Truss structure topological optimization method and system considering stability, stress and geometric feasibility

Abstract

The invention discloses a truss structure topological optimization method and system considering stability, stress and geometric feasibility. According to the method, firstly, the buckling load prediction precision is improved by adopting an improved geometric stiffness matrix; secondly, suppressing false stability and a suspension rod phenomenon caused by a middle density rod piece through nonlinear density projection; then, a unified segmented micro-constraint function is constructed, four states are smoothly switched according to the density and stress ratio of the rod piece, Euler buckling and tensile stress constraints are synchronously processed, and a density truncation strategy is effectively compatible; and further quantizing rod piece intersection and overlapping into continuous variables, and establishing strictly differentiable geometric constraints in combination with norm aggregation and the piecewise function. And finally, integrating all constraints through an enhanced Lagrangian method, and carrying out iterative solution by adopting a moving asymptote method. According to the method, the problems of stress singularity, false stability, geometry infeasibility and the like in traditional topological optimization are systematically solved, and stable, tough and manufacturable multi-target balance design is realized.

Description

Technical Field

[0001] This invention relates to a truss structure topology optimization technique based on the enhanced Lagrangian method, simultaneously integrating stability constraints, stress constraints, and geometric feasibility constraints, belonging to the field of structural optimization technology. This invention is particularly applicable to the innovative design of lightweight truss structures in the aerospace, civil engineering, and mechanical equipment fields where stringent requirements for safety, load-bearing efficiency, and manufacturability exist. Background Technology

[0002] Topology optimization is a key technique in advanced structural design, aiming to find the optimal material distribution path within a given design space and constraints. For truss structures, topology optimization based on the base structure method has been extensively studied. Traditional optimization methods often aim to minimize structural compliance (maximize stiffness) and are supplemented by material volume constraints. However, such simplified models often overlook several key factors that lead to structural failure in practical engineering, including elastic instability (buckling), material strength failure, and geometric unmanufacturability.

[0003] Existing technologies face a series of interconnected and unresolved numerical challenges and theoretical bottlenecks when dealing with these multiphysics constraints:

[0004] 1. Prediction accuracy of stability constraints and the problem of "spurious stability":

[0005] Global stability analysis heavily relies on the accuracy of the geometric stiffness matrix. Currently, the widely used geometric stiffness matrix formula suffers from unreliable predictive accuracy and lacks systematic validation, raising questions about the reliability of the optimization. More importantly, traditional density truncation strategies neglect members with densities below a threshold (e.g., c=0.01). However, while these "medium-density" members do not bear the main axial loads, they provide non-physical, illusory lateral support to the main load-bearing members, resulting in unmanufacturable "suspended rod" configurations in the optimization results and significantly overestimating the overall structural stability.

[0006] 2. Local stability and compatibility issues between stress constraint singularities and truncation strategies:

[0007] Local stability (Eulerian buckling) constraints and stress constraints present singularity problems in continuum optimization. Existing relaxation techniques, such as ε-relaxation (Cheng & Guo, 1997) and qp-relaxation (Bruggi, 2008), aim to alleviate this problem. However, when combined with density truncation strategies, they both exhibit fundamental flaws: ε-relaxation fails to effectively relax members below the truncation threshold, while qp-relaxation incorrectly relaxes members above the threshold. This leads to optimization results that either contain a large number of constraint-violation slender members or are overly conservative, failing to find truly lightweight topologies.

[0008] 3. The computability and scalability dilemma of geometric feasibility constraints:

[0009] In topologies generated by the base structure method, non-node intersections and overlaps between members are the main reasons for geometric infeasibility. Existing solutions mainly fall into two categories: First, heuristic algorithms (such as the genetic algorithm used by Cui et al.) or mixed-integer programming are employed. The former is prone to getting trapped in local optima and has low search efficiency, while the latter has high computational complexity and is difficult to apply to large-scale problems. Second, recent gradient-based methods (such as the work of Torii, Leng et al.) attempt to define differentiable geometric constraints, but their continuous approximations lack strict constraint enforcement, often leaving unacceptable geometric interferences in the final design.

[0010] 4. Computational burden and parameter sensitivity of multi-constraint ensemble frameworks:

[0011] When the above constraints are considered simultaneously, a massive number of local constraint functions are generated, making the solution of the optimization problem and the sensitivity analysis computationally expensive. To reduce the number of constraints, aggregation methods (such as KS functions and p-norms) are commonly used, but their performance is highly dependent on parameter selection. Overly large parameters can easily lead to oscillations and ill-conditioned convergence in the optimization process, while overly small parameters result in ineffective constraint enforcement. This inherent parameter sensitivity makes it difficult for the optimization process to robustly converge to a feasible solution.

[0012] In summary, there is an urgent need in this field for a novel, unified framework for topology optimization that must be able to: ① accurately predict global stability and eliminate spurious modes; ② rigorously and smoothly handle local constraints compatible with density truncation; ③ proactively avoid geometrically infeasible configurations in a differentiable and scalable manner; and ④ robustly and efficiently integrate all of the above constraints without resorting to difficult-to-tunable aggregation parameters. Currently, no solution has been found that can simultaneously meet all of these requirements. Summary of the Invention

[0013] The purpose of this invention is to overcome the shortcomings of existing technologies and provide a novel framework for topology optimization of truss structures. The core of this framework lies in proposing a multi-constraint integration strategy based on the augmented Lagrangian method, coupled with a series of innovative constraint processing techniques, thereby simultaneously addressing stability, stress, and geometric feasibility issues.

[0014] To achieve the above objectives, the present invention adopts the following technical solution:

[0015] A topology optimization method for truss structures considering stability, stress, and geometric feasibility includes the following steps:

[0016] S1. Problem Definition and Initialization: Define the design domain based on the basic structure method and initialize the density design variables of the members;

[0017] S2. Nonlinear density projection and finite element analysis: Through nonlinear density mapping relationship, the density design variables mentioned in step S1 are mapped to physical densities for physical analysis to suppress the false contribution of low-density members to structural stability; finite element analysis is performed based on the physical density to obtain the displacement field and member stress of the structure.

[0018] S3. Multi-constraint evaluation: The multi-constraint includes at least: global stability constraints based on the improved geometric stiffness matrix, local stability and stress constraints based on the unified piecewise differentiable function, and geometric feasibility constraints based on geometric conflict aggregation.

[0019] S4. Optimization Solution: The augmented Lagrangian method is used to integrate all the steps described in step S3 with stress constraints and geometric feasibility constraints into the objective function to construct the augmented Lagrangian function.

[0020] S5. Iteration and Output: The augmented Lagrangian function described in step S4 is iteratively solved using the moving asymptote method, and the density design variables are updated until the convergence condition is met, and the final topology is output.

[0021] Beneficial Effects: Compared with the prior art, the truss structure topology optimization method and system considering stability, stress, and geometric feasibility provided by this invention have the following significant beneficial effects:

[0022] First, it completely eliminates false stability and the "floating rod" phenomenon: The nonlinear density projection function introduced in this invention, through a density truncation threshold... , The physical density of intermediate-density members is drastically compressed to near zero, fundamentally depriving these members, which contribute little to mechanical strength, of their ability to provide spurious lateral stiffness. This prevents optimization algorithms from utilizing such non-physical support configurations to generate physically real, manufacturable self-supporting topologies, thus solving an inherent problem that has long plagued traditional density truncation strategies.

[0023] Second, the accuracy of global stability prediction is significantly improved: This invention is based on a global stability constraint using an improved geometric stiffness matrix, which more accurately reflects the large deformation geometric nonlinear effects of the structure. This makes the stability criteria relied upon in the optimization process more reliable, effectively overcoming the limitations of traditional methods. The inherent errors in the model when predicting critical buckling loads are eliminated, thus ensuring the safety of the optimized design in terms of global stability from the source.

[0024] Third, it achieves strict, smooth, and unified handling of local constraints: the constructed unified piecewise differentiable constraint function innovatively and automatically switches smoothly between four states based on the member density and stress ratio. This function is perfectly compatible with the density truncation strategy, which can impose strict constraints on high-density over-limit members while smoothly ignoring low-density members. It fundamentally solves the contradiction of "not relaxing when it should" or "relaxing when it shouldn't" in traditional ε-relaxation and qp-relaxation methods when handling truncation strategies, effectively avoiding stress singularities and local buckling failures, and ensuring the numerical stability of the gradient optimization process. The unified piecewise differentiable function is configured to automatically and smoothly switch between four states based on whether the member density is greater than the density truncation threshold and whether its stress ratio is greater than the allowable stress ratio, so as to simultaneously and compatiblely handle local Euler buckling constraints and tensile stress constraints.

[0025] Fourth, the geometric feasibility constraints combine rigor with computational scalability: by quantifying the intersection and overlap conflicts between members into continuous variables, and utilizing... By aggregating geometric constraints using norms and then substituting them into a unified piecewise function, this invention successfully transforms a large number of explosively combined discrete geometric constraints into a small number of strictly differentiable continuous constraints. This ensures the rigorous mathematical enforcement of geometric feasibility, avoiding the shortcomings of insufficient constraint force in existing gradient methods. Furthermore, its computational complexity is linearly related to the design variables, making it computationally feasible to handle geometric conflicts in large-scale basis structures, overcoming the bottleneck of methods such as mixed-integer programming in solving large-scale problems.

[0026] Fifth, the multi-constraint integrated framework is robust and efficient: The enhanced Lagrangian method (AL) is adopted as the core optimization framework, integrating all the aforementioned local constraints (stability, stress, geometry) into the objective function, and solving it using the moving asymptote method (MMA). This combined strategy inherently avoids the pitfalls of traditional aggregation functions (such as the KS function, ...). It addresses parameter sensitivity and approximate execution problems of the (-norm), and can accurately and robustly handle massive constraints, ensuring that the optimization process converges efficiently to a Pareto optimal solution that simultaneously satisfies the requirements of stability, strength, and geometric manufacturability.

[0027] As a preferred embodiment of the method described above, in step S2, the nonlinear density mapping relationship is configured such that when the density design variable is lower than a preset cutoff threshold, its physical density is projected to 10^-7.

[0028] As a preferred embodiment of the method described above, in step S2, the rod is calculated using a nonlinear density projection function. Design variables Mapped to physical density for physical analysis The projection function at the density cutoff threshold A cubic polynomial is used nearby for smooth transition:

[0029] ;

[0030] Among them, coefficient to By ensuring in and place Continuity is determined by the fact that function values ​​are equal and first derivatives are equal.

[0031] As a preferred embodiment of the method described above, step S3 specifically includes the following sub-steps:

[0032] S31: Global stability constraint evaluation:

[0033] Based on displacement field An improved element geometric stiffness matrix is ​​adopted. Assemble the global geometric stiffness matrix The The expression is:

[0034] ;

[0035] in, For Young's modulus, For rods cross-sectional area, For rods Length, For rods directional vector, For rods The displacement vector, This is an extended form of the identity matrix; subsequently, the generalized eigenvalue problem is solved:

[0036] ;

[0037] To obtain the accurate critical buckling load factor, where, This is the global material stiffness matrix. For the first 1 eigenvalue, For the first 1 eigenvector The structural degrees of freedom;

[0038] S32: Local stability and tensile stress constraint assessment:

[0039] For each member, its density and stress ratio are input into a unified piecewise differentiable function. For any member... Calculate its constraint function :

[0040] ;

[0041] in, , , Stress ratio, For unit density, The preset truncation threshold is used; This is the maximum value of the preset stress ratio limit; when used for local stability constraints, , The Euler stress ratio, For rods The stress value, For rods The Euler buckling stress limit; when used for tensile stress constraint. , For rods The tensile stress limit; It is the tensile stress ratio;

[0042] This function determines whether the rod density is greater than a preset cutoff threshold. And whether the stress ratio is greater than the maximum value of the preset stress ratio limit. It automatically and smoothly switches between four states, thereby simultaneously handling local Euler buckling constraints and tensile stress constraints, and is strictly compatible with density cutoff strategies.

[0043] S33: Geometric Feasibility Constraint Assessment

[0044] For any bar The program reads all the members that conflict with it from the pre-calculated conflict matrix. Design variables and utilize - Norms converge into a global conflict indicator:

[0045] ;

[0046] in, It is with rods Intersecting or overlapping rods Design variables, For aggregation parameters;

[0047] Aggregated conflict indicators Input a piecewise differentiable function with the same mathematical form as described in step S32, and calculate its bar intersection and overlap constraint values ​​to ensure that the geometric constraints are strictly differentiable:

[0048] ;

[0049] S34: Volume Constraint Assessment

[0050] Calculate the total volume of the structure and evaluate the volume constraint function.

[0051] As a preferred embodiment of the above method of the present invention, in step S5, a two-layer optimization strategy is used for iterative solution, wherein the outer loop manages the parameter update of the augmented Lagrangian function, and the inner loop uses the moving asymptote method to solve the unconstrained subproblem.

[0052] This invention further discloses a truss structure topology optimization system for implementing the aforementioned truss structure topology optimization method considering stability, stress, and geometric feasibility, comprising:

[0053] The processor; and the memory, which stores computer program instructions;

[0054] When the computer program instructions are executed by the processor, the system is configured to perform the method steps of any one of claims 1-6.

[0055] As a preferred embodiment of the above-mentioned optimized system technical solution, the system is further implemented through a modular architecture that works collaboratively, including:

[0056] The initialization and pre-calculation module is configured to define the design domain and initialize design variables.

[0057] The finite element analysis module is configured to perform finite element analysis based on the projected physical density;

[0058] The constraint integration processing module is configured to evaluate volumetric, global stability, local stability, stress, and geometric feasibility constraints.

[0059] The sensitivity analysis module is used to calculate the gradients of the objective function and constraint function with respect to the design variables using the adjoint variable method, and to provide gradient information to the optimization solver module.

[0060] The solver module is optimized and configured to perform iterative optimization based on the enhanced Lagrange method and the moving asymptote method.

[0061] The present invention further discloses a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the aforementioned truss structure topology optimization method considering stability, stress, and geometric feasibility.

[0062] The present invention further discloses a truss structure design data, which consists of the final topology data generated by the truss structure topology optimization method that considers stability, stress and geometric feasibility. The data is used to guide or directly drive the manufacturing process of the truss structure. Attached Figure Description

[0063] Figure 1 This is a general flowchart of the truss structure topology optimization method provided in the embodiments of the present invention;

[0064] Figure 2 This is a schematic diagram of the modular logical architecture of an embodiment of the optimization system of the present invention;

[0065] Figure 3 This is a schematic diagram of the design domain, boundary conditions, and loads for Example 1, which verifies the effect of nonlinear density projection.

[0066] Figure 4 This is a schematic diagram of the initial base structure corresponding to Embodiment 1;

[0067] Figure 5 This is the optimized topology result in Example 1 without using the nonlinear density projection strategy. The figure clearly shows the "floating rod" phenomenon.

[0068] Figure 6 In Example 1, the optimized topology results after adopting the nonlinear density projection strategy of the present invention show that the "suspended rod" has been eliminated and the structure is clear and feasible.

[0069] Figure 7 This is a schematic diagram of the analytical model of Example 2 (asymmetric double-bar model) used to verify the accuracy of the improved geometric stiffness matrix;

[0070] Figure 8 This is a schematic diagram of the analytical model of Example 3 (symmetric double-bar model) used to verify the accuracy of the improved geometric stiffness matrix;

[0071] Figure 9 The critical buckling load predicted by different geometric stiffness matrices and the nonlinear reference solution in Example 2 (with stiffness ratio) Comparison curves under (changes);

[0072] Figure 10 The critical buckling load predicted by different geometric stiffness matrices and the nonlinear reference solution in Example 3 (with angle) Comparison curves under (changes);

[0073] Figure 11 This is a schematic diagram of the unified piecewise differentiable constraint function proposed in this invention under specific parameters, which intuitively demonstrates its characteristic of smoothly switching between four states based on the member density and stress ratio;

[0074] Figure 12 This is a schematic diagram of the design domain, boundary conditions, and loads for Example 4, used to verify the effect of the unified constraint function;

[0075] Figure 13 This is a schematic diagram of the initial base structure corresponding to Example 4;

[0076] Figure 14 This is the optimized topology result obtained using the traditional qp-relaxation method in Example 4;

[0077] Figure 15 In Example 4, after optimization using the traditional qp-relaxation method, the Euler stress ratio distribution cloud map of the structure is shown, indicating the existence of an over-limit region. This indicates that the constraints were not strictly enforced;

[0078] Figure 16 This is the optimized topology result obtained using the unified constraint function of the present invention in Example 4;

[0079] Figure 17 In Example 4, after optimization using the unified constraint function of the present invention, the Euler stress ratio distribution cloud map of the structure is shown. The figure shows that all stress ratios are strictly controlled below 1.0, which proves the strictness and effectiveness of the constraint treatment of the present invention.

[0080] Figure 18 This is a graphical illustration of the geometric constraint (crossing / overlapping) function proposed in this invention under specific parameters, demonstrating its strictly differentiable property;

[0081] Figure 19 This is a schematic diagram of the design domain, boundary conditions, and loads for Example 5, used to verify the robustness of the fully constrained integrated framework;

[0082] Figure 20 This is a schematic diagram of the initial base structure corresponding to Example 5;

[0083] Figure 21 This is a three-dimensional view of the topology obtained after integrating all constraints for Example 5 (torque-loaded rectangular prism);

[0084] Figure 22 This is a front view of the optimization results of Example 5;

[0085] Figure 23 This is a top view of the optimization results of Example 5;

[0086] Figure 24 This is a cloud map showing the Euler buckling stress ratio distribution of the optimization results in Example 5;

[0087] Figure 25 This is a cloud map showing the tensile stress ratio distribution of the optimized results in Example 5;

[0088] Figure 26 This is a graph showing the convergence process of the objective function (compliance) during the optimization process in Example 5;

[0089] Figure 27 This is a graph showing the convergence process of the volume constraint and global stability constraint functions during the optimization process of Example 5;

[0090] Figure 28 This is a graph showing the convergence process of local stability and tensile stress constraint function during the optimization process of Example 5;

[0091] Figure 29 This is a graph showing the convergence process of the cross and overlap constraint functions during the optimization process of Example 5. Detailed Implementation

[0092] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be emphasized that the embodiments described are only used to fully demonstrate the technical solution of the present invention and the beneficial effects it can achieve, and are not intended to limit the scope of protection of the present invention. Various modifications and variations that can be made by those skilled in the art without creative effort based on the technical solution of the present invention should be included within the scope of protection of the present invention.

[0093] This invention provides a topology optimization method for truss structures that considers stability, stress, and geometric feasibility, comprising the following steps:

[0094] S1: Problem Initialization and Parameter Configuration

[0095] This step aims to set uniform initial conditions for the topology optimization problem, and the specific implementation is as follows:

[0096] S1.1: Define the design domain and boundary conditions

[0097] Input the geometric dimensions of the design domain, such as length, width, and height; boundary conditions, such as the location of fixed supports and hinged supports; and external loads, including the magnitude, direction, and point of application of concentrated or distributed forces.

[0098] Example 1, as Figure 3 , Figure 4 Used to verify the effect of nonlinear density projection, it adopts a columnar structure with a hinged bottom and pressure on the top;

[0099] Examples two and three, such as Figure 7 , Figure 8 : Used to verify the accuracy of the improved geometric stiffness matrix, employing a classic two-bar analytical model;

[0100] Example 4, as Figure 12 , Figure 13 : Used to verify the effect of the unified constraint function, a cantilever beam structure is adopted;

[0101] Example 5, as Figure 19 , Figure 20 : Used to verify the robustness of a fully constrained integrated framework, employing a three-dimensional torque loading structure.

[0102] S1.2: Generate the base structure and initialize the design variables

[0103] The initial design space is generated using the basic structure method, which involves arranging nodes within the design domain and connecting all possible nodes to form a set of members. The design variables for all members are then initialized. The initial value is usually set to a uniform value that satisfies the volume constraint.

[0104] S1.3: Set optimization parameters and material properties

[0105] This sub-step completes the configuration of all key parameters, which will be stored in the computer's memory for subsequent calculations. Unless otherwise specified in a specific embodiment, the uniform parameter settings shown in the table below are used. Note that these parameters are exemplary and not intended to limit the invention:

[0106]

[0107] S1.4: Pre-calculation and storage

[0108] This step is performed before the optimization iteration begins, and its purpose is to perform a one-time calculation to improve the efficiency of subsequent iterations.

[0109] S1.4.1: Pre-calculation of element stiffness matrix and geometric foundation

[0110] To ensure the efficiency of the optimization iteration process, the computer program performs the following one-time pre-calculation before executing the optimization loop, and stores the results in global memory:

[0111] 1. Calculate and store the cell direction vector. :

[0112] For each unit in the base structure The program calculates its direction vector based on its node coordinates. For a connected node and nodes Given a three-dimensional element, the program performs the following calculations:

[0113] Calculate the cell length: .

[0114] Calculate and store the direction vector:

[0115] .

[0116] This vector Used for subsequent rapid assembly of element stiffness matrix and geometric stiffness matrix.

[0117] 2. Calculate and store the constant part of the element stiffness matrix:

[0118] For each unit The program pre-calculates the constant part of its element stiffness matrix. In subsequent iterations, it is only necessary to consider the changing unit area. By scaling this constant component, the current element stiffness matrix can be quickly obtained. This greatly improves computational efficiency.

[0119] 3. Define the fundamental matrix of geometric stiffness. :

[0120] The program defines and stores the 6x6 extended identity matrix required for the geometric stiffness matrix in memory. :

[0121] ;

[0122] This matrix is ​​the assembly-improved geometric stiffness matrix. The foundation.

[0123] S1.4.2: Pre-calculation of geometric conflict relations

[0124] Before optimization begins, the program performs a one-time pre-calculation of the intersection and overlap relationships for all potential member pairs:

[0125] 1. The program iterates through all potential link pairs. .

[0126] 2. For each pair of members, based on their node coordinates, a computational geometry algorithm is used to determine the possibility of them intersecting or overlapping, and an initial conflict level metric is calculated. These measures are designed as continuous variables, with 0 indicating no conflict and positive values ​​indicating the severity of conflict.

[0127] 3. Store the conflict relationships and initial metric values ​​of all member pairs in a conflict matrix for quick retrieval during subsequent geometric constraint evaluation.

[0128] S2: Nonlinear density projection and finite element analysis

[0129] This step aims to map design variables to physical densities and complete the mechanical response analysis of the structure, specifically including the following sub-steps:

[0130] S2.1: Verification of Nonlinear Density Projection and Example 1

[0131] This is the core step in eliminating the phenomena of "false stability" and "suspended rod". In each optimization iteration, before performing finite element analysis, the program adjusts the current design variables. By applying a nonlinear projection function, the physical density is obtained for physical analysis. :

[0132] ;

[0133] Among them, coefficient to By ensuring in and place Continuity is determined by the equality of function values ​​and first derivatives. The program obtains the coefficients by solving the following system of linear equations:

[0134] exist Location:

[0135] The function values ​​are continuous;

[0136] The first derivative is continuous;

[0137] exist Location:

[0138] The function values ​​are continuous;

[0139] The first derivative is continuous;

[0140] Additional settings To ensure that when hour, It is projected onto 10^-7.

[0141] The program pre-calculates these coefficients during the initialization phase and calls them during each projection.

[0142] Example 1

[0143] This design is specifically intended to verify the effectiveness of this nonlinear projection function. This embodiment employs a columnar structural design domain (20m × 20m × 40m), with fixed supports at the bottom and axial loads at the top. Optimize as follows, such as Figure 3 , Figure 4 As shown.

[0144] The comparison scale did not use projection: the optimization results are as follows Figure 5 The diagram shows a typical "suspended rod" configuration. Its stability depends on a density slightly below a threshold. ,Right now The false lateral support provided by the rods makes the structure physically impossible to manufacture.

[0145] This invention uses projection: the optimization results are as follows Figure 6 As shown, the physical density of all medium-density rods is compressed to near zero, making it impossible to provide effective stiffness. The "suspended rods" are completely eliminated, resulting in a clear, self-supporting, and manufacturable topology.

[0146] The comparative results of Example 1 conclusively demonstrate that the nonlinear density projection function proposed in this invention is a key and effective technical means to eliminate "spurious stability" and ensure that the optimization results have physical authenticity and manufacturability.

[0147] S2.2: Finite Element Analysis

[0148] The computer program calculates the physical density after projection. Material property interpolation is performed and linear static finite element analysis is executed.

[0149] S2.2.1: Material Interpolation and Assembly of Material Stiffness Matrix

[0150] The program uses a penalty model for solid isotropic materials and is based on pre-calculated element orientation vectors. and constant part For each element, quickly calculate its current material stiffness matrix:

[0151] ;

[0152] Among them, the equivalent cross-sectional area The interpolation model according to claim 11 calculates:

[0153] ;

[0154] Subsequently, based on the node connection relationships of the elements, all elements were processed using the direct stiffness method. Assembled into a global material stiffness matrix .

[0155] S2.2.2: Solving the equilibrium equations and calculating stress

[0156] The program solves a system of linear equations:

[0157] ;

[0158] in, Let be the external load vector. Then, obtain the global nodal displacement vector of the structure. Based on the displacement results, the program calculates the value of each element. Axial stress :

[0159] ;

[0160] in, It is a unit The nodal displacement vector. Tensile stress is positive, and compressive stress is negative.

[0161] S3: Multi-constraint evaluation

[0162] This step aims to evaluate all constraint functions and is the core of the integrated optimization framework, which consists of four sub-steps.

[0163] S3.1: Global Stability Constraint Evaluation

[0164] This sub-step is based on displacement field An improved geometric stiffness matrix is ​​adopted. Perform linear buckling analysis.

[0165] S3.1.1: Assemble the improved geometric stiffness matrix

[0166] For each unit The program calculates the geometric stiffness matrix at the element level. :

[0167] ;

[0168] Among them, unit area An interpolation model different from the material stiffness matrix is ​​used:

[0169] ;

[0170] Subsequently, all units Assembled into a global geometric stiffness matrix .

[0171] S3.1.2: Solving the Generalized Eigenvalue Problem

[0172] The program constructs and solves the following generalized eigenvalue problem to evaluate the stability of the structure:

[0173] ;

[0174] In the formula, This is the global material stiffness matrix. For the structure of the first 1 eigenvalue, For the first structure 1 eigenvector Let be the structural degrees of freedom, where is the smallest eigenvalue. This is the critical buckling load factor of the structure. The critical buckling load is Stability requirements .

[0175] Examples two and three are specifically used to verify the methods used in this step. The accuracy of the formula.

[0176] S3.1.3: Accuracy Verification of Improved Geometric Stiffness Matrix

[0177] This invention selects two embodiments to verify the proposed... The advantages of the formula. For example... Figure 7 As shown, Example 2 is an asymmetric double-bar model. This model is used to test different material stiffness ratios. The prediction accuracy is as follows. Figure 8 As shown, Example 3 is a symmetrical double-bar model. This model is used to test different included angles. The prediction accuracy is reduced.

[0178] The program uses four different geometric stiffness matrix formulas. to The predicted results are compared with the nonlinear finite element benchmark solution considering large deformation effects. The results are as follows: Figure 9 , Figure 10 As shown. The results indicate that in all test cases, regardless of the structural parameters... or How it changes, the invention adopts The critical buckling load calculated by the formula has the highest degree of agreement with the nonlinear benchmark solution, which is significantly better than the other three traditional formulas.

[0179] S32: Local stability and tensile stress constraint assessment:

[0180] For each member, its density and stress ratio are input into a unified piecewise differentiable function. For any member... Calculate its constraint function :

[0181] ;

[0182] in, , , Stress ratio, For unit density, The preset truncation threshold is used; This is the maximum value of the preset stress ratio limit; when used for local stability constraints, , The Euler stress ratio, For rods The stress value, For rods The Euler buckling stress limit; when used for tensile stress constraint. , For rods The tensile stress limit; It is the tensile stress ratio;

[0183] This function determines whether the rod density is greater than a preset cutoff threshold. And whether the stress ratio is greater than the maximum value of the preset stress ratio limit. It automatically and smoothly switches between four states, thereby simultaneously handling local Euler buckling constraints and tensile stress constraints, and is strictly compatible with density cutoff strategies.

[0184] Figure 11 The Euler buckling stress and tensile stress constraint functions proposed in this embodiment of the invention are... , A schematic diagram of the time. This function is based on the cell density. and stress ratio Automatically and smoothly switch between four states:

[0185] Case 1 ( ): The element is important and the stress ratio exceeds the limit, so strict stress / buckling constraints should be applied.

[0186] Case 2 ( ): The unit is not important; only a density penalty is applied to make it evolve toward 0.

[0187] Case 3 ( ): The element is important and stress-safe, so it is allowed to be retained.

[0188] Case 4 ( ): The element is unimportant and the stress is safe; apply a relaxed joint constraint.

[0189] Example 4

[0190] This method is specifically designed to verify the superiority of unified constraint functions over traditional methods. This embodiment employs a rectangular design domain (50m × 30m), with a fixed support at the left end and a vertical load applied at the right end. Optimize as follows, such as Figure 12 , Figure 13 As shown.

[0191] Comparatively, the traditional QP-relaxation method: the optimization results show as follows Figure 14 The structure contains slender members that clearly violate Euler buckling constraints, with regions in the Euler stress ratio contour plot exceeding 1.5. This indicates that the constraints were not strictly enforced and there is a risk of local buckling instability. Figure 15 As shown.

[0192] This invention provides a unified constraint function: the optimization results have a clear structure, such as... Figure 16 As shown, the Euler stress ratio contour plots of all members were strictly controlled below 1.0, and the maximum stress ratio was significantly reduced. This proves that the method proposed in this invention can effectively constrain members with a cutoff density above a certain threshold, ensure local buckling stability, and obtain a safe and reliable design, such as... Figure 17 As shown.

[0193] This comparative study quantitatively demonstrates the significant advantages of the unified constraint function proposed in this invention in terms of compatible density truncation strategies and strict constraint enforcement.

[0194] S3.3: Geometric Feasibility Constraint Assessment

[0195] This sub-step handles the constraints of member intersections and overlaps.

[0196] S3.3.1: Quantification and Aggregation of Geometric Conflicts:

[0197] For any bar The program reads all the members that conflict with it from the pre-calculated conflict matrix. Design variables and utilize - Norms converge into a global conflict indicator:

[0198] ;

[0199] in, It is with rods Intersecting or overlapping rods Design variables, These are aggregation parameters.

[0200] S3.3.2: Calculation of geometric constraint functions:

[0201] Will As input, calculate the cross constraint and overlap constraint functions respectively:

[0202] ;

[0203] This design makes the geometric constraints strictly differentiable and compatible with optimizers.

[0204] Figure 18 The geometric constraints, intersections / overlaps, and functions proposed in this embodiment of the invention are... , A schematic diagram of the time. This function is based on the cell density. and global conflict indicators Automatically and smoothly switch between four states:

[0205] Case 1 ( ):unit and unit Overlapping / intersecting rods If they appear simultaneously in the topology, strict overlap / crossing constraints must be applied.

[0206] Case 2 ( ): Only with unit Overlapping / intersecting rods It appears in the topology and is allowed to be retained.

[0207] Case 3 ( ): Only units It appears in the topology and is allowed to be retained.

[0208] Case 4 ( ):unit and unit Overlapping / intersecting rods None of them appear in the topology, so a relaxed joint constraint is applied.

[0209] S3.4: Volume Constraint Assessment

[0210] This sub-step calculates the total volume of the structure. And evaluate the volume constraint function:

[0211] ;

[0212] in, It is the volume of the base structure when all design variables are 1.

[0213] S4: Optimization Solution

[0214] This step aims to integrate all local constraints by enhancing the Lagrange framework and to solve efficiently using the moving asymptote method.

[0215] S4.1: Constructing the augmented Lagrange function

[0216] To circumvent the parameter sensitivity of traditional aggregation functions, this invention employs the enhanced Lagrangian method as the core optimization framework. The program integrates all local constraint functions calculated in steps S3.2 and S3.3, including local stability, tensile stress, and cross and overlapping constraints, into the objective function to construct the augmented Lagrangian function:

[0217] ;

[0218] For each type of constraint It covers buckling, stress, intersection, and overlap. For the number of iterations, To design variable vectors, For Lagrange multipliers, As a penalty factor, The objective function is strain energy; penalty term. It is given by the following formula:

[0219] ;

[0220] In the formula, To effectively constrain violations. For rods The constraint type is constraint functions, Let be the number of links. This functional form ensures that the optimization subproblem maintains good numerical scaling throughout the iteration.

[0221] S4.2: Solution using the moving asymptote method and sensitivity analysis

[0222] The optimized solution logic unit 205 adopts a two-layer strategy to solve the above problem.

[0223] S4.2.1: Solving using the inner layer - moving asymptote method

[0224] Inner-layer moving asymptote solution logic 2052 receives the current augmented Lagrangian function constructed by outer-layer augmented Lagrangian framework logic 2051. .

[0225] S4.2.2: Gradient-driven

[0226] The sensitivity analysis logic unit 204 provides accurate gradient information for the moving asymptote method solver, including the sensitivity of the objective function and all constraint functions. This module employs the adjoint variable method, eigenvalue cluster mean sensitivity analysis, and analytical differentiation to efficiently complete this complex computational task.

[0227] S4.2.3: Subproblem Solving

[0228] The moving asymptote method solver will Treating the current unconstrained objective function as an example, and utilizing its gradient information, we construct and solve a series of convex approximation subproblems:

[0229] ;

[0230] ;

[0231] ;

[0232] ;

[0233] in, It is a global stability constraint; therefore, the moving asymptote solver can find a constraint that makes the global stability constraint true. Decreasing design variable update value .

[0234] S4.3: Update augmented Lagrange parameters

[0235] After completing one inner-layer solution, the program updates the augmented Lagrange method parameters based on the current solution, preparing for the next outer-layer iteration:

[0236] Update penalty factor: ;

[0237] Update the Lagrange multipliers: ;

[0238] This update rule makes it possible for constraints to be violated ( The multiplier increases accordingly, thus imposing a larger penalty in subsequent iterations, forcing the constraint to be satisfied.

[0239] S5: Iteration, Convergence, and Verification in Example 5

[0240] This step describes the optimized closed-loop iterative process and verifies the robustness of the process under fully constrained integration through Example 5.

[0241] S5.1: Automated Iterative Loop

[0242] Computer programs according to Figure 1 The overall flowchart shown automatically executes a closed-loop iterative process:

[0243] S5.1.1: Initialization

[0244] Initialized in step S1 , , Set an iteration counter as the starting point. .

[0245] S5.1.2: Core Steps of Sequence Execution

[0246] In each iteration, the program executes sequentially:

[0247] S2: Nonlinear density projection and finite element analysis.

[0248] S3: Multi-constraint evaluation.

[0249] S4: Optimize the solution, construct the augmented Lagrange function and solve using the moving asymptote method, and update the design variables and augmented Lagrange parameters.

[0250] S5.1.3: Iterative Progress

[0251] Update iterative counter .

[0252] S5.2: Comprehensive Convergence Judgment and Verification in Example 5

[0253] The program automatically performs a convergence check after each outer iteration. Convergence criteria include:

[0254] S5.2.1: Feasibility

[0255] All constraints on volume, global stability, local stability, stress, and geometry are satisfied.

[0256] S5.2.2: Stability

[0257] relative change of objective function or changes in design variables .

[0258] S5.2.3: Iteration Limit

[0259] ;

[0260] S5.3: Final Result Output

[0261] The optimization loop terminates when the convergence condition is met. The program then calls post-processing module 206 to design variables and truncation thresholds based on the final density. Generate a clear 0-1 discretized topology diagram and output stress cloud diagram, buckling mode and comprehensive performance report to fully achieve the preset invention purpose.

[0262] The full-constraint verification in Example 5 demonstrates the robustness of this framework under complex three-dimensional structures:

[0263] Final topology, Figures 21 to 23 Clear and free of geometric interference.

[0264] Stress cloud diagram, Figure 24 , Figure 25 This indicates that all stress ratios are within a safe range.

[0265] Convergence process graph Figures 26 to 29 This indicates that the optimization process converges smoothly and stably under all constraints.

[0266] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of the present invention. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. Moreover, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this specification, as well as the features of different embodiments or examples.

[0267] Any process or method description in the flowchart or otherwise herein can be understood as representing a module, segment, or portion of code comprising one or N executable instructions for implementing custom logic functions or processes, and the scope of preferred embodiments of the invention includes additional implementations in which functions may be performed not in the order shown or discussed, but in a substantially simultaneous manner or in the reverse order, as will be understood by those skilled in the art to which embodiments of the invention pertain.

[0268] Those skilled in the art will understand that all or part of the steps of the methods in the above embodiments can be implemented by a program instructing related hardware. The program can be stored in a computer-readable storage medium, and when executed, the program includes one or a combination of the steps of the method embodiments.

[0269] Furthermore, the functional modules in the various embodiments of the present invention can be integrated into a single processing module, or each member can exist physically separately, or two or more members can be integrated into a single module. The integrated module can be implemented in hardware or as a software functional module. If the integrated module is implemented as a software functional module and sold or used as an independent product, it can also be stored in a computer-readable storage medium.

Claims

1. A topology optimization method for truss structures considering stability, stress, and geometric feasibility, characterized in that, Includes the following steps: S1. Problem Definition and Initialization: Define the design domain based on the basic structure method and initialize the density design variables of the members; S2. Nonlinear density projection and finite element analysis: Through nonlinear density mapping relationship, the density design variables mentioned in step S1 are mapped to physical densities for physical analysis to suppress the false contribution of low-density members to structural stability; finite element analysis is performed based on the physical density to obtain the displacement field and member stress of the structure. S3. Multi-constraint evaluation: The multi-constraint includes at least global stability constraints based on the improved geometric stiffness matrix, local stability and stress constraints based on a unified piecewise differentiable function, and geometric feasibility constraints based on geometric conflict aggregation. S4. Optimization Solution: The augmented Lagrangian method is used to integrate all the local stability and stress constraints and geometric feasibility constraints mentioned in step S3 into the objective function to construct the augmented Lagrangian function; S5. Iteration and Output: The augmented Lagrangian function described in step S4 is iteratively solved using the moving asymptote method, and the density design variables are updated until the convergence condition is met, and the final topology is output.

2. The truss structure topology optimization method considering stability, stress, and geometric feasibility according to claim 1, characterized in that, In step S2, the nonlinear density mapping relationship is configured such that when the density design variable is lower than a preset cutoff threshold, its physical density is projected to 10^-7.

3. The truss structure topology optimization method considering stability, stress, and geometric feasibility according to claim 1, characterized in that, In step S2, the rod is calculated using a nonlinear density projection function. Design variables Mapped to physical density for physical analysis The projection function at the density cutoff threshold A cubic polynomial is used nearby for smooth transition: ； Among them, coefficient to By ensuring in and place Continuity is determined by the fact that function values ​​are equal and first derivatives are equal.

4. The truss structure topology optimization method considering stability, stress, and geometric feasibility according to claim 1, characterized in that, Step S3 specifically includes the following sub-steps: S31: Global stability constraint evaluation: Based on displacement field An improved element geometric stiffness matrix is ​​adopted. Assemble the global geometric stiffness matrix The The expression is: ； in, For Young's modulus, For rods cross-sectional area, For rods Length, For rods directional vector, For rods The displacement vector, This is an extended form of the identity matrix; subsequently, the generalized eigenvalue problem is solved to obtain the accurate critical buckling load factor. S32: Local stability and tensile stress constraint assessment: For each member, its density and stress ratio are input into a unified piecewise differentiable function. For any member... Calculate its constraint function : ； in, , , Stress ratio, For unit density, The preset truncation threshold is used; This is the maximum value of the preset stress ratio limit; when used for local stability constraints, , The Euler stress ratio, For rods The stress value, For rods The Euler buckling stress limit; when used for tensile stress constraint. , For rods The tensile stress limit; It is the tensile stress ratio; This function determines whether the rod density is greater than a preset cutoff threshold. And whether the stress ratio is greater than the maximum value of the preset stress ratio limit. It automatically and smoothly switches between four states, thereby simultaneously handling local Euler buckling constraints and tensile stress constraints, and is strictly compatible with density cutoff strategies. S33: Geometric Feasibility Constraint Assessment For any bar The program reads all the members that conflict with it from the pre-calculated conflict matrix. Design variables and utilize - Norms converge into a global conflict indicator: ； in, It is with rods Intersecting or overlapping rods Design variables, For aggregation parameters; Aggregated conflict indicators Input a piecewise differentiable function with the same mathematical form as described in step S32, and calculate its bar intersection and overlap constraint values ​​to ensure that the geometric constraints are strictly differentiable: ； S34: Volume Constraint Assessment Calculate the total volume of the structure and evaluate the volume constraint function.

5. The truss structure topology optimization method considering stability, stress, and geometric feasibility according to claim 1, characterized in that, In step S5, a two-layer optimization strategy is used for iterative solution. The outer loop manages the parameter update of the augmented Lagrangian function, while the inner loop uses the moving asymptote method to solve the unconstrained subproblems.

6. A truss structure topology optimization system, used to implement the truss structure topology optimization method considering stability, stress, and geometric feasibility as described in any one of claims 1-5, characterized in that, include: processor; as well as Memory, which stores computer program instructions; When the computer program instructions are executed by the processor, the system is configured to perform the method steps of any one of claims 1-6.

7. The truss structure topology optimization system according to claim 6, characterized in that, The system is further implemented through a modular architecture that works collaboratively, including: The initialization and pre-calculation module is configured to define the design domain and initialize design variables. The finite element analysis module is configured to perform finite element analysis based on the projected physical density; The constraint integration processing module is configured to evaluate volumetric, global stability, local stability, stress, and geometric feasibility constraints. The sensitivity analysis module is used to calculate the gradients of the objective function and constraint function with respect to the design variables using the adjoint variable method, and to provide gradient information to the optimization solver module. The solver module is optimized and configured to perform iterative optimization based on the enhanced Lagrange method and the moving asymptote method.

8. A computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements a truss structure topology optimization method considering stability, stress, and geometric feasibility as described in any one of claims 1-5.

9. A truss structure design data, characterized in that, The final topology data is generated by the truss structure topology optimization method considering stability, stress and geometric feasibility as described in any one of claims 1-5, and the data is used to guide or directly drive the manufacturing process of the truss structure.

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