Parameterized level set topology optimization method and system based on unstructured grid

Abstract

The present invention belongs to the technical field related to structural topology optimization, and discloses a parameterized level set topology optimization and system based on unstructured grids. The method includes the following steps: S1 obtains the support radius of the radial basis function of the topological structure to be processed; S2 solves the expansion coefficient of the time variable; solves the control equation of the design domain of the topological structure to be processed; S3 takes the minimum structural flexibility as the objective function, the maximum volume fraction as the constraint condition, and the expansion coefficient as the design variable to establish a parameterized level set topology optimization model based on an unstructured grid; S4 solves the optimization model until convergence, and outputs the current level set function corresponding to the convergence, and the structure corresponding to the current level set function is the optimized topological structure. Through the present invention, the dependence of the radial basis function on the grid in the parameterized level set method is eliminated, and the applicability of the parameterized level set topology optimization method to complex irregular structures is enhanced.

Description

Technical Field

[0001] The present invention belongs to the technical field related to structural topology optimization, and more specifically, relates to a parameterized level set topology optimization method and system based on unstructured grids. Background Art

[0002] Topology optimization is an effective design method for seeking high-performance, lightweight and multifunctional innovative structures. Among the many topology optimization methods, the level set method can accurately track the evolution of boundaries and has the advantage of obtaining clear and smooth structural boundaries. The parameterized level set method is an effective improvement on the level set method. It decouples the time variable and the space variable of the level set function by radial basis function interpolation, avoiding the numerical difficulties in directly solving the Hamilton-Jacobi partial differential equation and the problem that the structure cannot automatically open holes during the topology optimization process.

[0003] Currently, the parameterized level set method has shown its superiority in uniform regular grids, but there are still problems in non-uniform unstructured grids: the radial basis function support radius is affected by the density of the grid and shows grid dependence, and it is difficult to use a fixed value as the support radius. A support radius that is too small makes it difficult to search for enough control points. On the one hand, it will slow down the speed of iterative updates, resulting in increased computational costs, and on the other hand, it may also cause the results to appear jagged; a support radius that is too large will search for too many control points, which on the one hand may cause the interpolation matrix to be singular, and on the other hand, it may cause the results to lose local features. The present invention proposes a parameterized level set topology optimization method based on unstructured grids, which eliminates the dependence of the radial basis function on the grid, ensures that clear and smooth structural boundaries can still be obtained when applied to complex irregular structures, broadens the scope of application of the parameterized level set topology optimization method, and improves its engineering practicality. Summary of the invention

[0004] In view of the above defects or improvement needs of the prior art, the present invention provides a parameterized level set topology optimization and system based on unstructured grids, which eliminates the dependence of radial basis functions on grids in the parameterized level set method and enhances the applicability of the parameterized level set topology optimization method for complex irregular structures.

[0005] To achieve the above object, according to one aspect of the present invention, a parameterized level set topology optimization method based on unstructured grid is provided, and the method comprises the following steps:

[0006] S1 constructs a three-dimensional model of the topological structure to be processed and performs finite element mesh processing, determines the designable domain, the non-design domain, and the position of applying loads and constraints in the finite element mesh, and uses the finite element mesh to construct a level set function; constructs a neighborhood range of the radial basis function, and thereby obtains the support radius of the radial basis function;

[0007] S2 uses the radial basis function to decouple the level set function and solves to obtain the expansion coefficient of the time variable; uses the level set function to calculate the unit pseudo-density and interpolates it to the elastic tensor, thereby establishing a mapping relationship between the level set function and the elastic tensor, and uses the mapping relationship to solve the control equation of the topological structure design domain to be processed;

[0008] S3 takes the minimum structural flexibility as the objective function, the maximum volume fraction as the constraint condition, and the expansion coefficient as the design variable, and establishes a parameterized level set topology optimization model based on an unstructured grid, wherein the constraint condition of the maximum volume fraction is obtained by solving the control equation;

[0009] S4 solves the optimization model until convergence, and outputs a current level set function corresponding to convergence, wherein the structure corresponding to the current level set function is the optimized topological structure.

[0010] Further preferably, in step S1, the neighborhood range of the radial basis is performed according to the following relationship:

[0011]

[0012] Among them, R i is the neighborhood range of the adaptive grid, M is the total number of neighborhood control points, j is the number of the neighborhood control point, and x i is the coordinate of the i-th interpolation control point, x j is x i The coordinates of the neighborhood control points.

[0013] Further preferably, in step S1, the support radius of the radial basis is performed as follows:

[0014]

[0015] Among them, r(x) is the support radius of the adaptive radial basis function, x is the coordinate of the current sample point, and S is the scale factor, which is generally 2.0 to 4.0.

[0016] Further preferably, in step S2, the expansion coefficient is performed according to the following relationship:

[0017]

[0018] Among them, α(t) is the expansion coefficient related only to the time variable, Φ(x, t) is the level set function related to the time and space variables, is a radial basis function that is only related to the spatial variables.

[0019] Further preferably, in step S2, the mapping relationship between the level set function domain elastic tensor is performed according to the following relationship:

[0020]

[0021] in, is the elastic tensor field after unit pseudo-density interpolation, χe(Φ) is the pseudo-density field of the entire design domain, is the initial elastic tensor without interpolation.

[0022] Further preferably, in step S2, the control equation is performed according to the following relationship:

[0023]

[0024] Where Ω is the structural design domain, is the load boundary condition, ε is the structural strain tensor, is the elastic tensor field after unit pseudo-density interpolation, f is the structural volume force, is the structural boundary traction, dx is the volume integral, ds is the area integral, u is the structural displacement, v is the test function, and both u and v belong to the displacement space

[0025] Further preferably, in step S3, the topology optimization model is performed according to the following relationship:

[0026] Find:α=[α1,α2,...α n ] T

[0027]

[0028]

[0029] Among them, the design variable α is the expansion coefficient, the objective function J(Ω) is the structural flexibility, G(Ω) is the volume constraint, D is the entire design space, V max is the maximum volume fraction of the material specified, α(u, v) is the bilinear form of the energy, l(v) is the linear form of the load, is the structural boundary traction, u is the structural displacement, is the constraint boundary condition, is the load boundary condition, n is the number of control points, α nis the expansion coefficient value corresponding to the nth control point.

[0030] Further preferably, the energy bilinear form a(u, v) and the load linear form l(v) are carried out according to the following relationship:

[0031]

[0032]

[0033] Where ε is the structural strain tensor, is the elastic tensor field interpolated with the unit pseudo-density, It is the structural boundary traction.

[0034] Further preferably, in step S4, when solving the optimization model, the following steps are performed:

[0035] S41 calculates the sensitivity of the objective function and constraints with respect to the design variables;

[0036] S42 updates the expansion coefficient according to the sensitivity information;

[0037] S43 updates the level set function using the updated expansion coefficient;

[0038] S44 returns the updated level set function to step S2 until convergence.

[0039] According to another aspect of the present invention, a parameterized level set topology optimization system based on unstructured grids is provided. The system includes a processor, which executes the parameterized level set topology optimization method based on unstructured grids described above.

[0040] In general, the above technical solutions conceived by the present invention have the following beneficial effects compared with the prior art:

[0041] 1. The present invention adopts an adaptive radial basis function support radius strategy to enable the neighborhood range at each control point to adapt to the grid, thereby eliminating the grid dependency problem of the radial basis function in an irregular grid and avoiding the numerical problems caused by using a fixed support radius;

[0042] 2. The present invention calculates the shape derivatives of the objective function and constraint conditions with respect to the design variables based on the Hadamard boundary variation method and the adjoint analysis method, and combines the mature gradient-based optimization algorithm OC or MMA to update the design variables, thereby improving the accuracy of sensitivity analysis and the stability of iteration;

[0043] 3. The present invention is a parameterized level set topology optimization method based on unstructured grids. This method expands the scope of application of the parameterized level set topology optimization method, ensures that clear and smooth structural boundaries can still be obtained when applied to complex irregular structures, and enhances its engineering practicability. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 is a flow chart of a parameterized level set topology optimization method based on unstructured grids constructed according to a preferred embodiment of the present invention;

[0045] Figure 2 It is a three-dimensional structure diagram of the topological structure to be processed constructed according to the preferred embodiment of the present invention;

[0046] Figure 3 It is a structural diagram of the topological structure to be processed after finite element meshing constructed according to the preferred embodiment of the present invention;

[0047] Figure 4 It is a graph of the structural optimization objective function and constraint condition iteration process constructed according to the preferred embodiment of the present invention;

[0048] Figure 5 The diagrams are structural diagrams at different angles of the optimal configuration after structural optimization constructed according to the preferred embodiment of the present invention, wherein (a) is a front view, (b) is a bottom view, and (c) is a side view. DETAILED DESCRIPTION

[0049] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0050] like Figure 1 As shown, the present invention is a parameterized level set topology optimization method based on unstructured grids, comprising the following steps:

[0051] S1 builds a three-dimensional model, performs finite element meshing to determine the designable domain and non-design domain, and applies loads and constraints; the coordinates of each grid are used to construct the level set function using the coordinates on each node; an adaptive radial basis function support radius strategy is used to assemble radial basis functions that are only related to spatial variables. Specifically:

[0052] Wendland's C 2 The continuous radial basis function is considered to have a very simple form and robustness, and can meet the continuity and smoothness requirements of the level set function in the topology optimization process.2 The continuous radial basis function is expressed as:

[0053]

[0054] in r is the support radius, defined as:

[0055]

[0056] Where x represents the coordinate of the current sample point, x i represents the coordinates of the i-th (interpolation) control point; S is the scale factor, which is generally 2.0 to 4.0, R i Determine the neighborhood range of the current control point, S and R i Together they determine the influence range of the radial basis function at the control point.

[0057] In order to solve the grid dependency problem of the support radius, this paper proposes a radial basis function support radius method for adaptive grids. For each control point, its neighborhood range is the maximum value of the distance to all neighboring control points. Then the neighborhood range of the control point will change with the change of the grid. The neighborhood range will automatically become smaller in the dense grid area and larger in the sparse grid area, thereby realizing adaptive grid support. The new neighborhood range of the radial basis function is expressed as:

[0058]

[0059] Among them, x j is the neighborhood control point, and M is the number of neighborhood control points. Substituting (3) into (2) we can get the support radius of the adaptive radial basis function:

[0060]

[0061] Substituting (4) into (1) can assemble the adaptive global radial basis function matrix.

[0062] S2 decouples the level set function according to the radial basis function to obtain the expansion coefficient that is only related to the time variable; the unit pseudo-density is calculated according to the level set function and interpolated to the elastic tensor, and the mapping relationship between the level set function and the material property (elastic tensor) is established. The mapping relationship is used to solve the control equation of the design domain of the topological structure to be processed.

[0063] (1) Calculate the expansion factor

[0064] The parameterized level set method is an effective improvement on the traditional level set method. It decouples the time and space variables of the level set function by radial basis function interpolation, thereby transforming the complex partial differential equation into an ordinary differential equation that is easier to solve. Through radial basis function interpolation, the level set function can be approximately expressed as a set of radial basis functions. The product form of its expansion coefficient α(t):

[0065]

[0066] Substituting equation (1) into equation (5) can solve the time variable expansion coefficient α(t).

[0067] (2) Calculation of unit pseudo-density

[0068] In order to map the geometric information of the structural boundary into the mechanical model, the Heaviside function can be used to establish a mapping relationship between the level set function value and the structural material properties. The positive and negative values ​​of the level set function are used to describe the solid or hole inside the structure. The Φ(x) value in the area with material is greater than zero, Φ(x) is less than zero at the hole position, and Φ(x) is equal to zero on the boundary. The mathematical expression of the structural model can be re-expressed in a format that includes the Heaviside function:

[0069]

[0070] Since the exact Heaviside function is not differentiable, an approximate Heaviside function (smooth and differentiable) needs to be introduced instead in the topology optimization process:

[0071]

[0072] In formula (7), γ is a sufficiently small positive number, which is usually equal to twice the grid edge length. In order to facilitate the calculation of the objective function and volume fraction, the widely used “artificial material model” is used to calculate the unit pseudo-density χ e (Φ):

[0073]

[0074] Where τ represents a sufficiently small positive number, and the unit pseudo-density field of the entire design domain can be expressed as χe(Φ)=[χ1(Φ), χ2(Φ), ..., χ Ne (Φ)] T , so the Young's modulus field E after unit pseudo-density interpolation χ It is expressed as:

[0075] E χ =χe(Φ)E (9)

[0076] Here, E represents the uninterpolated Young's modulus.

[0077] Since Young's modulus is linearly related to the elastic tensor, the elastic tensor field after unit pseudo-density interpolation can be expressed as:

[0078]

[0079] in, Represents the uninterpolated elastic tensor.

[0080] (3) Solving the control equations

[0081] Design space for homogeneous isotropic materials The governing equation of the displacement field u of the linear elastic problem under small deformation conditions is expressed as:

[0082]

[0083] in, represents the Dirichlet boundary condition, which imposes a fixed constraint on this boundary; Represents the Neumann boundary condition, on which the load is applied; represents the boundary without traction; f represents the volume force; Represents the boundary The traction force; n represents The normal vector of the boundary. Substituting equation (10) into equation (11), after simplification, the weak form of the control equation after unit pseudo-density interpolation can be expressed as:

[0084]

[0085] Where ε represents the structural strain tensor, represents the fourth-order elastic tensor of the structure, v is a test function, and v belongs to the displacement space

[0086] S3 takes the minimum structural flexibility as the objective function, the maximum volume fraction as the constraint condition, and the expansion coefficient as the design variable to establish a parameterized level set topology optimization model based on unstructured grids;

[0087] Consider the most common flexibility optimization problem in structural topology optimization. Ignoring the body force f = 0, taking the minimum structural flexibility as the objective function and considering the maximum volume fraction as the constraint condition, a mathematical model for structural flexibility minimum topology optimization based on the parameterized level set method is established:

[0088] Find:α=[a1,α2,...α n ] T (13)

[0089]

[0090]

[0091] Among them, the expansion coefficient α is the design variable; the objective function is the structural flexibility; V max is the specified material volume fraction. The energy bilinear form a(u, v) and the load linear form l(v) are expressed as:

[0092]

[0093]

[0094] S4 calculates the shape derivatives of the objective function and constraints with respect to the design variables based on the Hadamard boundary variation method and adjoint analysis method, and uses the gradient-based optimization algorithm OC or MMA to update the expansion coefficients of the design variables. Finally, the level set function is recalculated based on the radial basis function and the expansion coefficient.

[0095] (1) Sensitivity analysis

[0096] In order to minimize the objective function (14), the Lagrangian function of the optimization objective function is established according to the equality constraints of the equilibrium equation and the inequality constraints of the structure volume fraction:

[0097]

[0098] Where λ is the Lagrangian multiplier of the volume constraint. Based on the Hadamard boundary variation method and adjoint analysis method, the shape derivative of the Lagrangian function can be obtained:

[0099]

[0100] where v(v∈U) is the solution to the following adjoint equation:

[0101]

[0102] According to the mapping relationship The boundary integral of equation (19) can be converted into the volume integral of the entire design domain:

[0103]

[0104] The first-order Hamilton-Jacobi equation based on parameterized level sets can be expressed as:

[0105]

[0106] According to formula (22), the normal velocity v of the structure boundary can be further obtainedn :

[0107]

[0108] Substituting equation (23) into equation (21), the Lagrangian function is further expressed as:

[0109]

[0110] The shape derivative of the Lagrangian function can be expressed as:

[0111]

[0112] Finally, combining equations (24) and (25), the sensitivity of the objective function and constraint conditions to the design variable (expansion coefficient) are expressed as:

[0113]

[0114]

[0115] (2) Update the expansion factor

[0116] According to the sensitivity information of equations (26) and (27), combined with the mature gradient-based optimization algorithms OC or MMA in topology optimization, the stable update of the design variable expansion coefficient α(t) can be achieved.

[0117] (3) Update the level set function

[0118] The updated expansion coefficient α(t) is combined with the adaptive radial basis function Substituting into equation (5) can update the level set function. The level set function value of the non-design domain is set to 1 / 2 of the current maximum value of the level set function.

[0119] In order to prove the effectiveness of the present invention, a specific example is used to verify the present invention as follows.

[0120] The optimization object is the GE aircraft engine bracket. The design space and boundary conditions of the model are as follows: Figure 2 As shown, the bottom of the bracket is fixed by four high-strength bolts. To simplify the calculation, a fixed constraint u=[0,0,0] is applied on the inner wall of the four bolt holes (red part) T , a load F = [0, 0, 1] is applied to the inner side of the two upper rings (green part) at an angle of 65° to the horizontal direction. T The constraint condition is that the structure volume fraction does not exceed 0.3, and the model is discretized using 1172879 tetrahedral mesh units.

[0121] The finite element model of GE aviation engine bracket is as follows: Figure 3As shown in the figure, the blue area represents the non-design domain and the gray area represents the design domain. The iterative process information of the objective function and the constraint conditions is shown in Figure 4 As shown in the figure, the red curve represents the structural flexibility and the blue curve represents the structural volume fraction. The whole process converges quickly first and then becomes stable after 50 iterations. The optimal topological structure of the bracket is shown in Figure 5 As shown in the figure, the model has clear and smooth boundaries and can be directly processed without post-processing.

[0122] In summary, the parameterized level set topology optimization method based on unstructured grid constructed by the present invention adopts the adaptive radial basis function support radius strategy, and the neighborhood range can automatically become smaller in the dense grid area and automatically become larger in the sparse grid area, thereby realizing adaptive grid support, eliminating the dependence of the radial basis function on the grid, and ensuring that a clear structural boundary can be obtained. In addition, the calculation of shape derivatives based on the Hadamard boundary variation method and the adjoint analysis method is adopted to improve the accuracy of sensitivity analysis; combined with the mature gradient-based optimization algorithm OC or MMA to update the design variables, the stability of iteration is enhanced.

[0123] It will be easily understood by those skilled in the art that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A parameterized level set topology optimization method based on unstructured grids, characterized in that: The method comprises the following steps: S1 constructs a three-dimensional model of the topological structure to be processed and performs finite element mesh processing, determines the designable domain, the non-design domain, and the position of applying loads and constraints in the finite element mesh, and uses the finite element mesh to construct a level set function; constructs a neighborhood range of the radial basis function, and thereby obtains the support radius of the radial basis function; The neighborhood range of the radial basis is carried out according to the following relationship: Among them, R i is the neighborhood range of the adaptive grid, M is the total number of neighborhood control points, j is the number of the neighborhood control point, and x i is the coordinate of the i-th interpolation control point, x j is x i Coordinates of neighborhood control points; The support radius of the radial basis is as follows: Among them, r(x) is the support radius of the adaptive radial basis function, x is the coordinate of the current sample point, and S is the scale factor, which is generally 2.0 to 4.0; S2 uses the radial basis function to decouple the level set function and solves to obtain the expansion coefficient of the time variable; uses the level set function to calculate the unit pseudo-density and interpolates it to the elastic tensor, thereby establishing a mapping relationship between the level set function and the elastic tensor, and uses the mapping relationship to solve the control equation of the topological structure design domain to be processed; The expansion factor is calculated according to the following relationship: Among them, α(t) is the expansion coefficient related only to the time variable, Φ(x, t) is the level set function related to the time and space variables, It is the radial basis function related only to the spatial variables; The control equation is carried out according to the following relationship: Where Ω is the structural design domain, is the load boundary condition, ε is the structural strain tensor, is the elastic tensor field after unit pseudo-density interpolation, f is the structural volume force, is the structural boundary traction, dx represents the volume integral, ds represents the area integral, u is the structural displacement, v is the test function, and both u and v belong to the displacement space S3 takes the minimum structural flexibility as the objective function, the maximum volume fraction as the constraint condition, and the expansion coefficient as the design variable, and establishes a parameterized level set topology optimization model based on an unstructured grid, wherein the constraint condition of the maximum volume fraction is obtained by solving the control equation; The topology optimization model is performed according to the following relationship: Find:α=[α1,α2,…α n ] T Among them, the design variable α is the expansion coefficient, the objective function J(Ω) is the structural flexibility, G(Ω) is the volume constraint, D is the entire design space, V max is the maximum volume fraction of the material specified, a(u, v) is the bilinear form of the energy, l(v) is the linear form of the load, is the structural boundary traction, u is the structural displacement, is the constraint boundary condition, is the load boundary condition, n is the number of control points, α n is the expansion coefficient value corresponding to the nth control point; S4 solves the optimization model until convergence, and outputs a current level set function corresponding to convergence, wherein the structure corresponding to the current level set function is the optimized topological structure.

2. The parameterized level set topology optimization method based on unstructured grids according to claim 1, characterized in that: In step S2, the mapping relationship between the level set function domain elastic tensor is performed according to the following relationship: in, is the elastic tensor field after unit pseudo-density interpolation, χ e (Φ) is the pseudo density field of the entire design domain, is the initial elastic tensor without interpolation.

3. The parameterized level set topology optimization method based on unstructured grids according to claim 1, characterized in that: The energy bilinear form a(u, v) and the load linear form l(v) are carried out according to the following relationship: in, ε is the structural strain tensor, is the elastic tensor field interpolated with the unit pseudo-density, It is the structural boundary traction.

4. The parameterized level set topology optimization method based on unstructured grids according to claim 1, characterized in that: In step S4, when solving the optimization model, the following steps are performed: S41 calculates the sensitivity of the objective function and constraints with respect to the design variables; S42 updates the expansion coefficient according to the sensitivity information; S43 updates the level set function using the updated expansion coefficient; S44 returns the updated level set function to step S2 until convergence.

5. A parameterized level set topology optimization system based on unstructured grids, characterized in that: The system comprises a processor, which executes the unstructured grid-based parameterized level set topology optimization method according to any one of claims 1 to 4.

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