A bridge machine truss optimization method and system based on a PGSA and NSGA-II fusion algorithm

Abstract

This invention discloses a method and system for optimizing bridge-building machine trusses based on a fusion algorithm of PGSA and NSGA-II. The method includes: determining the foundation configuration based on the design domain and boundary conditions; performing topology finding and morphological growth of the main truss using an improved simulated plant growth algorithm, where the sum of strain energy of the members connected to the nodes is used as the auxin concentration, a hybrid step-size parallel search is employed, and inefficient member removal, global exploration, and topology mutation mechanisms are introduced; the growth configuration is fine-tuned based on engineering experience to obtain an empirical configuration; starting from the empirical configuration, the NSGA-II algorithm is used for multi-objective collaborative optimization of node coordinates and member cross-sections. This invention also provides a corresponding optimization system. This invention integrates the topology growth capability of PGSA with the multi-objective parameter optimization capability of NSGA-II in a step-by-step chain manner, which can significantly reduce the amount of steel used in the structure while ensuring that the structural stiffness, strength, and stability meet the requirements of the specifications, and has significant engineering application value.

Description

Technical Field

[0001] This invention relates to the field of bridge construction equipment technology, specifically to an optimization method and system for the truss structure of a bridge-building machine used in cantilever casting construction. Background Technology

[0002] In recent years, with the widespread application of long-span continuous beam bridges, cantilever construction has become one of the mainstream techniques in bridge construction. The bridge-building machine (also known as a hanging basket), as the core equipment in cantilever construction, has its main truss structure design directly affecting construction safety and project cost. Intelligent bridge-building machines integrate automatic control systems such as jacks and hydraulic cylinders on the basis of traditional hanging baskets, resulting in a significant increase in structural weight. How to achieve lightweight design of the main truss while meeting safety regulations has become a pressing technical challenge in the engineering field.

[0003] In the field of structural optimization algorithms, the Plant Growth Simulation Algorithm (PGSA) is an intelligent optimization algorithm inspired by the phototropism mechanism of plants. Existing research has shown that PGSA can be effectively applied to the topology optimization problem of truss structures. To overcome the limitations of PGSA in complex structural optimization, Shi Kairong et al. proposed several improvement strategies, including an elite strategy for morphogenetic concentration calculation, an intelligent variable step size strategy, a limit mechanism for the set of gronomicable points, and a hybrid step size parallel search mechanism. Furthermore, the research team proposed a hybrid algorithm (PGSA-GA) combining PGSA and a Genetic Algorithm (GA), using GA for initial growth point selection, and verified its optimization effect through typical truss and single-layer reticulated shell examples. In the field of multi-objective optimization, the NSGA-II algorithm has been successfully applied to the multi-objective optimization design of the shape, size, and topology of truss structures.

[0004] Regarding the structural optimization of bridge-building machines, existing patents such as CN118780118A disclose a multi-objective optimization method for intelligent bridge-building machines based on response surface methodology, which combines response surface methodology with genetic algorithms for parameter optimization. However, this method only performs parameter-level optimization for existing topological configurations and fails to address the problem of topology discovery and generation for new configurations. Chinese patent application CN121110529A discloses a structural improvement scheme for bridge-building machines, but its optimization process relies on empirical design and lacks systematic intelligent optimization methods.

[0005] In summary, the main technical problems in the existing technology are: (1) the single PGSA algorithm is not capable enough in handling multi-objective fine optimization; (2) the hybrid of PGSA and GA is a single-stage parallel fusion, which does not realize the functional division of topology finding and parameter optimization; (3) the bridge building machine truss has the characteristics of large size, complex load and strict manufacturing constraints, and the general optimization algorithm is difficult to directly meet the engineering needs.

[0006] Therefore, there is an urgent need to develop a dedicated optimization method for bridge-building machine trusses that takes into account both topology finding capability and multi-objective parameter optimization capability. Summary of the Invention

[0007] The technical problem to be solved by the present invention is to provide a bridge-building machine truss optimization method and system based on the fusion algorithm of PGSA and NSGA-II, so as to overcome the defects of the existing technology that a single algorithm is difficult to take into account both topology finding and multi-objective fine optimization, the existing hybrid algorithm is a single-stage hybrid and fails to achieve functional division, and the general optimization algorithm is difficult to meet the engineering manufacturing constraints of bridge-building machine truss.

[0008] This invention provides a bridge-building machine truss optimization method based on the fusion algorithm of PGSA and NSGA-II, the core of which lies in a step-by-step fusion optimization strategy. The method includes the following steps: Step S1: Determining the basic configuration. Based on the design domain and boundary conditions of the main truss of the bridge-building machine, determine the basic nodes, and add auxiliary nodes to form an initial stable truss as the basic configuration. The basic nodes include at least support constraint nodes, load application nodes, and displacement control nodes. Step S2: Based on the initial configuration growth of the improved PGSA algorithm, starting from the basic configuration, the improved simulated plant growth algorithm is used to perform topology finding and morphological growth of the main truss to obtain the initial topology configuration. The improved simulated plant growth algorithm includes: using the sum of strain energy of all members connected to the node as the basis for calculating auxin concentration; employing a hybrid step-size parallel search strategy to search for candidate growth points; periodically removing members with the lowest stress utilization rate; and triggering a global exploration mechanism or topology mutation mechanism to escape local optima when optimization stalls. Step S3: Fine-tuning the structural configuration based on engineering experience. The initial topological configuration is fine-tuned according to engineering experience to obtain an empirical configuration. This fine-tuning includes adjusting node positions, adding necessary members, and / or setting hinge points. Step S4: Multi-objective collaborative optimization based on the NSGA-II algorithm. Starting from the empirical configuration, the NSGA-II algorithm is used to perform multi-objective collaborative optimization of node coordinates and member cross-sections to obtain the final optimized design scheme of the bridge-building machine truss. The optimization objectives of the NSGA-II algorithm include minimizing the total structural mass, controlling overall stiffness, controlling structural strength, and controlling member stability. Furthermore, each new member must undergo geometric stability verification to ensure structural feasibility.

[0009] Based on the above method, the present invention also provides a bridge-building machine truss optimization system based on the PGSA and NSGA-II fusion algorithm, the system comprising: The basic configuration determination module is used to determine the basic configuration based on the design domain and boundary conditions of the main truss of the bridge-building machine. The basic configuration includes at least support constraint nodes, load application nodes, and displacement control nodes. The PGSA topology-finding module, connected to the basic configuration determination module, is used to perform topology-finding and morphological growth of the main truss starting from the basic configuration, employing an improved simulated plant growth algorithm to obtain an initial topology configuration. The improved simulated plant growth algorithm includes: using the sum of strain energy of all members connected to a node as the basis for calculating auxin concentration; employing a hybrid step-size parallel search strategy to search for candidate growth points; periodically removing members with the lowest stress utilization rate; and triggering a global exploration mechanism or topology mutation mechanism to escape local optima when optimization stalls. An empirical configuration fine-tuning module, connected to the PGSA topology-finding module, is used to fine-tune the initial topology configuration based on engineering experience to obtain an empirical configuration. The fine-tuning includes adjusting node positions, adding necessary members, and / or setting hinge points. The NSGA-II multi-objective collaborative optimization module, connected to the empirical configuration fine-tuning module, is used to perform multi-objective collaborative optimization of node coordinates and member cross-sections using the NSGA-II algorithm, starting from the empirical configuration, to obtain the final optimized design scheme of the bridge-building machine truss. The optimization objectives of the NSGA-II algorithm include minimizing the total structural mass, controlling overall stiffness, controlling structural strength, and controlling member stability. The stability verification module, connected to the NSGA-II multi-objective collaborative optimization module, is used to perform geometric stability verification on each generation of new individuals, eliminate geometrically unstable individuals, and ensure that all solutions are feasible structures.

[0010] In the above methods and system schemes, the determination of the basic configuration can be achieved using an empirical method or a controlled stochastic method.

[0011] Furthermore, the formula for calculating the auxin concentration is C_i = ΣE_j / ΣE_total; where C_i represents the auxin concentration at node i, E_j represents the strain energy of the j-th member connected to node i, and ΣE_total represents the total strain energy of all members in the current truss structure.

[0012] Furthermore, the hybrid step-size parallel search strategy employs both large and small step-size parallel searches for candidate growth points in the early stages of optimization; and uses only small step-size for fine-grained search in the later stages of optimization.

[0013] Furthermore, the global exploration mechanism actively adds candidate growth points in the low strain energy region when optimization stagnates; the topology mutation mechanism randomly deletes non-critical force transmission path members and then reguides the growth process when optimization stagnates.

[0014] Furthermore, the multi-objective optimization model of the NSGA-II algorithm is Minimize F(x)=[W(x), d(x), σ_max(x), φ(x)], and satisfies the corresponding stress, displacement, slenderness ratio and feasible region constraints; wherein the stability index φ(x) is determined according to the slenderness ratio of the member or the overall stability coefficient.

[0015] Furthermore, the geometric stability verification uses the non-zero determinant criterion of the stiffness matrix to quickly determine the geometric stability and eliminate individuals with singular or nearly singular stiffness matrices.

[0016] Compared with the prior art, the present invention has the following beneficial effects: (1) This invention integrates the PGSA algorithm and the NSGA-II algorithm in a step-by-step chain, giving full play to the natural advantages of PGSA in topology finding and the powerful ability of NSGA-II in multi-objective trade-offs, forming a well-defined and interconnected optimization process. Unlike the single-stage hybrid strategy of PGSA and GA in the prior art, this invention realizes two-stage serial execution from topology finding to empirical fine-tuning and then to fine optimization, effectively solving the technical problem that a single algorithm cannot simultaneously take into account both topology finding and multi-objective fine optimization.

[0017] (2) This invention introduces multiple improvement mechanisms into the PGSA algorithm, such as periodic removal of inefficient members, global exploration, and topology mutation, which effectively prevents the algorithm from over-relying on the initial configuration and falling into local optimum traps, and significantly improves the global search capability and quality finding of topology optimization. At the same time, this invention applies the hybrid step-size parallel search mechanism to the bridge-building machine truss form finding problem for the first time, achieving a good balance between early global exploration and later fine search.

[0018] (3) The present invention incorporates an engineering experience fine-tuning step between PGSA growth and NSGA-II optimization, and makes rational adjustments to the pure numerical optimization results such as node regularization, member supplementation and hinge point setting, so that the optimization results are more in line with the actual needs of engineering manufacturing and construction safety requirements, and significantly improves the engineering feasibility and implementation of the optimization scheme.

[0019] (4) In the NSGA-II optimization stage, the present invention adopts a dual collaborative optimization strategy of node coordinates and member cross-sectional dimensions, and sets engineering-oriented constraints (such as processing module constraints, steel section library constraints, etc.), so that the optimization results can be directly used for engineering design and manufacturing, shortening the conversion cycle from optimization calculation to engineering implementation.

[0020] (5) This invention is specifically designed for the size and load characteristics of the main truss of a bridge-building machine, and provides a complete scheme for defining boundary nodes, design domain constraints and setting optimization variables, filling the technical gap in applying intelligent optimization algorithm systems to the field of bridge-building machine truss topology finding and multi-objective collaborative optimization. Attached Figure Description

[0021] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0022] Figure 1 This is a schematic diagram of the empirical method for determining the typical main truss foundation configuration in this invention.

[0023] Figure 2 This is a schematic diagram of the empirical method for determining the foundation configuration of the main truss in this invention.

[0024] Figure 3 This is an example of a basic configuration that is randomly generated under control in this invention.

[0025] Figure 4 This is a schematic diagram illustrating the effect of the hybrid step size growth strategy in this invention.

[0026] Figure 5 This is a flowchart illustrating the structural optimization implementation based on the fusion algorithm in this invention.

[0027] Figure 6 This is a diagram of the bridge-building machine truss optimization system architecture based on the PGSA and NSGA-II fusion algorithm in this invention. Detailed Implementation

[0028] To make the technical means, creative features, objectives and effects of this invention easier to understand, the invention will be further described below with reference to specific illustrations.

[0029] This invention addresses the main problems existing in the optimization methods for bridge-building machine trusses: the single PGSA algorithm is insufficient in multi-objective fine optimization; the hybrid algorithm of PGSA and GA is a single-stage parallel fusion that fails to achieve functional division; and the general optimization algorithm is difficult to meet the large size, complex load and engineering manufacturing constraints of bridge-building machine trusses. Therefore, it proposes a step-by-step chain fusion optimization scheme.

[0030] This invention first employs a step-by-step chain-like fusion architecture of PGSA and NSGA-II. The PGSA algorithm is positioned in the topology-finding stage, automatically discovering the optimal force transmission path and node connection relationships using its strain energy gradient-based growth mechanism in continuous space. The NSGA-II algorithm is positioned in the multi-objective parameter fine-tuning stage, simultaneously optimizing node coordinates and member cross-sections under a fixed topology configuration using its non-dominated sorting and elite retention strategies. The two stages are connected by "empirical configuration fine-tuning," achieving incremental optimization from the topology space to the parameter space.

[0031] Furthermore, the present invention optimizes the PGSA algorithm in the following ways: First, it uses the sum of strain energies of the members connected to the nodes as the basis for calculating the auxin concentration, so that the growth direction is directly driven by mechanical properties. Second, it adopts a hybrid step-size parallel search strategy, using both large and small step sizes to explore candidate growth points in the early stage, and switching to a single small step size for fine approximation in the later stage. Third, it introduces a periodic removal mechanism for inefficient members and a zero-member elimination mechanism in the later stage to gradually eliminate redundant members with low stress utilization. Fourth, it sets up two local optimum exit mechanisms: "global exploration" and "topology mutation." The former actively adds candidate growth points in low strain energy density regions, while the latter randomly deletes some members on non-critical force transmission paths and restarts the growth process. The synergistic effect of the above improvements is that they significantly enhance the convergence speed, global optimization capability, and topology economy of PGSA in complex truss topology optimization.

[0032] Furthermore, the present invention employs dual collaborative optimization variables of node coordinates and member cross-sectional dimensions in the NSGA-II stage, and sets strict engineering guidance constraints.

[0033] The specific implementation scheme of the method of the present invention will be described in detail below with reference to the accompanying drawings. Figure 1 and Figure 2 An example of the basic configuration determined by the empirical method is shown, with the support constraint nodes R1 and R2, load application nodes P1 and P2, and auxiliary nodes X1 and X2 labeled. Figure 3 Examples of several basic configurations generated by controlled randomization are shown; Figure 4 This illustration shows a comparison of the effects of large step size and small step size in the early stage of parallel search and the later stage of single fine search in the hybrid step size growth strategy; Figure 5 The entire optimization process is presented in the form of a flowchart.

[0034] Reference Figure 5 The execution flow of the bridge-building machine truss optimization method based on the PGSA and NSGA-II fusion algorithm provided by this invention is as follows.

[0035] Step 1: Determining the basic configuration.

[0036] This step first determines the necessary foundation growth nodes based on the design domain of the bridge-building machine's main truss, namely the allowable space range in the longitudinal, vertical, and lateral directions, as well as the mechanical force transmission path of the structure.

[0037] These nodes include: Support constraint nodes, such as support reaction points connected to piers or cast-in-place beams; Load application nodes, such as the force application points that bear the loads suspended at the front and rear ends of the hanging basket; And displacement control nodes, which typically specify the vertical displacement limit of the farthest node of the cantilever.

[0038] In addition to the aforementioned basic nodes, additional nodes are needed to form an initial stable truss. This invention further provides two complementary methods: the empirical method and the controlled stochastic method.

[0039] In the empirical method, designers add a few auxiliary nodes at key locations between basic nodes based on engineering experience, forming simple triangular trusses or parallel chord trusses as a starting point for growth, such as... Figure 1 As shown.

[0040] In the controlled stochastic method, the program randomly generates multiple sets of different basic node configurations under the premise of satisfying geometric stability constraints, i.e., the stiffness matrix is ​​nonsingular. These configurations serve as independent starting points for subsequent PGSA growth, and the optimal result is automatically selected. Figure 3 As shown.

[0041] This step leverages engineering experience to accelerate convergence and avoids over-reliance on a single initial configuration through random initialization.

[0042] Step 2: Initial configuration growth based on the improved PGSA algorithm.

[0043] This step starts with the basic configuration determined in step one and initiates the improved PGSA algorithm for topology finding.

[0044] This step specifically maps the phototropic growth mechanism of plants to the nodal growth process of a truss structure. First, the "auxin concentration" is calculated for all current nodes: the concentration C_i of node i is equal to the sum of the strain energies ΣE_j of all members connected to that node divided by the total strain energy ΣE_total of the entire structure, i.e.: C_i = ΣE_j / ΣE_total; Where C_i represents the auxin concentration at node i, E_j represents the strain energy of the j-th member connected to node i, and ΣE_total represents the total strain energy of all members in the current truss structure.

[0045] Based on this calculation model, it can be determined that the area with higher strain energy bears a greater load or undergoes more significant deformation. New nodes and new members should be added near this area to share the internal forces.

[0046] In each iteration, a corresponding "growth node" is selected probabilistically based on the auxin concentration of each node. Then, several candidate new node positions are generated around the selected node according to the current step size. These candidate positions must be located within the design domain and at a distance from existing nodes not less than a preset minimum spacing. Subsequently, the candidate nodes are connected to existing nodes through new members, and the structural mechanical response after the addition of the new members is calculated using the finite element method.

[0047] Regarding the control of the growth step size, this invention preferably employs a hybrid step size parallel search strategy. Specifically, in the early stages of optimization, two sets of parameters, one with a large step size and the other with a small step size, are used simultaneously to generate candidate growth points. The large step size ranges from 0.3m to 0.8m, and the small step size ranges from 0.05m to 0.2m. A large step size is beneficial for exploring new topological features within a larger spatial range, while a small step size can discover locally advantageous locations. In the later stages of optimization, the system switches to using only the small step size for fine-grained searching, such as... Figure 4 As shown.

[0048] During the growth process, an inefficient member removal operation is performed every preset generation period.

[0049] The cycle here can be determined according to actual needs, for example, 15 generations can be used.

[0050] In practice, the stress utilization rate of each member is calculated, which is the ratio of the actual axial stress to the material yield strength. Based on this, several members with the lowest stress utilization rate are removed. The removal ratio is usually 5% to 15%, but it is necessary to ensure that the structure still meets the geometric stability after removal.

[0051] In the later stages of optimization, when the improvement rate of the objective function in multiple consecutive iterations is lower than a preset threshold, virtually zero-value members, i.e. members with stress utilization rates below 5%, can be further removed to simplify the topology.

[0052] To prevent the algorithm from getting stuck in local optima, this invention sets up two active escape mechanisms: a global exploration mechanism and a topological mutation mechanism.

[0053] The first type of proactive exit mechanism is the global exploration mechanism: when the program detects that the improvement rate is lower than the threshold for multiple consecutive iterations, such as the improvement rate being lower than 2% for 10 consecutive generations, the program proactively adds candidate growth points in the low strain energy density region. These regions are defined as regions where the strain energy density is lower than the average value of the entire structure, in order to try to open up new force transmission paths.

[0054] If the global exploration fails to recover after a certain number of generations of improvement, the second mechanism, namely the topology mutation mechanism, is triggered: the program randomly deletes some members on non-critical force transmission paths in the current structure. Non-critical paths are defined as members with stress utilization rates below 20%, and the deletion ratio is usually 10% to 20%. Then, the growth process is restarted from this state.

[0055] By working together through the three mechanisms of inefficient member removal, global exploration, and topology mutation, the global optimization capability and convergence stability of PGSA can be significantly improved.

[0056] Furthermore, the objective function in this step is based on the total structural mass W (steel consumption), and a penalty term is introduced to handle stress, displacement, and slenderness ratio constraints, in the following form: F_obj = W + λ_1×P_stress + λ_2×P_displacement + λ_3×P_slenderness; When the improvement rate of the objective function is lower than the preset termination threshold for multiple consecutive iterations, for example, when the improvement rate is less than 1% for 20 consecutive generations, the PGSA growth process terminates and the current optimal initial topology is output.

[0057] Step 3: Fine-tuning the structural configuration based on engineering experience.

[0058] Although the initial topology generated in step two performs well in terms of steel consumption, its node coordinates are often non-modular (e.g., 5.537m), and the connection relationships of the members may be too singular or have insufficient redundancy. Directly using it for manufacturing will face processing difficulties, increased costs, and safety risks.

[0059] To address this, the present invention achieves the connection between the two optimization stages by fine-tuning the structural configuration based on engineering experience.

[0060] The first step in this empirical configuration fine-tuning process is to standardize the node positions, that is, to round the coordinates of all nodes to an integer multiple of the preset machining module, such as 5mm or 10mm. Secondly, the vertical height coordinates are discretized and constrained, meaning that the height value of each node is only allowed to take values ​​from a few pre-defined levels, such as 2.8m, 3.2m, 3.6m, and 4.0m. Furthermore, necessary members are added, namely, according to construction safety specifications, cross bracing members or longitudinal connecting members are automatically added in key areas such as between the cantilever end and the support point, and in the middle section of the cantilever end, to enhance structural redundancy. Finally, hinge points are set, that is, all nodes are uniformly set as hinged connections, which means releasing the node bending moment so that each member of the truss only bears axial tension and compression.

[0061] The "empirical configuration" obtained after the above fine-tuning not only retains the core force transmission path advantages of PGSA topology optimization, but also fully meets the practical requirements of engineering manufacturing, construction and installation, and safety redundancy.

[0062] Step 4: Multi-objective collaborative optimization based on the NSGA-II algorithm.

[0063] This step starts with the empirical configuration output from step three and initiates the NSGA-II algorithm for dual collaborative optimization of node coordinates and member cross-sections.

[0064] Specifically, this step optimizes two types of variables with different properties under the premise of fixed topological connection relationship: continuous node coordinates and discrete member cross-sectional dimensions. The node coordinates are limited by the processing module and vertical discrete level, while the member cross-sectional dimensions are selected from a predefined steel section library.

[0065] Two types of variables are encoded in the chromosome using a hybrid encoding: node coordinates are encoded with real numbers but are corrected to the most recent valid value before each generation evaluation, while bar sections are encoded with integers to directly index entries in the section library.

[0066] Furthermore, the optimization objectives in this step are four conflicting indices: minimizing the total structural mass W(x), minimizing the vertical displacement d(x) at the cantilever end, minimizing the maximum stress σ_max(x) in the members, and maximizing the stability index φ(x). Under the multi-objective minimization framework, this is equivalent to minimizing the ratio of 1 to the stability coefficient. The stability index φ(x) is determined based on the slenderness ratio of the members or the overall stability coefficient.

[0067] The multi-objective optimization model is configured as follows: Minimize F(x)=[W(x), d(x), σ_max(x), φ(x)], Where x represents a decision variable vector consisting of node coordinates and member cross-sectional dimensions.

[0068] The constraints include: the calculated stress σ_i of each member does not exceed the allowable stress of the material [σ]; the vertical displacement d at the cantilever end does not exceed the allowable limit [d]; the slenderness ratio λ of each member does not exceed the allowable limit [λ] of the specification; the decision variable x belongs to the feasible region, that is, the value of the node coordinates is limited by the preset discretization level, and the cross-sectional dimensions of the members are taken from the conventional steel section library. Here, σ_i represents the calculated stress of the i-th member, [σ] represents the allowable stress of the material, d represents the actual displacement at the cantilever end, [d] represents the allowable displacement limit, λ represents the slenderness ratio of the member, and [λ] represents the allowable slenderness ratio limit of the specification.

[0069] Based on this, the NSGA-II algorithm is configured to run according to the following steps: Initialize a population set of individuals with a size of N, where N is typically between 100 and 300; Finite element analysis is performed on each individual to calculate the values ​​of four objective functions and the degree of constraint violation. In each generation, fast non-dominated sorting is used to divide the population into multiple Pareto fronts, and crowding distance is used within the same front to maintain the diversity of solutions. Offspring populations are generated through binary tournament selection, simulated binary crossover, and polynomial mutation. After merging the parent and offspring generations, elites are retained.

[0070] During this process, after each generation completes individual evaluation, the stability verification module is immediately invoked to construct the overall stiffness matrix K for each new individual, calculate det(K) or its LDL decomposition. If the absolute value of det(K) is less than a preset threshold, such as 1e-8, then the individual is determined to be a geometrically variable system, i.e., a mechanism, and is removed from the population and replaced with a randomly generated feasible individual.

[0071] After iterating through a preset maximum number of generations, such as 200 to 500 generations, the algorithm converges to the Pareto optimal solution set. Designers can then select the individual solution that best meets their needs as the final optimized design scheme based on their engineering preferences.

[0072] In response to the above-mentioned bridge-building machine truss optimization method based on the fusion algorithm of PGSA and NSGA-II, the present invention also provides a bridge-building machine truss optimization system based on the fusion algorithm of PGSA and NSGA-II, which is used to automatically execute the above-mentioned method.

[0073] like Figure 6 As shown, the bridge-building machine truss optimization system based on the PGSA and NSGA-II fusion algorithm provided by the present invention is specifically composed of a basic configuration determination module 10, a PGSA topology finding module 20, an empirical configuration fine-tuning module 30, an NSGA-II multi-objective collaborative optimization module 40, and a stability verification module 50, which are sequentially combined.

[0074] The basic configuration determination module 10 in this system is located at the front end. It converts the user-input design domain parameters, constraint node coordinates and load conditions into an initial truss structure object. At the same time, the basic configuration determination module 10 sends the generated object to the PGSA topology finding module 20.

[0075] The PGSA topology-finding module 20, acting as the system's topology innovation engine, is configured to execute the improved PGSA algorithm to automatically discover the optimal node arrangement and member connection relationships, and generate an initial topology configuration. This PGSA topology-finding module 20 then passes the generated initial topology configuration to the empirical configuration fine-tuning module 30.

[0076] The empirical configuration fine-tuning module 30 is configured to fine-tune the initial topology configuration generated by the PGSA topology finding module 20, generating an "empirical configuration" that meets manufacturing feasibility, thus bridging the numerical optimization results with actual engineering manufacturing requirements. The empirical configuration generated by the empirical configuration fine-tuning module 30 is then input into the NSGA-II multi-objective collaborative optimization module 40.

[0077] The NSGA-II multi-objective collaborative optimization module 40 is configured to perform multi-objective fine optimization of node coordinates and member cross sections under a fixed topology, and forms a close collaborative relationship with the stability verification module 50: after each generation of candidate individuals is mechanically evaluated, the stability verification module 50 performs geometric stability discrimination, and after eliminating variable systems, the NSGA-II module 40 incorporates the remaining individuals into the population evolution.

[0078] The following section provides a further explanation of the specific structure and technical features of each module in this system.

[0079] In the specific implementation of the basic configuration determination module 10 in this system, input parameters are received through a graphical user interface or parameter file. These parameters include the design domain boundary, the coordinates of the support constraint nodes, the coordinates of the load application nodes and the load values, and the coordinates and allowable limits of the displacement control nodes.

[0080] Furthermore, the module is configured to generate an initial truss structure based on the user-selected working mode, namely experience mode or random mode.

[0081] In experience mode, the module has a built-in inference engine based on an engineering experience rule base. The rule base includes production rules such as "add an auxiliary node near the midpoint of the line connecting the fulcrum and the front suspension point". The module automatically triggers the corresponding rules based on the input design domain size to generate a deterministic basic configuration.

[0082] In random mode, the module adopts a graph theory-based random generation algorithm: with the basic set of nodes as the initial vertex set, additional vertices are randomly added and randomly connected to form edges under the condition of satisfying geometric stability constraints. This process is repeated until a connected and geometrically stable truss structure is generated, and multiple sets of random structure types are generated and stored in the queue to be optimized.

[0083] Based on the two working modes mentioned above, the standardized truss object output by the module includes arrays of nodes, members, boundary conditions, and loads. This module can encode professional design knowledge in the field of bridge construction machines in the form of a rule base, while retaining random initialization options to increase the diversity of starting points for topology optimization. This significantly reduces the dependence of the optimization process on user experience and improves the robustness of the algorithm's optimization.

[0084] In the specific implementation of the PGSA topology finding module 20 in this system, it is composed of multiple collaborative sub-units, including auxin concentration calculation unit 21, mixed step size parallel search unit 22, inefficient link removal unit 23, global exploration unit 24, topology mutation unit 25, and termination judgment unit 26.

[0085] After receiving the basic configuration, the PGSA topology finding module 20 first initializes the parameters, which include setting the growth step size (including large step size and small step size), the inefficient link removal cycle and removal ratio, the global exploration trigger threshold, the topology mutation trigger threshold, and the termination threshold.

[0086] Furthermore, during the operation of the PGSA topology finding module 20, its units are configured to work collaboratively in the following logical order.

[0087] First, the auxin concentration calculation unit 21 is configured to call the embedded finite element solver. The solver performs linear static analysis based on the direct stiffness method to obtain the axial force and deformation of each member, calculates the strain energy of each member E_j = (F_j^2·L_j) / (2E·A_j), and then calculates the auxin concentration of each node C_i = ΣE_j / ΣE_total.

[0088] Next, candidate growth points are generated by the hybrid step-size parallel search unit 22. This unit 22 is configured to select a step-size strategy based on the current optimization stage: In the early stages of optimization, two parallel threads, one with a large step size and the other with a small step size, are started simultaneously to generate candidate points around the selected node. These candidate points must satisfy the design domain and minimum spacing constraints. In the later stages of optimization, only small-step threads are used. This unit selects the optimal candidate points based on the strain energy gain rate and then adds them.

[0089] Next, the inefficient member removal unit 23 performs member simplification. This unit 23 is configured to calculate the stress utilization rate η_j = |σ_j| / σ_yield of each member every preset period, remove members with a ratio of R_remove before removal, and perform a geometric stability pre-check before removal; in the later stage of optimization, remove zero members with η_j < 5%.

[0090] Next, the global exploration unit 24 and the topology mutation unit 25 jointly execute the stagnation exit mechanism. Specifically, when the global exploration unit 24 detects that the improvement rate is lower than the threshold for multiple consecutive generations, it forcibly adds candidate growth points in inefficient regions where the strain energy density is lower than 30% of the average value. If the improvement rate does not recover within multiple generations after global exploration, the topology mutation unit 25 randomly deletes 15% of the members with stress utilization rate η_j < 20% and restarts the growth process.

[0091] The optimal termination condition is determined by the termination judgment unit 26. When the improvement rate is lower than the termination threshold for multiple consecutive generations, this unit outputs the current optimal initial topology configuration.

[0092] The collaborative work of the aforementioned units enables the PGSA topology-finding module 20 to automatically complete the search process from the initial configuration to the optimal topology without requiring manual parameter tuning by the user. The module's technical advantage lies in achieving fully automated configuration and execution of the PGSA algorithm through an embedded finite element solver and various improvement strategies.

[0093] In this system, the empirical configuration fine-tuning module 30 is specifically implemented by including a node regularization unit 31, a vertical height limiting unit 32, a member supplement unit 33, and a hinge point setting unit 34. These four units cooperate sequentially to realize the function of the empirical configuration fine-tuning module 30.

[0094] The node regularization unit 31 is configured to divide the coordinates of each node by a preset modulus, round them off, and then multiply them by the modulus.

[0095] The vertical height limiting unit 32 is configured to replace the Y coordinate of each node with the closest value in a preset height level set.

[0096] The member supplement unit 33 is configured to automatically execute based on supplement rules, such as "add cross bracing if there is no diagonal bracing between the cantilever end area and the fulcrum area".

[0097] The hinge point setting unit 34 is configured to modify the constraints on the degrees of freedom of all nodes to only constrain the translational degrees of freedom.

[0098] This empirical configuration fine-tuning module 30 automates the tedious operation that originally required manual adjustments one by one, and all adjustment rules can be customized by the user, achieving a unity of automation and customizability. This allows the highly irregular topology generated by PGSA to be transformed into a regular configuration that conforms to engineering specifications within seconds.

[0099] In this system, the NSGA-II multi-objective collaborative optimization module 40 and the stability verification module 50 are tightly coupled.

[0100] The NSGA-II multi-objective collaborative optimization module 40 includes a population initialization unit 41, a crossover and mutation unit 42, a finite element evaluation unit 43, a non-dominated sorting and crowding calculation unit 44, an elite retention unit 45, and an output unit 46.

[0101] After receiving the empirical configuration, the module first generates an initial population by the population initialization unit 41. The decision variables for each individual include the free node coordinates and the member section index. The node coordinates are constrained by the vertical height discrete set and the processing module. The member section index is selected from the predefined steel section library. The objective function is set to minimize four objectives, namely the total structural mass W, the vertical displacement d at the cantilever end, the maximum stress σ_max of the member, and the stability index φ (which is determined based on the member slenderness ratio or the overall stability coefficient).

[0102] Furthermore, during the optimization search process, the units in the NSGA-II multi-objective collaborative optimization module 40 are configured to work collaboratively according to the following logic.

[0103] Crossover and mutation unit 42 is configured to generate new candidate individuals. This unit uses simulated binary crossover and polynomial mutation operations to perform genetic operations on individuals in the current population to produce offspring. The discrete variable (bar section index) is mapped to the nearest valid value after mutation, thereby ensuring that each new individual satisfies the engineering constraints at the time of generation.

[0104] Finite element evaluation unit 43 is configured to perform mechanical performance analysis on each candidate individual. This unit calls the finite element solver to calculate the total structural mass W, vertical displacement d at the cantilever end, maximum stress σ_max of the members, stability index φ, and the degree of violation of each constraint condition (including stress constraint, displacement constraint, and slenderness ratio constraint) for each individual. At the same time, this unit passes the overall stiffness matrix of each individual to the stability verification module 50.

[0105] Correspondingly, the stability verification module 50 receives the stiffness matrix from the finite element evaluation unit 43 and uses the sparse LDL decomposition method to determine the positive definiteness of the matrix. If the decomposition fails or zero principal elements exist, the module returns an "unstable" flag, indicating that the individual is a geometrically variable system (mechanism); otherwise, it returns a "stable" flag. The technical feature of this module is that it can quickly and accurately identify and eliminate geometrically unstable individuals, ensuring that all individuals in the population are geometrically feasible structures.

[0106] The non-dominated sorting and crowding calculation unit 44 is configured to perform multi-objective sorting on the individuals retained after stability verification. This unit uses fast non-dominated sorting to divide individuals into multiple Pareto fronts and calculates the crowding distance between individuals within the same front to maintain the diversity of the solution set.

[0107] Elite retention unit 45 is configured to select the best individuals from the merged parent and offspring population to enter the next generation. This unit selects the top N individuals (N being the preset population size) based on non-dominance level and crowding distance. For individuals marked as "unstable" by the stability verification module, the elite retention unit removes them, and the population initialization unit randomly generates new feasible individuals (usually obtained by locally perturbing individuals selected from the current Pareto front) to replenish them, thus maintaining the population size unchanged.

[0108] Output unit 46 is configured to terminate the optimization process after the iteration reaches a preset maximum number of generations, output the final Pareto optimal solution set, and automatically select a recommended solution based on engineering preferences.

[0109] The collaborative work of the aforementioned units enables the NSGA-II multi-objective collaborative optimization module 40 to automatically complete multi-objective search and convergence while strictly satisfying engineering constraints. The resulting NSGA-II multi-objective collaborative optimization module 40 can directly embed engineering constraints (processing module, discrete height, steel profile library) into the crossover and mutation unit, ensuring that each generation of individuals satisfies the constraints at the time of generation; its tight coupling with the stability verification module 50 ensures that there are no geometrically variable individuals in the population.

[0110] The following section uses a specific example of a bridge-building machine truss optimization project to explain in detail the deployment method of the system, the detailed execution process of the method, and the technical characteristics and effects of the entire solution in practical applications.

[0111] (I) System Deployment and Instance Parameter Settings This example optimizes the main truss of a bridge-building machine used in the cantilever construction of a long-span continuous beam bridge for a highway. The system is deployed on an engineering workstation equipped with a multi-core processor and more than 32GB of memory. The software uses Python 3.11 combined with NumPy / SciPy for numerical calculations. The finite element solver is implemented in-house based on the direct stiffness method. The PGSA and NSGA-II algorithm modules are programmed using object-oriented programming, and the graphical user interface is developed based on PyQt5. The example design parameters are as follows: longitudinal length of the design domain L = 12.0m, vertical height range Y ∈ [2.5m, 4.5m], transverse width B = 6.0m. This example is simplified to a two-dimensional planar truss analysis; the coordinates of the foundation nodes are R1 (0,0), R2 (-1.5,0), P1 (12.0,3.0), and P2 (-1.0,0). The displacement control node coincides with P1, and the allowable displacement limit [d] = 20mm; the front-end hanging load is 200kN, and the rear-end hanging load is 80kN; the material is Q235 steel, with an elastic modulus E = 206GPa, yield strength σ_yield = 235MPa, and allowable stress [σ] = 215. MPa, slenderness ratio limit [λ]=150; 20 commonly used hot-rolled I-beam and H-beam sections are predefined in the steel section library; processing modulus is 5mm, vertical height dispersion levels are 2.8m, 3.2m, 3.6m, and 4.0m; PGSA parameters are set to large step size 0.6m, small step size 0.1m, removal cycle 15 generations, removal ratio 10%, global exploration threshold is improvement rate less than 2% for 10 consecutive generations, topological mutation threshold is improvement rate not recovered within 5 generations after global exploration, and termination threshold is improvement rate less than 1% for 20 consecutive generations; NSGA-II parameters are set to population size 200, maximum number of generations 300, crossover probability 0.9, and mutation probability 0.1.

[0112] (ii) Execution logic process Phase 1: Basic Configuration Determination. Designers input the above parameters in the graphical user interface and select "Experience Mode". The system's basic configuration determination module 10 automatically adds auxiliary nodes X1(5.5,2.5) and X2(7.5,3.0) near the midpoint of the R1P1 connection based on the built-in rule base, generating an initial simple truss with 6 nodes and 11 members.

[0113] Phase 2: PGSA Topology Finding. PGSA module 20 begins iteration after receiving the initial truss. The first 60 generations (early optimization phase) utilize parallel exploration with step sizes of 0.6m and 0.1m, gradually increasing the number of nodes from 6 to a peak of approximately 24. During this process, the auxin concentration in strain energy concentration areas remains consistently high, and newly added members are primarily arranged longitudinally and diagonally. Generation 72 triggers the removal of inefficient members for the first time, removing three redundant vertical members with stress utilization rates below 8%, reducing the number of nodes to 21. From generations 115 to 125, the objective function improvement rate remains below 2% for 10 consecutive generations, triggering a global exploration mechanism: two candidate growth points are forcibly added and connected in areas where strain energy density is below 30% of the average, and the improvement rate recovers to 2.5% within the next 5 generations. At generation 187, the termination condition of a 20-generation improvement rate below 1% is met, PGSA terminates, and an initial topology configuration with 18 nodes and 34 members is output, exhibiting an asymmetrical "fish belly" shape with diagonal bracing.

[0114] Phase 3: Experience-Based Configuration Fine-Tuning. The experience-based configuration fine-tuning module 30 normalizes the node coordinates, rounding them to 5mm, and maps the Y-coordinates of all nodes to the nearest discrete level, i.e., selecting the closest value from 2.8m, 3.2m, 3.6m, and 4.0m. The member supplement unit automatically adds two sets of cross bracing members according to the preset rule "if there is no diagonal bracing between the cantilever end (X>9m) and the support point (X<1m), add cross bracing". The hinge point setting unit marks all nodes as hinged. After fine-tuning, the number of nodes remains 18, the number of members increases to 38, all node coordinates are integer multiples of 5mm, and the heights are all preset level values.

[0115] Phase 4: NSGA-II Multi-Objective Co-optimization. Starting with an empirical configuration and fixing the topological connections, NSGA-II module 40 performs co-optimization on node coordinates and member cross-sections. Node coordinates are only allowed to be fine-tuned within discrete height levels, and member cross-sections are selected from 20 types of steel. After population initialization, stability verification module 50 eliminates approximately 8% of geometrically unstable individuals in the first generation. These individuals are mainly due to excessively small member cross-sections leading to excessively high slenderness ratios and near-singular stiffness matrices. As evolution progresses, the Pareto front gradually shifts outward. The algorithm converges in the 300th generation, and the system automatically selects the solution with the minimum steel usage from the final Pareto solution set as the recommended scheme. This scheme maintains a fish-belly topology, with node heights concentrated between 3.6m and 4.0m, and member cross-sections primarily using HM300×200 and I25a. The stress utilization rate of all members is distributed between 45% and 92%, with no individual member exceeding the allowable stress; the calculated displacement value of the cantilever end meets the specification limit; and the overall stability coefficient is greater than the minimum value required by the specification.

[0116] Phase 5: Output. The system generates three files: an optimization report, which includes a Pareto front diagram, node coordinate table, member section table, and mechanical verification results; a truss structure file in .truss format, which can be directly imported into finite element software; and manufacturing drawings in .dxf format, which are two-dimensional three-view drawings. The entire process, from input parameters to output files, requires no manual intervention.

[0117] As demonstrated by the above examples, this invention organically integrates the PGSA and NSGA-II algorithms and incorporates engineering experience judgments, forming a dedicated form-finding optimization process for bridge-building machine trusses. Compared to design methods that rely entirely on experience, this method can significantly reduce the amount of steel used in the structure while ensuring that the structural stiffness, strength, and stability meet the requirements of the specifications. The internal force distribution is more reasonable, and the utilization rate of the members is higher, demonstrating significant engineering application value and economic benefits.

[0118] 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 invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the present invention as claimed. The scope of protection of this invention is defined by the appended claims and their equivalents.

Claims

1. A method for optimizing the truss of a bridge-building machine based on a fusion algorithm of PGSA and NSGA-II, characterized in that, Includes the following steps: Step S1: Based on the design domain and boundary conditions of the main truss of the bridge-building machine, determine the foundation nodes, and form an initial stable truss as the basic configuration by adding auxiliary nodes on the basis of the foundation nodes; the foundation nodes include at least support constraint nodes, load application nodes and displacement control nodes; Step S2: Starting from the basic configuration, the improved simulated plant growth algorithm is used to perform topology finding and morphological growth of the main truss to obtain the initial topology configuration; wherein, the improved simulated plant growth algorithm includes: using the sum of strain energy of all members connected to the node as the basis for calculating the auxin concentration, using a hybrid step-size parallel search strategy to search for candidate growth points, periodically removing members with the lowest stress utilization rate, and triggering a global exploration mechanism or topology mutation mechanism to escape local optima when optimization stalls; Step S3: Fine-tune the initial topology configuration based on engineering experience to obtain an empirical configuration. The fine-tuning includes adjusting the node positions, adding necessary members, and / or setting hinge points. Step S4: Starting from the empirical configuration, the NSGA-II algorithm is used to perform multi-objective collaborative optimization of node coordinates and member cross-sections to obtain the final optimized design scheme of the bridge-building machine truss. The optimization objectives of the NSGA-II algorithm include minimizing the total mass of the structure, controlling the overall stiffness, controlling the structural strength, and controlling the stability of the members. Each new member must undergo geometric stability verification to ensure the feasibility of the structure.

2. The method according to claim 1, characterized in that, The formula for calculating the auxin concentration in step S2 is as follows: C_i = ΣE_j / ΣE_total Where C_i represents the auxin concentration at node i, E_j represents the strain energy of the j-th member connected to node i, and ΣE_total represents the total strain energy of all members in the current truss structure.

3. The method according to claim 1, characterized in that, The hybrid step-size parallel search strategy in step S2 is configured as follows: in the early stage of optimization, both large and small step sizes are used to search for candidate growth points in parallel; in the later stage of optimization, only small step sizes are used for fine-grained search.

4. The method according to claim 1, characterized in that, The global exploration mechanism in step S2 is configured as follows: when optimization stalls, candidate growth points are actively added in the low strain energy region for trial growth; the topology mutation mechanism is configured as follows: when optimization stalls, non-critical force transmission path members are randomly deleted and the growth process is restarted.

5. The method according to claim 1, characterized in that, The multi-objective optimization model of the NSGA-II algorithm in step S4 is as follows: Minimize F(x) = [W(x), d(x), σ_max(x), φ(x)]; Where x represents the decision variable vector consisting of node coordinates and member cross-sectional dimensions, W(x) represents the total mass of the structure, d(x) represents the vertical displacement at the cantilever end, σ_max(x) represents the maximum stress of the member, and φ(x) represents the stability index, which is determined based on the member slenderness ratio or the overall stability coefficient. The constraints are satisfied: σ_i ≤ [σ], d ≤ [d], λ ≤ [λ], and the values ​​of the node coordinates are limited by the preset discretization level. The cross-sectional dimensions of the members are taken from the conventional steel section library. Among them, σ_i represents the calculated stress of the i-th member, [σ] represents the allowable stress of the material, d represents the actual displacement of the cantilever end, [d] represents the allowable displacement limit, λ represents the slenderness ratio of the member, and [λ] represents the slenderness ratio limit allowed by the specification.

6. The method according to claim 1, characterized in that, The geometric stability verification in step S4 uses the non-zero determinant criterion of the stiffness matrix to quickly determine geometric stability and eliminate individuals with singular or nearly singular stiffness matrices.

7. A bridge-building machine truss optimization system based on the PGSA and NSGA-II fusion algorithm, characterized in that, The system includes: The basic configuration determination module (10) is used to determine the basic configuration based on the design domain and boundary conditions of the main truss of the bridge-building machine. The basic configuration includes at least support constraint nodes, load application nodes and displacement control nodes. The PGSA topology finding module (20) is connected to the basic configuration determination module (10) and is used to perform topology finding and morphological growth of the main truss starting from the basic configuration and using an improved simulated plant growth algorithm to obtain the initial topology configuration. The improved simulated plant growth algorithm includes: using the sum of strain energy of all members connected to the node as the basis for calculating the auxin concentration, using a hybrid step-size parallel search strategy to search for candidate growth points, periodically removing members with the lowest stress utilization rate, and triggering a global exploration mechanism or topology mutation mechanism to escape the local optimum when optimization stalls. The experience configuration fine-tuning module (30) is connected to the PGSA topology finding module (20) and is used to fine-tune the initial topology configuration based on engineering experience to obtain the experience configuration. The fine-tuning includes regularizing the node positions, adding necessary members and / or setting hinge points. The NSGA-II multi-objective collaborative optimization module (40) is connected to the empirical configuration fine-tuning module (30) and is used to perform multi-objective collaborative optimization of node coordinates and member cross-sections using the NSGA-II algorithm, starting from the empirical configuration, to obtain the final optimized design scheme of the bridge-building machine truss; wherein, the optimization objectives of the NSGA-II algorithm include minimizing the total mass of the structure, controlling the overall stiffness, controlling the structural strength and controlling the stability of the members; The stability verification module (50) is connected to the NSGA-II multi-objective collaborative optimization module (40) and is used to perform geometric stability verification on each generation of new individuals, eliminate geometrically unstable individuals, and ensure that all solutions are feasible structures.

8. The system according to claim 7, characterized in that, The PGSA topology finding module (20) is equipped with a hybrid step size parallel search unit (22). In the early stage of optimization, the hybrid step size parallel search unit (22) simultaneously uses large step size and small step size to search for candidate growth points in parallel, and in the later stage of optimization, it only uses small step size for fine search.

9. The system according to claim 7, characterized in that, The PGSA topology finding module (20) is equipped with a global exploration unit (24) and a topology mutation unit (25). The global exploration unit (24) is used to actively add candidate growth points in the low strain energy region for trial growth when optimization stalls. The topology mutation unit (25) is used to randomly delete non-critical force transmission path members and then guide the growth process again when optimization stalls.

10. The system according to claim 7, characterized in that, The stability verification module (50) uses the non-zero criterion of the stiffness matrix determinant to quickly determine the geometric stability and eliminate individuals with singular or nearly singular stiffness matrices.

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