A cross-scale robust topology optimization method and system considering multi-source interval uncertainty

Abstract

The present application provides a kind of cross-scale robustness topology optimization method and system considering multi-source interval uncertainty, for solving the problem of insufficient robustness of traditional cross-scale topology optimization when dealing with uncertainty.The method of the present application first establishes a unified representation scheme of multi-source uncertainty, models the size and direction of material elastic modulus and external load as interval variables, and formulates the optimization objective as a comprehensive function combining the mean value and deviation of the upper and lower bounds of structural response to balance performance stability and design robustness.For the non-monotonic relationship between structural response and multi-source uncertainty, an improved Taylor series expansion method is introduced to efficiently and accurately predict the upper and lower bounds of structural response.The present application method realizes the coupling of macro and micro scales through homogenization theory, using macro and micro relative densities as design variables, and finally obtains an optimized topology structure with high robustness, suitable for engineering design field.

Description

Technical Field

[0001] This invention relates to the field of structural topology optimization and uncertainty quantification, and in particular to a cross-scale robust topology optimization method and system that considers multi-source interval uncertainty. Background Technology

[0002] Structural optimization design is a key means to improve structural performance and achieve lightweighting, and it can be divided into size optimization, shape optimization, and topology optimization. Among them, topology optimization can find the optimal distribution of materials within a given design space and is the most promising method in the conceptual design stage. In recent years, with the development of porous materials and lattice materials, the research object of topology optimization has expanded from single solid materials to structures with multi-scale characteristics, namely, cross-scale topology optimization. Cross-scale topology optimization can simultaneously design the layout of macroscopic structures and the configuration of microscopic unit cells. Compared with traditional methods that use pre-defined unit cells, it has greater design freedom and is expected to achieve better overall performance.

[0003] However, most existing cross-scale topology optimization studies are based on deterministic frameworks. This means that the optimization process assumes that all parameters, such as material properties and external loads, are constant and that the resulting structure is in an ideal state. However, in real engineering environments, uncertainty is widespread and objectively present. For example, the magnitude and direction of the loads on a structure often vary due to environmental changes; in advanced processes such as additive manufacturing, material properties can fluctuate due to microscopic defects. If these uncertainties are ignored, the optimized configuration obtained based on deterministic assumptions may suffer severe performance degradation or even failure under the influence of actual uncertainties, leading to a significant gap between the designed performance and the actual results.

[0004] To ensure the reliable performance of structures under uncertainties, robustness optimization has become an important research direction. Currently, methods for handling uncertainty are mainly divided into two categories: probabilistic methods and non-probabilistic methods. Probabilistic methods require sufficient sample data to construct the probability distribution of uncertain parameters, which is often difficult to meet in the early stages of engineering where data is scarce. In contrast, non-probabilistic methods only require knowledge of the boundary information of uncertain parameters, making them more suitable for situations with incomplete information, and therefore have received widespread attention.

[0005] Significant progress has been made in the field of nonprobabilistic robust topology optimization. Some scholars have developed nonprobabilistic reliability or robustness topology optimization methods for continuum structures or considering load uncertainties using interval mathematics or convex model theory. At the cross-scale level, research has also focused on achieving synergistic robust design of materials and structures within a nonprobabilistic framework to minimize worst-case compliance.

[0006] Nevertheless, existing nonprobabilistic multi-scale robust topology optimization methods still face a significant common challenge and bottleneck: for uncertain parameters such as load direction that exhibit complex and non-monotonic relationships with structural response, there is a lack of efficient and accurate interval propagation analysis methods. Traditional interval vertex methods may suffer severe accuracy loss in such cases due to their inability to capture extreme points within the interval; while using high-precision numerical sampling methods would become prohibitively computationally expensive. This contradiction severely restricts the practicality and efficiency of nonprobabilistic methods in dealing with real-world complex multi-source uncertainties.

[0007] Therefore, there is an urgent need to develop a new non-probabilistic, cross-scale robust topology optimization method that can: 1) uniformly handle uncertainties in multiple regions, such as material properties, load amplitude, and direction; 2) efficiently and accurately solve the problem of interval propagation analysis caused by non-monotonic relationships; and 3) establish reasonable robustness indices to balance the average level of structural performance with robustness. This is precisely the technical problem that this invention aims to solve. Summary of the Invention

[0008] This invention addresses the shortcomings of existing technologies by providing a robust cross-scale interval topology optimization method and system based on improved Taylor series expansion and considering multi-source uncertainties. The range of the unit cell elastic tensor under material elastic modulus uncertainty is analyzed using a homogenization method. The influence of multiple uncertainties on structural response is calculated using Taylor series expansion. The Method of Moving Asymptotes (MMA) optimization method is used to update macro- and micro-design variables. The resulting design results better reflect real-world conditions, achieving a balance between engineering applicability, safety, and economy.

[0009] In a first aspect, the present invention provides a cross-scale robust topology optimization method considering multi-source interval uncertainties, applied to the topology optimization of integrated materials and structures, comprising:

[0010] Step 1: Given the macroscopic structural design domain, microscopic unit cell design domain, load and boundary conditions, and characterize the elastic modulus of the material, the amplitude and direction of the external load as interval uncertainty variables, then use the weighted sum of the upper and lower bounds of the macroscopic structural compliance response interval as the optimization objective, and use the macroscopic structural volume fraction and microscopic unit cell volume fraction as constraints to establish a cross-scale robust topology optimization model.

[0011] Step 2: Based on the homogenization method and the interval vertex method, solve for the upper and lower bounds of the unit cell equivalent elastic tensor considering the uncertainty of the material's elastic modulus interval;

[0012] Step 3: Based on the improved second-order Taylor expansion method, predict the upper and lower bounds of the structural compliance response under the influence of uncertainties in the load amplitude and direction intervals;

[0013] Step 4: Based on the adjoint method, calculate the sensitivity of the objective function in the cross-scale robust topology optimization model to macro and micro design variables;

[0014] Step 5: Apply sensitivity filtering technology and use the moving asymptote optimization algorithm to simultaneously update the macroscopic structural design and microscopic unit cell design variables;

[0015] Step 6: Determine whether the optimization process meets the convergence criterion; if it does, output the final macroscopic structural design and the macroscopic unit cell design variables corresponding to the macroscopic and microscopic topological configurations as the optimization result; if it does not meet the criterion, return to step 2 to continue the iteration.

[0016] Furthermore, the expression for the cross-scale robust topology optimization model is:

[0017] (1)

[0018] in, and These are the design variable vectors for the macroscopic and microscopic design domains, respectively. Represents the i-th macroscopic design variable. Let represent the j-th micro-design variable, and M and N represent the number of macro- and micro-design variables, respectively. and These represent the uncertain intervals of the magnitude and direction of the external load, respectively. This represents the uncertainty range of the material's elastic modulus. and These are the upper and lower bounds of the structural response under multi-source uncertainty. and It is a weighted coefficient for the mean and difference of structural flexibility. Indicates that in a given The global stiffness matrix under, Indicates that in a given The global displacement vector below, In the given The global force vector below, and These are the macroscopic and microscopic volume constraint functions, respectively. and These refer to the volume of materials used in the macro-design domain and the micro-design domain, respectively. and It is the allowable volume fraction of the macroscopic design domain and the microscopic design domain; and This represents the volume of macroscopic and microscopic unit cells. and Design variables representing macroscopic and microscopic unit cells, It is the minimum elastic modulus of the material. It is a penalty factor introduced to ensure a black-and-white solution. and They represent and p to the power of 0.

[0019] Furthermore, step 2 specifically includes:

[0020] Calculation of unit cell equivalent mechanical properties based on homogenization method:

[0021] (2)

[0022] in, Represents the local material elasticity tensor of a microscopic unit cell. express The equivalent unit cell elastic tensor, with the superscript H indicating the equivalent identifier. express The component with index ijkl in the middle. express The component with index ijpq. Represents the spatial or planar extent of a microstructure cell. The volume or area of ​​a microstructure cell; The overall representation of unit test strain, The overall representation of the sensed strain field, with superscripts 0 and 1. To distinguish between the two identifiers, the subscript pq indicates the spatial direction in the elastic tensor, and kl indicates the index number of the applied test load; sensing strain field Solve using the line equilibrium equations:

[0023] (3)

[0024] in, This represents the gradient of the virtual displacement with respect to the microscopic design variables;

[0025] Based on formula (2), when considering the influence of the uncertainty of the material's elastic modulus, the equivalent unit cell elastic tensor can be further expressed as:

[0026] (4)

[0027] in, This represents the uncertainty range of the material's elastic modulus. The equivalent unit cell elastic tensor interval represents the range of uncertainties in the elastic modulus interval of a material. The range of local material elastic tensors in a microscopic unit cell represents the range of uncertainties in the elastic modulus of a material.

[0028] Furthermore, step 3 specifically includes:

[0029] First, the load amplitude and load direction are quantized using interval uncertainty quantization. The quantized uncertainty characteristics include an upper bound. and the lower world and / or center value With radius Then, the load amplitude and load direction are used as comprehensive indicators. :

[0030] (5)

[0031] (6)

[0032] in, Indicators of comprehensive indicators Quantity, The number of uncertain parameters indicating the direction or magnitude of the load. This represents the i-th comprehensive index; It is the first The median of the composite indicators, It is the first The radius of a comprehensive indicator, This represents the range of magnitudes for the i-th load. Indicates the directional interval of the i-th load;

[0033] The structural response is expressed through a second-order Taylor series expansion. Compared with comprehensive indicators The relationship is described as follows:

[0034] (7)

[0035] in, Represents structural response In comprehensive indicators The second derivative at the midpoint, where N represents the dimension of the uncertain parameter;

[0036] Based on the above relational description, the structural response is expressed through a second-order Taylor series expansion with only diagonal elements. Compared with comprehensive indicators The relationship can be further described as follows:

[0037] (8)

[0038] Then, an improved second-order Taylor series expansion is used to represent the structural response. upper and lower boundaries and Describe respectively for:

[0039] (9)

[0040] (10).

[0041] Furthermore, step 4 specifically includes:

[0042] The gradient of the objective function with respect to the macroscopic design variables is expressed as:

[0043] (11)

[0044] (12)

[0045] in, This represents the upper bound of the structural response under multi-source uncertainty; It is the displacement matrix obtained when the upper bound of the structure is obtained under uncertainties in the magnitude and direction of the load. express transpose;

[0046] Among them, the global stiffness matrix Decomposed into:

[0047] (13)

[0048] in, Represents the element geometric matrix, Indicates the first The integration region of each unit, Represented by the elasticity matrix The first The element stiffness matrix of each element;

[0049] Combining formulas (12) and (13), we get:

[0050] (14)

[0051] in, and For the first The upper bound element displacement vector and the lower bound element displacement vector of each element;

[0052] The gradient of the objective function with respect to the micro-cell design variables is expressed as:

[0053] (15)

[0054] (16)

[0055] Combining formulas (13) and (16), we get:

[0056] (17)

[0057] in, It is the first The elasticity matrix after interpolation of the density of individual micro-cells It is the first The solid elastic matrix of a microscopic unit cell;

[0058] The sensitivity of the objective function to both the macroscopic structure and the volume constraints of the microscopic unit cell is an identity matrix.

[0059] Furthermore, step 5 specifically includes:

[0060] For macro scale, in macro units Define its neighborhood as the center. ,in For macroscopic filtering radius, This represents the spatial geometric distance between the center point of macroscopic unit i and the center point of another neighboring unit f; unit With neighboring units Weighting coefficients for:

[0061] (18)

[0062] Then macro design variables Sensitivity after filtration The calculation formula is:

[0063] (19)

[0064] in, The original sensitivity obtained from step 4, This indicates the sensitivity of the objective function to macroscopic design variables;

[0065] For the microscopic scale, a microscopic filtration radius is set. It adopts the same weighting coefficient form as the macro scale and is denoted as Then micro design variables Sensitivity after filtration The calculation formula is:

[0066] (20)

[0067] The smoothed sensitivity information obtained after filtering and It serves as input to the moving asymptote optimization algorithm for updating design variables.

[0068] Furthermore, in step 6, the convergence criterion is that the difference in material volume used in the two iterations of the macrostructure and microstructure is simultaneously less than a threshold; or the current iteration number reaches the maximum iteration number.

[0069] Secondly, the present invention provides a cross-scale robust topology optimization system considering multi-source interval uncertainties, applied to topology optimization of integrated materials and structures, comprising:

[0070] The topology optimization model construction module is used to provide a macroscopic structural design domain, a microscopic unit cell design domain, loads and boundary conditions, and to characterize the elastic modulus of the material, the amplitude and direction of the external load as interval uncertainty variables. Then, the weighted sum of the upper and lower bounds of the macroscopic structural flexibility response interval is used as the optimization objective, and the macroscopic structural volume fraction and the microscopic unit cell volume fraction are used as constraints to establish a cross-scale robust topology optimization model.

[0071] The unit cell equivalent elastic tensor constraint module is used to solve the upper and lower bounds of the unit cell equivalent elastic tensor considering the uncertainty of the material elastic modulus interval based on the homogenization method and the interval vertex method.

[0072] The structural flexibility response constraint module is used to predict the upper and lower bounds of the structural flexibility response under the influence of uncertainties in the load amplitude and direction range, based on the improved second-order Taylor expansion method.

[0073] The sensitivity calculation module is used to calculate the sensitivity of the objective function to macro and micro design variables in the cross-scale robust topology optimization model based on the adjoint method.

[0074] The design variable update module is used to apply sensitivity filtering technology and employs a moving asymptote optimization algorithm to synchronously update macroscopic structural design and microscopic unit cell design variables.

[0075] The convergence judgment module is used to determine whether the optimization process meets the convergence criteria. If it does, the final macroscopic structural design and the corresponding macroscopic and microscopic topological configurations of the microscopic unit cell design variables are output as the optimization result. If it does not meet the criteria, the iteration continues.

[0076] Thirdly, the present invention provides an electronic device including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor, when executing the program, implements the method as described in the first aspect.

[0077] Fourthly, the present invention provides a non-transitory computer-readable storage medium having a computer program stored thereon, wherein the computer program, when executed by a processor, implements the method described in the first aspect.

[0078] The beneficial effects of this invention are as follows:

[0079] This invention considers the influence of the interval uncertainty of material elastic modulus, load magnitude and direction on structural response, and takes into account the robustness of the structure. It establishes an optimization model with the goal of maximizing structural stiffness and the volume fraction of macroscopic structure and microscopic unit cell as constraints. Based on the optimization criterion method, it realizes the robust topology optimization design of cross-scale structure under multi-source interval uncertainty of materials, loads and other factors. This will greatly promote the application scope of material structure integration and design and manufacturing integration.

[0080] This invention presents a robust topology optimization method for cross-scale intervals, based on improved Taylor series expansion and considering multi-source uncertainties. It establishes a unified model for handling multi-source interval uncertainties such as material properties, load amplitude, and direction. The method constructs a weighted sum of the upper and lower bounds of the structural response interval as the robustness optimization objective. Based on asymptotic homogenization and interval vertex methods, it efficiently determines the equivalent elastic performance interval of the unit cell under the influence of material uncertainties. Addressing the non-monotonic relationship between load direction and structural response, an improved second-order Taylor series expansion method is proposed. By analyzing the concavity and convexity of the function and the location of extreme points, the boundary of the structural response interval is accurately predicted. Based on the adjoint variable method, the sensitivity of the interval robustness objective function and volume constraints to macro- and micro-density design variables is efficiently analyzed. Combining sensitivity filtering technology and moving asymptote optimization algorithms, the coordinated and stable updating of macro- and micro-design variables is achieved. This enables the comprehensive optimization and balance of the average performance, robustness, and manufacturability of the structure under uncertain environments. Attached Figure Description

[0081] Figure 1 A flowchart illustrating a cross-scale robust topology optimization method considering multi-source interval uncertainties provided in an embodiment of the present invention;

[0082] Figure 2 The geometric model and boundary condition diagram provided for a cross-scale topology optimization problem of cantilever beams are provided in this embodiment of the invention.

[0083] Figure 3 , Figure 4 , Figure 5 , Figure 6 , Figure 7 , Figure 8 , Figure 9 , Figure 10 , Figure 11 , Figure 12 , Figure 13 , Figure 14 , Figure 15 , Figure 16 , Figure 17 The configuration results and iterative process under various operating conditions are as follows; among which: Figure 3 , Figure 4 , Figure 5The macroscopic configuration, microscopic configuration, and iterative process of operating condition 1; Figure 6 , Figure 7 , Figure 8 The macroscopic configuration, microscopic configuration, and iterative process of operating condition 2; Figure 9 , Figure 10 , Figure 11 The macroscopic configuration, microscopic configuration, and iterative process of operating condition 3; Figure 12 , Figure 13 , Figure 14 This refers to the macroscopic configuration, microscopic configuration, and iterative process of operating condition 4. Figure 15 , Figure 16 , Figure 17 This refers to the macroscopic configuration, microscopic configuration, and iterative process of operating condition 5.

[0084] Figure 18 This is a structural block diagram of an electronic device provided in an embodiment of the present invention. Detailed Implementation

[0085] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of the embodiments of this invention will be clearly described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of this invention. All other embodiments obtained by those skilled in the art based on the embodiments of this invention without creative effort are within the scope of protection of this invention.

[0086] like Figure 1 As shown, this invention proposes a robust cross-scale topology optimization method based on improved Taylor series expansion, considering multi-source uncertainties. This method is used in the integrated material and structure topology optimization design to achieve robust cross-scale topology optimization design of cross-scale structures under concentrated loads and fixed / simply supported boundary constraints, under the influence of multi-source interval uncertainties. It reasonably characterizes and quantifies the comprehensive impact of interval uncertainties from materials and the environment on the structural configuration. The implementation steps are as follows:

[0087] S101: Establish a robust topology optimization model across scales.

[0088] Specifically, given the macroscopic structural design domain, the microscopic unit cell design domain, the load and boundary conditions, and the elastic modulus of the material, the amplitude and direction of the external load are characterized as interval uncertainty variables, the weighted sum of the upper and lower bounds of the macroscopic structural flexibility response interval is used as the optimization objective, and the macroscopic structural volume fraction and the microscopic unit cell volume fraction are used as constraints to establish a mathematical model for cross-scale robust topology optimization.

[0089] In this embodiment of the invention, a robust cross-scale topology optimization mathematical model based on density interpolation and combined with improved Taylor series expansion is established, specifically including the following formulas:

[0090] (1)

[0091] in, and These are the design variable vectors for the macroscopic and microscopic design domains, respectively. Represents the i-th macroscopic design variable. Let represent the j-th micro-design variable, and M and N represent the number of macro- and micro-design variables, respectively. and These represent the uncertain intervals of the magnitude and direction of the external load, respectively. This represents the uncertainty range of the material's elastic modulus. and These are the upper and lower bounds of the structural response under multi-source uncertainty. and It is a weighted coefficient for the mean and difference of structural flexibility. Indicates that in a given The global stiffness matrix under, Indicates that in a given The global displacement vector below, In the given The global force vector below, and These are the macroscopic and microscopic volume constraint functions, respectively. and These refer to the volume of materials used in the macro-design domain and the micro-design domain, respectively. and It is the allowable volume fraction of the macroscopic design domain and the microscopic design domain; and This represents the volume of macroscopic and microscopic unit cells. and Design variables representing macroscopic and microscopic unit cells, It is the minimum elastic modulus of the material to prevent singularity in the stiffness matrix; It is a penalty factor introduced to ensure a black-and-white solution. and They represent and p to the power of 0.

[0092] Equation (1) above indicates that the optimization model takes the combination of the upper and lower bounds of the structural flexibility as the optimization objective, the volume fraction of the macrostructure and the micro unit cell as the constraint, and the unit density of the macrostructure and the microstructure as the design variable. For the structural statics problem, considering the uncertainty of the elastic modulus inside the material and the influence of the load magnitude and direction uncertainty of the service environment, robust topology optimization design is carried out at both the macrostructure and micro unit cell scales.

[0093] S102: Based on the homogenization method and the interval vertex method, solve the upper and lower bounds of the unit cell equivalent elastic tensor considering the uncertainty of the material elastic modulus interval.

[0094] Specifically, the equivalent mechanical properties of a single cell are calculated based on a homogenization method:

[0095] (2)

[0096] in, Represents the local material elasticity tensor of a microscopic unit cell. express The equivalent unit cell elastic tensor, with the superscript H indicating the equivalent identifier. express The component with index ijkl in the middle. express The component indexed by ijpq is used in the elastic tensor, which is a fourth-order tensor. Therefore, four subscripts are used to index a specific component in the elastic tensor. Represents the spatial or planar extent of a microstructure cell. The volume or area of ​​a microstructure cell; The overall representation of unit test strain, The overall representation of the sensed strain field, with superscripts 0 and 1. To distinguish between the two identifiers, the subscript pq indicates the spatial direction in the elastic tensor, and kl indicates the index number of the applied test load; sensing strain field Solve using the line equilibrium equations:

[0097] (3)

[0098] in, This represents the gradient of the virtual displacement with respect to the microscopic design variables.

[0099] Based on formula (2), when considering the influence of the uncertainty of the material's elastic modulus, the equivalent unit cell elastic tensor is further expressed as:

[0100] (4)

[0101] in, This represents the uncertainty range of the material's elastic modulus. The equivalent unit cell elastic tensor interval represents the range of uncertainties in the elastic modulus interval of a material. The range of local material elastic tensors in a microscopic unit cell represents the range of uncertainties in the elastic modulus of a material.

[0102] Numerical simulation experiments revealed a monotonic functional relationship between the elastic modulus and structural flexibility. Therefore, the vertex method was used to calculate the upper and lower bounds of structural flexibility, with the calculation results at the endpoints of the interval serving as the upper and lower bounds of structural flexibility under the uncertainty of the material's elastic modulus interval. The uncertainty of the unit cell equivalent performance was calculated numerically from the interval uncertainty of the material's elastic modulus, and the uncertainty of the unit cell performance was used as a parameter input for structural design.

[0103] S103: Based on the improved second-order Taylor series expansion method, predict the upper and lower bounds of the structural flexibility response under the influence of uncertainties in the load amplitude and direction interval.

[0104] Specifically, considering the non-monotonic relationship between the load direction and the structural response, a second-order Taylor series expansion is performed at the midpoint of the interval, and the off-diagonal elements of the Hessian matrix are ignored to simplify the calculation. The first and second derivatives are calculated using numerical difference, and the concavity and vertex position of the resulting quadratic function are analyzed. Based on the decision table regarding the positional relationship between the vertex and the uncertainty interval, the interval variable values ​​that allow the structural response to achieve upper and lower bounds are determined. Combined with the equivalent unit cell elastic tensor under material uncertainty obtained in step S102, the upper and lower bounds of the structural compliance response under the combined influence of multiple source interval uncertainties are calculated. The specific process is as follows:

[0105] First, the load amplitude and load direction are quantized using interval uncertainty quantization. The quantized uncertainty characteristics include an upper bound. and the lower world and / or center value With radius Then, the load amplitude and load direction are used as comprehensive indicators. :

[0106] (5)

[0107] (6)

[0108] in, Indicators of comprehensive indicators Quantity, The number of uncertain parameters indicating the direction or magnitude of the load. This represents the i-th comprehensive index. It is an abstract concept that includes the magnitude and direction of the load, provided that each load has a corresponding range of magnitude and direction. It is the first The median of the composite indicators, It is the first The radius of a comprehensive indicator, This represents the range of magnitudes for the i-th load. This represents the directional range of the i-th load.

[0109] The structural response is expressed through a second-order Taylor series expansion. Compared with comprehensive indicators The relationship is described as follows:

[0110] (7)

[0111] in, Represents structural response In comprehensive indicators The second derivative at the midpoint, where N represents the dimension of the uncertain parameter;

[0112] Based on the above description of the relationship, the structural response is expressed by a second-order Taylor series expansion with only diagonal elements. Compared with comprehensive indicators The relationship can be further described as follows:

[0113] (8)

[0114] Then, an improved second-order Taylor series expansion is used to represent the structural response. upper and lower boundaries and Describe respectively for:

[0115] (9)

[0116] (10)

[0117] pass and The relationship between these factors is used to determine the approximate location of the stationary point in the structural response. The determination process is as follows:

[0118] The sign of ... The value of corresponds to the distance between the extreme point of the interval and the midpoint of the interval, while With interval radius The comparison is to determine whether the predicted extreme point is inside the interval.

[0119] S104: Based on the adjoint method, calculate the sensitivity of the objective function to macro and micro design variables in the cross-scale robust topology optimization model.

[0120] Specifically, based on the robust cross-scale topology optimization model considering multi-source uncertainties based on second-order Taylor series expansion given in step S101, the sensitivity information of the objective function and constraints in the optimization model is solved using the adjoint method.

[0121] The gradient of the objective function with respect to the macroscopic design variables is expressed as:

[0122] (11)

[0123] (12)

[0124] in, This represents the upper bound of the structural response under multi-source uncertainty, which is related to... It is a concept. This refers to the upper bound of the structural response that does not consider material uncertainties but only the uncertainties of load magnitude and direction. This concept means that the material is separated in the analysis process, and the elastic modulus endpoint is directly used for analysis in the final calculation (because of the monotonic relationship between the elastic modulus and the unit cell elastic tensor). It is the displacement matrix obtained when the upper bound of the structure is obtained under uncertainties in the magnitude and direction of the load. express The transpose of .

[0125] Among them, the global stiffness matrix Decomposed into:

[0126] (13)

[0127] in, Represents the element geometric matrix, Indicates the first The integration region of each unit, Represented by the elasticity matrix The first The element stiffness matrix of each element;

[0128] Combining formulas (12) and (13), we get:

[0129] (14)

[0130] in, and For the first The upper bound element displacement vector and the lower bound element displacement vector of each element;

[0131] The gradient of the objective function with respect to the micro-cell design variables is expressed as:

[0132] (15)

[0133] (16)

[0134] Combining formulas (13) and (16), we get:

[0135] (17)

[0136] in, It is the first The elasticity matrix after interpolation of the density of individual micro-cells It is the first The solid elastic matrix of a microscopic unit cell;

[0137] The sensitivity of the objective function to both the macroscopic structure and the volume constraints of the microscopic unit cell is an identity matrix.

[0138] S105: Apply sensitivity filtering technology and adopt a moving asymptote optimization algorithm to simultaneously update macroscopic structural design and microscopic unit cell design variables.

[0139] Specifically, after obtaining the sensitivity of the objective function with respect to macro and micro design variables using the adjoint vector method, sensitivity filtering technology is used to filter the sensitivity to reduce grayscale units.

[0140] For macro scale, in macro units Define its neighborhood as the center. ,in For macroscopic filtering radius, This represents the spatial geometric distance between the center point of macroscopic unit i and the center point of another neighboring unit f; unit With neighboring units Weighting coefficients for:

[0141] (18)

[0142] Then macro design variables Sensitivity after filtration The calculation formula is:

[0143] (19)

[0144] in, The original sensitivity obtained from step 4; This indicates the sensitivity of the objective function to macroscopic design variables.

[0145] For the microscopic scale, a microscopic filtration radius is set. It adopts the same weighting coefficient form as the macro scale and is denoted as Then micro design variables Sensitivity after filtration The calculation formula is:

[0146] (20)

[0147] The smoothed sensitivity information obtained after filtering and It serves as input to the moving asymptote optimization algorithm for updating design variables.

[0148] In this embodiment, the filtering operation is equivalent to local averaging of the design variable field, which can effectively eliminate the checkerboard pattern in the optimization results and indirectly introduce minimum length scale control, thereby enhancing the manufacturability and structural rationality of the final topology. The smoothing sensitivity information obtained after filtering... and The inputs will be used as inputs to the Moving Asymptote Optimization (MAO) algorithm to update the design variables. The MMA algorithm, a well-known optimization algorithm in the field, is used to update both macro and micro design variables.

[0149] S106: Determine whether the optimization process meets the convergence criterion; if it does, output the final macro- and micro-topological configurations as the optimization result; if it does not, return to step two to continue the iteration.

[0150] Specifically, in the convergence criteria, the first two conditions are that the difference in material volume used in the two iterations of the macroscopic structure and the difference in material volume used in the two iterations of the microscopic unit cell must both be less than the threshold; or the current iteration number reaches the maximum iteration number.

[0151] (twenty one)

[0152] This condition indicates that the difference between the macroscopic and microscopic volume fractions of the two structural optimizations must be less than the threshold or reach the maximum number of iterations at the same time. When this condition is met, the iteration terminates and the final topology optimization design of the structure is output. If the topology optimization iteration does not meet the convergence condition, steps S102 to S105 are repeated to continue driving structural changes to guide the iteration termination.

[0153] This invention discloses a robust cross-scale topology optimization method based on improved Taylor series expansion and considering multi-source uncertainties. In step S101, a cross-scale topology optimization model considering the influence of multi-source interval uncertainties is established, including constraints on macro- and micro-volume fractions. In step S102, equivalent unit cell performance is achieved based on a homogenization method, and the equivalent unit cell performance under the influence of material elastic modulus uncertainty is analyzed using numerical simulation. In step S103, considering the uncertainty characteristics of load amplitude and load direction, the structural mechanical behavior under the influence of interval uncertainties is analyzed using a second-order Taylor series expansion method. In step S104, the sensitivity of the objective function to macro- and micro-design variables, as well as macro- and micro-volume fractions, is analyzed based on the adjoint vector method. In step S105, the established macro- and micro-parallel robust topology optimization model is solved using the MMA optimization criterion.

[0154] Compared with existing technologies, this invention analyzes the impact of uncertainties in the elastic modulus range of materials on the dispersion of equivalent properties of unit cells based on the homogenization method, analyzes the impact of multi-source uncertainties on structural mechanical behavior, and uses the second-order Taylor series expansion method to analyze uncertainty indices, thereby reducing numerical calculation time while ensuring the accuracy of optimization results.

[0155] The robust cross-scale interval topology optimization method based on improved Taylor expansion and considering multi-source uncertainties disclosed in this invention is applicable to cross-scale topology optimization design of arbitrary planar structures, such as... Figure 2 The cross-scale topology optimization problem of the cantilever beam shown in this invention can be explained by employing a robust cross-scale interval topology optimization method based on improved Taylor series expansion and considering multi-source uncertainties, as proposed in this invention. This method can illustrate the influence of uncertainties of different magnitudes and types under different weighting coefficients in the topology optimization design. The design region is a rectangular region, macroscopically divided into 100×50 elements and microscopically into 50×50 elements, with the material elastic modulus ranging from [specific value range]. Poisson's ratio A fixed constraint is applied to the left boundary of the rectangle, and a concentrated load is applied to the lower right corner. The key parameter settings are shown in Table 1.

[0156] The configuration and iteration history curves of topology optimization under different operating conditions are as follows: Figures 3 to 17 As shown, where Figure 3 , Figure 4 , Figure 5 The macroscopic configuration, microscopic configuration, and iterative process of operating condition 1; Figure 6 , Figure 7 , Figure 8 The macroscopic configuration, microscopic configuration, and iterative process of operating condition 2; Figure 9 , Figure 10 , Figure 11 The macroscopic configuration, microscopic configuration, and iterative process of operating condition 3; Figure 12 , Figure 13 , Figure 14 This refers to the macroscopic configuration, microscopic configuration, and iterative process of operating condition 4. Figure 15 , Figure 16 , Figure 17 This refers to the macroscopic configuration, microscopic configuration, and iterative process of operating condition 5; among which... Figure 5 , Figure 8 , Figure 11 , Figure 14 , Figure 17 The variable It represents the number of iterations. Table 2 shows the weight coefficients, elastic tensors of the unit cell, and objective function data for topology optimization under different operating conditions.

[0157] Table 1 Key Parameter Settings

[0158]

[0159] Table 2 Comparison of Working Condition Results

[0160]

[0161] Results explanation:

[0162] Comparing the macroscopic topology under different operating conditions reveals a systematic change in the macroscopic structural topology when the weighting coefficient shifts from emphasizing average performance to emphasizing robustness. Specifically, this manifests as: the main load-bearing paths gradually thickening, the number of branch structures increasing, and the topological distribution becoming more uniform and symmetrical. This reflects the structural tendency to adopt more redundant and robust configurations to reduce performance fluctuations caused by uncertain loads.

[0163] The configuration of the micro-unit cell also evolved in tandem with the weighting coefficients. As the requirements for robustness increased, unit cell design tended to produce higher structural connectivity and isotropy, and the numerical range characteristics of its equivalent elastic tensor also changed accordingly, in order to support the robustness requirements of macroscopic structures from the material level.

[0164] The change in the objective function value is not monotonic; it reflects a comprehensive compromise between "average compliance" and "compliance variability" under specific weights. Condition 5 yielded the minimum objective function value, indicating that the design is optimal in terms of the single metric of "minimizing compliance variability," but its average performance may not be optimal.

[0165] This embodiment clearly demonstrates that, by adjusting the weighting coefficients, the method of this invention can provide designers with a series of continuous Pareto optimal solutions ranging from "high performance" to "high robustness". This proves that the interval robustness objective function constructed in this invention is effective, and that the optimization framework has the ability to flexibly generate optimal topology configurations that match different engineering robustness requirements.

[0166] This embodiment confirms the core feature of the method of the present invention by systematically changing the weighting factors—using the weighted sum of the upper and lower bounds of the interval response as the objective function, it can intuitively and quantitatively guide cross-scale topology optimization design to make a trade-off between "average performance" and "performance robustness", thereby meeting diverse engineering design requirements.

[0167] Based on the same inventive concept, this invention also provides a cross-scale robust topology optimization system that considers uncertainties in multiple source intervals, applied to the topology optimization of integrated materials and structures, including: a topology optimization model construction module, a unit cell equivalent elastic tensor constraint module, a structural flexibility response constraint module, a sensitivity calculation module, a design variable update module, and a convergence judgment module.

[0168] Specifically, the topology optimization model construction module is used to construct a robust cross-scale topology optimization model given the macroscopic structural design domain, microscopic unit cell design domain, loads, and boundary conditions. It characterizes the material's elastic modulus, the amplitude and direction of the external loads as interval uncertainty variables. Then, it uses the weighted sum of the upper and lower bounds of the macroscopic structural flexibility response interval as the optimization objective, and the macroscopic structural volume fraction and microscopic unit cell volume fraction as constraints. The unit cell equivalent elastic tensor constraint module is used to solve for the upper and lower bounds of the unit cell equivalent elastic tensor considering the interval uncertainty of the material's elastic modulus based on homogenization and interval vertex methods. The structural flexibility response constraint module is used for solving for the upper and lower bounds of the unit cell equivalent elastic tensor based on an improved second-order... The Taylor expansion method is used to predict the upper and lower bounds of the structural compliance response under the influence of uncertainties in the load amplitude and direction intervals. The sensitivity calculation module is used to calculate the sensitivity of the objective function in the cross-scale robust topology optimization model to macro and micro design variables based on the adjoint method. The design variable update module is used to apply sensitivity filtering technology and adopt the moving asymptote optimization algorithm to simultaneously update the macro-structural design and micro-unit cell design variables. The convergence judgment module is used to determine whether the optimization process meets the convergence criterion. If it does, the final macro-structural design and micro-unit cell design variables corresponding to the macro and micro topological configurations are output as the optimization result. If they do not meet the criteria, the iteration continues.

[0169] It should be noted that the cross-scale robust topology optimization system considering multi-source interval uncertainty provided in this embodiment of the invention is to implement the above method embodiments. Its specific functions can be referred to the above method embodiments, and will not be repeated here.

[0170] Figure 18 An example is a schematic diagram of the physical structure of an electronic device, such as... Figure 18As shown, the electronic device may include: a processor 1801, a communications interface 1802, a memory 1803, and a communications bus 1804, wherein the processor 1801, the communications interface 1802, and the memory 1803 communicate with each other through the communications bus 1804. Processor 1801 can call logic instructions in memory 1803 to execute a cross-scale robust topology optimization method considering multi-source interval uncertainties. This method includes: Step 1: Given a macroscopic structural design domain, a microscopic unit cell design domain, loads, and boundary conditions, and characterizing the material's elastic modulus, the amplitude and direction of external loads as interval uncertainty variables, then using the weighted sum of the upper and lower bounds of the macroscopic structural compliance response interval as the optimization objective, and using the macroscopic structural volume fraction and the microscopic unit cell volume fraction as constraints, establishing a cross-scale robust topology optimization model; Step 2: Based on the homogenization method and the interval vertex method, solving for the upper and lower bounds of the unit cell equivalent elastic tensor considering the interval uncertainty of the material's elastic modulus; Step 3: Based on the improved second-order Taylor expansion method, predicting the upper and lower bounds of the structural compliance response under the influence of interval uncertainties in load amplitude and direction; Step 4: Based on the adjoint method, calculating the sensitivity of the objective function in the cross-scale robust topology optimization model to macroscopic and microscopic design variables; Step 5: Sensitivity filtering technology is applied, and the moving asymptote optimization algorithm is used to simultaneously update the macroscopic structural design and microscopic unit cell design variables; Step 6: Determine whether the optimization process meets the convergence criterion; if it does, output the macroscopic and microscopic topological configurations corresponding to the final macroscopic structural design and microscopic unit cell design variables as the optimization result; if it does not meet the criterion, return to step 2 to continue the iteration.

[0171] Furthermore, when the logical instructions in the aforementioned memory 1803 are implemented as software functional units and sold or used as independent products, they can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present invention, in essence, or the part that contributes to the prior art, or a part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of the present invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0172] This invention also provides a computer program product, which includes a computer program stored on a non-transitory computer-readable storage medium. The computer program includes program instructions, and when the program instructions are executed by a computer, the computer can execute a cross-scale robust topology optimization method considering multi-source interval uncertainties provided in the above-described method embodiments.

[0173] This invention also provides a non-transitory computer-readable storage medium storing a computer program thereon, which, when executed by a processor, implements a cross-scale robust topology optimization method considering multi-source interval uncertainties provided in the above-described method embodiments.

[0174] Through the above description of the embodiments, those skilled in the art can clearly understand that each embodiment can be implemented by means of software plus necessary general-purpose hardware platforms, and of course, it can also be implemented by hardware. Based on this understanding, the above technical solutions, in essence or the part that contributes to the prior art, can be embodied in the form of a software product. This computer software product can be stored in a computer-readable storage medium, such as ROM / RAM, magnetic disk, optical disk, etc., and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute the methods described in the various embodiments or some parts of the embodiments.

[0175] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A cross-scale robust topology optimization method considering multi-source interval uncertainties, applied to the topology optimization of integrated material structures, characterized in that, include: Step 1: Given the macroscopic structural design domain, microscopic unit cell design domain, load and boundary conditions, and characterize the elastic modulus of the material, the amplitude and direction of the external load as interval uncertainty variables, then use the weighted sum of the upper and lower bounds of the macroscopic structural compliance response interval as the optimization objective, and use the macroscopic structural volume fraction and microscopic unit cell volume fraction as constraints to establish a cross-scale robust topology optimization model. Step 2: Based on the homogenization method and the interval vertex method, solve for the upper and lower bounds of the unit cell equivalent elastic tensor considering the uncertainty of the material's elastic modulus interval; Step 3: Based on the improved second-order Taylor expansion method, predict the upper and lower bounds of the structural compliance response under the influence of uncertainties in the load amplitude and direction intervals; Step 4: Based on the adjoint method, calculate the sensitivity of the objective function in the cross-scale robust topology optimization model to macro and micro design variables; Step 5: Apply sensitivity filtering technology and use the moving asymptote optimization algorithm to simultaneously update the macroscopic structural design and microscopic unit cell design variables; Step 6: Determine whether the optimization process meets the convergence criteria; If satisfied, the final macroscopic structural design and microscopic unit cell design variables corresponding to the macroscopic and microscopic topological configurations are output as the optimization results. If the condition is not met, return to step 2 and continue iterating.

2. The cross-scale robust topology optimization method considering multi-source interval uncertainties according to claim 1, characterized in that, The expression for the cross-scale robust topology optimization model is: （1） in, and These are the design variable vectors for the macroscopic and microscopic design domains, respectively. Represents the i-th macroscopic design variable. Let represent the j-th micro-design variable, and M and N represent the number of macro- and micro-design variables, respectively. and These represent the uncertain intervals of the magnitude and direction of the external load, respectively. This represents the uncertainty range of the material's elastic modulus. and These are the upper and lower bounds of the structural response under multi-source uncertainty. and It is a weighted coefficient for the mean and difference of structural flexibility. Indicates that in a given The global stiffness matrix under, Indicates that in a given The global displacement vector below, In the given The global force vector below, and These are the macroscopic and microscopic volume constraint functions, respectively. and These refer to the volume of materials used in the macro-design domain and the micro-design domain, respectively. and It is the allowable volume fraction of the macroscopic design domain and the microscopic design domain; and This represents the volume of macroscopic and microscopic unit cells. and Design variables representing macroscopic and microscopic unit cells, It is the minimum elastic modulus of the material. It is a penalty factor introduced to ensure a black-and-white solution. and They represent and p to the power of 0.

3. The cross-scale robust topology optimization method considering multi-source interval uncertainty according to claim 1, characterized in that, Step 2 specifically includes: Calculation of unit cell equivalent mechanical properties based on homogenization method: （2） in, Represents the local material elasticity tensor of a microscopic unit cell. express The equivalent unit cell elastic tensor, with the superscript H indicating the equivalent identifier. express The component with index ijkl in the middle. express The component with index ijpq. Represents the spatial or planar extent of a microstructure cell. The volume or area of ​​a microstructure cell; The overall representation of unit test strain, The overall representation of the sensed strain field, with superscripts 0 and 1. To distinguish between the two identifiers, the subscript pq indicates the spatial direction in the elastic tensor, and kl indicates the index number of the applied test load; sensing strain field Solve using the line equilibrium equations: （3） in, This represents the gradient of the virtual displacement with respect to the microscopic design variables; Based on formula (2), when considering the influence of the uncertainty of the material's elastic modulus, the equivalent unit cell elastic tensor can be further expressed as: （4） in, This represents the uncertainty range of the material's elastic modulus. The equivalent unit cell elastic tensor interval represents the range of uncertainties in the elastic modulus interval of a material. The range of local material elastic tensors in a microscopic unit cell represents the range of uncertainties in the elastic modulus of a material.

4. The cross-scale robust topology optimization method considering multi-source interval uncertainties according to claim 2, characterized in that, Step 3 specifically includes: First, the load amplitude and load direction are quantized using interval uncertainty quantization. The quantized uncertainty characteristics include an upper bound. and the lower world and / or center value With radius Then, the load amplitude and load direction are used as comprehensive indicators. : （5） （6） in, Indicators of comprehensive indicators Quantity, The number of uncertain parameters indicating the direction or magnitude of the load. This represents the i-th comprehensive index; It is the first The median of the composite indicators, It is the first The radius of a comprehensive indicator, This represents the range of magnitudes for the i-th load. Indicates the directional interval of the i-th load; The structural response is expressed through a second-order Taylor series expansion. Compared with comprehensive indicators The relationship is described as follows: （7） in, Represents structural response In comprehensive indicators The second derivative at the midpoint, where N represents the dimension of the uncertain parameter; Based on the above relational description, the structural response is expressed through a second-order Taylor series expansion with only diagonal elements. Compared with comprehensive indicators The relationship can be further described as follows: （8） Then, an improved second-order Taylor series expansion is used to represent the structural response. upper and lower boundaries and Describe respectively for: （9） （10）。 5. A cross-scale robust topology optimization method considering multi-source interval uncertainties according to claim 4, characterized in that, Step 4 specifically includes: The gradient of the objective function with respect to the macroscopic design variables is expressed as: （11） （12） in, This represents the upper bound of the structural response under multi-source uncertainty; It is the displacement matrix obtained when the upper bound of the structure is obtained under uncertainties in the magnitude and direction of the load. express transpose; Among them, the global stiffness matrix Decomposed into: （13） in, Represents the element geometric matrix, Indicates the first The integration region of each unit, Represented by the elasticity matrix The first The element stiffness matrix of each element; Combining formulas (12) and (13), we get: （14） in, and For the first The upper bound element displacement vector and the lower bound element displacement vector of each element; The gradient of the objective function with respect to the micro-cell design variables is expressed as: （15） （16） Combining formulas (13) and (16), we get: （17） in, It is the first The elasticity matrix after interpolation of the density of individual micro-cells It is the first The solid elastic matrix of a microscopic unit cell; The sensitivity of the objective function to both the macroscopic structure and the volume constraints of the microscopic unit cell is an identity matrix.

6. The cross-scale robust topology optimization method considering multi-source interval uncertainties according to claim 1, characterized in that, Step 5 specifically includes: For macro scale, in macro units Define its neighborhood as the center. ,in For macroscopic filtering radius, This represents the spatial geometric distance between the center point of macroscopic unit i and the center point of another neighboring unit f; unit With neighboring units Weighting coefficients for: （18） Then macro design variables Sensitivity after filtration The calculation formula is: （19） in, The original sensitivity obtained from step 4, This indicates the sensitivity of the objective function to macroscopic design variables; For the microscopic scale, a microscopic filtration radius is set. It adopts the same weighting coefficient form as the macro scale and is denoted as Then micro design variables Sensitivity after filtration The calculation formula is: （20） The smoothed sensitivity information obtained after filtering and It serves as input to the moving asymptote optimization algorithm for updating design variables.

7. The cross-scale robust topology optimization method considering multi-source interval uncertainties according to claim 1, characterized in that, In step 6, the convergence criterion is that the difference in material volume used in the two iterations of the macrostructure and microstructure is less than a threshold at the same time; or the current iteration number reaches the maximum iteration number.

8. A cross-scale robust topology optimization system considering multi-source interval uncertainties, applied to topology optimization of integrated materials and structures, characterized in that, include: The topology optimization model construction module is used to provide a macroscopic structural design domain, a microscopic unit cell design domain, loads and boundary conditions, and to characterize the elastic modulus of the material, the amplitude and direction of the external load as interval uncertainty variables. Then, the weighted sum of the upper and lower bounds of the macroscopic structural flexibility response interval is used as the optimization objective, and the macroscopic structural volume fraction and the microscopic unit cell volume fraction are used as constraints to establish a cross-scale robust topology optimization model. The unit cell equivalent elastic tensor constraint module is used to solve the upper and lower bounds of the unit cell equivalent elastic tensor considering the uncertainty of the material elastic modulus interval based on the homogenization method and the interval vertex method. The structural flexibility response constraint module is used to predict the upper and lower bounds of the structural flexibility response under the influence of uncertainties in the load amplitude and direction range, based on the improved second-order Taylor expansion method. The sensitivity calculation module is used to calculate the sensitivity of the objective function to macro and micro design variables in the cross-scale robust topology optimization model based on the adjoint method. The design variable update module is used to apply sensitivity filtering technology and employs a moving asymptote optimization algorithm to synchronously update macroscopic structural design and microscopic unit cell design variables. The convergence judgment module is used to determine whether the optimization process meets the convergence criteria; If satisfied, the final macroscopic structural design and microscopic unit cell design variables corresponding to the macroscopic and microscopic topological configurations are output as the optimization results. If the condition is not met, continue iterating.

9. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the program, it implements the method as described in any one of claims 1 to 7.

10. A non-transitory computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by a processor, it implements the method as described in any one of claims 1 to 7.

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