Multi-constraint geometrically nonlinear level set topology optimization method, apparatus, medium
By introducing displacement and frequency constraints into the Lagrangian function and combining them with nonlinear element force calculations, the optimization problem of the linear level set method under large deformations was solved, and the stability and reliability were improved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- GUANGZHOU UNIVERSITY
- Filing Date
- 2022-12-15
- Publication Date
- 2026-05-12
AI Technical Summary
The existing linear level set method is difficult to effectively optimize the component to be optimized under large deformation conditions, resulting in unsatisfactory optimization results.
By introducing displacement and frequency constraints, the Lagrangian function is extended, and nonlinear element unbalanced forces are combined to calculate element and overall thermal loads for topology optimization.
Under large deformation conditions, the stability and reliability of the optimization results were achieved, with displacement and frequency within limits, thus improving the effectiveness of topology optimization.
Smart Images
Figure CN116151062B_ABST
Abstract
Description
Technical Field
[0001] This application relates to the field of topology optimization technology, and in particular to methods, devices, and media for topology optimization of multi-constraint geometric nonlinear level sets. Background Technology
[0002] The level set method is one of the main centralized topology optimization methods. Its main advantage is that the optimization results have smooth boundaries. Compared with optimization results such as Solid Isotropic Material with Penalization (SIMP) and Bi-directional Evolutionary Structural Optimization (BESO), the level set method can avoid the problem of jagged boundaries or checkerboard pattern.
[0003] Currently, the level set method generally uses minimizing the compliance of the structure as the objective function under linear conditions, with the volume fraction of the structure as the constraint. Here, the volume fraction represents the ratio of the optimized solid material volume to the original design domain volume.
[0004] However, in actual engineering, the components to be optimized inevitably undergo large deformations, i.e., geometric nonlinearity. This situation cannot be considered by the current linear level set method, which leads to unsatisfactory optimization results. Summary of the Invention
[0005] This application provides a multi-constraint geometric nonlinear level set topology optimization method, device, and medium to solve the problem that the current linear level set method is difficult to optimize well under large deformation conditions of the component to be optimized.
[0006] This application provides a multi-constraint geometric nonlinear level set topology optimization method, including:
[0007] The design domain corresponding to the component to be optimized is divided into several finite element elements, and corresponding horizontal set radial basis functions based on volume constraints are established.
[0008] Based on the relationship between the element strain increment and the nodal displacement increment of the finite element, displacement constraint terms and frequency constraint terms are introduced to extend the Lagrange function;
[0009] Based on the nonlinear element unbalanced force, the element thermal load and the overall thermal load are calculated.
[0010] Based on the extended Lagrangian function and the overall thermal load of the component to be optimized at a set temperature, topology optimization is performed on the design domain.
[0011] In one example, the finite element element includes holes and mesh nodes; establishing the corresponding volume-constrained level set radial basis function specifically includes: establishing a level set radial basis function based on the distance between the holes and mesh nodes, combined with a two-dimensional multispline basis function; constraining the generalized expansion coefficients in the level set radial basis function based on the condition of the unique solution of the interpolation of the level set radial basis function, and updating the level set radial basis function according to the boundary constraint conditions; establishing a volume-constrained model based on the updated level set radial basis function to obtain the volume-constrained level set radial basis function.
[0012] In one example, before introducing displacement constraint terms and frequency constraint terms based on the relationship between the element strain increment and nodal displacement increment of the finite element element, the method further includes: determining the incremental strain matrix corresponding to the element strain increment and nodal displacement increment based on the virtual work equation of the finite element element; obtaining the element equilibrium equation based on the virtual work equation and the incremental strain matrix; and expressing the relationship between the element stress increment and strain increment using the incremental formulation method based on the element equilibrium equation, and determining the relationship between the element strain increment and nodal displacement increment.
[0013] In one example, the step of introducing displacement constraint terms and frequency constraint terms based on the relationship between the element strain increment and the nodal displacement increment of the finite element element, and extending the Lagrangian function, specifically includes: decomposing the incremental strain matrix to obtain a first matrix that is independent of the nodal displacement and a second matrix that is related to the nodal displacement; determining the incremental form of the element stiffness equation based on the first matrix and the second matrix, as well as the relationship between the element strain increment and the nodal displacement increment, and obtaining the global incremental equation.
[0014] In one example, the introduction of displacement and frequency constraints to extend the Lagrangian function specifically includes: introducing a displacement constraint term containing displacement Lagrange multipliers and displacement increments; introducing a frequency constraint term containing frequency Lagrange multipliers and frequency increments; and determining the extended Lagrangian function composed of the objective function, volume constraint term, displacement constraint term, and frequency constraint term.
[0015] In one example, the calculation of the element thermal load and the overall thermal load based on the nonlinear element unbalanced force specifically includes: determining the initial temperature strain; determining the element thermal load based on the overall element nodal force, the overall internal force of the structure, and the initial temperature strain in the calculation of the nonlinear element unbalanced force; and adding the element thermal loads to obtain the overall thermal load.
[0016] In one example, the component to be optimized is a deformable structure whose deformability exceeds a preset threshold.
[0017] In one example, the method further includes setting corresponding displacement limits and frequency limits based on the allowable deformability range of the application scenario of the deformable structure.
[0018] This application provides a multi-constraint geometric nonlinear level set topology optimization device, comprising: a memory, a processor, and a computer program stored in the memory and executable on the processor. When executed by the processor, the computer program implements the following method steps:
[0019] The design domain corresponding to the component to be optimized is divided into several finite element elements, and corresponding horizontal set radial basis functions based on volume constraints are established.
[0020] Based on the relationship between the element strain increment and the nodal displacement increment of the finite element, displacement constraint terms and frequency constraint terms are introduced to extend the Lagrange function;
[0021] Based on the nonlinear element unbalanced force, the element thermal load and the overall thermal load are calculated.
[0022] Based on the extended Lagrangian function and the overall thermal load of the component to be optimized at a set temperature, topology optimization is performed on the design domain.
[0023] This application provides a computer storage medium storing computer-executable instructions, characterized in that the computer-executable instructions are configured as follows:
[0024] The design domain corresponding to the component to be optimized is divided into several finite element elements, and corresponding horizontal set radial basis functions based on volume constraints are established.
[0025] Based on the relationship between the element strain increment and the nodal displacement increment of the finite element, displacement constraint terms and frequency constraint terms are introduced to extend the Lagrange function;
[0026] Based on the nonlinear element unbalanced force, the element thermal load and the overall thermal load are calculated.
[0027] Based on the extended Lagrangian function and the overall thermal load of the component to be optimized at a set temperature, topology optimization is performed on the design domain.
[0028] This application provides a multi-constraint geometric nonlinear level set topology optimization method, device, and medium. Based on the linear level set method, it considers displacement and frequency constraints under nonlinearity, extends the Lagrangian function, and obtains a multi-constraint geometric nonlinear level set topology optimization method. It can be applied to the component to be optimized under large deformation conditions, and obtains the best topology optimization result so that the displacement and frequency in practical applications do not exceed the limit, which is beneficial to increasing stability and reliability. Attached Figure Description
[0029] The accompanying drawings, which are included to provide a further understanding of this application and form part of this application, illustrate exemplary embodiments and are used to explain this application, but do not constitute an undue limitation of this application. In the drawings:
[0030] Figure 1 A flowchart of a multi-constraint geometric nonlinear level set topology optimization method provided in this application embodiment;
[0031] Figure 2 (a) is a schematic diagram of a simply supported beam structure design domain provided in an embodiment of this application;
[0032] Figure 2 (b) is a schematic diagram of the initial hole arrangement of a simply supported beam provided in an embodiment of this application;
[0033] Figure 3 A schematic diagram showing the topology optimization results of a simply supported beam based on volume constraints and thermal loads at different temperatures, as provided in the embodiments of this application.
[0034] Figure 4 A schematic diagram of the optimal topology and iterative process under the first multi-constraint condition provided in the embodiments of this application;
[0035] Figure 5 A schematic diagram of the optimal topology and iterative process under a second type of multi-constraint conditions provided in an embodiment of this application;
[0036] Figure 6 A schematic diagram of the optimal topology and iterative process under a third type of multi-constraint condition provided in this application embodiment;
[0037] Figure 7 A schematic diagram of the structure of a multi-constraint geometric nonlinear level set topology optimization device provided in an embodiment of this application. Detailed Implementation
[0038] To make the objectives, technical solutions, and advantages of this application clearer, the technical solutions of this application will be clearly and completely described below in conjunction with specific embodiments and corresponding drawings. Obviously, the described embodiments are only a part of the embodiments of this application, and not all of them. Based on the embodiments in this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.
[0039] Figure 1 The flowchart of the multi-constraint geometric nonlinear level set topology optimization method provided in this application embodiment specifically includes the following steps:
[0040] S101: Divide the design domain corresponding to the component to be optimized into several finite element elements, and establish the corresponding horizontal set radial basis functions based on volume constraints.
[0041] The component to be optimized represents an engineering structure that requires topology optimization, such as a simply supported beam or a cantilever beam. Finite element elements include holes and mesh nodes.
[0042] Establishing the corresponding level set radial basis functions based on volume constraints involves the following steps:
[0043] First, determine the location of the hole based on its center coordinates, calculate the distance between the hole center and the horizontal and vertical coordinates of each grid node, and establish a horizontal set function. Combine the horizontal and vertical coordinate distances between each grid node with the two-dimensional multiquadrics (MQ) spline basis function, as shown in the following formula (1):
[0044]
[0045] Where x represents the coordinates of the i-th node, c represents the shape parameter, and D represents the entire design area;
[0046] The radial basis functions of the level set at each center point can be obtained, as shown in the following formula (2):
[0047]
[0048] Where, α i (t) represents the generalized expansion coefficients, and p(x,t) is the coefficient of the first-order polynomial that varies with time, specifically expressed as formula (3):
[0049] p(x,t)=p0(t)+p1(t)x+p2(t)y (3)
[0050] Second, in order to ensure the unique solution of the radial basis function interpolation of the level set, the generalized expansion coefficients in the radial basis function of the level set are constrained based on this condition, as shown in formula (4):
[0051]
[0052] Combining formulas (2), (3), and (4), they can be rewritten in matrix form:
[0053] Gα(t)=φ(t) (5)
[0054] in,
[0055]
[0056]
[0057]
[0058] G represents the transition matrix, A represents the level set matrix, and P represents the node matrix.
[0059] The generalized expansion coefficients can then be converted into:
[0060] α(t)=G -1 φ(t) (9)
[0061] Correspondingly, the radial basis function of the level set represented by formula (2) can be simplified as:
[0062] φ(x,t)=g(x) T α(t) (10)
[0063] Where g(x)={g1(x)…g n (x), 1, x, y} T (11)
[0064] Substituting equation (11) into the Hamilton-Jacobi equation, we obtain the governing equations for the generalized expansion coefficients based on the level set radial basis functions:
[0065]
[0066] in,
[0067]
[0068]
[0069]
[0070] For time-dependent interpolation problems, the relevant constraints of formula (4) must be combined to ensure the correctness of the MQ spline conditions. In topology optimization problems, since the initial level set function values at the nodes are given and the matrix G is theoretically invertible, solving the n+3 linear equations yields the initial generalized expansion coefficients α(t0):
[0071] α(t0)=G -1 φ(t0) (16)
[0072] The approximate value obtained by using the first-order forward Euler method is:
[0073] α(t i+1 )=α(t i )+ΔtG -1 B(α(t i ),t i (17)
[0074] Among them, t i Let be the i-th time step, Δt be the time step, and B be the velocity vector.
[0075] In practical implementation, to prevent unnecessary numerical instability from formula (17), its speed is reiterated as follows:
[0076] B(t i )={V n (x1,t i )...V n (x n ,t i ),0,0,0} T (18)
[0077] To prevent near the boundary If the value is too large or too small, update the level set function φ for the entire design domain as follows:
[0078]
[0079] Since the relationship between φ and α is linear, α is updated to α. u :
[0080]
[0081] To avoid unbounded growth of the level set function, an approximate δ(φ) function is introduced into the evolution scheme of formula (17):
[0082]
[0083] Therefore, the updated equation is modified as follows:
[0084]
[0085] in It can be represented as t i The velocity vector at time t can be written as:
[0086]
[0087] The above-mentioned prevention near the boundary Values that are too large or too small, and to avoid unbounded growth of the level set function, can be used as boundary constraints to update the radial basis functions of the level set.
[0088] Third, based on the problem of minimizing the compliance of statically loaded linear elastic structures under volume constraints, a volume constraint model is established according to the updated level set radial basis functions, and the level set radial basis functions based on volume constraints are obtained.
[0089] Specifically, the mathematical model is expressed as:
[0090]
[0091]
[0092] The symbols in the above formulas (24) and (25) are defined as follows:
[0093]
[0094]
[0095] Where J(φ) is the objective function, u is the displacement field, ε is the linearized strain tensor, C is the Hooke elastic tensor, and u0, τ, and b represent the given displacement, traction force, and body force constants, respectively. The linear elastic equilibrium equation is expressed in energy form a(u,v,φ) and load form l(v,φ), respectively, where v represents the virtual displacement field in the allowable displacement field space U. G(φ) is the constraint limiting the use of materials, V max It is the maximum allowable volume of the design domain D. H(φ) represents the existence of matter at a point on the level set function value φ, and is defined as follows:
[0096]
[0097] For the minimum compliance problem, the normal acceleration can be derived:
[0098] V n =ε(u):C:ε(u)-λ (29)
[0099] Where λ is a Lagrange multiplier used to handle volume fraction constraints. Alternatively, the following augmented Lagrange update scheme can be used:
[0100]
[0101] Where μ and γ k Let γ be the parameter for the k-th iteration. k The update plan is as follows:
[0102] γ k+1 =min(γ) k +Δγ,γ max )k>n R (31)
[0103] Where Δγ is the increment, γ max This is the upper limit of γ. In level set-based optimization methods, the initial design volume fraction often does not meet the specified volume fraction. In the first iteration, the volume constraint is relaxed as follows:
[0104]
[0105] S102: Based on the relationship between the element strain increment and the nodal displacement increment of the finite element, displacement constraint terms and frequency constraint terms are introduced to extend the Lagrangian function.
[0106] In one embodiment, the incremental strain matrix corresponding to the element strain increment and nodal displacement increment is determined based on the virtual work equation of the finite element element. The element equilibrium equation is then obtained based on the virtual work equation and the incremental strain matrix. Based on the element equilibrium equation, the relationship between the element stress increment and strain increment is expressed using the incremental formulation method, and the relationship between the element strain increment and nodal displacement increment is determined.
[0107] Next, the incremental strain matrix is decomposed to obtain a first matrix that is independent of nodal displacements and a second matrix that is related to nodal displacements. Based on the first and second matrices, and the relationship between the element strain increment and the nodal displacement increment, the incremental form of the element stiffness equation is determined, and the global incremental equation is obtained.
[0108] Specifically, the principle of virtual work is universal for geometrically nonlinear problems. Based on the principle of virtual work, the virtual work equation of a unit can be expressed in the following form:
[0109]
[0110] Where {F} is the element nodal force vector, {ε *} e For the virtual strain of the element, {δ *} e This is the virtual displacement vector of the node.
[0111] The incremental strain-displacement relationship is as follows:
[0112]
[0113] Where, d{ε} e Represents the element nodal displacement {δ} e The differential.
[0114] Based on the formal similarity between variational and differential operations, it can be transformed into:
[0115]
[0116] Formulas (34) and (35) above The variable is the incremental strain matrix of the component to be optimized under large displacement, representing the relationship between the element strain increment and the nodal displacement increment. Under large displacement... It should be a function of nodal displacements.
[0117] The strain increment matrix is decomposed into a part [B0] (i.e., the first matrix) that is independent of nodal displacements and a part [B] that is related to nodal displacements. L (i.e., the second matrix) consists of two parts:
[0118]
[0119] Where [B0] is the strain matrix in general linear analysis.
[0120] Substituting equation (35) into equation (33), and considering the arbitrariness of the virtual displacements at the nodes, the element equilibrium equation can be expressed as:
[0121]
[0122] According to formula (37), a finite element formulation can be established for the entire structure. However, in geometric nonlinear analysis, the element and structural stiffness obtained by this formulation method are asymmetric, so the incremental formulation method can be used.
[0123] The equilibrium equation shown in formula (37) can be expressed in differential form:
[0124]
[0125] Because in geometrically nonlinear problems, the strain matrix and stress {σ} e Both are functions of nodal displacements, therefore we can obtain:
[0126]
[0127] Substituting formula (39) into formula (38), we get:
[0128]
[0129] There is a definite relationship between the stress increment and strain increment within an element, which can be expressed as an increment as follows:
[0130] d{σ} e =[D]d{ε} e (41)
[0131] Wherein, [D] is called the stress-strain relationship matrix, which is...
[0132]
[0133] Where E is the elastic modulus of the material, and nu is Poisson's ratio.
[0134] For linearly elastic materials, the following exists:
[0135] {σ} e=[D]({ε} e -{ε0} e )+{σ0} e (43)
[0136] Where, {ε0} e and {σ0} e These represent the initial strain and initial stress that may exist in the unit material, respectively.
[0137] Substituting equation (34) into equation (41), we can obtain the relationship between stress increment and element nodal displacement increment:
[0138]
[0139] Substituting formula (36) into formula (44), we get:
[0140] d{σ} e =[D]([B0]+[B L ])d{δ} e (45)
[0141] Therefore, the second term on the left-hand side of formula (40) can be expressed as:
[0142]
[0143] If let
[0144] Where [k0] is the element stiffness matrix in general linear analysis, which is independent of the element nodal displacements. Then, the terms in the second set of parentheses on the right-hand side of formula (46) can be expressed as:
[0145]
[0146] Among them, [k L ] is the element initial displacement matrix, which represents the effect of changes in element position on the element stiffness matrix.
[0147] Taking the first term on the left-hand side of formula (40) backtracking, considering the relationship with formula (36) and its independence from nodal displacements, its differential with respect to nodal displacements is zero. Therefore, the first term on the left-hand side of formula (40) can be expressed as:
[0148]
[0149] Among them, [k σ ] is the geometric stiffness matrix, which represents the influence of the stress present in the element on the element stiffness matrix.
[0150] The above [k0], [k L ] and [k σ Combining this with the above formula, we get:
[0151] ([k0]+[k σ ]+[k L ])d{δ} e =d{F} e (50)
[0152] remember
[0153] [k T ]=[k0]+[k σ ]+[k L (51)
[0154] Among them, [k T [ ] represents the tangent stiffness matrix. At this point, the element stiffness equation in incremental form is:
[0155] [k T ]d{δ} e =d{F} e (52)
[0156] The element tangent stiffness matrix is assembled to obtain:
[0157] [K T ]=∑[k T (53)
[0158] This leads to the overall incremental equation of the structure:
[0159] [K T ]d{δ} e =d{F} e (54)
[0160] In one embodiment, a displacement constraint term containing displacement Lagrange multipliers and displacement increments is introduced, and a frequency constraint term containing frequency Lagrange multipliers and frequency increments is introduced to determine an extended Lagrange function consisting of the objective function, volume constraint term, displacement constraint term, and frequency constraint term.
[0161] Specifically, the original Lagrange function is extended by adding corresponding frequency Lagrange multipliers and frequency constraints. The formula is as follows:
[0162]
[0163] Where C(x) is the objective function, and λ is the volume Lagrange multiplier. For volume constraints, μ1 is the displacement Lagrange multiplier, μ2 is the frequency Lagrange multiplier, and μ1(d j -d * ) represents the displacement constraint, μ2(ω) n -ω *) represents the frequency constraint, ω n Let ω be the natural frequency of the nth order of the structure. * Its upper limit.
[0164] S103: The element thermal load and the overall thermal load are calculated based on the nonlinear element unbalanced force.
[0165] In topology optimization, the influence of temperature on the optimization also needs to be considered. The initial temperature strain needs to be determined. In the calculation of nonlinear element unbalanced forces, the element thermal load is determined based on the overall element nodal forces, the overall internal forces of the structure, and the initial temperature strain. The overall thermal load is obtained by adding the element thermal loads together.
[0166] Specifically, under the influence of temperature T, a small length within an elastic body, if left unconstrained, will experience a normal strain αT, where α is the coefficient of thermal expansion. In an isotropic body, this normal strain is the same in all directions and is not accompanied by any shear strain. Therefore, for a two-dimensional elastic body, the initial strain due to temperature is:
[0167]
[0168] In the formula, The initial strain is caused by temperature.
[0169] In geometrically nonlinear finite element analysis for calculating nodal unbalanced forces, the total nodal force ∑{F} of the overall element is... e Including the overall structural internal forces ∑{F} generated by the initial nodal displacements {δ}1. m} e and the thermal load Σ{F due to temperature th} e .
[0170] ∑{F} e =∑{F m} e +∑{F th} e (57)
[0171] Unit thermal load {F th} e It can be represented as:
[0172]
[0173] Substituting equation (56) into equation (58), we can obtain the element thermal load {F} generated by temperature stress. th} e for:
[0174]
[0175] Finally, the thermal load contributions of each unit are summed to obtain the overall thermal load vector ∑{F} th} e .
[0176] It should be noted that the order in which S102 and S103 are written does not necessarily limit their actual execution order.
[0177] S104: Based on the extended Lagrangian function and the overall thermal load of the component to be optimized at a set temperature, perform topology optimization on the design domain.
[0178] This solution can be applied to any component requiring optimization that needs to consider displacement and frequency constraints, especially deformable structures whose deformability exceeds a preset threshold. Such deformable structures often experience large deformations in practical applications, necessitating restrictions on the range of displacement and frequency changes. Therefore, this solution is particularly effective in structures with large deformations.
[0179] Furthermore, based on the allowable deformability range of the application scenario of the deformable structure, corresponding displacement limits and frequency limits can be set to adapt to the needs of specific application scenarios.
[0180] In this embodiment, based on the linear level set method, displacement and frequency constraints under nonlinear conditions are considered, and the Lagrangian function is extended to obtain a multi-constraint geometric nonlinear level set topology optimization method. This method is applicable to the components to be optimized under large deformation conditions and obtains the best topology optimization results so that the displacement and frequency in practical applications do not exceed the limits, which is beneficial to increasing stability and reliability.
[0181] Example 1
[0182] Figure 2 (a) and (b) represent the foundation structural dimensions, boundary constraints, initial structure, and load information of the simply supported beam. Figure 2 (a) is the structural design domain. Figure 2 (b) is the initial hole arrangement.
[0183] like Figure 2 As shown in (a) and (b), the simply supported beam is divided into 120×30 element grids, and a concentrated load of F = -100 is applied at the midpoint of the upper end. The elastic modulus of the solid material is E0 = 1, and the elastic modulus of the material at the opening is 10. -9 Poisson's ratio nu = 0.3, and the volume fraction is 50%.
[0184] A topology optimization analysis was performed on a simply supported beam under volume constraints and thermal loads, with temperatures ranging from -10℃ to 20℃ in 10℃ intervals. Figure 3 The diagram shows the topology optimization results of a simply supported beam at different temperatures.
[0185] like Figure 3 As shown, the topology optimization results from left to right are as follows:
[0186] At -10℃, the compliance is 1.54051e10, the strain energy is 1.91238e10, and the displacement is -30954.553167.
[0187] At 0℃, the compliance is 1.56802e10, the strain energy is 2.00069e10, and the displacement is 23780.613432.
[0188] At 10℃, the compliance is 1.63058e10, the strain energy is 2.12485e10, and the displacement is 24804.910550.
[0189] At 20℃, the compliance is 1.75115e10, the strain energy is 2.27530e10, and the displacement is 79230.722700.
[0190] according to Figure 3 The optimized topology results and compliance and strength information at different temperatures show that as the temperature increases, the strain energy of the structure increases, and the differences in material distribution in the structural morphology become more and more obvious. In terms of material distribution, the main force transmission paths are consistent, but the number of branch force transmission paths differs significantly.
[0191] Example 2
[0192] Following the example in Example 1, Figure 2 The simply supported beams shown in (a) and (b) have basic structural dimensions, boundary constraints, initial structure, and material property settings. This embodiment presents the topology optimization of the structure with added displacement and frequency constraints at a temperature of 10℃. See Table 1 and... Figure 4 , Figure 5 , Figure 6 As shown;
[0193] Table 1
[0194]
[0195]
[0196] Table 1 presents three different constraints, along with the compliance, volume fraction, fundamental frequency, and displacement of the optimal topology under each constraint.
[0197] Figure 4 This is the optimal topology diagram for operating condition 1 and the corresponding iterative process.
[0198] Figure 5 This is the optimal topology diagram for working condition 2 and the corresponding iterative process.
[0199] Figure 6 This is the optimal topology diagram for working condition 3 and the corresponding iterative process.
[0200] Combined with Table 1 and Figures 4-6 It can be seen that the proposed scheme can satisfy different displacement and frequency constraints, and the algorithm's iterative process converges stably. This also verifies the rationality of adding a geometric nonlinear method for displacement and frequency constraints under thermal coupling in the level set method.
[0201] The above describes a multi-constraint geometric nonlinear level set topology optimization method provided in this application. Based on the same inventive concept, this application also provides a corresponding multi-constraint geometric nonlinear level set topology optimization device, such as... Figure 7 As shown.
[0202] Figure 7 The schematic diagram of the multi-constraint geometric nonlinear level set topology optimization device provided in this application embodiment specifically includes: a memory 70, a processor 72, and a computer program stored in the memory 70 and executable on the processor 72. When the computer program is executed by the processor 72, it implements the following method steps:
[0203] The design domain corresponding to the component to be optimized is divided into several finite element elements, and corresponding horizontal set radial basis functions based on volume constraints are established.
[0204] Based on the relationship between the element strain increment and the nodal displacement increment of the finite element, displacement constraint terms and frequency constraint terms are introduced to extend the Lagrange function;
[0205] Based on the nonlinear element unbalanced force, the element thermal load and the overall thermal load are calculated.
[0206] Based on the extended Lagrangian function and the overall thermal load of the component to be optimized at a set temperature, topology optimization is performed on the design domain.
[0207] The various embodiments in this application are described in a progressive manner. Similar or identical parts between embodiments can be referred to mutually. Each embodiment focuses on describing the differences from other embodiments. In particular, the device embodiments are basically similar to the method embodiments, so the description is relatively simple; relevant parts can be referred to the descriptions of the method embodiments.
[0208] It should also be noted that the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitation, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.
[0209] The above description is merely an embodiment of this application and is not intended to limit the scope of this application. Various modifications and variations can be made to this application by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this application should be included within the scope of the claims of this application.
Claims
1. A multi-constraint geometric nonlinear level set topology optimization method, characterized in that, include: The design domain corresponding to the component to be optimized is divided into several finite element elements, and corresponding horizontal set radial basis functions based on volume constraints are established. Based on the virtual work equation of the finite element, determine the incremental strain matrix corresponding to the element strain increment and the nodal displacement increment; Based on the virtual work equation and the incremental strain matrix, the element equilibrium equation is obtained; Based on the aforementioned element equilibrium equations, the relationship between element stress increment and strain increment is expressed using the incremental formulation method, and the relationship between element strain increment and nodal displacement increment is determined. Based on the relationship between the element strain increment and the nodal displacement increment of the finite element, displacement constraint terms and frequency constraint terms are introduced to extend the Lagrange function; Based on the nonlinear element unbalanced force, the element thermal load and the overall thermal load are calculated. Based on the extended Lagrangian function and the overall thermal load of the component to be optimized at a set temperature, topology optimization is performed on the design domain; The extension of the Lagrange function specifically includes: Decompose the incremental strain matrix to obtain a first matrix that is independent of nodal displacements and a second matrix that is related to nodal displacements; Based on the first and second matrices, and the relationship between the element strain increment and the nodal displacement increment, the incremental form of the element stiffness equation is determined, and the overall incremental equation is obtained.
2. The method according to claim 1, characterized in that, The finite element unit includes holes and mesh nodes; The establishment of the corresponding level set radial basis function based on volume constraints specifically includes: Based on the distance between the holes and the mesh nodes, and combined with the two-dimensional multispline basis functions, a horizontal set radial basis function is established; Based on the condition of the unique solution of the interpolation of the radial basis function of the level set, the generalized expansion coefficients in the radial basis function of the level set are constrained, and the radial basis function of the level set is updated according to the boundary constraint conditions. Based on the updated level set radial basis functions, a volume constraint model is established, and a level set radial basis function based on volume constraints is obtained.
3. The method according to claim 1, characterized in that, The introduction of displacement and frequency constraints to extend the Lagrange function specifically includes: Introduce a displacement constraint term that includes displacement Lagrange multipliers and displacement increments; Introduce a frequency constraint term that includes frequency Lagrange multipliers and frequency increments; Determine the extended Lagrangian function, which consists of the objective function, volume constraint term, displacement constraint term, and frequency constraint term.
4. The method according to claim 1, characterized in that, The calculation of the element thermal load and the overall thermal load based on the nonlinear element unbalanced force specifically includes: Determine the initial strain at temperature; In the calculation of nonlinear element unbalanced forces, the element thermal load is determined based on the overall element nodal forces, the overall internal forces of the structure, and the initial temperature strain. The overall thermal load is obtained by adding the individual thermal loads together.
5. The method according to claim 1, characterized in that, The component to be optimized is a deformable structure whose deformability exceeds a preset threshold.
6. The method according to claim 5, characterized in that, The method further includes: Based on the allowable deformability range of the application scenario of the deformable structure, set corresponding displacement limits and frequency limits.
7. A multi-constraint geometric nonlinear level set topology optimization device, characterized in that, include: A memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the computer program, when executed by the processor, performs the following method steps: The design domain corresponding to the component to be optimized is divided into several finite element elements, and corresponding horizontal set radial basis functions based on volume constraints are established. Based on the virtual work equation of the finite element, determine the incremental strain matrix corresponding to the element strain increment and the nodal displacement increment; Based on the virtual work equation and the incremental strain matrix, the element equilibrium equation is obtained; Based on the aforementioned element equilibrium equations, the relationship between element stress increment and strain increment is expressed using the incremental formulation method, and the relationship between element strain increment and nodal displacement increment is determined. Based on the relationship between the element strain increment and the nodal displacement increment of the finite element, displacement constraint terms and frequency constraint terms are introduced to extend the Lagrange function; Based on the nonlinear element unbalanced force, the element thermal load and the overall thermal load are calculated. Based on the extended Lagrangian function and the overall thermal load of the component to be optimized at a set temperature, topology optimization is performed on the design domain; The extension of the Lagrange function specifically includes: Decompose the incremental strain matrix to obtain a first matrix that is independent of nodal displacements and a second matrix that is related to nodal displacements; Based on the first and second matrices, and the relationship between the element strain increment and the nodal displacement increment, the incremental form of the element stiffness equation is determined, and the overall incremental equation is obtained.
8. A computer storage medium storing computer-executable instructions, characterized in that, The computer-executable instructions are set as follows: The design domain corresponding to the component to be optimized is divided into several finite element elements, and corresponding horizontal set radial basis functions based on volume constraints are established. Based on the virtual work equation of the finite element, determine the incremental strain matrix corresponding to the element strain increment and the nodal displacement increment; Based on the virtual work equation and the incremental strain matrix, the element equilibrium equation is obtained; Based on the aforementioned element equilibrium equations, the relationship between element stress increment and strain increment is expressed using the incremental formulation method, and the relationship between element strain increment and nodal displacement increment is determined. Based on the relationship between the element strain increment and the nodal displacement increment of the finite element, displacement constraint terms and frequency constraint terms are introduced to extend the Lagrange function; Based on the nonlinear element unbalanced force, the element thermal load and the overall thermal load are calculated. Based on the extended Lagrangian function and the overall thermal load of the component to be optimized at a set temperature, topology optimization is performed on the design domain; The extension of the Lagrange function specifically includes: Decompose the incremental strain matrix to obtain a first matrix that is independent of nodal displacements and a second matrix that is related to nodal displacements; Based on the first and second matrices, and the relationship between the element strain increment and the nodal displacement increment, the incremental form of the element stiffness equation is determined, and the overall incremental equation is obtained.