An integrated progressive structural optimization method for truss topology, shape, and size.

Abstract

This invention relates to an integrated progressive structural optimization method for truss topology, shape, and dimensions, belonging to the field of structural engineering optimization technology. The invention first initializes the model, setting geometry, loads, boundary conditions, and design variables, and performs initial finite element analysis to obtain strain energy and volume baseline values. Then, an iterative loop is entered: the sensitivity of design variables is calculated, and variables are adjusted synchronously to maintain structural feasibility; after updating, finite element analysis is performed to re-evaluate strain energy and volume; convergence is judged by stress uniformity and volume stability. If convergence is not achieved, the above steps are repeated until the maximum number of iterations is reached or the convergence condition is met. Finally, the optimization results are output, including node coordinates, element properties, and related performance indicators. The entire process progressively optimizes structural performance through iterative iteration, achieving efficient truss design.

Description

Technical Field

[0001] This invention belongs to the field of structural engineering optimization technology, and in particular relates to an integrated progressive structural optimization method for simultaneous optimization of topology, shape and size of truss structures. Background Technology

[0002] Truss structures, due to their simplicity, cost-effectiveness, and ease of construction, are widely used in civil engineering, mechanical engineering, and aerospace engineering. Correspondingly, the optimal design of truss structures has become an important research area in structural engineering. Traditional optimization methods are generally divided into three independent categories based on different design variables: size optimization, shape optimization, and topology optimization. Size optimization, based on a pre-defined topology and shape, uses mathematical programming methods to determine the optimal cross-sectional area of ​​each component. Shape optimization, under a fixed topology, adjusts node coordinates through gradient-based optimization methods, heuristics, or evolutionary algorithms to improve load distribution. Topology optimization aims to find the optimal material layout and often uses evolutionary algorithms such as genetic algorithms and particle swarm optimization to solve this combined optimization problem. In numerical implementation, the base structure method is one of the mainstream methods for handling truss topology optimization. This method pre-generates a dense network of members consisting of nodes and all possible connections within the design domain, transforming the topology selection problem into a continuous optimization problem of member cross-sectional dimensions or existence.

[0003] However, traditional methods have obvious limitations. First, the basis structure method has two significant drawbacks: (1) The quality of the optimization results is highly dependent on the number and distribution of the initial nodes, and it is difficult to generate high-quality initial meshes in irregular design domains.

[0004] (2) Optimization results often include a large number of slender members and complex nodes. Although theoretically efficient, they are costly to manufacture and have poor engineering practicality. Secondly, in the field of truss structure optimization, traditional methods usually separate the optimization problems of topology, shape, and size. This separate optimization strategy cannot fully consider the inherent coupling relationship between various design variables, resulting in the material efficiency and structural performance of the final design scheme not being fully explored. In order to overcome the limitations of separate optimization methods, the industry has begun to study integrated optimization methods that simultaneously optimize topology, shape, and size. However, existing integrated methods, especially non-gradient optimization algorithms, such as those using genetic algorithms and particle swarm optimization, are essentially unguided random searches. When dealing with high-dimensional design variables, they require massive finite element analysis, resulting in extremely high computational costs and making them difficult to apply to complex structures in actual engineering.

[0005] To address the aforementioned problems, this invention proposes an integrated progressive structural optimization method for truss topology, shape, and size. Summary of the Invention

[0006] The purpose of this invention is to propose an integrated progressive structural optimization method for truss topology, shape, and size to solve the following key technical problems: (1) Solving the problem of suboptimal performance of traditional separate optimization strategies: Existing methods treat topology, shape and size optimization separately, which cannot fully utilize the synergistic coupling effect between them. This invention aims to provide an integrated optimization framework to achieve synchronous synergistic optimization of the three types of design variables, thereby fundamentally discovering design schemes with better material efficiency and structural performance.

[0007] (2) Resolving the contradiction between "computational efficiency" and "convergence stability" in existing ensemble optimization methods: Non-gradient algorithms (such as genetic algorithms) have high computational costs due to global search; while existing gradient-based methods are prone to getting trapped in local optima and the optimization process is unstable. This invention aims to provide an efficient and stable gradient-based optimization scheme. Through sensitivity analysis, adaptive momentum gradient descent, and dynamic update strategies, it enhances the stability of the convergence process while ensuring computational efficiency and strives to find better local solutions for practical engineering applications.

[0008] (3) Solving the problem of poor engineering practicality of optimization results: Traditional topology optimization results often contain a large number of slender members, which are difficult to manufacture. This invention aims to systematically control the structural complexity during the optimization process by introducing a dynamic stress and truss member area penalty mechanism, automatically eliminating inefficient members, and guiding the final design towards a form with simple structure, clear nodes, and easy manufacturing.

[0009] To achieve the above objectives, the present invention adopts the following technical solution: An integrated progressive structural optimization method for truss topology, shape, and dimensions includes the following: S1. Initialization: Construct the truss optimization model, define the geometry, loads, boundary conditions and design variables of the initial truss base structure; set the node boundaries and initialize the design variables; S2. Initial Finite Element Analysis: Perform initial finite element analysis to calculate the total strain energy and total volume, which will serve as the baseline values ​​for subsequent iterations. S3. Sensitivity Calculation: Evaluate the objective function of the truss optimization model and calculate the sensitivity of the design variables; S4. Update design variables: Synchronously coordinate and adjust design variables to maintain the feasibility and stability of the structure; S5. Finite element analysis: Based on the updated design variables, perform finite element analysis to obtain nodal displacements and element stresses, and re-evaluate the total strain energy and volume change of the structure. S6. Convergence Assessment: Based on the preset number of iterations, the convergence is assessed through the stress uniformity and overall volume stability of the truss structure; if the convergence condition is not met, the next iteration begins. S7. Iteration loop: Repeat S3~S6 until the maximum number of iterations is reached or the specified convergence condition is met. S8. Output optimization results: After convergence, save the final optimized structure, including the updated node coordinates, element attributes and related indicators.

[0010] Preferably, the optimization objective of the truss optimization model is to minimize the overall strain energy of the truss structure to maximize the structural stiffness, and the mathematical expression of the objective function of the model is:

[0011] In the formula, The vector represents the total strain energy; f is the external load vector; the superscript T indicates the transpose operation; u is the global displacement vector generated by the applied load.

[0012] Preferably, the design variables include node coordinate variables and truss member cross-sectional area variables, specifically including: ①Spatial coordinates of truss nodes Represented as ,in They are nodes i The coordinates; ② Cross-sectional area of ​​the rod , recorded as ,in It is the first e The cross-sectional area of ​​the root member is expressed mathematically as follows:

[0013] In the formula, node coordinate variables are used for shape optimization; bar cross-sectional area variables are used for size and topology optimization; when Approaching When the contribution to structural stiffness is negligible, it is removed from the topology.

[0014] Preferably, the constraints of the truss optimization model are: ① Equilibrium equations: The structure satisfies static equilibrium under applied external loads:

[0015] In the formula, This is the global stiffness matrix; ② Design variable constraints: The coordinates and cross-sectional areas of truss members are limited to a specified range to ensure design feasibility; Let the initial coordinates of the truss nodes be:

[0016] Truss node spatial coordinates The following boundary conditions must be met:

[0017] In the formula, and These represent the lower and upper bounds of the coordinates of each node, respectively. The displacement of fixed nodes is set to zero, while the displacement of movable nodes is determined by an optimization algorithm. For the cross-sectional area of ​​truss members, the following boundary conditions apply:

[0018] In the formula, and These represent the minimum and maximum allowable cross-sectional areas, respectively. ③ Stress constraint: Each truss member is kept within the allowable tensile and compressive stress range to prevent yielding; the mathematical expression for stress constraint is:

[0019] In the formula, and These represent the allowable compressive stress and tensile stress of the material, respectively.

[0020] Preferably, S3 specifically includes the following: ① Calculate the sensitivity of the objective function to the nodal coordinates: For connection nodes i and j unit e Its length is:

[0021] right about and Differentiate:

[0022] Similarly, for , and , Use the same operation; unit e The strain under axial load is defined as:

[0023] In the formula, Representation unit e The initial length; For connection nodes i and j rod unit e strain energy about x, y, z The sensitivity expression for coordinates is:

[0024]

[0025] In the formula, It represents the elastic modulus.

[0026] strain energy about ， ， Coordinate sensitivity includes node i and j The contribution of all connected link elements; therefore, the sensitivity of the objective function to nodal coordinates. The expression is:

[0027] In the formula, and Representing the nodes respectively i and j A collection of connected rods; Calculate the sensitivity of the objective function to the cross-sectional area of ​​the member: To calculate the sensitivity of strain energy to cross-sectional area, for about Differentiating, we get:

[0028] Ignoring the effect of rod length on sensitivity, the expression for the sensitivity of the objective function to the cross-sectional area of ​​the rod is:

[0029] In the formula, This represents the sensitivity of the objective function to the cross-sectional area of ​​the rod.

[0030] Preferably, S4 specifically includes the following: Before updating the node coordinates, the original sensitivity is smoothed to minimize numerical instability. node i Smoothing sensitivity The expression is as follows:

[0031] In the formula, Indicates neighboring nodes j Scaling sensitivity, Based on nodes i and j Weighting factor of the distance between them It is the smooth radius The total number of internal nodes; where the weighting factor is... Defined as:

[0032] In the formula, It is a node i With nodes j The distance between them; This indicates the radius that controls the range of the smoothing effect.

[0033] The momentum method is used to update node coordinates to accelerate convergence and smooth the optimization path; the mathematical expressions for the momentum term and coordinate update are as follows:

[0034] In the formula, Indicates the current iteration step The momentum term at time; The momentum coefficient; This indicates the sensitivity after smoothing in the previous iteration; Before updating the cross-sectional area, the node coordinates are calibrated and updated using an adaptive mechanism based on the strain energy changes during the initial stage of the process. Current iteration step The expression for updating node coordinates at that time is:

[0035] In the formula, The updated node coordinates; It is the Heaviside step function; Adaptive learning rate; This is the amplification factor used to adjust the update amplitude; where the Heaviside step function is... The update direction is determined before applying changes to node coordinates, and it is defined as follows:

[0036] In the formula, This represents the change in total strain energy, which is calculated by tentatively updating the nodal coordinates based on initial sensitivity. If the energy change... Then the Heaviside step function will reverse its update direction, causing the algorithm to adaptively move towards the direction of minimizing strain energy in the next iteration; To further enhance convergence, the adaptive learning rate decays exponentially over time:

[0037] In the formula, The initial learning rate; The attenuation rate; This represents the current iteration step number; To ensure the stability of the numerical calculation, the displacement magnitude of the node coordinates is adjusted during the iteration process. Restrictions will be imposed:

[0038] like Exceeding the specified maximum displacement Then scale the update value proportionally to ensure The details are as follows:

[0039] To maintain physical feasibility, node coordinates are restricted to predefined boundaries. If the updated variable exceeds this range, it will be adjusted using soft constraints, the expression of which is:

[0040] In the formula, Used to control the steepness of the penalty term to ensure smooth and gradual adjustment near the critical value; To ensure controllable updates of design variables, each component of the node coordinate vector is constrained by the maximum displacement in each iteration. The displacement limit is proportional to the range of node coordinates.

[0041] In the formula, For a small fixed ratio; The updated node coordinates satisfy:

[0042] And it is further restricted within a pre-defined physical boundary:

[0043] The cross-sectional area of ​​truss members is updated based on the full stress criterion update mechanism, with the ideal full stress state as the target, and a sensitivity threshold is set:

[0044] When the rod unit e Sensitivity When the cross-sectional area of ​​the corresponding members is reduced, the material utilization rate is improved; conversely, when... In such cases, the cross-sectional area is increased to maintain structural safety; Dynamically adjust the step size to minimize unnecessary iterations while maintaining a smooth and stable optimization process. No. e Adaptive update factor for each truss member element The definition is as follows:

[0045] In the formula, For attenuation parameters; Number of iterations; exponential decay term Used to ensure It has always shown a decreasing trend; Attenuation parameter By convergence metric Determine by the following formula:

[0046] In the formula, Represents a convergence metric, used to measure how close a solution is to the optimal solution; when When the decay rate is more aggressive, when At that time, the decay rate slows down; The cross-sectional area of ​​truss members is updated according to the following rules: For members with increased cross-sectional area:

[0047] For members with reduced cross-sectional area:

[0048] In the formula, Indicates the first The truss unit in the first The updated cross-sectional area at the next iteration; and This represents the area of ​​the previous iteration; A dynamic penalty mechanism is introduced to actively suppress and eliminate thin, inefficient links. This dynamic penalty mechanism is applied at a certain number of iterations. Activated at time; Calculate the area penalty threshold for dynamic growth First, in the current iteration step, the maximum cross-sectional area of ​​all members is defined as the benchmark for applying the penalty:

[0049] The dynamic penalty threshold is defined as:

[0050] Simultaneously, a dynamic stress threshold is introduced, defined as:

[0051] For areas below the threshold And the absolute stress is less than the threshold. The members are penalized, and the penalty mechanism is based on the cross-sectional area of ​​the member and the median area of ​​all members. Implementation of the sigmoid function for the ratio:

[0052] The penalty value calculated using the above formula always falls between 0 and 1. Therefore, if both conditions are met... and The area of ​​the corresponding member is then updated as follows:

[0053] The above operations achieve topology simplification.

[0054] Preferably, S6 specifically includes the following: evaluating convergence through the stability of total volume, strain energy, and stress, specifically: In recent In each iteration, the relative changes in average total volume and strain energy satisfy:

[0055]

[0056] In the formula, Indicates the iteration step The average total volume at that time Indicates the preceding Average total volume over the next iteration; Indicates the iteration step The average total strain energy at that time Indicates recent The average total strain energy across the iterations; Indicates the number of backtracking iterations; This is the convergence threshold; Recent The average total volume and strain energy in each iteration are calculated using the following formulas:

[0057]

[0058] When all units satisfy When the conditions are met, stress convergence is achieved; once all optimization criteria are satisfied, the optimization process ends.

[0059] The present invention further protects a computer device, characterized in that the computer device includes a processor and a memory, wherein the memory stores at least one instruction, at least one program, code set or instruction set, and the instruction, program, code set or instruction set is loaded and executed by the processor to realize the above-mentioned integrated progressive structural optimization method for truss topology, shape and size.

[0060] The present invention further protects a computer-readable storage medium, characterized in that the computer-readable storage medium stores at least one instruction, at least one program, code set, or instruction set, wherein the instruction, program, code set, or instruction set is loaded and executed by a processor to implement the above-mentioned integrated progressive structural optimization method for truss topology, shape, and size.

[0061] Compared with the prior art, the present invention has the following beneficial effects: (1) Significantly improves structural performance and material efficiency: By simultaneously optimizing topology, shape and size, this invention fully explores the synergistic effect between design variables and can automatically discover excellent configurations that cannot be obtained by traditional separate optimization.

[0062] (2) Effective control of structural complexity and assurance of manufacturing feasibility: The dynamic stress and area penalty mechanism introduced in this invention can systematically identify and eliminate slender members that contribute little to the structural stiffness. This process is not a simple deletion, but rather guides the area to converge to the lower limit through a penalty function during optimization iteration, thereby achieving smooth evolution of the topology. The resulting truss structure has clear nodes, a reasonable number of members, and a simple layout, completely overcoming the drawbacks of complex and difficult-to-manufacture traditional topology optimization results, and greatly improving the engineering practical value of the design results.

[0063] (3) Combining high computational efficiency with excellent convergence stability: This invention adopts a sensitivity-based gradient optimization framework, whose computational cost is far lower than that of non-gradient algorithms (such as genetic algorithms) that require massive finite element analysis. At the same time, the introduction of adaptive momentum gradient descent and sensitivity smoothing techniques effectively alleviates the inherent oscillation problem and sensitivity to initial values ​​of the pure gradient method, ensuring that the optimization process converges quickly and stably.

[0064] (4) Producing high-performance and compliant designs: This invention combines the full stress criterion with dynamic stress constraint treatment, so that the stress of the members in the optimized structure can actively approach and strictly remain within the allowable stress range of the material, thus fundamentally ensuring structural safety while pursuing performance. Attached Figure Description

[0065] Figure 1 This is a boundary diagram of the design variables (truss node coordinates) for the optimization method proposed in this invention; Figure 2This is a flowchart of the optimization method proposed in this invention; Figure 3 The initial truss base structure setup (including dimensions, loads, and boundary conditions) for Embodiment 1 of the present invention. Figure 4 The results are the optimization results of Embodiment 1 of the present invention (displacement unit is mm, stress unit is MPa). Figure 5 This is the strain energy / volume evolution process of Embodiment 1 of the present invention; Figure 6 The initial truss base structure setup (including dimensions, loads, and boundary conditions) for Embodiment 2 of the present invention. Figure 7 The optimized results of Embodiment 2 of the present invention are shown (displacement unit is mm, stress unit is MPa). Figure 8 This is the strain energy / volume evolution process of Embodiment 2 of the present invention. Detailed Implementation

[0066] The following will provide a detailed description of an integrated progressive structural optimization method for truss topology, shape, and size, and its optimization scheduling method, which relates to the present invention. It should be emphasized that the following description is merely exemplary and not intended to limit the scope or application of the invention.

[0067] This invention provides an integrated evolutionary structural optimization method for truss structures. Its core lies in using a unified, gradient-based optimization framework to synchronously and automatically adjust the structure's topology, shape, and dimensions, thereby efficiently obtaining a high-performance, easily manufacturable design solution while satisfying all constraints. The specific working principle and process of the above method are as follows: 1.1 Mathematical Model of the Optimization Problem (1) Objective function The optimization objective is to minimize the total strain energy of the truss structure. To maximize structural stiffness, its mathematical expression is:

[0068] In the formula, For external load vector, The global displacement vector generated by the applied load.

[0069] (2) Design variables Design variables include two categories: nodal coordinate variables and member cross-sectional area variables. ①Spatial coordinates of truss nodes Represented as ,in They are nodes i The coordinates; ② Cross-sectional area of ​​the rod , recorded as ,in It is the first e The cross-sectional area of ​​the root member.

[0070]

[0071] In the formula, node coordinate variables are used for shape optimization, while member cross-sectional area variables are used for size and topology optimization. When Approaching At that point, the member will become very thin, so its contribution to the structural stiffness is negligible, and it will be considered to be removed from the topology.

[0072] (3) Constraints The optimization problem is subject to the following constraints: ① Equilibrium equations: The structure must satisfy static equilibrium under applied external loads:

[0073] In the formula, This is the global stiffness matrix.

[0074] ② Design variable constraints: The coordinates and cross-sectional areas of truss members must be limited within a specified range to ensure design feasibility.

[0075] Let the initial coordinates of the truss nodes be:

[0076] Truss node spatial coordinates The following boundary conditions must be met:

[0077] In the formula, and These represent the lower and upper bounds of the coordinates of each node, respectively. , , The allowed range of movement is defined in each coordinate direction, such as Figure 1 As shown.

[0078] The displacement of fixed nodes (outside the design domain) is set to zero, while the displacement of movable nodes is determined by the optimization algorithm.

[0079] For the cross-sectional area of ​​truss members, the following boundary conditions apply:

[0080] In the formula, and These represent the minimum and maximum allowable cross-sectional areas, respectively.

[0081] ③ Stress Constraints: To ensure structural safety, stress constraints must be applied. Each truss member must be kept within allowable tensile and compressive stress ranges to prevent yielding. The mathematical expression for stress constraints is:

[0082] In the formula, and These represent the allowable compressive stress and tensile stress of the material, respectively.

[0083] 1.2 Sensitivity Analysis To perform gradient-based optimization, the objective function needs to be differentiated with respect to the design variables to obtain sensitivity.

[0084] ① The sensitivity of the objective function to nodal coordinates For connection nodes i and j unit e Its length is:

[0085] right about and Differentiate:

[0086] Similarly, for , and , That's also true.

[0087] unit e The strain under axial load is defined as:

[0088] In the formula, It is a unit e The initial (undeformed) length.

[0089] For connection nodes i and j rod unit e strain energy about x, y, z The sensitivity expression for coordinates is:

[0090] strain energy about x, y, z Coordinate sensitivity includes node i and jThe contribution of all connected link elements. Therefore, the sensitivity of the objective function to nodal coordinates. The expression is:

[0091] In the formula, and Representing the nodes respectively i and j A collection of connected rods.

[0092] ② Sensitivity of the objective function to the cross-sectional area of ​​the rod: To calculate the sensitivity of strain energy to cross-sectional area, for about Taking the derivative, we get:

[0093] Since most members in a typical truss structure have inconsistent lengths, the formula (12) will be used in the following way. Eliminate and ignore the influence of rod length on sensitivity. Therefore, the expression for the sensitivity of the objective function to the cross-sectional area of ​​the rod is:

[0094] 1.3 Design Variable Updates Updating design variables is a step-by-step, adaptive iterative process that requires simultaneous and coordinated adjustments to node coordinates and cross-sectional areas to ensure efficient progress while maintaining structural feasibility and stability.

[0095] ① Truss node coordinate update (shape optimization) Before updating the node coordinates, the original sensitivity needs to be smoothed to minimize numerical instability.

[0096] node i Smoothing sensitivity The expression is as follows:

[0097] In the formula, Indicates neighboring nodes j Scaling sensitivity, Based on nodes i and j Weighting factor of the distance between them It is the smooth radius The total number of internal nodes.

[0098] Weighting factors Defined as:

[0099] In the formula, It is a node i With nodes j The distance between them; This represents the radius that controls the range of the smoothing effect. This technique ensures that the gradient at each node is more controlled by the radius. The influence of closer neighboring nodes is relatively large, while the influence of more distant nodes gradually weakens. It is generally recommended... It should be at least three times the minimum node spacing.

[0100] The momentum method is used to update node coordinates to accelerate convergence and smooth the optimization path. The mathematical expressions for the momentum term and coordinate update are as follows:

[0101] In the formula, Indicates the current iteration step The momentum term at time, This is the momentum coefficient (usually taken as 0.9). This is the sensitivity after smoothing in the previous iteration. This formula accelerates the convergence process by utilizing historical gradient information, effectively mitigating problems such as oscillation or slow convergence while ensuring a more stable and faster approach to the optimal design.

[0102] To ensure that the optimization process minimizes strain energy, the system calibrates the node coordinate updates using an adaptive mechanism based on strain energy changes during the initial stage of the process before updating the cross-sectional area. Current iteration step. The expression for updating node coordinates at that time is:

[0103] In the formula, For the updated node coordinates, For Heaviside step function, For adaptive learning rate, This is the amplification factor used to adjust the update amplitude.

[0104] Heaviside step function Before applying changes to node coordinates, determine the appropriate update direction, which is defined as follows:

[0105] In the formula, This represents the change in total strain energy, which is calculated by tentatively updating the nodal coordinates based on initial sensitivity. If the energy change... If this happens, the Heaviside step function will be updated in the opposite direction, causing the algorithm to adaptively move towards minimizing strain energy in the next iteration.

[0106] To further enhance convergence, the adaptive learning rate decays exponentially over time:

[0107] In the formula, The initial learning rate; The attenuation rate; This represents the current iteration step.

[0108] To ensure the stability of numerical calculations, the displacement magnitude of the nodal coordinates must be monitored during the iteration process. Restrictions will be imposed:

[0109] like Exceeding the specified maximum displacement Then scale the update value proportionally to ensure The details are as follows:

[0110] This method directly limits the displacement by applying a scaling factor only when the calculated displacement exceeds the maximum allowable value, thereby ensuring stability and keeping the evolution of the structural morphology controllable as it progresses toward the optimal solution.

[0111] To maintain physical feasibility, node coordinates must be restricted to predefined boundaries. If the updated variable exceeds this range, it will be adjusted using soft constraints, the expression of which is:

[0112] In the formula, The steepness of the penalty term is controlled to ensure a smooth and gradual adjustment near the critical value.

[0113] To ensure controllable updates of design variables, each component of the node coordinate vector is constrained by the maximum displacement in each iteration. The displacement limit is proportional to the range of node coordinates:

[0114] In the formula, It is a small, fixed ratio, usually taken as 0.01.

[0115] The updated node coordinates must satisfy:

[0116] And it is further restricted within a pre-defined physical boundary:

[0117] ② Truss member cross-sectional area update (dimension and topology optimization) Optimal truss design aims to achieve a full stress state, meaning that under applied load, all members reach their allowable stress limits essentially simultaneously. The cross-sectional area update of truss members employs an update mechanism based on the full stress criterion, which sets a sensitivity threshold with the ideal full stress state as the target.

[0118] Specifically, when the sensitivity of the rod element When the material utilization rate is high, it can be improved by reducing the cross-sectional area of ​​the corresponding members; conversely, when the material utilization rate is high, it can be improved by reducing the cross-sectional area of ​​the corresponding members. In such cases, the cross-sectional area needs to be increased to maintain structural safety. This method optimizes structural performance while ensuring efficient material utilization by adjusting the cross-sectional area of ​​the member elements. Furthermore, if the cross-sectional area of ​​a member decreases to the minimum allowable value... Since its contribution to stiffness is negligible, it is considered an invalid component and will be removed from the topology. This causes a topological change in the truss structure, which will be reconfigured based on the updated material distribution.

[0119] The cross-sectional area of ​​the truss members is updated using an adaptive update mechanism. The cross-sectional area is updated iteratively to gradually approach the optimal solution. In the early stages of the optimization iteration process, a larger step size is used to accelerate convergence, while the step size is gradually reduced as convergence approaches to enhance stability. To balance efficiency and accuracy, the adaptive update strategy dynamically adjusts the step size, minimizing unnecessary iterations while maintaining a smooth and stable optimization process.

[0120] No. e Adaptive update factor for each truss member element The definition is as follows:

[0121] In the formula, For attenuation parameters, The number of iterations. The exponential decay term. make sure The constant decreasing trend allows for finer adjustments to the optimization process as it approaches the optimal solution. This approach helps prevent over-adjustment and ensures a more controlled update process as the solution converges.

[0122] Attenuation parameter By convergence metric Determine by the following formula:

[0123] In the formula, This represents a convergence metric, used to measure how close a solution is to the optimal solution. When... At that time, the decay rate is more aggressive ( Larger solutions allow for faster updates when the solution is far from convergence. Conversely, when... At that time, the decay rate slows down ( (smaller) so that more fine-tuning can be done when the solution is close to the optimal solution.

[0124] The above adaptive method ensures the update factor Able to determine convergence metrics Dynamic adjustments are made to achieve rapid progress in the early stages of the optimization process and precise fine-tuning when convergence to the optimal solution, thereby achieving a balance between the two.

[0125] The cross-sectional area of ​​truss members is updated according to the following rules: For members that require an increase in cross-sectional area:

[0126] For members whose cross-sectional area needs to be reduced:

[0127] In the formula, Indicates the first The truss unit in the first The updated cross-sectional area at the next iteration, and It is the area of ​​the previous iteration.

[0128] Equations (29) and (30) ensure that the cross-sectional area is updated gradually and stably while satisfying design constraints. Update factor The adjustment process can be smoothly achieved, where equation (29) limits the area to To prevent the cross-sectional dimensions of the members from becoming too large, equation (30) is forced to satisfy... To avoid insufficient structural stiffness.

[0129] In truss topology optimization, controlling structural complexity is crucial for ensuring manufacturability and practical feasibility. Overly complex optimized structures, containing redundant components or slender members, increase manufacturing difficulty and drive up costs. By reducing complexity, especially by eliminating excessively thin and low-stress members, efficient and robust designs can be achieved without compromising load-bearing capacity.

[0130] To achieve near-optimal and directly manufacturable structures, the proposed method integrates two mechanisms: (i) imposing a penalty term on inefficient slender members; and (ii) adjusting the area of ​​overstressed members.

[0131] (i) Apply dynamic penalties to inefficient slender members First, in the current iteration step, the maximum cross-sectional area of ​​all members is defined as the benchmark for applying the penalty:

[0132] The dynamic penalty threshold is defined as:

[0133] With the number of iterations Increase from 100 to 200, Gradually from The penalty for the thin rod increases from 5% to 20%, thus imposing a stricter penalty on the thin rod in the middle of the optimization process, while gradually relaxing the constraint in the later stage.

[0134] At the same time, a dynamic stress threshold is introduced, defined as:

[0135] along with The value increases from 100 to 200. from The percentage increases from 20% to 80%. Formula (33) allows for relatively lenient penalties on bar elements with unreasonable stress distribution during intermediate optimization stages, while stricter constraints are implemented as the process progresses.

[0136] For the number of iterations In the case where the area of ​​the updated member is lower than the threshold Or its absolute stress is less than the threshold If the cross-sectional area of ​​the member is not specified, the member will be penalized. The penalty mechanism is based on the cross-sectional area of ​​the member and the median area of ​​all members. Implementation of the sigmoid function for the ratio:

[0137] The penalty value calculated using the above formula always falls between 0 and 1. Therefore, if both conditions are met... and The area of ​​the corresponding member is then updated as follows:

[0138] (ii) Adjust the cross-sectional area of ​​the overstressed member Imposing penalties on slender members may increase the stress on the remaining members of the truss structure; therefore, when At that time, for all members exceeding the stress limit Iterative strengthening is performed according to the following formula:

[0139] This smooth adjustment allows the structure to remain balanced at all times and ensures that the stress distribution of the truss members is always close to the full stress state.

[0140] 1.4 Convergence Criterion Convergence is evaluated through overall volume, strain energy, and stress stability. In recent years... In each iteration, the relative changes in average total volume and strain energy must satisfy:

[0141] In the formula, Indicates the iteration step The average total volume at that time Indicates the preceding Average total volume over the next iteration; Indicates the iteration step The average total strain energy at that time Indicates recent The average total strain energy across iterations. This indicates the number of backtracking iterations (usually 5). It is the convergence threshold (usually taken as 0.001).

[0142] Recent The average total volume and strain energy in each iteration can be calculated using the following formulas:

[0143] When all units satisfy When the conditions are met, stress convergence is achieved. The optimization process ends once all optimization criteria are satisfied.

[0144] 1.5 Optimize the process The iterative process of the method proposed in this invention follows these steps: (1) Initialization: Define the geometry, loads, boundary conditions and design variables of the initial truss base structure. Set the node boundaries and initialize the design variables.

[0145] (2) Initial finite element analysis: Perform initial finite element analysis to calculate the total strain energy and total volume, which serve as the reference values ​​for subsequent iterations.

[0146] (3) Sensitivity calculation: Evaluate the objective function and calculate the sensitivity of the design variables. Apply sensitivity smoothing to the truss nodes to ensure stability.

[0147] (4) Update spatial coordinates: Iteratively update the spatial coordinates of the nodes. A momentum-based update scheme is adopted to enhance convergence stability, and displacement is limited by a preset boundary to maintain structural feasibility.

[0148] (5) Update the cross-sectional area of ​​truss members: Adjust the cross-sectional area of ​​truss members based on the calculated sensitivity and sensitivity threshold.

[0149] (6) After the design variables are updated, perform finite element analysis: Based on the updated design variables, perform finite element analysis to obtain nodal displacements and element stresses, and re-evaluate the total strain energy and volume change of the structure.

[0150] (7) Convergence assessment: Based on the preset number of iterations, the convergence is assessed by the stress uniformity and overall volume stability of the truss structure. If the convergence condition is not met, the next iteration begins.

[0151] (8) Iteration loop: Repeat steps (3) to (7) until the maximum number of iterations is reached or the specified convergence condition is met.

[0152] (9) Output optimization results: After convergence, save the final optimized structure, including the updated node coordinates, element properties and related indicators (such as total strain energy and volume).

[0153] Figure 2 The process of the proposed integrated progressive structural optimization method for truss topology, shape, and dimensions is demonstrated. This algorithm supports implementation in general-purpose programming languages ​​such as Python and C / C++, and can be seamlessly embedded into mainstream finite element platforms such as Abaqus, ANSYS, and COMSOL. It is also suitable for open-source solvers. Through scripts, plugins, or multi-level API interfaces, it integrates with CAD / CAE workflows, achieving full-chain automation from modeling to solving to post-processing, thus improving optimization efficiency and engineering application level.

[0154] The specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings. This embodiment takes the optimization of two preferred application scenarios of the present invention, namely Embodiment 1 (two-dimensional short cantilever truss) and Embodiment 2 (space truss grid structure), as examples, but the application of the present invention is not limited to these.

[0155] 1. Parameter Settings (1)Material properties Both Example 1 and Example 2 use Q235 steel, with an allowable stress of 215 MPa and an elastic modulus of 206 GPa.

[0156] (2) Initial cross-sectional area and cross-sectional area constraints of the members The initial cross-sectional area of ​​the truss members in both Examples 1 and 2 is set to 100 mm². 2 The cross-sectional area constraint range is limited to to between.

[0157] (3) External loads The applied external force is set to P = 10 kN. For simplicity, the structure's self-weight is not considered.

[0158] (4) Optimize algorithm parameters Momentum coefficient Initial learning rate attenuation rate Sensitivity smoothing radius The value is set to 3 times the minimum node spacing; maximum displacement limit. Convergence threshold Number of backtracking iterations .

[0159] 2. Optimize the iterative process Step 1: Perform initial finite element analysis The generated .inp file is submitted to Abaqus for linear static analysis. After the analysis is complete, a Python script reads the result file (.odb) to obtain the displacement vectors of all nodes. and the axial stress of each member And calculate the initial total strain energy. and total volume .

[0160] Step 2: Sensitivity Calculation and Smoothing (1) Sensitivity to node coordinates: Based on the read displacement and stress, the nodes are calculated in batches according to Equations (10) and (11). i The original sensitivity. Then, a smoothing filter is applied, and according to equations (14) and (15), a linear weighting function is used to... Within range Smooth the surface to obtain the desired result. .

[0161] (2) Sensitivity to bar area: for bar elements e Calculate according to formula (13) Meanwhile, the total stress threshold is calculated according to equation (26). .

[0162] Step 3: Synchronously update two types of design variables: truss node coordinates and member cross-sectional areas. (1) Update the truss node coordinates to achieve shape optimization of the truss structure.

[0163] ① Calculate the momentum term: Obtain it according to equation (16) For the first iteration, .

[0164] ② Trial update and direction determination: Perform a trial update according to equation (17) and calculate the strain energy change. The Heaviside step function obtained through equation (18) Determine the final update direction.

[0165] ③ Formal Update and Boundary Control: A formal update is performed according to equation (17) and in conjunction with equation (19). Then, the nodal displacements are checked and limited according to equations (20) and (21) to ensure... Finally, the node coordinates are strictly limited to the preset boundaries according to equations (20) to (25). Inside.

[0166] (2) Update the cross-sectional area of ​​the members to achieve size and topology optimization of the truss structure.

[0167] ① Adaptive cross-sectional area update mechanism: Based on the full stress design criterion, the sensitivity of each member is updated. Compared with theoretical threshold A comparison is made. Based on the comparison results, the adaptive update factor of the member is calculated using equation (27). This factor is defined by the attenuation parameter in equation (28). Dynamic adjustments are implemented to ensure that the update step size is larger in the early stage of optimization to accelerate convergence, and the step size is reduced in the later stage for fine-tuning.

[0168] ② Cross-sectional area update rule: Based on the comparison results of the sensitivity of the members, the area is increased or decreased, and the update is strictly performed in accordance with the rules defined by equations (29) and (30). This rule automatically clamps the area value to the preset upper and lower limits while adjusting the area. and Within ) . When At the same time, inefficient members are removed and the topology is evolved.

[0169] ③ Structural Complexity Control: To actively suppress and eliminate slender, inefficient members and improve the engineering practicality of the final design, a dynamic penalty mechanism is introduced. This mechanism adjusts the penalty based on the number of iterations. Time activation, the dynamically growing area penalty threshold is calculated using equations (31) and (32). The stress penalty threshold is calculated using equation (33). For those that simultaneously meet the requirement of having an area less than And the absolute stress value is lower than The members will be penalized (area reduction) according to equations (34) and (35), forcing them out of the structure. This process systematically simplifies the topology and ensures the simplicity and manufacturability of the final solution.

[0170] Step 4: Convergence Assessment The average volume of the last 5 iterations was calculated using equations (38) and (39). and mean strain energy Check whether the relative change is less than 0.001 according to equations (40) and (41), and verify whether the stress of all members meets the constraints. If all are satisfied, the optimization converges and proceeds to the next step; otherwise, return to step 1 and perform the next iteration.

[0171] Step 5: Optimize Result Output and Post-processing After optimization and convergence, the node coordinates and member areas are exported, and these results can be imported into other software for detailed design or analysis.

[0172] 3. Example Analysis (1) Example 1: Two-dimensional short cantilever truss Example 1 relates to the optimization of a two-dimensional short cantilever truss, such as Figure 3 As shown. The design domain measures 2000 mm in length and 1000 mm in height, with a base grid size of 500 mm × 500 mm, comprising 8 base grids, 15 nodes, and 74 members. The initial truss base structure bears a concentrated load at the middle of its right end, while the left side is supported by three fixed hinges. Figure 3 (a) Showing the initial setup of the two-dimensional short cantilever truss base structure. Figure 3 (b) Demonstrates movable and fixed nodes. To further verify the effectiveness of the proposed ensemble method, two optimization schemes were designed using the same instance: (1) In Case A, the node position and cross-sectional area are optimized simultaneously, and the movable node is... x and y Allowable range of movement in coordinate direction (2) Case B restricts all nodes from shifting during the optimization process. Only cross-sectional area adjustment is allowed.

[0173] Figure 4 (a) to (d) show the final optimized structure of the short cantilever truss and its corresponding stress and displacement distribution. Figure 4 In Case A, as shown in (a) and (c), the optimized truss geometry undergoes a significant modification, resulting in a more compact structure that promotes efficient load transfer paths and improves stiffness. In contrast, Figure 4 Case B, shown in (b) and (d), retains the original node arrangement. Because the cantilever length is not shortened, the structural stiffness is relatively low. Displacement contour analysis shows that in Case A, node repositioning reduces overall deformation, and the structural stiffness increases by approximately 33.3% compared to Case B. The comparative results show that allowing node displacement (Case A) significantly improves load-bearing efficiency and structural stiffness through optimized configuration. Conversely, restricting node movement (Case B) reduces design flexibility, leading to increased deformation and decreased stiffness. This result highlights the advantages of integrated optimization methods in truss structure design.

[0174] Figure 5The evolution of total strain energy and total volume of a two-dimensional short cantilever truss throughout the optimization process is illustrated. As shown in the figure, the total volume decreases rapidly in the initial iteration stage and then tends to stabilize. In Case A, the total strain energy initially increases slightly, then gradually decreases and eventually converges; while in Case B, the total strain energy exhibits oscillations before reaching a stable state. The significantly lower final strain energy value in Case A indicates that allowing simultaneous optimization of node coordinate positions and member cross-sectional dimensions during the optimization process can effectively improve structural performance.

[0175] (2) Example 2: Space Truss Grid Structure Example 2 involves the optimization of a spatial truss grid structure, such as Figure 6 As shown, this spatial truss grid structure is 1000 mm long and wide, and 100 mm high. The basic grid size is 250 mm × 250 mm × 100 mm, containing 16 basic grids, 50 nodes, and 405 members. The bottom center of the grid structure bears a concentrated load, and fixed hinges are set at the four bottom corners for constraint. Figure 6 (a) through (c) show the initial setup of the space truss grid base structure, including dimensions, loading conditions and boundary constraints; Figure 6 (d) to (f) describe the fixed node, the specified active node, and their corresponding allowable displacement range.

[0176] To investigate the impact of the allowable displacement range on the optimization results, two scenarios were studied using the same case: (1) In Case A, the horizontal direction of the movable node is restricted ( ), but vertical displacement is allowed ( (2) In Case B, the movable node is allowed omnidirectional displacement and the restrictions in each direction are the same. ).

[0177] Figure 7 (a) to (f) present the final optimized structure and its corresponding stress and displacement distribution. Figure 7 In Case A shown in (a), nodal displacements are limited to the vertical direction, and the optimized structure basically maintains the original horizontal configuration. The adjustment only occurs in the vertical direction. z In the axial direction. In contrast, Case B allows for horizontal displacement, resulting in lateral repositioning of the nodes, thus forming... Figure 7 (d) shows a distinctly different structural form. The top view of Case B shows the top due to... z The structure tends to flatten towards the maximum permissible displacement, which limits further improvements in structural stiffness. Therefore, Figure 7 The displacement distributions in (b) and (e) indicate that Case A has smaller overall deformation and its structural stiffness is approximately 29.4% higher than that of Case B. Furthermore, as... Figure 7 As shown in (c) and (f), the normal stress in the optimized structural members is close to but does not exceed the specified stress limit.

[0178] Figure 8 The evolution of total strain energy and total volume of the spatial truss mesh structure throughout the optimization process is illustrated. As shown in the figure, the total volume decreases rapidly in the initial stage before stabilizing, while the total strain energy initially increases slightly and then gradually decreases oscillatingly until convergence. The final total strain energy of Case A is again significantly lower than that of Case B, further confirming the effectiveness of allowing simultaneous optimization of node coordinate positions and member cross-sectional dimensions during the optimization process in improving structural efficiency.

[0179] The above description is merely a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. An integrated progressive structural optimization method for truss topology, shape, and size, characterized in that, Includes the following: S1. Initialization: Construct the truss optimization model, define the geometry, loads, boundary conditions and design variables of the initial truss base structure; set the node boundaries and initialize the design variables; S2. Initial Finite Element Analysis: Perform initial finite element analysis to calculate the total strain energy and total volume, which will serve as the baseline values ​​for subsequent iterations. S3. Sensitivity Calculation: Evaluate the objective function of the truss optimization model and calculate the sensitivity of the design variables, including: ① Calculate the sensitivity of the objective function to the nodal coordinates: In the formula, Indicates the total strain energy; Representation unit e strain energy; and Representing the nodes respectively i and j A collection of connected rods; They are nodes i The coordinates; They are nodes j The coordinates; ② Calculate the sensitivity of the objective function to the cross-sectional area of ​​the member: For unit e strain energy Regarding the first e Cross-sectional area of ​​the root member Differentiating, we get: Ignoring the effect of rod length on sensitivity, the expression for the sensitivity of the objective function to the cross-sectional area of ​​the rod is: In the formula, Indicates the elastic modulus; Representation unit e Strain under axial load; Representation unit e Stress constraints; Representation unit e Length; This represents the sensitivity of the objective function to the cross-sectional area of ​​the rod; S4. Update design variables: Synchronously and coordinately adjust design variables to maintain the feasibility and stability of the structure, including: The momentum method is used to update node coordinates to accelerate convergence and smooth the optimization path; the mathematical expressions for the momentum term and coordinate update are as follows: In the formula, Indicates the current iteration step The momentum term at time; The momentum coefficient; This indicates the sensitivity after smoothing in the previous iteration; The cross-sectional area of ​​truss members is updated according to the following rules: For members with increased cross-sectional area: For members with reduced cross-sectional area: In the formula, Indicates the first The truss unit in the first The updated cross-sectional area at the next iteration; and This represents the area of ​​the previous iteration; Indicates the first e Adaptive update factor for each truss member element; and These represent the minimum and maximum allowable cross-sectional areas, respectively. A dynamic penalty mechanism is introduced to actively suppress and eliminate thin, inefficient links. This dynamic penalty mechanism is applied at a certain number of iterations. Activated at time; Calculate the area penalty threshold for dynamic growth First, in the current iteration step, the maximum cross-sectional area of ​​all members is defined as the benchmark for applying the penalty: The dynamic penalty threshold is defined as: Simultaneously, a dynamic stress threshold is introduced, defined as: For areas below the threshold And the absolute stress is less than the threshold. The members are penalized, and the penalty mechanism is based on the cross-sectional area of ​​the member and the median area of ​​all members. Implementation of the sigmoid function for the ratio: The penalty value calculated using the above formula always falls between 0 and 1; therefore, if both conditions are met... and The area of ​​the corresponding member is then updated as follows: The above operations achieve topology simplification; S5. Finite element analysis: Based on the updated design variables, perform finite element analysis to obtain nodal displacements and element stresses, and re-evaluate the total strain energy and volume change of the structure. S6. Convergence Assessment: Based on the preset number of iterations, the convergence is assessed through the stress uniformity and overall volume stability of the truss structure; if the convergence condition is not met, the next iteration begins. S7. Iteration loop: Repeat S3~S6 until the maximum number of iterations is reached or the specified convergence condition is met. S8. Output optimization results: After convergence, save the final optimized structure, including the updated node coordinates, element attributes and related indicators.

2. The integrated progressive structural optimization method for truss topology, shape, and size according to claim 1, characterized in that, The optimization objective of the truss optimization model is to minimize the total strain energy of the truss structure to maximize the structural stiffness. The mathematical expression of the objective function of the model is: In the formula, The vector represents the total strain energy; f is the external load vector; the superscript T indicates the transpose operation; u is the global displacement vector generated by the applied load.

3. The integrated progressive structural optimization method for truss topology, shape, and size according to claim 2, characterized in that, The design variables include node coordinate variables and truss member cross-sectional area variables, specifically including: ①Spatial coordinates of truss nodes Represented as ,in They are nodes i The coordinates; ② Cross-sectional area of ​​the rod , recorded as ,in It is the first e The cross-sectional area of ​​the root member is expressed mathematically as follows: In the formula, node coordinate variables are used for shape optimization; bar cross-sectional area variables are used for size and topology optimization; when Approaching When the contribution to structural stiffness is negligible, it is removed from the topology.

4. The integrated progressive structural optimization method for truss topology, shape, and size according to claim 3, characterized in that, The constraints of the truss optimization model are: ① Equilibrium equations: The structure satisfies static equilibrium under applied external loads: In the formula, This is the global stiffness matrix; ② Design variable constraints: The coordinates and cross-sectional areas of truss members are limited to a specified range to ensure design feasibility; Let the initial coordinates of the truss nodes be: Truss node spatial coordinates The following boundary conditions must be met: In the formula, and These represent the lower and upper bounds of the coordinates of each node, respectively. The displacement of fixed nodes is set to zero, while the displacement of movable nodes is determined by an optimization algorithm. For the cross-sectional area of ​​truss members, the following boundary conditions apply: In the formula, and These represent the minimum and maximum allowable cross-sectional areas, respectively. ③ Stress constraint: Each truss member is kept within the allowable tensile and compressive stress range to prevent yielding; the mathematical expression for stress constraint is: In the formula, and These represent the allowable compressive stress and tensile stress of the material, respectively.

5. The integrated progressive structural optimization method for truss topology, shape, and size according to claim 4, characterized in that, The specific derivation process for calculating the sensitivity of the objective function to the node coordinates described in S3 is as follows: For connection nodes i and j unit e Its length is: right about and Differentiate: Similarly, for , and , Use the same operation; unit e The strain under axial load is defined as: In the formula, Representation unit e The initial length; For connection nodes i and j rod unit e strain energy about x, y, z The sensitivity expression for coordinates is: In the formula, Indicates the elastic modulus; strain energy about ， ， Coordinate sensitivity includes node i and j The contribution of all connected link elements.

6. The integrated progressive structural optimization method for truss topology, shape, and size according to claim 5, characterized in that, S4 also includes the following: Before updating the node coordinates, the original sensitivity is smoothed to reduce numerical instability; node i Smoothing sensitivity The expression is as follows: In the formula, Indicates neighboring nodes j Scaling sensitivity, Based on nodes i and j Weighting factor of the distance between them It is the smooth radius The total number of internal nodes; where the weighting factor is... Defined as: In the formula, It is a node i With nodes j The distance between them; Indicates the radius that controls the range of the smoothing effect; Before updating the cross-sectional area, the node coordinates are calibrated and updated using an adaptive mechanism based on the strain energy changes during the initial stage of the process. Current iteration step The expression for updating node coordinates at that time is: In the formula, The updated node coordinates; It is the Heaviside step function; Adaptive learning rate; This is the amplification factor used to adjust the update amplitude; where the Heaviside step function is... The update direction is determined before applying changes to node coordinates, and it is defined as follows: In the formula, This represents the change in total strain energy, which is calculated by tentatively updating the nodal coordinates based on initial sensitivity. If the energy change... Then the Heaviside step function will reverse its update direction, causing the algorithm to adaptively move towards the direction of minimizing strain energy in the next iteration; To further enhance convergence, the adaptive learning rate decays exponentially over time: In the formula, The initial learning rate; The attenuation rate; This represents the current iteration step number; To ensure the stability of the numerical calculation, the displacement magnitude of the node coordinates is adjusted during the iteration process. Restrictions will be imposed: like Exceeding the specified maximum displacement Then scale the update value proportionally to ensure The details are as follows: To maintain physical feasibility, node coordinates are restricted to predefined boundaries. If the updated variable exceeds this range, it will be adjusted using soft constraints, the expression of which is: In the formula, Used to control the steepness of the penalty term to ensure smooth and gradual adjustment near the critical value; To ensure controllable updates of design variables, each component of the node coordinate vector is constrained by the maximum displacement in each iteration. The displacement limit is proportional to the range of node coordinates. In the formula, For a small fixed ratio; The updated node coordinates satisfy: And it is further restricted within a pre-defined physical boundary: The cross-sectional area of ​​truss members is updated based on the full stress criterion update mechanism, with the ideal full stress state as the target, and a sensitivity threshold is set: When the rod unit e Sensitivity When the cross-sectional area of ​​the corresponding members is reduced, the material utilization rate is improved; conversely, when... In such cases, the cross-sectional area is increased to maintain structural safety; Dynamically adjust the step size to minimize unnecessary iterations while maintaining a smooth and stable optimization process. No. e Adaptive update factor for each truss member element The definition is as follows: In the formula, For attenuation parameters; Number of iterations; exponential decay term Used to ensure It has always shown a decreasing trend; Attenuation parameter By convergence metric Determine by the following formula: In the formula, Represents a convergence metric, used to measure how close a solution is to the optimal solution; when When the decay rate is more aggressive, when At that time, the decay rate slows down.

7. The integrated progressive structural optimization method for truss topology, shape, and size according to claim 6, characterized in that, S6 specifically includes the following: evaluating convergence through the stability of total volume, strain energy, and stress, specifically as follows: In recent In each iteration, the relative changes in average total volume and strain energy satisfy: In the formula, Indicates the iteration step The average total volume at that time Indicates the preceding Average total volume over the next iteration; Indicates the iteration step The average total strain energy at that time Indicates recent The average total strain energy across the iterations; Indicates the number of backtracking iterations; This is the convergence threshold; Recent The average total volume and strain energy in each iteration are calculated using the following formulas: When all units satisfy Under certain conditions, stress convergence is achieved; The optimization process ends once all optimization criteria are met.

8. A computer device, characterized in that, The computer device includes a processor and a memory, the memory storing at least one instruction, at least one program, code set, or instruction set, the instruction, program, code set, or instruction set being loaded and executed by the processor to implement the integrated progressive structural optimization method for truss topology, shape, and size as described in any one of claims 1-7.

9. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores at least one instruction, at least one program, code set, or instruction set, which is loaded and executed by a processor to implement the integrated progressive structural optimization method for truss topology, shape, and size as described in any one of claims 1-7.

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