Calculation method for reactivity change caused by geometric deformation of a nuclear device

Abstract

The present application discloses a method for calculating the reactivity change caused by the geometric deformation of a nuclear device. The method includes the following steps: performing initial geometric modeling on the nuclear device; setting material properties for the modeling; creating an analysis step based on the initial model; setting contact properties for the reactivity model of the nuclear device after creating the analysis step; setting loads based on the reactivity model of the nuclear device with contact properties set; meshing the reactivity model of the nuclear device after setting the loads; and calculating the meshed reactivity model of the nuclear device to obtain a calculation result. The present application improves the efficiency of the impact dynamics analysis of the nuclear device, and the result processing after the operation analysis is convenient.

Description

Technical Field

[0001] This application belongs to the technical field of nuclear safety analysis, and particularly relates to a method for calculating reactivity change caused by geometric deformation of a nuclear device. Background Art

[0002] In the development process of nuclear engineering, evaluating the reactivity change of a nuclear device in shock dynamics is an important field in nuclear safety analysis. After a nuclear device is impacted, its structure will change, and the structure change leads to reactivity change. This process can be divided into shock dynamics calculation and neutronics calculation. Different software is used for the two calculations, and different software uses different file formats. Therefore, a technical solution is needed to transfer the results of the two. Currently, Sandia National Laboratories (SNL) in the United States has proposed a technical solution. First, the finite element simulation analysis software Presto is used to analyze the process of the nuclear device being impacted. Then, the DAGMC (Direct Accelerated Geometry Monte Carlo) program jointly developed by Sandia National Laboratories (SNL) and the University of Wisconsin (WISC) is used to perform neutronics calculation on the deformed core. This technical solution has done a good job in simulating small space reactors, but there are also some deficiencies. (1) Some of the software involved is not open to domestic users. (2) It is not convenient to operate and requires manual invocation of many software, including PRESTO, Cubit, sphgen3d, Fortran, DAGMC, and algebra. (3) It is not convenient to set the smoothed particle hydrodynamics (SPH) element in shock mechanics. The fluid grid needs to be processed by the sphgen3d program, and the Fortran program is required to handle the problem of grid boundary coincidence. In the study of the impact of shock dynamics on the safety of nuclear devices, there are two techniques, namely shock dynamics calculation and neutronics calculation. For shock dynamics calculation, it can be simulated with finite element analysis software, and there are already many mature software that can achieve it, including but not limited to ABAQUS, ANAYS, and COMSOL. For the neutronics calculation of a nuclear device after being impacted, it can be calculated by Monte Carlo software, including but not limited to MCNP, RMC, and OpenMC. During the research process, the following problems need to be solved. (1) How to perform shock dynamics simulation on a nuclear device to provide a deformation model for neutronics calculation. In shock dynamics calculation, the falling angle, material properties, and fluid grid settings need to be considered. (2) How to model the deformed core in neutronics calculation. Traditional Monte Carlo software models by inputting files, which is inefficient, error-prone, and cannot model deformed and irregular structures. (3) How to transfer the results of finite element analysis software and Monte Carlo software. The input and output files of the two software use different formats, and the file format needs to be converted during the coupling process. Summary of the Invention

[0003] The present invention aims to solve the deficiencies of the prior art and proposes a method for calculating the reactivity change caused by the geometric deformation of a nuclear device, including:

[0004] S1. Perform an initial geometric modeling on the nuclear device to obtain an initial geometric model;

[0005] S2. Set the material properties for the initial geometric model;

[0006] S3. Create an analysis step according to the initial geometric model;

[0007] S4. Set the contact properties and constraint conditions based on the initial geometric model after setting the material properties and creating the analysis step;

[0008] S5. Set the load based on the initial geometric model after setting the contact properties and constraint conditions, and divide the mesh for the initial geometric model after setting the load;

[0009] S6. Perform finite element mechanical calculation on the initial geometric model after dividing the mesh to obtain the mesh file after the geometric deformation of the nuclear device;

[0010] S7. Convert the mesh file into a neutronics model;

[0011] S8. Perform neutronics calculation on the neutronics model to obtain the reactivity change caused by the geometric deformation of the nuclear device.

[0012] Optionally, the material properties include material density, elastic modulus, Poisson's ratio, yield stress, and plastic strain.

[0013] Optionally, the contact properties include tangential behavior and normal behavior:

[0014] The tangential behavior is used to define the friction formula between the contact surfaces;

[0015] The normal behavior is used to set the relationship between the pressure and penetration on the contact surface.

[0016] Optionally, the constraint condition is used to firmly bond two surfaces in the model and they will not be separated during the analysis process.

[0017] Optionally, the load includes setting boundary conditions and physical field settings.

[0018] Optionally, the process of dividing the mesh for the initial geometric model after setting the load includes:

[0019] Directly divide the solid geometric model into a mesh;

[0020] Divide the liquid geometric model using smoothed particle hydrodynamics to obtain divided elements.

[0021] Optionally, the process of obtaining the mesh file after geometric deformation of the nuclear device includes:

[0022] After meshing, create a model in ABAQUS. In the Part module of the Module model, import the result file of the mechanical simulation, and select the deformed model at the corresponding time point to be imported.

[0023] Assemble the deformed model in the Module model;

[0024] In the job module of ABAQUS, export the mesh file of the deformed model in inp format;

[0025] If the model has many components, classify the components according to the material and export the mesh files in inp format respectively.

[0026] Optionally, the process of converting the mesh file into a neutronics model includes:

[0027] Import the mesh file in inp format of the generated deformed component in the Trelis program;

[0028] If the components are spatially continuous, directly export them as STL files;

[0029] If the components are not spatially continuous, use Boolean operations to merge the volumes and then export them as STL files;

[0030] Assign different material properties to the STL files;

[0031] Merge the STL files and convert them into a neutronics model.

[0032] Optionally, the process of performing neutronics calculations on the neutronics model includes:

[0033] Perform neutronics calculations on the neutronics model and the updated material density to obtain the reactivity change.

[0034] Optionally, the method for obtaining the updated material density includes:

[0035] First, import the generated mesh file in the Trelis program;

[0036] Classify the components according to the material;

[0037] Use Boolean operations to merge the components of the same material. In the program mode, select geometry, Boolean operation, merge volume in sequence, and then perform component volume merging after selecting the components;

[0038] Export the merged components and convert them into STL files;

[0039] Calculate the volume using the functions built into the Trelis program; based on the change in volume, obtain the change in density through the density formula.

[0040] Compared with the prior art, the beneficial effects of this application are as follows:

[0041] (1) Disclosed a process for converting the mesh file of ABAQUS into an h5m file that can be read by the Monte Carlo program.

[0042] (2) Improved the efficiency of calculating the reactivity after the nuclear device expands due to heat and is subjected to dynamic shock.

[0043] (3) Involves fewer programs in the operation process.

[0044] (4) The operation is convenient and the calculation efficiency is high, without the need to manually call a lot of software.

[0045] (5) Facilitates setting the fluid mesh as a smoothed particle hydrodynamics (SPH) element in shock dynamics.

[0046] (6) The post-processing of the results after operation and analysis is convenient. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] In order to more clearly illustrate the technical solutions of this application, the drawings required to be used in the embodiments are briefly introduced below. Obviously, the drawings in the following description are only some embodiments of this application. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0048] Figure 1 It is a method step diagram of a method for calculating the reactivity change caused by the geometric deformation of a nuclear device in an embodiment of this application;

[0049] Figure 2 It is a method step diagram of a method for calculating the reactivity change caused by thermal expansion in an embodiment of a method for calculating the reactivity change caused by the geometric deformation of a nuclear device in an embodiment of this application;

[0050] Figure 3 It is a method step diagram of a method for calculating the reactivity change caused by dynamic shock in an embodiment of a method for calculating the reactivity change caused by the geometric deformation of a nuclear device in an embodiment of this application;

[0051] Figure 4 It is a cross-sectional view of a fuel rod in an embodiment of a method for calculating the reactivity change caused by the geometric deformation of a nuclear device in an embodiment of this application;

[0052] Figure 5 It is a cross-sectional view of a small cylindrical reactor in an embodiment of a method for calculating the reactivity change caused by the geometric deformation of a nuclear device in an embodiment of this application;

[0053] Figure 6 For the C3D8R mesh diagram of the calculation method for reactivity change caused by geometric deformation of a nuclear device in an embodiment of the present application;

[0054] Figure 7 For the conversion diagram of converting the finite element (FEM) to the surface model (FBM) in the calculation method for reactivity change caused by geometric deformation of a nuclear device in an embodiment of the present application;

[0055] Figure 8 For the flow chart from the H5m file to neutronics calculation in the calculation method for reactivity change caused by geometric deformation of a nuclear device in an embodiment of the present application. Detailed implementation manners

[0056] Next, the technical solutions in the embodiments of the present application will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present application. Obviously, the described embodiments are only a part of the embodiments of the present application, rather than all of the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present application.

[0057] To make the above objects, features, and advantages of the present application more obvious and understandable, the present application will be further described in detail below in conjunction with the accompanying drawings and specific implementation manners.

[0058] Embodiment 1

[0059] In this embodiment, as Figure 1 shown, a calculation method for reactivity change caused by geometric deformation of a nuclear device specifically includes:

[0060] S1. Perform initial geometric modeling on the nuclear device to obtain an initial geometric model;

[0061] S2. Set material properties for the initial geometric model; the material properties include material density, elastic modulus, Poisson's ratio, yield stress, and plastic strain.

[0062] The contact properties include tangential behavior and normal behavior; the tangential behavior is used to define the friction formula between contact surfaces;

[0063] The normal behavior is used to set the relationship between pressure and penetration on the contact surface.

[0064] S3. Create an analysis step according to the initial geometric model;

[0065] S4. Set contact properties and constraint conditions based on the initial geometric model after setting material properties and creating an analysis step; the constraint conditions are used to firmly bond two surfaces in the model and they will not separate during the analysis process.

[0066] S5. Set the loads for the initial geometric model after setting the contact properties and constraint conditions, and mesh the initial geometric model after setting the loads; the loads include setting boundary conditions and physical field settings.

[0067] The process of meshing the initial geometric model after setting the loads includes:

[0068] Directly mesh the solid geometric model;

[0069] Use smoothed particle hydrodynamics to mesh the liquid geometric model to obtain the divided elements.

[0070] S6. Perform finite element mechanical calculations on the initial geometric model after meshing to obtain the mesh file after the geometric deformation of the nuclear device; optionally, the process of obtaining the mesh file after the geometric deformation of the nuclear device includes:

[0071] After meshing, create a model in ABAQUS, import the result file of the mechanical simulation in the Part module of the Module model, and select the deformed model at the corresponding time point to be imported;

[0072] Assemble the deformed model in the Module model;

[0073] In the job module of ABAQUS, export the mesh file in inp format of the deformed component;

[0074] If the model has many components, classify the components according to the material and export the mesh files in inp format respectively.

[0075] S7. Convert the mesh file into a neutronics model; the process of converting the mesh file into a neutronics model includes:

[0076] Import the inp format mesh file of the generated deformed component in the Trelis program;

[0077] If the components are spatially continuous, directly export them as STL files;

[0078] If the components are not spatially continuous, use Boolean operations to merge the volumes and then export them as STL files;

[0079] Assign different material properties to the STL files;

[0080] Merge the STL files and convert them into a neutronics model.

[0081] S8. Perform neutronics calculations on the neutronics model to obtain the reactivity change caused by the geometric deformation of the nuclear device.

[0082] The process of performing neutronics calculations on the neutronics model includes:

[0083] Perform neutronics calculations on the neutronics model and the updated material density to obtain the reactivity change.

[0084] The method for obtaining the updated material density includes:

[0085] First, import the generated mesh file into the Trelis program;

[0086] Classify the components according to the materials;

[0087] Merge the components of the same material using Boolean operations. In the program's mode, select geometry, Boolean operations, merge volumes in sequence, and perform component volume merging after selecting the components;

[0088] Export the merged components and convert them into STL files;

[0089] Use the built-in function of the Trelis program to calculate the volume; according to the change in volume, obtain the change in density through the density formula.

[0090] Example Two

[0091] In the example of the reactivity change caused by the thermal expansion of the fuel rod in the nuclear device, the flowchart is as Figure 2 shown and specifically includes:

[0092] Perform initial geometric modeling on the thermal expansion analysis of the fuel rod in the nuclear device. The fuel rod model is as Figure 4 shown. Determine the length unit as "cm", draw the model in the part module of ABAQUS, with a radius of 0.1 cm and a height of 38 cm;

[0093] Set the material properties for the fuel rod modeling, combine the thermal expansion initial model with the material properties to obtain the thermal expansion model. The thermal conductivity of uranium dioxide is 2 W / (m·K), the elastic modulus is 219 GPa, the Poisson's ratio is 0.345, the expansion coefficient is 1e-5, the density is 18.9 g / cm 3 ³, and the specific heat capacity is 330 J / (kg·K);

[0094] Create an analysis step for the fuel rod thermal expansion model. Select the coupled temperature-displacement analysis (coupled temp-displacement) for the thermal expansion model, set the total time of the thermal expansion analysis to 18000 s, and set the initial time increment step to 1 s and the temperature increment step to 15 K;

[0095] Set the contact properties for the fuel rod thermal expansion model. Set the ambient temperature of the thermal expansion model to 1000 K;

[0096] Set loads for the thermal expansion model. Set the boundary conditions and physical field settings of the thermal expansion model. The physical field setting of thermal expansion is the initial temperature of the nuclear device, and set the initial temperature of the nuclear device to 300K;

[0097] Mesh the fuel rod thermal expansion model. The mesh elements used are as Figure 6 shown. Set the element type of the thermal expansion model to coupled temperature-displacement element;

[0098] Perform thermal expansion calculations to obtain the calculation results.

[0099] Transfer the results of the above two geometric deformation calculations to the neutronics calculation software for neutronics calculations. The specific steps include:

[0100] Export the results of the geometric deformation calculations as a mesh (inp) file.

[0101] (1) First, create a model (model) in ABAQUS, select to import the result file (odb file) of the mechanical simulation, and select the components at the corresponding time points to be imported;

[0102] (2) Assemble each component according to the material in the part module and form an assembly in the assembly module;

[0103] (3) Create a task in the job module according to the assembled components and export the inp file of the deformed components.

[0104] Convert the inp file into a stereolithography (STL) file and calculate the density change of the nuclear device after geometric deformation for updating the material density in the neutronics calculation. Specifically include: (1) Import the generated inp file of the deformed components in the Trelis program; (2) If the components are continuous in space, directly export them as STL files; (3) If the components are not continuous in space, use Boolean operations to merge the volumes and then export them as STL files. If not done in this way, the complete components cannot be exported. Specifically, in the program mode, select geometry, Boolean operation, merge volume in sequence, and perform volume merging after selecting the components; (4) Use the built-in function of the Trelis program to calculate the volume of the deformed geometry. According to the change in volume, obtain the density change through the density formula. The purpose of calculating the density is to update the material density in the neutronics calculation.

[0105] Convert the STL file into a dagmc.h5m file (a file that can be recognized by Monte Carlo software), and then perform neutronics calculations. Specifically include:

[0106] (1)Based on the Python toolkit PyMOAB for MOAB, convert the STL file into a dagmc.h5m file that can be read by Monte Carlo codes such as OpenMC.

[0107] (2)Convert the STL file into a dagmc.h5m file through a Python script. The steps are as follows: 1) Add the STL files corresponding to each material and the graveyard.stl file to the conversion program together. 2) The Graveyard.stl file is a necessary file for OpenMC to run. This file is used to define the boundary conditions. When neutrons enter the graveyard, the transport history ends. 3) The graveyard space consists of the closed space between two concentric cubes. 4) In the script file, the STL files correspond one-to-one with the material numbers, and use the Python conversion program to generate the dagmc.h5m file.

[0108] (3)Perform neutronics calculations on the dagmc.h5m file using OpenMC. The flow chart of the neutronics calculation is as Figure 8 shown.

[0109] Example 3

[0110] In the example of reactivity change caused by the dynamic shock of a nuclear device, as Figure 3 shown, it specifically includes:

[0111] The calculation method for the reactivity change caused by the dynamic shock of the core of a nuclear device under transient conditions includes:

[0112] Perform initial geometric modeling of the dynamic shock of the nuclear device. Determine that the unit system adopted is cm, determine the structural dimensions of the nuclear device, the core radius is 35, the reflector radius is 45, the barrel radius is 50, and the core height is 100, and draw a sketch of the model;

[0113] Set the material properties for the modeling, combine the initial model of the dynamic shock of the nuclear device with the material properties to obtain the dynamic shock model of the nuclear device. Determine the material properties, material density, elastic modulus, Poisson's ratio, yield stress, and plastic strain;

[0114] Create an analysis step for the dynamic shock model of the nuclear device. Select to display the dynamic analysis and set the time of the dynamic shock analysis to 0.003 s;

[0115] Set the contact properties for the dynamic shock model of the nuclear device. Set the contact properties, including tangential behavior and normal behavior: the tangential behavior is used to define the friction formula between the contact surfaces; the normal behavior is used to set the relationship between the pressure and penetration on the contact surface. Then set the constraint conditions to fix the assembly of the core, reflector, and barrel;

[0116] Set the load for the nuclear device dynamic impact model. Set the fixed boundary condition for the ground where the cylindrical small reactor is impacted, and set the velocity field for the cylindrical small reactor, with a velocity of 240 m / s and the angle between the direction and the ground being 30 degrees;

[0117] The cross-sectional view of the said cylindrical small reactor is as Figure 5 shown.

[0118] Divide the grid cells for the nuclear device dynamic impact model. The grid cells adopted are as Figure 6 shown. It is characterized in that the element type is set. For the solid element, it is set as an explicit three-dimensional stress element;

[0119] Finally, conduct the impact dynamics calculation to obtain the calculation results.

[0120] Transfer the results of the above two geometric deformation calculations to the neutronics calculation software for neutronics calculation. The specific steps include:

[0121] Export the results of the geometric deformation calculation as a grid (inp) file.

[0122] (1) First, create a model (model) in ABAQUS, select to import the result file (odb file) of the mechanical simulation, and select the components at the corresponding time points to be imported;

[0123] (2) Assemble each component according to the material in the part module, and form an assembly in the assembly module;

[0124] (3) According to the assembled assembly, create a task in the job module and export the inp file of the deformed assembly.

[0125] Convert the inp file into a stereolithography (STL) file, and calculate the density change after the geometric deformation of the nuclear device for updating the material density in the neutronics calculation. Specifically include:

[0126] (1) Import the generated inp file of the deformed assembly in the Trelis program;

[0127] (2) If the assembly is continuous in space, directly export it as an STL file;

[0128] (3) If the assembly is not continuous in space, use the Boolean operation to merge the volumes and then export it as an STL file. If not done like this, the complete assembly cannot be exported. Specifically include sequentially selecting geometry, Boolean operation, merge volume, and selecting the assembly for volume merging in the mode of the program;

[0129] (4) Use the functions provided in the Trelis program to calculate the volume of the small cylindrical pile. Based on the change in volume, the change in density is obtained through the density formula. The purpose of calculating the density is to update the density of materials in neutronics calculations.

[0130] Convert the STL file into a dagmc.h5m file (a file that can be recognized by Monte Carlo software), and then perform neutronics calculations. Specifically, it includes:

[0131] (1) Based on the Python toolkit PyMOAB of MOAB, convert the STL file into a dagmc.h5m file that can be read by Monte Carlo programs such as OpenMC.

[0132] (2) Convert the STL file into a dagmc.h5m file through a Python script. The steps are as follows: 1) Add the STL files corresponding to each material and the graveyard.stl file to the conversion program together. 2) The Graveyard.stl file is a necessary file for OpenMC to run. This file is used to define the boundary conditions. After neutrons enter the graveyard, the transport history ends. 3) The graveyard space consists of the closed space between two concentric cubes. 4) In the script file, the STL files correspond one by one to the material numbers, and a dagmc.h5m file is generated using the Python conversion program.

[0133] (3) OpenMC performs neutronics calculations on the dagmc.h5m file. The flowchart of the neutronics calculation is as Figure 8 shown.

[0134] OpenMC performs neutronics calculations on the deformed model. The flowchart of the neutronics calculation is as Figure 8 shown.

[0135] For finite element programs, the model to be calculated is divided into grids one by one, and then the forces and displacements at the nodes of these grids are calculated. However, for fluids, this method cannot simulate well because fluids and solids are different, and there will be splashes during the collision process. To better simulate fluids, this application uses smoothed particle hydrodynamics. The units divided by this method are called smoothed particle hydrodynamics units. The conversion diagram is as Figure 7 shown. In the traditional method, sph units need to be converted using sphgen3d, but the traditional method requires two software, one for drawing the solid model and one for drawing the fluid. According to the method of this application, both sph units and solid structure units only need to be drawn using ABAQUS. Reducing the number of software used makes the operation more convenient and the applicability stronger.

[0136] The embodiments described above are only descriptions of the preferred embodiments of this application, and do not limit the scope of this application. Without departing from the design spirit of this application, various deformations and improvements made by those of ordinary skill in the art to the technical solutions of this application shall fall within the protection scope determined by the claims of this application.

Claims

1. A method for calculating the reactivity change caused by the geometric deformation of a nuclear device, characterized in that, Including: S1. Conduct an initial geometric modeling of the nuclear device to obtain an initial geometric model; S2. Set the material properties for the initial geometric model; S3. Create an analysis step according to the initial geometric model; S4. Set the contact properties and constraint conditions based on the initial geometric model after setting the material properties and creating the analysis step; S5. Set the load based on the initial geometric model after setting the contact properties and constraint conditions, and divide the mesh for the initial geometric model after setting the load; S6. Conduct finite element mechanical calculations on the initial geometric model after mesh division to obtain a mesh file after geometric deformation of the nuclear device; The process of obtaining the mesh file after geometric deformation of the nuclear device includes: Create a model in ABAQUS, select to import the result file of the mechanical simulation, and select the components at the corresponding time point to be imported; Assemble each component according to the material under the part module, and form an assembly under the assembly module; Create a task in the job module according to the assembled components, and export the mesh file in inp format of the deformed components; S7. Convert the mesh file into a neutronics model; The process of converting the mesh file into a neutronics model includes: Import the inp format mesh file of the generated deformed components in the Trelis program; If the components are continuous in space, directly export them as STL files; If the components are not continuous in space, use Boolean operations to merge the volumes and then export them as STL files; Assign different material properties to the STL files; Merge the STL files, and convert the STL files into a neutronics model based on PyMOAB, and the format of the neutronics model is h5m; S8. Conduct neutronics calculations on the neutronics model to obtain the reactivity change caused by the geometric deformation of the nuclear device.

2. The method for calculating the reactivity change caused by the geometric deformation of the nuclear device according to claim 1, characterized in that, The material properties include material density, elastic modulus, Poisson's ratio, yield stress, and plastic strain.

3. The method for calculating the reactivity change caused by the geometric deformation of the nuclear device according to claim 1, characterized in that, The contact properties include tangential behavior and normal behavior: The tangential behavior is used to define the friction formula between the contact surfaces; The normal behavior is used to set the relationship between the pressure and penetration on the contact surface.

4. The method for calculating the reactivity change caused by the geometric deformation of the nuclear device according to claim 1, characterized in that The constraint conditions are used to firmly bond two surfaces in the model and they will not be separated during the analysis process.

5. The method for calculating the reactivity change caused by the geometric deformation of the nuclear device according to claim 1, wherein The load includes setting boundary conditions and physical field settings.

6. The method for calculating the reactivity change caused by the geometric deformation of the nuclear device according to claim 1, characterized in that, The process of dividing the mesh for the initial geometric model after setting the load includes: Directly divide the solid geometric model into a mesh; Use smoothed particle hydrodynamics to divide the liquid geometric model to obtain divided elements.

7. The method for calculating the reactivity change caused by the geometric deformation of the nuclear device according to claim 1, characterized in that, The process of conducting neutronics calculations on the neutronics model includes: Conduct neutronics calculations on the neutronics model and the updated material density to obtain the reactivity change.

8. The method for calculating the reactivity change caused by the geometric deformation of the nuclear device according to claim 7, characterized in that, The method for obtaining the updated material density includes: First, import the generated mesh file in the Trelis program; Classify the components according to the material; Use Boolean operations to merge the components of the same material. In the program mode, select geometry, Boolean operation, merge volume in sequence, and then perform component volume merging after selecting the components; Export the merged components and convert them into STL files; Use the built-in function of the Trelis program to calculate the volume; According to the change in volume, the change in density is obtained through the density formula.

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