An Isogeometric Transient Temperature Field Analysis Method Integrating Birth and Death Element Technique

Abstract

This application discloses an isogeometric transient temperature field analysis method integrating birth and death element technology, relating to the field of computational mechanics. The method includes establishing an isogeometric NURBS model, calculating the element heat transfer matrix, element thermal melting matrix, and element load array; determining whether the current calculation element is in the computation time domain; if not, setting the element heat transfer matrix and element load array to 0; if so, restoring the loads corresponding to the element heat transfer matrix and element load array, and removing the loads shared by the element and other elements in the computation time domain at their boundary; assembling the element heat transfer matrix, element thermal melting matrix, and element load array into a global matrix, and constructing the heat transfer equation based on the global matrix; calculating the temperature of the control points based on the heat transfer equation, and determining the temperature field of the final solution domain. This application considers the realization of an isogeometric transient temperature field analysis method under complex temperature load conditions including time-series problems, promoting the application of isogeometric analysis in engineering.

Description

Technical Field

[0001] This application relates to the field of computational mechanics, and in particular to an isogeometric transient temperature field analysis method that incorporates birth and death element techniques. Background Technology

[0002] Compared to traditional finite element analysis (FEM) methods, isogeometric analysis directly utilizes non-uniform rational B-splines (NURBS) functions for physical modeling. The model automatically generates control points, eliminating the need for complex and time-consuming mesh generation. Furthermore, due to the consistency between the geometric and analytical models and the high-order nature of NURBS basis functions, isogeometric analysis can achieve high-order continuity across the entire domain, resulting in higher solution accuracy and computational efficiency compared to FEM.

[0003] Currently, temperature field solutions are mostly performed using finite element method (FE) software, while isogeometric analysis remains largely confined to theoretical examples and is rarely applied in problems involving complex loads and computational domains with time-sequential requirements (such as dam concrete pouring). Therefore, there is an urgent need for a temperature field analysis method that incorporates isogeometric analysis to leverage its advantages in engineering problems. Summary of the Invention

[0004] The purpose of this application is to provide an isogeometric transient temperature field analysis method that integrates birth and death element technology, which solves the problem of difficulty in analyzing isogeometric transient temperature fields with complex load forms and time-sequence requirements for the computational domain.

[0005] To achieve the above objectives, this application provides the following solution:

[0006] This application provides an isogeometric transient temperature field analysis method that integrates birth and death element technology, including:

[0007] Establish a NURBS model for isogeometric analysis and calculate the element heat transfer matrix, element thermal melting matrix, and element load array.

[0008] Determine whether the current computing unit is in the computing time domain, and process the unit heat transfer matrix, the unit thermal melting matrix, and the unit load array;

[0009] If not, set the unit heat transfer matrix and the unit load array to 0;

[0010] If so, restore the loads corresponding to the unit heat transfer matrix and the unit load array, and remove the loads in the unit load array and the unit heat transfer matrix that share the boundary with the units in the calculation time domain.

[0011] The unit heat transfer matrix, the unit thermal melting matrix, and the unit load array are assembled into an overall matrix, and the heat transfer equation is constructed based on the overall matrix.

[0012] The temperature of the control point is determined based on the heat transfer equation, and the temperature field of the solution domain is determined.

[0013] In one embodiment, using Define the NURBS model; where χ is the geometric field; u(χ) is the physical field; R m For NURBS basis functions; m is the control point label; ξ represents the parametric coordinates; P m U represents the control point; N represents the total number of control points; u m For physical quantities at control points;

[0014] Based on the NURBS model, using Determine the element heat transfer matrix; where, k is the element heat transfer matrix; x R is the thermal conductivity coefficient; R is the shape function matrix formed by the NURBS basis functions; k y The thermal conductivity coefficient; The convective heat transfer coefficient between the object and the surrounding medium; To integrate the area within the cell; To integrate over the boundary under the third type of boundary conditions; T denotes the transpose matrix; x is the x-coordinate of the control point; y is the y-coordinate of the control point; A e The indicator shows that the area is integrated over the unit area;

[0015] use Determine the element thermal melting matrix; where, Here, ρ is the element thermal fusion matrix; c is the material density; T Specific heat capacity;

[0016] use Determine the element load array; where, is the element load array; Q is the intensity of the internal heat source of the object; The boundary heat flux density; T ∞ The ambient temperature; This is a second type of boundary condition; This is a third type of boundary condition.

[0017] In one embodiment, the unit heat transfer matrix, the unit thermal melting matrix, and the unit load array are assembled into an integral matrix;

[0018] Based on the central point difference method, using Construct the heat transfer equation for the overall matrix; where Δt is the time step; K T For the overall heat transfer matrix; CT The overall thermal melting matrix is ​​represented by k; k represents the time node; q represents the overall thermal melting matrix. k and q k+1 These are the overall temperature arrays corresponding to time nodes k and k+1, respectively; and These are the overall load arrays corresponding to time nodes k and k+1, respectively.

[0019] In one implementation, a high-dimensional priority transformation method is used to transform the two-dimensional index arrays of local control points and global control points into one-dimensional index arrays.

[0020] Store the mapping relationship of the one-dimensional index array into a new mapping index array;

[0021] The unit heat transfer matrix, the unit thermal melting matrix, and the unit load array are assembled into an integral matrix according to the new mapping index array.

[0022] In one embodiment, θ(τ)=θ0(1-e -cτ The temperature rise function of concrete is determined by θ(τ), where θ(τ) is the expression for the temperature rise function of concrete; θ0 is the final adiabatic temperature rise of concrete; c is a constant related to the type of cement and the amount of cement per cubic meter in concrete; and τ is the calculation time point.

[0023] In one implementation, a visual network is constructed based on the NURBS model;

[0024] Based on the shape function matrix and the temperature array of the control points, the temperature value at the control point is converted into the temperature at the vertex position of the visualization grid;

[0025] Extract the temperature values ​​at the vertex positions of the visualized mesh, map the temperature values ​​to the range corresponding to the linear color gradient, and determine the temperature field of the solution domain.

[0026] According to the specific embodiments provided in this application, the following technical effects are disclosed:

[0027] This application provides an isogeometric transient temperature field analysis method that integrates birth and death element technology. It determines whether the current computational element is in the computation time domain. If not, the element heat transfer matrix and element load array are set to 0. If in the computation time domain, the loads corresponding to the element heat transfer matrix and element load array are restored, and the loads shared by the element and other elements in the computation time domain are removed. This enables the analysis of isogeometric transient temperature fields for temperature load problems with time-series requirements. This application considers the implementation of isogeometric transient temperature field analysis methods under complex temperature load conditions, promoting the application of isogeometric analysis in engineering. Attached Figure Description

[0028] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0029] Figure 1 This is a flowchart illustrating an isogeometric transient temperature field analysis method that incorporates birth and death element technology in one embodiment of this application.

[0030] Figure 2 This is a schematic diagram of an isogeometric discrete model provided in an embodiment of this application.

[0031] Figure 3 This is a schematic diagram of the birth and death unit processing flow provided in an embodiment of this application.

[0032] Figure 4 This is a schematic diagram of the isogeometric analysis visualization process provided in one embodiment of this application.

[0033] Figure 5 This is a schematic diagram of the temperature field calculation results provided in an embodiment of this application.

[0034] Figure 6 This is a schematic diagram of the calculation process for an isogeometric transient temperature field provided in an embodiment of this application. Detailed Implementation

[0035] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0036] To make the above-mentioned objectives, features and advantages of this application more apparent and understandable, the application will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0037] The purpose of this application is to propose an isogeometric transient temperature field analysis method that integrates the birth and death element technique. It establishes an isogeometric discrete model of foundation and multi-layer concrete that can automatically generate control points without mesh generation. At the same time, by applying the birth and death element technique, isogeometric analysis is applied to simulate the concrete pouring process, which can improve modeling and calculation efficiency and achieve more efficient temperature field solution.

[0038] like Figure 1 As shown in the figure, this application provides an isogeometric transient temperature field analysis method that integrates birth and death element technology, the details of which are as follows.

[0039] S1: Establish a NURBS model for isogeometric analysis and calculate the element heat transfer matrix, element thermal melting matrix, and element load array.

[0040] S2: Determine whether the current computing unit is in the computing time domain, and process the unit heat transfer matrix, the unit thermal melting matrix, and the unit load array.

[0041] S3: If not, set the unit heat transfer matrix and the unit load array to 0.

[0042] S4: If so, restore the loads corresponding to the unit heat transfer matrix and the unit load array, and remove the loads in the unit load array and the unit heat transfer matrix that share the boundary with the unit in the calculation time domain.

[0043] S5: Assemble the unit heat transfer matrix, the unit thermal melting matrix, and the unit load array into an overall matrix, and construct the heat transfer equation based on the overall matrix.

[0044] S6: Determine the temperature of the control point based on the heat transfer equation, and determine the temperature field of the solution domain.

[0045] Furthermore, in an exemplary embodiment, S1 can be replaced by the following steps.

[0046] S101: Utilization Define the NURBS model; where χ is the geometric field; u(χ) is the physical field; R m For NURBS basis functions; m is the control point label; ξ represents the parametric coordinates; P m U represents the control point; N represents the total number of control points; u m This refers to the physical quantity at the control point.

[0047] like Figure 2 As shown, a geometrically discrete model of the foundation and concrete was established. The model consists of 8 patches, 162 elements, 290 control points, 18 free boundaries, and 7 shared boundaries. The model materials include the foundation and roller-compacted concrete. Different NURBS patches were numbered to facilitate subsequent assembly matrix and element birth / death operations.

[0048] S102: Based on the NURBS model, using Determine the element heat transfer matrix; where, k is the element heat transfer matrix; x R is the thermal conductivity coefficient; R is the shape function matrix formed by the NURBS basis functions; k y The thermal conductivity coefficient; The convective heat transfer coefficient between the object and the surrounding medium; To integrate the area within the cell; To integrate over the boundary under the third type of boundary conditions; T denotes the transpose matrix; x is the x-coordinate of the control point; y is the y-coordinate of the control point; A e The identifier indicates that the area is integrated within the cell.

[0049] Where, k x k y ρ is the thermal conductivity coefficient, with units of W / (m·K); ρ is the material density, with units of kg / m³. 3 ;c T is the specific heat capacity, expressed in J / (kg·℃); Q is the intensity of the internal heat source of the object, expressed in W / kg; Boundary heat flux density, in W / m³ 2 ; The convective heat transfer coefficient between the object and the surrounding medium, expressed in W / (m²). 2 ·K); T ∞ The ambient temperature is expressed in °C.

[0050] S103: Exploit Determine the element thermal melting matrix; where, Here, ρ is the element thermal fusion matrix; c is the material density; T Specific heat capacity.

[0051] S104: Utilization Determine the element load array; where, is the element load array; Q is the intensity of the internal heat source of the object; The boundary heat flux density; T ∞ The ambient temperature; This is a second type of boundary condition; This is a third type of boundary condition.

[0052] The method of birth and death of elements is used to deal with the temporal issues in transient heat transfer processes. The specific operation is as follows: for elements outside the calculation time domain, their heat transfer matrix is ​​set to 0 and all loads on the element boundary are removed, i.e., the element is "killed". When the element calculation time is reached, the heat transfer matrix and the corresponding load are restored, and the loads at the boundary shared by the element and the previous element are removed, thus "activating" the element.

[0053] like Figure 3 As shown, further, in an exemplary embodiment, S5 can be replaced by the following steps.

[0054] S501: Assemble the unit heat transfer matrix, the unit thermal melting matrix, and the unit load array into an integral matrix.

[0055] S502: Based on the central point difference method, using Construct the heat transfer equation for the overall matrix; where Δt is the time step; K T For the overall heat transfer matrix; C T The overall thermal melting matrix is ​​represented by k; k represents the time node; q represents the overall thermal melting matrix. k and q k+1 These are the overall temperature arrays corresponding to time nodes k and k+1, respectively; and These are the overall load arrays corresponding to time nodes k and k+1, respectively.

[0056] Furthermore, S501 specifically includes:

[0057] S5011: The high-dimensional priority transformation method is adopted to transform the two-dimensional index arrays of local control points and global control points into one-dimensional index arrays.

[0058] S5012: Store the mapping relationship of the one-dimensional index array into a new mapping index array.

[0059] S5013: Assemble the unit heat transfer matrix, the unit thermal melting matrix, and the unit load array into an integral matrix according to the new mapping index array.

[0060] The assembly process of the unit matrix adopts a "high-dimensional priority" transformation method, which transforms the two-dimensional index arrays of control points in local units and global control points into two one-dimensional index arrays. The mapping relationship between the two one-dimensional arrays is stored in a new mapping array, and the unit matrix is ​​assembled into the whole through the mapping array.

[0061] The element matrix is ​​calculated and assembled, in which the killed concrete material elements are "activated" one by one in a bottom-up order.

[0062] Furthermore, S501 also includes:

[0063] Using θ(τ)=θ0(1-e -cτ The temperature rise function of concrete is determined by θ(τ), where θ(τ) is the expression for the temperature rise function of concrete; θ0 is the final adiabatic temperature rise of concrete; c is a constant related to the type of cement and the amount of cement per cubic meter in concrete; and τ is the calculation time point.

[0064] The time step and boundary conditions are set. The foundation surface is subjected to convective heat transfer boundary conditions, the bottom and both sides are subjected to adiabatic boundary conditions, the outer surface of the concrete block is subjected to convective heat transfer boundary conditions, and the interior is subjected to hydration heat. The adiabatic temperature rise of the concrete is fitted according to an exponential function. In this application, θ0 is 23℃, m is 0.4, the initial temperature of the concrete is 12℃, and the outside temperature and the initial temperature of the foundation are 5℃.

[0065] like Figure 5 and Figure 6As shown, the control point temperature is obtained by solving the heat transfer equation. The temperature field of the entire solution domain is calculated based on the control point temperature. It can be clearly seen that the temperature of the concrete block increases from bottom to top and the stratification is obvious. The highest temperature calculated by the structure gradually tends to the initial temperature of the concrete plus the adiabatic temperature rise. The temperature at the junction of the foundation and the concrete gradually rises, which is in line with the theoretical law and proves the reliability of the calculation results.

[0066] Furthermore, in an exemplary embodiment, S6 can be replaced by the following steps.

[0067] S601: Construct a visual network based on the NURBS model.

[0068] S602: Based on the shape function matrix and the temperature array of the control points, the temperature value at the control point is converted into the temperature at the vertex position of the visualization grid.

[0069] S603: Extract the temperature values ​​at the vertex positions of the visualized mesh, map the temperature values ​​to the range corresponding to the linear color gradient, and determine the temperature field of the solution domain.

[0070] like Figure 4 As shown, since control points in isogeometric analysis are usually not on the physical model, the temperature at the control points needs to be processed to obtain the temperature field of the solution domain. The specific operation is as follows: First, construct a visual mesh based on the NURBS surface. Then, use the shape function matrix and the control point temperature array to convert the temperature values ​​at the control points into the temperatures at the mesh vertex positions. After extracting the mesh vertex data, map the data to the range corresponding to the linear color gradient to obtain the temperature field contour map.

[0071] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0072] This document uses specific examples to illustrate the principles and implementation methods of this application. The descriptions of the above embodiments are only for the purpose of helping to understand the methods and core ideas of this application. Furthermore, those skilled in the art will recognize that, based on the ideas of this application, there will be changes in the specific implementation methods and application scope. Therefore, the content of this specification should not be construed as a limitation of this application.

Claims

1. An isogeometric transient temperature field analysis method fused with birth-death element technology, characterized by, The application relates to a method for calculating the temperature field of a heat transfer system, and belongs to the technical field of heat transfer calculation. The application relates to a method for calculating the temperature field of a heat transfer system, and belongs to the technical field of heat transfer calculation. Utilizing , determining a NURBS model; wherein, is a geometric field; is a physical field; is a NURBS basis function; m is a control point index; denotes a parametric coordinate; is a control point; N is the total number of control points; is a physical quantity at the control point; According to the NURBS model, a unit heat transfer matrix is determined by using , wherein, is the unit heat transfer matrix; is the heat conduction coefficient; R is a shape function matrix composed of NURBS base functions; is the heat conduction coefficient; is the convection heat transfer coefficient of the object and the surrounding medium; is the integral of the area within the unit range; is the integral of the boundary within the third type of boundary condition; T represents the transpose matrix; x is the horizontal coordinate of the control point; y is the vertical coordinate of the control point; indicates the integral of the area within the unit range; Utilizing , a unit heat fusion matrix is determined; wherein, is the unit heat fusion matrix; is the material density; is the specific heat capacity; Utilizing , a unit load array is determined; wherein, is the unit load array; is the internal heat source intensity of the object; is the boundary heat flux density; is the ambient temperature; is the second type of boundary condition; is the third type of boundary condition; The application relates to a method for calculating the temperature field of a heat transfer system, and belongs to the technical field of heat transfer calculation. The application relates to a method for calculating the temperature field of a heat transfer system, and belongs to the technical field of heat transfer calculation. The application relates to a method for calculating the temperature field of a heat transfer system, and belongs to the technical field of heat transfer calculation. The application relates to a method for calculating the temperature field of a heat transfer system, and belongs to the technical field of heat transfer calculation. The application relates to a method for calculating the temperature field of a heat transfer system, and belongs to the technical field of heat transfer calculation.

2. The iso-geometric transient temperature field analysis method of claim 1, wherein, The application relates to a method for calculating the temperature field of a heat transfer system, and belongs to the technical field of heat transfer calculation. The application relates to a method for calculating the temperature field of a heat transfer system, and belongs to the technical field of heat transfer calculation. According to the center point difference method, a heat transfer equation of an overall matrix is constructed by using , wherein, is a time step; is an overall heat transfer matrix; is an overall heat melting matrix; is a time node; and are overall temperature column arrays corresponding to time nodes k and k+ 1, respectively; and are overall load column arrays corresponding to time nodes k and k+ 1, respectively.

3. The iso-geometric transient temperature field analysis method of claim 2, wherein, The application relates to a method for calculating the temperature field of a heat transfer system, and belongs to the technical field of heat transfer calculation. The application relates to a method for calculating the temperature field of a heat transfer system, and belongs to the technical field of heat transfer calculation. The application relates to a method for calculating the temperature field of a heat transfer system, and belongs to the technical field of heat transfer calculation. The application relates to a method for calculating the temperature field of a heat transfer system, and belongs to the technical field of heat transfer calculation.

4. The iso-geometric transient temperature field analysis method of claim 2, wherein, The application relates to a method for calculating the temperature field of a heat transfer system, and belongs to the technical field of heat transfer calculation. Utilizing determining a temperature rise function of the concrete; wherein, is a temperature rise function expression of the concrete; is a final adiabatic temperature rise of the concrete; c is a constant related to a cement variety and a cement consumption per unit in the concrete; is a calculation time point.

5. The iso-geometric transient temperature field analysis method of claim 1, wherein, The application relates to a method for calculating the temperature field of a heat transfer system, and belongs to the technical field of heat transfer calculation. The application relates to a method for calculating the temperature field of a heat transfer system, and belongs to the technical field of heat transfer calculation. The application relates to a method for calculating the temperature field of a heat transfer system, and belongs to the technical field of heat transfer calculation. The application relates to a method for calculating the temperature field of a heat transfer system, and belongs to the technical field of heat transfer calculation. The application relates to a method for calculating the temperature field of a heat transfer system, and belongs to the technical field of heat transfer calculation. The application relates to a method for calculating the temperature field of a heat transfer system, and belongs to the technical field of heat transfer calculation. The application relates to a method for calculating the temperature field of a heat transfer system, and belongs to the technical field of heat transfer calculation. The application relates to a method for calculating the temperature field of a heat transfer system, and belongs to the technical field of heat transfer calculation. The application relates to a method for calculating the temperature field of a heat transfer system, and belongs to the technical field of heat transfer calculation. The application relates to a method for calculating the temperature field of a heat transfer system, and belongs to the technical field of heat transfer calculation. The application relates to a method for calculating the temperature field of a heat transfer system, and belongs to the technical field of heat transfer calculation. 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