Thermal-mechanical coupling topology optimization method considering material temperature dependence
By constructing a temperature-dependent thermo-coupled topology optimization model on the ABAQUS platform, the optimization error problem of aero-engine turbine disks under large temperature gradients was solved, and efficient structural performance prediction and optimization design were achieved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- HUNAN UNIV
- Filing Date
- 2022-12-07
- Publication Date
- 2026-08-04
AI Technical Summary
Existing thermo-mechanically coupled topology optimization methods fail to effectively consider the temperature dependence of materials under large temperature gradients when dealing with structures such as aero-engine turbine disks, resulting in large analysis and optimization errors.
A structural model was constructed using ABAQUS, a material interpolation model was constructed based on the variable density method, and a temperature-dependent thermo-mechanical coupling topology optimization model was constructed. Sensitivity analysis was performed by solving the nonlinear heat conduction equation and the thermo-mechanical coupling equation, combined with the Newton-Raphson iteration method and the adjoint equation method, and the gradient moving asymptote algorithm was used for optimization.
It achieves efficient thermo-mechanical coupling topology optimization of aero-engine turbine disk structures under large temperature gradients, provides more accurate structural performance prediction and optimization design tools, and improves the accuracy and efficiency of design.
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Figure CN115935740B_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of thermo-coupling topology optimization design, and relates to a thermo-coupling topology optimization method that considers the temperature dependence of materials. Background Technology
[0002] Topology optimization is a structural optimization method that seeks the optimal distribution of materials within a given design domain under given constraints, thereby maximizing the performance of the structure. Topology optimization methods typically focus on the presence or absence of structural materials (continuous topology optimization) or the node layout of truss structures (discrete topology optimization), and are generally applied in the conceptual design phase of structures. Therefore, compared to traditional structural optimization methods such as size and shape optimization, topology optimization offers more design freedom and a larger design space, effectively shortening the structural design cycle, improving material utilization efficiency, reducing development costs, and significantly enhancing structural performance. It represents a vital and challenging research direction.
[0003] In recent decades, advancements in modern processing methods and technologies have made the fabrication of complex structures possible. Simultaneously, topology optimization theory and methods have been continuously improved and developed. Therefore, topology optimization methods have gradually become an important tool in current product design and development, and have been widely applied in fields such as vehicles, ships, bridges, and aerospace. Early topology optimization typically considered only the thermal or mechanical properties of the structure. However, in the design of modern high-end equipment, structures usually bear both mechanical and thermal loads simultaneously. Thermodynamic coupling significantly impacts structural performance; designs considering only a single physics field may lead to structural failure, thereby impairing the performance of the equipment structure.
[0004] Aero-engines are the crown jewel of industry, facing extremely high temperatures and loads during operation, with structural temperatures varying across a wide range from low to high. Under these conditions, mechanical and thermal material properties (elastic modulus, thermal conductivity, coefficient of thermal expansion, etc.) will change significantly with large temperature gradients. Currently, thermo-mechanical coupled structural topology optimization is typically based on the assumption of constant material properties. However, when the temperature gradient across the entire structure is large, this assumption of invariant material properties becomes unreasonable, leading to significant analysis and optimization errors. Therefore, conducting thermo-mechanical coupled structural topology optimization design that considers the temperature dependence of materials for critical aero-engine structures such as turbine disks is of great significance. Summary of the Invention
[0005] To address the limitations of material property constant assumptions in traditional thermo-coupled topology optimization, this invention aims to provide a thermo-coupled topology optimization method that considers the temperature dependence of materials, capable of solving thermo-coupled topology optimization problems under large temperature gradients in aero-engine turbine disk structures. First, the preprocessing function of ABAQUS is used to analyze the model and derive its physical information. A material interpolation model considering temperature variations is then presented based on the variable density method. Subsequently, a temperature-dependent thermo-coupled topology optimization model is constructed, and the structural performance response is calculated by solving the nonlinear heat conduction equation and the thermo-coupled equation. Finally, sensitivity analysis is performed, and the gradient-based moving asymptote (MMA) algorithm is used for optimization. This method can take into account the actual influence of temperature on the thermodynamic parameters of materials during thermo-coupled topology optimization, providing an effective tool for the optimization design of multi-field coupled structures in engineering.
[0006] This invention provides a thermo-mechanically coupled topology optimization method considering the temperature dependence of materials, which can solve thermo-mechanically coupled topology optimization problems considering large temperature gradients. This invention includes the following steps:
[0007] Step 1: Construct a structural model using ABAQUS and obtain the model's physical information, including mesh data, stress conditions, constraint boundaries, and heat transfer boundaries;
[0008] Step 2: Based on the density interpolation model, construct the material elastic modulus of the e-th element of the structure. as follows:
[0009]
[0010] in, The elastic modulus of a solid material. Let be the elastic modulus of the empty phase material, p be the penalty factor, and N be the number of finite elements. This represents the element physical density obtained after density filtering and Heaviside projection filtering. The density filtering function can be expressed as:
[0011]
[0012] in Let e be the set of elements within the filtration radius. and The weight function is given by the weight function for the volume and density of the i-th unit in the aforementioned set. It can be represented as:
[0013]
[0014] in The distance from the center of element e to the center of element i. The filter radius. After density filtering, the intermediate variable... It also needs to be filtered by Heaviside projection. It can be calculated using the following expression:
[0015]
[0016] in Filter density threshold These are projection parameters;
[0017] Step 3: Construct a temperature-dependent material parameter interpolation model, and determine the thermal conductivity of the e-th element of the structure. With coefficient of thermal expansion as follows:
[0018]
[0019]
[0020] in, The thermal conductivity coefficient of a solid material as a function of temperature T. This represents the coefficient of thermal expansion of a solid material as a function of temperature T.
[0021] Step 4: Construct a thermo-mechanically coupled topology optimization model that considers the temperature dependence of materials:
[0022]
[0023] in Indicates structural flexibility. For nodal displacement, The overall stiffness matrix, For node temperature, The temperature load is calculated based on the given thermal boundary conditions. The total heat transfer coefficient matrix is shown below. Represents equivalent thermal load, topology design variable This represents the density of the e-th unit. Indicates the initial volume of the structure. Indicates the structural volume. It is the volume fraction;
[0024] Step 5: Perform nonlinear heat transfer finite element analysis to obtain the overall heat transfer coefficient matrix. Through the unit heat transfer coefficient matrix Assembled to obtain:
[0025]
[0026] In the formula The element thermal strain matrix, The heat transfer coefficient matrix of the unit cell. Given that the heat transfer coefficient matrix is a constant matrix, and noting the nonlinear part of the heat transfer coefficient matrix, the Newton-Raphson iterative method is applied to solve the nonlinear residual equation:
[0027]
[0028] Step 6: For the thermo-coupling finite element equations, the equivalent thermal load It can be obtained through the following methods:
[0029] (20)
[0030] in and Let e be the strain-displacement matrix and elasticity matrix of the e-th element. It is the elastic matrix of the solid material;
[0031] (twenty one)
[0032] in For thermal strain, The coefficient of thermal expansion is For reference temperature, Let be the unit assembly matrix. Since the elastic properties and coefficient of thermal expansion of the material are both related to the physical density, equation (20) can be expressed as:
[0033] (twenty two)
[0034] Step 7: Perform thermo-mechanical coupled finite element analysis:
[0035] (twenty three)
[0036] Step 8: Perform sensitivity analysis on the thermo-coupled topology optimization model:
[0037] (twenty four)
[0038] right Taking the derivative of both ends with respect to the topological variables, we get:
[0039] (31)
[0040] Substituting equation (31) into equation (24), we can obtain the sensitivity of the objective function to the physical density:
[0041] (32)
[0042] Step 9: Considering the effect of filtering, derive the sensitivity of the objective function to the design variables using the chain rule:
[0043] (34)
[0044] Step 10: Based on the sensitivity information of the objective function with respect to the design variables, the design variables are updated and iterated using a gradient-based MMA optimization algorithm;
[0045] Step 11: Determine convergence. If it does not converge, return to step 2 until the calculation converges and the optimal topology of the thermo-coupled structure is obtained.
[0046] Furthermore, step 8 involves solving for the sensitivity information of the objective function with respect to the design variables. The specific steps include:
[0047] The derivative of the equivalent thermal load with respect to the physical density is:
[0048] (25)
[0049] Its expansion can be further expressed as:
[0050] (26)
[0051] To solve the unknown terms in the equation First, we need to use the nonlinear residual equation (9) from step 5 to obtain:
[0052] (27)
[0053] Therefore, we can conclude that:
[0054] (28)
[0055] in Let the Lagrange multipliers be the adjoint variables, and based on the method of adjoint variables, let:
[0056] (29)
[0057] Therefore, equation (28) can be transformed into:
[0058] (30)
[0059] Substituting equations (30) and (26) into equation (32), we obtain the sensitivity of the topology optimization objective function to physical density:
[0060] (33)
[0061] in This represents the element displacement matrix.
[0062] The beneficial effects of this invention are:
[0063] 1. This invention takes into account the characteristics of material property changes in real physical fields in structural topology optimization design, and constructs a thermo-mechanical coupling topology optimization model that considers temperature dependence;
[0064] 2. This invention combines the adjoint equation method and the Newton-Raphson iteration method to derive the sensitivity information of the topology design variables for this nonlinear topology optimization problem, and proposes an efficient thermo-mechanical coupled topology optimization solution algorithm;
[0065] 3. This invention addresses the thermo-mechanical coupling topology optimization design problem that considers the temperature dependence of materials. It uses the ABAQUS secondary development function module to perform thermo-mechanical coupling simulation calculations and topology optimization design. This method can handle problems in arbitrary irregular design domains and provides an effective tool for structural optimization design in practical applications. Attached Figure Description
[0066] Figure 1 This is a flowchart of the present invention.
[0067] Figure 2 This is a schematic diagram of the structural design domain and boundary conditions.
[0068] Figure 3 This is a schematic diagram of the thermo-mechanical coupling topology optimization results for material temperature dependence.
[0069] Figure 4 This is a schematic diagram of the thermo-coupling topology optimization iteration curve for material temperature dependence. Detailed Implementation
[0070] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.
[0071] This embodiment proposes a thermo-coupling topology optimization method considering the temperature dependence of materials. First, an interpolation model for relevant material coefficients is defined based on the temperature variability of the material, and a topology optimization model considering the temperature dependence of materials is constructed. Then, the nonlinear heat transfer equation is solved using the Newton-Raphson iterative method, and the target compliance value of the thermo-coupling structure is calculated. Finally, the sensitivity of the objective function to the structural topology design variables is derived using the adjoint equation method, and a gradient-based optimization algorithm is employed to solve the problem, ultimately obtaining the topology configuration of the thermo-coupling structure. The invention will be further described in detail below with reference to the accompanying drawings and a specific example of topology optimization design for an aero-engine turbine disk:
[0072] Figure 3 The design domain for a turbine disk with an inner bore radius of 0.6m, a hub radius of 2.4m, and a thickness of 0.01m is given. The gray area represents the non-design domain. The external surface medium temperature is 300℃, and the internal medium temperature of the hole is 20℃. The specific calculation steps of the method of this invention are detailed below:
[0073] Step 1: Build a model using ABAQUS and obtain the model's physical information, including mesh data, stress conditions, constraint boundaries, and heat transfer boundaries.
[0074] In this step: Preprocessing is performed using ABAQUS to export the relevant physical information of the turbine disk model, resulting in five txt files: elemnode, nodecoor, mat, disp, and thermal. These files contain the model's mesh elements; the design domain mesh is divided into 8240 plane stress (CPS4) elements; node coordinate information; material properties (elastic modulus E0 = 210 GPa, Poisson's ratio v = 0.3, element thickness t = 0.01 m); boundary constraint information; a fixed constraint at the design domain center; and a concentrated force F with a magnitude of 100 kN.
[0075] Step 2: Based on the density interpolation model, construct the material elastic modulus of the e-th element of the turbine disk structure. as follows:
[0076] (1)
[0077] in, The elastic modulus of the solid material is 210 GPa. The elastic modulus of the empty phase material. =10 -5 p is the penalty factor p=3, and N is the number of finite elements, i.e., 8240. This represents the element physical density obtained after density filtering and Heaviside projection filtering. The density filtering function can be expressed as:
[0078] (2)
[0079] in As an intermediate variable, Let e be the set of elements within the filtration radius. and The weight function is given by the weight function for the volume and density of the i-th unit in the aforementioned set. It can be represented as:
[0080] (3)
[0081] in The distance from the center of element e to the center of element i. The filter radius is set to 3. After density filtering, the intermediate variable... It also needs to be filtered by Heaviside projection. It can be calculated using the following expression:
[0082] (4)
[0083] in Filter density threshold This is the projection parameter, and its initial value is set to 1;
[0084] Step 3: Construct a temperature-dependent material parameter interpolation model, and determine the thermal conductivity of the e-th element of the turbine disk structure. With coefficient of thermal expansion as follows:
[0085] (5)
[0086] (6)
[0087] thermal conductivity W / (m·K), coefficient of thermal expansion / ℃;
[0088] Step 4: Construct a thermo-mechanically coupled topology optimization model for the turbine disk structure, taking into account the material temperature dependence:
[0089] (7)
[0090] in Indicates softness, For nodal displacement, The overall stiffness matrix, For node temperature, The temperature load is calculated based on the given thermal boundary conditions. The total heat transfer coefficient matrix is shown below. Represents equivalent thermal load, topology design variable This represents the density of the e-th unit. Indicates the initial volume of the structure. Indicates the structural volume. The volume fraction is set to 0.4;
[0091] Step 5: Interpolate the material parameters The data is imported into ABAQUS and subjected to nonlinear heat transfer finite element analysis. The overall heat transfer coefficient matrix is obtained. Through the unit heat transfer coefficient matrix The assembly yielded:
[0092] (8)
[0093] In the formula The element thermal strain matrix, The heat transfer coefficient matrix of the unit cell. Let be a constant matrix of the heat transfer coefficient matrix. Noting the nonlinear part of the heat transfer coefficient matrix, we apply the Newton-Raphson iterative method to solve the nonlinear residual equation:
[0094] (9)
[0095] The nonlinear equation can be solved as follows:
[0096] (10)
[0097] In the k-th iteration step, the solution It may not be able to fit the equation exactly, i.e. To obtain the solution for the (k+1)th iteration, the residual is linearly approximated using a first-order Taylor series expansion:
[0098] (11)
[0099] in The tangent stiffness matrix can be used as follows: express. It is possible to pass Correction for calculating Taylor expansion equal to 0:
[0100] (12)
[0101] To obtain the tangent stiffness matrix, the derivative of the nonlinear residual equation is:
[0102] (13)
[0103] In order to solve ,structure matrix:
[0104] (14)
[0105] The element in the k-th row and l-th column It can be represented as:
[0106] (15)
[0107] It can be further written as:
[0108] (16)
[0109] Temperature The temperature at the center of the e-th element can be obtained from the node temperatures:
[0110] (17)
[0111] in , It is the shape function of the nth node. Using... The expression can The expression is rewritten as:
[0112] (18)
[0113] Ultimately, it can be obtained matrix:
[0114] (19)
[0115] Based on the above solutions, the temperature field of the turbine disk in the current iteration step can be obtained;
[0116] Step 6: For the thermo-coupling finite element equations, the equivalent thermal load... It can be obtained through the following methods:
[0117] (20)
[0118] in and Let the strain-displacement matrix and elasticity matrix of the e-th element be respectively. It is the elastic matrix of the solid material.
[0119] (twenty one)
[0120] in For thermal strain, The coefficient of thermal expansion is For reference temperature, For the unit assembly matrix, since the elastic properties and coefficient of thermal expansion of the material are both related to the physical density, the equivalent thermal load can be further expressed as:
[0121] (twenty two)
[0122] Step 7: Re-interpolate the material parameters , Import the data into ABAQUS and perform a thermo-mechanical coupled finite element analysis:
[0123] (twenty three)
[0124] The displacement field of the turbine disk in the current iteration step can be obtained;
[0125] Step 8: Perform sensitivity analysis on the thermo-coupled topology optimization model:
[0126] (twenty four)
[0127] The derivative of the equivalent thermal load with respect to the physical density is:
[0128] (25)
[0129] Its expansion can be further expressed as:
[0130] (26)
[0131] Solve using the nonlinear residual equation in equation (5)
[0132] (27)
[0133] Simplified solution yields:
[0134] (28)
[0135] in Let the Lagrange multipliers be the adjoint variables, and based on the method of adjoint variables, let:
[0136] (29)
[0137] Therefore, we obtain The expression:
[0138] (30)
[0139] right Taking the derivative of both ends with respect to the topological variables, we get:
[0140] (31)
[0141] Substituting the above equation into the expression for the derivative of the objective function, we can obtain the expression for the sensitivity of the objective function.
[0142] (32)
[0143] Then Substituting these values into the expression for the derivative of the objective function, we obtain the sensitivity of the objective function to the physical density:
[0144] (33)
[0145] in This represents the element displacement matrix.
[0146] Step 9: Derive the complete objective function sensitivity based on the chain rule of multi-field filtering:
[0147] (34)
[0148] Step 10: Based on the sensitivity information of the objective function to the design variables, the design variables are updated and iterated using the gradient-based MMA optimization algorithm.
[0149] Step 11: Determine convergence. If convergence fails, return to step 3. Continue until convergence is achieved, obtaining the optimal topology for the thermo-coupled structure.
[0150] Figure 3 The middle gives Figure 2 The topology optimization iterative process and design results considering material temperature dependence in the design domain are shown, where Figure 3 In the diagram, 'd' represents the final topology configuration obtained by this method. The iteration history shows that the proposed thermo-mechanically coupled topology optimization method, which considers the temperature dependence of materials, exhibits good stability and convergence. The iterative process of topology optimization demonstrates that the overall structural configuration emerges in fewer iterations, and subsequent iterations further refine the details of the configuration, illustrating the computational efficiency of the solution method. The optimized structure obtained using this method better meets the performance requirements of aero-engine turbine disks operating under conditions of large temperature variations. Furthermore, this method can fully utilize the powerful pre- and post-processing functions and finite element analysis capabilities of ABAQUS for topology optimization design of large and complex structures.
[0151] This invention addresses the issue of material property parameters varying with temperature by proposing a thermo-mechanically coupled topology optimization method that considers the temperature dependence of materials. First, a structural model and thermodynamic boundary conditions are defined based on ABAQUS, and relevant physical information is derived. Second, a material parameter interpolation model considering temperature variations and a corresponding thermo-mechanically coupled topology optimization model are constructed. Then, the topology optimization objective function is calculated by solving the temperature-dependent nonlinear heat conduction equation and the thermo-mechanical coupling equation. Subsequently, the sensitivity information of the topology design variables is derived based on the adjoint variable method. Finally, a gradient-based optimization algorithm is used to update the design variables, ultimately obtaining the topology configuration of the thermo-mechanically coupled structure.
[0152] The above description, in conjunction with specific preferred embodiments, provides a further detailed explanation of the present invention. It should not be construed that the specific implementation of the present invention is limited to these descriptions. For those skilled in the art, several equivalent substitutions or obvious modifications can be made without departing from the concept of the present invention, and all such modifications, achieving the same performance or purpose, should be considered within the scope of protection of the present invention.
Claims
1. A thermo-mechanically coupled topology optimization method considering material temperature dependence, characterized in that, Includes the following steps: Step 1: Construct a structural model using ABAQUS and obtain the model's physical information, including mesh data, stress conditions, constraint boundaries, and heat transfer boundaries; Step 2: Based on the density interpolation model, construct the material elastic modulus of the e-th element of the structure. as follows: （1） in, The elastic modulus of a solid material. Let be the elastic modulus of the empty phase material, p be the penalty factor, and N be the number of finite elements. This represents the element physical density obtained after density filtering and Heaviside projection filtering. The density filtering function can be expressed as: （2） in Let e be the set of elements within the filtration radius. and The weight function is given by the weight function for the volume and density of the i-th unit in the aforementioned set. It can be represented as: （3） in The distance from the center of element e to the center of element i. The filter radius is the intermediate variable after density filtering. It also needs to be filtered by Heaviside projection. It can be calculated using the following expression: （4） in Filter density threshold These are projection parameters; Step 3: Construct a temperature-dependent material parameter interpolation model, and determine the thermal conductivity of the e-th element of the structure. With coefficient of thermal expansion as follows: （5） （6） in, The thermal conductivity coefficient of a solid material as a function of temperature T. This represents the coefficient of thermal expansion of a solid material as a function of temperature T. Step 4: Construct a thermo-mechanically coupled topology optimization model that considers the temperature dependence of materials: （7） in Indicates structural flexibility. For nodal displacement, The overall stiffness matrix, For node temperature, The temperature load is calculated based on the given thermal boundary conditions. (T) is the overall heat transfer coefficient matrix. Represents equivalent thermal load, topology design variable This represents the density of the e-th unit. Indicates the initial volume of the structure. Indicates the structural volume. It is the volume fraction; Step 5: Perform nonlinear heat transfer finite element analysis to obtain the overall heat transfer coefficient matrix. Through the unit heat transfer coefficient matrix Assembled to obtain: （8） In the formula The element thermal strain matrix, The heat transfer coefficient matrix of the unit cell. Given that the heat transfer coefficient matrix is a constant matrix, and noting the nonlinear part of the heat transfer coefficient matrix, the Newton-Raphson iterative method is applied to solve the nonlinear residual equation: ； Step 6: For the thermo-coupling finite element equations, the equivalent thermal load It can be obtained through the following methods: (20) in and These are the strain-displacement matrix and elasticity matrix of the e-th element, respectively. It is the elastic matrix of the solid material; (21) in For thermal strain, The coefficient of thermal expansion is For reference temperature, As the unit assembly matrix, since the elastic properties and coefficient of thermal expansion of the material are both related to the physical density, equation (20) can be expressed as: (22) Step 7: Perform thermo-mechanical coupled finite element analysis: (23) Step 8: Perform sensitivity analysis on the thermo-coupled topology optimization model: (24) right Taking the derivative of both ends with respect to the topological variables, we get: (31) Substituting equation (31) into equation (24), we can obtain the sensitivity of the objective function to the physical density: (32) Step 9: Considering the effect of filtering, derive the sensitivity of the objective function to the design variables using the chain rule: (34) Step 10: Based on the sensitivity information of the objective function with respect to the design variables, the design variables are updated and iterated using a gradient-based MMA optimization algorithm; Step 11: Determine convergence. If it does not converge, return to step 2 until the calculation converges and the optimal topology of the thermo-coupled structure is obtained.
2. The thermo-coupling topology optimization method considering material temperature dependence according to claim 1, characterized in that: Step 8 involves solving for the sensitivity information of the objective function with respect to the design variables. The specific steps include: The derivative of the equivalent thermal load with respect to the physical density is: (25) Its expansion can be further expressed as: (26) To solve the unknown terms in the equation First, we need to use the nonlinear residual equation (9) from step 5 to obtain: (27) Therefore, we can conclude that: (28) in Let the Lagrange multipliers be the adjoint variables, and based on the method of adjoint variables, let: (29) Therefore, equation (28) can be transformed into: (30) Substituting equations (30) and (26) into equation (32), we obtain the sensitivity of the topology optimization objective function to physical density: (33) in This represents the element displacement matrix.