Method for calculating temperature and heat flow of an isotropic plate containing a heat source at a point near the boundary

Abstract

The application discloses a kind of isotropic plate containing heat source near boundary inner point temperature and heat flow calculation method, including 1 by double reciprocal boundary integral equation solution containing heat source of isotropic plate boundary element node on the temperature value and heat flow value;2 by proximity identification near boundary inner point, and using analytic algorithm to calculate the near singularity integral appearing in boundary integral equation;3 in double reciprocal boundary element method, application near singularity integral analytic algorithm, calculate the temperature and heat flow of isotropic plate near boundary inner point.The application can avoid the domain integral that appears when calculating the temperature and heat flow of boundary point of isotropic plate containing heat source in boundary element method, only need to calculate boundary integration, thereby effectively reduce the amount of calculation, and can accurately calculate the temperature and heat flow of inner point closer to boundary in isotropic plate containing heat source.

Description

Technical Field

[0001] This invention belongs to the field of solid heat conduction calculation, specifically a method for calculating the temperature and heat flow at the near-boundary point of an isotropic plate containing a heat source. Background Technology

[0002] Calculating the temperature at points near the boundary has always been an important technique in engineering applications such as analyzing the thermal insulation effect of thermal barrier coatings. Only by accurately calculating the temperature and heat flow at points near the boundary can the thermal insulation effectiveness of the coating be clearly determined. The boundary element method (BEM) has advantages over domain-specific methods like the finite element method (FEA) in reducing dimensionality and computational complexity, as it only requires dividing the object's boundary into elements. However, when using the BEM to analyze the temperature and heat flow within an object containing a heat source, the Poisson equation needs to be solved. This results in domain integrals in the boundary integral equations. If these domain integrals are not converted to boundary integrals, the BEM loses its dimensionality reduction advantage. Furthermore, when calculating the temperature and heat flow at points near the boundary, the boundary integral equations present the challenge of calculating near-singular integrals. Conventional Gaussian integrals gradually become ineffective or even distorted when calculating near-singular integrals as the points near the boundary become increasingly closer to the boundary, making it impossible to accurately calculate the temperature and heat flow at points near the boundary of an isotropic plate containing a heat source. Summary of the Invention

[0003] This invention addresses the shortcomings of existing technologies by proposing a method for calculating the temperature and heat flow at near-boundary interior points of an isotropic plate containing a heat source. The method aims to transform the domain integral in the temperature boundary integral equation into a boundary integral, and by calculating the near-singular integral in the temperature boundary integral equation containing the heat source, it can effectively calculate the temperature and heat flow at near-boundary interior points of an isotropic plate containing a heat source. This significantly reduces the computational load and improves the accuracy and efficiency of the calculations.

[0004] To achieve the above-mentioned objectives, the present invention adopts the following technical solution:

[0005] The method for calculating the near-boundary interior point temperature and heat flow of an isotropic plate containing a heat source, as described in this invention, is performed according to the following steps:

[0006] Step 1: Discretize the boundary of the isotropic plate:

[0007] A two-dimensional coordinate system oxy is established with the center point o of the isotropic plate as the origin, the horizontal rightward direction as the positive x-axis, and the vertical upward direction as the positive y-axis.

[0008] The boundary of the isotropic plate is divided into linear elements in a counterclockwise direction to obtain N boundary elements and N boundary nodes.

[0009] Choose any node P on the boundary as the source point, whose coordinates are (x P ,yP );

[0010] Select the first node Q of the nth boundary element n,1 and the last node Q n,2 As two field points, their coordinates are respectively (x n,1 ,y n,1 ) and (x n,2 ,y n,2 );

[0011] Select L interior points near the boundary within the isotropic plate, and denote the coordinates of any interior point P′ as (x′, y′).

[0012] Step 2: Establish the two-dimensional boundary integral equations of the double reciprocal boundary element method, and calculate the unknown boundary temperature and heat flux of the isotropic plate:

[0013] Step 2.1: Establish the temperature boundary integral equation of the source point P with respect to the nth boundary element. After discretization, we can obtain equation (1):

[0014]

[0015] In equation (1), c P The coefficient u represents the smoothness at the source point P. P This represents the temperature value at the source point P; Indicates field point Q n,1 Temperature value at that location, Indicates field point Q n,2 Temperature value at that location, Indicates field point Q n,1 The heat flux value at that location, Indicates field point Q n,2 The heat flux value at that location; It is the source point P to the field point Q n,1 Temperature influence coefficient, It is the source point P to the field point Q n,2 Temperature influence coefficient, It is the source point P to the field point Q n,1 The heat flux influence coefficient, It is the source point P to the field point Q n,2 The heat flux influence coefficient; α represents the heat conduction coefficient of the isotropic plate, N+L represents the total number of collocation points in the double reciprocal boundary element method, B k Let represent the heat source term at the k-th collocation point within the isotropic plate, and we have:

[0016]

[0017] In equation (2), β k It is the coefficient vector of the kth collocation point that approximates the real heat source; Indicates the field point Q under the influence of the heat source.n,1 The equivalent particular solution temperature for the kth collocation point, Indicates the field point Q under the influence of the heat source. n,2 The equivalent particular solution temperature for the kth collocation point, Indicates the field point Q under the influence of the heat source. n,1 The heat flux of the equivalent particular solution at the k-th collocation point, Indicates the field point Q under the influence of the heat source. n,2 The equivalent particular solution heat flow for the kth collocation point;

[0018] Step 2.2: Using equations (3.a), (3.b), (3.c), and (3.d), calculate the field point Q of the nth boundary element. n,1 The equivalent particular solution temperature of the kth collocation point and equivalent specific heat flow Field Q n,2 The equivalent particular solution temperature of the kth collocation point and equivalent specific heat flow

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[0023] In equations (3.a), (3.b), (3.c), and (3.d), It is field point Q n,1 The radial function to the kth collocation point, It is field point Q n,2 The radial function to the kth collocation point, γ nx γ is the direction cosine of the outward normal of the nth boundary element in the x-direction. ny It is the direction cosine of the outward normal of the nth boundary element in the y-direction;

[0024] Step 2.3: Based on the source point P, write the temperature boundary integral equations for each of the N boundary elements, and construct the boundary integral equations using equation (4):

[0025]

[0026] By introducing known boundary conditions, the boundary integral equations are solved to obtain the temperature and heat flux of all boundary nodes on the boundary element.

[0027] Step 3: Calculate the temperature and heat flux at points near the boundary of the isotropic plate containing the heat source;

[0028] Step 3.1: Calculate the proximity e from the interior point P′ to the nth boundary cell using equation (5). P′,n :

[0029]

[0030] In equation (5), s n ε represents the length of the nth boundary cell. P′,n s represents the distance from interior point P′ to the center of the nth boundary cell. nx s represents the length of the nth boundary cell in the x-direction. ny ε represents the length of the nth boundary element in the y-direction. P′,nx ε represents the distance in the x-direction from interior point P′ to the center of the nth boundary cell. P′,ny This represents the distance in the y-direction from the interior point P′ to the center of the nth boundary cell;

[0031] Step 3.2, when the proximity e P′,n When ≤2, the interior point P′ is a near-boundary interior point, and the near-singular integral appearing in the influence coefficient of temperature and heat flux of the near-boundary interior point of the isotropic plate is analytically calculated using equations (6.a), (6.b), (6.c), (6.d), (6.e), (6.f), (6.g), (6.h), (6.i), (6.j), (6.k), and (6.l).

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[0044] In equations (6.a), (6.b), (6.c), (6.d), (6.e), (6.f), (6.g), (6.h), (6.i), (6.j), (6.k), and (6.l), ξ is the integration variable, and I n,01 I n,02 Let I represent two first-order near-singular integrals. n,11 I n,12 I n,13 I n,14 I n,15 I n,16 I represents six second-order near-singular integrals; n,21 I n,22 I n,23 I n,24 Represents four third-order near-singular integrals; a n b n c n a n,11 a n,12 a n,13 a n,14 a n,15 a n,16 b n,11 b n,12 b n,13 b n,14 b n,15 b n,16 c n,11 c n,12 c n,13 c n,14 c n,15 c n,16 a n,21 a n,22 a n,23 a n,24 b n,21 b n,22 b n,23 b n,24 c n,21 c n,22 c n,23 c n,24 Both are the coordinates (x′, y′) of the interior point P′ and the field point Q. n,1 Q n,2 coordinates (x) n,1 ,y n,1 ), (x n,2 ,y n,2 The relevant coefficients are:

[0045] a n =(x n,2 -x n,1 )2 +(y n,2 -y n,1 ) 2 (7.a)

[0046]

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[0048] a n,11 =0 (7.d.1)

[0049] a n,12 =0 (7.d.2)

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[0054] b n,11 =-1 (7.e.1)b n,12 =1 (7.e.2)

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[0059] c n,11 =1 (7.f.1)c n,12 =1 (7.f.2)

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[0076] Step 3.3: Using equations (8.a), (8.b), (8.c), (8.d), (8.e), (8.f), (8.g), (8.h), (8.i), (8.j), (8.k), and (8.l), the obtained near-singular integral analytical calculation results are assigned to each influence coefficient:

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[0089] In equations (8.a), (8.b), (8.c), (8.d), (8.e), (8.f), (8.g), (8.h), (8.i), (8.j), (8.k), and (8.l), The interior point P′ is relative to the field point Q. n,1 Temperature influence coefficient, The interior point P′ is relative to the field point Q. n,2 Temperature influence coefficient, The interior point P′ is relative to the field point Q. n,1 The heat flux influence coefficient, The interior point P′ is relative to the field point Q. n,2 The heat flux influence coefficient; yes Partial derivative with respect to the x-coordinate, yes The partial derivative with respect to the y-coordinate, yes Partial derivative with respect to the x-coordinate, yes The partial derivative with respect to the y-coordinate, yes Partial derivative with respect to the x-coordinate, yes The partial derivative with respect to the y-coordinate, yes Partial derivative with respect to the x-coordinate, yes Partial derivative with respect to the y-coordinate; λ n These are coefficients related to the coordinates of the interior points and field points, and we have:

[0090] λ n =x n,1 y′-y n,1 x′+x n,2 y n,1 -x n,2 y′+y n,2 x′-y n,2 x n,1 (9)

[0091] Step 3.4: Calculate the temperature u at point P′ near the boundary of the isotropic plate containing the heat source according to equation (10). P′ :

[0092]

[0093] In equation (10), Let represent the equivalent particular temperature of interior point P′ with respect to the k-th collocation point under the influence of the heat source, and we have:

[0094]

[0095] In equation (11), r P′,k This represents the distance from interior point P′ to the kth collocation point;

[0096] Step 3.5: Calculate the heat flux component q in the x-direction at the interior point near the boundary of the isotropic plate containing the heat source according to equations (12.a) and (12.b). P′x And the heat flow component q in the y direction P′y :

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[0099] In equations (12.a) and (12.b), yes Partial derivative with respect to the x-coordinate, yes The partial derivative with respect to the y-coordinate is:

[0100]

[0101]

[0102] In equations (13.a) and (13.b), r P′,kx r represents the distance in the x-direction from interior point P′ to the k-th collocation point. P′,ky This represents the distance in the y-direction from the interior point P′ to the kth collocation point.

[0103] The present invention provides an electronic device, comprising a memory and a processor, wherein the memory is used to store a program that supports the processor in executing a method for calculating the near-boundary in-point temperature and heat flow of an isotropic plate containing a heat source, and the processor is configured to execute the program stored in the memory.

[0104] The present invention discloses a computer-readable storage medium, wherein a computer program is stored on the computer-readable storage medium, and the computer program, when executed by a processor, performs the steps of the method for calculating the temperature and heat flow of the in-point near the boundary of the isotropic plate containing a heat source.

[0105] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0106] 1. This invention transforms the domain integral in the boundary integral equation into the boundary integral using the double reciprocity method, and solves for the temperature and heat flux values ​​of the nodes on the boundary element of an isotropic plate containing a heat source. This allows the boundary element method to calculate the temperature and heat flux values ​​of each boundary node without dividing the domain into elements when studying heat conduction problems containing heat sources, thus reducing the complexity and computational load of the boundary element method when dealing with domain integrals.

[0107] 2. This invention applies an analytical algorithm of near-singular integral in the double reciprocal boundary element method, which can effectively reduce the proximity of the temperature and heat flow at the near boundary interior point to be calculated, thereby enabling more accurate calculation of the temperature and heat flow at the near boundary interior point in an isotropic plate containing a heat source. Detailed Implementation

[0108] In this embodiment, a method for calculating the near-boundary interior point temperature and heat flow of an isotropic plate containing a heat source is performed according to the following steps:

[0109] Step 1: Discretize the boundary of the isotropic plate;

[0110] A two-dimensional coordinate system oxy is established with the center point o of the isotropic plate as the origin, the horizontal rightward direction as the positive x-axis, and the vertical upward direction as the positive y-axis.

[0111] The boundary of the isotropic plate is divided into linear elements in a counterclockwise direction to obtain N boundary elements and N boundary nodes.

[0112] Choose any node P on the boundary as the source point, whose coordinates are (x P ,y P );

[0113] Select the first node Q of the nth boundary element n,1 and the last node Q n,2 As two field points, their coordinates are respectively (x n,1 ,y n,1 ) and (x n,2 ,y n,2 In the boundary element method, any boundary node can be used as a source point, and the first and last nodes on any boundary element can be used as field points.

[0114] Choose L interior points near the boundary within the isotropic plate, and denote the coordinates of any interior point P′ as (x′, y′).

[0115] Step 2: Establish the two-dimensional boundary integral equations of the double reciprocal boundary element method, and calculate the unknown boundary temperature and heat flux of the isotropic plate:

[0116] Step 2.1: Establish the temperature boundary integral equation of the source point P with respect to the nth boundary element. After discretization, we can obtain equation (1):

[0117]

[0118] In equation (1), c P The coefficient u represents the smoothness at the source point P. P This represents the temperature value at the source point P; Indicates field point Q n,1 Temperature value at that location, Indicates field point Q n,2 Temperature value at that location, Indicates field point Q n,1 The heat flux value at that location, Indicates field point Q n,2 The heat flux value at the point; here, the temperature value and heat flux value may have known and unknown differences due to boundary conditions, but this will not affect the establishment of the discrete double reciprocal boundary integral equation; It is the source point P to the field point Q n,1 Temperature influence coefficient, It is the source point P to the field point Q n,2 Temperature influence coefficient, It is the source point P to the field point Q n,1 The heat flux influence coefficient, It is the source point P to the field point Q n,2 The heat flux influence coefficient; α represents the heat conduction coefficient of the isotropic plate, N+L represents the total number of collocation points in the double reciprocal boundary element method, B k Let represent the heat source term at the k-th collocation point within the isotropic plate, and we have:

[0119]

[0120] In equation (2), β k It is the coefficient vector of the kth collocation point that approximates the real heat source; Indicates the field point Q under the influence of the heat source. n,1 The equivalent particular solution temperature for the kth collocation point, Indicates the field point Q under the influence of the heat source. n,2 The equivalent particular solution temperature for the kth collocation point, Indicates the field point Q under the influence of the heat source. n,1 The heat flux of the equivalent particular solution at the k-th collocation point, Indicates the field point Q under the influence of the heat source. n,2 The equivalent particular solution heat flow for the kth collocation point.

[0121] Step 2.2: Using equations (3.a), (3.b), (3.c), and (3.d), calculate the field point Q of the nth boundary element. n,1 The equivalent particular solution temperature of the kth collocation point and equivalent specific heat flow Field Q n,2 The equivalent particular solution temperature of the kth collocation point and equivalent specific heat flow

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[0126] In equations (3.a), (3.b), (3.c), and (3.d), It is field point Q n,1 The radial function to the kth collocation point, It is field point Q n,2 The radial function to the kth collocation point, γ nx γ is the direction cosine of the outward normal of the nth boundary element in the x-direction. ny It is the direction cosine of the outward normal of the nth boundary element in the y-direction;

[0127] Step 2.3: Based on the source point P, write the temperature boundary integral equations for each of the N boundary elements, and construct the boundary integral equations using equation (4):

[0128]

[0129] By introducing known boundary conditions, the boundary integral equations are solved to obtain the temperature and heat flux of all boundary nodes on the boundary element. The boundary integral equations include both the temperature and heat flux known from the boundary conditions and the unknown temperature and heat flux. By solving this equation system, the unknown temperature and heat flux of all boundary nodes can be obtained. Using the temperature and heat flux of all boundary nodes, the temperature and heat flux of the in-points near the boundary of the isotropic plate can be accurately calculated through step three.

[0130] Step 3: Calculate the temperature and heat flux at points near the boundary of the isotropic plate containing the heat source;

[0131] Step 3.1: Calculate the proximity e from the interior point P′ to the nth boundary cell using equation (5). P′,n :

[0132]

[0133] In equation (5), s n ε represents the length of the nth boundary cell. P′,n s represents the distance from interior point P′ to the center of the nth boundary cell. nx s represents the length of the nth boundary cell in the x-direction. ny ε represents the length of the nth boundary element in the y-direction. P′,nx ε represents the distance in the x-direction from interior point P′ to the center of the nth boundary cell. P′,ny This represents the distance in the y-direction from the interior point P′ to the center of the nth boundary cell;

[0134] Step 3.2, when the proximity eP′,n When ≤2, the interior point P′ is a near-boundary interior point. At this time, near-singular integrals will appear in the boundary equations for temperature and heat flux at the interior point. Conventional Gaussian integrals cannot accurately calculate near-singular integrals. Through equations (6.a), (6.b), (6.c), (6.d), (6.e), (6.f), (6.g), (6.h), (6.i), (6.j), (6.k), and (6.l), the near-singular integrals appearing in the influence coefficients of temperature and heat flux at the interior point near the boundary of an isotropic plate are analytically calculated.

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[0147] In equations (6.a), (6.b), (6.c), (6.d), (6.e), (6.f), (6.g), (6.h), (6.i), (6.j), (6.k), and (6.l), ξ is the integration variable, and I n,01 I n,02 Let I represent two first-order near-singular integrals. n,11 I n,12 I n,13 I n,14 I n,15 I n,16 I represents six second-order near-singular integrals; n,21 I n,22 I n,23 I n,24 Represents four third-order near-singular integrals; an b n c n a n,11 a n,12 a n,13 a n,14 a n,15 a n,16 b n,11 b n,12 b n,13 b n,14 b n,15 b n,16 c n,11 c n,12 c n,13 c n,14 c n,15 c n,16 a n,21 a n,22 a n,23 a n,24 b n,21 b n,22 b n,23 b n,24 c n,21 c n,22 c n,23 c n,24 Both are the coordinates (x′, y′) of the interior point P′ and the field point Q. n,1 Q n,2 coordinates (x) n,1 ,y n,1 ), (x n,2 ,y n,2 The relevant coefficients are:

[0148] a n =(x n,2 -x n,1 ) 2 +(y n,2 -y n,1 ) 2 (7.a)

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[0151] a n,11 =0 (7.d.1)

[0152] a n,12 =0 (7.d.2)

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[0157] b n,11 =-1 (7.e.1)

[0158] b n,12 =1 (7.e.2)

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[0163] c n,11 =1 (7.f.1)

[0164] c n,12 =1 (7.f.2)

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[0181] Step 3.3: Using equations (8.a), (8.b), (8.c), (8.d), (8.e), (8.f), (8.g), (8.h), (8.i), (8.j), (8.k), and (8.l), the obtained near-singular integral analytical calculation results are assigned to each influence coefficient:

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[0194] In equations (8.a), (8.b), (8.c), (8.d), (8.e), (8.f), (8.g), (8.h), (8.i), (8.j), (8.k), and (8.l), The interior point P′ is relative to the field point Q. n,1 Temperature influence coefficient, The interior point P′ is relative to the field point Q. n,2 Temperature influence coefficient, The interior point P′ is relative to the field point Q. n,1 The heat flux influence coefficient, The interior point P′ is relative to the field point Q. n,2 The heat flux influence coefficient; yes Partial derivative with respect to the x-coordinate, yes The partial derivative with respect to the y-coordinate, yes Partial derivative with respect to the x-coordinate, yes The partial derivative with respect to the y-coordinate, yes Partial derivative with respect to the x-coordinate, yes The partial derivative with respect to the y-coordinate, yes Partial derivative with respect to the x-coordinate, yes Partial derivative with respect to the y-coordinate; λ n These are coefficients related to the coordinates of the interior points and field points, and we have:

[0195] λ n =x n,1 y′-y n,1 x′+x n,2 y n,1 -x n,2 y′+y n,2 x′-y n,2 x n,1 (9)

[0196] Step 3.4: Calculate the temperature u at point P′ near the boundary of the isotropic plate containing the heat source according to equation (10). P′ :

[0197]

[0198] In equation (10), Let represent the equivalent particular temperature of interior point P′ with respect to the k-th collocation point under the influence of the heat source, and we have:

[0199]

[0200] In equation (11), r P′,k This represents the distance from interior point P′ to the kth collocation point.

[0201] Step 3.5: Calculate the heat flux component q in the x-direction at the interior point near the boundary of the isotropic plate containing the heat source according to equations (12.a) and (12.b). P′x And the heat flow component q in the y direction P′y :

[0202]

[0203]

[0204] In equations (12.a) and (12.b), yes Partial derivative with respect to the x-coordinate, yes The partial derivative with respect to the y-coordinate is:

[0205]

[0206]

[0207] In equations (13.a) and (13.b), r P′,kx r represents the distance in the x-direction from interior point P′ to the k-th collocation point. P′,ky This represents the distance in the y-direction from the interior point P′ to the kth collocation point.

[0208] In this embodiment, an electronic device includes a memory and a processor. The memory stores a program that supports the processor in executing the above-described method, and the processor is configured to execute the program stored in the memory.

[0209] In this embodiment, a computer-readable storage medium stores a computer program, which is executed by a processor to perform the steps of the above method.

Claims

1. A method for calculating the near-boundary interior point temperature and heat flux of an isotropic plate containing a heat source, characterized in that, The procedure is as follows: Step 1: Discretize the boundary of the isotropic plate: A two-dimensional coordinate system oxy is established with the center point o of the isotropic plate as the origin, the horizontal rightward direction as the positive x-axis, and the vertical upward direction as the positive y-axis. The boundary of the isotropic plate is divided into linear elements in a counterclockwise direction to obtain N boundary elements and N boundary nodes. Choose any node P on the boundary as the source point, whose coordinates are (x P ,y P ); Select the first node Q of the nth boundary element n,1 and the last node Q n,2 As two field points, their coordinates are respectively (x n,1 ,y n,1 ) and (x n,2 ,y n,2 ); Select L interior points near the boundary within the isotropic plate, and denote the coordinates of any interior point P′ as (x′, y′). Step 2: Establish the two-dimensional boundary integral equations of the double reciprocal boundary element method, and calculate the unknown boundary temperature and heat flux of the isotropic plate: Step 2.1: Establish the temperature boundary integral equation of the source point P with respect to the nth boundary element. After discretization, we can obtain equation (1): In equation (1), c P The coefficient representing the smoothness at the source point P, u P This represents the temperature value at the source point P; Indicates field point Q n,1 Temperature value at that location, Indicates field point Q n,2 Temperature value at that location, Indicates field point Q n,1 The heat flux value at that location, Indicates field point Q n,2 The heat flux value at that location; It is the source point P to the field point Q n,1 Temperature influence coefficient, It is the source point P to the field point Q n,2 Temperature influence coefficient, It is the source point P to the field point Q n,1 The heat flux influence coefficient, It is the source point P to the field point Q n,2 The heat flux influence coefficient; α represents the heat conduction coefficient of the isotropic plate, N+L represents the total number of collocation points in the double reciprocal boundary element method, B k Let represent the heat source term at the k-th collocation point within the isotropic plate, and we have: In equation (2), β k It is the coefficient vector of the kth collocation point that approximates the real heat source; Indicates the field point Q under the influence of the heat source. n,1 The equivalent particular solution temperature for the kth collocation point, Indicates the field point Q under the influence of the heat source. n,2 The equivalent particular solution temperature for the kth collocation point, Indicates the field point Q under the influence of the heat source. n,1 The heat flux of the equivalent particular solution at the k-th collocation point, Indicates the field point Q under the influence of the heat source. n,2 The equivalent particular solution heat flow at the k-th collocation point; Step 2.2: Using equations (3.a), (3.b), (3.c), and (3.d), calculate the field point Q of the nth boundary element. n,1 The equivalent particular solution temperature of the kth collocation point and equivalent specific heat flow Field Q n,2 The equivalent particular solution temperature of the kth collocation point and equivalent specific heat flow In equations (3.a), (3.b), (3.c), and (3.d), It is field point Q n,1 The radial function to the kth collocation point, It is field point Q n,2 The radial function to the kth collocation point, γ nx γ is the direction cosine of the outward normal of the nth boundary element in the x-direction. ny It is the direction cosine of the outward normal of the nth boundary element in the y-direction; Step 2.3: Based on the source point P, write the temperature boundary integral equations for each of the N boundary elements, and construct the boundary integral equations using equation (4): By introducing known boundary conditions, the boundary integral equations are solved to obtain the temperature and heat flux of all boundary nodes on the boundary element. Step 3: Calculate the temperature and heat flux at points near the boundary of the isotropic plate containing the heat source; Step 3.1: Calculate the proximity e from the interior point P′ to the nth boundary cell using equation (5). P′,n : In equation (5), s n ε represents the length of the nth boundary cell. P′,n s represents the distance from interior point P′ to the center of the nth boundary cell. nx s represents the length of the nth boundary cell in the x-direction. ny ε represents the length of the nth boundary element in the y-direction. P′,nx ε represents the distance in the x-direction from interior point P′ to the center of the nth boundary cell. P′,ny This represents the distance in the y-direction from the interior point P′ to the center of the nth boundary cell; Step 3.2, when the proximity e P′,n When ≤2, the interior point P′ is a near-boundary interior point, and the near-singular integral appearing in the influence coefficient of temperature and heat flux of the near-boundary interior point of the isotropic plate is analytically calculated using equations (6.a), (6.b), (6.c), (6.d), (6.e), (6.f), (6.g), (6.h), (6.i), (6.j), (6.k), and (6.l). In equations (6.a), (6.b), (6.c), (6.d), (6.e), (6.f), (6.g), (6.h), (6.i), (6.j), (6.k), and (6.l), ξ is the integration variable, and I n,01 I n,02 Let I represent two first-order near-singular integrals. n,11 I n,12 I n,13 I n,14 I n,15 I n,16 Represents six second-order near-singular integrals; I n,21 I n,22 I n,23 I n,24 Represents four third-order near-singular integrals; a n b n c n a n,11 a n,12 a n,13 a n,14 a n,15 a n,16 b n,11 b n,12 b n,13 b n,14 b n,15 b n,16 c n,11 c n,12 c n,13 c n,14 c n,15 c n,16 a n,21 a n,22 a n,23 a n,24 b n,21 b n,22 b n,23 b n,24 c n,21 c n,22 c n,23 c n,24 Both are the coordinates (x′, y′) of the interior point P′ and the field point Q. n,1 Q n,2 coordinates (x) n,1 ,y n,1 ), (x n,2 ,y n,2 The relevant coefficients are: to n =(x n,2 -x n,1 ) 2 +(and n,2 -and n,1 ) 2 (7.a) a n,11 =0 (7.d.1) a n,12 =0 (7.d.2) b n,11 =-1 (7.e.1) b n,12 =1 (7.e.2) c n,11 =1 (7.f.1) c n,12 =1 (7.f.2) Step 3.3: Using equations (8.a), (8.b), (8.c), (8.d), (8.e), (8.f), (8.g), (8.h), (8.i), (8.j), (8.k), and (8.l), the obtained near-singular integral analytical calculation results are assigned to each influence coefficient: In equations (8.a), (8.b), (8.c), (8.d), (8.e), (8.f), (8.g), (8.h), (8.i), (8.j), (8.k), and (8.l), The interior point P′ is relative to the field point Q. n,1 Temperature influence coefficient, The interior point P′ is relative to the field point Q. n,2 Temperature influence coefficient, The interior point P′ is relative to the field point Q. n,1 The heat flux influence coefficient, The interior point P′ is relative to the field point Q. n,2 The heat flux influence coefficient; yes Partial derivative with respect to the x-coordinate, yes The partial derivative with respect to the y-coordinate, yes Partial derivative with respect to the x-coordinate, yes The partial derivative with respect to the y-coordinate, yes Partial derivative with respect to the x-coordinate, yes The partial derivative with respect to the y-coordinate, yes Partial derivative with respect to the x-coordinate, yes Partial derivative with respect to the y-coordinate; λ n These are coefficients related to the coordinates of the interior points and field points, and we have: λ n =x n,1 y′-y n,1 x′+x n,2 and n,1 -x n,2 y′+y n,2 x′-y n,2 x n,1 (9) Step 3.4: Calculate the temperature u at point P′ near the boundary of the isotropic plate containing the heat source according to equation (10). P′ : In equation (10), Let represent the equivalent particular temperature of interior point P′ with respect to the kth collocation point under the influence of the heat source, and we have: In equation (11), r P′,k This represents the distance from interior point P′ to the kth collocation point; Step 3.5: Calculate the heat flux component q in the x-direction at the interior point near the boundary of the isotropic plate containing the heat source according to equations (12.a) and (12.b). P′x And the heat flow component q in the y direction P′y : In equations (12.a) and (12.b), yes Partial derivative with respect to the x-coordinate, yes The partial derivative with respect to the y-coordinate is: In equations (13.a) and (13.b), r P′,kx r represents the distance in the x-direction from interior point P′ to the k-th collocation point. P′,ky This represents the distance in the y-direction from the interior point P′ to the kth collocation point.

2. An electronic device, comprising a memory and a processor, characterized in that, The memory is used to store a program that supports the processor in executing the method for calculating the near-boundary in-point temperature and heat flow of an isotropic plate containing a heat source as described in claim 1, and the processor is configured to execute the program stored in the memory.

3. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is run by the processor, it executes the steps of the method for calculating the temperature and heat flow at the near-boundary interior point of an isotropic plate containing a heat source as described in claim 1.

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