Energy resolution cluster dynamics solving method for multi-principal-element alloy irradiation damage

Abstract

The invention discloses an energy resolution cluster dynamics solving method for multi-principal-element alloy irradiation damage. The method comprises the following steps: establishing dynamics ordinary differential equations of various defect clusters in physical time through a reaction mechanism; converting the concentration variable of the cluster in the dynamic ordinary differential equation set from a concentration domain to a logarithm domain to obtain a logarithm domain ordinary differential equation set; performing time marching calculation on the log-domain ordinary differential equation set by adopting a numerical integration method, and updating the defect concentration change rate in the log-domain ordinary differential equation set in the solving process to obtain a defect evolution result; according to the method, the cluster dynamics simulation process can be accelerated, the evolution process of defect clusters can be accurately simulated, the method is widely suitable for defect behavior research of various irradiation materials, and the method has remarkable calculation efficiency and physical accuracy.

Description

Technical Field

[0001] This invention relates to the interdisciplinary field of materials science and computer numerical simulation, specifically to an energy-resolved cluster dynamics simulation method for the evolution of irradiation defects in multi-principal alloys. Background Technology

[0002] In nuclear energy structural materials such as high-entropy alloys, austenitic stainless steel, and iron-chromium alloys, irradiation can induce the generation, migration, aggregation, and dissociation of numerous point defects (vacancies, interstitial atoms) and their clusters. The evolution of these microscopic defects directly affects macroscopic properties, such as hardening, embrittlement, and irradiation swelling.

[0003] Existing methods for simulating cluster dynamics typically employ sets of Ordinary Differential Equations (ODEs) to describe the change in defect cluster concentration over time. For example, the paper "Cluster Dynamics Simulation of Electron Irradiation Defect Evolution in BCC Iron and Iron-Copper Alloys, Gao Chao, Master's Thesis" points out that the interactions between various defects in a crystal can be reduced to a set of highly coupled and complex dynamic differential equations, the numerical solution of which can obtain the evolution of the number density of point defects and their clusters over time. This type of method performs well in iron-copper alloy systems, and the reaction rate constant can be parameterized based on the averaged migration energy and formation energy. However, existing methods still have the following shortcomings when simulating the cluster dynamics of irradiation defect evolution in multi-component master alloys: (1) Poor numerical stability: The defect concentration spans more than 20 orders of magnitude. Traditional ODE systems that solve directly in the concentration domain are prone to having some cluster concentrations close to the lower limit of the computer's double-precision floating-point, which can easily lead to numerical underflow or overflow. For example, if the cluster concentration value is too small, it will be rounded to zero when it exceeds the lower limit of the floating-point, causing the relevant reaction rate terms in the system to become unbalanced, resulting in increased rigidity and amplified error. This makes it impossible to adjust the integration step size stably, ultimately leading to the integrator crashing or the result being distorted.

[0004] (2) Low computational efficiency: large-scale cluster systems (the number of variables can reach 10 ... - The sparse Jacobian matrix is ​​not fully utilized because Jacobian construction relies on complex reaction rate formulas, making sparse pattern recognition and index management difficult. At the same time, existing automatic differentiation tools and general frameworks often generate dense matrices, which are difficult to directly adapt to sparse linear solvers.

[0005] (3) Incomplete energy representation: Traditional models usually use averaged migration energy and formation energy, which fail to reflect the statistical distribution characteristics of the local chemical environment. In actual irradiation environments, there are significant local energy perturbations in the element distribution and defect structure of multi-principal alloys. Ignoring this factor will cause deviations between simulation results and experiments.

[0006] (4) Unreasonable output: Existing programs usually output results in linear time steps. The resolution of high-frequency dynamic processes in the early stage is insufficient, while the changes in the later stage are slower but generate a large amount of redundant data, resulting in low data utilization. Summary of the Invention

[0007] The technical problem to be solved by this invention is how to improve the computational efficiency of numerical simulation of the evolution dynamics of defect clusters in multi-principal alloys and high-entropy systems.

[0008] The present invention solves the above-mentioned technical problems through the following technical means: A method for solving energy-resolved cluster dynamics of irradiation damage in multi-principal element alloys is proposed, the method comprising: A set of ordinary differential equations for the kinetics of various defect clusters over physical time was established based on the reaction mechanism. The concentration variable of the cluster in the kinetic ordinary differential equation system is transformed from the concentration domain to the logarithmic domain to obtain the logarithmic domain ordinary differential equation system. The logarithmic domain ordinary differential equation system is calculated using a numerical integration method with time progression. During the solution process, the defect concentration change rate in the logarithmic domain ordinary differential equation system is updated to obtain the defect evolution results.

[0009] Furthermore, the system of ordinary differential equations for the kinetics of various defect clusters over physical time, established through the reaction mechanism, is expressed as follows:

[0010] In the formula, and Indicates in Time, defect size and defect concentration, and Indicates the size is The defect generation rate and the absorption rate of defect traps; Indicates the size is The defect dimension is The generation rate coefficient of defect transformation, Represented as size The defect was sized as The rate coefficient of defect consumption, and This indicates two different defect sizes.

[0011] Furthermore, the process of converting the concentration variable of the cluster in the kinetic ordinary differential equation system from the concentration domain to the logarithmic domain to obtain the logarithmic domain ordinary differential equation system includes: A logarithmic transformation is performed on the concentration variable of the cluster to convert it into the logarithmic domain, as expressed by the formula:

[0012] In the formula, Represents the concentration variable in the logarithmic field. Indicates time step The concentration of defective clusters at that time; Based on the concentration variable in the logarithmic domain, a system of ordinary differential equations in the logarithmic domain is obtained, expressed as follows:

[0013] In the formula, .

[0014] Furthermore, the numerical integration method is used to perform time-progressive calculations on the logarithmic domain ordinary differential equation system, and the defect concentration change rate in the logarithmic domain ordinary differential equation system is updated during the solution process to obtain the defect evolution results, including: The logarithmic domain ordinary differential equations are solved by implicit integration using the BDF method according to the set time steps, and the defect evolution results are obtained. The defect concentration change rate for each time step is updated in segments according to the time window.

[0015] Furthermore, the implicit integration method using BDF (Browser-Defined Functions) is employed according to a set time step to solve the logarithmic domain ordinary differential equation system in a time-progressive manner, obtaining the defect evolution results, including: At each time step, the Newton iteration method is used to solve the logarithmic domain ordinary differential equation system nonlinearly according to the set number of iterations. At each iteration, the SPGMR solver or KLU solver is used to determine whether the defect concentration vector obtained by the iteration has converged. If not, exit the simulation process; If so, proceed to the next iteration until the termination condition is met; The iteration continues until all time steps are solved, at which point the defect evolution result is obtained.

[0016] Furthermore, the step of using the Newton iteration method to progressively solve the logarithmic domain ordinary differential equation system nonlinearly at each time step according to a set number of iterations includes: Based on the concentration variable in the logarithmic domain ordinary differential equation system, the nonlinear equation is constructed at each time step as follows:

[0017] In the formula, This is the nonlinear residual vector at the current time step. For the current time step ( The cluster defect concentration vector, where each component corresponds to the defect cluster concentration in different energy and size states. For order, For time step, The rate of change of defect concentration. A baseline rate parameter for scaling the reaction rate term. For the historical weight parameters of the multi-step integral format, For the front Logarithmic concentration vector at each historical time step; The Newton iteration method was used to iteratively solve the nonlinear equations to obtain the defect concentration vector.

[0018] Furthermore, the process of updating the defect concentration change rate in segments according to time windows includes: The adjacent time steps are divided into several time windows, and the migration energy and free energy of the defects are perturbed when the time windows are switched. The updated defect concentration change rate is calculated by using the perturbed migration energy and free energy in conjunction with the pre-calculated initialization parameters. The initialization parameters include the cluster geometric radius, coordination number factor, and baseline reaction rate.

[0019] Furthermore, the perturbation of the migration energy and free energy of the defect during time window switching includes: When switching time windows, the migration energy and free energy are revalued according to the Gaussian distribution, as expressed by the formula:

[0020]

[0021] In the formula, This represents the instantaneous migration energy of the cluster within the energy-resolved framework during the current time window. The mean of the migration energy, Standard deviation Represents a standard normal distribution; The instantaneous free energy of the cluster within the current time window. The mean free energy.

[0022] Furthermore, the step of using the SPGMR solver or KLU solver to determine whether the defect concentration vector obtained in each iteration converges includes: When dealing with small-to-medium scale sparse systems, the SPGMR solver is used to determine whether the defect concentration vector obtained by iteration converges. When the logarithmic domain ordinary differential equation system corresponds to a large-scale sparse system, the KLU solver is used to determine whether the defect concentration vector obtained by iteration converges.

[0023] Furthermore, the method also includes: The defect evolution results are sampled and output using a logarithmic time distribution.

[0024] Furthermore, this invention also proposes a cluster dynamics simulation system for the evolution of irradiation defects in multi-principal alloys, the system comprising: The equation-building module is used to establish a set of kinetic ordinary differential equations for various defect clusters in physical time through reaction mechanisms; The logarithmic transformation module is used to convert the concentration variable of the cluster in the system of kinetic ordinary differential equations from the concentration domain to the logarithmic domain, thus obtaining the system of ordinary differential equations in the logarithmic domain. The numerical solution module is used to perform time-progression calculations on the logarithmic domain ordinary differential equation system using numerical integration methods, and to update the defect concentration change rate in the logarithmic domain ordinary differential equation system during the solution process to obtain the defect evolution results.

[0025] Furthermore, the present invention also proposes a computer-readable storage medium storing a computer program thereon, which, when executed by a processor, implements the energy-resolved cluster dynamics solution method for irradiation damage of multi-principal alloys as described above.

[0026] The advantages of this invention are: (1) This invention avoids numerical underflow by using logarithmic domain variables to represent the concentration of defect clusters, thereby improving the computational efficiency, stability and accuracy of numerical simulation. In addition, since the reaction rate constant of the concentration of multi-principal alloy clusters is randomly changing, an energy segmented constant update mechanism is introduced to update the rate of change of defect concentration in the equation set, reflecting the fluctuation of the local chemical environment during irradiation and enhancing the physical reality of the model.

[0027] (2) The present invention uses a numerical integration method to perform time-progression calculations on the logarithmic domain ordinary differential equation system. The specific integration algorithm is the BDF (Backward Differentiation Formula) method, which is a multi-step implicit backward difference scheme and is particularly suitable for the rigid problems commonly found in the irradiation defect dynamics model.

[0028] (3) By using the optional SPGMR / KLU linear solver, a unified processing framework for small-scale and large-scale problems is realized. During the calculation process, the linear solver SPGMR or KLU can be selected to adapt to small-scale or large-scale problems, so that the method can adapt to different scales of calculation tasks. It can efficiently process small-scale systems and ensure calculation accuracy and efficiency in large-scale systems.

[0029] (4) In order to further improve computational efficiency, the present invention pre-calculates key parameters in the initialization stage and directly calls them by table lookup, which significantly reduces the amount of computation during runtime.

[0030] Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0031] The accompanying drawings, which form part of this specification, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an undue limitation thereof. Figure 1 This is a flowchart illustrating an energy-resolved cluster dynamics solution method for irradiation damage in multi-principal alloys, as proposed in an embodiment of the present invention. Figure 2 This is a schematic diagram illustrating the mapping relationship between logarithmic domain variables and concentration domain variables in one embodiment of the present invention; Figure 3 This is a schematic diagram of the energy segment constant update mechanism in one embodiment of the present invention; Figure 4 This is a schematic diagram of the defect evolution result output file in one embodiment of the present invention. Detailed Implementation

[0032] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below in conjunction with the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0033] like Figures 1 to 2 As shown, the first embodiment of the present invention proposes an energy-resolved cluster dynamics solution method for irradiation damage in multi-principal element alloys, the method comprising the following steps: S10. Establish a set of ordinary differential equations for the kinetics of various defect clusters in physical time through reaction mechanisms; S20. Convert the concentration variable of the cluster in the kinetic ordinary differential equation system from the concentration domain to the logarithmic domain to obtain the logarithmic domain ordinary differential equation system. S30. The logarithmic domain ordinary differential equation system is calculated using a numerical integration method, and the defect concentration change rate in the logarithmic domain ordinary differential equation system is updated during the solution process to obtain the defect evolution results.

[0034] It should be noted that this embodiment avoids numerical underflow by using a logarithmic domain variable to represent the defect cluster concentration, thereby improving the computational efficiency, stability, and accuracy of the numerical simulation. Furthermore, since the reaction rate constant of the multi-principal alloy cluster concentration varies randomly, an energy-segmented constant update mechanism is introduced to update the defect concentration change rate in the equations, reflecting the fluctuations in the local chemical environment during irradiation and enhancing the physical realism of the model. This approach aims to address the problems of numerical instability, low computational efficiency, and incomplete energy representation in existing methods, and is particularly suitable for studying the irradiation defect evolution of nuclear energy structural materials such as high-entropy alloys, austenitic stainless steel, and iron-chromium alloys.

[0035] As a further preferred technical solution, step S10 involves establishing a set of kinetic ordinary differential equations for various defect clusters over physical time based on the reaction mechanism. The equations are expressed as follows:

[0036] In the formula, and Indicates in Time, defect size and defect concentration, and Indicates the size is The defect generation rate and the absorption rate of defect traps; Indicates the size is The defect dimension is The generation rate coefficient of defect transformation, Represented as size The defect was sized as The rate coefficient of the defect consumption.

[0037] As a further preferred technical solution, step S20: converting the concentration variable of the cluster in the kinetic ordinary differential equation system from the concentration domain to the logarithmic domain to obtain the logarithmic domain ordinary differential equation system, specifically includes the following steps: S21. Perform a logarithmic transformation on the concentration variable of the cluster to convert it into the logarithmic domain. The formula is as follows:

[0038] In the formula, Represents the concentration variable in the logarithmic field. The variable representing the defect cluster concentration at time step t; S22. Based on the concentration variable in the logarithmic domain, a system of ordinary differential equations in the logarithmic domain is obtained, expressed as follows:

[0039] In the formula, .

[0040] It should be noted that extremely low concentration values ​​(such as...) To address the numerical instability caused by underflow, this embodiment converts the concentration variation of each cluster to the logarithmic domain. Logarithmic domain variables can better handle large-scale concentration differences and avoid loss of numerical precision.

[0041] As a further preferred technical solution, step S30: using a numerical integration method to perform time-progression calculations on the logarithmic domain ordinary differential equation system, and updating the defect concentration change rate in the logarithmic domain ordinary differential equation system during the solution process to obtain the defect evolution results, specifically including: S30'. Using the implicit integration method of BDF according to the set time step, the logarithmic domain ordinary differential equation system is solved in time-progression to obtain the defect evolution result; The defect concentration change rate for each time step is updated in segments according to the time window.

[0042] It should be noted that after obtaining the logarithmic domain ordinary differential equation system, i.e., the logarithmic domain cluster dynamics equation system, this embodiment uses the CVODE library in the SUNDIALS software package as the core numerical solution engine. CVODE has a built-in local truncation error (LTE) monitoring mechanism, and the program automatically adjusts the integration step size and order according to the error magnitude, realizing adaptive control of the step size and order, improving the overall computational efficiency while ensuring controllable error. Furthermore, this library has high stability and versatility, and can effectively handle large-scale, highly rigid ordinary differential equation systems. The integration algorithm selected is the BDF method, a multi-step implicit backward difference scheme, which is particularly suitable for rigid problems commonly found in irradiation defect dynamics models.

[0043] As a further preferred technical solution, step S30': using the BDF implicit integration method to solve the logarithmic domain ordinary differential equation system in a time-progressive manner according to a set time step, and obtaining the defect evolution result, specifically includes the following steps: S31' At each time step, the Newton iteration method is used to solve the logarithmic domain ordinary differential equation system nonlinearly according to the set number of iterations. In each iteration, the SPGMR solver or KLU solver is used to determine whether the defect concentration vector obtained by the iteration has converged. If it has, step S32' is executed; otherwise, step S34' is executed. S32', Execute the next iteration until the termination condition is met; S33', proceed with the iterative solution of the next time step, repeat steps S31'-S32' until all time steps are iteratively solved, and the defect evolution result is obtained; S34', Exit the simulation process.

[0044] Specifically, in this embodiment, when performing time integration using the BDF implicit multi-step method, the general form of the BDF is:

[0045] in, This represents the reaction rate vector in the energy-resolved cluster kinetics equations, describing the generation, adsorption, emission, migration, and energy transition processes of clusters of different sizes and energy states. and This can be understood as the defect concentration vector being two points on the curve corresponding to the logarithmic domain ordinary differential equation system. For order, For time step, The reaction rate scaling factor (i.e., the baseline reaction rate) ), The coefficients of the BDF multistep scheme are determined by the selected order. And the discrete format is determined.

[0046] When using BDF for implicit integration, the nonlinear equations are constructed at each time step as follows:

[0047] In the formula, The residual function for the current time step is used for error analysis.

[0048] Next, the constructed nonlinear equations are solved. The solution process includes a nonlinear solution stage and a linear solution stage. In the nonlinear solution stage, the Newton iteration method is used to iteratively solve the nonlinear equations to obtain the defect concentration vector. The results of each iteration are then linearized.

[0049] Update in each iteration This continues until the residuals meet the convergence criteria (e.g., the set number of iterations is reached). To solve the linearized equation.

[0050] Then, the convergence of the defect concentration vector obtained by the iteration is determined using the SPGMR (Scalable Preconditioned GMRES) solver or the KLU solver. For example, when the equation system is large, a Jacobian mode of the sparse matrix needs to be provided, and the sparse LU decomposition method is used for acceleration. When the equation system is of medium to small scale, the convergence is determined using the SPGMR solver, which does not require explicit Jacobian mode and only depends on the right-hand side (RHS). The iteration convergence tolerance is set to... .

[0051] It should be noted that time-progressed calculations of the logarithmic domain ODE system are performed using numerical integration methods (such as the CVODE solver based on the SUNDIALS framework). Since the equation system is a stiff system, an implicit algorithm based on the Backward Differentiation Formula (BDF) and Newton's iteration is employed to ensure computational stability and high accuracy. During this process, Jacobian matrices are constructed using automatic differentiation or sparse matrix techniques to accelerate the solution and improve the computational efficiency of large-scale systems.

[0052] As a further preferred technical solution, the process of updating the defect concentration change rate in segments according to a time window specifically includes: The adjacent time steps are divided into several time windows, and the migration energy and free energy of the defects are perturbed when the time windows are switched. The updated defect concentration change rate is calculated by using the perturbed migration energy and free energy in conjunction with the pre-calculated initialization parameters. The initialization parameters include the cluster geometric radius, coordination number factor, and baseline reaction rate. These parameters are used together to construct the reaction rate matrix and transition probability terms of the energy-resolved cluster kinetic model, specifically: Cluster geometric radius : Used to calculate the adsorption cross-section, adsorption probability, and geometric trapping coefficient with free point defects between clusters of different sizes, where, , Atomic volume This represents the number of atoms in the cluster.

[0053] Coordination factor The coordination number factor describes the number of effective atomic sites on the cluster surface that can participate in reactions, and it determines the surface emissivity, trapping rate, and interaction strength with the substrate lattice. The coordination number factor is typically used as a scaling parameter for the exponential term before energy state transitions. Values .

[0054] Reference reaction rate The baseline reaction rate, as a global scaling parameter of the rate matrix, is used to normalize the highly rigid reaction network, thereby maintaining the stability of the energy-resolved equations during numerical solutions. , Let the radius of the defect be . Let be the diffusion coefficient of the defect. The concentration of defects, This is the atomic volume.

[0055] Specifically, such as Figure 3 As shown, in this embodiment, several time windows are obtained by dividing the time between adjacent time steps t and time step t+1. In each time window The internal energy values ​​(including defect migration energy (Emi, Emv) and free energy (Efreei, Efreev)) remain unchanged. When the time window changes, the energy values ​​are reassigned according to the Gaussian distribution. By introducing a time-segmented constant window (changetime) within the above differential equation solution framework, the defect migration energy (Emi, Emv) and free energy (Efreei, Efreev) are statistically perturbed, thereby updating the defect concentration change rate. The heterogeneity of defects in complex alloys is captured by energy segmentation, and the simulation is made adaptive as the defect evolves through dynamic updates.

[0056] Specifically, when switching time windows, the migration energy and free energy are revalued according to the Gaussian distribution, as expressed by the formula:

[0057]

[0058] In the formula, This represents the instantaneous migration energy of the cluster within the energy-resolved framework during the current time window. The mean of the migration energy, Standard deviation Represents a standard normal distribution; The instantaneous free energy of the cluster within the current time window. The mean free energy.

[0059] This embodiment introduces random perturbations to reflect the fluctuations in the local chemical environment during irradiation. This mechanism can reflect the modulation of energy by the local chemical environment.

[0060] In addition, this embodiment performs a large number of pre-calculations during the initialization phase to obtain some fixed parameters, including: geometric factor: the geometric radius of different defect clusters ( ) and coordination factor ( ); Baseline reaction rate β: Rate constant for calculating defective reactions; Energy difference index The energy difference between defect migration and reaction, where For defect free energy, Boltzmann's constant, For temperature.

[0061] This embodiment pre-stores these fixed parameters for direct table lookup during simulation, significantly reducing runtime computational overhead. Specifically, for high-entropy alloys (such as FeCoCrNiMn), the system can directly generate an energy perturbation table and perform simulation by inputting defect energy distribution data; for austenitic stainless steel, experimentally measured migration energy and formation energy can be imported through table lookup to simulate its irradiation stability.

[0062] As a further preferred technical solution, the defect evolution results obtained from the simulation are sampled and output through a logarithmic time distribution, including a concentration matrix, a logarithmic concentration matrix, a total defect number density and a logarithmic time series. According to the simulation requirements, it can provide high-resolution output of the early process and effective compression of the late process.

[0063] Furthermore, taking the simulation of defect evolution in FeCr alloys as an example (small to medium scale, SPGMR solver): (1) Model settings A Fe-20%Cr binary alloy was selected as the simulation object, and the temperature was set to 600 K; the number of vacancy clusters was... = 1000, number of interstitial clusters = 1000, total number of states =2000; initial defect concentration is set to 2000; By initializing the number field, we obtain And the concentration variable is transformed into a logarithmic domain variable.

[0064] (2) Physical parameters Migration energy: average value , Standard deviation Free energy data is provided dimension-by-dimension and stored in a lookup table file. It stores thermodynamic constants, including the Boltzmann constant. .

[0065] (3) Numerical integration The logarithmic domain ordinary differential equations are solved by implicit integration using the BDF method at a set time step, and the defect evolution results are obtained.

[0066] (4) Energy segmentation and random perturbation Set the time window changetime=0.01 seconds, and monitor the migration energy within each time window. and free energy Apply random perturbation; the random perturbation uses a standard Gaussian distribution with a standard deviation set to 1. .

[0067] Each time the migration energy is updated, the energy value is reassigned according to a Gaussian distribution to ensure that the simulation can capture the fluctuations in the local chemical environment. After each energy update, the newly calculated migration energy and free energy are recorded and applied to subsequent defect evolution calculations.

[0068] (5) Simulation results output The simulation results will show the concentration changes of defects within each time step, as well as the trend of these changes after the introduction of random perturbations. Specifically, the output step size is sampled according to a logarithmic time distribution, with the minimum time interval within the first second being [value missing]. The maximum interval after 10 seconds is 10 seconds; the output files include: defect concentration matrix concentration.csv, log concentration logC.csv, and total defect number density Nd.csv.

[0069] In addition, the scale of the defect evolution in the FeCr alloy was expanded to a larger scale, and large-scale simulations and the KLU sparse direct solver were performed. The process is as follows: (1) Model size expansion In large-scale systems, set the number of vacancy clusters. = 50000, number of interstitial clusters = 50000, total number of states =10000; the initial concentration is still set to 10000; The logarithm field is initialized to ; The KLU sparse direct solver is used to accelerate computation, and is particularly suitable for large-scale sparse systems. Set the time window changetime to 0.01 seconds, and the total simulation time to 3000 seconds.

[0070] (2) Simulation process: Numerical integration was performed using the BDF method and Newton's iterative method from the SUNDIALSCVODE library. The Jacobian matrix is ​​a sparse matrix with finite bandwidth, approximately 100. For large-scale systems, LU decomposition is performed using a KLU sparse direct solver, and computation is accelerated by providing a Jacobian mode for the sparse matrix, reducing the computational complexity to [value missing]. .

[0071] (3) Simulation output: The output includes the defect concentration of each size, such as Figure 4 As shown, the system automatically generates CSV files and corresponding PDF reports, including concentration-time curves, defect distribution histograms, and energy perturbation-time curves; users can quickly compare simulation results under different conditions through a graphical interface.

[0072] It should be noted that this embodiment encapsulates the energy-resolved cluster dynamics solution method for irradiation damage in multi-principal element alloys into a simulation software package, supporting YAML configuration file input. Users only need to input parameters such as material type, temperature, and maximum cluster number in the configuration file, and the system will automatically run the simulation. This method is particularly suitable for the following irradiation simulation scenarios: (1) Radiation damage kinetics of multi-principal alloys: such as the evolution of vacancies, interstitials, He atoms and their complexes in systems like FeCr, WTaCrV, NiCoCr, NbZrTi, and HEA; (2) Energy-resolved defect rate simulation: maintaining computational stability in cases where the energy barrier distribution is wide and the reaction rate spans multiple time scales; (3) Study on the interaction between bubbles and dislocations: prevent numerical divergence of explicit algorithms during the rapid formation or dissolution of defect clusters; (4) High-throughput accelerated simulation task: It can still converge stably under multi-parameter scans (temperature, flux, alloy composition).

[0073] It should be noted that the computer-readable medium disclosed in this embodiment may be a computer-readable signal medium or a computer-readable storage medium, or any combination thereof. A computer-readable storage medium may be, for example,—but not limited to—an electrical, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, or device, or any combination thereof. More specific examples of a computer-readable storage medium may include, but are not limited to: an electrical connection having one or more wires, a portable computer disk, a hard disk, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), optical fiber, and portable compact disk read-only memory (CD-ROM). ROM, optical storage devices, magnetic storage devices, or any suitable combination thereof. In this disclosure, a computer-readable storage medium can be any tangible medium containing or storing a program that can be used by or in connection with an instruction execution system, apparatus, or device. In this disclosure, a computer-readable signal medium can include a data signal propagated in baseband or as part of a carrier wave, carrying computer-readable program code. Such propagated data signals can take various forms, including but not limited to electromagnetic signals, optical signals, or any suitable combination thereof. A computer-readable signal medium can also be any computer-readable medium other than a computer-readable storage medium, which can send, propagate, or transmit a program for use by or in connection with an instruction execution system, apparatus, or device. The program code contained on the computer-readable medium can be transmitted using any suitable medium, including but not limited to: wires, optical fibers, RF (radio frequency), etc., or any suitable combination thereof.

[0074] The aforementioned computer-readable medium may be included in the aforementioned electronic device; or it may exist independently and not assembled into the electronic device. The aforementioned computer-readable medium carries one or more programs, which, when executed by the electronic device, cause the electronic device to perform a zero-sample image anomaly detection method according to the above embodiments.

[0075] Computer program code for performing the operations of this disclosure can be written in one or more programming languages ​​or a combination thereof, including object-oriented programming languages ​​such as Java, Smalltalk, and C++, and conventional procedural programming languages ​​such as the "C" language or similar programming languages. The program code can be executed entirely on the user's computer, partially on the user's computer, as a standalone software package, partially on the user's computer and partially on a remote computer, or entirely on a remote computer or server.

[0076] In cases involving remote computers, the remote computer can be connected to the user's computer via any type of network—including a local area network (LAN) or a wide area network (WAN), or it can be connected to an external computer (e.g., via the Internet using an Internet service provider).

[0077] It should be understood that various parts of the present invention can be implemented in hardware, software, firmware, or a combination thereof. In the above embodiments, multiple steps or methods can be implemented in software or firmware stored in memory and executed by a suitable instruction execution system. For example, if implemented in hardware, as in another embodiment, it can be implemented using any one or a combination of the following techniques known in the art: discrete logic circuits having logic gates for implementing logical functions on data signals, application-specific integrated circuits (ASICs) having suitable combinational logic gates, programmable gate arrays (PGAs), field-programmable gate arrays (FPGAs), etc.

[0078] In the description of this specification, references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of the invention. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples.

[0079] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of indicated technical features. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one of that feature. In the description of this invention, "a plurality of" or "several" means at least two, such as two, three, etc., unless otherwise explicitly specified.

[0080] Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those skilled in the art can make changes, modifications, substitutions and variations to the above embodiments within the scope of the present invention.

Claims

1. A method for energy-resolved cluster dynamics solution of irradiation damage in multi-principal element alloys, characterized in that, The method comprises the following steps: a kinetic ordinary differential equation group in physical time of various defect clusters is established by a reaction mechanism; a concentration variable of the cluster in the kinetic ordinary differential equation group is converted from a concentration domain to a logarithmic domain to obtain a logarithmic domain ordinary differential equation group; a numerical integration method is used to perform time advancing calculation on the logarithmic domain ordinary differential equation group, and a defect concentration change rate in the logarithmic domain ordinary differential equation group is updated during the solving process to obtain a defect evolution result.

2. The energy-resolved cluster dynamics solution method for irradiation damage of multi-principal element alloys as described in claim 1, characterized in that, The kinetic ordinary differential equation group in physical time of various defect clusters established by the reaction mechanism is expressed by a formula as follows: wherein and denote the concentration of defects of size at time t, the size of the defects being and the concentration of defects of size and denote the generation rate of defects of size and the absorption rate of defects of size denote the generation rate coefficient of conversion of defects of size into defects of size , denote the rate coefficient of consumption of defects of size by defects of size .

3. The method of claim 1, wherein the multi-principal element alloy is irradiation damaged. The concentration variable of the cluster in the kinetic ordinary differential equation group is converted from the concentration domain to the logarithmic domain to obtain the logarithmic domain ordinary differential equation group, which comprises the following steps: The concentration variable of the cluster is logarithmically transformed to convert the concentration variable into the logarithmic domain, which is expressed by a formula as follows: wherein denotes the concentration variable in the logarithmic domain, denotes the concentration variable of defect clusters at time step t; Based on the concentration variable in the logarithmic domain, the logarithmic domain ordinary differential equation group is obtained, which is expressed by a formula as follows: In the formulae, .

4. The method of claim 1, wherein the multi-principal element alloy is irradiation damaged. 4 The numerical integration method is used to perform time advancing calculation on the logarithmic domain ordinary differential equation group, and the defect concentration change rate in the logarithmic domain ordinary differential equation group is updated during the solving process to obtain the defect evolution result, which comprises the following steps: The logarithmic domain ordinary differential equation group is solved by using the BDF implicit integration method at a set time step to obtain the defect evolution result; The defect concentration change rate corresponding to each time step is updated according to a time window.

5. The method of claim 4, wherein the multi-principal element alloy is irradiation damaged. 5 The logarithmic domain ordinary differential equation group is solved by using the BDF implicit integration method at a set time step to obtain the defect evolution result, which comprises the following steps: At each time step, the logarithmic domain ordinary differential equation group is solved by using the Newton iteration method step by step according to a set number of iterations, and whether the defect concentration vector obtained by iteration converges is judged by using the SPGMR solver or the KLU solver at each iteration; If not, the simulation process is exited; If yes, the next iteration is performed until a termination condition is reached; The iteration solving of the next time step is performed, and the defect evolution result is obtained after the iteration solving of all time steps is completed.

6. The method of claim 5, wherein the multi-principal element alloy is irradiation damaged. 5 The logarithmic domain ordinary differential equation group is solved by using the Newton iteration method step by step according to a set number of iterations at each time step, which comprises the following steps: Based on the concentration variable in the logarithmic domain ordinary differential equation group, a nonlinear equation is constructed at each time step as follows: wherein is the residual function of the current time step, and may be understood as two points on the curve corresponding to the defect concentration vector, i.e. to the system of ordinary differential equations in the logarithmic domain, is the order, is the time step, is the reaction rate scale factor (reference reaction rate), is a coefficient of the BDF multi-step scheme determined by the selected order and the discretization scheme. The Newton iteration method is used to iteratively solve the nonlinear equation to obtain a defect concentration vector result.

7. The method of claim 4, wherein the multi-principal element alloy is irradiation damaged. 7 The process of updating the defect concentration change rate according to a time window comprises the following steps: Adjacent time steps are divided into a plurality of time windows, and the migration energy and the free energy of the defect are disturbed when the time window is switched; The updated defect concentration change rate is calculated by using the disturbed migration energy and free energy and combining the pre-calculated initialization parameters; The initialization parameters comprise a cluster geometric radius, a coordination number factor and a benchmark reaction rate, and the initialization parameters are used to construct a reaction rate matrix and a transition probability item of an energy resolution cluster kinetic model.

8. The method of claim 7, wherein the multi-principal element alloy is irradiation damaged. 8 The migration energy and the free energy of the defect are disturbed when the time window is switched, which comprises the following steps: The migration energy and the freedom are revalued according to Gaussian distribution at time window switching, and the formula is expressed as: wherein is the instantaneous migration energy of the cluster in the energy-resolved frame for the current time window, is the mean value of the migration energy, is the standard deviation, denotes the standard normal distribution; is the instantaneous free energy of the cluster for the current time window, is the mean value of the free energy.

9. The energy-resolved cluster dynamics solution method for irradiation damage of multi-principal element alloys as described in claim 5, characterized in that, The SPGMR solver or the KLU solver is used to judge whether the iteration obtained defect concentration vector converges or not at each iteration, and the judgment includes: The SPGMR solver is used to judge whether the iteration obtained defect concentration vector converges or not when the logarithmic field ordinary differential equation group corresponds to a small-scale sparse system; The KLU solver is used to judge whether the iteration obtained defect concentration vector converges or not when the logarithmic field ordinary differential equation group corresponds to a large-scale sparse system.

10. The method of claim 1, wherein, The method further includes: The defect evolution result is sampled and output through logarithmic time distribution.

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