A method for calculating and optimizing thermal stress sensitivity of a tail nozzle based on an adjoint method

Abstract

The present application belongs to the field of aero-engine nozzle structure design, and discloses a nozzle thermal stress sensitivity calculation and optimization method based on an adjoint method, which comprises generating a nozzle initial model and dividing a finite element grid; establishing a heat conduction equation and a thermal elastic equation; constructing an objective function; solving an adjoint equation and obtaining sensitivity; using a gradient descent method for nozzle design optimization; and performing iteration to obtain a final nozzle model. The nozzle thermal stress sensitivity calculation and optimization method based on the adjoint method quickly and accurately calculates the sensitivity of the maximum thermal stress to the control point coordinates by introducing an adjoint equation and an adjoint variable, and iteratively optimizes by using the gradient descent method, so as to obtain a nozzle shape meeting the performance requirements, and has engineering practical value.

Description

Technical Field

[0001] This invention belongs to the field of aero-engine tail nozzle structure design, specifically involving a method for calculating and optimizing the thermal stress sensitivity of tail nozzles based on the adjoint method. Background Technology

[0002] In the design of aero-engine exhaust systems, the tail nozzle is a key component for the discharge of high-temperature gases. The geometry and material properties of the tail nozzle directly affect the engine's efficiency, safety, and service life. During operation, the tail nozzle must withstand complex thermal and mechanical loads, making accurate calculations of its structural strength and thermal stress distribution crucial.

[0003] Traditional methods for calculating thermal stress typically rely on finite element analysis to modify design parameters one by one and perform trial calculations. While this method can provide some results, it is inefficient, especially in complex design optimization problems, and suffers from the following shortcomings: First, the computational cost is high, requiring multiple iterations to adjust design variables, and each time the temperature and displacement fields of the entire field need to be re-solved, resulting in long computation time and high resource consumption. Second, it is difficult to obtain sensitivity information. Traditional trial-and-error methods cannot directly provide the accurate derivative of the objective function with respect to design variables (sensitivity), which limits the optimization efficiency. Third, it is difficult to handle multi-disciplinary coupling problems. In high-temperature environments, the design of the tail nozzle needs to consider the coupling effects of multiple physical fields such as heat conduction, thermoelastic deformation, and geometry. Traditional methods cannot efficiently integrate these factors.

[0004] Currently, there is an urgent need to develop a method for calculating and optimizing the thermal stress sensitivity of the tail nozzle based on the adjoint method. Summary of the Invention

[0005] The technical problem to be solved by the present invention is to provide a method for calculating and optimizing the thermal stress sensitivity of the tail nozzle based on the adjoint method, so as to overcome the defects of the prior art.

[0006] The method for calculating and optimizing the thermal stress sensitivity of the tailpipe based on the adjoint method of the present invention includes the following steps:

[0007] S10. Generate the initial model of the tail nozzle and generate a finite element mesh;

[0008] According to the definition of Bézier curves, at least 5 control variables are taken as initial design parameters along the nozzle's travel direction. Given the nozzle wall thickness, the nozzle geometry is generated, the initial nozzle model is obtained, and a finite element mesh is generated.

[0009] S20. Establish the heat conduction equation and the thermoelastic equation;

[0010] The temperature field of the tail nozzle is solved using the heat conduction equation and the thermoelastic equation. and displacement field Then calculate the maximum thermal stress of the tail nozzle. and minimizing polymerization stress ;

[0011] S30. Construct the objective function;

[0012] Using minimized polymerization stress Calculate the objective function, where the objective function is the maximum thermal stress of the exhaust nozzle. ;

[0013] S40. Solve the adjoint equation and obtain the sensitivity;

[0014] Sensitivity Update control point coordinates based on gradient descent method The shape of the exhaust nozzle is optimized to reduce the maximum thermal stress.

[0015] S50. Optimize tail nozzle design using gradient descent method;

[0016] Update control point coordinates using gradient descent. This enables optimization of the tail nozzle design.

[0017] S60. Perform iterations to obtain the final model of the tail nozzle;

[0018] Based on minimizing the maximum thermal stress of the tail nozzle The design goal was to iterate repeatedly from S10 to S50 until convergence was determined, resulting in the final model of the tail nozzle.

[0019] Furthermore, the heat conduction equation for S20 is as follows:

[0020] ;

[0021] in, For gradient, The thermal conductivity coefficient, The computational domain for the tail nozzle; For the temperature field, The internal temperature of the tail nozzle. Ambient temperature; The convective heat transfer coefficient is... The boundary normal vector; The inner wall boundary of the tail nozzle; The outer wall boundary of the tail nozzle;

[0022] The thermoelastic equation is as follows:

[0023] ;

[0024] in, For stress tensor, For strain tensor, The coefficient of thermal expansion is... For reference temperature, Unit tensor; The elastic tensor is determined by the elastic modulus and Poisson's ratio. ":" indicates the tensor double dot product. The strain tensor... From displacement field Defined by geometric equations as follows: , For displacement field;

[0025] Temperature field The steady-state temperature distribution is solved using the heat conduction equation and its boundary conditions through the finite element method. ; temperature field Substitute into the thermoelastic equation to calculate the thermal strain. Combining displacement boundary conditions including fixed constraints, the strain tensor With displacement field Relationship: Solve the equilibrium equations Obtain the displacement field .

[0026] Furthermore, the objective function of S30 is as follows:

[0027] ;

[0028] in, This refers to the stress concentration parameter; To minimize polymerization stress, ; For the deviatoric stress tensor, ; For stress tensor; For the computational domain; The KS function represents the polymerization stress.

[0029] Furthermore, the adjoint equations of S40 include temperature adjoint equations and displacement adjoint equations;

[0030] Introducing the tail nozzle temperature field and displacement field and temperature-related variables and displacement associated variables Write the adjoint equation of the Lagrange function:

[0031] ;

[0032] After finite element discretization, the state equation is:

[0033] ;

[0034] in, For Lagrange operators, For the residuals of the heat conduction equation, The residuals of the thermoelastic equation; These are the coordinates of the control points, which are design variables.

[0035] The adjoint equation of the Lagrange function then transforms into:

[0036] ;

[0037] in, This is the transpose of the temperature-related variable. This is the transpose of the displacement-associated variable;

[0038] The adjoint equation of the Lagrange function with respect to the coordinates of the control points The total derivative equation is:

[0039] ;

[0040] After reorganization:

[0041] ;

[0042] Eliminate the implicit derivative term and let the temperature associated variable and displacement associated variables Satisfies the adjoint equation of the Lagrange function:

[0043] ;

[0044] Simplifying the total derivative equation, we obtain the simplified total derivative equation:

[0045] ;

[0046] The discrete objective function is:

[0047] ;

[0048] in, It is element stress. Polymer stress; Unit number; The total number of units;

[0049] The thermal stress in the exhaust nozzle depends on both the displacement field and the temperature field. In discrete form, the objective function, the maximum thermal stress in the exhaust nozzle, is calculated. derivative and The temperature adjoint equation and displacement adjoint equation obtained are as follows:

[0050] ;

[0051] A coupled adjoint equation is constructed by combining the temperature adjoint equation and the displacement adjoint equation. The coupled adjoint equation is as follows:

[0052] ;

[0053] The Jacobian matrix of the coupled adjoint equation is:

[0054] ;

[0055] in, Here is the thermal conductivity stiffness matrix. Here is the elastic stiffness matrix. It is a thermo-mechanical coupling matrix. , , After transposition:

[0056] ;

[0057] For the upper triangular equation system obtained after transposition, first solve the displacement adjoint equation: The displacement co-occurrence quantity is obtained. Then solve the temperature adjoint equation: The temperature accompanying quantity was obtained. Substituting the simplified total derivative equation, we obtain the sensitivity. Sensitivity Used for updating control point coordinates using gradient descent. The shape of the exhaust nozzle is optimized to reduce maximum thermal stress.

[0058] Furthermore, the gradient descent method in S50 is used to update the control point coordinates. The updated formula is as follows:

[0059] ;

[0060] in, To optimize the coordinates of the control points, To optimize the coordinates of the control points, The step size.

[0061] The adjoint method-based method for calculating and optimizing the thermal stress sensitivity of the tail nozzle in this invention introduces adjoint equations and adjoint variables to quickly and accurately calculate the sensitivity of the maximum thermal stress to the coordinates of the control point, and uses the gradient descent method for iterative optimization to obtain the tail nozzle shape that meets the performance requirements.

[0062] The core of the adjoint method-based method for calculating and optimizing the thermal stress sensitivity of the tail nozzle in this invention lies in establishing the heat conduction equation and the thermoelastic equation, constructing a method to minimize the cohesive stress to approximate the objective function of the maximum thermal stress of the tail nozzle, and deriving the sensitivity information by coupling the adjoint method. This significantly reduces the computational cost and improves the design optimization efficiency. It is applicable to the thermal stress optimization design of complex multi-physics field coupling, effectively reducing computation time and resource consumption, and providing technical support for the efficient design of aero-engine tail nozzles. Attached Figure Description

[0063] Figure 1 This is a flowchart of the tail nozzle thermal stress sensitivity calculation and optimization method based on the adjoint method of the present invention.

[0064] Figure 2 The iterative process of the maximum thermal stress in the embodiment;

[0065] Figure 3 This is a schematic diagram comparing the geometry of the initial model and the final model of the tail nozzle obtained for the example. Detailed Implementation

[0066] The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0067] Example: This example is based on the development of an optimization platform for the tail nozzle thermal stress sensitivity calculation and optimization method based on the adjoint method of the present invention, which is implemented by a self-written program. The programming language is Python. The heat conduction equation, thermoelastic equation and adjoint equation are based on the open source program Fenics, and the open source software Gmsh is used for finite element mesh generation.

[0068] like Figure 1 As shown, the tail nozzle thermal stress sensitivity calculation and optimization method based on the adjoint method in this embodiment includes the following steps:

[0069] S10. Generate the initial model of the tail nozzle and generate a finite element mesh;

[0070] According to the definition of Bézier curves, at least 5 control variables are taken as initial design parameters along the nozzle's travel direction. Given the nozzle wall thickness, the nozzle geometry is generated, the initial nozzle model is obtained, and a finite element mesh is generated.

[0071] This embodiment n The expression for a Bezier curve is:

[0072] ;

[0073] in, The coordinates of the control points are as follows: there are 9 control points; the length of the tail nozzle is 2.0m and the thickness is 2mm; the throat of the tail nozzle is at 40% of its length.

[0074] S20. Establish the heat conduction equation and the thermoelastic equation;

[0075] The temperature field of the tail nozzle is solved using the heat conduction equation and the thermoelastic equation. and displacement field Then calculate the maximum thermal stress of the tail nozzle. and minimizing polymerization stress ;

[0076] S30. Construct the objective function;

[0077] Using minimized polymerization stress Calculate the objective function, where the objective function is the maximum thermal stress of the exhaust nozzle. ;

[0078] S40. Solve the adjoint equation and obtain the sensitivity;

[0079] Sensitivity Update control point coordinates based on gradient descent method The shape of the exhaust nozzle is optimized to reduce the maximum thermal stress.

[0080] S50. Optimize tail nozzle design using gradient descent method;

[0081] Update control point coordinates using gradient descent. This enables optimization of the tail nozzle design.

[0082] S60. Perform iterations to obtain the final model of the tail nozzle;

[0083] Based on minimizing the maximum thermal stress of the tail nozzle The design goal was to iterate repeatedly from S10 to S50 until convergence was determined, resulting in the final model of the tail nozzle.

[0084] Furthermore, to simplify calculations, the pressure and temperature distribution on the inner wall of the nozzle in this embodiment is determined by the total inlet pressure of the nozzle, which is 2.5e. 6 Pa and total temperature of 1200 K were obtained through gas dynamics equations; the ambient pressure of 1 e was given on the outer wall of the tailpipe. 5 Pa and temperature 300K; the material properties of the tail nozzle are: elastic modulus 2e 11 Pa, Poisson's ratio 0.3, coefficient of thermal expansion 1.2e -5 / K, thermal conductivity 15.0W / Mk;

[0085] The heat conduction equation for S20 is as follows:

[0086] ;

[0087] in, For gradient, The thermal conductivity coefficient, The computational domain for the tail nozzle; For the temperature field, The internal temperature of the tail nozzle. Ambient temperature; The convective heat transfer coefficient is... The boundary normal vector; The inner wall boundary of the tail nozzle; The outer wall boundary of the tail nozzle;

[0088] The thermoelastic equation is as follows:

[0089] ;

[0090] in, For stress tensor, For strain tensor, The coefficient of thermal expansion is... For reference temperature, Unit tensor; The elastic tensor is determined by the elastic modulus and Poisson's ratio. ":" indicates the tensor double dot product. The strain tensor... From displacement field Defined by geometric equations as follows: , For displacement field;

[0091] Temperature field The steady-state temperature distribution is solved using the heat conduction equation and its boundary conditions through the finite element method. ; temperature field Substitute into the thermoelastic equation to calculate the thermal strain. Combining displacement boundary conditions including fixed constraints, the strain tensor With displacement field Relationship: Solve the equilibrium equations Obtain the displacement field .

[0092] Furthermore, the objective function of S30 is as follows:

[0093] ;

[0094] in, This refers to the stress concentration parameter; To minimize polymerization stress, ; For the deviatoric stress tensor, ; For stress tensor; For the computational domain; The KS function represents the polymerization stress.

[0095] Furthermore, the adjoint equations of S40 include temperature adjoint equations and displacement adjoint equations;

[0096] Introducing the tail nozzle temperature field and displacement field and temperature-related variables and displacement associated variables Write the adjoint equation of the Lagrange function:

[0097] ;

[0098] After finite element discretization, the state equation is:

[0099] ;

[0100] in, For Lagrange operators, For the residuals of the heat conduction equation, The residuals of the thermoelastic equation; These are the coordinates of the control points, which are design variables.

[0101] The adjoint equation of the Lagrange function then transforms into:

[0102] ;

[0103] in, This is the transpose of the temperature-related variable. This is the transpose of the displacement-associated variable;

[0104] The adjoint equation of the Lagrange function with respect to the coordinates of the control points The total derivative equation is:

[0105] ;

[0106] After reorganization:

[0107] ;

[0108] Eliminate the implicit derivative term and let the temperature associated variable and displacement associated variables Satisfies the adjoint equation of the Lagrange function:

[0109] ;

[0110] Simplifying the total derivative equation, we obtain the simplified total derivative equation:

[0111] ;

[0112] Note that during the tailpipe optimization process, the control point coordinates... These are the control points for the nozzle geometry, and their shape derivatives need to be calculated. This is because control point coordinates are used. As a design variable, and since the finite element mesh is generated from control points, the shape derivative reflects the change of mesh node coordinates with the coordinates of control points.

[0113] The discrete objective function is:

[0114] ;

[0115] in, It is element stress. Polymer stress; Unit number; The total number of units;

[0116] The thermal stress in the exhaust nozzle depends on both the displacement field and the temperature field. In discrete form, the objective function, the maximum thermal stress in the exhaust nozzle, is calculated. derivative and The temperature adjoint equation and displacement adjoint equation obtained are as follows:

[0117] ;

[0118] A coupled adjoint equation is constructed by combining the temperature adjoint equation and the displacement adjoint equation. The coupled adjoint equation is as follows:

[0119] ;

[0120] The Jacobian matrix of the coupled adjoint equation is:

[0121] ;

[0122] in, Here is the thermal conductivity stiffness matrix. Here is the elastic stiffness matrix. It is a thermo-mechanical coupling matrix. , , After transposition:

[0123] ;

[0124] For the upper triangular equation system obtained after transposition, first solve the displacement adjoint equation: The displacement co-occurrence quantity is obtained. Then solve the temperature adjoint equation: The temperature accompanying quantity was obtained. Substituting the simplified total derivative equation, we obtain the sensitivity. Sensitivity Used for updating control point coordinates using gradient descent. The shape of the exhaust nozzle is optimized to reduce maximum thermal stress.

[0125] Furthermore, the gradient descent method in S50 is used to update the control point coordinates. The updated formula is as follows:

[0126] ;

[0127] in, To optimize the coordinates of the control points, To optimize the coordinates of the control points, The step size.

[0128] Furthermore, S60 is based on minimizing the maximum thermal stress of the tail nozzle. The design goal was to iterate repeatedly from S10 to S50 until convergence was determined, resulting in the final model of the tail nozzle.

[0129] The iterative process of the maximum thermal stress in this embodiment is shown in [link to example]. Figure 2 ,from Figure 2 It can be seen that, by using the maximum stress optimization method of the present invention, after about 8 iterations, the cohesive stress and maximum equivalent stress of the tail nozzle are stabilized at a fixed value, and the decrease is significant.

[0130] This embodiment obtains as follows: Figure 3 The diagram showing the geometric comparison between the initial and final models of the tailpipe is provided. Figure 3 It can be seen that the profile of the contraction section of the tail nozzle has changed significantly after optimization, bulging outward compared to the initial tail nozzle contraction section, while the profile of the expansion section remains basically unchanged.

[0131] Although the embodiments of the present invention have been disclosed above, they are not limited to the applications listed in the specification and embodiments. For those skilled in the art, all features disclosed in the present invention, or all steps in all methods or processes disclosed, except for mutually exclusive features and / or steps, can be combined in any way without departing from the principles of the present invention. The present invention is not limited to the specific details and illustrations shown and described herein.

Claims

1. A method for calculating and optimizing the thermal stress sensitivity of a tailpipe based on the adjoint method, characterized in that, The aforementioned method for calculating and optimizing the thermal stress sensitivity of the tailpipe includes the following steps: S10. Generate the initial model of the tail nozzle and generate a finite element mesh; According to the definition of Bézier curves, at least 5 control variables are taken as initial design parameters along the nozzle's travel direction. Given the nozzle wall thickness, the nozzle geometry is generated, the initial nozzle model is obtained, and a finite element mesh is generated. S20. Establish the heat conduction equation and the thermoelastic equation; The heat conduction equation is as follows: ； in, For gradient, The thermal conductivity coefficient, The computational domain for the tail nozzle; For temperature field, The internal temperature of the tail nozzle. The ambient temperature; The convective heat transfer coefficient is... The boundary normal vector; The inner wall boundary of the tail nozzle; The outer wall boundary of the tail nozzle; The thermoelastic equation is as follows: ； in, For stress tensor, For strain tensor, The coefficient of thermal expansion is For reference temperature, Unit tensor; The elastic tensor is determined by the elastic modulus and Poisson's ratio. ":" indicates the tensor double dot product. The strain tensor... From displacement field Defined by geometric equations as follows: , For displacement field; Temperature field The steady-state temperature distribution is solved using the heat conduction equation and its boundary conditions through the finite element method. T ; temperature field Substitute into the thermoelastic equation to calculate the thermal strain. Combining displacement boundary conditions including fixed constraints, the strain tensor With displacement field Relationship: Solve the equilibrium equations Obtain the displacement field ; The temperature field of the tail nozzle is solved using the heat conduction equation and the thermoelastic equation. and displacement field Then calculate the maximum thermal stress of the tail nozzle. and minimizing polymerization stress ; S30. Construct the objective function; Using minimized polymerization stress Calculate the objective function, which is the maximum thermal stress of the exhaust nozzle. ; The objective function is as follows: ； in, This refers to the stress concentration parameter; To minimize polymerization stress, ; For the deviatoric stress tensor, ; For stress tensor; For the computational domain; The KS function represents the polymeric stress; S40. Solve the adjoint equation and obtain the sensitivity; The adjoint equations include the temperature adjoint equation and the displacement adjoint equation; Introducing the tail nozzle temperature field and displacement field and temperature-related variables and displacement associated variables Write the adjoint equation of the Lagrange function: ； After finite element discretization, the state equation is: ； in, For Lagrange operators, For the residuals of the heat conduction equation, The residuals of the thermoelastic equation; These are the coordinates of the control points, which are design variables. The adjoint equation of the Lagrange function then transforms into: ； in, This is the transpose of the temperature-related variable. This is the transpose of the displacement-associated variable; The adjoint equation of the Lagrange function with respect to the coordinates of the control points The total derivative equation is: ； After reorganization: ； Eliminate the implicit derivative term and let the temperature associated variable and displacement associated variables Satisfies the adjoint equation of the Lagrange function: ； Simplifying the total derivative equation, we obtain the simplified total derivative equation: ； The discrete objective function is: ； in, It is element stress. Polymer stress; Unit number; The total number of units; The thermal stress in the exhaust nozzle depends on both the displacement field and the temperature field. In discrete form, the objective function, the maximum thermal stress in the exhaust nozzle, is calculated. derivative and The temperature adjoint equation and displacement adjoint equation obtained are as follows: ； A coupled adjoint equation is constructed by combining the temperature adjoint equation and the displacement adjoint equation. The coupled adjoint equation is as follows: ； The Jacobian matrix of the coupled adjoint equation is: ； in, Here is the thermal conductivity stiffness matrix. Here is the elastic stiffness matrix. It is a thermo-mechanical coupling matrix. , , After transposition: ； For the upper triangular equation system obtained after transposition, first solve the displacement adjoint equation: The displacement co-occurrence quantity is obtained. Then solve the temperature adjoint equation: The temperature accompanying quantity was obtained. Substituting the simplified total derivative equation, we obtain the sensitivity. Sensitivity Used for updating control point coordinates using gradient descent. The shape of the exhaust nozzle is optimized to reduce the maximum thermal stress. Sensitivity Update control point coordinates based on gradient descent method The shape of the exhaust nozzle is optimized to reduce the maximum thermal stress. S50. Optimize tail nozzle design using gradient descent method; Update control point coordinates using gradient descent. This enables optimization of the tail nozzle design. S60. Perform iterations to obtain the final model of the tail nozzle; Based on minimizing the maximum thermal stress of the tail nozzle The design goal was to iterate repeatedly from S10 to S50 until convergence was determined, resulting in the final model of the tail nozzle.

2. The method for calculating and optimizing the thermal stress sensitivity of the tailpipe based on the adjoint method according to claim 1, characterized in that, The gradient descent method in S50 is used to update the control point coordinates. The updated formula is as follows: ； in, To optimize the coordinates of the control points, To optimize the coordinates of the control points, The step size.

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