An adaptive isogeometric topology optimization design method and device for shell structure

Abstract

The present application belongs to the technical field of structural topology optimization, and discloses a self-adaptive isometric topology optimization design method and device for shell structures, which comprises the following steps: S11, constructing a parameterized model 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 complete self-adaptive strategy, and performing topology optimization design on the shell structure to obtain the density design variable and active element distribution of the iteration step; S13, using the complete self-adaptive strategy to locally refine and coarsen the hierarchical grid of the shell structure to be optimized according to the maximum change of the density design variable of the continuous two iteration steps and the density difference of the adjacent active elements, and then completing the self-adaptive isometric topology optimization design of the shell structure. The present application improves the precision and efficiency.

Description

Technical Field

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

[0002] Traditional manual design has its limitations, relying on years of design experience and knowledge reserves. In today's manufacturing industry, there are high-precision and high-efficiency design requirements, which traditional manual design can no longer meet. Structural optimization techniques based on numerical analysis methods can use computers to calculate superior structural design schemes under given design constraints, making it one of the main ways to achieve intelligent design in industrial products. Shell structures, with their lightweight, high-efficiency load-bearing capacity and versatility, are widely used in various industrial fields, such as automotive, aerospace, and shipbuilding. A well-designed shell structure can improve its load-bearing capacity to meet more industrial demands. In recent years, topology optimization has become a highly efficient design tool in the field of structural optimization design. It can find an optimal material layout within a given constraint and design domain to meet the expected structural performance. Therefore, topology optimization methods are widely used in the optimization design of shell structures.

[0003] In structural topology optimization models, the definition of optimization variables depends on the mesh discretization of the design domain, and the accuracy of sensitivity analysis of optimization variables depends on the solution accuracy of numerical analysis. The finite element method (FEM) is the most commonly used numerical analysis method for topology optimization, used to obtain the unknown physical field of the design domain under different material distributions, driving iterative updates of optimization design variables. Regarding accuracy, FEM-based structural topology optimization methods use NURBS to represent the CAD model and interpolation approximation using Lagrange polynomials as basis functions to represent the CAE model. Errors can occur during format conversion, and the use of linear Lagrange elements reduces the accuracy of the optimization process, exhibiting numerical instability phenomena such as checkerboard patterns and mesh dependence. In terms of solution efficiency, repeated conversion between the CAD and CAE models during optimization iterations often consumes significant time. Furthermore, the low continuity of C0 between finite element elements and the low efficiency between higher-order elements limit efficient structural optimization solutions. These two shortcomings restrict 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-surfaces corresponding to node intervals as computational units, achieving integration of design and analysis. Compared to the classical finite element method, isogeometric analysis has the following characteristics: 1) Since the shape functions are high-order basis functions, they exhibit good convergence and C1 high-order continuity, making them very suitable for numerical analysis of thin shell structures; 2) The unification of design and analysis models eliminates geometric errors caused by mesh discretization of thin shell structures; 3) The computational domain geometry can be kept invariant through node insertion or order increase methods. Based on these characteristics, isogeometric analysis has been gradually applied to shell structure topology optimization to improve optimization efficiency and accuracy. However, limited by the structural characteristics of NURBS tensor products, existing structural topology optimization methods based on isogeometric analysis cannot adaptively mesh the structure, thus affecting optimization accuracy and efficiency. Summary of the Invention

[0005] To address the aforementioned deficiencies or improvement needs of existing technologies, this invention provides an adaptive isogeometric topology optimization design method and apparatus for shell structures. It aims to solve the problem that existing structural topology optimization methods based on isogeometric analysis cannot adaptively mesh the structure, resulting in low optimization accuracy and efficiency.

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

[0007] S11. Based on the material parameters of the shell structure to be optimized and the truncated hierarchical B-spline basis function, construct a parameterized model of the shell structure to be optimized, and then discretize 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. Construct a topology optimization model for the shell structure to be optimized based on the isotropic material penalty method with a fully adaptive strategy. 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 element distribution of the iteration step.

[0009] S13. Based on the maximum change of the density design variables in two consecutive iterations and the density difference between adjacent active units, a fully adaptive strategy is used to locally refine and coarsen the hierarchical mesh of the shell structure to be optimized, 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 the continuous iteration steps is greater than the first threshold and the number of iterations is less than the second threshold. If yes, proceed to step S12; otherwise, complete the adaptive geometric topology optimization design of the shell structure.

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

[0012]

[0013] In the formula, where The coordinates of the control points of the geometric model. The basis functions for truncating hierarchical B-splines, These are the coordinates of a point on the geometric model.

[0014] Furthermore, by discretizing the displacement field of the shell structure using the truncated hierarchical B-spline basis functions in the parametric model expression, the solution for any point can be obtained. displacement The expression is:

[0015] .

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

[0017]

[0018] In the formula, It is the density design variable of the hierarchical grid. It is the highest level. It is the first The number of active units in the layer, , For the flexibility of the structure, It is the external load vector. It is the global displacement vector of all active control points. It is the first The first layer Young's modulus of each active unit, , and Let these represent the local displacement vector, volume, and element stiffness matrix of the active element, respectively. It is the volume allowed by the structure. express The allowable space.

[0019] Furthermore, the expression for the density interpolation model of the isotropic material penalty method with a fully adaptive strategy is as follows:

[0020]

[0021] in Taking a very small value ensures that the stiffness matrix does not become singular. It is the initial Young's modulus; It is a penalty factor used to eliminate intermediate density units.

[0022] Furthermore, in step S13, the mathematical description of the fully adaptive policy is as follows:

[0023]

[0024] in Indicates the first The logical value that determines whether refinement and coarsening are triggered in each optimization iteration step. This represents the upper limit of the relative error, and is updated to half of its current value after each refinement and coarsening operation.

[0025] Furthermore, the formula for calculating the maximum value of the density difference of all active units is: 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 includes a memory and a processor. The memory stores a computer program, and the processor executes the computer program to perform the adaptive isogeometric topology optimization design method for shell structures as described above.

[0027] The present invention also provides a computer-readable storage medium storing machine-executable instructions, which, when invoked and executed by a processor, cause the processor to implement the adaptive geometric topology optimization design method for shell structures as described above.

[0028] In summary, compared with the prior art, the adaptive geometric topology optimization design method and device for shell structures provided by this invention have the following advantages:

[0029] 1. Adaptive isogeometric topology optimization techniques based on truncated hierarchical B-splines were applied to Kirchhoff-Love shell structures, successfully solving the topology optimization problem of minimizing flexibility / maximizing stiffness. The hierarchical mesh was adaptively refined only in the topological boundary region and coarsened in the principle boundary region, fully demonstrating the advanced nature of hierarchical B-spline adaptive techniques in the field of shell structure topology optimization. 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, fully demonstrating the ease of use and effectiveness of the THB spline method in the field of adaptive topology optimization of shell structures, and promoting 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 of non-uniform rational B-splines (NURBS) that can only perform global analysis mesh refinement, reducing the amount of computation and improving optimization efficiency. On the other hand, it avoids complex topology structures caused by too many design variables to a certain extent, and enhances the manufacturability of the optimization model. Attached Figure Description

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

[0032] Figure 2 This is a flowchart of the method for constructing a truncated hierarchical B-spline provided in this invention.

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

[0034] Figure 4 These are the geometric models of the shell structure constructed using truncated hierarchical B-splines in this invention, namely the physical domain and the parametric domain;

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

[0036] Figure 6 This is the geometric model and boundary conditions of the saddle-shaped shell constructed using the B-spline of the 0th truncated layer in this invention example;

[0037] Figure 7 These are the optimization results of the saddle-shaped shell under different mesh levels in the examples of this invention;

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

[0039] Figure 9 In the diagram, (a), (b), and (c) represent the parametric domain mesh with a maximum mesh density of 30*30 and the quarter-optimization model based on THB and NURBS, respectively; (d), (e), and (f) represent the parametric domain mesh with a maximum mesh density of 60*60 and the quarter-optimization model based on THB and NURBS, respectively; and (h), (i), and (j) represent the parametric domain mesh with a maximum mesh density of 120*120 and the quarter-optimization model based on THB and NURBS, respectively.

[0040] Figure 10This is the convergence curve of the objective function for saddle-shaped shell optimization using THB splines and NURBS meshes of different densities in an example of this invention. Detailed Implementation

[0041] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be 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 illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0042] Please see Figure 1 This invention provides an adaptive isogeometric topology optimization design method for shell structures. This method is easy to implement. Compared to the fixed mesh of non-uniform rational B-splines, the adaptive hierarchical mesh using truncated hierarchical B-splines (THBs) can solve the topology optimization problem of shell structures with fewer degrees of freedom and design variables, demonstrating the effectiveness of the adaptive topology optimization method for shell structures and providing possibilities for further development in this research direction. Specifically, truncated hierarchical B-splines are used instead of non-uniform rational B-splines (NURBSs) for the adaptive refinement and coarsening of the shell structure hierarchical mesh.

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

[0044] S11. Based on the material parameters of the shell structure to be optimized and the truncated hierarchical B-spline basis functions, construct a parameterized model of the shell structure to be optimized, and then discretize 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, and geometric parameters.

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

[0047]

[0048] In the formula, where The coordinates of the control points of the geometric model. The basis functions for truncating hierarchical B-splines, Let be the coordinates of a point on the geometric model. Based on the isoparametric concept in the finite element method, the isogeometric method can be expressed as discretizing the displacement field of the shell structure using the truncated hierarchical B-spline basis functions in the parameterized model expression, thus solving for any point... displacement The expression is:

[0049] .

[0050] The construction steps for a truncated hierarchical B-spline are as follows:

[0051] S111, Given a sequence of node vectors , No. Node vectors From the conduct The second binary search yields the result, then the first... Layer function space The B-spline basis functions are defined by the Cox-de Boor recurrence formula:

[0052]

[0053] in, node vector The basis functions and These represent the order of the basis functions and the number of control points, respectively.

[0054] S112. Given a nested sequence of parameter fields ,in By the The first layer of the labeling refinement unit consists of the first layer of the labeling refinement unit. The layer parameter domain; thus, the linearly independent truncated layer B-spline function space. Defined as:

[0055] initialization: ;

[0056] Recursive form: ,in and ,and ;

[0057] .

[0058] S113. Based on the two-scale relationship of the basis functions of adjacent levels, the first... Hierarchical B-spline function space any basis function Can be expressed as the first Hierarchical B-spline function space The linear combination of basis functions is given by the following formula:

[0059]

[0060] basis functions Regarding the first The truncation operation of layer basis functions is defined as: truncating the basis functions that are fully supported in the parameter domain. The Layer basis function removal, this operation can be defined as:

[0061] .

[0062] S12. Construct a topology optimization model for the shell structure to be optimized using the isotropic material penalty method based on a fully adaptive strategy. 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 element distribution for the iteration step.

[0063] The mathematical expression for the topology optimization model is:

[0064]

[0065] In the formula, It is the density design variable of the hierarchical grid. It is the highest level. It is the first The number of active units in the layer, , For the flexibility of the structure, It is the external load vector. It is the global displacement vector of all active control points. It is the first The first layer Young's modulus of each active unit, , and Let these represent the local displacement vector, volume, and element stiffness matrix of the active element, respectively. It is the volume allowed by the structure. express The allowable space.

[0066] The expression for the density interpolation model of the isotropic material penalty method with a fully adaptive strategy is as follows:

[0067]

[0068] in Taking a very small value ensures that the stiffness matrix does not become singular. It is the initial Young's modulus; It is a penalty factor used to eliminate intermediate density units.

[0069] S13. Based on the maximum change of the density design variable in two consecutive iterations and the density difference between adjacent active cells, a fully adaptive strategy is used to locally refine and coarsen the hierarchical mesh of the shell structure to be optimized.

[0070] The mathematical description of a fully adaptive policy is:

[0071]

[0072] in Indicates the first The logical value that determines whether refinement and coarsening are triggered in each optimization iteration step. This represents the upper limit of the relative error, and is updated to half of its current value after each refinement and coarsening operation.

[0073] S14. Determine whether the optimization result has converged based on the given iterative convergence conditions: 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 yes, proceed to step S12; otherwise, complete the adaptive isogeometric topology optimization design of the shell structure.

[0074] The formula for calculating the maximum value of the density difference of all active units is: In one implementation, the first threshold is 0.01 and the second threshold is 100.

[0075] The present invention also provides an adaptive isogeometric topology optimization design system for shell structures. The system includes a memory and a processor. The memory stores a computer program, and the processor executes the computer program to perform the adaptive isogeometric topology optimization design method for shell structures as described above.

[0076] The present invention also provides a computer-readable storage medium storing machine-executable instructions, which, when invoked and executed by a processor, cause the processor to implement the adaptive geometric topology optimization design method for shell structures as described above.

[0077] The present invention will be further described in detail below with reference to specific embodiments. The topology optimization design method provided by the embodiments of the present invention includes the following steps:

[0078] S11. Based on the material parameters of the saddle-shaped shell structure (including Young's modulus, Poisson's ratio, geometric parameters, etc.), construct a parametric model based on truncated hierarchical B-spline basis functions, such as... Figures 4-6 As shown, the displacement field of the shell structure is then discretized based on the parameterized model to obtain the displacement of each point of the shell structure.

[0079] S12. Construct a topology optimization model for the shell structure to be optimized using the isotropic material penalty method based on a fully adaptive strategy. Based on the obtained displacement and the topology optimization model, perform topology optimization design on the saddle shell structure to obtain the density design variables and active element distribution for the iteration step.

[0080] S13. Based on the maximum change of the density design variable in two consecutive iterations and the density difference between adjacent active cells, a fully adaptive strategy is used to achieve local refinement and coarsening of the hierarchical mesh.

[0081] S14. Determine whether the optimization result has converged based on the given iterative convergence condition: Check whether the maximum value of the density difference of all active units in consecutive iteration steps is reached. Greater than If the threshold and the number of iterations are less than the maxiter threshold, proceed to step S12; otherwise, complete the adaptive isogeometric topology optimization design of the saddle shell structure.

[0082] Furthermore, to address the problem of finding a material layout that meets constraints within a defined design domain to minimize the structural flexibility, the topology optimization model can be expressed as:

[0083]

[0084] in, It is the density design variable of the hierarchical grid. It is the highest level. It is the first The number of active units in the layer, For the flexibility of the structure, It is the external load vector. It is the global displacement vector of all active control points. It is the first The first layer Young's modulus of each active unit, , and These represent the local displacement vector, volume, and element stiffness matrix of the active element, respectively. It is the volume allowed by the structure. express The allowable space.

[0085] Furthermore, the displacement field of the shell structure is discretized using truncated hierarchical B-spline basis functions, and the displacement at any point is... The expression is:

[0086]

[0087] Furthermore, the density interpolation model for the isotropic material penalty method is as follows:

[0088]

[0089] in Taking a very small value ensures that the stiffness matrix does not become singular. It is the initial Young's modulus. It is a penalty factor used to eliminate intermediate density units.

[0090] Furthermore, the mathematical description of a fully adaptive policy is as follows:

[0091]

[0092] in Indicates the first The logical value that determines whether refinement and coarsening are triggered in each optimization iteration step. This represents the upper limit of the relative error, and is updated to half of its current value after each refinement and coarsening operation.

[0093] Furthermore, the flowchart for constructing truncated hierarchical B-splines is as follows: Figure 2 As shown, the construction steps are as follows:

[0094] S111, Given a sequence of node vectors , No. Node vectors From the conduct The second binary search yields the result, then the third... Layer function space The B-spline basis functions are defined by the Cox-de Boor recurrence formula:

[0095]

[0096] in, node vector The basis functions and These represent the order of the basis functions and the number of control points, respectively.

[0097] S112. Given a nested sequence of parameter fields ,in By the The first layer of the labeling refinement unit consists of the first layer of the labeling refinement unit. The layer parameter domain; thus, the linearly independent layer B-spline function space Defined as:

[0098] initialization: ;

[0099] Recursive form: ,in and ,and; ;

[0100] .

[0101] S113. Based on the two-scale relationship of the basis functions of adjacent levels, the first... Hierarchical B-spline function space any basis function It can be expressed as the first Hierarchical B-spline function space The linear combination of basis functions is shown in the following equation:

[0102]

[0103] basis functions Regarding the first The truncation operation of layer basis functions is defined as: truncating the basis functions that are fully supported in the parameter domain. The Layer basis function removal, this operation can be defined as:

[0104]

[0105] The final truncation level B-spline basis function curve is as follows: Figure 3 As shown.

[0106] The topology optimization results for the saddle-shaped shell structure using truncated hierarchical B-splines are as follows: Figure 7 As shown. Figure 8 These are the convergence curves of the objective function and constraint functions during the optimization process. In iterations 18 and 31, the analysis mesh underwent local refinement. Figure 9 This is a comparison of the parametric domain meshes and configurations optimized using THB splines and NURBS under different mesh densities. Figure 10 These are the convergence curves of the objective functions optimized using locally adaptively refined THB splines and NURBS splines with different mesh densities. It can be seen that, at the same resolution, the objective function values ​​optimized using the two splines differ by 14.02%. Figure 9 It is evident that the main reason for the significant difference in the objective function values ​​is that global refinement increases the number of design variables, making it easier to approach the optimal solution. On the other hand, inconsistencies in the number of design variables lead to substantial differences in the material layout of the optimized shell structure. However, with the increase in design variables and degrees of freedom, the computational load also surges. Table 1 compares the maximum number of degrees of freedom and design variables for both methods during the optimization process. This example demonstrates that, compared to 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, fully demonstrating the ease of use and effectiveness of the THB spline method in the field of adaptive topology optimization of shell structures.

[0107] Table 1 Comparison of maximum degrees of freedom and number of design variables for THB splines and NURBS during the optimization process.

[0108]

[0109] Those skilled in the art will readily understand 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 within the scope of protection of the present invention.

Claims

1. A method for adaptive isogeometric topology optimization design for shell structures, characterized in that: The method includes the following steps: S11. Based on the material parameters of the shell structure to be optimized and the truncated hierarchical B-spline basis function, construct a parameterized model of the shell structure to be optimized, and then discretize the displacement field of the shell structure based on the parameterized model to obtain the displacement of each point of the shell structure. S12. Construct a topology optimization model for the shell structure to be optimized based on the isotropic material penalty method with a fully adaptive strategy. 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 element distribution of the iteration step. S13. Based on the maximum change of the density design variables in two consecutive iterations and the density difference between adjacent active units, a fully adaptive strategy is used to locally refine and coarse the hierarchical mesh of the shell structure to be optimized, thereby completing the adaptive isogeometric topology optimization design of the shell structure. The mathematical expression for the topology optimization model is: In the formula, It is the density design variable of the hierarchical grid. It is the highest level. It is the first The number of active units in the layer, , For the flexibility of the structure, It is the external load vector. It is the global displacement vector of all active control points. It is the first The first layer Young's modulus of each active unit , and They represent the first The first layer Local displacement vectors, volumes, and element stiffness matrices of each active element. It is the volume allowed by the structure. express The allowable space.

2. The adaptive isogeometric topology optimization design method for shell structures as described in 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 the continuous iteration steps is greater than the first threshold and the number of iterations is less than the second threshold. If yes, proceed to step S12; otherwise, complete the adaptive geometric topology optimization design of the shell structure.

3. The adaptive isogeometric topology optimization design method for shell structures as described in claim 1, characterized in that: The mathematical expression for the parameterized model is: In the formula, where The coordinates of the control points of the geometric model. The basis functions for truncating hierarchical B-splines, These are the coordinates of a point on the geometric model.

4. The adaptive isogeometric topology optimization design method for shell structures as described in claim 3, characterized in that: Discretizing the displacement field of the shell structure using the truncated hierarchical B-spline basis functions in the parametric model expression, then solving for any point... displacement The expression is: 。 5. The adaptive isogeometric topology optimization design method for shell structures as described in claim 1, characterized in that: The expression for the density interpolation model of the isotropic material penalty method with a fully adaptive strategy is as follows: in Taking a very small value ensures that the stiffness matrix does not become singular. It is the initial Young's modulus; It is a penalty factor used to eliminate intermediate density units.

6. The adaptive isogeometric topology optimization design method for shell structures as described in claim 1, characterized in that: In step S13, the mathematical description of the fully adaptive policy is as follows: in Indicates the first The logical value that determines whether refinement and coarsening are triggered in each optimization iteration step. This represents the upper limit of the relative error, and is updated to half of its current value after each refinement and coarsening operation.

7. The adaptive isogeometric topology optimization design method for shell structures as described in claim 2, characterized in that: The formula for calculating the maximum value of the density difference of all active units is: ； The first threshold is 0.01, and the second threshold is 100.

8. An adaptive isogeometric topology optimization design system for shell structures, characterized in that: The system includes a memory and a processor. The memory stores a computer program, and when the processor executes the computer program, it performs the adaptive isogeometric topology optimization design method for shell structures as described in any one of claims 1-7.

9. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores machine-executable instructions, which, when invoked and executed by a processor, cause the processor to implement the adaptive isogeometric topology optimization design method for shell structures as described in any one of claims 1-7.

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