Phase field simulation method and device for nuclear fuel grain growth

Abstract

The invention discloses a phase field simulation method and device for nuclear fuel grain growth, and relates to the technical field of irradiation microstructure evolution simulation. The method comprises the steps that on the basis of the phase field theory, a system total free energy equation of the nuclear fuel is constructed, and the system total free energy equation comprises phase field variables corresponding to all crystal grains of the nuclear fuel; based on the total free energy equation of the system and the phase field variables corresponding to the crystal grains, dynamic evolution equations of the crystal grains are constructed respectively; and based on the dynamic evolution equation, carrying out parallel evolution on the growth form of each crystal grain so as to simulate the real morphology change of the nuclear fuel crystal grain growth. According to the method, the calculated amount in the process of simulating the evolution of the polycrystalline system can be reduced, the evolution efficiency is improved, and meanwhile, the real morphology change of grain growth can be simulated.

Description

Technical Field

[0001] The present application relates to the technical field of irradiation microstructure evolution simulation, and in particular to a phase field simulation method and device for nuclear fuel grain growth. Background Art

[0002] Many accident tolerant fuels (ATFs) have been proposed by researchers to replace traditional UO2 fuels, such as U3Si2. Compared with traditional UO2 fuels, ATF fuels are required to have better or at least equivalent performance in normal operation and better performance in accident scenarios. ATF fuels have higher thermal conductivity and higher uranium density than UO2. The high thermal conductivity and higher uranium density of ATF fuels relative to UO2 make it a very attractive fuel candidate compound.

[0003] The physical properties of ATF fuel, such as corrosion resistance and thermal conductivity, and mechanical properties, such as strength and ductility, all depend on the size and distribution of its grains. Therefore, it is of great significance to study the evolution of ATF fuel grain structure.

[0004] At present, numerical simulation methods such as the Monte Carlo-Pots model, frontier tracking method, and cellular automaton are usually used to simulate the evolution of grain structure. However, these methods need to track the boundaries of individual grains and process sharp interfaces during the simulation process, which leads to excessive calculation and low calculation efficiency. In addition, these numerical simulation methods mostly process grain by grain during simulation, so when the number of grains is large, the error accumulation will become larger and larger, making it difficult to simulate the real morphological changes of grain growth. Summary of the invention

[0005] In view of this, the present application provides a phase field simulation method and device for nuclear fuel grain growth, which mainly aims to reduce the amount of calculation in the simulation process, improve the evolution efficiency, and simulate the actual morphological changes of grain growth.

[0006] According to a first aspect of the present application, a phase field simulation method for nuclear fuel grain growth is provided, the method comprising:

[0007] Based on the phase field theory, constructing a total free energy equation of the nuclear fuel system, wherein the total free energy equation of the system includes phase field variables corresponding to each grain of the nuclear fuel;

[0008] Based on the total free energy equation of the system and the phase field variables corresponding to each grain, construct the dynamic evolution equation of each grain respectively;

[0009] Based on the kinetic evolution equation, the growth morphology of each grain is evolved in parallel to simulate the actual morphological changes of the nuclear fuel grain growth.

[0010] According to a second aspect of the present application, a phase field simulation device for nuclear fuel grain growth is provided, the device comprising:

[0011] A first construction unit is used to construct a system total free energy equation of the nuclear fuel based on phase field theory, wherein the system total free energy equation contains phase field variables corresponding to each grain of the nuclear fuel;

[0012] A second construction unit is used to construct the dynamic evolution equation of each grain based on the total free energy equation of the system and the phase field variables corresponding to each grain;

[0013] The simulation unit is used to evolve the growth morphology of each grain in parallel based on the kinetic evolution equation to simulate the actual morphological changes of the nuclear fuel grain growth.

[0014] According to a third aspect of the present application, a storage medium is provided, on which a computer program is stored, and when the program is executed by a processor, the phase field simulation method of nuclear fuel grain growth is implemented.

[0015] According to the fourth aspect of the present application, an electronic device is provided, comprising a storage medium, a processor, and a computer program stored on the storage medium and executable on the processor, wherein the processor implements the above-mentioned phase field simulation method of nuclear fuel grain growth when executing the program.

[0016] By means of the above technical solution, the present application provides a phase field simulation method and device for nuclear fuel grain growth. Through phase field theory, the total free energy equation of the nuclear fuel system is constructed, and based on the total free energy equation of the system, the kinetic evolution equation of each grain is respectively constructed, and the growth morphology of each grain can be evolved based on the kinetic evolution equation. Since the phase field simulation method of the present application does not need to track the interface of a single grain, it can reduce the amount of calculation and improve the evolution efficiency. At the same time, by constructing the kinetic evolution equation of each grain, the present application can process multiple grains in parallel, thereby preventing error accumulation and ensuring the reliability of the evolution results, and then simulating the real morphological changes of nuclear fuel grain growth.

[0017] The above description is only an overview of the technical solution of the present application. In order to more clearly understand the technical means of the present application, it can be implemented in accordance with the contents of the specification. In order to make the above and other purposes, features and advantages of the present application more obvious and easy to understand, the specific implementation methods of the present application are listed below. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] The drawings described herein are used to provide a further understanding of the present application and constitute a part of the present application. The illustrative embodiments of the present application and their descriptions are used to explain the present application and do not constitute an improper limitation on the present application. In the drawings:

[0019] Figure 1 A schematic flow chart of a phase field simulation method for nuclear fuel grain growth provided in an embodiment of the present application is shown;

[0020] Figure 2 The grain evolution diagram provided in the embodiment of the present application with the evolution time of t=100min under the conditions of N=256, L=2.739μm is shown;

[0021] Figure 3 The grain evolution diagram provided in the embodiment of the present application under the conditions of N=256, L=2.739 μm and the evolution time is t=120 min is shown;

[0022] Figure 4 The grain evolution diagram provided in the embodiment of the present application with the evolution time of t=130min under the conditions of N=256, L=2.739μm is shown;

[0023] Figure 5 The grain evolution diagram provided in the embodiment of the present application under the conditions of N=256, L=2.739 μm and the evolution time is t=140 min is shown;

[0024] Figure 6 The grain evolution diagram provided in the embodiment of the present application under the conditions of N=256, L=2.739 μm and the evolution time is t=150 min is shown;

[0025] Figure 7 The grain evolution diagram provided in the embodiment of the present application with the evolution time of t=200min under the conditions of N=256, L=2.739μm is shown;

[0026] Figure 8 The grain evolution diagram provided in the embodiment of the present application with the evolution time of t=500min under the conditions of N=256, L=2.739μm is shown;

[0027] Fig. 9 The figure shows the grain evolution diagram provided in the embodiment of the present application under the conditions of N=256, L=2.739 μm and the evolution time is t=1000 min;

[0028] Fig.10 A schematic diagram of a process for analyzing the grain growth evolution law provided in an embodiment of the present application is shown;

[0029] Fig.11 The graphs showing the dynamic changes of the grain area and grain radius under the conditions of N=256, L=2.739 μm and N=512, L=5.478 μm provided in the embodiments of the present application are shown;

[0030] Fig.12A graph showing the change in the number of grains under the conditions of N=256, L=2.739 μm and N=512, L=5.478 μm provided in the embodiment of the present application is shown;

[0031] Fig.13 A schematic diagram of a process for analyzing grain size distribution provided in an embodiment of the present application is shown;

[0032] Fig.14 The grain size distribution at different times under the conditions of N=512, L=5.478 μm provided in the embodiment of the present application is shown.

[0033] Fig.15 A schematic structural diagram of a phase field simulation device for nuclear fuel grain growth provided in an embodiment of the present application is shown. DETAILED DESCRIPTION

[0034] The present application will be described in detail below with reference to the accompanying drawings and in combination with embodiments. It should be noted that the embodiments and features in the embodiments of the present application can be combined with each other without conflict.

[0035] At present, numerical simulation methods usually need to track single grain boundaries and process sharp interfaces when simulating the grain evolution of polycrystalline materials, which will lead to excessive calculation and low evolution efficiency. In addition, these numerical simulation methods mostly process each grain one by one during simulation. Therefore, when the number of grains is large, the error accumulation will become larger and larger, making it difficult to simulate the actual morphological changes of grain growth.

[0036] In order to solve the above problems, an embodiment of the present invention provides a phase field simulation method for nuclear fuel grain growth, such as Figure 1 As shown, the method includes:

[0037] Step 101: Based on phase field theory, construct a total free energy equation for the nuclear fuel system.

[0038] Among them, the total free energy equation of the system contains the phase field variables corresponding to each grain of the nuclear fuel. The nuclear fuel can specifically be U3Si2. It should be noted that the nuclear fuel of the embodiment of the present invention is not limited to U3Si2, and can also be other types of nuclear fuel. The embodiment of the present invention is mainly suitable for simulating the growth process of nuclear fuel grains. The execution subject of the embodiment of the present invention is a device or equipment that can simulate the growth process of nuclear fuel grains, which can be specifically set on the server side.

[0039] For the embodiment of the present invention, in order to perform phase field simulation on the growth process of nuclear fuel, it is necessary to pre-construct the total free energy equation of the nuclear fuel system. For this process, step 101 specifically includes: based on the phase field variables corresponding to each grain, as well as the gradient energy coefficient and the interface energy coefficient, respectively constructing the gradient energy term and the interface energy term; according to the gradient energy term and the interface energy term, as well as the free energy barrier coefficient, constructing the total free energy equation of the nuclear fuel system. The specific form of the total free energy equation of the nuclear fuel system is as follows:

[0040]

[0041] Where F is the total free energy of the system, is the interface energy term, is the gradient energy term, F0 is the free energy barrier coefficient, r is the spatial position, M is the number of grains, η i and η j represents the order parameter of any two grains i and j, i.e., the phase field variable, is the Hamiltonian operator, κ is the gradient energy coefficient, γ ij is the interfacial energy coefficient.

[0042] Step 102: construct the dynamic evolution equation of each grain based on the total free energy equation of the system and the phase field variables corresponding to each grain.

[0043] For the embodiment of the present invention, after constructing the total free energy equation of the system, the dynamic evolution equation of each grain is constructed according to the total free energy equation of the system and the phase field variables corresponding to each grain. For this process, step 102 specifically includes: performing phase field variable explicit processing on the expression form of the total free energy equation of the system to obtain the total free energy equation of the system after explicit processing; based on the phase field variables of each grain, respectively taking partial derivatives of the total free energy equation of the system after explicit processing to obtain the free energy equation of each grain after partial derivative; according to the free energy equation of each grain after partial derivative, as well as the mobility and thermal fluctuation terms, respectively construct the dynamic evolution equation of each grain.

[0044] When the expression form of the total free energy equation of the system is explicitly processed by phase field variables, the specific grain boundary energy equation of any two grains in the grains is constructed; according to the variational principle, the extreme value function corresponding to the total free energy equation of the system is determined; based on the extreme value function, an equivalent integral equation is constructed; based on the boundary conditions, the equivalent integral equation is transformed to obtain the transformed equivalent integral equation; the equivalent integral equation, the transformed equivalent integral equation and the specific grain boundary energy equation are combined and solved to obtain the expression of the interface energy coefficient, wherein the expression of the interface energy coefficient is related to the gradient energy coefficient, the free energy barrier coefficient and the phase field variables of the any two grains; based on the expression of the interface energy coefficient and the relationship between the gradient energy coefficient, the interface energy coefficient, the free energy barrier coefficient and the grain boundary interaction length, the total free energy equation of the system after explicit processing is obtained.

[0045] Specifically, according to the phase field variable η of any two grains i and η j , construct the specific grain boundary energy equation of any two grains, as follows:

[0046]

[0047] Where x is the normal coordinate of the grain boundary. According to the variational principle, the extreme value function of the total free energy equation of the system satisfies the following relationship:

[0048]

[0049] Furthermore, the equivalent integral equation can be obtained from formula (3), as follows:

[0050]

[0051] When considering the boundary conditions, formula (4) can be transformed into:

[0052]

[0053] Furthermore, by combining equations (4), (5) and (2) and solving them, we can obtain the interfacial energy coefficient γ ij The expression is as follows:

[0054]

[0055] In the process of determining the kinetic evolution equation, the embodiment of the present invention needs to continuously find the relationship between the parameter variables to reduce the parameter variables of the total free energy equation of the system, that is, to use the phase field variable η as much as possible. i and η jThe total free energy equation of the system is expressed as: ij With the gradient energy coefficient κ, free energy barrier coefficient F0, phase field variable η i and η j The relationship between.

[0056] Furthermore, combined with references, the free energy barrier coefficient F0, the interface energy γ between grains, the gradient energy coefficient κ of the grain boundary, and the grain boundary interaction length l are obtained. int The relationship between them is as follows:

[0057]

[0058] in,

[0059] Further, the interface profile is obtained by considering the symmetric case of any two grains i and j, in which case η j =1-η i , we can get the following formula:

[0060]

[0061] application We can further get:

[0062]

[0063] The phase field variable η can be expressed as i and η j The relationship between the free energy barrier coefficient F0 and the gradient energy coefficient κ of the grain boundary is expressed, so as to achieve the purpose of reducing the number of parameters, that is, the phase field variable η is finally used i and η j represents the total free energy equation of the system, and because η j =1-η i , so the final system total free energy equation can be explicitly expressed using only one phase field variable.

[0064] Furthermore, after the total free energy equation of the system is explicitly processed, the partial derivatives of the total free energy equation of the system after explicit processing are obtained based on the phase field variables of each grain, and the free energy equation after partial derivative is obtained. Then, according to the free energy equation after partial derivative of each grain, as well as the mobility and thermal fluctuation terms, the Allen-Cahn kinetic evolution equation of each grain is constructed. Taking grain i as an example, the specific form of the kinetic evolution equation of grain i is as follows:

[0065]

[0066] Among them, L ηi is the mobility, is the average grain boundary mobility, and γ is the interfacial energy between grains. In order to reflect thermal fluctuations, the thermal fluctuation term ξ determined by the wave dissipation theorem is added to the above equation v Therefore, according to formula (11), the dynamic evolution equation of each grain can be obtained.

[0067] Step 103: Based on the kinetic evolution equation, the growth morphology of each grain is evolved in parallel to simulate the actual morphological changes of the nuclear fuel grain growth.

[0068] For the embodiment of the present invention, after determining the kinetic evolution equation, the kinetic evolution equation is solved to obtain the growth morphology of each grain when the total free energy F is minimum. int =ωδ G , δ G is the grain boundary width, ω is the constant of the diffusion grain boundary, and in the calculation process, the grid size is preferably Δl = 3.33δ G The values ​​of F0 and κ can be determined by consulting references. For example, the simulation parameters are F0 = 6.84 × 10 8 J / m 3 ,κ=1.92×10 -8 J / m, ω=10, Δl=10.7nm, l int =20.7nm,

[0069] Then, the implicit Fourier transform method is used to simulate the grid with a characteristic length of L = NΔL (ΔL = 10.7nm, N = 256) at a temperature of 600K to obtain the grain evolution diagram at different times, such as Figure 2-9 shown.

[0070] It should be noted that the above simulation parameters, characteristic lengths and temperatures given in the embodiments of the present invention are only preferred and are not limited thereto.

[0071] The embodiment of the present invention provides a phase field simulation method for nuclear fuel grain growth. The total free energy equation of the nuclear fuel system is constructed through phase field theory, and based on the total free energy equation of the system, the dynamic evolution equation of each grain is respectively constructed, and the growth morphology of each grain can be evolved based on the dynamic evolution equation. Since the phase field simulation method of the embodiment of the present invention does not need to track the interface of a single grain, it can reduce the amount of calculation and improve the evolution efficiency. At the same time, the embodiment of the present invention can process multiple grains in parallel by constructing the dynamic evolution equation of each grain, thereby preventing error accumulation and ensuring the reliability of the evolution result, and then simulating the real morphological changes of nuclear fuel grain growth.

[0072] Furthermore, after obtaining the evolution result of grain growth through phase field simulation, statistical analysis can be performed based on the evolution result to obtain the grain growth evolution law. Based on this, this embodiment also provides an analysis method for the grain growth evolution law, such as Fig.10 As shown, the method includes:

[0073] Step 201: construct a grain size dynamics equation.

[0074] For the embodiments of the present invention, grain boundary energy and mobility are two main parameters that need to be determined for grain growth of nuclear fuel (such as U3Si2). During grain growth, the pressure exerted on the grain boundary can be given by the following formula:

[0075]

[0076] Among them, P a is the pressure exerted on the grain boundary, γ is the interfacial energy between the grains, and R is the radius of curvature of the grain.

[0077] Furthermore, the grain growth rate can be expressed as:

[0078]

[0079] Among them, the constant M is related to the mobility, P r is the resistance. The sources of this resistance include: voids, pores and precipitates that cannot diffuse along the boundary. As a result of grain growth, when the driving pressure P a Down to P r When the value is reached, there is no net pressure and grain growth stops at this limit.

[0080] Further, the average grain size D is defined as:

[0081]

[0082] in, is the average radius of curvature of the grain, and the grain topology is included in the geometric constant α. According to formulas (12), (13) and (14), the grain size dynamics equation is constructed as follows:

[0083]

[0084] Among them, M and represent the average grain boundary mobility and energy, respectively.

[0085] Step 202: Based on the grain size kinetic equation, determine the change pattern of the grain size over time.

[0086] For the embodiment of the present invention, after constructing the grain size kinetic equation, based on the grain size kinetic equation, the change law of the grain size over time is determined. For this process, step 202 specifically includes: when considering the situation where the grains have no resistance, constructing an equation about the rate constant; solving the target parameter value according to the equation about the rate constant, the temperature dependence of the grain boundary energy, the temperature dependence of the rate constant, and the temperature dependence of the mobility; bringing the target parameter value into the grain size kinetic equation to obtain the change law of the grain size over time.

[0087] Specifically, when considering the case of no resistance, the equation for the rate constant can be obtained, which is as follows:

[0088]

[0089] Through molecular dynamics simulation, the temperature dependence of nuclear fuel grain boundary energy can be obtained as follows:

[0090]

[0091] H(T)=1.21×10 -7 T 2 -3.46×10 -5 T+0.85Jm 2 (18)

[0092] S(T)=4.58×10 -7 T-1.72×10 -4 Jm 2 K(19)

[0093] The temperature dependence of the rate constant and the mobility can be measured experimentally:

[0094] K = 8.77 × 10 -18 exp(-0.33eVk B T)m 2 s(20)

[0095]

[0096] Furthermore, the target parameter value can be solved by combining equations (16), (17), (20) and (21), such as α = 0.96. Then, the target parameter value is substituted into the grain size dynamics equation (15), and the change law of grain size over time can be obtained.

[0097] Step 203: Based on the growth morphology, determine the variation pattern of the number of grains over time.

[0098] For the embodiment of the present invention, the grain evolution morphology obtained by phase field simulation is as follows: Figure 2-9 , the number of grains at different times can be counted, so as to determine the change pattern of the number of grains over time.

[0099] Step 204: Determine the change pattern of the grain area over time based on the change pattern of the grain size over time and the change pattern of the grain quantity over time.

[0100] For the embodiment of the present invention, the grain area at different times can be calculated based on the grain size and the number of grains at different times, so that the change law of the grain area over time can be obtained. The statistical results of the system sizes L = 2.739 μm and L = 5.478 μm are as follows: Fig.11 and Fig.12 shown.

[0101] An analysis method for the grain growth evolution law provided by an embodiment of the present invention can determine the change law of grain size over time by constructing a grain size kinetic equation. At the same time, based on the grain growth evolution results of phase field simulation, it can determine the change law of grain number over time and the change law of grain area over time, thereby solving the problems of long cycle, high cost and high risk in experimental analysis of nuclear fuel grain growth law.

[0102] Furthermore, the embodiment of the present invention can also analyze the grain size distribution according to the change law of the grain size over time. Based on this, the present embodiment also provides a grain size distribution analysis method, such as Fig.13 As shown, the method includes:

[0103] Step 301: construct an equation about grain growth and relaxation according to the variation law of the grain size over time.

[0104] For the embodiment of the present invention, due to the change in the number of grain sides, the size of some grains will decrease. During the system evolution, grains with less than 6 sides disappeared. This process takes place after a relaxation time τ RIt will reappear later, so the embodiment of the present invention introduces a grain size distribution function Ψ(R), which gives the density of grains with radius R. Ψ(R)dR is the total number of grains in the grain radius range between R and R+dR, and dR is a small increment of the grain radius. The growth and relaxation of grains can be described by the following equations respectively:

[0105]

[0106]

[0107] Wherein, t is time. Since the variation law of the grain size (diameter D) with time has been determined in Example 2, the relationship between the grain radius R and the time t in equation (22) is obtained.

[0108] Step 302: Based on the equations regarding grain growth and relaxation, determine a basic expression for a grain size distribution function.

[0109] For the present embodiment, the total net change in grain count is:

[0110]

[0111] The number of grains in equilibrium can be expressed as:

[0112]

[0113] Furthermore, by solving formula (24), the basic expression of the grain size distribution function can be obtained as follows:

[0114]

[0115]

[0116] Where v(R) represents the grain growth rate and C0 is the integral constant of the total number of grains.

[0117] Step 303: Substitute the grain growth rate formula into the basic expression to obtain the Rayleigh distribution function.

[0118] For the embodiment of the present invention, the grain growth rate formula (13) is substituted into equation (25) to obtain the Rayleigh distribution function, as shown below:

[0119]

[0120] Step 304: Determine the grain size distribution according to the Rayleigh distribution function.

[0121] For an embodiment of the present invention, after obtaining the Rayleigh distribution function, the grain size distribution is determined based on the distribution function. For this process, step 304 specifically includes: constructing a Fourier transform expression corresponding to the Rayleigh distribution function, wherein the Fourier transform expression contains a first unknown function and a second unknown function about a time variable; solving the first unknown function and the second unknown function in the Fourier transform expression to obtain an expression of the first unknown function and an expression of the second unknown function; substituting the expression of the first unknown function and the expression of the second unknown function into the Fourier transform expression to obtain the grain size distribution.

[0122] Specifically, since the Rayleigh distribution function (26) does not include the time variable, it is necessary to perform Fourier transform on it to introduce the time variable. The Fourier transform expression corresponding to the Rayleigh distribution function is as follows:

[0123]

[0124] Where y is the grain radius expressed as a(t), ψ(t) and are the first unknown function and the second unknown function respectively.

[0125] Furthermore, the variation of grain radius with time is as follows:

[0126] dR / dt=θ(t)σ(y) (28)

[0127] Here the time correlation and particle size derivative defined by θ(t) satisfy:

[0128] By replacing the time derivative and space derivative with: d / dR = d / ady, the part describing time and size can be separated into the following forms:

[0129]

[0130] Furthermore, by setting both sides of equation (29) to the same constant -C and C>0, two independent time and particle size equations can be obtained:

[0131]

[0132]

[0133] According to equations (29), (30) and (31), the specific expressions of the first unknown function and the second unknown function can be solved as follows:

[0134] ψ(t)=ψ0[a(t)] -C exp(-t / τR ) (32)

[0135]

[0136] Among them, ψ0 and is the integral constant, and exp represents an exponential function with the natural constant e as the base. Then, by substituting the expression (32) of the first unknown function and the expression (33) of the second unknown function into formula (27), the Rayleigh distribution function expression including the time variable can be determined, so that the grain size distribution can be obtained, as shown in Fig.14 shown.

[0137] In some embodiments, other relevant indicators about grain growth can be further obtained. For this process, the method includes: according to the expression of the first unknown function and the expression of the second unknown function, and the Rayleigh distribution function, deriving the function of the change of the number of grains with time, the expression of the initial grain number parameter, the function of the change of the grain radius with time, and the expression of the initial grain radius parameter.

[0138] Specifically, according to formulas (32), (33) and (26), the following related indicators about grain growth can be derived, as follows:

[0139]

[0140] Among them, N(t) is the function of the number of grains changing with time, N0 is the initial grain number parameter, R(t) is the function of the grain radius changing with time, R0 is the initial grain radius parameter, and z is the power exponent.

[0141] Through the above formula, it can be found that the Rayleigh distribution function is more suitable for describing the size distribution of nuclear fuel grains during their growth process. At the same time, it is found that the evolution law of the grain radius and area is independent of the size of the phase field simulation system, and the size of the system only affects the change in the number of grains.

[0142] Further, as Figure 1 , Fig.10 and Fig.13 The specific implementation of the method shown in the embodiment provides a phase field simulation device for nuclear fuel grain growth, such as Fig.15 As shown, the device includes: a first building unit 41, a second building unit 42 and a simulation unit 43.

[0143] The first construction unit 41 can be used to construct a system total free energy equation of the nuclear fuel based on phase field theory, wherein the system total free energy equation contains phase field variables corresponding to each grain of the nuclear fuel.

[0144] The second construction unit 42 can be used to construct the dynamic evolution equation of each grain based on the total free energy equation of the system and the phase field variables corresponding to each grain.

[0145] The simulation unit 43 can be used to evolve the growth morphology of each grain in parallel based on the kinetic evolution equation to simulate the actual morphological changes of the growth of the nuclear fuel grains.

[0146] In some embodiments, the first construction unit 41 can be specifically used to construct gradient energy terms and interface energy terms based on the phase field variables corresponding to each grain, as well as the gradient energy coefficient and the interface energy coefficient, respectively; and construct the total free energy equation of the nuclear fuel system based on the gradient energy terms and the interface energy terms, as well as the free energy barrier coefficient.

[0147] In some embodiments, the second construction unit 42 includes: a processing module, a calculation module and a construction module.

[0148] The processing module can be used to perform phase field variable explicit processing on the expression form of the total free energy equation of the system to obtain the total free energy equation of the system after explicit processing.

[0149] The calculation module can be used to calculate partial derivatives of the total free energy equation of the system after the explicit processing based on the phase field variables of each grain, so as to obtain the free energy equation of each grain after the partial derivative is calculated.

[0150] The construction module can be used to construct the dynamic evolution equation of each grain according to the free energy equation after partial derivative of each grain, as well as the mobility and thermal fluctuation terms.

[0151] In some embodiments, the processing module can be specifically used to construct a specific grain boundary energy equation of any two grains among the grains; determine the extreme value function corresponding to the total free energy equation of the system according to the variational principle; construct an equivalent integral equation based on the extreme value function; transform the equivalent integral equation based on the boundary conditions to obtain the transformed equivalent integral equation; combine the equivalent integral equation, the transformed equivalent integral equation and the specific grain boundary energy equation, and solve them to obtain an expression of the interface energy coefficient, wherein the expression of the interface energy coefficient is related to the gradient energy coefficient, the free energy barrier coefficient and the phase field variables of the any two grains; based on the expression of the interface energy coefficient, and the relationship between the gradient energy coefficient, the interface energy coefficient, the free energy barrier coefficient and the grain boundary interaction length, the total free energy equation of the system after explicit processing is obtained.

[0152] In some embodiments, the simulation unit 43 can be specifically used to solve the kinetic evolution equation to obtain the growth morphology of each grain when the total free energy is minimum.

[0153] In some embodiments, the apparatus further comprises: a determining unit.

[0154] The first construction unit 41 can also be used to construct a grain size dynamics equation.

[0155] The determination unit may be used to determine a change pattern of the grain size over time based on the grain size kinetic equation.

[0156] The determination unit may also be used to determine a law of change of the number of grains over time based on the growth morphology.

[0157] The determination unit may also be used to determine a law of change of the grain area over time based on a law of change of the grain size over time and a law of change of the grain quantity over time.

[0158] In some embodiments, the determination unit can be specifically used to construct an equation about the rate constant when taking into account the situation where the grains have no resistance; solve the target parameter value according to the equation about the rate constant, the temperature dependence of grain boundary energy, and the temperature dependence of the rate constant and the temperature dependence of the mobility; bring the target parameter value into the grain size kinetic equation to obtain the change pattern of the grain size over time.

[0159] In some embodiments, the first construction unit 41 may also be used to construct equations regarding grain growth and relaxation according to the variation pattern of the grain size over time.

[0160] The determination unit may also be used to determine a basic expression of a grain size distribution function based on the equations regarding grain growth and relaxation.

[0161] The determination unit may also be used to substitute the grain growth rate formula into the basic expression to obtain the Rayleigh distribution function.

[0162] The determining unit may also be configured to determine grain size distribution according to the Rayleigh distribution function.

[0163] In some embodiments, the determination unit can be specifically used to construct a Fourier transform expression corresponding to the Rayleigh distribution function, wherein the Fourier transform expression includes a first unknown function and a second unknown function related to a time variable; solving the first unknown function and the second unknown function in the Fourier transform expression to obtain an expression of the first unknown function and an expression of the second unknown function; substituting the expression of the first unknown function and the expression of the second unknown function into the Fourier transform expression to obtain a grain size distribution.

[0164] In some embodiments, the determination unit can also be used to derive the function of the change of the number of grains over time, the expression of the initial grain number parameter, the function of the change of the grain radius over time, and the expression of the initial grain radius parameter according to the expression of the first unknown function, the expression of the second unknown function, and the Rayleigh distribution function.

[0165] It should be noted that for other corresponding descriptions of the functional units involved in the phase field simulation device for nuclear fuel grain growth provided in this embodiment, reference can be made to Figure 1 , Fig.10 and Fig.13 The corresponding description in will not be repeated here.

[0166] Based on the above Figure 1 , Fig.10 and Fig.13 The method shown in the embodiment also provides a storage medium on which a computer program is stored. When the program is executed by a processor, the above-mentioned Figure 1 , Fig.10 and Fig.13 Phase-field simulation method for nuclear fuel grain growth shown.

[0167] Based on this understanding, the technical solution of the present application can be embodied in the form of a software product, which can be stored in a non-volatile storage medium (which can be a CD-ROM, USB flash drive, mobile hard disk, etc.), including a number of instructions for enabling an electronic device (which can be a personal computer, server, or network device, etc.) to execute the methods of various implementation scenarios of the present application.

[0168] Based on the above Figure 1 , Fig.10 and Fig.13 The method shown, and Fig.15In order to achieve the above-mentioned purpose, the embodiment of the present application also provides an electronic device, which can be a personal computer, a tablet computer, a server, or other network equipment, etc. The device includes a storage medium and a processor; the storage medium is used to store a computer program; the processor is used to execute the computer program to achieve the above-mentioned Figure 1 , Fig.10 and Fig.13 Phase-field simulation method for nuclear fuel grain growth shown.

[0169] Optionally, the above-mentioned physical device may also include a user interface, a network interface, a camera, a radio frequency (RF) circuit, a sensor, an audio circuit, a WI-FI module, etc. The user interface may include a display, an input unit such as a keyboard, etc., and the optional user interface may also include a USB interface, a card reader interface, etc. The network interface may optionally include a standard wired interface, a wireless interface (such as a WI-FI interface), etc.

[0170] Those skilled in the art will appreciate that the above-mentioned physical device structure provided in this embodiment does not constitute a limitation on the physical device, and may include more or fewer components, or a combination of certain components, or different arrangements of components.

[0171] The storage medium may also include an operating system and a network communication module. The operating system is a program that manages the hardware and software resources of the above-mentioned physical device, and supports the operation of the information processing program and other software and / or programs. The network communication module is used to realize the communication between the components inside the storage medium, and the communication with other hardware and software in the information processing physical device.

[0172] Through the description of the above implementation methods, those skilled in the art can clearly understand that the present application can be implemented by means of software plus a necessary general hardware platform, or by hardware.

[0173] The embodiment of the present invention constructs the total free energy equation of the nuclear fuel system through phase field theory, and based on the total free energy equation of the system, constructs the kinetic evolution equation of each grain, and can evolve the growth morphology of each grain based on the kinetic evolution equation. Since the phase field simulation method of the embodiment of the present invention does not need to track the interface of a single grain, it can reduce the amount of calculation and improve the evolution efficiency. At the same time, the embodiment of the present invention can process multiple grains in parallel by constructing the kinetic evolution equation of each grain, thereby preventing error accumulation and ensuring the reliability of the evolution results, and then can simulate the real morphological changes of nuclear fuel grain growth.

[0174] Those skilled in the art will appreciate that the accompanying drawings are only schematic diagrams of a preferred implementation scenario, and the modules or processes in the accompanying drawings are not necessarily necessary for implementing the present application. Those skilled in the art will appreciate that the modules in the devices in the implementation scenario can be distributed in the devices of the implementation scenario according to the description of the implementation scenario, or can be changed accordingly and located in one or more devices different from the present implementation scenario. The modules of the above-mentioned implementation scenario can be combined into one module, or can be further split into multiple submodules.

[0175] The above serial numbers of this application are only for description and do not represent the advantages and disadvantages of the implementation scenarios. The above disclosure is only a few specific implementation scenarios of this application, but this application is not limited to them, and any changes that can be thought of by technicians in this field should fall within the scope of protection of this application.

Claims

1. A phase field simulation method for nuclear fuel grain growth, characterized in that: include: Based on the phase field theory, constructing a total free energy equation of the nuclear fuel system, wherein the total free energy equation of the system includes phase field variables corresponding to each grain of the nuclear fuel; Based on the total free energy equation of the system and the phase field variables corresponding to each grain, construct the dynamic evolution equation of each grain respectively; Based on the kinetic evolution equation, the growth morphology of each grain is evolved in parallel to simulate the actual morphological changes of the nuclear fuel grain growth.

2. The method according to claim 1, characterized in that Based on phase field theory, the total free energy equation of the nuclear fuel system is constructed, including: Based on the phase field variables corresponding to each grain, as well as the gradient energy coefficient and the interface energy coefficient, a gradient energy term and an interface energy term are constructed respectively; Constructing a total free energy equation of the nuclear fuel system according to the gradient energy term, the interface energy term, and the free energy barrier coefficient; and / or Based on the kinetic evolution equation, the growth morphology of each grain is evolved in parallel, including: Solving the kinetic evolution equation to obtain the growth morphology of each grain when the total free energy is minimum; and / or The method further comprises: Constructing grain size kinetic equations; Based on the grain size kinetic equation, determining the change pattern of grain size over time; Based on the growth morphology, determining the change pattern of the number of grains over time; Based on the change pattern of the grain size over time and the change pattern of the grain quantity over time, the change pattern of the grain area over time is determined.

3. The method according to claim 2, characterized in that Based on the total free energy equation of the system and the phase field variables of each grain, the dynamic evolution equations of each grain are constructed respectively, including: Performing phase field variable explicit processing on the expression form of the total free energy equation of the system to obtain the total free energy equation of the system after explicit processing; Based on the phase field variables of each of the grains, partial derivatives of the total free energy equation of the system after the explicit processing are respectively obtained to obtain the free energy equation of each of the grains after partial derivatives are obtained; The dynamic evolution equations of the grains are constructed according to the free energy equations after partial derivatives of the grains, as well as the mobility and thermal fluctuation terms.

4. The method according to claim 3, characterized in that The expression form of the total free energy equation of the system is subjected to phase field variable explicit processing to obtain the total free energy equation of the system after explicit processing, including: Constructing a specific grain boundary energy equation of any two grains in the grains; According to the variational principle, determining the extreme value function corresponding to the total free energy equation of the system; Based on the extreme value function, construct an equivalent integral equation; Based on the boundary conditions, the equivalent integral equation is transformed to obtain a transformed equivalent integral equation; The equivalent integral equation, the transformed equivalent integral equation and the specific grain boundary energy equation are combined and solved to obtain an expression of the interface energy coefficient, wherein the expression of the interface energy coefficient is related to the gradient energy coefficient, the free energy barrier coefficient and the phase field variables of any two grains; Based on the expression of the interface energy coefficient, and the relationship between the gradient energy coefficient, the interface energy coefficient, the free energy barrier coefficient, and the grain boundary interaction length, the total free energy equation of the system after explicit processing is obtained.

5. The method according to claim 2, characterized in that: Based on the grain size kinetic equation, the variation law of grain size over time is determined, including: Construct an equation for the rate constant when considering the case where the grains have no resistance; Solving for target parameter values ​​according to the equations for the rate constant, the temperature dependence of the grain boundary energy, the temperature dependence of the rate constant and the temperature dependence of the mobility; Substituting the target parameter value into the grain size kinetic equation to obtain the change pattern of the grain size over time; and / or The method further comprises: According to the change law of the grain size over time, construct equations about grain growth and relaxation; Determine a basic expression for a grain size distribution function based on the equations for grain growth and relaxation; Substituting the grain growth rate formula into the basic expression, the Rayleigh distribution function is obtained; According to the Rayleigh distribution function, the grain size distribution is determined.

6. The method according to claim 5, characterized in that Determining the grain size distribution according to the Rayleigh distribution function includes: Constructing a Fourier transform expression corresponding to the Rayleigh distribution function, wherein the Fourier transform expression includes a first unknown function and a second unknown function with respect to a time variable; Solving the first unknown function and the second unknown function in the Fourier transform expression to obtain an expression of the first unknown function and an expression of the second unknown function; Substitute the expression of the first unknown function and the expression of the second unknown function into the Fourier transform expression to obtain the grain size distribution.

7. The method according to claim 6, characterized in that The method further comprises: According to the expression of the first unknown function, the expression of the second unknown function, and the Rayleigh distribution function, the function of the change of the number of grains over time, the expression of the initial grain number parameter, the function of the change of the grain radius over time, and the expression of the initial grain radius parameter are derived respectively.

8. A phase field simulation device for nuclear fuel grain growth, characterized in that: include: A first construction unit is used to construct a system total free energy equation of the nuclear fuel based on phase field theory, wherein the system total free energy equation contains phase field variables corresponding to each grain of the nuclear fuel; A second construction unit is used to construct the dynamic evolution equation of each grain based on the total free energy equation of the system and the phase field variables corresponding to each grain; The simulation unit is used to evolve the growth morphology of each grain in parallel based on the kinetic evolution equation to simulate the actual morphological changes of the nuclear fuel grain growth.

9. A storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the method according to any one of claims 1 to 7 is implemented.

10. An electronic device comprising a storage medium, a processor, and a computer program stored in the storage medium and executable on the processor, characterized in that: When the processor executes the computer program, the method according to any one of claims 1 to 7 is implemented.

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