Design object structure topological optimization method based on adaptive material field series expansion

Abstract

The invention provides a design object structure topological optimization method based on adaptive material field series expansion. The method has a self-adaptive correlation function which considers spatial correlation and load correlation between field points in the material field at the same time, the correlation of the material field is described more accurately, and the series expansion effect of the material field is improved. A topological optimization adaptive material field series expansion method with double circulation is provided based on an adaptive correlation function, and the optimization efficiency is improved while the design variable scale is reduced and the checkerboard phenomenon and the grid dependence are avoided.

Description

Technical Field

[0001] The present invention relates to the technical field of structural topology optimization, and particularly relates to a structural topology optimization method for a design object based on the series expansion of an adaptive material field. Background Art

[0002] Thin plate structures are widely used in engineering practice. Due to their low stiffness and damping, they are prone to vibration during operation. Laying free damping materials can achieve vibration reduction and noise reduction of thin plate structures, and the operation is convenient and the cost is low. However, laying free damping materials throughout significantly increases the mass of the thin plate structure, and it is necessary to perform topological optimization of the damping material layout.

[0003] The variable density method widely used in structural topology optimization has problems such as a high dimension of design variables and a large scale of the mathematical model. To overcome this problem of the variable density method, the Material Field Series Expansion (MFSE) method uses the theory of random field series expansion to convert the uncertainty of the material field representing the structure topology into the uncertainty of the expansion coefficients, greatly reducing the scale of design variables and avoiding the checkerboard phenomenon and mesh dependence at the same time. The correlation function between field points in the material field series expansion method is used to describe the correlation between material field points and is a key factor for the dimensionality reduction series expansion of the material field. However, the stationary correlation function established by the material field series expansion method only considers the spatial correlation between field points, and the correlation description is not accurate enough. The stationary correlation function needs to be further improved to improve the dimensionality reduction expansion accuracy of the material field. At the same time, the stationary correlation function does not consider the characteristics of the continuous evolution of the material field representing the structure topology during the topological optimization process, and the topological optimization efficiency also needs to be improved.

[0004] Therefore, it is of great significance to develop a structural topology optimization method for a design object based on the series expansion of an adaptive material field. Summary of the Invention

[0005] The purpose of the present invention is to provide a structural topology optimization method for a design object based on the series expansion of an adaptive material field to solve the problems existing in the prior art.

[0006] The technical solution adopted to achieve the purpose of the present invention is as follows: A structural topology optimization method for a design object based on the series expansion of an adaptive material field includes the following steps:

[0007] 1) Predefine the design domain Ω for the topological optimization of the design object des , finite element mesh division, material properties, and boundary conditions.

[0008] 2) Take each unit as a field point of the material field, and the field point position is represented by z iIt is indicated that the material field function is φ(z). For the established bounded uncertain material field φ(z) ∈ [-1, 1], the mapping of the structural topology and the material distribution field function is carried out, and the mapping relationship between the structural topology and the material field function value at the field point z i is defined as follows:

[0009]

[0010] In the formula, Ω void and Ω solid respectively represent the solid elements and void elements in the topology optimization design domain Ω des .

[0011] 3) Adaptive series expansion of the bounded material field. Based on the adaptive material field series expansion strategy, an adaptive correlation function C r with spatial correlation and load correlation is introduced to describe the correlation between the field points of the material field. For the field points z a and z b , the adaptive correlation function C r is:

[0012]

[0013] In the formula, ‖·‖ represents the 2-norm. l c is defined as the correlation length between the field points of the material field, and its size controls the minimum size in the optimization result. x a and x b are the relative density values of the elements at the field points z a and z b respectively, and they are functions of the projected field function value .

[0014] By analogy with the K-L series expansion commonly used in the random field model, the series expansion of the established bounded material field is carried out. For the bounded uncertain material field with N e field points, its series expansion form is:

[0015]

[0016] In the formula, is the vector composed of the adaptive correlation functions between the field point z in the uncertain material field and other field points. η k is the uncertain coefficient of the expansion of the uncertain material field. λ k and ψ k are the eigenvalues and eigenvectors of the correlation function matrix C r , which can be obtained through the following eigenvalue problem:

[0017]

[0018] In the formula, the eigenvalue λ of the correlation matrix k is arranged in descending order.

[0019] 4) Dimensionality reduction of the expansion coefficients of the material field series. Define the truncation criterion as:

[0020]

[0021] In the formula, ε t is the given truncation error. K is the number of eigenvectors retained after truncation.

[0022] The dimensionality-reduced series expansion of the material field after truncation is:

[0023]

[0024] Equation (6) is rewritten in matrix form as:

[0025]

[0026] In the formula, the coefficient vector η = {η 1 η 2 … η K}. T The eigenvalue diagonal matrix Λ = diag(λ 1 λ 2 … λ K ). The eigenvector matrix

[0027] 5) Obtain the structural topology optimization mathematical model based on the series expansion of the material field. Use the Heaviside function to project the material field function φ(z) after dimensionality-reduced series expansion:

[0028]

[0029] In the formula, β is the smoothing parameter of the projection function. When β approaches infinity, the projected material field is approximately a discrete material field of -1 or 1.

[0030] Taking the element as the material field point, map the material field point value to the relative density value x of the element i :

[0031]

[0032] Adopt the interpolation function in the SIMP method, and the relationship between the projected material field and the Young's modulus of the material is defined as:

[0033]

[0034] In the formula, p is the penalty factor for suppressing intermediate density elements, taken as 3. Emin The minimum value selected to avoid numerical problems.

[0035] Based on the above projection and mapping relationships, the established structural topology optimization mathematical model is:

[0036]

[0037] In the formula, f(η k ) is the topology optimization objective function. g(η k ) is the equality constraint function. h(η k ) is the inequality constraint function. Compared with the topology optimization model established based on the variable density method, in Equation (11). The dimension of the design variables in the topology optimization model is reduced from N e to K.

[0038] 6) Perform topology optimization based on the double-loop adaptive material field series expansion method. The outer loop calculates the adaptive correlation function matrix of the current material field Based on it, a corresponding topology optimization mathematical model is established by material field series expansion, and then passed to the inner loop. In the inner loop, under the current optimization mathematical model, the MMA optimizer is used to search for the optimal solution. When the inner loop converges, the obtained optimal design variables are passed back to the outer loop to reconstruct a new material field, thereby updating the adaptive correlation function matrix and the topology optimization model.

[0039] 7) According to the optimized material field design variables, project to obtain the layout form of the design object, and obtain the finally optimized topological configuration.

[0040] Furthermore, the design object is a free damping plate. The free damping plate includes a base plate and a damping layer. For the material layout of the damping layer, the method described in Claim 1 is used to perform topology optimization for maximizing the structural modal loss factor.

[0041] The technical effects of the present invention are beyond doubt: considering both the spatial correlation and load correlation between field points in the material field, more accurately describing the material field correlation, and improving the effect of material field series expansion. While reducing the scale of design variables and avoiding checkerboard phenomena and mesh dependence, the optimization efficiency is improved. Brief Description of the Drawings

[0042] Figure 1 is the double-loop adaptive material field series expansion method;

[0043] Figure 2 is the comparison of the effects of material field series expansion under different correlation functions;

[0044] Figure 3 is a topology optimization example of a free damping plate;

[0045] Figure 4Optimal damping distribution of a cantilever free damping plate under different methods;

[0046] Figure 5 Convergence curve of the topological optimization objective function for a cantilever free damping plate;

[0047] Figure 6 Optimal damping distribution of a simply supported free damping plate under different methods;

[0048] Figure 7 Convergence curve of the topological optimization objective function for a simply supported free damping plate. Detailed implementation manners

[0049] The present invention will be further described below in conjunction with embodiments, but it should not be understood that the above-mentioned subject matter scope of the present invention is limited to the following embodiments. Without departing from the above technical idea of the present invention, various substitutions and modifications made according to the common general knowledge and customary means in the art should be included within the protection scope of the present invention.

[0050] Embodiment 1:

[0051] This embodiment provides a structural topology optimization method for a design object based on the adaptive material field series expansion, including the following steps:

[0052] 1) Establish a bounded material field for structural topology optimization:

[0053] Regarding each unit in the variable density method as a field point of the material field, the position of the field point is represented by z i and the material field function is Since the distribution information of the material field is unknown, a bounded material field is established to describe the uncertainty of the material field based on the bounded uncertain field theory. For the established bounded uncertain material field the mapping relationship between the structural topology at the field point z i and the material field function value is defined as

[0054]

[0055] wherein, Ω void and Ω solid respectively represent the solid elements and void elements in the topological optimization design domain Ω des

[0056] 2) Establish an adaptive correlation function of the material field:

[0057] ​In the series expansion of Equation (3), there are material field-related functions to be determined. For the established bounded material field, its correlation is described. First, considering that the material field of topology optimization is continuously distributed and gradually changing, there is spatial correlation between the field points of the material field. The correlation is related to the distance between the field points. The closer the distance, the higher the correlation degree between the field points; conversely, the weaker the correlation. Second, during the evolution of the material field, the relative density values of the elements at important field points gradually tend to 1, while those at unimportant field points gradually tend to 0. As solid field points and void field points continuously emerge, the structure topology gradually forms according to the needs of bearing the load (generalized). Therefore, there is load correlation between solid field points due to jointly bearing the load, while void field points cannot bear the load and there is no load correlation between them. Considering the spatial correlation and load correlation between the field points of the material field comprehensively, an adaptive correlation function that describes the correlation between field points more comprehensively is proposed. For field points z a and z b , the adaptive correlation function C r is as follows:

[0058]

[0059] In the formula, ‖·‖ represents the 2-norm; l c is defined as the correlation length between the field points of the material field, and its size controls the minimum size in the optimization result; x a and x b are the relative density values of the elements at field points z a and z b respectively, and they are functions of the projected field function value .

[0060] When the elements at field points z a and z b are close to each other and the relative density values are large, the spatial correlation and load correlation between the field points are strong, and the value of the adaptive correlation function C r is large; when the elements at field points z a and z b are close to each other or the relative density values are large, the spatial correlation or load correlation between the field points is strong, and the value of the adaptive correlation function C r is slightly smaller; when the elements at field points z a and z b are far from each other and the relative density values are small, the spatial correlation and load correlation between the field points are weak, and the value of the adaptive correlation function C r is small. At the same time, since the proposed adaptive correlation function C r contains the relative density values of the elements that change continuously with the progress of topology optimization, during the topology optimization process, the adaptive correlation function changes adaptively with the evolution of the material field to realize the tracking of the structure evolution

[0061] 3) Adaptive series expansion of the bounded material field:

[0062] By analogy with the K-L series expansion commonly used in the random field model, the established bounded material field is expanded in series. For a bounded uncertain material field with N e field points, its series expansion form is:

[0063]

[0064] where is the vector composed of the adaptive correlation functions between the field point z in the uncertain material field and other field points; η k is the uncertain coefficient for the expansion of the uncertain material field; λ k and ψ k are the eigenvalues and eigenvectors of the correlation function matrix C r and can be obtained through the following eigenvalue problem:

[0065]

[0066] where the eigenvalues λ k of the correlation matrix are arranged in descending order.

[0067] 4) Dimension reduction of the coefficients of the material field series expansion:

[0068] Through the series expansion of Equation (3), the uncertainty of the spatial distribution of the bounded uncertain material field is transformed into the uncertainty of the expansion coefficients in another orthogonal space. The dimension of the expansion coefficients is usually the same as the dimension of the field points in the original material field, and all the information of the expanded material field is retained. Usually, the terms with larger eigenvalues in Equation (3) contribute more to the expanded uncertain field than the terms with smaller eigenvalues. Therefore, on the premise of trying to maintain the expansion accuracy, by truncating the terms with smaller eigenvalues, the dimension reduction of the expansion coefficients can be achieved, and the expansion of the high-dimensional material field in a lower-dimensional orthogonal space can be realized. To measure the truncation degree of the eigenvalues, the truncation criterion is defined as the percentage of the sum of the retained eigenvalues in the sum of all eigenvalues as

[0069]

[0070] where ε t is the given truncation error; K is the number of retained eigenvectors after truncation.

[0071] The dimension-reduced series expansion formula of the truncated material field is:

[0072]

[0073] Equation (6) rewritten in matrix form is:

[0074]

[0075] In the formula, the coefficient vector η = {η 1 η 2 … η K} T ; the eigenvalue diagonal matrix Λ = diag(λ 1 λ 2 … λ K ); the eigenvector matrix

[0076] 5) Structural topology optimization mathematical model based on the series expansion of the material field:

[0077] In order to obtain a topological structure with clear boundaries, the Heaviside function is used to project the material field function φ(z) after the dimensionality reduction series expansion:

[0078]

[0079] In the formula, β is the smoothing parameter of the projection function. When β approaches infinity, the projected material field is approximately a discrete material field of -1 or 1.

[0080] The projected material field and the Young's modulus of the material need to further establish a functional relationship. Taking the element as the material field point, the material field point value is mapped to the relative density value x of the element i :

[0081]

[0082] Using the interpolation function in the SIMP method, the relationship between the projected material field and the Young's modulus of the material is defined as:

[0083]

[0084] In the formula, p is the penalty factor for suppressing intermediate density elements, taken as 3; E min is the minimum value selected to avoid numerical problems.

[0085] Combining the above projection and mapping relationships, the established structural topology optimization mathematical model is:

[0086]

[0087] In the formula, f(η k ) is the topological optimization objective function; g(η k ) is the equality constraint function; h(η k ) is the inequality constraint function. Compared with the topological optimization model established based on the variable density method, the dimension of the design variable in the topological optimization model in Equation (11) is Ne Reduce to K.

[0088] 6) Topology optimization adaptive material field series expansion method with double loops:

[0089] Using the adaptive correlation function constructed in step 2, a reduced-dimensional expanded structural topology optimization mathematical model was established through steps 3, 4, and 5 in sequence. Since the adaptive correlation function changes adaptively with the evolution of the material field during the topology optimization process, the mathematical model in equation (11) also changes as the optimization progresses, thus always maintaining an accurate expansion effect on the evolving material field. Aiming at the change of the mathematical model during the optimization process, the topology optimization adaptive material field series expansion method with double loops as shown in Figure 1 is proposed. In the figure, it in and it out represent the iteration steps of the inner and outer loops respectively. The outer loop calculates the adaptive correlation function matrix of the current material field and based on it, a material field series expansion is carried out to establish the corresponding topology optimization mathematical model, and then it is passed to the inner loop; the inner loop searches for the optimal solution using the MMA optimizer under the current optimization mathematical model; when the inner loop converges, the obtained optimal design variables (expansion coefficients) are passed back to the outer loop to reconstruct a new material field, thereby updating the adaptive correlation function matrix and the topology optimization model. Due to the more accurate expansion effect and adaptive update of the adaptive correlation function, the accuracy of the expanded material field used in the optimization process is improved, thus accelerating the convergence speed of the topology optimization.

[0090] In the topology optimization adaptive material field series expansion method with double loops, the number of times of eigenvalue decomposition of the adaptive correlation function matrix is the same as the number of outer loop times. When appropriate convergence accuracies ε in and ε out are selected, the topology optimization adaptive material field series expansion method requires fewer outer loops, thus reducing the computational amount brought by the eigenvalue decomposition of the adaptive correlation function matrix. The inner loop convergence accuracy ε in is generally set to be less than the outer loop convergence accuracy ε out , usually ε in is taken as 0.1 times of ε out .

[0091] Example 2:

[0092] The main content of this example is the same as that of Example 1. Among them, in this example, the reduced-dimensional material field series expansion effects are compared. Taking four different distributed material fields as examples, under the same truncation error ε tPerform dimensionality reduction expansion at =0.01, and compare the dimensionality reduction expansion effects of using material field series expansion and adaptive material field series expansion. The four material fields are as follows: Material field A, a 40×20 uniform material field; Material field B, a 40×20 random material field; Material field C, a regularly distributed material field of 50×50; Material field D, a regularly distributed material field of 45×15. The correlation length l between field points during the dimensionality reduction expansion of each material field c is taken as 0.2 times the shorter side length.

[0093] Table 1 Number of eigenvectors retained after dimensionality reduction expansion of material fields under different methods

[0094]

[0095] The expansion effects of the four material fields are as Figure 2 shown. The number of eigenvectors retained after dimensionality reduction expansion by the two methods is shown in Table 1. For the uniform material field A, the correlation function matrix in the adaptive material field series expansion is a scalar multiple matrix of the correlation function matrix in the material field series expansion. Therefore, the expanded material fields are the same, and the number of eigenvectors retained is also equal. For the non-uniformly distributed material fields B, C, and D, the gray cells of the material fields after dimensionality reduction expansion using the adaptive material field series expansion are significantly reduced, and the original material fields are restored more accurately, with the expansion effect being significantly improved. Under the same truncation error, the number of eigenvectors retained using the adaptive material field series expansion is slightly more than that using the material field series expansion. However, compared with the number of field points in the original material field, the adaptive material field series expansion also shows an obvious dimensionality reduction effect. Therefore, the proposed adaptive material field series expansion that takes into account the spatial correlation and load correlation between field points has a better dimensionality reduction expansion effect of the material field while reducing the dimension of the expansion coefficients.

[0096] Example 3:

[0097] The main content of this example is the same as that of Example 1. Among them, in this example, the topology optimization of a free damping plate is carried out. Thin plate structures are widely used in engineering practice. Due to their low stiffness and damping, they are prone to vibration during operation. By laying free damping materials, vibration reduction and noise reduction of thin plate structures can be achieved, and the operation is convenient and the cost is low. However, laying free damping materials throughout significantly increases the mass of the thin plate structure. Therefore, topology optimization of the layout of free damping materials can improve the damping characteristics of the thin plate structure while reducing the mass increase. For the layout of damping materials for thin plate structures, the proposed topology optimization adaptive material field series expansion (AMFSE) method is used to perform topology optimization to maximize the structural modal loss factor.

[0098] The example of the free damping plate is as Figure 3As shown, the size of the plate is 0.5×0.4m, and it is divided into 50×40 two-dimensional finite element meshes. The material properties of the base plate and the damping layer are shown in the table, and the volume fraction constraint limit of the damping material is 0.5.

[0099] Table 2 Material properties of the free damping plate

[0100]

[0101] For Figure 3 a fixed displacement boundary constraint is applied to a short side of the free damping plate in , and the other three sides are free boundaries, forming a cantilever free damping plate. The material field series expansion (MFSE) method and the adaptive material field series expansion (AMFSE) method are respectively used to perform topology optimization on the cantilever free damping plate to maximize the first-order modal loss factor, and the truncation error of the series expansion is taken as ε t = 0.01, and the correlation length l c of the material field is taken as 0.1 times the minimum side length of the design domain. Taking the change amount of the objective function between adjacent iteration steps as the convergence judgment condition, the convergence accuracy of the material field series expansion (MFSE) method is 0.01, and the convergence accuracies of the inner and outer loops in the adaptive material field series expansion (AMFSE) method are both set to 0.01.

[0102] To measure the proportion of gray elements in the optimization results, the discreteness M nd is used as an index, and the calculation method is:

[0103]

[0104] As can be seen from the above formula, the closer the relative density x i of the element is to 0 or 1, the smaller the value of the discreteness M nd , indicating that the fewer gray-scale elements and the clearer the structure topology; otherwise, the more intermediate-density elements, the larger the value of the discreteness M nd , and the more blurred the structure topology.

[0105] The optimization results of the cantilever free damping plate using the two methods are shown in the table. The relative change in the table refers to the change ratio of the optimization result of the adaptive material field series expansion (AMFSE) method relative to the optimization result of the material field series expansion (MFSE) method. The optimal layout of the free damping material in the two methods is as Figure 4 described. As can be seen from Table 3, the first-order modal loss factors of the cantilever free damping plates optimized by the adaptive material field series expansion (AMFSE) method and the material field series expansion (MFSE) method are similar, Figure 4The optimal damping material layouts are also similar, verifying the effectiveness of using the Adaptive Material Field Series Expansion (AMFSE) method for structural damping topology optimization. In the optimization results of both methods, no checkerboard phenomenon occurs. Comparing the number of design variables of the two methods, the number of expansion bases used in the Adaptive Material Field Series Expansion (AMFSE) method (207, 207, 210) is slightly larger than that of the Material Field Series Expansion (MFSE) method (207). Compared with the number of field points in the material field, a significant compression of the design variable dimension is achieved, and the reduction effect on the scale of the topology optimization model is obvious. In addition, compared with the Material Field Series Expansion (MFSE) method, the number of iterations in the topology optimization using the Adaptive Material Field Series Expansion (AMFSE) method is reduced by 72.62%, and the iteration time is reduced by 48.83%. Therefore, the Adaptive Material Field Series Expansion (AMFSE) method has higher optimization efficiency and less computational effort in topology optimization compared with the Material Field Series Expansion (MFSE) method.

[0106] Table 3 Topology optimization results of a cantilever free damping thin plate under different methods

[0107]

[0108] The convergence curves of the damping topology optimization objective function of the cantilever free damping plate under the two methods are as Figure 5 shown. It can be seen from the figure that the convergence curve of the objective function of the Adaptive Material Field Series Expansion (AMFSE) method drops faster, the convergence speed is better than that of the Material Field Series Expansion (MFSE) method, and the convergence curve is smoother. Change the boundary conditions of the free damping plate, apply fixed constraints to its four sides to form a four-sided clamped free damping plate. The Material Field Series Expansion (MFSE) method and the Adaptive Material Field Series Expansion (AMFSE) method are respectively used to perform damping topology optimization for maximizing the first-order modal loss factor of the four-sided clamped free damping plate. The damping topology optimization results of the four-sided clamped free damping plate using the Material Field Series Expansion (MFSE) method and the Adaptive Material Field Series Expansion (AMFSE) method are shown in Table 4, and the optimal layouts of the free damping material in the two methods are as Figure 6 shown.

[0109] Table 4 Topology optimization results of a four-sided clamped free damping plate under different methods

[0110]

[0111] In the topology optimization of the first-order modal loss factor of a four-sided clamped free damping plate, the effectiveness and convergence rate of the adaptive material field series expansion (AMFSE) method are also verified. The first-order modal loss factor optimized by the adaptive material field series expansion (AMFSE) method is almost the same as that optimized by the material field series expansion (MFSE) method, while the number of iteration steps and iteration duration of the topology optimization are reduced by 70.45% and 47.47% respectively. Figure 6 In both methods, the layout of the damping materials obtained by optimization is generally the same, and the material distribution is in good agreement with the first-order modal vibration mode of the four-sided clamped free damping plate. The free damping material is mainly distributed in the regions with larger modal strain energy. Figure 7 The convergence curves of the topology optimization objective function of the four-sided clamped free damping plate using the two methods are shown. The convergence curves of the objective function indicate that the damping topology optimization using the adaptive material field series expansion (AMFSE) method has a faster convergence rate. During the optimization process of the four-sided clamped free damping thin plate, the number of design variables used in the adaptive material field series expansion (AMFSE) method (207, 240, 250) is slightly larger than that in the material field series expansion (MFSE) method (207). Compared with the number of field points in the material field, a significant reduction in the dimension of the design variables is achieved, and the reduction effect on the scale of the topology optimization model is obvious. In addition, there is no checkerboard phenomenon in the damping optimization results of the two methods.

Claims

1. A method for topological optimization of a design object structure based on adaptive material field series expansion, characterized in that: The following steps are involved: 1) Predefine the design domain Ω for topology optimization of the design object des , finite element meshing, material properties and boundary conditions; 2) Each unit is regarded as a field point of the material field, and the field point position is represented by z i It means that the material field function is φ(z); For the established bounded uncertain material field φ(z)∈[-1,1], the mapping between the structural topology and the material distribution field function is performed, and the field point z i The mapping relationship between the structural topology and the material field function value at is defined as: In the formula, Ω void and Ω solid They represent the topology optimization design domain Ω des Solid elements and void elements in ; 3) Adaptive series expansion of bounded material fields; Based on the adaptive material field series expansion strategy, an adaptive correlation function C with spatial correlation and load correlation is introduced. r Describe the correlation between material field points; for field point z a and z b , adaptive correlation function c r for: In the formula, ‖·‖ represents the 2-norm; l c Defined as the correlation length between material field points, its size controls the minimum size in the optimization result; x a and x b are field points z a and z b The relative density value of the unit at is the field function value after projection Function of Analogous to the KL series expansion commonly used in random field models, the established bounded material field is expanded in series; for N e The bounded uncertain material field of field points has the series expansion form: In the formula, is the vector formed by the adaptive correlation function between the field point z and other field points in the uncertain material field; η k is the uncertainty coefficient of the uncertain material field expansion; λ k and ψ k is the correlation function matrix C r The eigenvalues ​​and eigenvectors of can be obtained through the following eigenvalue problem: In the formula, the eigenvalue λ of the correlation matrix is k Sort in descending order; 4) Dimension reduction of material field series expansion coefficients; define the truncation criterion as: In the formula, ε t is the given truncation error; K is the number of eigenvectors retained after truncation; The dimension reduction series expansion of the truncated material field is: Formula (6) is rewritten into matrix form as follows: In the formula, the coefficient vector η={η1η2…η K } T ; The eigenvalue diagonal matrix Λ=diag(λ1λ2…λ K ) ; eigenvector matrix 5) Obtain a mathematical model for structural topology optimization based on material field series expansion; Use the Heaviside function to project the material field function φ(z) after the dimensionality reduction series expansion: Where β is the smoothness parameter of the projection function; when β tends to infinity, the material field after projection Discrete material fields approximately -1 or 1; Take the element as the material field point and map the material field point value to the element relative density value x i : Using the interpolation function in the SIMP method, the projected material field The relationship between and the Young's modulus of the material is defined as: Where p is the penalty factor for suppressing the intermediate density unit, which is taken as 3; E min The minimum value was chosen to avoid numerical problems; Based on the above projection and mapping relationships, the established mathematical model of structural topology optimization is: In the formula, f(η k ) is the topology optimization objective function; g(η k ) is the equality constraint function; h(η k ) is the inequality constraint function. 6) Perform topology optimization based on the double-loop adaptive material field series expansion method; the outer loop calculates the adaptive correlation function matrix of the current material field Based on the material field series expansion, the corresponding topology optimization mathematical model is established and then passed to the inner loop; The inner loop uses the MMA optimizer to search for the optimal solution under the current optimization mathematical model. When the inner loop converges, the optimal design variables are passed back to the outer loop to reconstruct the new material field, and then update the adaptive correlation function matrix and topology optimization model. 7) According to the material field design variables at the time of optimization convergence, the layout form of the design object is projected to obtain the final optimized topological configuration.

2. The method for topological optimization of a design object structure based on adaptive material field series expansion according to claim 1, characterized in that: The design object is a free damping plate; the free damping plate includes a base plate and a damping layer; with respect to the material arrangement of the damping layer, the method described in claim 1 is used to perform topological optimization for maximizing the structural modal loss factor.

3. The method for topological optimization of a design object structure based on adaptive material field series expansion according to claim 2, characterized in that: A fixed displacement boundary constraint is imposed on one short side of the free damping plate, and the other three sides are free boundaries to form a cantilever free damping plate.

Images