A compact topology optimization method for porous models based on Voronoi partitioning

Abstract

The present invention discloses a compact topology optimization method for porous models based on Voronoi partitioning, which directly uses explicit topology and geometric parameters as design degrees of freedom to construct a compact design space for porous model design. Different from the traditional voxel-based model representation, the porous model is represented by a continuous distance field function, which can accurately extract the model features; and this representation method remains unified during the optimization process, avoiding unstable model conversion. The proposed compact topology optimization method can optimize the global topological structure and local size distribution of the porous model based on effective gradient calculation, and can generate high-quality porous models for various complex free surface models. In addition, the present invention proposes a numerical coarsening method to construct a general polygon / polyhedron shape function. Compared with the classical numerical homogenization method, this method improves the simulation accuracy by an order of magnitude at a similar computational cost.

Description

Technical Field

[0001] The invention relates to the technical field of structural optimization, and in particular to a compact topology optimization method for a porous model based on Voronoi partitioning. Background Art

[0002] Porous models are widely used in industry due to their relatively low density and multifunctional properties. They have played a unique and outstanding industrial value in important equipment or instruments such as aerospace components, medical implants, and energy absorption protection devices. However, due to their extremely complex structures, it is very challenging to automatically design porous models with specified target performance. The dual-scale topology optimization method is a porous model design method that is currently widely studied. Under this framework, the design domain is usually discretized into a set of very fine voxel units to capture the complex structural distribution of the porous model. These discrete units are used to represent the basic geometry of the model, as well as the design parameters and finite element meshes for performance optimization. Under this representation, the porous model design problem is expressed as the optimal voxel unit distribution under certain physical or geometric constraints to meet specific design goals. This method involves repeated structural simulation and re-modification until convergence.

[0003] For complex porous models, the voxel-based discrete representation brings a large number of design degrees of freedom, corresponding to a dense design space. The inherent challenge is that the explicit geometric feature information of the model will be lost during the optimization process, often resulting in invalid porous models that cannot be actually manufactured, which may contain broken, narrow or extremely small pore distributions. Therefore, this method requires the imposition of cumbersome geometric constraints to generate a valid model or meet specific shape control objectives. Moreover, under this discrete representation, the shape of the structure also depends on the resolution of the voxel grid, and it is very challenging to extract fine features during the optimization process. In addition, a large number of voxel units as the underlying finite element grid poses a severe challenge to the computational efficiency of the simulation. Using the classic numerical homogenization method can improve efficiency, but it will result in up to 20 times the loss of accuracy.

[0004] Another commonly used method is to embed some type of porous elements according to the specified material or physical field. This strategy can easily control the shape of the model, but it greatly limits the type of elements used, which in turn limits the physical properties that can be achieved by the porous model. The embedded elements can be expanded through the inverse homogenization method or the parameter reconstruction process. Although these methods can generate a large number of porous elements, when the adjacent elements are required to maintain geometric connection, the candidate elements become very limited. More importantly, such methods are usually only applicable to conventional design domains, such as cuboids. Summary of the invention

[0005] In view of the problems existing in the design of the above-mentioned porous structure, the present invention provides a compact topology optimization method for porous models based on Voronoi partitioning. The effect of porous model design depends largely on the design space that describes the porous model. An ideal design space needs to meet three properties: explicit geometric control, a wide spatial range, and efficient performance simulation. The topological structure and geometric shape of the porous model can be adjusted by a set of clear geometric parameters and do not depend on a specific grid resolution. The parameterized space can cover a wide range of design candidates to meet specific property / functional requirements. Therefore, based on Voronoi partitioning, the present invention defines explicit topological and geometric control parameters, establishes a compact design space to describe the porous model, and proposes a compact topology optimization method for designing porous models.

[0006] The technical solution adopted by the present invention is as follows:

[0007] A compact topology optimization method for a porous model based on Voronoi partitioning includes the following steps:

[0008] 1) In a given design domain, Voronoi division is performed according to the input seed points to construct a porous model based on Voronoi division. The seed points of Voronoi division are used as topological control parameters, each of its edges is regarded as a rod, and the rod diameter is used as a geometric control parameter, namely the Voronoi porous model. Further, an implicit expression is established for each rod, and implicit fitting is performed for all connected rods to obtain an overall smooth, continuous, and naturally self-connected porous model;

[0009] 2) Construct general polygonal / polyhedral material-aware shape functions, establish a mapping relationship between generalized coarse grid unit nodes, i.e., curve bridge nodes, and model internal nodes, and achieve high-precision simulation of porous models on coarse grids;

[0010] 3) Based on the gradient optimization of the Voronoi porous model, the seed points and rod diameters of the Voronoi partition are used as design variables. Guided by physical properties (minimizing structural flexibility), the geometric shape of the porous model is constrained by the classic centroid Voronoi partitioning method to achieve topological optimization of the porous model in a compact design space.

[0011] In the above technical solution, the step 1) is specifically as follows:

[0012] In a given design domain, the initial seed point distribution is obtained by random sampling. After Voronoi division based on the initial seed point, each edge corresponds to each rod in the Voronoi porous model. Each rod is uniquely determined by a vector. In the two-dimensional case Where (x j ,y j ) is the center point of the rod, Lj is the length of the rod, α j is the inclination angle of the rod (counterclockwise angle about the horizontal x-axis), is the radius of the rod; in three dimensions Where (x j ,y j , z j ) is the center point of the rod, α j ,θ j , γ j is the angle between the local coordinate system of the rod and the global coordinate system. Here, according to the right-hand rule, the long side of the rod is taken as the x-axis to establish the local coordinate system, and the global coordinate system is the real coordinate system;

[0013] Each seed point corresponds to a rod diameter variable, that is, r = {r i , i = 1...N}, N is the number of seed points, for the jth edge in the Voronoi partition, the width of the edge Determined by the average value of the rod diameter variables of its adjacent seed points,

[0014]

[0015] in, is the set of adjacent seed points, is the number of adjacent seed points, and k represents the kth adjacent seed point of the jth edge.

[0016] For any point x in the design domain, the topological description function φ j (x) describes each rod, and an implicit representation of each rod is obtained. The topological description of the overall structure is the union of the topological description functions of all rods, that is, Φ(x) = max(φ1, ..., φ n ), n is the number of rods. In actual calculations, Φ(x) is approximated by a continuous and smooth function,

[0017]

[0018] Where p=2.

[0019] From the perspective of numerical implementation, a step function H(Φ(x)) is introduced for regularization.

[0020]

[0021] Where ∈ is a parameter that controls the size of the regularization, α = 1e -3 Take a very small positive value to avoid singularity of the global stiffness matrix.

[0022] Furthermore, the step 2) constructs a general polygonal / polyhedral material perception shape function, which is specifically as follows:

[0023] For any finite element mesh D divided into polygons or polyhedrons, it contains M coarse mesh finite elements D α , for each coarse grid cell D α , and the same finite element division is performed to obtain m α Fine mesh finite element D e Some of the upper boundary nodes of the fine grid are selected as bridge nodes. Two Bezier control points are inserted into every two bridge nodes in the polygonal unit. Multi-sided generalized Bezier control points are inserted into the polyhedral unit according to the generalized centroid coordinates. The bridge nodes and Bezier control points are collectively called curve bridge nodes. The displacement of the curve bridge nodes is used as the degree of freedom of the coarse grid simulation calculation.

[0024] Definition N α (x) is the shape function required, which is obtained by the Bessel interpolation matrix Ψ and the boundary-internal transformation matrix composition,

[0025]

[0026] Where N h (x) is the classic linear interpolation function. Boundary-internal transformation matrix The mapping relationship between boundary nodes and internal nodes can be established by solving the coarse grid unit D α The local finite element problem is obtained.

[0027] The Bezier interpolation matrix Ψ establishes the mapping relationship between the curve bridge nodes and the boundary nodes. For polygonal cells, Ψ is based on the Bezier basis corresponding to all boundary nodes on each edge of the polygon. get,

[0028]

[0029] where χ b is the set of all Bezier control points on edge b, the Bezier basis of all boundary nodes Arrange in rows.

[0030] For polyhedral units, each face is a polygon with an uncertain number of edges, which is represented by a polygonal generalized Bezier surface. Define ω t (x) is the generalized centroid basis function of the irregular polygon P, n is the number of all generalized Bezier control points in the polygon P, and for the tth control point, the basis function yes The polynomial expansion of

[0031]

[0032] in For multiple indexes, are polynomial coefficients, and in this method the degree is d=2.

[0033] Similarly, the Bezier interpolation matrix Ψ of a polyhedral cell is composed of the Bezier basis corresponding to all boundary nodes on each face of the polyhedron get

[0034]

[0035] where χ P is the set of all Bezier control points on the surface P.

[0036] According to the shape function N α (x), high-precision simulation on a coarse grid;

[0037] The global stiffness matrix K is

[0038]

[0039] Element stiffness matrix K α for,

[0040]

[0041] The penalty factor q = 2, H q (Φ(x)) is used to represent the material density at point x, [LN h ] is N h (x) is the partial derivative with respect to x, where D0 is the elasticity tensor of the default material of the porous model.

[0042] Furthermore, for a more complex design domain Ω, such as a curved surface, an embedded simulation framework is introduced based on the shape function constructed in step 2), and simulation is performed on a coarsened background grid. The Dirichlet boundary conditions of the background grid are applied in a weak form based on the Nitsche method, which can extend the simulation method to a design domain with complex free-form surfaces.

[0043] Further, the step 3) is as follows:

[0044] This method innovatively uses the seed point X of the Voronoi partition and the rod diameter r defined at the point as design variables. In the optimization process, in addition to minimizing the target structural compliance C(X, r), from the perspective of practical manufacturability, the classic centroid Voronoi partition energy function S(X) is introduced to control the geometry of the Voronoi partition so that the inner hole is as close to a regular polygon as possible. Note that the energy function S(X) is only related to the seed point X.

[0045]

[0046] Represents each seed point X i To the centroid of the current Voronoi cell The Euclidean distance is , and N is the number of seed points.

[0047] The final optimization problem is expressed as,

[0048]

[0049]

[0050] Where Q is the background grid node displacement vector to be calculated, K(X, r) is the overall stiffness matrix of the porous model, f represents the node external force load vector, w is the weight that controls the proportion of the two objective functions, v is the target volume fraction, V0 is the total volume of the design domain Ω, V(X, r) is the volume of the Voronoi porous model, C(X, r) is the compliance of the structure,

[0051]

[0052] C(X, r) = f T Q

[0053] where y α (X, r) is the coarse grid unit D α volume of material.

[0054] The beneficial effects of the present invention are:

[0055] 1) A compact design space is constructed based on Voronoi partitioning, which covers a wide range of design spaces with a small number of parameters and can explicitly control the topology and geometry of the porous model.

[0056] 2) A continuous implicit representation of the porous model is established, which can extract extremely fine features, provide a unified representation for downstream simulation and optimization, and avoid unstable complex model conversion.

[0057] 3) A numerical coarsening method based on polygon / polyhedron shape functions in matrix form is proposed. Compared with classical numerical homogenization, the simulation accuracy is improved by an order of magnitude at a similar computational cost.

[0058] 4) Through effective gradient calculation, the global topological structure and local size distribution of the porous model can be optimized simultaneously. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] Figure 11 is a flow chart of the method of the present invention; (a) problem definition and background grid → (b) initial seed point distribution → (c) initial Voronoi porous model → (d) simulation of the Voronoi porous model → (e) optimization of the Voronoi porous model, return to (b), until convergence.

[0060] Figure 2 This is a simulation example of a cantilever beam porous model: a. The simulation result of this method; b. The simulation result of the classical numerical homogenization method (the gray structure is the benchmark result). DETAILED DESCRIPTION

[0061] The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments.

[0062] The present invention provides a method for compact topology optimization of porous models based on Voronoi partitioning. The effect of porous model design depends largely on the design space that describes the porous model. An ideal design space needs to meet three properties: explicit geometric control, a wide design space, and efficient performance simulation. The topological structure and geometric shape of the porous model can be adjusted by a set of clear geometric parameters and do not depend on a specific grid resolution. The parameterized space can cover a wide range of design candidates to meet specific property / functional requirements. Therefore, based on Voronoi partitioning, the present invention defines explicit topological and geometric control parameters, establishes a compact design space to describe the porous model, and proposes a compact topology optimization method for designing a porous model. Specifically, it may include the following steps:

[0063] 1) Construct a porous model based on Voronoi partitioning, using the seed point of Voronoi partitioning as the topological control parameter, each of its edges as a rod, and the rod diameter as the geometric control parameter

[0064] In a given design domain, the initial seed point distribution is obtained by random sampling. After Voronoi division according to the initial seed point, each edge corresponds to each rod in the Voronoi porous model. The jth rod is represented by the vector m j The only certainty is that in two dimensions Where (x j ,y j ) is the center point of the rod, L j is the length of the rod, α j is the inclination angle of the rod (counterclockwise angle about the horizontal x-axis), is the radius of the rod; in three dimensions Where (x j ,y j , z j ) is the center point of the rod, α j ,θ j, γ j is the angle between the local coordinate system of the rod and the global coordinate system. Here, according to the right-hand rule, the long side of the rod is taken as the x-axis to establish the local coordinate system, and the global coordinate system is the real coordinate system;

[0065] Each seed point corresponds to a rod diameter variable, that is, r = {r i , i = 1...N}, N is the number of seed points, for the jth edge in the Voronoi partition, the width of the edge Determined by the average value of the rod diameter variables of its adjacent seed points,

[0066]

[0067] in, is the set of adjacent seed points, is the number of adjacent seed points, k represents the kth adjacent seed point of the jth edge;

[0068] For any point x, the topological description function φ j (x) describes each rod, and obtains an implicit representation of each rod. In the two-dimensional case, φ j (x) is defined as,

[0069]

[0070] where q = 6, and

[0071]

[0072] In three dimensions, φ j (x) is defined as,

[0073]

[0074] where q = 6, and

[0075]

[0076]

[0077] The topological description of the overall structure is the union of the topological description functions of all the rods, that is, Φ(x) = max(φ1, ..., φ n ), n is the number of rods. In actual calculations, Φ(x) is approximated by a continuous and smooth function,

[0078]

[0079] Where p=2.

[0080] From the perspective of numerical implementation, a step function H(Φ(x)) is introduced for regularization.

[0081]

[0082] Where ∈ is a parameter that controls the size of the regularization, α = 1e -3 Take a very small positive value to avoid singularity of the global stiffness matrix.

[0083] 2) Construct general polygonal / polyhedral material perception shape functions and establish a mapping relationship between generalized coarse grid unit nodes, i.e., curve bridge nodes, and model internal nodes

[0084] For any finite element mesh D divided into polygons or polyhedrons, it contains M coarse mesh finite elements D α , for each coarse grid cell D α , and the same finite element division is performed to obtain m α Fine mesh finite element D e Some of the upper boundary nodes of the fine grid are selected as bridge nodes. Two Bezier control points are inserted into every two bridge nodes in the polygonal unit. Multi-sided generalized Bezier control points are inserted into the polyhedral unit according to the generalized centroid coordinates. The bridge nodes and Bezier control points are collectively called curve bridge nodes. The displacement of the curve bridge nodes is used as the degree of freedom of the coarse grid simulation calculation.

[0085] Definition N α (x) is the shape function required, which is obtained by the Bessel interpolation matrix Ψ and the boundary-internal transformation matrix composition,

[0086]

[0087] Where N h (x) is the classic linear interpolation function. Boundary-internal transformation matrix The mapping relationship between boundary nodes and internal nodes can be established by solving the coarse grid unit D α The local finite element problem is obtained.

[0088] The Bezier interpolation matrix Ψ establishes the mapping relationship between the curve bridge nodes and the boundary nodes. For polygonal cells, Ψ is based on the Bezier basis corresponding to all boundary nodes on each edge of the polygon. get,

[0089]

[0090] where χ b is the set of all Bezier control points on edge b, the Bezier basis of all boundary nodes Arrange in rows.

[0091] For polyhedral units, each face is a polygon with an uncertain number of edges, which is represented by a polygonal generalized Bezier surface. Define ω t (x) is the generalized centroid basis function of the irregular polygon P, n is the number of all generalized Bezier control points in the polygon P, and for the tth control point, the basis function yes The polynomial expansion of

[0092]

[0093] in For multiple indexes, are polynomial coefficients, and in this method the degree is d=2.

[0094] Similarly, the Bezier interpolation matrix Ψ of a polyhedral cell is composed of the Bezier basis corresponding to all boundary nodes on each face of the polyhedron get

[0095]

[0096] where χ P is the set of all Bezier control points on the surface P.

[0097] According to the shape function N α (x), high-precision simulation on a coarse grid;

[0098] The global stiffness matrix K is

[0099]

[0100] Element stiffness matrix K α for,

[0101]

[0102] The penalty factor q = 2, H q (Φ(x)) is used to represent the material density at point x, [LN h ] is N h (x) is the partial derivative with respect to x, where D0 is the elasticity tensor of the default material of the porous model.

[0103] Furthermore, for a more complex design domain Ω, such as a curved surface, an embedded simulation framework is introduced based on the shape function constructed in step 2), and simulation is performed on a coarsened background grid. The Dirichlet boundary conditions of the background grid are applied in a weak form based on the Nitsche method, which can extend the simulation method to a design domain with complex free-form surfaces.

[0104] 3) Based on the gradient optimization of the Voronoi porous model, the seed points and rod diameters of the Voronoi division are used as design variables to minimize the structural compliance while constraining the geometric shape of the porous model

[0105] The seed point X of the Voronoi partition and the rod diameter r defined at the point are used as design variables. In the optimization process, in addition to minimizing the target structural compliance C(X, r), from the perspective of actual manufacturability, the classic centroid Voronoi partition energy function S(X) is introduced to control the geometric shape of the Voronoi partition so that the inner hole is as close to a regular polygon as possible. Note that the energy function S(X) is only related to the seed point x.

[0106]

[0107] Represents each seed point X i To the centroid of the current Voronoi cell The Euclidean distance is , and N is the number of seed points.

[0108] The final optimization problem is expressed as,

[0109]

[0110]

[0111] Where Q is the background grid node displacement vector to be calculated, K(X, r) is the overall stiffness matrix of the porous model, f represents the node external force load vector, w is the weight that controls the proportion of the two objective functions, v is the target volume fraction, V0 is the total volume of the design domain Ω, V(X, r) is the volume of the Voronoi porous model, C(X, r) is the compliance of the structure,

[0112]

[0113] C(X, r) = f T Q

[0114] Where V α (X, r) is the coarse grid unit D α volume of material.

[0115] According to the chain rule, the sensitivity of the objective function C(X, r) is derived as follows:

[0116]

[0117] where a can be X i or i ,

[0118]

[0119] as well as,

[0120]

[0121] Here is the guide The calculation is obtained by difference.

[0122] The sensitivity of other functions S(X), V(X, r) to the design variables is,

[0123]

[0124]

[0125] in is the coarse grid unit D α The unit volume of .

[0126] The following is a specific solution case using the method of the present invention:

[0127] Figure 1 (a) is a femur model with a complex shell obtained by real 3D scanning. The left end is subject to an upward force, the right end is subject to a downward force, and the lower end is fixed. Since the femur model has a free shape, this example constructs a tetrahedral background grid surrounding the entire model outside the femur. Each coarse grid unit is divided into a finer tetrahedral grid for the calculation of the polyhedral material-aware shape function. (b) is the distribution of the initial randomly sampled seed points, and (c) is the initial Voronoi porous model corresponding to the seed point. According to the constructed polyhedral material-aware shape function and the embedded simulation framework, the simulation deformation result is shown in (d). With the seed points and the rod diameters distributed on the seed points as design variables, the goal is to minimize the structural compliance (maximize the stiffness), and under certain volume constraints, the Voronoi porous model is optimized (e). After each iteration, a new seed point and rod diameter are obtained, and then return to (b) until convergence to obtain the final optimized porous model. The obtained results can meet the design requirements of the femur to effectively support the human body under a relatively light weight. The present invention describes the porous model based on Voronoi partitioning, optimizes physical properties and porous geometric shapes at the same time, can ensure natural self-connection of the structure, and finally obtains a porous model with optimized physical properties and controllable geometry.

[0128] Figure 2 This is a cantilever beam porous model, the left end is fixed on the wall, and the lower right end is subjected to a downward force. (a) is the simulation result of this method, with an error of 5e -3 , (b) is the simulation result of the classic numerical homogenization method, with an error of 0.6, and the gray structure is the benchmark result.

[0129] The present invention directly uses explicit topology and geometric parameters as design degrees of freedom to construct a compact design space for porous model design. Unlike traditional voxel-based model representation, the porous model is represented by a continuous distance field function, which can accurately extract model features; and this representation method remains unified during the optimization process, avoiding unstable model conversion. The proposed compact topology optimization method can optimize the global topological structure and local size distribution of the porous model based on effective gradient calculations, and can generate high-quality porous models for various complex free surface models. In addition, the present invention proposes a numerical coarsening method to construct a general polygon / polyhedron shape function. Compared with the classic numerical homogenization method, this method improves the simulation accuracy by an order of magnitude at a similar computational cost.

Claims

1. A compact topology optimization method for porous models based on Voronoi partitioning, characterized in that: The steps include: 1) In a given design domain, Voronoi division is performed according to the input seed points to construct a porous model based on Voronoi division. The seed points of Voronoi division are used as topological control parameters, each edge is regarded as a rod, and the rod diameter is used as a geometric control parameter. An implicit expression is established for each rod, and implicit fitting is performed on all connected rods to obtain an overall smooth, continuous, and naturally self-connected porous model; 2) Construct general polygonal / polyhedral material-aware shape functions, establish a mapping relationship between generalized coarse grid unit nodes, i.e., curve bridge nodes, and model internal nodes, and achieve high-precision simulation of porous models on coarse grids; 3) Based on the gradient optimization of the Voronoi porous model, the seed points and rod diameters of the Voronoi partition are used as design variables, the physical properties are used as a guide to minimize the structural compliance, and the geometric shape of the porous model is constrained by the classic centroid Voronoi partitioning method to achieve the topological optimization of the porous model in a compact design space; The step 1) is as follows: In a given design domain, the initial seed point distribution is obtained by random sampling. After Voronoi division according to the initial seed point, each edge corresponds to each rod in the Voronoi porous model. The jth rod is represented by the vector m j The only certainty is that in two dimensions Where (x j ,y j ) is the center point of the rod, L j is the length of the rod, α j is the inclination angle of the rod, that is, the counterclockwise angle about the horizontal x-axis, is the radius of the rod; in three dimensions Where (x j ,y j ,z j ) is the center point of the rod, α j ,θ j ,γ j is the angle between the local coordinate system of the rod and the global coordinate system. Here, according to the right-hand rule, the long side of the rod is taken as the x-axis to establish the local coordinate system, and the global coordinate system is the real coordinate system; Each seed point corresponds to a rod diameter variable, that is, r = {r i , i=1…N}, N is the number of seed points. For the jth edge in the Voronoi partition, the width of the edge is Determined by the average value of the rod diameter variables of its adjacent seed points, in, is the set of adjacent seed points, is the number of adjacent seed points, k represents the kth adjacent seed point of the jth edge; For any point x, the topological description function φ j (x) describes each rod, and obtains an implicit representation of each rod. The topological description of the overall structure is the union of the topological description functions of all rods, that is, Φ(x) = max(φ1,…,φ n ), n is the number of rods; in actual calculations, Φ(x) is approximated by a continuous and smooth function, Where p = 2; From the perspective of numerical implementation, a step function H(Φ(x)) is introduced for regularization Where ∈ is a parameter that controls the size of the regularization, α = 1e -3 Take a very small positive value to avoid singularity of the global stiffness matrix.

2. The method for compact topology optimization of porous models based on Voronoi partitioning according to claim 1 is characterized in that: In the step 2), a general polygonal / polyhedral material perception shape function is constructed as follows: For any finite element mesh D divided into polygons or polyhedrons, it contains M coarse mesh finite elements D α , for each coarse grid cell D α , and the same finite element division is performed to obtain m α Fine mesh finite element D e ; Select some upper boundary nodes of the fine grid as bridge nodes, insert two Bezier control points between every two bridge nodes in the polygonal unit, and insert polygonal generalized Bezier control points according to the generalized centroid coordinates in the polyhedral unit; the bridge nodes and Bezier control points are collectively called curve bridge nodes, and the displacement of the curve bridge nodes is used as the degree of freedom of the coarse grid simulation calculation; Definition N α (x) is the shape function to be sought, which is obtained by the Bessel interpolation matrix Ψ and the boundary-internal transformation matrix composition, Where N h (x) is the classic linear interpolation function, boundary-internal transformation matrix Establish the mapping relationship between boundary nodes and internal nodes, and solve the coarse grid unit D α The local finite element problem is obtained; The Bezier interpolation matrix Ψ establishes the mapping relationship between the curve bridge nodes and the boundary nodes. For polygonal units, Ψ is based on the Bezier basis corresponding to all boundary nodes on each edge of the polygon. get, where χ b is the set of all Bezier control points on edge b, the Bezier basis of all boundary nodes Arrange by row; For polyhedral units, each face is a polygon with an uncertain number of edges, which is represented by a polygonal generalized Bezier surface. Define ω t (x) is the generalized centroid basis function of the irregular polygon P, n is the number of all generalized Bezier control points in the polygon P, and for the tth control point, the basis function yes The polynomial expansion of in For multiple indexes, are polynomial coefficients, degree d = 2; Similarly, the Bezier interpolation matrix Ψ of a polyhedral cell is composed of the Bezier basis corresponding to all boundary nodes on each face of the polyhedron get where χ P is the set of all Bezier control points on the surface P; According to the shape function N α (x), high-precision simulation on a coarse grid; The global stiffness matrix K is Element stiffness matrix K α for, The penalty factor q = 2, H q (Φ(x)) is used to represent the material density at point x, [LN h ] is N h (x) partial derivative with respect to x, D0 is the elastic tensor of the default material of the porous model; For a more complex design domain Ω, i.e., a curved surface, an embedded simulation framework can be introduced for simulation.

3. The method for compact topology optimization of porous models based on Voronoi partitioning according to claim 1, characterized in that: The step 3) is as follows: The seed point X of the Voronoi partition and the rod diameter r defined at the point are used as design variables. In the optimization process, in addition to minimizing the target structural compliance C(X,r), from the perspective of actual manufacturability, the classic centroid Voronoi partition energy function S(X) is introduced to control the geometric shape of the Voronoi partition so that the inner hole is as close to a regular polygon as possible. The energy function S(X) is only related to the seed point X. represents the i-th seed point X i To the centroid of the current Voronoi cell The Euclidean distance, N is the number of seed points; The final optimization problem is expressed as, Where Q is the background grid node displacement vector to be calculated, K(X,r) is the overall stiffness matrix of the porous model, f represents the node external force load vector, W is the weight that controls the proportion of the two objective functions, v is the target volume fraction, V0 is the total volume of the design domain Ω, V(X,r) is the volume of the Voronoi porous model, and C(X,r) is the compliance of the structure. C(X,r)=f T Q Where V α (X,r) is the coarse grid element D α volume of material.

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