Topological optimization method of high-performance Voronoi homogeneous gradient porous structure

Abstract

The invention relates to a topological optimization method for a high-performance Voronoi homogeneous gradient porous structure, which is applied to advanced construction. Voronoi gradient sampling points are generated by adopting a gradient Poisson disk sampling method, the spacing of the Voronoi porous unit sampling points is used as the Poisson disk radius, the Poisson disk radius is used as a design variable, and the porous structure is constructed according to the optimal Young modulus obtained according to the minimum structural flexibility, so that the gradient change of material distribution can be reflected on the whole, and the optimal Young modulus can be obtained according to the design variable. And local areas with similar Young modulus are uniformly distributed, so that the calculation precision is improved, and the reasonability of material distribution is ensured.

Description

Technical Field

[0001] The invention belongs to the technical field of new materials and new structures, and in particular relates to a topological optimization method for a high-performance Voronoi homogeneous gradient porous structure, which is used in advanced construction. Background Art

[0002] A porous structure refers to an interconnected network of solid rods and plates that form the edges and walls of the holes. Due to its advantages of light weight, high strength, energy absorption and buffering, sound insulation and noise reduction, and heat insulation, it has attracted much attention in recent years and has been widely used in advanced construction fields such as architecture and metal. The main manufacturing processes for porous structures include melt foaming, sheet metal forming, and 3D printing.

[0003] Under a given manufacturing process, the mechanical properties of porous structures depend on the characteristics of the pore units. When a structure is subjected to a specific load, the stress inside the structure is often unevenly distributed, and the material properties cannot be fully utilized. By rationally changing the geometry and arrangement of the porous structure, the mechanical properties of the structure can be effectively enhanced. In addition, the complex geometry inside the porous structure leads to poor constructability. When the unit wall thickness is too small, the porous unit cannot be manufactured or the forming quality is reduced. These construction defects will also reduce the mechanical properties of the component.

[0004] Gradient porous structures can be divided into heterogeneous and homogeneous gradient porous structures, both of which can optimize material distribution according to the stress conditions of the model. The wall thickness of the pore unit of the heterogeneous gradient porous structure is not fixed, and there is a disadvantage that thin-walled units cannot be manufactured; while the wall thickness of the porous unit of the homogeneous gradient porous structure can be set as a fixed constant according to the construction technology, which can ensure the manufacturability of thin-walled units and has better controllability of the forming quality.

[0005] Document 1 "Zhang, et al, Efficient design-optimization of variable-density hexagonal cellular structure by additive manufacturing: theory and validation. Journal of Manufacturing Science and Engineering, 2015.137(2):021004." discloses a homogenization method that can optimize material distribution according to the stress conditions of the model and generate a heterogeneous gradient porous structure. However, there are problems such as the inability to manufacture units with small wall thickness and the difficulty in controlling the forming quality of the model.

[0006] The inventor's pending patent 2023116433045 of document 2 discloses a 3D printing gradient porous structure path planning method based on a Voronoi polygon skeleton, which has relatively low calculation accuracy and relatively poor mechanical properties of the optimized structure. Summary of the invention

[0007] In view of the problem that the existing gradient porous structure has poor forming quality or low mechanical properties, the technical problem to be solved by the present invention is to propose a topology optimization method for a high-performance Voronoi homogeneous gradient porous structure.

[0008] The present invention solves the technical problem by adopting the following technical solution:

[0009] A topology optimization method for a high-performance Voronoi homogeneous gradient porous structure comprises the following steps:

[0010] Step 1: Given the initial design domain, specify the construction parameters of the Voronoi gradient porous structure: including the porous unit wall thickness t, the preset filling rate w, and the minimum value r of the porous unit sampling point spacing r min ;

[0011] Step 2: Establish the relationship between the porous unit wall thickness, sampling point spacing and Young's modulus:

[0012] E=(ρ w / ρ v ) n =(2t / r) n (1)

[0013] Where n is a constant, ranging from 1 to 3; ρ w is the packing density of the porous unit; ρ v is the density when the porous unit is fully filled; E is the Young's modulus of the porous unit;

[0014] When r takes the minimum value r min When E reaches its maximum value E max , E max =(2t / r min ) n ;

[0015] Step 3: With the goal of minimizing the structural flexibility, the initial design domain is discretized into a number of quadrilateral mesh units. The row and column numbers of the quadrilateral mesh units are represented by i and j respectively. The optimal Young's modulus E of each quadrilateral mesh unit in the design domain is obtained according to the stress conditions of the structure. ij ;

[0016] Step 4: Generate gradient sampling points:

[0017] Step 4-1 Set the Young's modulus threshold δ, 0<δ <E max , delete the quadrilateral mesh elements whose optimal Young's modulus value is less than δ, generate a new design domain according to the boundaries of the remaining quadrilateral mesh elements, and smooth the boundaries of the new design domain;

[0018] Step 4-2 uses the gradient Poisson disk sampling method to arrange the gradient sampling points. Specifically, uniform sampling points with a spacing less than the side length of the quadrilateral grid unit are arranged in the new design domain; according to the E of the quadrilateral grid unit where the uniform sampling points are located, ij The value is calculated using formula (1) and r is used as the radius of the Poisson disk;

[0019] Each uniform sampling point corresponds to a Poisson disk radius. Find the sampling point corresponding to the minimum Poisson disk radius among the current uniform sampling points. Draw a circle with the radius of the minimum Poisson disk radius among the current uniform sampling points with the sampling point as the center point. Store the center point and clear other uniform sampling points in the circle.

[0020] Repeat the process of removing uniform sampling points until all uniform sampling points in the new design domain are deleted, at which point the stored center point is the gradient sampling point;

[0021] Step 5: Construct a gradient porous structure construction area and adjust the filling rate:

[0022] Step 5-1 Generate a Voronoi gradient porous unit skeleton in the new design domain according to the gradient sampling points; offset the skeleton to both sides by t / 2 to generate the porous unit construction area R C , offset the new design domain inward by t to generate the shell structure region R S , optimized porous structure construction area R P For R C and R S The union of;

[0023] The current fill rate w (k) For R P The ratio of the initial design domain area;

[0024] Set the filling rate error to ε and determine the current filling rate w (k) Is the absolute value of the difference from the preset filling rate w less than ε? If not, execute step 5-2; if |w (k) -w|<ε, the iterative calculation is stopped to obtain the optimized high-performance Voronoi homogeneous gradient porous structure;

[0025] Step 5-2 Set an interval [vol1,vol2], and the initial values ​​of vol1 and vol2 are 0 and E respectively. max; By narrowing the range of the interval [vol1,vol2] and continuously changing f, the filling rate w at the kth iteration is (k) Gradually approach the preset filling rate w; the calculation formula of f at each update is formula (2):

[0026]

[0027] In the formula, the superscript (k) indicates the kth iteration calculation, and f is the sum of all E ij The sum of the values ​​and E ij Ratio of quantity;

[0028] When f changes, the optimal Young's modulus E of each quadrilateral mesh element is recalculated. ij , repeat steps 4 and 5-1 to obtain the current filling rate for the next iteration.

[0029] Furthermore, in step 3, the mathematical model for obtaining the optimal Young's modulus is expressed as:

[0030] Solve: E=(E 11 ,E 12 ,E 13 ,……,E ij )

[0031] Minimize: c(E) = F T U=U T KU

[0032] Constraint: KU = F

[0033] S=f·S0=∑ i ∑ j E ij ·s ij

[0034] 0≤E ij ≤E max

[0035] Where E is the Young's modulus vector of the quadrilateral mesh element, and all E ij The initial value of is 0.5; c(E) is the structural flexibility; F and U are the load vector and displacement vector of all quadrilateral mesh elements respectively; K is the overall stiffness matrix of the structure; S and S0 are the area of ​​the optimized structure and the area of ​​the initial design domain respectively, s ij is the area of ​​the quadrilateral grid cell.

[0036] Compared with the prior art, the present invention has the following beneficial effects:

[0037] (1) The present invention proposes a gradient Poisson disk sampling method to generate Voronoi gradient sampling points, and uses the spacing between the porous unit sampling points of Voronoi as the radius of the Poisson disk, which is used as a design variable. The porous structure is constructed according to the optimal Young's modulus obtained by minimizing the structural flexibility. This method can not only reflect the gradient change of material distribution in general, but also make the local areas with similar Young's modulus evenly distributed, thereby improving the calculation accuracy and ensuring the rationality of material distribution.

[0038] (2) In combination with the construction technology characteristics of existing new materials and structures such as 3D printing, the wall thickness of the porous unit is set as a constant, and a mechanical model of the porous unit filling density and Young's modulus is established. Then, the relationship between the wall thickness of the porous unit, the sampling point spacing and the Young's modulus is established. A gradient porous structure is constructed according to the optimal Young's modulus. The theoretical derivation is rigorous, which can not only improve the mechanical properties of the structure but also ensure the construction quality.

[0039] (3) The method of the present invention can be used by users to adjust the minimum distance between porous units, thereby improving manufacturability and mechanical properties while taking into account functions such as structural thermal insulation, sound absorption and noise reduction. The method can be used in the design optimization of functional porous structures such as house structure design, automobile manufacturing, and aerospace. BRIEF DESCRIPTION OF THE DRAWINGS

[0040] Figure 1 It is a schematic diagram of the overall process of the topology optimization method of the high-performance Voronoi homogeneous gradient porous structure of the present invention;

[0041] Figure 2 Schematic diagram of the construction parameters of the porous structure;

[0042] Figure 3 is the distribution diagram of the load-bearing components and Young's modulus;

[0043] Figure 4 is the new design domain boundary map;

[0044] Figure 5 This is a schematic diagram of uniform sampling points in Example 1;

[0045] Figure 6 This is a schematic diagram of the gradient Poisson disk sampling method of Example 1;

[0046] Figure 7 This is the gradient sampling point distribution diagram of Example 1;

[0047] Figure 8 Schematic diagram of Voronoi gradient porous unit skeleton and gradient porous structure;

[0048] Fig. 9 It is a schematic diagram of the load and constraint conditions of the three-point bending simply supported beam model;

[0049] Fig.10Comparison chart of optimization results and 3D printing results of different optimization methods;

[0050] Fig.11 This is a physical picture of the mechanical test experiment;

[0051] Fig.12 Comparison of load-displacement curves of mechanical tests of printed components obtained using different optimization methods. DETAILED DESCRIPTION

[0052] The following is a combination of the accompanying drawings and specific embodiments. The specific embodiments are only used to introduce the technical solutions of the present invention in detail, and are not intended to limit the protection scope of the present application.

[0053] The topology optimization method of the high performance Voronoi homogeneous gradient porous structure of the present invention (see Figure 1 ), including the following steps:

[0054] Step 1: Given the initial design domain (i.e., the design space occupied by the structure), the user specifies the construction parameters of the Voronoi gradient porous structure:

[0055] like Figure 2 As shown in the figure, the construction parameters of the porous structure include the porous unit wall thickness t, the preset filling rate w, and the porous unit sampling point spacing r. The minimum value of the porous unit sampling point spacing r is represented by r min Indicates. The user can specify r min The value is used to adjust the minimum spacing of the porous elements, r min The value setting can take into account functional requirements such as sound absorption and noise reduction, thermal insulation, etc. The larger the value, the larger the spacing between porous units.

[0056] Step 2: Establish the relationship between the porous unit wall thickness, sampling point spacing and Young's modulus:

[0057] Step 2-1 Voronoi porous unit has no fixed geometric shape. Its shape is closest to a regular hexagon, so regular hexagon approximation is used for calculation. Each porous unit contains a sampling point. The relative density of the porous unit corresponding to the sampling point is calculated as: ρ w / ρ v =s w / s v =2t / r,

[0058] Among them, ρ w and w are the packing density and area of ​​the porous unit respectively; ρ v and v are the density and area of ​​the porous unit when it is fully filled;

[0059] Step 2-2 The relationship between Young's modulus and packing density is E = (ρ w / ρv ) n =(2t / r) n , n ranges from 1 to 3 and can also be specified by the user to adjust the density distribution of porous units. The larger n is, the more porous units are distributed in the area with higher Young's modulus. From the above relationship, we can conclude that When r takes the minimum value r min When E reaches its maximum value E max , E max =(2t / r min ) n .

[0060] Step 3: Calculate the optimal Young's modulus based on the stress conditions of the structure:

[0061] Step 3-1 takes the minimum structural flexibility as the goal, and discretizes the initial design domain into a number of small quadrilateral grid units. In this embodiment, the side length of the quadrilateral grid unit is 1 and has no unit. The row and column numbers of the quadrilateral grid unit are represented by i and j respectively. The mathematical model is expressed as:

[0062] Solve: E=(E 11 ,E 12 ,E 13 ,……,E ij )

[0063] Minimize: c(E) = F T U=U T KU

[0064] Constraint: KU = F

[0065] S=f·S0=∑ i ∑ j E ij ·s ij

[0066] 0≤E ij ≤E max

[0067] Where E is the Young's modulus vector of the quadrilateral mesh element, and all E ij The initial value of is 0.5; c(E) is the structural flexibility; F and U are the load vector and displacement vector of all quadrilateral mesh elements respectively; K is the overall stiffness matrix of the structure, F is given by the user, and K and U are obtained through existing calculations; S and S0 are the area of ​​the optimized structure and the area of ​​the initial design domain respectively, and S changes with the change of f; f is the area of ​​all E ij The sum of the values ​​and E ij Ratio of quantity; s ij is the area of ​​the quadrilateral mesh unit, which remains unchanged during the optimization process;

[0068] Step 3-2: Based on the initial value of E, change its value in different directions, and use finite element analysis to calculate the structural flexibility c(E) under different E. When c(E) is reduced to the point where it no longer changes, the optimal Young's modulus E of each quadrilateral mesh unit is obtained. ij ;

[0069] Figure 3 This is a cloud diagram of the Young's modulus of a cantilever beam structure. The darker the color, the larger the Young's modulus value. The left boundary of the structure is completely fixed, and the middle of the right boundary is subject to a vertical downward force.

[0070] Step 4: Generate gradient sampling points:

[0071] Step 4-1 Set the Young's modulus threshold δ (0 < δ <E max ), delete the quadrilateral mesh elements with the optimal Young's modulus value less than δ in the initial design domain, generate a new design domain according to the boundaries of the remaining quadrilateral mesh elements, and smooth the boundaries of the new design domain, as shown in Figure 4 Indicated by the dashed line;

[0072] Step 4-2 uses the gradient Poisson disk sampling method to arrange the gradient sampling points. Specifically, uniform sampling points with a spacing less than the side length of the quadrilateral grid unit are arranged in the new design domain, such as Figure 5 As shown; according to the E of the quadrilateral grid unit where the uniform sampling point is located ij The value is calculated using the formula E = (ρ w / ρ v ) n =(2t / r) n Calculate r, which is the radius of the corresponding Poisson disk;

[0073] The process of removing uniform sampling points: Each uniform sampling point corresponds to a Poisson disk radius. Find the sampling point corresponding to the minimum Poisson disk radius among the current uniform sampling points, draw a circle with the radius of the minimum Poisson disk radius among the current uniform sampling points with the sampling point as the center point, store the center point and remove other uniform sampling points in the circle, such as Figure 6 As shown;

[0074] Repeat this process of removing uniform sampling points until all uniform sampling points in the new design domain are deleted. At this time, the stored center point is the gradient sampling point, such as Figure 7 As shown, at this time, relatively uniform and locally varying gradient sampling points are generated in the new design domain according to the optimal Young's modulus.

[0075] Step 5: Construct a gradient porous structure construction area and adjust the filling rate:

[0076] Step 5-1 generates a gradient Voronoi porous unit skeleton in the new design domain according to the stored gradient sampling points, such as Figure 8 As shown by the black line in the middle. The skeleton is offset to both sides by t / 2 to generate the porous unit construction area R C , offset the new design domain inward by t to generate the shell structure region R S , optimized porous structure construction area R P For R C and R S The union of;

[0077] The current fill rate w (k) For R P The ratio of the initial design domain area;

[0078] Set the filling rate error to ε and determine the current filling rate w (k) Is the absolute value of the difference from the preset filling rate w less than ε? If not, execute step 5-2; if |w (k) -w|<ε, the iterative calculation is stopped to obtain the optimized high-performance Voronoi homogeneous gradient porous structure;

[0079] Step 5-2 Set an interval [vol1,vol2], and the initial values ​​of vol1 and vol2 are 0 and E respectively. max ; By narrowing the range of the interval [vol1,vol2] and continuously changing f, the filling rate w at the kth iteration is (k) Gradually approach the preset filling rate w. The calculation formula of f at each update is:

[0080]

[0081] In the formula, the superscript (k) indicates the kth iteration calculation, and f is the sum of all E ij The sum of the values ​​and E ij Ratio of quantity;

[0082] The filling rate is related to the sampling point density, which is determined by the deletion circle radius r ij Decision, r ij The size depends on E ij , by adjusting f to achieve the optimal Young's modulus value E ij The radius of the deletion circle is the minimum value of the Poisson disk radius in the current uniform sampling point. When f changes through formula (2), the optimal Young's modulus value E of each quadrilateral mesh unit is recalculated. ij , repeat steps 4 and 5-1 to obtain the filling rate for the next iteration.

[0083] Figure 8The calculation results of the gradient porous structure are given. As can be seen from the figure, the method of the present invention can generate a gradient porous structure that adapts to mechanical properties according to the stress conditions of the structure (calculate the Young's modulus according to the stress conditions, and then optimize the porous structure according to the Young's modulus), optimize the material distribution, and the wall thickness of the porous unit is homogeneous. During the optimization process, the porous unit wall thickness t does not change after being set.

[0084] Example 2

[0085] This embodiment uses the topology optimization method of Embodiment 1 to optimize the three-point bending simply supported beam structure.

[0086] Comparative Example 1 is the optimization method of Document 1 in the background technology, and Comparative Example 2 is the optimization method of the Voronoi polygon skeleton in Patent 202311643304.5.

[0087] use Fig. 9 The three-point bending simply supported beam structure shown compares the mechanical properties of components optimized by different methods. The component size is 160×40mm. Fig.10 The optimization results and 3D printing results of different methods under the same filling rate conditions are given. The thickness of the 3D printed component is 10 mm. A universal testing machine (with a range of 10 kN, Shenzhen Sansi Zongheng Technology Co., Ltd.) was used to conduct mechanical tests on the printed components. Fig.11 . Fig.12 The load displacement curve within the linear elastic range is given. The average value of the slope of the curve at three points with loading displacements of 0.5mm, 1.0mm and 1.5mm is taken as the stiffness comparison of the component, and the calculation results are as follows: Comparative Example 1 is 225N / mm, Comparative Example 2 is 251N / mm, and Example 2 is 327N / mm. The optimization results of Comparative Example 1 show that some thin-walled units cannot be manufactured and the forming quality is poor, so the stiffness is less than that of Example 2; although Comparative Example 2 can guarantee the printing quality, the porous units have stronger randomness, and the stiffness of the component is still less than that of Example 2.

[0088] Any matters not described in the present invention are applicable to the prior art.

Claims

1. A topology optimization method for high-performance Voronoi homogeneous gradient porous structure, characterized in that: The topology optimization method comprises the following steps: Step 1: Given the initial design domain, specify the construction parameters of the Voronoi gradient porous structure: including the porous unit wall thickness t, the preset filling rate w, and the minimum value r of the porous unit sampling point spacing r min ; Step 2: Establish the relationship between the porous unit wall thickness, sampling point spacing and Young's modulus: （1）， Where n is a constant; is the packing density of the porous unit; is the density when the porous unit is fully filled; E is the Young's modulus of the porous unit; When r takes the minimum value r min When E reaches its maximum value E max , E max =(2t / r min ) n ; Step 3: With the goal of minimizing the structural flexibility, the initial design domain is discretized into a number of quadrilateral mesh units. The row and column numbers of the quadrilateral mesh units are represented by i and j respectively. The optimal Young's modulus E of each quadrilateral mesh unit in the design domain is obtained according to the stress conditions of the structure. ij ; Step 4: Generate gradient sampling points: Step 4-1 Set the Young's modulus threshold δ, 0<δ <E max , delete the quadrilateral mesh elements whose optimal Young's modulus value is less than δ, generate a new design domain according to the boundaries of the remaining quadrilateral mesh elements, and smooth the boundaries of the new design domain; Step 4-2 uses the gradient Poisson disk sampling method to arrange the gradient sampling points. Specifically, uniform sampling points with a spacing less than the side length of the quadrilateral grid unit are arranged in the new design domain; according to the E of the quadrilateral grid unit where the uniform sampling points are located, ij The value of r is calculated using formula (1), and r is used as the radius of the Poisson disk; Each uniform sampling point corresponds to a Poisson disk radius. Find the sampling point corresponding to the minimum Poisson disk radius among the current uniform sampling points. Draw a circle with the radius of the minimum Poisson disk radius among the current uniform sampling points with the sampling point as the center point. Store the center point and remove other uniform sampling points in the circle. Repeat the process of removing uniform sampling points until all uniform sampling points in the new design domain are deleted. At this time, the stored center point is the gradient sampling point; Step 5: Construct a gradient porous structure construction area and adjust the filling rate: Step 5-1 Generate a Voronoi gradient porous unit skeleton in the new design domain according to the gradient sampling points; offset the skeleton to both sides by t / 2 to generate the porous unit construction area R C , offset the new design domain inward by t to generate the shell structure region R S , optimized porous structure construction area R P For R C and R S The union of; The current fill rate w (k)为RP The ratio of the initial design domain area; Set the fill rate error to , determine the current filling rate w (k)与预设填充率w的差的绝对值是否小于 , if it is not less than, then execute step 5-2; if , then the iterative calculation is stopped, and the optimized high-performance Voronoi homogeneous gradient porous structure is obtained; Step 5-2 Set an interval [vol1,vol2], and the initial values ​​of vol1 and vol2 are 0 and E respectively. max ; By narrowing the range of the interval [vol1,vol2] and continuously changing f, the filling rate w at the kth iteration is (k)逐步逼近预设填充率w；每次更新时f的计算公式为式（2）： （2） In the formula, the superscript (k) indicates the kth iteration calculation, and f is the sum of all E ij The sum of the values ​​and E ij Ratio of quantity; When f changes, the optimal Young's modulus E of each quadrilateral mesh element is recalculated. ij , repeat steps 4 and 5-1 to obtain the current filling rate for the next iteration.

2. The method according to claim 1, characterized in that In step 3, the mathematical model for obtaining the optimal Young's modulus is expressed as: ， Where E is the Young's modulus vector of the quadrilateral mesh element, and all E ij The initial value of is 0.5; c(E) is the structural flexibility; F and U are the load vector and displacement vector of all quadrilateral mesh elements respectively; K is the overall stiffness matrix of the structure; S and S0 are the area of ​​the optimized structure and the area of ​​the initial design domain respectively, s ij is the area of ​​the quadrilateral grid cell.

3. The method according to claim 1, characterized in that: The value range of n is 1-3.

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