Method and device for determining field domain temperature based on weak form generalized finite difference method

Abstract

The invention relates to the technical field of heat conduction simulation, in particular to a method and device for determining field domain temperature based on a weak form generalized finite difference method. The method provided by the invention comprises the following steps: establishing a geometric structure model of a to-be-analyzed object; discretizing the geometric structure model into a plurality of nodes to obtain a discrete model; giving initial conditions and boundary conditions to the discrete model; converting the heat conduction differential control equation into a local weak form of the control equation; constructing an integral domain of each node, and arranging Gaussian points according to the local weak form of the control equation and the integral domain; approximating the temperature and the temperature derivative of any point in the field domain through a generalized finite difference method; determining a discrete matrix equation according to the temperature and the temperature derivative of any point in the field domain and the local weak form and the boundary condition of a control equation; based on initial conditions and boundary conditions, the temperature of each node at any moment is calculated by using a discrete matrix equation so as to improve the precision and stability of heat conduction solution.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of heat conduction simulation, in particular to a method and device for determining temperature of a field based on a weak form generalized finite difference method. BACKGROUND

[0002] As a basic way of heat transfer, heat conduction exists widely in nature and engineering, from macro building energy saving to micro electronic device heat dissipation, and its accurate simulation is crucial for the optimal design of thermal management system. Numerical solution of heat conduction process can predict the temperature field distribution of target objects in complex scenarios, providing basis for engineering thermal design. Traditional heat conduction problems are described by heat conduction partial differential equations to evolve temperature field, and numerical solution is mainly based on grid method. As a classical meshless method, the generalized finite difference method is easier to discretize complex objects than the grid method because it does not need grid. It is simple and efficient in calculation. However, due to its main use of strong form, it is prone to problems such as insufficient numerical stability, poor adaptability to complex boundaries, and limited precision in processing derivative boundary conditions when facing complex objects and boundaries, which restricts its application in high-precision thermal analysis. SUMMARY

[0003] Therefore, the present application provides a method and device for determining temperature of a field based on a weak form generalized finite difference method, which can more simply discretize target objects and improve the precision and stability of heat conduction solution.

[0004] Specifically, the present application is realized by the following technical solutions:

[0005] The first aspect of the present application provides a method for determining temperature of a field based on a weak form generalized finite difference method, which comprises:

[0006] establishing a geometric structure model of an object to be analyzed, and discretizing the geometric structure model into a plurality of nodes to obtain a discrete model corresponding to the geometric structure model; the geometric structure model is used to describe heat sources, shapes, positions and material parameters of the heat sources in the field;

[0007] assigning initial conditions and boundary conditions to the discrete model;

[0008] converting the heat conduction differential control equation corresponding to the discrete model into a local weak form of the control equation; the local weak form of the control equation is used to describe the energy conservation relationship among the temperature changes, boundary heat exchange and internal heat sources in the field;

[0009] constructing an integral domain for each node based on the average node spacing, and arranging Gauss points on the boundary of each integral domain according to the local weak form of the control equation and the integral domain;

[0010] approximate the temperature and temperature derivative of any point in the field based on the coordinates of each Gauss point; wherein the temperature and temperature derivative of any point are represented by a linear combination of the node temperature parameters in the nebula;

[0011] determine a discrete matrix equation corresponding to heat conduction based on the temperature and temperature derivative of any point in the field, the local weak form of the control equation and the boundary conditions; the discrete matrix equation is used to describe how the temperature at all discrete points in the field changes over time;

[0012] based on the initial conditions and the boundary conditions, the temperature of each node at any time is obtained by time iteration calculation using the discrete matrix equation.

[0013] The second aspect of the application provides a device for determining the temperature of a field based on a weak form of a generalized finite difference method, the device comprising a modeling module, a generating module, a processing module and an updating module; wherein,

[0014] The modeling module is used to establish a geometric structure model of an object to be analyzed, and discretize the geometric structure model into a plurality of nodes to obtain a discrete model corresponding to the geometric structure model; the geometric structure model is used to describe the heat source, shape, position and material parameters of the heat source in the field;

[0015] The generating module is used to assign initial conditions and boundary conditions to the discrete model;

[0016] The processing module is used to convert the heat conduction differential control equation corresponding to the discrete model into a local weak form of the control equation; the local weak form of the control equation is used to describe the temperature change, boundary heat exchange and energy conservation relationship between internal heat sources in the field;

[0017] The processing module is used to construct an integral domain for each node based on the average node spacing, and arrange Gauss points on the boundary of each integral domain according to the local weak form of the control equation and the integral domain;

[0018] The processing module is used to approximate the temperature and temperature derivative of any point in the field based on the coordinates of each Gauss point according to the generalized finite difference method; wherein the temperature and temperature derivative of any point are represented by a linear combination of the node temperature parameters in the nebula;

[0019] The processing module is used to determine a discrete matrix equation corresponding to heat conduction based on the temperature and temperature derivative of any point in the field, the local weak form of the control equation and the boundary conditions; the discrete matrix equation is used to describe how the temperature at all discrete points in the field changes over time;

[0020] The updating module is configured to calculate the temperature of each node at any time based on the initial condition and the boundary condition by using the discrete matrix equation through time iteration.

[0021] The third aspect of the present application provides a device for determining the temperature of a field based on a weak form generalized finite difference method, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor implements the steps of any one of the methods provided in the first aspect of the present application.

[0022] The fourth aspect of the present application provides a computer readable storage medium having a computer program stored thereon, wherein the program, when executed by a processor, implements the steps of any one of the methods provided in the first aspect of the present application.

[0023] The method and device for determining the temperature of a field based on a weak form generalized finite difference method provided by the present application can accurately describe the heat source, shape, position and material parameters of the heat source in the field, discretize the geometric structure model into multiple nodes and assign initial conditions and boundary conditions, and convert the heat conduction differential control equation of the discrete model into a local weak form of the control equation to represent the energy conservation relationship between the temperature change in the field, the boundary heat exchange and the internal heat source. Based on the average node spacing, an integral domain is constructed for each node, and a Gaussian point is arranged on the boundary according to the local weak form of the control equation and the characteristics of the integral domain. Then, using the generalized finite difference method, the temperature and temperature derivative of each Gaussian point can be represented by a linear combination of the node temperature parameters in the nebula. Combining the approximate temperature and temperature derivative of the generalized finite difference method, the local weak form of the control equation and the boundary conditions, a discrete matrix equation is established, and finally based on the initial conditions and boundary conditions, the temperature of each node at any time is obtained by iterative calculation of the discrete matrix equation. This technical solution adapts the boundary heat exchange through the integral form, relies on meshless discretization, high-precision Gaussian point integration and adaptive integral domain control, significantly improves the numerical stability and solving accuracy, and provides a solution with physical reality and engineering practicality for thermal conduction simulation under complex geometric structures and boundary conditions. BRIEF DESCRIPTION OF DRAWINGS

[0024] Figure 1 A flowchart of the method for determining the temperature of a field based on a weak form generalized finite difference method according to Embodiment One of the present application;

[0025] Figure 2 A distribution diagram of the integral domain of each node according to an exemplary embodiment of the present application;

[0026] Figure 3 A local weak form diagram of the generalized finite difference method according to an exemplary embodiment of the present application;

[0027] Figure 4 A hardware structure diagram of a device for determining a field temperature based on a weak form generalized finite difference method according to an exemplary embodiment of the present application;

[0028] Figure 5 A schematic diagram of a device for determining a field temperature based on a weak form generalized finite difference method according to an exemplary embodiment of the present application. DETAILED DESCRIPTION

[0029] The exemplary embodiments will be described in detail herein with reference to the accompanying drawings. In the following description, unless otherwise indicated, like numbers in the different drawings represent similar or identical elements. The following exemplary embodiments described in the detailed description section are not meant to be an all-inclusive description of all embodiments of the application.

[0030] The terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the application. As used herein, the singular forms "a", "an" and "the" are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms "comprises" and / or "comprising," when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and / or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and / or groups thereof. As used herein, the word "if' can be construed to mean "when" or "when a condition is fulfilled" or "in response to the determination of" that a certain condition is fulfilled.

[0031] Heat conduction, as one of the basic modes of heat transfer, has wide existence and importance in the natural world and engineering applications. From macro-scale building energy-saving design to micro-scale electronic device heat dissipation, heat conduction phenomena exist universally in solid, liquid and gaseous media, and accurate simulation of heat conduction plays a key role in the optimization design of thermal management systems such as thermal sensors and heat dissipation devices. Solving the heat conduction process by numerical methods can effectively predict the temperature field distribution of material and structural systems under complex geometric structures and boundary conditions, and provide a theoretical basis for engineering thermal design.

[0032] The mathematical description of traditional heat conduction problems relies on the heat conduction partial differential equation, which establishes the quantitative relationship between the time and space evolution of the temperature field. Current numerical solution methods are mainly divided into two categories: grid-discrete-based methods and meshless methods. Compared with the former, meshless methods have unique advantages in dealing with grid distortion, large deformation and moving boundaries. Among them, the generalized finite difference method, as a classic meshless method, combines the algorithm simplicity and computational efficiency of the finite difference method, and is widely used in engineering practice. However, its interpolation scheme based on the strong form leads to three significant defects: (1) insufficient numerical stability; (2) poor adaptability to complex boundaries; (3) limited precision in handling derivative boundary conditions, which seriously restricts the application of the method in high-precision thermal analysis.

[0033] The application provides a method and device for determining field temperature based on a weak form generalized finite difference method, which is used for more simple discrete target objects and improves the accuracy and stability of heat conduction solving.

[0034] The application introduces a weak form integral characteristic into a generalized finite difference framework, effectively solves the stability and accuracy bottleneck of a traditional generalized finite difference method in heat conduction simulation by establishing a new type of generalized finite volume format, and can more stably and accurately calculate heat conduction problems in engineering practice.

[0035] The application provides a method and device for determining field temperature based on a weak form generalized finite difference method from a bottom layer discrete format. The overall idea of the application is to integrate the control volume concept in the finite volume method into the generalized finite difference method. In the local weak form, the control volume is introduced as an integral domain, in which a Heaviside function is used as a weight function. However, unlike the central cell control volume used in the finite volume method, in the application, the integral domain is centered on the vertex center. The trial function is constructed by a moving least square approximation based on Taylor series expansion. By applying the divergence theorem, the integral of the local weak form of the control equation in the control volume is converted into a boundary integral, thereby reducing the continuity requirements of the trial function and the thermal conductivity. The method provided by the application is free from the dependence on the grid, has simple discretization, strong stability and high accuracy. Since it does not require a grid, it is particularly suitable for processing heat transfer problems of objects with complex shapes, complex boundaries and large deformations.

[0036] The following specific embodiments are given to introduce the technical scheme of the application in detail.

[0037] Figure 1 A flowchart of the method for determining field temperature based on a weak form generalized finite difference method according to the first embodiment of the application is provided. Figure 1 The method for determining field temperature based on a weak form generalized finite difference method provided by the application can include the following steps.

[0038] S101, a geometric structure model of an object to be analyzed is established, and the geometric structure model is discretized into a plurality of nodes to obtain a discrete model corresponding to the geometric structure model; the geometric structure model is used to describe a heat source, a shape, a position and material parameters of the heat source in a field.

[0039] It should be noted that the object to be analyzed is any one of the following devices: an electronic device, an integrated circuit board or other devices with heat conduction; wherein the self-heating object on the electronic device, the integrated circuit board or other devices with heat conduction is a heat source.

[0040] Specifically, the geometric model is a digital description of the spatial form and physical properties of the object, which is used to define the physical space where heat conduction occurs and the related parameters, i.e. the shape, position and material parameters of the object, such as the density, specific heat capacity, thermal conductivity, etc. These parameters directly affect the rate and direction of heat conduction and are the core inputs of the heat conduction differential control equation.

[0041] Specifically, the node is a discrete point for field energy and temperature calculation. The continuous geometric model is discretized into multiple nodes, which become the basic unit for subsequent numerical calculation.

[0042] Further, the heat conduction differential control equation, as the core mathematical expression describing the physical law of heat transfer, is the dynamic balance relationship between heat diffusion, energy storage and internal heat source excitation in the field.

[0043] In the specific implementation, the heat conduction differential control equation corresponding to the discrete model in the established heat conduction model is:

[0044]

[0045] Specifically, k x and k y represent the thermal conductivity in the x direction and the y direction, respectively; represents the first derivative of temperature in the x direction; represents the first derivative of temperature in the y direction; ρ represents the density; c represents the heat capacity; Q represents the heat source in the field; T represents the temperature in the field; and the Q and the T are functions of time and space.

[0046] It can be understood that the heat source refers to a physical factor that generates or consumes heat in the research domain. In the specific implementation, in this embodiment, Q(x, t) is used to represent the heat generation rate per unit volume at a certain point x in space and time t.

[0047] S102, initial conditions and boundary conditions are assigned to the discrete model.

[0048] In the specific implementation, an initial temperature is set for each node as the starting point for deriving and solving the heat conduction differential control equation, and the temperature field will be calculated and updated based on these initial conditions subsequently.

[0049] Specifically, the temperature distribution at the initial time is:

[0050] T = T0;

[0051] Further, the boundary condition of the field refers to the temperature or heat flow constraint on the boundary of the heat conduction field, which is used to limit the transfer mode of heat at the boundary. In the specific implementation, in this example, the explicit boundary condition is divided into three categories:

[0052] (1) Internal boundary conditions

[0053] Specifically, Γ sI Representing the integration field Γ s The portion of the boundary that does not intersect with the object boundary Γ, i.e., the internal boundary, is used to study the interfaces between different materials or subdomains within a field, such as the interlayer interfaces of composite materials and the contact interfaces of assemblies. It must satisfy the condition of heat flow continuity. Ensuring energy conservation during heat transfer at the interfaces of different media is a key boundary for heat conduction in complex domains.

[0054] (2) Dirichlet boundary conditions

[0055] Specifically, Γ sD Representing the integration field Γ s The portion where the boundary intersects with the object's Dirichlet boundary is used to directly specify the temperature value on the boundary, and its settings are as follows:

[0056]

[0057] (3) Newman boundary conditions

[0058] Specifically, Γ sN Representing the integration field Γ s The portion where the boundary intersects with the Newman boundary of the object is used to specify the heat flux density (heat transfer rate per unit area) on the boundary, and it is set as follows:

[0059]

[0060] Specifically, n x n represents the x-component of the normal vector at the boundary; y This represents the y-component of the normal vector at the boundary. The heat flow applied at the boundary, the It is a function of time and space.

[0061] It is understandable that by describing the energy interaction process between the heat source and boundary conditions within the field through a geometric structural model, the heat conduction in the real situation can be reproduced, providing a spatial range and defining conditions for numerical calculation, and supporting the transformation of the differential control equation of heat conduction into a discrete matrix equation.

[0062] S103. The heat conduction differential control equation corresponding to the discrete model is transformed into a local weak form of the control equation; the local weak form of the control equation is used to describe the energy conservation relationship between temperature change, boundary heat exchange, and internal heat sources within the field.

[0063] Specifically, in one possible implementation, the differential governing equations for heat conduction corresponding to the discrete model are transformed into locally weak forms of the governing equations, including:

[0064] Step 1, constructing a residual based on the heat conduction differential control equation.

[0065] Specifically, to realize the conversion from the differential control equation to the discrete matrix equation, referring to the foregoing description, the residual is constructed based on the heat conduction differential control equation formula, as follows:

[0066]

[0067] It can be understood that R is the residual constructed, and introducing the residual constructs the strong form equation into the weak form equation, avoiding the situation that the strong form equation cannot be directly numerically solved.

[0068] Step 2, constructing a weak form initial heat conduction equation based on the pre-constructed test function and the residual; wherein the weak form initial heat conduction equation is used to describe that the weighted integral of the residual on the test function space is 0.

[0069] Specifically, the test function v is the core tool connecting the heat conduction differential control equation and the discrete matrix equation, and the test function can be a Dirac function, an error function, a Heaviside step function, a trial function, etc. In the embodiment, the Heaviside step function is taken as the test function, so that the integral action range is reduced from the field domain to the integral domain. The weak form initial heat conduction equation is a weak form equation constructed by the residual and the test function, and is used to describe that the weighted integral of the residual on the test function space is 0.

[0070] It can be understood that the weak form initial heat conduction equation is constructed, and through the weighted integral of the residual on the test function space being 0, both the requirement for the smoothness of the temperature field is reduced to adapt to the discontinuous scene in engineering, and the boundary condition is naturally integrated, laying a foundation for the subsequent generalized finite difference method.

[0071] In specific implementation, the weak form initial heat conduction equation constructed is as follows:

[0072]

[0073] Specifically, Ω represents the field domain; v represents the test function; ρ represents the density; c represents the heat capacity; represents the first order derivative of temperature with respect to time; represents the first order derivative of temperature in the x direction; represents the first order derivative of temperature in the y direction; k x and k y respectively represent the thermal conductivity in the x direction and the y direction; Q represents the heat source in the field domain; T represents the temperature in the field domain; the Q and the T are functions of time and space;

[0074] Further, the test function is:

[0075]

[0076] where Ω s is the integration domain of node x I ; v is equal to 1 if point x is in the integration domain of node x I , otherwise, v is equal to 0.

[0077] Step 3, based on the distribution integral rule and the divergence theorem, the weak form of the initial heat conduction equation is converted into the local weak form of the control equation.

[0078] Specifically, the distribution integral rule is a rule for solving the derivative of the product of two functions, which is used to expand the derivative of the product of the residual of the heat conduction differential control equation and the test function, and decompose the derivative of the complex composite function into separable terms. The divergence theorem relates the volume integral of a vector field to the area integral of the vector field on the boundary, which is used to convert the integral in the weak form of the initial heat conduction equation into a boundary integral, thereby explicitly introducing the boundary condition in the equation. The local weak form of the control equation is the final weak form equation obtained by conversion through the distribution integral rule and the divergence theorem.

[0079] It can be understood that the local weak form of the control equation explicitly describes the energy conservation relationship in the field domain, where the left side of the equation is the change in energy storage caused by the change in temperature in the field domain, and the right side of the equation is the boundary heat exchange and internal heat source. It is the direct basis for subsequent construction of discrete matrix equations, ensuring that the numerical solution satisfies the energy conservation relationship.

[0080] In specific implementation, through the distribution integral rule, the complex partial derivative term in the weak form of the initial heat conduction equation is decomposed into two simpler terms, as follows:

[0081]

[0082] Specifically, where v represents the test function; represents the first-order derivative of the test function in the x direction; represents the first-order derivative of the test function in the y direction.

[0083] It can be understood that based on the above distribution integral rule, the test function is substituted into the weak form of the initial heat conduction equation, and the divergence theorem is used to derive:

[0084]

[0085] Specifically, where Γ s represents the integration domain; the formula of the divergence theorem is In the above formula, and

[0086] Further, the internal boundary condition, the Dirichlet boundary condition and the Neumann boundary condition are substituted into the boundary integral in the above equation, and a local weak form of the control equation is obtained as follows:

[0087]

[0088] where Γ sI represents a part of the boundary of the integral domain Γ s which is not intersected with the object boundary Γ sD represents a part of the boundary of the integral domain Γ s which is intersected with the Dirichlet boundary of the object; Γ sN represents a part of the boundary of the integral domain Γ s which is intersected with the Neumann boundary of the object; n x represents a component of the normal vector at the boundary in the x-axis; n y represents a component of the normal vector at the boundary in the y-axis; represents a heat flow applied on the boundary, and the heat flow q is a function of time and space.

[0089] In this step, the discrete model corresponding to the heat conduction differential control equation is converted into a local weak form of the control equation. The differential equation is converted into an integral form by using a test function and combining a distribution integral rule, and the local weak form of the control equation is obtained based on the boundary condition. A basis for the operation of the generalized finite difference method is provided, thereby forming a local conservation law.

[0090] S104, based on the average node spacing, an integral domain of each node is constructed, and a Gaussian point is arranged on the boundary of each integral domain according to the local weak form of the control equation and the integral domain.

[0091] Figure 2 A distribution diagram of the integral domain of each node shown in an exemplary embodiment of the present application is shown. Please refer to Figure 2 , in Figure 2 , the field domain Ω is an irregular region enclosed by a black broken line; the integral domain Ω s of the node I is a circular region with the node I as the center and the pink region; the internal boundary Γ sI is a part of the boundary of the integral domain Γ s which is not intersected with the object boundary Γ sD represents a part of the boundary of the integral domain Γ s which is intersected with the Dirichlet boundary of the object; the Neumann boundary Γ sN represents a part of the boundary of the integral domain Γ s which is intersected with the Neumann boundary of the object.

[0092] In the embodiment, the integral domain of each node is determined by taking the node as the center and the node nebula radius as the radius.

[0093] Specifically, the integral domain of each node is constructed based on the average node distance, including:

[0094] Step 1, for any two nodes in all discrete nodes, the distance between the two nodes is calculated, and the average distance between the nodes is determined according to all the calculated distances.

[0095] Specifically, the distance between any two nodes reflects the density of the field domain. By traversing all nodes, the distance between two nodes is calculated respectively, and then the average distance between all two nodes is obtained. The average distance between nodes is the core basis for determining the integral domain radius, which ensures that the integral domain size can adapt to the field.

[0096] For example, in the embodiment, the distance between node a(x a , y a ) and node b(x b , y b ) is The average distance between nodes is Where A s is the estimated support domain volume. is the estimated total number of support domain nodes.

[0097] Step 2, according to the average distance and the preset coefficient, the node nebula radius is determined.

[0098] Specifically, the node nebula radius R determines the spatial coverage of the integral domain, and the preset coefficient a is a constant value according to the field characteristics and accuracy requirements, for example, a>1 is taken in the area with rapid heat flow change, and a<1 is taken in the flat area. If a is too small, the integral domain cannot cover the heat exchange area of adjacent nodes, resulting in the omission of energy conservation calculation; if a is too large, the integral domain overlaps too much, increasing the calculation amount and possibly introducing false heat exchange.

[0099] In the embodiment, the node nebula radius formula is as follows:

[0100]

[0101] Step 3, for each node, the circular region with the node as the center and the node nebula radius as the radius is determined as the integral domain of the node.

[0102] Specifically, the integral domain is constructed by taking the node as the center and the node nebula radius R as the radius, and the integral domain formula is as follows:

[0103] Ω s ={(x, y) | (x-x i) 2 + (y - y i ) 2 ≤ R 2};

[0104] It can be understood that constructing the integral domain of each node based on the average node spacing can more accurately reflect the spatial relationship between the nodes. It helps to optimize the accuracy of the finite volume method when solving problems, especially when dealing with complex geometric structures or distributions.

[0105] Specifically, according to the local weak form of the control equation and the arrangement of the Gauss points on the boundary of each integral domain.

[0106] It can be understood that the Gauss point is a specific sampling point selected in the Gaussian quadrature method to improve the accuracy of integral calculation. The Gaussian quadrature method is to multiply the function value of the discrete point by the weight, so that it can approximate the integral of the continuous function in the interval with high precision.

[0107] Specifically, through the Gauss point, the same or higher integral accuracy is achieved with fewer quantities, the error caused by insufficient sampling is reduced, and the calculation stability can be maintained when the boundary shape of the integral domain is complex.

[0108] In specific implementation, the core formula of the Gaussian quadrature method is as follows:

[0109]

[0110] Specifically, wherein c and d represent the two endpoints of the interval; M represents the number of discrete points X I ; f(X) represents the continuous function of the discrete point X I ; w I represents the Gaussian weighting factor of X I .

[0111] Further, for the integral in the local weak form of the control equation, the above Gaussian quadrature method is adopted, and the formula is converted to:

[0112]

[0113] Specifically, wherein represents the length of the internal boundary Γ sI ; N i represents the number of Gauss points on the internal boundary Γ sI ; A represents the Gaussian weighting factor of x I-i ; x I-i represents the spatial coordinates of the i-th Gauss point on the internal boundary of the node I; n x (x I-i ) and n y (x I-i ) represent the unit normal vector of the internal boundary of the node I at x I-iand y I-i the direction cosine of the outward normal of the boundary surface of the integral domain.

[0114] It should be noted that the specific implementation principle and implementation process of the local weak form according to the control equation and the arrangement of the Gauss point on the boundary of each integral domain can refer to the description in the related art, and will not be repeated here.

[0115] S105, based on the coordinates of each Gauss point, the temperature and temperature derivative of any point in the field are approximated by the generalized finite difference method; wherein the temperature and temperature derivative of any point are represented by a linear combination of node temperature parameters in the nebula.

[0116] Figure 3 The local weak form of the generalized finite difference method shown in an exemplary embodiment of the present application is shown in the figure. Please refer to Figure 3 , including: the red area is the integral domain; the blue area is the nebula; the center point is the red center of the integral domain in the nebula; the Gauss point is the blue point on the integral domain; the adjacent node is the r green point selected in the nebula.

[0117] In specific implementation, for example, in the present embodiment, r takes the value of 7, that is, the nebula centered on the center point is composed of 7 nodes x i =(x i , y i ), whose serial numbers are 1, 2, …, 7. The number of Gauss points is 4, whose serial numbers are 7-1, 7-2, 7-3 and 7_4.

[0118] Specifically, on the boundary of each integral domain, the Gauss point is arranged, based on the coordinates of each Gauss point, the temperature and temperature derivative of any point in the field are approximated by the generalized finite difference method, including:

[0119] (1) Construct a local temperature approximation model; the local temperature approximation model uses Taylor series to describe the node temperature parameters in the neighborhood of the center point to approximate the temperature of the center point, and expresses the temperature as the inner product of the local basis function vector and the derivative vector; the local basis function vector is determined based on the coordinate difference between any node in the node nebula and the center point; the derivative vector contains the temperature and the derivative of the temperature at the center point.

[0120] In specific implementation, the spatial derivative at the Gauss point is approximated and calculated by using the generalized finite difference method with adjacent nodes.

[0121] Specifically, the test function is approximated by a moving least squares method based on Taylor series expansion. Consider a s-order differential operator L in a two-dimensional region. The operator at any point The value at the center point can be replaced by a corresponding differential operator, which involves a combination of all the node values within the star.

[0122] Reference Figure 3 A star centered at a point is composed of r nodes x i = (x i , y i ), where i is the node number within each star, i = 1, 2, …, r. The expression of the local approximation model is based on the Taylor series expansion, which can be written as:

[0123]

[0124] Specifically, s represents the order of the local approximation; p represents the order of the Taylor series term; T (x ) represents the temperature value at the center point; the coordinate difference of the nodes within the defined domain is defined as and represents the transpose of the local basis function vector; represents the derivative vector, which includes the temperature at the center point and the derivatives of the temperature of each order.

[0125] Further, the local basis function vector formula is as follows:

[0126]

[0127] Specifically, the derivative vector formula is as follows:

[0128]

[0129] Specifically, m represents the number of unknown coefficients; represents the vector of local basis functions. is a vector that includes all derivatives of order not exceeding s.

[0130] It can be understood that by approximating the temperature and temperature derivatives using the generalized finite difference method, the continuous temperature field within the field domain is converted into a derivative vector, thereby facilitating numerical solution. The generalized finite difference method provides a discrete representation of the temperature field and its derivatives through Gaussian point coordinates, which can provide higher accuracy for numerical solution.

[0131] (2) Based on the pre-constructed objective function, the linear equation between the derivative vector, the local basis function vector, and the preset weight vector is obtained by minimizing the objective function; the objective function is constructed based on the weighted two-norm.

[0132] Specifically, the weight vector is a value related to the distance between the Gaussian point and the node. The closer the node is to the Gaussian point, the higher the weight vector, ensuring the reliability of the local approximation.

[0133] Furthermore, by summing the weighted squares of the errors at each node, the objective function is obtained as follows:

[0134]

[0135] Specifically, among them, r represents the point Node x in the central nebula i Temperature value on; Indicates with points point x in the central nebula i The relevant weighting function.

[0136] As can be understood from the preceding description, the objective function is constructed based on the weighted L2 norm.

[0137] Furthermore, in this embodiment, a quartic spline is used as the weighting function:

[0138]

[0139] Among them, d i It is node x i Time distance; r i It is the farthest node in the nebula. The distance. Furthermore, by solving for the extrema of the objective function, we obtain... and The linear equation is as follows:

[0140]

[0141] It is understandable that in linear equations:

[0142]

[0143]

[0144] Matrix expressions for temperature and its derivative are derived from linear equations.

[0145] (3) Determine the temperature and temperature derivative at any point in the field based on the linear equation.

[0146] Specifically, by solving linear equations We can obtain:

[0147]

[0148] Specifically, the above formula is expanded as follows:

[0149]

[0150] Specifically, wherein, is the element of the i-th row and j-th column of the difference matrix

[0151] Further, the generalized finite difference method is used to approximate the temperature T and its first-order partial derivatives and The generalized finite difference method approximated temperature T and its first-order partial derivatives and Any point approximated by the generalized finite difference method can be obtained by the temperature and temperature derivative, which is as follows:

[0152]

[0153] M1(x) is the generalized finite difference approximation temperature vector, which is the first row vector of the difference matrix; M2(x) is the generalized finite difference approximation temperature x partial derivative vector, which is the second row vector of the difference matrix; M3(x) is the generalized finite difference approximation temperature y partial derivative vector, which is the third row vector of the difference matrix.

[0154] Specifically, when the Gaussian quadrature method is used for calculation, the derivative of the test function at each Gaussian point is approximated by its corresponding star cloud. Referring to Figure 3 , points 7-1, 7-2, 7-3 and 7_4 are Gaussian points inside the boundary of the center point 7. Using the generalized finite difference method to approximate the temperature and temperature derivative, the first-order derivative of the temperature at Gaussian point 7-1 and can be approximated as follows:

[0155]

[0156] Specifically, wherein, indicates the temperature value at node x 7-1 in the star cloud centered at point 7-1.

[0157] S106, according to the temperature and temperature derivative of any point in the field, the local weak form of the control equation and the boundary condition, determine the discrete matrix equation corresponding to heat conduction; the discrete matrix equation is used to describe how the temperature at all discrete points in the field changes with time.

[0158] Specifically, the spatial derivatives at each Gaussian point are combined by the weak form to form a global discrete matrix equation.

[0159] ​In a specific implementation, the temperature and temperature derivative of any point in the field are substituted into the local weak form of the control equation to obtain a discrete matrix equation as follows:

[0160]

[0161] Specifically, C represents a heat capacity matrix; K represents a thermal conductivity matrix; and F represents a heat flux vector.

[0162]

[0163] Specifically, M1, J represents the first row and the Jth column of the difference matrix M; M 2，J represents the second row and the Jth column of the difference matrix M; M 3，J represents the third row and the Jth column of the difference matrix M.

[0164] It can be understood that by using the local weak form of the control equation and combining the temperature and temperature derivative of any point in the field, the heat conduction process and its interaction with external heat sources can be clearly mathematized. The form of the discrete matrix equation simplifies the complex heat conduction problem and clearly describes how heat propagates and distributes.

[0165] S107, based on the initial condition and the boundary condition, the temperature of each node at any time is obtained by time iteration calculation using the discrete matrix equation.

[0166] Specifically, in order to solve the semi-discrete form of the discrete matrix equation, the backward Euler method is used for time discretization processing, and the discrete matrix equation becomes:

[0167]

[0168] Specifically, Δt is the time step, which is used to control the precision and calculation efficiency of time discretization. The smaller the time step, the more accurate the capture of transient temperature changes in theory.

[0169] It can be understood that by fixing the time step and continuously iterating to the termination time, the numerical temperature of all nodes at the termination time can be obtained. The dynamic process of the temperature of the nodes in the field can be tracked time by time, and finally the numerical temperature of all nodes at the termination time can be obtained, which provides accurate results for transient heat conduction analysis and effectively supports the simulation and analysis of complex transient heat scenes such as fire heat spread and electronic device transient heat dissipation.

[0170] Specifically, the above formula represents the iterative relationship between the temperature distribution T(t+Δt) at the next moment and the temperature distribution T(t) at the current moment. On the premise of obtaining T(t+Δt), T(t+Δt) is replaced by T(t) as the temperature distribution at the current moment, and the temperature distribution T(t+2Δt) at the next moment can be obtained. Similarly, after obtaining T(t+2Δt), T(t+3Δt), T(t+4Δt), and T(t+5Δt) can be iteratively solved. Such repeated solving is performed until the iteration reaches the termination time, and finally the temperature of each node at any time is obtained.

[0171] Specifically, the advantages of the present application compared with the prior art are:

[0172] (1) Compared with the traditional generalized finite difference method, the weak form is innovatively introduced in the present method, which significantly improves the accuracy and stability of numerical calculation at the theoretical level.

[0173] (2) Through the systematic introduction of the weak form, the present method can strictly meet the mathematical expression requirements of the natural boundary condition, and exhibits better numerical performance in boundary treatment, providing a unified mathematical framework for modeling complex boundary problems.

[0174] (3) The method in the present application is based on the combination of control volume and local weak form, which retains the potential for extending the method to a fully conservative format.

[0175] The method for determining the temperature of a field based on the weak form generalized finite difference method provided in the present embodiment accurately describes the heat source, shape, position, and material parameters of the heat source in the field, discretizes the geometric structure model into multiple nodes and assigns initial conditions and boundary conditions, and converts the discrete model corresponding to the heat conduction differential control equation into a local weak form of the control equation to represent the energy conservation relationship between the temperature change in the field, the boundary heat exchange, and the internal heat source. Based on the average node spacing, an integral domain is constructed for each node, and Gauss points are arranged on the boundary according to the local weak form of the control equation and the characteristics of the integral domain. Then, using the generalized finite difference method, the temperature and temperature derivative of any point in the field can be represented by a linear combination of the node temperature parameters in the star cloud based on the coordinates of each Gauss point. Combining the temperature and temperature derivative of any point in the field, the local weak form of the control equation, and the discrete matrix equation describing the global balance of energy change, heat conduction, and external heat input in the field, the initial temperature of the nodes and the Gauss points are finally obtained by iterative calculation of the discrete matrix equation through Gaussian quadrature, and the temperature of each node at each time is obtained. This technical solution adapts to boundary heat exchange through the natural global conservation characteristics of the integral form, relies on meshless discretization, high-precision Gaussian point integration, and adaptive integral domain control, significantly improves the numerical stability and solving accuracy, and provides a solution that combines physical reality and engineering practicality for heat conduction simulation under complex geometric structures and boundary conditions.

[0176] Corresponding to the foregoing embodiment of the method for determining field temperature based on the weak form of the generalized finite difference method, the application further provides an embodiment of an apparatus for determining field temperature based on the weak form of the generalized finite difference method.

[0177] The embodiment of the apparatus for determining field temperature based on the weak form of the generalized finite difference method can be applied to the equipment for determining field temperature based on the weak form of the generalized finite difference method. The apparatus embodiment can be realized by software, or realized by hardware or a combination of software and hardware. Taking the software realization as an example, as a logical apparatus, it is formed by reading the corresponding computer program instructions in the non-volatile memory into the memory for running by the processor of the equipment for determining field temperature based on the weak form of the generalized finite difference method. From the hardware level, Figure 4 The hardware structure diagram of the equipment for determining field temperature based on the weak form of the generalized finite difference method, in which the apparatus for determining field temperature based on the weak form of the generalized finite difference method is located, shown in the exemplary embodiment of the application, in addition to the processor, the memory, the network interface, and the non-volatile memory shown, Figure 4 the equipment for determining field temperature based on the weak form of the generalized finite difference method, in which the apparatus for determining field temperature based on the weak form of the generalized finite difference method is located, usually further includes other hardware according to the actual functions of the apparatus for determining field temperature based on the weak form of the generalized finite difference method, which will not be described here.

[0178] Figure 5 The schematic diagram of the apparatus embodiment one provided by the application for determining field temperature based on the weak form of the generalized finite difference method. Please refer to Figure 5 The apparatus provided by the embodiment includes a modeling module 510, a generating module 520, a processing module 530, and an updating module 540; wherein,

[0179] The modeling module 510 is configured to establish a geometric structure model of an object to be analyzed, and discretize the geometric structure model into a plurality of nodes to obtain a discrete model corresponding to the geometric structure model; the geometric structure model is configured to describe heat sources, shapes, positions, and material parameters of the heat sources in a field;

[0180] The generating module 520 is configured to assign initial conditions and boundary conditions to the discrete model;

[0181] The processing module 530 is configured to convert a heat conduction differential control equation corresponding to the discrete model into a local weak form of the control equation; the local weak form of the control equation is configured to describe temperature changes, boundary heat exchanges, and energy conservation relationships between internal heat sources in the field;

[0182] The processing module 530 is configured to construct an integral domain of each node based on the average node spacing, and arrange Gaussian points on the boundary of each integral domain according to the local weak form of the control equation and the integral domain.

[0183] The processing module 530 is configured to approximate the temperature and temperature derivative of any point in the field domain by the generalized finite difference method based on the coordinates of each Gaussian point, wherein the temperature and temperature derivative of any point are represented by a linear combination of the node temperature parameters in the nebula.

[0184] The processing module 530 is configured to determine a discrete matrix equation corresponding to heat conduction according to the temperature and temperature derivative of any point in the field domain, the local weak form of the control equation and the boundary condition; the discrete matrix equation is used to describe how the temperature of all discrete points in the field domain changes over time; and the updating module 540 is configured to calculate the temperature of each node at any time by time iteration based on the initial condition and the boundary condition, and by using the discrete matrix equation.

[0185] Please continue to refer to Figure 5 The application further provides a device for determining the temperature of a field domain based on a weak form generalized finite difference method, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor implements the steps of any of the methods provided in the first aspect of the application when executing the program.

[0186] The application further provides a computer readable storage medium having a computer program stored thereon, wherein the program implements the steps of any of the methods provided in the application when executed by a processor.

[0187] The implementation process of the functions and roles of each unit in the above device is specifically described in the implementation process of the corresponding steps in the above method, which will not be repeated here.

[0188] For the device embodiment, since it basically corresponds to the method embodiment, the relevant parts are described in the method embodiment. The above described device embodiment is only illustrative, wherein the units described as separate components can be or can not be physically separated, and the components displayed as units can be or can not be physical units, i.e., they can be located in one place or distributed on multiple network units. According to actual needs, some or all of the modules can be selected to achieve the purpose of the application. Those skilled in the art can understand and implement without creative labor.

[0189] The above only is the preferred embodiment of the present application, and is not used to limit the present application, any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present application should be included in the protection scope of the present application.

Claims

1. A method of determining a temperature of a field domain based on a weak form generalized finite difference method, the method comprising: The method comprises: establishing a geometric model of an object to be analyzed, and discretizing the geometric model into a plurality of nodes to obtain a discrete model corresponding to the geometric model; the geometric model is used to describe a heat source, a shape, a position and material parameters of the heat source in a field; assigning initial conditions and boundary conditions to the discrete model; converting a heat conduction differential control equation corresponding to the discrete model into a local weak form of the control equation; the local weak form of the control equation is used to describe temperature variation, boundary heat exchange and energy conservation relationship between internal heat sources in the field; constructing an integral domain of each node based on an average node spacing, and arranging Gauss points on the boundary of each integral domain according to the local weak form of the control equation and the integral domain; approximating temperature and temperature derivative of any point in the field by using a generalized finite difference method based on coordinates of each Gauss point; wherein the temperature and the temperature derivative of any point are represented by a linear combination of node temperature parameters in a star cloud; determining a discrete matrix equation corresponding to heat conduction according to the temperature and the temperature derivative of any point in the field, the local weak form of the control equation and the boundary conditions; the discrete matrix equation is used to describe how the temperature at all discrete points in the field changes over time; obtaining the temperature of each node at any time by time iteration calculation based on the initial conditions and the boundary conditions and by using the discrete matrix equation.

2. The method of claim 1, wherein, The method comprises: constructing a residual based on the heat conduction differential control equation; constructing an initial heat conduction equation of the weak form based on a pre-constructed test function and the residual; wherein the initial heat conduction equation of the weak form is used to describe that the weighted integral of the residual on the test function space is 0; converting the initial heat conduction equation of the weak form into the local weak form of the control equation based on a distribution integral rule and a divergence theorem.

3. The method of claim 2, wherein, The method comprises: constructing a local temperature approximation model; the local temperature approximation model describes the node temperature parameters in the neighborhood of the center point by using Taylor series to approximate the temperature of the center point, and represents the temperature as an inner product of a local basis function vector and a derivative vector; the local basis function vector is determined based on the coordinate difference between any node in the node star cloud and the center point; the derivative vector contains the temperature and the derivatives of the temperature at the center point; obtaining a linear equation between the derivative vector, the local basis function vector and a preset weight vector by minimizing a pre-constructed objective function; the objective function is constructed based on a weighted two norm; determining the temperature and the temperature derivative of any point in the field based on the linear equation.

4. The method of claim 1, wherein, The method comprises: calculating the distance between any two nodes in all discrete nodes, and determining the average distance between the nodes according to all the calculated distances; determining a node star cloud radius according to the average distance and a preset coefficient; For each node, a circular region with the node as the center and the node nebula radius as the radius is determined as the integral domain of the node.

5. The method of claim 2, wherein, The initial heat conduction equation in weak form is: where Ω represents a field domain; v represents a test function; p represents a density; and c represents a heat capacity; represents a first derivative of temperature with respect to time; represents a first derivative of temperature in the x direction; represents a first derivative of temperature in the y direction; x and k y represent thermal conductivities in the x and y directions, respectively; Q represents a heat source within a field domain; and T represents a temperature within the field domain; each of Q and T being a function of time and space. The test function is: where Ω s is the integration domain of node x I ; v represents 1 if point x is within the integration domain of node x I , otherwise 0. The local weak form of the control equation is: where Γ sI denotes the part of the boundary of the integration domain Γ s which does not intersect the object boundary Γ sD denotes the part of the boundary of the integration domain Γ s which intersects the Dirichlet boundary of the object; Γ sN denotes the part of the boundary of the integration domain Γ s which intersects the Neumann boundary of the object; n x denotes the component of the normal vector at the boundary in the x-axis; n y denotes the component of the normal vector at the boundary in the y-axis; denotes the heat flux applied at the boundary, said is a function of time and space.

6. The method of claim 5, wherein, The discrete matrix equation is: Wherein, C is the heat capacity matrix; K is the thermal conductivity matrix; F is the heat flux vector; where J is the global number of nodes; M 1,J represents the vector of J nodal temperatures approximated by the generalized finite difference method; M 2，J represents the vector of the partial derivative of the J nodal temperatures with respect to x approximated by the generalized finite difference method; M 3，J represents the vector of the partial derivative of the J nodal temperatures with respect to y approximated by the generalized finite difference method.

7. The method of claim 1, wherein, The object to be analyzed is any one of the following devices: electronic devices, integrated circuit boards or other devices with heat conduction; wherein the self-heating object on the electronic device, the integrated circuit board or other devices with heat conduction is the heat source.

8. A device for determining the temperature of a field based on a weak-form generalized finite difference method, characterized in that, The device comprises a modeling module, a generating module, a processing module and an updating module; wherein, The modeling module is used to establish a geometric structure model of the object to be analyzed, and discretize the geometric structure model into a plurality of nodes to obtain a discrete model corresponding to the geometric structure model; the geometric structure model is used to describe the heat source, the shape, position and material parameters of the heat source in the field; The generating module is used to assign initial conditions and boundary conditions to the discrete model; The processing module is used to convert the heat conduction differential control equation corresponding to the discrete model into a local weak form of the control equation; the local weak form of the control equation is used to describe the temperature change in the field, the boundary heat exchange and the energy conservation relationship between the internal heat sources; The processing module is used to construct the integral domain of each node based on the average node spacing, and arrange Gaussian points on the boundary of each integral domain according to the local weak form of the control equation and the integral domain; The processing module is used to approximate the temperature and temperature derivative of any point in the field based on the coordinates of each Gaussian point according to the generalized finite difference method; wherein the temperature and temperature derivative of any point are represented by a linear combination of node temperature parameters in the nebula; The processing module is used to determine the discrete matrix equation corresponding to the heat conduction according to the temperature and temperature derivative of any point in the field, the local weak form of the control equation and the boundary conditions; the discrete matrix equation is used to describe how the temperature at all discrete points in the field changes with time; The updating module is used to obtain the temperature of each node at any time based on the initial conditions and the boundary conditions by using the discrete matrix equation through time iteration calculation.

9. A device for determining the temperature of a field based on the weak-form generalized finite difference method, characterized in that, A computer program product comprising a memory, a processor and a computer program stored on the memory and executable on the processor, characterized in that the processor implements the steps of the method of any one of claims 1-7 when executing the program.

10. A computer-readable storage medium having stored thereon a computer program, characterized in that, The program is executed by the processor to implement the steps of the method of any one of claims 1-7.

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