Three-dimensional layered semi-infinite foundation transient heat transfer stress coupling numerical analysis method

Abstract

A three-dimensional layered semi-infinite foundation transient heat transfer stress coupling numerical analysis method comprises the steps that a semi-infinite domain is divided into a near-field area and a far-field area, coordinate transformation is established based on geometric similarity between similar surfaces and boundary surfaces, and a proportional boundary finite element coordinate system is formed; deriving a heat conduction and thermal stress control equation under an SBFEM coordinate system based on a variational principle and a virtual work principle, and establishing a dimensionless heat conduction matrix under a frequency domain; solving a heat conduction matrix by adopting a continuous fractional expansion method and Scherr decomposition, and realizing time domain integration through an improved fine time-history integration method (MPTSIM); under a sequential coupling framework, firstly solving a transient temperature field, and then applying the transient temperature field as a thermal load to a mechanical equilibrium equation to obtain a displacement field and a stress field; according to the method, the calculation scale and resource consumption are remarkably reduced while high precision is guaranteed, and the method is suitable for efficient analysis of the transient thermal-mechanical coupling problem in the three-dimensional layered semi-infinite foundation.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of three-dimensional layered semi-infinite foundation transient heat transfer stress, and particularly relates to a three-dimensional layered semi-infinite foundation transient heat transfer stress coupling numerical analysis method. BACKGROUND

[0002] In engineering practice, the transient heat conduction and thermal stress coupling analysis in three-dimensional layered semi-infinite foundation is a key problem. Temperature change in semi-infinite media such as deep rock mass, layered foundation, etc. will trigger a complex and time-evolving stress field, which directly affects the stability, durability and safety of engineering structures. Accurate prediction of such thermal-mechanical coupling behavior is crucial for geothermal resource development, nuclear waste geological disposal and large foundation design. However, the traditional numerical method often faces the challenge of balancing the calculation accuracy and efficiency when simulating such complex problems involving unbounded domain, material heterogeneity and transient process.

[0003] For the thermal-mechanical coupling problem of three-dimensional layered semi-infinite foundation, numerical methods and analytical methods are the main research means. Numerical methods such as finite element method can handle complex geometry, material nonlinearity and transient process, but it needs to be discretized in the whole calculation domain. When dealing with semi-infinite domain, it is often necessary to introduce artificial boundary for truncation, which not only may introduce error, but also significantly increases the model size, leading to high computational cost and low efficiency. On the other hand, analytical methods can provide accurate solutions and high computational efficiency, but they usually rely on highly simplified and idealized assumptions such as homogeneous material, regular geometry and simple boundary conditions, which have limited applicability for layered, heterogeneous media and complex thermal load conditions commonly encountered in practical engineering. Therefore, it is of urgent need and great value to develop a method that can efficiently and accurately analyze the transient thermal-mechanical coupling response in three-dimensional layered semi-infinite foundation. SUMMARY

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

[0005] To achieve the above-mentioned purpose, the present application provides a three-dimensional layered semi-infinite foundation transient heat transfer stress coupling numerical analysis method, which comprises the following steps: S1, based on geometric similarity between similar surfaces and boundary surfaces, coordinate expressions of any point in a three-dimensional layered semi-infinite foundation are established, the similar surfaces and the boundary surfaces are discretized in the same way, the coordinate transformation from the Cartesian coordinate system to the scaled boundary finite element (SBFEM) coordinate system is established through the Jacobian matrix, the inverse matrix of the Jacobian matrix is further calculated, and the differential operators of the heat conduction problem and the thermal stress problem are obtained, and the temperature and temperature gradient expressions of any point in the three-dimensional layered semi-infinite foundation in the SBFEM coordinate system are obtained according to the temperature interpolation shape function.

[0006] S2, a total energy functional including heat dissipation potential energy and heat storage potential energy is constructed, a nonlinear first-order ordinary differential equation about temperature is derived by taking the first variation of the total energy functional and making it zero, a temperature field control equation in the SBFEM coordinate system is obtained, a dimensionless frequency and a dimensionless heat conduction matrix are introduced, the heat conduction matrix is derived with respect to the radial coordinate and the frequency respectively, the control equation about the heat conduction matrix is obtained according to the relationship between the derivatives, and the control equation of the displacement field in the SBFEM coordinate system is derived by considering the influence of thermal strain through the geometric equation and the constitutive relation.

[0007] S3, the frequency in the heat conduction control equation is set to zero to obtain the control equation of the steady-state heat conduction problem, the control equation is expressed in the form of Schur decomposition as an eigenvalue problem, so that the steady-state heat conduction matrix tending to infinity is solved; the frequency is converted into a normalized format, and the heat conduction control equation is converted into a form similar to the elastic dynamics equation, the continuous fraction method is used to expand the transient heat conduction matrix, the function about the normalized frequency is obtained in the descending order of power, and the result expression of the transient heat conduction matrix tending to infinity is obtained through cyclic recursion.

[0008] S4, the heat flow-temperature relationship in the frequency domain is converted into the relationship in the time domain through inverse Fourier transform to obtain a standard time domain equation, the relationship between the temperatures corresponding to adjacent time points is obtained based on the general solution of the equation; the improved precise integration method is used to obtain the accurate solution of the exponential matrix, the Taylor expansion of the exponential matrix at the time point is performed and the numerical cycle is executed to obtain the solution of the homogeneous term, the non-homogeneous term is solved by using Gauss integral, and finally the temperature values at each time point are obtained by superimposing the solution of the homogeneous term and the solution of the non-homogeneous term.

[0009] S5, based on the obtained temperature values at each time point, the expression of the temperature change rate is derived, and the coupling equation of the thermal-mechanical system is expressed in the form of a block matrix; the temperature values at each time point in the whole domain are obtained through the temperature interpolation shape function, and the temperature values are applied as loads to the mechanical equilibrium equation, and the displacement field and the thermal stress field are solved according to the constitutive relation.

[0010] The beneficial effects of the application are: ​​(1) The application constructs a similar surface of the proportional boundary finite element method by introducing a new coordinate transformation based on geometric similarity, converts a three-dimensional semi-infinite domain problem into a form combining two-dimensional discretization on the boundary and analytical solution in the radial direction, significantly reduces the calculation dimension and resource consumption while ensuring simulation accuracy.

[0011] (2) The application uses continued fraction expansion to construct the heat conduction matrix, and combines the improved fine time integration method for time domain solving, realizes high-precision and high-efficiency decoupling calculation of the transient heat conduction and thermal stress coupling system, and greatly reduces the calculation freedom and CPU time on the premise of obtaining similar accuracy to the traditional finite element method.

[0012] (3) The application is successfully applied to the thermal force coupling analysis of the layered semi-infinite domain through the verified numerical framework, reveals the dominant role of the surface mechanical constraint on the thermal stress distribution, and clearly defines the internal trade-off mechanism between the stress field and the displacement field, providing an effective simulation tool for thermal-force coupling design and analysis in complex layered media. BRIEF DESCRIPTION OF DRAWINGS

[0013] Figure 1 is a flowchart of the three-dimensional layered semi-infinite foundation transient heat stress coupling numerical analysis method of the application; Figure 2 is Figure 1 the schematic diagram shown; Figure 3 is a schematic diagram of the calculation model of the first embodiment; Figure 4 is a schematic diagram of the substructure division and element discretization of the first embodiment; Figure 5 is a schematic diagram of the calculation model of the second embodiment; Figure 6 is a schematic diagram of the substructure division and element discretization of the second embodiment; Figure 7 is Figure 4 a cross-sectional schematic diagram of the similar surface and the boundary surface; Figure 8 is Figure 6 element mapping of the similar surface and the boundary surface. DETAILED DESCRIPTION

[0014] It should be understood that the specific embodiments described herein are only used to explain the application and not to limit the application. The implementation, functional features and advantages of the application will be further described with reference to the embodiments and the accompanying drawings.

[0015] As shown with reference to the drawings, Figure 1 is a flowchart of the three-dimensional layered semi-infinite foundation transient heat stress coupling numerical analysis method of the application; Figure 2 isFigure 1 schematic diagram; Figure 3 schematic diagram of the calculation model of the first embodiment; Figure 4 schematic diagram of the substructure division and unit discretization of the first embodiment; Figure 5 schematic diagram of the calculation model of the second embodiment; Figure 6 schematic diagram of the substructure division and unit discretization of the second embodiment; Figure 7 schematic diagram of the Figure 4 schematic diagram of the cross section of the similar surface and the boundary surface; Figure 8 schematic diagram of the Figure 6 schematic diagram of the unit mapping of the similar surface and the boundary surface. The detailed method specifically comprises the following steps: S1, based on the geometric similarity between the similar surface and the boundary surface, the coordinate expression of any point in the three-dimensional layered semi-infinite foundation is established, the similar surface and the boundary surface are discretized in the same way, the coordinate transformation from the Cartesian coordinate system to the scaled boundary finite element (SBFEM) coordinate system is established through the Jacobian matrix, the inverse matrix of the Jacobian matrix is further calculated, and the differential operators of the heat conduction problem and the thermal stress problem are obtained, and the temperature and temperature gradient expressions of any point in the three-dimensional layered semi-infinite foundation in the SBFEM coordinate system are obtained according to the temperature interpolation shape function.

[0016] S2, a total energy functional including heat dissipation potential energy and heat storage potential energy is constructed, a nonlinear first-order ordinary differential equation about temperature is derived by taking the first variation of the total energy functional and setting it to zero, a temperature field control equation in the SBFEM coordinate system is obtained, a dimensionless frequency and a dimensionless heat conduction matrix are introduced, the heat conduction matrix is derived with respect to the radial coordinate and the frequency respectively, the control equation about the heat conduction matrix is obtained according to the relationship between the derivatives, and the control equation of the displacement field in the SBFEM coordinate system is derived by considering the influence of thermal strain through the geometric equation and the constitutive relation.

[0017] S3, the frequency in the heat conduction control equation is set to zero to obtain the control equation of the steady-state heat conduction problem, the control equation is expressed in the form of Schur decomposition as an eigenvalue problem, so as to solve the steady-state heat conduction matrix tending to infinity with respect to the radial coordinate ; the frequency is converted into a normalized format, the heat conduction control equation is converted into a form similar to the elastic dynamics equation, the transient heat conduction matrix is expanded by using the continuous fractional method, the function about the normalized frequency is obtained in the descending order of power, and the result expression of the transient heat conduction matrix tending to infinity is obtained through cyclic recursion .

[0018] S4, the heat flow-temperature relationship in the frequency domain is converted into the relationship in the time domain by inverse Fourier transform to obtain a standard time domain equation, a relationship between temperatures corresponding to adjacent time points is obtained based on a general solution of the equation, an accurate solution of an exponential matrix is obtained by using an improved precise integration method, the exponential matrix at the time points is subjected to Taylor expansion and numerical loop to obtain a solution of the homogeneous term, and the non-homogeneous term is solved by using Gauss integral, and finally the temperature value at each time point is obtained by superimposing the solution of the homogeneous term and the solution of the non-homogeneous term.

[0019] S5, based on the obtained temperature value at each time point, an expression of the temperature change rate is derived, and the coupling equation of the thermal-mechanical system is expressed in the form of a block matrix; the temperature value at each time point in the whole domain is obtained by using a temperature interpolation shape function, and the temperature value is applied as a load to the mechanical equilibrium equation, and the displacement field and the thermal stress field are solved according to the constitutive relation.

[0020] Further, in the step S1, a coordinate expression of an arbitrary point in the three-dimensional layered semi-infinite foundation is established based on the geometric similarity between the similar surface and the boundary surface, and the specific expression is: wherein, 、 、 is the coordinate of an arbitrary point in the three-dimensional layered semi-infinite foundation, 、 、 is the coordinate of a point on the similar surface, 、 、 is the coordinate of a point on the boundary surface, is the radial coordinate.

[0021] The similar surface and the boundary surface are discretized in the same way: wherein, 、 、 is the node coordinate vector of the element after the boundary surface is discretized, 、 、 is the node coordinate vector of the element after the similar surface is discretized, is the shape function of the surface element, 、 is the hoop coordinate.

[0022] The coordinate transformation from the Cartesian coordinate system to the SBFEM coordinate system is established by using the Jacobian matrix, and the expression of the Jacobian matrix is: wherein, is the Jacobian matrix, and the subscript , denote the circumferential coordinate , The first derivative is taken.

[0023] The inverse of the Jacobian matrix is calculated: where the superscript -1 denotes the inverse of the matrix, denotes the determinant of the Jacobian matrix; the subscript denote the radial coordinate The first derivative is taken. denotes all vectors of the inverse Jacobian matrix.

[0024] The differential operator for the heat conduction problem is obtained as where, , , denote the partial derivative with respect to , , The partial derivative is taken. , , denote the partial derivative with respect to ξ , η , ζ The partial derivative is taken; the superscript T denotes the transpose of the vector; denotes the differential operator for the heat conduction problem; , , denotes the sub-coefficient matrix for the heat conduction problem, which is given as The differential operator for the thermal stress problem is obtained as where, denotes the differential operator for the thermal stress problem; , , denotes the sub-coefficient matrix for the thermal stress problem, which is given as .

[0025] The temperature and temperature gradient at any point in the three-dimensional layered semi-infinite foundation in the SBFEM coordinate system are obtained according to the temperature interpolation shape function: where, denotes 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.

[0026] , Furthermore, 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, This indicates integration over the computational domain.

[0027] 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. 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.

[0028] 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.

[0029] 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.

[0030] 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: , .

[0031] 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: in, for The steady-state heat conduction matrix that approaches infinity.

[0032] 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.

[0033] The above formula is expressed as follows: in, Represents the identity matrix; , , , Represents the eigenvector submatrix; This represents the eigenvalue submatrix.

[0034] Thus, the solution is obtained Steady-state heat conduction matrix approaching infinity: Convert the frequency to a normalized format: in, This represents the normalized frequency.

[0035] 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 Find the derivative.

[0036] The transient heat conduction matrix is ​​expanded using the continuous fractional method, and the expression is as follows: in, Indicates about of m Order function; , This is the coefficient matrix in the power series expansion.

[0037] Obtain the function of normalized frequency by sorting in descending order of powers: 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.

[0038] 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.

[0039] 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: in, This is the time-domain representation of temperature. W The state matrix, It is an external source of income.

[0040] 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: in, t For any historical point in time; , The first p and pthe time point corresponding to the time step of +1; , the first p and p the temperature of the time point corresponding to the time step of +1; dt is the time integration; exp is the exponential function; is the exponential matrix, is the time step.

[0041] The improved precise integration method is used to obtain the accurate solution of the exponential matrix: wherein, n is the Taylor expansion series; is the time interval.

[0042] The Taylor expansion of the exponential matrix within the time step is obtained: wherein, is a high-order infinitesimal of , the exclamation mark represents the factorial operation.

[0043] The numerical loop is executed to obtain the solution of the homogeneous term, and the expression of the numerical loop is as follows: wherein, is the iteration number; N is the total iteration number; The non-homogeneous term is solved using the Gaussian integral: wherein, and are the weight and Gaussian point of the Gaussian integral, is the summation from 1 to l , l indicates the number of integration points.

[0044] Finally, the solution of the homogeneous term and the solution of the non-homogeneous term are superimposed to obtain the temperature value of each time step, and the expression is as follows: .

[0045] Further, in the step S5, based on the obtained temperature value of each time point, the expression of the temperature change rate is derived as: wherein, is the temperature change rate.

[0046] The coupling equation of the thermal-mechanical system is expressed in the form of a block matrix: where, is the time-domain representation of displacement; , , is the stiffness sub-matrix of the thermal-mechanical coupling system; is the heat flux; is the thermal stress.

[0047] The temperature value at each time point in the entire domain is obtained by temperature interpolation shape function: where, is the temperature value at each time step in the entire domain.

[0048] Again, is applied to the mechanical equilibrium equation as a thermal load, and the displacement field and thermal stress field are solved according to the constitutive relation, and the specific expression is as follows: where, is the displacement field of the entire domain, is the thermal stress field of the entire domain.

[0049] In this embodiment, by introducing a new coordinate transformation based on geometric similarity, the semi-infinite domain is effectively mapped to a finite calculation domain; high-order spectral element shape functions are used to interpolate and approximate field variables, combined with coordinate transformation and application of virtual work principle and energy stationary principle, the proportional boundary finite element control equation of the heat conduction problem and the thermal stress problem is systematically derived; in terms of solution strategy, for the heat conduction equation, the continuous fractional expansion method is used to efficiently construct the frequency domain solution, and the improved precise time step integration method (MPTSIM) is combined for high-precision time domain integration; for the thermal stress problem, the temperature field obtained by solving is applied to the mechanical equilibrium equation as a thermal load; based on the domain decomposition strategy and the interface compatibility condition, a complete fluid-structure coupling system solution scheme is integrated. This method significantly reduces the calculation scale and cost of three-dimensional layered semi-infinite thermal-mechanical coupling problems, greatly improves the calculation efficiency while ensuring high accuracy.

[0050] Further, in order to verify the accuracy and applicability of the present application in predicting transient heat conduction and the resulting thermal stress, two benchmark examples are proposed.

[0051] The first embodiment is a three-layer medium transient heat conduction model, as shown in Figure 3The model is a three-layer medium with orthotropic thermal conductivity for each layer (see Table 1 for layer parameters), which simulates the non-uniformity in practical engineering. The whole domain is at a uniform temperature of 293.15 K. A Gaussian distributed heat source with a radius of 1 m (total power of 20 kW) is applied on the top surface, and both the top and bottom surfaces are set as convective heat transfer boundary conditions. As shown in Figure 4 and Figure 7 The whole model is discretized using only five 15-node SBFEM elements, which demonstrates the high efficiency of the method of "boundary discretization". The results of the present application are compared with the semi-analytical solution of Reitzle et al. (2019). The comparison points are the transient temperature changes within 60 seconds. As shown in Table 2, the present application shows very high accuracy at five different locations. The determination coefficient (R 2 ) is between 0.97 and 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%, and the root mean square error (RMSE) is also at a very low level, proving the correctness and applicability of the present application.

[0052] Table 1: Material parameters of different layers Table 2: Comparison of transient temperature (K) at selected points by different methods Further, in order to comprehensively verify the accuracy and efficiency of the method in the transient heat-stress coupling of semi-infinite foundation, a second embodiment - thermal-mechanical coupling verification in a homogeneous semi-infinite domain is given.

[0053] As shown in Figure 5 , a homogeneous semi-infinite domain has a rectangular area on its surface subjected to a time-varying temperature load (rising from 0°C to 80°C), with a length of 200 m and a width of 80 m. The far field is discretized by SBFEM on the similar surface, using only three 15-node SBFEM elements, as shown in Figure 6 and Figure 8 The results of SBFEM are compared with the detailed simulation results of the traditional finite element method (FEM). The internal point A (-30, 0, -40) and the surface point B (60, 0, 0) are selected to compare their temperature, stress and displacement at different time points (Tables 3 and 4). The results show that the determination coefficients R 2 of all variables are very high (for example, the R 2The average relative error (MRE) is very small (MRE=1.20% for the stress field of point B). In addition, the present application only needs 29163 degrees of freedom, while a FEM model with equivalent precision needs 82636 degrees of freedom, and the CPU computing time of SBFEM is 13.6 minutes, while that of FEM is 28.8 minutes, and the computing efficiency is improved by about 52.8%. Thus, the present application can greatly improve the computing efficiency while ensuring the computing precision.

[0054] Table 3: Comparison of the calculation results of point A under different methods Table 4: Comparison of the calculation results of point B under different methods .

Claims

1. A three-dimensional layered semi-infinite foundation transient heat transfer stress coupling numerical analysis method, characterized by, The method comprises the following steps: S1, based on the geometric similarity between the similar surface and the boundary surface, the coordinate expression of any point in the three-dimensional layered semi-infinite foundation is established, the similar surface and the boundary surface are discretized in the same way, the coordinate transformation from the Cartesian coordinate system to the SBFEM coordinate system is established through the Jacobian matrix, the inverse matrix of the Jacobian matrix is further calculated, and the differential operators of the heat conduction problem and the thermal stress problem are obtained, and the temperature and temperature gradient expressions of any point in the three-dimensional layered semi-infinite foundation in the SBFEM coordinate system are obtained according to the temperature interpolation shape function; S2, a total energy functional including heat dissipation potential energy and heat storage potential energy is constructed, a nonlinear first-order ordinary differential equation about temperature is derived by taking the first variation of the total energy functional and setting it to zero, and the temperature field control equation in the SBFEM coordinate system is obtained, a dimensionless frequency and a dimensionless heat conduction matrix are introduced, the heat conduction matrix is differentiated with respect to the radial coordinate and the frequency respectively, the control equation about the heat conduction matrix is obtained according to the relationship between the derivatives, and the control equation of the displacement field in the SBFEM coordinate system is derived through the geometric equation and the constitutive relation, considering the influence of thermal strain; S3, set the frequency in the heat conduction control equation to zero, get the control equation of the steady-state heat conduction problem, express the control equation in the form of Schur decomposition as an eigenvalue problem, thereby solving the radial coordinate The steady-state heat conduction matrix tends to infinity; convert the frequency into a normalized format, and convert the heat conduction control equation into a form similar to the elastic dynamics equation, expand the transient heat conduction matrix using the continuous fraction method, arrange the functions about the normalized frequency in descending order of power, and get the result expression of the transient heat conduction matrix tending to infinity through recursive loop The steady-state heat conduction matrix tends to infinity; convert the frequency into a normalized format, and convert the heat conduction control equation into a form similar to the elastic dynamics equation, expand the transient heat conduction matrix using the continuous fraction method, arrange the functions about the normalized frequency in descending order of power, and get the result expression of the transient heat conduction matrix tending to infinity through recursive loop S4, the heat flow-temperature relationship in the frequency domain is converted into the relationship in the time domain through inverse Fourier transform, the standard time domain equation is obtained, the relationship between the temperatures corresponding to adjacent time points is obtained based on the general solution of the equation; the accurate solution of the exponential matrix is obtained by using the improved precise integration method, the Taylor expansion of the exponential matrix at the time point is carried out and the numerical loop is executed to obtain the solution of the homogeneous term, and the non-homogeneous term is solved by using Gauss integral, and finally the temperature values at each time point are obtained by superimposing the solutions of the homogeneous term and the non-homogeneous term; S5, based on the obtained temperature values at each time point, the expression of the temperature change rate is derived, and the coupling equation of the thermal-mechanical system is expressed in the form of a block matrix; the temperature values at each time point in the whole domain are obtained through the temperature interpolation shape function, and then the temperature values are applied as loads to the mechanical equilibrium equation, and the displacement field and the thermal stress field are solved according to the constitutive relation.

2. The three-dimensional layered semi-infinite foundation transient heat transfer stress coupling numerical analysis method of claim 1, wherein, In the step S1, based on the geometric similarity between the similar surface and the boundary surface, the coordinate expression of any point in the three-dimensional layered semi-infinite foundation is established, and the specific expression is: wherein , , is the coordinate of an arbitrary point in the three-dimensional layered semi-infinite foundation, , , is the coordinate of a point on the similar plane, , , is the coordinate of a point on the boundary plane, is the radial coordinate; The similar surface and the boundary surface are discretized in the same way: wherein, , , is the node coordinate vector of the discretized element for the boundary surface, , , is the node coordinate vector of the discretized element for the similar surface, is the shape function of the surface element, , is the circumferential coordinate; The coordinate transformation from the Cartesian coordinate system to the SBFEM coordinate system is established through the Jacobian matrix, and the expression of the Jacobian matrix is: wherein, is the Jacobian matrix, the subscript , denote the first derivatives with respect to the toroidal coordinate , the first derivatives with respect to the poloidal coordinate The inverse matrix of the Jacobian matrix is calculated: where the superscript -1 denotes the inverse of the matrix, denotes the determinant of the Jacobian matrix; the subscript denotes the radial coordinate the first derivative is taken; denotes all vectors of the inverse Jacobian matrix; The differential operator of the heat conduction problem is obtained, and the expression is: wherein , , respectively denote the partial derivatives with respect to , , , , respectively denote the partial derivatives with respect to The differential operator of the thermal stress problem is: , The temperature and temperature gradient expressions of any point in the three-dimensional layered semi-infinite foundation in the SBFEM coordinate system are obtained according to the temperature interpolation shape function: , In the step S2, the total energy functional including heat dissipation potential energy and heat storage potential energy is constructed as: denotes the differential operator of the heat conduction problem; , , denotes the sub-coefficient matrix of the heat conduction problem, and the specific expression is as follows:​​ Through the first variation of the total energy functional and setting it to zero, a nonlinear first-order ordinary differential equation about temperature is derived, and the temperature field control equation in the SBFEM coordinate system is obtained, and the specific expression is as follows: wherein a differential operator representing the thermal stress problem; , , a sub-coefficient matrix representing the thermal stress problem, which is expressed as follows: ； The dimensionless frequency and the dimensionless heat conduction matrix are introduced: wherein, denotes the temperature value at an arbitrary point; denotes the temperature interpolation form in the SBFEM coordinate system; denotes the temperature gradient; denotes the temperature interpolation form function; and denotes the coordinate transformation matrix of the heat conduction problem; , 。 3. A three-dimensional layered semi-infinite foundation transient heat transfer stress coupling numerical analysis method according to claim 2, characterized in that, The heat conduction matrix is differentiated with respect to the radial coordinate and the frequency respectively, and the specific expression is: wherein, is the total energy, is the heat dissipation potential energy, is the heat storage potential energy, is the is a short hand for, is a short hand for, is a short hand for, is the heat conduction coefficient, is the imaginary number, is the frequency, is the density, is the specific heat capacity, is the entire computational domain, denotes integration over the computational domain; ​ wherein denotes the radial coordinate the second derivative is taken with respect to the radial coordinate , , is the coefficient matrix of the heat conduction problem; is the mass matrix of the heat conduction problem, which is expressed as: wherein is the thermal conductivity matrix, , denotes the circumferential coordinate , integration, denotes the summation over the entire domain; ​ wherein, is a dimensionless frequency, r is the area of the circumferential interface, is the original heat conduction matrix, is the dimensionless heat conduction matrix, denotes any diagonal element in the heat conduction coefficient matrix; ​ wherein is the abbreviation for , is the partial derivative of the frequency with respect to time. The governing equation of heat conduction matrix is obtained according to the relationship between derivatives: wherein , , is the coefficient matrix of the steady-state heat conduction problem tending to infinity; is the mass matrix of the steady-state heat conduction problem tending to infinity, the subscript denotes the first derivative with respect to the frequency ; The governing equation of displacement field in SBFEM coordinate system is derived by considering the effect of thermal strain through geometric equation and constitutive relation: where, represents the displacement interpolation form in the SBFEM coordinate system; , , is the coefficient matrix of the thermal stress problem, is the mass matrix of the thermal stress problem, and is specifically expressed as: where represents the elastic matrix; represents the displacement interpolation shape function; and represents the coordinate transformation matrix for the thermal stress problem, and is expressed as follows: , 。 4. The three-dimensional layered semi-infinite foundation transient heat conduction stress coupling numerical analysis method of claim 3, wherein, In the step S3, the frequency in the governing equation of heat conduction is set to zero to obtain the governing equation of steady heat conduction problem: wherein is steady-state heat conduction matrix that tends to infinity; The governing equation is expressed in the form of Schur decomposition as an eigenvalue problem: wherein, is a matrix of eigenvectors, is a matrix of eigenvalues, is a Hamiltonian matrix; The above formula is expressed as follows: wherein denotes the identity matrix; , , , denotes the eigenvector submatrix; denotes the eigenvalue submatrix; Thus, the solution is Steady-state heat conduction matrix tending to infinity: The frequency is converted into normalized format: wherein denotes the normalized frequency; And the governing equation of heat conduction is converted into the form similar to the elastic dynamics equation: wherein is the transient heat conduction matrix tending to infinity, is the mass matrix of the transient heat conduction problem tending to infinity, denotes the derivative with respect to the normalized frequency ; The transient heat conduction matrix is expanded by using continuous fraction method, and the expression is as follows: wherein represents a polynomial of degree of degree m order; , is a coefficient matrix in the power series expansion; The function of normalized frequency is obtained according to the descending order of power: wherein, represents m the initial value of the represents m the recursively computed value of the represents the +1 order function with respect to m +1 order function; By cyclic recurrence we obtain Resulting expression for the transient heat conduction matrix that tends to infinity wherein, represents an initial value of a 1st order function; represents a recursively calculated value of the 1st order function; represents an initial value of a 2nd order function; represents a recursively calculated value of the 2nd order function.

5. The three-dimensional layered semi-infinite foundation transient heat conduction stress coupling numerical analysis method of claim 4, wherein, In the step S4, the heat flow-temperature relationship in the frequency domain is converted into the relationship in the time domain by inverse Fourier transform to obtain the standard time domain equation: wherein is a time domain representation of temperature, W is a state matrix, is an exogenous sink term; The relationship between the temperatures corresponding to adjacent time points is obtained based on the general solution of the equation, and the specific expression is as follows: wherein, t is the arbitrary historical time point; , are the time points corresponding to the time steps p and p + 1, respectively; , are the temperatures at the time points corresponding to the time steps p and p + 1, respectively; dt is the time integration; exp is the exponential function; is the exponential matrix, is the time step; The accurate solution of exponential matrix is obtained by using improved precise integration method: wherein n is the Taylor expansion series; is the time interval; The Taylor expansion of the exponential matrix in the time step is obtained as follows: wherein is a higher-order infinitesimal, the exclamation mark denotes the factorial operation; The solution of homogeneous term is obtained by executing numerical loop, and the expression of numerical loop is as follows: wherein is the number of iterations; N is the total number of iterations; The non-homogeneous term is solved by using Gauss integral: in, and These are the weights and Gaussian points of the Gaussian integral, respectively. From 1 to l Summation, l Indicates the number of integration points; The temperature value of each time step is obtained by finally superimposing the solution of homogeneous term and the solution of non-homogeneous term, and the expression is as follows: 。 6. A three-dimensional layered semi-infinite foundation transient heat conduction stress coupling numerical analysis method according to claim 5, characterized in that, In the step S5, the expression of temperature change rate is derived based on the obtained temperature value of each time point as follows: wherein, is the rate of temperature change; The coupling equation of heat-force system is expressed in the form of block matrix: wherein, is the time domain representation of the displacement; , , is the stiffness submatrix of the thermal-mechanical coupling system; is the heat flux; is the thermal stress; The temperature value of each time point in the whole domain is obtained by temperature interpolation shape function: wherein, T is the temperature value for each time step within the entire domain; Again As thermal load applied to the mechanical equilibrium equation, the displacement field and thermal stress field are solved according to the constitutive relation, and the specific expression is as follows: wherein, is the displacement field of the entire domain, is the thermal stress field of the entire domain.

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