Steel pipe concrete arch rib parameter design method and system based on multi-objective optimization

Abstract

This invention relates to the field of civil engineering bridge structure optimization technology, specifically a method and system for designing steel-concrete composite arch rib parameters based on multi-objective optimization. The method includes: determining multiple optimization variables and multiple objective functions based on pre-acquired preliminary design data of the steel-concrete composite bridge; generating a uniformly distributed initial population using chaotic mapping according to the value range of the optimization variables; dividing the initial population into regions using the objective functions, and obtaining the number of individuals in each sub-region based on the region division results; and iteratively obtaining the optimized parameters of the steel-concrete composite arch rib using a multi-objective optimization algorithm based on the number of individuals in each sub-region. This invention solves the problems of low efficiency and inaccurate optimization results in existing multi-objective optimization methods for steel-concrete composite arch rib parameters. By improving the multi-objective optimization algorithm, and employing scientific spatial partitioning and efficient search strategies, this invention significantly improves the efficiency and accuracy of steel-concrete composite arch rib parameter design.

Description

Technical Field

[0001] This invention relates to the field of civil engineering bridge structure optimization technology, specifically to a method and system for designing parameters of steel-concrete composite arch ribs based on multi-objective optimization. Background Technology

[0002] Concrete-filled steel tube arch bridges are widely used in the design and construction of long-span arch bridges due to their superior performance, including good seismic resistance, high structural stiffness, and low cost. Their number and span are constantly increasing. Currently, eight of the world's ten longest arch bridges are located in China. However, with the increase in design span, the design complexity and construction difficulty of arch bridges have significantly increased. The arch rib, as the core load-bearing component of an arch bridge, directly affects the structural performance and economy. Traditional design methods mainly rely on engineers' experience, involving repeated modifications to design parameters and iterative calculations. Model adjustments are time-consuming and labor-intensive, and the design results often have high redundancy, making it difficult to fully utilize material properties.

[0003] Optimizing complex structures like steel-concrete composite arch bridges involves numerous design variables and is a high-dimensional, multi-objective nonlinear problem. Traditional optimization algorithms suffer from the following shortcomings when dealing with high-dimensional or nonlinear complex problems: First, the population initialization is highly random, resulting in an uneven distribution of initial solutions and affecting global search capabilities. Second, crossover and mutation operations lack effective utilization of historical evolutionary information, leading to low search efficiency. Third, the uniformity of the solution set distribution during population evolution is difficult to guarantee, especially when the target space is large, which can easily lead to local convergence.

[0004] Therefore, there is an urgent need for a parameter design method for concrete-filled steel tube arch ribs that offers both high optimization efficiency and better optimization results. Summary of the Invention

[0005] To address the shortcomings of existing technologies, one aspect of the present invention provides a method and system for designing steel-concrete composite arch rib parameters based on multi-objective optimization, which solves the problems of low efficiency and inaccurate optimization results in existing technologies for steel-concrete composite arch rib parameters through multi-objective optimization.

[0006] To achieve the above objectives, this invention provides a multi-objective optimization-based method for designing parameters of steel-concrete composite arch ribs. The method includes: determining multiple optimization variables and multiple objective functions based on pre-acquired preliminary design data of a steel-concrete composite bridge; generating a uniformly distributed initial population using chaotic mapping according to the value range of the optimization variables; dividing the initial population into regions using the objective functions, and obtaining the number of individuals in each sub-region based on the region division results; and iteratively obtaining the optimized parameters of the steel-concrete composite arch rib using a multi-objective optimization algorithm based on the number of individuals in each sub-region.

[0007] This invention ensures the design is targeted by accurately determining the optimization variables and objective function; the initial population generated by chaotic mapping is evenly distributed, improving the global search capability; the region is divided according to the objective function and the number of individuals in the sub-region is balanced, enhancing the diversity of the solution set; the multi-objective optimization algorithm is iteratively optimized, combined with individual migration, crossover mutation and elite retention strategies, to efficiently converge to the optimal solution, thus improving the efficiency and accuracy of the parameter design of steel tube concrete arch ribs.

[0008] Optionally, determining multiple optimization variables and multiple objective functions based on pre-acquired preliminary design data of the steel-concrete composite bridge includes: constructing a first finite element model based on the pre-acquired preliminary design data of the steel-concrete composite bridge; performing stress analysis on the first finite element model to determine multiple optimization objectives; acquiring the parameter variables of the steel-concrete composite arch bridge and performing sensitivity analysis on the parameter variables to obtain optimization variables; obtaining multiple combinations of structural design parameters through orthogonal experiments using Latin hypercube sampling based on the value range of the optimization variables; constructing multiple second finite element models based on the combinations of structural design parameters; calculating multiple optimization objective values ​​using the second finite element models and the optimization objectives; and performing response surface function fitting on the combinations of structural design parameters and the optimization objective values ​​to obtain multiple objective functions.

[0009] This invention constructs a first finite element model and performs stress analysis based on preliminary design data to accurately locate structural stress shortcomings and determine optimization objectives. It screens key optimization variables through sensitivity analysis to avoid interference from invalid variables, and combines orthogonal experiments with Latin hypercube sampling to efficiently generate representative combinations of structural design parameters. Then, a second finite element model is built to calculate the target value, and the target function is obtained by fitting the response surface function, thus improving the scientific nature and accuracy of the target function.

[0010] Optionally, the step of dividing the initial population into regions using the objective function and obtaining the number of individuals in each sub-region based on the result of the region division includes: calculating the objective function value of each individual based on the initial population using the objective function; setting a target space based on the number of individuals in the objective function; setting a reference point; calculating the Euclidean distance between the normalized objective function value and the reference point; and dividing the initial population into sub-regions based on the Euclidean distance to obtain the number of individuals in each sub-region.

[0011] This invention calculates the objective function value of individuals in the initial population using an objective function, and accurately sets the target space based on the number of individuals with the objective function, laying the foundation for region division. By scientifically setting reference points, combined with normalization processing and Euclidean distance calculation, it eliminates dimensional interference, accurately measures the spatial relationship between individuals and reference points, and then divides the initial population according to distance to obtain the number of individuals in sub-regions. Overall, this makes the initial population region division more objective and uniform, avoids imbalance in individual distribution, ensures that each sub-region fully covers the target space, and improves the scientific nature of the initial population division.

[0012] Optionally, setting the reference point includes: setting a unit hypersphere and a unit simplex based on the number of objective functions; generating uniformly distributed sampling points on the unit hypersphere using the golden section spiral sampling method; and projecting the sampling points onto the unit simplex generation reference point.

[0013] Based on the number of objective functions, a unit hypersphere and a unit simplex are constructed to ensure that the spatial dimension is adapted to the optimization requirements. Then, by using the golden ratio spiral sampling, uniformly distributed sampling points are generated on the unit hypersphere to avoid the local concentration problem that is easy to occur in traditional sampling. Finally, the sampling points are projected onto the unit simplex to generate reference points, which improves the feasibility and accuracy of reference point calculation.

[0014] Optionally, the step of iteratively obtaining the optimized parameters of the steel-concrete composite arch rib using a multi-objective optimization algorithm based on the number of individuals in the sub-region includes: performing individual migration between sub-regions of the initial population based on the number of individuals in the sub-region; performing fast non-dominated sorting on the migrated initial population to obtain the non-dominated level of individuals; performing crossover and mutation on the initial population based on the non-dominated level of individuals to form the next generation population; iterating based on the next generation population until the prediction termination condition is met, and outputting the optimized parameters of the steel-concrete composite arch rib.

[0015] This invention uses the number of individuals in a sub-region to migrate individuals, balances the individual density in each region, and avoids search bias caused by local overcrowding or sparseness of individuals. Then, it uses rapid non-dominated sorting to clarify the optimization priority of individuals, providing a scientific basis for subsequent evolutionary operations. Based on the non-dominated level of individuals, it carries out crossover and mutation, which not only preserves high-quality genes but also introduces new characteristics, improves population diversity, and enhances the efficiency and accuracy of calculating optimization parameters.

[0016] Optionally, the migration of individuals between sub-regions of the initial population based on the number of individuals in the sub-region includes: setting a threshold for the number of individuals in a region, and determining the region to be migrated into and the region to be migrated out based on a comparison between the threshold for the number of individuals in the region and the number of individuals in the sub-region; calculating the crowding distance of individuals in the region to be migrated out; determining individuals to be migrated out based on the crowding distance, and randomly migrating the individuals to be migrated out to the region to be migrated into.

[0017] This invention accurately determines areas to be migrated into and out of by setting a threshold for the number of individuals in each region, avoiding excessive density or sparseness of individuals in sub-regions. It then calculates the crowding distance of individuals in areas to be migrated out, providing an objective basis for screening individuals to be migrated and ensuring priority migration of densely distributed individuals. Finally, individuals to be migrated out are randomly migrated into areas requiring replenishment. Overall, this approach balances the individual density across sub-regions, preventing population aggregation in local target spaces that could limit the search, maintaining population diversity, and improving the comprehensiveness and reliability of the population.

[0018] Optionally, the step of crossover and mutation of the initial population based on the individual non-dominance level to form the next generation population includes: performing a crossover operation based on the individual non-dominance level; introducing elite individuals to guide mutation; and using the elite individuals to guide mutation based on the result of the crossover operation to perform a mutation operation to generate the next generation population.

[0019] This invention utilizes crossover operations at the individual non-dominant level to prioritize gene exchange between individuals with high optimization potential, preserving superior population characteristics and avoiding interference from inferior genes in low-level individuals. Elite individuals are then introduced to guide mutations, ensuring that the mutation direction aligns more closely with the optimal solution exploration direction. This reduces ineffective searches caused by blind mutations and improves search efficiency.

[0020] Optionally, the crossover operation based on the individual's non-dominance level includes: randomly selecting individuals to be crossovered from the initial population; determining crossover targets based on the individuals to be crossovered and their corresponding non-dominance levels; the crossover targets satisfying the following formula: ,in, The individual non-dominant level of the individuals to be crossed is The set to which the cross-objects belong. The individual non-dominance level is respectively The corresponding set of individuals; perform cross-operation on the individuals to be crossed and the cross-operation objects.

[0021] This invention determines the set of crossover targets by using a formula based on the non-dominant level of the individuals to be crossovered, limiting the crossover targets to individuals with higher or equal levels, avoiding interference from inferior genes of lower-level individuals, and further improving search efficiency.

[0022] Optionally, the step of generating the next generation population by using the elite individual-guided mutation based on the result of the crossover operation includes: randomly selecting individuals to be mutated based on the result of the crossover operation, and performing a mutation operation on the individuals to be mutated using the elite individual-guided mutation; generating the next generation population based on the result of the mutation operation using an elite preservation strategy; the mutation operation satisfies the following formula: ,in, For the mutated individual The solution vector, For individuals The solution vector, This is the solution vector in the first non-dominated level. This is the learning rate.

[0023] This invention guides mutation by using elite individuals as the target, guiding the individual to evolve towards a high-quality solution, reducing blind mutation, improving mutation effectiveness, and finally using an elite retention strategy to generate the next generation population, retaining high-quality individuals, further improving search efficiency.

[0024] Another aspect of the present invention provides a multi-objective optimization-based steel-concrete composite arch rib parameter design system, comprising: a processor, an input device, an output device, and a memory, wherein the processor, the input device, the output device, and the memory are interconnected, wherein the memory is used to store a computer program, the computer program including program instructions, and the processor is configured to call the program instructions to execute the multi-objective optimization-based steel-concrete composite arch rib parameter design method according to any one of the preceding aspects of the present invention.

[0025] The present invention provides a multi-objective optimization-based steel-concrete composite arch rib parameter design system, which is compact, stable in performance, highly integrated and simple in construction. It can stably execute the multi-objective optimization-based steel-concrete composite arch rib parameter design method provided in the foregoing aspect of the present invention, further improving the overall applicability and practical application capability of the present invention. Attached Figure Description

[0026] Figure 1 This is a flowchart of a method for designing parameters of a steel-concrete composite arch rib based on multi-objective optimization, according to an embodiment of the present invention. Figure 2 This is a schematic diagram of the structural design system for the parameter design of steel-concrete composite arch ribs based on multi-objective optimization, according to an embodiment of the present invention. Detailed Implementation

[0027] Specific embodiments of the present invention will now be described in detail. It should be noted that the embodiments described herein are for illustrative purposes only and are not intended to limit the invention. In the following description, numerous specific details are set forth in order to provide a thorough understanding of the invention. However, it will be apparent to those skilled in the art that these specific details are not necessary to practice the invention. In other instances, well-known circuits, software, or methods have not been specifically described to avoid obscuring the invention.

[0028] Throughout this specification, references to "an embodiment," "an embodiment," "an example," or "an example" mean that a particular feature, structure, or characteristic described in connection with that embodiment or example is included in at least one embodiment of the invention. Therefore, the phrases "in an embodiment," "in an embodiment," "an example," or "an example" appearing in various places throughout the specification do not necessarily refer to the same embodiment or example. Furthermore, specific features, structures, or characteristics can be combined in one or more embodiments or examples in any suitable combination and / or sub-combination. Moreover, those skilled in the art will understand that the illustrations provided herein are for illustrative purposes and are not necessarily drawn to scale.

[0029] Please see Figure 1 To address the shortcomings of the prior art, in one alternative embodiment, such as Figure 1 The parameter design method for steel-concrete composite arch ribs based on multi-objective optimization, as shown, includes the following steps: Step S1: Based on the preliminary design data of the pre-acquired steel-concrete composite bridge, determine multiple optimization variables and multiple objective functions.

[0030] The process of determining multiple optimization variables and objective functions based on pre-acquired preliminary design data for steel-concrete composite bridges includes the following sub-steps: Step S101: Construct the first finite element model based on the pre-acquired preliminary design data of the steel-concrete composite bridge.

[0031] In this embodiment, the preliminary design data for the steel-concrete composite bridge includes detailed design drawings, material selection reports, and preliminary load planning. The core geometric features of the bridge, the cross-sectional form and dimensions of the arch ribs, and material performance parameters are extracted from the preliminary design data. Initial load conditions are determined according to existing bridge design specifications such as the "General Specifications for Highway Bridge and Culvert Design." Then, professional finite element analysis software is used for geometric modeling, restoring the structural spatial form at a 1:1 scale. Boundary constraints are applied according to the support forms in the preliminary design. Dead load, live load, wind load, and temperature load are applied in detail according to load specifications. Mesh optimization is then performed: stress concentration areas such as the arch foot and arch crown are denser, and the mesh size can be appropriately enlarged for straight arch rib sections. Mesh quality checks ensure calculation accuracy. Finally, model verification is conducted, calculating the vertical deflection of the arch crown, the axial force and bending moment at the arch foot under self-weight, and comparing these with manually calculated estimates from the preliminary design. Geometric modeling deviations, element selection errors, or omissions in load application are identified, ultimately forming a first finite element model that accurately reflects the structural stress characteristics of the preliminary design.

[0032] Step S102: Perform stress analysis on the first finite element model to determine multiple optimization objectives.

[0033] In this embodiment, when performing stress analysis on the first finite element model, loads such as self-weight and live load are first applied to the model based on the actual engineering situation and bridge design specifications. Key indicators such as arch rib stress, main beam vertical deflection, main arch elastic overall stability coefficient, and arch rib bending strain energy are calculated to accurately locate the stress shortcomings in the preliminary design. Then, combined with economic requirements, relevant indicators of arch rib material usage are incorporated to form multiple optimization objectives for both stress performance and economy. This ensures that the optimization objectives not only meet the specifications but also conform to the actual stress requirements of the structure.

[0034] Step S103: Obtain the parameter variables of the steel-concrete composite arch bridge, and perform sensitivity analysis on the parameter variables to obtain optimization variables.

[0035] In this embodiment, parameters that affect the optimization objective, such as cross-sectional dimensions and arch rib shape, are selected. Numerical simulation methods are used to perform sensitivity analysis on the selected parameters, and the sensitivity index of each parameter is calculated. Sensitivity indices greater than 1 are selected as high-sensitivity indices and included in the optimization variables. For multiple optimization objectives, there are numerous variables. If the sensitivity value of a variable is less than 1 for all optimization objectives, it is discarded. If the sensitivity index of any optimization objective is greater than 1, that variable is added to the optimization variables. The sensitivity analysis uses the elasticity coefficient method, and the sensitivity index is defined as: , In the formula, For design parameters, For parameter changes, For target response, This represents the change in the target response.

[0036] Step S104: Based on the value range of the optimization variables, orthogonal experiments are conducted using Latin hypercube sampling to obtain multiple combinations of structural design parameters.

[0037] In this embodiment, the range of values ​​for the optimization variables is determined based on bridge engineering experience and bridge design specifications. This ensures that the range satisfies both structural safety and construction feasibility, while also covering the critical intervals where the variables may affect the optimization objective. Subsequently, the Latin hypercube sampling method is used to perform stratified random sampling within the value range of each optimization variable. This ensures that the sample points for each variable dimension are evenly distributed, avoiding the local sample concentration problem that is prone to occur in traditional sampling, and fully covering the variable value space. On this basis, the sampling points of each variable are combined according to the orthogonal experimental design principle, so that the generated combination of structural design parameters can reflect the interaction of different variable levels and has good representativeness and balance. Finally, multiple combinations of structural design parameters are obtained.

[0038] Step S105: Construct multiple second finite element models based on the combination of the structural design parameters.

[0039] In this embodiment, based on the key parameters of the arch ribs specified in each combination of structural design parameters, multiple second finite element models are obtained by modeling according to the construction method of the first finite element model.

[0040] Step S106: Calculate multiple optimization target values ​​using the second finite element model and the optimization target.

[0041] In this embodiment, based on the bridge design conditions, corresponding loads are precisely applied to the model and matching boundary constraints are set. Then, for the determined optimization objectives (such as arch rib bending strain energy, arch rib material price, steel pipe and concrete stress values, main beam vertical deflection, main arch elastic overall stability coefficient, etc.), calculations are carried out using finite element analysis tools to obtain multiple sets of optimization objective values ​​corresponding to each combination of structural design parameters.

[0042] Step S107: Perform response surface function fitting on the combination of structural design parameters and the optimization target value to obtain multiple target functions.

[0043] In this embodiment, the parameter variables and optimization target values ​​in the optimization of steel-concrete composite arch ribs exhibit a nonlinear relationship. A response surface function form containing quadratic cross terms is selected, and regression analysis is performed on the data using fitting methods such as the least squares method to construct the functional relationship between the optimization variables and individual optimization target values. During the fitting process, the accuracy of the function needs to be verified to ensure that the function can accurately approximate the calculation results of the finite element model. Finally, for each optimization target, the response surface function fitting is completed separately to obtain multiple objective functions that can explicitly express the optimization variables and corresponding optimization target values.

[0044] In an alternative embodiment, the objective function includes the flexural strain energy response surface function of the steel-concrete composite arch rib and the response surface function obtained by fitting the arch rib material price.

[0045] The surface function of the bending strain energy response of a steel-concrete composite arch rib satisfies the following formula: , The response surface function obtained by fitting the price of the arch rib material satisfies the following formula: , in The bending strain energy of the steel-concrete composite arch rib. Price of steel-concrete composite arch rib material. For the inclination angle of the arch rib, For the arch axis coefficient, The height inside the arch foot surface, This represents the variation coefficient of the arch crown and arch foot.

[0046] Step S2: Generate a uniformly distributed initial population using chaotic mapping based on the value range of the optimization variables.

[0047] In this embodiment, the initial values ​​of the sequence are first randomly generated within the range of (0,1). ( ), To optimize the number of variables.

[0048] For the The first variable Chaotic sequence value corresponding to each individual ( , (where the population size is 1), the next value is generated using a standard chaotic mapping without any perturbation. : , Divide [0, 1] into equal parts The number of sub-intervals, the first The intervals are ,but The index of the interval is: (Note: To round down, Statistics up to the [number]th When there are 1 element, The number of elements in the corresponding interval is (Initially set to 0, updated every iteration), then the probability density estimate for this interval is... for: , The perturbation amplitude is calculated based on the probability density of the interval and limited to a given perturbation threshold.

[0049] , For the disturbance amplitude, For the maximum disturbance amplitude, in the formula It is a random number within the interval (0, 1). The adjustment coefficient can be selected within the range of [0.5, 2], and the final disturbance term is... .

[0050] Finally, the perturbation term is superimposed on the standard mapping, and truncation is used to ensure that the generated value is in the range [0,1]. , For the first The first optimization variable is the... A chaotic sequence value, For the first The optimization variable is the first The amplitude of the perturbation of each element, For the first The optimization variable is the first The random perturbation coefficient of each element.

[0051] After obtaining the chaotic sequence value, based on the input of the first... Upper and lower bounds of input variables Map the chaotic sequence to the decision space. The resulting initial population contains the [missing information - likely a specific type of sequence]. The first variable Individual values The calculation formula is as follows: , For the first The first variable The chaotic sequence value corresponding to each individual.

[0052] Step S3: Divide the initial population into regions using the objective function, and obtain the number of individuals in each sub-region based on the result of the region division.

[0053] The process of dividing the initial population into regions using the objective function and obtaining the number of individuals in each sub-region based on the region division results specifically includes the following sub-steps: Step S301: Calculate the objective function value of an individual based on the objective function according to the initial population.

[0054] In this embodiment, the objective function value for each individual is obtained by substituting the initial population into each objective function.

[0055] To achieve the desired optimization effect, the objective function value can be adjusted according to pre-set constraints.

[0056] In an optional embodiment, the constraints are set as follows: Strength constraints: For the truss arch rib section, calculated as an integral section under eccentric compression, its bearing capacity should meet the following requirements: , in, This is the structural importance coefficient; The design value for axial force of the 1 / 4 span arch rib; This is the slenderness ratio reduction factor; This is the moment reduction factor; This is the initial stress reduction factor for the steel pipe; This is the reduction factor for concrete voids; This is the design value of the axial compressive strength of the steel-concrete composite section; This represents the cross-sectional area of ​​the steel-concrete composite structure.

[0057] The stress in the steel should meet the following requirements: , The concrete stress should meet the following requirements: , in For steel stress, For concrete stress, For the allowable stress of steel, This represents the allowable compressive stress in concrete.

[0058] Stiffness requirement: The vertical deflection of its main beam under the static and live load of the train should not exceed [a certain value]. ,Right now: , in The maximum vertical deflection of the main beam. The main beam span.

[0059] Stability requirement: The overall elastic stability coefficient of the main arch shall not be less than 4.0.

[0060] Substitute the constraints into the calculation. If the constraints are not met, add 10 to the power of 6 to the calculated objective function value.

[0061] Step S302: Set the target space according to the number of objective functions.

[0062] In this embodiment, the target space is a structured mathematical space used to carry and analyze the numerical values ​​of optimization objectives in a multi-objective optimization problem. Its core feature is that the number of dimensions of the space is uniquely determined by the number of optimization objectives defined by the optimization problem. That is, if there are n independent optimization objectives, an n-dimensional target space is constructed accordingly. Each dimension of the space corresponds to an optimization objective, and the numerical range of the dimension must cover all possible values ​​of the optimization objective within the problem solution range. Its core function is to serve as a mapping carrier for the objective function values ​​of individual optimizations (corresponding to combinations of design parameters) during the optimization process.

[0063] Step S303: Set the reference point.

[0064] Setting a reference point specifically includes the following sub-steps: Step S30301: Set the unit hypersphere and unit simplex based on the number of objective functions.

[0065] In this embodiment, the unit hypersphere is an n-dimensional spatial geometric structure with the same dimension as the number of objective functions (denoted as n). Its core feature is that the distance from all points in the space to the origin is equal to 1, mathematically expressed as the sum of the squares of the coordinate values ​​of all points in the space equals 1. It can accommodate the dimensional requirements corresponding to n optimization objectives, providing a basic spatial carrier for subsequent uniform sampling. The unit simplex is also an n-dimensional spatial geometric structure with a dimension matching the number of objective functions n. Its core feature is that the coordinate values ​​of all points in the space are non-negative and the sum of all coordinate values ​​equals 1. It is mainly used to receive the projection of the sampling points from the hypersphere.

[0066] Step S30302: Generate uniformly distributed sampling points on the unit hypersphere using the golden section spiral sampling method.

[0067] In this embodiment, to ensure the diversity of individuals in the population during the optimization process and to avoid the population getting trapped in local optima, the target space is divided into Q sub-regions during the optimization process. Each sub-region focuses on optimizing a certain part of the target space, thereby improving the diversity of the solution set. Reference points are generated in each sub-region. The reference points are generated by uniformly sampling in high-dimensional space using the golden section spiral and then projecting onto the simplex of the target space to ensure that the reference points are evenly distributed.

[0068] To reduce the computational complexity of traditional combinatorial methods in high-dimensional spaces, the spatial problem is transformed into a uniform sampling problem on a hypersphere. In 3D space, a unit hypersphere Points on Satisfying the formula: , In order to achieve hyperspherical surfaces Q points are obtained in a uniform distribution. The golden section spiral method is used for calculation. For the i-th point... Reference points ( ),calculate Each spherical angle: , , In the formula For the last angle, It is the golden ratio. For other angles, among which .

[0069] Then the first The coordinates of a reference point on the unit hypersphere can be expressed as: , , , , , For the first The first-dimensional rectangular coordinates of each sampling point, For the first The second-dimensional rectangular coordinates of each sampling point, For the first The third-dimensional rectangular coordinates of each sampling point For the first The sampling point of the first sampling point 3D rectangular coordinates For the first The sampling point of the first sampling point 3D rectangular coordinates The first polar angle corresponding to the unit hypersphere. The second polar angle corresponding to the unit hypersphere. The third polar angle corresponding to the unit hypersphere. The first corresponding to the unit hypersphere One polar angle, The first corresponding to the unit hypersphere Angle of polarity.

[0070] Step S30303: Project the sampling points onto the unit simplex generation reference point.

[0071] In this embodiment, the sampling points on the obtained unit hypersphere are projected onto the unit simplex to generate the final reference points. ,in Its projection formula is as follows.

[0072] , For the first The first-dimensional coordinates of each reference point, For the first The second-dimensional coordinates of each reference point, For the first The first reference point 3D coordinates For the first The first reference point 3D coordinates For the first The sampling point of the first sampling point 3D rectangular coordinates.

[0073] Step S304: Calculate the Euclidean distance between the normalized objective function value and the reference point.

[0074] In this implementation, the objective function values ​​of different individuals are normalized to eliminate the influence of their dimensions.

[0075] , in, For the first Individual in the population The normalized objective function value of the optimization objective. For the first The individual The objective function value of the optimization objective. For the first The global minimum of an optimization objective. For the first The global maximum value of each optimization objective. , , To avoid the numerator having a zero term.

[0076] , , Calculate the next Individual normalized target value With all reference points Euclidean distance: , For the first The normalized objective function value of each individual in the population and the first... Euclidean distance between reference points For the first The normalized objective function value corresponding to each individual in the population. No. The value of each reference point For the first Individual in the population The normalized objective function value of the optimization objective. For the first The first reference point Dimensional coordinates.

[0077] Step S305: Spatial division of the initial population based on the Euclidean distance to obtain the number of individuals in sub-regions.

[0078] In this implementation, the target space is first divided into sub-regions based on the reference points projected onto the unit simplex: since each reference point is a uniformly distributed positioning marker in the target space, the target space is divided into sub-regions with the same number of reference points as the center of each reference point. The range of the sub-region is set as the spatial area enclosed by the perpendicular bisector of the line connecting the reference point and the adjacent reference point, ensuring that each sub-region covers the target space without overlap, and each sub-region uniquely corresponds to one reference point.

[0079] After completing the initial division of the target space, the reference point closest to the individual is determined based on the calculated Euclidean distance. The individual is then assigned to the sub-region corresponding to the nearest reference point. This completes the initial population space division based on Euclidean distance. Finally, the total number of individuals contained in each sub-region is counted to obtain the number of individuals corresponding to each sub-region.

[0080] Step S4: Based on the number of individuals in the sub-region, the optimized parameters of the steel-concrete composite arch rib are obtained by iteratively using a multi-objective optimization algorithm.

[0081] The process of obtaining the optimized parameters of the steel-concrete composite arch rib by iteratively using a multi-objective optimization algorithm based on the number of individuals in the sub-region specifically includes the following sub-steps: Step S401: Perform individual migration between sub-regions of the initial population based on the number of individuals in the sub-region.

[0082] Specifically, the migration of individuals between sub-regions of the initial population based on the number of individuals in the sub-region includes the following sub-steps: Step S40101: Set a threshold for the number of individuals in a region, and determine the region to be moved into and the region to be moved out based on the comparison result between the threshold for the number of individuals in the region and the number of individuals in the sub-region.

[0083] In this embodiment, the threshold for the number of individuals in a region is set according to a balanced distribution logic, combining the total number of individuals in the initial population with the total number of sub-regions. For example, the threshold is based on the initial total number of individuals divided by the total number of regions. It can be fine-tuned according to optimization needs to ensure that the threshold can both prevent overcrowding in a single sub-region and prevent insufficient exploration of the target direction due to too few individuals in the sub-region. Then, the actual number of individuals in each sub-region is compared with the threshold one by one: if the number of individuals in a certain sub-region is greater than the threshold, it means that the individuals in the region are densely distributed, which may cause the population to get stuck in the optimization bottleneck in the local target space. It is determined to be a region to be migrated out. If the number of individuals in a certain sub-region is less than the threshold, it means that the individuals in the region are sparsely distributed, which may miss the high-quality solutions in the corresponding target direction. It is determined to be a region to be migrated into.

[0084] Step S40102: Calculate the crowding distance of individuals in the area to be relocated.

[0085] In this embodiment, all individuals within the region to be emigrated are first extracted to form a set of individuals for calculating congestion distance. The normalized objective function value of each individual in this set is then retrieved. Next, the target dimensions involved in the calculation are determined based on the number of objective functions. For each target dimension, the individuals within the region to be emigrated are sorted in ascending order according to the objective function value corresponding to that dimension. For individuals at both ends of the sorted dimension, their congestion distance component in that dimension is directly set to infinity. For individuals in the middle of the sorted dimension, their congestion distance component in that dimension is calculated by subtracting the objective function value of the individual to the right from the individual to the left, and then dividing by the range of the objective function values ​​of all individuals in that dimension (i.e., the difference between the maximum and minimum values ​​of the objective function values ​​in that dimension). This yields the congestion distance component of the individual in the current dimension. Finally, the congestion distance components of each individual in all target dimensions are summed, and the result is the total congestion distance of the individual in the region to be emigrated.

[0086] Step S40103: Determine the individuals to be relocated based on the congestion distance, and randomly relocate the individuals to be relocated to the area to be relocated to.

[0087] In this embodiment, based on the principle that the smaller the crowding distance, the denser the distribution of individuals within the region, and the more priority should be given to relocating them, all individuals in the region to be relocated are first sorted by crowding distance from smallest to largest. Based on the sum of the differences between the threshold of the region to be relocated and the actual number of individuals, the top N individuals (N equals the sum of the differences) are selected from the sorting results as individuals to be relocated. Then, a random allocation method is used to randomly assign the selected individuals to be relocated to each region to be relocated, ensuring that the number of individuals received by each region to be relocated matches its shortage, thus completing the individual migration to balance the individual density of each sub-region.

[0088] Step S402: Perform a fast non-dominated sort on the migrated initial population to obtain the individual non-dominated levels.

[0089] In this embodiment, for each individual in the population, the number of other individuals who can dominate that individual is counted and recorded as the domination count. At the same time, a list of other individuals that the individual can dominate is recorded. Individuals with a domination count of 0 are assigned to the first non-dominant level. Then, the domination list of individuals in this level is traversed, and the domination count of each individual in the list is decremented by 1. If the domination count of an individual is reduced to 0, it is assigned to the next non-dominant level. This process is repeated until all individuals in the population are assigned to the corresponding non-dominant level, and finally the individual non-dominant level of each individual is obtained.

[0090] Step S403: Based on the individual non-dominance level, crossover and mutation are performed on the initial population to form the next generation population.

[0091] The process of crossover and mutation of the initial population based on the individual non-dominance level to form the next generation population specifically includes the following sub-steps: Step S40301: Perform crossover operations based on the individual's non-dominance level.

[0092] The crossover operation based on the individual's non-dominance level includes: Step S4030101: Randomly select individuals to be crossovered from the initial population.

[0093] In this embodiment, a random number is obtained. ,if Less than the given crossover probability Then apply the crossover strategy to that individual, if Greater than If the crossover strategy is not executed, then no crossover strategy will be implemented.

[0094] Generate a random number belonging to the interval [0,1] for each individual. , Then, use this random number Crossover probability with a pre-set value If a comparison is made, Less than If rand is greater than or equal to 0, then the individual is determined to be an individual to be crossed and added to the set of individuals to be crossed. If the crossover strategy is not executed, the individual will not be added to the set of individuals to be crossed. By performing this random judgment on each individual in the initial population, the final set of individuals to be crossed for subsequent crossover operations is obtained.

[0095] Step S4030102: Determine the crossing object based on the individual to be crossed and the individual non-dominance level corresponding to the individual to be crossed.

[0096] In this embodiment, based on the non-dominance level of the individual to be crossed, individuals with a higher non-dominance level are first selected as the set of crossing objects. Then, an individual is randomly selected from the set of crossing objects as the crossing object of the individual to be crossed. This ensures the stability of the population hierarchy while promoting information exchange between individuals with different optimization potentials and maintaining population diversity.

[0097] The intersection object satisfies the following formula: , The individual non-dominant level of the individuals to be crossed is The set to which the cross-objects belong. The individual non-dominance level is respectively The corresponding set of individuals.

[0098] Step S4030103: Perform a crossover operation on the individual to be crossed and the crossover object.

[0099] In this embodiment, the crossover operation refers to the core evolutionary operation in NSGA-II that simulates biological gene recombination. It is used to generate offspring individuals by exchanging gene information between parent individuals, thereby enhancing the diversity of solutions while preserving high-quality genes in the population and promoting the convergence and exploration of the Pareto optimal frontier by the algorithm.

[0100] Step S40302: Introduce elite individuals to guide mutation, and use the elite individuals to guide mutation to generate the next generation population based on the result of the crossover operation.

[0101] In this embodiment, elite-individual guided mutation is an intelligent optimization mechanism used to improve search efficiency and solution quality during population evolution. Its core idea is to allow poorly performing individuals (low-level individuals) in the population to learn from high-performing elite individuals (high-level individuals), thereby transferring the superior genetic information of the elite individuals to themselves to guide their own evolutionary direction.

[0102] Specifically, the generation of the next generation population by using the elite individuals to guide mutation based on the results of the crossover operation includes the following sub-steps: Step S4030201: Based on the result of the crossover operation, randomly select individuals to be mutated, and use the elite individuals to guide the mutation to perform mutation operations on the individuals to be mutated.

[0103] In this embodiment, to avoid getting trapped in local optima, a migration mutation strategy is implemented on a subset of individuals (lower-level individuals learn from higher-level individuals), and random numbers are re-acquired. ,if If the value is greater than 0.5, then the normal mutation strategy is executed. If the value is less than 0.5, then the elite individual-guided mutation is used to perform mutation operation on the individual to be mutated. Here, the individual to be mutated specifically refers to the individual that needs to undergo elite individual-guided mutation.

[0104] The mutation operation satisfies the following formula: , in, For the mutated individual The solution vector, For individuals The solution vector, This is the solution vector in the first non-dominated level. This is the learning rate.

[0105] Step S4030202: Based on the result of the mutation operation, an elite retention strategy is used to generate the next generation population.

[0106] In this embodiment, the offspring population generated by crossover and mutation operations is first merged with the parent population to form a temporary population with increased size. Then, the temporary population is subjected to rapid non-dominated sorting, and individuals are stratified according to their non-dominated level. The crowding distance of individuals within the same non-dominated level is calculated. Then, individuals are selected into the next generation population layer by layer in descending order of non-dominated level. When the population size exceeds the preset size if all individuals in a certain non-dominated level are added, individuals within that level are sorted from largest to smallest according to their crowding distance, and individuals with larger crowding distances are selected first, until the predetermined size of the next generation population is filled. This preserves excellent individuals while maintaining the distribution diversity of the population at the Pareto front.

[0107] Step S404: Iterate based on the next generation population until the prediction termination condition is met, and output the optimized parameters of the steel-concrete composite arch rib.

[0108] In this embodiment, iterative calculations are performed using the NSGA-II algorithm. The iteration termination condition must simultaneously meet two requirements: first, the preset maximum number of iterations must be reached; second, the distribution variation of individuals in the target space within the Pareto optimal solution set for multiple consecutive generations must be less than a set threshold. When the above-mentioned prediction termination condition is met, the iteration stops and the optimized parameters of the steel-concrete composite arch rib in the final Pareto optimal solution set are output. Specifically, these parameters include the specific values ​​of optimized variables such as the arch rib inclination angle, arch axis coefficient, in-plane height of the arch foot, and arch crown-arch foot variation coefficient, as well as the corresponding objective function values ​​such as arch rib bending strain energy and material price, providing a parameter scheme that combines load-bearing performance and economy for engineering design.

[0109] The calculation results show that, compared with the traditional multi-objective particle swarm optimization algorithm, the multi-objective optimization-based parameter design method for steel-concrete composite arch ribs proposed in this invention solves the multi-objective optimization model of steel-concrete composite arch ribs. The results are closer to the Pareto front in the objective space and more uniformly distributed, that is, the convergence and diversity are better.

[0110] like Figure 2 As shown, in another aspect, the present invention also provides a multi-objective optimization-based steel-concrete composite arch rib parameter design system, comprising: a processor, an input device, an output device, and a memory, wherein the processor, the input device, the output device, and the memory are interconnected, wherein the memory is used to store a computer program, the computer program includes program instructions, and the processor is configured to call the program instructions to execute the relevant steps of the relevant embodiments of the multi-objective optimization-based steel-concrete composite arch rib parameter design method of the present invention.

[0111] This invention provides a multi-objective optimization-based parameter design system for steel-concrete composite arch ribs. The functional components can be integrated into a single processing unit, exist as separate physical entities, or be integrated into a single unit. These integrated components can be implemented in hardware or as software functions.

[0112] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features therein. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention, and they should all be covered within the scope of the claims and specification of the present invention.

Claims

1. A parameter design method for steel-concrete composite arch ribs based on multi-objective optimization, characterized in that, The method includes: Based on the preliminary design data of the pre-acquired steel-concrete composite bridge, multiple optimization variables and multiple objective functions were determined. Based on the range of values ​​of the optimization variables, a uniformly distributed initial population is generated using chaotic mapping; The initial population is divided into regions using the objective function, and the number of individuals in each sub-region is obtained based on the results of the region division. The optimized parameters of the steel-concrete composite arch rib are obtained by iteratively using a multi-objective optimization algorithm based on the number of individuals in the sub-region.

2. The method for designing parameters of steel-concrete composite arch ribs based on multi-objective optimization according to claim 1, characterized in that, The determination of multiple optimization variables and multiple objective functions based on pre-acquired preliminary design data of steel-concrete composite bridges includes: The first finite element model was constructed based on the preliminary design data of the pre-acquired steel-concrete composite bridge. Force analysis was performed on the first finite element model to determine multiple optimization objectives; Obtain the parameter variables of the steel-concrete composite bridge, and perform sensitivity analysis on the parameter variables to obtain optimization variables; Based on the value range of the optimization variables, orthogonal experiments were conducted using Latin hypercube sampling to obtain multiple combinations of structural design parameters; Multiple second finite element models are constructed based on the combination of the structural design parameters; Multiple optimization objective values ​​are calculated using the second finite element model and the optimization objective; Multiple objective functions are obtained by fitting the combination of structural design parameters and the optimization objective value to a response surface function.

3. The method for designing parameters of steel-concrete composite arch ribs based on multi-objective optimization according to claim 1, characterized in that, The process of dividing the initial population into regions using the objective function and obtaining the number of individuals in each sub-region based on the results of the region division includes: The objective function value of an individual is calculated using the objective function based on the initial population. Set the target space according to the number of the objective functions; Set a reference point; Calculate the Euclidean distance between the normalized objective function value and the reference point; The initial population is spatially divided into sub-regions based on the Euclidean distance to obtain the number of individuals in each sub-region.

4. The method for designing parameters of steel-concrete composite arch ribs based on multi-objective optimization according to claim 3, characterized in that, The reference points for setting include: Based on the number of objective functions, set a unit hypersphere and a unit simplex; The golden section spiral sampling method is used to generate uniformly distributed sampling points on the unit hypersphere. The sampling points are projected onto the unit simplex generation reference point.

5. The method for designing parameters of steel-concrete composite arch ribs based on multi-objective optimization according to claim 1, characterized in that, The optimization parameters for the steel-concrete composite arch rib obtained by iteratively using a multi-objective optimization algorithm based on the number of individuals in the sub-region include: Based on the number of individuals in the sub-region, the initial population is used to migrate individuals between sub-regions; The initial population after migration is subjected to a fast non-dominated sorting to obtain the individual non-dominated levels. Based on the individual non-dominance level, the initial population is crossovered and mutated to form the next generation population; Based on the next generation population, iterate until the prediction termination condition is met, and output the optimized parameters of the steel-concrete composite arch rib.

6. The method for designing parameters of steel-concrete composite arch ribs based on multi-objective optimization according to claim 5, characterized in that, The process of migrating individuals between sub-regions of the initial population based on the number of individuals in the sub-region includes: Set a threshold for the number of individuals in a region, and determine the regions to be moved into and the regions to be moved out based on the comparison between the threshold for the number of individuals in the region and the number of individuals in the sub-regions; Calculate the crowding distance for individuals in the area to be relocated; Based on the crowding distance, individuals to be relocated are identified, and these individuals are randomly moved to the receiving area.

7. The method for designing parameters of steel-concrete composite arch ribs based on multi-objective optimization according to claim 5, characterized in that, The process of crossover and mutation of the initial population based on the individual non-dominance level to form the next generation population includes: Cross operations are performed based on the individual's non-dominance level; Elite individuals are introduced to guide mutation, and the next generation population is generated by using the elite individuals to guide mutation based on the results of the crossover operation.

8. The method for designing parameters of steel-concrete composite arch ribs based on multi-objective optimization according to claim 7, characterized in that, The crossover operation based on the individual's non-dominance level includes: Randomly select individuals to be crossovered from the initial population; The crossover target is determined based on the individual to be crossed and the non-dominance level of the individual corresponding to the individual to be crossed; The intersection object satisfies the following formula: ， The individual non-dominant level of the individuals to be crossed is The set to which the cross-objects belong, The individual non-dominance level is respectively The corresponding set of individuals; Perform a crossover operation on the individual to be crossed and the crossover object.

9. The method for designing parameters of steel-concrete composite arch ribs based on multi-objective optimization according to claim 7, characterized in that, The process of generating the next generation population by using the elite individuals to guide mutation based on the results of the crossover operation includes: Based on the results of the crossover operation, individuals to be mutated are randomly selected, and the elite individuals are used to guide the mutation to perform the mutation operation on the individuals to be mutated. Based on the results of the mutation operation, an elite preservation strategy is used to generate the next generation population. The mutation operation satisfies the following formula: ， in, For the mutated individual The solution vector, For individuals The solution vector, This is the solution vector in the first non-dominated level. This is the learning rate.

10. A parameter design system for steel-concrete composite arch ribs based on multi-objective optimization, characterized in that, include: The system includes a processor, an input device, an output device, and a memory, all interconnected. The memory stores a computer program, which includes program instructions. The processor is configured to invoke the program instructions to execute the multi-objective optimization-based steel-concrete composite arch rib parameter design method as described in any one of claims 1 to 9.

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