A topology optimization method for multi-material turbine disks considering gradient interfaces

Abstract

The present invention provides a multi-material turbine disk topology optimization method that considers gradient interfaces. This method, based on practical engineering problems, defines a turbine disk radial cross-section design domain; establishes a stress-minimizing mathematical model for multi-material structural topology optimization that considers gradient interfaces; uses a piecewise Heaviside projection function to obtain the density field projection of the gradient material phase; establishes a multi-gradient material elastic modulus and stress interpolation model based on the standard SIMP method and the piecewise Heaviside function; obtains a displacement matrix of the turbine disk cross section based on finite element calculations; further calculates the unit von Mises stress and the global stress of the objective function based on the interpolation model and the displacement matrix; solves the global stress sensitivity using the adjoint method; calculates the mass sensitivity; updates the design variables using the moving asymptote method; and filters the design variables to obtain the multi-gradient material turbine disk design. This method improves the design efficiency of multi-material turbine disks, realizes multi-gradient material transitions between multi-material interfaces, and is suitable for additive manufacturing.

Description

Technical Field

[0001] The present invention relates to the technical field of engineering structure design and analysis, and in particular to a multi-material turbine disk topology optimization method considering gradient interfaces. Background Art

[0002] Multi-material topology optimization enables high-performance designs and is widely used in fields such as aerospace. By introducing gradient materials, gradient transitions between multi-material interfaces are achieved, avoiding sudden changes in material properties caused by direct connection, improving the overall strength and durability of the design, and adapting to additive manufacturing technologies.

[0003] At present, multi-material interpolation models considering gradient interfaces are mainly divided into continuous gradient models and discontinuous gradient models. On the one hand, the continuous gradient material model (functional gradient material) forms a continuous transition of mutual diffusion between multi-material interfaces. However, current manufacturing technology cannot realize the structure of such a diffuse material. On the other hand, the discontinuous and inhomogeneous gradient material model combines structural interface performance and manufacturability, providing new ideas for high-performance design. The methods for dealing with discontinuous gradient interfaces mainly include filtering method, random field method and volume fraction constraint method, among which filtering method and random field method are often used to identify single gradient materials. The volume fraction constraint method represents the gradient material by introducing multiple design variables and adjusting the volume fraction of the parent material, which increases the computational complexity.

[0004] In view of this, this application is filed. Summary of the Invention

[0005] The present invention provides a multi-material turbine disk topology optimization method considering gradient interfaces, which can at least partially improve the above-mentioned problems.

[0006] To achieve the above object, the present invention adopts the following technical solutions:

[0007] A multi-material turbine disk topology optimization method considering gradient interfaces includes:

[0008] Obtain the predefined turbine disk radial section design domain and design domain boundary conditions;

[0009] Establishing a stress-minimizing multi-gradient material structure topology optimization mathematical model based on the turbine disk radial section design domain and the design domain boundary conditions;

[0010] Based on the stress-minimized multi-gradient material structure topology optimization mathematical model, the distribution of the gradient material phase in a single material field is described using a piecewise Heaviside projection function to obtain a density field projection of the gradient material phase;

[0011] The elastic modulus and stress interpolation model of multi-gradient materials is established based on the density field projection of gradient material phase, standard SIMP method and piecewise Heaviside projection function;

[0012] Perform finite element calculations to obtain the displacement matrix of the turbine disk section;

[0013] Calculating the unit von Mises stress and the objective function global stress according to the displacement matrix and the multi-gradient material elastic modulus and stress interpolation model;

[0014] The adjoint method is used to solve the global stress sensitivity of the objective function global stress and simultaneously calculate the mass sensitivity;

[0015] The design variables are updated according to the global stress sensitivity and mass sensitivity by the moving asymptote method;

[0016] The updated design variables are filtered and the objective function value is judged to have reached the convergence condition. Based on the judgment result, the segmented Heaviside projection function is used again to describe the distribution of the gradient material phase in a single material field, or to visualize the multi-gradient material turbine disk design.

[0017] In summary, the proposed multi-material turbine disk topology optimization method considering gradient interfaces is based on practical engineering problems. It defines the radial cross-section design domain of the turbine disk; establishes a mathematical model for multi-material structural topology optimization that minimizes stress by considering gradient interfaces; uses a piecewise Heaviside projection function to obtain the density field projection of the gradient material phase; establishes a multi-gradient material elastic modulus and stress interpolation model based on the standard SIMP method and the piecewise Heaviside function; derives the displacement matrix of the turbine disk cross-section through finite element calculations; further calculates the unit von Mises stress and the global stress of the objective function based on the interpolation model and displacement matrix; solves the global stress sensitivity using the adjoint method; calculates the mass sensitivity; updates the design variables using the moving asymptote method; and filters the design variables to obtain a multi-material turbine disk design. This method improves the design efficiency of multi-material turbine disks and achieves multi-gradient material transitions between multi-material interfaces, making it suitable for additive manufacturing. In short, the proposed multi-material turbine disk topology optimization method considering gradient interfaces controls the number and properties of gradient materials by adjusting the gradient ratio of the parent material, resulting in a multi-material turbine disk design with gradient interfaces, improving structural performance and manufacturing feasibility. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Figure 1 1 is a schematic flow chart of a multi-material turbine disk topology optimization method considering a gradient interface provided by the first aspect of the present invention;

[0019] Figure 21 is a flow chart of a multi-material turbine disk topology optimization method considering a gradient interface provided by the second aspect of the present invention;

[0020] Figure 3 Schematic diagram of the turbine disk cross-section design domain and boundary conditions provided by an embodiment of the present invention;

[0021] Figure 4 is a diagram of the material properties of a turbine disk provided by an embodiment of the present invention;

[0022] Figure 5 These are the topology optimization results of single-gradient, double-gradient, and triple-gradient turbine disks provided by the embodiments of the present invention. DETAILED DESCRIPTION

[0023] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention will be further described in detail below in conjunction with the 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.

[0024] refer to Figure 1 、 Figure 2 As shown, the first embodiment of the present invention discloses a multi-material turbine disk topology optimization method considering a gradient interface, which specifically includes the following steps:

[0025] S1, obtain the predefined turbine disk radial section design domain and design domain boundary conditions;

[0026] Specifically, in this embodiment, the turbine disk radial section design domain is defined in combination with actual engineering problems, and the design domain boundary conditions are defined. The initial design domain, boundary conditions and load conditions of the turbine disk section are as follows: Figure 3 In the dimensionless design domain, the length is 2 and the width is 1. The turbine disk design domain is divided into 200 100 finite element mesh elements with an element size of 0.005 0.005. The elongated rectangular area on the right boundary is the non-designable area. To avoid stress concentration at the load application point, the vertical external force Applied to the six middle nodes on the right side of the design domain. The left side of the design domain is the rotation axis. A fixed displacement constraint is applied on the rotating axis side of the turbine disk.

[0027] S2, establishing a stress-minimizing multi-gradient material structure topology optimization mathematical model based on the turbine disk radial section design domain and the design domain boundary conditions;

[0028] Specifically, step S2 includes: the formula of the stress-minimized multi-gradient material structure topology optimization mathematical model is: ,in, is the design variable of the e-th unit in the design domain, is the total number of units, is the global stress aggregated by the p-norm function, is the unit von Mises stress, is the aggregation parameter, is the volume of the e-th unit, For quality constraints, is the overall stiffness matrix, is the overall displacement vector, is the load vector, is the minimum value of the design variable.

[0029] In this embodiment, a mathematical model for topological optimization of multi-gradient material structures with stress minimization is established. Specifically, for the optimization problem with the goal of minimizing p-norm stress and the structural mass as a constraint, the mathematical model can be described as follows: Where, is the design variable of the e-th unit in the design domain; Nel is the total number of units; is the volume of the e-th unit; M is the mass constraint, which is 0.3 in this embodiment; K is the overall stiffness matrix, which is obtained from the unit stiffness matrix Assembly formation; U is the overall displacement vector; F is the load vector; is the minimum value of the design variable to avoid numerical singularity, .

[0030] S3, based on the stress-minimized multi-gradient material structure topology optimization mathematical model, using a piecewise Heaviside projection function to describe the distribution of the gradient material phase in a single material field, and obtaining a density field projection of the gradient material phase;

[0031] Specifically, step S3 includes: obtaining a z-gradient material field, wherein the z-gradient material field includes two parent phase materials, z gradient materials, and one empty material;

[0032] According to the z-gradient material field, the design variables in the design domain are The normalized continuous material field Divide into continuous intervals, where ;

[0033] The material field is projected using the piecewise Heaviside projection function Mapping into a piecewise, discrete function with clear material classification;

[0034] For material i, the design variables in the local interval are mapped to the global interval [0,1] by translating the X axis. The material field function of material i is The formula is: ;

[0035] Convert the material field variables in each local interval into smoothed material density values , the formula is: ,in, To control the sharpness parameter of the transition area projection, is the projection threshold parameter;

[0036] Density value of the smoothed material Perform Y-axis translation and map it back to the global interval [0,1] to obtain the density field projection of the gradient material phase, and assign the variable of the material field less than the first dividing point or greater than the last dividing point to 0 or 1 respectively, where the piecewise Heaviside projection function The formula is: .

[0037] In this embodiment, the segmented Heaviside projection function is used to describe the distribution of the gradient material phase in a single material field, and the density field projection of the gradient material phase is obtained. For a z-gradient material field, it contains two parent materials, z gradient materials, and 1 empty material, with a total of z+3 material phases. For any design variable x in the design domain, the normalized continuous material field , , is divided into The piecewise Heaviside projection function transforms the material field into Mapped to a piecewise, discrete function with clear material classification.

[0038] For each material phase i, the design variables in the local interval are mapped to the global interval [0,1] by translating the X axis. The material field function of material i is Expressed as: ;Convert the material field variables in each local interval into smoothed material density values : , where Parameters to control the sharpness of the projection in the transition area; is the projection threshold parameter, which is set to 0.5.

[0039] To ensure that the mapped density value falls within the correct material range, the calculated Perform Y-axis translation and map it back to the global interval [0,1]. If the variable of the material field is less than the first cutoff point or greater than the last cutoff point, it is assigned a value of 0 or 1 respectively. Piecewise Heaviside mapping function Expressed as:

[0040] S4, based on the density field projection of the gradient material phase, the standard SIMP method and the piecewise Heaviside projection function, the elastic modulus and stress interpolation model of the multi-gradient material is established;

[0041] Specifically, step S4 includes: in the gradient material model, the strong material MAT1 and the weak material MAT2 are uniformly mixed in a gradient ratio, and the Represents the gradient ratio corresponding to the kth material, the formula is: ;

[0042] The density formula of the kth material is: ,in, is the actual density value of the strong material MAT1, is the actual density value of the weak material MAT2;

[0043] When the Poisson's ratio of the strong material MAT1 to the weak material MAT2 is determined to be 1 / 3, the gradient ratio corresponding to the kth material The mathematical expression of the lower limit Young's modulus model of isotropic material is: ,in, is the lower bound of the Hashin-Shtrikmann model, is the elastic modulus of the strong material MAT1, is the elastic modulus of the soft material MAT2;

[0044] The gradient ratio corresponding to the kth material The mathematical expression of the upper limit Young's modulus model of isotropic material is: ,in, is the upper bound of the Hashin-Shtrikmann model;

[0045] The elastic modulus of the gradient material is expressed by the average value of the upper and lower limits, and the formula is: ;

[0046] A multi-gradient material interpolation model is established based on the standard SIMP method and the piecewise Heaviside projection function. For the z-gradient material structure, the material properties are normalized and mapped to the [0,1] interval, converted into dimensionless values. The material property expression of the i-th material after normalization is: ,in, is the normalized density of material i, is the normalized elastic modulus of material i, is the true density of material i, is the true elastic modulus of material i, is the maximum value of the density of all materials, is the maximum value of the elastic modulus of all materials, is the total number of materials;

[0047] The material field of the z gradient is expressed as a gradient ratio, and the material property expressions of different material phases are: , is the normalized density of the solid material, is the normalized elastic modulus of the solid material;

[0048] According to the formula, the material density field For linear conversion, the formula is: Among them, the design variables after function projection are ;

[0049] The power law penalty is performed according to the preset standard SIMP method, and the elastic modulus interpolation with a specific gradient ratio is realized according to the formula. The formula of the elastic modulus interpolation mathematical model is: , where p is the penalty index;

[0050] The formula of the stress interpolation model based on the ordered SIMP method is: , where q is the relaxation coefficient.

[0051] In this embodiment, a multi-gradient material elastic modulus and stress interpolation model is established based on the standard SIMP method and the piecewise Heaviside function. In the gradient material model, the strong material MAT1 and the weak material MAT2 are uniformly mixed according to the gradient ratio (volume fraction). represents the volume fraction of MAT1 in the mixed material, . Accordingly, (1- ) is the volume fraction of MAT2 in the mixed material. The z gradient material is evenly distributed between the two base materials MAT1 and MAT2. Gradient ratio is divided into z+1 intervals (segmented from MAT1 to MAT2), thus forming z+2 materials. Represents the gradient ratio corresponding to the kth material, then It can be defined as: When z = 1, that is, single gradient material, ; When z = 2, that is, double gradient material, ; When z = 3, that is, three-gradient material, .

[0052] In the gradient material model, the density of the kth material can be expressed as: , where and are the actual density values ​​of MAT1 and MAT2 respectively. Hashin-Shtrikmam Boundary is applicable to gradient materials with random composition and reflects the real mechanical behavior of gradient materials under various loading conditions. When the Poisson's ratio of the two materials is 1 / 3, the gradient ratio is The upper and lower bounds of the Hashin-Shtrikmann Young's modulus model for an isotropic material are expressed as: , , where and are the elastic moduli of the strong material MAT1 and the soft material MAT2, .when When the value of is close to 1, the upper and lower limits of the Hashin-Shtrikmam model are , all gradient materials are strong materials; when When the value of is close to 0, the upper and lower limits of the Hashin-Shtrikmam model are , all gradient materials are weak materials. Therefore, the elastic modulus of the gradient material is expressed as the average value of the upper and lower limits:

[0053] A new multi-gradient material interpolation model is established based on the standard SIMP method and the piecewise Heaviside function. For the z-gradient material structure, the material properties (elastic modulus, density, etc.) are normalized and mapped to the [0,1] interval, converting them into dimensionless values. Therefore, the material properties of the i-th material after normalization are described as: , where and are the normalized density and elastic modulus of material i, and are the true density and elastic modulus of material i, and is the maximum value of the density and elastic modulus of all materials, and z+3 is the total number of materials.

[0054] The solid material (two parent phase materials and z gradient materials) is represented by a gradient ratio, and the material properties of different material phases are described as: , where and are the normalized density and elastic modulus of the solid material, Indicates the gradient ratio corresponding to the k-th solid material. When i = 1, it indicates an empty material; when i = 2 and k = 1, it indicates a weak MAT2; when and When , it corresponds to the gradient material GRA; when and , it indicates strong material MAT2.

[0055] To ensure that the gradient change of material properties transitions according to the specified gradient ratio, the material density field is calculated according to the formula Perform a linear transformation: , where the design variables after function projection are .

[0056] The power law penalty is performed according to the standard SIMP method, and the elastic modulus interpolation with a specific gradient ratio is realized according to the formula. The mathematical model of elastic modulus interpolation is: , where p is the penalty index, which is generally set to 3. The material properties of the relevant materials in this embodiment are as follows Figure 4 shown.

[0057] The stress interpolation model based on the ordered SIMP method is: , where q is the relaxation coefficient, and its value range is .

[0058] S5, perform finite element calculation to obtain the displacement matrix of the turbine disk section;

[0059] Specifically, step S5 includes: the displacement matrix of the turbine disc cross section is expressed as: ,in, is the stress column vector at the center point of the e-th element, is the material elastic matrix, is the strain displacement matrix, is the unit displacement vector.

[0060] In this embodiment, the displacement matrix of the turbine disc section is obtained based on finite element calculation. The relaxed element stress vector is expressed as Where, is the stress column vector at the center point of the e-th element, expressed as , where is the material elastic matrix; is the strain displacement matrix; is the unit displacement vector.

[0061] S6, performing calculations based on the displacement matrix and the multi-gradient material elastic modulus and stress interpolation model to calculate the unit von Mises stress and the objective function global stress;

[0062] Specifically, step S6 includes: for the two-dimensional continuum structure plane unit, the displacement matrix includes the normal stress components in the X and Y directions 、 , and the shear stress components in the xy plane , the expression is: ;

[0063] The calculation formula of the unit von Mises stress is: ;

[0064] The p-norm function is used to aggregate the von Mises stresses of all elements into a global stress, which is approximately equivalent to the maximum von Mises stress and is expressed as: , where P is the aggregation parameter.

[0065] In this embodiment, the unit von Mises stress is further calculated based on the interpolation model and the displacement matrix. And the objective function global stress For a two-dimensional continuum structural plane element, the stress vector contains normal stress components in the X and Y directions and , and the shear stress component in the xy plane , expressed as: , the calculation formula of the unit vonMises stress is: ; The p-norm function is used to aggregate the von Mises stresses of all elements into a global stress, which is approximately equivalent to the maximum von Mises stress and can be expressed as , where P is the aggregation parameter.

[0066] S7, the global stress sensitivity of the objective function global stress is solved using the adjoint method, and the mass sensitivity is calculated at the same time;

[0067] Specifically, step S7 includes: processing the global stress of the objective function using the moving asymptote method, and obtaining the global stress relative to the projected unit design variable according to the chain rule. The sensitivity is: ;

[0068] Global stress relative to the design variable of the e-th element in the design domain The sensitivity formula is: ,in, is the accompanying variable, is the element stiffness matrix, for.

[0069] Structural mass M is the design variable of the e-th element in the design domain The sensitivity formula is: .

[0070] In this embodiment, the global stress sensitivity is solved by the adjoint method; the mass sensitivity is calculated. The moving asymptotes method (MMA) is used to solve the optimization problem. The global stress relative to the projected unit design variables can be obtained according to the chain rule. Sensitivity: ; Global stress relative to the original element design variables The sensitivity is: , where is the accompanying variable .

[0071] Quality versus original unit design variables The sensitivity is .

[0072] S8, updating the design variables according to the global stress sensitivity and mass sensitivity by the moving asymptote method;

[0073] In this embodiment, the moving asymptote method is used to update the design variables. Based on the sensitivity of global stress and mass, an approximate model of the objective function and constraints is constructed. The MMA algorithm is used to iteratively update the design variables, ensuring that each update satisfies the constraints while gradually approaching the optimal solution.

[0074] S9, filter the updated design variables and determine whether the objective function value meets the convergence condition. Based on the determination result, re-use the segmented Heaviside projection function to describe the distribution of the gradient material phase within a single material field, or perform visualization to obtain a multi-gradient material turbine disk design.

[0075] Specifically, step S9 includes: the filtered design variables are: , ,in, The distance between the center of unit e is less than the filter radius The set of all units of is the weight factor, for The wth unit in is the distance between the wth unit and the eth unit;

[0076] If so, perform visualization to obtain a turbine disk design with multi-gradient materials;

[0077] If not, the piecewise Heaviside projection function is used again to describe the distribution of the gradient material phase within a single material field.

[0078] In this embodiment, the design variables are filtered and it is determined whether the objective function value reaches the convergence condition. If converged, visualization is performed to obtain the multi-gradient material turbine disk design. Otherwise, the process returns to step S3. The filtered design variables are: , where The distance from the center of unit e is less than the filter radius The set of all units of ; is the weight factor, expressed as .

[0079] Determine whether the objective function value has met the convergence criteria. The iteration process terminates when the difference between the objective function values ​​of two consecutive iterations does not exceed 0.001 or when the maximum number of iterations, 197, has been exceeded. If convergence has been achieved, visualization is performed to obtain the multi-gradient material turbine disk design. Otherwise, the process returns to step S3. If the convergence criteria have been met, the iteration process exits and visualization is performed to obtain the multi-gradient material turbine disk design.

[0080] In this embodiment, the multi-material turbine disk topology optimization method considering gradient interfaces is used to perform topology optimization on single-gradient, double-gradient, and triple-gradient turbine disks, and the multi-material turbine disk structure design considering gradient interfaces is obtained. The topology results of turbine disks with different gradient numbers are shown in the figure. Figure 5 As shown. The total mass of this embodiment does not exceed 0.3, strictly complying with the mass constraint conditions.

[0081] Figure 5 (a) shows a single-gradient turbine disk. Three materials are distributed along the gradient, with gradient material GRA1 positioned between MAT1 and MAT2, achieving a gradient transition between the multi-material interfaces. MAT1 is fixedly assigned to the right of the design domain. The clear structure and concentrated material distribution facilitate additive manufacturing.

[0082] Figure 5 (b) is a dual-gradient turbine disk. The material stiffness gradually increases from the outer edge to the inside. The intermediate gradient materials GRA2 and GRA3 are distributed between MAT1 and MAT2, avoiding the direct transition of the parent phase material and forming a topological design with clear boundaries and gradual interface, which is suitable for additive manufacturing.

[0083] Figure 5 (c) is a three-gradient turbine disk. The material stiffness also gradually increases from the outer edge to the inside. The intermediate gradient materials GRA4, GRA5 and GRA6 are distributed between MAT1 and MAT2, obtaining a topological design with clear boundaries and gradual interfaces.

[0084] Compared to traditional multi-material turbine disks, this method, which considers gradient interfaces, achieves an efficient distribution of materials with varying gradient quantities, optimizing the disk's performance and structural stability. This method is adaptable to additive manufacturing, improving design feasibility and material utilization efficiency, and providing innovative solutions for the aerospace industry.

[0085] The above is a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principles of the present invention. These improvements and modifications are also considered to be within the scope of protection of the present invention.

Claims

1. A multi-material turbine disk topology optimization method considering gradient interfaces, characterized in that: include: Obtain the predefined turbine disk radial section design domain and design domain boundary conditions; Establishing a stress-minimizing multi-gradient material structure topology optimization mathematical model based on the turbine disk radial section design domain and the design domain boundary conditions; Based on the stress-minimized multi-gradient material structure topology optimization mathematical model, the distribution of the gradient material phase in a single material field is described using a piecewise Heaviside projection function to obtain a density field projection of the gradient material phase; The elastic modulus and stress interpolation model of multi-gradient materials is established based on the density field projection of the gradient material phase, the standard SIMP method and the piecewise Heaviside projection function, specifically: In the gradient material model, the strong material MAT1 and the weak material MAT2 are mixed evenly in a gradient ratio. Represents the gradient ratio corresponding to the kth material, the formula is: ; Assume that the density formula of the kth material is: ,in, is the actual density value of the strong material MAT1, is the actual density value of the weak material MAT2; When the Poisson's ratio of the strong material MAT1 to the weak material MAT2 is determined to be 1 / 3, the gradient ratio corresponding to the kth material The mathematical expression of the lower limit Young's modulus model of isotropic material is: ,in, is the lower bound of the Hashin-Shtrikmann model, is the elastic modulus of the strong material MAT1, is the elastic modulus of the soft material MAT2; The gradient ratio corresponding to the kth material The mathematical expression of the upper limit Young's modulus model of isotropic material is: ,in, is the upper bound of the Hashin-Shtrikmann model; The elastic modulus of the gradient material is expressed by the average value of the upper and lower limits, and the formula is: ; A multi-gradient material interpolation model is established based on the standard SIMP method and the piecewise Heaviside projection function. For the z-gradient material structure, the material properties are normalized and mapped to the [0,1] interval, converted into dimensionless values. The material property expression of the i-th material after normalization is: ,in, is the normalized density of material i, is the normalized elastic modulus of material i, is the true density of material i, is the true elastic modulus of material i, is the maximum value of the density of all materials, is the maximum value of the elastic modulus of all materials, is the total number of materials; The material field of the z gradient is expressed as a gradient ratio, and the material property expressions of different material phases are: , is the normalized density of the solid material, is the normalized elastic modulus of the solid material; According to the formula, the material density field For linear conversion, the formula is: , where the unit design variables after function projection are ; The power law penalty is performed according to the preset standard SIMP method, and the elastic modulus interpolation with a specific gradient ratio is realized according to the formula. The formula of the elastic modulus interpolation mathematical model is: , where p is the penalty index; The formula of the stress interpolation model based on the ordered SIMP method is: , where q is the relaxation coefficient; Perform finite element calculations to obtain the displacement matrix of the turbine disk section; Calculating the unit von Mises stress and the objective function global stress according to the displacement matrix and the multi-gradient material elastic modulus and stress interpolation model; The adjoint method is used to solve the global stress sensitivity of the objective function global stress and simultaneously calculate the mass sensitivity; The design variables are updated according to the global stress sensitivity and mass sensitivity by the moving asymptote method; The updated design variables are filtered and the objective function value is judged to have reached the convergence condition. Based on the judgment result, the segmented Heaviside projection function is used again to describe the distribution of the gradient material phase in a single material field, or to visualize the multi-gradient material turbine disk design.

2. A multi-material turbine disk topology optimization method considering gradient interfaces according to claim 1, characterized in that: The formula of the mathematical model for topology optimization of multi-gradient material structure with minimal stress is: ,in, is the design variable of the e-th unit in the design domain, is the total number of units, is the global stress aggregated by the p-norm function, is the unit von Mises stress, is the aggregation parameter, is the volume of the e-th unit, For quality constraints, is the overall stiffness matrix, is the overall displacement vector, is the load vector, is the minimum value of the design variable.

3. The multi-material turbine disk topology optimization method considering gradient interface according to claim 2 is characterized in that: Based on the stress-minimized multi-gradient material structure topology optimization mathematical model, the segmented Heaviside projection function is used to describe the distribution of the gradient material phase in a single material field, and the density field projection of the gradient material phase is obtained, specifically: Acquire a z-gradient material field, wherein the z-gradient material field includes two parent phase materials, z gradient materials, and one empty material; According to the z-gradient material field, the design variables in the design domain are The normalized continuous material field Divide into continuous intervals, where ; The material field is projected using the piecewise Heaviside projection function Mapping into a piecewise, discrete function with clear material classification; For material i, the design variables in the local interval are mapped to the global interval [0,1] by translating the X axis. The material field function of material i is The formula is: ; Convert the material field variables in each local interval into smoothed material density values , the formula is: ,in, To control the sharpness parameter of the transition area projection, is the projection threshold parameter; Density value of the smoothed material Perform Y-axis translation and map it back to the global interval [0,1] to obtain the density field projection of the gradient material phase, and assign the variable of the material field less than the first dividing point or greater than the last dividing point to 0 or 1 respectively, where the piecewise Heaviside projection function The formula is: .

4. The method for topology optimization of a multi-material turbine disk considering a gradient interface according to claim 3, characterized in that: Finite element calculation is performed to obtain the displacement matrix of the turbine disk section, which is: The displacement matrix of the turbine disk section is expressed as: ,in, is the stress column vector at the center point of the e-th element, is the material elastic matrix, is the strain displacement matrix, is the unit displacement vector.

5. The method for topology optimization of a multi-material turbine disk considering a gradient interface according to claim 4, characterized in that: The unit vonMises stress and the objective function global stress are calculated based on the displacement matrix, the multi-gradient material elastic modulus, and the stress interpolation model, specifically: For a two-dimensional continuum structure plane element, the displacement matrix contains the normal stress components in the X and Y directions. 、 , and the shear stress components in the xy plane , the expression is: ; The calculation formula of the unit von Mises stress is: ; The p-norm function is used to aggregate the von Mises stresses of all elements into a global stress, which is approximately equivalent to the maximum von Mises stress and is expressed as: , where P is the aggregation parameter.

6. The method for topology optimization of a multi-material turbine disk considering a gradient interface according to claim 5, characterized in that: The adjoint method is used to solve the global stress sensitivity of the objective function global stress, specifically: The global stress of the objective function is processed by the moving asymptote method, and the global stress relative to the projected unit design variable is obtained according to the chain rule. The sensitivity is: ; Global stress relative to the design variable of the e-th element in the design domain The sensitivity formula is: ,in, is the accompanying variable, is the element stiffness matrix.

7. The method for topology optimization of a multi-material turbine disk considering a gradient interface according to claim 6, characterized in that: Calculate the mass sensitivity as: Structural mass M is the design variable of the e-th element in the design domain The sensitivity formula is: .

8. The method for topology optimization of a multi-material turbine disk considering gradient interfaces according to claim 7, characterized in that: Filter the updated design variables and determine whether the objective function value meets the convergence condition. Based on the determination result, re-use the piecewise Heaviside projection function to describe the distribution of the gradient material phase within a single material field, or visualize it to obtain the multi-gradient material turbine disk design. Specifically: The filtered design variables are: , ,in, The distance between the center of unit e is less than the filter radius The set of all units of is the weight factor, for The wth unit in is the distance between the wth unit and the eth unit; Determine whether the objective function value meets the convergence condition; If so, perform visualization to obtain a turbine disk design with multi-gradient materials; If not, the piecewise Heaviside projection function is used again to describe the distribution of the gradient material phase within a single material field.

Images