A heterogeneous superstructure template generation method based on implicit expression

Abstract

The application discloses a heterogeneous superstructure template generation method based on implicit expression, which combines the numerical homogenization theory and the variable density method topological optimization to construct a superstructure inverse homogenization topological optimization method; the superstructure inverse homogenization topological optimization method is combined with a design automation platform ISIGHT to obtain different superstructure configurations with the same equivalent elastic tensor by taking optimization parameters as variables and minimizing the solid material volume as an objective function; then, a configuration with good manufacturability and regular shape is selected, a level set function is used to construct a superstructure template, and good connection between adjacent superstructures is realized by controlling the structure between the cutting function value to be equal; finally, a function relationship between the level set control parameter and the equivalent performance tensor component is established; and for specific material performance requirements, the level set control parameter is inversely solved according to the function relationship; and the application can effectively improve the design efficiency of the predetermined attribute heterogeneous structured material.

Description

Technical Field

[0001] This invention belongs to the field of heterogeneous structured material design technology, specifically relating to a method for generating heterogeneous superstructure templates based on implicit expression. Background Technology

[0002] In modern industrial technology, the requirements for material properties are constantly increasing. Relying solely on homogeneous materials for macroscopic structural design is increasingly insufficient to meet structural performance demands. Heterogeneous structured materials possess advantages such as lightweight, high strength, high specific stiffness, and strong designability. Therefore, it is necessary to develop design methods for heterogeneous structured materials. In continuum structure design, topology optimization is a design method with high design freedom. Given boundary conditions, optimization objectives, and constraint functions, it can automatically generate structural configurations that meet design requirements. Applying topology optimization to heterogeneous structure design can effectively obtain material superstructures with special properties such as negative Poisson's ratio and limiting elastic tensor.

[0003] However, the design results of topology optimization are constrained by the optimization parameters. Inappropriate parameter values ​​may make the design difficult to manufacture or even prevent the optimization process from converging. To design clear, manufacturable structures with specific properties, it is necessary to continuously adjust the optimization parameters to escape local optima and obtain the ideal configuration. This makes the structural topology optimization design process limited by the designer's optimization experience and also greatly increases the design time. Therefore, there is an urgent need for an efficient design method for heterogeneous structured materials with predetermined properties. Summary of the Invention

[0004] To address the above technical problems, this invention provides a method for generating heterogeneous superstructure templates based on implicit expression, which can effectively improve the design efficiency of heterogeneous structured materials with predetermined properties.

[0005] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0006] A method for generating heterogeneous superstructure templates based on implicit expression includes the following steps:

[0007] 1) Combining numerical homogenization theory with variable density topology optimization, a superstructure inverse homogenization topology optimization method is constructed;

[0008] 2) The superstructure inverse homogenization topology optimization method is combined with the design automation platform ISIGHT, with optimization parameters as design variables and minimization of solid material volume as the objective function, to obtain different superstructure configurations with the same equivalent elastic tensor.

[0009] 3) Select a superstructure configuration with good manufacturability and regular shape, use a horizontal set function to construct a superstructure template, and achieve good connection between adjacent structures by controlling the cut function values ​​between superstructures to be equal;

[0010] 4) Establish the functional relationship between the level set control parameters and the equivalent elastic tensor components; for specific material performance requirements, the level set control parameters can be derived inversely based on the functional relationship.

[0011] In step 1), a superstructure inverse homogenization topology optimization method is constructed, and the equivalent elastic tensor E of the superstructure is... H ijkl Represented as:

[0012]

[0013] In the formula, V is the entire superstructure space, and E pqrs It is a local elastic tensor. It is a macroscopic strain field. The local strain field is obtained by applying test strain to each element; after finite element discretization, the equivalent elastic tensor E is solved. H ij The expression is:

[0014]

[0015] In the formula, N is the total number of discrete elements, e is the element number, and k is the number of discrete elements. e The element stiffness matrix, It is a macroscopic displacement field. It is a microscopic displacement field, V e It is a superstructure space described by the finite element method.

[0016] In step 1), the superstructure inverse homogenization topology optimization method is constructed, and the element elastic tensor is calculated by the following formula:

[0017] E(x)=E min +(E max -E min )x p

[0018] In the formula, E(x) is the element elasticity tensor obtained by interpolation, E min To assess the elastic properties of the hole, a value close to 0 is chosen to prevent matrix singularities, E max is the elastic modulus of the solid phase, x is the relative density of the element, and p is the penalty factor;

[0019] The material uses a two-dimensional orthogonal anisotropic tensor E 0 For ideal performance, optimize the model writing:

[0020]

[0021] In the formula, c is the objective function, and E H 1111 E H1122 E H 1212 and E 0 1111 E 0 1122 E 0 1212 These are the three free components of the actual elastic tensor and the target elastic tensor of the structure in two dimensions, respectively. U is the structural displacement field, F is the external force, K is the stiffness matrix, and V(x) is the volume fraction. It is the average relative density of the unit cell, f max It is the maximum allowed volume fraction, where i is the element number and N is the total number of elements in the finite element subdivision;

[0022] Taking the first derivative of the objective function, we obtain its sensitivity expression:

[0023]

[0024]

[0025] dc is the first derivative of the objective function c, dE H 1111 dE H 1122 dE H 1212 It is E H 1111 E H 1122 E H 1212 The first derivative of , dx is the first derivative of the density variable x;

[0026] The density variable is iterated using the moving asymptote method, and the checkerboard effect is suppressed using the sensitivity filtering technique. The convergence condition is that the change of the objective function c between two iterations is less than a given value. The iteration stops when the iteration condition is met or the number of iterations reaches the specified maximum number of iterations, loopmax.

[0027] Step 2) specifically refers to:

[0028] 2.1) Adjust the superstructure inverse homogenization topology optimization program on the MATLAB platform, reserving input and output parameters for interfacing with the design automation platform ISIGHT;

[0029] 2.2) The design automation platform ISIGHT calls the MATLAB module and the optimization module. The MATLAB module is used to call the MATLAB platform. In ISGHT, the interface parameters corresponding to the control parameters of the superstructure inverse homogenization topology optimization program are defined.

[0030] 2.3) In the optimization module of the ISIGHT platform, call the superstructure inverse homogenization topology optimization program, set the penalty factor, sensitivity filter radius, and radius of the circular hole in the initial configuration as optimization variables, take the number of iterations at the end of the topology optimization program as less than the maximum number of iterations loopmax as the optimization constraint, and minimize the volume fraction of solid material in the superstructure. In the optimization module, set the number of calculation steps, number of islands, and mutation rate parameters, etc. The optimization module saves the topology optimization interface parameters and topology optimization results after each step.

[0031] In step 3), among all the topology optimization convergent structures obtained by continuous calculation and solution on the ISIGHT platform, a superstructure configuration with good manufacturability and regular shape is selected based on the criteria of clear boundaries, simple and regular shape, few gray units, and no checkerboard phenomenon. The structure template is then constructed using the level set function.

[0032] The level set expression Φ of the superstructure template c (n,t):

[0033]

[0034] In the formula, n is the coordinate vector of each discrete element in the hyperstructure design domain, and t is the cutting function C. c The "time" variable of (n,t), D c Ω represents the entire design domain. c It is the internal region of the superstructure. It is a superstructure boundary;

[0035] Φ p (n) is the signed distance function:

[0036]

[0037] C c (n,t) is the cutting function, C c (n,t) and Φ p The intersection of (n) in high-dimensional space is the shape of the superstructure. b These are the unit coordinates at the structural boundary.

[0038] For each superstructure, define a "height" variable h, then the global height vector H(t) of the lattice is expressed as:

[0039] H(t)=[h1(t) h2(t) h3(t) ... h m (t)] T

[0040] In the formula, m is the total number of hyperstructures, and h(t) is the height function of each hyperstructure. For a hyperstructure in a two-dimensional configuration, if it is connected to four surrounding structures, then when calculating the cutting function of a single hyperstructure, the relevant height vector h is... c (t) Writing:

[0041] h c (t)=[h1(t) h2(t) h3(t) h4(t)] T =S c H(t)

[0042] In the formula S c It is the selection matrix for choosing superstructures surrounding the current superstructure. The cutting function is obtained by interpolating the "height" of each superstructure, and the interpolation function N is... c (x) is represented as:

[0043] N c (n)=[N1(n) N2(n) N3(n) N4(n)] T

[0044] In the formula, N1(n), N2(n), N3(n), and N4(n) are the interpolation functions of the four surrounding superstructures, respectively;

[0045] Then the cutting function C c The expression for (n,t) is:

[0046]

[0047] When the cutting function C at the superstructure boundary c When (n,t) is continuous, the structure has at least C0 continuity.

[0048] Step 4) establishing the functional relationship between the level set control parameters and the equivalent performance tensor components specifically involves: uniformly selecting several discrete points within the range of the superstructure control parameter h, substituting them into the level set expression to obtain the corresponding superstructure, and using numerical homogenization theory to solve the equivalent elastic tensor of each superstructure. Data fitting is performed on each free component of the tensor to obtain the functional relationship between each tensor component and the control parameters, thereby realizing a one-to-one mapping between the control parameters, the superstructure, and the macroscopic equivalent performance.

[0049] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0050] Because this invention constructs an inverse homogenization topology optimization method for heterogeneous structured material superstructures and combines it with the ISIGHT platform optimization module for automatic continuous calculation, it eliminates the empirical dependence of the design results on penalty factors, filtering radii, and initial configurations during the topology optimization process. From a large number of automatically designed results, the most suitable result is selected based on the manufacturability of the structure, saving time spent manually adjusting optimization parameters to obtain manufacturable superstructures. For superstructures with simple structures and good manufacturability, this invention extracts their shape expressions using the level set method and obtains the macroscopic equivalent performance of all superstructures corresponding to this template through discrete point data fitting, achieving a one-to-one mapping between control parameters, superstructures, and equivalent elastic tensors. Therefore, it effectively improves the efficiency of designing superstructures with specific performance characteristics. Attached Figure Description

[0051] Figure 1 This is a flowchart of an embodiment of the present invention.

[0052] Figure 2 This embodiment of the invention describes the interface for calling MATLAB modules and optimization modules on the ISIGHT design automation platform.

[0053] Figure 3 The equivalent elastic tensor of this embodiment of the invention is Several different superstructures with the same macroscopic equivalent properties were obtained through automatic continuous calculation.

[0054] Figure 4 This is a three-dimensional level set function image of the two-dimensional superstructure template in an embodiment of the present invention.

[0055] Figure 5 This is a CO continuous connection image between superstructures with the same template but different "heights" in embodiments of the present invention.

[0056] Figure 6 This represents the functional relationship between the components of the equivalent elastic tensor of the superstructure template in this embodiment of the invention and the control parameters. Detailed Implementation

[0057] To better illustrate the technical solution, objectives, and advantages of the present invention, the present invention will be further described below in conjunction with the accompanying drawings and embodiments. Furthermore, the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.

[0058] like Figure 1 As shown, a method for generating heterogeneous superstructure templates based on implicit expression includes the following steps:

[0059] 1) Combining numerical homogenization with variable density topology optimization, a superstructure inverse homogenization topology optimization method is constructed;

[0060] This embodiment uses a structure with a circular hole in the center of a rectangular design domain as the initial configuration for topology optimization, and constructs a superstructure inverse homogenization topology optimization method. Homogenization theory separates the macroscopic and microscopic scales of the structure, and calculates the macroscopic equivalent material properties through microstructural analysis. According to numerical homogenization theory, the equivalent elastic tensor E of the superstructure is... H ijkl It can be represented as:

[0061]

[0062] In the formula, V is the entire superstructure space, and E pqrs It is a local elastic tensor. It is a macroscopic strain field. The local strain field is obtained by applying test strain to each element; after finite element discretization, the equivalent elastic tensor E is solved. H ij The expression is:

[0063]

[0064] In the formula, N is the total number of discrete elements, e is the element number, and k is the number of discrete elements. e The element stiffness matrix, It is a macroscopic displacement field. It is a microscopic displacement field, V e It is a hyperstructure space described by the finite element method;

[0065] Continuously varying element material properties are obtained by interpolation using the element relative density x, which varies continuously between 0 and 1, where 0 represents voids and 1 represents solid materials. A penalty factor p is used to force the material properties to approximate those of solid materials or voids. The element elastic modulus is calculated by the following formula:

[0066] E(x)=E min +(E max -E min )x p

[0067] In the formula, E(x) is the element elastic modulus obtained by interpolation, E min To account for the elastic properties of the hole, a decimal close to 0 is used to prevent matrix singularities, E. max is the elastic modulus of the solid phase, x is the relative density of the element, and p is the penalty factor;

[0068] The material uses a two-dimensional orthogonal anisotropic tensor E 0 For ideal performance, the optimization model can be written as:

[0069]

[0070] In the formula, c is the objective function, and E H1111 E H 1122 E H 1212 and E 0 1111 E 0 1122 E 0 1212 These are the three free components of the actual elastic tensor and the target elastic tensor of the structure in two dimensions, respectively. U is the structural displacement field, F is the external force, K is the stiffness matrix, and V(x) is the volume fraction. It is the average relative density of the unit cell, f max It is the maximum allowed volume fraction, where i is the element number and N is the total number of elements in the finite element subdivision;

[0071] Taking the first derivative of the objective function, we obtain its sensitivity expression:

[0072]

[0073]

[0074] dc is the first derivative of the objective function c, dE H 1111 dE H 1122 dE H 1212 It is E H 1111 E H 1122 E H 1212 The first derivative of , dx is the first derivative of the density variable x;

[0075] After solving for the objective function and its sensitivity, and the constraint function and its sensitivity, the density variable is iterated using the moving asymptote method. Sensitivity filtering is used to suppress the checkerboard phenomenon. The convergence condition is that the change of the objective function c between two iterations is less than a given value. The iteration stops when the iteration condition is met or the number of iterations reaches the specified maximum number of iterations, loopmax.

[0076] 2) For the target performance tensor of the superstructure, determine the discretization of the topology optimization design domain, the magnitude of loads and constraints, and boundary conditions such as material properties according to requirements; call the MATLAB module and optimization module on the design automation platform ISIGHT. The MATLAB module is used to call the MATLAB platform. In ISGHT, the topology optimization penalty factor, filter radius, and initial configuration are used as optimization variables, and minimizing the volume fraction of the superstructure solid material is used as the optimization objective to calculate multiple superstructures with the same equivalent performance; the specific steps include:

[0077] 2.1) Adjust the inverse homogenization topology optimization program on the MATLAB platform, reserving input and output parameters for interface with the design ISIGHT platform. Input parameters include: penalty factor, sensitivity filter radius, and radius of the circular hole in the initial configuration. Output parameters include: number of iterations and volume fraction of solid material.

[0078] 2.2) The ISIGHT automation platform is used to call the MATLAB module and the optimization module. The MATLAB module is used to call the MATLAB platform. In ISIGHT, the interface parameters corresponding to the control parameters of the superstructure inverse homogenization topology optimization method are defined.

[0079] 2.3) In the optimization module, the penalty factor, sensitivity filter radius, and radius of the circular hole in the initial configuration are set as optimization variables. The optimization constraint is that the number of iterations at the end of the topology optimization program is less than the maximum number of iterations loopmax. The optimization objective function is to minimize the volume fraction of solid material in the superstructure. The number of calculation steps, number of islands, and mutation rate are set in the optimization module. The optimization module saves the topology optimization interface parameters and topology optimization results after each step.

[0080] 3) Among all the topology optimization convergent structures obtained by continuous optimization on the ISIGHT platform, the superstructure configuration with simple structural features is selected based on the criteria of clear boundaries, simple and regular shape, few gray units, and no checkerboard phenomenon. The structural template is constructed using the level set function.

[0081] Extract the level set expression Φ of its structural shape c (n,t):

[0082]

[0083] In the formula, n is the coordinate vector of each discrete element in the hyperstructure design domain, and t is the cutting function C. c The "time" variable of (n,t), D c Ω represents the entire design domain. c It is the internal region of the superstructure. It is a superstructure boundary; Φ p (n) is the signed distance function:

[0084]

[0085] C c (n,t) is the cutting function, C c (n,t) and Φ p The intersection of (n) in high-dimensional space is the shape of the superstructure. b These are the unit coordinates at the structural boundary;

[0086] When connecting superstructures corresponding to the same type of level set functions, to prevent boundary mismatch between adjacent superstructures from degrading the performance of the entire lattice structure, a "height" variable h is defined for each superstructure. Then, the global height vector H(t) of the lattice can be expressed as:

[0087] H(t)=[h1(t) h2(t) h3(t) … h m (t)] T

[0088] In the formula, m is the total number of hyperstructures, and h(t) is the height function of each hyperstructure. For a hyperstructure in a two-dimensional configuration, if it is connected to four surrounding structures, then when calculating the cutting function of a single structure, the relevant height vector h is... c (t) Writing:

[0089] h c (t)=[h1(t) h2(t) h3(t) h4(t)] T =S c H(t)

[0090] In the formula S c It is a selection matrix that selects the structures surrounding the current superstructure. The cutting function is obtained by interpolating the "height" of each superstructure. The interpolation function N c (x) is represented as:

[0091] N c (n)=[N1(n) N2(n) N3(n) N4(n)] T

[0092] In the formula, N1(n), N2(n), N3(n), and N4(n) are the interpolation functions of the four surrounding superstructures, respectively;

[0093] Then the cutting function C c The expression for (n,t) is:

[0094]

[0095] When the cutting function C at the superstructure boundary c When (n,t) is continuous, the structure has at least C0 continuity;

[0096] 4) Select several discrete points uniformly within the range of the superstructure control parameter h, substitute them into the level set expression to obtain the corresponding superstructure, and use numerical homogenization theory to solve the equivalent elastic tensor of each superstructure. Perform data fitting on each free component of the tensor to obtain the functional relationship between each tensor component and the control parameter, realize the one-to-one mapping between control parameters, superstructure, and macroscopic equivalent performance. For a specific target elastic tensor, use this mapping relationship to efficiently obtain a superstructure with satisfactory performance and good manufacturability.

[0097] In this embodiment, the square design domain is discretized into 100×100, the Young's modulus of the solid material is set to 1, and the Poisson's ratio is set to 0.3. Using the macroscopically equivalent elastic tensor as the target, a superstructure inverse homogenization topology optimization method was constructed. Then, a large number of different superstructures were obtained through continuous computation using ISIGHT. Figure 2 It is a model combining the ISIGHT platform's joint optimization module and the MATLAB module. Figure 3 Images of three superstructures are given. The second structure is selected to extract the superstructure template, resulting in... Figure 4 The distance function shown; for two superstructures with the same template but different "heights", the cutting function is made continuous at the boundary to improve the continuity between structures, resulting in... Figure 5 The structure shown; discrete points are taken within the range of the level set control parameters, and homogenized to obtain its equivalent elastic tensor. The three free components E of the elastic tensor of the orthogonal anisotropic structure are then analyzed. 1111 E 1122 E 1212 Perform polynomial fitting to obtain the functional relationship between each component and the control parameter, and plot the relationship, as shown below. Figure 6 As shown.

[0098] The above-described embodiments were completed under specific design conditions. This invention should include, but is not limited to, the described embodiments. Various changes made by those skilled in the art based on the disclosed design ideas and spirit of this invention, without departing from the essence of this invention, are still within the scope of protection of this invention.

Claims

1. A method for generating heterogeneous superstructure templates based on implicit expression, characterized in that, Includes the following steps: 1) Combining numerical homogenization theory with variable density topology optimization, a superstructure inverse homogenization topology optimization method is constructed; 2) By combining the superstructure inverse homogenization topology optimization method with the design automation platform ISIGHT, using optimization parameters as variables and minimizing the volume of solid material as the objective function, different superstructure configurations with the same equivalent elastic tensor are obtained; specifically: 2.1) Adjust the inverse homogenization topology optimization program on the MATLAB platform, reserving input and output parameters for interfacing with the design automation platform ISIGHT; 2.2) The design automation platform ISIGHT calls the MATLAB module and the optimization module. The MATLAB module is used to call the MATLAB platform. In ISGHT, the interface parameters corresponding to the control parameters of the superstructure inverse homogenization topology optimization program are defined. 2.3) In the optimization module of the ISIGHT platform, call the superstructure inverse homogenization topology optimization program, set the penalty factor, sensitivity filter radius, and radius of the circular hole in the initial configuration as optimization variables, take the number of iterations at the end of the topology optimization program as less than the maximum number of iterations loopmax as the optimization constraint, the optimization objective function is to minimize the volume fraction of solid material in the superstructure, and set the calculation steps, island number and mutation rate parameters in the optimization module. The optimization module saves the topology optimization interface parameters and topology optimization results after each step. 3) Select a superstructure configuration with good manufacturability and regular shape, use a horizontal set function to construct a superstructure template, and achieve good connection between adjacent structures by controlling the cut function values ​​between structures to be equal; 4) Establish the functional relationship between the level set control parameters and the equivalent performance tensor components; for specific material performance requirements, inversely calculate the level set control parameters based on the functional relationship; The establishment of the functional relationship between the level set control parameters and the equivalent performance tensor components specifically involves: [The text abruptly shifts to a different topic] ...in the superstructure control parameters... h Several discrete points are uniformly selected within the range of values, and the corresponding superstructures are obtained by substituting them into the level set expression. The equivalent elastic tensor of each superstructure is solved using numerical homogenization theory. Data fitting is performed on each free component of the tensor to obtain the functional relationship between each tensor component and the control parameters, thereby realizing a one-to-one mapping between control parameters, superstructures, and macroscopic equivalent performance.

2. The method according to claim 1, characterized in that, In step 1), the equivalent elastic tensor of the superstructure in the superstructure inverse homogenization topology optimization method is constructed. E H ijkl Represented as: In the formula V It is the entire superstructure space. E pqrs It is a local elastic tensor. , It is a macroscopic strain field. , The local strain field is obtained by applying test strain to each element; after finite element discretization, the equivalent elastic tensor is solved. E H ij The expression is: In the formula N The total number of units obtained by discretization. e k is the unit number. e The element stiffness matrix, , It is a macroscopic displacement field. , It is a microscopic displacement field. V e It is a superstructure space described by the finite element method.

3. The method according to claim 2, characterized in that, In step 1), the element elastic modulus in the superstructure inverse homogenization topology optimization method is calculated by the following formula: In the formula E ( x () represents the element elasticity tensor obtained through interpolation. E min To account for the elastic properties of the holes, a decimal close to 0 is used to prevent matrix singularities. E max It is the elastic modulus of the solid phase. x It is the relative density of the unit cell. p As a penalty factor; Material with two-dimensional orthogonal anisotropic tensor E 0 For ideal performance, optimize the model writing: In the formula c Let be the objective function. E H 1111 , E H 1122 , E H 1212 and E 0 1111 , E 0 1122 , E 0 1212 These are the three free components of the actual elastic tensor and the target elastic tensor of the structure in two dimensions. U For structural displacement field, F As an external force, K Here is the stiffness matrix. V ( x () represents the volume fraction. It is the average relative density of the unit cells. f max It is the maximum allowed volume fraction. i For unit numbering, N The total number of elements in the finite element method; Taking the first derivative of the objective function, we obtain its sensitivity expression: DC It is the objective function c The first derivative, dE H 1111 , dE H 1122 , dE H 1212 yes E H 1111 , E H 1122 , E H 1212 The first derivative, dx It is a density variable x The first derivative.

4. The method according to claim 3, characterized in that, The density variable is iterated using the moving asymptote method, and the checkerboard effect is suppressed using sensitivity filtering. The convergence condition is the objective function. c The iteration stops when the change between two iterations is less than a given value, or when the iteration condition is met or the number of iterations reaches the specified maximum number of iterations, loopmax.

5. The method according to claim 1, characterized in that, In step 3), among all the topology optimization convergent structures obtained by ISIGHT continuous solution, a superstructure configuration with good manufacturability and regular shape is selected based on the criteria of clear boundaries, simple and regular shape, few gray units, and no checkerboard phenomenon. The structure template is constructed using the level set function.

6. The method according to claim 5, characterized in that, The level set expression of the hyperstructure template Φ c ( n , t ): In the formula, n It is the coordinate vector of each discrete element in the superstructure design domain. t For cutting function C c ( n , t The "time" variable, D c Represents the entire design domain. Ω c It is the internal region of the superstructure. Ω c It is a superstructure boundary; Φ p ( n ) is the signed distance function: C c ( n , t ) is the cutting function. C c ( n , t )and Φ p ( n The intersection of these elements in higher-dimensional space constitutes the shape of the superstructure. n b These are the unit coordinates at the structural boundary.

7. The method according to claim 6, characterized in that, Define a "height" variable for each superstructure. h Then the global height vector of the lattice H ( t ) is represented as: In the formula m It is the total number of superstructures. h ( t Let be the height function of each superstructure. For a superstructure in a two-dimensional configuration, if it is connected to four surrounding structures, then when calculating the cut function of a single structure, the relevant height vector is... h c ( t )writing: In the formula S c It is a selection matrix that selects the structures surrounding the current superstructure. The cutting function is obtained by interpolating the "height" of each superstructure. The interpolation function is... N c ( x ) is represented as: In the formula N 1( n ), N 2( n ), N 3( n ), N 4( n These are the interpolation functions for the four surrounding superstructures; Then the cutting function C c ( n , t The expression for ) is: When the cutting function at the superstructure boundary C c ( n , t When continuous, the structure has at least C0 continuity.

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