A numerical analysis method for transient heat transfer stress coupling of three-dimensional layered semi-infinite foundation

Abstract

A numerical analysis method for transient heat transfer and stress coupling in three-dimensional layered semi-infinite foundations is proposed. This method involves dividing the semi-infinite domain into near-field and far-field regions, establishing a coordinate transformation based on the geometric similarity between similar surfaces and boundary surfaces, and forming a scaled boundary finite element coordinate system. Based on variational and virtual work principles, the governing equations for heat conduction and thermal stress are derived in the SBFEM coordinate system, and a dimensionless heat conduction matrix in the frequency domain is established. The heat conduction matrix is ​​solved using continuous fractional expansion and Schul decomposition, and time-domain integration is achieved through an improved refined time history integration method (MPTSIM). Within a sequential coupling framework, the transient temperature field is first solved, and then applied as a thermal load to the mechanical equilibrium equations to obtain the displacement and stress fields. This method significantly reduces computational scale and resource consumption while maintaining high accuracy, and is suitable for the efficient analysis of transient thermo-mechanical coupling problems in three-dimensional layered semi-infinite foundations.

Description

Technical Field

[0001] This invention relates to the field of transient heat transfer stress in three-dimensional layered semi-infinite foundations, and particularly to a coupled numerical analysis method for transient heat transfer stress in three-dimensional layered semi-infinite foundations. Background Technology

[0002] In engineering practice, the analysis of transient heat conduction and thermal stress coupling in three-dimensional layered semi-infinite foundations is a key challenge. Temperature changes in semi-infinite media such as deep rock masses and layered foundations induce complex and time-evolving stress fields, directly affecting the stability, durability, and safety of engineering structures. Accurately predicting this thermo-mechanical coupling behavior is crucial for fields such as geothermal resource development, nuclear waste geological disposal, and large-scale foundation design. However, traditional numerical methods often face the challenge of balancing computational accuracy and efficiency when simulating complex problems involving unbounded domains, material heterogeneity, and transient processes.

[0003] For the thermo-mechanical coupling problem of three-dimensional layered semi-infinite foundations, numerical and analytical methods are the main research approaches. Numerical methods, such as the finite element method, can handle complex geometries, material nonlinearities, and transient processes, but they require discretization of the entire computational domain. When dealing with semi-infinite domains, it is often necessary to introduce artificial boundaries for truncation, which can not only introduce errors but also significantly increase the model size, leading to high computational costs and reduced efficiency. On the other hand, while analytical methods can provide accurate solutions and are computationally efficient, they usually rely on highly simplified and idealized assumptions, such as homogeneous materials, regular geometries, and simple boundary conditions, limiting their applicability to layered, heterogeneous media and complex thermal loads commonly encountered in practical engineering. Therefore, developing a computational method that can efficiently and accurately analyze the transient thermo-mechanical coupling response in three-dimensional layered semi-infinite foundations is urgently needed and of significant value. Summary of the Invention

[0004] The present invention aims to provide an efficient numerical analysis method for the coupling of transient heat conduction and thermal stress in a three-dimensional layered semi-infinite foundation, so as to overcome the limitations of existing methods in terms of insufficient accuracy and low computational efficiency when dealing with heterogeneous half-space problems.

[0005] To achieve the above objectives, this invention provides a numerical analysis method for transient heat transfer and stress coupling in a three-dimensional layered semi-infinite foundation, the method comprising the following steps:

[0006] S1. Based on the geometric similarity between similar surfaces and boundary surfaces, establish the coordinate representation of any point in the three-dimensional layered semi-infinite foundation. Perform the same discretization on similar surfaces and boundary surfaces. Establish the coordinate transformation from the Cartesian coordinate system to the scaled boundary finite element SBFEM coordinate system through the Jacobian matrix. Further calculate the inverse of the Jacobian matrix and obtain the differential operators for the heat conduction problem and the thermal stress problem. Obtain the temperature and temperature gradient representation of any point in the three-dimensional layered semi-infinite foundation in the SBFEM coordinate system based on the temperature interpolation shape function.

[0007] S2. Construct a total energy functional that includes heat dissipation potential energy and heat storage potential energy. By taking the first variation of the total energy functional and setting it to zero, derive the nonlinear first-order ordinary differential equation with respect to temperature, and obtain the temperature field control equation in the SBFEM coordinate system. Introduce dimensionless frequency and dimensionless heat conduction matrix, and differentiate the heat conduction matrix with respect to radial coordinate and frequency respectively. Obtain the control equation with respect to the heat conduction matrix based on the relationship between the derivatives. Through geometric equations and constitutive relations, and considering the influence of thermal strain, derive the control equation of the displacement field in the SBFEM coordinate system.

[0008] S3. Setting the frequency in the heat conduction governing equation to zero, we obtain the governing equation for the steady-state heat conduction problem. Using Schur decomposition, we express the governing equation as an eigenvalue problem, thereby solving for the radial coordinates. The steady-state heat conduction matrix approaches infinity; the frequency is transformed into a normalized form, and the heat conduction governing equation is transformed into a form similar to the elastic dynamics equation. The transient heat conduction matrix is ​​expanded using the continuous fractional method, and the function with respect to the normalized frequency is obtained by arranging the equations in descending order of powers. This process is then recursively repeated to obtain... The resulting expression for the transient heat conduction matrix approaching infinity.

[0009] S4. By using inverse Fourier transform, the heat flow-temperature relationship in the frequency domain is transformed into a relationship in the time domain, resulting in a standard time-domain equation. Based on the general solution of the equation, the relationship between the temperatures corresponding to adjacent time points is obtained. An improved fine integration method is used to obtain the exact solution of the exponential matrix. Taylor expansion is performed on the exponential matrix within the time point, and numerical loops are executed to obtain the solution of the homogeneous term. Gaussian integration is used to solve the non-homogeneous term. Finally, the solutions of the homogeneous term and the non-homogeneous term are superimposed to obtain the temperature value of each time point.

[0010] S5. Based on the temperature values ​​obtained at each time point, derive the expression for the rate of temperature change and express the coupled equation of the thermo-mechanical system in a block matrix form; obtain the temperature value at each time point in the entire domain through the temperature interpolation shape function, and then apply the temperature value as a load to the mechanical equilibrium equation, and solve the displacement field and thermal stress field according to the constitutive relation.

[0011] The beneficial effects of this invention are:

[0012] (1) This invention introduces a novel coordinate transformation based on geometric similarity to construct a similarity surface for the proportional boundary finite element method, transforming the three-dimensional semi-infinite domain problem into a combination of two-dimensional discretization on the boundary and analytical solution in the radial direction. While ensuring simulation accuracy, it significantly reduces computational dimension and resource consumption.

[0013] (2) The present invention uses continuous fraction expansion to construct the heat conduction matrix and combines the improved fine time history integration method to solve the time domain, realizing high-precision and high-efficiency decoupling calculation of the transient heat conduction and thermal stress coupling system. Under the premise of obtaining similar accuracy as the traditional finite element method, the computational degree of freedom and CPU time are greatly reduced.

[0014] (3) This invention has been successfully applied to the thermo-mechanical coupling analysis of layered semi-infinite domains through the verified numerical framework, revealing the dominant role of surface mechanical constraints on thermal stress distribution, clarifying the intrinsic trade-off mechanism between stress field and displacement field, and providing an effective simulation tool for thermo-mechanical coupling design and analysis in complex layered media. Attached Figure Description

[0015] Figure 1 This is a flowchart illustrating the numerical analysis method for transient heat transfer and stress coupling in a three-dimensional layered semi-infinite foundation according to the present invention.

[0016] Figure 2 for Figure 1 The schematic diagram shown below;

[0017] Figure 3 This is a schematic diagram of the computational model in the first embodiment;

[0018] Figure 4 This is a schematic diagram of the substructure division and unit discretization in the first embodiment;

[0019] Figure 5 This is a schematic diagram of the computational model in the second embodiment;

[0020] Figure 6 This is a schematic diagram of the substructure division and unit discretization in the second embodiment;

[0021] Figure 7 for Figure 4 Cross-sectional schematic diagram of similar surfaces and boundary surfaces;

[0022] Figure 8 for Figure 6 Element mapping between similar surfaces and boundary surfaces. Detailed Implementation

[0023] It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. The realization of the invention's objectives, functional characteristics, and advantages will be further explained in conjunction with the embodiments and with reference to the accompanying drawings.

[0024] As shown in the attached diagram Figure 1 This is a flowchart illustrating the numerical analysis method for transient heat transfer and stress coupling in a three-dimensional layered semi-infinite foundation according to the present invention. Figure 2 for Figure 1 The schematic diagram shown below; Figure 3 This is a schematic diagram of the computational model in the first embodiment; Figure 4 This is a schematic diagram of the substructure division and unit discretization in the first embodiment; Figure 5 This is a schematic diagram of the computational model in the second embodiment; Figure 6 This is a schematic diagram of the substructure division and unit discretization in the second embodiment; Figure 7 for Figure 4 Cross-sectional schematic diagram of similar surfaces and boundary surfaces; Figure 8 for Figure 6 Element mapping between similar surfaces and boundary surfaces. The detailed method includes the following steps:

[0025] S1. Based on the geometric similarity between similar surfaces and boundary surfaces, establish the coordinate representation of any point in the three-dimensional layered semi-infinite foundation. Perform the same discretization on similar surfaces and boundary surfaces. Establish the coordinate transformation from the Cartesian coordinate system to the scaled boundary finite element SBFEM coordinate system through the Jacobian matrix. Further calculate the inverse of the Jacobian matrix and obtain the differential operators for the heat conduction problem and the thermal stress problem. Obtain the temperature and temperature gradient representation of any point in the three-dimensional layered semi-infinite foundation in the SBFEM coordinate system based on the temperature interpolation shape function.

[0026] S2. Construct a total energy functional that includes heat dissipation potential energy and heat storage potential energy. By taking the first variation of the total energy functional and setting it to zero, derive the nonlinear first-order ordinary differential equation with respect to temperature, and obtain the temperature field control equation in the SBFEM coordinate system. Introduce dimensionless frequency and dimensionless heat conduction matrix, and differentiate the heat conduction matrix with respect to radial coordinate and frequency respectively. Obtain the control equation with respect to the heat conduction matrix based on the relationship between the derivatives. Through geometric equations and constitutive relations, and considering the influence of thermal strain, derive the control equation of the displacement field in the SBFEM coordinate system.

[0027] S3. Setting the frequency in the heat conduction governing equation to zero, we obtain the governing equation for the steady-state heat conduction problem. Using Schur decomposition, we express the governing equation as an eigenvalue problem, thereby solving for the radial coordinates. The steady-state heat conduction matrix approaches infinity; the frequency is transformed into a normalized form, and the heat conduction governing equation is transformed into a form similar to the elastic dynamics equation. The transient heat conduction matrix is ​​expanded using the continuous fractional method, and the function with respect to the normalized frequency is obtained by arranging the equations in descending order of powers. This process is then recursively repeated to obtain... The resulting expression for the transient heat conduction matrix approaching infinity.

[0028] S4. By using inverse Fourier transform, the heat flow-temperature relationship in the frequency domain is transformed into a relationship in the time domain, resulting in a standard time-domain equation. Based on the general solution of the equation, the relationship between the temperatures corresponding to adjacent time points is obtained. An improved fine integration method is used to obtain the exact solution of the exponential matrix. Taylor expansion is performed on the exponential matrix within the time point, and numerical loops are executed to obtain the solution of the homogeneous term. Gaussian integration is used to solve the non-homogeneous term. Finally, the solutions of the homogeneous term and the non-homogeneous term are superimposed to obtain the temperature value of each time point.

[0029] S5. Based on the temperature values ​​obtained at each time point, derive the expression for the rate of temperature change and express the coupled equation of the thermo-mechanical system in a block matrix form; obtain the temperature value at each time point in the entire domain through the temperature interpolation shape function, and then apply the temperature value as a load to the mechanical equilibrium equation, and solve the displacement field and thermal stress field according to the constitutive relation.

[0030] Furthermore, in step S1, based on the geometric similarity between the similar surface and the boundary surface, a coordinate representation of any point in the three-dimensional layered semi-infinite foundation is established, specifically as follows:

[0031]

[0032] in, , , Let be the coordinates of any point in a three-dimensional layered semi-infinite foundation. , , Let these be the coordinates of points on similar surfaces. , , Let these be the coordinates of a point on the boundary surface. For radial coordinates.

[0033] Similar surfaces and boundary surfaces are discretized in the same way:

[0034]

[0035] in, , , The node coordinate vectors of the element after the boundary surface is discretized. , , Let be the nodal coordinate vector of the element after discretization of the similar surface. Let be the shape function of the surface element. , For circumferential coordinates.

[0036] The coordinate transformation from the Cartesian coordinate system to the SBFEM coordinate system is established using the Jacobian matrix, where the expression for the Jacobian matrix is:

[0037] in, For Jacobian matrices, subscripts are... , Representing the circumferential coordinates respectively , Find the first derivative.

[0038] Calculate the inverse of the Jacobian matrix:

[0039]

[0040] Where the superscript -1 denotes the inverse of the matrix, Represents the determinant of a Jacobian matrix; subscript Represents radial coordinates Find the first derivative; Let represent all vectors that represent the Jacobian inverse matrix.

[0041] The differential operator for the heat conduction problem is obtained, and its expression is:

[0042]

[0043] in, , , They represent respectively to , , Find the partial derivative; , , They represent respectively to ξ , η , ζ Find the partial derivative; the superscript T denotes the transpose of the vector; Differential operators representing heat conduction problems; , , The sub-coefficient matrix representing the heat conduction problem is expressed as follows:

[0044]

[0045] The differential operator for the thermal stress problem is obtained as follows:

[0046]

[0047] in, Differential operators representing thermal stress problems; , , The sub-coefficient matrix representing the thermal stress problem is expressed as follows:

[0048] .

[0049] The temperature and temperature gradient at any point in the SBFEM coordinate system are obtained based on the temperature interpolation shape function:

[0050]

[0051]

[0052] in, This represents the temperature value at any point. This represents the temperature interpolation form in the SBFEM coordinate system. Represents the temperature gradient; Represents the shape function of temperature interpolation; and The coordinate transformation matrix represents the heat conduction problem.

[0053] ,

[0054] Furthermore, in step S2, the total energy functional comprising heat dissipation potential energy and heat storage potential energy is constructed as follows:

[0055]

[0056]

[0057]

[0058] in, For total energy, For heat dissipation potential energy, To store potential energy for heat, for abbreviation, for abbreviation, The thermal conductivity coefficient, It is an imaginary number. For frequency, For density, For specific heat capacity, For the entire computational domain, This indicates integration over the computational domain.

[0059] By taking the first variation of the total energy functional and setting it to zero, the nonlinear first-order ordinary differential equation with respect to temperature is derived, resulting in the temperature field control equation in the SBFEM coordinate system, as shown in the following expression:

[0060]

[0061] in, Represents radial coordinates Find the second derivative; , , This is the coefficient matrix for the heat conduction problem; The mass matrix for the heat conduction problem is specifically expressed as:

[0062]

[0063]

[0064]

[0065]

[0066] in, Here is the thermal conductivity matrix. , Represents the circumferential coordinates , integral, This indicates summation over the entire field;

[0067] Introducing dimensionless frequency and dimensionless heat conduction matrix:

[0068]

[0069]

[0070] in, It is a dimensionless frequency. r It is the area of ​​the circumferential boundary surface. This is the original heat conduction matrix. It is a dimensionless thermal conductivity matrix. This represents any diagonal element in the thermal conductivity matrix.

[0071] The specific expression for differentiating the heat conduction matrix with respect to the radial coordinate and frequency is as follows:

[0072]

[0073] in, for abbreviation, For frequency Find the partial derivative.

[0074] The governing equations for the heat conduction matrix are obtained based on the relationship between the derivatives:

[0075]

[0076] in, , , for The coefficient matrix of a steady-state heat conduction problem approaching infinity; for Mass matrix of steady-state heat conduction problem approaching infinity, subscript Indicates frequency Find the first derivative.

[0077] By using geometric equations and constitutive relations, and considering the influence of thermal strain, the governing equations of the displacement field in the SBFEM coordinate system are derived:

[0078] in, This represents the displacement interpolation form in the SBFEM coordinate system; , , The coefficient matrix for the thermal stress problem, The mass matrix for the thermal stress problem is specifically expressed as:

[0079]

[0080]

[0081]

[0082]

[0083] in, Represents the elasticity matrix; Represents the shape function of displacement interpolation; and The coordinate transformation matrix representing the thermal stress problem is expressed as follows:

[0084] , .

[0085] Furthermore, in step S3, by setting the frequency in the heat conduction control equation to zero, the control equation for the steady-state heat conduction problem is obtained:

[0086]

[0087] in, for The steady-state heat conduction matrix that approaches infinity.

[0088] The governing equations are expressed as an eigenvalue problem in the form of Schur decomposition:

[0089]

[0090] in, It is the eigenvector matrix. It is an eigenvalue matrix. It is a Hamiltonian matrix.

[0091] The above formula is expressed as follows:

[0092]

[0093] in, Represents the identity matrix; , , , Represents the eigenvector submatrix; This represents the eigenvalue submatrix.

[0094] Thus, the solution is obtained Steady-state heat conduction matrix approaching infinity:

[0095]

[0096] Convert the frequency to a normalized format:

[0097]

[0098] in, This represents the normalized frequency.

[0099] The heat conduction governing equation is then transformed into a form similar to that of the elastic dynamics equation:

[0100]

[0101] in, for Transient heat conduction matrix approaching infinity for The mass matrix of a transient heat conduction problem approaching infinity. express normalized frequency Find the derivative.

[0102] The transient heat conduction matrix is ​​expanded using the continuous fractional method, and the expression is as follows:

[0103]

[0104] in, Indicates about of m Order function; , This is the coefficient matrix in the power series expansion.

[0105] Obtain the function of normalized frequency by sorting in descending order of powers:

[0106]

[0107] in, express m Initial value of the first-order function; express m The recursive calculation value of the order function; Indicates about of m +1 order function.

[0108] After iterative recursion, we obtain... The resulting expression for the transient heat conduction matrix approaching infinity:

[0109]

[0110] in, This represents the initial value of a first-order function; Represents the recursive calculated value of a first-order function; Represents the initial value of a second-order function; This represents the recursive calculated value of a second-order function.

[0111] Furthermore, in step S4, the heat flow-temperature relationship in the frequency domain is transformed into a relationship in the time domain through inverse Fourier transform, resulting in the standard time-domain equation:

[0112]

[0113] in, This is the time-domain representation of temperature. W The state matrix, It is an external source of income.

[0114] The relationship between temperatures at adjacent time points is obtained based on the general solution of the equation, and the specific expression is as follows:

[0115]

[0116] in, t For any historical point in time; , The firstp and p +1 corresponds to the time point in time; , The first p and p The temperature at the time point corresponding to the +1 time step; dt This is the integral over time; exp is the exponential function. It is an exponential matrix. For time step.

[0117] An improved refined integration method is used to obtain the exact solution of the exponent matrix:

[0118]

[0119] in, n Expand the Taylor series; For time intervals.

[0120] Taylor expansion of the exponent matrix within the time step yields:

[0121]

[0122] in, for Higher-order infinitesimals, exclamation mark This represents the factorial operation.

[0123] The solution for the homogeneous terms is obtained by executing a numerical loop. The expression for the numerical loop is as follows:

[0124]

[0125] in, This represents the number of iterations. N This represents the total number of iterations.

[0126] Solve for non-homogeneous terms using Gaussian integrals:

[0127]

[0128] in, and These are the weights and Gaussian points of the Gaussian integral, respectively. From 1 to l Summation, l This indicates the number of integration points.

[0129] Finally, by superimposing the solutions of the homogeneous and non-homogeneous terms, the temperature value at each time step is obtained, as shown in the following expression:

[0130] .

[0131] Furthermore, in step S5, based on the obtained temperature values ​​at each time point, the expression for the rate of temperature change is derived as follows:

[0132]

[0133] in, This represents the rate of temperature change.

[0134] The coupling equations of the thermo-mechanical system are expressed in block matrix form:

[0135]

[0136] in, This is the time-domain representation of the displacement; , , For the stiffness submatrix of the thermo-mechanical coupling system; For heat flux; This is thermal stress.

[0137] The temperature value at each time point within the entire domain is obtained using a temperature interpolation shape function:

[0138]

[0139] in, This represents the temperature value at each time step within the entire domain.

[0140] Then As a thermal load applied to the mechanical equilibrium equations, the displacement field and thermal stress field are solved according to the constitutive relation, and the specific expressions are as follows:

[0141]

[0142] in, Let be the displacement field of the entire domain. This represents the thermal stress field across the entire domain.

[0143] In this embodiment, a novel coordinate transformation based on geometric similarity is introduced to effectively map the semi-infinite domain to a finite computational domain. Higher-order spectral element shape functions are used to interpolate and approximate the field variables. Combining coordinate transformation with the principles of virtual work and stationary energy, the proportional boundary finite element governing equations for heat conduction and thermal stress problems are systematically derived. For the solution strategy, a continuous fractional expansion method is used to efficiently construct the frequency domain solution for the heat conduction equation, and an improved precise time-step integration method (MPTSIM) is used for high-precision time-domain integration. For the thermal stress problem, the solved temperature field is applied as a thermal load to the mechanical equilibrium equations through sequential coupling. Based on the domain decomposition strategy and interface compatibility conditions, a complete solution scheme for fluid-structure interaction systems is integrated. This method significantly reduces the computational scale and cost of three-dimensional layered semi-infinite domain thermo-mechanical coupling problems, greatly improving computational efficiency while maintaining high accuracy.

[0144] Furthermore, to verify the accuracy and applicability of the present invention in predicting transient heat conduction and the resulting thermal stress, two benchmark examples are presented.

[0145] The first embodiment is a three-layer dielectric transient heat conduction model, such as Figure 3 As shown in the figure. This model is a three-layer medium, with each layer having orthotropic thermal conductivity (material parameters for each layer are shown in Table 1), simulating the non-uniformity in actual engineering. The entire domain is at a uniform temperature of 293.15 K. A Gaussian distributed heat source (total power 20 kW) with a radius of 1 m is applied to the top surface, and both the top and bottom surfaces are set as convective heat transfer boundary conditions. Figure 4 and Figure 7 As shown, the entire model was discretized using only five 15-node SBFEM elements, demonstrating the efficiency of this method's "boundary discretization." The calculation results of this invention were compared with the semi-analytical solution of Reitzle et al. (2019). The transient temperature changes at the comparison points over 60 seconds were analyzed. As shown in Table 2, this invention exhibited extremely high accuracy at five different locations. The coefficient of determination (R²) is... 2 The values ​​range from 0.97 to 0.99, indicating a very strong linear correlation between the predicted and reference values. The mean relative error (MRE) is very low, with a maximum of only 0.74%, while the root mean square error (RMSE) is also at an extremely low level, demonstrating the correctness and applicability of the invention.

[0146] Table 1: Material parameters of different layers

[0147]

[0148] Table 2: Comparison of transient temperatures (K) at selected points using different methods

[0149]

[0150] Furthermore, in order to fully verify the accuracy and efficiency of this method in transient heat transfer stress coupling on a semi-infinite foundation, a second embodiment—thermal-mechanical coupling verification in a homogeneous semi-infinite domain—is presented.

[0151] like Figure 5 As shown, a rectangular region on a homogeneous semi-infinite domain surface is subjected to a time-varying temperature load (from 0°C to 80°C). The rectangle is 200m long and 80m wide. The far field is discretized using SBFEM to create similar surfaces, employing only three 15-node SBFEM elements, as shown below. Figure 6 and Figure 8 As shown, the results of SBFEM are comprehensively compared with the detailed simulation results of the traditional finite element method (FEM). Internal point A (-30, 0, -40) and surface point B (60, 0, 0) are selected, and the calculated values ​​of temperature, stress, and displacement at different time points are compared (Tables 3 and 4). The results show that the coefficient of determination R for all variables is... 2 Extremely high (e.g., R of the stress field at point B) 2 =0.9992), and the mean relative error (MRE) is very small (MRE of stress field at point B = 1.20%). Furthermore, this invention requires only 29,163 degrees of freedom, while the FEM model with equivalent accuracy requires 82,636 degrees of freedom. The CPU computation time for SBFEM is 13.6 minutes, while that for FEM is 28.8 minutes, representing an improvement in computational efficiency of approximately 52.8%. Therefore, this invention significantly improves computational efficiency while maintaining computational accuracy.

[0152] Table 3: Comparison of calculation results for point A using different methods

[0153]

[0154] Table 4: Comparison of calculation results for point B using different methods

[0155] .

Claims

1. A numerical analysis method for transient heat transfer and stress coupling in a three-dimensional layered semi-infinite foundation, characterized in that, The method includes the following steps: S1. Based on the geometric similarity between similar surfaces and boundary surfaces, establish the coordinate representation of any point in the three-dimensional layered semi-infinite foundation. Perform the same discretization on similar surfaces and boundary surfaces. Establish the coordinate transformation from the Cartesian coordinate system to the scaled boundary finite element SBFEM coordinate system through the Jacobian matrix. Further calculate the inverse matrix of the Jacobian matrix and obtain the differential operators for the heat conduction problem and the thermal stress problem. Obtain the temperature and temperature gradient representation of any point in the three-dimensional layered semi-infinite foundation in the SBFEM coordinate system based on the temperature interpolation shape function. S2. Construct a total energy functional that includes heat dissipation potential energy and heat storage potential energy. By taking the first variation of the total energy functional and setting it to zero, derive the nonlinear first-order ordinary differential equation about temperature, and obtain the temperature field control equation in the SBFEM coordinate system. Introduce dimensionless frequency and dimensionless heat conduction matrix. Differentiate the heat conduction matrix with respect to radial coordinate and frequency respectively. Obtain the control equation about the heat conduction matrix based on the relationship between the derivatives. Through geometric equations and constitutive relations, considering the influence of thermal strain, derive the control equation of the displacement field in the SBFEM coordinate system. S3. Setting the frequency in the heat conduction governing equation to zero, we obtain the governing equation for the steady-state heat conduction problem. Using Schur decomposition, we express the governing equation as an eigenvalue problem, thereby solving for the radial coordinates. The steady-state heat conduction matrix approaches infinity; the frequency is transformed into a normalized form, and the heat conduction governing equation is transformed into a form similar to the elastic dynamics equation. The transient heat conduction matrix is ​​expanded using the continuous fractional method, and the function with respect to the normalized frequency is obtained by arranging the equations in descending order of powers. This process is then recursively repeated to obtain... The resulting expression for the transient heat conduction matrix approaching infinity; S4. By using inverse Fourier transform, the heat flow-temperature relationship in the frequency domain is transformed into a relationship in the time domain, resulting in a standard time domain equation. Based on the general solution of the equation, the relationship between the temperatures corresponding to adjacent time points is obtained. An improved fine integration method is used to obtain the exact solution of the exponential matrix. The exponential matrix within the time point is expanded by Taylor and numerical loop is executed to obtain the solution of the homogeneous term. The non-homogeneous term is solved using Gaussian integration. Finally, the solutions of the homogeneous term and the non-homogeneous term are superimposed to obtain the temperature value of each time point. S5. Based on the temperature values ​​obtained at each time point, derive the expression for the rate of temperature change and express the coupled equation of the thermo-mechanical system in a block matrix form; obtain the temperature value at each time point in the entire domain through the temperature interpolation shape function, and then apply the temperature value as a load to the mechanical equilibrium equation, and solve the displacement field and thermal stress field according to the constitutive relation.

2. The numerical analysis method for transient heat transfer and stress coupling in a three-dimensional layered semi-infinite foundation as described in claim 1, characterized in that, In step S1, based on the geometric similarity between the similar surface and the boundary surface, a coordinate representation of any point in the three-dimensional layered semi-infinite foundation is established, specifically as follows: in, , , Let be the coordinates of any point in a three-dimensional layered semi-infinite foundation. , , Let these be the coordinates of points on similar surfaces. , , Let these be the coordinates of a point on the boundary surface. Radial coordinates; Similar surfaces and boundary surfaces are discretized in the same way: in, , , The node coordinate vectors of the element after the boundary surface is discretized. , , Let be the nodal coordinate vector of the element after discretization of the similar surface. Let be the shape function of the surface element. , Circular coordinates; The coordinate transformation from the Cartesian coordinate system to the SBFEM coordinate system is established using the Jacobian matrix, where the expression for the Jacobian matrix is: in, For Jacobian matrices, subscripts are... , Representing the circumferential coordinates respectively , Find the first derivative; Calculate the inverse of the Jacobian matrix: Where the superscript -1 denotes the inverse of the matrix, Represents the determinant of a Jacobian matrix; subscript Represents radial coordinates Find the first derivative; All vectors representing the Jacobian inverse matrix; The differential operator for the heat conduction problem is obtained, and its expression is: in, , , They represent respectively to , , Find the partial derivative; , , These represent partial derivatives with respect to ξ, η, and ζ, respectively; the superscript T indicates the transpose of the vector. Differential operators representing heat conduction problems; , , The sub-coefficient matrix representing the heat conduction problem is expressed as follows: The differential operator for the thermal stress problem is obtained as follows: in, Differential operators representing thermal stress problems; , , The sub-coefficient matrix representing the thermal stress problem is expressed as follows: ； The temperature and temperature gradient at any point in the SBFEM coordinate system are obtained based on the temperature interpolation shape function: in, This represents the temperature value at any point. This represents the temperature interpolation form in the SBFEM coordinate system. Represents the temperature gradient; Represents the shape function of temperature interpolation; and The coordinate transformation matrix representing the heat conduction problem; , 。 3. The numerical analysis method for transient heat transfer and stress coupling in a three-dimensional layered semi-infinite foundation as described in claim 2, characterized in that, In step S2, the total energy functional comprising heat dissipation potential energy and heat storage potential energy is constructed as follows: in, For total energy, For heat dissipation potential energy, To store potential energy for heat, for abbreviation, for abbreviation, The thermal conductivity coefficient, It is an imaginary number. For frequency, For density, For specific heat capacity, For the entire computational domain, Indicates integration over the computational domain; By taking the first variation of the total energy functional and setting it to zero, the nonlinear first-order ordinary differential equation with respect to temperature is derived, resulting in the temperature field control equation in the SBFEM coordinate system, as shown in the following expression: in, Represents radial coordinates Find the second derivative; , , This is the coefficient matrix for the heat conduction problem; The mass matrix for the heat conduction problem is specifically expressed as: in, Here is the thermal conductivity matrix. , Represents the circumferential coordinates , integral, This indicates summation over the entire field; Introducing dimensionless frequency and dimensionless heat conduction matrix: in, It is a dimensionless frequency, and r is the area of ​​the circumferential boundary surface. This is the original heat conduction matrix. It is a dimensionless thermal conductivity matrix. This represents any diagonal element in the thermal conductivity matrix; The specific expression for differentiating the heat conduction matrix with respect to the radial coordinate and frequency is as follows: in, for abbreviation, For frequency Find the partial derivative; The governing equations for the heat conduction matrix are obtained based on the relationship between the derivatives: in, , , for The coefficient matrix of a steady-state heat conduction problem approaching infinity; for Mass matrix of steady-state heat conduction problem approaching infinity, subscript Indicates frequency Find the first derivative; By using geometric equations and constitutive relations, and considering the influence of thermal strain, the governing equations of the displacement field in the SBFEM coordinate system are derived: in, This represents the displacement interpolation form in the SBFEM coordinate system; , , The coefficient matrix for the thermal stress problem, The mass matrix for the thermal stress problem is specifically expressed as: in, Represents the elasticity matrix; Represents the shape function of displacement interpolation; and The coordinate transformation matrix representing the thermal stress problem is expressed as follows: , 。 4. The numerical analysis method for transient heat transfer and stress coupling in a three-dimensional layered semi-infinite foundation as described in claim 3, characterized in that, In step S3, the frequency in the heat conduction control equation is set to zero to obtain the control equation for the steady-state heat conduction problem: in, for Steady-state heat conduction matrix approaching infinity; The governing equations are expressed as an eigenvalue problem in the form of Schur decomposition: in, It is the eigenvector matrix. It is an eigenvalue matrix. It is a Hamiltonian matrix; The eigenvalue problem can be further represented in the form of a submatrix: in, Represents the identity matrix; , , , Represents the eigenvector submatrix; Represents the eigenvalue submatrix; Thus, the solution is obtained Steady-state heat conduction matrix approaching infinity: Convert the frequency to a normalized format: in, Indicates the normalized frequency; The heat conduction governing equation is then transformed into a form similar to that of the elastic dynamics equation: in, for Transient heat conduction matrix approaching infinity for The mass matrix of a transient heat conduction problem approaching infinity. express normalized frequency Differentiate; The transient heat conduction matrix is ​​expanded using the continuous fractional method, and the expression is as follows: in, Indicates about an m-order function; , This is the coefficient matrix in the power series expansion; Obtain the function of normalized frequency by sorting in descending order of powers: in, Represents the initial value of an m-th order function; This represents the recursive calculated value of an m-th order function; Indicates about A function of order m+1; After iterative recursion, we obtain... The resulting expression for the transient heat conduction matrix approaching infinity: in, This represents the initial value of a first-order function; Represents the recursive calculated value of a first-order function; Represents the initial value of a second-order function; This represents the recursive calculated value of a second-order function.

5. The numerical analysis method for transient heat transfer and stress coupling in a three-dimensional layered semi-infinite foundation as described in claim 4, characterized in that, In step S4, the heat flow-temperature relationship in the frequency domain is transformed into a relationship in the time domain through inverse Fourier transform, resulting in the standard time-domain equation: in, Let W be the time-domain representation of temperature, and W be the state matrix. For external sources; The relationship between temperatures at adjacent time points is obtained based on the general solution of the equation, and the specific expression is as follows: Where t is any historical time point; , These are the time points corresponding to the p-th and p+1-th time steps, respectively; , ... It is an exponential matrix. For time step; An improved refined integration method is used to obtain the exact solution of the exponent matrix: Where n is the Taylor series; For time intervals; Taylor expansion of the exponent matrix within the time step yields: in, for Higher-order infinitesimals, exclamation mark This represents the factorial operation; The solution for the homogeneous terms is obtained by executing a numerical loop. The expression for the numerical loop is as follows: in, The iteration number is denoted as N; the total number of iterations is denoted as N. Solve for non-homogeneous terms using Gaussian integrals: in, and These are the weights and Gaussian points of the Gaussian integral, respectively. The summation is performed from 1 to l, where l represents the number of integration points; Finally, by superimposing the solutions of the homogeneous and non-homogeneous terms, the temperature value at each time step is obtained, as shown in the following expression: 。 6. The numerical analysis method for transient heat transfer and stress coupling in a three-dimensional layered semi-infinite foundation as described in claim 5, characterized in that, In step S5, based on the obtained temperature values ​​at each time point, the expression for the rate of temperature change is derived as follows: in, The rate of temperature change; The coupling equations of the thermo-mechanical system are expressed in block matrix form: in, This is the time-domain representation of the displacement; , , For the stiffness submatrix of the thermo-mechanical coupling system; For heat flux; Thermal stress; The temperature value at each time point within the entire domain is obtained using a temperature interpolation shape function: in, This represents the temperature value at each time step within the entire domain. Then As a thermal load applied to the mechanical equilibrium equations, the displacement field and thermal stress field are solved according to the constitutive relation, and the specific expressions are as follows: in, Let be the displacement field of the entire domain. This represents the thermal stress field across the entire domain.

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