A parameterized level set topology optimization method for porous structures considering geometric nonlinearity

Abstract

This application belongs to the field of structural optimization technology and specifically discloses a parameterized level set topology optimization method for porous structures that takes geometric nonlinearity into account. The method comprises: constructing a level set function to implicitly describe the structural boundary, and using compactly supported radial basis function spline interpolation to obtain the parameterized level set function; establishing an equilibrium equation for geometric nonlinear topology optimization, and performing geometric nonlinear finite element analysis to solve the structural response under large deformation conditions; approximating and simplifying local volume constraints through local constraint domains and p-norm functions; establishing a topology optimization model based on maximizing the geometric nonlinear porous filling stiffness; performing sensitivity analysis through the adjoint method, and calculating the 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 maximizing the geometric nonlinear porous filling stiffness; and performing iterative optimization to obtain the optimized topology structure. This application can realize the topology optimization of porous fillings under geometric nonlinear conditions.

Description

Technical Field

[0001] The present application belongs to the field of structural optimization technology, 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 exhibit 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 the design of porous structures, current filling designs are mostly based on the assumption of linear elastic deformation and do not consider geometric nonlinearity. For materials such as soft rubber or slender structures, when subjected to large deformations, the strain and displacement fields exhibit a strong nonlinear relationship, demonstrating significant geometric nonlinearity. Using traditional linear finite element analysis to optimize porous structures results in poor topology optimization accuracy.

[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 response to the defects of the existing technology, the purpose of this application is to provide a parameterized level set porous structure topology optimization method that takes into account geometric nonlinearity, aiming to solve the problem of insufficient accuracy and adaptability of porous structure topology optimization in the existing technology when dealing with complex working conditions due to the continued use of linear finite element analysis.

[0006] To achieve the above objectives, in a first aspect, the present application provides a parameterized level set topology optimization method for porous structures considering geometric nonlinearity, comprising:

[0007] A level set function is constructed to implicitly describe the structural boundary, and a parameterized level set function is obtained by using compactly supported radial basis function spline interpolation.

[0008] Establish the equilibrium equations for geometrically nonlinear topology optimization and use geometrically nonlinear finite element analysis to solve the structural response under large deformation conditions;

[0009] 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.

[0010] A topology optimization model based on geometrically nonlinear porous filling stiffness maximization is established to minimize structural flexibility.

[0011] Sensitivity analysis is performed using the adjoint method to calculate the partial derivatives of the objective function of the topology optimization model with respect to the expansion coefficients to determine the optimization direction of each finite element in the geometrically nonlinear porous filling stiffness maximization problem.

[0012] 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. It is iterated continuously until the convergence condition is met, and the final topology optimization structure is obtained.

[0013] Optionally, the topological implicit description of the parameterized level set includes:

[0014] The zero isosurface of the level set function is used to implicitly represent the solid, boundary and hole of the structure;

[0015] Determining a pseudo-time parameter, using the pseudo-time parameter to describe the evolution of the structure shape over time, and establishing a partial differential equation of the level set function and time to track the dynamic changes of the structure boundary over time;

[0016] The level set function is used to represent the evolution of the structural boundary;

[0017] constructing a parameterized level set function through spline interpolation of a spatial shape function, and calculating the normal velocity of the structure boundary according to the spatial shape function and the expansion coefficient function;

[0018] The normal velocity is used to update the level set function to simulate the evolution of the structure shape.

[0019] Optionally, the specific steps of the geometric nonlinear finite element analysis include:

[0020] Constructing an equilibrium equation for geometric nonlinear topology optimization; 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, body force, and traction, and is established based on the mechanical state and external action;

[0021] The equilibrium equation is linearized 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;

[0022] 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 for linear expression.

[0023] Optionally, the aggregation strategy of the local volume constraint includes:

[0024] determining an improved first function and a second function, wherein the first function represents a volume fraction of each finite element and the second function represents a material ratio in a prescribed local constraint domain; the local constraint domain is determined by establishing a circular region of a specified radius around each finite element;

[0025] The maximum value of the local volume fraction is set as the upper bound of the material ratio, multiple local constraints are simplified into a single constraint function, and the constraint function is approximated using a p-norm function;

[0026] 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.

[0027] Optionally, the method for establishing the topology optimization model includes:

[0028] Acquiring an actual displacement field and a virtual displacement field of a target space, and determining a strain energy density according to the actual displacement field and the virtual displacement field;

[0029] Establishing an objective function corresponding to a topology optimization model according to the strain energy density and the Heaviside function, and minimizing structural flexibility based on the objective function to maximize geometrically nonlinear porous filling stiffness;

[0030] 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 a linear elastic equilibrium equation, which is determined based on an energy function and a load function.

[0031] Optionally, the sensitivity analysis process includes the following steps:

[0032] Calculating the derivative of the energy function and the derivative of the load function to obtain a derivative expression;

[0033] 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;

[0034] 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;

[0035] Determine the Lagrange multiplier value to ignore the effect of the displacement field on the rate of change of the design variables;

[0036] 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.

[0037] Optionally, the iterative optimization process includes:

[0038] During the iteration process, according to the current objective function and constraints, strictly convex approximation subproblems are generated and solved, and the generation of the subproblems is controlled by the moving asymptotes;

[0039] Solving the strictly convex approximation subproblem to obtain updated design variables;

[0040] Substituting the updated design variables into the level set function to determine whether the set convergence conditions are met;

[0041] It is determined that the convergence condition is met, the iteration is terminated and the optimized topology structure is output.

[0042] The present application also provides a parameterized level set porous structure topology optimization device considering geometric nonlinearity, comprising:

[0043] Implicit description module, used to construct the level set function to implicitly describe the structure boundary, and the parameterized level set function is obtained by compactly supported radial basis function spline interpolation;

[0044] Finite element analysis module, used to establish the equilibrium equations for geometrically nonlinear topology optimization and solve the structural response under large deformation conditions through geometrically nonlinear finite element analysis;

[0045] 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;

[0046] The flexibility minimization module is used to establish a topology optimization model based on geometrically nonlinear porous filling stiffness maximization to minimize structural flexibility;

[0047] Sensitivity analysis module, which is used to perform sensitivity analysis using the adjoint method and calculate the 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;

[0048] The iterative optimization module 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 and perform finite element analysis again, and iterate continuously until the convergence conditions are met, until the final topology optimization structure is obtained.

[0049] In general, the above technical solutions conceived by this application have the following beneficial effects compared with the existing technologies:

[0050] (1) This application proposes a parameterized level set topology optimization method for porous structures that takes geometric nonlinearity into account, which can improve the accuracy and applicability of topology optimization. By using a parameterized level set method that separates space and time and controls topological structure changes through expansion coefficients, more accurate optimization results can be achieved when dealing with complex geometrically nonlinear porous structures, improving the accuracy and adaptability of topology optimization and making it applicable to large-deformation structures under various design requirements.

[0051] (2) The parameterized level set topology optimization method for porous structures, which takes geometric nonlinearity into consideration, is proposed in this application and can improve 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, the optimization process can meet local structural requirements while effectively controlling numerical instability, thereby reducing the number of iterations and improving computational efficiency.

[0052] (3) The parameterized level set topology optimization method for porous structures, which takes geometric nonlinearity into account, provided in this application, can expand its application scope and engineering applicability. It demonstrates excellent performance under static analysis and various design parameters, meeting the design requirements of high-strength, lightweight structures, and is suitable for promotion and application in demanding engineering fields such as aerospace and construction. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 This is one of the flow charts of the parameterized level set porous structure topology optimization method considering geometric nonlinearity provided in the embodiments of the present application;

[0054] 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;

[0055] Figure 3 Schematic diagram of structural topology and static analysis of classical topology optimization by introducing geometric nonlinearity in an embodiment of the present application;

[0056] Figure 4 This is a schematic diagram of the topological and static analysis results of the topological optimization of geometrically nonlinear porous filling under different external forces in an embodiment of the present application.

[0057] Figure 5 This is a schematic diagram of the porous structure generated by local volume confinement polymerization in GN-ITO of the embodiment of the present application;

[0058] Figure 6 It is a structural schematic diagram of a parameterized level set porous structure topology optimization device considering geometric nonlinearity provided in an embodiment of the present application. DETAILED DESCRIPTION

[0059] In order to make the purpose, technical solutions and advantages of this application more clear, the following further describes this application in detail with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain this application and are not intended to limit this application.

[0060] The term "and / or" in this application describes an association relationship between associated objects, indicating that three relationships can exist. For example, A and / or B can represent: A exists alone, A and B exist simultaneously, or 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.

[0061] In the specification and claims of this application, the terms "first" and "second" are used to distinguish different objects, rather than to describe a specific order of objects. For example, "first response message" and "second response message" are used to distinguish different response messages, rather than to describe a specific order of response messages.

[0062] In the embodiments of this 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 this application should not be interpreted as being preferred or advantageous over other embodiments or designs. Rather, the use of words such as "exemplary" or "for example" is intended to present the relevant concepts in a concrete manner.

[0063] In the description of the embodiments of the present application, unless otherwise specified, "multiple" means two or more, for example, multiple processing units means two or more processing units, etc.; multiple elements means two or more elements, etc.

[0064] The embodiments of the present application are described below in conjunction with the drawings in the embodiments of the present application.

[0065] Reference Figure 1 , this application provides a parameterized level set porous structure topology optimization method considering geometric nonlinearity, including:

[0066] S101. Construct a level set function to implicitly describe the structural boundary, and use compactly supported radial basis function spline interpolation to obtain a parameterized level set function;

[0067] S102. Establish the equilibrium equations for geometrically nonlinear topology optimization and perform geometrically nonlinear finite element analysis to determine the structural response under large deformation conditions.

[0068] S103. Using the improved Heaviside function to represent the volume fraction of each finite element, and approximating and simplifying the local volume constraint by using the local constraint domain and the p-norm function;

[0069] S104. Establish a topology optimization model based on geometrically nonlinear porous filling stiffness maximization to minimize structural flexibility.

[0070] S105. Perform sensitivity analysis using the adjoint method to calculate the partial derivatives of the objective function of the topology optimization model with respect to the expansion coefficients to determine the optimization direction of each finite element in the geometrically nonlinear porous filling stiffness maximization problem.

[0071] S106. Update the level set discrete value by the moving asymptote method, bring the updated level set discrete value into the objective function and perform finite element analysis again, iterating continuously until the convergence condition is met, until the final topology optimization structure is obtained.

[0072] The following is a detailed description of each step of this application.

[0073] The process of constructing the level set function in step S101 is:

[0074] S11. Implicit description of structural boundaries: Use level set functions to define and represent the boundaries of structures.

[0075] S12. Simulating Boundary Evolution: Apply partial differential equations (PDEs) to simulate the time evolution of structural boundaries.

[0076] S13. Parameterized level set function: The parameterized level set function is obtained by using compactly supported radial basis function spline interpolation technique.

[0077] 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.

[0078] In step S102, the process of geometric nonlinear finite element analysis is as follows:

[0079] S21. Establishing equilibrium equations: Geometrically nonlinear finite element analysis to establish equilibrium equations related to large structural displacements.

[0080] S22. Linearization: Linearize the equilibrium equations to effectively solve large displacement problems and ensure the stability and accuracy of numerical calculations.

[0081] In step S103, the improved Heaviside function is used to represent the volume fraction of each finite element. By introducing a local constraint domain and a p-norm function, the local volume constraint is approximated and simplified to meet the design requirements.

[0082] In step S104, the process of establishing a topology optimization model is as follows:

[0083] S41. Maximizing Stiffness of Porous Infills: Develop a topology optimization model to maximize the stiffness of geometrically nonlinear porous infills.

[0084] S42. Minimize Structural Flexibility: Minimize the flexibility of a structure through an optimization process to improve its overall performance.

[0085] In step S105, the sensitivity analysis process is as follows:

[0086] Adjoint Sensitivity Analysis: This analysis calculates the partial derivatives of the objective function of the topology optimization model with respect to the expansion coefficients. This allows for sensitivity analysis of maximizing the stiffness of geometrically nonlinear porous infills, providing essential sensitivity information for the optimization process.

[0087] 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 topology is obtained.

[0088] The complete steps of the embodiment of this application can be referred to Figure 2 , Figure 2 The process includes:

[0089] Enter initial parameters;

[0090] Define material properties, loads, and boundary conditions;

[0091] Establish geometrically nonlinear finite element mesh;

[0092] Initialize the level set function and expansion coefficient;

[0093] Establishing local constraint domain of structure;

[0094] Apply local volume constraints to design variables;

[0095] Geometrically nonlinear finite element analysis to calculate residual forces;

[0096] Calculate the objective function and constraint function;

[0097] Perform sensitivity analysis on objective function and constraint function;

[0098] Interpolate and filter objective and constraint functions;

[0099] Use the MMA method to update the level set expansion coefficients;

[0100] Determine the convergence conditions;

[0101] Output a geometrically nonlinear porous topology optimization model.

[0102] Optionally, the topological implicit description of the parameterized level set includes:

[0103] The zero isosurface of the level set function is used to implicitly represent the solid, boundary and hole of the structure;

[0104] Determining a pseudo-time parameter, using the pseudo-time parameter to describe the evolution of the structure shape over time, and establishing a partial differential equation of the level set function and time to track the dynamic changes of the structure boundary over time;

[0105] The level set function is used to represent the evolution of the structural boundary;

[0106] constructing a parameterized level set function through spline interpolation of a spatial shape function, and calculating the normal velocity of the structure boundary according to the spatial shape function and the expansion coefficient function;

[0107] The normal velocity is used to update the level set function to simulate the evolution of the structure shape.

[0108] 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.

[0109] The implicit zero isosurface is obtained by cutting a cross section of the three-dimensional (3D) level set function. Similar to a contour line, the cross section (contour line) is at z = 0, hence the name "zero isosurface." This representation method of obtaining a 2D result by solving a 3D curve is called an "implicit representation."

[0110] Furthermore, entities, boundaries, and holes are expressed by the following formulas:

[0111]

[0112] in is the level set function, In the design domain At any point in the Represents the evolution of the structural shape, Represents the material domain. In order to find the dynamic change 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 of the structural boundary (i.e., the zero level set) over time:

[0113]

[0114] make , then the normal velocity of the structure boundary is Can be:

[0115]

[0116] 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:

[0117]

[0118] In the level set method, the traditional approach to solving the HJ PDE is to directly update the structural 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 and can be expressed as:

[0119]

[0120] 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:

[0121]

[0122] The above formula decouples the level set function from space and time. The components of CSRBF are expressed as:

[0123]

[0124] The close support radius is expressed as:

[0125]

[0126] In the formula For the current node and the central node in the local support region The Euclidean distance between . The radius of the local support domain represents the scale of the local support domain. =3. To avoid decimals with zero numerators ( =10 -4 ),prevent The numerator of is 0. By introducing CSRBF into HJ PDE, the normal velocity can be obtained:

[0127]

[0128] Where, 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.

[0129] Optionally, the specific steps of the geometric nonlinear finite element analysis include:

[0130] Constructing an equilibrium equation for geometric nonlinear topology optimization; 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, body force, and traction, and is established based on the mechanical state and external action;

[0131] The equilibrium equation is linearized 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;

[0132] 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 for linear expression.

[0133] Specifically, geometrically nonlinear finite element analysis can solve large displacement problems of structures. Displacement interpolation is expressed as:

[0134]

[0135] in, is the displacement, is the shape function. Based on the definition of Green-Lagrange strain tensor, Lagrange strain Written as:

[0136]

[0137] In the above formula is the identity matrix, is the deformation gradient, expressed as:

[0138]

[0139] Second-order Piola-Kirchhoff stress for:

[0140]

[0141] in, is the elastic constitutive matrix. Therefore, the equilibrium equation for geometric nonlinear topology optimization is:

[0142]

[0143] 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:

[0144]

[0145] in, and are the internal force vector and the external force vector respectively. It is expressed as:

[0146]

[0147] The calculation formula is as follows:

[0148]

[0149] In the formula is the geometrically nonlinear strain-displacement matrix. According to the Newton-Raphson method, the linearized form of the equilibrium equation is:

[0150]

[0151] is the strain-displacement matrix caused by the stress state, is the displacement increment. The formula is:

[0152]

[0153] Therefore, the updated displacement can be The tangent stiffness matrix in the above formula is obtained. for:

[0154]

[0155] The matrix mentioned above 、 and They are:

[0156]

[0157]

[0158]

[0159] in, is the Piola-Kirchhoff stress component. is the shape function Node coordinates The derivative of is the deformation gradient The weight.

[0160] Optionally, the aggregation strategy of the local volume constraint includes:

[0161] determining an improved first function and a second function, wherein the first function represents a volume fraction of each finite element and the second function represents a material ratio in a prescribed local constraint domain; the local constraint domain is determined by establishing a circular region of a specified radius around each finite element;

[0162] The maximum value of the local volume fraction is set as the upper bound of the material ratio, multiple local constraints are simplified into a single constraint function, and the constraint function is approximated using a p-norm function;

[0163] 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.

[0164] Specifically, the embodiment of the present application 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:

[0165]

[0166] Where, is the area of ​​each finite element, The Heaviside function that represents the existence of the material at any point in the level set function is:

[0167]

[0168] Create a constraint radius around each finite element. The circular area is called the local constraint domain :

[0169]

[0170] 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 a prescribed local constraint domain:

[0171]

[0172] To solve the problem of a large number of constraints in the structure, the local volume fraction Set to The upper bound of :

[0173]

[0174] The above formula can be used to combine a large number of constraints, namely , reduced to a constraint. Since the constraint formula is non-differentiable and not suitable for numerical optimization, use p Norm function To approximate the function:

[0175]

[0176] 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:

[0177]

[0178] In the design domain, the volume constraint formula of the overall structure can be summarized as follows: is the number of units.

[0179]

[0180] Optionally, the method for establishing the topology optimization model includes:

[0181] Acquiring an actual displacement field and a virtual displacement field of a target space, and determining a strain energy density according to the actual displacement field and the virtual displacement field;

[0182] Establishing an objective function corresponding to a topology optimization model according to the strain energy density and the Heaviside function, and minimizing structural flexibility based on the objective function to maximize geometrically nonlinear porous filling stiffness;

[0183] 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 a linear elastic equilibrium equation, which is determined based on an energy function and a load function.

[0184] Specifically, in this embodiment, the geometrically nonlinear porous structure topology optimization model is:

[0185]

[0186] 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:

[0187]

[0188] in, is the Hooke elastic tensor, is the linearized strain tensor.

[0189] Optionally, the sensitivity analysis process includes the following steps:

[0190] Calculating the derivative of the energy function and the derivative of the load function to obtain a derivative expression;

[0191] 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;

[0192] 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;

[0193] Determine the Lagrange multiplier value to ignore the effect of the displacement field on the rate of change of the design variables;

[0194] 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.

[0195] For the topology optimization model, geometric nonlinear finite element sensitivity analysis is performed. Among them, the energy function and load function The derivatives of are:

[0196]

[0197]

[0198] In the formula is the Dirac function, Depend on Based on the residual force vector , use the adjoint method to modify the objective function and use the Lagrange multiplier Introduced into the objective function:

[0199]

[0200] According to the equivalence relation and , it can be deduced that:

[0201]

[0202] Therefore, the shape derivative of the objective function is:

[0203]

[0204] To derive the partial derivative of the objective function with respect to the expansion coefficient , the improved objective function is the Lagrangian function, recorded as , its sensitivity is:

[0205]

[0206] Among them, the shape function equal:

[0207]

[0208] The Lagrangian function is:

[0209]

[0210] To completely ignore the Impact, multiplier Pick:

[0211]

[0212] Therefore, the sensitivity of the objective function is expressed as:

[0213]

[0214] Across the entire structural design domain, the improved target sensitivity is:

[0215]

[0216] Node flexibility sensitivity , defined as the mean of the sum of the flexibility sensitivities of the elements that make up the node.

[0217]

[0218] Indicates the total number of elements with common nodes.

[0219] Optionally, the iterative optimization process includes:

[0220] During the iteration process, according to the current objective function and constraints, strictly convex approximation subproblems are generated and solved, and the generation of the subproblems is controlled by the moving asymptotes;

[0221] Solving the strictly convex approximation subproblem to obtain updated design variables;

[0222] Substituting the updated design variables into the level set function to determine whether the set convergence conditions are met;

[0223] It is determined that the convergence condition is met, the iteration is terminated and the optimized topology structure is output.

[0224] 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 topology optimization configuration is obtained.

[0225] The following further combines Figure 3 A specific embodiment is shown to describe the above steps and results of the present application in detail:

[0226] like Figure 3 As shown in the first part of the paper, linear and geometric nonlinearities were considered for cantilever beam topology optimization and static analysis, respectively. The relevant parameters were consistent: the design domain dimensions were 280 mm × 40 mm, the number of elements was 280 × 40, the volume fraction was 0.45, and the external force was 200 kN. The left end of the cantilever beam was fully constrained, and a downward external force was applied to the midpoint of the right end. Compared to traditional cantilever beam structures, the structural flexibility achieved by considering geometric nonlinearity is reduced. It can be seen that the linear result contains a large number of holes, while the geometrically nonlinear structure meets the design requirements with a small but robust distribution of members. As the external force increases, the geometrically nonlinear structure exhibits asymmetry, even forming a protruding member on the far right. The asymmetry of the optimized structure due to the nonuniform boundary conditions is consistent with the characteristics of geometric nonlinearity. It is important to note that if the maximum allowable range of the structure is exceeded, the Newton-Raphson iteration (NR) method will fail to converge, resulting in failure of the nonlinear finite element analysis and the inability to obtain an optimized structure. Figure 3 The static analysis in the first part was performed using Abaqus, and the maximum displacement difference between the linear and geometrically nonlinear structures was 9.09%. Figure 3The second section presents the static analysis results of an MBB beam with a design domain size of 240 mm × 40 mm, a cell count of 240 × 40, a volume fraction of 0.45, and an external force of 200 kN, using both linear and geometrically nonlinear optimizations. While the linear structure has uniform material distribution at the bottom, the geometrically nonlinear structure forms independent, symmetrical members, better accommodating large loads. This is because, in this case, the applied external force is directed vertically upward, resulting in symmetrical restraint effects on both sides of the MBB beam. Static analysis shows that the maximum displacement of the linear structure increases by 9.40% compared to the geometrically nonlinear structure. Figure 3 The third section presents the topology optimization and static analysis results for a compactly supported beam with a design domain size of 240mm×40mm, a cell count of 240×40, a volume fraction of 0.45, and an external force of 100kN, both under linear and geometric nonlinear conditions. The linear optimization results generate multiple internal triangular structures on both sides of the structure to satisfy the constraints; the geometric nonlinear results, on the other hand, generate smaller triangular holes within the structure and form sturdy members on both sides to satisfy the boundary conditions. The geometrically nonlinear structure contains fewer members with larger cross-sections, thereby enhancing the structure's support capacity under vertical external forces. In the static analysis, the deformation trends of the two structures were observed to be consistent, with a maximum displacement difference of 12.96%.

[0227] The following combination Figure 4 A specific embodiment is shown to describe the above steps and results of the present application in detail:

[0228] 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 4 Three different optimized designs and their static analysis results are presented. To examine the effect of external forces on the topology, the parameters of the three cases remain the same: a design domain size of 480 mm × 120 mm, a cell count of 480 × 120, and a volume fraction of 0.5. Figure 4The first part shows the design of two cantilever beams subjected to loads of 10kN and 18kN respectively. Due to the vertical downward external force, the cantilever beam presents an asymmetric structure, and the internal rods extend along the direction of force flow. This asymmetry is due to 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 meet 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 as follows Figure 4 As shown in the second part of the figure, the topology 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 primarily due to the fact that under large external forces, the boundary conditions of the MBB beam tend to become bilaterally symmetrical, resulting in a symmetrical structural distribution. Furthermore, unlike traditional structures, the GN-ITO-optimized MBB beam lacks significant individual members, which may be consistent with the aforementioned analysis. Figure 4 The third section shows the structural topology changes of the clamped beam under external forces of 10 kN and 18 kN. Due to the rigid constraints on both sides, the optimized clamped beam generates a higher density of pores on both sides. The internal dense structure changes to accommodate the increased external force, primarily manifesting in the formation of a ring-shaped porous structure around the central point of the clamped beam.

[0229] Reference Figure 6 The present application also provides a parameterized level set porous structure topology optimization device considering geometric nonlinearity, comprising:

[0230] 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 using compactly supported radial basis function spline interpolation;

[0231] Finite element analysis module 620, used to establish equilibrium equations for geometrically nonlinear topology optimization, and solve the structural response under large deformation conditions through geometrically nonlinear finite element analysis;

[0232] a volume constraint module 630 for expressing the volume fraction of each finite element using a modified Heaviside function, and approximating and simplifying the local volume constraint by a local constraint domain and a p-norm function;

[0233] A flexibility minimization module 640 is used to establish a topology optimization model based on geometrically nonlinear porous filling stiffness maximization to minimize structural flexibility;

[0234] Sensitivity analysis module 650, for performing sensitivity analysis by adjoint method, calculating 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 maximizing the stiffness of geometrically nonlinear porous filling;

[0235] 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.

[0236] Optionally, the topological implicit description of the parameterized level set includes:

[0237] The zero isosurface of the level set function is used to implicitly represent the solid, boundary and hole of the structure;

[0238] Determining a pseudo-time parameter, using the pseudo-time parameter to describe the evolution of the structure shape over time, and establishing a partial differential equation of the level set function and time to track the dynamic changes of the structure boundary over time;

[0239] The level set function is used to represent the evolution of the structural boundary;

[0240] constructing a parameterized level set function through spline interpolation of a spatial shape function, and calculating the normal velocity of the structure boundary according to the spatial shape function and the expansion coefficient function;

[0241] The normal velocity is used to update the level set function to simulate the evolution of the structure shape.

[0242] Optionally, the specific steps of the geometric nonlinear finite element analysis include:

[0243] Constructing an equilibrium equation for geometric nonlinear topology optimization; 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, body force, and traction, and is established based on the mechanical state and external action;

[0244] The equilibrium equation is linearized 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;

[0245] 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 for linear expression.

[0246] Optionally, the aggregation strategy of the local volume constraint includes:

[0247] determining an improved first function and a second function, wherein the first function represents a volume fraction of each finite element and the second function represents a material ratio in a prescribed local constraint domain; the local constraint domain is determined by establishing a circular region of a specified radius around each finite element;

[0248] The maximum value of the local volume fraction is set as the upper bound of the material ratio, multiple local constraints are simplified into a single constraint function, and the constraint function is approximated using a p-norm function;

[0249] 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.

[0250] Optionally, the method for establishing the topology optimization model includes:

[0251] Acquiring an actual displacement field and a virtual displacement field of a target space, and determining a strain energy density according to the actual displacement field and the virtual displacement field;

[0252] Establishing an objective function corresponding to a topology optimization model according to the strain energy density and the Heaviside function, and minimizing structural flexibility based on the objective function to maximize geometrically nonlinear porous filling stiffness;

[0253] 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 a linear elastic equilibrium equation, which is determined based on an energy function and a load function.

[0254] Optionally, the sensitivity analysis process includes the following steps:

[0255] Calculating the derivative of the energy function and the derivative of the load function to obtain a derivative expression;

[0256] 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;

[0257] 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;

[0258] Determine the Lagrange multiplier value to ignore the effect of the displacement field on the rate of change of the design variables;

[0259] 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.

[0260] Optionally, the iterative optimization process includes:

[0261] During the iteration process, according to the current objective function and constraints, strictly convex approximation subproblems are generated and solved, and the generation of the subproblems is controlled by the moving asymptotes;

[0262] Solving the strictly convex approximation subproblem to obtain updated design variables;

[0263] Substituting the updated design variables into the level set function to determine whether the set convergence conditions are met;

[0264] It is determined that the convergence condition is met, the iteration is terminated and the optimized topology structure is output.

[0265] 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 and will not be repeated here.

[0266] It will be understood that the various numerical numbers involved in the embodiments of the present application are merely distinctions for the convenience of description and are not intended to limit the scope of the embodiments of the present application.

[0267] It is easy for those skilled in the art to understand that the above 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 should be included in the scope of protection of the present application.

Claims

1. A parameterized level set topology optimization method for porous structures considering geometric nonlinearity, characterized by: include: A level set function is constructed to implicitly describe the structural boundary, and a parameterized level set function is obtained by using compactly supported radial basis function spline interpolation. Establish the equilibrium equations for geometrically nonlinear topology optimization and use geometrically 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 using the adjoint method to calculate the partial derivatives of the objective function of the topology optimization model with respect to the expansion coefficients 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; Among them, the Heaviside function is shown in the following formula: Where, is the area of ​​each finite element, The Heaviside function that represents the existence of the material at any point in the level set function is: 。 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 hole of the structure; Determining a pseudo-time parameter, using the pseudo-time parameter to describe the evolution of the structure shape over time, and establishing 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; constructing a parameterized level set function through spline interpolation of a spatial shape function, and calculating the normal velocity of the structure boundary according to the spatial shape function and the expansion coefficient function; The normal velocity is used to update the level set function 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 an equilibrium equation for geometric nonlinear topology optimization; 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, body force, and traction, and is established based on the mechanical state and external action; The equilibrium equation is linearized 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 geometric nonlinear conditions.

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 a volume fraction of each finite element and the second function represents a material ratio in a prescribed 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 bound of the material ratio, multiple local constraints are simplified into a single constraint function, and the constraint function is approximated using 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: Acquiring an actual displacement field and a virtual displacement field of a target space, and determining a strain energy density according to the actual displacement field and the virtual displacement field; Establishing an objective function corresponding to a topology optimization model according to the strain energy density and the Heaviside function, and minimizing structural flexibility based on the objective function to maximize geometrically nonlinear porous filling stiffness; 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 a linear elastic equilibrium equation, which is determined based on an energy function and a 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 the displacement field on the rate of change of the 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: During the iteration process, according to the current objective function and constraints, strictly convex approximation subproblems are generated and solved, and the generation of the subproblems 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 the parameterized level set function is obtained by compactly supported radial basis function spline interpolation; Finite element analysis module, used to establish the equilibrium equations for geometrically nonlinear topology optimization and solve the structural response under large deformation conditions through geometrically 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 using the adjoint method and calculate the 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; Iterative optimization module is used to update the level set discrete value by moving asymptote method, bring the updated level set discrete value into the objective function and perform finite element analysis again, and iterate continuously until the convergence condition is met, until the final topology optimization structure is obtained; Among them, the Heaviside function is shown in the following formula: Where, is the area of ​​each finite element, The Heaviside function that represents the existence of the material at any point in the level set function is: 。

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