Hexahedral element-based transient heat conduction simulation method, system, device and medium

Abstract

The present application relates to the technical field of transient heat conduction simulation in thermal stress simulation, and discloses a hexahedral element-based transient heat conduction simulation method, system, device and medium, wherein the method comprises: taking any one node in a hexahedral linear element in a region to be simulated as an object node, calculating a single volume based on the cross-sectional area of a flow and the distance from the object node to the flow cross section, accumulating the single volumes corresponding to the object node in the hexahedral linear element to obtain a virtual node domain of the object node; based on the virtual node domain volume of each node, correcting load distribution to obtain a corrected spatial discretization algebraic equation; using the mean value theorem to discretize the time partial derivative term in the corrected spatial discretization algebraic equation to obtain a new discretization algebraic equation, outputting the node temperature data at each time in transient heat conduction simulation, and finally performing thermal stress simulation according to the temperature simulation result. The present application can significantly reduce the simulation error of low-quality elements.

Description

Technical Field

[0001] This invention relates to the field of transient heat conduction simulation technology, and in particular to a transient heat conduction simulation method, system, device and medium based on hexahedral elements. Background Technology

[0002] Electronic components experience temperature changes due to internal heat generation and environmental variations. These temperature changes cause thermal expansion and contraction of the heat carrier, leading to stress due to differences in the magnitude of temperature changes, varying coefficients of thermal expansion (copper's coefficient of thermal expansion is 6.3 times that of silicon), or constrained displacement. This stress can result in warping, microcracks, and interface delamination, causing localized loss of contact, jeopardizing normal operation, and even posing safety hazards. For example, rapid temperature changes such as cold starts, or localized high heat flux densities like chip hotspots, can generate high-value stresses that can lead to failure.

[0003] Meanwhile, to meet the ever-increasing demands for high efficiency, multifunctionality, lightweight, and compact size in electronic components, it is necessary to accurately grasp and control stress and deformation through optimized design. The aforementioned stress and deformation originate from temperature changes. Stress and deformation caused by temperature changes are typically obtained through thermo-structural coupling simulation. In this simulation, the nodal temperature changes and thermal expansion coefficients at each time step are applied to the stress analysis in the form of nodal loads. It can be considered that the primary issue in achieving optimized design lies in accurately grasping instantaneous temperature changes, i.e., the transient temperature values ​​of each node at each time step.

[0004] Temperature changes are a result of the conservation laws governing heat generation, conduction, and dissipation. Accurately capturing instantaneous temperature changes through simulation requires meeting these conservation principles at high-order precision. Describing the propagation of heat in complex stacked structures with varying material parameters and sizes, such as chips, bonding wires, solder joints, heat conductors, substrates, and heat sinks, as well as in trench structures with extremely high aspect ratios, requires simulation prediction methods that are robust to low-quality cells and can improve upon the mesh dependency inherent in traditional methods.

[0005] The Galerkin finite element method (GFEM), a classic numerical solution for differential equations, has become a commonly used technique in transient heat conduction simulations due to its clear error assessment and wide applicability. However, the traditional GFEM still faces several problems in practical applications, limiting the improvement of simulation accuracy and applicability. Specifically, it exhibits significant mesh dependency when dealing with different mesh patterns, leading to simulation result deviations for the same physical model due to different meshing methods. When the element quality is low (e.g., sharp or obtuse angles significantly deviating from right angles, or significant distortion), the numerical error increases dramatically, making it difficult to meet the requirements of high-precision simulations. More importantly, when considering heat flux with second-order accuracy, linear elements exhibit inconsistencies between systems considering heat inflow and outflow around nodes and systems considering transient and source terms, failing to satisfy the law of conservation of heat conduction.

[0006] Existing technologies have proposed improvement schemes for different unit types, but their applicability is obviously limited. For example, patent document CN118350257B discloses a transient heat conduction simulation method and system based on triangular units. Although it uses triangular units to carry out transient heat conduction simulation, it partially solves the system inconsistency problem in two-dimensional scenes, but it is limited by the unit dimension and cannot adapt to the simulation requirements of three-dimensional structures.

[0007] Patent document CN119720678A discloses a transient heat conduction simulation method and system for quadrilateral units based on virtual node domains. This simulation method also focuses on two-dimensional space. When facing three-dimensional engineering structures (such as mechanical parts, building components, electronic device cavities, etc.), it is difficult to accurately reproduce the heat conduction path and temperature change law in three-dimensional space.

[0008] Patent document CN119720677A discloses a transient diffusion simulation method and system based on tetrahedral elements, and patent document CN121744760B discloses a method and device for predicting temperature changes in electronic components based on pentahedral elements. While using tetrahedral or pentahedral elements for three-dimensional transient diffusion simulation or temperature change prediction overcomes dimensional limitations, tetrahedral elements, due to their accuracy limitations, are mostly used in the initial design stage or simulations where speed is the primary requirement. Pentahedral elements also have limitations in accuracy and geometry, limiting their applicability to specific situations. Furthermore, the computational logic of their virtual node domains is designed for the four-node geometry of tetrahedrons and the six-node geometry of pentahedrons. The definitions of related temperature functions and the methods of element integration are completely different from those of hexahedral elements, making them unsuitable for hexahedral elements with eight nodes.

[0009] Furthermore, existing technologies have not yet developed a systematic optimization scheme for the structural characteristics of hexahedral linear elements. This makes it difficult to improve the problems faced by hexahedral elements in transient heat conduction simulation, such as the mismatch between load distribution and heat assessment system, and the deep dependence of simulation results on mesh patterns. Consequently, it is difficult to fully leverage the comprehensive advantages of hexahedral elements in terms of computational accuracy and speed in three-dimensional numerical simulation, and it is impossible to meet the demand for high-precision and high-efficiency transient heat conduction simulation in complex three-dimensional engineering scenarios. Summary of the Invention

[0010] To address the aforementioned issues, this invention proposes a transient heat conduction simulation method, system, device, and medium based on hexahedral elements. This method achieves second-order accuracy while satisfying conservation laws, improving mesh dependency and maintaining simulation accuracy even with low-quality elements. Using the transient temperature obtained by this invention as the starting point for thermal-structural coupling simulation provides a reliable basis for planning optimal solutions and achieving optimized designs. This overcomes the bottleneck of thermal design, promoting technological advancement while ensuring the safety and stability of electronic components and extending their service life.

[0011] The technical solution adopted in this invention is as follows: A transient heat conduction simulation method based on hexahedral elements, comprising: Take any node in the hexahedral linear element within the simulation area as the object node, calculate the single volume based on the cross-sectional area of ​​the flow and the distance from the object node to the cross-sectional area, and sum the single volumes corresponding to the object node within the hexahedral linear element to obtain the virtual node domain of the object node. Based on the principle of distributing the load of the hexahedral linear element according to the virtual node domain volume of each node, the load of the Galerkin finite element method is modified to obtain the modified spatial discretized algebraic equation. The time partial derivative terms in the corrected spatial discretized algebraic equations are discretized in the time direction using the mean value theorem, resulting in the discretized algebraic equations for transient heat conduction of hexahedral linear elements. The nodal temperature data at each moment in the transient heat conduction simulation are output, and finally, thermal stress simulation is performed based on the temperature simulation results.

[0012] Furthermore, the calculation of a single volume based on the cross-sectional area of ​​the flow path and the distance from the object node to the cross-sectional area includes: Determine the spatial straight-line distance between the object node and any of the other seven nodes in the hexahedral linear element. Set a plane perpendicular to the line at the midpoint of the line as a flow section. The area of ​​the flow section is the product of the corresponding coefficient value obtained by the element integration after the isoparametric transformation of the hexahedral linear element by the Galerkin finite element method and the spatial straight-line distance between the nodes. Using the flow cross-section as the bottom and the object node as the vertex, the distance from the vertex to the bottom is determined to be half of the straight-line distance in space. Based on the flow cross-section area and the distance from the vertex to the bottom, the individual volume corresponding to the object node and any of the other seven nodes is calculated.

[0013] Further, the virtual node domain of the object node is obtained by summing the individual volumes corresponding to the object nodes within the accumulated hexahedral linear units, including: The object node within the hexahedral linear element is designated as node number 1. The individual volumes corresponding to node 1 and the other seven nodes numbered 2 to 8 within the hexahedral linear element are calculated sequentially. The seven individual volumes are then summed to obtain the virtual node domain of node number 1 as the object node. Each node numbered 2 to 8 within the hexahedral linear element is sequentially treated as an object node. The individual volumes corresponding to each object node and the other seven nodes within the element are calculated and summed to obtain the virtual node domain of each node. The sum of the volumes of the virtual node domains of the eight nodes within the hexahedral linear element is equal to the actual volume of the hexahedral linear element.

[0014] Furthermore, the load distribution on the hexahedral linear element based on the virtual node domain volume of each node includes: Based on the virtual node domain volume of each node, the larger the virtual node domain volume, the more load is allocated to the node, and the sum of the loads allocated to all nodes in the hexahedral linear element is equal to the total load generated in the hexahedral linear element, so as to satisfy the conservation law of heat conduction. The load is obtained by integrating the hexahedral linear elements after isoparametric transformation, and is the total load generated by the transient term and the source term.

[0015] Furthermore, the load modification of the Galerkin finite element method yields the modified spatial discretized algebraic equations, including: Based on the spatial discretization equations obtained by the Galerkin finite element method for the three-dimensional transient heat conduction problem, a correction term is introduced according to the principle of distributing the load of the hexahedral linear element based on the virtual node domain volume of each node. The correction term is the difference between the virtual node domain volume of the object node and the element integral of the hexahedral element of the Galerkin weight function, multiplied by the sum of the transient term and the source term at the geometric center of the hexahedral linear element. After introducing the correction term, a new spatial discretization algebraic equation is obtained as the corrected spatial discretization algebraic equation. In the corrected spatial discretization algebraic equation, the sum of the changes in the load distribution of each node in the element is zero, and the heat conduction conservation law at the element level is still maintained. At the same time, the treatment of transient and source terms is matched with the second-order accurate heat inflow and outflow evaluation system.

[0016] Furthermore, the discretization of the time partial derivatives in the modified spatial discretized algebraic equations using the mean value theorem in the time direction includes: The time step and time parameter are set. The time step is the time difference between time n and time n-1. The time parameter is used to adjust the weight of the time partial derivative terms at time n and time n-1 in the time discretization process, and the value range is between 0 and 1. The time partial derivative term is the partial derivative of temperature with respect to time. Calculate the product of the time partial derivative term at time n and the time parameter, and calculate the difference between the time partial derivative term at time n-1 multiplied by 1 and the time parameter. Summing these values ​​yields the discretized time partial derivative term.

[0017] Furthermore, the discretized algebraic equation for transient heat conduction of the hexahedral linear element is obtained, the nodal temperature data at each moment in the transient heat conduction simulation are output, and finally, thermal stress simulation is performed based on the temperature simulation results, including: Select the simulation region for transient heat conduction and complete the mesh generation of hexahedral linear elements. Input the initial temperature conditions and initialize the time-related parameters. Based on the discretized algebraic equations of transient heat conduction of hexahedral linear elements, the element matrix and the global matrix are generated respectively. The simulation time is advanced according to the set time step, and the node temperature and source term data of the previous moment are substituted to generate a system of simultaneous equations. Substitute the boundary condition data to solve the simultaneous equations, output the current node temperature data for thermal stress simulation, and repeat the time progression and solution process until the simulation ends when the simulation ends.

[0018] A transient heat conduction simulation system based on hexahedral elements, comprising: The virtual node domain calculation module is configured to take any node in the hexahedral linear element within the simulation area as the object node, calculate the single volume based on the cross-sectional area of ​​the flow and the distance from the object node to the cross-sectional area, and accumulate the single volumes corresponding to the object node in the hexahedral linear element to obtain the virtual node domain of the object node. The load distribution correction module is configured to correct the load of the Galerkin finite element method according to the principle of distributing the load of the hexahedral linear element based on the virtual node domain volume of each node, and obtain the corrected spatial discretized algebraic equation. The discretization and output module is configured to discretize the time partial derivative terms in the modified spatial discretization algebraic equation using the mean value theorem in the time direction, thereby obtaining the discretized algebraic equation for transient heat conduction of hexahedral linear elements, and outputting the nodal temperature data at each moment in the transient heat conduction simulation for thermal stress simulation.

[0019] A computer device includes a memory and a processor, the memory storing a computer program, and the processor executing the computer program to implement the transient heat conduction simulation method based on hexahedral elements.

[0020] A computer-readable storage medium storing a computer program that, when executed by a processor, implements the transient heat conduction simulation method based on hexahedral elements.

[0021] The beneficial effects of this invention are as follows: 1. This invention enables accurate simulation of three-dimensional transient heat conduction: Targeting the eight-node spatial geometry of hexahedral elements, this invention constructs a scientific virtual nodal domain calculation method. By clarifying the calculation logic and accumulation rules for individual volumes, it ensures a strict match between the virtual nodal domain volume and the actual element volume, providing a precise geometric basis for load distribution. Combined with load distribution principles and correction schemes based on virtual nodal domain volumes, it achieves precise matching between the processing of transient and source terms and the second-order accuracy heat inflow / outflow assessment system. This fundamentally solves the system inconsistency problem of the traditional Galerkin finite element method. While strictly satisfying the heat conduction conservation law at the element level, it also satisfies the heat conduction conservation law with high accuracy at the node level, significantly improving the accuracy of temperature simulation results and also significantly improving the accuracy of thermal stress simulation results.

[0022] 2. This invention reduces mesh dependency and low-quality element errors: The corrected discretized algebraic equation converges to the original differential equation in an average sense in three-dimensional space, effectively weakening the influence of mesh generation mode on simulation results. The variation amplitude of simulation errors under different mesh modes is significantly reduced. Simultaneously, this method exhibits good adaptability to irregular, low-quality hexahedral linear elements. Through reasonable virtual node domain definition and load correction, it can reduce numerical errors caused by element quality defects, broadening the applicability of the simulation method to complex mesh scenarios. Attached Figure Description

[0023] Figure 1 This is a schematic diagram of the flow area of ​​a quadrangular prism unit.

[0024] Figure 2 This is a schematic diagram of the flow area of ​​the cuboid element node.

[0025] Figure 3 It is a local mesh diagram composed of 16 quadrangular prism units.

[0026] Figure 4 This is a schematic diagram of a single volume of a virtual node domain.

[0027] Figure 5 This is a flowchart of the finite element simulation analysis process.

[0028] Figure 6 These are schematic diagrams of the seven types of grid structures used.

[0029] Figure 7 This is a bar chart showing the error results for simulation problem 1.

[0030] Figure 8 This is a bar chart showing the error results for simulation problem 2. Detailed Implementation

[0031] To provide a clearer understanding of the technical features, objectives, and effects of the present invention, specific embodiments are now described. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention; that is, the described embodiments are only a part of the embodiments of the invention, not all of them. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.

[0032] Example 1 This embodiment provides a transient heat conduction simulation method based on hexahedral elements, including: Take any node in the hexahedral linear element within the simulation area as the object node, calculate the single volume based on the cross-sectional area of ​​the flow and the distance from the object node to the cross-sectional area, and sum the single volumes corresponding to the object node within the hexahedral linear element to obtain the virtual node domain of the object node. Based on the principle of distributing the load of the hexahedral linear element according to the virtual node domain volume of each node, the load of the Galerkin finite element method is modified to obtain the modified spatial discretized algebraic equation. The time partial derivative terms in the corrected spatial discretized algebraic equations are discretized in the time direction using the mean value theorem, resulting in the discretized algebraic equations for transient heat conduction of hexahedral linear elements. The nodal temperature data at each moment in the transient heat conduction simulation are output, and finally, thermal stress simulation is performed based on the temperature simulation results.

[0033] It should be noted that this method optimizes the Galerkin finite element method in terms of virtual node domain definition and load distribution correction, solving the conservation problem of inconsistent heat assessment system in transient heat conduction simulation. At the same time, it improves the convergence problem of finite element equation to differential equation, reduces the impact of mesh generation mode dependency, and reduces simulation errors caused by low-quality hexahedral linear elements. It enables accurate three-dimensional transient heat conduction simulation under the premise that hexahedral element generation can be easily implemented even if the shape of the simulation area is very complex.

[0034] Preferably, the calculation of a single volume based on the cross-sectional area of ​​the flow path and the distance from the object node to the cross-sectional area includes: determining the spatial straight-line distance between the object node and any of the other seven nodes within the hexahedral linear element; setting a plane perpendicular to the connecting line at the midpoint of the connecting line as the flow path; the area of ​​the flow path is the product of the corresponding coefficient value obtained by the element integration after isoparametric transformation of the hexahedral linear element using the Galerkin finite element method and the spatial straight-line distance between the nodes; using the flow path as the bottom surface and the object node as the vertex, determining that the distance from the vertex to the bottom surface is half of the spatial straight-line distance, and calculating the single volume corresponding to the object node and any of the other seven nodes based on the cross-sectional area of ​​the flow path and the distance from the vertex to the bottom surface.

[0035] Specifically, for each object node in the hexahedral linear element, the spatial straight-line distance between it and the other seven nodes in the element is calculated. At the midpoint of the line connecting two nodes, a plane perpendicular to the line is constructed and used as a flow section. The area of ​​this flow section is the product of the corresponding coefficient value obtained by integrating the element after performing isoparametric transformation on the hexahedral linear element and the spatial straight-line distance between the nodes. Then, this flow section is used as the bottom surface, and the object node is used as the vertex. The distance from the vertex to the bottom surface is determined to be half of the aforementioned spatial straight-line distance. Based on the basic calculation rules of geometric volume, the individual volume of the object node and the corresponding node is calculated by combining the area of ​​the flow section and this distance.

[0036] It should be noted that the calculation method for a single volume is based on the second-order precision heat flux to determine the core geometric parameters, which ensures the matching between the calculation of the single volume and the heat inflow and outflow assessment system. This gives the definition of the single volume a clear physical meaning of heat conduction, lays a solid geometric foundation for the accurate calculation of the subsequent virtual node domain, and improves the rationality and scientificity of the definition of the virtual node domain.

[0037] Preferably, the virtual node domain of the object node is obtained by accumulating the individual volumes corresponding to the object nodes within the hexahedral linear unit, including: designating the object node within the hexahedral linear unit as node number 1, sequentially calculating the individual volumes corresponding to node 1 and the remaining seven nodes numbered 2 to 8 within the hexahedral linear unit, and accumulating the seven individual volumes to obtain the virtual node domain of node number 1 as the object node; sequentially designating each node numbered 2 to 8 within the hexahedral linear unit as an object node, calculating the individual volumes corresponding to each object node and the remaining seven nodes within the unit, and accumulating them to obtain the virtual node domain of each node; and the sum of the volumes of the virtual node domains of the eight nodes within the hexahedral linear unit is equal to the actual volume of the hexahedral linear unit.

[0038] Specifically, the eight nodes of the hexahedral linear element are first numbered sequentially according to the rules of the finite element method. The first object node to be calculated is marked as node number 1. The individual volumes corresponding to this node and the seven nodes numbered 2 to 8 are calculated in turn. The seven individual volumes are summed to obtain the virtual node domain of node number 1 as the object node. Then, nodes numbered 2 to 8 are taken as new object nodes in turn, and the above operation of calculating and accumulating individual volumes is repeated to obtain the virtual node domain corresponding to each node. In the whole calculation process, through theoretical derivation and element integration verification, it is ensured that the sum of the volumes of the virtual node domains of the eight nodes in the element is completely consistent with the actual volume of the hexahedral linear element.

[0039] It should be noted that this method realizes the systematic and standardized calculation of all virtual node domains within the hexahedral linear element, while ensuring the matching of the total volume of the virtual node domain with the actual volume of the element. This provides a volume basis that conforms to the law of conservation of heat conduction for the subsequent reasonable distribution of loads, and ensures the conservation of load distribution at the element level from a geometric perspective.

[0040] Preferably, the load allocation principle for the hexahedral linear element is based on the virtual node domain volume of each node. This includes: using the virtual node domain volume of each node as the core basis, nodes with larger virtual node domain volumes receive more loads, and the sum of the loads allocated to all nodes within the hexahedral linear element equals the total load generated within that hexahedral linear element, satisfying the conservation law of heat conduction at the element level; the load is obtained by element integration after isoparametric transformation of the hexahedral linear element, and is the total load generated by both transient and source terms.

[0041] It should be noted that this load distribution principle takes the virtual node domain volume as the core basis for distribution, realizes the accurate matching between load distribution and the second-order accurate heat input / output assessment system, and strictly satisfies the conservation law of heat conduction at the element level. It fundamentally solves the core problem of inconsistency between the load distribution system around the node and the heat input / output assessment system in the Galerkin finite element method, and lays a conservation foundation for subsequent load correction and equation optimization.

[0042] Preferably, the load of the Galerkin finite element method is modified according to the principle of distributing the load of the hexahedral linear element based on the virtual node domain volume of each node, resulting in a modified spatial discretization algebraic equation. This modification includes: introducing a correction term based on the spatial discretization equation obtained by the Galerkin finite element method for the three-dimensional transient heat conduction problem. The correction term is the difference between the virtual node domain volume of the object node and the element integral of the hexahedral element with the Galerkin weight function, multiplied by the sum of the transient and source terms at the geometric center of the hexahedral linear element. After introducing the correction term, a new spatial discretization algebraic equation is obtained as the modified spatial discretization algebraic equation. In the modified spatial discretization algebraic equation, the sum of the changes in the load distribution of each node of the element is zero, maintaining the heat conduction conservation law at the element level, while ensuring that the treatment of the transient and source terms matches the second-order accuracy heat inflow and outflow assessment system.

[0043] Specifically, we first obtain the original spatial discretization equation derived by the Galerkin finite element method for the three-dimensional transient heat conduction problem, calculate the difference between the virtual nodal domain volume of the object node and the element integral of the hexahedral element of the Galerkin weight function, then measure the sum of the transient term and the source term at the geometric center of the element, and introduce the product of the two as a correction term into the original spatial discretization equation to construct a new spatial discretization algebraic equation, which is the corrected spatial discretization algebraic equation.

[0044] It should be noted that the introduction of the correction term enables accurate and quantitative correction of the loads in the Galerkin finite element method, making the treatment of transient and source terms fully compatible with the second-order accurate heat input and output evaluation system. At the same time, since the sum of the changes in load distribution at each node of the element is zero, the heat conduction conservation law at the element level can still be strictly maintained after correction. In addition, this correction can also improve the convergence of the finite element equation to the original differential equation of three-dimensional transient heat conduction, providing a reliable mathematical and physical basis for subsequent improvement of simulation accuracy.

[0045] Preferably, the time partial derivative terms in the corrected spatial discretized algebraic equation are discretized in the time direction using the mean value theorem, including: setting a time step and a time parameter, wherein the time step is the time difference between time n and time (n-1); the time parameter is used to adjust the weights of the time partial derivative terms at time n and time (n-1) in the time discretization process, and the value range is between 0 and 1; the time partial derivative term is the partial derivative of temperature with respect to time; the product of the time partial derivative term at time n and the time parameter is calculated, and the difference between the time partial derivative term at time (n-1) multiplied by 1 and the time parameter is calculated, and the sum is obtained to obtain the discretized time partial derivative term.

[0046] Specifically, based on the actual requirements of three-dimensional transient heat conduction simulation, a fixed time step is first set, which is the time difference between adjacent times n and n-1. At the same time, a time parameter with a value between 0 and 1 is set to adjust the weight ratio of the time partial derivative term in the discretization process at different times. After determining that the time partial derivative term is the temperature partial derivative with respect to time, the product of the time partial derivative term at time n and the time parameter, and the product of the time partial derivative term at time n-1 and the difference between 1 and the time parameter are calculated respectively. The two product results are summed to obtain the time partial derivative term after time discretization.

[0047] Preferably, the discretized algebraic equations of transient heat conduction in hexahedral linear elements are obtained, and the nodal temperature data at each moment in the transient heat conduction simulation are finally output. This includes: selecting the simulation region for transient heat conduction and completing the mesh generation of hexahedral linear elements; inputting the initial temperature conditions and initializing the time-related parameters; generating the element matrix and the global matrix based on the discretized algebraic equations of transient heat conduction in hexahedral linear elements; advancing the simulation time according to the set time step and substituting the nodal temperature and source term data from the previous moment to generate a system of simultaneous equations; substituting the boundary condition-related data to solve the system of simultaneous equations and outputting the nodal temperature data at the current moment for thermal stress simulation; if the simulation termination condition is not met, the time advancement and solution are repeated until the simulation termination condition is met and the simulation ends.

[0048] Specifically, the simulation region is first determined based on the actual requirements of the three-dimensional transient heat conduction simulation. This region is then meshed using hexahedral linear elements to form several interconnected hexahedral linear elements. Next, the initial temperature conditions of all nodes within the simulation region are input, and time-related parameters such as time step, initial time, and time count are initialized. Based on the derived hexahedral linear element transient heat conduction discretization algebraic equations, corresponding element matrices and global matrices are generated. The simulation time is progressively advanced according to the set time step, substituting the node temperature data from the previous time step, along with the source term data from the current and previous time steps, into the equations to generate a corresponding system of simultaneous equations. Subsequently, the boundary condition data related to heat conduction are substituted into this system of simultaneous equations, and the equations are solved using numerical calculation methods to obtain the node temperature data at the current time step. The node temperature data at the current time step is output for thermal stress simulation. Finally, it is determined whether the preset simulation termination condition has been met. If not, the time count is updated, and the steps of time advancement and equation solving are repeated. If the condition is met, the simulation ends.

[0049] It should be noted that this implementation process constructs a complete transient heat conduction simulation process in three-dimensional thermal stress simulation. It deeply integrates the modified discretized algebraic equations with the actual simulation operation process in engineering, realizing an orderly and standardized operation from the determination of the simulation area to the output of temperature data. The cyclic execution characteristic of the process enables continuous transient simulation over time, ensuring the continuity and integrity of the simulation results. At the same time, the standardized step design greatly improves the efficiency of the simulation operation, reduces the error of manual operation, and adapts to the transient heat conduction simulation needs of large-scale and complex areas.

[0050] Accordingly, this embodiment also provides a transient heat conduction simulation system based on hexahedral elements, including: The virtual node domain calculation module is configured to take any node in the hexahedral linear element within the simulation area as the object node, calculate the single volume based on the cross-sectional area of ​​the flow and the distance from the object node to the cross-sectional area, and accumulate the single volumes corresponding to the object node in the hexahedral linear element to obtain the virtual node domain of the object node. The load distribution correction module is configured to correct the load of the Galerkin finite element method according to the principle of distributing the load of the hexahedral linear element based on the virtual node domain volume of each node, and obtain the corrected spatial discretized algebraic equation. The discretization and output module is configured to discretize the time partial derivatives in the modified spatial discretized algebraic equations using the mean value theorem in the time direction, thereby obtaining the discretized algebraic equations of transient heat conduction in hexahedral linear elements, and outputting the nodal temperature data at each moment in the transient heat conduction simulation for use in thermal stress simulation.

[0051] Specifically, the virtual node domain calculation module receives externally input geometric parameter data of hexahedral linear element nodes, automatically selects and calibrates object nodes, calculates the individual volumes corresponding to object nodes and other nodes in sequence, obtains the virtual node domain of each node through cumulative calculation, and transmits the relevant data of the virtual node domain to the load distribution correction module in real time. The load distribution correction module automatically calculates correction terms according to the received virtual node domain volume data and introduces them into the original spatial discretization equation of the Galerkin finite element method. After completing the load correction, the corrected spatial discretization algebraic equation is obtained, and the equation data is transmitted to the discretization processing and result output module. The discretization processing and result output module automatically discretizes the time partial derivative terms of the corrected equation, derives the transient heat conduction discretization algebraic equation applicable to hexahedral linear elements, and then automatically receives externally input simulation data such as initial conditions and boundary conditions and substitutes them into the equation to complete the generation and solution of the simultaneous equation system, and outputs the node temperature data for thermal stress simulation.

[0052] It should be noted that the system implements the core functions of virtual node domain calculation, load distribution correction, time discretization processing, and result output separately through modular functional division. Standardized and orderly data transmission is achieved between modules, realizing the fully automated processing of three-dimensional transient heat conduction simulation and greatly improving the simulation calculation efficiency. At the same time, the modular design mode facilitates subsequent functional expansion, module upgrades, and system maintenance, and can flexibly adapt to three-dimensional transient heat conduction simulation applications with different industrial scenarios and simulation requirements.

[0053] Example 2 This embodiment provides a transient heat conduction simulation method based on hexahedral elements to solve the inconsistency between the system considering heat inflow and outflow around nodes and the system considering increase, decrease, generation and cancellation when using hexahedral linear elements to perform transient heat conduction simulations and examining the Galerkin finite element method with second-order accurate heat flux. It also improves the problem that the finite element equation does not converge to the differential equation. The specific explanation is as follows.

[0054] 1. Galerkin's Finite Element Method and Its Problems The Galerkin finite element method is a widely used numerical method for solving differential equations. However, problems such as its mesh generation mode dependence and increased numerical errors due to low element quality remain unresolved. This embodiment describes a method to solve these problems using hexahedral linear elements in 3D simulations.

[0055] The governing equation for the three-dimensional transient heat conduction problem is shown in equation (1).

[0056] (1) in, It is the Nabla operator. " is the vector inner product, and λ is the thermal conductivity coefficient; It's temperature. It is time. The partial derivative of temperature with respect to time is called the transient term. It is the source term; take a rectangular coordinate system (x, y, z).

[0057] Applying the Galerkin finite element method to equation (1), we obtain the integral equation (2) for the object node P under consideration.

[0058] (2) in, Let e ​​be the Galerkin weight function, and let e be any element surrounding the object node P. Σ is the integral over the element, and Σ is the sum of the integrals of all surrounding elements. Applying the integration by parts theorem and Gauss-Green theorem to the element integral on the left side of equation (2), we obtain the algebraic equation (3) as the spatial discretization of the Galerkin finite element method.

[0059] (3) For hexahedral elements, the finite element method typically begins by transforming the coordinate system from (x, y, z) to (y, z) coordinate system. , , The transformation of the arbitrary hexahedral element into a cubic isoparametric element with the center at the origin and a side length of 2 is performed. Then, the element is integrated, and the result (3) becomes (4).

[0060]

[0061] (4) in, The equation is a Jacobian determinant, and the right side is referred to as the load. Since the treatment of natural boundary conditions is irrelevant to this invention, it will not be discussed here.

[0062] The conservation and convergence of the integral result of equation (4) obtained by Galerkin finite element method are examined.

[0063] 1.1 Examination based on the law of conservation by Figure 1 Taking the square prism unit shown as an example, it consists of units located at... Parallelograms on a plane directional stretching thickness The resulting parallelogram has a base that begins at the origin, coincides with the y-axis, and has a length of h. The z-axis represents the height of the parallelogram, with a height of h, and slopes towards the negative y-axis. Angle. This element has eight nodes. The node located at the origin is numbered 1, the node on the x-axis is numbered 2, the node on the y-axis is numbered 4, and the remaining nodes are numbered according to the rules of the finite element method.

[0064] First, taking node 1 as the object node P, the unit integration result on the left side of equation (4) is equation (5).

[0065] (5) like Figure 1 As shown in subgraph (a), using To represent nodes and The midpoint of the line is used To represent the distance between two nodes, , Let i and j be the temperatures of nodes i and j, respectively. Midpoint From the node Flow to node The second-order precision of the heat flux.

[0066] Therefore, it can be considered that the first term on the right side of equation (5) is The brackets [ ] represent dots. The second-order accurate heat flux from node 1 to node 2 at point 1 The second-order accurate heat flux from node 4 to node 3 at point 3, The second-order accurate heat flux from node 5 to node 6 at point 6 and point 6 The weighted average of the second-order accurate heat flux flowing from node 8 to node 7. Coefficient hh It is the flow area. The product of [ ] and the coefficient is the outflowing heat flux. This flow surface is perpendicular to the heat flux and passes through the midpoint. .

[0067] Similarly, consider the second term on the right-hand side of equation (5) to be a point. The second-order accurate heat flux from node 1 to node 4 at point 4 and point 4 The weighted average of the second-order accurate heat flux from node 2 to node 3 at the flow area, after passing through the flow area The outflowing heat flux. This flow surface is perpendicular to the heat flux and passes through the midpoint. .

[0068] Similarly, the third term on the right side of equation (5) is considered to be a point. The second-order accurate heat flux from node 1 to node 8 at point 8 and point 8 The weighted average of the second-order accurate heat flux from node 2 to node 7 at the flow area, after passing through the flow area The outflowing heat flux. The flow surface is perpendicular to the heat flux and passes through the midpoint. .

[0069] Next, taking node 4 as the object node P for examination, the unit integral on the left side of equation (4) is equation (6).

[0070] (6) Similar to the discussion on the right-hand side of equation (5), the physical meaning of each term on the right-hand side of equation (6) is the heat flux estimated by second-order precision heat flux. Details are as follows: Figure 1 The subgraph (b) is shown.

[0071] Comparing the right side of equation (5) with the right side of equation (6), it is confirmed that the third term on the right side of equation (5) does not exist in equation (6), that is, the flow area of ​​the heat flux flowing out of node 1 is greater than that of node 4. .

[0072] On the other hand, let V represent the element volume in the xyz coordinate system, and perform element integration on the right side of equation (4). When the transient term and the source term are constant within the element, the integration result is as shown in equation (7), where the load is evenly distributed to each node.

[0073] (7) It is evident that the system used by Galerkin's finite element method to evaluate the inflow and outflow of heat around nodes is inconsistent with the system used to consider the increase or decrease of heat over time, and the generation or dissipation of heat. From the perspective of second-order accuracy, it does not satisfy the conservation law.

[0074] like Figure 2 As shown, only when the element shape is a cube or cuboid, the flow area of ​​each node is equal, and the enclosed volume is V / 8, satisfying the conservation law from the perspective of second-order accuracy. Cube elements are generally considered to have the highest simulation accuracy, which also supports the rationality of examining the conservation of heat flux based on second-order accuracy.

[0075] 1.2 Examination of the convergence of differential equations Composed of sixteen hexahedral units Figure 3 The local mesh shown is used as an example for examination. Each element is a quadrangular prism as described in the conservation study. Eight elements form a layer, stacked in two layers along the x-axis. The element surface composed of nodes 1, 2, 3, and 4 as described in the conservation study is now placed on the xy or xz plane. Taking the center node of this local mesh as P, the integral result of equation (4) can be written as equation (8) after Taylor expansion and other simplifications.

[0076] (8) At this point, the transient term and the source term are assumed to be constants. Equation (8) converges only to the original differential equation (1) in the x-direction, but does not converge in other directions.

[0077] To reduce numerical errors caused by conservation and convergence issues, a significant amount of manpower is required for mesh generation during simulation to obtain high-quality elements that approximate cube or cuboid shapes. Despite this substantial investment of manpower, the presence of low-quality elements is still unavoidable due to limitations imposed by the geometry of the simulation region and the required mesh density.

[0078] 2. Technical issues This embodiment aims to address the inconsistency between the system considering heat inflow and outflow around nodes and the system considering heat generation and depletion when using second-order accurate heat flux in transient heat conduction simulations of thermal stress using hexahedral linear elements in the Galerkin finite element method. It also aims to improve the problem of the finite element equations not converging to differential equations. The approach involves proposing new methods for handling transient and source terms.

[0079] 3. Technical Solution 3.1 A new concept of virtual node domain based on the definition of hexahedral linear units is proposed. For a linear hexahedral element of arbitrary shape, the temperature within the element is determined by the simultaneous nodal temperatures via... Calculate N, where i T is a shape function defined on a hexahedral element. i Let i be the temperature value of node i. Take any node as the object node P, and let its number in the element be 1. Then the element integral on the left side of equation (4) can be written as equation (9).

[0080]

[0081] (9) Regarding the coefficient part on the right side of equation (9), equation (10) is theoretically proven.

[0082] (10) Using equations (9) and (10), and introducing the distance between nodes 1 and i. The unit integral on the right side of equation (4) can be written as equation (11).

[0083]

[0084] (11) The right side of equation (11) The physical meaning is that it is located at the midpoint I of the line segment connecting nodes 1 and i. 1i The second-order accurate heat flux from node 1 to node i at a given point depends on its coefficient. Let I be the area of ​​the cross-section through which the flow passes. 1i The plane is perpendicular to the heat flux, i.e., perpendicular to the line segment connecting nodes 1 and i; their product becomes the heat flux of the flow area with second-order accuracy. (The flow cross-section is used for this purpose.) Let be the base, and node 1 be the vertex. The distance from the vertex to the base is given. Define the volume calculated by equation (12). , Figure 4 This is a schematic diagram of the volume.

[0085] (12) in, Define the sum of the seven volumes calculated by equation (12). For the virtual node domain of node 1, since object node P and node 1 are the same thing, it is calculated according to equation (13).

[0086] (13) Nodes numbered 2, 3, 4, 5, 6, 7, and 8 are respectively taken as object nodes P, and their virtual node domains are defined according to equation (13). , , , , , , The virtual node domain is a volume concept, and its proportion in the element volume is related to the element shape. For example, the virtual node domain of each node in a cube or cuboid element is V / 8. Figure 2 The flow cross-section shown is closed and joined; while in the conservation case, the virtual node domain of node 1 in the quadrangular prism unit is 2V / 12 and the virtual node domain of node 4 is V / 12.

[0087] 3.2 A novel method is proposed to distribute loads to nodes in a reference virtual node domain that strictly satisfies the conservation law at the element level. When distributing loads within an element to each node, the load distribution principle is based on the virtual node domain volume of each node, allocating the loads to the hexahedral linear element. Simultaneously, to ensure strict adherence to conservation laws at the element level, the sum of the loads distributed to each node must equal the loads generated within that element. Therefore, the virtual node domain, serving as the load distribution benchmark, must satisfy the prerequisite that the sum of the virtual node domains of each node equals the element volume.

[0088] Given the unit volume V and the Jacobian determinant There exists the relationship shown in equation (14). The components of the Jacobian determinant on the left side of equation (15) are represented using the corresponding simplified symbols on the right side. For example, using... express .

[0089] (14) (15) According to equations (13) and (12) and the definition of the Galerkin weight function, the sum of the virtual node domains of the eight nodes of the same unit is calculated by equation (16).

[0090]

[0091]

[0092] (16) The partial derivatives of the shape function with respect to the coordinates in equation (16) are represented by equation (17).

[0093] (17) Substituting equation (17) into equation (16), and utilizing the properties of shape functions... The summation calculation part of equation (16) is obtained by rearranging: (18) The right side of equation (18) is The sign of the division is 1, that is, the integral function of equation (16) is Referring to equation (14), we obtain the result that the sum of the virtual node domains of the eight nodes shown in equation (16) is equal to the unit volume, i.e., equation (19) holds.

[0094] (19) This indicates that the new method of assigning loads to nodes by referring to the virtual node domain can guarantee that the conservation law is strictly satisfied at the element level.

[0095] 3.3 Load Correction The Galerkin finite element method is modified by a new method that strictly satisfies the conservation law at the element level and distributes loads to nodes according to the size of the virtual node domain. This makes the treatment of transient phases and source terms compatible with a heat input / output assessment system with second-order accuracy, while improving the problems of finite element equations not converging to differential equations and mesh dependency. Equation (20) is now proposed to replace equation (4).

[0096]

[0097] (20) in, This embodiment proposes a modification to the Galerkin method, where the subscript 'O' indicates the value at the geometric center of the element. Let P be the virtual node domain of node P. Nodes with larger virtual node domains will be assigned more loads. Although the load distribution of nodes is changed due to the introduction of the correction term, as shown in equation (19) ,and The sum of the changes in load distribution at each node of the element is zero, that is, the property of Galerkin method strictly satisfying the conservation law at the element level has not changed due to the introduction of the modification term. Equation (20) strictly satisfies the conservation law at the element level, that is, the sum of the loads distributed to each node of the element is equal to the load generated in the element.

[0098] 3.4 Theoretical Support for Load Correction 3.4.1 Conservation of Node Level For cube-shaped units The correction term in equation (20) is zero. This type of element itself does not have a conservation problem. For elements of general shape, the first term within the brackets [] on the right-hand side of equation (20) is... The resulting nodal loads can be considered to originate from the load on the element volume V. P The weighted transient and source terms. The correction term introduced by the Galerkin finite element method, namely the second term in the brackets on the right side of equation (20), is the term from the virtual node domain. Subtract The resulting volume is multiplied by the transient and source terms at the geometric center, and the resulting loads distributed to each node vary with the element shape. In particular, when the transient phase and source terms are constants within the element, the area within the brackets on the right-hand side of equation (20) is directly simplified to... This means that the system volume considering transient and source terms is equal to the volume of the system enclosed by the flow cross-section when considering heat inflow and outflow around the nodes with second-order precision. Compared to the Galerkin method, it satisfies the conservation law with higher precision at the node level.

[0099] 3.4.2 Convergence of the differential equation For simplicity, it is assumed that the transient term and the source term are constants in the local mesh around node P, and the element integral on the right side of equation (20) can be simplified to equation (21).

[0100] (twenty one) For a structure composed of sixteen hexahedral units Figure 3 The local mesh shown has each element being a quadrangular prism as described in the conservation consideration. Eight elements form a layer, stacked in two layers along the x-axis. The element surface formed by nodes 1, 2, 3, and 4, as described in the conservation consideration, is now placed on the xy or xz plane. Let P be the center node of this local mesh. Figure 1 Node 4 in the table corresponds to its virtual node field. Substituting it into the right side of equation (21), and then performing the calculation of equation (20), after Taylor expansion and other simplifications, we obtain equation (22).

[0101] (twenty two) The coefficient 4V / 3 on the right side of equation (22) originates from sixteen units, i.e. It is evident that the introduction of the correction term makes equation (22) converge to the differential equation in the sense of the average of the x, y, and z directions, thus improving the convergence compared to the Galerkin method. The convergence test is conducted when the element size approaches zero, so assuming that the transient and source terms are constants does not affect the convergence conclusion.

[0102] 4. Discretization in the time direction Let the time step be Δ The time parameter is Using the mean value theorem of equation (23), the time partial derivative term in equation (20) is discretized. .

[0103] (twenty three) in, The superscript n indicates time n, and the superscript n-1 indicates time n-1. Finally, the discretized algebraic equation (24) of the transient heat conduction equation of the hexahedral element is obtained.

[0104] (twenty four) 5. Simulation Process like Figure 5 As shown, this embodiment provides a finite element simulation analysis process, including: 1) Select the simulation area; 2) Mesh the simulation area; 3) Input initial conditions and set... n=1; 4) Generate the unit matrix: the {} on the left side of equation (24) and the {} in the first term on the right side; 5) Generate the overall matrix: the first term on the left and right sides of equation (24); 6) Implementation time: ; 7) Temperature at the previous moment Substitute the first term on the right side of equation (24); 8) Source Item and Substitute the second term on the right side of equation (24) to generate the nodal loads; 9) Generate a system of simultaneous equations; 10) Substitute the boundary conditions; 11) Solve the system of equations to obtain the temperature. ; 12) Based on temperature Perform thermal stress simulation; 13) If not finished, Proceed to step 6). 14) End.

[0105] 6. Verification Example 6.1 Validation Issues This study aims to solve for the temperature of two types of three-dimensional transient heat conduction problems with boundary conditions frequently encountered in practical engineering, thereby verifying the improvement in numerical simulation accuracy brought about by load correction based on virtual nodal domains. To ensure the universality of the verification conclusions, a dimensionless mathematical physics model is adopted, and the heat conduction coefficient is taken as... The simulation object is cube-shaped. The heat source. To accurately evaluate the simulation error, a transient heat conduction problem with analytical temperature solution is used to eliminate the influence of factors such as measurement errors.

[0106] As simulation problem 1, Equation (25) is the heat source term, and Equation (26) is the analytical solution of temperature.

[0107] (25) (26) Will Substituting into equation (26) yields the initial conditions, and respectively... , , Substituting into equation (26) yields the first type of boundary conditions on the six boundary surfaces, i.e., the boundary conditions for simulation problem 1 are known temperature conditions.

[0108] As simulation problem 2, Equation (27) is the heat source term, and Equation (28) is the analytical solution of temperature.

[0109] (27) (28) Will Substituting into equation (28) yields the expression as the initial condition. Then, respectively... , , Substituting into equation (28), the resulting expression serves as the first type of boundary condition on the three surfaces. On the remaining three boundary surfaces, the temperature gradient in the outward normal direction of equation (28) is taken as the second type of boundary condition. That is, simulation problem 2 has both known temperature and known heat flux boundaries.

[0110] As an evaluation index for numerical simulation error, the maximum relative value at the end of the simulation, as shown in equation (29), is adopted.

[0111] ×100 (%) (29) Here , , These are the nodes at the end of the simulation. Temperature analytical solution, nodes The numerical solution and the maximum temperature value of the analytical solution are obtained. Numerical calculations are performed using double precision and eight-point numerical integration.

[0112] 6.2 Using a grid Figure 6 Seven different mesh types were demonstrated. The mesh generation method involves... Quadrilateral grid on a plane Eight layers of equal thickness are stretched in the direction, and the values ​​in parentheses represent the total number of points. Figure 6 In the diagram, the grid of subgraphs (a) to (f) is only shown due to symmetry. , The part is indicated by "○". The position where z=1. The number of cells around the node in subgraph (b) ranges from 6 to 16; the meshes of subgraphs (c) to (e) contain elements with an angle of... The subgraph (a) has low-quality flat-angled elements; the subgraph (e) has elongated elements; the subgraph (f) has a mesh composed of low-quality elements with closely spaced nodes; the subgraph (g) has a mesh with varying density. For the meshes of subgraphs (a) to (d) and (g), the time step is taken as... Take the grid of Figures (e) and (f) All time parameters are taken as follows: At the end of the time .

[0113] 6.3 Simulation Results and Observations Figure 7 The numerical error of simulation problem 1 is shown. Figure 8 The numerical error of simulation problem 2 is shown. The horizontal axis represents the grid pattern, and (a), (b), ..., (g) represent the results of... Figure 6 The results obtained from the grid of neutron maps (a), (b), ... (g); the vertical axis represents the numerical error, specifically the error defined by equation (29). (%)and The product of To summarize the points, This represents the number of nodes in the cube's unit cell. Because... It can be approximated as the reciprocal of the element size. Theoretically, the error of a linear element is proportional to the square of the element size. Therefore, the error representation method on the vertical axis eliminates the influence of the number of nodes on the error, making... Figure 7 and Figure 8 Mesh dependency analysis can also be performed between methods using a square cell mesh as a reference. "GFE" represents the result of the Galerkin method, and "Vnd" represents the result of the method of this invention.

[0114] for Figure 6 For the cubic cells of the neutron map (a), since the correction term is zero, the results from both methods are the same. For the same reason... Figure 6 The results for the neutron map (b) mesh are the same under the first type of boundary condition. In other cases, the introduction of the correction term reduces the error, with significant reductions of up to half the error.

[0115] Comparing the error variations of different meshes, the Galerkin method results in an error of approximately 2.7 times; while the method of this invention shows an error variation of less than 2.0 (simulation problem 1) and 1.5 (simulation problem 2), significantly improving mesh dependency, even when angles are involved. Even with low-quality elements and low-quality elements with very close nodes, the accuracy is still guaranteed. This makes the time-consuming simulation step of mesh generation easier and increases the reliability of the simulation results.

[0116] 7. Conclusion 1) For the finite element method numerical simulation of transient heat conduction problems using hexahedral linear elements, a virtual nodal domain is defined based on second-order accurate heat flux. The load distribution generated by transient and source terms is corrected so that the conservation law at the nodal level is satisfied.

[0117] 2) Theoretical analysis confirms that the introduction of the correction term satisfies the conservation law not only at the node level but also at the element level, and improves the convergence of the finite element equation to the differential equation.

[0118] 3) Numerical simulation results using several mesh patterns show that, compared with the commonly used Galerkin finite element method, the method of this invention can reduce the simulation error of low-quality elements, with the error being halved in some cases. The error variation caused by different meshes is also reduced from 2.7 times to less than 2.0 and 1.5 times, significantly improving the mesh dependency problem.

[0119] Example 3 This embodiment is based on embodiment 1: This embodiment provides a computer device, including a memory and a processor. The memory stores a computer program, and the processor executes the computer program to implement the transient heat conduction simulation method based on hexahedral elements as described in Embodiment 1. The computer program can be in the form of source code, object code, executable file, or some intermediate form.

[0120] Specifically, a pre-written computer program is stored in the computer device's memory. This program contains all the execution steps, operational logic, and data processing rules for implementing a transient heat conduction simulation method based on hexahedral elements. When the processor calls and executes this computer program, it will sequentially execute core steps in the simulation process, such as virtual node domain calculation, load allocation and correction, time discretization processing, and simulation process execution, according to the preset logic. The processor's high-speed computing capabilities enable various operations such as matrix generation, equation solving, geometric parameter calculation, and data iteration. During the simulation process, the memory simultaneously undertakes the task of storing various simulation parameters, intermediate calculation data, and final node temperature data, providing data support for the processor's operations. As time progresses, the node temperature data in the memory is output for thermal stress simulation.

[0121] It should be noted that by solidifying the simulation method into a computer program and executing it on a computer device, the simulation method is digitized and automated. The high-speed computing power of the computer device is used to greatly improve the speed and accuracy of simulation calculations. It can efficiently handle transient heat conduction simulation calculations of complex simulation regions and a large number of hexahedral linear elements, and is suitable for the large-scale, high-precision complex simulation needs in engineering practice. At the same time, the operability of the computer device greatly reduces the application threshold of the simulation method, making it easier for engineering technicians to use in practice.

[0122] Example 4 This embodiment is based on embodiment 1: This embodiment provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the transient heat conduction simulation method based on hexahedral elements described in Embodiment 1. The computer program can be in the form of source code, object code, executable file, or some intermediate form. The storage medium includes any entity or device capable of carrying computer program code, a recording medium, computer memory, read-only memory (ROM), random access memory (RAM), electrical carrier signals, telecommunication signals, and software distribution media, etc.

[0123] Specifically, the completed computer program that implements the transient heat conduction simulation method based on hexahedral elements is compiled using an appropriate computer coding method and then stored in a computer-readable storage medium. This storage medium can interact with the processors of various computer devices that have data reading and processing capabilities. When the processor reads and executes the computer program from the storage medium, it will strictly follow the preset logic and execution steps in the program to complete all calculation, data processing, and result output operations for the transient heat conduction simulation based on hexahedral elements.

[0124] It should be noted that this computer-readable storage medium enables program storage and cross-device portability of the simulation method, allowing the simulation program to be flexibly executed on different computer devices, greatly improving the applicability and flexibility of the simulation method; at the same time, the stable storage characteristics of the storage medium can ensure the integrity and security of program data, facilitating the technical promotion, engineering application and cross-scenario deployment of the simulation method.

[0125] The above description is merely a preferred embodiment of the present invention. It should be understood that the present invention is not limited to the forms disclosed herein and should not be construed as excluding other embodiments. It can be used in various other combinations, modifications, and environments, and can be altered within the scope of the concept described herein through the above teachings or related technologies or knowledge. Modifications and variations made by those skilled in the art that do not depart from the spirit and scope of the present invention should be within the protection scope of the appended claims.

[0126] It should be noted that, for the sake of simplicity, the foregoing method embodiments are described as a series of actions. However, those skilled in the art should understand that this application is not limited to the described order of actions, as some steps may be performed in other orders or simultaneously according to this application. Furthermore, those skilled in the art should also understand that the embodiments described in the specification are preferred embodiments, and the actions and modules involved are not necessarily essential to this application.

Claims

1. A transient heat conduction simulation method based on hexahedral elements, characterized in that, include: Take any node in the hexahedral linear element within the simulation area as the object node, calculate the single volume based on the cross-sectional area of ​​the flow and the distance from the object node to the cross-sectional area, and sum the single volumes corresponding to the object node within the hexahedral linear element to obtain the virtual node domain of the object node. Based on the principle of distributing the load of the hexahedral linear element according to the virtual node domain volume of each node, the load of the Galerkin finite element method is modified to obtain the modified spatial discretized algebraic equation. The time partial derivative terms in the corrected spatial discretized algebraic equations are discretized in the time direction using the mean value theorem, resulting in the discretized algebraic equations for transient heat conduction of hexahedral linear elements. The nodal temperature data at each moment in the transient heat conduction simulation are output, and finally, thermal stress simulation is performed based on the temperature simulation results.

2. The transient heat conduction simulation method based on hexahedral elements according to claim 1, characterized in that, The calculation of a single volume based on the cross-sectional area of ​​the flow path and the distance from the object node to the cross-sectional area includes: Determine the spatial straight-line distance between the object node and any of the other seven nodes in the hexahedral linear element. Set a plane perpendicular to the line at the midpoint of the line as a flow section. The area of ​​the flow section is the product of the corresponding coefficient value obtained by the element integration after the isoparametric transformation of the hexahedral linear element by the Galerkin finite element method and the spatial straight-line distance between the nodes. Using the flow cross-section as the bottom and the object node as the vertex, the distance from the vertex to the bottom is determined to be half of the straight-line distance in space. Based on the flow cross-section area and the distance from the vertex to the bottom, the individual volume corresponding to the object node and any of the other seven nodes is calculated.

3. The transient heat conduction simulation method based on hexahedral elements according to claim 2, characterized in that, The virtual node domain of the object node is obtained by summing the individual volumes corresponding to the object nodes within the accumulated hexahedral linear units, including: The object node within the hexahedral linear element is designated as node number 1. The individual volumes corresponding to node 1 and the other seven nodes numbered 2 to 8 within the hexahedral linear element are calculated sequentially. The seven individual volumes are then summed to obtain the virtual node domain of node number 1 as the object node. Each node numbered 2 to 8 within the hexahedral linear element is sequentially treated as an object node. The individual volumes corresponding to each object node and the other seven nodes within the element are calculated and summed to obtain the virtual node domain of each node. The sum of the volumes of the virtual node domains of the eight nodes within the hexahedral linear element is equal to the actual volume of the hexahedral linear element.

4. The transient heat conduction simulation method based on hexahedral elements according to claim 3, characterized in that, The load distribution on the hexahedral linear element based on the virtual node domain volume of each node includes: Based on the virtual node domain volume of each node, the larger the virtual node domain volume, the more load is allocated to the node, and the sum of the loads allocated to all nodes in the hexahedral linear element is equal to the total load generated in the hexahedral linear element, so as to satisfy the conservation law of heat conduction. The load is obtained by integrating the hexahedral linear elements after isoparametric transformation, and is the total load generated by the transient term and the source term.

5. The transient heat conduction simulation method based on hexahedral elements according to claim 4, characterized in that, The load correction for the Galerkin finite element method yields the corrected spatial discretized algebraic equations, including: Based on the spatial discretization equations obtained by the Galerkin finite element method for the three-dimensional transient heat conduction problem, a correction term is introduced according to the principle of distributing the load of the hexahedral linear element based on the virtual node domain volume of each node. The correction term is the difference between the virtual node domain volume of the object node and the element integral of the hexahedral element of the Galerkin weight function, multiplied by the sum of the transient term and the source term at the geometric center of the hexahedral linear element. After introducing the correction term, a new spatial discretization algebraic equation is obtained as the corrected spatial discretization algebraic equation. In the corrected spatial discretization algebraic equation, the sum of the changes in the load distribution of each node in the element is zero, and the heat conduction conservation law at the element level is still maintained. At the same time, the treatment of transient and source terms is matched with the second-order accurate heat inflow and outflow evaluation system.

6. The transient heat conduction simulation method based on hexahedral elements according to claim 5, characterized in that, The time partial derivative terms in the modified spatial discretized algebraic equations are discretized in the time direction using the mean value theorem, including: The time step and time parameter are set. The time step is the time difference between time n and time n-1. The time parameter is used to adjust the weight of the time partial derivative terms at time n and time n-1 in the time discretization process, and the value range is between 0 and 1. The time partial derivative term is the partial derivative of temperature with respect to time. Calculate the product of the time partial derivative term at time n and the time parameter, and calculate the difference between the time partial derivative term at time n-1 multiplied by 1 and the time parameter. Summing these values ​​yields the discretized time partial derivative term.

7. The transient heat conduction simulation method based on hexahedral elements according to claim 6, characterized in that, The discretized algebraic equations for transient heat conduction of the hexahedral linear element are obtained, and the nodal temperature data at each moment in the transient heat conduction simulation are output. Finally, thermal stress simulation is performed based on the temperature simulation results, including: Select the simulation region for transient heat conduction and complete the mesh generation of hexahedral linear elements. Input the initial temperature conditions and initialize the time-related parameters. Based on the discretized algebraic equations of transient heat conduction of hexahedral linear elements, the element matrix and the global matrix are generated respectively. The simulation time is advanced according to the set time step, and the node temperature and source term data of the previous moment are substituted to generate a system of simultaneous equations. Substitute the boundary condition data to solve the simultaneous equations, output the current node temperature data for thermal stress simulation, and repeat the time progression and solution process until the simulation ends when the simulation ends.

8. A transient heat conduction simulation system based on hexahedral elements, characterized in that, include: The virtual node domain calculation module is configured to take any node in the hexahedral linear element within the simulation area as the object node, calculate the single volume based on the cross-sectional area of ​​the flow and the distance from the object node to the cross-sectional area, and accumulate the single volumes corresponding to the object node in the hexahedral linear element to obtain the virtual node domain of the object node. The load distribution correction module is configured to correct the load of the Galerkin finite element method according to the principle of distributing the load of the hexahedral linear element based on the virtual node domain volume of each node, and obtain the corrected spatial discretized algebraic equation. The discretization and output module is configured to discretize the time partial derivative terms in the modified spatial discretization algebraic equation using the mean value theorem in the time direction, thereby obtaining the discretized algebraic equation for transient heat conduction of hexahedral linear elements, and outputting the nodal temperature data at each moment in the transient heat conduction simulation for thermal stress simulation.

9. A computer device comprising a memory and a processor, wherein the memory stores a computer program, characterized in that, When the processor executes the computer program, it implements the transient heat conduction simulation method based on hexahedral elements as described in any one of claims 1-7.

10. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by the processor, it implements the transient heat conduction simulation method based on hexahedral elements as described in any one of claims 1-7.

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