An intelligent design method for multi-scale optimization of special-shaped structures

Abstract

The present invention relates to a multi-scale optimization intelligent design method for special-shaped structures, belonging to the field of additive manufacturing of spacecraft structures and mechanisms. It includes: constructing a special-shaped lattice structure using a level set function and performing interpolation and correction; setting relative density sample points and performing polynomial interpolation to establish a geometric surrogate model; obtaining a constitutive matrix through numerical homogenization and establishing a constitutive surrogate model; based on the constitutive surrogate model and the isogeometric topology optimization method, iteratively solving with the goal of minimizing the compliance of the special-shaped sandwich structure to obtain the optimal density distribution; substituting the optimal density distribution into the geometric surrogate model to output interpolation coefficients, and generating an implicit model of the special-shaped sandwich structure by gradient lattice filling. This method solves the problem of the lack of a multi-scale optimization design method for special-shaped sandwich structures filled with gradient lattices by constructing an implicit model of a special-shaped sandwich structure filled with a conforming continuous gradient special-shaped lattice structure and generating a special-shaped sandwich structure by additive manufacturing based on the implicit model.

Description

Technical Field

[0001] The present invention relates to the field of additive manufacturing of spacecraft structures and mechanisms, and particularly to an intelligent design method for multi-scale optimization of special-shaped structures. Background Art

[0002] With the increasingly urgent weight reduction requirements for spacecraft models, the application of lightweight structures based on additive manufacturing in spacecraft has gradually increased. The special-shaped sandwich structure is a typical lightweight structure with high stiffness and strength, and is widely used in the spacecraft field. The performance of the special-shaped sandwich structure mainly depends on the type of the sandwich, including random sandwiches (such as foams) and periodic sandwiches (such as lattices). Although random sandwiches are easy to manufacture, their mechanical properties are unpredictable, while periodic sandwiches are composed of repeatedly arranged porous units, so they have predictable mechanical properties. By adjusting the geometric shape and distribution of the lattice, better performance control can be achieved.

[0003] With the development of additive manufacturing technology, the design space of special-shaped sandwich structures has been greatly expanded. However, the existing sandwiches are usually designed based on regular shapes and are difficult to be directly applied to special-shaped sandwich structures; and when designing periodic sandwiches, they are usually based on classical orthogonal envelope lattices, there are discontinuous connections between the lattices, and there is a geometric mismatch between the lattices and the special-shaped sandwich structure, which not only leads to an increase in the local stress concentration and a decrease in strength of the structure, but also affects the overall performance of the structure.

[0004] It can be seen that providing an intelligent design method for multi-scale optimization of special-shaped structures to realize the optimized design and preparation of gradient lattice-filled special-shaped sandwich structures is an urgent problem to be solved. Summary of the Invention

[0005] In view of the above analysis, the present invention aims to provide an intelligent design method for multi-scale optimization of special-shaped structures to solve the problem of the lack of an optimized design and preparation method for gradient lattice-filled special-shaped sandwich structures.

[0006] The present invention provides an intelligent design method for multi-scale optimization of special-shaped structures, and the method includes the following steps:

[0007] Construct a special-shaped lattice structure by using a level set function, and obtain a conforming gradient special-shaped lattice structure through interpolation and correction of the special-shaped lattice structure; set relative density sample points, calculate the relationship between the interpolation coefficients and the relative density sample points based on the conforming gradient special-shaped lattice structure, and establish a geometric surrogate model by performing polynomial interpolation on the relative density sample points; obtain the constitutive matrix corresponding to the relative density sample points through numerical homogenization based on the conforming gradient special-shaped lattice structure, and establish a constitutive surrogate model by performing polynomial interpolation on the relative density sample points;

[0008] Based on the constitutive surrogate model and the isogeometric topology optimization method, iteratively solve with the goal of minimizing the compliance of the special-shaped sandwich structure to obtain the optimal density distribution;

[0009] Substitute the optimal density distribution into the geometric surrogate model to output the interpolation coefficients, correct the interpolation coefficients through the NURBS basis functions, and generate the implicit model of the special-shaped sandwich structure by gradient lattice filling.

[0010] Furthermore, construct the geometric space of the special-shaped lattice structure, obtain the constitutive matrix corresponding to the interpolated relative density sample points through polynomial interpolation of the relative density sample points, construct the constitutive matrix space of the special-shaped lattice structure, and determine the mapping relationship from the geometric space to the constitutive matrix space to establish the constitutive surrogate model.

[0011] Furthermore, input the constitutive surrogate model into CAD software, construct an isogeometric model based on the NURBS basis functions, refine the isogeometric model through k-refinement, define the boundary conditions and constraints, establish a density distribution function, establish an objective function with the goal of minimizing the compliance of the special-shaped sandwich structure, perform sensitivity analysis based on the density distribution function, and iteratively solve the objective function through the MMA algorithm to obtain the optimal density distribution.

[0012] Furthermore, calculate the relationship between the interpolation coefficients, the aspect ratio, and the relative density sample points through formulas (8)-(10):

[0013] , (8)

[0014] , (9)

[0015] , (10)

[0016] where is the Heaviside function, is the relative density of the conforming gradient special-shaped lattice structure, is the aspect ratio of the conforming gradient special-shaped lattice structure, is the interpolation coefficient of the conforming gradient special-shaped lattice structure, is the polynomial interpolation coefficient, is the interpolation order, is the maximum interpolation order, represents the level set function, represents the design domain;

[0017] The expression of the geometric surrogate model is as follows:

[0018] , (11)

[0019] Among them, represents a geometric surrogate model.

[0020] Furthermore, the implicit modeling expression of a single special-shaped lattice structure is:

[0021] , (3)

[0022] Among them, Void represents the void of the special-shaped lattice structure, Boundary represents the boundary of the special-shaped lattice structure, and Solid represents the area where the special-shaped lattice is filled with material.

[0023] Furthermore, substitute the optimal density distribution into the interpolation coefficients output by the geometric surrogate model established by formulas (10) and (11), correct the interpolation coefficients through the NURBS basis function to obtain the corrected interpolation coefficients, and construct a conformal continuous gradient special-shaped lattice structure based on the corrected interpolation coefficients and the conformal gradient special-shaped lattice structure using the level set function. Output the implicit model of the special-shaped sandwich structure filled with the conformal continuous gradient special-shaped lattice structure through formula (3).

[0024] Furthermore, the corrected interpolation coefficients are obtained through the following formula:

[0025] , (20)

[0026] Among them, is the corrected interpolation coefficient.

[0027] Furthermore, a conformal continuous gradient special-shaped lattice structure is constructed through the following formula:

[0028] ,

[0029] , (21)

[0030] Among them, is the global level set function of the conformal continuous gradient special-shaped lattice structure, C is the Cth special-shaped lattice structure, N is the number of special-shaped lattice structures, and X is the set of all special-shaped lattice structures in physical space.

[0031] Furthermore, establish a geometric surrogate model by calculating the relationship between the interpolation coefficients, aspect ratio, and interpolated relative density through polynomial interpolation of relative density sample points.

[0032] Furthermore, based on the generated implicit model, generate a special-shaped sandwich structure through additive manufacturing.

[0033] Compared with the prior art, the present invention can at least achieve one of the following beneficial effects:

[0034] 1. The present invention constructs an implicit model of a special-shaped sandwich structure filled with a conformal continuous gradient special-shaped lattice structure, and generates a special-shaped sandwich structure by additive manufacturing based on the implicit model, solving the problem of the lack of a multi-scale optimization design method for gradient lattice-filled special-shaped sandwich structures.

[0035] 2. The present invention extends the density distribution of a small number of control points to the density distribution of a large number of parameter points through an interpolation coefficient, corrects the interpolation coefficient through a NURBS basis function, and updates the LSF function based on the corrected interpolation coefficient, realizing a smooth transition between local LSF functions, thereby realizing the continuous connection between lattice structures, effectively solving the connectivity problem between gradient lattice structures under orthogonal envelopes, and significantly improving the quality of the special-shaped sandwich structure design and its manufacturability in engineering.

[0036] 3. The present invention constructs a constitutive surrogate model, which can quickly evaluate the mechanical properties of special-shaped lattice structures under different key geometric parameters, avoiding calculating the constitutive matrix of each lattice one by one, reducing the calculation amount, and improving the calculation efficiency. Moreover, through polynomial interpolation, efficient and accurate prediction of the mechanical properties of special-shaped lattice structures with different relative densities, aspect ratios, and deflection angles is realized.

[0037] 4. The present invention implicitly constructs special-shaped lattice structures of various shapes through a level set function. Not only can special-shaped lattice structures of various topological forms be obtained, but also gradient special-shaped lattice structures with the same topological configuration but different characteristic sizes can be obtained through interpolation. The present invention constructs a conformally varying special-shaped lattice structure by correcting the level set function corresponding to the gradient special-shaped lattice structure through a NURBS basis function, realizing the adaptive matching between the lattice structure and the special-shaped sandwich structure.

[0038] In the present invention, the above technical solutions can also be combined with each other to realize more preferred combination schemes. Other features and advantages of the present invention will be described in the subsequent specification, and some advantages can be made obvious from the specification or understood by implementing the present invention. The objectives and other advantages of the present invention can be realized and obtained through the content specifically pointed out in the specification and the drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] The drawings are only for the purpose of showing specific embodiments and are not considered as limiting the present invention. Throughout the drawings, the same reference numerals represent the same components;

[0040] Figure 1 is a flowchart of the intelligent design method for multi-scale optimization of the special-shaped structure in the embodiment of the present invention;

[0041] Figure 2 is a schematic diagram of constructing a special-shaped lattice structure using the LSF function in the embodiment of the present invention;

[0042] Figure 3 Figure (a) is a schematic diagram of the special-shaped BCC lattice structure and its envelope domain according to an embodiment of the present invention; Figure 3 Figure (b) is a schematic diagram of the key geometric feature parameters of the special-shaped BCC lattice structure according to an embodiment of the present invention;

[0043] Figure 4 is the constitutive matrix corresponding to the special-shaped BCC lattice structure at a relative density of 30% according to an embodiment of the present invention;

[0044] Figure 5 Figure (a) is a schematic diagram of the geometric space of the special-shaped BCC lattice structure according to an embodiment of the present invention; Figure 5 Figure (b) is a schematic diagram of the constitutive surrogate model according to an embodiment of the present invention; Figure 5 Figure (c) is an embodiment of the present invention 、 The mapping relationship from the geometric space of the special-shaped BCC lattice structure to the constitutive matrix space; Figure 5 Figure (d) is an embodiment of the present invention 、 The mapping relationship from the geometric space of the special-shaped BCC lattice structure to the constitutive matrix space;

[0045] Figure 6 Figure (a) is the global LSF function of the discontinuous special-shaped BCC lattice structure according to an embodiment of the present invention; Figure 6 Figure (b) is a schematic diagram of the control point density distribution of the arc sandwich structure and the parameter point density distribution before interpolation according to an embodiment of the present invention; Figure 6 Figure (c) is the global LSF function of the connected special-shaped BCC lattice structure according to an embodiment of the present invention; Figure 6 Figure (d) is a schematic diagram of the discontinuous special-shaped BCC lattice structure according to an embodiment of the present invention; Figure 6 Figure (e) is a schematic diagram of the connected special-shaped BCC lattice structure according to an embodiment of the present invention; Figure 6 Figure (f) is a schematic diagram of the control point density distribution of the three-dimensional arc sandwich structure and the parameter point density distribution after interpolation according to an embodiment of the present invention; Figure 6 Figure (g) is a schematic diagram of the conforming continuous gradient BCC lattice structure filled in the arc sandwich structure according to an embodiment of the present invention; Figure 6 Figure (h) is a schematic diagram of the plate-based BCC lattice structure filled in the arc sandwich structure according to an embodiment of the present invention;

[0046] Figure 7 Figure (a) is the design domain and boundary conditions of the three-point bending semi-circular arc sandwich structure according to an embodiment of the present invention; Figure 7 Figure (b) is the uniform special-shaped lattice structure according to an embodiment of the present invention; Figure 7 Figure (c) is a partial enlarged view of the gradient special-shaped lattice structure along the YOZ plane according to an embodiment of the present invention;Figure 7 Figure (d) is a partial enlarged view of the gradient-shaped lattice structure according to an embodiment of the present invention along the XOY plane; Figure 7 Figure (e) is an overall schematic diagram of the gradient-shaped lattice structure according to an embodiment of the present invention; Figure 7 Figure (f) is a schematic diagram of the gradient-shaped lattice structure according to an embodiment of the present invention with the first layer of the lattice structure removed; Figure 7 Figure (g) is a schematic diagram of the gradient-shaped lattice structure according to an embodiment of the present invention after removing 1 / 4 of the overall lattice structure;

[0047] Figure 8 Figure (a) is the output of the gradient lattice structure optimization model according to an embodiment of the present invention; Figure 8 Figure (b) is the three-point bending test and DIC characterization of the orthogonal uniform lattice structure according to an embodiment of the present invention; Figure 8 Figure (c) is the three-point bending test and DIC characterization of the shaped uniform lattice structure according to an embodiment of the present invention; Figure 8 Figure (d) is the three-point bending test and DIC characterization of the shaped gradient uniform lattice structure according to an embodiment of the present invention; Figure 8 Figure (e) is the force-displacement curves of the semi-circular orthogonal uniform, shaped uniform, and shaped gradient lattice structures according to an embodiment of the present invention; Figure 8 Figure (f) is the bending modulus and ultimate strength of the semi-circular orthogonal uniform, shaped uniform, and shaped gradient lattice structures according to an embodiment of the present invention;

[0048] Figure 9 Schematic diagram of the design, preparation, and test of the shaped gradient lattice structure of the rudder plate according to an embodiment of the present invention;

[0049] Figure 10 Figure (a) is the test structure of the orthogonal uniform lattice structure according to an embodiment of the present invention; Figure 10 Figure (b) is the test structure of the shaped gradient lattice structure according to an embodiment of the present invention; Figure 10 Figure (c) is the internal schematic diagram of the orthogonal uniform lattice rudder plate and the fracture diagram of the rudder shaft according to an embodiment of the present invention; Figure 10 Figure (d) is the internal schematic diagram of the shaped gradient lattice rudder plate and the enlarged view of the rudder shaft according to an embodiment of the present invention. Detailed implementation manners

[0050] The preferred embodiments of the present invention will be specifically described below with reference to the accompanying drawings. The accompanying drawings form a part of this application and are used together with the embodiments of the present invention to explain the principles of the present invention, rather than to limit the scope of the present invention.

[0051] Embodiment 1

[0052] A specific embodiment of the present invention discloses a multi-scale optimization intelligent design method for a shaped structure. As Figure 1 shown, the method includes the following steps:

[0053] S1. Construct a special-shaped lattice structure using a level set function. Obtain a conforming gradient special-shaped lattice structure by interpolating and correcting the special-shaped lattice structure. Set relative density sample points, calculate the relationship between the interpolation coefficients and the relative density sample points based on the conforming gradient special-shaped lattice structure, and establish a geometric surrogate model by performing polynomial interpolation on the relative density sample points. Based on the conforming gradient special-shaped lattice structure, obtain the constitutive matrix corresponding to the relative density sample points through numerical homogenization, and establish a constitutive surrogate model by performing polynomial interpolation on the relative density sample points.

[0054] S2. Based on the constitutive surrogate model and the isogeometric topology optimization method, perform iterative solution with the goal of minimizing the compliance of the special-shaped sandwich structure to obtain the optimal density distribution.

[0055] S3. Substitute the optimal density distribution into the geometric surrogate model to output the interpolation coefficients, correct the interpolation coefficients through NURBS basis functions, and generate an implicit model of the special-shaped sandwich structure by gradient lattice filling.

[0056] Specifically, in step S1, the expression for constructing a single special-shaped lattice structure using a local level set function is:

[0057] , (1)

[0058] where represents the set of physical space coordinate points of a single special-shaped lattice structure, represents the sampling points of the local level set function, represents the local level set function corresponding to a single special-shaped lattice structure, represents the boundary of a single special-shaped lattice structure, represents defining the level set function using the signed distance function.

[0059] Specifically, interpolate the special-shaped lattice structure through the following formula:

[0060] , (2)

[0061] where is the interpolation coefficient.

[0062] Specifically, the implicit modeling expression of a single special-shaped lattice structure obtained from formulas (1) and (2) is:

[0063] , (3)

[0064] where Void represents the void of the special-shaped lattice structure, Boundary represents the boundary of the special-shaped lattice structure, and Solid represents the region where the special-shaped lattice is filled with material.

[0065] It should be noted that the special-shaped sandwich structure is filled with a large number of lattice structures. The lattice structure is the basic building unit of the special-shaped sandwich structure, and the modeling method of the lattice structure has a significant impact on the performance of the special-shaped sandwich structure. Therefore, it is crucial to select an appropriate lattice structure modeling method. The present invention adopts an implicit modeling method of a level set function (LSF) constructed based on a signed distance function (SDF). The boundary of the special-shaped lattice structure is implicitly represented by the zero isosurface of the high-dimensional LSF. Compared with the model generated by traditional modeling software, the implicit model has higher accuracy and smaller data volume, and can conveniently realize the batch modeling of a large number of lattice structures, improving the modeling efficiency. The present invention can implicitly construct special-shaped lattice structures of various shapes through the level set function (LSF). Therefore, based on the above formulas (1)-(3), not only can special-shaped lattice structures of various topological forms be obtained, but also gradient special-shaped lattice structures with the same topological configuration but different characteristic sizes can be obtained by interpolation.

[0066] Exemplarily, several typical special-shaped lattice structures are constructed as Figure 2 shown, including the two-dimensional X-shaped structure and X-Box structure, and the three-dimensional body-centered cubic (BCC) structure and face-centered cubic (FCC) structure. Among them, "Prototype LSF" represents the special-shaped lattice structure constructed by using the level set function, and "Update LSF" represents the corresponding special-shaped lattice structure obtained by interpolating the special-shaped lattice structure. From Figure 2 it can be seen that the zero isosurface projection of the LSF function corresponding to "Prototype LSF" is updated after interpolation, so as to generate "Update LSF" with the same topological configuration but different characteristic sizes.

[0067] Furthermore, the constructed and interpolated special-shaped lattice structures are corrected by NURBS basis functions to obtain conforming gradient special-shaped lattice structures.

[0068] Specifically, the expression of the B-spline basis function is:

[0069] , (4)

[0070] where represents the i-th knot of the NURBS basis function; represents the NURBS basis function; represents the knot vector, which is a non-decreasing ordered parameter set; n represents the number of B-spline basis functions, that is, the number of control points; p is the polynomial order.

[0071] It should be noted that the B-spline curve is defined by the B-spline basis function. The numerical range of each B-spline basis function is [0,1]. Therefore, it is non-zero within a specific interval and zero outside this interval, enabling the B-spline basis function to flexibly control the curve. Each parameter in the knot vector of the B-spline basis function represents a knot. All the knots divide the entire domain into multiple sub-intervals, and each sub-interval corresponds to the non-zero interval of one or more B-spline basis functions, thus determining the local control characteristics of the curve.

[0072] Specifically, the expression of the NURBS basis function is:

[0073] , (5)

[0074] It should be noted that the NURBS basis function is obtained by weighted averaging of the B-spline basis function within the unit partition framework. The NURBS (Non-Uniform Rational B-Spline) basis function has stronger generalization ability compared to the B-spline basis function, and can better control the curve shape and improve flexibility.

[0075] It should be noted that the special-shaped lattice structure constructed by the level set function is an orthogonal geometric envelope. When filling the special-shaped sandwich configuration, only the redundant lattice structure can be trimmed through Boolean operations, resulting in a geometric mismatch between the lattice structure and the special-shaped sandwich structure. In the present invention, the NURBS basis function is used to correct formula (1) to construct a special-shaped lattice structure with conformal variation, realizing the adaptive matching between the lattice structure and the special-shaped sandwich structure.

[0076] Exemplarily, according to formula (1), a three-dimensional special-shaped lattice structure is constructed by using the local level set function, and the three-dimensional special-shaped lattice structure is corrected by the NURBS basis function. The expression of the conformal gradient three-dimensional special-shaped lattice structure obtained is:

[0077] , (6)

[0078] , (7)

[0079] Among them, represents the sampling points in the parameter space, represents the physical coordinates corresponding to the sampling points of the local level set function, is the LSF function corresponding to the corrected special-shaped lattice structure. R represents the basis function extended from one dimension to three dimensions in formula (5), and P is the coordinate of the control point.

[0080] Furthermore, relative density sample points are set, and key geometric feature parameters are extracted according to the geometric changes of the conformal gradient heterogeneous lattice structure. The key geometric feature parameters include the relative density, aspect ratio, and deflection angle of the conformal gradient heterogeneous lattice structure. The relationship between the interpolation coefficient and the aspect ratio and relative density sample points is calculated using the Heaviside function.

[0081] Specifically, the relative density sample points refer to the relative density values set according to experience and actual requirements. The relationship between the interpolation coefficient and the aspect ratio and relative density sample points is calculated through formulas (8)-(10).

[0082] , (8)

[0083] , (9)

[0084] , (10)

[0085] wherein, is the Heaviside function, is the relative density of the conformal gradient heterogeneous lattice structure, is the aspect ratio of the conformal gradient heterogeneous lattice structure, is the interpolation coefficient of the conformal gradient heterogeneous lattice structure, is the polynomial interpolation coefficient, is the interpolation order, is the maximum interpolation order, represents the level set function, represents the design domain.

[0086] Preferably, in this application, b is taken as 2.

[0087] Exemplarily, the heterogeneous BCC lattice structure is selected as the conformal gradient heterogeneous lattice structure. The heterogeneous BCC lattice structure and its envelope domain are shown in Figure 3 (a). Based on the geometric changes of the heterogeneous BCC lattice structure, its key geometric feature parameters are obtained, as shown in Figure 3 (b).

[0088] It should be noted that the BCC lattice structure can withstand multi-axial forces from different directions, has high specific stiffness and high specific strength, and is easy to be realized by additive manufacturing processes. As can be seen from Figure 3 (b), the interpolation coefficient is related to the relative density and aspect ratio, and is independent of the deflection angle . Therefore, the relationship between and and is calculated.

[0089] Further, by performing polynomial interpolation on the relative density sample points to calculate the relationship between the interpolation coefficients, aspect ratio, and interpolated relative density, a geometric surrogate model is established. The expression of the geometric surrogate model is as follows:

[0090] , (11)

[0091] where, represents the geometric surrogate model.

[0092] It should be noted that the geometric surrogate model includes the relationship between the interpolation coefficients and the set as well as the interpolated relative density sample points. Therefore, through the geometric surrogate model, the relationship between the interpolation coefficients and any relative density can be obtained.

[0093] Further, based on the conformal gradient anisotropic lattice structure, each component of the elastic tensor of the anisotropic lattice structure is calculated through numerical homogenization, so as to obtain the constitutive matrix corresponding to the anisotropic lattice structure at the relative density sample points.

[0094] Specifically, each component of the elastic tensor is calculated by the following formula:

[0095] , (12)

[0096] where, 、 are custom uniform strain fields, 、 are the true strains generated by the custom strain fields, is the volume of the anisotropic lattice structure, represents the Young's modulus of the anisotropic lattice structure.

[0097] Specifically, the constitutive matrix is an elastic coefficient matrix composed of each component of the elastic tensor.

[0098] Exemplarily, the anisotropic BCC lattice structure is an anisotropic structure with 1 elastic symmetry plane. Based on the anisotropic BCC lattice structure, through numerical homogenization, the constitutive matrix corresponding to it at a relative density of 30% is solved, as Figure 4 shown. From Figure 4 it can be seen that when the angle of the anisotropic BCC lattice structure deflects, tensile-shear coupling occurs, resulting in non-zero elements corresponding to the tensile-shear coupling terms in the constitutive matrix, and the absolute value of the elements corresponding to the tensile-shear coupling terms increases with the increase of the deflection angle. In addition, as the aspect ratio of the anisotropic BCC lattice structure decreases, making it more flattened, it significantly affects the elements corresponding to the second direction in the constitutive matrix.

[0099] Furthermore, the constitutive matrix corresponding to the interpolated relative density sample points is obtained by performing polynomial interpolation on the relative density sample points, thereby establishing a constitutive surrogate model.

[0100] Specifically, a geometric space of the special-shaped lattice structure is constructed, and the constitutive matrix space of the special-shaped lattice structure is constructed by performing polynomial interpolation on the relative density sample points to obtain the constitutive matrix corresponding to the interpolated relative density sample points, and the mapping relationship from the geometric space to the constitutive matrix space is determined to establish a constitutive surrogate model.

[0101] Specifically, the geometric space is a three-dimensional space composed of relative density, aspect ratio, and deflection angle, and each element in this geometric space represents a special-shaped lattice structure with specific geometric characteristics. The expression of the geometric space is as follows:

[0102] ,

[0103] ,

[0104] ,

[0105] , (13)

[0106] where, represents the geometric space of the special-shaped lattice structure.

[0107] Specifically, polynomial interpolation is performed through the following formula:

[0108] , (14)

[0109] where, s is the row in the constitutive matrix; t is the column in the constitutive matrix; m is the order of polynomial interpolation; represents when 、 are fixed, the interpolation coefficient of the sth row, tth column, and (m - r)th term of the constitutive matrix.

[0110] Specifically, each element in the constitutive matrix space represents the constitutive matrix of a special-shaped lattice structure with specific geometric characteristics. The expression of the constitutive matrix space is as follows:

[0111] , (15)

[0112] where, represents the constitutive matrix space of the special-shaped lattice structure.

[0113] Specifically, the expression of the constitutive surrogate model is as follows:

[0114] , (16)

[0115] Among them, represents the constitutive surrogate model.

[0116] Exemplarily, the geometric space of the special-shaped BCC lattice structure is as shown in Figure 5 (a), and the constitutive surrogate model of the special-shaped BCC lattice structure under different relative densities, deflection angles and aspect ratios is predicted by polynomial interpolation as shown in Figure 5 (b). As can be seen from Figure 5 (b), there are differences in the sensitivities of the elements in the constitutive matrix to different geometric parameters. On the premise of the large symmetry of the constitutive matrix, the sensitivities of the main diagonal elements to the aspect ratio and relative density are greater than the deflection angle, while the sensitivities of the elements corresponding to the tensile-shear coupling coefficient to the aspect ratio and relative density are less than the deflection angle. As can be seen from Figure 5 (c) and 5(d), when the aspect ratio and deflection angle are fixed, the mapping relationship from the geometric space of the special-shaped BCC lattice structure to the constitutive matrix space satisfies the continuous differentiable relationship with the relative density.

[0117] It should be noted that if the constitutive matrix of each lattice is repeatedly calculated every time during the multi-scale optimization process of the special-shaped lattice structure, the calculation amount is large and the time consumption is long. Therefore, the present invention can quickly evaluate the mechanical properties of the special-shaped lattice structure under different key geometric parameters by constructing a constitutive surrogate model, avoiding calculating the constitutive matrix of each lattice one by one, reducing the calculation amount and improving the calculation efficiency. And through polynomial interpolation, the mechanical properties of the special-shaped lattice structure with different relative densities, aspect ratios and deflection angles can be predicted efficiently and accurately.

[0118] Specifically, in step S2, the constitutive surrogate model is input into the CAD software, an isogeometric model is constructed based on the NURBS basis function, the isogeometric model is refined by k-refinement, boundary conditions and constraints are defined, a density distribution function is established, an objective function is established with the goal of minimizing the compliance of the special-shaped sandwich structure, sensitivity analysis is performed based on the density distribution function, and the optimal density distribution is obtained by iteratively solving the objective function through the MMA algorithm.

[0119] It should be noted that k-refinement is a local refinement method in isogeometric analysis (IGA). By increasing the order of the NURBS basis function, a specific area of the isogeometric model can be refined without changing the overall topology of the lattice, so as to improve the accuracy and efficiency of isogeometric analysis.

[0120] Specifically, the density distribution function is as follows:

[0121] , (17)

[0122] Among them, represents the density distribution function, , , are the coordinates of the control points in the isogeometric model parameter space, , , , represents the NURBS basis function, represents the density distribution of the control points, where l, m, and n are the maximum number of nodes in the three parameter directions respectively.

[0123] Specifically, an objective function is established with the goal of minimizing the compliance of the special-shaped sandwich structure as follows:

[0124] ,

[0125] ,

[0126] ,

[0127] ,

[0128] , (18)

[0129] Among them, represents the compliance of the special-shaped sandwich structure, is the strain, is the constitutive matrix, is the displacement column vector, is the global force vector of the special-shaped sandwich structure, is the volume fraction of the special-shaped sandwich structure, represents the constitutive stiffness matrix of the special-shaped sandwich structure, is the optimized set volume fraction.

[0130] Specifically, sensitivity analysis is carried out through the following formula:

[0131] ,

[0132] , (19)

[0133] Among them, , , are the coordinates of the Gauss points in the isogeometric model parameter space, represents the stiffness matrix of the special-shaped lattice structure in finite element analysis, represents the displacement vector of the special-shaped lattice structure.

[0134] It should be noted that the control points refer to the control points of the NURBS basis functions. In the present invention, the constitutive matrix is calculated by the Gaussian integration method, so the corresponding Gaussian points are obtained through the control points.

[0135] Specifically, in step S3, the optimal density distribution is substituted into the geometric surrogate model established by formula (10) to output the interpolation coefficients. The interpolation coefficients are corrected by the NURBS basis functions to obtain the corrected interpolation coefficients. Based on the corrected interpolation coefficients and the conforming gradient cellular structure, a conforming continuous gradient cellular structure is constructed using the level set function, and an implicit model of the special-shaped sandwich structure filled with the conforming continuous gradient cellular structure is output through formula (3).

[0136] Furthermore, the corrected interpolation coefficients are obtained through the following formula:

[0137] , (20)

[0138] where is the corrected interpolation coefficient.

[0139] Furthermore, a conforming continuous gradient cellular structure is constructed through the following formula:

[0140] ,

[0141] , (21)

[0142] where is the global level set function of the conforming continuous gradient cellular structure, C is the Cth cellular structure, N is the number of cellular structures, and X is the set of all cellular structures in the physical space.

[0143] It should be noted that the interpolation coefficients are scalar field interpolation coefficients. The level set function for constructing a single special-shaped cellular structure is a local level set function. In order to generate a special-shaped sandwich structure, it is necessary to connect N special-shaped cellular structures, so it is necessary to combine N local level set functions to construct a global level set function.

[0144] Exemplarily, 4 2D X-shaped gradient lattice structures with gradually increasing relative densities are filled in the arc-shaped sandwich structure, and its global LSF function is as Figure 6 (a) shown. Through the zero-level surface projection of the global LSF function, a conforming variable special-shaped gradient lattice structure is generated, as Figure 6 (d) shown. However, as Figure 6As shown in (b), it can be seen from the density distribution of control points and the density distribution of parameter points (the size of the points represents the magnitude of the equivalent density) that there are discontinuous connection phenomena between the special-shaped gradient lattice structures, resulting in severe stress concentration and prone to structural failure. By modifying the interpolation coefficients with NURBS basis functions, the LSF function is updated based on the modified interpolation coefficients as shown in Figure 6 (c), realizing the smooth transition between local LSF functions, and achieving the continuous connection between special-shaped gradient lattice structures through low-dimensional projection as shown in Figure 6 (e). As shown in Figure 6 (f) and Figure 6 (g), this method can be easily extended to three-dimensional lattice structures, such as BCC and P-FCC lattice structures, to solve the connectivity problem between lattice structures.

[0145] It can be understood that the present invention expands the density distribution of a small number of control points to the density distribution of a large number of parameter points through interpolation coefficients, modifies the interpolation coefficients with NURBS basis functions, and updates the LSF function based on the modified interpolation coefficients, realizing the smooth transition between local LSF functions, thereby achieving the continuous connection between lattice structures, effectively solving the connectivity problem between gradient lattice structures under orthogonal envelopes, and significantly improving the quality of the special-shaped sandwich structure design and its manufacturability in engineering.

[0146] Embodiment 2

[0147] Another specific embodiment of the present invention is to generate a special-shaped sandwich structure by additive manufacturing based on the implicitly generated model above.

[0148] Specifically, an STL format file is output based on the implicit model, and the STL format file is input into a 3D printer for additive manufacturing to print and generate a special-shaped sandwich structure.

[0149] It should be noted that compared with the models generated by traditional modeling software, the implicit model has higher accuracy and smaller data volume, and is more suitable for engineering manufacturing.

[0150] Embodiment 3

[0151] Another specific embodiment of the present invention realizes the multi-scale optimization design and experimental verification of a three-point bending semi-circular arc sandwich structure.

[0152] Specifically, the design domain and boundary conditions of the three-point bending semi-circular arc sandwich structure are as shown in Figure 7 (a), the distributed force of the external load is applied to the middle position of the upper surface of the semi-circular arc, and the middle positions of both bottom ends are fixed respectively, and displacement constraints are applied in three directions to restrict the degrees of freedom. The Young's modulus of the base material , Poisson's ratio , density Outer diameter of the design domain , inner diameter , width , the design domain is discretized into elements for the optimized filling of the special-shaped BCC lattice structure. During the optimization iteration process, when the maximum value of the change in the control point density distribution is less than 0.01 or the maximum number of iterations reaches 300, the iteration is terminated. The objective function is set to minimize the structural compliance, and the relative volume fraction is limited to 0.3, Figure 7 (b) shows the semi-circular arc uniform lattice structure generated at the same relative density. The results of the multi-scale optimized design of the gradient special-shaped lattice structure are as shown in Figure 7 (e). It can be seen that the high-density special-shaped BCC lattice structure forms a continuous force transmission path from the middle of the top of the semi-circular arc to the support end, while a large number of medium- and low-density special-shaped BCC lattices are distributed on the secondary force transmission paths, which can effectively resist shear deformation and stably support the upper and lower solid panels. In addition, as shown in Figure 7 (c) and Figure 7 (d), the special-shaped BCC lattice structures are continuously connected. After removing the first layer of lattice structure of the optimized design gradient special-shaped lattice structure and observing its internal structure, as shown in Figure 7 (f), it can be found that the high-density lattice structures are clearly distributed on the force transmission paths to resist structural deformation and enhance the structural stiffness. Figure 7 (g) shows a schematic diagram of the semi-circular arc gradient lattice structure after removing 1 / 4 of the overall lattice structure. It can be seen that there are four force transmission paths forming an angle of about 30° with the XOY plane, which improves the stability of the sandwich structure.

[0153] Specifically, as shown in Figure 8 (a), it shows the connectivity between the lattice structures of the gradient lattice structure optimization model. In order to verify the performance of the gradient lattice structure provided by the present invention, the semi-circular arc sandwich structures filled with orthogonal uniform, special-shaped uniform, and special-shaped gradient lattice structures were prepared by 3D printing and subjected to a three-point bending test. All specimens were printed by the same printer under the same conditions and subjected to a three-point bending test on a Instron universal testing machine. First, a preloading force of was applied, and then the loading was carried out at a speed of until all specimens were broken. When the indenter displacement was 5 mm, the three-point bending test and DIC results of the orthogonal uniform lattice structure, special-shaped uniform lattice structure, and special-shaped gradient lattice structure are shown in Figure 8 (b)- Figure 8As shown in (d); it can be seen that there is strain concentration near the load application point in the orthogonal uniform lattice structure. Since the low-density region is prone to buckling and softening under the action of the load, it leads to the overall failure of the structure. The strain distribution of the special-shaped uniform lattice structure is relatively more uniform, but there are still local high-strain regions near the load application point and the support end, and the risk of local failure is still relatively large. The strain distribution of the special-shaped gradient lattice structure is more uniform, and the high-density region is concentrated on the main force transmission path, effectively enhancing the stiffness of the load transmission path and reducing the risk of local buckling. As Figure 8 shown in (e), at the same volume fraction, the strength and stiffness of the semi-circular arc gradient lattice structure (GLSS) are both superior to those of the semi-circular arc uniform lattice structure (CLSS) and the orthogonal uniform lattice structure (ULSS). The slope of the load-displacement curve of GLSS is 89.8% higher than that of ULSS, and the ultimate strength is 18.7% higher. At the same time, obvious local buckling occurs in ULSS during the loading process, while GLSS maintains relatively high stability. As Figure 6 shown in (f), the bending modulus and ultimate load of GLSS are both superior to those of CLSS and ULSS.

[0154] The test results show that compared with the traditional orthogonal uniform lattice structure filling, the lattice structure intelligent filling method provided by the present invention significantly improves the bearing capacity, flexural stiffness and tensile strength of the semi-circular arc sandwich structure by reasonably optimizing the density distribution of the lattice structure, setting high-density lattices on the main force-bearing paths to improve the bearing capacity, and using low-density lattices as auxiliary support structures at the same time. Thereby improving the stability of the overall structure, and effectively reducing the risk of local strain concentration and buckling failure, having broad engineering application potential.

[0155] Example 4

[0156] Another specific embodiment of the present invention realizes the optimized design and preparation of the windward rudder at the tail of the aircraft.

[0157] Specifically, the windward rudder surface is distributed in a '+' shape or an 'x' shape, and has a pre-contracted wedge-shaped cross-section to reduce aerodynamic drag. The rudder surface is designed with a hexahedron lattice filling and optimized under the condition of wind pressure load. As Figure 9 shown, the optimized design results of the special-shaped gradient lattice structure for the rudder plate structure show that the high-density lattice structure is mainly concentrated in the area extending from the fixed end to the force-bearing end, forming the main load-bearing path, effectively alleviating the local buckling deformation, and improving the stiffness and strength of the overall structure. In addition, the high-density lattice structure is concentrated near the rudder shaft and on both sides of the rudder surface, increasing the moment of inertia of the bending section, and significantly improving the bending performance of the rudder plate.

[0158] To verify the performance of the above-optimized rudder plate structure design, the present invention uses metal additive manufacturing to prepare a rudder plate filled with an orthorhombic uniform lattice structure (ULSS-R) and a shaped gradient lattice structure (CGLSS-R). Using an L-PBF machine, equipped with a fiber laser with a maximum laser power of , a spot size of , and metal powder of for preparation.

[0159] As shown in Figure 10 (a) and Figure 10 (b), the two-point loading test results of the rudder plate structures filled with the orthorhombic uniform lattice structure and the shaped gradient lattice structure are respectively shown; the set initial pressure of the hydraulic support rod is 2000 N, the gain coefficient is 5, and the custom peak load is 10000 N; in the rudder plate structure filled with the uniform lattice, when the peak forces at the loading points reach 7620.5 N and 6820.2 N respectively, the rudder shaft breaks, and the corresponding displacements of the loading points are 14000 and 7000 ; while in the rudder plate structure filled with the gradient lattice, when the peak forces at the loading points are 9700 N and 9500 N respectively, the corresponding displacements of the loading points are 4800 and 12000 ; it can be seen that the peak force of the rudder plate structure filled with the gradient lattice is about 45.6% higher than that of the rudder plate structure filled with the uniform lattice, and the displacement is reduced by about 28.7%. As shown in Figure 10 (c) and Figure 10 (d), in the rudder plate structure filled with the orthorhombic uniform lattice, significant fractures occurred in the lattice structure within the arc transition region of the rudder shaft, while no fractures occurred in the rudder plate structure filled with the gradient lattice.

[0160] The test results show that no cracks or microcracks appeared in the transition region between the rudder shaft and the rudder surface of the gradient lattice structure, significantly improving the flexural stiffness and tensile strength of the rudder plate structure, effectively reducing the risk of local buckling and crack propagation, and thus enhancing the overall reliability and service life of the structure.

[0161] Compared with the prior art, the beneficial effects of the multi-scale optimized intelligent design method for the shaped structure provided by the present invention are as follows:

[0162] 1. By constructing an implicit model of a shaped sandwich structure filled with a conformal continuous gradient shaped lattice structure and generating a shaped sandwich structure by additive manufacturing based on the implicit model, the present invention solves the problem of the lack of a multi-scale optimized design method for a gradient lattice-filled shaped sandwich structure.

[0163] 2. The present invention expands the density distribution of a small number of control points to that of a large number of parameter points through interpolation coefficients, modifies the interpolation coefficients through NURBS basis functions, and updates the LSF function based on the modified interpolation coefficients, achieving a smooth transition between local LSF functions, thereby realizing the continuous connection between lattice structures, effectively solving the connectivity problem between gradient lattice structures under orthogonal envelopes, and significantly improving the quality of the design of special-shaped sandwich structures and their manufacturability in engineering.

[0164] 3. By constructing a constitutive surrogate model, the present invention can quickly evaluate the mechanical properties of special-shaped lattice structures under different key geometric parameters, avoiding the calculation of the constitutive matrix of each lattice one by one, reducing the computational amount, and improving the computational efficiency. Moreover, through polynomial interpolation, efficient and accurate prediction of the mechanical properties of special-shaped lattice structures with different relative densities, aspect ratios, and deflection angles is realized.

[0165] 4. The present invention implicitly constructs special-shaped lattice structures of various shapes through the level set function. Not only can special-shaped lattice structures of various topological forms be obtained, but also gradient special-shaped lattice structures with the same topological configuration but different characteristic sizes can be obtained through interpolation. The present invention constructs a conformally varying special-shaped lattice structure by modifying the level set function corresponding to the gradient special-shaped lattice structure through NURBS basis functions, realizing the adaptive matching between the lattice structure and the special-shaped sandwich structure.

[0166] Those skilled in the art can understand that all or part of the processes of implementing the methods in the above embodiments can be completed by instructing relevant hardware through a computer program, and the program can be stored in a computer-readable storage medium. Among them, the computer-readable storage medium is a magnetic disk, an optical disk, a read-only memory, or a random access memory, etc.

[0167] As mentioned above, the above are only the preferred specific embodiments of the present invention, but the protection scope of the present invention is not limited thereto. Any changes or substitutions that can be easily thought of by those skilled in the art within the technical scope disclosed by the present invention should be covered by the protection scope of the present invention.

Claims

1. An intelligent design method for multi-scale optimization of special-shaped structures, characterized in that, The method includes the following steps: Construct a special-shaped lattice structure using a level set function, and obtain a conforming gradient special-shaped lattice structure through interpolation and correction of the special-shaped lattice structure; set relative density sample points, calculate the relationship between the interpolation coefficients and the relative density sample points based on the conforming gradient special-shaped lattice structure, and establish a geometric surrogate model by performing polynomial interpolation on the relative density sample points; based on the conforming gradient special-shaped lattice structure, obtain the constitutive matrix corresponding to the relative density sample points through numerical homogenization, and establish a constitutive surrogate model by performing polynomial interpolation on the relative density sample points. Based on the constitutive surrogate model and the isogeometric topology optimization method, iteratively solve with the goal of minimizing the compliance of the special-shaped sandwich structure to obtain the optimal density distribution. Substitute the optimal density distribution into the geometric surrogate model to output the interpolation coefficients, correct the interpolation coefficients through NURBS basis functions, and generate an implicit model of the special-shaped sandwich structure by filling with gradient lattices.

2. The intelligent design method for multi-scale optimization of special-shaped structures according to claim 1, characterized in that Construct the geometric space of the special-shaped lattice structure, obtain the constitutive matrix corresponding to the interpolated relative density sample points by performing polynomial interpolation on the relative density sample points, construct the constitutive matrix space of the special-shaped lattice structure, and determine the mapping relationship from the geometric space to the constitutive matrix space to establish a constitutive surrogate model.

3. The intelligent design method for multi-scale optimization of special-shaped structures according to claim 1, characterized in that, Input the constitutive surrogate model into CAD software, construct an isogeometric model based on NURBS basis functions, refine the isogeometric model through k-refinement, define boundary conditions and constraints, establish a density distribution function, establish an objective function with the goal of minimizing the compliance of the special-shaped sandwich structure, perform sensitivity analysis based on the density distribution function, and iteratively solve the objective function through the MMA algorithm to obtain the optimal density distribution.

4. The multi-scale optimization intelligent design method for special-shaped structures according to claim 1, wherein, Calculate the relationship between the interpolation coefficients and the aspect ratio and the relative density sample points through formulas (8)-(10): ， (8) ， (9) ， (10) wherein, is the Heaviside function, is the relative density of the conformally graded lattice structure with variable cross-section, is the aspect ratio of the conformally graded lattice structure with variable cross-section, is the interpolation coefficient of the conformally graded lattice structure with variable cross-section, is the polynomial interpolation coefficient, is the interpolation order, is the maximum interpolation order, represents the level set function, represents the design domain, represents the physical coordinates corresponding to the sampling points of the local level set function; The expression of the geometric surrogate model is as follows: ， (11) Among them, represents a geometric surrogate model.

5. The intelligent design method for multi-scale optimization of special-shaped structures according to claim 4, characterized in that The implicit modeling expression of a single special-shaped lattice structure is: ， （3） Among them, Void represents the void of the special-shaped lattice structure, Boundary represents the boundary of the special-shaped lattice structure, and Solid represents the area where the special-shaped lattice is filled with materials; , represents the sampling points of the local level set function, D represents the set of physical space coordinate points of a single special-shaped lattice structure, represents the local level set function corresponding to a single special-shaped lattice structure.

6. The multi-scale optimization intelligent design method for special-shaped structures according to claim 5, wherein, Substitute the optimal density distribution into the geometric surrogate model established by formulas (10) and (11) to output the interpolation coefficients, correct the interpolation coefficients through NURBS basis functions to obtain the corrected interpolation coefficients, construct a conforming continuous gradient special-shaped lattice structure using a level set function based on the corrected interpolation coefficients and the conforming gradient special-shaped lattice structure, and output the implicit model of the special-shaped sandwich structure filled with the conforming continuous gradient special-shaped lattice structure through formula (3).

7. The intelligent design method for multi-scale optimization of special-shaped structures according to claim 6, characterized in that Obtain the corrected interpolation coefficients through the following formula: ，(20) Among them, is the corrected interpolation coefficient, where l, m, and n are the maximum number of nodes in the three parameter directions respectively, represents the NURBS basis function.

8. The multi-scale optimized intelligent design method for special-shaped structures according to claim 6, wherein Construct a conforming continuous gradient special-shaped lattice structure through the following formula: ， ， (21) Among them, is the global level set function of the conformally continuous gradient heterogeneous lattice structure, C is the C-th heterogeneous lattice structure, N is the number of heterogeneous lattice structures, and X is the set of all heterogeneous lattice structures in physical space, is the corrected interpolation coefficient.

9. The multi-scale optimized intelligent design method for special-shaped structures according to claim 1, wherein Calculate the relationship between the interpolation coefficients and the aspect ratio and the interpolated relative density by performing polynomial interpolation on the relative density sample points, thereby establishing a geometric surrogate model.

10. The intelligent design method for multi-scale optimization of special-shaped structures according to claim 1, wherein Based on the generated implicit model, generate a special-shaped sandwich structure through additive manufacturing.

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