Radioactive nuclide activity calculation method considering multistage decay based on Pade approximation

Abstract

The invention provides a radionuclide activity calculation method considering multistage decay based on Pade approximation. The radionuclide activity calculation method comprises the steps of S1, constructing a complete decay matrix containing all nuclides; the step S1 comprises the following steps: S11, receiving a decay chain data structure as input, and constructing an n * n square matrix A; s12, initializing all elements of the whole matrix into zero values; s13, setting elements in the ith row and the ith column as a negative value of the decay constant of the ith nuclide, namely A (i, i) =; s14, traversing all possible nuclide pairs (i, j); s2, calculating a matrix index by adopting a matrix norm driven adaptive scaling algorithm in combination with a Pade approximation matrix; and S3, executing corresponding radionuclide activity calculation according to the calculation mode. According to the method, the complete decay matrix is constructed, contribution of all nuclides is considered, and truncation errors are avoided. A self-adaptive scaling algorithm is combined with a Pade approximation matrix solution to realize rigid system matrix calculation, so that numerical value overflow and rounding errors are effectively avoided, and the stability of a calculation process is ensured.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of nuclear radiation, in particular to a radionuclide activity calculation method considering multi-stage decay based on Padé approximation. BACKGROUND

[0002] In the Level 3 Probabilistic Safety Assessment (PSA) of nuclear power plants, the activity evolution of radioactive substances after release is the core of public dose assessment and radiation consequence analysis. At present, the Bateman formula performs well in the two-stage decay scene containing only parent nuclei and first-generation daughter nuclei due to its simplicity and efficiency, and has certain calculation accuracy and efficiency.

[0003] However, complex accident scenarios are often accompanied by multi-stage decay chains, with a large number of nuclides and complex coupling relationships. At this time, traditional analytical solutions or numerical integrals are difficult to balance accuracy and efficiency, especially in the presence of a large number of nuclides and complex coupling relationships, the calculation accuracy is lost, the time is too long, and the numerical instability is highlighted.

[0004] To solve the model description of multi-stage decay under complex conditions, introducing a description matrix is a common approach. However, multi-stage chains often have large differences in parent and daughter half-lives, complex branching paths, and long time scales, resulting in a highly rigid mathematical model. Traditional numerical integration methods, such as the Runge-Kutta numerical integration method, have significant error accumulation during high-rigidity, long-time integration, and the accuracy rapidly degrades.

[0005] In recent years, Padé approximation has shown excellent numerical stability and convergence speed in rigid systems and matrix exponent calculations due to its high-order rational function approximation. Compared with traditional Taylor series, Padé approximation can achieve high-precision approximation over a wider time interval, especially for systems with large decay constant spans and long step lengths.

[0006] However, there is still a lack of a systematic Padé approximation calculation method for multi-stage radioactive decay chains that can fully integrate theoretical models and numerical calculation methods, especially in terms of adapting theoretical and numerical solving methods and efficiency control. Its application in Level 3 PSA analysis of nuclear power plants has not yet formed an engineering implementation path.

[0007] Therefore, the present application inventors designed a radionuclide activity calculation method considering multi-stage decay based on Padé approximation to overcome the above technical problems. SUMMARY

[0008] The technical problem solved by the present application is to overcome the defects of incomplete processing of complex decay chain, low calculation accuracy and poor stability in the prior art, and to provide a radionuclide activity calculation method considering multi-stage decay based on Padé approximation.

[0009] The present application solves the above technical problems by the following technical solutions:

[0010] A radionuclide activity calculation method considering multi-stage decay based on Padé approximation, characterized in that the radionuclide activity calculation method comprises:

[0011] S1, constructing a complete decay matrix containing all nuclides;

[0012] The step S1 of constructing the matrix comprises:

[0013] S 11 , receiving a decay chain data structure as input, and constructing an n*n square matrix A, where n is the total number of nuclides in the decay chain;

[0014] S 12 , initializing all elements of the entire matrix to zero values;

[0015] S 13 , setting the element of the i-th row and the i-th column to the negative value of the decay constant of the i-th nuclide, i.e.

[0016] A(i,i) = , where is the decay constant of the i-th nuclide;

[0017] S 14 , traversing all possible nuclide pairs (i,j), when i is greater than j and the branching ratio of the i-th nuclide to the j-th nuclide is greater than zero, calculating the non-diagonal element, and setting A(i,j) to ;

[0018] S2, using a matrix norm driven adaptive scaling algorithm to calculate the matrix exponential in combination with Padé approximation;

[0019] S3, performing corresponding radionuclide activity calculation according to the calculation mode.

[0020] According to an embodiment of the present application, the decay matrix uses a multi-stage decay matrix equation, and the decay calculation is based on the radionuclide activity;

[0021] For a decay chain containing n nuclides, the differential equation of the decay matrix is:

[0022]

[0023] where the structure of the decay matrix A is:

[0024]

[0025] where diagonal elements represent spontaneous decay of the ith nuclide; lower triangular elements represent the product of branching ratio from the ith nuclide to the jth nuclide and decay constant of the ith nuclide; upper triangular elements 0 represent no inverse decay.

[0026] According to an embodiment of the present application, the Padé approximation method comprises matrix norm calculation, order selection, matrix scaling and inverse scaling.

[0027] According to an embodiment of the present application, the matrix exponential calculation in step S2 comprises:

[0028] S 21 , performing matrix norm calculation: calculating the infinity norm of matrix A ||A||∞;

[0029] S 22 , order selection, dynamically selecting Padé approximation order according to the size of matrix norm:

[0030] S 23 , performing matrix scaling: when ||A||∞ > 1, performing binary scaling: ; where s is the scaling number;

[0031] S 24 , using Padé approximation calculation: calculating numerator polynomial matrix N(n) and denominator polynomial matrix D(n), n is the order of the matrix; through get exp(A't);

[0032] S 25 , performing inverse scaling, restoring the original scale by repeating the square operation:

[0033] .

[0034] According to an embodiment of the present application, the order selection comprises:

[0035] When ||A||∞ < 0.015: select 3 orders;

[0036] When 0.015 ≤ ||A||∞ < 0.25: select 5 orders;

[0037] When 0.25 ≤ ||A||∞ < 0.95: select 7 orders;

[0038] When 0.95 ≤ ||A||∞ < 2.1: select 9 orders;

[0039] When ||A||∞ ≥ 2.1: select 13 order.

[0040] According to one embodiment of the present application, the step S 23 The selection strategy of matrix scaling is:

[0041] When ||A||∞ × t > 1, scaling is performed.

[0042] According to one embodiment of the present application, the radionuclide activity calculation in step S3 comprises: based on the nucleon number, the radionuclide activity is calculated;

[0043] The formula of the nucleon number of each nuclide is: ;

[0044] Wherein, is the element of the i-th row and the j-th column of the matrix index, is the initial nucleon number of the j-th nuclide.

[0045] According to one embodiment of the present application, the step S 11 In the step S, the decay chain data structure comprises the total number of nuclides, the name of the nuclide, the decay constant, the branching ratio and the initial activity.

[0046] The present application also provides an electronic device, characterized in that the electronic device comprises a processor and a memory, the memory stores programs or instructions executable on the processor, and the programs or instructions are executed by the processor to realize the radionuclide activity calculation method considering multi-stage decay based on Padé approximation as described above.

[0047] The present application also provides a readable storage medium, characterized in that the readable storage medium stores programs or instructions, and the programs or instructions are executed by a processor to realize the radionuclide activity calculation method considering multi-stage decay based on Padé approximation as described above.

[0048] The positive progress effect of the present application is that:

[0049] The radionuclide activity calculation method considering multi-stage decay based on Padé approximation has the following advantages:

[0050] I. By constructing a complete decay matrix, the contribution of all nuclides is considered, and the truncation error of the traditional method is avoided.

[0051] II. The radionuclide activity calculation method proposed in the present application can obtain the radioactivity of all nuclides in a decay chain through one calculation, greatly improving the solving rate of radionuclides under complex conditions.

[0052] III. The adaptive Padé order and scaling strategy driven by matrix norm is proposed. The Padé approximation order and scaling times are dynamically selected based on the matrix infinity norm. A systematic range and order parameter matching scheme is established according to the stiffness characteristics of the decay chain, which can adaptively match the matrix size and stiffness characteristics to ensure numerical stability and high-precision calculation.

[0053] IV. The complete technical link from input data to activity result output is formed through the collaborative optimization of physical modeling (full matrix construction and sub-matrix extraction), numerical algorithm (adaptive Padé approximation), and time advancing strategy (dynamic interval division). This scheme can quickly and stably complete the post-accident activity evolution calculation for hundreds to thousands of nuclides, providing reliable technical support for the rapid assessment and emergency response of off-site radiation consequences.

[0054] V. Support for complex multi-branch and multi-level decay chains, suitable for various nuclear physics calculation scenarios. BRIEF DESCRIPTION OF DRAWINGS

[0055] The above and other features, properties, and advantages of the present application will become more apparent by the following description with reference to the accompanying drawings and embodiments, in which the same reference numerals are used throughout the drawings and represent the same features, and wherein:

[0056] Fig. 1 The flowchart of the radionuclide activity calculation method considering multi-level decay based on Padé approximation of the present application.

[0057] Fig. 2 The flowchart of the radionuclide activity calculation method considering multi-level decay based on Padé approximation of the present application, the decay chain calculation matrix construction.

[0058] Fig. 3 The flowchart of the radionuclide activity calculation method considering multi-level decay based on Padé approximation of the present application, the matrix solving algorithm. DETAILED DESCRIPTION

[0059] In order to make the above-mentioned purposes, features and advantages of the present application more obvious and easy to understand, the specific embodiments of the present application will be described in detail below with reference to the accompanying drawings.

[0060] The embodiments of the present application will now be described in detail with reference to the accompanying drawings. The preferred embodiments of the present application will now be described in detail with reference to the drawings, examples of which are shown in the drawings. In all the drawings, the same reference numerals will be used to represent the same or similar parts, if possible.

[0061] Also, although the terms used in the present application are selected from publicly-known and common terms, some of the terms mentioned in the specification of the present application can be selected by the applicant in his or her judgment, the detailed meanings of which are described in the relevant portions of the description herein.

[0062] Also, the present application is required to be understood not only by the actual terms used, but also by the meanings implied by each term.

[0063] As Figs. 1-3 indicated, the present application discloses a radionuclide activity calculation method considering multi-stage decay based on Padé approximation, which comprises the following steps:

[0064] Step S1, constructing a complete decay matrix containing all nuclides.

[0065] The present application adopts a matrix construction form and method for completely describing the radionuclide decay process of a multi-stage, complex decay chain. Through equation conversion, the radionuclide decay activity of a complex decay chain is made feasible for numerical calculation, and the complex multi-stage decay calculation is converted into sparse matrix solving.

[0066] For example, as Fig. 2 indicated, the matrix construction form and construction method in the present embodiment are as follows:

[0067] I. Matrix form:

[0068] The basic equation of multi-stage decay considers multi-generation decay, and the decay formula form for activity calculation is as follows:

[0069] Parent nucleus: ;

[0070] Child nucleus: ;

[0071] wherein, indicates the nuclide number, dimensionless; indicates the activity of nuclide i at time t, Bq; j indicates the nuclide number of the nuclide producing nuclide i by decay, dimensionless; indicates the activity of nuclide j at time t, Bq; indicates the decay constant of nuclide i, 1 / s; indicates the branching ratio of nuclide j decaying into nuclide i, dimensionless.

[0072] The multi-stage decay matrix equation needs to be based on the number of nucleons to describe the decay relationship. In the process of calculating the activity of a three-stage PSA, the decay calculation is usually directly based on the activity of the radionuclide.

[0073] If a description matrix equation is used, the activity needs to be converted to the number of nucleons in the calculation process.

[0074] For example, for a decay chain containing n nuclides, the differential equation of the decay matrix is:

[0075]

[0076] where the structure of the decay matrix A is:

[0077]

[0078] where the diagonal element represents the spontaneous decay of the ith nuclide; the lower triangular element represents the product of the branching ratio from the ith nuclide to the jth nuclide and the decay constant of the ith nuclide; and the upper triangular element 0 represents no back decay.

[0079] The above decay chain matrix can completely describe the decay chain relationship between nuclides at each level in a decay chain, and is not affected by the length of the decay chain and the type of nuclides. Based on the matrix equation, the residual activity of all radioactive nuclides in the decay chain at any time can be solved at one time.

[0080] Preferably, the step S1 of constructing the matrix comprises:

[0081] Step S 11 , receiving a decay chain data structure as input, and constructing an n x n square matrix A, where n is the total number of nuclides in the decay chain.

[0082] Here, the decay chain data structure includes information such as the total number of nuclides, the name of the nuclide, the decay constant, the branching ratio, and the initial activity.

[0083] Step S 12 , initializing all elements of the entire matrix to zero.

[0084] Step S 13 , setting the element in the ith row and the ith column to the negative value of the decay constant of the ith nuclide, i.e.

[0085] A(i,i) = , where is the decay constant of the ith nuclide.

[0086] Step S 14 , traversing all possible nuclide pairs (i, j), and when i is greater than j and the branching ratio of the jth nuclide to the ith nuclide is greater than zero, calculating the off-diagonal element and setting A(i, j) to .

[0087] Step S2, the matrix norm driven adaptive scaling algorithm is used to calculate the matrix exponential in combination with the Padé approximation matrix. The solving method of the Padé approximation matrix preferably can include matrix norm calculation, order selection, matrix scaling and reverse scaling.

[0088] The transformed decay activity equation is a linear constant differential equation, and the general solution of the equation is:

[0089]

[0090] wherein, is the exponential of matrix A, defined as the power series expansion:

[0091]

[0092] Preferably, the matrix exponential calculation in the step S2 includes:

[0093] Step S 21 , matrix norm calculation is performed: the infinite norm ||A||∞ of the matrix A is calculated.

[0094] Step S 22 , order selection, the Padé approximation order is dynamically selected according to the size of the matrix norm.

[0095] For example, the order selection can preferably include:

[0096] When ||A||∞ < 0.015: select 3 orders;

[0097] When 0.015 ≤ ||A||∞ < 0.25: select 5 orders;

[0098] When 0.25 ≤ ||A||∞ < 0.95: select 7 orders;

[0099] When 0.95 ≤ ||A||∞ < 2.1: select 9 orders;

[0100] When ||A||∞ ≥ 2.1: select 13 orders.

[0101] Step S 23 , matrix scaling is performed: when ||A||∞ > 1, binary scaling is performed: ; wherein s is the scaling number.

[0102] Step S 24 , Padé approximation calculation: calculate the numerator polynomial matrix N(n) and the denominator polynomial matrix D(n), n is the matrix order; through get exp(A't).

[0103] Step S 25, inverse scaling is performed to restore the original scale by repeating the square operation:

[0104] .

[0105] Step S3, performing the corresponding radionuclide activity calculation according to the calculation mode.

[0106] After the step S2 of solving the exponential matrix is completed, the radionuclide nucleon number is further calculated based on the radionuclide nucleon number formula. And based on the nucleon number, the radionuclide activity is calculated.

[0107] Preferably, the radionuclide activity calculation in the step S3 includes: calculating the radionuclide activity based on the nucleon number;

[0108] The nucleon number formula of each radionuclide is: ;

[0109] Wherein, is the element of the i-th row and the j-th column of the matrix exponential, is the initial nucleon number of the j-th radionuclide.

[0110] According to the above method step description, the radionuclide activity calculation method based on Padé approximation considering multi-stage decay of the present application is further described in detail through a specific embodiment as follows:

[0111] I. Introduction of the embodiment

[0112] This embodiment selects the SR-91 four-generation decay chain as a typical representative of a complex decay chain. The decay chain contains four radionuclides: SR-91 as the parent nucleus, which produces Y-91m metastable radionuclide through decay, Y-91m further decays to produce Y-91, and finally Y-91 decays to form stable Zr-91 radionuclide. This embodiment further illustrates the implementation process of the present patent. The initial activity of Sr-91 in this embodiment is 1.0×10 12 Bq, and the initial activity of other sub-nuclei is 0.

[0113] Wherein, the radionuclide material parameters are:

[0114] The decay constant of SR-91 is 1.9994×10 -5 per second, corresponding to a half-life of 9.6 hours;

[0115] The decay constant of Y-91m is 2.3240×10 -4 per second, corresponding to a half-life of 49.8 minutes, and the decay branch ratio is 57.234%;

[0116] The decay constant of Y-91 is 1.3711×10 -7Per second, corresponding half-life 58.5 days, decay branch ratio 42.766%;

[0117] Zr-91 is a stable nuclide, decay constant is 0;

[0118] This decay chain has the following characteristics: the difference in decay constant reaches 10 6 orders of magnitude; contains metastable nuclide Y-91m; contains stable nuclide Zr-91; the decay branch relationship is complex. -7 -4

[0119] II. Calculation process

[0120] 2.1, Decay matrix construction;

[0121] According to the decay chain information, the initialization matrix A is constructed;

[0122] Step 1, initialize the matrix to 0;

[0123]

[0124] Step 2, fill in the diagonal elements (decay constant);

[0125] A(0,0) = -λ1 = -1.9994×10 -5

[0126] A(1,1) = -λ2= -2.3240×10 -4

[0127] A(2,2) = -λ3= -1.3711×10 -7

[0128] A(3,3) = -λ4= 0

[0129] Step 3, fill in the non-diagonal elements (branch ratio);

[0130] A(1,0) = λ1 × b 21 = 1.9994×10 -5 × 0.57234 = 1.1446×10 -5

[0131] A(2,0) = λ1 × b 31 = 1.9994×10 -5 × 0.42766 = 8.5480×10 -6

[0132] A(2,1) = λ2 × b 32 ​​= 2.3240×10 -4 × 1.00000 = 2.3240 × 10 -4

[0133] A(3,2) = λ3 × b 43 = 1.3711×10 -7 × 1.00000 = 1.3711 × 10 -7

[0134] Step 4, Final decay matrix;

[0135]

[0136] 2.2 Matrix calculation based on Padé approximation;

[0137] 2.2.1 Calculation of matrix norm;

[0138] Step 1: Calculate the infinite norm of matrix A

[0139] ||A||∞ = max(1.9994×10 -5 , 2.4385×10 -4 , 2.4095×10 -4 1.3711×10 -7 )

[0140] = 2.4385×10 -4

[0141] 2.2.2 Selection of the Padé approximation order;

[0142] In this embodiment, the calculation time is 10,000 seconds, approximately 27.8 hours.

[0143] The scaled norm = ||A||∞ × t = 2.4385 × 10 -4 × 100000 =24.385.

[0144] Based on the order selection strategy, the 13th-order Padé approximation is chosen.

[0145] In this embodiment, the order selection strategy is as follows:

[0146] When ||A||∞ < 0.015: Choose the 3rd order;

[0147] When 0.015 ≤ ||A||∞ < 0.25: Choose order 5;

[0148] When 0.25 ≤ ||A||∞ < 0.95: choose the 7th order;

[0149] When 0.95 ≤ ||A||∞ < 2.1: select 9 orders;

[0150] When ||A||∞ ≥ 2.1: select 13 orders.

[0151] The specific steps of the matrix scaling strategy in the embodiment are as follows:

[0152] Step 1: Determine the scaling factor;

[0153] When ||A||∞ × t > 1, perform binary scaling, and finally determine the scaling factor as 5 orders.

[0154] Step 2: Construct the scaling matrix;

[0155] Matrix scaled_A = A × (t / 2^s) = A × (100000 / 2^5) = A × 3125

[0156] Scaled_A =

[0157] By the binary scaling strategy in the embodiment, ||A||∞ × t < 1 is ensured, and the convergence of the approximate calculation method is ensured.

[0158] Meanwhile, the order selection strategy suitable for the rigidity characteristics of the radionuclide decay chain matrix is determined based on a large number of sensitivity analyses in the present application, including the matrix norm range and the corresponding approximate order, which can well ensure the calculation efficiency and calculation accuracy of the radionuclide decay matrix under various time scales and decay chain compositions.

[0159] 2.2.3, Padé approximation calculation;

[0160] The Padé approximation calculation process in the embodiment specifically includes the following steps:

[0161] Step 1: Initialization: Create two n×n matrices N and D, and initialize them as unit matrices I; at the same time, set matrix X=A, and set the initial coefficient c=1.0;

[0162] Step 2: Loop expansion calculation: iterate from k=1 to m in turn, and gradually calculate the numerator and denominator polynomials: calculate the iteration coefficient: ;

[0163] Update the numerator polynomial: ;

[0164] Update the denominator polynomial: ;

[0165] Step 3: If k < m, update the power matrix ;

[0166] Step 4: Return the coefficient matrix ;

[0167] In this embodiment, the 13-order Padé approximation coefficient calculation result is as follows:

[0168] k = 1: c = 1 x (13-1+1) / (1 x (26-1+1)) = 13 / 26 = 0.5

[0169] k = 2: c = 0.5 x (13-2+1) / (2 x (26-2+1)) = 0.5 x 12 / 50 = 0.12

[0170] k = 3: c = 0.12 x (13-3+1) / (3 x (26-3+1)) = 0.12 x 11 / 72 = 0.01833 ...

[0172] k = 13: c =... = 1.234 x 10 -6 ;

[0173] The final result is:

[0174] exp(A') =

[0175] 2.2.4, Matrix inversion;

[0176] In this embodiment, the inverse matrix of the denominator polynomial D is calculated by using the augmented matrix method, and the specific steps include:

[0177] Step 1, construct an augmented matrix: create a matrix [A|I], where I is an identity matrix;

[0178] Step 2, Gaussian-Jordan elimination:

[0179] Select the element with the largest absolute value in each column as the pivot element;

[0180] If the pivot element is not on the diagonal, perform row exchange;

[0181] Perform linear combination operation on each row to eliminate non-diagonal elements;

[0182] Normalize each row;

[0183] Step 3, extract the inverse matrix: after elimination, the right half of the augmented matrix is the inverse matrix.

[0184] In the matrix inversion process of this embodiment, the pivot selection strategy is adopted, which can effectively avoid numerical instability under the rigid matrix.

[0185] 2.2.5, reverse scaling calculation

[0186] In this embodiment, the result matrix result is calculated 5 times by repeated squaring, each squaring operation doubles the time parameter, and finally returns to the original time. This method can avoid directly calculating the matrix exponent of a large time parameter, improve numerical stability and calculation accuracy, and is particularly important when calculating ultra-long time scales (such as 10 years or even 1 million years) in three-level PSA.

[0187] In this embodiment, the result of each matrix calculation is as follows:

[0188] First squaring operation:

[0189]

[0190] Second squaring operation:

[0191]

[0192] Third squaring operation:

[0193]

[0194] Fourth squaring operation:

[0195]

[0196] Fifth squaring operation:

[0197]

[0198] After 5 squaring calculations, the final result matrix is obtained:

[0199]

[0200] III. Radionuclide activity conversion

[0201] In the present application, radionuclide activity conversion is achieved by the following steps:

[0202] 1. Construct an initial nucleon number vector, convert the initial input of nuclide activity from the second PSA to the initial nucleon number vector (N = A / λ, where N is the nucleon number, A is the activity, and λ is the decay constant).

[0203] In the embodiment, the initial nucleon number vector is as follows:

[0204]

[0205] 2. Calculate the nucleon number result after decay based on the matrix coefficient.

[0206] The specific nucleon number result vector calculation formula of the present application is as follows:

[0207]

[0208] The calculation result in the embodiment is:

[0209]

[0210]

[0211]

[0212]

[0213] 3. Final activity calculation.

[0214]

[0215]

[0216]

[0217]

[0218]

[0219] The radioactive activity matrix calculation method based on the Padé approximation method in the embodiment can reach 10 -12 , and the single calculation time without considering parallelism only needs hundreds of μ s .

[0220] In addition, the above-mentioned embodiments of the present application can be replaced by the following alternatives:

[0221] I. Replace the adaptive scaling strategy with an iterative scaling strategy

[0222] The scaling factor is gradually replaced and adjusted through the iterative scaling strategy until the convergence condition is met.

[0223] II. Replace the full matrix form with a sparse vector linked list structure

[0224] For a system with a very large number of nuclides, the traditional n x n matrix will cause memory waste. At this time, the sparse matrix linked list or CSR compressed storage structure can be used for matrix construction, only storing non-zero elements and recording their positions, thereby saving memory and speeding up matrix operation, which is suitable for embedded platforms or high-performance parallel computing systems.

[0225] III. Replace the adaptive order selection strategy with an iterative order selection strategy.

[0226] The application further provides an electronic device, characterized in that the electronic device comprises a processor and a memory, the memory stores programs or instructions executable on the processor, and the programs or instructions are executed by the processor to realize the radionuclide activity calculation method considering multi-stage decay based on Padé approximation.

[0227] The application further provides a readable storage medium, characterized in that the readable storage medium stores programs or instructions, and the programs or instructions are executed by a processor to realize the radionuclide activity calculation method considering multi-stage decay based on Padé approximation.

[0228] In summary, the radionuclide activity calculation method considering multi-stage decay based on Padé approximation can uniformly describe the full-link conversion relationship between nuclides, has good engineering calculation efficiency while ensuring numerical stability and high precision, and meets the actual needs of accident source term modeling in three-level PSA.

[0229] The above disclosure of the application is only used as an example and does not constitute a limitation on the application. Although the application is not explicitly described herein, those skilled in the art can make various modifications, improvements and corrections to the application. Such modifications, improvements and corrections are suggested in the application, so such modifications, improvements and corrections still belong to the spirit and scope of the exemplary embodiments of the application.

[0230] Meanwhile, specific words are used in the application to describe the embodiments of the application. For example, "one embodiment", "an embodiment", and / or "some embodiments" means a certain feature, structure or characteristic related to at least one embodiment of the application. Therefore, it should be emphasized and noted that the "one embodiment" or "one alternative embodiment" mentioned in different positions in the specification does not necessarily refer to the same embodiment. In addition, some features, structures or characteristics in one or more embodiments of the application can be properly combined.

[0231] Some aspects of this application can be executed entirely by hardware, entirely by software (including firmware, resident software, microcode, etc.), or by a combination of hardware and software. The aforementioned hardware or software may be referred to as a "data block," "module," "engine," "unit," "component," or "system." The processor may be one or more application-specific integrated circuits (ASICs), digital signal processors (DSPs), digital signal processing devices (DAPDs), programmable logic devices (PLDs), field-programmable gate arrays (FPGAs), processors, controllers, microcontrollers, microprocessors, or combinations thereof. Furthermore, aspects of this application may manifest as computer products residing in one or more computer-readable media, including computer-readable program code. For example, computer-readable media may include, but are not limited to, magnetic storage devices (e.g., hard disks, floppy disks, magnetic tapes, etc.), optical discs (e.g., compressed CDs, digital multifunction DVDs, etc.), smart cards, and flash memory devices (e.g., cards, sticks, key drives, etc.).

[0232] A computer-readable medium may contain a propagated data signal containing computer program code, for example, on baseband or as part of a carrier wave. This propagated signal may take various forms, including electromagnetic, optical, and so on, or suitable combinations thereof. A computer-readable medium can be any computer-readable medium other than a computer-readable storage medium, which can be connected to an instruction execution system, apparatus, or device to enable communication, propagation, or transmission of a program for use. The program code located on the computer-readable medium can be propagated through any suitable medium, including radio, cable, fiber optic cable, radio frequency signals, or similar media, or any combination of the above media.

[0233] Similarly, it should be noted that, in order to simplify the description of the embodiments disclosed in this application and thus aid in the understanding of one or more embodiments of the invention, the foregoing description of the embodiments of this application sometimes combines multiple features into a single embodiment, drawing, or description thereof. However, this disclosure method does not imply that the subject matter of this application requires more features than those mentioned in the claims. In fact, the embodiments have fewer features than all the features of a single embodiment disclosed above. Some embodiments use numbers describing the number of components or attributes; it should be understood that such numbers used in the description of embodiments are modified in some examples by the terms "approximately," "about," or "generally."

[0234] Although the specific embodiments of the present application have been described above, it is understood by those skilled in the art that these are merely illustrative, and the scope of protection of the present application is defined by the appended claims. Those skilled in the art can make various changes or modifications to the embodiments without departing from the principles and the essence of the present application, and such changes and modifications fall within the scope of protection of the present application.

Claims

1. A method for calculating the activity of radionuclides taking into account multi-stage decay based on Padé approximation, characterized in that, The radionuclide activity calculation method comprises: S1, constructing a complete decay matrix containing all radionuclides; The step S1 of constructing a matrix comprises: S 11 , receiving a decay chain data structure as input, constructing an n x n square matrix A, where n is the total number of nuclides in the decay chain; S 12 , initializing all elements of the entire matrix to a zero value; S 13 , setting the element of the i-th row and i-th column to the negative of the i-th nuclide decay constant, i.e. A(i,i) = 1 - exp(-λiΔt) where is the decay constant of the ith nuclide; S 14 , traversing all possible pairs of nuclides (i, j), when i is greater than j and the branching ratio of the ith nuclide to the jth nuclide is greater than zero, calculating the off-diagonal element, setting A(i, j) to ; S2, using a matrix norm driven adaptive scaling algorithm to calculate the matrix exponential in combination with a Padé approximation matrix; S3, performing corresponding radionuclide activity calculation according to the calculation mode.

2. The method for calculating the activity of radionuclides considering multi-stage decay based on Padé approximation according to claim 1, characterized in that, The decay matrix uses a multi-level decay matrix equation to perform decay calculation based on radionuclide activity; For a decay chain containing n radionuclides, the differential equation of the decay matrix is: Wherein, the structure of the decay matrix A is: where the diagonal elements represent the spontaneous decay of the ith nuclide; the lower triangular elements represent the product of the branching ratio to the jth nuclide from the ith nuclide and the decay constant of the ith nuclide; the upper triangular elements 0 represent no inverse decay.

3. The method for calculating the activity of radionuclides considering multi-stage decay based on Padé approximation according to claim 1, characterized in that, The Padé approximation method comprises matrix norm calculation, order selection, matrix scaling and reverse scaling.

4. The method for calculating the activity of radionuclides considering multi-stage decay based on Padé approximation according to claim 1, characterized in that, The matrix exponential calculation in the step S2 comprises: S 21 , performing a matrix norm calculation: calculating an infinity norm of the matrix A, ||A||∞; S 22 , order selection, dynamically select the order of Padé approximation according to the size of the matrix norm: S 23 , matrix scaling: when ||A||∞ > 1, perform binary scaling: ; where s is the number of scaling times; S 24 , using Padé approximation: calculating a numerator polynomial matrix N(n) and a denominator polynomial matrix D(n), n being the matrix order; by obtaining exp(A't); S 25 , and reverse scaling by repeating squaring operations to recover the original scale: 。 5. The method for calculating the activity of radionuclides considering multi-stage decay based on Padé approximation according to claim 4, characterized in that, The order selection comprises: When ||A||∞ < 0.015: select 3 orders; When 0.015 ≤ ||A||∞ < 0.25: select 5 orders; When 0.25 ≤ ||A||∞ < 0.95: select 7 orders; When 0.95 ≤ ||A||∞ < 2.1: select 9 orders; When ||A||∞ ≥ 2.1: select 13 orders.

6. The method for calculating the activity of radionuclides considering multi-stage decay based on Padé approximation according to claim 4, characterized in that, The step S 23 The selection strategy of the middle matrix scaling is: Scaling is performed when ||A||∞ × t > 1.

7. The method for calculating the activity of radionuclides considering multi-stage decay based on Padé approximation according to claim 1, characterized in that, The radionuclide activity calculation in the step S3 comprises: calculating the radionuclide activity based on the number of nucleons; The formula for the number of nucleons of each nuclide is: ; wherein, is the element in the i-th row and j-th column of the matrix exponential, is the initial number of nucleons of the j-th nuclide.

8. The method for calculating the activity of radionuclides considering multi-stage decay based on Padé approximation according to claim 1, characterized in that, The step S 11 In this case, the decay chain data structure includes the total number of nuclides, the nuclide name, the decay constant, the branching ratio, and the initial activity.

9. An electronic device, comprising: The electronic device comprises a processor and a memory, wherein the memory stores programs or instructions executable on the processor, and the programs or instructions are executed by the processor to realize the radionuclide activity calculation method considering multi-level decay based on Padé approximation according to any one of claims 1-8.

10. A readable storage medium, characterized by, The readable storage medium stores programs or instructions, and when the programs or instructions are executed by the processor, the radionuclide activity calculation method considering multi-level decay based on Padé approximation according to any one of claims 1-8 is realized.

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