Multi-material turbine disc topological optimization method considering gradient interface

Abstract

The invention provides a multi-material turbine disc topological optimization method considering a gradient interface, and the method comprises the steps: defining a radial section design domain of a turbine disc in combination with an actual engineering problem; establishing a multi-material structure topological optimization mathematical model with minimum stress and considering the gradient interface; obtaining a density field projection of a gradient material phase by using a segmented Heaviside projection function; establishing a multi-gradient material elasticity modulus and stress interpolation model based on a standard SIMP method and a segmented Heaviside function; according to finite element calculation, obtaining a displacement matrix of the section of the turbine disc; according to the interpolation model and the displacement matrix, further calculating unit von Mises stress and target function global stress; solving global stress sensitivity by adopting an adjoint method; calculating mass sensitivity; the design variables are updated through a moving asymptote method; and filtering the design variables to obtain the multi-gradient material turbine disc design. The design efficiency of the multi-material turbine disc can be improved, multi-gradient material transition among multi-material interfaces is achieved, and the method is suitable for additive manufacturing.

Description

Technical Field

[0001] The 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 can achieve high-performance design and is widely used in aerospace and other fields. By introducing gradient materials, gradient transitions between multi-material interfaces can be achieved, avoiding sudden changes in material properties caused by direct connection of materials, improving the overall strength and durability of the design, and adapting to additive manufacturing technology.

[0003] At present, the multi-material interpolation model considering the gradient interface is mainly divided into continuous gradient model and discontinuous gradient model. On the one hand, the continuous gradient material model (functional gradient material) forms a continuous transition of mutual diffusion between multi-material interfaces. However, the current manufacturing technology cannot realize the structure of such a diffuse material. On the other hand, the discontinuous and inhomogeneous gradient material model combines the structural interface performance and manufacturability, providing a new idea 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 taking into account 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 interface, comprising:

[0008] Obtain a 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 according to the turbine disk radial section design domain and design domain boundary conditions;

[0010] Based on the stress-minimized multi-gradient material structure topology optimization mathematical model, a piecewise Heaviside projection function is used to describe the distribution of the gradient material phase in a single material field 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] Calculate 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] Filter the updated design variables and determine whether the objective function value reaches the convergence condition. Based on the determination result, re-use the segmented Heaviside projection function to describe the distribution of the gradient material phase in a single material field, or perform visualization to obtain a multi-gradient material turbine disk design.

[0017] In summary, the multi-material turbine disk topology optimization method considering gradient interface combines practical engineering problems, defines the design domain of the turbine disk radial section; establishes a mathematical model of multi-material structural topology optimization considering gradient interface with stress minimization; uses the segmented Heaviside projection function to obtain the density field projection of the gradient material phase; establishes the elastic modulus and stress interpolation model of multi-gradient materials based on the standard SIMP method and the segmented Heaviside function; obtains the displacement matrix of the turbine disk section according to the finite element calculation; further calculates the unit von Mises stress and the global stress of the objective function according to the interpolation model and the displacement matrix; uses the adjoint method to solve the global stress sensitivity; calculates the mass sensitivity; updates the design variables by the moving asymptote method; filters the design variables to obtain the multi-gradient material turbine disk design. It can improve the design efficiency of multi-material turbine disks, realize the multi-gradient material transition between multi-material interfaces, and is suitable for additive manufacturing. In short, the multi-material turbine disk topology optimization method considering gradient interface controls the number and properties of gradient materials by adjusting the gradient ratio of the parent phase material, obtains the multi-material turbine disk design with gradient interface, and improves the structural performance and manufacturing feasibility. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Figure 1 It 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 2It is a schematic 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 It is a schematic diagram of the design domain and boundary conditions of the turbine disk cross section 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 in the embodiments of the present invention. DETAILED DESCRIPTION

[0023] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used 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, obtaining the predefined turbine disk radial section design domain and design domain boundary conditions;

[0026] Specifically, in this embodiment, in combination with actual engineering problems, the turbine disk radial section design domain is defined, 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 As shown. 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 units with a unit 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, a vertical external force F=1N is applied to the middle 6 nodes on the right end of the design domain. The left end of the design domain is the rotation axis. The rotating axis side of the turbine disk is a fixed displacement constraint.

[0027] S2, establishing a stress-minimizing multi-gradient material structure topology optimization mathematical model according to 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: Among them, x e is the design variable of the e-th unit in the design domain, Nel is the total number of units, σ PN is the global stress aggregated by the p-norm function, is the unit von Mises stress, P is the cohesion parameter, V e is the volume of the e-th unit, M is the mass constraint, K is the overall stiffness matrix, U is the overall displacement vector, F is the load vector, and x min is the minimum value of the design variable.

[0029] In this embodiment, a mathematical model for topological optimization of multi-gradient material structures with minimal stress 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: In the formula, x e is the design variable of the e-th unit in the design domain; Nel is the total number of units; V e 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 assembled from the unit stiffness matrix K0; U is the overall displacement vector; F is the load vector; x min is the minimum value of the design variable to avoid numerical singularity, x min =0.001.

[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 1 empty material;

[0032] According to the material field of the z gradient, the normalized continuous material field of the design variable x in the design domain is is divided into z+2 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 ​​Θ i (x), the formula is: Among them, β is the sharpness parameter that controls the projection of the transition area, and η is the projection threshold parameter;

[0036] For the smoothed material density value θ i (x) is translated along the Y axis and mapped back to the global interval [0,1] to obtain the density field projection of the gradient material phase, and the variables of the material field that are less than the first dividing point or greater than the last dividing point are assigned values ​​of 0 or 1, respectively. 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 to obtain the density field projection of the gradient material phase. For a material field with a z gradient, 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 z+2 continuous intervals. The piecewise Heaviside projection function transforms the material field 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. Material field function of material i It is expressed as: Convert the material field variables in each local interval into smoothed material density values ​​Θ i (x): Where x is the sharpness parameter that controls the projection of the transition area; η is the projection threshold parameter, which is taken as 0.5.

[0039] To ensure that the mapped density value falls within the correct material interval, the calculated Θ i (x) is translated along the Y axis and mapped back to the global interval [0,1]. If the material field variable 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 It is 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 multi-gradient materials is established;

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

[0042] The density formula of the kth material is: ρ(ξ k )=ξ k ρ1+(1-ξ k )ρ2, where ρ1 is the actual density value of the strong material MAT1, and ρ2 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 ξ corresponding to the kth material k The mathematical expression of the lower limit Young's modulus model of isotropic material is: in, is the lower limit of the Hashin-Shtrikmann model, E1 is the elastic modulus of the strong material MAT1, and E2 is the elastic modulus of the soft material MAT2;

[0044] The gradient ratio ξ corresponding to the kth material k The mathematical expression of the upper limit Young's modulus model of isotropic material is: in, It 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 and 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, ρ i is the true density of material i, E i is the true elastic modulus of material i, ρ max is the maximum value of the density of all materials, E max is the maximum value of the elastic modulus of all materials, z+3 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 of 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 evenly mixed according to the gradient ratio (volume fraction). The gradient ratio ξ represents the volume fraction of MAT1 in the mixed material, ξ∈[0,1]. Correspondingly, (1-ξ) is the volume fraction of MAT2 in the mixed material. The z gradient materials are evenly distributed between the two base materials MAT1 and MAT2. The gradient ratio ξ is evenly divided into z+1 intervals (segmented from MAT1 to MAT2), thus forming z+2 materials. Assume ξ k represents the gradient ratio corresponding to the kth material, then ξ k It can be defined as: When z=1, that is, single gradient material, ξ1=0, ξ3=1; when z=2, that is, double gradient material, ξ1=0, ξ4=1; when z=3, that is, three-gradient material, ξ1=0, ξ5=1.

[0052] In the gradient material model, the density of the kth material can be expressed as: ρ(ξ k )=ξ k ρ1+(1-ξ k )ρ2, where ρ1 and ρ2 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 load conditions. When the Poisson's ratio of the two materials is 1 / 3, the gradient ratio is ξ k The upper and lower bounds of the Hashin-Shtrikmann Young's modulus model for isotropic materials are expressed as:

[0053] Wherein, E1 and E2 are the elastic moduli of the strong material MAT1 and the soft material MAT2 respectively, and E1 > E2. When the value of ξ is close to 1, both the upper and lower bounds of the Hashin-Shtrikmam model are E1, and the gradient material is entirely composed of the strong material; when the value of ξ is close to 0, both the upper and lower bounds of the Hashin-Shtrikmam model are E2, and the gradient material is entirely composed of the weak material. Therefore, the elastic modulus of the gradient material is represented by the average value of the upper and lower bounds:

[0054] A new interpolation model for multi-gradient materials is established based on the standard SIMP method and the piecewise Heaviside function. For the z-gradient material structure, the material properties (such as elastic modulus, density, etc.) are normalized and mapped to the interval [0, 1], and converted into dimensionless values. Therefore, the material properties of the i-th material after normalization are described as: Wherein, and are the density and elastic modulus of the normalized material i respectively, ρ i and E i are the true density and elastic modulus of the material i respectively, ρ max and E max are the maximum values of the density and elastic modulus of all materials, and z + 3 is the total number of materials.

[0055] The solid materials (two matrix materials and z gradient materials) are represented by gradient ratios, and the material properties of different material phases are described as: Wherein, and are the density and elastic modulus of the normalized solid material respectively, and ξ k represents the gradient ratio corresponding to the k-th solid material. When i = 1, it represents the void material; when i = 2 and k = 1, it represents the weak MAT2; when 2 < i < z + 3 and 1 < k < z + 2, it corresponds to the gradient material GRA; when i = z + 3 and k = z + 2, it represents the strong material MAT2.

[0056] To ensure that the gradient change of the material properties transitions according to the specified gradient ratio, the material density field is linearly transformed according to the formula: Wherein, the design variable after function projection

[0057] is subjected to power-law penalty with reference to the standard SIMP method, and the elastic modulus interpolation for a specific gradient ratio is realized according to the formula. The mathematical model of elastic modulus interpolation is: Wherein, p is the penalty exponent, and generally takes the value of 3. The material properties of the relevant materials in this embodiment are as Figure 4 shown.

[0058] The stress interpolation model based on the ordered SIMP method is: Where q is the relaxation coefficient, which ranges from 0 to <q<1。

[0059] S5, perform finite element calculation to obtain the displacement matrix of the turbine disk section;

[0060] Specifically, step S5 includes: the displacement matrix of the turbine disk cross section is expressed as: in, is the stress column vector at the center point of the e-th unit, D0 is the material elastic matrix, B is the strain displacement matrix, u e is the unit displacement vector.

[0061] In this embodiment, the displacement matrix of the turbine disk section is obtained based on finite element calculation. The relaxed unit stress vector is expressed as In the formula, is the stress column vector at the center point of the e-th unit, expressed as Where D0 is the material elastic matrix; B is the strain displacement matrix; u e is the unit displacement vector.

[0062] S6, 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;

[0063] Specifically, step S6 includes: for the two-dimensional continuum structure plane unit, the displacement matrix includes normal stress components σ in the X and Y directions ex , σ ey , and the shear stress component τ in the xy plane exy , the expression is: e =[σ ex ,σ ey ,τ exy ] T ;

[0064] The calculation formula for the unit von Mises stress is:

[0065] 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: Among them, P is the aggregation parameter.

[0066] 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 σ PN For a two-dimensional continuum structural plane element, the stress vector contains normal stress components σ in the X and Y directions ex and σ ey , and the shear stress component τ in the xy plane exy , expressed as: e =[σ ex ,σ ey ,τ exy ] T , 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 can be expressed as Where P is the aggregation parameter.

[0067] S7, the adjoint method is used to solve the global stress sensitivity of the objective function global stress, and the mass sensitivity is calculated at the same time;

[0068] Specifically, step S7 includes: using the moving asymptote method to process the global stress of the objective function, and obtaining the global stress relative to the projected unit design variable according to the chain rule The sensitivity is:

[0069] Global stress with respect to the design variable x of the e-th element in the design domain e The sensitivity formula is:

[0070] Among them, λ T is the accompanying variable, K0 is the unit stiffness matrix, for.

[0071] Structural mass M for the design variable x of the e-th element in the design domain e The sensitivity formula is:

[0072] In this embodiment, the global stress sensitivity is solved by the adjoint method; the mass sensitivity is calculated. The moving asymptote method (MMA) is used to solve the optimization problem. According to the chain rule, the global stress relative to the projected unit design variable can be obtained. Sensitivity: Global stress relative to the original element design variable x e The sensitivity is:

[0073] In the formula, λ T is the accompanying variable [λ] Nel′1 .

[0074] Quality versus original unit design variable x e The sensitivity is

[0075] S8, updating the design variables according to the global stress sensitivity and mass sensitivity by the moving asymptote method;

[0076] In this embodiment, the design variables are updated by the moving asymptote method. According to the sensitivity of global stress and mass, an approximate model of the objective function and the constraint conditions is constructed. The design variables are iteratively updated using the MMA algorithm to ensure that each update can meet the constraint conditions and gradually approach the optimal solution.

[0077] S9, filter the updated design variables, and determine whether the objective function value reaches the convergence condition. According to the determination result, re-use the segmented Heaviside projection function to describe the distribution of the gradient material phase in a single material field, or perform visualization to obtain a multi-gradient material turbine disk design.

[0078] Specifically, step S9 includes: the filtered design variables are: H ew =max(0,r min -r ew ), where N e The distance between the center of unit e is less than the filter radius r min The set of all units of H ew is the weight factor, x w N e The wth unit in r ew is the distance between the wth unit and the eth unit;

[0079] If so, perform visualization and obtain the turbine disk design with multi-gradient materials;

[0080] If not, the piecewise Heaviside projection function is used again to describe the distribution of the gradient material phase within a single material field.

[0081] 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 N e The distance from the center of unit e is less than the filter radius r min The set of all units of ew is the weight factor, denoted as H ew=max(0,r min -r ew ).

[0082] Determine whether the objective function value meets the convergence condition. When the difference between the objective function values ​​of two adjacent cycles does not exceed 0.001 or exceeds the maximum number of iterations of 197, the iteration process stops. If converged, visualize the multi-gradient material turbine disk design, otherwise, return to step S3. That is, if the convergence condition is met, exit the loop for visualization and obtain the multi-gradient material turbine disk design.

[0083] In this embodiment, the multi-material turbine disk topology optimization method considering gradient interface 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 interface is obtained. The topology results of turbine disks with different gradient numbers are shown in Figure 2. Figure 5 As shown. The total mass of this embodiment does not exceed 0.3, strictly complying with the mass constraint conditions.

[0084] Figure 5 (a) is a single gradient turbine disk. The three materials are distributed along the gradient, that is, the gradient material GRA1 is distributed between MAT1 and MAT2, realizing the gradient transition between the multi-material interfaces. The right side of the design domain is the fixedly allocated MAT1. The structure is clear and the material distribution is concentrated, which is conducive to additive manufacturing.

[0085] Figure 5 (b) is a double-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 interfaces, which is suitable for additive manufacturing.

[0086] 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.

[0087] Compared with the traditional multi-material turbine disk, the multi-material turbine disk topology optimization method considering the gradient interface realizes the effective distribution of materials with different gradient quantities and optimizes the performance and structural stability of the turbine disk by proposing a multi-material turbine disk topology optimization method considering the gradient interface. This method is suitable for additive manufacturing, improves the feasibility of design and material utilization efficiency, and provides an innovative solution for the aerospace field.

[0088] The above is a preferred embodiment of the present invention. It should be pointed out that a person skilled in the art can make several improvements and modifications without departing from the principle 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 interface, characterized in that: include: Obtain a predefined turbine disk radial section design domain and design domain boundary conditions; Establishing a stress-minimizing multi-gradient material structure topology optimization mathematical model according to the turbine disk radial section design domain and design domain boundary conditions; Based on the stress-minimized multi-gradient material structure topology optimization mathematical model, a piecewise Heaviside projection function is used to describe the distribution of the gradient material phase in a single material field 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 gradient material phase, standard SIMP method and piecewise Heaviside projection function; Perform finite element calculations to obtain the displacement matrix of the turbine disk section; Calculate 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; Filter the updated design variables and determine whether the objective function value reaches the convergence condition. Based on the determination result, re-use the segmented Heaviside projection function to describe the distribution of the gradient material phase in a single material field, or perform visualization to obtain a multi-gradient material turbine disk design.

2. A multi-material turbine disk topology optimization method considering gradient interface according to claim 1, characterized in that: The formula of the mathematical model of topology optimization of multi-gradient material structure with minimal stress is: Among them, x e is the design variable of the e-th unit in the design domain, Nel is the total number of units, σ PN is the global stress aggregated by the p-norm function, is the unit von Mises stress, P is the cohesion parameter, V e is the volume of the e-th unit, M is the mass constraint, K is the overall stiffness matrix, U is the overall displacement vector, F is the load vector, and x min is the minimum value of the design variable.

3. A multi-material turbine disk topology optimization method considering gradient interface according to claim 2, 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 1 empty material; According to the material field of the z gradient, the normalized continuous material field of the design variable x in the design domain is is divided into z+2 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 ​​Θ i (x), the formula is: Among them, β is the sharpness parameter that controls the projection of the transition area, and η is the projection threshold parameter; For the smoothed material density value θ i (x) is translated along the Y axis and mapped back to the global interval [0,1] to obtain the density field projection of the gradient material phase, and the variables of the material field that are less than the first dividing point or greater than the last dividing point are assigned values ​​of 0 or 1, respectively. The formula is:

4. A multi-material turbine disk topology optimization method considering gradient interface according to claim 3, characterized in that: 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 uniformly mixed according to the gradient ratio. k Represents the gradient ratio corresponding to the kth material, the formula is: Assume that the density formula of the kth material is: ρ(ξ k )=ξ k ρ1+(1-ξ k )ρ2, where ρ1 is the actual density value of the strong material MAT1, and ρ2 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 ξ corresponding to the kth material k The mathematical expression of the lower limit Young's modulus model of isotropic material is: in, is the lower limit of the Hashin-Shtrikmann model, E1 is the elastic modulus of the strong material MAT1, and E2 is the elastic modulus of the soft material MAT2; The gradient ratio ξ corresponding to the kth material k The mathematical expression of the upper limit Young's modulus model of isotropic material is: in, It 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 and 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, ρ i is the true density of material i, E i is the true elastic modulus of material i, ρ max is the maximum value of the density of all materials, E max is the maximum value of the elastic modulus of all materials, z+3 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: ρ ξk 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: Among them, the design variables after function projection are The power law penalty is performed according to the preset standard SIMP method, and the elastic modulus interpolation of 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.

5. A multi-material turbine disk topology optimization method considering gradient interface according to claim 4, 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 unit, D0 is the material elastic matrix, B is the strain displacement matrix, u e is the unit displacement vector.

6. A multi-material turbine disk topology optimization method considering gradient interface according to claim 5, characterized in that: The calculation is performed according to the displacement matrix and the multi-gradient material elastic modulus and stress interpolation model to calculate the unit vonMises stress and the objective function global stress, specifically: For a two-dimensional continuum structural plane element, the displacement matrix contains the normal stress components σ in the X and Y directions ex , σ ey , and the shear stress component τ in the xy plane exy , the expression is: e =[σ ex ,σ ey ,τ exy ] T ; The calculation formula for 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: Among them, P is the aggregation parameter.

7. A multi-material turbine disk topology optimization method considering gradient interface according to claim 6, 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 with respect to the design variable x of the e-th element in the design domain e The sensitivity formula is: Among them, λ T is the accompanying variable, K0 is the unit stiffness matrix, for.

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

9. A multi-material turbine disk topology optimization method considering gradient interface according to claim 8, characterized in that: Filter the updated design variables and determine whether the objective function value reaches the convergence condition. Based on the determination result, re-use the segmented Heaviside projection function to describe the distribution of the gradient material phase in a single material field, or visualize it to obtain the multi-gradient material turbine disk design, specifically: The filtered design variables are: H ew =max(0,r min -r ew ), where N e The distance between the center of unit e is less than the filter radius r min The set of all units of H ew is the weight factor, x w N e The wth unit in r ew 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 and obtain the 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.

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