A topology optimization method based on clear smooth boundary

Abstract

The present application relates to a kind of topology optimization method based on clear smooth boundary.Step includes the construction volume constraint under minimum flexibility structural topology optimization model;Definition initialization parameter, complete structure finite element analysis setting and the weight coefficient calculation of each filter;Enter the main cycle of this optimization method, execute finite element analysis to structure, calculate sensitivity number and average sensitivity number;Using improved proportional topology optimization algorithm updates unit volume fraction, carries out filter processing to it;Calculate grid point density, and realize the real / empty design of grid point;Using level set function generates clear smooth topology boundary;Iteration calculation termination condition judgment;Output optimization design result.The present application has the advantages that: this optimization method can quickly realize the topology optimization design of structure, solve the sawtooth boundary and grey area problem of prior art, improve structural optimization design effect, obtain the optimization structure with clear smooth boundary, provide convenience for the processing and manufacturing of subsequent optimization structure.

Description

Technical Field

[0001] This invention relates to the field of structural optimization design technology, and more specifically to a topology optimization method based on clear and smooth boundaries. Background Technology

[0002] Topology optimization, as a higher-level structural optimization design method, essentially aims to achieve optimal or near-optimal structural performance by rationally allocating materials within the structural design domain. Despite its complexity, topology optimization is widely used to explore and solve optimal structural design problems in various fields due to its advantages, such as fundamentally altering the force transmission path, improving overall structural performance, minimizing structural weight, and providing novel structural configurations with diverse performance characteristics. These problems include minimizing structural weight, maximizing structural frequency and stress, ensuring structural reliability and robustness, designing multi-material structures, and multi-scale structures.

[0003] Solving structural topology optimization problems requires the use of topology optimization methods. Through decades of research and development by scholars both domestically and internationally, various effective topology optimization methods have been proposed, including homogenization methods, progressive structural optimization methods, density methods, level set methods, independent continuous mapping methods, moving deformable component methods, and improved versions of these methods, as well as other novel methods. For example, Chinese Patent 202311732916.1 discloses a multi-scale topology optimization design method for geometrically-material asymmetric sandwich structures, and 202511101014.7 discloses a design optimization method for free-form curve features.

[0004] Although existing topology optimization methods can solve various structural optimization design problems, the resulting optimized structures generally suffer from jagged boundaries and grayscale defects. This not only hinders the subsequent processing and manufacturing of the optimized structures but has also been a persistent technical challenge for researchers in the field of structural design. Summary of the Invention

[0005] The technical problem to be solved by the present invention is to provide a topology optimization method based on clear and smooth boundaries, which can solve the problem that the optimized structure obtained in the prior art has jagged boundaries and gray areas, which is not conducive to the subsequent processing and manufacturing of the optimized structure.

[0006] To solve the above-mentioned technical problems, the technical solution of the present invention is: a topology optimization method based on clear and smooth boundaries, characterized by comprising the following steps: S1. Construct a topology optimization model that minimizes flexibility under volume constraints; S2: Define initialization parameters and complete the finite element analysis settings for the structure and the calculation of the weight coefficients for each filter; S3: Enter the main loop of the topology optimization method, perform finite element analysis on the structure, and calculate the sensitivity number and average sensitivity number based on this analysis; S4: Update the cell volume fraction using an improved proportional topology optimization algorithm and then filter it using a filter. S5: Calculate the grid point density and use the grid point density to implement the solid and empty grid point design; S6: Use level set functions to generate clean and smooth topological boundaries; S7: Determine the termination condition for iterative calculation. If the termination condition is met, the iterative calculation terminates, and step S8 is executed; otherwise, step S3 is executed. S8: Output the optimized design results.

[0007] Furthermore, step S1 constructs a structural topology optimization model that minimizes flexibility under volume constraints as follows: Minimize: , The constraints to be satisfied are: , , , In the formula, C and V f These are the structural flexibility and the allowable material volume fraction, respectively. K and F These are the stiffness matrix and load vector of the structure, respectively; U and M These are the displacement vector of the structure and the total number of elements, respectively. V e and X e They are units e Volume and volume fraction; r min It is the minimum value of the grid point density; The unit volume fraction X e The mathematical expression is: , In the formula, N It is a unit e The number of grid points distributed on the top; r e,g It is a unit e The first g Density of grid points.

[0008] Furthermore, in step S2, the initialization parameters that need to be defined include: allowable material volume fraction. V fHistorical coefficients Scaling factor l Filter cutoff radius r min Filter radius for generating node density c Young's modulus of the material E 1 Compared to Poisson u ; The filter has two weighting coefficients: The mathematical expression for the first weighting coefficient is: , In the formula, d e,j It is a unit e , j The distance between; r min and These are the filter cutoff radius and the weighting coefficient, respectively. The mathematical expression for the second weighting coefficient is: , In the formula, d n,e It is a unit e and nodes n The distance between; c and x n,e These are the filter radius and weight coefficients used to generate node density. x n,e Its function is to calculate the node density.

[0009] Furthermore, the calculation of the sensitivity count and average sensitivity count in step S3 is as follows: The expression for the sensitivity number is: , In the formula, α e It is a unit e The sensitivity number; u e and K e 1 They are units e The displacement vector and stiffness matrix vector; The average sensitivity number can be calculated from the sensitivity number, and the mathematical expression for the average sensitivity number is: , In the formula, α e k and α e k-1These are the sensitivity numbers obtained in the current iteration and the previous iteration, respectively; It is the average sensitivity number.

[0010] Furthermore, updating the cell volume fraction in step S4 includes two parts: The first part updates and improves the cell volume fraction in the inner loop of the proportional topology optimization algorithm, using the following mathematical expression: , In the formula, It is the updated cell volume fraction in the inner loop of the improved proportional topology optimization algorithm; X j It is a unit j volume fraction, V RM It is the volume of the remaining material; It is the ratio of the average sensitivity number after filtration, calculated. The expression is: , In the formula, It is a unit j The average sensitivity ratio can be calculated using the following expression: , In the formula, It is a unit j The average sensitivity number; λ is the scaling factor; The second part updates and improves the cell volume fraction in the main loop of the proportional topology optimization algorithm, using the following mathematical expression: , In the formula, It is the updated cell volume fraction in the main loop of the improved proportional topology optimization algorithm; It is the unit volume fraction obtained from the previous main loop iteration. It's a historical coefficient; also, regarding... The mathematical expression used for filtering is: , In the formula, It is the volume fraction of the new unit after filtration.

[0011] Furthermore, the detailed process of calculating the grid point density and using the grid point density to implement the solid and void design of the grid points in step S5 is as follows: Before calculating the grid point density, the node density must first be calculated, and its mathematical expression is as follows: , In the formula, r n It is the first n The density of nodes; When the structural design domain is discretized into elements with four nodes, the mesh point density can be solved using the node density. The mathematical expression used is: , In the formula, r ( g , or )and r ( x , y ) are respectively in local coordinates ( g , or ) and global coordinates ( x , y ) grid point density; and P κ ( g , or These represent the node density and a shape function, respectively. The mathematical expression required to achieve the solid and void design of grid points using grid point density is: , In the formula, ε and β are the threshold and control parameter, respectively; σ( ) indicates the relationship with variables The relevant functions; a, b, c and All are constant parameters; when r e,g = r min At that time, unit e The first distribution of the upper distribution g One grid point was deleted; when r e,g When =1, unit e The first distribution of the upper distribution g Each grid point is retained, and based on this rule, the design of the grid points, whether solid or empty, can be realized.

[0012] Furthermore, the level set function used in step S6 to generate a clear and smooth topological boundary F ( x , y The mathematical expression for ) is: .

[0013] Furthermore, there are two termination conditions in step S7: First termination condition: The total variation in the topology design is less than the allowable variation. d Its mathematical expression is: , In the formula, Xe k and X e k-1 These are the unit volume fractions obtained in the current iteration and the previous iteration, respectively; The second termination condition is that the error of the topological boundary is less than the allowable error. t Its mathematical expression is: , In the formula, N e It is the total number of intermediate units that are not on the boundary.

[0014] Furthermore, the optimized design results output in step S8 include the structural flexibility, topology, and historical iteration diagram.

[0015] The advantages of this invention are as follows: by effectively integrating element volume fraction, sensitivity number, average sensitivity number, improved proportional topology optimization algorithm, and level set function, it solves the finite element analysis problem of structures, the element volume fraction update problem, and the problem of generating clear and smooth boundaries of optimized structures. Specifically, it solves the finite element analysis problem of structures by using element volume fraction; it solves the element volume fraction update problem by effectively combining sensitivity number, average sensitivity number, and improved proportional topology optimization algorithm; and it solves the problem of generating clear and smooth boundaries of optimized structures by effectively combining element volume fraction and level set function. The optimization method of this invention can quickly realize the topology optimization design of the structure, solve the jagged boundary and gray area problems that are common in the optimized structures obtained by existing topology optimization methods, improve the optimization design effect of the structure, and obtain an optimized structure with clear and smooth boundaries, which facilitates the subsequent processing and manufacturing of the optimized structure. Attached Figure Description

[0016] Figure 1 This is a flowchart of an embodiment of a topology optimization method based on clear and smooth boundaries according to the present invention; Figure 2 This invention provides the design domain and boundary conditions for a cantilever beam, as an embodiment of a topology optimization method based on clear and smooth boundaries. Figure 3 This is a schematic diagram of the topology obtained after optimization of a cantilever beam, according to an embodiment of the topology optimization method based on clear and smooth boundaries of the present invention. Figure 4 This is a historical iteration diagram during the optimization design of a cantilever beam, based on a topology optimization method with clear and smooth boundaries according to an embodiment of the present invention. Detailed Implementation

[0017] 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 specific embodiments. The following embodiments will enable those skilled in the art to more fully understand this invention, but do not limit the invention to the scope of the described embodiments.

[0018] The specific implementation method adopts the following technical solution: as shown in the appendix. Figure 1 The diagram shown illustrates a topology optimization method based on clear and smooth boundaries, comprising the following steps: S1: Construct a topology optimization model that minimizes the flexibility under volume constraints; The topology optimization model that minimizes the flexibility under volume constraints is as follows: Minimize: , The constraints to be satisfied are: , , , In the formula, C and V f These are the structural flexibility and the allowable material volume fraction, respectively. K and F These are the stiffness matrix and load vector of the structure, respectively; U and M These are the displacement vector of the structure and the total number of elements, respectively. V e and X e They are units e Volume and volume fraction; r min It is the minimum value of the grid point density; In addition, calculate the unit volume fraction. X e The mathematical expression is: , In the formula, N It is a unit e The number of grid points distributed on the top; r e,g It is a unit e The first g Density of grid points.

[0019] S2: Define initialization parameters and complete the finite element analysis settings for the structure and the calculation of the weight coefficients for each filter; The initialization parameters that need to be defined mainly include: allowable material volume fraction. V f Historical coefficients Scaling factor l Filter cutoff radius r min Filter radius for generating node density c Young's modulus of the material E 1 Compared to Poisson u ; The parameters are set as follows in this embodiment: V f =0.33; =0.3; l =2.0; r min =1.5mm; c =1.0mm; E 1 =1.0 MPa; u =0.3.

[0020] Taking the design of a cantilever beam as an example, attached Figure 2 This defines the design domain and boundary conditions for the cantilever beam. (See attached diagram.) Figure 2 It can be seen that the design domain (length × width × thickness) of the cantilever beam is 120mm × 60mm × 1mm. Furthermore, the left end of the structure is subject to a fixed constraint, while the middle of its right end bears a downward vertical load of 1N. (The remaining text appears to be incomplete and requires further context.) Figure 2 During the topology optimization design of the cantilever beam shown, the structural design is divided into units of nx=120 and ny=60 in the length and width directions, respectively, with a total of 120×60 units and each unit having a size of 1mm×1mm×1mm.

[0021] In addition, the filter used has two weighting coefficients: The mathematical expression for the first weighting coefficient is: , In the formula, d e,j It is a unit e , j The distance between; r min and These are the filter cutoff radius and the weighting coefficient, respectively. The mathematical expression for the second weighting coefficient is: , In the formula, d n,e It is a unit e and nodes n The distance between; c and x n,e These are the filter radius and weight coefficients used to generate node density. x n,e Its function is to calculate the node density.

[0022] S3: Enter the main loop of the topology optimization method. With the help of the topology optimization model provided in step S1, perform finite element analysis on the structure to obtain the element compliance and structural compliance results required for the improved proportional topology optimization algorithm, and calculate the sensitivity number and average sensitivity number based on this. The expression for the sensitivity number is: , In the formula, α e It is a unit e The sensitivity number; u e and K e 1 They are units e The displacement vector and stiffness matrix vector; In addition, the average sensitivity number can be calculated from the sensitivity number, and the mathematical expression for the average sensitivity number is: , In the formula, α e k and α e k-1 These are the sensitivity numbers obtained in the current iteration and the previous iteration, respectively; It is the average sensitivity number.

[0023] S4: Update the cell volume fraction using an improved proportional topology optimization algorithm and then filter it using a filter. The improved proportional topology optimization algorithm for updating the unit volume fraction consists of two parts: The first part updates and improves the cell volume fraction in the inner loop of the proportional topology optimization algorithm, using the following mathematical expression: , In the formula, It is the updated cell volume fraction in the inner loop of the improved proportional topology optimization algorithm; X j It is a unit j volume fraction, V RM It is the volume of the remaining material; It is the ratio of the average sensitivity number after filtration, calculated. The expression is: , In the formula, It is a unit jThe average sensitivity ratio can be calculated using the following expression: , In the formula, It is a unit j The average sensitivity number; λ is the scaling factor; The second part updates and improves the cell volume fraction in the main loop of the proportional topology optimization algorithm, using the following mathematical expression: , In the formula, It is the updated cell volume fraction in the main loop of the improved proportional topology optimization algorithm; It is the unit volume fraction obtained from the previous main loop iteration. It's a historical coefficient; also, regarding... The mathematical expression used for filtering is: , In the formula, It is the volume fraction of the new unit after filtration.

[0024] S5: Calculate the grid point density and use the grid point density to implement the solid and empty grid point design; Before calculating the grid point density, the node density must first be calculated, and its mathematical expression is as follows: , In the formula, r n It is the first n The density of nodes; Furthermore, when the structural design domain is discretized into elements with four nodes, the mesh point density can be solved using the node density. The mathematical expression used is: , In the formula, r ( g , or )and r ( x , y ) are respectively in local coordinates ( g , or ) and global coordinates ( x , y ) grid point density; and P κ ( g , or These represent the node density and a shape function, respectively. Furthermore, the mathematical expression required to achieve the solid and void design of grid points using grid point density is: , In the formula, ε and β are the threshold and control parameter, respectively; σ( ) indicates the relationship with variables The relevant functions; a, b, c and All are constant parameters; when r e,g = r min At that time, unit e The first distribution of the upper distribution g One grid point was deleted; when r e,g When =1, unit e The first distribution of the upper distribution g Each grid point is retained, and based on this rule, the design of the grid points, whether solid or empty, can be realized.

[0025] S6: Use level set functions to generate clean and smooth topological boundaries; Among them, the level set function used to generate clear and smooth topological boundaries F ( x , y The mathematical expression for ) is: .

[0026] S7: Determine the termination condition for iterative calculation. If the termination condition is met, the iterative calculation terminates, and step S8 is executed; otherwise, step S3 is executed. There are two termination conditions: First termination condition: The total variation in the topology design is less than the allowable variation. d Its mathematical expression is: , In the formula, X e k and X e k-1 These are the unit volume fractions obtained in the current iteration and the previous iteration, respectively; The second termination condition is that the error of the topological boundary is less than the allowable error. t Its mathematical expression is: , In the formula, N e It is the total number of intermediate units that are not on the boundary.

[0027] S8: Output optimization design results: The optimization design results include the flexibility of the structure, topology, and historical iteration diagram.

[0028] Appendix Figure 3 This is a schematic diagram of the optimized topology of the cantilever beam. (See attached diagram.) Figure 3It is evident that the topology optimization method based on clear and smooth boundaries provided by this invention can obtain the topology of a cantilever beam with clear and smooth boundaries. This indicates that the method proposed in this invention can not only solve the topology optimization problem of minimizing flexibility under volume constraints and the jagged boundaries and gray areas that are common in the optimized structures obtained by existing topology optimization methods, but also improve the optimization design effect of the structure and provide convenience for the subsequent processing and manufacturing of the optimized structure.

[0029] Appendix Figure 4 This is a historical iteration diagram of the cantilever beam optimization design process. (See attached...) Figure 4 As can be seen, the topology optimization method based on clear and smooth boundaries provided by this invention can obtain a suboptimal topology for the cantilever beam in the 10th iteration, with a compliance of 93.90 J. In the 15th iteration, the obtained topology is significantly improved compared to the 10th iteration, with a compliance of 87.65 J. In the 30th iteration, the obtained topology is similar to that of the 15th iteration, with a compliance of 87.17 J. In the 102nd iteration, the obtained topology is similar to that of the 30th iteration, with a compliance of 87.10 J. Furthermore, comparing the results of the 15th, 30th, and 102nd iterations reveals that the obtained topologies are generally similar, and the differences in compliance results are minimal. This indicates that subsequent iterations have little impact on the topology optimization design of the structure, merely serving to meet the termination condition. The above analysis results further confirm that the topology optimization method based on clear and smooth boundaries provided by this invention can quickly realize the topology optimization design of the structure.

[0030] The foregoing has shown and described the basic principles and main features of the present invention, as well as its advantages. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the present invention as claimed. The scope of protection of this invention is defined by the appended claims and their equivalents.

Claims

1. A topology optimization method based on clear and smooth boundaries, characterized in that: Includes the following steps: S1. Construct a topology optimization model that minimizes flexibility under volume constraints; S2: Define initialization parameters and complete the finite element analysis settings for the structure and the calculation of the weight coefficients for each filter; S3: Enter the main loop of the topology optimization method, perform finite element analysis on the structure, and calculate the sensitivity number and average sensitivity number based on this analysis; S4: Update the cell volume fraction using an improved proportional topology optimization algorithm and then filter it using a filter. S5: Calculate the grid point density and use the grid point density to implement the solid and empty grid point design; S6: Use level set functions to generate clean and smooth topological boundaries; S7: Determine the termination condition for iterative calculation. If the termination condition is met, the iterative calculation terminates, and then proceed to step S8. Conversely, proceed to step S3; S8: Output the optimized design results.

2. The topology optimization method based on clear and smooth boundaries according to claim 1, characterized in that: The structural topology optimization model that minimizes flexibility under volume constraints in step S1 is as follows: Minimize: , The constraints to be satisfied are: , ， ， In the formula, C and V f These are the structural flexibility and the allowable material volume fraction, respectively. K and F These are the stiffness matrix and load vector of the structure, respectively; U and M These are the displacement vector of the structure and the total number of elements, respectively. V e and X e They are units e Volume and volume fraction; ρ min It is the minimum value of the grid point density; The unit volume fraction X e The mathematical expression is: , In the formula, N It is a unit e The number of grid points distributed on the top; ρ e,g It is a unit e The first g Density of grid points.

3. The topology optimization method based on clear and smooth boundaries according to claim 1, characterized in that: In step S2, the initialization parameters that need to be defined include: allowable material volume fraction. V f Historical coefficients Scaling factor λ Filter cutoff radius r min Filter radius for generating node density γ Young's modulus of the material E 1 Compared to Poisson υ ; The filter has two weighting coefficients: The mathematical expression for the first weighting coefficient is: , In the formula, d e,j It is a unit e , j The distance between; r min and These are the filter cutoff radius and the weighting coefficient, respectively. The mathematical expression for the second weighting coefficient is: , In the formula, d n,e It is a unit e and nodes n The distance between; γ and ξ n,e These are the filter radius and weight coefficients used to generate node density. ξ n,e Its function is to calculate the node density.

4. The topology optimization method based on clear and smooth boundaries according to claim 1, characterized in that: The calculation of the sensitivity number and average sensitivity number in step S3 is as follows: The expression for the sensitivity number is: , In the formula, α e It is a unit e The sensitivity number; u e and K e 1 They are units e The displacement vector and stiffness matrix vector; The average sensitivity number can be calculated from the sensitivity number, and the mathematical expression for the average sensitivity number is: ， In the formula, α e k and α e k-1 These are the sensitivity numbers obtained in the current iteration and the previous iteration, respectively; It is the average sensitivity number.

5. The topology optimization method based on clear and smooth boundaries according to claim 1, characterized in that: The step S4 of updating the cell volume fraction includes two parts: The first part updates and improves the cell volume fraction in the inner loop of the proportional topology optimization algorithm, using the following mathematical expression: , In the formula, It is the updated cell volume fraction in the inner loop of the improved proportional topology optimization algorithm; X j It is a unit j volume fraction, V RM It is the volume of the remaining material; It is the ratio of the average sensitivity number after filtration, calculated. The expression is: , In the formula, It is a unit j The average sensitivity ratio can be calculated using the following expression: , In the formula, It is a unit j The average sensitivity number; λ is the scaling factor; The second part updates and improves the cell volume fraction in the main loop of the proportional topology optimization algorithm, using the following mathematical expression: , In the formula, It is the updated cell volume fraction in the main loop of the improved proportional topology optimization algorithm; It is the unit volume fraction obtained from the previous main loop iteration. It's a historical coefficient; also, regarding... The mathematical expression used for filtering is: , In the formula, It is the volume fraction of the new unit after filtration.

6. The topology optimization method based on clear and smooth boundaries according to claim 1, characterized in that: The detailed process of calculating the grid point density and using the grid point density to implement the solid and void design of the grid points in step S5 is as follows: Before calculating the grid point density, the node density must first be calculated, and its mathematical expression is as follows: ， In the formula, ρ n It is the first n The density of nodes; When the structural design domain is discretized into elements with four nodes, the mesh point density can be solved using the node density. The mathematical expression used is: ， In the formula, ρ ( ζ , η )and ρ ( x , y ) are respectively in local coordinates ( ζ , η ) and global coordinates ( x , y ) grid point density; and P κ ( ζ , η These represent the node density and a shape function, respectively. The mathematical expression required to achieve the solid and void design of grid points using grid point density is: ， In the formula, ε and β are the threshold and control parameter, respectively; σ( ) indicates the relationship with variables The relevant functions; a, b, c, and ς are all constant parameters; when ρ e,g = ρ min At that time, unit e The first distribution of the upper distribution g One grid point was deleted; when ρ e,g When =1, unit e The first distribution of the upper distribution g Each grid point is retained, and based on this rule, the design of the grid points, whether solid or empty, can be realized.

7. The topology optimization method based on clear and smooth boundaries according to claim 1, characterized in that: The level set function used in step S6 to generate a clear and smooth topological boundary Φ ( x , y The mathematical expression for ) is: 。 8. The topology optimization method based on clear and smooth boundaries according to claim 1, characterized in that: There are two termination conditions in step S7: First termination condition: The total variation in the topology design is less than the allowable variation. δ Its mathematical expression is: ， In the formula, X e k and X e k-1 These are the unit volume fractions obtained in the current iteration and the previous iteration, respectively; The second termination condition is that the error of the topological boundary is less than the allowable error. τ Its mathematical expression is: ， In the formula, N e It is the total number of intermediate units that are not on the boundary.

9. The topology optimization method based on clear and smooth boundaries according to claim 1, characterized in that: The optimized design results output in step S8 include the structural flexibility, topology, and historical iteration diagram.

Images