NSGA-II multi-objective optimization method of truss structure by combining SDR with niche

Abstract

The application discloses a kind of NSGA-II multi-objective truss structure optimization methods of fusing SDR and habitat.The method comprises the following steps: according to the mathematical model of truss structure under model constraint is established using finite element analysis method according to different parameters of truss structure;Same preset external load is applied to the mathematical model of each truss structure, then the mass and flexibility of the truss structure in each mathematical model are optimized using the multi-objective genetic NSGA-II method of fusing SDR and habitat, and the final optimized design scheme is obtained to realize the multi-objective optimization of truss structure.The method of the application can be used for the truss structure with mass-flexibility multi-index coexistence and accompanied by displacement, stress and other engineering constraints, and can be used in the truss optimization scene with complex constraints and multi-index coexistence, considering diversity and convergence, reducing redundant evaluation overhead, obtaining more balanced and better convergent optimization solution set, and improving engineering usability and robustness.

Description

Technical Field

[0001] This invention relates to a truss structure optimization method, and to the field of engineering application technology at the intersection of intelligent optimization and computational mechanics. Specifically, it relates to an NSGA-II multi-objective truss structure optimization method that integrates SDR and microclimate. Background Technology

[0002] Truss structure optimization design is a comprehensive engineering problem that combines optimization theory with computer simulation. As engineering structures develop towards lightweight, long-life, and high-reliability designs, designers need to balance weight reduction with stiffness, strength, and dynamic performance, while also satisfying multiple constraints related to displacement and member buckling stress. The purpose of truss structure optimization research is to achieve optimal performance, stiffness, mass, and cost of the structure while meeting various standards and regulations.

[0003] Currently, multi-objective evolutionary algorithms have been widely used in structural optimization. However, in practical truss problems, the above methods still have the following shortcomings: First, when the objective dimensions differ significantly and the scale span is large, simply relying on Euclidean distance or crowding distance can easily lead to dimensional deviations, resulting in the repeated retention of solutions with similar directions and insufficient directional coverage; Second, when engineering constraints are strong and the feasible region is narrow, the standard feasibility-first strategy and fixed penalty function often lead to premature convergence of the population or fluctuations in the feasibility rate, making it difficult to stably reach the feasible frontier; Third, under complex loads or multi-objective coupling, the algorithm needs to continuously balance global diversity and local convergence, and traditional static parameter settings are difficult to take into account both early expansion and later convergence.

[0004] Meanwhile, in engineering practice, there are also challenges at the computational process level: each candidate solution requires a finite element solution to obtain nodal displacements, member stresses, and structural flexibility, which is computationally expensive; if the algorithm retains a large number of solutions with similar directions and redundant information in the environment selection stage, it will significantly increase the cost of subsequent evaluation and iteration, and also reduce the probability of obtaining a high-quality solution set with a limited number of iterations. Summary of the Invention

[0005] To address the problems existing in the background art, the present invention provides a multi-objective truss structure optimization method that integrates SDR and microhabitat.

[0006] The technical solution adopted in this invention is:

[0007] The present invention provides a multi-objective truss structure optimization method for NSGA-II that integrates SDR and microhabitat principles, comprising:

[0008] Step 1: Based on the truss structures with different parameters, use the finite element analysis method to establish a mathematical model of the truss structure under model constraints, and define the boundary and load conditions of the model to determine the model constraints.

[0009] Step 2: Apply the same preset external load to the mathematical model of each truss structure, and then use the multi-objective genetic algorithm NSGA-II (Nondominated Sorting Genetic Algorithm-II) that combines SDR and microenvironment to optimize the mass and flexibility of the truss structure in each mathematical model to obtain the final optimized design scheme.

[0010] Step 3: Design the truss structure based on the final optimized design scheme to achieve multi-objective optimization of the truss structure.

[0011] In step 1, the cross-sectional areas of the members are different, but the material density and length of the members are the same. The model constraints include preset nodal displacements and preset member stresses. The connection point between two members in the truss structure is used as a node.

[0012] Step 2 is described in detail below:

[0013] Step 2.1: Based on the M different target individuals of the truss structure and the adaptive angle threshold, obtain N microhabitats.

[0014] Step 2.2: Compare within and between each microhabitat, and rank and screen target individuals in each microhabitat according to the reinforcement dominance relation (SDR). At the same time, repair and compensate each microhabitat to finally obtain several elite solutions.

[0015] Step 2.3: Establish a parent pool based on each elite solution set, and then perform selection, crossover, and mutation to generate a child set.

[0016] Step 2.4: Repeat steps 2.1-2.3 for iterative solution of the child set until a final Pareto solution is obtained to determine the final optimized design scheme.

[0017] In step 2.1, the different target individuals are different initial design schemes for the truss structure. Each initial design scheme includes the material density and length of the truss members, the cross-sectional area of ​​the members, and the overall mass and flexibility of the truss structure. After normalizing and unitizing the mass and flexibility of the M target individuals, M direction vectors are obtained. For each direction vector, the smallest angle among the M-1 angles between the current direction vector and each of the other direction vectors is selected to obtain M minimum angles. The median of the M minimum angles is taken as the adaptive angle threshold. When dividing the niche, firstly, for each direction vector and its M-1 angles, After comparing the included angle with the adaptive angle threshold, one included angle and its direction vector that are less than or equal to the adaptive angle threshold are randomly selected. For the selected direction vector and its M-1 included angles, the comparison with the adaptive angle threshold is continued until there are no included angles less than or equal to the adaptive angle threshold. The target individuals corresponding to the direction vectors of all selected included angles are formed into a small habitat. Based on the last direction vector selected in the previous small habitat, the direction vector with the smallest included angle and its M-1 included angles are selected, and the next small habitat is divided until all direction vectors are divided, resulting in N small habitats.

[0018] In step 2.2, when comparing within each niche, for each niche and each target individual within it, the convergence index of each target individual is obtained and sorted from smallest to largest. Simultaneously, the shared fitness of each target individual is obtained as follows:

[0019] ;

[0020] ;

[0021] ;

[0022] in, and These are the final shared fitness and the initial shared fitness of the i-th target individual, respectively. For diversity weight, ; and These are the current iteration number and the maximum iteration number, respectively. and Let represent the minimum and maximum angular distances between the direction vector of the i-th target individual and the direction vectors of all other target individuals, respectively. To avoid the smallest first positive number that is divisible by zero; Let be the fitness of the i-th target individual; Let be the shared density factor for the i-th target individual; Let i be the set of all target individuals except the i-th target individual; The attenuation sensitivity coefficient, ; Let be the angle between the direction vectors of the i-th and j-th target individuals. The adaptive angle threshold is obtained from each current target individual.

[0023] Within a niche, each target individual is ranked according to its convergence index. When the number of target individuals within a niche is greater than M / N, the M / N target individuals with high shared fitness are selected for cross-niche comparison.

[0024] When comparing different niches, the convergence of each target individual is weighted and compared based on the M / N target individuals selected in each niche by introducing an angle ratio correction factor, as follows:

[0025] ;

[0026] in, and These are the convergence indices for the i-th and j-th target individuals, respectively.

[0027] Then at this time, the j-th target individual Better than the i-th target individual The elite solution set and its order SDR are obtained by sorting each target individual in sequence. niche .

[0028] In step 2.2, when performing the overall sorting of each niche, for each niche, if the number of target individuals in the niche is greater than M / N, then the selected M / N target individuals are used as non-dominant individuals within the niche for cross-niche comparison. Niches with fewer than M / N target individuals are repaired to have M / N target individuals, obtaining the unit direction vector for each niche. For each niche to be repaired, several target individuals ranked higher are copied from the niches with the nearest unit direction vector to the current niche to be repaired until M / N target individuals are reached, thus obtaining a total of more than M target individuals. Then, a second comparison and sorting is performed across niches using enhanced dominance judgment and secondary crowding screening. Finally, M target individuals are retained under the condition of uniform distribution and minimal redundancy. Specifically, the dominance relationship between the M / N target individuals being compared in each niche is determined, and the individuals are sorted according to the dominance relationship, as follows:

[0029] When the i-th non-dominant individual Dominates the j-th non-dominated individual At that time, the specific details are as follows:

[0030] ;

[0031] ;

[0032] ;

[0033] ;

[0034] in, This indicates the overall score in strengthening the dominant relationship; This represents the dominance level of a non-dominant individual x within its territory, where x is the i-th non-dominant individual. or the j-th non-dominated individual , Scaling factor denoted as the average directional sparsity of the non-dominated individual x; Crowding score for non-dominant individual x; As a moderating factor, it controls the degree of influence of directional differences in the scoring; , and These represent the maximum angle, minimum angle, and average angle between the non-dominant individual x and all non-dominant individuals x within the current microhabitat, respectively. To prevent the smallest second positive number with a denominator of zero; and The directional crowding degree of each non-dominated individual x is represented by the following. The maximum and minimum values; The number of non-dominant individuals x within the niche; For non-dominated individual x and the j-th non-dominated individual The angle between them.

[0035] In step 2.2, boundary elite compensation is performed on each niche, and for each target individual in the niche, the extreme value solution of the objective function of the m-th objective of the target individual is obtained. The objective function includes quality and flexibility. The extreme solutions of each non-dominated individual are used to construct a candidate solution set for the boundary, and the target individual with the most crowded niche is selected. When the most crowded target individuals Average directional sparsity Greater than or equal to the average directional sparsity of each non-dominated individual x The minimum value should remain unchanged; otherwise, the most crowded target individual will be... The boundary replacement is applied as follows:

[0036]

[0037] Where P represents the restored microhabitat.

[0038] In step 2.3, during the iterative solution process, a parent population is generated using the elite solution set obtained from the previous iteration through a global binary tournament selection. Then, the elite solution set obtained in the current iteration and the previous parent population are combined to form a temporary population. In the temporary population, two candidate solutions are drawn with equal probability. When the first candidate solution and the second candidate solution satisfy: , and They are from the p-th niche k p candidate solutions and the candidate solution of the q-th niche The shared fitness of the first candidate solution is... Winning sets are added to the parent pool; otherwise, the second candidate set is added. Add the solutions to the parent pool until the number of candidate solutions in the parent pool is M. Then, perform crossover and mutation on each candidate set in the parent pool to obtain the offspring set.

[0039] In step 2.4, during the iterative solution process, the final candidate Pareto solution is output as the final optimized design scheme until the preset number of iterations or the preset convergence condition (termination condition) is reached.

[0040] The present invention provides an NSGA-II multi-objective truss structure optimization system that integrates SDR and microhabitat principles, comprising:

[0041] The model building module uses the finite element analysis method to establish a mathematical model of the truss structure under model constraints, based on different parameters.

[0042] The design optimization module applies the same preset external load to the mathematical model of each truss structure, and then uses the multi-objective genetic NSGA-II method that integrates SDR and microenvironment to optimize the mass and flexibility of the truss structure in each mathematical model, so as to obtain the final optimized design scheme.

[0043] The structural design module designs the truss structure based on the final optimized design scheme, achieving multi-objective optimization of the truss structure.

[0044] The beneficial effects of this invention are:

[0045] This invention addresses truss structures with multiple quality and flexibility indices, accompanied by engineering constraints such as displacement and stress. Through a collaborative approach involving objective normalization and directionality processing, adaptive niche partitioning based on nearest-neighbor directional differences, shared density adjustment, internal and external sorting to strengthen dominance relationships, secondary crowding truncation, and boundary elite compensation, the algorithm balances diversity and convergence in complex truss optimization scenarios with multiple constraints and indices. This reduces redundant evaluation overhead and improves the engineering usability and stability of the solution set. Furthermore, this invention achieves a more balanced and convergent optimization solution set under displacement and buckling stress constraints in multi-objective trusses, enhancing engineering usability and robustness. Attached Figure Description

[0046] Figure 1 This is a flowchart of the method of the present invention;

[0047] Figure 2 This is a schematic diagram illustrating an example of 25 members in a truss according to an embodiment of the present invention;

[0048] Figure 3 This is a schematic diagram illustrating an example of 72 members in a truss according to an embodiment of the present invention;

[0049] Figure 4 This is a schematic diagram illustrating an example of a 120-bar truss in an embodiment of the present invention;

[0050] Figure 5 This is an optimized Pareto front diagram of the truss with 120 members in an embodiment of the present invention;

[0051] Figure 6 This is a comparison chart of the Inverted Generational Distance (IGD) index results for the optimized 120 members of the truss in this embodiment of the invention.

[0052] Figure 7 This is a diagram showing the optimized result of 25 members of the truss in an embodiment of the present invention;

[0053] Figure 8 This is a diagram showing the stress constraint results of the 72 members of the truss in an embodiment of the present invention. Detailed Implementation

[0054] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0055] like Figure 1 As shown, the NSGA-II multi-objective truss structure optimization method integrating SDR and niche is as follows:

[0056] This invention uses member cross-sections or grouping parameters as design variables. While satisfying the upper limits of nodal displacements and member tensile and compressive stresses, it also considers quality objectives, deformation, and flexibility, overcoming the shortcomings of traditional methods in terms of differences in target dimensions, narrow feasible regions, and uneven front distribution. The overall process of this invention is as follows: A mathematical model of the truss and a target-constraint system are established; boundary conditions and loads are applied, and finite element analysis is performed to obtain displacements and stresses; multiple target values ​​are normalized and directionalized, and niches are divided using the target direction difference as a metric; a shared function is calculated within each niche to obtain local congestion, and fitness is density-diluted; a strengthened dominance relationship is used to achieve ranking both within and across niches; when non-dominated individuals in a niche exceed the limit, a secondary congestion threshold is used to maintain directional uniformity; simultaneously, boundary elite compensation and adaptive parameter adjustment are introduced to maintain the representativeness of the front ends and key directions; through niche-aware tournament selection, crossover, and mutation iterations until the termination condition is met, a Pareto solution set is output.

[0057] This invention applies a multi-objective optimization method for truss structures to engineering examples with multiple constraints, such as... Figure 2 , Figure 3 and Figure 4 The diagram shows truss structures with 25, 72, and 120 members, along with the relationships between nodes and members in each typical structure. The numbers of each member and node are also labeled. Figure 3 In the model, h1=3.00m, h2=5.85m, h3=7.00m, d1=6.94m, d2=12.04m, and d3=15.89m. First, a mathematical model of the truss structure under model constraints is established using the finite element method based on the different parameters. The boundary and load conditions of the model are defined to determine the model constraints. For truss structures with different parameters, the cross-sectional area of ​​the members is different, while the material density and length of the members are the same. "Different" and "same" refer to different truss structures, not different members within the same truss structure. Geometric and physical property parameters such as material density, member cross-section, and length are determined when establishing the model. Model constraints include preset nodal displacements and preset member stresses. The connection point between two members in the truss structure is used as a node.

[0058] Then, the same preset external load was applied to the mathematical model of each truss structure. The mass and flexibility of the truss structure in each mathematical model were then optimized using the multi-objective genetic NSGA-II method that integrates SDR and niche approaches to obtain the final optimized design scheme, as follows:

[0059] First, N niches are obtained based on M different target individuals of the truss structure and an adaptive angle threshold. The different target individuals represent different initial design schemes for the truss structure. Each initial design scheme includes the material density and length of the truss members, their cross-sectional area, and the overall mass and flexibility of the truss structure. After normalizing and unitizing the mass and flexibility of the M target individuals, M direction vectors are obtained. These vectors are defined by using mass as the x-axis and flexibility as the y-axis, and then connecting M coordinate points to the origin based on the M target individuals. For each direction vector, the smallest angle among the M-1 angles between the current direction vector and all other direction vectors is selected, resulting in M ​​minimum angles. The median of these M minimum angles is then taken as the adaptive angle. Threshold: When dividing niches, firstly, for each direction vector and its M-1 included angles, the included angles are compared with an adaptive angle threshold. Then, one included angle and its direction vector that is less than or equal to the adaptive angle threshold are randomly selected. For the selected direction vector and its M-1 included angles, the comparison with the adaptive angle threshold continues until no included angle is less than or equal to the adaptive angle threshold. The target individuals corresponding to all the selected included angles constitute a niche. Based on the last selected direction vector in the previous niche, the direction vector with the smallest included angle and its M-1 included angles are selected, and the next niche is divided, until all direction vectors are divided, resulting in N niches. A graph connectivity method is used to divide directionally consistent niches according to the connectivity relationship of "angle ≤ threshold".

[0060] The design variables for the truss are the cross-sectional area parameters of each member. Its objective function as follows:

[0061]

[0062]

[0063]

[0064] in, They are respectively the 1st, 2nd, ..., i, ..., nth b Cross-sectional area of ​​each member; and These are the objective functions for quality and compliance, respectively. and These are the target quality and the target compliance, respectively. Let the material density of the i-th member be . Let be the length of the i-th member; This is the external load vector; This refers to the nodal displacement response under the current design. The total stiffness matrix of the truss structure is established by the finite element method, which is composed of the stiffness matrices of each element spliced ​​together according to the assembly rules.

[0065] The target normalization and unitization are performed as follows:

[0066]

[0067] in, and Let represent the direction vectors of the i-th target individual before and after normalization, respectively; This represents the minimum value of all objectives in the current population. This represents the maximum value of each objective in the current population. The included angle is also defined. as follows:

[0068]

[0069] Adaptive Angle Threshold Take the median of the nearest neighbor angle in the current population.

[0070] Niche segmentation uses target orientation difference as a unified metric, with the threshold adaptively determined based on the statistical measure of the orientation difference between current nearest neighbors. Compared to clustering methods based on Euclidean distance or fixed radius, this method is more robust to targets of different dimensions and scales, avoiding the problems of over-preservation of individuals with the same orientation and insufficient orientation coverage.

[0071] Then, comparisons are made within and between each niche, and target individuals within each niche are ranked and screened according to the Reinforcement Domination Relationship (SDR). Simultaneously, each niche is repaired and compensated, ultimately yielding several elite solutions. During the comparisons within each niche, for each niche and each target individual within it, the convergence index of each target individual is obtained and ranked from smallest to largest. The shared fitness of each target individual is also obtained as follows:

[0072] ;

[0073] ;

[0074] ;

[0075] in, and These are the final shared fitness and the initial shared fitness of the i-th target individual, respectively. For diversity weight, ; and These are the current iteration number and the maximum iteration number, respectively. and Let represent the minimum and maximum angular distances between the direction vector of the i-th target individual and the direction vectors of all other target individuals, respectively. To avoid the smallest first positive number that is divisible by zero; Let be the fitness of the i-th target individual; Let be the shared density factor for the i-th target individual; Let i be the set of all target individuals except the i-th target individual; The attenuation sensitivity coefficient, ; Let be the angle between the direction vectors of the i-th and j-th target individuals. The adaptive angle threshold is obtained from each current target individual.

[0076] Within a small region, a sharing function is calculated based on the directional differences between individuals to characterize the local crowding. This function is then used to dilute or reweight individual fitness to obtain the sharing density and adjust individual fitness, prioritizing solutions with sparser orientations and stronger representativeness when convergence is similar. A diversity weighting mechanism is introduced into the sharing fitness function.

[0077] Convergence index Specifically as follows:

[0078]

[0079] in, M1 is the normalized objective value of the m-th optimization objective in the i-th objective individual; M1 is the number of optimization objectives in the current problem, which is 2, and the optimization objectives include quality and flexibility.

[0080] Within a niche, each target individual is ranked according to its convergence index. When the number of target individuals within a niche is greater than M / N, the M / N target individuals with high shared fitness are selected for cross-niche comparison.

[0081] When comparing different niches, the convergence of each target individual is weighted and compared based on the M / N target individuals selected in each niche by introducing an angle ratio correction factor, as follows:

[0082] ;

[0083] in, and These are the convergence indices for the i-th and j-th target individuals, respectively.

[0084] Then at this time, the j-th target individual Better than the i-th target individual The elite solution set and its order SDR are obtained by sorting each target individual in sequence.niche .

[0085] This invention introduces an exponential shared fitness for local density dilution within each small habitat and performs a two-level ranking using convergence indices and reinforcement dominance relationships.

[0086] When ranking each niche as a whole, for each niche, if the number of target individuals in the niche is greater than M / N, then the selected M / N target individuals are used as non-dominant individuals within the niche for cross-niche comparison. Niches with fewer than M / N target individuals are repaired to have M / N target individuals, obtaining the unit direction vector for each niche. For each niche to be repaired, several target individuals ranked higher are copied from the niche with the nearest unit direction vector to the current niche to be repaired until M / N target individuals are reached, thus obtaining a total of more than M target individuals. Then, a second comparison and ranking across niches is performed using reinforced dominance judgment and secondary crowding screening. Finally, M target individuals are retained under the condition of uniform distribution and minimal redundancy. Specifically, the dominance relationship between the M / N target individuals compared in each niche is determined, and the ranking is performed according to the dominance relationship, as follows:

[0087] When the i-th non-dominant individual Dominates the j-th non-dominated individual At that time, the specific details are as follows:

[0088] ;

[0089] ;

[0090] ;

[0091] ;

[0092] in, This indicates the overall score in strengthening the dominant relationship; This represents the dominance level of a non-dominant individual x within its territory, where x is the i-th non-dominant individual. or the j-th non-dominated individual , Scaling factor denoted as the average directional sparsity of the non-dominated individual x; Crowding score for non-dominant individual x; As a moderating factor, it controls the degree of influence of directional differences in the scoring; , and These represent the maximum angle, minimum angle, and average angle between the non-dominant individual x and all non-dominant individuals x within the current microhabitat, respectively. To prevent the smallest second positive number with a denominator of zero; and The directional crowding degree of each non-dominated individual x is represented by the following. The maximum and minimum values; The number of non-dominant individuals x within the niche; For non-dominated individual x and the j-th non-dominated individual The angle between them.

[0093] Shared density adjustment is used to suppress local crowding: within the same small habitat, the shared value is calculated based on the directional difference between an individual and its neighborhood and accumulated as the shared density; the algorithm dilutes or reweights the shared density during sorting or fitness evaluation, so that individuals with sparser orientations and stronger representativeness are retained first when convergence is similar. The influence radius and shape parameters of the shared function can be set fixedly or slightly adaptively adjusted with the number of iterations to achieve the strategy of "early redistribution and later convergence".

[0094] Strengthening the dominance relationship is used to unify the comparison rules between intra- and inter-niche areas. Within an intra-niche area, the convergence index is prioritized and the feasibility-first principle is followed. When comparing across niche areas, a correction factor related to the direction difference is introduced to weight the convergence, avoiding bias between different directional clusters, thereby maintaining the directional equilibrium of the frontier while ensuring convergence.

[0095] When the number of non-dominant individuals in a niche exceeds the limit, this invention employs a secondary crowding degree based on the included angle to retain representative solutions with sparse orientations. The secondary crowding degree, centered on the orientation difference, is calculated and truncated accordingly. Representative solutions with more uniform orientation distribution and less redundancy are preferentially retained. The secondary crowding degree is characterized by the sum of orientation differences between an individual and other individuals in the same cluster, and representative solutions are retained based on sparsity priority. This ensures that the two ends of the global front are not excessively deleted to maintain the integrity and balanced coverage of the Pareto front. An elite compensation strategy is used for feasibility repair. To avoid excessive deletion at both ends of the front, the algorithm performs boundary elite checks after truncation and performs boundary compensation using soft replacement when necessary.

[0096] After sorting, truncation is performed when selecting parent individuals. Only when there are more than M target individuals will the sorting be performed. The last few individuals are removed and the top M individuals are combined with the previous generation's parent population (M individuals) to form a temporary population of 2M individuals. The population is then sorted by fitness and crowding. The top M individuals are selected as the parents of this generation for crossover and mutation selection.

[0097] Then, boundary elite compensation is performed on each niche, and for each target individual in the niche, the extreme solution of the objective function of the m-th objective of the target individual is obtained. The objective function includes quality and flexibility. The extreme solutions of each non-dominated individual are used to construct a candidate solution set for the boundary, and the target individual with the most crowded niche is selected. When the most crowded target individuals Average directional sparsity Greater than or equal to the average directional sparsity of each non-dominated individual x The minimum value should remain unchanged; otherwise, the most crowded target individual will be... The boundary replacement is applied as follows:

[0098]

[0099]

[0100] Where P represents the restored microhabitat.

[0101] Boundary elite compensation uses the extreme individuals of each objective dimension as candidates, and the minimum value of each objective as the candidate. The corresponding individual is considered a candidate for the boundary elite; if they do not enter the final elite set... The replacement is determined in the elite solution set. The method uses a combination of directional similarity and sparsity to perform soft replacements on crowded and directionally redundant individuals to ensure the integrity of the solution set boundary. Boundary elite compensation is used to restore and maintain the representativeness of the Pareto front on each target.

[0102] The parameter settings of this invention are engineering-operable: the niche threshold is determined by the median or quantile of the nearest neighbor direction difference and allows the setting of scaling factors; the influence radius and shape parameters of the shared function can be initialized according to empirical values; the tradeoff coefficients in the reinforcement domination can be adjusted algebraically linearly or piecewise; when the feasibility is low or fluctuates greatly, dynamic penalty or simple repair rules can be enabled to improve the feasibility and convergence stability.

[0103] Then, a parent pool is established based on each elite solution set, and selection, crossover, and mutation are performed to generate the offspring set. During the iterative solution process, a global binary tournament selection is used to generate the parent population from the elite solution set obtained in the previous iteration. Then, the elite solution set obtained in the current iteration and the previous parent population are combined to form a temporary population. In the temporary population, two candidate solutions are drawn with equal probability. When the first candidate solution and the second candidate solution satisfy: , and They are from the p-th niche kp candidate solutions and the candidate solution of the q-th niche The shared fitness of the first candidate solution is... Winning sets are added to the parent pool; otherwise, the second candidate set is added. The solutions are added to the parent pool until the number of candidate solutions in the parent pool reaches M. Then, crossover and mutation are performed on each candidate set in the parent pool to obtain the offspring set. In the parent selection and elite retention stages, this invention uses a niche-aware tournament and boundary compensation strategy to ensure the directional coverage and boundary integrity of the solution set.

[0104] Then, the child set is iteratively solved by repeating steps 2.1-2.3. During the iterative solution, the preset number of iterations or the preset convergence condition (termination condition) is reached, and the final candidate Pareto solution is output as the final optimized design scheme. Finally, the truss structure is designed according to the final optimized design scheme to achieve multi-objective optimization of the truss structure.

[0105] In practical implementation, this invention takes a 25-bar space truss as an example, where the mass density of all materials used in the 25-bar reference truss structure is 0.1 lb / in. 3 The elastic modulus is 10000 ksi. The 25-bar truss is symmetrical about the X and Y axes, and it is divided into 8 design variables. The members are also subject to stress constraints. The minimum and maximum allowable values ​​for the cross-section members of the 25-bar truss structure are set to 0.01 in. 2 and 3.4in 2 The free nodes of a 25-bar truss are allowed displacements of ±0.35 inches in the X, Y, and Z directions. The problem considers 124 nonlinear design constraints. Fixed constraints are applied at nodes 7, 8, 9, and 10, and external loads are applied at nodes 1 and 2 in the 25-bar space truss; eight sets of cross-sectional areas are used as design variables; the objectives include mass and flexibility, and the constraints are upper limits on multi-node displacements and member stress limits. Using the method of this invention, a more balanced solution set coverage is obtained under the same computational budget, and representative solutions are maintained in the critical directions. The final optimization result is 544.041 lbs. Figure 7 As shown, this allows engineers to make informed choices about which solutions to adopt.

[0106] This invention takes a 72-bar space truss as an example. The optimization problem of the 72-bar truss is grouped into 16 area variables, and the material density is 0.1 lb / in. 3The elastic modulus of the material used in the 72-bar truss is 10000 ksi. The members of the 72-bar truss are constrained by a stress of 25 ksi in both tension and compression. The nodal displacements of nodes 17, 18, 19, and 20 are constrained to ±0.25 inches in the X and Y directions. The allowable stress magnitude for all members in the 72-bar truss structure is ±25 ksi. The optimization problem consists of 320 nonlinear design constraints. Using the method of this invention, a more balanced solution set coverage is obtained under the same computational budget, and representative solutions are maintained in the critical directions. The final optimization result is 379.549 psi. Figure 8 As shown, the actual stress distribution of the unit members of the 72-bar truss of the present invention is within the upper and lower limits of the allowable stress constraint.

[0107] This invention takes a 120-bar dome truss with grouped design as an example. In the 120-bar spatial dome truss, fixed loads are applied at nodes 38-49, and external loads are applied at the remaining free nodes. The geometric representation of the 120-bar dome truss is as follows: Figure 3 As shown, the structure is symmetrical about the X and Y axes. The 120 member elements are grouped, and the truss elements are divided into 7 groups through connection nodes. Seven groups of cross-sectional dimensions are independently set as optimization parameters for each group. For stress constraints, the maximum stress constraint... Take 400 MPa, the density of the material 0.288 lb / in 3 The elastic modulus is 30450 kSi. The 120 member elements are divided into several groups according to symmetry and stress characteristics, with the group cross-section as the design variable; the objectives include mass and flexibility, and the constraints are upper limits for multi-node displacement and member stress limits. Using the method of this invention, a more balanced solution set coverage is obtained under the same computational budget, and representative solutions are maintained in key directions, facilitating engineers in selecting appropriate solutions. Figure 5 As shown, the Pareto front distribution of a 120-bar spatial dome truss after optimization using the NSGA-II / SDR-niche non-dominated sorting genetic algorithm of this invention, which employs niche and reinforced dominance collaboration. Figure 6 As shown, a comparison of the reverse generation distance results for a 120-bar truss using the multi-objective heat transfer search algorithm, the non-dominated sorting gray wolf optimization algorithm, the multi-objective water circulation algorithm, the multi-objective grasshopper optimization algorithm, and the niche and enhanced dominant cooperative non-dominated sorting genetic algorithm NSGA-II / SDR-niche of the present invention shows that the present invention has the best effect.

[0108] This invention can also be used under different load conditions, different target combinations, or different constraint weights; it does not limit the selection of crossover and mutation operators and parameters, and common simulated binary crossover and polynomial mutation can be used; it does not strictly limit the normalization method of the target, feasibility judgment and repair strategy, as long as the engineering rationality and algorithm consistency are met, they all fall within the protection scope of this invention.

[0109] This invention also designs a multi-objective truss structure optimization system integrating SDR and niche genetics (NSGA-II). The system includes a model building module, a design optimization module, and a structural design module. The model building module uses the finite element analysis method to establish a mathematical model of the truss structure under model constraints based on the truss structure with different parameters. The design optimization module applies the same preset external load to the mathematical model of each truss structure, and then uses the multi-objective genetic NSGA-II method integrating SDR and niche genetics to optimize the mass and flexibility of the truss structure in each mathematical model to obtain the final optimized design scheme. The structural design module designs the truss structure according to the final optimized design scheme to achieve multi-objective optimization of the truss structure.

[0110] This invention can be deployed as software on computer devices or integrated with structural analysis programs to automate data preprocessing, niche partitioning, shared density calculation, reinforcement dominance sorting, secondary crowding truncation, boundary compensation, and evolutionary operations. For scenarios requiring batch optimization, parallel finite element solutions and caching mechanisms can reduce the computation time per generation.

[0111] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of this application, and not to limit them. Although this application 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 of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of this application.

Claims

1. A method for NSGA-II multi-objective optimization of truss structure by fusing SDR with niche, characterized in that, The application relates to a method for multi-objective optimization of a truss structure. The method comprises the following steps: Step 1: a mathematical model of the truss structure under model constraints is established by using a finite element analysis method according to different parameters of the truss structure; Step 2: the same preset external load is applied to the mathematical model of each truss structure, then the mass and flexibility of the truss structure in each mathematical model are optimized by using a multi-objective genetic NSGA-II method combining SDR and niches, and finally an optimized design scheme is obtained; Step 3: the truss structure is designed according to the final optimized design scheme, and the multi-objective optimization of the truss structure is realized. The step 2 specifically comprises the following steps: Step 2.1: N niches are obtained by dividing according to M different target individuals of the truss structure and an adaptive angle threshold value; Step 2.2: comparison is carried out in each niche and between niches, each target individual in each niche is sorted and screened according to a reinforced dominance relation SDR, each niche is repaired and compensated, and finally a plurality of elite solution sets are obtained; Step 2.3: a parent pool is established according to each elite solution set, and a child set is generated by selection, crossover and mutation; Step 2.4: the child set is repeatedly subjected to step 2.1-2.3 for iterative solution until a final Pareto solution is obtained to determine the final optimized design scheme; ； ； ； wherein, and are the final shared fitness and the initial shared fitness of the ith target individual, respectively; is a diversity weight; and are the current iteration number and the maximum iteration number, respectively; and denote the minimum and maximum angular distance between the direction vector of the ith target individual and the direction vectors of the other target individuals, respectively; is a first positive number; is the fitness of the ith target individual; is the sharing density factor of the ith target individual; is the set of target individuals other than the ith target individual; is a decay sensitivity coefficient; is the angle between the direction vectors of the ith and jth target individuals, is an adaptive angle threshold value; In step 2.2, when comparison is carried out in each niche, the convergence index of each target individual is obtained and sorted from small to large for each niche and each target individual in the niche, and the shared fitness of each target individual is obtained as follows: When the number of target individuals in the niche is greater than M / N, the M / N target individuals with high shared fitness are selected for comparison between niches; ； wherein, and are the convergence indicators for the i-th and j-th target individual, respectively. then the jthtarget individual at this time is superior to the ithtarget individual , and the elite solution set and its ranking order SDR are obtained by sequentially ranking each target individual niche .

2. The NSGA-II multi-objective truss structure optimization method fusing SDR with niche according to claim 1, characterized in that: When comparison is carried out between niches, the convergence of each target individual is weighted and compared by introducing an angle proportion correction factor according to the M / N target individuals selected from each niche, as follows: In step 1, the cross-sectional areas of the rod members are different between the truss structures with different parameters, and the material density and length of the rod members are the same; the model constraints include preset node displacement and preset rod stress, and the connecting points between two rod members in the truss structure are regarded as nodes.

3. The NSGA-II multi-objective truss structure optimization method fusing SDR with niche according to claim 1, characterized in that: In step 2.1, different target individuals are different initial design schemes of the truss structure, and each initial design scheme includes the material density and length of the rod of the truss structure, the cross-sectional area of the rod, the mass and flexibility of the whole truss structure; the mass and flexibility in the M target individuals are normalized and unitized to obtain M direction vectors; for each direction vector, the minimum angle among M-1 angles between the current direction vector and each of the other direction vectors is selected to obtain M minimum angles, and the median of the M minimum angles is taken as an adaptive angle threshold; when the niches are divided, firstly, for each direction vector and its M-1 angles, the angles are compared with the adaptive angle threshold, and then one angle less than or equal to the adaptive angle threshold and its direction vector are randomly selected; for the selected direction vector and its M-1 angles, the comparison with the adaptive angle threshold is continued until there is no angle less than or equal to the adaptive angle threshold; the target individuals of the direction vectors corresponding to all the selected angles form a niche, and according to the last selected direction vector in the previous niche, one direction vector with the minimum angle between the last selected direction vector and the direction vector and its M-1 angles are selected, and the division of the next niche is continued until all the direction vectors are divided, and N niches are obtained.

4. The NSGA-II multi-objective truss structure optimization method fusing SDR with habitat according to claim 3, characterized in that: In step 2.2, when the overall sorting of each niche is performed, for each niche, if the number of target individuals in the niche is greater than M / N, the selected M / N target individuals are taken as non-dominated individuals in the niche for comparison across niches, and the niches with the number of target individuals less than M / N are repaired to M / N target individuals, the unit direction vectors of each niche are obtained, for each niche to be repaired, a number of target individuals in front of the order are copied from the niche most adjacent to the unit direction vector to the current niche to be repaired until M / N target individuals are reached, so that the overall number of target individuals is greater than M, and then the comparison and sorting across niches are further performed by using the strengthened dominance judgment and the secondary crowding degree screening, specifically, the dominance relationship between the M / N target individuals in each niche is judged, and the sorting is performed according to the dominance relationship, as follows: When the ith non-dominated individual dominates the jth non-dominated individual is as follows: ； ； ； ； wherein, denotes the dominance relation integrated score; denotes the dominance rank of non-dominated individual x within its niche, non-dominated individual x being the ith non-dominated individual or the jth non-dominated individual , is a scaling factor, is the average directional sparsity of non-dominated individual x; is the crowding score of non-dominated individual x; is a tuning factor; , and are respectively the maximum, minimum and average of the angle between non-dominated individual x and all non-dominated individuals x within the current niche; is a second positive number; and are respectively the maximum and minimum of the directional crowding of each non-dominated individual x; is the number of non-dominated individuals x within the niche; is the angle between non-dominated individual x and the jth non-dominated individual .

5. The NSGA-II multi-objective truss structure optimization method fusing SDR with habitat according to claim 3, characterized in that: In step 2.2, boundary elite compensation is performed on each niche, and for each target individual in the niche, the extreme value solution of the objective function of the m-th objective of the target individual is obtained. The objective function includes quality and flexibility. The extreme solutions of each non-dominated individual are used to construct a candidate solution set for the boundary, and the target individual with the most crowded niche is selected. When the most crowded target individuals Average directional sparsity Greater than or equal to the average directional sparsity of each non-dominated individual x The minimum value should remain unchanged; otherwise, the most crowded target individual will be... The boundary replacement is applied as follows: ； Wherein, P is the repaired niche.

6. The NSGA-II multi-objective truss structure optimization method fusing SDR with habitat according to claim 1, characterized in that: In step 2.3, during the iterative solution process, a parent population is generated using the elite solution set obtained from the previous iteration through a global binary tournament selection. Then, the elite solution set obtained in the current iteration and the previous parent population are combined to form a temporary population. In the temporary population, two candidate solutions are drawn with equal probability. When the first candidate solution and the second candidate solution satisfy: , and They are from the p-th niche k p candidate solutions and the candidate solution of the q-th niche The shared fitness of the first candidate solution is... Winning sets are added to the parent pool; otherwise, the second candidate set is added. Add the solutions to the parent pool until the number of candidate solutions in the parent pool is M. Then, perform crossover and mutation on each candidate set in the parent pool to obtain the offspring set.

7. The NSGA-II multi-objective truss structure optimization method fusing SDR with habitat according to claim 1, characterized in that: In step 2.4, when the iterative solution is performed, until the preset iteration number is reached or the preset convergence condition is reached, a final candidate Pareto solution is output as a final optimized design scheme.

8. An optimization system for implementing the NSGA-II multi-objective truss structure optimization method of fusing SDR and niche as claimed in any one of claims 1 to 7, characterized in that, It comprises: a model construction module, which establishes a mathematical model of the truss structure under model constraints by using a finite element analysis method according to different parameters of the truss structure; a design optimization module, which applies the same preset external load to each mathematical model of the truss structure, and then optimizes the mass and flexibility of the truss structure in each mathematical model by using a multi-objective genetic NSGA-II method combining SDR and niches to obtain a final optimized design scheme; The structural design module designs the truss structure according to the final optimized design scheme, and realizes multi-objective optimization of the truss structure.

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