Lattice structure based on topology optimization, design method and application

Abstract

The application relates to the technical field of lattice structure optimization, and discloses a lattice structure based on topological optimization, a design method and application, a topological optimization model is constructed, and physical parameters and optimization parameters of the topological optimization model are initialized; a structure design domain corresponding to the topological optimization model, a load and boundary conditions are defined, and a finite element model is established; the defined load and boundary conditions are applied to the topological optimization model, finite element analysis is carried out, displacement responses of each node on the design domain are obtained, and the sensitivity of structural flexibility and volume to design variables is analyzed; the sensitivity is filtered, and the material density represented by the design variable is filtered by adopting a projection method based on a Heaviside function to inhibit the chessboard phenomenon; the design variable is updated according to the filtered sensitivity and the material density; whether the current iteration result meets a convergence condition is checked, if yes, the result is output, and if not, iteration is continued.

Description

Technical Field

[0001] This invention relates to the field of lattice structure optimization technology, and in particular, to a lattice structure, design method and application based on topology optimization. Background Technology

[0002] Due to their excellent specific strength, specific stiffness, and multifunctional properties, crystal lattice structures have shown broad application prospects in lightweight design in industrial fields such as aerospace, automotive manufacturing, and biomedicine. Currently, conventional crystal lattice structure designs mostly adopt predefined uniform lattice types, such as body-centered cubic lattices, face-centered cubic lattices, or simple combinations thereof. Although such predefined lattices offer certain conveniences in modeling and manufacturing, their unit cell configurations are not specifically optimized for the actual stress state of the specific structure, resulting in material distribution that cannot fully adapt to actual load conditions and making it difficult to maximize material utilization efficiency.

[0003] Topology optimization, as an advanced structural design method, can seek the optimal material distribution within the design domain under given loads and constraints, thereby obtaining a lightweight configuration with excellent performance. However, in current technical practices, topology optimization methods and lattice structure design are often independent of each other, failing to achieve effective synergy. Common design approaches can be broadly divided into two categories: First, after completing the main structural design, pre-set uniform lattice units are simply filled into non-critical areas. While this method utilizes the lightweight properties of lattices, it does not incorporate topology optimization to personalize the lattice units, limiting further improvements in structural performance. Second, the optimal layout of the solid material is directly obtained through topology optimization. Although this can achieve high material utilization, it fails to introduce the unique lightweight and multifunctional potential of lattice structures, limiting the design results in further lightweighting and functional integration.

[0004] Neither of the above two methods has achieved a deep integration between topology optimization and lattice structure design. As a result, the existing lattice structures still have significant shortcomings in terms of load-bearing efficiency, lightweight level and functional adaptability, making it difficult to meet the increasingly demanding requirements of high-performance equipment for comprehensive structural performance. Summary of the Invention

[0005] This invention provides a crystal structure, design method, and application based on topology optimization, which can effectively coordinate topology optimization and crystal configuration design. Through multi-condition topology optimization, a high-performance crystal structure is generated to achieve a synergistic improvement in the overall performance and lightweight level of the structure. This solves the technical problem in the prior art where topology optimization design and crystal structure design are disconnected, resulting in low utilization of structural materials and difficulty in further improving the lightweight level and comprehensive performance.

[0006] According to one aspect of the present invention, a design method for a crystal structure based on topology optimization is provided, comprising the following steps: S100, constructing a topology optimization model with minimizing structural flexibility as the optimization objective and structural volume as the constraint, and initializing the physical parameters and optimization parameters of the topology optimization model; S200, defining the structural design domain, loads, and boundary conditions corresponding to the topology optimization model according to the crystal design requirements, and establishing a finite element model; S300, applying the defined loads and boundary conditions to the topology optimization model, performing finite element analysis, obtaining the displacement response of each node in the design domain, and analyzing the sensitivity of structural flexibility and volume to design variables; S400, filtering the sensitivity and filtering the material density represented by the design variables using a projection method based on the Heaviside function to suppress the checkerboard phenomenon; S500, updating the design variables using a moving asymptotic method based on the filtered sensitivity and material density; S600, checking whether the current iteration result meets the convergence condition, outputting the result if it does, and returning to step S300 to continue the iteration if it does not meet the condition.

[0007] Furthermore, the topology optimization model in step S100 is as follows:

[0008] ;

[0009] in, Indicates the density of different units, e Indicates the unit number; C Indicates the flexibility of the structure. T Represents the transpose of a matrix. F Represents the global load vector. U Represents the global displacement vector. K Represents the overall stiffness matrix; N Indicates the total number of units. The elastic modulus representing the interpolation. Represents the displacement of the element. This represents the element stiffness matrix that does not include the elastic modulus. Indicates the assembly of units; Indicates constraints. This represents the volume of the structure after topology optimization. Indicates the original volume of the structure. Represents the volume of a solid unit. Indicates volume constraints. This represents the minimum unit density.

[0010] Furthermore, the finite element analysis in step 300 specifically includes: S301, defining the global load vector based on the defined loads and boundary conditions. F S302. Assemble the stiffness matrix of each element according to the node number. ,in The global stiffness matrix is ​​obtained by interpolating the elastic modulus. K S303, via global load vector F and global stiffness matrix K Solving the global displacement field U , .

[0011] Further, in step S302, the interpolated elastic modulus Specifically:

[0012] ;

[0013] in, The elastic modulus of the solid element. To prevent numerical singularity, the minimum elastic modulus is defined as , and is taken as . ; The penalty factor is set to 3.

[0014] Further, in step S300, the sensitivity of structural flexibility and volume to design variables is analyzed, specifically as follows:

[0015] ;

[0016] .

[0017] Furthermore, step S400 specifically includes:

[0018] ;

[0019] ;

[0020] In the formula, This represents the cell density after sensitivity filtering has been applied. This indicates the implementation of Heaviside density projection; Indicates the sharpness of the projection. The threshold representing the projection; This represents the weighting factor.

[0021] Furthermore, weighting factors Specifically:

[0022] ;

[0023] in, This represents the defined filter radius. Representation unit e unit i The distance between them.

[0024] Furthermore, according to the chain rule, the sensitivity of structural flexibility and volume to design variables is as follows:

[0025] ;

[0026] .

[0027] According to another aspect of the present invention, a topology-optimized lattice structure is also provided, which is designed and prepared using the above-described topology-optimized lattice structure design method.

[0028] According to another aspect of the present invention, an application of a topology-optimized lattice structure is also provided, wherein the above-described topology-optimized lattice structure is applied to aerospace components, automotive parts, or artificial bone implants.

[0029] The present invention has the following beneficial effects:

[0030] 1. Significantly improves the material utilization efficiency and lightweight level of the structure: With the goal of minimizing structural flexibility (i.e. maximizing stiffness) and with the structural volume as a strict constraint, the optimization process automatically seeks the material distribution mode that maximizes the structural stiffness under a given amount of material. Through iterative optimization from S300 to S600, the material is intelligently removed from inefficient regions and redistributed to efficient load-bearing regions, thereby avoiding material waste caused by uniform lattice or simple filling, and thus solving the problem of material not being maximized. This results in higher structural stiffness under the same weight, or lighter structural weight under the same performance requirements.

[0031] 2. Ensures the physical feasibility and numerical stability of the optimization results: In step S400, by filtering the sensitivity and using the Heaviside function for density projection, common numerical instabilities (such as checkerboard phenomenon and mesh dependence) in the optimization process are effectively suppressed, ensuring that the final material distribution cloud map is clear and smooth, and easy to convert into a manufacturable three-dimensional lattice model, laying the foundation for subsequent additive manufacturing and other processes.

[0032] 3. A performance-driven gradient lattice structure is generated: The final output of the method of this invention is a non-uniform, functionally gradient lattice structure in which the morphological parameters (such as relative density) of the lattice units are continuously varied. This variation strictly follows the mechanical laws determined by the load conditions, thereby substantially improving the overall mechanical properties (such as stiffness) of the structure compared with the uniform lattice structure.

[0033] In addition to the objectives, features, and advantages described above, the present invention has other objectives, features, and advantages. The invention will now be described in further detail with reference to the figures. Attached Figure Description

[0034] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an undue limitation of the invention. In the drawings:

[0035] Figure 1 This is a flowchart of a preferred embodiment of the lattice design method based on topology optimization of the present invention;

[0036] Figure 2 These are schematic diagrams of three types of topology-optimized lattice designs according to a preferred embodiment of the present invention, wherein... Figure 2 (a) To apply force loads at the eight corner points of the design domain, Figure 2 (b) Apply force loads at the centers of the six faces of the design domain, respectively. Figure 2 (c) Apply force loads at the midpoints of the twelve lines in the design domain;

[0037] Figure 3 This is a diagram showing the topology-optimized lattice result of a preferred embodiment of the present invention, wherein... Figure 3 (a) is Figure 2 (a) The final lattice configuration corresponding to the load, Figure 3 (b) is Figure 2 (b) The final lattice configuration corresponding to the load. Figure 3 (c) is Figure 2 (c) The final lattice configuration corresponding to the load.

[0038] Legend:

[0039] 1. Design domain; 2. Force load boundary conditions. Detailed Implementation

[0040] The embodiments of the present invention will be described in detail below with reference to the accompanying drawings. However, the present invention can be implemented in many different ways as defined and covered below.

[0041] like Figure 1As shown, the topology-optimized lattice structure design method of this embodiment includes the following steps: S100, constructing a topology optimization model with minimizing structural flexibility as the optimization objective and structural volume as the constraint, and initializing the physical and optimization parameters of the topology optimization model; S200, defining the structural design domain, loads, and boundary conditions corresponding to the topology optimization model according to the lattice design requirements, and establishing a finite element model; S300, applying the defined loads and boundary conditions to the topology optimization model, performing finite element analysis, obtaining the displacement response of each node in the design domain, and analyzing the sensitivity of structural flexibility and volume to design variables; S400, filtering the sensitivity and using a projection method based on the Heaviside function to filter the material density represented by the design variables to suppress the checkerboard phenomenon; S500, updating the design variables using the moving asymptotic method based on the filtered sensitivity and material density; S600, checking whether the current iteration result meets the convergence condition. If it does, output the result; otherwise, return to step S300 to continue the iteration. This invention presents a topology-optimized lattice structure design method that aims to minimize structural flexibility (i.e., maximize stiffness) while strictly constraining structural volume. The optimization process automatically seeks the material distribution that maximizes structural stiffness within a given material quantity. Through iterative optimization steps S300 to S600, material is intelligently removed from inefficient regions and redistributed to high-efficiency load-bearing regions. This avoids material waste associated with uniform lattices or simple filling, thus solving the problem of maximizing material utilization. This results in higher structural stiffness for the same weight or lighter structural weight for the same performance requirements. In step S400, sensitivity filtering and density projection using the Heaviside function effectively suppress common numerical instabilities (such as checkerboard patterns and mesh dependence) during optimization, ensuring that the final material distribution cloud map is clear, smooth, and easily converted into a manufacturable three-dimensional lattice model, laying the foundation for subsequent additive manufacturing processes. The final output of this invention is a non-uniform, functionally graded lattice structure whose morphological parameters (such as relative density) of lattice units vary continuously, strictly following mechanical laws determined by load conditions. This results in a substantial improvement in the overall mechanical properties (such as stiffness) of the structure compared to a uniform lattice structure. This invention's topology-optimized lattice structure design method, through a series of synergistic optimization steps, deeply embeds topology optimization theory into the lattice structure design and generation process. It automatically generates a non-uniform lattice structure with optimal material distribution and superior mechanical properties under macroscopic load conditions, thereby fundamentally resolving the contradiction between low material utilization and insufficient lightweight potential in traditional design methods, and achieving a synergistic improvement in the overall structural performance.

[0042] In this embodiment, the topology optimization model in step S100 is as follows:

[0043] ;

[0044] in, Indicates the density of different units, e Indicates the unit number; C Indicates the flexibility of the structure. T Represents the transpose of a matrix. F Represents the global load vector. U Represents the global displacement vector. K Represents the overall stiffness matrix; N Indicates the total number of units. The elastic modulus representing the interpolation. Represents the displacement of the element. This represents the element stiffness matrix that does not include the elastic modulus. Indicates the assembly of units; Indicates constraints. This represents the volume of the structure after topology optimization. Indicates the original volume of the structure. Represents the volume of a solid unit. Indicates volume constraints. This represents the minimum unit density. The optimization objective of the model is to minimize the structural flexibility C (i.e., The physical meaning of compliance C is the strain energy of a structure under load. Minimizing compliance is equivalent to maximizing the overall stiffness of the structure under a given load. This objective function drives the optimization process to automatically redistribute material to the region that contributes most to stiffness; constraints V(x) / V 0 ≤f This ensures that the total amount of material used is strictly limited to a fraction of the initial volume. f Within this framework, the mathematical setting of this "objective-constraint" ensures that the optimization process actively seeks the most efficient material layout under strict lightweight requirements. The model utilizes equilibrium equations... KU=F The optimization process is linked to the mechanical response of the structure (global displacement vector). U Strong correlation indicates that the calculation of the objective function and constraints in each iteration is based on the actual mechanical state, ensuring that the optimization direction always follows physical laws; design variables x (Unit density) through interpolation of elastic modulus E e It directly affects the element stiffness matrix, and thus the overall stiffness matrix. K This establishes a direct mathematical bridge from "material distribution" to "structural performance." This physics-based modeling approach ensures that the final material distribution scheme is mechanically sound and feasible. Constraint condition 0 ≤ x min ≤x ≤1, set a minimum relative density. x min This avoids the element density becoming zero during optimization, which would lead to a singular element stiffness matrix (i.e., the matrix is ​​not invertible), thus ensuring the integrity of the finite element equations. KU=F There is always a solution; constraining the design variables to the [0,1] interval provides a clear search space for the optimization algorithm, which is the mathematical basis for the stable convergence and effective results of the iterative optimization process. By establishing a precise mathematical model with "minimizing compliance" as the objective function and "volume fraction" and "equilibrium equation" as the core constraints, the optimization process can automatically and efficiently seek the optimal distribution of materials in the design domain from both mathematical principles and physical mechanisms. This guides the generation of a material layout pattern with maximum stiffness under a given weight constraint, providing a scientific and reliable theoretical basis and input data for the subsequent generation of high-performance non-uniform lattice structures, thereby fundamentally achieving a synergistic improvement in the utilization efficiency and performance of structural materials.

[0045] In this embodiment, the finite element analysis in step 300 specifically includes: S301, defining a global load vector based on the defined loads and boundary conditions. F S302. Assemble the stiffness matrix of each element according to the node number. ,in The global stiffness matrix is ​​obtained by interpolating the elastic modulus. K S303, via global load vector F and global stiffness matrix K Solving the global displacement field U , Steps S301-S303 constitute a complete linear static finite element analysis process, which involves solving the equilibrium equations. The global displacement field continuously distributed within the design domain was obtained. U It accurately characterizes the current material distribution (implied in the global stiffness matrix). K (In the middle) Under certain external loads and boundary conditions, the mechanical response of the structure to the global displacement field. U This is the subsequent calculation of the objective function (such as compliance). And the only input data for sensitivity analysis; without this precise global displacement field U The solution and optimization process loses its direction. Global stiffness matrix. K In the assembly, the stiffness matrix of each element is determined by the interpolated elastic modulus, which is a function of the element design variable (density). This means that any tiny change in material distribution will be reflected in the global stiffness matrix. KUpdates directly alter the displacement field, affecting the overall macroscopic performance of the structure (such as flexibility). This rigorous mathematical-physical model ensures that the optimization algorithm can accurately assess the impact of each design change on performance, thus achieving "performance-driven" design. The essence of finite element analysis is solving the governing equations of mechanics, and the solutions must satisfy equilibrium conditions. By treating this solution process as the inner loop of each optimization iteration, it is mandatory that every possible material distribution scheme explored in the optimization must satisfy the mechanical equilibrium relationship. This fundamentally eliminates the possibility of producing physically unrealistic and unrealizable optimization results, ensuring that the final design is not only mathematically optimal but also physically reasonable.

[0046] In this embodiment, in step S302, the interpolated elastic modulus Specifically:

[0047] ;

[0048] in, The elastic modulus of the solid element. To prevent numerical singularity, the minimum elastic modulus is defined as , and is taken as . ; The penalty factor is set to 3. When calculating the elastic modulus of the element, an elastic modulus much smaller than that of the solid material is introduced. minimum elastic modulus (Values) ); when the unit density x e When the elastic modulus approaches 0 during optimization (i.e., the element tends to be deleted), its elastic modulus... It will approach a very small non-zero value Instead of absolute zero, it ensures that the global stiffness matrix is ​​assembled. K Even for empty regions, the element stiffness matrix will not be completely zero, thus preventing the entire stiffness matrix from becoming zero. K The occurrence of singularities (i.e., a determinant of zero and a non-invertible matrix) ensures the balance of the equations. KU=F There is always a solution, allowing the optimization process to run stably until convergence. In the elastic modulus interpolation formula, for the design variables... x e A penalty factor was introduced. p (Taking a value of 3), this factor penalizes intermediate density values ​​that are neither 0 nor 1, because when p When >1, x e p The value will be much smaller than x eThis inherent characteristic makes the relative stiffness-to-weight ratio of intermediate density units highly uneconomical. During optimization aimed at maximizing stiffness, the algorithm tends to eliminate these inefficient intermediate density units, leading to a convergence of the material distribution to a well-defined, almost gray-area 0-1 binary distribution. This is beneficial for generating well-defined, manufacturable lattice or solid structures. (Modulus interpolation formula) E e =f(x e ) It is about design variables x e The continuous and differentiable nature of the function allows for the analytical solution of the sensitivity (derivative) of the objective function (such as compliance) to design variables. This continuous and differentiable dependence enables the effective application of gradient-based optimization algorithms (such as Moving Average Mutual Aspects Analysis), using sensitivity information to guide the update direction of design variables and ultimately find the optimal solution. The elastic modulus interpolation model introduces a minimum elastic modulus. and penalty factor These two key parameters work together to not only eliminate the potential for singularities in numerical calculations mathematically, ensuring the robustness of finite element analysis, but also drive the optimization results to converge toward a clear topological form that is feasible in engineering through a continuous penalty mechanism of mechanical properties. This core computational link provides technical support for the numerical stability and feasibility of the entire topology optimization process.

[0049] In this embodiment, step S300 involves analyzing the sensitivity of structural flexibility and volume to design variables, specifically as follows:

[0050] ;

[0051] .

[0052] Flexibility and sensitivity formula Quantitative analysis reveals that when a unit's design variables... x e Even a small change in (material density) will affect the overall structural flexibility. C The degree of change, with a negative value indicating that adding material reduces flexibility (increases stiffness), directly indicates the efficiency or value of each element in improving structural stiffness. Optimization algorithms (such as the moving progressive method) utilize this information to intelligently transfer material from inefficient regions with low sensitivity to efficient regions with high sensitivity, thereby optimizing material distribution. Volume sensitivity formula. ,unit e Material density x e For each additional unit, the total volume of the structure...V Just increase the volume of one unit v 0, which allows the optimization algorithm to accurately quantify the impact of each material redistribution on the total volume, thereby strictly controlling the total volume within the preset constraints during iteration (i.e., V(x) / v 0 ≤f Sensitivity analysis (SSA) is the mathematical foundation for achieving controllable lightweight design. It provides gradient information of the objective function (compliance) and constraint function (volume) with respect to design variables. It is based on this precise sensitivity information that gradient-based mathematical programming algorithms (such as MMA) can be effectively applied to solve this complex, high-dimensional engineering optimization problem. Sensitivity analysis is the bridge connecting finite element analysis (the forward problem) and topology optimization (the inverse problem), transforming "how to modify the design to improve performance" into mathematically computable gradient vectors, thereby driving the design to evolve automatically and efficiently toward the optimal solution.

[0053] In this embodiment, step S400 specifically includes:

[0054] ;

[0055] ;

[0056] In the formula, This represents the cell density after sensitivity filtering has been applied. This indicates the implementation of Heaviside density projection; Indicates the sharpness of the projection. The threshold representing the projection; This represents the weighting factor. The first formula (sensitivity filtering) is obtained by applying weights to units... e A weighted average of the density of all cells within a certain filter radius centered on the filter element (based on weighting factors). This effectively eliminates numerical instability issues such as checkerboard patterns and mesh dependencies that may occur during the optimization process, resulting in a smoother and physically more reasonable cell density. (Intermediate density field) provides a high-quality foundation for subsequent projection operations. The second formula (Heaviside density projection) is a nonlinear transformation based on the hyperbolic tangent function tanh, which filters the unit density containing a large amount of intermediate density. (Intermediate density field), sharpened to a threshold For a binary density field where the boundary is defined and the result infinitely approaches 0 or 1, the sharpness of the projection... It controls the sharpness of the projection curve. The larger the value, the closer the projection result is to the ideal 0 / 1 distribution. The resulting clear boundaries define the regions of lattice rods and pores, directly determining the specific morphology of the final non-uniform lattice structure, making it easier to realize through additive manufacturing and other techniques. Threshold and sharpness As an adjustable parameter, it provides designers with a way to intervene in the optimization results. By adjusting... This allows control over the overall proportion of materials in the final structure; by adjusting... This allows for control over the abruptness of the transition from porous to dense regions. This tunability enables the method to balance numerical convergence with design intent, adapting to the requirements of different manufacturing processes for minimum feature size and smooth transition of the structure. Step S400, through the synergistic operation of sensitivity filtering and Heaviside density projection, first numerically stabilizes the optimization process, and then physically drives the material distribution to evolve towards a clear, manufacturable topological configuration. The continuous density field with gray regions generated by topology optimization is ultimately transformed into a well-defined, clearly defined, and nearly binarized material layout scheme, ensuring that the entire optimization method can be transformed from a theoretically optimal solution into a high-performance non-uniform lattice structure that can be practically fabricated in engineering.

[0057] In this embodiment, the weighting factor Specifically:

[0058] ;

[0059] in, This represents the defined filter radius. Representation unit e unit i The distance between elements. A weighting relationship is defined that decreases linearly with the distance between elements, for the central element. e Only considering its surrounding filtration radius Neighboring units within the range i The influence of the weighting factor is significant, and the closer the unit is, the greater its weight. By introducing the correlation between units, the independent update mode of each unit in the original optimization model is effectively broken, thereby fundamentally suppressing numerical instability phenomena such as checkerboard patterns caused by independent mutations of units. The calculation depends on the actual geometric distance. Instead of using the element's mesh index, the sensitivity filtering effect is decoupled from the finite element mesh generation method (such as mesh density and shape regularity). Regardless of mesh variations, as long as the filtering radius... The filter radius is fixed, meaning its influence on the physical space is definite. This ensures that the optimal topology obtained through optimization will not change significantly with mesh refinement or re-partitioning, exhibiting mesh independence and making the optimization results more reliable. As an adjustable parameter, its magnitude directly determines the range of motion for smoothing operations; a larger value... This implies a broader smoothing effect, which suppresses the emergence of fine features in the optimization results. This indirectly controls the minimum size of the final generated lattice structure members or pores, providing designers with a means to control the geometric features of the structure. This allows them to actively avoid generating difficult-to-manufacture ultra-fine features, enhancing the manufacturability of the design results. The weighting factor calculation formula, through a linear weighting rule based on geometric distance, constructs an effective spatial smoothing mechanism. This is not only a key mathematical tool for achieving sensitivity filtering, ensuring numerical stability in the optimization process, and achieving grid independence in the results, but also provides the possibility of indirectly controlling the geometric feature size of the final lattice structure. The scientific definition of this weighting factor is fundamental to ensuring that the entire topology optimization process can produce physically reasonable, geometrically smooth, and easily manufacturable non-uniform lattice structures.

[0060] In this embodiment, according to the chain rule, the sensitivity of structural flexibility and volume to design variables is as follows:

[0061] ;

[0062] .

[0063] In calculating the objective function / constraints (compliance) C ,volume V ) For the original design variables x e Sensitivity At that time, instead of calculating directly, the original variables were fully considered. x e After projection, the variable Then, the filtered variables The complete mathematical transformation chain. Through the chain rule. By continuously multiplying the local derivatives of each transformation step, the contribution of filtering and projection operations to the final sensitivity value is accurately reflected. This ensures that the gradient information passed to the optimization algorithm accurately describes the true impact of modifying the original design variables on the overall performance. Optimization algorithms such as the moving average method directly operate on the original design variables. x e However, what actually affects the structural mechanical performance is the physical density field after filtering and projection processing. The chain rule backpropagates the sensitivity of the objective function with respect to the final physical density and maps it to the sensitivity with respect to the original design variables, ensuring that the optimization algorithm...x e Each search and update in the space is accurately transmitted to the physical performance space through a defined mathematical path, ensuring the mathematical consistency of the entire optimization model and the correctness of the solution. The chain rule is used to calculate sensitivity, constructing a precise mathematical bridge that seamlessly and rigorously connects the nonlinear variable transformation process, including filtering and projection, with the gradient-based optimization algorithm. This ensures that the gradient direction upon which the optimization algorithm relies truly and completely reflects the combined effect of all numerical processing steps. This is a key technical step in ensuring that the entire topology optimization process can still converge stably and produce the correct optimal solution under complex mathematical transformations, providing a reliable mathematical guarantee for the final generation of high-performance non-uniform lattice structures.

[0064] This invention directly generates highly customized non-classical lattice structures that perfectly adapt to specific boundary conditions through multi-condition topology optimization. This allows materials to be optimally distributed according to the actual mechanical path, thus overcoming the problem of insufficient material utilization in traditional uniform lattices. This invention intrinsically defines the lattice structure as a solution to topology optimization through an optimization model that aims to minimize compliance and constrains volume. This ensures that the final lattice configuration inherently possesses optimal mechanical properties under a given lightweight objective. This invention derives the sensitivity of the objective function and volume constraints and employs sensitivity filtering and Heaviside density projection to ensure clear boundary definition during the topology-optimized lattice process.

[0065] The topology-optimized lattice structure in this embodiment was designed and fabricated using the aforementioned topology-optimized lattice structure design method.

[0066] The application of the topology-optimized lattice structure in this embodiment applies the aforementioned topology-optimized lattice structure to aerospace components, automotive parts, or artificial bone implants.

[0067] In implementation, a lattice design method based on topology optimization is provided, and the process of this design method is as follows: Figure 1 As shown, the specific steps include:

[0068] Step 1: Construct a topology optimization model, where the optimization objective is to minimize the structural flexibility, the constraint is the structural volume, and initialize the physical parameters and optimization parameters of the model.

[0069] Furthermore, in step 1, the topology optimization model is as follows:

[0070] ;

[0071] In the formula, Indicates the density of different units, eIndicates the unit number; C Indicates the flexibility of the structure. T Represents the transpose of a matrix. F Represents the global load vector. U Represents the global displacement vector. K Represents the overall stiffness matrix; N Indicates the total number of units. The elastic modulus representing the interpolation. Represents the displacement of the element. This represents the element stiffness matrix that does not include the elastic modulus. Indicates the assembly of units; Indicates constraints. This represents the volume of the structure after topology optimization. Indicates the original volume of the structure. Represents the volume of a solid unit. Indicates volume constraints. This represents the minimum unit density.

[0072] Step 2: Based on the lattice design requirements, specify the design domain, loads, and boundary conditions of the structure, and establish a finite element model.

[0073] Step 3: Apply the boundary conditions described in Step 2 to the topology optimization model, and obtain the displacement response of each node in the design domain by finite element method.

[0074] Furthermore, in step 3, the finite element solution includes the following steps:

[0075] Step 3.1: Define the global load vector based on the boundary conditions described in Step 2. F .

[0076] Step 3.2: Assemble the stiffness matrix of each element according to the node number. Obtain the global stiffness matrix K .

[0077] In the formula The interpolated elastic modulus is specifically:

[0078] ;

[0079] in, The elastic modulus of the solid element. To prevent numerical singularity, the minimum elastic modulus is defined as , and is taken as . ; The penalty factor is set to 3.

[0080] Step 3.3: Through the global load vector F and global stiffness matrix KSolving the global displacement field U Specifically:

[0081] .

[0082] Step 4: Analyze the sensitivity of structural flexibility and volume to design variables.

[0083] Furthermore, in step 4, the sensitivity of the structural flexibility and volume to design variables is as follows:

[0084] ;

[0085] .

[0086] Step 5: Solve for the flexibility of the structure and the sensitivity described in Step 4.

[0087] Step 6: To prevent the checkerboard effect, sensitivity is filtered and density is projected separately. The density projection uses the Heaviside projection method.

[0088] Furthermore, in step 6, the sensitivity filtering and Heaviside density projection are as follows:

[0089] ;

[0090] ;

[0091] In the formula, This represents the cell density after sensitivity filtering has been applied. This indicates the implementation of Heaviside density projection; Indicates the sharpness of the projection. The threshold representing the projection; The weighting factors are as follows:

[0092] ;

[0093] in, This represents the defined filter radius. Representation unit e unit i The distance between them.

[0094] Furthermore, according to the chain rule, the sensitivities of structural flexibility and volume to design variables are as follows:

[0095] ;

[0096] .

[0097] Step 7: Update the design variables using the moving progressive method.

[0098] Step 8: Check if the iteration stopping condition is met. If it is, output the result; otherwise, return to step 3 and repeat the above steps.

[0099] The design domain and boundary conditions in this embodiment are set as follows: Figure 2 As shown, this example provides boundary conditions for three types of force loads as follows:

[0100] 1. Apply force loads at the eight corner points of the design domain.

[0101] 2. Apply force loads at the center of each of the six faces of the design domain.

[0102] 3. Apply force loads at the midpoints of the twelve lines in the design domain.

[0103] The technical implementation method disclosed in this invention is as follows:

[0104] Step 1: Determine the lattice design domain of the cube and divide it into finite element meshes of 100×100×100, with each element mesh having a size of 1×1×1 and volume constraints. Set the elastic modulus of the material. Poisson's ratio A mathematical model for the optimization problem is established with compliance as the objective function.

[0105] Step 2: According to Figure 2 Based on the force load boundary conditions shown, a finite element model is established.

[0106] Step 3: Apply the boundary conditions described in Step 2 to the topology optimization model, and obtain the displacement response of each node in the design domain by finite element method.

[0107] Steps 4-6: Calculate the structural flexibility and sensitivity, then filter and apply Heaviside density projection to suppress numerical instability issues such as checkerboard patterns while ensuring optimization convergence; the sharpness of the projection... The initial value is 1, multiplied by 2 every 100 iterations, with an upper limit of 32; the threshold of the projection. Set to 0.5; Filter radius Set it to 4.

[0108] Step 7: Update the design variables using the moving progressive method.

[0109] Step 8: Check if the iteration stopping condition is met: the maximum change of the design variable is less than 0.01. If this condition is met, output the result, such as... Figure 3 As shown; otherwise, return to step 3 and repeat the above steps.

[0110] Figure 3 The diagram illustrates the three-dimensional non-uniform lattice structure generated by the topology optimization method according to a preferred embodiment of the present invention. Three different configurations are presented from specific angles, as follows:

[0111] Figure 3 (a) is Figure 2 (a) The final lattice configuration corresponding to the load: its lines are clear and its shape is regular, indicating that the optimization process converged to a topological form with stable material distribution and clear geometric characteristics under the given boundary conditions. The spatial arrangement of the rods intuitively reflects the optimal force transmission path of the material under this working condition.

[0112] Figure 3 (b) is Figure 2 (b) Final lattice configuration corresponding to the load: A notable feature is the formation of a clear hole at the center of the structure, with all the rod lines converging at the central point, forming a highly symmetrical shape. This structure indicates that the optimization algorithm automatically identified the central region as a low-stress area, thereby removing the material there to achieve lightweighting, while efficiently transferring the load through a symmetrical rod layout.

[0113] Figure 3 (c) is Figure 2 (c) The final lattice configuration corresponding to the load: its lines intersect, forming a dense network structure. (and) Figure 3 (b) In contrast, this structure has a denser distribution of members and more complex connections, suggesting that it may correspond to more complex multi-condition loads. The optimization algorithm generates this highly connected network to ensure the overall stiffness and stability of the structure under various loads.

[0114] Figure 3 The core effectiveness of the method described in this invention was verified through three typical configurations:

[0115] 1. Effectiveness: The method of the present invention can successfully handle design domains of different shapes and generate clear and realizable lattice structures.

[0116] 2. Intelligence: The optimization results (such as the generation of central holes and the density distribution of the network) directly reflect the performance-driven material layout optimization, rather than simple geometric filling.

[0117] 3. Potential: The results show that the method can generate non-uniform lattices ranging from relatively simple to highly complex, demonstrating its great potential in achieving lightweight structures and customized performance.

[0118] In summary, Figure 3 This strongly demonstrates that the lattice design method based on topology optimization of this invention can automatically generate high-performance, non-uniform lattice structures, solving the problem of low utilization rate of traditional uniform lattice materials.

[0119] Matters not covered in this invention are common knowledge.

[0120] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0121] The embodiments described above are merely examples of several implementations of the present invention, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of the invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the scope of protection of the present invention.

[0122] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A design method of a lattice structure based on topology optimization, characterized by, The method comprises the following steps: S100, constructing a topology optimization model with the optimization objective of minimizing structural flexibility and the constraint condition of structural volume, and initializing physical parameters and optimization parameters of the topology optimization model; The topology optimization model in step S100 is as follows: wherein, denotes the density of different elements, e denotes the element number; C denotes the flexibility of the structure, T denotes the transpose of a matrix, F denotes the global load vector, U denotes the global displacement vector, K denotes the global stiffness matrix; N denotes the total number of elements, denotes the interpolated elastic modulus, denotes the displacement of an element, denotes the element stiffness matrix without elastic modulus, denotes the assembly of elements; denotes the constraint condition, denotes the structure volume after topology optimization, denotes the original volume of the structure, denotes the volume of a solid element, denotes the volume constraint, denotes the minimum element density; S200, defining a structural design domain, a load and boundary conditions corresponding to the topology optimization model according to lattice design requirements, and establishing a finite element model; S300, applying the defined load and boundary conditions to the topology optimization model, performing finite element analysis, obtaining displacement responses of each node on the design domain, and analyzing the sensitivity of structural flexibility and volume with respect to design variables; In step S300, the sensitivity of structural flexibility and volume with respect to design variables is analyzed, specifically as follows: ； S400, filtering the sensitivity, and filtering the material density represented by the design variable using a projection method based on a Heaviside function to suppress the checkerboard phenomenon; According to the chain rule of differentiation, the sensitivity of structural flexibility and volume with respect to design variables is as follows: ； S500, updating the design variable according to the filtered sensitivity and material density using a moving asymptotic method; S600, checking whether the current iteration result meets the convergence condition, and if yes, outputting the result, and if not, returning to step S300 to continue iteration.

2. The method of designing a lattice structure based on topology optimization according to claim 1, wherein, The finite element analysis in step 300 is specifically as follows: S301, define a global load vector according to the defined load and boundary conditions F ; S302, assemble each unit stiffness matrix according to node number wherein is the interpolated elastic modulus, obtaining the global stiffness matrix K ; S303、Through the global load vector F and global stiffness matrix K Solve the global displacement field U , .

3. The method of designing a lattice structure based on topology optimization according to claim 2, wherein, In step S302, the interpolated elastic modulus , specifically: wherein, E is the elastic modulus of the solid element, Emin is the minimum elastic modulus defined to prevent numerical singularity, taken as ; P is a penalty factor, taken as 3.

4. The method of designing a lattice structure based on topology optimization according to claim 1, wherein, Step S400 is specifically as follows: wherein denotes the cell density after implementing sensitivity filtering, denotes the implementation of the Heaviside density projection; denotes the sharpness of the projection, denotes the threshold of the projection; denotes the weight factor.

5. The method of designing a lattice structure based on topology optimization according to claim 4, wherein, weight factor in particular: wherein, represents the defined filter radius, represents the unit e unit i distance between.

6. A lattice structure based on topology optimization, characterized in that, The lattice structure based on topology optimization is designed and prepared by using the design method of the lattice structure based on topology optimization in any one of claims 1 to 5.

7. Use of a lattice structure based on topology optimization, characterized in that The lattice structure based on topology optimization in claim 6 is applied to aerospace components, automobile parts or artificial bone implants.

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