A topology optimization method for gradient lattice structures with boundary matching of multiple microstructures

Abstract

The present invention belongs to the technical field related to structural topology optimization, and discloses a gradient lattice structure topology optimization method for matching the boundaries of multiple types of microstructures, comprising the following steps: (1) constructing a parameterized level set function of multiple types of microstructure cells; (2) obtaining a microstructure topological configuration containing multiple sub-microstructure cells; (3) obtaining a global parameterized level set function of the lattice structure; (4) decoupling the Hamilton-Jacobi partial differential equation that controls the evolution of the level set function; (5) defining a finite element analysis grid; (6) defining a global weight coefficient at the vertex of each lattice structure cell; (7) establishing a mathematical model for topology optimization of a functional gradient lattice structure; (8) performing a finite element analysis on the lattice structure; (9) performing a sensitivity analysis; and (10) updating global design variables to obtain a gradient lattice structure in which multiple types of microstructures are distributed within the same design domain. The present invention ensures a continuous transition of the geometric boundaries of microstructures of different types and different volume fractions.

Description

Technical Field

[0001] The present invention belongs to the technical field related to structural topology optimization, and more specifically, relates to a gradient lattice structure topology optimization method for multi-type microstructure boundary matching. Background Art

[0002] Functionally graded lattice structures exhibit spatially gradient microstructural distribution and mechanical properties, resulting in superior mechanical properties compared to periodic, uniform porous structures, such as high specific stiffness / strength, energy absorption and vibration reduction, and high thermal conductivity. These structures are widely used in aerospace, rail transit, marine, and other fields. However, the diverse geometric forms and complex functional-structure coupling of lattice structures complicate their optimization and design. Therefore, expanding the design and optimization space for functionally graded lattice structures and fully realizing their potential remains a hot topic in the field of structural optimization and design.

[0003] As an intelligent design method, topology optimization provides enormous design freedom for innovative structural configuration design. Compared with single-scale topology optimization methods, multi-scale topology optimization methods can comprehensively optimize the distribution of macro-structural materials and their spatially varying micro-structural configurations, and are the main design methods for topological optimization of functionally gradient porous structures. Depending on the optimization strategy, it can be divided into layer-by-layer, domain-by-domain, and point-by-point multi-scale design methods. However, problems such as the singleness of micro-structural distribution types and poor continuity of micro-structural geometric boundaries in the design of gradient lattice structures restrict the design space and application potential of lattice structures. Therefore, how to balance the diversity of micro-structural configurations and the continuity of geometric boundaries is a challenging problem in the engineering design of high-performance gradient lattice structures. Summary of the Invention

[0004] In response to the above-mentioned defects or improvement needs of the prior art, the present invention provides a gradient lattice structure topology optimization method for matching the boundaries of multiple types of microstructures. By presetting multiple types of microstructure unit cell configurations, a parameterized level set function is constructed to achieve the topological design of gradient lattice structures with diversified microstructure configuration distributions, and ensure the continuous transition of microstructure geometric boundaries of different types and different volume fractions.

[0005] To achieve the above objectives, according to one aspect of the present invention, a gradient lattice structure topology optimization method for multi-type microstructure boundary matching is provided, the method mainly comprising the following steps:

[0006] (1) At the microstructure modeling level, the signed distance function is used to construct the level set function of the microstructure unit cell, and then the parameterized level set functions of various types of microstructure unit cells are constructed;

[0007] (2) At the local unit cell level of the lattice structure, the parameterized level set functions of multiple sub-microstructure unit cells are subjected to a Boolean sum operation, and the level set function obtained by the Boolean sum operation is cut through the zero level plane to obtain the latest microstructure topological configuration containing multiple sub-microstructure unit cells;

[0008] (3) At the global level of the lattice structure, the parameterized level set function of the local unit cell is defined, and then the global parameterized level set function of the lattice structure is obtained, and the lattice structure is obtained by cutting the zero level plane;

[0009] (4) Substitute the parameterized level set function into the Hamilton-Jacobi partial differential equation that controls the evolution of the level set function to obtain the ordinary differential equation;

[0010] (5) Define the finite element analysis grid in the spatial coordinate system so that the geometric model of the structural expression matches the calculated finite element model;

[0011] (6) Define the global weight coefficient at the vertex of each lattice structure unit cell, and obtain the local weight coefficient inside the microstructure unit cell through linear interpolation of the shape function;

[0012] (7) A mathematical model for topology optimization of functional gradient lattice structures is established, with the global weight coefficient defined on the unit cell node as the global design variable, the maximization of structural stiffness as the objective function, and the volume fraction of the lattice structure as the constraint condition;

[0013] (8) Perform finite element analysis on the lattice structure to obtain the unit strain energy of the unit in the design domain;

[0014] (9) Derive the derivatives of the objective function and constraints with respect to the global design variables and perform sensitivity analysis;

[0015] (10) A gradient-based optimization algorithm is used to solve the optimization model to update the global design variables, thereby obtaining a gradient lattice structure with various types of microstructures distributed in the same design domain, and then obtaining the optimal geometric topology optimization configuration of the gradient multi-lattice structure.

[0016] Furthermore, in step (10), it is determined whether the iterative optimization process satisfies the given convergence conditions. If the convergence conditions are met, the optimization ends and the optimal functional gradient multi-lattice structure geometric topology optimization configuration is output; if the convergence conditions are not met, the process goes to step (6).

[0017] Furthermore, the convergence criterion is defined as follows:

[0018] or n≥n max

[0019] Where δ is the given threshold, n max Represents the maximum number of iterations allowed during optimization.

[0020] Furthermore, the parameterized level set function of the microstructure unit cell The definition is as follows:

[0021]

[0022] In the formula, the coefficient is the weight coefficient with respect to pseudo-time t, which specifically means the global design variable of the i-th microstructure on the k-th unit cell in the design domain, and its value range is [0 1]; and is the unit cell level set function of the predefined type i microstructure at the kth unit cell position in the design domain. According to the definition of the local level set function, the predefined unit cell is periodically distributed in the design domain. and Can also write and γ i (x), and γ i (x) is the predefined type i microstructure, specifically representing the level set function of the same type of microstructure unit cell at different volume fractions, r is the number of predefined unit cell types, and M is the total number of unit cells after the design domain is discretized.

[0023] Furthermore, by constructing parameterized level set functions of various types of microstructure cells, multiple sub-microstructure cell topological configurations are obtained based on the zero-level plane cutting of the level set function. The implicit level set modeling is expressed as follows:

[0024]

[0025] Where, The area represents the solid area of ​​the microstructure, The area represents the void area in the microstructure. It is the boundary between the solid area and the void area of ​​the microstructure.

[0026] Furthermore, the Boolean sum operation is as follows:

[0027]

[0028] Where, Φ k (x, t) represents the parameterized level set function of the kth microstructure unit cell in the design domain;

[0029] The parameterized level set function is defined on the design domain unit cell. According to the number of pre-divided units in the design domain space, the corresponding number of level set functions is defined. The parameterized level set functions of all local unit cells are assembled to obtain the global level set function of the lattice structure, which is defined as follows:

[0030]

[0031] Where M represents the total number of unit cells after the design domain is discretized, r represents the number of types of microstructures contained in the design domain; the area where Φ(x, t)>0 represents the solid area in the lattice structure, and the area where Φ(x, t)<0 represents the void area in the lattice structure. It represents the boundary between the solid area and the void area in the lattice structure.

[0032] Furthermore, the Hamilton-Jacobi partial differential equation expresses the level set function in the velocity field v n The evolutionary form driven by is defined as follows:

[0033]

[0034] Substitute the defined parameterized level function into the original Hamilton-Jacobi partial differential equation and transform it into an ordinary differential equation. The expression is as follows:

[0035]

[0036] Where, is the modulus of the gradient of the level set function.

[0037] Furthermore, the implicit geometric description method of parameterized level set function is used to express the geometric model of the structure, and the finite element method is used to calculate the mechanical response of the structure.

[0038] Furthermore, a global weight coefficient is defined at each node of the discrete unit cell, which is the global design variable in the optimization process:

[0039] T i (t)=[T i 1 (t) T i 2 (t) T i 3 (t) … T i m (t)] Τ

[0040] Where, T i (t) represents the global design variable of the predefined i-th microstructure in the design domain, and m is the total number of unit cell nodes contained in the discretized design domain;

[0041] For the local weight coefficients at all nodes of the unit cell, that is, the local design variables, the shape function N(x) is used for interpolation.

[0042] Furthermore, the mathematical expression of the topology optimization mathematical model of the functionally graded lattice structure is:

[0043] find:T i k (t)

[0044]

[0045]

[0046] Where J(Φ) represents the total flexibility of the lattice structure, ε represents the strain field of the structure, u is the displacement field of the structure, E is the equivalent elastic matrix of the predefined material, and a Ф (u,v)=l Ф (v) is the weak form of the elastic equilibrium equation, v represents a virtual displacement domain in the dynamically allowed displacement domain space U, g(Φ) is the volume constraint function of the structure, ζ represents the volume fraction of the structure, represents the volume of the design domain.

[0047] In general, compared with the prior art, the above technical solutions conceived by the present invention have the following advantages over the prior art:

[0048] Beneficial effects:

[0049] 1. The present invention uses the global weight coefficient defined on the microstructure unit cell as the global design variable, constructs a parameterized level set function, and transforms the original Hamilton-Jacobi partial differential equation into an ordinary differential equation. It can be combined with mature optimization algorithms in the field of optimization, thereby improving the solution efficiency of the level set method and reducing the complexity of the solution.

[0050] 2. The present invention introduces multiple types of microstructure unit cells. Through the proposed parameterized level set topology optimization method, it can achieve a reasonable gradient distribution of diversified microstructures in the lattice structure, and obtain a functional gradient lattice structure with excellent mechanical properties. Compared with the gradient lattice structure with a single type of microstructure distribution, the present invention effectively expands the design space and design freedom of the lattice structure, and can greatly improve the design performance of the lattice structure.

[0051] 3. The present invention ensures the continuous transition of geometric boundaries of microstructures of different types and different volume fractions with the help of the full-scale finite element method, guarantees the mechanical properties and subsequent manufacturability of the functionally gradient lattice structure, and provides a feasible technical solution for the optimization design of high-performance functionally gradient lattice structures. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 This is a structural topology optimization flow chart related to a gradient lattice structure topology optimization method for multi-type microstructure boundary matching provided by the present invention;

[0053] Figure 2 It is a schematic diagram of implicit modeling of an X-shaped microstructure unit cell involved in the present invention;

[0054] Figure 3 This is a schematic diagram of implicit modeling of a U-shaped microstructure unit cell involved in the present invention;

[0055] Figure 4 This is a schematic diagram of implicit modeling of a cross-shaped microstructure unit cell involved in the present invention;

[0056] Figure 5 Schematic diagram of the process of parameterization and Boolean summation of different types of microstructures of the present invention;

[0057] Figure 6 It is a schematic diagram of the two-dimensional design and grid discretization and design variable definition of the present invention;

[0058] Figure 7 Schematic diagram of a two-dimensional cantilever beam structure according to an embodiment of the present invention;

[0059] Figure 8 This is the final optimized configuration diagram of the two-dimensional functional gradient lattice structure in Example 1 of the present invention;

[0060] Figure 9 1 are gradient distribution sub-configuration diagrams of the X-shaped microstructure and the U-shaped microstructure in Example 1 of the present invention, (a) is the gradient distribution sub-configuration diagram of the X-shaped microstructure, and (b) is the gradient distribution sub-configuration diagram of the U-shaped microstructure;

[0061] Figure 10 : is the objective function and volume fraction optimization iteration curve diagram in Example 1 of the present invention;

[0062] Figure 11 This is a diagram of the optimized configuration of the two-dimensional functional gradient lattice structure considering only the X-shaped microstructure in Example 1 of the present invention;

[0063] Figure 12 This is the final optimized configuration diagram of the two-dimensional functional gradient lattice structure in Example 2 of the present invention;

[0064] Figure 13 Figures 1 and 2 are the gradient distribution sub-configuration graphs and optimization iteration curves of the X-shaped, U-shaped, and cross-shaped microstructures in Example 2 of the present invention. (a) is the gradient distribution sub-configuration graph of the X-shaped microstructure; (b) is the gradient distribution sub-configuration graph of the U-shaped microstructure; (c) is the gradient distribution sub-configuration graph of the cross-shaped microstructure; and (d) is the objective function and volume fraction optimization iteration curve.

[0065] Figure 14 Schematic diagram of implicit modeling of a three-dimensional BCC microstructure unit cell in Example 3 of the present invention;

[0066] Figure 15 Schematic diagram of implicit modeling of a three-dimensional SC microstructure unit cell in Example 3 of the present invention;

[0067] Figure 16 is a schematic diagram of a three-dimensional cantilever beam structure in Example 3 of the present invention;

[0068] Figure 17 This is the final optimized configuration diagram of the three-dimensional functional gradient lattice structure in Example 3 of the present invention;

[0069] Figure 18 1 are the gradient distribution sub-configuration diagrams of the BCC microstructure and the SC microstructure in Example 3 of the present invention, (a) is the gradient distribution sub-configuration diagram of the BCC microstructure; (b) is the gradient distribution sub-configuration diagram of the SC microstructure;

[0070] Figure 19 3 is an iterative graph of the objective function and volume fraction optimization in Example 3 of the present invention. DETAILED DESCRIPTION

[0071] In order to make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely for the purpose of explaining the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.

[0072] See also Figure 1 、 Figure 2 、 Figure 3 、 Figure 4 、 Figure 5 and Figure 6 The present invention provides a gradient lattice structure topology optimization method for multi-type microstructure boundary matching, the method comprising the following steps:

[0073] S1. At the microstructure modeling level, the signed distance function is used to construct the level set function of the microstructure unit cell, and then the parameterized level set functions of various types of microstructure unit cells are constructed.

[0074] Parameterized level set function of microstructure unit cell The definition is as follows:

[0075]

[0076] In the formula, the coefficient is the weight coefficient with respect to pseudo-time t, which specifically means the global design variable of the i-th microstructure on the k-th unit cell in the design domain, and its value range is [0 1]; and is the unit cell level set function of the predefined type i microstructure at the kth unit cell position in the design domain. According to the definition of the local level set function, the predefined unit cell is periodically distributed in the design domain. and You can also write and γ i (x), and γ i (x) is the predefined type i microstructure, specifically representing the level set function of the same type of microstructure unit cell at different volume fractions, r is the number of predefined unit cell types, and M is the total number of unit cells after the design domain is discretized.

[0077] By constructing parameterized level set functions of various types of microstructure cells, multiple sub-microstructure cell topological configurations are obtained based on the zero-level plane cutting of the level set function. The implicit level set modeling is expressed as follows:

[0078]

[0079] Where, The area represents the solid area of ​​the microstructure, The area represents the void area in the microstructure. It is the boundary between the solid area and the void area of ​​the microstructure.

[0080] S2, at the local unit cell level of the lattice structure, the parameterized level set functions of multiple sub-microstructure unit cells are subjected to a Boolean sum operation, and the level set function obtained by the Boolean sum operation is cut through the zero level plane to obtain the latest microstructure topological configuration containing multiple sub-microstructure unit cells.

[0081] The obtained multiple microstructures are sub-microstructures of the lattice structure microstructure. After obtaining the parameterized level set function of the sub-microstructure unit cell, a Boolean sum operation is performed on it. Then, the level set function after the Boolean sum operation is cut through the zero level plane to obtain the final microstructure topological configuration containing multiple sub-microstructure unit cells. The Boolean sum operation is as follows:

[0082]

[0083] Where, Φ k (x,t) represents the parameterized level set function of the kth microstructure unit cell in the design domain.

[0084] S3, at the global level of the lattice structure, define the parameterized level set function of the local unit cell, then obtain the global parameterized level set function of the lattice structure, and obtain the lattice structure by cutting the zero horizontal plane.

[0085] The parameterized level set function is defined on the design domain unit cell. According to the number of pre-divided units in the design domain space, the corresponding number of level set functions is defined. The parameterized level set functions of all local unit cells are assembled to obtain the global level set function of the lattice structure, which is defined as follows:

[0086]

[0087] Where M represents the total number of unit cells after the design domain is discretized, r represents the number of types of microstructures contained in the design domain; the area where Φ(x, t)>0 represents the solid area in the lattice structure, and the area where Φ(x, t)<0 represents the void area in the lattice structure. It represents the boundary between the solid area and the void area in the lattice structure.

[0088] S4, the parameterized level set function is introduced into the Hamilton-Jacobi partial differential equation that controls the evolution of the level set function to obtain the ordinary differential equation, realizing the spatiotemporal decoupling of the Hamilton-Jacobi partial differential equation.

[0089] The Hamilton-Jacobi partial differential equation expresses the level set function in the velocity field v n The evolutionary form driven by is defined as follows:

[0090]

[0091] Substitute the defined parameterized level function into the original Hamilton-Jacobi partial differential equation and transform it into an ordinary differential equation. The expression is as follows:

[0092]

[0093] Where, is the modulus of the gradient of the level set function, which can be abbreviated as

[0094] S5, define the finite element analysis mesh in the spatial coordinate system so that the geometric model of the structural expression matches the calculated finite element model.

[0095] According to the number of unit cells divided into the design domain and the size of the corresponding microstructure, the same number of finite element analysis grids are defined in the spatial coordinate system so that the geometric model expressed by the structure matches the calculated finite element model.

[0096] According to the predetermined number of unit cells and the size of the corresponding microstructure, the design domain D is discretized, and the same number of finite element analysis grids are defined in the spatial coordinate system so that the geometric model of the structural expression matches the calculated finite element model. The implicit geometric description method of the parameterized level set function is used to express the geometric model of the structure, and the finite element method is used to calculate the mechanical response of the structure.

[0097] S6, define the global weight coefficient at the vertex of each lattice structure unit cell, and obtain the local weight coefficient inside the microstructure unit cell through linear interpolation of the shape function.

[0098] According to the preset number of microstructure types, the same number of global weight coefficients are defined on the vertices of each lattice structure unit cell, and the local weight coefficients inside the microstructure unit cell are obtained by linear interpolation of the shape function.

[0099] Define the global weight coefficient at each node of the discrete unit cell, which is the global design variable in the optimization process:

[0100] T i (t)=[T i 1 (t) T i 2 (t) T i 3 (t) … T i m (t)] Τ

[0101] Where, T i (t) represents the global design variable of the predefined i-th microstructure in the design domain, and m is the total number of unit cell nodes contained in the discretized design domain.

[0102] For any unit cell structure k in the discrete design domain, which contains 4 (2D structure) or 8 (3D structure) macro design variables defined at the nodes, the global design variables of unit cell k obtained from the global design variables are operated as follows:

[0103] T i k (t) = S k T i (t)

[0104] Where S k Select the matrix for the global design variables of the unit cell.

[0105] For the local weight coefficients at all nodes of the unit cell, that is, the local design variables, the shape function N(x) is used for interpolation.

[0106] The shape function of a two-dimensional quadrilateral element is defined as follows:

[0107]

[0108] Then the shape function of the two-dimensional microstructure unit cell is defined as:

[0109] N(x)=[N1(x) N2(x) N3(x) N4(x)]

[0110] The shape function of a three-dimensional hexahedral element is defined as follows:

[0111]

[0112] Then the shape function of the three-dimensional microstructure unit cell is defined as:

[0113] N C (x)=[N1(x) N2(x) N3(x) N4(x) N5(x)N6(x) N7(x) N8(x)]

[0114] Where X, Y, and Z are the local coordinate system coordinates of the parent element of the hexahedral element.

[0115] Then the local weight coefficients on all unit nodes contained in the unit cell are:

[0116] T C (t) = N(x)T i k (t) = N(x)S k T i k (t).

[0117] S7, with the global weight coefficient defined on the unit cell node as the global design variable, the maximization of structural stiffness as the objective function, and the volume fraction of the lattice structure as the constraint condition, a mathematical model for topology optimization of functional gradient lattice structure is established.

[0118] A mathematical model for topology optimization of functionally graded lattice structures is established, with the global weight coefficient defined on the unit cell node of the lattice structure as the global design variable, minimization of structural flexibility as the objective function, and the volume fraction of the multi-lattice structure as the constraint condition:

[0119] find:T i k (t)

[0120]

[0121]

[0122] Where J(Φ) represents the total flexibility of the lattice structure, ε represents the strain field of the structure, u is the displacement field of the structure, E is the equivalent elastic matrix of the predefined material, and a Ф (u,v)=l Ф (v) is the weak form of the elastic equilibrium equation, v represents a virtual displacement domain in the dynamically allowed displacement domain space U, g(Φ) is the volume constraint function of the structure, ζ represents the volume fraction of the structure, represents the volume of the design domain.

[0123] S8, perform finite element analysis on the lattice structure and solve to obtain the unit strain energy of the unit in the design domain.

[0124] The finite element method is used to solve the given lattice structure design domain. After obtaining the unit strain energy, the unit strain energy is further converted into the unit node strain energy for subsequent sensitivity calculation. Ф (u,v)=l Ф (v) Obtain:

[0125] a Φ (u,v)=∫ Ω ε Τ (u)Eε(v)dΩ

[0126]

[0127] Where δ(Ф) is the derivative of the Heaviside function with respect to the Ф function, f is the traction force applied to the boundary of the finite element model structure, and p is the volume force inside the structure.

[0128] S9, derive the derivatives of the objective function and constraints with respect to the global design variables to complete the sensitivity analysis.

[0129] The derivative of the objective function with respect to the time variable t is:

[0130]

[0131] Where G(Ф) is the strain energy of the unit node, is the magnitude of the gradient of the level set function.

[0132] The normal velocity v of the level set function boundary evolution in step S4 n for:

[0133]

[0134] Substituting it into the derivative expression of the objective function with respect to the time variable t, we get:

[0135]

[0136] According to the chain rule:

[0137]

[0138] By comparing the above mathematical expressions, the derivative of the objective function with respect to the global design variable can be obtained as:

[0139]

[0140] Where, is the sensitivity of the objective function with respect to the global design variables.

[0141] Similarly, the derivative of the constraint function with respect to the global design variable can be derived as:

[0142]

[0143] Where, is the sensitivity of the constraint with respect to the global design variable.

[0144] S10, a gradient-based optimization algorithm is used to solve the optimization model to update the global design variables, thereby obtaining a gradient lattice structure with various types of microstructures distributed in the same design domain.

[0145] A gradient-based optimization algorithm can be used to solve the mathematical model and update the global design variables, such as the optimal criterion method or the moving progressive algorithm. In this embodiment, the optimal criterion method is used.

[0146] S11, judging whether the iterative optimization process meets the given convergence conditions. If the convergence conditions are met, the optimization ends and the optimal functional gradient multi-lattice structure geometric topology optimization configuration is output; if the convergence conditions are not met, go to step S6.

[0147] The convergence criterion is defined as follows:

[0148] or n≥n max

[0149] Wherein, δ is a given threshold value, in this embodiment, δ=0.001, n max Represents the maximum number of iterations allowed during optimization. In this implementation, n max =200.

[0150] The present invention is further described in detail below with reference to several embodiments.

[0151] Example 1

[0152] See also Figure 7 、 Figure 8 、 Figure 9 、 Figure 10 and Figure 11 As shown, this example constructs a two-dimensional microstructure based on the implicit modeling method of the level set function, including X-shaped, square-shaped and cross-shaped, as shown in Figure 2-Figure 4 As shown in the figure, the structural design domain is a two-dimensional cantilever beam with dimensions of L×H=400×200. The left boundary of the design domain is fixed, and a vertical downward concentrated load F=1N is applied to the lower right corner. Figure 7As shown in the figure, the design domain is discretized into 20×10 quadrilateral cells, each of which is further discretized into 20×20 four-node planar quadrilateral elements. X-shaped and square-shaped cells are used as the preset microstructure cells, with a minimum cell volume fraction of 0 and a maximum cell volume fraction of 0.7.

[0153] During the optimization, the optimization goal was to minimize the structural flexibility, and the structural volume fraction constraint was 0.45. After the optimization, the functional gradient lattice topology configuration containing two types of microstructure gradient distributions was obtained as follows Figure 8 As shown. Among them, the gradient distribution sub-configurations of X-shaped and square-shaped microstructures are as follows Figure 9 As shown in (a) and (b) in Figure 2, it can be found that the geometric boundary transition of microstructures with different configurations and different volume fractions is continuous, and there is no geometric step feature. The objective function and volume fraction iteration curve are shown in Figure 2. Figure 10 As shown in the figure, the structural flexibility converges to 128.2, the volume fraction converges to 0.45, and the algorithm iteration is smooth, showing good stability.

[0154] To further illustrate the superiority of the present invention, only the X-shaped microstructure is considered as the preset unit cell for functional gradient lattice structure optimization, and the volume fraction is set to 0.45, which is consistent with the optimization example considering two microstructures. The final topology optimization configuration is as follows Figure 11 As shown in the figure, the structure only contains X-shaped microstructures and has a structural flexibility of 178.6. By comparison, it can be found that the method of the present invention significantly improves the structural stiffness under the same volume fraction constraint, demonstrating the effectiveness and superiority of the gradient lattice structure topology optimization method provided by the present invention with boundary matching of multiple microstructures.

[0155] Example 2

[0156] See also Figure 12 and Figure 13 As shown in the figure, this example uses three types of microstructures, namely X-shaped, square-shaped and cross-shaped, as preset unit cells, with the minimum volume fraction of the unit cell being 0.1 and the maximum volume fraction being 0.7. Figure 7 The cantilever beam structure shown in the figure was optimized. After 49 iterations, the optimization was completed and the structural flexibility converged to 114.2. The functional gradient lattice topology configuration containing three types of microstructure distribution was obtained, as shown in the figure. Figure 12 The distribution sub-configurations of the three microstructures in the functionally graded lattice structure are shown in Figure 2. Figure 13 As shown in (a) to (c) in the figure, the optimization iteration curve is as follows Figure 13 This preferred embodiment further illustrates the effectiveness of the gradient lattice structure topology optimization method for multi-type microstructure boundary matching provided by the present invention in considering three types of microstructure topology optimization in the gradient lattice structure.

[0157] Example 3

[0158] See also Figure 14 、 Figure 15 、 Figure 16 、 Figure 17 、 Figure 18 and Figure 19 This example is based on a multi-class microstructure boundary matching gradient lattice structure topology optimization method provided by the present invention. Two types of three-dimensional microstructures are used to optimize the three-dimensional cantilever beam structure. According to the level set function implicit modeling method, three-dimensional microstructures are constructed, including two types of microstructures: BCC (Body-Centered Cubic) and SC (Simple Cubic). Figure 14 and Figure 15 The design domain of the three-dimensional cantilever beam is shown in Figure 16 As shown, the dimensions are: L×H×W=168×84×14, the left side of the design domain is fixed, and a vertical downward concentrated load F=10N is applied to the center of the right side. The design domain is discretized into 12×6×1 hexahedral units, and each unit cell can be further discretized into 14×14×14 hexahedral units. Figure 16 The three-dimensional cantilever beam structure shown in the figure is optimized. After the optimization is completed, a functional gradient lattice topology configuration containing two three-dimensional microstructure distributions is obtained, as shown in FIG. Figure 17 The distribution sub-configurations of BCC and SC microstructures in the functionally graded lattice structure are shown in Figure 2. Figure 18 The objective function and volume fraction iteration curve are shown in (a) and (b). Figure 19 As shown, the structural flexibility converges to 879.4, the volume fraction converges to 0.45, the algorithm iteration is smooth, and it still demonstrates good stability in three-dimensional numerical examples. This preferred embodiment further illustrates the effectiveness of the gradient lattice structure topology optimization method for multi-class microstructure boundary matching provided by the present invention in considering two types of microstructure topology optimization in three-dimensional gradient lattice structures.

[0159] It will be easily understood by those skilled in the art that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A gradient lattice structure topology optimization method for multi-type microstructure boundary matching, characterized in that: The method comprises the following steps: (1) At the microstructure modeling level, the signed distance function is used to construct the level set function of the microstructure unit cell, and then the parameterized level set functions of various types of microstructure unit cells are constructed; (2) At the local unit cell level of the lattice structure, the parameterized level set functions of multiple sub-microstructure unit cells are subjected to a Boolean sum operation, and the level set function obtained by the Boolean sum operation is cut through the zero level plane to obtain the latest microstructure topological configuration containing multiple sub-microstructure unit cells; (3) At the global level of the lattice structure, the parameterized level set function of the local unit cell is defined, and then the global parameterized level set function of the lattice structure is obtained, and the lattice structure is obtained by cutting the zero level plane; (4) Substitute the parameterized level set function into the Hamilton-Jacobi partial differential equation that controls the evolution of the level set function to obtain the ordinary differential equation; (5) Define the finite element analysis grid in the spatial coordinate system so that the geometric model of the structural expression matches the calculated finite element model; (6) Define the global weight coefficient at the vertex of each lattice structure unit cell, and obtain the local weight coefficient inside the microstructure unit cell through linear interpolation of the shape function; (7) A mathematical model for topology optimization of functional gradient lattice structures is established, with the global weight coefficient defined on the unit cell node as the global design variable, the maximization of structural stiffness as the objective function, and the volume fraction of the lattice structure as the constraint condition; (8) Perform finite element analysis on the lattice structure to obtain the unit strain energy of the unit in the design domain; (9) Derive the derivatives of the objective function and constraints with respect to the global design variables and perform sensitivity analysis; (10) A gradient-based optimization algorithm is used to solve the optimization model to update the global design variables, thereby obtaining a gradient lattice structure with various types of microstructures distributed in the same design domain, and then obtaining the optimal geometric topology optimization configuration of the gradient multi-lattice structure.

2. The gradient lattice structure topology optimization method for multi-type microstructure boundary matching according to claim 1, characterized in that: In step (10), it is determined whether the iterative optimization process satisfies a given convergence condition. If the convergence condition is satisfied, the optimization is terminated and the optimal functional gradient multi-lattice structure geometric topology optimization configuration is output; If the convergence condition is not met, go to step (6).

3. The gradient lattice structure topology optimization method for multi-type microstructure boundary matching according to claim 2, characterized in that: The convergence criterion is defined as follows: or n≥n max Where δ is the given threshold, n max Represents the maximum number of iterations allowed during optimization.

4. The gradient lattice structure topology optimization method for multi-type microstructure boundary matching according to claim 1, characterized in that: Parameterized level set function of microstructure unit cell The definition is as follows: In the formula, the coefficient is the weight coefficient with respect to pseudo-time t, which specifically means the global design variable of the i-th microstructure on the k-th unit cell in the design domain, and its value range is [0 1]; and is the unit cell level set function of the predefined type i microstructure at the kth unit cell position in the design domain. According to the definition of the local level set function, the predefined unit cell is periodically distributed in the design domain. and Can also write and γ i (x), and γ i (x) is the predefined type i microstructure, specifically representing the level set function of the same type of microstructure unit cell at different volume fractions, r is the number of predefined unit cell types, and M is the total number of unit cells after the design domain is discretized.

5. The gradient lattice structure topology optimization method for multi-type microstructure boundary matching according to claim 4, characterized in that: By constructing parameterized level set functions of various types of microstructure cells, multiple sub-microstructure cell topological configurations are obtained based on the zero-level plane cutting of the level set function. The implicit level set modeling is expressed as follows: Where, The area represents the solid area of ​​the microstructure, The area represents the void area in the microstructure. It is the boundary between the solid area and the void area of ​​the microstructure.

6. The gradient lattice structure topology optimization method for multi-type microstructure boundary matching according to claim 5, characterized in that: The Boolean sum operation is as follows: Where, Φ k (x, t) represents the parameterized level set function of the kth microstructure unit cell in the design domain; The parameterized level set function is defined on the design domain unit cell. According to the number of pre-divided units in the design domain space, the corresponding number of level set functions is defined. The parameterized level set functions of all local unit cells are assembled to obtain the global level set function of the lattice structure, which is defined as follows: Where M represents the total number of unit cells after the design domain is discretized, r represents the number of types of microstructures contained in the design domain, the region with Φ(x, t)>0 represents the solid region in the lattice structure, and the region with Φ(x, t)<0 represents the void region in the lattice structure.

7. The gradient lattice structure topology optimization method for multi-type microstructure boundary matching according to claim 6, characterized in that: The Hamilton-Jacobi partial differential equation expresses the level set function in the velocity field v n The evolutionary form driven by is defined as follows: Substitute the defined parameterized level function into the original Hamilton-Jacobi partial differential equation and transform it into an ordinary differential equation. The expression is as follows: Where, is the modulus of the gradient of the level set function.

8. The gradient lattice structure topology optimization method for multi-type microstructure boundary matching according to claim 7, characterized in that: The implicit geometric description method of parameterized level set function is used to express the geometric model of the structure, and the finite element method is used to calculate the mechanical response of the structure.

9. The gradient lattice structure topology optimization method for multi-type microstructure boundary matching according to claim 7, characterized in that: Define the global weight coefficient at each node of the discrete unit cell, which is the global design variable in the optimization process: T i (t)=[T i 1 (t)T i 2 (t)T i 3 (t)…T i m (t)] Τ Where, T i (t) represents the global design variable of the predefined i-th microstructure in the design domain, and m is the total number of unit cell nodes contained in the discretized design domain; For the local weight coefficients at all nodes of the unit cell, that is, the local design variables, the shape function N(x) is used for interpolation.

10. The gradient lattice structure topology optimization method for multi-type microstructure boundary matching according to claim 9, characterized in that: The mathematical expression of the topology optimization mathematical model of functionally graded lattice structure is: find:T i k (t) min: Where J(Φ) represents the total flexibility of the lattice structure, ε represents the strain field of the structure, u is the displacement field of the structure, E is the equivalent elastic matrix of the predefined material, and a Ф (u,v)=l Ф (v) is the weak form of the elastic equilibrium equation, v represents a virtual displacement domain in the dynamically allowed displacement domain space U, g(Φ) is the volume constraint function of the structure, ζ represents the volume fraction of the structure, represents the volume of the design domain.

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