Method for calculating heat flow intensity factor based on singularity asymptotic expansion and equigeometric boundary element

Abstract

The application discloses a heat flow intensity factor calculation method based on singularity asymptotic expansion and an equivalent geometric boundary element, and comprises the following steps: 1, dividing an isotropic V-shaped notched plate into a sector structure containing a notched tip and a remaining structure; 2, performing singularity asymptotic expansion analysis on the sector structure containing the notched tip; 3, performing equivalent geometric boundary element analysis on the remaining structure; and 4, establishing a system of equations to solve the heat flow intensity factor by combining the singularity asymptotic expansion analysis and the equivalent geometric boundary element analysis. The application can accurately describe the heat flow singular state of the notched tip, accurately simulate a complex boundary by using fewer units, accurately calculate the heat flow intensity factor of the isotropic V-shaped notched plate, and thus can reduce the calculation amount and improve the calculation efficiency.

Description

Technical Field

[0001] This invention belongs to the field of solid heat conduction calculation, specifically a method for calculating the heat flux intensity factor based on singular asymptotic expansion and equal geometric boundary elements. Background Technology

[0002] V-shaped notches are widely present in various engineering structures. When the ambient temperature changes abruptly, a singular heat flow field exists near the notch tip, which can easily induce crack initiation and threaten structural safety. Therefore, studying the heat flow singularity at the notch tip and calculating the heat flow intensity factor characterizing this singularity is of great significance for components with notches. Although the conventional boundary element method only requires dividing the structure surface into elements, the element division process is still time-consuming and laborious for complex geometries, especially curved boundary shapes. Due to the heat flow singularity at the V-shaped notch tip, conventional boundary element interpolation cannot describe the severe heat flow concentration at the notch tip, and further subdivision of elements increases the computational load. Summary of the Invention

[0003] This invention addresses the shortcomings of existing technologies by utilizing the equal geometric boundary element method to simulate complex structural boundaries with fewer elements, and by employing singular asymptotic expansion to describe the singular state of heat flow at the cut tip. It proposes a method for calculating the heat flow intensity factor based on singular asymptotic expansion and the equal geometric boundary element method. This method aims to accurately simulate complex boundaries with fewer elements and precisely describe the singular state of heat flow at the cut tip, thereby accurately calculating the heat flow intensity factor of an isotropic V-shaped cut plate, thus reducing computational load while improving computational efficiency.

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

[0005] This invention discloses a method for calculating the heat flux intensity factor based on singular asymptotic expansion and isogeometric boundary elements. The method is used to calculate the heat flux intensity factor Ω of an isotropic V-shaped notch plate. The method is characterized by the following steps:

[0006] Step 1: Discretize the isotropic V-shaped notch plate Ω using elements;

[0007] The opening angle of the V-shaped notch plate Ω is denoted as α; with the tip of the notch as the origin O, the horizontal rightward direction as the positive x1 axis, and the vertical upward direction as the positive x2 axis, construct the Ox1x2 coordinate system;

[0008] Centered on the origin O, a sector-shaped structure with radius R and angle α is divided at the tip of the V-shaped cut plate Ω, thus dividing the V-shaped cut plate Ω into a sector-shaped structure Ω1 containing the tip of the cut and the remaining structure Ω2; the arc boundary of the sector-shaped structure Ω1 is Γ1, and the boundary of the remaining structure Ω2 is Γ2, wherein Γ2 includes the arc boundary Γ formed by Ω dividing the sector-shaped structure Ω1. 2Y and non-circular boundary Γ 2F ;

[0009] The circular arc boundary Γ1 of Ω1 is meshed using quadratic boundary elements, resulting in N1 boundary nodes and M. 1Y Unit;

[0010] Using quadratic boundary elements for the circular arc boundary Γ of Ω2 2Y The cells are divided into N1 boundary nodes and M. 2Y Units, of which M 2Y =M 1Y ;

[0011] Using quadratic boundary elements for the non-circular boundary Γ of Ω2 2F The cells are divided into N2 boundary nodes and M. 2F Unit;

[0012] The total number of boundary nodes on the boundary Γ2 of Ω2 is N = N1 + N2, and the total number of elements is M2 = M 2Y +M 2F ;

[0013] Step 2: Perform singular asymptotic expansion analysis on the sector structure Ω1 containing the notch tip;

[0014] Using equations (1.a) and (1.b), the temperature u of any boundary node P on the circular arc boundary Γ1 of the sector structure Ω1 is constructed. P and heat flow q P The asymptotic expansion of singularity:

[0015]

[0016]

[0017] In equations (1.a) and (1.b), L is the number of series terms truncated in the singular asymptotic expansion; θ P λ is the angle between the line connecting the origin O and the boundary node P and the positive direction of the x1 axis, with counterclockwise rotation being positive; k The k-th order heat flux singularity index of the cut; The k-th order temperature characteristic angle function at θ P The value at point A; k The k-th order heat flux intensity factor;

[0018] Using equations (2.a) and (2.b), the singular asymptotic expansions of temperature and heat flux at N1 boundary nodes on the circular arc boundary Γ1 of the sector structure Ω1 are obtained:

[0019] U A =[D1 D2 … D k … D L ]{A1 A2 … A k … A L} T =DA (2.a)

[0020] Q A =[E1 E2 … E k … E L ]{A1 A2 … A k … A L} T =EA (2.b)

[0021] In formula (2.a), U A D is a column vector consisting of the temperatures of N1 boundary nodes on the circular arc boundary Γ1 of the sector structure Ω1. k The k-th order heat flux intensity factor A is the temperature at each point on Γ1. k The column vector consisting of the corresponding coefficients, D L The Lth order heat flux intensity factor A is the temperature at each point on Γ1. L The column vector D, consisting of the corresponding coefficients, is composed of column vectors D1, D2, ..., D... k ... D L The matrix formed;

[0022] In equation (2.b), Q A E is a column vector consisting of the heat flow of N1 boundary nodes on the circular arc boundary Γ1 of the sector structure Ω1. k The k-th order heat flux intensity factor A is the heat flux at each point on Γ1. k The corresponding column vector of coefficients, E L The Lth order heat flux intensity factor A is the heat flux at each point on Γ1. L The column vector E, consisting of the corresponding coefficients, is composed of column vectors E1, E2, ..., E... k ..., E L The matrix formed;

[0023] In equations (2.a) and (2.b), A represents the heat flux intensity factors A1, A2, ..., A of each order. k A L The column vector formed;

[0024] Step 3: Perform isogeometric boundary element analysis on the remaining structure Ω2;

[0025] Step 3.1: Convert the temperature and heat flux of the boundary nodes and control points;

[0026] Step 3.1.1: Set control points and weighting coefficients;

[0027] Set the circular arc boundary Γ of Ω2 2Y The first and last control points of each unit coincide with the first and last boundary nodes of each unit, and the middle control point of each unit is the intersection of the tangents of the arc boundary at the first and last boundary nodes of each unit.

[0028] Set the non-circular boundary Γ of Ω2 2F The control points of each unit coincide with the boundary nodes of each unit;

[0029] Let Ξ = {ξ1, ξ2, ..., ξ} be a monotonically non-decreasing sequence of real numbers. m ,…,ξ s}, where ξ m Let m represent the m-th element, and s = 2M² + 4 be the total number of elements;

[0030] The rules for the real numbers in Ξ are: ξ1=0, ξ s =M2, when m≠1 and m≠s, in, Indicates rounding down;

[0031] Set the circular arc boundary Γ of Ω2 2Y The weighting coefficient of the intermediate control point of each unit is Set the weight coefficient of any other control point to 1;

[0032] Step 3.1.2: Use equation (3) to obtain the configuration parameter η corresponding to the m′-th boundary node on Ω2. m′ :

[0033]

[0034] In equation (3), ξ m′+i It is the m′+ith element in Ξ;

[0035] Define an integration parameter ζ that takes values ​​in the interval [0, M2]. Each value of the integration parameter corresponds to a unique boundary node on the boundary Γ2 of the remaining structure Ω2. Use equation (4) to obtain the natural coordinates of any integration parameter on the j-th element of Ω2.

[0036]

[0037] In equation (4), η 2j-1η is the value of the configuration parameter corresponding to the first boundary node of the j-th unit. 2(j+1)-1 The value of the configuration parameter corresponding to the last boundary node of the j-th unit;

[0038] Step 3.1.3: Interpolate the temperature and heat flux at all control points using equations (5.a) and (5.b) to obtain the temperature u(ζ) and heat flux q(ζ) of a boundary node corresponding to the integral parameter ζ:

[0039]

[0040]

[0041] In equations (5.a) and (5.b), u i q is the temperature at the i-th control point; i R is the heat flow at the i-th control point; i,p (ζ) is the function value of the i-th p-th non-uniform rational B-spline interpolation basis function of the integration parameter ζ, and is obtained from equation (6):

[0042]

[0043] In equation (6), p is the order of the element, ω i ω is the weight coefficient corresponding to the i-th control point. j N represents the weight coefficient corresponding to the j-th control point; j,p (ζ) represents the function value of the j-th p-th B-spline interpolation basis function with integral parameter ζ, N i,p (ζ) represents the function value of the i-th p-th B-spline interpolation basis function with integral parameter ζ, N i,p (ζ) is obtained recursively from equations (7.a) and (7.b):

[0044]

[0045]

[0046] In equations (7.a) and (7.b), t is a positive integer with a maximum value of p, and ξ i Let ξ represent the i-th element in Ξ. i+1 ξ represents the (i+1)th element in Ξ. i+t Let ξ represent the (i+t)th element in Ξ. i+t+1 N represents the (i+t+1)th element in Ξ. i,0 (ζ) represents the function value of the i-th zero-order B-spline interpolation basis function with integral parameter ζ, N i,t (ζ) is the function value of the i-th t-th B-spline interpolation basis function with integral parameter ζ, N i,t-1(ζ) represents the function value of the i-th (t-1)th B-spline interpolation basis function of the integration parameter ζ, N i+1,t-1 (ζ) is the function value of the (i+1)th (t-1)th B-spline interpolation basis function of the integration parameter ζ; in equation (7.b), when the denominator is 0, the numerator is also 0;

[0047] Step 3.1.4: Calculate the temperature and heat flow of the control points using the temperature and heat flow of the boundary nodes;

[0048] Using equations (5.a) and (5.b), the temperature and heat flux of each boundary node on the remaining structure Ω2 are expressed in terms of the temperature and heat flux of the corresponding control points, thus obtaining equations (8.a) and (8.b):

[0049]

[0050]

[0051] In formula (8.a), U BY It is the circular arc boundary Γ of the remaining structure Ω2 2Y A column vector consisting of the temperatures of each boundary node, U BF The non-circular boundary Γ of the remaining structure Ω2 2F A column vector consisting of the temperatures of each boundary node; U CY It is the circular arc boundary Γ of the remaining structure Ω2 2Y The column vector consisting of the temperatures of each control point, U CF The non-circular boundary Γ of the remaining structure Ω2 2F A column vector consisting of the temperatures of each control point;

[0052] In equation (8.b), Q BY It is the circular arc boundary Γ of the remaining structure Ω2 2Y Q is a column vector composed of the heat flow of each boundary node. BF The non-circular boundary Γ of the remaining structure Ω2 2F The column vector consisting of the heat flow of each boundary node; Q CY It is the circular arc boundary Γ of the remaining structure Ω2 2Y The column vector of heat flux at each control point, Q CF The non-circular boundary Γ of the remaining structure Ω2 2F A column vector composed of the heat flow at each control point;

[0053] In equations (8.a) and (8.b), C YY The circular arc boundary Γ represents Ω2. 2Y The upper boundary nodes and the circular arc boundary Γ 2Y The transformation matrix of each control point, C YF The circular arc boundary Γ represents Ω2. 2YThe upper boundary nodes and the non-circular boundary Γ 2F The transformation matrix of each control point, C FY Γ represents the non-circular boundary of Ω2 2F The upper boundary nodes and the circular arc boundary Γ 2Y The transformation matrix of each control point, C FF Γ represents the non-circular boundary of Ω2 2F The upper boundary nodes and the non-circular boundary Γ 2F The transformation matrices for each control point are obtained from equations (9.a), (9.b), (9.c), and (9.d), respectively:

[0054]

[0055]

[0056]

[0057]

[0058] In equations (9.a), (9.b), (9.c), and (9.d), These are the configuration parameters corresponding to the I1th boundary node. These are the configuration parameters corresponding to the N1th boundary node. These are the configuration parameters corresponding to the (N1+I2)th boundary node. These are the configuration parameters corresponding to the Nth boundary node;

[0059] In equation (9.a), It is the circular arc boundary Γ of the remaining structure Ω2 2Y Configuration parameters corresponding to the I1th boundary node The function value of the I1th p-th non-uniform rational B-spline interpolation basis function. It is Γ 2Y Configuration parameters corresponding to the N1th boundary node The function value of the N1th p-th non-uniform rational B-spline interpolation basis function;

[0060] In equation (9.b), It is the circular arc boundary Γ of the remaining structure Ω2 2Y Configuration parameters corresponding to the I1th boundary node The function value of the N1+I2th p-th non-uniform rational B-spline interpolation basis function. It is Γ 2Y Configuration parameters corresponding to the N1th boundary node The function value of the Nth p-th non-uniform rational B-spline interpolation basis function;

[0061] In equation (9.c), The non-circular boundary Γ of the remaining structure Ω2 2F Configuration parameters corresponding to the I2th boundary node The function value of the I1th p-th non-uniform rational B-spline interpolation basis function. It is Γ 2F Configuration parameters corresponding to the N2nd boundary node The function value of the N1th p-th non-uniform rational B-spline interpolation basis function;

[0062] In equation (9.d), The non-circular boundary Γ of the remaining structure Ω2 2F Configuration parameters corresponding to the I2th boundary node The function value of the N1+I2th p-th non-uniform rational B-spline interpolation basis function. It is Γ 2F Configuration parameters corresponding to the N2nd boundary node The function value of the Nth p-th non-uniform rational B-spline interpolation basis function;

[0063] C is calculated from equations (9.b) and (9.c). YF With C FY Since both are zero matrices, equations (8.a) and (8.b) can be rewritten as equations (10.a) and (10.b):

[0064]

[0065]

[0066] In equations (10.a) and (10.b), 0 YF Representing dimension and C YF The same zero matrix, 0 FY Representing dimension and C FY The same zero matrix;

[0067] The temperature and heat flux at the control point on Ω2 are obtained using equations (11.a) and (11.b):

[0068]

[0069]

[0070] In equations (11.a) and (11.b), It is C YY The inverse matrix, It is C FF The inverse matrix;

[0071] Using equations (12.a) and (12.b), the circular arc boundary Γ of the remaining structure Ω2 is obtained. 2Y Temperature U at each boundary node BY and heat flow Q BY :

[0072] U BY =U A =DA (12.a)

[0073] Q BY =-Q A =-EA (12.b)

[0074] Using equations (13.a) and (13.b), the circular arc boundary Γ of the remaining structure Ω2 is obtained. 2Y Temperature U at each control point CY and heat flow Q CY :

[0075]

[0076]

[0077] Step 3.2: Establish integral equations with equal geometric boundaries on the remaining structure Ω2;

[0078] The fundamental solutions for temperature and heat flux of the boundary integral equations are obtained using equations (14.a) and (14.b), respectively:

[0079]

[0080]

[0081] In equations (14.a) and (14.b), x is any field point, x 1x The x1 coordinate of the field point x, x 2x The x-coordinate of the field point x is x²; y is any source point, x 1y The x1 coordinate of the source point y is x. 2y y is the x2 coordinate of the source point y; π is pi, r is the distance from the source point y to the field point x, n1 is the direction cosine of the outward normal of the boundary at the field point x in the x1 direction, and n2 is the direction cosine of the outward normal of the boundary at the field point x in the x2 direction.

[0082] Based on equations (14.a) and (14.b), the integral equation of the isogeometric boundary at the source point y is established using equation (15):

[0083]

[0084] In equation (15), C(y) is the singularity coefficient at the source point y, and in, The angle between the front and rear boundary tangents at the source point y; ζ represents the element number of the element containing the source point y, and e represents the element number of the element containing the field point x; x Let ζ be the integration parameter corresponding to the field point x. y The integral parameter corresponding to the source point y; For the integration parameter ζ x The natural coordinates of R; 2e-1+l,p (ζ x ) is the basis function of the p-th non-uniform rational B-spline interpolation of the 2e-1+lth degree corresponding to the 1st boundary node in the ith element, with the integration parameter ζ. x The function value at that point, It is the first The first boundary node in the first unit corresponds to the first boundary node. The basis functions of the p-th degree non-uniform rational B-spline interpolation have the integration parameter ζ. y The function value at u; le and q le These represent the temperature and heat flux at the l-th control point in the e-th unit, respectively. for Jacobi, and obtained from equation (16):

[0085]

[0086] Step 4: Establish the system of equations for solving the heat flux intensity factor;

[0087] Step 4.1: Take the non-circular arc boundary Γ of the remaining structure Ω2. 2F Taking each boundary node on the boundary as the source point, establish the integral equation of the isogeometric boundary shown in equation (15), with H as the source point. FY U CY The corresponding non-circular boundary Γ 2F The temperature coefficient matrix above, with H FF U CF The corresponding non-circular boundary Γ 2F The temperature coefficient matrix on, with G FY Q represents CY The corresponding non-circular boundary Γ 2F The heat flux coefficient matrix on G FF Q represents CF The corresponding non-circular boundary Γ 2F The heat flux coefficient matrix is ​​obtained from the above, thus yielding equation (17):

[0088]

[0089] Substituting equations (13.a) and (13.b) into equation (17), we obtain equation (18):

[0090]

[0091] In equation (18), H FA Γ represents the non-circular boundary corresponding to A. 2F The temperature coefficient matrix on, and G FA Γ represents the non-circular boundary corresponding to A. 2F The heat flux coefficient matrix on, and

[0092] Step 4.2, at the circular arc boundary Γ of the remaining structure Ω2 2Y Among the non-first and last boundary nodes, arbitrarily select L boundary nodes as source points and establish the integral equation of the isogeometric boundary shown in equation (15) to obtain equation (19):

[0093]

[0094] In equation (19), H YA Γ represents the circular arc boundary corresponding to A. 2Y The temperature coefficient matrix on, H YF U CF Corresponding circular boundary Γ 2Y The temperature coefficient matrix on, G YA Γ represents the circular arc boundary corresponding to A. 2Y The heat flux coefficient matrix on, G YF Q represents CF Corresponding circular boundary Γ 2Y The heat flux coefficient matrix on;

[0095] Based on equations (18) and (19), the system equations are obtained using equation (20):

[0096]

[0097] Step 5: Solve the system equations using Gaussian elimination to obtain the heat flux intensity factor vector A of the isotropic V-shaped notch plate.

[0098] 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 the heat flux intensity factor calculation method based on singular asymptotic expansion and isogeometric boundary elements, and the processor is configured to execute the program stored in the memory.

[0099] The present invention discloses a computer-readable storage medium on which a computer program is stored, wherein the computer program, when executed by a processor, performs the steps of the method for calculating the heat flux intensity factor based on singular asymptotic expansion and isogeometric boundary elements.

[0100] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0101] 1. This invention uses non-uniform rational B-splines to interpolate the structural boundary. Compared with the traditional boundary element method, it uses fewer elements and provides a more accurate description of complex boundaries. This reduces the number of elements and computational cost when the boundary element method is used to study isotropic V-shaped notched plate problems.

[0102] 2. This invention divides the cut structure into two parts: a fan-shaped region at the cut tip and the remaining structure. It characterizes the temperature field and heat flow field at the cut tip through singularity feature analysis, and establishes the boundary integral equation of the remaining structure through the equal geometric boundary element method. The combination of the two establishes a system equation set, which can accurately calculate the heat flow intensity factor of the isotropic V-shaped cut plate, avoiding the huge amount of computation required by the traditional boundary element method to subdivide the elements at the cut tip. Detailed Implementation

[0103] In this embodiment, a method for calculating the heat flux intensity factor based on singular asymptotic expansion and isogeometric boundary elements is performed according to the following steps:

[0104] Step 1: Discretize the isotropic V-shaped notch plate Ω using elements;

[0105] The opening angle of the V-shaped notch plate Ω is denoted as α; with the tip of the notch as the origin O, the horizontal rightward direction as the positive x1 axis, and the vertical upward direction as the positive x2 axis, construct the Ox1x2 coordinate system;

[0106] Centered on the origin O, a sector-shaped structure with radius R and angle α is divided at the tip of the V-shaped cut plate Ω, thus dividing the V-shaped cut plate Ω into a sector-shaped structure Ω1 containing the tip of the cut and the remaining structure Ω2; the arc boundary of the sector-shaped structure Ω1 is Γ1, and the boundary of the remaining structure Ω2 is Γ2, wherein Γ2 includes the arc boundary Γ formed by Ω dividing the sector-shaped structure Ω1. 2Y and non-circular boundary Γ 2F ;

[0107] The circular arc boundary Γ1 of Ω1 is meshed using quadratic boundary elements, resulting in N1 boundary nodes and M. 1Y Unit;

[0108] Using quadratic boundary elements for the circular arc boundary Γ of Ω2 2Y The cells are divided into N1 boundary nodes and M. 2Y Units, of which M 2Y =M 1Y The non-circular boundary Γ of Ω2 is constructed using quadratic boundary elements. 2F The cells are divided into N2 boundary nodes and M. 2F Unit;

[0109] The total number of boundary nodes on the boundary Γ2 of Ω2 is N = N1 + N2, and the total number of elements is M2 = M 2Y +M 2F ;

[0110] Step 2: Perform singular asymptotic expansion analysis on the sector structure Ω1 containing the notch tip;

[0111] The cut tip exhibits severe thermal flow singularity. Using equations (1.a) and (1.b), the temperature u at any boundary node P on the circular arc boundary Γ1 of the sector structure Ω1 is constructed. P and heat flow q P The asymptotic expansion of singularity:

[0112]

[0113]

[0114] In equations (1.a) and (1.b), L is the number of series terms truncated in the singular asymptotic expansion; θ P λ is the angle between the line connecting the origin O and the boundary node P and the positive direction of the x1 axis, with counterclockwise rotation being positive; k The k-th order heat flux singularity index of the cut; The k-th order temperature characteristic angle function at θ P The value at point A; k The k-th order heat flux intensity factor;

[0115] Using equations (2.a) and (2.b), the singular asymptotic expansions of temperature and heat flux at N1 boundary nodes on the circular arc boundary Γ1 of the sector structure Ω1 are obtained:

[0116] U A =[D1 D2 … D k … D L ]{A1 A2 … A k … A L} T =DA (2.a)

[0117] Q A =[E1 E2 … E k … E L ]{A1 A2 … A k … A L} T =EA (2.b)

[0118] In formula (2.a), U A D is a column vector consisting of the temperatures of N1 boundary nodes on the circular arc boundary Γ1 of the sector structure Ω1.k The k-th order heat flux intensity factor A is the temperature at each point on Γ1. k The column vector consisting of the corresponding coefficients, D L The Lth order heat flux intensity factor A is the temperature at each point on Γ1. L The column vector D, consisting of the corresponding coefficients, is composed of column vectors D1, D2, ..., D... k ... D L The matrix formed;

[0119] In equation (2.b), Q A E is a column vector consisting of the heat flow of N1 boundary nodes on the circular arc boundary Γ1 of the sector structure Ω1. k The k-th order heat flux intensity factor A is the heat flux at each point on Γ1. k The corresponding column vector of coefficients, E L The Lth order heat flux intensity factor A is the heat flux at each point on Γ1. L The column vector E, consisting of the corresponding coefficients, is composed of column vectors E1, E2, ..., E... k ..., E L The matrix formed;

[0120] In equations (2.a) and (2.b), A represents the heat flux intensity factors A1, A2, ..., A of each order. k A L The column vector formed;

[0121] Step 3: Perform isogeometric boundary element analysis on the remaining structure Ω2;

[0122] Step 3.1: Convert the temperature and heat flux of the boundary nodes and control points;

[0123] Step 3.1.1: Set control points and weighting coefficients;

[0124] Set the circular arc boundary Γ of Ω2 2Y The first and last control points of each unit coincide with the first and last boundary nodes of each unit, and the middle control point of each unit is the intersection of the tangents of the arc boundary at the first and last boundary nodes of each unit.

[0125] Set the non-circular boundary Γ of Ω2 2F The control points of each unit coincide with the boundary nodes of each unit;

[0126] According to the definition of control points, the total number of control points is the same as the total number of boundary nodes on the boundary Γ2 of Ω2, which is also N;

[0127] Let Ξ = {ξ1, ξ2, ..., ξ} be a monotonically non-decreasing sequence of real numbers. m ,...,ξ s}, where ξ mLet m represent the m-th element, and s = 2M² + 4 be the total number of elements;

[0128] The rules for the real numbers in Ξ are: ξ1=0, ξ s =M2, when m≠1 and m≠s, in, Indicates rounding down;

[0129] Set the circular arc boundary Γ of Ω2 2Y The weighting coefficient of the intermediate control point of each unit is Set the weight coefficient of any other control point to 1; since each control point corresponds to a weight coefficient, the total number of weight coefficients is also N;

[0130] Step 3.1.2: Use equation (3) to obtain the configuration parameter η corresponding to the m′-th boundary node on Ω2. m′ :

[0131]

[0132] In equation (3), ξ m′+i It is the m′+ith element in Ξ;

[0133] Define an integration parameter ζ that takes values ​​in the interval [0, M2]. Each value of the integration parameter corresponds to a unique boundary node on the boundary Γ2 of the remaining structure Ω2. Use equation (4) to obtain the natural coordinates of any integration parameter on the j-th element of Ω2.

[0134]

[0135] In equation (4), η 2j-1 η is the value of the configuration parameter corresponding to the first boundary node of the j-th unit. 2(j+1)-1 Let be the value of the configuration parameter corresponding to the final boundary node of the j-th element; the value range of any integral parameter on the j-th element is [η]. 2(j+1)-1 ,η 2j-1 ], natural coordinates of the integration parameters The range of its value is [-1, 1];

[0136] Step 3.1.3: Interpolate the temperature and heat flux at all control points using equations (5.a) and (5.b) to obtain the temperature u(ζ) and heat flux q(ζ) of a boundary node corresponding to the integral parameter ζ:

[0137]

[0138]

[0139] In equations (5.a) and (5.b), ui q is the temperature at the i-th control point; i R is the heat flow at the i-th control point; i,p (ζ) is the function value of the i-th p-th non-uniform rational B-spline interpolation basis function of the integration parameter ζ, and is obtained from equation (6):

[0140]

[0141] In equation (6), p is the order of the element. Since all elements used in this invention are quadratic boundary elements, p = 2; ω i ω is the weight coefficient corresponding to the i-th control point. j N represents the weight coefficient corresponding to the j-th control point; j,p (ζ) represents the function value of the j-th p-th B-spline interpolation basis function with integral parameter ζ, N i,p (ζ) represents the function value of the i-th p-th B-spline interpolation basis function with integral parameter ζ, N i,p (ζ) is obtained recursively from equations (7.a) and (7.b):

[0142]

[0143]

[0144] In equations (7.a) and (7.b), t is a positive integer with a maximum value of p, and ξ i Let ξ represent the i-th element in Ξ. i+1 ξ represents the (i+1)th element in Ξ. i+t Let ξ represent the (i+t)th element in Ξ. i+t+1 N represents the (i+t+1)th element in Ξ. i,0 (ζ) represents the function value of the i-th zero-order B-spline interpolation basis function with integral parameter ζ, N i,t (ζ) is the function value of the i-th t-th B-spline interpolation basis function with integral parameter ζ, N i,t-1 (ζ) represents the function value of the i-th (t-1)th B-spline interpolation basis function of the integration parameter ζ, N i+1,t-1 (ζ) is the function value of the (i+1)th t-1th B-spline interpolation basis function of the integration parameter ζ; in equation (7.b), when the denominator is 0, the numerator is also 0, that is, for the cases of 0 / 0 or 1 / 0, 0 is used as the value of the numerator to continue calculating the function value;

[0145] Step 3.1.4: Calculate the temperature and heat flow of the control points using the temperature and heat flow of the boundary nodes;

[0146] Using equations (5.a) and (5.b), the temperature and heat flux of each boundary node on the remaining structure Ω2 are expressed in terms of the temperature and heat flux of the corresponding control points, thus obtaining equations (8.a) and (8.b):

[0147]

[0148]

[0149] In formula (8.a), U BY It is the circular arc boundary Γ of the remaining structure Ω2 2Y A column vector consisting of the temperatures of each boundary node, U BF The non-circular boundary Γ of the remaining structure Ω2 2F A column vector consisting of the temperatures of each boundary node; U CY It is the circular arc boundary Γ of the remaining structure Ω2 2Y The column vector consisting of the temperatures of each control point, U CF The non-circular boundary Γ of the remaining structure Ω2 2F A column vector consisting of the temperatures of each control point;

[0150] In equation (8.b), Q BY It is the circular arc boundary Γ of the remaining structure Ω2 2Y Q is a column vector composed of the heat flow of each boundary node. BF The non-circular boundary Γ of the remaining structure Ω2 2F The column vector consisting of the heat flow of each boundary node; Q CY It is the circular arc boundary Γ of the remaining structure Ω2 2Y The column vector of heat flux at each control point, Q CF The non-circular boundary Γ of the remaining structure Ω2 2F A column vector composed of the heat flow at each control point;

[0151] In equations (8.a) and (8.b), C YY The circular arc boundary Γ represents Ω2. 2Y The upper boundary nodes and the circular arc boundary Γ 2Y The transformation matrix of each control point, C YF The circular arc boundary Γ represents Ω2. 2Y The upper boundary nodes and the non-circular boundary Γ 2F The transformation matrix of each control point, C FY Γ represents the non-circular boundary of Ω2 2F The upper boundary nodes and the circular arc boundary Γ 2Y The transformation matrix of each control point, C FF Γ represents the non-circular boundary of Ω2 2F The upper boundary nodes and the non-circular boundary Γ 2F The transformation matrices for each control point are obtained from equations (9.a), (9.b), (9.c), and (9.d), respectively:

[0152]

[0153]

[0154]

[0155]

[0156] In equations (9.a), (9.b), (9.c), and (9.d), These are the configuration parameters corresponding to the I1th boundary node. These are the configuration parameters corresponding to the N1th boundary node. These are the configuration parameters corresponding to the (N1+I2)th boundary node. These are the configuration parameters corresponding to the Nth boundary node;

[0157] In equation (9.a), It is the circular arc boundary Γ of the remaining structure Ω2 2Y Configuration parameters corresponding to the I1th boundary node The function value of the I1th p-th non-uniform rational B-spline interpolation basis function. It is Γ 2Y Configuration parameters corresponding to the N1th boundary node The function value of the N1th p-th non-uniform rational B-spline interpolation basis function;

[0158] In equation (9.b), It is the circular arc boundary Γ of the remaining structure Ω2 2Y Configuration parameters corresponding to the I1th boundary node The function value of the N1+I2th p-th non-uniform rational B-spline interpolation basis function. It is Γ 2Y Configuration parameters corresponding to the N1th boundary node The function value of the Nth p-th non-uniform rational B-spline interpolation basis function;

[0159] In equation (9.c), The non-circular boundary Γ of the remaining structure Ω2 2F Configuration parameters corresponding to the I2th boundary node The function value of the I1th p-th non-uniform rational B-spline interpolation basis function. It is Γ 2F Configuration parameters corresponding to the N2nd boundary node The function value of the N1th p-th non-uniform rational B-spline interpolation basis function;

[0160] In equation (9.d), The non-circular boundary Γ of the remaining structure Ω22F Configuration parameters corresponding to the I2th boundary node The function value of the N1+I2th p-th non-uniform rational B-spline interpolation basis function. It is Γ 2F Configuration parameters corresponding to the N2nd boundary node The function value of the Nth p-th non-uniform rational B-spline interpolation basis function;

[0161] C is calculated from equations (9.b) and (9.c). YF With C FY Since both are zero matrices, equations (8.a) and (8.b) can be rewritten as equations (10.a) and (10.b):

[0162]

[0163]

[0164] In equations (10.a) and (10.b), 0 YF Representing dimension and C YF The same zero matrix, 0 FY Representing dimension and C FY The same zero matrix;

[0165] Based on equations (10.a) and (10.b), the temperature and heat flux at the control point on Ω2 are obtained using equations (11.a) and (11.b):

[0166]

[0167]

[0168] In equations (11.a) and (11.b), It is C YY The inverse matrix, It is C FF The inverse matrix;

[0169] The temperature U at the upper boundary node of the arc-shaped boundary Γ1 of the fan-shaped structure Ω1 A Heat flow Q A The circular arc boundary Γ of the remaining structure Ω2 2Y Temperature U at the upper boundary node BY Heat flow Q BY It has a continuity relationship; using equations (12.a) and (12.b), the circular arc boundary Γ of the remaining structure Ω2 is obtained. 2Y Temperature U at each boundary node BY and heat flow Q BY :

[0170] UBY =U A =DA (12.a)

[0171] Q BY =-Q A =-EA (12.b)

[0172] Substituting equation (12.a) into equation (11.a) and equation (12.b) into equation (11.b), we can then use equations (13.a) and (13.b) to obtain the circular arc boundary Γ of the remaining structure Ω2. 2Y Temperature U at each control point CY and heat flow Q CY :

[0173]

[0174]

[0175] Step 3.2: Establish integral equations with equal geometric boundaries on the remaining structure Ω2;

[0176] For the heat conduction problem of isotropic materials, the fundamental solutions of temperature and heat flow for the boundary integral equations are obtained using equations (14.a) and (14.b), respectively:

[0177]

[0178]

[0179] In equations (14.a) and (14.b), x is any field point, x 1x The x1 coordinate of the field point x, x 2x The x-coordinate of the field point x is x²; y is any source point, x 1y The x1 coordinate of the source point y is x. 2y y is the x2 coordinate of the source point y; π is pi, r is the distance from the source point y to the field point x, n1 is the direction cosine of the outward normal of the boundary at the field point x in the x1 direction, and n2 is the direction cosine of the outward normal of the boundary at the field point x in the x2 direction.

[0180] Based on equations (14.a) and (14.b), the integral equation of the isogeometric boundary at the source point y is established using equation (15):

[0181]

[0182] In equation (15), C(y) is the singularity coefficient at the source point y, and in, The angle between the front and rear boundary tangents at the source point y; ζ represents the element number of the element containing the source point y, and e represents the element number of the element containing the field point x; x Let ζ be the integration parameter corresponding to the field point x. y The integral parameter corresponding to the source point y; For the integration parameter ζ x The natural coordinates of R; 2e-1+l,p (ζ x ) is the basis function of the p-th non-uniform rational B-spline interpolation of the 1-th boundary node in the 1-th element, with the integration parameter ζ. x The function value at that point, It is the first The first boundary node in the first unit corresponds to the first boundary node. The basis functions of the p-th degree non-uniform rational B-spline interpolation have the integration parameter ζ. y The function value at u; le and q le These represent the temperature and heat flux at the l-th control point in the e-th unit, respectively. for Jacobi, and obtained from equation (16):

[0183]

[0184] Step 4: Establish the system of equations for solving the heat flux intensity factor;

[0185] Step 4.1: Take the non-circular arc boundary Γ of the remaining structure Ω2. 2F Taking each boundary node on the boundary as the source point, establish the integral equation of the isogeometric boundary shown in equation (15), with H as the source point. FY U CY The corresponding non-circular boundary Γ 2F The temperature coefficient matrix above, with H FF U CF The corresponding non-circular boundary Γ 2F The temperature coefficient matrix on, with G FY Q represents CY The corresponding non-circular boundary Γ 2F The heat flux coefficient matrix on G FF Q represents CF The corresponding non-circular boundary Γ 2F The heat flux coefficient matrix is ​​obtained from the above, thus yielding equation (17):

[0186]

[0187] Substituting equations (13.a) and (13.b) into equation (17), we obtain equation (18):

[0188]

[0189] In equation (18), H FA Γ represents the non-circular boundary corresponding to A. 2F The temperature coefficient matrix on, and G FA Γ represents the non-circular boundary corresponding to A. 2F The heat flux coefficient matrix on, and

[0190] Step 4.2, at the circular arc boundary Γ of the remaining structure Ω2 2Y Among the non-first and last boundary nodes, arbitrarily select L boundary nodes as source points and establish the integral equation of the isogeometric boundary shown in equation (15) to obtain equation (19):

[0191]

[0192] In equation (19), H YA Γ represents the circular arc boundary corresponding to A. 2Y The temperature coefficient matrix on, H YF U CF Corresponding circular boundary Γ 2Y The temperature coefficient matrix on, G YA Γ represents the circular arc boundary corresponding to A. 2Y The heat flux coefficient matrix on, G YF Q represents CF Corresponding circular boundary Γ 2Y The heat flux coefficient matrix on;

[0193] Based on equations (18) and (19), the system equations are obtained using equation (20):

[0194]

[0195] Step 5: Solve the system equations using Gaussian elimination to obtain the heat flux intensity factor vector A of the isotropic V-shaped notch plate.

[0196] The system equation set (20) includes both known temperature and heat flow, as well as unknown temperature and heat flow. During calculation, the known temperature and heat flow are moved to the right side of the equation set, and the unknown temperature, heat flow, and heat flow intensity factor are moved to the left side of the equation set. By solving this equation set, the unknown temperature, heat flow, and heat flow intensity factor can be obtained.

[0197] 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.

[0198] 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 heat flux intensity factor based on singular asymptotic expansion and isogeometric boundary elements, used to calculate the heat flux intensity factor Ω of an isotropic V-shaped notch plate, characterized in that, The heat flux intensity factor is calculated according to the following steps: Step 1: Discretize the isotropic V-shaped notch plate Ω using elements; The opening angle of the V-shaped notch plate Ω is denoted as α; with the tip of the notch as the origin O, the horizontal rightward direction as the positive x1 axis, and the vertical upward direction as the positive x2 axis, construct the Ox1x2 coordinate system; Centered on the origin O, a sector-shaped structure with radius R and angle α is divided at the tip of the V-shaped cut plate Ω, thus dividing the V-shaped cut plate Ω into a sector-shaped structure Ω1 containing the tip of the cut and the remaining structure Ω2; the arc boundary of the sector-shaped structure Ω1 is Γ1, and the boundary of the remaining structure Ω2 is Γ2, wherein Γ2 includes the arc boundary Γ formed by Ω dividing the sector-shaped structure Ω1. 2Y and non-circular boundary Γ 2F ; The circular arc boundary Γ1 of Ω1 is meshed using quadratic boundary elements, resulting in N1 boundary nodes and M. 1Y Units; Using quadratic boundary elements for the circular arc boundary Γ of Ω2 2Y The cells are divided into N1 boundary nodes and M. 2Y Units, of which M 2Y =M 1Y ; Using quadratic boundary elements for the non-circular boundary Γ of Ω2 2F The cells are divided into N2 boundary nodes and M. 2F Units; The total number of boundary nodes on the boundary Γ2 of Ω2 is N = N1 + N2, and the total number of elements is M2 = M 2Y +M 2F ; Step 2: Perform singular asymptotic expansion analysis on the sector structure Ω1 containing the notch tip; Using equations (1.a) and (1.b), the temperature u of any boundary node P on the circular arc boundary Γ1 of the sector structure Ω1 is constructed. P and heat flow q P The singularity asymptotic expansion: In equations (1.a) and (1.b), L is the number of series terms truncated in the singular asymptotic expansion; θ P λ is the angle between the line connecting the origin O and the boundary node P and the positive direction of the x1 axis, with counterclockwise rotation being positive; k The k-th order heat flux singularity index of the cut; The k-th order temperature characteristic angle function at θ P The value at point A; k The k-th order heat flux intensity factor; Using equations (2.a) and (2.b), the singular asymptotic expansions of temperature and heat flux at N1 boundary nodes on the circular arc boundary Γ1 of the sector structure Ω1 are obtained: U A =[D1 D2 … D k … D L ]{A1 A2 … A k … A L } T =DA (2.a) Q A =[E1 E2 … E k … AND L ]{A1 A2 … A k … TO L } T =EA (2.b) In formula (2.a), U A D is a column vector consisting of the temperatures of N1 boundary nodes on the circular arc boundary Γ1 of the sector structure Ω1. k The k-th order heat flux intensity factor A is the temperature at each point on Γ1. k The column vector consisting of the corresponding coefficients, D L The Lth order heat flux intensity factor A is the temperature at each point on Γ1. L The column vector D, consisting of the corresponding coefficients, is composed of column vectors D1, D2, ..., D... k ... D L The matrix formed; In equation (2.b), Q A E is a column vector consisting of the heat flow of N1 boundary nodes on the circular arc boundary Γ1 of the sector structure Ω1. k The k-th order heat flux intensity factor A is the heat flux at each point on Γ1. k The corresponding column vector of coefficients, E L The Lth order heat flux intensity factor A is the heat flux at each point on Γ1. L The column vector E, consisting of the corresponding coefficients, is composed of column vectors E1, E2, ..., E... k ... E L The matrix formed; In equations (2.a) and (2.b), A represents the heat flux intensity factors A1, A2, ..., A of each order. k A L The column vector formed; Step 3: Perform isogeometric boundary element analysis on the remaining structure Ω2; Step 3.1: Convert the temperature and heat flux of the boundary nodes and control points; Step 3.1.1: Set control points and weighting coefficients; Set the circular arc boundary Γ of Ω2 2Y The first and last control points of each unit coincide with the first and last boundary nodes of each unit, and the middle control point of each unit is the intersection of the tangents of the circular arc boundary at the first and last boundary nodes of each unit. Set the non-circular boundary Γ of Ω2 2F The control points of each unit coincide with the boundary nodes of each unit; Let Ξ = {ξ1, ξ2, ..., ξ} be a monotonically non-decreasing sequence of real numbers. m ,…,ξ s }, where ξ m Let m represent the m-th element, and s = 2M² + 4 be the total number of elements; The rules for the real numbers in Ξ are: ξ1=0, ξ s =M2, when m≠1 and m≠s, in, Indicates rounding down; Set the circular arc boundary Γ of Ω2 2Y The weighting coefficient of the intermediate control point of each unit is Set the weight coefficient of any other control point to 1; Step 3.1.2: Use equation (3) to obtain the configuration parameter η corresponding to the m′-th boundary node on Ω2. m′ : In equation (3), ξ m′+i It is the m′+ith element in Ξ; Define an integration parameter ζ that takes values ​​in the interval [0, M2]. Each value of the integration parameter corresponds to a unique boundary node on the boundary Γ2 of the remaining structure Ω2. Use equation (4) to obtain the natural coordinates of any integration parameter on the j-th element of Ω2. In equation (4), η 2j-1 η is the value of the configuration parameter corresponding to the first boundary node of the j-th unit. 2(j+1)-1 The value of the configuration parameter corresponding to the last boundary node of the j-th unit; Step 3.1.3: Interpolate the temperature and heat flux at all control points using equations (5.a) and (5.b) to obtain the temperature u(ζ) and heat flux q(ζ) of a boundary node corresponding to the integral parameter ζ: In equations (5.a) and (5.b), u i q is the temperature at the i-th control point; i R is the heat flow at the i-th control point; i,p (ζ) is the function value of the i-th p-th non-uniform rational B-spline interpolation basis function of the integration parameter ζ, and is obtained from equation (6): In equation (6), p is the order of the element, ω i ω is the weight coefficient corresponding to the i-th control point. j N represents the weight coefficient corresponding to the j-th control point; j,p (ζ) is the function value of the j-th p-th B-spline interpolation basis function with integral parameter ζ, N i,p (ζ) is the function value of the i-th p-th B-spline interpolation basis function with integral parameter ζ, N i,p (ζ) is obtained recursively from equations (7.a) and (7.b): In equations (7.a) and (7.b), t is a positive integer with a maximum value of p, and ξ i Let ξ represent the i-th element in Ξ. i+1 ξ represents the (i+1)th element in Ξ. i+t Let ξ represent the (i+t)th element in Ξ. i+t+1 N represents the (i+t+1)th element in Ξ. i,0 (ζ) represents the function value of the i-th zero-order B-spline interpolation basis function with integral parameter ζ, N i,t (ζ) represents the function value of the i-th t-th B-spline interpolation basis function with integral parameter ζ, N i,t-1 (ζ) represents the function value of the i-th (t-1)th B-spline interpolation basis function of the integration parameter ζ, N i+1,t-1 (ζ) is the function value of the (i+1)th (t-1)th B-spline interpolation basis function of the integration parameter ζ; in equation (7.b), when the denominator is 0, the numerator is also 0; Step 3.1.4: Calculate the temperature and heat flow of the control points using the temperature and heat flow of the boundary nodes; Using equations (5.a) and (5.b), the temperature and heat flux of each boundary node on the remaining structure Ω2 are expressed in terms of the temperature and heat flux of the corresponding control points, thus obtaining equations (8.a) and (8.b): In formula (8.a), U BY It is the circular arc boundary Γ of the remaining structure Ω2 2Y A column vector consisting of the temperatures of each boundary node, U BF The non-circular boundary Γ of the remaining structure Ω2 2F A column vector consisting of the temperatures of each boundary node; U CY It is the circular arc boundary Γ of the remaining structure Ω2 2Y The column vector consisting of the temperatures of each control point, U CF The non-circular boundary Γ of the remaining structure Ω2 2F A column vector consisting of the temperatures of each control point; In equation (8.b), Q BY It is the circular arc boundary Γ of the remaining structure Ω2 2Y Q is a column vector composed of the heat flow of each boundary node. BF The non-circular boundary Γ of the remaining structure Ω2 2F The column vector consisting of the heat flow of each boundary node; Q CY It is the circular arc boundary Γ of the remaining structure Ω2 2Y The column vector of heat fluxes at each control point, Q CF The non-circular boundary Γ of the remaining structure Ω2 2F A column vector composed of heat flow at each control point; In equations (8.a) and (8.b), C YY The circular arc boundary Γ represents Ω2. 2Y The upper boundary nodes and the circular arc boundary Γ 2Y The transformation matrix of each control point, C YF The circular arc boundary Γ represents Ω2. 2Y The upper boundary nodes and the non-circular boundary Γ 2F The transformation matrix of each control point, C FY Γ represents the non-circular boundary of Ω2 2F The upper boundary nodes and the circular arc boundary Γ 2Y The transformation matrix of each control point, C FF Γ represents the non-circular boundary of Ω2 2F The upper boundary nodes and the non-circular boundary Γ 2F The transformation matrices for each control point are obtained from equations (9.a), (9.b), (9.c), and (9.d), respectively: In equations (9.a), (9.b), (9.c), and (9.d), These are the configuration parameters corresponding to the I1th boundary node. These are the configuration parameters corresponding to the N1th boundary node. These are the configuration parameters corresponding to the (N1+I2)th boundary node. These are the configuration parameters corresponding to the Nth boundary node; In equation (9.a), It is the circular arc boundary Γ of the remaining structure Ω2 2Y Configuration parameters corresponding to the I1th boundary node The function value of the I1th p-th non-uniform rational B-spline interpolation basis function. It is Γ 2Y Configuration parameters corresponding to the N1th boundary node The function value of the N1th p-th non-uniform rational B-spline interpolation basis function; In equation (9.b), It is the circular arc boundary Γ of the remaining structure Ω2 2Y Configuration parameters corresponding to the I1th boundary node The function value of the N1+I2th p-th non-uniform rational B-spline interpolation basis function. It is Γ 2Y Configuration parameters corresponding to the N1th boundary node The function value of the Nth p-th non-uniform rational B-spline interpolation basis function; In equation (9.c), The non-circular boundary Γ of the remaining structure Ω2 2F Configuration parameters corresponding to the I2th boundary node The function value of the I1th p-th non-uniform rational B-spline interpolation basis function. It is Γ 2F Configuration parameters corresponding to the N2nd boundary node The function value of the N1th p-th non-uniform rational B-spline interpolation basis function; In equation (9.d), The non-circular boundary Γ of the remaining structure Ω2 2F Configuration parameters corresponding to the I2th boundary node The function value of the N1+I2th p-th non-uniform rational B-spline interpolation basis function. It is Γ 2F Configuration parameters corresponding to the N2nd boundary node The function value of the Nth p-th non-uniform rational B-spline interpolation basis function; C is calculated from equations (9.b) and (9.c). YF With C FY Since both are zero matrices, equations (8.a) and (8.b) can be rewritten as equations (10.a) and (10.b): In equations (10.a) and (10.b), 0 YF Representing dimension and C YF The same zero matrix, 0 FY Representing dimension and C FY The same zero matrix; The temperature and heat flux at the control point on Ω2 are obtained using equations (11.a) and (11.b): In equations (11.a) and (11.b), It is C YY The inverse matrix, It is C FF The inverse matrix; Using equations (12.a) and (12.b), the circular arc boundary Γ of the remaining structure Ω2 is obtained. 2Y Temperature U at each boundary node BY and heat flow Q BY : The BY =U A =DA(12.a) Q BY =-Q A =-EA(12.b) Using equations (13.a) and (13.b), the circular arc boundary Γ of the remaining structure Ω2 is obtained. 2Y Temperature U at each control point CY and heat flow Q CY : Step 3.2: Establish integral equations with equal geometric boundaries on the remaining structure Ω2; The fundamental solutions for temperature and heat flux of the boundary integral equations are obtained using equations (14.a) and (14.b), respectively: In equations (14.a) and (14.b), x is any field point, x 1x The x1 coordinate of the field point x, x 2x The x-coordinate of the field point x is x²; y is any source point, x 1y The x1 coordinate of the source point y is x. 2y y is the x2 coordinate of the source point y; π is pi, r is the distance from the source point y to the field point x, n1 is the direction cosine of the outward normal of the boundary at the field point x in the x1 direction, and n2 is the direction cosine of the outward normal of the boundary at the field point x in the x2 direction. Based on equations (14.a) and (14.b), the integral equation of the isogeometric boundary at the source point y is established using equation (15): In equation (15), C(y) is the singularity coefficient at the source point y, and in, The angle between the front and rear boundary tangents at the source point y; ζ represents the element number of the element containing the source point y, and e represents the element number of the element containing the field point x; x Let ζ be the integration parameter corresponding to the field point x. y The integral parameter corresponding to the source point y; For the integration parameter ζ x The natural coordinates of R; 2e-1+l,p (ζ x ) is the basis function of the p-th non-uniform rational B-spline interpolation of the 2e-1+lth degree corresponding to the l-th boundary node in the e-th element, with the integration parameter ζ. x The function value at that point, It is the first The first boundary node in the first unit corresponds to the first boundary node. The basis functions of the p-th degree non-uniform rational B-spline interpolation have the integration parameter ζ. y The function value at u; le and q le These represent the temperature and heat flux at the l-th control point in the e-th unit, respectively. for Jacobi, and obtained from equation (16): Step 4: Establish the system of equations for solving the heat flux intensity factor; Step 4.1: Take the non-circular arc boundary Γ of the remaining structure Ω2. 2F Taking each boundary node on the boundary as the source point, establish the integral equation of the isogeometric boundary shown in equation (15), with H as the source point. FY U CY The corresponding non-circular boundary Γ 2F The temperature coefficient matrix above, with H FF U CF The corresponding non-circular boundary Γ 2F The temperature coefficient matrix on, with G FY Q represents CY The corresponding non-circular boundary Γ 2F The heat flux coefficient matrix on G FF Q represents CF The corresponding non-circular boundary Γ 2F The heat flux coefficient matrix is ​​obtained from the above, thus yielding equation (17): Substituting equations (13.a) and (13.b) into equation (17), we obtain equation (18): In equation (18), H FA Γ represents the non-circular boundary corresponding to A. 2F The temperature coefficient matrix on, and G FA Γ represents the non-circular boundary corresponding to A. 2F The heat flux coefficient matrix on, and Step 4.2, at the circular arc boundary Γ of the remaining structure Ω2 2Y Among the non-first and last boundary nodes, arbitrarily select L boundary nodes as source points and establish the isogeometric boundary integral equation shown in equation (15) to obtain equation (19): In equation (19), H YA Γ represents the circular arc boundary corresponding to A. 2Y The temperature coefficient matrix on, H YF U CF Corresponding circular boundary Γ 2Y The temperature coefficient matrix on, G YA Γ represents the circular arc boundary corresponding to A. 2Y The heat flux coefficient matrix on, G YF Q represents CF Corresponding circular boundary Γ 2Y The heat flux coefficient matrix on; Based on equations (18) and (19), the system equations are obtained using equation (20): Step 5: Solve the system equations using Gaussian elimination to obtain the heat flux intensity factor vector A of the isotropic V-shaped notch plate.

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 heat flux intensity factor calculation method based on singular asymptotic expansion and isogeometric boundary elements 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, The computer program, when run by the processor, executes the steps of the heat flux intensity factor calculation method based on singular asymptotic expansion and isogeometric boundary elements as described in claim 1.

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