A method for controlling the complexity of structural topology optimization based on morphological feedback

Abstract

The present application relates to a kind of structural topology optimization complexity control method based on morphological feedback, comprising the following steps: modeling reaction diffusion equation level set structural topology optimization problem, and volume fraction constraint is executed;Solving structural topology optimization problem, and sensitivity analysis is carried out, and assembly physical drive item under defined target and constraint;Judge whether control is activated, if control is activated, then measure structural morphological complexity, determine corresponding control mode, regulate structural morphological complexity, obtain revised drive item, replace original physical drive item in reaction diffusion equation with revised drive item, and update reaction diffusion equation;Otherwise, keep original physical drive item and update reaction diffusion equation;Only when the convergence of target and constraint of structural topology optimization problem and the number of structure hole equal target value, iteration terminates, output structure based on the solving result of structural topology optimization problem and end process, otherwise, carry out next iteration.Compared with prior art, the present application has the advantages of no initial layout dependence and accurate control of structural complexity.

Description

Technical Field

[0001] This invention relates to the field of structural topology optimization, and in particular to a method for controlling the complexity of structural topology optimization based on morphological feedback. Background Technology

[0002] In structural topology optimization, controlling structural complexity (i.e., the number of holes) is of great significance. For example, in additive manufacturing, the number of holes directly affects the manufacturing feasibility, structural performance, and cost-effectiveness of the product. In some applications, such as bolt holes in mechanical assemblies or door and window holes in building components, the design needs to meet a specific number of holes.

[0003] In existing technologies, methods for controlling structural complexity are mainly divided into implicit and explicit methods. Implicit methods, such as sensitivity filtering and scale control, adjust structural complexity indirectly. While these methods can effectively suppress numerical artifacts, they cannot directly control the number of holes. Explicit methods, on the other hand, directly constrain the number of holes in the topology. Typical methods include explicit control strategies based on graphs and set theory. For example, graph and set theory methods represent the topology by constructing nodes and edges of a graph and directly constrain the number of holes. However, these methods usually require an initial layout of a specific number of holes, and the number of holes can only satisfy a certain range of constraints, failing to precisely achieve the predetermined target. Another method is the hole-filling method, which approximates the target number of holes by gradually filling existing holes. Although it can precisely control the number of holes, its complexity can only decrease monotonically, requiring optimization starting from the design of a specific number of holes.

[0004] Chinese patent CN112765856A discloses a hybrid level set method for topology optimization of functionally graded porous structures. This method uses level set functions for geometric description and constructs a hybrid level set function for the unit cell of the functionally graded porous structure through a hybrid level set method. It then combines this with the Hamilton-Jacobi equation for spatiotemporal decoupling, optimizing global design variables to maximize structural stiffness and achieving a gradient distribution of geometry and function in space. While this method avoids the initial layout limitations of explicit control methods, it cannot precisely control the number of pores and is computationally complex.

[0005] Therefore, there is an urgent need for a new technology that is independent of the initial layout and can precisely control the structural complexity. Summary of the Invention

[0006] The purpose of this invention is to provide a structural topology optimization complexity control method based on morphological feedback, which achieves precise control of structural complexity through morphological feedback mechanism and physical hole counting.

[0007] The objective of this invention can be achieved through the following technical solutions: A complexity control method for structural topology optimization based on morphological feedback includes the following steps: Model the structural topology optimization problem of the level set of the reaction-diffusion equation and apply volume fraction constraints; Solve the structural topology optimization problem and perform sensitivity analysis, assembling the physical driving terms under the defined objectives and constraints; Determine whether control is activated. If control is activated, measure the structural morphological complexity, determine the corresponding control mode based on the measurement result, adjust the structural morphological complexity based on the control mode to obtain the modified driving term, replace the original physical driving term in the reaction-diffusion equation with the modified driving term, and update the reaction-diffusion equation; otherwise, keep the original physical driving term and update the reaction-diffusion equation. The iteration terminates only when the objective and constraints of the structural topology optimization problem converge and the number of holes in the structure equals the objective value. The structure is then output based on the solution to the structural topology optimization problem, and the process ends. Otherwise, the process returns to re-execute the volume fraction constraints and solve the structural topology optimization problem for the next iteration.

[0008] The structural topology optimization problem of the level set of the modeling reaction-diffusion equation is specifically as follows: In a fixed area Internal design structure, with boundaries defined by a Dirichlet portion of specified displacement. and the Neumann section that applies external traction Composition, material distribution is determined by the level set function The implicit representation is as follows: ; Furthermore, the relevant feature function is defined by a hard threshold, as shown in the following equation: ; in, Let D be a point within a fixed region. For structural entity regions, The boundary of the structural entity region, It is the characteristic function; The structural solid region and the void region are respectively composed of and Give; The structural complexity is quantified by the number of internal holes, as shown in the following formula: ; In the formula, Represents the pore region The connected component set, For counting functions, For structural morphological complexity, if a pore component V Not with outer boundary If they intersect, they are counted as internal holes; The structural topology optimization problem is formulated as a compliance minimization problem under volume fraction constraints and strict structural complexity constraints, as shown in the following equation: ; In the formula, For displacement field, To allow virtual displacements in Hilbert space, To linearize the strain tensor, For elastic tensors, For Newman boundary conditions, As an external force, For Dirichlet boundary conditions, The upper limit of the allowed volume fraction is an integer. It is the specified target number of holes; the volume fraction constraint is passed. To enforce it.

[0009] The specific implementation of the volume fraction constraint is as follows: Evaluate the volume fraction constraint function using a continuous relaxation method: ; In the formula, It is the volume fraction of the initial design. Indicates the current iteration number. To specify the number of iterations for gradually tightening the relaxation boundary, in After the last iteration, the relaxation term disappears. G Simplify to the original volume constraint; The volume fraction constraint is implemented using a multiplier-based Lagrange-type scheme, where the Lagrange multiplier λ is updated as follows: ; In the formula, the parameters p and d Control the responsiveness of exponential updates, scalar G max This is a normalization parameter used to constrain the volume fraction function in the exponent. G Scaling / Scaling It is the topological derivative.

[0010] The sensitivity analysis specifically refers to: Sensitivity information about reaction-diffusion evolution is extracted from the topological derivative, which measures the first-order effect of introducing an infinitesimal hole into the current solid domain. Let... Let represent the solid region induced by the level set description, and consider removing a radius of at any point in Ω. small spherical holes , It is a small regularization constant, where , For the aperture boundary, the objective functional topological derivative Defined by normalizing the target perturbation relative to the measure of removal: ; In the formula, It is a measure function; The level set is updated using the reaction-diffusion equation, introducing a dummy time variable. The level set function evolves by re-initializing the equation using the level set. The diffusion term provides regularization to enhance smoothness and numerical robustness, while the reaction term provides the driving force constructed from topological derivative information. Homogeneous Dirichlet conditions are specified on the outer boundary to strictly restrict the evolution within a fixed domain. The update equation based on the reaction-diffusion equation is as follows: ; In the formula, It is a regularization parameter. It is a physics-driven term. It is the Laplace operator. It is a level set function. It is time. It is a partial differential operator. C These are normalization coefficients introduced to reduce sensitivity to the scale of the problem. C Defined by the following formula: ; Semi-implicit backward Euler discretization is used in virtual time to implicitly handle the diffusion term to improve stability, while keeping the reaction term explicit. The updated results are as follows: ; In the formula, Indicates the virtual time step.

[0011] The measurement of structural morphological complexity specifically includes the following steps: Level set function of consecutive nodes Mapping to unit-level feature functions This function defines a discrete entity domain. : ; in, It is a unit e The average set value, Finite element mesh; For discrete entity domains Perform connectivity component analysis and identify the largest connectivity component as the main structure. : ; in yes The set of all disjoint connected components in the set. Representation Component Volume; For the main structure Apply the hole-filling operation to fill all internal hole areas completely surrounded by the main structure, producing the filled area. ; Physical holes H Defined as the Boolean difference between the filled region and the original main structure: ; in, It is the complement of the main structure; The number of holes is calculated by counting the number of connected components in the hole mask. The connected components are determined using four-neighbor connectivity: ; in, This is a function for counting connected components.

[0012] The measurement of structural morphological complexity also includes the processing of per-hole attributes, specifically: for each identified cavity... Record its element index set Scalar area A i and its interface neighborhood Among them, the neighborhood pass The shape is obtained by expansion.

[0013] The specific steps for determining the corresponding control mode based on the structural morphological complexity measurement result are as follows: Define the difference between the current number of holes and the target number of holes as the topology error: ; in, N cur This is the current number of holes. N target It refers to the number of target holes; Based on topological error The symbol determines the corresponding control mode: if If so, the topology is considered balanced, and feedback regulation remains in an idle state; if This is the first control mode, where holes are filled; if This is the second control mode, which involves pore nucleation.

[0014] The specific method of adjusting the structural morphological complexity based on the control mode is as follows: Define two masks associated with the two control modes. For hole filling, the mask... Coverage target set The expansion space domain of redundant holes, for hole nucleation, mask Coverage to select the optimal nucleation site Centered on, with radius The circular region, wherein the target set From physical holes based on geometric importance scores H Choose the lowest ranking A hole is formed. The expression is as follows: ; in, The geometric importance score is determined by the area of ​​the aperture. Defined and normalized using the largest aperture area in the current design: ; in It is the largest area among all current holes; Optimal nucleation site The strain energy density field was determined to be within the candidate domain. The minimum spatial location is represented by the following formula: ; in Candidate entity regions; Use indicator functions Define two masks as follows: ; in, The spatial domain representing the hole, Indicates the use of structural elements Perform morphological dilation. It is the specified nucleation radius; To ensure that manual adjustments match physical driving forces throughout the optimization process, based on physical driving terms... The average size introduces an adaptive reference scale. : ; in It is a small regularization constant. It is a structural entity region; Based on topological error The symbol, constructing a two-way regulating force : ; in It is a dimensionless fundamental coefficient. and These are the gain multipliers for filling and nucleation, respectively.

[0015] The modified driving term is represented as: ; in, It is the driving factor for the correction; With the modified driver Replacing the original driving term in the reaction-diffusion equation yields the adjusted evolution equation: ; In the formula, It is a regularization parameter. It is the Laplace operator. It is a level set function. It is time. It is a partial differential operator. C It is a normalization coefficient introduced to reduce sensitivity to the scale of the problem. D It is a fixed area.

[0016] In the hole nucleation process, at most one hole is nucleated in each iteration.

[0017] Compared with the prior art, the present invention has the following beneficial effects: (1) This invention measures and controls the structural morphological complexity through a morphological feedback mechanism and physical void counting, and feeds this information back into the reaction-diffusion equation, thereby achieving precise control of structural complexity. Compared with traditional implicit or explicit methods, this invention can autonomously complete the nucleation and elimination of voids without relying on a complex initial layout, thus ensuring that the optimization results meet the predetermined goals. This technical effect makes structural optimization more practical and reliable in additive manufacturing and high-precision design.

[0018] (2) This invention improves the flexibility and stability of topology optimization by introducing a gradient-based soft adjustment strategy, which can dynamically switch between hole nucleation and elimination as needed. Through precise hole counting and feedback adjustment, numerical artifacts are effectively suppressed, ensuring the quality of the design, and the increased time cost is negligible.

[0019] (3) The strong smoothness and stability of the reaction-diffusion equation level set framework for topology optimization enable the scheme of this invention to converge stably in complex design spaces and ensure the reliability of the final design in practical engineering applications. This invention provides an innovative and efficient solution for the field of complex structure topology optimization, with broad application prospects and commercial value. Attached Figure Description

[0020] Figure 1 This is a flowchart of the method of the present invention; Figure 2 This is a schematic diagram of the cantilever beam design in Example 2; Figure 3 This section presents the optimized cantilever beam design and iteration history for different target complexities in Example 2. Figure 4 This is a schematic diagram of the simply supported beam design in Example 3; Figure 5 This section presents the optimized simply supported beam design and iteration history for different target complexities in Example 3. Detailed Implementation

[0021] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments. These embodiments are based on the technical solution of the present invention and provide detailed implementation methods and specific operating procedures. However, the scope of protection of the present invention is not limited to the following embodiments.

[0022] Example 1 This embodiment provides a method for controlling the complexity of structural topology optimization based on morphological feedback, such as... Figure 1 As shown, it includes the following steps: Model the structural topology optimization problem of the level set of the reaction-diffusion equation and apply volume fraction constraints; Solve the structural topology optimization problem and perform sensitivity analysis, assembling the physical driving terms under the defined objectives and constraints; Determine whether control is activated. If control is activated, measure the structural morphological complexity, determine the corresponding control mode based on the measurement result, adjust the structural morphological complexity based on the control mode to obtain the modified driving term, replace the original physical driving term in the reaction-diffusion equation with the modified driving term, and update the reaction-diffusion equation; otherwise, keep the original physical driving term and update the reaction-diffusion equation. The iteration terminates only when the objective and constraints of the structural topology optimization problem converge and the number of holes in the structure equals the objective value. The structure is then output based on the solution to the structural topology optimization problem, and the process ends. Otherwise, the process returns to re-execute the volume fraction constraints and solve the structural topology optimization problem for the next iteration.

[0023] Specifically, the structural topology optimization problem for modeling the level set of the reaction-diffusion equation is as follows: In a fixed area Internal design structure, with boundaries defined by a Dirichlet portion of specified displacement. and the Neumann section that applies external traction Composition, material distribution is determined by the level set function The implicit representation is as follows: ; Furthermore, the relevant feature function is defined by a hard threshold, as shown in the following equation: ; in, Let D be a point within a fixed region. For structural entity regions, The boundary of the structural entity region, It is the characteristic function.

[0024] The structural solid region and the void region are respectively composed of and Provided.

[0025] The structural complexity is quantified by the number of internal holes, as shown in the following formula: ; In the formula, Represents the pore region The connected component set, For counting functions, For structural morphological complexity, if a pore component V Not with outer boundary If they intersect, they are counted as internal holes.

[0026] The structural topology optimization problem is formulated as a compliance minimization problem under volume fraction constraints and strict structural complexity constraints, as shown in the following equation: ; In the formula, For displacement field, To allow virtual displacements in Hilbert space, To linearize the strain tensor, For elastic tensors, For Newman boundary conditions, As an external force, For Dirichlet boundary conditions, The upper limit of the allowed volume fraction is an integer. It is the specified target number of holes; the volume fraction constraint is passed. To enforce this, an elastic tensor is constructed using an alternative material interpolation model on a fixed mesh. .

[0027] Subsequently, volume fraction constraints are applied. To enhance early robustness in the event of an initial design violation of the specified volume limits, a continuous relaxation method is used to evaluate the volume fraction constraint function: ; In the formula, It is the volume fraction of the initial design. Indicates the current iteration number. To specify the number of iterations for gradually tightening the relaxation boundary, in After the last iteration, the relaxation term disappears. G Simplified to the original volume constraint.

[0028] The volume fraction constraint is implemented using a multiplier-based Lagrange-type scheme, where the Lagrange multiplier λ is updated as follows: ; In the formula, the parameters p and d Control the responsiveness of exponential updates, scalar G max This is a normalization parameter used to constrain the volume fraction function in the exponent. G Scaling is performed, which is considered as the current allowable volume level in relation to the volume constraint, so that the ratio remains dimensionless and well scaled throughout convergence. It is the topological derivative.

[0029] Then, the structural topology optimization problem is solved using finite element analysis, and the sensitivity is analyzed.

[0030] Specifically, the sensitivity information of the reaction-diffusion evolution is extracted from the topological derivative, which measures the first-order effect of introducing an infinitesimal hole into the current solid domain. Let... Let represent the solid region induced by the level set description, and consider removing a radius of at any point in Ω. small spherical holes , It is a small regularization constant, where , For the aperture boundary, the objective functional topological derivative Defined by normalizing the target perturbation relative to the measure of removal: ; In the formula, Let be the measure function.

[0031] The level set is updated using the reaction-diffusion equation, introducing a dummy time variable. The level set function evolves by re-initializing the equation using the level set. The diffusion term provides regularization to enhance smoothness and numerical robustness, while the reaction term provides the driving force constructed from topological derivative information. Homogeneous Dirichlet conditions are specified on the outer boundary to strictly restrict the evolution to a fixed domain. The update equation based on the reaction-diffusion equation is: ; In the formula, It is a regularization parameter. It is a physics-driven term. It is the Laplace operator. It is a level set function. It is time. It is a partial differential operator. C These are normalization coefficients introduced to reduce sensitivity to the scale of the problem. C Defined by the following formula: .

[0032] Semi-implicit backward Euler discretization is used in virtual time to implicitly handle the diffusion term to improve stability, while keeping the reaction term explicit. The updated results are as follows: ; In the formula, Indicates the virtual time step.

[0033] In each iteration step, the driving field The evaluation is performed using nodal topological derivative information and volume multipliers on the background mesh. After each update, the nodal values ​​are truncated to satisfy the boundary constraints of the level set function, and subsequent material variables are updated by thresholding.

[0034] To achieve closed-loop feedback control in topology optimization, it is crucial to introduce a reliable mechanism to detect the number of holes within the main structure. While level set functions While the structure's boundaries are implicitly defined, directly calculating unconnected hole regions can be unstable due to the presence of external background holes and isolated regions unrelated to surrounding entities. To address this issue, a morphological feedback mechanism is proposed, combining connected component labeling and binary image reconstruction. This mechanism accurately identifies hole morphology, effectively filters out numerical artifacts, and extracts the current hole number. And the geometric properties of each hole. This process is performed on a discretized finite element mesh. Run on.

[0035] The first step is to set the level function of consecutive nodes. Mapping to unit-level feature functions This function defines a discrete entity domain. : ; in, It is a unit e The average set value, The result is a finite element mesh. This discretization produces binary representations of solid and void regions, which will be used for subsequent morphological operations.

[0036] Next, the main structure is extracted from the entity domain. Because topology optimization may result in disconnected fragments or small, isolated entity regions, this is done by optimizing the discrete entity domain. Perform connectivity component analysis and identify the largest connectivity component as the main structure. The main structure is defined as the largest connected component, which is determined by maximizing the component volume: ; in yes The set of all disjoint connected components in the set. Representation Component The volume. This step ensures that isolated islands of material are excluded during the hole counting process.

[0037] After identifying the main structure, hole extraction is performed. For the main structure... Apply the hole-filling operation to fill all internal hole areas completely surrounded by the main structure, producing the filled area. .

[0038] Physical holes H Defined as the Boolean difference between the filled region and the original main structure: ; in, It is the complement of the main structure. This operation isolates cavities in the structure that represent potential voids.

[0039] The number of holes is calculated by counting the number of connected components in the hole mask. The connected components are determined using four-neighbor connectivity: ; in, This is a function for counting connected components.

[0040] This process ensures To accurately reflect the topological genus of the structure, each physically closed hole region is treated as a separate hole. (In addition to global counting...) In addition, the morphological feedback mechanism also includes the processing of per-cavity attributes, specifically: for each identified cavity Record its element index set Scalar area A i (defined as base number) ) and its interface neighborhood Among them, the neighborhood pass The morphological expansion is obtained, providing the space needed to evaluate the local sensitivity and strain energy around each specific hole.

[0041] This method of analyzing each hole individually allows the feedback mechanism to selectively fill or nucleate holes based on their size, importance, and the energy of the surrounding material.

[0042] After completion After calculation and processing of hole-by-hole properties, the deviation from the design intent is evaluated to determine the appropriate control mode.

[0043] Define the difference between the current number of holes and the target number of holes as the topology error: ; in, N cur This is the current number of holes. N target It represents the number of target holes.

[0044] Based on topological error The symbol determines the corresponding control mode: if If so, the topology is considered balanced, and feedback regulation remains in an idle state; if This is the first control mode, where holes are filled; if This is the second control mode, which involves pore nucleation.

[0045] In the mechanism of void filling, when redundant voids exist, a key issue is selecting cavities to be removed in a consistent and stable manner. Random elimination is not optimal because it may remove established geometric features and cause unnecessary topological fluctuations. Therefore, a geometric importance score is introduced. To the set Sort the existing holes in the array. Geometric importance score. From the area of ​​the hole Defined and normalized using the largest aperture area in the current design: ; in It is the largest area among all current holes. Larger The higher score indicates a more geometrically prominent and stable cavity, while a lower score corresponds to a less significant hole feature that can be eliminated with limited impact on the global layout. Therefore, the system prioritizes removing the holes with the lowest scores.

[0046] Target set for elimination Through physical holes H Select Lowest ranking Formed by a hole, The expression is as follows: .

[0047] In the mechanism of pore nucleation, it is necessary to introduce New pores are introduced to increase complexity. To ensure that material removal has minimal impact on structural stiffness, nucleation should occur in regions of lowest strain energy density. Candidate nucleation domains are defined by excluding the boundary layer and regions near existing pores. This is to prevent immediate merging or boundary violations. Optimal nucleation location. The strain energy density field was determined to be within the candidate domain. The minimum spatial location is represented by the following formula: ; in This is a candidate entity region. This energy-driven selection strategy ensures that new topological features are initiated in mechanically inefficient regions, thus aligning topological adjustment with the compliance minimization objective.

[0048] Subsequently, a two-way soft regulation strategy is introduced to regulate evolution by injecting spatially localized artificial driving forces into the reaction-diffusion update. The key idea is to keep the original reaction-diffusion equation framework unchanged and incorporate topological control only through driving terms, thereby achieving topological transformation through the inherently smooth evolution of the reaction-diffusion mechanism.

[0049] To localize the adjustment, two masks are defined for the two control modes. For hole filling, the mask... Coverage target set The expansion space domain of redundant holes, for hole nucleation, mask Coverage to select the optimal nucleation site Centered on, with radius A circular region. Consistent with the implementation method, even... Each iteration only forms one hole.

[0050] Use indicator functions Define two masks as follows: ; in, The spatial domain representing the hole, Indicates the use of structural elements Perform morphological dilation. It is the defined nucleation radius.

[0051] To ensure that manual adjustments match physical driving forces throughout the optimization process, based on physical driving terms... The average size introduces an adaptive reference scale. : ; in It is a small regularization constant. It is a structural entity region.

[0052] Based on topological error The symbol, constructing a two-way regulating force : ; in It is a dimensionless fundamental coefficient. and These are the gain multipliers for filling and nucleation, respectively.

[0053] Therefore, the modified driving term is represented as: ; in, It is the driver for the correction.

[0054] With the modified driver Replacing the original driving term in the reaction-diffusion equation yields the adjusted evolution equation: ; This reaction diffusion structure is achieved through... Injecting regulation rather than direct change Topological transformations are achieved through the inherent smoothness of reaction-diffusion evolution, producing geometrically regular changes while maintaining numerical stability.

[0055] Example 2 This embodiment, based on the method of Embodiment 1, uses a representative cantilever beam for benchmark testing to evaluate the proposed method for precise structural complexity control in topology optimization. All tests assume the material has... and Linear elastic isotropic materials, with the void phase replacing stiffness Modeling. In the evolution of the reaction-diffusion equation, the virtual time step is set to... The maximum volume limit is fixed in all cases. .pass The continuation phase of the next iteration and its use and The exponential multiplier update is used to enforce volume control, by Initialization. The structural complexity control regulator is in... Activated after the next iteration. The interface for local adjustments is... Definition, nucleation radius Soft adjustment uses the basic gain. and multiplier and The maximum number of iterations is capped at 1. Any situation that fails to converge within this constraint is considered to be non-convergent.

[0056] The optimization is considered to have converged when the following conditions are met simultaneously: (1) the iterative index exceeds the continuation phase ( (2) The volume fraction does not exceed the specified limit of 0.005; (3) The relative change of the objective function in the last five iterations is less than 1%; (4) When structural complexity control is enabled, the structural complexity error is less than 1%. .

[0057] The design domain is discretized into a uniform shape. A quadrilateral mesh with a completely fixed left boundary, and a strength of [value missing] applied across six cell spans at the midpoint of the right edge. A uniformly distributed vertical line load, such as Figure 2 As shown. In the reaction-diffusion equation framework, the regularization parameter Used as a global parameter, it qualitatively biases towards structural complexity. This example studies two values, namely... and .

[0058] When the structural complexity control speed regulator is enabled, as described above... Repeatedly perform topology optimization, while also... The value changed from 0 to 15. The results show that... This will affect the achievable complexity range. A larger one... It facilitates simpler layouts but makes it difficult to achieve high complexity, while smaller... This facilitates richer topologies. In these cases, which depend on... Within the trend range, the proposed structural complexity control framework can achieve quantitative complexity targets. Hole-free or extremely low-hole designs become feasible, and more complex layouts can be obtained than without structural complexity control.

[0059] Figure 3 Showing The representative optimization results and process are shown below. The left side shows the optimized layout, and the right side shows the iteration history. All designs exactly meet the specified target number of holes. The results also demonstrate pure complexity tuning. This method primarily changes the topology, i.e., the number of holes, without altering the component dimensions determined by compliance minimization. Figure 3 In (e), the material is still concentrated in the main load-bearing skeleton, while the small holes are located in a mechanically reasonable position and have a limited impact on the main force path.

[0060] Figure 3 The right side reports the iterative curves for pore number, volume fraction, and compliance. The topology adjuster is activated at iteration 21, causing only brief transient changes in the compliance and volume curves. After this adjustment, the compliance reduction and volume adjustment remain stable and converge within a relatively small number of iterations, thanks to the soft adjustment mechanism. The level of oscillation in the pore number trajectory reflects the difficulty of setting the structural complexity control target. Figure 3 (b) and the forced implementation of extremely low complexity, including the case of zero holes, produces large oscillations in the complexity iteration process, but still achieves the accurate goal. Figure 3 (d) Demonstrates real-time bidirectional adjustment. During iterations 21–26 and 32, when Nucleation of pores occurs at that time, and then when The timing triggers hole filling, and it can still guarantee during convergence. .exist Figure 3 In (f), the number of pores shows an overall upward trend, indicating a state dominated by nucleation. The corresponding compliance curve remains smooth with only slight perturbations, supporting numerical stability and the effectiveness of autonomous nucleation.

[0061] Example 3 This embodiment is based on the method of Embodiment 1, and selects a classic simply supported beam model for experimentation. The loads and boundary conditions are as follows: Figure 4 As shown. The material properties of the simply supported beam, the upper limit of volume, regulator activation condition, number of iterations, and other parameters in the reaction-diffusion equation evolution, as well as the optimization convergence condition, are the same as in Example 2. Due to symmetry, only half of the design domain is modeled. The concentrated load is set to... The design domain is discretized into a uniform shape. Grid. Two operating conditions for the regularization parameters of the reaction-diffusion equation were studied, namely... and .

[0062] Figure 5 Showing Representative designs and their corresponding iteration history. Figure 5(c) shows that the proposed structural complexity control mechanism can drive the design towards higher complexity, generating a large number of holes. Despite the large number of holes, the material distribution in space remains quite reasonable. Most of the material remains on the main truss-like skeleton, while the additional small holes are located in less critical areas. The topology adjuster is activated at the 21st iteration, introducing a brief adjustment phase, while subsequent evolution continues to improve the objective and satisfy volume constraints without significant interference. The bidirectional nature of the adjustment... Figure 5 It is evident in (b). When the number of holes exceeds... When the number of holes drops below the target, excess holes are suppressed; when the number of holes drops below the target, new holes are introduced to restore the target complexity.

[0063] The preferred embodiments of the present invention have been described in detail above. It should be understood that those skilled in the art can make numerous modifications and variations based on the concept of the present invention without creative effort. Therefore, all technical solutions that can be obtained by those skilled in the art based on the concept of the present invention through logical analysis, reasoning, or limited experimentation on the basis of existing technology should be within the scope of protection defined by the claims.

Claims

1. A method for controlling the complexity of structural topology optimization based on morphological feedback, characterized in that, Includes the following steps: Model the structural topology optimization problem of the level set of the reaction-diffusion equation and apply volume fraction constraints; Solve the structural topology optimization problem and perform sensitivity analysis, assembling the physical driving terms under the defined objectives and constraints; Determine whether control is activated. If control is activated, measure the structural morphological complexity. Determine the corresponding control mode based on the structural morphological complexity measurement result. Adjust the structural morphological complexity based on the control mode to obtain the modified driving term. Replace the original physical driving term in the reaction-diffusion equation with the modified driving term and update the reaction-diffusion equation. Otherwise, retain the original physical driving terms and update the reaction-diffusion equation; The iteration terminates only when the objective and constraints of the structural topology optimization problem converge and the number of holes in the structure equals the objective value. The structure is then output based on the solution to the structural topology optimization problem, and the process ends. Otherwise, the process returns to re-execute the volume fraction constraints and solve the structural topology optimization problem for the next iteration.

2. The structural topology optimization complexity control method based on morphological feedback according to claim 1, characterized in that, The structural topology optimization problem of the level set of the modeling reaction-diffusion equation is specifically as follows: In a fixed area Internal design structure, with boundaries defined by a Dirichlet portion of specified displacement. and the Neumann section that applies external traction Composition, material distribution is determined by the level set function The implicit representation is as follows: ； Furthermore, the relevant feature function is defined by a hard threshold, as shown in the following equation: ； in, Let D be a point within a fixed region. For structural entity regions, The boundary of the structural entity region, It is the characteristic function; The structural solid region and the void region are respectively composed of and Give; The structural complexity is quantified by the number of internal holes, as shown in the following formula: ； In the formula, Represents the pore region The connected component set, For counting functions, For structural morphological complexity, if a pore component V Not with outer boundary If they intersect, they are counted as internal holes; The structural topology optimization problem is formulated as a compliance minimization problem under volume fraction constraints and strict structural complexity constraints, as shown in the following equation: ； In the formula, For displacement field, To allow virtual displacements in Hilbert space, To linearize the strain tensor, For elastic tensors, For Newman boundary conditions, As an external force, For Dirichlet boundary conditions, The upper limit of the allowed volume fraction is an integer. It is the specified target number of holes; the volume fraction constraint is passed. To enforce it.

3. The structural topology optimization complexity control method based on morphological feedback according to claim 2, characterized in that, The specific implementation of the volume fraction constraint is as follows: Evaluate the volume fraction constraint function using a continuous relaxation method: ； In the formula, It is the volume fraction of the initial design. Indicates the current iteration number. To specify the number of iterations for gradually tightening the relaxation boundary, in After the last iteration, the relaxation term disappears. G Simplified to the original volume constraint; The volume fraction constraint is implemented using a multiplier-based Lagrange-type scheme, where the Lagrange multiplier λ is updated as follows: ； In the formula, the parameters p and d Control the responsiveness of exponential updates, scalar G max This is a normalization parameter used to constrain the volume fraction function in the exponent. G Scaling / Scaling It is the topological derivative.

4. The structural topology optimization complexity control method based on morphological feedback according to claim 2, characterized in that, The sensitivity analysis specifically refers to: Sensitivity information about reaction-diffusion evolution is extracted from the topological derivative, which measures the first-order effect of introducing an infinitesimal hole into the current solid domain. Let... Let represent the solid region induced by the level set description, and consider removing a radius of at any point in Ω. small spherical holes , It is a small regularization constant, where , For the aperture boundary, the objective functional topological derivative Defined by normalizing the target perturbation relative to the measure of removal: ； In the formula, It is a measure function; The level set is updated using the reaction-diffusion equation, introducing a dummy time variable. The level set function evolves by re-initializing the equation using the level set. The diffusion term provides regularization to enhance smoothness and numerical robustness, while the reaction term provides the driving force constructed from topological derivative information. Homogeneous Dirichlet conditions are specified on the outer boundary to strictly restrict the evolution within a fixed domain. The update equation based on the reaction-diffusion equation is as follows: ； In the formula, It is a regularization parameter. It is a physics-driven term. It is the Laplace operator. It is a level set function. It is time. It is a partial differential operator. C These are normalization coefficients introduced to reduce sensitivity to the scale of the problem. C Defined by the following formula: ； Semi-implicit backward Euler discretization is used in virtual time to implicitly handle the diffusion term to improve stability, while keeping the reaction term explicit. The updated results are as follows: ； In the formula, Indicates the virtual time step.

5. The structural topology optimization complexity control method based on morphological feedback according to claim 1, characterized in that, The measurement of structural morphological complexity specifically includes the following steps: Level set function of consecutive nodes Mapping to unit-level feature functions This function defines a discrete entity domain. : ； in, It is a unit e The average set value, Finite element mesh; For discrete entity domains Perform connectivity component analysis and identify the largest connectivity component as the main structure. : ； in yes The set of all disjoint connected components in the set. Representation Component Volume; For the main structure Apply the hole-filling operation to fill all internal hole areas completely surrounded by the main structure, producing the filled area. ; Physical holes H Defined as the Boolean difference between the filled region and the original main structure: ； in, It is the complement of the main structure; The number of holes is calculated by counting the number of connected components in the hole mask. The connected components are determined using four-neighbor connectivity: ； in, This is a function for counting connected components.

6. The structural topology optimization complexity control method based on morphological feedback according to claim 5, characterized in that, The measurement of structural morphological complexity also includes the processing of per-hole attributes, specifically: for each identified cavity... Record its element index set Scalar area A i and its interface neighborhood Among them, the neighborhood pass The shape is obtained by expansion.

7. The structural topology optimization complexity control method based on morphological feedback according to claim 1, characterized in that, The specific steps for determining the corresponding control mode based on the structural morphological complexity measurement result are as follows: Define the difference between the current number of holes and the target number of holes as the topology error: ； in, N cur This is the current number of holes. N target It refers to the number of target holes; Based on topological error The symbol determines the corresponding control mode: if If so, the topology is considered balanced, and feedback regulation remains in an idle state; if This is the first control mode, where holes are filled; if This is the second control mode, which involves pore nucleation.

8. The structural topology optimization complexity control method based on morphological feedback according to claim 7, characterized in that, The specific method of adjusting the structural morphological complexity based on the control mode is as follows: Define two masks associated with the two control modes. For hole filling, the mask... Coverage target set The expansion space domain of redundant holes, for hole nucleation, mask Coverage to select the optimal nucleation site Centered on, with radius The circular region, wherein the target set From physical holes based on geometric importance scores H Choose the lowest ranking A hole is formed. The expression is as follows: ； in, The geometric importance score is determined by the area of ​​the aperture. Defined and normalized using the largest aperture area in the current design: ； in It is the largest area among all current holes; Optimal nucleation site The strain energy density field was determined to be within the candidate domain. The minimum spatial location is represented by the following formula: ； in Candidate entity regions; Use indicator functions Define two masks as follows: ； in, The spatial domain representing the hole, Indicates the use of structural elements Perform morphological dilation. It is the specified nucleation radius; To ensure that manual adjustments match physical driving forces throughout the optimization process, based on physical driving terms... The average size introduces an adaptive reference scale. : ； in It is a small regularization constant. It is a structural entity region; Based on topological error The symbol, constructing a two-way regulating force : ； in It is a dimensionless fundamental coefficient. and These are the gain multipliers for filling and nucleation, respectively.

9. The structural topology optimization complexity control method based on morphological feedback according to claim 8, characterized in that, The modified driving term is represented as: ； in, It is the driving factor for the correction; With the modified driver Replacing the original driving term in the reaction-diffusion equation yields the adjusted evolution equation: ； In the formula, It is a regularization parameter. It is the Laplace operator. It is a level set function. It is time. It is a partial differential operator. C It is a normalization coefficient introduced to reduce sensitivity to the scale of the problem. D It is a fixed area.

10. The structural topology optimization complexity control method based on morphological feedback according to claim 7, characterized in that, In the hole nucleation process, at most one hole is nucleated in each iteration.

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