A geometric topology optimization method and device capable of processing embedded domains with complex boundaries
By embedding the STL model into an equigeometric background domain and utilizing NURBS basis functions and optimality criteria, the adaptability and accuracy issues of traditional topology optimization under complex boundaries are solved, achieving efficient and stable generation of topology optimization results.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- HUAZHONG UNIV OF SCI & TECH
- Filing Date
- 2026-05-18
- Publication Date
- 2026-07-14
AI Technical Summary
Traditional topology optimization methods face challenges when dealing with complex geometries, including difficulties in generating finite element meshes, large geometric approximation errors, jagged boundaries in the optimization results, and heavy computational burden. They are particularly unsuitable for complex topological features with multiple holes or separated regions.
Using isogeometric analysis, the STL model of the structure to be optimized is embedded into an isogeometric background domain based on NURBS basis functions. By constructing a global stiffness matrix, calculating sensitivity and filtering, and combining the optimality criterion method, the design variables are iteratively updated to reconstruct a high-precision three-dimensional density field and generate optimization results with smooth boundaries.
It improves the accuracy and adaptability of complex boundary optimization, shortens the design and manufacturing cycle, avoids numerical instability and oscillation, and achieves efficient and stable conversion of topology optimization results.
Smart Images
Figure CN122389486A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the technical field of material structure optimization, and more specifically, relates to a method and device for geometric topology optimization, such as embedded domains, capable of handling complex boundaries. Background Technology
[0002] Topology optimization is a key technique in structural design. Given loads, boundary conditions, and constraints, it explores the optimal distribution of materials within a design domain to achieve specific structural performance goals. With the increasing demands for structural performance and complexity in modern engineering, topology optimization has been widely applied in numerous high-tech fields such as aerospace, automotive manufacturing, and biomedicine.
[0003] In traditional topology optimization methods, structures are typically discretized using finite element analysis (FEM). This method divides the design domain into a large number of independent elements, iteratively optimizing the structure by changing the density or presence of these elements. However, FEM-based topology optimization faces significant challenges when dealing with complex geometries, especially those with curved boundaries and fine features. First, generating the FEM mesh is itself a time-consuming and specialized task, particularly for complex geometries. Second, geometric approximation errors exist between the FEM mesh and the geometric model, often resulting in jagged boundaries in the optimization results, making direct conversion to CAD models for manufacturing difficult. Obtaining smooth boundaries often requires complex post-processing or refinement techniques, increasing the computational burden and design cycle.
[0004] In recent years, isogeometric analysis (IGA), as an emerging numerical method, has attracted widespread attention for its direct use of basis functions such as NURBS (non-uniform rational B-splines) or T-splines commonly used in CAD models for geometric description and analysis. IGA can provide accurate geometric representation and high-order continuity, greatly reducing geometric approximation errors and simplifying the process from design to analysis. Introducing isogeometric analysis into topology optimization, namely isogeometric topology optimization, allows for optimization directly on accurate geometric models, generating optimization results with smooth boundaries, thus effectively solving the complex conversion between design and optimization.
[0005] However, applying isogeometric topology optimization to complex CAD models with multiple topologies still faces significant challenges. Traditional isogeometric methods rely on accurate NURBS geometric modeling, but for CAD models containing complex topological features such as multiple holes, bridging, or separating regions, directly constructing a single or even multi-faceted NURBS model is extremely difficult, or even impossible, severely limiting the adaptability of isogeometric topology optimization in exploring a wider design space. Summary of the Invention
[0006] To address the aforementioned deficiencies or improvement needs of existing technologies, this invention provides a method and apparatus for optimizing embedded domains with complex boundaries, aiming to solve the problem of low accuracy caused by poor adaptability of isogeometric topology optimization when dealing with complex multi-topology models.
[0007] To achieve the above objectives, according to one aspect of the present invention, a geometric topology optimization method capable of handling complex boundaries, such as embedded domains, is provided, comprising the following steps: (1) Embed the STL model of the structure to be optimized into the isogeometric background domain constructed based on NURBS basis functions to obtain the complex geometric boundary, solid element, clipping element and control points of the isogeometric background domain of the structure to be optimized; (2) Design variables defined at the control points of the isogeometric background domain General element stiffness matrix template for solid elements Element stiffness matrix of the trimmed element Assemble the global stiffness matrix And by solving the structural equilibrium equations Obtain the global displacement field of the structure to be optimized ;in, This is the applied force vector; (3) Based on global displacement field The adjoint sensitivity method is used to calculate the objective function of the optimization model relative to the design variables. Original sensitivity The original sensitivity is filtered using a pre-calculated filtering matrix H to obtain the filtered sensitivity. ; (4) Using the optimality criterion method, based on the sensitivity after filtering Maximum volume fraction Movement restrictions and physical volume contribution weights for each control point Iteratively update design variables The optimal control point density distribution is obtained by using tensor product interpolation to reconstruct a continuous three-dimensional density field using the inherent tensor product properties of NURBS basis functions. Based on the reconstructed three-dimensional density field, the final STL model of the structure to be optimized is generated by isosurface extraction algorithm.
[0008] Furthermore, establish the element elastic modulus With design variables The relationship between them, and thus the construction of structural flexibility C Minimize as the objective function, and with the maximum volume fraction The topology optimization model is defined by the constraint function, and its mathematical expression is as follows:
[0009] In the formula, For structural flexibility, For the applied force vector, This is the global displacement vector; The total volume of the current material. The total number of control points. For the first i The physical volume contribution weight of each control point; This represents the initial total volume of the design domain; This represents the maximum allowed volume percentage.
[0010] Furthermore, the unit elastic modulus With design variables The relationship between them is represented using the SIMP material interpolation model, and the corresponding expression is:
[0011] In the formula, Density is the design variable for the i-th control point; As a penalty factor; The elastic modulus of a solid material; This represents the elastic modulus of porous materials.
[0012] Furthermore, the physical volume contribution weight of each control point. It is obtained by accumulating the NURBS basis function values at Gaussian points within each background cell of the equige geometric background domain and the corresponding Gaussian integral weights, and then distributing them to the corresponding control points.
[0013] Furthermore, structural flexibility affects design variables. The derivative is:
[0014] In the formula, For structural flexibility, For the applied force vector, K is the global displacement vector; K is the global stiffness matrix. abbreviation; The design variable for the i-th control point; Utilizing pre-calculated and This was obtained by accumulating contributions unit by unit using the chain rule:
[0015] in, .
[0016] Furthermore, the filtering employs a pre-computed filtering matrix H and a normalized vector HS, which are based on a preset filtering radius. The filtered sensitivity is pre-calculated on a 3D control point mesh. The calculation formula is:
[0017] in, For the design variables of the i-th control point, Let be the original sensitivity vector, ⊙ denotes element-wise multiplication of the vector, and γ be the numerical stability constant.
[0018] Furthermore, during the iterative update process, the update criterion of the optimality criterion method incorporates a pre-calculated physical volume contribution weight. The expression for the update criterion is:
[0019] in, The new density after updating cell i, For the current density of cell i, This is the minimum density. To update the operator for the OC method, The damping coefficient to ensure convergence; where , For the corrected compliance sensitivity of unit i, The Lagrange multipliers corresponding to the volume constraint are solved iteratively using the bisection method to ensure that the total volume does not exceed a preset upper limit during the iteration process. Let be the volume weight of unit i.
[0020] Furthermore, in each iteration, based on the current design variables... , , Assemble the global stiffness matrix During assembly, the corresponding pre-calculated stiffness matrix is selected according to the element type, and the elastic modulus of the element is scaled in conjunction with the SIMP interpolation model.
[0021] The present invention also provides a geometric topology optimization system capable of handling complex boundaries such as embedded domains. The system includes a memory and a processor. The memory stores a computer program, and the processor executes the computer program to perform the geometric topology optimization method for handling complex boundaries such as embedded domains as described above.
[0022] The present invention also provides a computer-readable storage medium storing machine-executable instructions, which, when invoked and executed by a processor, cause the processor to implement the geometric topology optimization method described above, which can handle complex boundaries such as embedded domains.
[0023] In summary, compared with the prior art, the geometric topology optimization method and device for handling complex boundaries such as embedded domains provided by this invention have the following advantages: 1. This invention embeds the STL model of the structure to be optimized into an isogeometric background domain constructed based on NURBS basis functions, fundamentally solving the significant difficulties of multi-topology NURBS modeling or reparameterization in traditional isogeometric topology optimization. The underlying NURBS basis function analysis maintains high-order geometric continuity throughout, avoiding the CAD-FEA geometric gap and ensuring accuracy from geometric input to result reconstruction, thus improving adaptability and flexibility. Simultaneously, solid element reuse templates... The clipping unit is calculated independently using an exact Gaussian integral. This significantly improves the computational efficiency of complex embedded domain structures; sensitivity analysis and design variables. During the update, the physical volume contribution weight of each control point is accurately utilized. This invention ensures a physically accurate mapping of local material distribution to global performance, thereby improving the physical accuracy and convergence stability of the optimization results. Utilizing the tensor product interpolation characteristics of NURBS basis functions, the optimal control point density distribution is rapidly reconstructed into a continuous three-dimensional density field through efficient matrix operations. Based on this high-precision continuous density field, a high-precision STL model is directly generated through isosurface extraction, achieving seamless and high-precision conversion from topology optimization results to parametric CAD models. This significantly shortens the design and manufacturing cycle and demonstrates high adaptability.
[0024] 2. This invention employs the embedded domain method for topology optimization, specifically by using the STL model of arbitrarily complex topologies as the actual design domain. The boundaries of the design domain are described in the form of a regular background grid. This method fundamentally solves the problem that traditional isogeometric topology optimization requires parameterizing complex geometry into a single or multi-faceted NURBS model, enabling structures with complex topological features such as multiple holes or separations to be directly analyzed and optimized within an isogeometric framework.
[0025] 3. The update criterion of the aforementioned optimality criterion method incorporates a pre-calculated physical volume contribution weight. , Introduction By correcting the uneven volume proportion caused by geometric damage to the clipping unit and using weighted correction, the "fragmented" control points at the boundary of the embedding domain can be identified. The design variables are adjusted according to the actual physical domain proportion of the control points, thereby avoiding the numerical instability and oscillation phenomenon that traditional embedding domain methods are prone to at the boundary.
[0026] 4. Employing an isogeometric background domain to handle complex STL models provides a carrier for embedded domain isogeometric analysis and optimization, freeing it from the dependence of traditional meshes on the meshing of complex irregular boundaries; the control points of the isogeometric background domain define the design variables. It serves as the fundamental solution unit for subsequent topological iterations, enabling continuous characterization of material distribution across the geometric domain; element elastic modulus With design variables The mapping relationship establishes the physical correlation between material density and structural stiffness, providing a constitutive basis for steps such as element stiffness matrix assembly and structural response solution.
[0027] 5. When dealing with topology optimization problems in embedded domains defined by complex STL models, this invention exhibits faster convergence speed and higher optimization result quality, effectively suppresses numerical oscillations and checkerboard phenomena, and has stronger robustness and reliability.
[0028] 6. The method fully utilizes the precision and high-order continuity of isogeometric analysis in geometric description, and combined with the fine processing of the embedded domain, it significantly improves the boundary clarity and computational efficiency of the topology optimization results for complex geometric structures, while ensuring the adaptability of the optimization process to non-uniform discretization. Attached Figure Description
[0029] Figure 1 This is a flowchart of a geometric topology optimization method for embedding domains and other complex boundaries provided by an embodiment of the present invention; Figure 2 This is a schematic diagram illustrating the construction of the embedded domain geometric model of the present invention; Figure 3 This is a schematic diagram of the design domain, boundary conditions, and loads of a cantilever beam provided in an embodiment of the present invention; Figure 4 These are the control point density map, 3D rendering map, and actual exported STL model map of the final topology structure optimized by geometric topology optimization methods such as embedded domains in the embodiments of the present invention. Figure 5 This is a grayscale comparison diagram of the optimization results of geometric topology optimization such as embedded domain and standard ITO and standard SIMP methods in an embodiment of the present invention. Detailed Implementation
[0030] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.
[0031] This invention provides an isogeometric topology optimization method for embedded domains that can handle complex boundaries. The method fully utilizes the precision and high-order continuity of isogeometric analysis in geometric description, and combined with fine processing of the embedded domain, it significantly improves the boundary clarity and computational efficiency of the topology optimization results for complex geometric structures, while ensuring the adaptability of the optimization process to non-uniform discretization.
[0032] The proposed method overcomes the inherent difficulties of traditional isogeometric topology optimization in handling complex multi-topology CAD model frameworks, and improves the accuracy of integrals of complex geometric boundaries and the physical accuracy of optimization results, achieving more stable, efficient, and high-precision structural topology optimization.
[0033] Please see Figure 1 and Figure 2 The method mainly includes the following steps: Step 1: Constructing an embedded geometric model and optimization problem: Embedding the STL model of the structure to be optimized into an isogeometric background domain constructed based on NURBS basis functions, obtaining the complex geometric boundaries, solid elements, clipped elements, and control points of the isogeometric background domain of the structure to be optimized; defining design variables at the control points of the isogeometric background domain. Establish the unit elastic modulus With design variables The relationship between them, and thus the construction of structural flexibility C Minimize as the objective function, and with the maximum volume fraction This is a topology optimization model with constraint functions.
[0034] Employing an isogeometric background domain to handle complex STL models provides a carrier for embedded domain isogeometric analysis and optimization, freeing it from the dependence of traditional meshes on the meshing of complex irregular boundaries; the control points of the isogeometric background domain define the design variables. It serves as the fundamental solution unit for subsequent topological iterations, enabling continuous characterization of material distribution across the geometric domain; element elastic modulus With design variables The mapping relationship was established, and the physical correlation between material density and structural stiffness was established, providing a constitutive basis for steps such as unit stiffness matrix assembly and structural response solution. A topology optimization model with minimum flexibility as the objective function and volume fraction as the constraint function was constructed, clarifying the optimization direction and boundary constraints of topology optimization.
[0035] The unit elastic modulus With design variables The relationship is represented using the SIMP material interpolation model, and its expression is:
[0036] In the formula, For the design variable at the i-th control point, density, As a penalty factor; The elastic modulus of a solid material; This is the elastic modulus of the porous material, used to prevent the stiffness matrix from becoming singular.
[0037] The mathematical expression for the topology optimization model is:
[0038] In the formula, For structural flexibility, For the applied force vector, This is the global displacement vector; The total volume of the current material. The total number of control points. For the first i The physical volume contribution weight of each control point; This represents the initial total volume of the design domain; This represents the maximum allowed volume percentage.
[0039] In one implementation, a regular, isogeometric background domain containing design space is first established. The geometry of these geometric background domains is determined by NURBS basis functions. , , It means that, among them For control point index, Let be the order of NURBS. Any physical point in the isogeometric background domain. It can be obtained through interpolation using its control points and NURBS basis functions:
[0040] in, For three-dimensional NURBS basis functions, These are coordinates in parameter space. , , These represent the number of control points in the three directions.
[0041] The embodiments of this invention employ an embedded domain method, using an STL model with arbitrarily complex topologies as the actual design domain. The boundaries of the design domain are described in the form of a regular background grid. See also Figure 2 ,Should Figure 2 This paper intuitively demonstrates the construction method of NURBS background domain and STL design domain embedded with isogeometric background domain. This method fundamentally solves the problem that traditional isogeometric topology optimization requires parameterizing complex geometry into a single or multi-faceted NURBS model, enabling structures with complex topological features such as multiple holes or separations to be directly analyzed and optimized within the isogeometric framework.
[0042] Design variables Each control point is defined on the control points of the equal geometric background domain. Associate a design variable , representing the relative density of the material at that control point. Choosing a small value close to zero avoids singularity in the stiffness matrix. Take 1.
[0043] The relationship between material properties and design variables is established using a SIMP interpolation model. For background elements in a homogeneous geometries... equivalent elastic modulus and its equivalent density The relationship is as follows:
[0044] in, The elastic modulus of a solid material. The elastic modulus of porous materials, The SIMP penalty factor is usually taken as... Equivalent density of background cells The design variables of the control points affected by this background element can be used. The weighted average is obtained.
[0045] The optimization objective of this embodiment is to minimize structural compliance. and based on the total volume of the material Not exceeding the upper limit As constraints, the mathematical expression of the topology optimization model is:
[0046] In the formula, The objective function is... The total integral of the material; It is the displacement vector; Here is the stiffness matrix; It represents the volume fraction of the material. For the first i The physical volume contribution weight of each control point; The initial volume of the design domain is given; Dirichlet boundary conditions are implemented using the Nitsche method, rather than directly constraining the degrees of freedom of the control points.
[0047] Step 2, Preprocessing and Global Response Calculation: Pre-calculate the general element stiffness matrix template for the solid elements. The element stiffness matrix of the trimmed element is pre-calculated independently. ; Calculate the physical volume contribution weight for each control point Based on current design variables , , Assemble the global stiffness matrix And by solving the structural equilibrium equations Obtain the global displacement field of the structure to be optimized ;in, This is the applied force vector.
[0048] Physical volume contribution weight of each control point It is obtained by accumulating the NURBS basis function values at Gaussian points within each background cell of the equige geometric background domain and the corresponding Gaussian integral weights, and then distributing them to the corresponding control points.
[0049] For the complete solid element pre-calculation general element stiffness matrix template For the clipped elements cut by the boundary of the STL model, Gaussian integrals are pre-calculated independently and accurately to obtain their element stiffness matrix. Accurately calculate the physical volume contribution weight of each control point. The physical volume contribution weight is obtained by accumulating the NURBS basis function values at the Gaussian points inside the corresponding control points and clipping units, and multiplying them by the corresponding integral weights; in each iteration, based on the current design variables... , , Efficient assembly of global stiffness matrix During assembly, the corresponding pre-calculated stiffness matrix is selected according to the element type, and the element elastic modulus is scaled using the SIMP interpolation model. The penalty and consistency terms introduced by the Nitsche method are also simultaneously assembled into the global stiffness matrix. Finally, the global equilibrium equations are solved. To obtain the global displacement field of the structure .
[0050] Step 3, Sensitivity Analysis and Filtering: Based on Global Displacement Field The objective function of the optimization model relative to the design variables is calculated using the adjoint sensitivity method. Original sensitivity The original sensitivity is filtered using a pre-calculated filtering matrix H to obtain the filtered sensitivity. .
[0051] Structural flexibility on design variables The derivative is:
[0052] In the formula, For structural flexibility, For the applied force vector, K is the global displacement vector; K is the global stiffness matrix; Let be the design variable for the i-th control point. Utilizing pre-calculated and This was obtained by accumulating contributions unit by unit using the chain rule:
[0053] in, .
[0054] To ensure optimization stability, the obtained raw sensitivity is filtered. The filtering uses a pre-calculated filtering matrix H and a normalized vector HS, which are based on a preset filtering radius. The filtered sensitivity is pre-calculated on a 3D control point mesh. The calculation formula is:
[0055] in, For the design variables of the i-th control point, Let be the original sensitivity vector, ⊙ denotes element-wise multiplication of the vector, and γ be the numerical stability constant.
[0056] The Lagrange multiplier λ is determined using the optimality criterion method via binary search, and the result is based on the filtered sensitivity. Current value of design variables Physical volume contribution weight Adjustment coefficient η Design variable lower bound Upper limit And update design variables using the maximum movement limit step size. .
[0057] Step 4, Iteratively update design variables: Using the optimality criterion method, based on the filtered sensitivity... Maximum volume fraction Movement restrictions and physical volume contribution weights for each control point Iteratively update design variables Until design variables Once the convergence condition is met, the optimal control point density distribution is obtained. Using the tensor product interpolation method and leveraging the inherent tensor product properties of the NURBS basis functions, the final optimal control point density distribution is reconstructed into a continuous three-dimensional density field. Based on the reconstructed continuous three-dimensional density field, the final STL model of the structure to be optimized is generated using an isosurface extraction algorithm.
[0058] In design variables After the convergence condition is met, the density field of the control points is efficiently reconstructed using the tensor product interpolation method to obtain a continuous three-dimensional density field. Based on the continuous three-dimensional density field, a high-precision STL geometric model is generated using an isosurface extraction algorithm and then visualized.
[0059] Updating design variables using the OC method This update process precisely considers the physical volume contribution weight of each control point i. After updating the design variables, the system checks whether the convergence condition is met. If not, it returns to step two to continue iterating; otherwise, the optimization process ends.
[0060] After optimization and convergence, this embodiment of the invention utilizes the inherent tensor product properties of NURBS basis functions through tensor product interpolation to rapidly and accurately reconstruct the density field of the final discrete control points into a continuous three-dimensional density field. Based on this reconstructed continuous three-dimensional density field, an isosurface extraction algorithm is used to generate the final STL model, which can be directly used in CAD systems.
[0061] During the iterative update process, the key lies in how to handle the non-uniform physical contribution caused by the embedding domain cutting. The optimality criterion method introduces a pre-calculated physical volume contribution weight into its update criterion. The expression for the update criterion is:
[0062] in, The new density after updating cell i, For the current density of cell i, This is the minimum density. To update the operator for the OC method, The damping coefficient to ensure convergence; where , For the corrected compliance sensitivity of unit i, The Lagrange multipliers corresponding to the volume constraint are solved iteratively using the bisection method to ensure that the total volume does not exceed a preset upper limit during the iteration process. Let be the volume weight of unit i.
[0063] Introduced here By correcting the uneven volume proportions of clipping units caused by geometric damage, the optimizer can identify "fragmented" control points at the boundary of the embedding domain through weighted correction. It can then adjust the design variables according to the actual physical domain proportion of the control points, thereby avoiding the numerical instability and oscillation phenomena that are easily generated at the boundary by traditional embedding domain methods.
[0064] In order to use a gradient-based optimizer, the compliance objective function must be computed precisely. C Relative to design variables The sensitivity, based on the accompanying sensitivity analysis:
[0065] Based on the hybrid pre-calculation strategy in step two, the element stiffness matrix... The derivative calculation is divided into two cases: solid elements using template matrices. ,have For the trimming unit: using the independent integral matrix ,have .
[0066] in, The equivalent density of a cell is the weighted integral of the control point design variables associated with that cell through the basis function. The sensitivity is obtained through weighted calculation. This sensitivity calculation fully utilizes the geometric information from the preprocessing stage, ensuring mathematical consistency between the objective function and its gradient.
[0067] In each iteration, "ghost nodes" are also automatically identified. The physical volume contribution weight of a control point is also considered. If the threshold is extremely small, the node is determined to be completely outside the design domain, and its density is forcibly cleaned. It does not participate in sensitivity filtering, thereby further improving computational efficiency and ensuring the clarity of the embedded domain boundary.
[0068] After calculating the updated values of the design variables, the physical volume parameters and compliance information are updated, and the next iteration begins until the convergence condition is met.
[0069] The present invention will be further described in detail below with reference to specific embodiments.
[0070] like Figure 3 As shown, a cantilever beam example is used to verify the method provided by the present invention.
[0071] The STL model is defined as follows: the left end is fixed, and a downward mechanical load F is applied to the center of the right end, with an upper limit for the volume fraction. .
[0072] Figure 4The final topology obtained using the embedded domain isogeometric topology optimization method of this invention is shown. It can be seen that the final structure perfectly conforms to the complex boundaries defined by the original STL, without exhibiting the jagged boundaries commonly seen in traditional finite element topology optimization. The material distribution follows the force transmission path, forming a clear hollow truss structure, fully demonstrating the advantages of the high-order continuity of the isogeometric method.
[0073] Figure 5 The results of three different topology optimization methods are compared under the same boundary conditions and volume constraints: The method of this invention achieves smooth grayscale transitions at structural edges and within the structure, and incorporates physical volume weighting. The optimization process exhibits a smooth compliance curve with no significant numerical oscillations, resulting in a final configuration with inherent high precision.
[0074] Standard geometric topology optimization: Smooth structural edges, relying on parametric modeling of the initial design domain. However, when faced with complex STL irregular geometric boundaries, it is difficult to construct a matching global analysis model, resulting in poor modeling flexibility and applicability.
[0075] The standard SIMP method is slightly faster in computation, but the structure edges exhibit a noticeable stepped, jagged appearance, and the optimization process is unstable. The final convergence result has poor boundary quality, making it difficult to apply directly.
[0076] As can be seen from the comparison, the method of the present invention significantly improves the flexibility of topology optimization when dealing with complex boundary problems, and at the same time performs well in terms of convergence stability and grayscale smoothness of the results.
[0077] The present invention also provides a system for implementing the above method, the system comprising: a modeling and definition module for constructing an isogeometric background domain and introducing an external STL model, defining the embedded domain and its physical boundaries; and a hybrid pre-computation module for distinguishing between solid elements and clipped elements, and pre-computing their stiffness matrix templates and physical volume contribution weights respectively. The global response solution module is used to assemble the global stiffness matrix and solve the displacement field, while also handling load integrals at the boundaries. The sensitivity analysis and filtering module is used to calculate the original sensitivity based on physical volume weighting and to perform smoothing using a tensor filtering matrix. The optimization solution module is used to update the design variables using the OC update criterion with weight correction and to perform convergence checks. The reconstruction and export module is used to reconstruct the three-dimensional density field using tensor product interpolation and export it as a high-precision STL model.
[0078] The present invention also provides a geometric topology optimization system capable of handling complex boundaries such as embedded domains. The system includes a memory and a processor. The memory stores a computer program, and the processor executes the computer program to perform the geometric topology optimization method for handling complex boundaries such as embedded domains as described above.
[0079] The present invention also provides a computer-readable storage medium storing machine-executable instructions, which, when invoked and executed by a processor, cause the processor to implement the geometric topology optimization method described above, which can handle complex boundaries such as embedded domains.
[0080] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements 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 geometric topology optimization method capable of handling complex boundaries such as embedded domains, characterized in that, The steps are as follows: (1) Embed the STL model of the structure to be optimized into the isogeometric background domain constructed based on NURBS basis functions to obtain the solid elements, clipping elements and control points of the isogeometric background domain of the structure to be optimized; (2) Design variables defined at the control points of the isogeometric background domain General element stiffness matrix template for solid elements Element stiffness matrix of the trimmed element Assemble the global stiffness matrix And by solving the structural equilibrium equations Obtain the global displacement field of the structure to be optimized ;in, This is the applied force vector; (3) Based on global displacement field The adjoint sensitivity method is used to calculate the objective function of the optimization model relative to the design variables. Original sensitivity The original sensitivity is filtered using a pre-calculated filtering matrix H to obtain the filtered sensitivity. ; (4) Using the optimality criterion method, based on the sensitivity after filtering Maximum volume fraction Movement restrictions and physical volume contribution weights for each control point Iteratively update design variables The optimal control point density distribution is obtained by using tensor product interpolation to reconstruct a continuous three-dimensional density field using the inherent tensor product properties of NURBS basis functions. Based on the reconstructed three-dimensional density field, the final STL model of the structure to be optimized is generated by isosurface extraction algorithm.
2. The geometric topology optimization method for embedded domains and other structures capable of handling complex boundaries as described in claim 1, characterized in that: Establish the unit elastic modulus With design variables The relationship between them, and thus the construction of structural flexibility C Minimize as the objective function, and with the maximum volume fraction The topology optimization model is defined by the constraint function, and its mathematical expression is as follows: In the formula, For structural flexibility, For the applied force vector, This is the global displacement vector; The total volume of the current material. The total number of control points. For the first i The physical volume contribution weight of each control point; This represents the initial total volume of the design domain; This represents the maximum allowed volume percentage.
3. The geometric topology optimization method for embedded domains and other structures capable of handling complex boundaries as described in claim 2, characterized in that: The unit elastic modulus With design variables The relationship between them is represented using the SIMP material interpolation model, and the corresponding expression is: In the formula, Density is the design variable for the i-th control point; As a penalty factor; The elastic modulus of a solid material; This represents the elastic modulus of porous materials.
4. The geometric topology optimization method for embedded domains and other structures capable of handling complex boundaries as described in claim 1, characterized in that: Physical volume contribution weight of each control point It is obtained by accumulating the NURBS basis function values at Gaussian points within each background cell of the equige geometric background domain and the corresponding Gaussian integral weights, and then distributing them to the corresponding control points.
5. The geometric topology optimization method for embedded domains and other structures capable of handling complex boundaries as described in any one of claims 1-4, characterized in that: Structural flexibility on design variables The derivative is: In the formula, For structural flexibility, For the applied force vector, K is the global displacement vector; K is the global stiffness matrix. abbreviation; The design variable for the i-th control point; Utilizing pre-calculated and This was obtained by accumulating contributions unit by unit using the chain rule: in, .
6. The geometric topology optimization method for embedded domains and other structures capable of handling complex boundaries as described in claim 5, characterized in that: The filtering uses a pre-computed filtering matrix H and a normalized vector HS, which are based on a preset filtering radius. The filtered sensitivity is pre-calculated on a 3D control point mesh. The calculation formula is: in, For the design variables of the i-th control point, Let be the original sensitivity vector, ⊙ denotes element-wise multiplication of the vector, and γ be the numerical stability constant.
7. The geometric topology optimization method for embedded domains and other methods capable of handling complex boundaries as described in any one of claims 1-4, characterized in that: During the iterative update process, the update criterion of the optimality criterion method incorporates a pre-calculated physical volume contribution weight. The expression for the update criterion is: in, The new density after updating cell i, For the current density of cell i, This is the minimum density. To update the operator for the OC method, The damping coefficient to ensure convergence; where , For the corrected compliance sensitivity of unit i, The Lagrange multipliers corresponding to the volume constraint are solved iteratively using the bisection method to ensure that the total volume does not exceed a preset upper limit during the iteration process. Let be the volume weight of unit i.
8. The geometric topology optimization method for embedded domains and other structures capable of handling complex boundaries as described in any one of claims 1-4, characterized in that: In each iteration, based on the current design variables , , Assemble the global stiffness matrix During assembly, the corresponding pre-calculated stiffness matrix is selected according to the element type, and the elastic modulus of the element is scaled in conjunction with the SIMP interpolation model.
9. A geometric topology optimization system capable of handling complex boundaries, such as embedded domains, characterized in that: The system includes a memory and a processor. The memory stores a computer program, and when the processor executes the computer program, it performs the geometric topology optimization method, such as the embedded domain method for handling complex boundaries, as described in any one of claims 1-8.
10. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores machine-executable instructions, which, when invoked and executed by a processor, cause the processor to implement the geometric topology optimization method, such as the embedded domain method for handling complex boundaries, as described in any one of claims 1-8.