Gas diffusion-deformation-fracture coupling simulation method in nuclear fuel UOservice process, model construction, electronic equipment and program product

Abstract

The invention discloses a gas diffusion-deformation-fracture coupling simulation model in the nuclear fuel UOS service process. ; ; . The problem that an existing simulation model can only consider one or two of gas diffusion, deformation and fracture in nuclear fuel UO2 is solved, and the simulation model considers three physical elements of gas diffusion, deformation and fracture at the same time.

Description

Technical Field

[0001] This disclosure belongs to the field of nuclear material design technology, and specifically relates to a gas diffusion-deformation-fracture coupling simulation method, model building method, nuclear fuel rod design, electronic equipment and program products during the service of nuclear fuel UO2. Background Technology

[0002] Uranium dioxide (UO2) has been a cornerstone of the commercial nuclear power industry for decades, serving as the primary fuel for light water reactors (LWRs) and fast breeder reactors (FBRs) worldwide. During the reactor's operational life, the fuel is subjected to a combination of extreme conditions that induce complex changes in microstructure and material properties, ultimately affecting its performance. These conditions include factors such as neutron irradiation and stress, which trigger numerous different physical processes. Under irradiation, fission gases such as xenon (Xe) and krypton (Kr) are produced in the nuclear fuel. These gases have extremely low solubility in the UO2 matrix and therefore migrate via diffusion. This migration and subsequent aggregation lead to the nucleation and growth of intragranular and intergranular fission bubbles. Once the bubbles have grown and connected to a certain threshold, gaseous fission products are released from the fuel pellets into the pellet-cladding interstitial space. This release of fission gases reduces the thermal conductivity of the interstitial space and exerts significant pressure on the inner cladding wall, which can lead to creep deformation and failure. Under stress conditions, such as those caused by temperature gradients, UO2 deforms, and cracks may form and propagate within the fuel pellet. The formation and propagation of these cracks are heavily influenced by the presence and distribution of fission bubbles. The formation of crack networks reduces the thermal conductivity within the fuel pellet, leading to fuel-cladding interactions and causing fuel repositioning. Furthermore, crack networks may alter the interconnected flow paths of fission gas release and change gas diffusion behavior. These phenomena indicate that these physical processes, including diffusion, deformation, and fracture, are far from independent but are tightly coupled.

[0003] Given the extreme conditions of nuclear fuel service, it is virtually impossible to experimentally study the coupling effects of these factors on the performance of nuclear fuels such as UO2 in situ. Therefore, numerous numerical methods and models have been developed to address gas diffusion, deformation, and fracture issues in UO2 at various scales. For example, at the microscopic scale, molecular dynamics methods can be used to study the diffusion behavior of different types of atoms and vacancies in the nuclear fuel matrix. At the mesoscopic scale, phase-field methods are widely used to study the evolution of gas concentration fields and the growth and connection of bubbles; deformation studies involve the development of material constitutive models considering factors such as irradiation and creep under different conditions; fracture problems at the microscopic and atomic scales can be studied using molecular dynamics methods, while crack evolution at the macroscopic and mesoscopic scales can be studied using phase-field fracture methods, cohesive elements, and extended finite element models. Summary of the Invention

[0004] One aspect of this disclosure is a finite element multilevel cross-optimization method for the gas diffusion-deformation-fracture coupling problem in nuclear fuel UO2. The gas diffusion-deformation-fracture coupling problem during the service of nuclear fuel UO2 is simulated using a simulation model, and the simulation process is optimized using finite element multilevel cross-optimization. The simulation model is described by the following set of governing equations (7)-(10).

[0005]

[0006]

[0007] One aspect of this disclosure is a simulation method for gas diffusion-deformation-fracture coupling during the service of nuclear fuel UO2. The simulation method uses the above-mentioned simulation model to simulate the gas diffusion-deformation-fracture coupling problem during the service of nuclear fuel UO2, and optimizes the simulation process through multi-level cross-optimization of finite element method.

[0008] In one aspect of this disclosure, an electronic device includes a memory, a processor, and a computer program stored in the memory and executable on the processor, the processor running the computer program to implement the above-described gas diffusion-deformation-fracture coupling simulation method during the service of nuclear fuel UO2.

[0009] In one aspect of this disclosure, a storage medium storing a computer program that, when executed by a processor, implements the aforementioned gas diffusion-deformation-fracture coupling simulation method during the service of nuclear fuel UO2.

[0010] In one aspect of this disclosure, a computer program product includes a computer program that is executed by a processor to implement the above-described gas diffusion-deformation-fracture coupling simulation method during the service of nuclear fuel UO2. Attached Figure Description

[0011] The above and other objects, features, and advantages of exemplary embodiments of the present invention will become readily apparent from the following detailed description taken in conjunction with the accompanying drawings. Several embodiments of the invention are illustrated in the drawings by way of example, not limitation, in which:

[0012] Figure 1 A geometric configuration and mesh diagram of a double-bubble model according to one embodiment of the present disclosure.

[0013] Figure 2 A schematic diagram of the evolution of a critically connected-inflated channel for two bubbles according to one embodiment of this disclosure. Detailed Implementation

[0014] While existing modeling methods have greatly facilitated and deepened the study of diffusion, deformation, and fracture behavior of nuclear fuel gases, they often only address one or at most two aspects. Therefore, there is currently no unified scheme that can comprehensively consider all three factors—diffusion, deformation, and fracture—and their coupling mechanisms for studying performance changes in nuclear fuel under complex conditions. The phase-field method, an energy-based simulation scheme, constructs the system's free energy and correctly considers the physical fields of interest. The phase-field method can naturally handle the evolution of each field and their coupling mechanisms, making it an ideal tool for addressing such complex phenomena and coupled evolution in nuclear fuels.

[0015] This disclosure proposes a unified framework based on phase-field theory that comprehensively considers the gas diffusion, deformation, and crack evolution of nuclear fuel UO2. It utilizes fundamental thermodynamic theorems to constrain the evolution of free energy, thereby deriving the system's governing equations. Furthermore, a multi-level finite element cross-optimization algorithm is designed based on this framework and used to solve the coupled equations.

[0016] According to one or more embodiments, this disclosure proposes a finite element multi-level cross-optimization algorithm model construction method for the coupled problem of nuclear fuel UO2 diffusion, deformation and fracture.

[0017] The model disclosed in this paper simultaneously considers gas diffusion, deformation, and fracture phenomena in nuclear fuel UO2. Based on the theory of continuous medium thermodynamics, it describes the system state through four basic field variables: displacement field. Xenon atomic concentration Non-conservative order parameters characterizing the bubble phase and non-conservative order parameters characterizing crack phases The derivation of the governing equations strictly follows the first and second laws of thermodynamics, conservation laws, and the theory of micro-force equilibrium. In the mass conservation equation, a source term characterizing a local increase in gas concentration is introduced. This is because when a crack forms near an existing bubble, xenon atoms migrate rapidly to the crack core region, causing a local increase in concentration within the crack. Therefore, the increase in gas concentration within the local crack region is calculated accordingly. Saturation sources defined as threshold activation:

[0018] in, express The generation rate, This is the Heaviside function.

[0019] The total Helmholtz free energy Ψ is defined as the integral of the energy density Ψ over the domain, and consists of three main components:

[0020] In the completely intact region where mechanical properties have not been degraded due to bubbles or cracks, the elastic energy density is To describe the degradation effect of mechanical properties, two different degradation functions are introduced: one for bubbles and one for air bubbles. And for cracks Both functions are defined on the interval [0, 1] and monotonically decrease from 1 (representing a healthy state) to 0 (representing a completely degraded state). However, to reflect the different effects of bubbles and cracks on mechanical equilibrium and fracture driving forces, these functions must be applied to different parts of the elastic strain energy. Therefore, the elastic energy density... Addition is decomposed into positive parts related to stretching and expansion. And the negative part related to compression and contraction. . It is only permitted to act on the positive portion of the elastic deformation energy to prevent compression-induced fracture and the penetration of crack surfaces. Considering the circular geometry of the bubble, this disclosure assumes that their opposing surfaces will not come into contact. Therefore, Simultaneously acting on both the positive and negative parts of the elastic energy, this avoids non-physical compressive stresses that may occur when the relative bubble surfaces approach each other. Based on this, the elastic Helmholtz free energy functional... Defined as:

[0021] To achieve continuous medium characterization, the surface area of ​​sharp cracks Through the following area density functional Perform regularized approximation:

[0022] in It is a regularization parameter that controls the width of the dispersed crack characterization. Characterizing the crack phase field (0: intact, 1: complete fracture). (Item) This is a normalization constant used to ensure the tolerance towards the sharp crack limit. -convergence: Therefore, the functional expression for the crack free energy can be obtained as follows:

[0023] Diffusion energy It includes the contribution of a double-well potential and interface energy to regulate phase separation behavior.

[0024] Based on the constitutive assumptions, the governing equations are derived strictly following the conservation laws and dissipation inequalities. The system is described by the following set of coupled equations:

[0025]

[0026] in, Indicates stress, Representing the chemical potential. This set of equations captures the following key coupling mechanisms: the degradation of material mechanical properties due to bubble formation, and the degradation through the source term. This allows gas to be released into the crack region. Note that... , , , , , The specific functional form of etc. can be specified according to the actual physics, and is not affected by the governing equations (7)-(10).

[0027] The governing equations (8)-(11) involve the coupling of four different physical fields, and their energy landscape is obviously non-convex. Therefore, it is difficult to solve the four equations directly using the overall Newton method. This disclosure considers separating these four equations into a diffusion subproblem and a displacement-crack subproblem, and solving the two subproblems in the same time step to obtain the solution of (7)-(10). Before the algorithm design, since the governing equations (7)-(10) are in strong form, they cannot be directly solved numerically by the finite element method. Therefore, they need to be transformed into the following weak form defined by the residuals:

[0028]

[0029]

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[0031]

[0032]

[0033] Given the large number of symbols involved, for clarity, the symbols and their meanings are summarized below:

[0034]

[0035] For the numerical solution of the weak forms (11)-(15) of the governing equations, this disclosure presents a multi-level finite element cross-optimization algorithm. The table below shows the flow and logic of the algorithm.

[0036]

[0037] The function solver in the table above represents a multi-level finite element cross-optimization algorithm according to an embodiment of this disclosure. This is a numerical method for solving multiphysics coupling problems. The algorithm ensures the convergence and accuracy of the solution by performing multi-level iterative solutions at each time step. The algorithm consists of two levels of iterative processes: an inner-level iteration (Newton solver) is used to solve the nonlinear equations of diffusion and displacement-crack respectively, ensuring that the solutions of the diffusion field and displacement-crack field converge at the current time step (when other variables are fixed); and an outer-level iteration (multi-level coupling solution and cross-optimization) is used to ensure that the solution error of the diffusion-displacement-crack equations, i.e., equations (7)-(10), is within an acceptable range. Specifically,

[0038] The computational steps of this multi-level finite element cross-optimization algorithm can be divided into the following stages:

[0039] Step 1, Initialization: Set the time step size for each time step. Set time step Initialize the iteration counter, and set the multi-level iteration counter. Initialize to 0; initialize the solution vector, taking the solution from the previous time step. As an initial guess at the current time step ( ;

[0040] Step 2, Outer Loop (Multi-level Iteration): Enter the outer loop and begin multi-level iteration until the global convergence condition is met; initialize and define the state vector in each outer iteration step. In the definition, the diffusion state vector is... including concentration field Chemical potential field Phase field with bubbles ;

[0041] Step 3, Diffusion Subsystem Solver: Solve the diffusion subsystem using the residual-based Newton's method until one of the following convergence criteria is met:

[0042] , ;

[0043] Step 4, Inner Loop (Displacement-Crack Sub-Solver): Initialize the inner loop counter, setting the inner loop counter n to 0; Fix the diffusion state, and process the mechanical field problem under the fixed diffusion state; Update the displacement field, using a residual-based Newton solver to update the displacement field U until one of the following convergence criteria is met:

[0044] ,or

[0045] Update the crack phase field. Under the premise of a fixed displacement field U, update the crack phase field α using a variational inequality solver until one of the following convergence criteria is satisfied:

[0046] ,or

[0047] Update the inner loop counter. ;

[0048] Once the overall convergence requirement is met, exit the loop.

[0049] Step 5, Update the solution and iteration counter: Update the physics solution ( Update the multi-level iterative counter. ;

[0050] Step 6, Termination Condition: The outer loop reaches convergence; Algorithm End Time Step Update: Update the time step. ;

[0051] The algorithm described above ensures the convergence and accuracy of the solution to the multiphysics coupling problem by performing multi-level iterative solutions at each time step. Inner iterations solve the nonlinear equations, while outer iterations ensure that the coupling errors between different physical fields are within acceptable limits. Through this multi-level iterative solution method, the algorithm provides a robust finite element numerical approximation of the coupled system equations, thus serving as a numerical solution to the governing equations.

[0052] Each time step The computational flow is organized by an outer loop. In each outer iteration step (m+1), the solution process is interleaved into two main stages. First, the diffuser solver uses a residual-based Newton's method to calculate the concentration field. Chemical potential field and bubble phase field The process then proceeds to the displacement-crack sub-solver, which handles the mechanical field problem through an internal staggered loop. Under the condition that the diffusion state remains unchanged, this internal loop first updates the displacement field using a residual-based Newton solver. Next, in a fixed displacement field Under the premise of updating the crack phase field using PETSc's SNES variational inequality solver. The internal staggered loop will continue until the displacement-fracture convergence criterion is met. The external loop then repeats this two-stage process until global convergence is achieved. The state at the time step is then finally determined.

[0053] The Newton-Raphson method is used to solve the nonlinear equations of a control diffusion and displacement subsystem. For a nonlinear system... The algorithm performs an iterative process: Updated later .in, The uniform tangent operator is used for the diffuse subsystem. For the displacement subsystem During the crack solving phase, the variational inequality solver dynamically identifies the active constraint set to update the crack phase field and strictly enforces the nodal irreversibility condition during the iteration process. All linearized equations are solved using the direct method of LU decomposition. The numerical implementation of the algorithm is completed within the FEniCSx open-source framework. After each sub-solver and outer loop converges, the final solution provides a robust finite element numerical approximation of the coupled system equations (6)-(10), thus serving as the numerical solution of the governing equations (3)-(6).

[0054] According to one or more embodiments, a finite element multi-level cross-optimization method for the coupled problem of nuclear fuel UO2 diffusion, deformation, and fracture specifically includes,

[0055] Step 1: Specify the constitutive relations in the governing equations (8)-(11) based on the physical phenomena of nuclear fuel under different conditions: such as stress-strain relations, softening behavior and diffusion behavior.

[0056]

[0057]

[0058]

[0059]

[0060] Step 2: Transform the strong form obtained in Step 1 into a weak form (12)-(16) using the standard integration method:

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[0065]

[0066] Step 3: Implement the algorithm using the finite element method according to the algorithm flow of the multi-level cross-optimization finite element solver of this disclosure embodiment.

[0067] Step 4: Set initial and boundary conditions, and use the implemented finite element solver to solve the field variables at different time steps.

[0068] like Figure 1 The diagram shows the geometric configuration and mesh of the double-bubble model. Figure 1 (a) represents the initial geometry of the bubble and the applied stress boundary conditions. Figure 1 (b) indicates the rotation angle The overall grid at that time. For example... Figure 2 As shown, it is and far-field stress Crack evolution and gas channel dynamics under certain conditions. Figure 2 (ac) is the gas concentration at the critical moment before crack initiation. Bubble Phase Field and maximum principal stress distributed. Figure 2 (df) The three stages of crack evolution between bubbles, with the white dashed lines indicating the bubble boundaries. Figure 2 (gi) is through the bubble phase field The subsequent evolution of the captured inflation channels over time. For example... Figure 2 Under a far-field tension of 35 MPa, the region between two submillimeter-sized bubbles expands and stress concentrates due to gas supersaturation. Once the bubble wall spacing is compressed to a critical size of ≈1.2 µm, the crack immediately penetrates along the bubble connection perpendicular to the tensile stress direction, forming an instantaneously opened gas-filled channel. Subsequent gas continues to permeate along this channel, causing the crack-bubble system to enter a rapid propagation stage.

[0069] This disclosure implements a multi-level cross-optimized finite element solver using steps one through four, and uses this solver to study a typical two-bubble model in UO2 considering gas diffusion, deformation, and fracture. The two-bubble model comprises two bubbles embedded in a solid matrix and subjected to far-field uniaxial tensile stress. Function. The line connecting the centers of the two bubbles forms an angle with the transverse direction perpendicular to the loading axis. (Radians). The initial radius of each bubble is 250 nm, and the center-to-center distance between two bubbles is 4000 nm. A schematic diagram of this configuration is shown below. Figure 1 As shown.

[0070] To minimize the influence of the domain boundary on the stress field, a large-size computational domain is used. nm. Within this domain, the center is defined. A rectangular region with a unit size of 20 nm is used. To accurately resolve the crack-bubble interface, a uniformly structured fine mesh with a unit size of 20 nm is used within the rectangular region, while a coarser mesh is used outside the region to reduce computational cost. The grid of time, such as Figure 1 As shown in (b).

[0071] In the simulation, according to Figure 1 In configuration (a), two bubbles are initialized in the central region of the computational domain. (The bubble phase field inside the bubble...) and gas concentration Set to 1, and the initial value of the surrounding matrix is ​​set to... and A matrix gas concentration significantly exceeding the equilibrium value was set to drive continuous bubble growth. Driven by the supersaturated gas concentration, the bubbles continued to grow until a critical state was reached, after which rapid crack propagation occurred. At this critical point, the bubble edge spacing measured along a line connecting the bubble centers was approximately 1200 nm. , The distribution of the maximum principal stress is shown in [the diagram]. Figure 2 (a)-(c). Once the critical state is reached, the crack initiates between the two bubbles and propagates along their connecting line (i.e., perpendicular to the loading direction). The three stages of crack evolution are as follows: Figure 2 As shown in (d)-(f), the white dashed lines indicate the bubble boundaries. Subsequent diffusion development is as follows... Figure 2 As shown in (g)-(i).

[0072] Therefore, the beneficial effect of this disclosure lies in addressing the incompleteness of existing models and simulation methods, which can only consider one or two of the gas diffusion, deformation, and fracture problems in nuclear fuel UO2. This disclosure proposes a framework that can simultaneously consider all three physical elements—gas diffusion, deformation, and fracture—and a corresponding multi-level cross-optimization finite element algorithm. The solver involved in this disclosure can simultaneously solve multi-field solutions for gas diffusion, deformation, and fracture problems at different time points, further addressing the problem that some physical elements may be ignored by existing models. In particular, this framework and algorithm are theoretically not limited by the specific material constitutive choice, thus possessing considerable generalization potential.

[0073] It should be understood that in the embodiments of the present invention, the term "and / or" is merely a description of the relationship between associated objects, indicating that three relationships can exist. For example, A and / or B can represent: A existing alone, A and B existing simultaneously, and B existing alone. Furthermore, the character " / " in this document generally indicates that the preceding and following associated objects have an "or" relationship.

[0074] If the integrated unit is implemented as a software functional unit and sold or used as an independent product, it can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present invention, in essence, or the part that contributes to the prior art, or all or part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of the present invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0075] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any person skilled in the art can easily conceive of various equivalent modifications or substitutions within the technical scope disclosed in the present invention, and such modifications or substitutions should all be covered within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A method for constructing a coupled simulation model of gas diffusion-deformation-fracture during the service of nuclear fuel UO2, characterized in that, The simulation model is described by the following set of control equations (7)-(10).

2. The method according to claim 1, characterized in that, The weak form of the governing equations (8)-(11) includes the following set of governing equations (11)-(15):

3. The method according to claim 2, characterized in that, The control equations (11)-(15) are solved using the finite element multi-level cross-optimization algorithm.

4. The method according to claim 3, characterized in that, The computational flow of the finite element multi-level cross-optimization algorithm is organized by an outer loop. In each iteration step of the outer loop, the computation process is interleaved into two steps. (1) The concentration field is calculated using the residual Newton method through the diffuser solver. Chemical potential field and bubble phase field ; (2) The mechanical field is calculated by internal interleaved loop processing through the displacement-crack sub-solver.

5. The method according to claim 4, characterized in that, In step (2), under the condition that the diffusion state remains unchanged, the inner loop first updates the displacement field using a Newton-based residual solver. ; Next, in a fixed displacement field Under the premise of updating the crack phase field using PETSc's SNES variational inequality solver. .

6. A coupled simulation method for gas diffusion-deformation-fracture during the service of nuclear fuel UO2, characterized in that, The gas diffusion-deformation-fracture coupling problem during the service of nuclear fuel UO2 is simulated using the model described in claim 1, and the simulation process is optimized by multi-level cross-optimization of the finite element method.

7. The method according to claim 6, characterized in that, This method is used for the design of nuclear fuel rods with a pellet-cladding structure.

8. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, The processor executes the computer program to implement the method as described in any one of claims 1 to 7.

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

10. A computer program product, comprising a computer program, characterized in that, The computer program is executed by a processor to implement the method according to any one of claims 1 to 7.

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