Self-adaptive isogeometric topology optimization design method and device for shell structure

Abstract

The invention belongs to the related technical field of structure topological optimization, and discloses a shell structure-oriented adaptive isogeometric topological optimization design method and equipment, and the method comprises the following steps: S11, constructing a parameterized model based on the material parameters of a to-be-optimized shell structure and a truncation level B-spline basis function, and constructing a parameterized model; discretizing the displacement field of the shell structure based on the parameterized model to obtain the displacement of each point of the shell structure; s12, constructing a topological optimization model of a to-be-optimized shell structure based on an isotropic material penalty method of a complete adaptive strategy, and performing topological optimization design on the shell structure to obtain a density design variable and active unit distribution of an iterative step; and S13, according to the maximum variable quantity of the density design variables of two continuous iteration steps and the density difference of adjacent active units, performing local refining and coarsening on the hierarchical grids of the to-be-optimized shell structure by using a complete self-adaptive strategy so as to complete the self-adaptive isogeometric topology optimization design of the shell structure. The precision and the efficiency are improved.

Description

Technical Field

[0001] The present invention belongs to the technical field related to structural topology optimization, and more specifically, relates to an adaptive isogeometric topology optimization design method and device for shell structures. Background Art

[0002] Traditional manual design has its limitations and relies on the designer's years of design experience and knowledge reserves. In today's era, the manufacturing industry has high-precision and high-efficiency design requirements, and traditional manual design can no longer meet this requirement. Structural optimization technology based on numerical analysis methods can obtain structural design solutions with better performance through computer calculations under given design constraints, and is one of the main ways to achieve intelligent design of industrial products. Shell structures are widely used in various industrial fields, such as automobiles, aerospace, and ships, due to their lightweight and efficient load-bearing capacity and versatility. Reasonable shell structure design can improve the load-bearing capacity of the structure to meet more industrial needs. In recent years, topology optimization has become an efficient design tool in the field of structural optimization design. It can find a material layout that optimizes the problem under given constraints and in the design domain to meet the expected structural performance. Therefore, topology optimization methods are widely used in the optimization design of shell structures.

[0003] In the structural topology optimization model, the definition of optimization variables depends on the mesh discretization of the design domain, and the accuracy of the sensitivity analysis of optimization variables depends on the accuracy of the numerical analysis solution. The finite element method is the most commonly used structural numerical analysis method for topology optimization. It is used to obtain the unknown physical field of the design domain under different material distribution conditions and drive the iterative update of the optimization design variables. In terms of accuracy, the structural topology optimization method based on the finite element method uses NURBS to express the CAD model, and uses the interpolation approximation of Lagrangian polynomials as the basis function to express the CAE model. There will be errors in the format conversion process between the two, and the use of linear Lagrangian units will reduce the accuracy of the optimization process. There are numerical instability phenomena such as checkerboard and grid dependence. In terms of solution efficiency, the repeated conversion between the CAD model and the CAE model during the optimization iteration process often takes a lot of time. At the same time, the low C0 continuity between finite element units and the low efficiency between high-order units will limit the efficient solution of structural optimization. The above two defects have limited the application of topology optimization in the structural optimization design of industrial products.

[0004] Isogeometric analysis uses spline basis functions as shape functions and sub-faces corresponding to node intervals as calculation units, realizing the integration of design and analysis. Compared with the classical finite element method, isogeometric analysis has the following characteristics: 1) Since the shape function is a high-order basis function, it exhibits good convergence and C1 high-order continuity, which is very suitable for the numerical analysis of thin shell structures; 2) The unification of design and analysis models eliminates the geometric errors caused by mesh discretization of thin shell structures; 3) The geometry of the computational domain can be kept unchanged by node insertion or order increase. Based on the above characteristics, isogeometric analysis has been gradually applied to the topology optimization of shell structures to improve the efficiency and accuracy of optimization. However, limited by the characteristics of the NURBS tensor product structure, the existing structural topology optimization methods based on isogeometric analysis cannot perform adaptive meshing of the structure, which in turn affects the optimization accuracy and efficiency. Summary of the invention

[0005] In view of the above defects or improvement needs of the prior art, the present invention provides an adaptive isogeometric topology optimization design method and device for shell structures, which aims to solve the problem that the existing structural topology optimization method based on isogeometric analysis cannot adaptively mesh the structure, resulting in low optimization accuracy and efficiency.

[0006] To achieve the above object, according to one aspect of the present invention, there is provided an adaptive isogeometric topology optimization design method for shell structures, the method comprising the following steps:

[0007] S11, constructing a parameterized model of the shell structure to be optimized based on the material parameters of the shell structure to be optimized and the truncated hierarchical B-spline basis function, and then discretizing the displacement field of the shell structure based on the parameterized model to obtain the displacement of each point of the shell structure;

[0008] S12. constructing a topology optimization model of the shell structure to be optimized based on the isotropic material penalty method of the fully adaptive strategy, and performing topology optimization design on the shell structure based on the obtained displacement and the topology optimization model to obtain density design variables and active unit distribution of the iteration step;

[0009] S13. According to the maximum change of the density design variable in two consecutive iteration steps and the density difference of adjacent active units, the hierarchical grid of the shell structure to be optimized is locally refined and coarsened using a fully adaptive strategy, thereby completing the adaptive isogeometric topology optimization design of the shell structure.

[0010] Furthermore, after local refinement and coarsening, the following steps are also included: check whether the maximum value of the density difference of all active units in consecutive iteration steps is greater than a first threshold and the number of iterations is less than a second threshold, if so, go to step S12; if not, complete the adaptive isogeometric topology optimization design of the shell structure.

[0011] Furthermore, the mathematical expression of the parameterized model is:

[0012]

[0013] In the formula, are the coordinates of the control points of the geometric model, is the basis function of the truncated hierarchical B-spline, and S(ξ,η) is the coordinate of the point on the geometric model.

[0014] Furthermore, the displacement field of the shell structure is discretized using the truncated hierarchical B-spline basis function in the parameterized model expression, and the displacement u of any point S(ξ,η) is solved: h The expression is:

[0015]

[0016] Furthermore, the mathematical expression of the topology optimization model is:

[0017]

[0018] In the formula, is the density design variable of the hierarchical grid, N0 is the maximum level, N l is the number of active elements in layer l, c is the flexibility of the structure, f is the external load vector, u is the global displacement vector of all active control points, is the Young's modulus of the e-th active unit in the l-th layer, and denote the local displacement vector, volume and element stiffness matrix of the active element, respectively. is the volume allowed by the structure, χ represents of allowable space.

[0019] Furthermore, the density interpolation model of the isotropic material penalty method of the fully adaptive strategy is expressed as:

[0020]

[0021] Where E min A very small value is taken to ensure that the stiffness matrix will not be singular. E0 is the initial Young's modulus; p is the penalty factor used to eliminate intermediate density units.

[0022] Further, in step S13, the mathematical description of the fully adaptive strategy is:

[0023]

[0024] Among them, M triggerin The logical value indicating whether the refinement and coarsening of the kth optimization iteration step is triggered or not, tol AMRIt represents the upper limit of the relative error and is updated to half of its current value after each refinement and coarsening.

[0025] Furthermore, the maximum value of all active cell density differences is calculated as: The first threshold is 0.01 and the second threshold is 100.

[0026] The present invention also provides an adaptive isogeometric topology optimization design system for shell structures, the system comprising a memory and a processor, the memory storing a computer program, and the processor executing the adaptive isogeometric topology optimization design method for shell structures as described above when executing the computer program.

[0027] The present invention also provides a computer-readable storage medium, which stores machine-executable instructions. When the machine-executable instructions are called and executed by a processor, the machine-executable instructions prompt the processor to implement the adaptive isogeometric topology optimization design method for shell structures as described above.

[0028] In general, compared with the prior art, the above technical solutions conceived by the present invention mainly have the following beneficial effects:

[0029] 1. The adaptive and geometric topology optimization technology based on truncated hierarchical B-splines was applied to the Kirchhoff-Love shell structure, successfully solving the topology optimization problem of minimizing flexibility / maximizing stiffness. The hierarchical mesh was adaptively refined only in the topological boundary area and coarsened in the principle boundary area, which fully demonstrated the advancement of the adaptive technology of hierarchical B-splines in the field of topological optimization of shell structures; compared with the global refinement method of NURBS, the local adaptive refinement of THB splines is more suitable for the optimization design of shell structures. A high-resolution topology optimization model was obtained with fewer degrees of freedom and design variables, which fully demonstrated the ease of use and effectiveness of the THB spline method in the field of adaptive topology optimization of shell structures, and promoted the development of truncated hierarchical B-splines in the topology optimization of complex shell structures.

[0030] 2. The application of truncated hierarchical B-splines in shell structure topology optimization eliminates the limitation that non-uniform rational B-splines (NURBS) can only perform global analysis mesh refinement, reduces the amount of calculation and improves the optimization efficiency; on the other hand, it avoids the complex topological structure caused by too many design variables to a certain extent and enhances the manufacturability of the optimization model. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] Figure 1 It is a flow chart of an adaptive isogeometric topology optimization design method for shell structures provided by the present invention;

[0032] Figure 2 It is a flow chart of a construction method of truncated hierarchical B-spline provided by an example of the present invention;

[0033] Figure 3 It is a truncated hierarchical B-spline basis function curve in the example of the present invention, with the hierarchical B-spline on the left and the truncated hierarchical B-spline on the right;

[0034] Figure 4 is a geometric model of a shell structure constructed using truncated hierarchical B-splines in an example of the present invention, which are a physical domain and a parameter domain;

[0035] Figure 5 It is an example diagram of displacement before and after deformation of an isogeometric shell based on Kirchhoff-Love theory in an example of the present invention;

[0036] Figure 6 It is the geometric model and boundary conditions of the saddle shell constructed using the 0th level truncated level B-spline in the example of the present invention;

[0037] Figure 7 It is the optimization result of the saddle shell in the example of the present invention under different levels of grids;

[0038] Figure 8 is a convergence curve diagram of strain energy and volume fraction for topological optimization of saddle shell using truncated hierarchical B-splines in an example of the present invention;

[0039] Fig. 9 (a), (b), and (c) are the parameter domain grids with a maximum grid density of 30*30 and a quarter optimization model based on THB and NURBS, respectively; (d), (e), and (f) are the parameter domain grids with a maximum grid density of 60*60 and a quarter optimization model based on THB and NURBS, respectively; (h), (i), and (j) are the parameter domain grids with a maximum grid density of 120*120 and a quarter optimization model based on THB and NURBS, respectively;

[0040] Fig.10 It is the objective function convergence curve of the saddle shell optimization using THB splines and NURBS grids of different densities in the example of the present invention. DETAILED DESCRIPTION

[0041] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain 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 can be combined with each other as long as they do not conflict with each other.

[0042] See also Figure 1 The present invention provides an adaptive isogeometric topology optimization design method for shell structures. The design method is easy to implement. Compared with the fixed grid of non-uniform rational B-spline, the adaptive hierarchical grid using truncated hierarchical B-spline (THB) can solve the topology optimization problem of shell structures with fewer degrees of freedom and design variables, which reflects the effectiveness of the adaptive topology optimization method for shell structures and provides the possibility for the in-depth development of this research direction. Among them, truncated hierarchical B-spline is used instead of non-uniform rational B-spline (NURBS) for adaptive refinement and coarsening of the hierarchical grid of shell structures.

[0043] The design method mainly includes the following steps:

[0044] S11. Constructing a parameterized model of the shell structure to be optimized based on the material parameters of the shell structure to be optimized and the truncated hierarchical B-spline basis function, and then discretizing the displacement field of the shell structure based on the parameterized model to obtain the displacement of each point of the shell structure.

[0045] Material parameters include Young's modulus, Poisson's ratio, geometric parameters, etc.

[0046] The mathematical expression of the parameterized model is:

[0047]

[0048] In the formula, are the coordinates of the control points of the geometric model, is the basis function of the truncated hierarchical B-spline, and S(ξ,η) is the coordinate of the point on the geometric model. According to the isoparametric idea in the finite element method, the isogeometric method can be expressed as using the truncated hierarchical B-spline basis function in the parameterized model expression to discretize the displacement field of the shell structure, and then solve the displacement u of any point S(ξ,η) h The expression is:

[0049]

[0050] The construction steps of truncated hierarchical B-spline are:

[0051] S111, given a node vector sequence {Ξ 0 ,Ξ 2 ,…Ξ N-1}, the lth node vector Ξ l By 0 Perform l binary divisions to obtain the l-th layer function space B l The B-spline basis function is defined by the Cox-de Boor recursion formula:

[0052]

[0053]

[0054] in, is the node vector is the i-th basis function, p and n represent the order of the basis function and the number of control points respectively.

[0055] S112. Given a nested parameter domain sequence in The parameter domain of the first layer consists of the l-1th layer of labeled refinement units; thus, the linearly independent truncated hierarchical B-spline function space is defined as:

[0056] initialization:

[0057] Recursive form: ,in and and

[0058]

[0059] S113, according to the two-scale relationship of the basis functions of adjacent levels, the B-spline function space B of the lth level l Any basis function s can be expressed as the l+1th level B-spline function space B l+1 The linear combination of basis functions, the corresponding formula is:

[0060]

[0061] The truncation operation of the basis function s with respect to the l+1th basis function is defined as: The l+1th layer basis function is removed, and the operation can be defined as:

[0062]

[0063] S12. Construct a topology optimization model of the shell structure to be optimized based on the isotropic material penalty method of the fully adaptive strategy, and perform topology optimization design on the shell structure based on the obtained displacement and the topology optimization model to obtain the density design variables and active unit distribution of the iterative step.

[0064] The mathematical expression of the topology optimization model is:

[0065]

[0066] In the formula, is the density design variable of the hierarchical grid, N0 is the maximum level, N lis the number of active elements in layer l, c is the flexibility of the structure, f is the external load vector, u is the global displacement vector of all active control points, is the Young's modulus of the e-th active unit in the l-th layer, and denote the local displacement vector, volume and element stiffness matrix of the active element, respectively. is the volume allowed by the structure, χ represents of allowable space.

[0067] The density interpolation model of the isotropic material penalty method of the fully adaptive strategy is expressed as:

[0068]

[0069] Where E min A very small value is taken to ensure that the stiffness matrix will not be singular. E0 is the initial Young's modulus; p is the penalty factor used to eliminate intermediate density units.

[0070] S13. Based on the maximum change of the density design variable between two consecutive iterations and the density difference between adjacent active cells, a fully adaptive strategy is used to locally refine and coarsen the hierarchical grid of the shell structure to be optimized.

[0071] The mathematical description of the fully adaptive strategy is:

[0072]

[0073] Among them, M triggerin The logical value indicating whether the refinement and coarsening of the kth optimization iteration step is triggered or not, tol AMR It represents the upper limit of the relative error and is updated to half of its current value after each refinement and coarsening.

[0074] S14. Determine whether the optimization result converges according to the given iterative convergence condition: check whether the maximum value of the density difference of all active units in the continuous iteration steps is greater than the first threshold and the number of iterations is less than the second threshold. If so, go to step S12; if not, complete the adaptive isogeometric topology optimization design of the shell structure.

[0075] The maximum value of all active cell density differences is calculated as: In one embodiment, the first threshold is 0.01 and the second threshold is 100.

[0076] The present invention also provides an adaptive isogeometric topology optimization design system for shell structures, the system comprising a memory and a processor, the memory storing a computer program, and the processor executing the adaptive isogeometric topology optimization design method for shell structures as described above when executing the computer program.

[0077] The present invention also provides a computer-readable storage medium, which stores machine-executable instructions. When the machine-executable instructions are called and executed by a processor, the machine-executable instructions prompt the processor to implement the adaptive isogeometric topology optimization design method for shell structures as described above.

[0078] The present invention is further described in detail with reference to a specific embodiment below. The topology optimization design method provided in the embodiment of the present invention comprises the following steps:

[0079] S11. According to the material parameters of the saddle shell structure (including Young's modulus, Poisson's ratio, geometric parameters, etc.), a parameterized model based on truncated hierarchical B-spline basis functions is constructed, such as Figure 4-Figure 6 As shown, the displacement field of the shell structure is discretized based on the parameterized model to obtain the displacement of each point of the shell structure.

[0080] S12. A topology optimization model of the shell structure to be optimized is constructed based on the isotropic material penalty method of the fully adaptive strategy. The saddle shell structure is topologically optimized based on the obtained displacement and the topology optimization model to obtain the density design variables and active unit distribution of the iterative step.

[0081] S13. Based on the maximum change of the density design variable in two consecutive iteration steps and the density difference between adjacent active cells, a fully adaptive strategy is used to achieve local refinement and coarsening of the hierarchical grid.

[0082] S14. Determine whether the optimization result converges according to the given iterative convergence condition: Check whether the maximum value of the density difference of all active units in consecutive iterations is is greater than the tol threshold and the number of iterations is less than the maxiter threshold; if so, go to step S12; if not, complete the adaptive isogeometric topology optimization design of the saddle shell structure.

[0083] Furthermore, in order to solve the problem of finding a material layout that meets the constraints within the specified design domain to minimize the flexibility of the structure, the topology optimization model can be expressed as:

[0084]

[0085] in, is the density design variable of the hierarchical grid, N0 is the maximum level, N l is the number of active elements in layer l, c is the flexibility of the structure, f is the external load vector, u is the global displacement vector of all active control points, is the Young's modulus of the e-th active unit in the l-th layer, and denote the local displacement vector, volume and element stiffness matrix of the active element, respectively. is the volume allowed by the structure, χ represents of allowable space.

[0086] Furthermore, the displacement field of the shell structure is discretized using the truncated hierarchical B-spline basis function. The displacement u of any point is h The expression is:

[0087]

[0088] Furthermore, the density interpolation model of the isotropic material penalty method is:

[0089]

[0090] Where E min A small value is taken to ensure that the stiffness matrix will not be singular. E0 is the initial Young's modulus. p is a penalty factor used to eliminate intermediate density units.

[0091] Furthermore, the mathematical description of the fully adaptive strategy is:

[0092]

[0093] Among them, M triggerin The logical value indicating whether the refinement and coarsening of the kth optimization iteration step is triggered or not, tol AMR It represents the upper limit of the relative error and is updated to half of its current value after each refinement and coarsening.

[0094] Furthermore, the flowchart of the construction method of truncated hierarchical B-spline is as follows: Figure 2 As shown, the construction steps are as follows:

[0095] S111, given a node vector sequence {Ξ 0 ,Ξ 2 ,…Ξ N-1}, the lth node vector Ξ l By 0 Perform l binary divisions to obtain the l-th layer function space B l The B-spline basis function is defined by the Cox-de Boor recursion formula:

[0096]

[0097]

[0098] in, is the node vector is the i-th basis function, p and n represent the order of the basis function and the number of control points respectively.

[0099] S112. Given a nested parameter domain sequence in The parameter domain of the first layer consists of the l-1th layer of labeled thin units; thus, the linearly independent hierarchical B-spline function space is defined as:

[0100] initialization:

[0101] Recursive form: ,in and and;

[0102]

[0103]

[0104] S113, according to the two-scale relationship of the basis functions of adjacent levels, the B-spline function space B of the lth level l Any basis function s can be expressed as the l+1th level B-spline function space B l+1 The linear combination of basis functions is shown below:

[0105]

[0106] The truncation operation of the basis function s with respect to the l+1th basis function is defined as: The l+1th layer basis function is removed, and the operation can be defined as:

[0107]

[0108] The final truncated hierarchical B-spline basis function curve is as follows: Figure 3 shown.

[0109] The topology optimization results of saddle shell structure using truncated hierarchical B-spline are shown in Figure 7 shown. Figure 8 is the convergence curve of the objective function and constraint function in the optimization process. In the 18th and 31st iterations, the analysis grid is locally refined. Fig. 9 Comparison of parameter domain meshes and configurations optimized using THB splines and NURBS at different mesh densities. Fig.10 This is the convergence curve of the objective function optimized using the local adaptive refinement THB spline and NURBS with different mesh densities. It can be seen that at the same resolution, the objective function values ​​of the two splines differ by 14.02%. Fig. 9It can be seen that the main reason for the large difference in the objective function value is that after global refinement, the design variables increase, which will make it easier to approach the optimal solution. On the other hand, the inconsistent number of design variables will result in a significant difference in the material layout of the optimized shell structure. However, with the increase in the number of design variables and degrees of freedom, the amount of calculation will also surge. Table 1 shows the comparison of the maximum number of degrees of freedom and the number of design variables in the optimization process. This example shows that compared with the global refinement method of NURBS, the local adaptive refinement of THB splines is more suitable for the optimization design of shell structures. A high-resolution optimization model was obtained with fewer degrees of freedom and design variables, which fully demonstrated the ease of use and effectiveness of the THB spline method in the field of adaptive topology optimization of shell structures.

[0110] Table 1 Comparison of the maximum number of degrees of freedom and the number of design variables between THB spline and NURBS in the optimization process

[0111]

[0112] It will be easily understood by those skilled in the art that the above description is only 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 protection scope of the present invention.

Claims

1. An adaptive isogeometric topology optimization design method for shell structures, characterized by: The method comprises the following steps: S11, constructing a parameterized model of the shell structure to be optimized based on the material parameters of the shell structure to be optimized and the truncated hierarchical B-spline basis function, and then discretizing the displacement field of the shell structure based on the parameterized model to obtain the displacement of each point of the shell structure; S12. constructing a topology optimization model of the shell structure to be optimized based on the isotropic material penalty method of the fully adaptive strategy, and performing topology optimization design on the shell structure based on the obtained displacement and the topology optimization model to obtain density design variables and active unit distribution of the iteration step; S13. According to the maximum change of the density design variable in two consecutive iteration steps and the density difference of adjacent active units, the hierarchical grid of the shell structure to be optimized is locally refined and coarsened using a fully adaptive strategy, thereby completing the adaptive isogeometric topology optimization design of the shell structure.

2. The method for adaptive isogeometric topology optimization design of shell structures according to claim 1, characterized in that: After local refinement and coarsening, the following steps are also included: check whether the maximum value of the density difference of all active units in consecutive iteration steps is greater than a first threshold and the number of iterations is less than a second threshold. If so, go to step S12; if not, complete the adaptive isogeometric topology optimization design of the shell structure.

3. The method for adaptive isogeometric topology optimization design of shell structures according to claim 1, characterized in that: The mathematical expression of the parameterized model is: In the formula, are the coordinates of the control points of the geometric model, is the basis function of the truncated hierarchical B-spline, and S(ξ,η) is the coordinate of the point on the geometric model.

4. The method for adaptive isogeometric topology optimization design of shell structures according to claim 3, characterized in that: The displacement field of the shell structure is discretized using the truncated hierarchical B-spline basis function in the parameterized model expression, and the displacement u of any point S(ξ,η) is solved h The expression is:

5. The method for adaptive isogeometric topology optimization design of shell structures according to claim 1, characterized in that: The mathematical expression of the topology optimization model is: In the formula, is the density design variable of the hierarchical grid, N0 is the maximum level, N l is the number of active elements in layer l, c is the flexibility of the structure, f is the external load vector, u is the global displacement vector of all active control points, is the Young's modulus of the e-th active unit in the l-th layer, and denote the local displacement vector, volume and element stiffness matrix of the active element, respectively. is the volume allowed by the structure, χ represents of allowable space.

6. The method for adaptive isogeometric topology optimization design of shell structures according to claim 5, characterized in that: The density interpolation model of the isotropic material penalty method of the fully adaptive strategy is expressed as: Where E min A very small value is taken to ensure that the stiffness matrix will not be singular. E0 is the initial Young's modulus; p is the penalty factor used to eliminate intermediate density units.

7. The method for adaptive isogeometric topology optimization design of shell structures according to claim 1, characterized in that: In step S13, the mathematical description of the fully adaptive strategy is: Among them, M triggerin The logical value indicating whether the refinement and coarsening of the kth optimization iteration step is triggered or not, tol AMR It represents the upper limit of the relative error and is updated to half of its current value after each refinement and coarsening.

8. The method for adaptive isogeometric topology optimization design of shell structures according to claim 2, characterized in that: The maximum value of all active cell density differences is calculated as: The first threshold is 0.01 and the second threshold is 100.

9. An adaptive isogeometric topology optimization design system for shell structures, characterized by: The system includes a memory and a processor, wherein the memory stores a computer program, and when the processor executes the computer program, the adaptive isogeometric topology optimization design method for shell structures according to any one of claims 1 to 8 is executed.

10. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores machine-executable instructions. When the machine-executable instructions are called and executed by the processor, the machine-executable instructions prompt the processor to implement the adaptive isogeometric topology optimization design method for shell structures as described in any one of claims 1-8.

Images