalent before and after optimization; F not equal to 0 indicates that the topology has changed, the optimization scheme is rejected and a penalty is imposed; Step 5: Establish a joint optimization formula: U = Bn × SE (3) Among them, SE(3) is a Lie group for rigid body motion in three-dimensional space, and the state of the Lie group describes the geometric changes in space; Step 6: Establish a multi-objective collaborative optimization model, the formula is: Where L is the path length cost, C is the collision cost, T is the topology change cost, and S is the structural stress cost. These are the weighting coefficients; Step 7: Braid group genetic search optimization; Using a genetic algorithm for optimization search, the formula for generating chromosome X, i.e., individuals in the population, is: Iterative optimization of chromosome code X yields X ; Step 8: Output the optimal piping layout; Based on X optimized in step 7 The optimal solution is obtained using the following formula: X =( ,P1 , P2 ,…,Pn ) Among them, Pi Let Qi,j be the optimal pipeline space curve for the i-th pipeline. Let be the optimal spatial coordinates of the j-th control point of the i-th pipeline, where j ∈ m and m is the total number of control points; The final optimized piping layout is obtained using the following formula: Among them, P This is the set of all optimized pipeline space curves.

2. The multi-pipeline cooperative topology layout optimization method based on braid group theory according to claim 1, characterized in that, The three-dimensional spatial model in step S2 adopts a CAD solid model, which specifically includes: exporting solid model files in STEP and IGES formats from CAD software such as SolidWorks, CATIA, UG, and NX, and obtaining the outer contour surface, internal cavity, and obstacle boundary information of the equipment by parsing the model file.

3. The multi-pipeline cooperative topology layout optimization method based on braid group theory according to claim 1, characterized in that, The topological invariants in step S4 include at least one of the following: braided index, Burau matrix representation, or Alexander polynomial.

4. The multi-pipeline cooperative topology layout optimization method based on braid group theory according to claim 1, characterized in that, Step S5, Lie group space optimization, employs transformation matrix operations, with the following formula: in, Let R be the transformation matrix, R be a 3×3 orthogonal matrix with a determinant of 1, and t be the translation vector. Continuous optimization is achieved through exponential mapping. in, , Let ξ be the spatial pose transformation matrix for the (k+1)th and kth iterations of optimization, where ξ is an element of the Lie algebra containing rotation and translation vector components.

5. The multi-pipeline cooperative topology layout optimization method based on braid group theory according to claim 1, characterized in that, The collision cost C is determined based on the available space Ω. When the control point of the pipeline is within the available space Ω, C is 0. When the control point of the pipeline is not within the available space Ω, C is infinite.

6. The multi-pipeline cooperative topology layout optimization method based on braid group theory according to claim 1, characterized in that, The optimization process in step 7 includes: The initial population is generated in the following way: Step a: For each pipeline, randomly generate intermediate control points between the start and end points. These control points are located within the available space Ω. Calculate the set of spatial curves representing the pipeline based on these intermediate control points. ; Step b: Calculate the corresponding topological state braid group term β' based on the spatial intersection relationship between the pipelines; Based on step a 1. Obtain β' from step b, generate individual X', perform feasibility verification on individual X', and remove individuals that collide with obstacles; Step c: Crossover operator calculation: A topological crossover operator based on braid group generators is adopted. For two parent individuals, a crossover position k is randomly selected, and the first k braid group generators of the topological state braid group word βa in the first parent individual are concatenated with the braid group generators of the latter half of the topological state braid group word βb in the second parent individual to obtain the offspring braid group βc. Correspondingly, for the pipeline space curves Pa and Pb of the two parent individuals, when the generator fragments of the topological state braid group word βa are exchanged at position k, the pipeline also exchanges the control point fragments at the same position. A crossover position l is randomly selected, and the first l control points of Pa are concatenated with the latter half of Pb to obtain the offspring pipeline space curve Pc. The offspring inherits the topological category of one of the parents through topological invariant detection. Step d: Mutation operator calculation; Using the topology-preserving mutation operator, randomly select the braid group generator σi and execute: in, The transformed braided words are obtained by performing conjugate transformations on the braided words to maintain homotopy equivalence. Apply random perturbations to the coordinates of the control points of the pipeline, using the following formula: Qi,j'=Qi,j+Δ Where Qi,j is the initial spatial coordinate of the j-th control point of the i-th pipeline, Qi,j' is the mutated spatial coordinate of the j-th control point of the i-th pipeline, and Δ is a small-amplitude random vector that satisfies a Gaussian distribution; In addition, a spatial constraint correction operator is used: the mutated control points are checked, and if there is a collision with an obstacle, that is, the mutated control points are not within Ω, the control points are pushed away from the obstacle surface along the collision normal direction by the local gradient descent method until the safety distance requirement is met. The optimization proceeds iteratively until the termination condition is met, i.e., the maximum number of iterations is reached or the fitness change is less than a threshold, thus obtaining the optimal solution.

7. The multi-pipeline cooperative topology layout optimization method based on braid group theory according to claim 1, characterized in that, Braid group generator generation methods include: Choose the main extension direction of the pipeline as the projection direction, and project the pipeline spatial curve. Projecting the image onto a two-dimensional plane perpendicular to the projection direction yields the projection curve; Detect all intersections between pipes on the projection plane and sort the intersections according to the spatial order along the projection direction; For the intersection of the i-th pipeline and its adjacent (i+1)-th pipeline, the braid group generator of the two adjacent pipelines is determined based on their actual spatial elevation relationship along the projection direction at the intersection. Specific methods include: The i-th pipeline is located above and in front of the (i+1)-th pipeline. These two adjacent pipelines are defined as having a positive intersection, with the intersection direction being... =+1, the generator of the braid group is denoted as the positive crossover. ; The i-th pipeline is located below and behind the (i+1)-th pipeline. These two adjacent pipelines are defined as crossing in opposite directions. =-1, the generator of the braid group is denoted as the reverse crossover. .

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