Enhanced structure design method and system for three-period minimal curved surface

Abstract

The invention belongs to the technical field of material structure optimization, and discloses an enhanced structure design method and system for a three-period minimal curved surface, and the method comprises the steps: modeling a geometric model of the three-period minimal curved surface, and exporting the model; expanding a design domain of the geometric model to obtain a corresponding expanded design space of the three-period minimal curved surface; a design space file is converted into an analysis data structure, a general expression mode is constructed for the three-period minimum curved surface, an equal geometric element stiffness matrix is solved by using a Gaussian integral method and the elastic performance of each point in a design domain and assembled into a global stiffness matrix, and the structural response of the model is solved; constructing a local density distribution function in the Bayesian unit, and assembling all LDDFs to obtain a global density distribution function for presenting structural topology; and constructing a stiffness maximization topological optimization mathematical model for the TPMS, calculating an objective function and analyzing sensitivity, and carrying out iterative updating on design variables in the optimization model based on an optimal criterion method to obtain a reinforced structure configuration.

Description

Technical Field

[0001] The present invention belongs to but is not limited to the technical field of material structure optimization, and in particular relates to a method and system for designing an enhanced structure for a three-periodic minimal surface. Background Art

[0002] In recent years, triply periodic minimal surface (TPMS) has attracted great attention from scholars due to its mathematically controllable geometric topology and excellent mechanical properties. Due to its special geometric properties such as zero mean curvature, full connectivity and open unit topology, TPMS structure has better mechanical properties than traditional lattice structures such as truss lattice. In other words, TPMS structure has attracted much attention due to its ideal properties such as high porosity, large specific surface area, excellent energy absorption, high specific strength and stiffness, and fatigue resistance, and is widely used in lightweight stents, exchangers, and medical implants.

[0003] The TPMS design methods widely used at present are implicit method and explicit method. The implicit method refers to the description of TPMS structure by a three-variable single-valued function composed of trigonometric functions. This method is highly efficient, so it is currently the first choice for constructing shell TPMS lattices. However, only a few implicit functions of TPMS structures are known, and in the implicit expression, the structural geometry can only be changed by adjusting a single parameter, which greatly limits the design flexibility and performance improvement of TPMS. In the explicit method, the explicitly constructed facets are iterated by minimizing the surface area and are subject to boundary constraints of the periodic combination of the facets. The difficulty of this method lies in balancing the boundary flexibility of the combination with the explicit facet calculation of the given boundary. With the development of optimization technology, shape optimization design methods have gradually been applied to the design of TPMS structures with specific performance improvements. Although the shell lattice based on shape optimization has a series of excellent characteristics, its topological structure usually changes less, and the adjustable size is limited to shape and thickness. Therefore, it is of great significance to introduce topological optimization methods into the research of TPMS structural design. The TPMS structure can be naturally combined with the density-based topology optimization method, in which the equivalent mechanical performance of the structural unit cell is expressed as a function of the relative density of the given structure through a homogenization algorithm, and the relative density is used as a design variable in the optimization process. Although many research works have demonstrated the superiority of topology optimization methods in improving TPMS performance, the geometric characteristics of the TPMS lattice itself lead to fewer adjustable parameters, and some holes generated by topology optimization methods even destroy the integrity of the structure, which greatly limits the further improvement of its mechanical properties and makes it difficult to give full play to the advantages of topology optimization in TPMS lattice design.

[0004] In view of the above analysis, the technical problems that urgently need to be solved in the existing technology are: the geometric characteristics of the TPMS lattice itself lead to fewer adjustable parameters, and even some holes generated by topology optimization methods will destroy the integrity of the structure, which greatly limits the further improvement of its mechanical properties and makes it difficult to give full play to the advantages of topology optimization in TPMS lattice design. Summary of the invention

[0005] In response to the problems existing in the prior art, the present invention provides an enhanced structural design method and system for three-periodic minimal surfaces, with the aim of addressing multiple limitations in the design of three-periodic minimal surfaces, such as the few adjustable parameters and the lack of design flexibility, exploring the room for improving the target performance, discovering new configurations of three-periodic minimal surfaces, and realizing the enhancement of the mechanical properties of the new structure.

[0006] The present invention is implemented as follows: an enhanced structure design method for three-periodic minimal surfaces, comprising:

[0007] S1. Model the geometric model of the three-periodic minimal surface in CAD software (Rhinoceros 3D and Grasshopper, T-splines plug-in) based on T-splines and export the model;

[0008] S2, based on the double-layer offset strategy, the design domain of the geometric model is expanded to obtain the corresponding extended design space of the three-periodic minimal surface;

[0009] S3, read the T-spline-based extended design space file and convert it into an analysis data structure, build a universal expression for the three-periodic minimal surface, and establish an analysis model based on Bayesian extraction. Use the Gaussian integral method and the elastic properties of each point in the design domain to solve the geometric unit stiffness matrix and assemble it into a global stiffness matrix to solve the structural response of the model;

[0010] S4, constructing a local density distribution function in the Bayesian unit, and assembling all local density distribution functions to obtain a global density distribution function for presenting a structural topology;

[0011] S5. Taking the maximization of structural stiffness as the optimization goal, a mathematical model of stiffness maximization topology optimization for TPMS unit cells is constructed;

[0012] S6. Calculate the objective function and analyze the sensitivity, iteratively update the design variables in the optimization model based on the optimal criterion method, and obtain the enhanced structural configuration.

[0013] Further, the implementation of step S1 includes:

[0014] The TPMS geometry model is created in the Grasshopper plug-in in the Rhinoceros 3D software, and the model is converted into a T-splines-based surface geometry model in combination with the T-splines plug-in. The combination of the Grasshopper plug-in and the Rhinoceros 3D software can realize the visual process of the geometry modeling steps, which is conducive to the transformation and modification of the model. The construction of the three-periodic minimal surface geometry model mainly includes the following steps:

[0015] (1) Constructing a three-periodic minimal surface structure lattice: divide the unidirectional region (-π, π) into m equal sub-regions, and then orthogonally form an m×m×m point array from the unidirectional regions, with the spacing between each adjacent point being 2π / m;

[0016] (2) Constructing a unit cube based on the structural lattice: Taking the coordinate distance between two adjacent points in the lattice as the side length and each point as the center point, a uniform unit cube is generated that extends to the initial boundary of the design;

[0017] (3) Decomposition of the cube matrix based on the implicit function: Substitute the coordinates of all center points into the surface function and divide the array into φ according to the function value P (x,y,z)>0 and φ P (x, y, z) < 0, where points greater than 0 are located on the outside of the surface, and points less than 0 are located on the inside of the surface. By indexing each point to the corresponding cube, we can get the inner and outer parts consisting of multiple cubes;

[0018] (4) Decomposing the matrix surface based on the center point of the cube: extract the center point of each face of the corresponding cube and find the two nearest neighboring points of the center point; then correspond these center points to the faces they are on, and splice all the faces to obtain an approximate surface;

[0019] (5) Smoothing the original surface based on the Laplace operator: The original mesh surface is smoothed by the Laplace operator to obtain the final minimum surface structure.

[0020] Further, the implementation of step S2 includes:

[0021] The implicit function standard value C = 0 of the three-periodic minimal surface is used as the initial surface, and C is offset along the normal direction of the initial surface. 1 , offset C in the opposite direction of the normal 2 , the above surfaces together constitute the design space. By adding a series of adjustable parameters, such as C 1 / C 2 , undesignable domain and other design parameters, greatly expanding the design space.

[0022] Further, in step S3, it is characterized in that:

[0023] S3.1, combining T-spline and Bayesian extraction technology to construct a unified general expression for the surface, then the T-spline hybrid function can be expressed as

[0024] N e (ξ,η)=C e B(ξ,η)

[0025] In the formula, is a vector containing the T-spline mixing functions for support element e, where is the T-spline blending function of support unit e, a represents the relative local index of the control point, and n is the total number of control points. is the unit extraction operator of unit e, and the polynomial order of the present invention in each direction is the same, and the dimension is n×(p+1) 2 . is the vector that defines the Bernstein polynomial. Let is the vector of T-spline mixing functions of unit e, and the T-spline mixing function of unit e can be expressed as

[0026] and

[0027] In the formula, and w e are two forms of expression of the weight of the control point of unit e. The first and second order differentials of the T-spline mixing function are

[0028]

[0029] In the formula, The T-spline basis function form based on Bayesian extraction will be further applied in IGA analysis to solve the unknown fields in the design domain.

[0030] S3.2. The three-periodic minimal surface is described by Kirchhoff-Löf thin shell theory. Its deformation is represented by the shell mid-surface. Then the displacement vector of any point on the mid-surface is

[0031]

[0032] In the formula, θ 1 and θ 2 are the curvilinear coordinates, x and are the position vectors of the material point in the deformed (actual) and undeformed (reference) states, respectively. According to the minimum potential energy principle, the internal virtual work δW can be obtained int The expression is

[0033]

[0034] Where E is Young's modulus, t is the thickness of the shell structure, and ν is Poisson's ratio. ε and κ define the membrane tensor and bending strain tensor, respectively, and the corresponding ε αβ and κ αβ Expressed as

[0035]

[0036] C (h) It can be expressed as

[0037]

[0038] Based on the analysis model of T-spline and general surface expression, the corresponding high-order isogeometric analysis numerical solution model of the surface shell is established, and the high-precision isogeometric unit numerical integration technology is integrated for analysis and solution. The unit displacement field can be discretized using an unstructured T-spline mixed function, expressed as

[0039]

[0040] Where N p is the total number of T-spline blending functions. is the displacement of the control point. Assuming that the displacement u and strain ε vary linearly along the thickness direction of the shell structure, it can be expressed as

[0041] u=u (m) (θ 1 ,θ 2 )+θ 3 u (b) (θ 1 ,θ 2 )

[0042] ε=ε (m) (u)+θ 3 ε (b) (u)

[0043] u (m) and u (b) are membrane displacement and bending displacement, ε (m) and ε (b) are membrane strain and bending strain, θ 1 and θ 2 represents the curvilinear coordinates in the shell midplane, θ 3 is the third coordinate along the normal direction in the range [-1,1]. By introducing the above formula into the principle of virtual work, the discrete equilibrium equation can be expressed as

[0044] KU=F

[0045] Where K is the global stiffness matrix and F is the global force matrix, calculated as

[0046]

[0047] The global stiffness matrix is ​​assembled from the element stiffness matrices

[0048]

[0049] Where E is Young's modulus, ν is Poisson's ratio, t represents the thickness of the shell TPMS structure, which is defined as uniform thickness in this paper, and C (h) represents the elastic matrix of the matrix material. In order to simplify the formula, a new character g(u) is introduced and expressed as

[0050]

[0051] The element stiffness matrix is ​​calculated based on the Gaussian orthogonal method:

[0052]

[0053] in, Represents Gaussian quadrature point The density of g is the weight of the corresponding Gaussian point, N g is the total number of Gaussian orthogonal points, and J represents the Jacobian matrix mapping from parameter space to physical space. Similarly, the global force vector F is also obtained by the unit force vector F e Calculated in the same way.

[0054] Further, in step S4, it is characterized in that, as can be seen from S3, the refined Bayesian unit shares p+1 control points with the adjacent unit, wherein the two refined units contain p-1 control points in the shared control points. This ensures that the coordinates of the control points on the boundaries of the two adjacent units are the same, and the control point density used to construct the density distribution function (DDF) is also the same. Therefore, the topological description model established using the Bayesian unit mainly includes the following parts:

[0055] (1) Define the initial density of Bayesian control points, that is, the Bayesian control density φ i {i=1,...,(p+1) 2};

[0056] (2) Introduce a smoothing mechanism to improve the smoothness of the Bayesian control density of each unit. The Shepard function is used to construct the smoothing mechanism, and the corresponding mathematical formula is:

[0057]

[0058] In the formula, ψ(φ i ) is at the control point Q i The Shepard function value of represents the smoothed Bayesian control density.

[0059] (3) The local density distribution function is constructed based on the linear combination of the Bernstein polynomials and the smoothing control design variables in the corresponding Bayesian unit:

[0060]

[0061] In the formula, is a vector of smoothing control density.

[0062] (4) Adopting the implicit description mechanism of the structural boundary in the classical level set method, the isovalue contour of the local DDF is used to represent the structural boundary of the corresponding topology:

[0063]

[0064] In the formula, Φ ISO represents the isocontour value of the local DDF, is the structural topology of the Bayesian unit.

[0065] (5) A single local density distribution function is used to describe the structural topology of a single Bayesian unit. The structural topology of the entire design domain can be described by the global DDF.

[0066]

[0067] In the formula, represents the local density distribution function of the ith Bayesian unit, Φ g is the global DDF representing the structural topology.

[0068] The corresponding structural topology can be expressed as

[0069]

[0070] In the formula, Represents the structural topology of the design domain.

[0071] Further, step S5 is characterized in that the stiffness maximization topology optimization model of the enhanced design is expressed as:

[0072]

[0073] Among them, x i is the design variable, which represents the density of the control points. n is the number of control points in the discrete design domain. J is the flexibility of the structure, which measures the overall deformation of the structure. V represents the material volume constraint, υ 0 is the volume fraction of the solid, V 0 is the volume limit imposed by the design domain.

[0074] Further, step S6 includes:

[0075] S6.1, Substitute the design variables into the S3 analysis model, and calculate the displacement field by KU=F, so as to calculate the objective function and sensitivity;

[0076] S6.2. Update the control density φ using the optimal criterion method i That is, the design variables are repeatedly executed until the iteration termination condition is reached, and the topological optimization configuration of the strengthened structure is obtained according to the objective function and sensitivity calculated by S6.2 in the last iteration.

[0077] The sensitivity analysis of the three-period minimal surface optimization model requires calculating the first-order derivative of the objective function relative to the DDF:

[0078]

[0079] According to the construction principle of global DDF, the first-order derivative of DDF with respect to the design variable can be obtained as

[0080]

[0081] In the formula, B(ξ,η) represents the Shepard function value at the control point. According to the above derivation, the sensitivity analysis of the objective function and the volume constraint function with respect to the design variables is:

[0082]

[0083] Where the calculation point (ξ,η) corresponds to the Gaussian orthogonal point in the design domain, which is different from the control point.

[0084] Another object of the present invention is to provide an enhanced structure design system for three-periodic minimal surfaces that implements the enhanced structure design method for three-periodic minimal surfaces, comprising:

[0085] The geometric model building module is used to model the geometric model of three-periodic minimal surfaces in CAD software (Rhinoceros 3D and Grasshopper, T-splines plug-in) based on T-splines and export the model;

[0086] The design space construction module expands the design domain of the geometric model based on the double-layer offset strategy, obtains the corresponding extended design space of the three-periodic minimal surface, and determines its design domain;

[0087] The structural response solving module converts the T-spline-based extended design space file into an analysis data structure, builds a universal expression for three-periodic minimal surfaces, establishes an analysis model based on Bayesian extraction, and uses the Gaussian integral method and the elastic properties of each point in the design domain to solve the geometric unit stiffness matrix and assemble it into a global stiffness matrix to solve the structural response of the model;

[0088] The structural topology description module constructs a local density distribution function based on the Bayesian unit and assembles all local DDFs to obtain a global density distribution function for presenting structural topology.

[0089] The optimization model building module takes the maximization of structural stiffness as the optimization goal and builds a stiffness maximization topology optimization mathematical model for the TPMS lattice;

[0090] The sensitivity analysis and update module calculates the objective function and analyzes the sensitivity, and iteratively updates the design variables in the optimization model based on the optimal criterion method to obtain the enhanced structural configuration.

[0091] Another object of the present invention is to provide a computer device, which includes a memory and a processor, wherein the memory stores a computer program, and when the computer program is executed by the processor, the processor executes the steps of the enhanced structure design method for three-periodic minimal surfaces.

[0092] Another object of the present invention is to provide a computer-readable storage medium storing a computer program, which, when executed by a processor, enables the processor to execute the steps of the enhanced structure design method for three-periodic minimal surfaces.

[0093] Another object of the present invention is to provide an information data processing terminal, which includes the enhanced structure design system for three-periodic minimal surfaces.

[0094] In combination with the above technical solutions and the technical problems solved, the advantages and positive effects of the technical solutions to be protected by the present invention are as follows:

[0095] First, the technical solution conceived by the present invention can achieve the following beneficial effects:

[0096] (1) The enhanced structural design method for three-periodic minimal surfaces provided by the present invention develops a double-layer offset strategy to increase the optimizable parameters in order to solve the problem that the geometric characteristics of TPMS result in fewer adjustable parameters. The optimization domain is expanded by controlling the offset value of the shell TPMS lattice along the normal direction or the opposite direction, and a series of optimizable parameters are added, providing more possibilities for the topological optimization strategy based on the isogeometric framework. At the same time, the design domain of the structural design is determined according to the application scenario of the lattice, and the integrity of the structure is ensured while retaining the geometric characteristics of the TPMS structure, further improving the mechanical properties of the lattice;

[0097] (2) The enhanced structural design method for three-periodic minimal surfaces provided by the present invention is aimed at the problems that the current topology optimization method destroys the integrity of the structure and limits the improvement of its mechanical properties. An enhanced structural design method based on isogeometric topology optimization technology is developed. T-splines with local refinement characteristics can accurately describe the geometry of arbitrarily complex three-periodic minimal surfaces. At the same time, Bayesian extraction technology can provide a unified general expression for surface structures with zero curvature everywhere. The two are combined to construct a three-periodic minimal surface analysis model and solution space. A mathematical model of isogeometric topology optimization with maximized stiffness is constructed based on a hybrid function of the Bayesian expression to maximize the structural bearing capacity, and the same numerical scheme and optimization algorithm are used to obtain the structural topological configuration, which solves the problems of low numerical accuracy and low analysis efficiency caused by previous classical methods in the optimization process, thereby improving the effectiveness and efficiency of the method;

[0098] (3) The enhanced structure design method for three-periodic minimal surfaces provided by the present invention has strong applicability and can be applied to the new structure mining of any three-periodic minimal surface. The new enhanced unit cell obtained based on this method not only has better geometric properties (increased surface area / volume ratio), but also has more superior mechanical properties (more uniform stress distribution, higher stiffness and strength), and the energy absorption characteristics under geometric deformation are also greatly improved, which shows that this method has strong effectiveness and applicability.

[0099] Second, (1) the expected benefits and commercial value of the technical solution of the present invention after transformation are as follows: the enhanced structural design method for three-periodic minimal surfaces of the present invention can significantly improve the design flexibility and performance of the TPMS structure, expand the optimization design space by introducing a double-layer offset strategy, and combine with Bayesian extraction technology to overcome the numerical accuracy and computational efficiency problems faced by traditional methods in the topology optimization process. The implementation of this technology is expected to improve the mechanical properties of TPMS structures in the fields of lightweight stents, medical implants, heat exchangers, etc., and broaden its application prospects in high-tech fields such as industry, medicine, and aerospace. The commercial value is reflected in the following aspects:

[0100] The increase in optimizable design parameters can significantly improve structural stiffness, strength and fatigue resistance, meeting the needs of high strength and high reliability.

[0101] Through optimized design methods, material waste can be reduced, manufacturing efficiency can be improved, and production costs can be reduced.

[0102] It has broad application potential in aerospace, medical, automotive, construction and other fields, promoting technological innovation and product upgrades in these industries.

[0103] (2) The technical solution of the present invention fills the technical gap in the industry at home and abroad: At present, although the TPMS structure has significant advantages in mechanical properties, in actual design, it is limited by geometric characteristics and has poor design flexibility. Especially in the topology optimization process, traditional methods often destroy the integrity of the structure, resulting in limited performance improvement. The existing technology mainly focuses on implicit and explicit modeling methods, and lacks a systematic method that takes into account both design flexibility and structural performance improvement. By adopting the dual-layer offset strategy and the optimization method based on Bayesian extraction of the present invention, the present invention fills this technical gap and provides an innovative design method that can maintain the geometric characteristics of TPMS and significantly improve structural performance. Therefore, the present invention has unique innovation in the field of TPMS enhanced structure design and fills the technical gap in this field at home and abroad.

[0104] (3) The technical solution of the present invention solves a technical problem that people have been eager to solve but have never been able to solve successfully: the existing TPMS design methods face two key problems: first, the geometric characteristics of the TPMS structure lead to fewer adjustable parameters, and second, in the topology optimization process, the optimization results often destroy the integrity of the structure. Traditional optimization methods make it difficult to improve the mechanical properties while ensuring the integrity of the structure. Especially in complex three-periodic minimal surface structures, it is difficult to find a method that can effectively improve the stiffness and maintain structural stability. For a long time, the industry has been full of demand for how to solve the contradiction between the flexibility and performance improvement of TPMS structural design, but has never been able to come up with an effective solution. The present invention successfully solves this technical problem through an innovative double-layer offset strategy and Bayesian extraction technology, breaking the limitations of traditional design methods and achieving the dual improvement of performance optimization and structural integrity of TPMS structural design.

[0105] (4) The technical solution of the present invention overcomes technical prejudice: Traditional TPMS design methods are often restricted by factors such as geometric shape and topological structure, resulting in insufficient design flexibility, and some classic topological optimization methods are prone to structural damage during the optimization process, and cannot bring out the performance potential of the TPMS structure. Especially in the industry, it is often believed that the geometric characteristics of TPMS are the bottleneck of design flexibility, and the application potential of topological optimization in TPMS structure is limited. In this regard, the present invention breaks this prejudice by introducing a double-layer offset strategy, providing more optimization parameters for the TPMS structure and greatly enhancing the design flexibility; at the same time, combined with Bayesian extraction and isogeometric topological optimization technology, structural damage can be avoided during the optimization process, the original geometric characteristics are maintained, and the superior performance of the TPMS structure is fully utilized. This technological breakthrough not only solves the problems that traditional optimization methods cannot overcome, but also promotes the technological innovation of TPMS structural design methods and overcomes the technical prejudice of the industry towards TPMS structural topological optimization. BRIEF DESCRIPTION OF THE DRAWINGS

[0106] Figure 1 A flow chart of an enhanced structure design method for three-periodic minimal surfaces provided by an embodiment of the present invention;

[0107] Figure 2 A schematic diagram of the main steps of modeling a TPMS unit cell based on the Grasshopper plug-in provided in an embodiment of the present invention;

[0108] Figure 3 A schematic diagram of the construction of a double-layer offset strategy provided by an embodiment of the present invention;

[0109] Figure 4 Schematic diagram of the preferred embodiment P, G, D, I-WP type strengthening structure and its performance provided by the embodiment of the present invention, wherein the design parameters are C 1 =0.08, C 2 =-0.05, volume fraction = 62.5% of the strengthening structure configuration;

[0110] Figure 5 It is a structural diagram of an enhanced structure design system for three-periodic minimal surfaces provided by an embodiment of the present invention;

[0111] Figure 6 Schematic diagram of numerical simulation test of the technical effect of the embodiment of the present invention, wherein (a) is a simulation model, (b) is a stress distribution and deformation diagram of four different resolutions under the same compression displacement of 3 mm, and (c) is a load-displacement curve of different resolutions under a compression displacement of 3 mm;

[0112] Figure 7 1 is a schematic diagram comparing the technical effects of the examples of the present invention, including: (a) a comparison diagram of the surface area / volume ratio of the reinforced structure and the original unit cell, (b) a comparison diagram of the maximum stress and the stress distribution under the same compressive displacement, (c) a comparison diagram of the stress-strain curves of the reinforced and original unit cells, (d) a comparison diagram of the equivalent strength and Young's modulus of the reinforced and original unit cells, and (e) a comparison diagram of the energy absorption characteristics of the reinforced and original unit cells under the same compressive displacement. DETAILED DESCRIPTION

[0113] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0114] The enhanced structure design method for three-periodic minimal surfaces provided by the present invention has the following process: Figure 1 As shown, the following steps are included:

[0115] (1) Create a TPMS geometric model in the Grasshopper plug-in in the Rhinoceros 3D software. The specific modeling steps are as follows: Figure 2 As shown, it mainly includes:

[0116] ① Construct a three-periodic minimal surface structure lattice: divide the unidirectional region (-π, π) into m sub-regions, and then orthogonally form an m×m×m point array from the unidirectional region, with a spacing of 2π / m between adjacent points;

[0117] ② Construct a unit cube based on the structural lattice: take the coordinate distance between two adjacent points in the lattice as the side length, and take each point as the center point to generate a uniform unit cube that extends to the initial boundary of the design;

[0118] ③ Decompose the cube matrix based on the implicit function: bring the coordinates of all center points into the surface function and divide the array into φ according to the function value P (x,y,z)>0 and φ P (x, y, z) < 0, where points greater than 0 are located on the outside of the surface, and points less than 0 are located on the inside of the surface. By indexing each point to the corresponding cube, we can get the inner and outer parts consisting of multiple cubes;

[0119] ④ Decompose the matrix surface based on the center point of the cube: extract the center point of each face in the corresponding cube, and find the two nearest neighboring points of the center point, then correspond these center points to the faces they are on, and splice all the faces to get an approximate surface;

[0120] ⑤ Smooth the original surface based on the Laplace operator: Smooth the original mesh surface through the Laplace operator to obtain the final minimum surface structure.

[0121] And combined with the Autodesk T-splines plug-in, the model is converted into a T-spline-based surface geometry model.

[0122] (2) Expand the design space based on the double-layer offset strategy, such as Figure 3 As shown. Taking the implicit function standard value C = 0 of the three-periodic minimal surface as the initial surface, the normal direction of the initial surface is offset by C 1 , offset C in the opposite direction of the normal 2 , the above surfaces together constitute the design space. In order to meet the integrity and sealing requirements of the TPMS structure, the design space after the double-layer offset strategy is divided into a design domain and an undesignable domain. In the optimization process, the area and thickness of the undesignable domain need to be set. That is to say, after obtaining the geometric model of the expanded design space, the design domain needs to be set manually.

[0123] (3) According to the design space expansion in step (2), a unified general expression is constructed for the surface by combining T-spline and Bayesian extraction technology. Then the T-spline hybrid function can be expressed as

[0124] N e (ξ,η)=C e B(ξ,η)

[0125] In the formula, is a vector containing the T-spline mixing functions for support element e, where is the T-spline blending function of support unit e, a represents the relative local index of the control point, and n is the total number of control points. is the unit extraction operator of unit e, and the polynomial order of the present invention in each direction is the same, and the dimension is n×(p+1) 2 . is the vector that defines the Bernstein polynomial. Let is the vector of T-spline mixing functions of unit e, and the T-spline mixing function of unit e can be expressed as

[0126] and

[0127] In the formula, and w e are two forms of expression of the weight of the control point of unit e. The first and second order differentials of the T-spline mixing function are

[0128]

[0129] In the formula,

[0130] Based on the analysis model of T-spline and general expression of surface, the corresponding high-order isogeometric analysis numerical solution model of surface shell is established, and the high-precision isogeometric unit numerical integration technology is integrated for analysis and solution. The unit displacement field can be discretized by unstructured T-spline mixed function, which is expressed as

[0131]

[0132] Where N p is the total number of T-spline blending functions. is the displacement of the control point. Assuming that the displacement u and strain ε vary linearly along the thickness direction of the shell structure, it can be expressed as

[0133] u=u (m) (θ 1 ,θ 2 )+θ 3 u (b) (θ 1 ,θ2 )

[0134] ε=ε (m) (u)+θ 3 ε (b) (u)

[0135] u (m) and u (b) are membrane displacement and bending displacement, ε (m) and ε (b) are membrane strain and bending strain, θ 1 and θ 2 represents the curvilinear coordinates in the shell midplane, θ 3 is the third coordinate along the normal direction in the range [-1,1]. By introducing the above formula into the principle of virtual work, the discrete equilibrium equation can be expressed as

[0136] KU=F

[0137] Where K is the global stiffness matrix and F is the global force matrix, calculated as

[0138]

[0139] The global stiffness matrix is ​​assembled from the element stiffness matrices

[0140]

[0141] Where E is Young's modulus, ν is Poisson's ratio, t represents the thickness of the shell TPMS structure, which is defined as uniform thickness in this paper, and C (h) represents the elastic matrix of the matrix material. In order to simplify the formula, a new character g(u) is introduced and expressed as

[0142]

[0143] The element stiffness matrix is ​​calculated based on the Gaussian orthogonal method:

[0144]

[0145] in, Represents Gaussian quadrature point The density of g is the weight of the corresponding Gaussian point, N g is the total number of Gaussian orthogonal points, and J represents the Jacobian matrix mapping from parameter space to physical space. Similarly, the global force vector F can be obtained by the unit force vector F e Calculated in the same way.

[0146] (4) By utilizing the spatial consistency characteristics of the Bayesian unit, the coordinates of the control points on the boundaries of two adjacent units are ensured to be the same, so that the density of the control points used to construct the DDF is also the same, thereby ensuring the continuity of the LDDF combination into the global DDF. The establishment of the topological description model includes:

[0147] ①Define the initial density of Bayesian control points, that is, the Bayesian control density φ i {i=1,...,(p+1) 2};

[0148] ② Construct a smoothing mechanism based on the Shepard function. The corresponding mathematical formula is:

[0149]

[0150] In the formula, ψ(φ i ) is at the control point Q i The Shepard function value of represents the smoothed Bayesian control density.

[0151] ③ Construct LDDF based on the linear combination of Bernstein polynomials and smooth control design variables in the corresponding Bayesian unit:

[0152]

[0153] In the formula, is a vector of smoothing control density.

[0154] ④ Adopt the implicit description mechanism of the structural boundary in the classical level set method, and use the equivalent contour of the local DDF to represent the structural boundary of the corresponding topology:

[0155]

[0156] In the formula, Φ ISO represents the isocontour value of the local DDF, is the structural topology of the Bayesian unit.

[0157] ⑤ A single LDDF is used to describe the structural topology of a single Bayesian unit. The structural topology of the entire design domain can be described by the global DDF assembled from all local DDFs:

[0158]

[0159] In the formula, represents the LDDF of the ith Bayesian unit, Φ g is the global DDF representing the structural topology. The corresponding structural topology can be expressed as

[0160]

[0161] In the formula, Represents the structural topology of the design domain.

[0162] (5) Based on the model established in the above steps, the stiffness maximization topology optimization model is expressed as:

[0163]

[0164] In the formula, x i is the design variable, which represents the density of the control points. n is the number of control points in the discrete design domain. J is the flexibility of the structure, which measures the overall deformation of the structure. V represents the material volume constraint, υ 0 is the volume fraction of the solid, V 0 is the volume limit imposed by the design domain.

[0165] (6) The sensitivity analysis of the three-period minimal surface optimization model requires calculating the first-order derivative of the objective function relative to the DDF:

[0166]

[0167] According to the construction principle of global DDF, the first-order derivative of DDF with respect to the design variable can be obtained as

[0168]

[0169] In the formula, B(ξ,η) represents the Shepard function value at the control point. According to the above derivation, the sensitivity analysis of the objective function and the volume constraint function with respect to the design variables is:

[0170]

[0171] Where the calculation point (ξ,η) corresponds to the Gaussian orthogonal point in the design domain, which is different from the control point.

[0172] (7) Update the control density φ using the OC method i That is, the design variables. The updated design variables are substituted into the updated topological description model representing the global design domain. The displacement field is updated by the updated model, so as to update the objective function and sensitivity calculation, and obtain new control variables. It is determined whether the convergence condition is met (the difference in node density between two consecutive iterations is less than a certain value). If not, the above iterative process is continued. If it is met, the topological optimization configuration of the strengthened structure is obtained.

[0173] Combine the following Figure 4 A specific embodiment shown is used to describe the above steps and results of the present invention in detail:

[0174] (1) The initial model of the classic three-periodic minimal surface is modeled in the Grasshopper plug-in based on the following implicit function:

[0175] φ P (x,y,z)=cos(ω x x)+cos(ω y y)+cos(ω z z)=C

[0176] φ G (x,y,z)=sin(ω x x)cos(ω y y)+sin(ω z z)cos(ω x x)+sin(ω y y)cos(ω z z)=C

[0177] φ D (x,y,z)=cos(ω x x)cos(ω y y)cos(ω z z)-sin(ω x x)sin(ω y y)sin(ω z z)=C

[0178] φ I-WP (x,y,z)=2[cos(ω x x)cos(ω y y)+cos(ω y y)cos(ω z z)+cos(ω z z)cos(ω x x)】

[0179] -【cos(2ω x x)+cos(2ω y y)+cos(2ω z z)]=C

[0180] And transform it into a T-spline geometric model. The specific steps are as follows Figure 2 As shown;

[0181] (2) Import the model file, use the standard value C=0 in the implicit function as the initial surface, and offset C along the normal direction of the initial surface. 1 , offset C in the opposite direction of the normal 2 , the above surfaces together constitute the design space, such as Figure 3 As shown, in order to clearly show the configuration of the strengthening structure, the design parameters of this example are selected as shown in Table 1;

[0182] Table 1 Design parameters of the classic three-periodic minimal surface

[0183]

[0184] (3) Combining T-spline and Bayesian extraction technology to construct a unified general expression for the surface, and integrating numerical integration technology to establish the corresponding surface shell high-order geometric analysis numerical solution model:

[0185] The T-spline mixing function can be expressed as

[0186] N e (ξ,η)=C e B(ξ,η)

[0187] In the formula, is a vector containing the T-spline mixing functions for support element e, where is the T-spline blending function of support unit e, a represents the relative local index of the control point, and n is the total number of control points. is the unit extraction operator of unit e, and the polynomial order of the present invention in each direction is the same, and the dimension is n×(p+1) 2 . is the vector that defines the Bernstein polynomial. Let is the vector of T-spline mixing functions of unit e, and the T-spline mixing function of unit e can be expressed as

[0188] and

[0189] In the formula, and w e are two forms of expression of the weight of the control point of unit e. The first and second order differentials of the T-spline mixing function are

[0190]

[0191] In the formula,

[0192] The unit displacement field can be discretized using an unstructured T-spline mixed function, expressed as

[0193]

[0194] Where N p is the total number of T-spline blending functions. is the displacement of the control point. Assuming that the displacement u and strain ε vary linearly along the thickness direction of the shell structure, it can be expressed as

[0195] u=u (m)(θ 1 ,θ 2 )+θ 3 u (b) (θ 1 ,θ 2 )

[0196] ε=ε (m) (u)+θ 3 ε (b) (u)

[0197] u (m) and u (b) are membrane displacement and bending displacement, ε (m) and ε (b) are membrane strain and bending strain, θ 1 and θ 2 represents the curvilinear coordinates in the shell midplane, θ 3 is the third coordinate along the normal direction in the range [-1,1]. By introducing the above formula into the principle of virtual work, the discrete equilibrium equation can be expressed as

[0198] KU=F

[0199] Where K is the global stiffness matrix and F is the global force matrix, calculated as

[0200]

[0201] The global stiffness matrix is ​​assembled from the element stiffness matrices

[0202]

[0203] Where E is Young's modulus, ν is Poisson's ratio, t represents the thickness of the shell TPMS structure, which is defined as uniform thickness in this paper, and C (h) represents the elastic matrix of the matrix material. In order to simplify the formula, a new character g(u) is introduced and expressed as

[0204]

[0205] The element stiffness matrix is ​​calculated based on the Gaussian orthogonal method:

[0206]

[0207] in, Represents Gaussian quadrature point The density of g is the weight of the corresponding Gaussian point, N g is the total number of Gaussian orthogonal points, and J represents the Jacobian matrix mapping from parameter space to physical space.

[0208] (4) Using Bayesian units to establish a topological description model includes the following steps

[0209] ①Define the initial density of Bayesian control points, that is, the Bayesian control density φ i {i=1,...,(p+1) 2};

[0210] ② Apply Shepard function to construct smoothing mechanism, the corresponding mathematical formula is:

[0211]

[0212] In the formula, ψ(φ i ) is at the control point Q i The Shepard function value of represents the smoothed Bayesian control density.

[0213] ③ Construct LDDF based on the linear combination of Bernstein polynomials and smooth control design variables in the corresponding Bayesian unit:

[0214]

[0215] In the formula, is a vector of smoothing control density.

[0216] ④ Adopt the implicit description mechanism of the structural boundary in the classical level set method, and use the equivalent contour of the local DDF to represent the structural boundary of the corresponding topology:

[0217]

[0218] In the formula, Φ ISO represents the isocontour value of the local DDF, is the structural topology of the Bayesian unit.

[0219] ⑤ A single LDDF is used to describe the structural topology of a single Bayesian unit. The structural topology of the entire design domain can be described by the global DDF assembled from all local DDFs:

[0220]

[0221] In the formula, represents the LDDF of the ith Bayesian unit, Φ g is the global DDF representing the structural topology. The corresponding structural topology can be expressed as

[0222]

[0223] In the formula, Represents the structural topology of the design domain.

[0224] (5) The mathematical model of isogeometric topology optimization for three-periodic minimal surfaces with maximum stiffness is as follows:

[0225]

[0226] In the formula, x i is the design variable, which represents the density of the control points. n is the number of control points in the discrete design domain. J is the flexibility of the structure, which measures the overall deformation of the structure. V represents the material volume constraint, υ 0 is the volume fraction of the solid, V 0 is the volume limit imposed by the design domain.

[0227] (6) The sensitivity analysis of the three-period minimal surface optimization model requires calculating the first-order derivative of the objective function relative to the DDF:

[0228]

[0229] According to the construction principle of global DDF, the first-order derivative of DDF with respect to the design variable can be obtained as

[0230]

[0231] In the formula, B(ξ,η) represents the Shepard function value at the control point. According to the above derivation, the sensitivity analysis of the objective function and the volume constraint function with respect to the design variables is:

[0232]

[0233] Where the calculation point (ξ,η) corresponds to the Gaussian orthogonal point in the design domain, which is different from the control point.

[0234] (7) Update the control density φ using the OC method i That is, the design variables, the updated design variables are substituted into the updated topological description model representing the complex structure design domain, and the displacement field is updated by the updated model, so as to update the objective function and sensitivity calculation, obtain new control variables, and determine whether the convergence conditions are met. If not, continue the above iteration process, and if it is met, the optimized topological optimization configuration is obtained. The convergence criterion is that the difference in node density between two consecutive iterations is less than 1%, or the maximum iteration step length of 150 is reached. The topological configuration of P, G, D, and IWP type strengthening structures is as follows: Figure 4 As shown in the figure, the material gradually transitions from being distributed circumferentially to being distributed along the normal direction of the TPMS structure, and the main structural elements gradually evolve from shell structures to a configuration combining plate structures and beam structures. This phenomenon can be explained from the perspective of metamaterials. The beam structure provides more targeted stiffness under unilateral loads, and the increase in overall thickness will enhance the stiffness and strength of the new reinforced structure.

[0235] like Figure 5 As shown in the figure, the enhanced structure design system for three-periodic minimal surfaces includes:

[0236] The geometric model building module is used to model the geometric model of three-periodic minimal surfaces in CAD software (Rhinoceros 3D and Grasshopper, T-splines plug-in) based on T-splines and export the model;

[0237] The design space construction module expands the design domain of the geometric model based on the double-layer offset strategy, obtains the corresponding extended design space of the three-periodic minimal surface, and determines its design domain;

[0238] The structural response solving module converts the T-spline-based extended design space file into an analysis data structure, builds a universal expression for three-periodic minimal surfaces, establishes an analysis model based on Bayesian extraction, and uses the Gaussian integral method and the elastic properties of each point in the design domain to solve the geometric unit stiffness matrix and assemble it into a global stiffness matrix to solve the structural response of the model;

[0239] The structural topology description module constructs a local density distribution function based on the Bayesian unit and assembles all local DDFs to obtain a global density distribution function for presenting structural topology.

[0240] The optimization model building module takes the maximization of structural stiffness as the optimization goal and builds a stiffness maximization topology optimization mathematical model for the TPMS lattice;

[0241] The sensitivity analysis and update module calculates the objective function and analyzes the sensitivity, and iteratively updates the design variables in the optimization model based on the optimal criterion method to obtain the enhanced structural configuration.

[0242] An application embodiment of the present invention provides a computer device, which includes a memory and a processor. The memory stores a computer program. When the computer program is executed by the processor, the processor executes the steps of an enhanced structure design method for three-periodic minimal surfaces.

[0243] An application embodiment of the present invention provides a computer-readable storage medium storing a computer program. When the computer program is executed by a processor, the processor executes the steps of an enhanced structure design method for three-periodic minimal surfaces.

[0244] An application embodiment of the present invention provides an information data processing terminal, which includes an enhanced structure design system for three-periodic minimal surfaces.

[0245] 1. Specific application fields or related products of the present invention.

[0246] (1) Aerospace lightweight brackets: In the field of aerospace, light weight and high strength are the core requirements of supporting structures. The enhanced TPMS structure design method of the present invention can be applied to the design of supporting frames, bulkheads, wing frames and other components in aerospace. By introducing the double-layer offset strategy of the present invention, the adjustable parameters of the TPMS structure are increased, so that the stiffness and strength of the structure can be improved while meeting the lightweight requirements. It is particularly suitable for lightweight design of spacecraft and aircraft that need to withstand complex loads. Its technical value is reflected in: through optimized design, the weight of aerospace vehicles is reduced, the material utilization rate is improved, the fatigue resistance and impact resistance of the structure are enhanced, and the safety and reliability of spacecraft are improved.

[0247] (2) TPMS enhanced orthopedic implants: In the field of medical devices, especially in the design of orthopedic implants (such as artificial joints, skeleton support structures, etc.), the mechanical properties of materials and the biocompatibility of structures are crucial. By adopting the enhanced TPMS structure design method of the present invention, orthopedic implants with optimized mechanical properties can be designed. These implants can not only provide higher load-bearing capacity, but also provide better mechanical adaptability during use, reducing the occurrence of postoperative complications. Its technical value is reflected in: the technical solution of the present invention can improve the mechanical properties and biocompatibility of orthopedic implants, increase the service life of implants and reduce the patient's rehabilitation cycle, while reducing material waste and reducing costs. The optimization of the TPMS structure can achieve a perfect fit between the implant and the bone, reduce stress concentration, and promote bone healing.

[0248] (3) Core components of heat exchangers: Heat exchangers are widely used in the energy, chemical, air-conditioning and other industries, and their core requirements are efficient heat conduction and excellent mechanical properties. Using the TPMS structure optimization method of the present invention, a heat exchanger material with a large specific surface area, high porosity and high thermal conductivity can be designed. This structure can effectively improve the heat exchange efficiency while reducing weight, and is suitable for air and liquid cooling systems. Its technical value is reflected in: by optimizing the geometric characteristics of the TPMS structure, the thermal conductivity and fluid flow efficiency of the heat exchanger are improved, while the amount of material used is reduced and the manufacturing cost is reduced. This technology can significantly improve the efficiency and reliability of the heat exchanger and promote the innovative development of the energy industry.

[0249] 2. Relevant evidence of the technical effects obtained by the embodiments of the present invention.

[0250] The mechanical properties and energy absorption characteristics of the original TPMS unit cell and the enhanced unit cell are compared through numerical simulation tests to verify the technical effect of the present invention. Figure 6As shown in (a), two discrete rigid plates are in contact with the upper and lower surfaces of the unit cell structure respectively. The contact mode adopts general contact interaction, in which "hard" contact describes normal behavior, and the friction coefficient in the tangential direction is 0.12. The two reference points are located on two rigid plates at corresponding positions. These reference points are convenient for extracting compressive stress and reaction force, thereby generating a simulated load-displacement curve. During the compression process, the lower plate is fixed, all degrees of freedom of the bottom reference point are fixed, and the top reference point follows the upper plate to compress downward at a constant rate of 1m / s until the compression displacement reaches the specified displacement. At the same time, in order to ensure the reliability of the numerical simulation results, the analysis results of the original unit cell structure with 4 different resolutions and a compression displacement of 3mm were tested. The stress and deformation results are shown in the figure. Figure 6 (b) shows the corresponding load-displacement curve. Figure 6 (c). Figure 6 It can be found that under the same compression displacement, the stress distribution of different mesh resolutions is almost the same, and the load displacement curve also shows the same trend, indicating the correctness of the numerical simulation analysis results. Different resolutions will lead to different maximum stress values ​​because a larger number of meshes will increase the accuracy and thus increase the stress value. However, as the mesh resolution increases, the calculation time also increases significantly. Therefore, when the results converge, the second resolution will be used in subsequent analysis. At the same time, considering that the initial design parameters of the strengthening unit cell are different, the specified compression displacement is uniformly set to 4mm.

[0251] and Figure 7 (a) The surface area to volume (SA / V) ratio of the original unit cell (-0) and the reinforced lattice (Re-) of the classic P, G, D, and I-WP is compared. It can be found that the reinforced unit cell has a higher SA / V ratio at the same density. The combination of isogeometric topology optimization and double-layer offset strategy can effectively increase the surface area of ​​the reinforced unit cell and thus increase the SA / V ratio, which also makes the reinforced unit cell more suitable for medical implant applications such as artificial tissue filling. Figure 7 (b) shows the stress maximum and stress distribution comparison of the original unit cell with the same density and the reinforced structure under the same specified compression displacement. It is obvious that the stress maximum of the reinforced structure is reduced and the stress distribution is more uniform. Therefore, it can be concluded that the reinforced structure of the application example of the present invention can effectively distribute stress uniformly and alleviate stress concentration while having the geometric performance of the forehead guard. The load-displacement curve obtained from the compression process is divided by the structural contact area to obtain the stress-strain curve, as shown in FIG. Figure 7 As shown in (c), they all experienced elastic deformation, plastic platform and densification stages, indicating the effectiveness and correctness of the stress-strain curve of the strengthened structure during compression. And by extracting the slope and endpoint of the elastic stage of the stress-strain curve, the Young's modulus and yield strength of the structure are calculated, as shown in Figure 7(d) is shown, which is also an effective concept for describing the mechanical properties of structural materials. By comparing the stress-strain curves between the reinforced structure and the original structure, it can be found that the Young's modulus and yield strength of the reinforced structure are both higher than those of the original unit cell, indicating that the stiffness and strength of the reinforced structure obtained by the present invention are enhanced. The results show that the enhanced structural design proposed by the present invention is helpful to improve the mechanical properties of the TPMS structure. Figure 7 (c) It can be found that the stress-strain curves of the original unit cell and the strengthened structure both present a stress plateau stage, indicating that both the original unit cell and the strengthened structure have effective energy absorption capabilities. The energy absorption characteristics of the structure are characterized by the energy absorbed per unit volume, such as Figure 7 As shown in (e), the energy absorption characteristics of the reinforced structure are significantly higher than those of the original unit cell, indicating that the present invention has important significance in terms of energy absorption characteristics. The above data all illustrate the technical effects of the present invention.

[0252] It should be noted that the embodiments of the present invention can be implemented by hardware, software, or a combination of software and hardware. The hardware part can be implemented using dedicated logic; the software part can be stored in a memory and executed by an appropriate instruction execution system, such as a microprocessor or dedicated design hardware. It can be understood by a person of ordinary skill in the art that the above-mentioned devices and methods can be implemented using computer executable instructions and / or contained in a processor control code, such as a carrier medium such as a disk, CD or DVD-ROM, a programmable memory such as a read-only memory (firmware), or a data carrier such as an optical or electronic signal carrier. Such code is provided on the carrier medium. The device and its modules of the present invention can be implemented by hardware circuits such as very large-scale integrated circuits or gate arrays, semiconductors such as logic chips, transistors, etc., or programmable hardware devices such as field programmable gate arrays, programmable logic devices, etc., can also be implemented by software executed by various types of processors, and can also be implemented by a combination of the above-mentioned hardware circuits and software, such as firmware.

[0253] The above description is only a specific implementation mode of the present invention, but the protection scope of the present invention is not limited thereto. Any modifications, equivalent substitutions and improvements made by any technician familiar with the technical field within the technical scope disclosed by the present invention and within the spirit and principle of the present invention should be covered by the protection scope of the present invention.

Claims

1. An enhanced structure design method for three-periodic minimal surfaces, characterized in that: include: S1. Model the geometric model of the three-periodic minimal surface in CAD software based on T-spline and export the model; S2, based on the double-layer offset strategy, the design domain of the geometric model is expanded to obtain the corresponding extended design space of the three-periodic minimal surface; S3, convert the T-spline-based extended design space file into an analysis data structure, construct a general expression for three-periodic minimal surfaces, establish an analysis model based on Bayesian extraction, use Gaussian integration method and elastic properties of each point in the design domain to solve the geometric unit stiffness matrix and assemble it into a global stiffness matrix to solve the structural response of the model; S4, constructing a local density distribution function in the Bayesian unit, and assembling all LDDFs to obtain a global density distribution function for presenting a structural topology; S5. Taking the maximization of structural stiffness as the optimization goal, a stiffness maximization topology optimization mathematical model for TPMS is constructed; S6. Calculate the objective function and analyze the sensitivity, iteratively update the design variables in the optimization model based on the optimal criterion method, and obtain the enhanced structural configuration.

2. The enhanced structure design method for three-periodic minimal surfaces according to claim 1, characterized in that: The implementation of step S1 includes: The TPMS geometry model is created in the Grasshopper plug-in in the Rhinoceros 3D software, and the model is converted into a T-spline-based surface geometry model in combination with the Autodesk T-splines plug-in. The combination of the Grasshopper plug-in and the Rhinoceros 3D software can realize the visual process of the geometry modeling steps, which is conducive to the transformation and modification of the model. The construction of the three-periodic minimal surface geometry model mainly includes the following steps: (1) Constructing a three-periodic minimal surface structure lattice: divide the unidirectional region (-π, π) into m equal sub-regions, and then orthogonally form an m×m×m point array from the unidirectional regions, with the spacing between each adjacent point being 2π / m; (2) Constructing a unit cube based on the structural lattice: Taking the coordinate distance between two adjacent points in the lattice as the side length and each point as the center point, a uniform unit cube is generated that extends to the initial boundary of the design; (3) Decomposition of the cube matrix based on the implicit function: Substitute the coordinates of all center points into the surface function and divide the array into φ according to the function value P (x,y,z)>0 and φ P (x, y, z) < 0, where points greater than 0 are located outside the surface and points less than 0 are located inside the surface; by indexing each point to the corresponding cube, we can get the inner and outer parts consisting of multiple cubes; (4) Decomposing the matrix surface based on the center point of the cube: extract the center point of each face of the corresponding cube and find the two nearest neighboring points of the center point; then correspond these center points to the faces they are on, and splice all the faces to obtain an approximate surface; (5) Smoothing the original surface based on the Laplace operator: The original mesh surface is smoothed by the Laplace operator to obtain the final minimum surface structure.

3. The enhanced structure design method for three-periodic minimal surfaces according to claim 1, characterized in that: The implementation of step S2 includes: The standard value C=0 of the implicit function of the three-period minimal surface is taken as the initial surface, and C1 is offset along the normal direction of the initial surface, and C2 is offset along the opposite direction of the normal. The above surfaces together constitute the design space. By adding a series of adjustable parameters, such as C1 / C2, undesignable domain and other design parameters, the design space is greatly expanded.

4. The enhanced structure design method for three-periodic minimal surfaces according to claim 1, characterized in that: In step S3, the feature is that S3.1, combining T-spline and Bayesian extraction technology to construct a unified general expression for the surface, then the T-spline hybrid function can be expressed as N e (ξ,η)=C e B(ξ,η) In the formula, is a vector containing the T-spline mixing functions for support element e, where is the T-spline blending function of support unit e, a represents the relative local index of the control point, and n is the total number of control points; is the unit extraction operator of unit e, and the polynomial order of the present invention in each direction is the same, and the dimension is n×(p+1) 2 ; is the vector that defines the Bernstein polynomial; let is the vector of T-spline mixing functions of unit e, and the T-spline mixing function of unit e can be expressed as In the formula, and w e are two forms of expression of the weights of the control points of unit e; and the first and second order differentials of the T-spline mixing function are In the formula, And the T-spline basis function form based on Bayesian extraction will be further applied in IGA analysis to solve the unknown field of the design domain; S3.

2. The three-periodic minimal surface is described by Kirchhoff-Löf thin shell theory. Its deformation is represented by the shell mid-surface. Then the displacement vector of any point on the mid-surface is In the formula, θ 1 and θ 2 are the curvilinear coordinates, x and are the position vectors of the material point in the deformed and undeformed states respectively; according to the minimum potential energy principle, the internal virtual work δW can be obtained int The expression is Where E represents Young's modulus, t is the thickness of the shell structure, ν is Poisson's ratio; ε and κ define the membrane tensor and bending strain tensor respectively, and the corresponding ε αβ and κ αβ Expressed as Based on the analysis model of T-spline and general surface expression, the corresponding high-order isogeometric analysis numerical solution model of the surface shell is established, and the high-precision isogeometric unit numerical integration technology is integrated for analysis and solution; the unit displacement field can be discretized using an unstructured T-spline mixed function, expressed as Where N p is the total number of T-spline mixing functions; is the displacement of the control point; assuming that the displacement u and strain ε vary linearly along the thickness direction of the shell structure, it can be expressed as u=u (m) (i 1 ,i 2 )+θ 3 you (b) (i 1 ,i 2 ) e=e (m) (u)+θ 3 e (b) (u) u (m) and u (b) are membrane displacement and bending displacement, ε (m) and ε (b) are membrane strain and bending strain, θ 1 and θ 2 represents the curvilinear coordinates in the shell midplane, θ 3 is the third coordinate along the normal direction in the range [-1,1]; by introducing the above formula into the principle of virtual work, the discrete equilibrium equation can be expressed as KU=F Where K is the global stiffness matrix and F is the global force matrix, calculated as The global stiffness matrix is ​​assembled from the element stiffness matrices Where E is Young's modulus, ν is Poisson's ratio, t represents the thickness of the shell TPMS structure, which is defined as uniform thickness in this paper, and C (h) represents the elastic matrix of the matrix material; in order to simplify the formula, a new character g(u) is introduced and expressed as The element stiffness matrix is ​​calculated based on the Gaussian orthogonal method: in, Represents Gaussian quadrature point The density of g is the weight of the corresponding Gaussian point, N g is the total number of Gaussian orthogonal points, J represents the Jacobian matrix mapping from parameter space to physical space; similarly, the global force vector F is also obtained by the unit force vector F e Calculated in the same way.

5. The enhanced structure design method for three-periodic minimal surfaces according to claim 1, characterized in that: In step S4, it is characterized in that, as can be seen from S3, the refined Bayesian unit shares p+1 control points with the adjacent unit, wherein the two refined units contain p-1 control points in the shared control points; thereby ensuring that the coordinates of the control points on the boundaries of the two adjacent units are the same, and the control point density used to construct the density distribution function is also the same; therefore, using the Bayesian unit to establish a topological description model mainly includes the following parts: (1) Define the initial density of Bayesian control points, that is, the Bayesian control density φ i {i=1,...,(p+1) 2 }; (2) Introduce a smoothing mechanism to improve the smoothness of the Bayesian control density of each unit; apply the Shepard function to construct a smoothing mechanism, and the corresponding mathematical formula is: In the formula, ψ(φ i ) is at the control point Q i The Shepard function value of represents the smoothed Bayesian control density; (3) Construct the LDDF based on the linear combination of the Bernstein polynomials and smoothing control design variables in the corresponding Bayesian unit: In the formula, is the vector of smoothing control density; (4) Adopting the implicit description mechanism of the structural boundary in the classical level set method, the isovalue contour of the local DDF is used to represent the structural boundary of the corresponding topology: In the formula, Φ ISO represents the isocontour value of the local DDF, is the structural topology of the Bayesian unit; (5) A single LDDF is used to describe the structural topology of a single Bayesian unit. The structural topology of the entire design domain can be described by the global DDF assembled from all local DDFs: In the formula, represents the LDDF of the ith Bayesian unit, Φ g is the global DDF representing the structural topology; the corresponding structural topology can be expressed as In the formula, Represents the structural topology of the design domain.

6. The enhanced structure design method for three-periodic minimal surfaces according to claim 1, characterized in that: Step S5, characterized in that the stiffness maximization topology optimization model of the enhanced design is expressed as: Among them, x i is the design variable, which represents the density of the control points; n is the number of control points in the discrete design domain; J is the flexibility of the structure, which measures the overall deformation of the structure; V represents the material volume constraint, υ0 is the volume fraction of the entity, and V0 is the volume limit specified in the design domain.

7. The enhanced structure design method for three-periodic minimal surfaces according to claim 1, characterized in that: Step S6 includes: S6.1, Substitute the design variables into the S3 analysis model, and calculate the displacement field by KU=F, so as to calculate the objective function and sensitivity; S6.

2. Update the control density φ using the optimal criterion method i That is, the design variables are repeatedly executed until the iteration termination condition is reached, and the topological optimization configuration of the strengthened structure is obtained according to the objective function and sensitivity calculated by S6.2 in the last iteration; The sensitivity analysis of the three-period minimal surface optimization model requires calculating the first-order derivative of the objective function relative to the DDF: According to the construction principle of global DDF, the first-order derivative of DDF with respect to the design variable can be obtained as Where B(ξ,η) represents the Shepard function value at the control point. According to the above derivation, the sensitivity analysis of the objective function and the volume constraint function with respect to the design variables is: Where the calculation point (ξ,η) corresponds to the Gaussian orthogonal point in the design domain, which is different from the control point.

8. An enhanced structure design system for three-periodic minimal surfaces that implements the enhanced structure design method for three-periodic minimal surfaces as claimed in any one of claims 1 to 7, characterized in that: include: The geometric model building module is used to model the geometric model of three-periodic minimal surfaces in CAD software (Rhinoceros 3D and Grasshopper, T-splines plug-in) based on T-splines and export the model; The design space construction module expands the design domain of the geometric model based on the double-layer offset strategy, obtains the corresponding extended design space of the three-periodic minimal surface, and determines its design domain; The structural response solving module converts the T-spline-based extended design space file into an analysis data structure, builds a universal expression for three-periodic minimal surfaces, establishes an analysis model based on Bayesian extraction, and uses the Gaussian integral method and the elastic properties of each point in the design domain to solve the geometric unit stiffness matrix and assemble it into a global stiffness matrix to solve the structural response of the model; The structural topology description module constructs a local density distribution function based on the Bayesian unit and assembles all local DDFs to obtain a global density distribution function for presenting structural topology. The optimization model building module takes the maximization of structural stiffness as the optimization goal and builds a stiffness maximization topology optimization mathematical model for the TPMS lattice; The sensitivity analysis and update module calculates the objective function and analyzes the sensitivity, and iteratively updates the design variables in the optimization model based on the optimal criterion method to obtain the enhanced structural configuration.

9. A computer device, comprising a memory and a processor, wherein the memory stores a computer program, and when the computer program is executed by the processor, the processor executes the steps of the enhanced structure design method for three-periodic minimal surfaces as described in any one of claims 1 to 7.

10. An information data processing terminal, comprising the enhanced structure design system for tri-periodic minimal surfaces according to claim 8.

Images