A space-time coupled method for anisotropic thermal analysis of functionally graded materials
Patent Information
- Application Number
- CN202610846245.9
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2026-06-12
- Publication Date
- 2026-10-20
AI Technical Summary
然而，对于瞬态热传导问题，传统广义有限差分法仍然依赖时间推进格式（如显式或隐式时间步进）
[0039]与现有技术相比，本发明将时间作为额外坐标维度，将瞬态热传导问题转化为时空联合域上的边界值问题，一次性求解全部时空点的温度值，从根本上消除了传统时间步进法的误差累积和稳定性限制；采用无网格点云离散，无需生成网格，能够直接处理复杂几何边界；空间变化的密度、比热容及各向异性导热系数张量被自然纳入离散格式，特别适合功能梯度材料的连续梯度特征；数据点与配点解耦，求解后可直接输出任意时空位置的温度值，无需后处理插值；可采用二阶、三阶或四阶精度格式，具有较大的灵活性，为功能梯度材料构件的热设计及热可靠性评估提供了高精度、强鲁棒的数值分析手段。
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Abstract
Description
Technical Field
[0001] This invention belongs to the field of thermal analysis technology of functionally graded materials, and relates to a spatiotemporal joint method for anisotropic thermal analysis of functionally graded materials. Background Technology
[0002] Functionally graded materials (FJCTs) are advanced composite materials whose composition and microstructure vary continuously in space. Their thermophysical properties (such as density, specific heat capacity, and thermal conductivity) transition smoothly with location, making them widely used in high-tech fields such as aerospace thermal barrier coatings, electronic packaging heat dissipation components, and biomedical implants. Because parameters such as thermal conductivity within FJCTs are spatially continuous and often exhibit anisotropic thermal conductivity in actual service environments, their heat conduction process is controlled by partial differential equations with spatially dependent coefficients, making analytical solutions extremely difficult and typically requiring numerical methods for analysis.
[0003] Currently, numerical methods for heat conduction problems in functionally graded materials (FGMs) mainly include the finite element method (FEM) and the finite difference method (FD). The FEM can handle complex geometries, but to accurately capture the continuous gradient changes in material properties, it often requires extremely fine meshes, resulting in high computational costs. The FD is typically limited to regular meshes, making it difficult to adapt to the complex boundary shapes and anisotropic thermal conduction behaviors commonly found in FGMs. Meshless methods developed in recent years, such as the generalized finite difference method (GFD), avoid mesh generation by constructing derivative schemes through weighted least-squares approximation on local point clouds, demonstrating advantages in handling irregular regions and variable coefficient problems. However, for transient heat conduction problems, the traditional GFD still relies on time-progression schemes (such as explicit or implicit time steps). Explicit schemes are constrained by stability conditions, requiring very small time steps; implicit schemes, while more stable, require iterative solutions at each time step. More importantly, time-progression methods gradually accumulate errors during long simulations, affecting computational accuracy. Furthermore, existing methods typically require solving for all spatiotemporal points first, and then interpolating to obtain the value at the point of interest, failing to directly provide the temperature response at arbitrary spatiotemporal locations. Therefore, a novel numerical method is urgently needed that can overcome these shortcomings and is applicable to the anisotropic transient heat conduction analysis of functionally graded materials. Summary of the Invention
[0004] 1. The technical problem to be solved:
[0005] How to directly solve transient anisotropic heat conduction problems without relying on traditional time-progression schemes, avoid error accumulation, and adapt to complex geometric boundaries, spatially varying material parameters, and anisotropic thermal conductivity characteristics.
[0006] 2. Technical Solution:
[0007] To address the above problems, this invention provides a spatiotemporal joint method for anisotropic thermal analysis of functionally graded materials, comprising the following steps:
[0008] Step 1: Obtain the spatial domain, time domain, spatially varying material parameters, and anisotropic thermal conductivity tensor of the functionally graded material component.
[0009] Step 2: Unify the spatial and temporal domains into a single structure. dimensional spatiotemporal joint domain Data point sets are respectively set within the spatiotemporal joint domain. and matching dot set ,in For the internal collocation set, Provide a set of points on the boundary that satisfy... , The dimension of the spatiotemporal joint domain; These are the internal collocation sets. Boundary collocation set Middle, data point set The points used for calculation.
[0010] Step 3: For each collocation point From the data point set Selected from The nearest neighbors form a local star set. Let the order of the Taylor expansion be p, where p takes the value 2, 3, or 4, corresponding to second-order, third-order, and fourth-order precision schemes, respectively. Then, construct the partial derivatives of each order based on the weighted least squares method. linear combination coefficients ,satisfy:
[0011] ,
[0012] In the formula, Let α represent the α-order partial derivative of the temperature field T at ξc with respect to the spacetime coordinates.
[0013] Step 4: Using the aforementioned linear combination coefficients, discretize the anisotropic heat conduction control equations and boundary conditions of the functionally graded material into a globally sparse linear system. ,
[0014] In the formula, This represents the global sparse coefficient matrix generated by the discrete scheme; This represents a column vector consisting of the unknown temperature values at all data points. It is a column vector of right-hand terms, whose components include internal heat source terms, boundary conditions, and initial conditions.
[0015] Step 5: Solve the global sparse linear system To obtain all spatiotemporal data points The temperature value is directly output as the thermal response result at any time and space location.
[0016] In step 1, the spatially varying material parameters include density, specific heat capacity, and thermal conductivity tensor, wherein the thermal conductivity tensor is a symmetric positive definite matrix whose components change continuously with spatial position, used to characterize the anisotropic thermal conductivity of functionally graded materials.
[0017] In step 3, the local star set The selection is based on the Euclidean distance in the spatiotemporal joint domain, and the diagonal weight matrix Weights in A function that monotonically decreases with increasing distance, including spline functions, exponential functions, or radial basis functions.
[0018] In step 3, the solution formula for the weighted least squares method is:
[0019] ,
[0020] In the formula, For the Vandermonde matrix, It is a diagonal weight matrix. Indicates the location of points The temperature function Its own value and all of them until A column vector composed of the partial derivatives of each order of the order of the first element. It is the Vandermonde matrix. Represents a local point set A column vector consisting of the temperature function values at each data point.
[0021] In step 4, for internal collocation points The discrete form of the governing equations is:
[0022] ,
[0023] In the formula, This represents the product of density and specific heat capacity at the collocation point. Indicates the summation index, This represents the set of data point indices for a local point set. , , This represents the combination coefficients of the derivatives obtained by the weighted least squares method. The components representing the thermal conductivity tensor. Representing data points Temperature value at that location, This represents the heat source term at the spatiotemporal coordinate point.
[0024] For boundary collocation points The discrete form of the boundary conditions is:
[0025] Dirichlet Boundary:
[0026]
[0027] ,
[0028] In the formula, Indicates the point at the spacetime boundary The given temperature value.
[0029] Neumann boundary:
[0030] ,
[0031] In the formula, Indicates the point at the spacetime boundary The given heat flux magnitude (i.e., the known value of the normal heat flux density in the Neumann boundary condition). Denotes the first unit vector of the outward normal at the boundary point. Each component.
[0032] Third type of boundary:
[0033] ,
[0034] In the formula, This represents the convective heat transfer coefficient at the boundary point.
[0035] For the initial plane The collocation of points on the coordinate plane, in discrete form, is:
[0036] ,
[0037] In the formula, This represents the initial temperature field.
[0038] 3. Beneficial effects:
[0039] Compared with existing technologies, this invention uses time as an additional coordinate dimension, transforming the transient heat conduction problem into a boundary value problem in a spatiotemporal joint domain. It solves the temperature values of all spatiotemporal points in one go, fundamentally eliminating the error accumulation and stability limitations of traditional time-stepping methods. It uses meshless point cloud discretization, eliminating the need for mesh generation and enabling direct handling of complex geometric boundaries. Spatially varying density, specific heat capacity, and anisotropic thermal conductivity tensors are naturally incorporated into the discretization scheme, making it particularly suitable for the continuous gradient characteristics of functionally graded materials. Data points and collocation points are decoupled, allowing direct output of temperature values at any spatiotemporal location after solving, without the need for post-processing interpolation. It can use second-, third-, or fourth-order precision schemes, providing great flexibility and offering a high-precision and robust numerical analysis method for the thermal design and thermal reliability assessment of functionally graded material components. Attached Figure Description
[0040] Figure 1 This is a flowchart of the present invention.
[0041] Figure 2 This is a schematic diagram of the point set distribution and local star set of the spatiotemporal joint method provided by the present invention in the 1D space + time domain, wherein (a) is a global data point and matching point distribution diagram, and (b) is a schematic diagram of local star set selection.
[0042] Figure 3 This is a schematic diagram of the geometric model of the I-shaped functional graded material component used in the invention implementation case.
[0043] Figure 4 The temperature field distribution cloud map is obtained by solving the anisotropic heat conduction problem of I-shaped functionally graded material components using the method of the present invention in the embodiments of the invention.
[0044] Figure 5 This is a diagram showing the absolute error distribution between the temperature field obtained by the method of this invention and the analytical solution in an embodiment of this invention. Detailed Implementation
[0045] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.
[0046] like Figure 1 As shown, a spatiotemporal joint method for anisotropic thermal analysis of functionally graded materials includes the following steps:
[0047] Step 1: Obtain the spatial domain, time domain, spatially varying material parameters, and anisotropic thermal conductivity tensor of the functionally graded material component.
[0048] Step 2: Unify the spatial and temporal domains into a single structure. dimensional spatiotemporal joint domain Data point sets are respectively set within the spatiotemporal joint domain. and matching dot set ,in For the internal collocation set, Provide a set of points on the boundary that satisfy... , The dimension of the spatiotemporal joint domain; These are the internal collocation sets. Boundary collocation set Middle, data point set The points used for calculation.
[0049] Step 3: As Figure 2 As shown, the spatial and temporal domains are first unified into a spatiotemporal joint domain. Within this joint domain, a data point set and a collocation point set are established. For each collocation point, a local star-shaped set is selected from its neighboring data points. Linear combination coefficients of the derivative are constructed based on the weighted least squares method. Specifically, for each collocation point… From the data point set Selected from The nearest neighbors form a local star set. Let the order of the Taylor expansion be p, where p takes the value 2, 3, or 4, corresponding to second-order, third-order, and fourth-order precision schemes, respectively. Then, construct the partial derivatives of each order based on the weighted least squares method. linear combination coefficients ,satisfy:
[0050] ,
[0051] In the formula, in the formula, This represents the α-order partial derivative of the temperature field T at ξc with respect to the spacetime coordinates.
[0052] Step 4: Using the aforementioned linear combination coefficients, discretize the anisotropic heat conduction control equations and boundary conditions of the functionally graded material into a globally sparse linear system. ,
[0053] In the formula, This represents the global sparse coefficient matrix generated by the discrete scheme; This represents a column vector consisting of the unknown temperature values at all data points. It is a column vector of terms on the right side, whose components include internal heat source terms, boundary conditions, and initial conditions.
[0054] Step 5: Solve the global sparse linear system To obtain all spatiotemporal data points The temperature value is directly output as the thermal response result at any time and space location.
[0055] Example
[0056] For example Figure 3Taking the functionally graded material component with an I-shaped cross-section as an example, a numerical simulation of its temperature field in a transient anisotropic heat conduction problem is performed. The I-shaped cross-section consists of three parts: the upper flange region... Web region and the lower wing edge area The component is made of ceramic-metal gradient composite material, and its material parameters change continuously with spatial position.
[0057] Step 1: Obtain the spatial domain of functionally graded material components Time domain Spatially varying material parameters and anisotropic thermal conductivity tensor And obtain the thermal boundary conditions and initial conditions.
[0058] In this embodiment, the spatial domain For the above-mentioned I-shaped cross-section, the time domain is taken as... Material parameters are set as follows: The thermal conductivity tensor is in orthotropic form:
[0059] ;
[0060] heat source item Based on the manufacturing analytical solution settings, the analytical solution is taken as:
[0061]
[0062] Initial conditions: .
[0063] The boundary conditions are of type Robin: ,in Derived from the analytical solution, specifically at different boundaries:
[0064] ,
[0065] The symbol is selected based on the outward normal direction.
[0066] Step 2: Spatial Domain With the time domain Unified construction as dimensional spatiotemporal joint domain Data point sets are respectively set within the spatiotemporal joint domain. and matching dot set .
[0067] In this embodiment, Spatiotemporal joint domain Generate a non-uniformly distributed set of data points within the I-shaped domain. A total of approximately 19,000 points are generated; a set of matching points is generated at the boundaries and within the spatiotemporal domain. There are approximately 20,000 points in total, ensuring appropriate densification in geometrically singular regions such as concave corners, and satisfying [certain conditions]. .
[0068] Step 3: For each collocation point From the data point set Selecting neighboring points to form a local star set Constructing partial derivatives of each order based on weighted least squares method linear combination coefficients .
[0069] This embodiment uses a fourth-order Taylor expansion ( ), number of local star-shaped set points The weighting function uses a cubic spline function:
[0070]
[0071] in For spacetime distance, Let the radius be the local support radius. Solve the weighted least squares problem:
[0072]
[0073] The combination coefficients of partial derivatives of each order are obtained. ,satisfy:
[0074] .
[0075] Step 4: Using the aforementioned linear combination coefficients, discretize the anisotropic heat conduction control equations and boundary conditions of the functionally graded material into a globally sparse linear system. .
[0076] Governing equations:
[0077]
[0078] Internal distribution points The discrete form of it is:
[0079]
[0080] Set up points at the Robin border. The discrete form of it is:
[0081]
[0082] The collocation of points on the initial plane satisfies:
[0083]
[0084] Step 5: Solve the global sparse linear system This allows us to obtain the temperature values of all spatiotemporal data points and directly output the thermal response results at any spatiotemporal location.
[0085] Because the system is overdetermined ( The least squares method is used to solve this problem. Calculate the results separately. Temperature fields at two different moments. Figure 4 shows... Temperature field distribution at time, Figure 5 The corresponding absolute error distribution is given. The results show that, except for the area near the concave corner of the I-shaped cross-section where the error is slightly larger due to geometric singularities (approximately...), the error is... Apart from that, the error throughout the entire domain is within... Below. Quantitatively, relatively The error is: internal spacetime point The final moment for .
[0086] This embodiment fully demonstrates the superior performance of the method of the present invention in handling non-convex complex geometries, spatially varying anisotropic materials, non-zero heat sources, and Robin boundary conditions, and exhibits no error accumulation during long-term simulations. For functionally graded components with complex geometries, the accurate prediction of the temperature field by the present invention can effectively assess the thermal stress distribution, guide thermal protection design, and increase the service life of engineering components.
Claims
1. A spatiotemporal joint method for anisotropic thermal analysis of functionally graded materials, characterized in that: Includes the following steps: Step 1: Obtain the spatial domain, time domain, spatially varying material parameters, and anisotropic thermal conductivity tensor of the functionally graded material component; Step 2: Unify the spatial and temporal domains into a single structure. dimensional spatiotemporal joint domain Data point sets are respectively set within the spatiotemporal joint domain. and matching dot set ,in For the internal collocation set, Provide a set of points on the boundary that satisfy... , The dimension of the spatiotemporal joint domain; These are the internal collocation sets. Boundary collocation set Middle, data point set The points used for calculation; Step 3: For each collocation point From the data point set Selected from The nearest neighbors form a local star set. Let the order of the Taylor expansion be p, where p takes the value 2, 3, or 4, corresponding to second-order, third-order, and fourth-order precision schemes, respectively. Then, construct the partial derivatives of each order based on the weighted least squares method. linear combination coefficients ,satisfy: ， In the formula, This represents the α-th order partial derivative of the temperature field T at ξc with respect to the spacetime coordinates; Step 4: Using the aforementioned linear combination coefficients, discretize the anisotropic heat conduction control equations and boundary conditions of the functionally graded material into a globally sparse linear system. , In the formula, This represents the global sparse coefficient matrix generated by the discrete scheme; This represents a column vector consisting of the unknown temperature values at all data points. It is a column vector of terms on the right side, whose components include internal heat source terms, boundary conditions, and initial conditions; Step 5: Solve the global sparse linear system To obtain all spatiotemporal data points The temperature value is directly output as the thermal response result at any time and space location.
2. The spatiotemporal joint method for anisotropic thermal analysis of functionally graded materials as described in claim 1, characterized in that: In step 1, the spatially varying material parameters include density, specific heat capacity, and thermal conductivity tensor, wherein the thermal conductivity tensor is a symmetric positive definite matrix whose components change continuously with spatial position, used to characterize the anisotropic thermal conductivity of functionally graded materials.
3. The spatiotemporal joint method for anisotropic thermal analysis of functionally graded materials as described in claim 2, characterized in that: In step 3, the local star set The selection is based on the Euclidean distance in the spatiotemporal joint domain, and the diagonal weight matrix Weights in A function that monotonically decreases with increasing distance, including spline functions, exponential functions, or radial basis functions.
4. The spatiotemporal joint method for anisotropic thermal analysis of functionally graded materials as described in claim 3, characterized in that: In step 3, the solution formula for the weighted least squares method is: ， In the formula, For the Vandermonde matrix, It is a diagonal weight matrix. Indicates the location of points The temperature function Its own value and all of them until A column vector composed of the partial derivatives of each order of the order of the first element. It is the Vandermonde matrix. Represents a local point set A column vector consisting of the temperature function values at each data point.
5. The spatiotemporal joint method for anisotropic thermal analysis of functionally graded materials as described in claim 4, characterized in that: In step 4, for internal collocation points The discrete form of the governing equations is: ， In the formula, This represents the product of density and specific heat capacity at the collocation point. Indicates the summation index, This represents the set of data point indices for a local point set. , , This represents the combination coefficients of the derivatives obtained by the weighted least squares method. The components representing the thermal conductivity tensor. Representing data points Temperature value at that location, This represents the heat source term at the spatiotemporal coordinate point.
6. The spatiotemporal joint method for anisotropic thermal analysis of functionally graded materials as described in claim 5, characterized in that: For boundary collocation points The discrete form of the boundary conditions is: Dirichlet Boundary: ， In the formula, Indicates the point at the spacetime boundary The given temperature value; Neumann boundary: ， In the formula, Indicates the point at the spacetime boundary Given the magnitude of the heat flux, Denotes the first unit vector of the outward normal at the boundary point. Each component. Third type of boundary: ， In the formula, This represents the convective heat transfer coefficient at the boundary point; For the initial plane The collocation of points on the coordinate plane, in discrete form, is: ， In the formula, This represents the initial temperature field.