A functionally graded material structure composition optimization method based on temperature-dependent properties

Abstract

The application discloses a kind of functionally graded material structure component optimization method based on temperature-dependent property, belong to thermal engineering and structural optimization design technical field.The application selects the matrix material of different thermal conductivity, introduces material volume fraction variable, constructs the function model of thermal conductivity, specific heat capacity with temperature variation, obtains temperature-dependent equivalent thermal physical property parameter by multi-phase material interpolation, solves heat conduction equation and obtains temperature field.Structure thermal performance is taken as target, material volume fraction is taken as design variable, two-dimensional collaborative optimization of in-plane topological optimization and thickness direction parameterized gradient optimization is carried out, and optimal material distribution is obtained.The application considers the temperature effect of material thermal physical property, improves the accuracy of high-temperature heat transfer analysis, can reduce the maximum temperature of structure, improve temperature uniformity, reduce thermal stress, improve the performance and manufacturability of thermal protection structure, and is suitable for aerospace, high-temperature equipment and other high heat flux density scenes.

Description

Technical Field

[0001] This invention belongs to the field of thermal engineering and structural optimization design technology, specifically relating to a method for optimizing the structural composition of functionally graded materials based on temperature-related properties. Background Technology

[0002] In engineering fields such as aerospace, high-temperature energy equipment, and high-power electronic devices, structures typically operate for extended periods in environments with high heat flux density and large temperature gradients. Traditional thermal protection and thermal management designs often rely on homogeneous materials or layered composite structures, adding insulation or thermally conductive layers to cope with thermal loads. However, this type of design method generally suffers from the following shortcomings: Firstly, the thermal properties of materials are usually simplified to constants, ignoring their nonlinear characteristics as a function of temperature, resulting in insufficient accuracy in heat transfer prediction under high-temperature conditions; secondly, structural and material design are often disconnected, making it difficult to achieve proactive control of thermal loads as a whole.

[0003] Functionally graded materials (FGMs), due to their continuous spatial variation in material composition and properties, are considered an important approach to solving thermal-structural coupling problems under large temperature gradients. Existing FGM heat conduction designs often employ fixed thermophysical parameters, neglecting the influence of temperature on properties, or using only single-dimensional optimization methods. The former leads to distorted temperature field calculations under large temperature gradient conditions due to drastic temperature changes in thermophysical properties, resulting in optimization results that deviate from actual operating conditions. The latter is limited to in-plane topology optimization or parametric design in a single thickness direction, failing to balance the rationality of in-plane material layout with the continuity of thickness gradients, making it difficult to meet the high-efficiency heat transfer / insulation requirements under complex high-temperature conditions. Furthermore, many optimization design methods remain at the numerical simulation level, lacking manufacturability and experimental verification support, thus limiting their engineering applications.

[0004] Therefore, there is an urgent need for a design method that can optimize the heat transfer performance of functionally graded structures by taking into account the temperature-related thermophysical parameters of materials, so as to improve the temperature control capability and thermal management reliability under large temperature gradient conditions. Summary of the Invention

[0005] The purpose of this invention is to overcome the shortcomings of the prior art and provide a method for optimizing the structural composition of functionally graded materials based on temperature-related properties, so as to solve the problems of neglecting the change of material thermal properties with temperature, disconnect between structure and material design, single optimization dimension, and poor engineering manufacturability in the prior art.

[0006] To achieve the above objectives, the present invention employs the following technical solution: A method for optimizing the structural composition of functionally graded materials based on temperature-related properties includes the following steps: Establish a structural design domain that can withstand thermal loads with large temperature gradients, and construct a heat conduction analysis model within the design domain; At least two matrix materials with different thermal conductivity are selected, and a material volume fraction variable is introduced in the design domain to describe the volume fraction distribution of one of the matrix materials at a spatial location. The volume fraction of the other matrix material is the complement of the material volume fraction variable. Based on experimental data or material databases, establish a functional model of the thermophysical parameters of the matrix material as a function of temperature; Based on the material volume fraction variable and the temperature-related thermophysical parameter function model, the equivalent thermophysical parameters that change with temperature at any location within the design domain are obtained using a multiphase material interpolation method. Considering the equivalent thermal property parameters, the thermal conduction control equation of the structure is established and solved to obtain the temperature field distribution of the structure under given thermal boundary conditions. Using structural thermal performance as the optimization objective and the material volume fraction variable as the design variable, a heat transfer optimization model is constructed. Under preset constraints, a two-dimensional collaborative optimization solution is performed to obtain the optimal material distribution of the functionally graded structure. The two-dimensional collaborative optimization includes in-plane topology optimization and thickness-direction parameterized gradient optimization.

[0007] A further improvement of the present invention is that: Preferably, the in-plane topology optimization includes: taking the minimization of the highest temperature within a specified region of the structure as the objective, introducing a p-norm function to smoothly approximate the highest temperature; obtaining a smoothed design variable field through a density filtering model based on Helmholtz-type partial differential equations; and obtaining the optimal in-plane material volume fraction distribution by iteratively updating the design variables within the design variable field until the convergence condition is met.

[0008] Preferably, the in-plane topology optimization process is subject to volume fraction constraints and temperature distribution uniformity indices; the volume fraction constraints are:

[0009] Among them, C max This is the preset upper limit of material volume fraction. This represents the smoothed material volume fraction. Design domain volume; The temperature distribution uniformity index is:

[0010] Among them, Ω h For the structural design domain, |Ω h| represents the area of ​​the design domain; T(x) represents the local temperature at any location x within the design domain; T represents the average temperature within the design domain, C. T,max This is the preset upper limit for temperature variance.

[0011] Preferably, the convergence condition is:

[0012] Where k is the number of iterations, J is the optimization objective function defined based on the temperature field, and ε is the preset convergence threshold.

[0013] Preferably, the thickness-direction parameterized gradient optimization includes: optimizing the thickness-direction parameterized gradient using a normalized global one-dimensional gradient function related to thickness, wherein the optimization objective function is: ; Where θ is the parameter to be optimized for the gradient function in the thickness direction. This represents the equivalent temperature after averaging the temperature field in the XOY plane at the thickness location z, where H is the total thickness of the structure, and C... z This is the integral of the square of the temperature gradient along the thickness.

[0014] Preferably, during the optimization process, at any planar position (x, y), the integral average of the material volume fraction along the thickness direction is equal to the volume fraction c(x, y) obtained at that position through planar optimization, and the mathematical expression is:

[0015] Among them, c A (x,y,z) represents the volume fraction of material A, and c B (x,y,z) represents the volume fraction of material B, and c(z) is the thickness gradient function that satisfies the normalization condition.

[0016] Preferably, during the optimization process, c(x,y) remains unchanged, and the expression for the final material volume fraction at any spatial location (x,y,z) of the three-dimensional structure is: c A (x,y,z)=c(x,y)·c(z) c B (x,y,z)=1 c A (x,y,z) Among them, c A (x,y,z) represents the volume fraction of material A, and c B (x,y,z) represents the volume fraction of material B.

[0017] Preferably, the multiphase material interpolation method is the volume averaging method, and the equivalent thermal conductivity keff and equivalent specific heat capacity cp,eff at any location within the design domain are expressed as follows: k eff (T,x)=c(x)k1(T)+[1-c(x)]k2(T) Where k1(T) and k2(T) represent the functions of thermal conductivity of the two matrix materials as a function of temperature; c p,eff (T,x)=c(x)c p,1 (T)+[1-c(x)]c p,2 (T) Among them, c p,1 (T) and c p,2 (T) represents the specific heat capacity of the two matrix materials as a function of temperature; c(x) is the volume fraction distribution of a matrix material at a spatial location.

[0018] Preferably, the heat conduction control equation is divided into heat conduction control equation under steady-state conditions and heat conduction control equation under transient conditions; The heat conduction control equation under the steady-state condition is:

[0019] Where T(x) is the temperature field distribution function, and q(x) is the volumetric heat source term. Let be the equivalent thermal conductivity at any location within the domain; Under the aforementioned transient condition, the heat conduction control equation is:

[0020] Where ρ(x) is the equivalent density of the structure at position x. Let be the temperature field distribution function at time t. Let be the volumetric heat source term at time t.

[0021] Preferably, the method further includes numerical or experimental verification of the functionally graded structure based on the optimization results. The numerical verification includes comparing and analyzing the temperature field results obtained using a constant material thermophysical parameter model and a temperature-dependent material thermophysical parameter model. The experimental verification includes measuring the surface or internal temperature field of the functionally graded structure sample using infrared thermography under high-temperature heat flux loading conditions.

[0022] Compared with the prior art, the present invention has the following beneficial effects: This invention discloses a method for optimizing the structural composition of functionally graded materials (FJTs) based on temperature-dependent properties. This method addresses the problems of insufficient heat transfer prediction accuracy due to significant temperature variations in material thermophysical parameters under large temperature gradient conditions, and the difficulty of balancing thermal performance and manufacturability using traditional single-dimensional optimization. It introduces a material volume fraction variable within the structural design domain to describe the continuous distribution of different matrix materials, and establishes functional models of material thermal conductivity and specific heat capacity as a function of temperature based on experimental data or material databases. Equivalent thermophysical parameters considering temperature effects are obtained through multiphase material interpolation. Based on this, a heat conduction analysis model is constructed, and a two-dimensional collaborative optimization design is carried out, using in-plane topology optimization and thickness-direction parameterized gradient optimization as the optimization objective. This method can effectively reduce the maximum temperature of the structure and improve the uniformity of the temperature field distribution under large temperature gradient conditions, suppress thermal stress in the thickness direction, and significantly improve the service reliability and manufacturability of thermal protection structures, demonstrating good engineering applicability.

[0023] This invention organically couples temperature-dependent thermal property modeling, equivalent thermal property parameter calculation, in-plane topology optimization, and thickness-direction parameterized gradient optimization into a complete process. Each step is sequentially connected and mutually supportive, forming a closed-loop system from high-precision heat transfer analysis to engineering structural design. Specifically, the introduction of temperature-dependent material properties effectively improves the accuracy of temperature field calculations under high-temperature conditions. In-plane topology optimization is used to determine the optimal material distribution within the plane, providing a core layout basis for overall heat transfer performance optimization. Thickness-direction parameterized gradient optimization serves as an extension and engineering implementation of in-plane optimization. By constructing a material gradient function that continuously varies along the thickness, the two-dimensional optimal layout is extended into a three-dimensional continuous gradient structure, resulting in a smoother transition in material distribution and adaptability to additive manufacturing processes. This solves the problems of traditional optimization methods being limited to a single dimension, having discontinuous gradients, and being difficult to implement in engineering.

[0024] This invention introduces a temperature-dependent material model, a multiphase material interpolation method, and a two-dimensional optimization strategy driven by thermal performance. While ensuring in-plane temperature uniformity, it effectively reduces the temperature gradient and thermal stress in the thickness direction, taking into account the thermal protection performance, mechanical reliability, and manufacturability of the structure. It achieves precise control of the temperature field distribution of the structure, thereby effectively reducing the maximum temperature of the structure, reducing the temperature gradient, and improving the heat flow distribution characteristics. Attached Figure Description

[0025] Figure 1 This is a schematic diagram of the functional gradient structure heat transfer optimization design process of the present invention; Figure 2 This is a schematic diagram of the continuous distribution of material volume fraction in a functionally graded structure. Figure 3 This is a graph showing the change in thermal conductivity of the two-phase matrix material selected for this invention as a function of temperature. Among them, (a) is the thermal conductivity-temperature change curve of the low thermal conductivity phase Inconel 625 high-temperature alloy, and (b) is the thermal conductivity-temperature change curve of the high thermal conductivity phase Cu metal.

[0026] Figure 4 The diagram shows the heat transfer optimization results of the functional gradient structure under different external operating conditions according to an embodiment of the present invention. In the figure, (a) constant temperature difference between the left and right ends; (b) constant heat flow q on the left and convective heat dissipation on the right; (c) uniform heat source within the domain + convection along the entire edge; (d) high-intensity heat source at a single point (small circle) + convection along the entire edge; (e) multiple randomly distributed point sources + convection; (f) constant temperature on the left + segmented convection on the right.

[0027] Figure 5 This is a comparison chart of the planar optimization results of different thermal property models under constant temperature difference conditions in this invention after heat transfer optimization. In the figure, (a) the volume fraction distribution of the material under a fixed thermal conductivity at room temperature of 300K; (b) the volume fraction distribution of the material under a fixed thermal conductivity at high temperature of 800K; (c) the volume fraction distribution of the material under the condition that the thermal conductivity changes with temperature; (d) the structural temperature field distribution under a fixed thermal conductivity at room temperature of 300K; (e) the structural temperature field distribution under a fixed thermal conductivity at high temperature of 800K; and (f) the structural temperature field distribution under the condition that the thermal conductivity changes with temperature.

[0028] Note: In the material distribution cloud map, the red area represents a higher proportion of Cu volume fraction (high thermal conductivity), and the blue area represents a higher proportion of Inconel 625 volume fraction (low thermal conductivity); the temperature field cloud map is measured in thermodynamic temperature (K). Detailed Implementation

[0029] Hereinafter, the terms "first," "second," "third," and "fourth" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Therefore, a feature defined as "first," "second," "third," or "fourth" may explicitly or implicitly include one or more of that feature.

[0030] The method provided in this application can be applied to mobile phones, tablets, wearable devices, in-vehicle devices, augmented reality (AR) / virtual reality (VR) devices, laptops, and ultra-mobile personal computers. In this application, the specific type of terminal device is not limited to terminal devices such as mobile personal computers (UMPCs), netbooks, and personal digital assistants (PDAs).

[0031] It should be noted that the terms "first," "second," etc., used in the specification and drawings of this invention are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate so that the embodiments of the invention described herein can be implemented in orders other than those illustrated or described herein. Furthermore, the terms "comprising" and "having," and any variations thereof, are intended to cover non-exclusive inclusion; for example, a process, method, system, product, or apparatus that comprises a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.

[0032] This invention discloses a method for optimizing the structural composition of functionally graded materials based on temperature-dependent properties, comprising the following steps: S1. Establish a structural design domain subjected to a large temperature gradient thermal load, and construct a heat conduction analysis model within the design domain; wherein, a large temperature gradient refers to a temperature gradient of 10. 2 -10 6 K / m.

[0033] S2. Select at least two matrix materials with different thermal conductivity. In the design domain, introduce a material volume fraction variable c(x)∈[0,1] to describe the volume fraction distribution of one matrix material at spatial location x, and the volume fraction of the other matrix material is 1. c(x); S3. Based on experimental data or material databases, establish functional models of the thermal properties of all materials as a function of temperature, so that the thermal properties of materials are expressed as a continuous function of temperature. The specific thermal properties of the material include thermal conductivity and specific heat capacity. Continuous function models of the thermal conductivity and specific heat capacity of the matrix material as a function of temperature are established respectively.

[0034] Furthermore, the functional model of the material's thermophysical parameters changing with temperature is established using a polynomial function, an exponential function, or a piecewise continuous function.

[0035] For example, the relationship between the thermal conductivity of a material and temperature can be expressed as a polynomial function: k i (T)=a i,0 +a i,1 T+a i,2 T 2 +…+a i,n T n Where, k i(T) represents the thermal conductivity of the i-th material at temperature T, a i,j The coefficients are the polynomials obtained by fitting experimental or literature data, where T is the absolute temperature of the material and n represents the number of terms in the thermal conductivity polynomial.

[0036] The relationship between the specific heat capacity of a material and temperature is expressed as follows: C p,i (T)=b i,0 +b i,1 T+b i,2 T 2 +…+b i,m T m ; Among them, C p,i (T) represents the isobaric heat capacity of the i-th material at temperature T, b i,j is the fitting coefficient for the specific heat capacity of the material, and m represents the number of terms in the specific heat capacity polynomial.

[0037] S4. Based on the material volume fraction variable and temperature-related material property model, a multiphase material interpolation method is used to obtain the equivalent thermal property parameters that change with temperature at any location within the design domain; the multiphase material interpolation method is the volume averaging method, the mixing law, or an improved form thereof, so as to achieve a continuous transition of the material thermal property parameters in space.

[0038] As a preferred method, a multiphase material interpolation method is used to obtain the equivalent thermophysical property parameters that vary with temperature at any location within the design domain; Based on the aforementioned material volume fraction variable, the equivalent thermal conductivity k at any location within the domain is designed. eff Represented as: k eff (T,x)=c(x)k1(T)+[1-c(x)]k2(T) Where k1(T) and k2(T) represent the functions of thermal conductivity of the two matrix materials as a function of temperature; Similarly, the equivalent specific heat capacity can be expressed as: c p,eff (T,x)=c(x)c p,1 (T)+[1-c(x)]c p,2 (T) Among them, c p,1 (T) and c p,2 (T) represents the specific heat capacity of the two matrix materials as a function of temperature.

[0039] The above interpolation method is one implementation of the volume averaging method. Other hybrid law forms that satisfy continuity and differentiability can also be used.

[0040] S5. Considering the equivalent thermal property parameters, establish and solve the thermal conduction control equation of the structure to obtain the temperature field distribution of the structure under given thermal boundary conditions; the thermal conduction control equation is a steady-state thermal conduction equation or a transient thermal conduction equation.

[0041] Under steady-state conditions, the governing equation for heat conduction within the structure is expressed as:

[0042] Where T(x) is the temperature field distribution function and q(x) is the volume heat source term.

[0043] Under transient conditions, the governing equation for heat conduction is expressed as:

[0044] Where ρ(x) is the equivalent density of the structure at position x, and its value is determined by the local composition distribution of the material through interpolation (such as the linear mixing rule), ρ(x)=c(x)ρ1+[1 c(x)]ρ2, where ρ1 and ρ2 are the densities of the two-phase material, and ρ(x) is used to characterize the mass properties of functionally graded materials as they change along spatial coordinates. It is a key equivalent thermophysical property parameter describing the thermal inertia of materials in transient heat conduction analysis. Let be the temperature field distribution function at time t. Let be the volumetric heat source term at time t.

[0045] It should be noted that steady-state conditions refer to a situation where the temperature field at all locations within the structure does not change over time, the heat transfer within the system reaches dynamic equilibrium, and temperature is only a function of spatial coordinates, independent of time. In contrast, transient conditions refer to a situation where the temperature field within the structure changes continuously over time, the heat within the system has not reached equilibrium, temperature is a function of both spatial coordinates and time, and there is a process of internal energy accumulation or release over time.

[0046] S6. Using the structural thermal performance as the optimization objective, the material volume fraction variable is used as the design variable to construct a heat transfer optimization model, and the optimization solution is performed under preset constraints to obtain the optimal material distribution of the functionally graded structure.

[0047] The objectives for optimizing thermal performance include at least one of the following: minimizing the maximum structural temperature, minimizing the average structural temperature, minimizing the temperature gradient, or homogenizing the heat flux distribution.

[0048] In a specific example, the optimization process first constructs an optimization objective function. In the plane, the objective is to minimize the maximum temperature, with the material volume fraction as the design variable. The maximum temperature is represented in a smooth approximation form using the p-norm. Topology optimization is performed through density filtering to obtain the optimal material distribution in the plane. In the thickness direction, the objective is to minimize the square integral of the temperature gradient. The gradient function parameters (such as linear or exponential forms) are optimized to achieve a material gradient that changes continuously along the thickness.

[0049] In this invention, the optimization target is preferably a specified structural region Ω. h The objective function for minimizing the highest temperature within the interior can be expressed as:

[0050] To avoid directly optimizing the non-smooth max(T) function, a p-norm function is introduced to provide a smooth approximation of the maximum temperature value. A smooth approximation is used; for example, the optimization function can be expressed in p-norm form: (p>>1) During the solution process, the design variable field c(x) is solved in the XOY plane. The smoothed design variable field is obtained using a density filtering model based on Helmholtz-type partial differential equations. Its mathematical expression is:

[0051] Where r is the filtering radius, used to control the smoothness and minimum feature size of the material volume fraction distribution, avoiding numerical oscillations and meeting engineering fabrication requirements. A gradient-driven optimization algorithm, such as the moving asymptote method, is used to iteratively solve the design variable field. In each iteration, the current design variable field is first... Calculate the equivalent thermal properties and solve the heat conduction equation to obtain the temperature field. Then calculate the optimal function value. .

[0052] The optimization described in this invention is an optimization of the continuous distribution of functionally graded material composition, such as... Figure 2 As shown; the entire design domain is composed of solid materials, and only the volume fraction distribution of the two-phase materials is optimized, without involving topology optimization of structural solids and voids.

[0053] Planar topology optimization employs a sensitivity optimization algorithm commonly used in this field for iterative solution. Using the material volume fraction c(x,y) as the design variable, each iteration sequentially solves for the heat conduction field, calculates the objective function, performs sensitivity analysis on the design variables, and updates and corrects the design variables using the moving asymptote method (MMA). This iterative process is repeated until the objective function of two adjacent iterations satisfies the following convergence criterion, at which point the optimization process is considered convergent:

[0054] Where k is the number of iterations, J is the optimization objective function defined based on the temperature field, and ε is the preset convergence threshold.

[0055] Through the above filtering and projection process, a functionally graded material distribution form with continuously varying and clearly defined boundaries within the XOY plane is obtained. This physical material distribution variable is further coupled with a temperature-dependent material thermophysical property model to construct a heat conduction analysis model and solve for the structural temperature field.

[0056] Furthermore, during the optimization process, material volume fraction constraints are applied to control the overall proportion of a certain phase matrix material within the design domain:

[0057] Among them, C max This is the preset upper limit of material volume fraction. This represents the smoothed material volume fraction. To design the domain volume.

[0058] Furthermore, in the process of planar topology optimization, to suppress local high-temperature concentration in the structure, a performance index C based on temperature distribution uniformity is introduced. T It is defined as the spatial variance of the temperature field within the design domain relative to the average temperature. Temperature uniformity constraints are applied during the planar optimization process, and the constraint expression is:

[0059] Among them, Ω h For the structural design domain, |Ω h | represents the area of ​​the design domain; T(x) represents the local temperature at any location x within the design domain; The average temperature within the design domain, C T,max This is a preset upper limit for temperature variance, used to limit temperature fluctuations within the structure and ensure a uniform temperature field. The smaller the value, the more uniform the temperature distribution of the structure and the weaker the local high temperature concentration phenomenon.

[0060] The above method enables the topology optimization design of functionally graded structures that takes into account the influence of temperature-related material properties in the XOY plane. The heat transfer optimization is carried out in a two-dimensional plane. By optimizing the material volume fraction distribution in the plane, the in-plane heat transfer performance of the functionally graded structure is optimized.

[0061] It should be noted that the in-plane topology optimization process only optimizes the material volume fraction distribution c(x,y) within the XOY plane. During optimization, the material composition is constrained to be uniformly distributed along the structural thickness direction, achieving only optimal heat transfer performance in the two-dimensional plane, without considering the design of material gradient patterns along the thickness direction. After planar optimization, the resulting material distribution is uniformly spread across the entire thickness domain, failing to spontaneously form a continuous composition gradient along the z-direction. This leads to severe temperature gradients and excessive thermal stress along the thickness direction, making it difficult to meet the reliability requirements for high-temperature service and additive manufacturing processes.

[0062] Therefore, this invention, based on planar optimization, further conducts parameterized gradient optimization along the thickness direction. This step uses the optimal in-plane material layout obtained from planar optimization as a basic constraint. By optimizing the parameters of the thickness gradient function, a globally unified component distribution function c(z) along the thickness direction is solved, extending the two-dimensional in-plane distribution into a three-dimensional, fully continuous gradient field. While retaining the optimal in-plane heat transfer layout, a smooth gradient of material composition along the thickness direction is achieved, effectively suppressing the temperature gradient along the thickness direction, reducing structural thermal stress, and ensuring that the material distribution meets the requirements for continuous, abrupt fabrication, thus forming a complete three-dimensional functionally graded material optimization scheme.

[0063] The heat transfer optimization is performed in the thickness direction. By introducing a normalized global one-dimensional gradient function that is only related to the thickness coordinate and optimizing the parameters of the gradient function, the heat transfer performance of the functionally graded structure along the thickness direction is optimized.

[0064] Specifically, a normalized gradient function c(z) is defined along the thickness direction z of the structure to describe the relative proportional relationship between the two matrix materials in the thickness direction. The normalized gradient function satisfies:

[0065] c(z) can be any one or more of the following: linear function, power-law function, exponential function, quadratic function, trigonometric function, or hyperbolic function.

[0066] For example, the normalized gradient function includes, but is not limited to, the following forms: Linear gradient function:

[0067] Power-law gradient function:

[0068] Exponential gradient function:

[0069] in, Here, z is an adjustable gradient parameter, z is the coordinate in the thickness direction, z=0 corresponds to the high-temperature heated surface, z=H corresponds to the low-temperature heat dissipation surface, and H is the total thickness of the structure.

[0070] Using the aforementioned parameterized function form, a continuously varying functionally graded material distribution can be constructed along the thickness direction without the need for discrete material layering. Based on the thickness-direction normalized gradient function, the equivalent thermal properties at any location on the structure are determined by its spatial location and local temperature. By introducing the gradient parameters as optimization variables into a temperature-dependent material property model, the gradient distribution along the thickness direction can be parameterized.

[0071] In the thickness-direction gradient optimization process, the core objective is to minimize the temperature gradient in the thickness direction, and the objective function is constructed as follows: ; Where θ is the parameter to be optimized for the gradient function in the thickness direction. This represents the equivalent temperature after averaging the temperature field in the XOY plane at the thickness location z, where H is the total thickness of the structure. z It is the integral of the square of the temperature gradient along the thickness direction, used to characterize the severity of temperature change along the thickness direction. The smaller the value, the more gradual the temperature distribution along the thickness direction and the smaller the thermal stress.

[0072] By incorporating this objective function into the optimization process, on the basis of achieving in-plane temperature uniformity through planar optimization, the temperature gradient in the thickness direction can be further suppressed, the thermal stress of the structure can be reduced, and the synergistic optimization of the planar and thickness dimensions can be achieved. At the same time, the material gradient is ensured to transition continuously along the thickness, thereby improving the reliability and manufacturability of the structure.

[0073] The gradient parameters are updated iteratively. When the change of the objective function in the thickness direction in two adjacent iterations meets the preset convergence condition, the gradient parameters are considered to be optimized. Through the above method, the parameterized optimization design of the functional gradient structure considering the influence of temperature-related material properties in the thickness direction is realized.

[0074] After the in-plane topology optimization is completed, the material volume fraction distribution c(x,y) at each location in the XOY plane has been obtained. This in-plane basic value remains fixed in the subsequent thickness optimization process, fully inheriting the optimal in-plane heat transfer layout.

[0075] For the subsequent thickness optimization process, the core constraint of this method is: at any planar position (x, y), the integral average of the material volume fraction along the thickness direction is strictly equal to the volume fraction c(x, y) obtained at that position through planar optimization. The mathematical expression is:

[0076] Based on the above constraints, the planar material distribution is spatially coupled with the normalized thickness gradient function to obtain the final material volume fraction at any spatial location (x, y, z) of the three-dimensional structure: c A (x,y,z)=c(x,y)·c(z) c B (x,y,z)=1 c A (x,y,z) Among them, c A (x,y,z) represents the volume fraction of material A, and c B (x,y,z) represents the volume fraction of material B, and c(z) is the thickness gradient function that satisfies the normalization condition.

[0077] The above coupling method achieves gradual change through thickness redistribution with a constant total amount: planar optimization determines the total amount of material used at each location, and thickness optimization only redistributes this total amount in the thickness direction. The relative proportion between points in the plane remains unchanged, thus completely avoiding the problem of the planar optimization layout being destroyed.

[0078] The in-plane optimization determines the optimal material volume fraction distribution within the XOY plane, enriching high-insulating phase materials in high-heat-flux regions and high-conductivity phase materials in heat-dissipating regions, thereby improving the uniformity of in-plane temperature distribution. The thickness-direction optimization, based on the material distribution pattern obtained from the in-plane optimization, further constructs a continuously varying material gradient along the thickness direction, ensuring that the high-temperature side is dominated by insulating phases and the low-temperature side by conductive phases, achieving thermal stress mitigation and improved heat transfer efficiency along the thickness direction. The in-plane and thickness-direction optimizations act on different dimensions, progressing sequentially and supporting each other without optimization conflicts, together forming a complete three-dimensional optimization system for functionally graded material structures.

[0079] S7. Based on the optimization results, the functional gradient structure is numerically or experimentally verified to evaluate its heat transfer performance under large temperature gradient conditions.

[0080] The numerical verification includes a comparative analysis of the temperature field results obtained using a constant material thermophysical parameter model and a temperature-dependent material thermophysical parameter model.

[0081] Furthermore, the experimental verification includes measuring the surface or internal temperature field of the functionally graded structure specimen under high-temperature heat flux loading conditions.

[0082] Furthermore, the temperature field measurement employs infrared thermography, heat flow meter measurement, or a combination of both.

[0083] The functionally graded structure is prepared by an additive manufacturing process, which includes a multi-material additive manufacturing process.

[0084] This method is applicable to the heat transfer optimization design of aerospace thermal protection structures, high-temperature energy equipment, or high heat flux density engineering structures.

[0085] In practical applications, a high-temperature structural component can be selected as the research object, which bears a strong heat flux load from the high-temperature side during operation. A metal material with excellent high-temperature strength but low thermal conductivity and a metal material with high thermal conductivity are selected as the matrix materials of the functionally graded structure.

[0086] The following description, in conjunction with specific embodiments, provides further details.

[0087] Example See Figure 3 Combining experimental data and data from research papers, this invention establishes a continuous function model for the thermophysical parameters of a two-phase matrix material as a function of thermodynamic temperature. The evolution of the material's thermal conductivity with temperature is shown below. Figure 3 As shown in Figure (a), the thermal conductivity of the low-thermal-conductivity phase Inconel 625 high-temperature alloy increases monotonically with increasing temperature; the thermal conductivity of the material at room temperature (300K) is approximately 10 W / (m²). The thermal conductivity of the material at 800K is approximately 17.5 W / (m²). In the entire service temperature range, the thermal conductivity of the material continuously increases with increasing temperature, and the traditional constant thermal property assumption cannot characterize this dynamic change.

[0088] As shown in Figure (b), the thermal conductivity of the high thermal conductivity phase Cu metal decreases monotonically with increasing temperature; the thermal conductivity of the material at room temperature (300K) is approximately 365 W / (m²). The thermal conductivity of the material at 800K is approximately 400 W / (m²). K), the thermal conductivity of the material continuously decreases as the temperature increases throughout the entire service temperature range.

[0089] In the figure, the red dashed line corresponds to the fixed thermal conductivity value at 800K high temperature, and the blue dashed line corresponds to the fixed thermal conductivity value at 300K room temperature, representing the constant physical property parameters used in traditional optimization methods. The orange solid line represents the thermal conductivity curve related to the actual temperature of the material. Comparing the curves, it can be seen that the traditional constant thermal property modeling method can only select fixed values ​​at a single temperature node, completely ignoring the inherent property of the nonlinear evolution of thermal conductivity with temperature in two-phase materials: for Inconel 625, the fixed physical property value at room temperature significantly underestimates the thermal conductivity of the material in the high-temperature range, and the fixed physical property value at high temperature overestimates the thermal conductivity of the material in the low-temperature range; for Cu, the fixed physical property value at room temperature significantly underestimates the thermal conductivity of the material across the entire temperature range, and the fixed physical property value at high temperature overestimates the thermal conductivity of the material in the low-temperature range. These deviations in physical property parameters directly lead to distortion in the solution of the heat transfer control equation, resulting in the material layout obtained from planar optimization deviating from the actual service conditions of the structure, ultimately causing defects such as temperature field prediction errors and localized thermal stress concentration.

[0090] Therefore, based on the above-mentioned real temperature-related thermal conductivity curves, this invention constructs a functional model of the thermal properties of the matrix material changing with temperature, and combines the multiphase material volume average interpolation method to solve the equivalent thermal property parameters of the structure's global position dynamically changing with temperature, thus fundamentally solving the problem of simulation distortion under high-temperature conditions in traditional constant property models.

[0091] This embodiment uses a rectangular functionally graded thermal protection structure with dimensions of 100mm × 50mm as the research object. Inconel 625, a material with excellent high-temperature strength, and Cu, a high thermal conductivity metal, were selected. Temperature-dependent material properties were adopted, and the volume fraction variable c(x) was introduced as the volume fraction of Cu, while 1-c(x) was used as the volume fraction of Inconel 625. Optimization simulations were conducted under six typical thermal load conditions to verify the adaptability and effectiveness of the proposed method for heat transfer optimization under complex conditions. The optimization results are as follows: Figure 4 As shown, the material cloud map displays the distribution of the two materials in the plane and the material percentage at each point. Different colors are used to distinguish them in the plane, and the scale on the right is the volume fraction of Cu, c(x)∈[0,1].

[0092] The first working condition is a constant temperature difference between the left and right ends, with a high-temperature boundary T applied to the left end of the structure. h Apply a low-temperature boundary T to the right end c No volume heat source is set within the computational region. The optimized material volume fraction distribution is shown in (a). The high thermal conductivity phase Cu is mainly distributed in the direct heat flow path, while the low thermal conductivity phase Inconel625 is concentrated in the central insulation region, forming a highly efficient thermal barrier structure. This effectively suppresses the direct heat flow driven by temperature difference and reduces cross-regional heat transfer, indicating that the method of this invention can form an optimal in-plane material distribution that matches the heat flow path under constant temperature difference boundary conditions.

[0093] The second operating condition is a constant heat flux at the left end and a convective heat transfer condition at the right end. A constant heat flux density q is applied to the left end of the structure, and a convective heat transfer boundary is applied to the right end. There is no internal heat source in the computational region. The optimization results are shown in Figure (b). The high thermal conductivity phase Cu is enriched in the heat input and heat dissipation regions, while the low thermal conductivity phase Inconel625 is distributed in the middle heat insulation region, forming a distribution pattern of heat insulation in the middle and heat conduction on both sides. This can achieve a matching balance between heat input and heat dissipation capacity, verifying the heat flux control capability of the method of the present invention under asymmetric thermal boundary conditions.

[0094] The third operating condition is the uniform volume heat source and global convection heat dissipation condition, where a uniform volume heat source q is applied within the structural computational domain. v The structure has convective heat dissipation boundaries on all four outer surfaces. The optimization results are shown in Figure (c). The low thermal conductivity phase Inconel 625 is concentrated in the high-temperature region at the center of the structure, while the high thermal conductivity phase Cu is distributed in the surrounding heat dissipation region. This can form thermal insulation protection in the internal heat-generating region and enhance heat dissipation capacity in the external region, indicating that the method of the present invention can achieve heat flow optimization and temperature uniformity control under internal heat source conditions.

[0095] The fourth operating condition is a single-point high-intensity heat source and a global convection cooling condition. A single-point high-intensity heat source is set at the center of the structure, and the outer surface of the structure is equipped with a convection cooling boundary. The optimization result is shown in Figure (d). The low thermal conductivity phase Inconel 625 surrounds the point heat source to form a heat insulation ring, and the high thermal conductivity phase Cu is distributed in the outer heat dissipation area. This can suppress the diffusion of local hot spots and enhance the heat dissipation of the periphery, demonstrating the precise control effect of the method of this invention on local high heat flux areas.

[0096] The fifth operating condition is a multi-random point heat source and global convection heat dissipation condition. Multiple randomly distributed point heat sources are set inside the structure, and the outer surface is equipped with a convection heat dissipation boundary. The optimization result is shown in Figure (e). The low thermal conductivity phase Inconel 625 forms a continuous thermal insulation channel between the heat sources, while the high thermal conductivity phase Cu is distributed in non-heat source areas to ensure overall heat dissipation. This indicates that the method of the present invention can adapt to complex thermal load environments with multiple heat sources and achieve overall temperature uniformity optimization under the coupling effect of multiple heat sources.

[0097] The sixth operating condition is a isothermal boundary on the left and a segmented convection boundary on the right. An isothermal boundary Th is applied to the left end of the structure, while a segmented convection heat transfer boundary is used on the right end. Different regions correspond to different convection heat transfer coefficients. The optimization results are shown in Figure (f). The material volume fraction exhibits an asymmetric distribution. The low thermal conductivity phase Inconel 625 is concentrated in the high-temperature region and the strong convection region, while the high thermal conductivity phase Cu is distributed in the heat dissipation advantage region. The material distribution can be adaptively adjusted according to the non-uniform boundary conditions to achieve thermal protection optimization under complex non-uniform boundaries.

[0098] In addition, this embodiment conducts simulation verification for constant temperature difference service conditions. A rectangular thermal protection structure with a size of 50mm×50mm is selected as the research object. The two phase matrix materials are also selected as high temperature alloy Inconel625 as the heat insulation phase material and high thermal conductivity metal copper Cu as the heat conduction phase material. The material volume fraction variable c(x) is introduced as the volume fraction of Cu, and 1-c(x) is the volume fraction of Inconel625.

[0099] To verify the superiority of the temperature-related thermophysical parameter model constructed in this invention, three sets of comparative optimization schemes were set up in this embodiment. Under the condition of a constant temperature difference boundary between the upper and lower end faces, the in-plane composition distribution optimization process was carried out, and the material distribution law and structural heat transfer performance differences under different thermophysical modeling methods were compared. The optimization results are as follows: Figure 5 As shown, the material cloud map displays the distribution of the two materials in the plane and the material percentage at each point. Different colors are used to distinguish them in the plane. The scale on the right is the volume fraction of Cu, c(x)∈[0,1]. At the same time, the optimized material plane temperature field distribution map is compared. The scale on the right is the steady-state temperature at each point, in K.

[0100] The first set of control schemes uses a constant thermal property model at room temperature. Fixed thermal conductivity parameters for both materials are selected throughout the process at 300K, ignoring the nonlinear evolution of material thermal properties with temperature. The solution is obtained using the planar optimization process of this invention. The final material volume fraction distribution in the XOY plane is shown in Figure (a), and the corresponding steady-state heat transfer temperature field distribution is shown in Figure (d). The second set of control schemes uses a constant thermal property model at high temperature. Fixed thermal conductivity parameters for both materials are selected throughout the process at 800K, again ignoring the temperature dependence of material thermal properties. Planar optimization is completed under the same boundary conditions. The resulting in-plane material volume fraction distribution is shown in Figure (b), and the corresponding steady-state temperature field distribution is shown in Figure (e). The third group is the temperature-related thermal property model scheme proposed in this invention. Based on the material experimental database, a continuous function model of the thermal conductivity and specific heat capacity of Inconel625 and Cu two-phase materials as a function of temperature is established. The equivalent thermal property parameters of the whole domain are solved by combining the volume average multiphase material interpolation method. Under the same constant temperature difference boundary conditions, the in-plane composition distribution optimization is completed. The in-plane material volume fraction distribution results are shown in Figure (c), and the corresponding steady-state temperature field distribution of the structure is shown in Figure (f).

[0101] A performance comparison analysis was conducted based on the optimization results in the attached figures: The upper half of the figures (a), (b), and (c) are cloud maps of the planar material volume fraction distribution corresponding to different modeling schemes. The red areas in the cloud maps represent the areas where the high thermal conductivity phase Cu is enriched in volume, and the blue areas represent the areas where the low thermal conductivity phase Inconel 625 is enriched in volume. The lower half of the figures (d), (e), and (f) are cloud maps of the global steady-state temperature field distribution of the structure under the corresponding modeling schemes. The color values ​​correspond to the local thermodynamic temperatures of the structure, and the units are K. The comparison of material distribution results shows that the in-plane material layouts obtained by the two traditional fixed property models, namely constant thermal conductivity at room temperature and constant thermal conductivity at high temperature, are similar, with only slight differences in the composition transition boundary region. The material distribution cannot adapt to the dynamic changes in material properties under wide temperature range and large temperature gradient conditions. The material distribution obtained by the temperature-related thermal property model of this invention presents a symmetrical doubly connected domain layout, with a smooth and continuous transition interface between the two phases. The arrangement of the heat dissipation channel of the high thermal conductivity phase and the heat insulation region of the low thermal conductivity phase is highly matched with the heat transfer path under real working conditions, and can fully adapt to the nonlinear changes in the thermal properties of the two-phase materials with temperature fluctuations.

[0102] The comparison of temperature field distributions reveals that traditional constant thermal property modeling methods exhibit significant heat transfer distortion across a wide temperature range: room-temperature fixed property models overestimate the thermal insulation performance in high-temperature regions, while high-temperature fixed property models underestimate the heat transfer performance in low-temperature regions. Furthermore, the internal temperature stratification uniformity is poor, with significant localized high-temperature concentrations, easily inducing potential thermal stress concentrations within the structure. In contrast, the temperature-dependent thermal property model of this invention provides a smoother and more uniform temperature field distribution, effectively narrows the high-temperature region across the entire structure, significantly reduces the temperature gradient along the path, and achieves a high degree of accuracy in temperature field simulation solutions that closely match the actual service conditions of the structure.

[0103] The comparative results above verify the necessity of introducing temperature-related thermal property modeling in this invention: traditional constant material property assumptions cannot adapt to service conditions with large temperature gradients and wide temperature ranges, which can easily lead to distortion in heat transfer analysis and material layout obtained by in-plane optimization deviating from engineering reality; this invention, by establishing a functional relationship between the thermal properties of the matrix material and temperature, combined with in-plane composition optimization strategies, can solve for the optimal in-plane material distribution that fits the actual working conditions, effectively smoothing the internal temperature gradient of the structure, suppressing local high-temperature concentration defects, and significantly improving the overall service reliability of functionally graded thermal protection structures.

[0104] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0105] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0106] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0107] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the specific implementation of the present invention. Any modifications or equivalent substitutions that do not depart from the spirit and scope of the present invention should be covered within the scope of protection of the claims of the present invention.

Claims

1. A method for optimizing the structural composition of functionally graded materials based on temperature-dependent properties, characterized in that, Includes the following steps: Establish a structural design domain that can withstand thermal loads with large temperature gradients, and construct a heat conduction analysis model within the design domain; At least two matrix materials with different thermal conductivity are selected, and a material volume fraction variable is introduced in the design domain to describe the volume fraction distribution of one of the matrix materials at a spatial location. The volume fraction of the other matrix material is the complement of the material volume fraction variable. Based on experimental data or material databases, establish a functional model of the thermophysical parameters of the matrix material as a function of temperature; Based on the material volume fraction variable and the temperature-related thermophysical parameter function model, the equivalent thermophysical parameters that change with temperature at any location within the design domain are obtained using a multiphase material interpolation method. Considering the equivalent thermal property parameters, the thermal conduction control equation of the structure is established and solved to obtain the temperature field distribution of the structure under given thermal boundary conditions. Using structural thermal performance as the optimization objective and the material volume fraction variable as the design variable, a heat transfer optimization model is constructed. Under preset constraints, a two-dimensional collaborative optimization solution is performed to obtain the optimal material distribution of the functionally graded structure. The two-dimensional collaborative optimization includes in-plane topology optimization and thickness-direction parameterized gradient optimization.

2. The method for optimizing the structural composition of functionally graded materials based on temperature-related properties according to claim 1, characterized in that, The in-plane topology optimization includes: minimizing the highest temperature within a specified region of the structure, introducing a p-norm function to smoothly approximate the highest temperature; obtaining a smoothed design variable field through a density filtering model based on Helmholtz-type partial differential equations; and obtaining the optimal in-plane material volume fraction distribution by iteratively updating the design variables within the design variable field until the convergence condition is met.

3. The method for optimizing the structural composition of functionally graded materials based on temperature-related properties according to claim 2, characterized in that, The in-plane topology optimization process is subject to volume fraction constraints and temperature distribution uniformity indices; the volume fraction constraints are as follows: Among them, C max This is the preset upper limit of material volume fraction. This represents the smoothed material volume fraction. Design domain volume; The temperature distribution uniformity index is: Among them, Ω h For the structural design domain, |Ω h | represents the area of ​​the design domain; T(x) represents the local temperature at any location x within the design domain; T represents the average temperature within the design domain, C. T,max This is the preset upper limit for temperature variance.

4. The method for optimizing the structural composition of functionally graded materials based on temperature-related properties according to claim 2, characterized in that, The convergence condition is: Where k is the number of iterations, J is the optimization objective function defined based on the temperature field, and ε is the preset convergence threshold.

5. The method for optimizing the structural composition of functionally graded materials based on temperature-related properties according to claim 1, characterized in that, The thickness-direction parameterized gradient optimization includes: optimizing the thickness-direction parameterized gradient using a normalized global one-dimensional gradient function related to thickness, wherein the objective function is: ； Where θ is the parameter to be optimized for the gradient function in the thickness direction. This represents the equivalent temperature after averaging the temperature field in the XOY plane at the thickness location z, where H is the total thickness of the structure, and C... z This is the integral of the square of the temperature gradient along the thickness.

6. The method for optimizing the structural composition of functionally graded materials based on temperature-related properties according to claim 5, characterized in that, During the optimization process, at any planar location (x, y), the integral average of the material volume fraction along the thickness direction is equal to the volume fraction c(x, y) obtained at that location through planar optimization. The mathematical expression is: Where c(x,y,z) is the volume fraction of the material at any position in three-dimensional space.

7. The method for optimizing the structural composition of functionally graded materials based on temperature-related properties according to claim 6, characterized in that, During the optimization process, c(x,y) remains unchanged, and the expression for the final material volume fraction at any spatial location (x,y,z) of the three-dimensional structure is: c A (x,y,z)=c(x,y)·c(z) c B (x,y,z)=1 c A (x,y,z) Among them, c A (x,y,z) represents the volume fraction of material A, and c B (x,y,z) represents the volume fraction of material B, and c(z) is the thickness gradient function that satisfies the normalization condition.

8. The method for optimizing the structural composition of functionally graded materials based on temperature-related properties according to claim 1, characterized in that, The multiphase material interpolation method is the volume averaging method. The equivalent thermal conductivity keff and equivalent specific heat capacity cp,eff at any location within the design domain are expressed as follows: k eff (T,x)=c(x)k1(T)+[1-c(x)]k2(T) Where k1(T) and k2(T) represent the functions of thermal conductivity of the two matrix materials as a function of temperature; c p,eff (T,x)=c(x)c p,1 (T)+[1-c(x)]c p,2 (T) Among them, c p,1 (T) and c p,2 (T) represents the specific heat capacity of the two matrix materials as a function of temperature; c(x) is the volume fraction distribution of a matrix material at a spatial location.

9. The method for optimizing the structural composition of functionally graded materials based on temperature-related properties according to claim 1, characterized in that, The heat conduction control equations are divided into heat conduction control equations under steady-state conditions and heat conduction control equations under transient conditions. The heat conduction control equation under the steady-state condition is: Where T(x) is the temperature field distribution function, and q(x) is the volumetric heat source term. The equivalent thermal conductivity at any location within the design domain; Under the aforementioned transient condition, the heat conduction control equation is: Where ρ(x) is the equivalent density of the structure at position x. Let be the temperature field distribution function at time t. Let be the volumetric heat source term at time t.

10. A method for optimizing the structural composition of functionally graded materials based on temperature-related properties according to any one of claims 1-9, characterized in that, The method further includes numerical or experimental verification of the functionally graded structure based on the optimization results. The numerical verification includes comparing and analyzing the temperature field results obtained by using a constant material thermophysical parameter model and a temperature-dependent material thermophysical parameter model. The experimental verification includes measuring the surface or internal temperature field of the functionally graded structure sample using infrared thermography under high-temperature heat flux loading conditions.

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