Parameterized level set porous structure topological optimization method considering geometric nonlinearity

Abstract

The invention belongs to the technical field of structure optimization, and particularly discloses a geometric nonlinearity-considered parameterized level set porous structure topological optimization method, which comprises the following steps of: constructing a level set function to implicitly describe a structure boundary, and performing spline interpolation by adopting a tightly-supported radial basis function to obtain a parameterized level set function; establishing an equilibrium equation of geometric nonlinear topological optimization, and solving structural response under the condition of large deformation through geometric nonlinear finite element analysis; approximating and simplifying local volume constraint through a local constraint domain and a p norm function; establishing a topological optimization model based on geometric nonlinear porous filling rigidity maximization; performing sensitivity analysis through an adjoint method, and calculating a partial derivative of an objective function of the topological optimization model to an expansion coefficient so as to determine an optimization direction of each finite element in geometric nonlinear porous filling stiffness maximization; and carrying out iterative optimization to obtain an optimized topological structure. According to the method and the device, porous filling topological optimization under geometric nonlinear conditions can be realized.

Description

Technical Field

[0001] The present application belongs to the technical field of structural optimization, and more specifically, relates to a parameterized level set porous structure topology optimization method considering geometric nonlinearity. Background Art

[0002] The design inspiration of porous structures comes from nature, such as bones and plant stems. These structures have gradually adapted to the influence of external unstable loads during thousands of years of evolution. Compared with traditional optimization results, porous structures show excellent mechanical properties, such as lightweight, enhanced strength-to-weight ratio and robustness in the face of material defects. In addition, topology optimization technology has further promoted the development of porous structure design and has been maturely applied in a variety of design problems, such as flexibility minimization, thermal flexibility minimization and natural frequency maximization. At present, the topology optimization method of porous structures is mainly implemented in the following ways: forming a porous structure by constraining the upper limit of the volume of local materials; generating a porous filling structure by homogenization; indirectly generating a porous structure through the idea of ​​fiber composite materials; and using a coated porous structure composed of an outer shell and an inner porous filling.

[0003] However, despite the increasing application of topology optimization in porous structure design, current filling designs are mostly based on the assumption of linear elastic deformation and have not yet considered geometric nonlinearity. For soft rubber materials or slender structures, when subjected to large deformations, there is a strong nonlinear relationship between the strain and displacement field, showing significant geometric nonlinear characteristics. If traditional linear finite element analysis theory is used to optimize porous structures, the accuracy of topology optimization will be poor.

[0004] Therefore, how to target geometric nonlinear characteristics, cope with complex working conditions such as large deformation and large strain, and improve the accuracy and adaptability of porous design topology optimization is a technical problem that needs to be solved urgently. Summary of the invention

[0005] In view of the defects of the prior art, the purpose of this application is to provide a parameterized level set porous structure topology optimization method taking into account geometric nonlinearity, aiming to solve the problem of insufficient accuracy and adaptability of porous structure topology optimization in the prior art when dealing with complex working conditions due to the still use of linear finite element analysis.

[0006] To achieve the above objectives, in a first aspect, the present application provides a parameterized level set porous structure topology optimization method considering geometric nonlinearity, comprising: The level set function is constructed to implicitly describe the structural boundary, and the parameterized level set function is obtained by using compactly supported radial basis function spline interpolation. Establish the equilibrium equation of geometric nonlinear topology optimization, and use geometric nonlinear finite element analysis to solve the structural response under large deformation conditions; The improved Heaviside function is used to represent the volume fraction of each finite element, and the local volume constraint is approximated and simplified by the local constraint domain and p-norm function. A topology optimization model based on geometrically nonlinear porous filling stiffness maximization is established to minimize structural flexibility. Sensitivity analysis is performed by adjoint method to calculate the partial derivatives of the objective function of the topology optimization model with respect to the expansion coefficients, so as to determine the optimization direction of each finite element in the geometrically nonlinear porous filling stiffness maximization problem. The level set discrete value is updated by the moving asymptote method, and the updated level set discrete value is brought into the objective function to perform finite element analysis again, and it is iterated continuously until the convergence condition is met, until the final topology optimization structure is obtained.

[0007] Optionally, the topological implicit description of the parameterized level set includes: The zero isosurface of the level set function is used to implicitly represent the solid, boundary and holes of the structure. Determine a pseudo-time parameter, use the pseudo-time parameter to describe the evolution of the structure shape over time, and establish a partial differential equation of the level set function and time to track the dynamic changes of the structure boundary over time; The level set function is used to represent the evolution of the structural boundary; A parameterized level set function is constructed by spline interpolation of a spatial shape function, and a normal velocity of a structure boundary is calculated according to the spatial shape function and an expansion coefficient function; The level set function is updated using the normal velocity to simulate the evolution of the structure shape.

[0008] Optionally, the specific steps of the geometric nonlinear finite element analysis include: Constructing a geometrically nonlinear topological optimization equilibrium equation; the equilibrium equation determines the mechanical state of the structure through the variational form of Piola Kirchhoff stress and strain, determines the external action of the structure through virtual displacement and body force and traction, and is established based on the mechanical state and external action; The equilibrium equation is linearized by using Newton's method, and a linearized equation including a geometrically nonlinear strain-displacement matrix, a strain-displacement matrix caused by a stress state, and an elastic constitutive matrix is ​​established; The displacement increment and the residual force vector are solved based on the linearized equation to obtain the displacement state of the structure within one iteration step under the condition of geometric nonlinearity for linear expression.

[0009] Optionally, the aggregation strategy of the local volume constraint includes: Determining an improved first function and a second function, wherein the first function represents the volume fraction of each finite element, and the second function represents the material proportion in a specified local constraint domain; the local constraint domain is determined by establishing a circular region of a specified radius around each finite element; The maximum value of the local volume fraction is set as the upper limit of the material ratio, multiple local constraints are simplified into a single constraint function, and the constraint function is approximated by a p-norm function; A structural volume constraint formula is determined based on the approximated constraint function, and the structural volume is constrained based on the volume constraint formula.

[0010] Optionally, the method for establishing the topology optimization model includes: Acquire an actual displacement field and a virtual displacement field of a target space, and determine a strain energy density according to the actual displacement field and the virtual displacement field; An objective function corresponding to the topology optimization model is established according to the strain energy density and the Heaviside function, and structural flexibility is minimized based on the objective function to maximize the geometrically nonlinear porous filling stiffness; Among them, the objective function satisfies energy balance constraints, volume constraint formulas, internal and external vector balances, and boundary constraints; the energy balance constraints are determined based on the linear elastic equilibrium equation, and the linear elastic equilibrium equation is determined based on the energy function and the load function.

[0011] Optionally, the sensitivity analysis process includes the following steps: Calculating the derivative of the energy function and the derivative of the load function to obtain a derivative expression; According to the equivalence relationship between the energy function and the load function and the derivative expression, the shape derivative of the objective function is determined; According to the residual force vector being 0, the objective function is corrected by the adjoint method, and the Lagrangian multiplier is introduced into the objective function to obtain an improved Lagrangian function; Determine the Lagrange multiplier value to ignore the effect of displacement field on the rate of change of design variables; The sensitivity function of the objective function is obtained according to the Lagrange multiplier value, and the sensitivity of the flexibility of the finite element node is obtained according to the sensitivity function.

[0012] Optionally, the iterative optimization process includes: In the iterative process, according to the current objective function and constraints, strictly convex approximation sub-problems are generated and solved, and the generation of the sub-problems is controlled by the moving asymptotes; Solving the strictly convex approximation subproblem to obtain updated design variables; Substituting the updated design variables into the level set function to determine whether the set convergence conditions are met; It is determined that the convergence condition is met, the iteration is terminated and the optimized topology structure is output.

[0013] The present application also provides a parameterized level set porous structure topology optimization device considering geometric nonlinearity, comprising: Implicit description module, used to construct the level set function to implicitly describe the structure boundary, and use compactly supported radial basis function spline interpolation to obtain the parameterized level set function; Finite element analysis module, used to establish the equilibrium equations of geometric nonlinear topology optimization, and solve the structural response under large deformation conditions through geometric nonlinear finite element analysis; Volume constraint module, which is used to express the volume fraction of each finite element using the improved Heaviside function, and approximate and simplify the local volume constraint through the local constraint domain and p-norm function; The flexibility minimization module is used to establish a topology optimization model based on geometrically nonlinear porous filling stiffness maximization to minimize structural flexibility; Sensitivity analysis module, which is used to perform sensitivity analysis by adjoint method and calculate the partial derivative of the objective function of the topology optimization model with respect to the expansion coefficient to determine the optimization direction of each finite element in the geometric nonlinear porous filling stiffness maximization; The iterative optimization module is used to update the discrete value of the level set by the moving asymptote method, bring the updated discrete value of the level set into the objective function to perform finite element analysis again, and continuously iterate until the convergence conditions are met, until the final topology optimization structure is obtained.

[0014] In general, the above technical solutions conceived by this application have the following beneficial effects compared with the prior art: (1) The parameterized level set porous structure topology optimization method considering geometric nonlinearity provided in this application can improve the accuracy and applicability of topology optimization. By using the parameterized level set method, space and time are separated and the topological structure change is controlled by the expansion coefficient. When dealing with complex geometric nonlinear porous structures, more accurate optimization results can be achieved, the accuracy and adaptability of topology optimization can be improved, and it is suitable for large deformation structures under various design requirements.

[0015] (2) The parameterized level set porous structure topology optimization method considering geometric nonlinearity provided in this application can improve the optimization efficiency and numerical stability. By introducing local volume constraints and Heaviside functions for aggregate layout, and combining the Lagrangian adjoint method to perform sensitivity analysis on the objective function and, in the optimization process, it is possible to meet local structural requirements and effectively control numerical instability, thereby reducing the number of iterations and improving computational efficiency.

[0016] (3) The parameterized level set porous structure topology optimization method considering geometric nonlinearity provided in this application can expand the scope of application and engineering applicability. It shows excellent performance under static analysis and various design parameters, meets the design requirements of high-strength and lightweight structures, and is suitable for promotion and application in high-requirement engineering fields such as aerospace and construction. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] Figure 1 It is one of the flow diagrams of the parameterized level set porous structure topology optimization method considering geometric nonlinearity provided in the embodiments of the present application; Figure 2 This is the second flow chart of the parameterized level set porous structure topology optimization method considering geometric nonlinearity provided in the embodiment of the present application; Figure 3 It is a schematic diagram of structural topology and static analysis of introducing geometric nonlinearity to classical topology optimization in an embodiment of the present application; Figure 4 It is a schematic diagram of topological and static analysis results of topological optimization of geometrically nonlinear porous filling under different external forces in an embodiment of the present application.

[0018] Figure 5 This is a schematic diagram of the generation of a porous structure by local volume constrained polymerization in GN-ITO of an embodiment of the present application; Figure 6 It is a schematic structural diagram of a parameterized level set porous structure topology optimization device considering geometric nonlinearity provided in an embodiment of the present application. DETAILED DESCRIPTION

[0019] In order to make the purpose, technical solution and advantages of the present application more clearly understood, the present application 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 application and are not used to limit the present application.

[0020] The term "and / or" in this application is a description of the association relationship of associated objects, indicating that there can be three relationships. For example, A and / or B can represent: A exists alone, A and B exist at the same time, and B exists alone. The symbol " / " in this application indicates that the associated objects are in an or relationship, for example, A / B means A or B.

[0021] The terms "first" and "second" in the specification and claims of this application are used to distinguish different objects rather than to describe a specific order of objects. For example, a first response message and a second response message are used to distinguish different response messages rather than to describe a specific order of response messages.

[0022] In the embodiments of the present application, words such as "exemplary" or "for example" are used to indicate examples, illustrations or descriptions. Any embodiment or design described as "exemplary" or "for example" in the embodiments of the present application should not be interpreted as being more preferred or more advantageous than other embodiments or designs. Specifically, the use of words such as "exemplary" or "for example" is intended to present related concepts in a specific way.

[0023] In the description of the embodiments of the present application, unless otherwise specified, "multiple" means two or more than two. For example, multiple processing units refer to two or more processing units, etc.; multiple elements refer to two or more elements, etc.

[0024] The embodiments of the present application are described below in conjunction with the drawings in the embodiments of the present application.

[0025] Reference Figure 1 , the present application provides a parameterized level set porous structure topology optimization method considering geometric nonlinearity, comprising: S101. Construct a level set function to implicitly describe the structure boundary, and use compactly supported radial basis function spline interpolation to obtain a parameterized level set function; S102. Establish the equilibrium equation of geometric nonlinear topology optimization, and solve the structural response under large deformation conditions through geometric nonlinear finite element analysis; S103. Using the improved Heaviside function to represent the volume fraction of each finite element, and approximating and simplifying the local volume constraint by the local constraint domain and the p-norm function; S104. Establish a topology optimization model based on geometric nonlinear porous filling stiffness maximization to minimize structural flexibility; S105. Perform sensitivity analysis by adjoint method and calculate partial derivatives of the objective function of the topology optimization model with respect to the expansion coefficient to determine the optimization direction of each finite element in the geometrically nonlinear porous filling stiffness maximization problem; S106. Update the discrete value of the level set by the moving asymptote method, bring the updated discrete value of the level set into the objective function and perform finite element analysis again, and continue to iterate until the convergence condition is met, until the final topology optimization structure is obtained.

[0026] The following is a detailed description of the various steps of this application.

[0027] The process of constructing the level set function in step S101 is: S11. Implicit description of structural boundaries: Use level set functions to define and represent the boundaries of structures.

[0028] S12. Simulating boundary evolution: Apply partial differential equations (PDEs) to simulate the evolution of structural boundaries over time.

[0029] S13. Parameterized level set function: The parameterized level set function is obtained by using compactly supported radial basis function spline interpolation technology.

[0030] S14 Calculate Normal Velocity: Based on the parameterized level set function, the normal velocity of the structure boundary is calculated to update the boundary during the optimization process.

[0031] In step S102, the process of geometric nonlinear finite element analysis is as follows: S21. Establishing equilibrium equations: Geometrically nonlinear finite element analysis to establish equilibrium equations related to large structural displacements.

[0032] S22. Linearization: Linearize the equilibrium equations to effectively solve large displacement problems and ensure the stability and accuracy of numerical calculations.

[0033] In step S103, the improved Heaviside function is used to represent the volume fraction of each finite element. By introducing the local constraint domain and the p-norm function, the local volume constraint is approximated and simplified to meet the design requirements.

[0034] In step S104, the process of establishing a topology optimization model is as follows: S41. Maximizing the Stiffness of Porous Infills: A topology optimization model is developed to maximize the stiffness of geometrically nonlinear porous infills.

[0035] S42. Minimize structural flexibility: Minimize the flexibility of the structure through the optimization process to improve its overall performance.

[0036] In step S105, the process of sensitivity analysis is: Adjoint sensitivity analysis: The partial derivatives of the objective function of the topology optimization model with respect to the expansion coefficients are calculated through adjoint sensitivity analysis. This allows for sensitivity analysis of geometrically nonlinear porous filling stiffness maximization, providing the necessary sensitivity information for the optimization process.

[0037] In step S106, the moving asymptote method MMA is used to update the discrete value of the level set. The updated discrete value of the level set is substituted into the level set function for iterative optimization until the final optimized topological structure is obtained.

[0038] The complete steps of the embodiment of this application can be referred to Figure 2 , Figure 2 The process includes: Enter initial parameters; Define material properties, loads, and boundary conditions; Establish geometrically nonlinear finite element mesh; Initialize the level set function and expansion coefficient; Establishing local constraint domain of structure; Apply local volume constraints to design variables; Geometrically nonlinear finite element analysis to calculate residual forces; Calculate objective function and constraint function; Conduct sensitivity analysis on objective function and constraint function; Interpolate and filter objective and constraint functions; Use the MMA method to update the level set expansion coefficients; Determine the convergence conditions; Output a geometrically nonlinear porous topology optimization model.

[0039] Optionally, the topological implicit description of the parameterized level set includes: The zero isosurface of the level set function is used to implicitly represent the solid, boundary and holes of the structure. Determine a pseudo-time parameter, use the pseudo-time parameter to describe the evolution of the structure shape over time, and establish a partial differential equation of the level set function and time to track the dynamic changes of the structure boundary over time; The level set function is used to represent the evolution of the structural boundary; A parameterized level set function is constructed by spline interpolation of a spatial shape function, and a normal velocity of a structure boundary is calculated according to the spatial shape function and an expansion coefficient function; The level set function is updated using the normal velocity to simulate the evolution of the structure shape.

[0040] Specifically, refer to Figure 3 and Figure 4 ,In the topology description of this embodiment, the structural boundary is generally implicitly represented by the zero isosurface of the level set function.

[0041] The zero isosurface is obtained by cutting the cross section of the three-dimensional (3D) level set function. Similar to the contour line, the cross section (contour line) is z=0, so it is called the "zero isosurface". This representation method of obtaining a 2D result by solving a 3D curve is called "implicit representation".

[0042] Furthermore, entities, boundaries, and holes are expressed by the following formulas:

[0043] in is the level set function, In the design domain Any point in the Represents the evolution of the structural shape, Represents the material domain. In order to find the dynamic change process of the level set function over time, it is necessary to establish the partial differential of the level set function with time, so as to find the evolution process of the structural boundary (i.e., the zero level set) over time:

[0044] make , then the normal velocity of the structure boundary is Can be:

[0045] in is the normal direction of the boundary, is the gradient of the level set function. Therefore, the Hamilton-Jacobi partial differential equation (HJ-PDE) can be derived:

[0046] In the level set method, the traditional method for solving HJ PDE is generally to directly update the structure boundary using the upwind method. On the other hand, the parametric level set function is obtained by interpolating several compactly supported radial basis function (CSRBF) splines with C2 continuity, which can be expressed as:

[0047] in Indicates the number of nodes, It is the compactly supported radial basis function, also known as the spatial shape function. is the expansion coefficient function. and Respectively expressed as:

[0048] The above formula realizes the decoupling of space and time for the level set function. The components of CSRBF are expressed as:

[0049] The close support radius is expressed as:

[0050] In the formula For the current node and the central node located in the local support region The Euclidean distance between . The radius of the local support domain indicates the size of the local support domain. =3. is to avoid decimals with zero numerators ( =10 -4 ),prevent The numerator of is 0. By introducing CSRBF into HJ PDE, we can get the normal velocity:

[0051] In the formula, It is related to the time derivative of the expansion coefficient. The level set function is then evaluated on the refined grid inside each cell using bilinear interpolation.

[0052] Optionally, the specific steps of the geometric nonlinear finite element analysis include: Constructing a geometrically nonlinear topological optimization equilibrium equation; the equilibrium equation determines the mechanical state of the structure through the variational form of Piola Kirchhoff stress and strain, determines the external action of the structure through virtual displacement and body force and traction, and is established based on the mechanical state and external action; The equilibrium equation is linearized by using Newton's method, and a linearized equation including a geometrically nonlinear strain-displacement matrix, a strain-displacement matrix caused by a stress state, and an elastic constitutive matrix is ​​established; The displacement increment and the residual force vector are solved based on the linearized equation to obtain the displacement state of the structure within one iteration step under the condition of geometric nonlinearity for linear expression.

[0053] Specifically, geometrically nonlinear finite element analysis can solve large displacement problems of structures. Displacement interpolation is expressed as:

[0054] in, is the displacement, is the shape function. Based on the definition of the Green-Lagrange strain tensor, the Lagrangian strain Written as:

[0055] In the above formula is the identity matrix, is the deformation gradient, expressed as:

[0056] Second-order Piola-Kirchhoff stress for:

[0057] in, is the elastic constitutive matrix. Therefore, the equilibrium equation for geometric nonlinear topology optimization is:

[0058] is the variational form of strain, is the virtual displacement, For physical strength, is the traction force. Here we need to calculate the residual of the geometric nonlinear process:

[0059] in, and are the internal force vector and the external force vector respectively. The residual force vector It is expressed as:

[0060] The calculation formula is as follows:

[0061] In the formula is the geometrically nonlinear strain-displacement matrix. According to Newton-Raphson method, the linearized form of the equilibrium equation is:

[0062] is the strain-displacement matrix caused by the stress state, is the displacement increment. The formula is:

[0063] Therefore, the updated displacement can be The tangent stiffness matrix in the above formula is for:

[0064] The matrix mentioned above , and They are:

[0065]

[0066]

[0067] in, is the Piola-Kirchhoff stress component. is the shape function Node coordinates The derivative of is the deformation gradient The amount.

[0068] Optionally, the aggregation strategy of the local volume constraint includes: Determining an improved first function and a second function, wherein the first function represents the volume fraction of each finite element, and the second function represents the material proportion in a specified local constraint domain; the local constraint domain is determined by establishing a circular region of a specified radius around each finite element; The maximum value of the local volume fraction is set as the upper limit of the material ratio, multiple local constraints are simplified into a single constraint function, and the constraint function is approximated by a p-norm function; A structural volume constraint formula is determined based on the approximated constraint function, and the structural volume is constrained based on the volume constraint formula.

[0069] Specifically, the present application embodiment is to generate a porous structure, such as Figure 5 As shown, through an improved Heaviside function To express the volume fraction of each finite element:

[0070] In the formula, is the area of ​​each finite element, The Heaviside function that represents the existence of a material at any point in the level set function is:

[0071] Create a constraint radius around each finite element. The circular area is called the local constraint domain :

[0072] in, represents the central cell in the local constraint domain, represents any unit in the local constraint domain. Then the improved Heaviside function Represents the material proportions in the prescribed local constraint domain:

[0073] To solve the large number of constraints in the structure, the local volume fraction Set to The upper bound of :

[0074] The above formula can be used to represent a large number of constraints, namely , reduced to a constraint. Since the constraint formula is non-differentiable and not suitable for numerical optimization, we use p Norm function To approximate the function:

[0075] when When it approaches infinity, the above formula takes the equal sign, and in this method, =16. To improve the accuracy of the approximation, apply p norm function, so the constraint can be written as:

[0076] In the design domain, the volume constraint formula of the overall structure can be summarized as follows: is the number of units.

[0077]

[0078] Optionally, the method for establishing the topology optimization model includes: Acquire an actual displacement field and a virtual displacement field of a target space, and determine a strain energy density according to the actual displacement field and the virtual displacement field; An objective function corresponding to the topology optimization model is established according to the strain energy density and the Heaviside function, and structural flexibility is minimized based on the objective function to maximize the geometrically nonlinear porous filling stiffness; Among them, the objective function satisfies energy balance constraints, volume constraint formulas, internal and external vector balances, and boundary constraints; the energy balance constraints are determined based on the linear elastic equilibrium equation, and the linear elastic equilibrium equation is determined based on the energy function and the load function.

[0079] Specifically, in this embodiment, the geometrically nonlinear porous structure topology optimization model is:

[0080] in, is the objective function requiring minimum flexibility, is the strain energy density. It's space The displacement field in is the virtual displacement field. is the volume constraint, Represents a given displacement. are the internal force vector and the external force vector respectively. The linear elastic equilibrium equation is expressed in the energy bilinear form Written in weak variational form, the load is in linear form Defined as:

[0081] in, is the Hooke elastic tensor, is the linearized strain tensor.

[0082] Optionally, the sensitivity analysis process includes the following steps: Calculating the derivative of the energy function and the derivative of the load function to obtain a derivative expression; According to the equivalence relationship between the energy function and the load function and the derivative expression, the shape derivative of the objective function is determined; According to the residual force vector being 0, the objective function is corrected by the adjoint method, and the Lagrangian multiplier is introduced into the objective function to obtain an improved Lagrangian function; Determine the Lagrange multiplier value to ignore the effect of displacement field on the rate of change of design variables; The sensitivity function of the objective function is obtained according to the Lagrange multiplier value, and the sensitivity of the flexibility of the finite element node is obtained according to the sensitivity function.

[0083] For the topology optimization model, a geometric nonlinear finite element sensitivity analysis is performed. Among them, the energy function and load function The derivatives of are:

[0084]

[0085] In the formula is the Dirac function, Depend on Based on the residual force vector , the objective function is modified by the adjoint method, and the Lagrange multiplier Introduced into the objective function:

[0086] According to the equivalence relation and , it can be deduced that:

[0087] Therefore, the shape derivative of the objective function is:

[0088] To derive the partial derivative of the objective function with respect to the expansion coefficients , the improved objective function is the Lagrangian function, denoted as , its sensitivity is:

[0089] Among them, the shape function equal:

[0090] The Lagrangian function is:

[0091] To completely ignore The impact, multiplier Pick:

[0092] Therefore, the sensitivity of the objective function is expressed as:

[0093] Across the entire structural design domain, the improved target sensitivity is:

[0094] Node flexibility sensitivity , defined as the mean of the sum of the flexibility sensitivities of the elements that make up the node.

[0095]

[0096] Represents the total number of cells with common nodes.

[0097] Optionally, the iterative optimization process includes: In the iterative process, according to the current objective function and constraints, strictly convex approximation sub-problems are generated and solved, and the generation of the sub-problems is controlled by the moving asymptotes; Solving the strictly convex approximation subproblem to obtain updated design variables; Substituting the updated design variables into the level set function to determine whether the set convergence conditions are met; It is determined that the convergence condition is met, the iteration is terminated and the optimized topology structure is output.

[0098] In this embodiment, the MMA method is used to update the discrete values ​​of the level set, i.e., the design variables. The updated design variables are substituted into the updated parameterized level set function to determine whether the convergence conditions are met. If not, the above iterative process is continued. If it is met, the optimized topological optimization configuration is obtained.

[0099] The following further combines Figure 3 A specific embodiment is shown to describe the above steps and results of the present application in detail: like Figure 3As shown in the first part of the paper, linear and geometric nonlinear problems are considered for cantilever beam topology optimization and static analysis respectively, and the relevant parameters are consistent: the design domain size is 280mm×40mm, the number of elements is 280×40, the volume fraction is 0.45, and the external force is 200kN. The left end of the cantilever beam is fully constrained, and a downward external force is applied to the midpoint of the right end. Compared with the traditional cantilever beam structure, the structural flexibility considering the geometric nonlinear effect is reduced. It can be seen that the linear result contains more holes, while the geometric nonlinear structure meets the design requirements by distributing a small number of sturdy rods. As the external force increases, the geometric nonlinear structure shows asymmetry, and even forms a convex component on the far right. The optimized structure shows asymmetry due to the uneven boundary conditions, which is consistent with the characteristics of geometric nonlinearity. It should be noted that when the maximum range allowed by the structure is exceeded, the Newton-Raphson iteration (NR) method will not converge, resulting in the failure of the nonlinear finite element analysis and the inability to obtain the optimized structure. Figure 3 The static analysis in the first part was performed by Abaqus and the maximum displacement difference between the linear and geometrically nonlinear structures was 9.09%. Figure 3 The second part of the paper shows the static analysis results of an MBB beam with a design domain size of 240mm×40mm, 240×40 elements, a volume fraction of 0.45, and an external force of 200kN under linear and geometric nonlinear optimization. The linear structure has a uniform material distribution at the bottom, while the geometric nonlinear structure forms independent bars that are symmetrical on the left and right to better adapt to large loads. This is because in this case, the direction of the applied external force is vertically upward, resulting in a symmetrical constraint effect on both sides of the MBB beam. The static analysis shows that the maximum displacement of the linear structure increases by 9.40% compared to the geometric nonlinear structure. Figure 3 The third part of the paper shows the topological optimization and static analysis results of a compactly supported beam with a design domain size of 240mm×40mm, 240×40 elements, a volume fraction of 0.45, and an external force of 100kN under linear and geometric nonlinearity. In the linear optimization results, multiple internal triangular structures are generated on both sides of the structure to meet the constraints; while the geometric nonlinear results generate smaller triangular holes inside the structure and form thick bars on both sides to meet the boundary conditions. The geometric nonlinear structure contains fewer bars with larger cross-sections, which enhances the support capacity of the structure under vertical external forces. In the static analysis, it was observed that the deformation trends of the two structures were consistent, with a maximum displacement difference of 12.96%.

[0100] Combine the following Figure 4 A specific embodiment is shown to describe the above steps and results of the present application in detail: Since external force is a key factor affecting the finite element analysis of geometric nonlinear problems, this paper mainly discusses the impact of external force changes on the optimization results of this method (GN-ITO). Figure 4Three different optimization designs and their static analysis results are shown. To investigate the effect of external forces on the topological structure, the parameters of the three cases are consistent, with a design domain size of 480 mm × 120 mm, a cell number of 480 × 120, and a volume fraction of 0.5. Figure 4 The first part shows two cantilever beam designs subjected to loads of 10kN and 18kN respectively. Due to the external force acting vertically downward, the cantilever beam presents an asymmetric structure, and the internal rods extend in the direction of force flow. This asymmetry originates from geometric nonlinear factors, where the tangent stiffness matrix in the finite element analysis is calculated using an asymmetric matrix, which shows asymmetry in the optimization results. As the load changes, the distribution of the rods will be adaptively adjusted, showing a high degree of adaptability. It is worth noting that as the external force increases, only slender rods and small holes are formed in the structure, and no significant thick rods appear. During the optimization process, if a rod breaks, the structure will maintain stability by adding thin rods and satisfy local volume constraints, showing the characteristics of a porous structure. However, when the external force exceeds the structural bearing limit, the geometric nonlinear finite element analysis will fail and the topological structure cannot be generated. MBB beam structure design under different external forces is shown in Figure 2. Figure 4 As shown in the second part of the figure, the topological structure changes accordingly to adapt to the increased external force as the internal dense structure adjusts. Unlike the cantilever beam, the MBB beam structure does not show obvious asymmetry, similar to Figure 3 This is mainly because under the action of large external forces, the boundary conditions of the MBB beam tend to be left-right symmetrical, resulting in symmetrical structural distribution. In addition, unlike the traditional structure, no significant single rod appears in the MBB beam after GN-ITO optimization, which may be consistent with the above analysis. Figure 4 The third part shows the structural topology changes of the clamped beam when the external force is 10kN and 18kN. Due to the strong constraints on both sides, a higher density of holes is generated on both sides of the optimized clamped beam. The internal dense structure changes to adapt to the increased external force, which is mainly reflected in the formation of a ring-shaped porous structure around the external force application point in the center of the clamped beam.

[0101] Reference Figure 6 The present application also provides a parameterized level set porous structure topology optimization device considering geometric nonlinearity, comprising: An implicit description module 610 is used to construct a level set function to implicitly describe the structure boundary, and to obtain a parameterized level set function by using compactly supported radial basis function spline interpolation; The finite element analysis module 620 is used to establish the equilibrium equation of geometric nonlinear topology optimization, and to solve the structural response under large deformation conditions through geometric nonlinear finite element analysis; A volume constraint module 630, for expressing the volume fraction of each finite element using an improved Heaviside function, and approximating and simplifying the local volume constraint through a local constraint domain and a p-norm function; A flexibility minimization module 640 is used to establish a topology optimization model based on geometric nonlinear porous filling stiffness maximization to minimize structural flexibility; A sensitivity analysis module 650 is used to perform sensitivity analysis by adjoint method, calculate partial derivatives of the objective function of the topology optimization model with respect to the expansion coefficient, so as to determine the optimization direction of each finite element in the geometric nonlinear porous filling stiffness maximization; The iterative optimization module 660 is used to update the level set discrete value by the moving asymptote method, bring the updated level set discrete value into the objective function to perform finite element analysis again, and continuously iterate until the convergence condition is met, until the final topology optimization structure is obtained.

[0102] Optionally, the topological implicit description of the parameterized level set includes: The zero isosurface of the level set function is used to implicitly represent the solid, boundary and holes of the structure. Determine a pseudo-time parameter, use the pseudo-time parameter to describe the evolution of the structure shape over time, and establish a partial differential equation of the level set function and time to track the dynamic changes of the structure boundary over time; The level set function is used to represent the evolution of the structural boundary; A parameterized level set function is constructed by spline interpolation of a spatial shape function, and a normal velocity of a structure boundary is calculated according to the spatial shape function and an expansion coefficient function; The level set function is updated using the normal velocity to simulate the evolution of the structure shape.

[0103] Optionally, the specific steps of the geometric nonlinear finite element analysis include: Constructing a geometrically nonlinear topological optimization equilibrium equation; the equilibrium equation determines the mechanical state of the structure through the variational form of Piola Kirchhoff stress and strain, determines the external action of the structure through virtual displacement and body force and traction, and is established based on the mechanical state and external action; The equilibrium equation is linearized by using Newton's method, and a linearized equation including a geometrically nonlinear strain-displacement matrix, a strain-displacement matrix caused by a stress state, and an elastic constitutive matrix is ​​established; The displacement increment and the residual force vector are solved based on the linearized equation to obtain the displacement state of the structure within one iteration step under the condition of geometric nonlinearity for linear expression.

[0104] Optionally, the aggregation strategy of the local volume constraint includes: Determining an improved first function and a second function, wherein the first function represents the volume fraction of each finite element, and the second function represents the material proportion in a specified local constraint domain; the local constraint domain is determined by establishing a circular region of a specified radius around each finite element; The maximum value of the local volume fraction is set as the upper limit of the material ratio, multiple local constraints are simplified into a single constraint function, and the constraint function is approximated by a p-norm function; A structural volume constraint formula is determined based on the approximated constraint function, and the structural volume is constrained based on the volume constraint formula.

[0105] Optionally, the method for establishing the topology optimization model includes: Acquire an actual displacement field and a virtual displacement field of a target space, and determine a strain energy density according to the actual displacement field and the virtual displacement field; An objective function corresponding to the topology optimization model is established according to the strain energy density and the Heaviside function, and structural flexibility is minimized based on the objective function to maximize the geometrically nonlinear porous filling stiffness; Among them, the objective function satisfies energy balance constraints, volume constraint formulas, internal and external vector balances, and boundary constraints; the energy balance constraints are determined based on the linear elastic equilibrium equation, and the linear elastic equilibrium equation is determined based on the energy function and the load function.

[0106] Optionally, the sensitivity analysis process includes the following steps: Calculating the derivative of the energy function and the derivative of the load function to obtain a derivative expression; According to the equivalence relationship between the energy function and the load function and the derivative expression, the shape derivative of the objective function is determined; According to the residual force vector being 0, the objective function is corrected by the adjoint method, and the Lagrangian multiplier is introduced into the objective function to obtain an improved Lagrangian function; Determine the Lagrange multiplier value to ignore the effect of displacement field on the rate of change of design variables; The sensitivity function of the objective function is obtained according to the Lagrange multiplier value, and the sensitivity of the flexibility of the finite element node is obtained according to the sensitivity function.

[0107] Optionally, the iterative optimization process includes: In the iterative process, according to the current objective function and constraints, strictly convex approximation sub-problems are generated and solved, and the generation of the sub-problems is controlled by the moving asymptotes; Solving the strictly convex approximation subproblem to obtain updated design variables; Substituting the updated design variables into the level set function to determine whether the set convergence conditions are met; It is determined that the convergence condition is met, the iteration is terminated and the optimized topology structure is output.

[0108] It should be understood that the above-mentioned device is used to execute the method in the above-mentioned embodiment. The implementation principle and technical effect of the corresponding program module in the device are similar to those described in the above-mentioned method. The working process of the device can refer to the corresponding process in the above-mentioned method, which will not be repeated here.

[0109] It should be understood that the various numerical numbers involved in the embodiments of the present application are only used for the convenience of description and are not used to limit the scope of the embodiments of the present application.

[0110] It will be easily understood by those skilled in the art that the above description is only a preferred embodiment of the present application and is not intended to limit the present application. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present application shall be included in the scope of protection of the present application.

Claims

1. A parameterized level set porous structure topology optimization method considering geometric nonlinearity, characterized in that: include: The level set function is constructed to implicitly describe the structural boundary, and the parameterized level set function is obtained by using compactly supported radial basis function spline interpolation. Establish the equilibrium equation of geometric nonlinear topology optimization, and use geometric nonlinear finite element analysis to solve the structural response under large deformation conditions; The improved Heaviside function is used to represent the volume fraction of each finite element, and the local volume constraint is approximated and simplified by the local constraint domain and p-norm function. A topology optimization model based on geometrically nonlinear porous filling stiffness maximization is established to minimize structural flexibility. Sensitivity analysis is performed by adjoint method to calculate the partial derivatives of the objective function of the topology optimization model with respect to the expansion coefficients, so as to determine the optimization direction of each finite element in the geometrically nonlinear porous filling stiffness maximization problem. The level set discrete value is updated by the moving asymptote method, and the updated level set discrete value is brought into the objective function to perform finite element analysis again, and it is continuously iterated until the convergence condition is met, until the final topology optimization structure is obtained.

2. The parameterized level set porous structure topology optimization method considering geometric nonlinearity according to claim 1, characterized in that: The topological implicit description of the parameterized level set includes: The zero isosurface of the level set function is used to implicitly represent the solid, boundary and holes of the structure. Determine a pseudo-time parameter, use the pseudo-time parameter to describe the evolution of the structure shape over time, and establish a partial differential equation of the level set function and time to track the dynamic changes of the structure boundary over time; The level set function is used to represent the evolution of the structural boundary; A parameterized level set function is constructed by spline interpolation of a spatial shape function, and a normal velocity of a structure boundary is calculated according to the spatial shape function and an expansion coefficient function; The level set function is updated using the normal velocity to simulate the evolution of the structure shape.

3. The parameterized level set porous structure topology optimization method considering geometric nonlinearity according to claim 1, characterized in that: The specific steps of the nonlinear finite element analysis include: Constructing a geometrically nonlinear topology optimization equilibrium equation; the equilibrium equation determines the mechanical state of the structure through the variational form of Piola Kirchhoff stress and strain, determines the external action of the structure through virtual displacement and body force and traction, and is established based on the mechanical state and external action; The equilibrium equation is linearized by using Newton's method, and a linearized equation including a geometrically nonlinear strain-displacement matrix, a strain-displacement matrix caused by a stress state, and an elastic constitutive matrix is ​​established; The displacement increment and the residual force vector are solved based on the linearized equation to obtain the displacement state of the structure within one iteration step under the geometric nonlinear condition.

4. The parameterized level set porous structure topology optimization method considering geometric nonlinearity according to claim 1, characterized in that: The aggregation strategy of the local volume constraint includes: Determining an improved first function and a second function, wherein the first function represents the volume fraction of each finite element, and the second function represents the material proportion in a specified local constraint domain; the local constraint domain is determined by establishing a circular region of a specified radius around each finite element; The maximum value of the local volume fraction is set as the upper limit of the material ratio, multiple local constraints are simplified into a single constraint function, and the constraint function is approximated by a p-norm function; A structural volume constraint formula is determined based on the approximated constraint function, and the structural volume is constrained based on the volume constraint formula.

5. The parameterized level set porous structure topology optimization method considering geometric nonlinearity according to claim 1, characterized in that: The method for establishing the topology optimization model includes: Acquire an actual displacement field and a virtual displacement field of a target space, and determine a strain energy density according to the actual displacement field and the virtual displacement field; An objective function corresponding to the topology optimization model is established according to the strain energy density and the Heaviside function, and structural flexibility is minimized based on the objective function to maximize the geometrically nonlinear porous filling stiffness; Among them, the objective function satisfies energy balance constraints, volume constraint formulas, internal and external vector balances, and boundary constraints; the energy balance constraints are determined based on the linear elastic equilibrium equation, and the linear elastic equilibrium equation is determined based on the energy function and the load function.

6. The parameterized level set porous structure topology optimization method considering geometric nonlinearity according to claim 5, characterized in that: The sensitivity analysis process includes the following steps: Calculating the derivative of the energy function and the derivative of the load function to obtain a derivative expression; According to the equivalence relationship between the energy function and the load function and the derivative expression, the shape derivative of the objective function is determined; According to the residual force vector being 0, the objective function is corrected by the adjoint method, and the Lagrangian multiplier is introduced into the objective function to obtain an improved Lagrangian function; Determine the Lagrange multiplier value to ignore the effect of displacement field on the rate of change of design variables; The sensitivity function of the objective function is obtained according to the Lagrange multiplier value, and the sensitivity of the finite element node flexibility is obtained according to the sensitivity function.

7. The parameterized level set porous structure topology optimization method considering geometric nonlinearity according to claim 1, characterized in that: The iterative optimization process includes: In the iterative process, according to the current objective function and constraints, strictly convex approximation sub-problems are generated and solved, and the generation of the sub-problems is controlled by the moving asymptotes; Solving the strictly convex approximation subproblem to obtain updated design variables; Substituting the updated design variables into the level set function to determine whether the set convergence conditions are met; It is determined that the convergence condition is met, the iteration is terminated and the optimized topology structure is output.

8. A parameterized level set porous structure topology optimization device considering geometric nonlinearity, characterized in that: include: Implicit description module, used to construct the level set function to implicitly describe the structure boundary, and use compactly supported radial basis function spline interpolation to obtain the parameterized level set function; Finite element analysis module, used to establish the equilibrium equations of geometric nonlinear topology optimization, and solve the structural response under large deformation conditions through geometric nonlinear finite element analysis; Volume constraint module, which is used to express the volume fraction of each finite element using the improved Heaviside function, and approximate and simplify the local volume constraint through the local constraint domain and p-norm function; The flexibility minimization module is used to establish a topology optimization model based on geometrically nonlinear porous filling stiffness maximization to minimize structural flexibility; Sensitivity analysis module, which is used to perform sensitivity analysis by adjoint method and calculate the partial derivative of the objective function of the topology optimization model with respect to the expansion coefficient to determine the optimization direction of each finite element in the geometric nonlinear porous filling stiffness maximization; The iterative optimization module is used to update the discrete value of the level set by the moving asymptote method, bring the updated discrete value of the level set into the objective function to perform finite element analysis again, and continuously iterate until the convergence conditions are met, until the final topology optimization structure is obtained.

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