Characteristic line determination theory electron transport calculation method based on continuous moderation approximation
By employing a characteristic line deterministic method based on the continuous slowing approximation, the electron transport problem is transformed into a multi-group neutral particle transport problem, which solves the computational efficiency and stability issues under complex geometric conditions and achieves efficient electron transport computation.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2026-01-14
- Publication Date
- 2026-04-14
AI Technical Summary
Existing methods for calculating electron transport are inefficient and produce unstable results under complex geometric conditions, making it difficult to meet the needs of engineering analysis. In particular, the Monte Carlo method is computationally expensive, the discrete ordinate method is not suitable for complex geometry, and existing deterministic methods lack applicability.
We adopt a characteristic line determination theory method based on the continuous slowing approximation. By discretizing the energy of the continuous slowing approximation operator and constructing multi-group pseudo-section data, we can transform the electron transport problem into a multi-group neutral particle transport problem. We then introduce the characteristic line method solution framework for integral solution, which avoids statistical noise and is applicable to complex geometries.
It achieves stable and deterministic electron transport calculation results, reduces engineering implementation costs, is applicable to analysis under complex geometric conditions, and integrates with existing neutron and photon transport analysis workflows.
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Figure CN121859682A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of radiation transport calculation and nuclear engineering numerical analysis technology, specifically involving a characteristic line deterministic electron transport calculation method based on the continuous slowing approximation. Background Technology
[0002] Electron transport processes are important research subjects in engineering and scientific fields such as reactor physics, radiation protection, medical physics, and radiation damage analysis. Due to their small mass, frequent collisions, and the relatively small energy loss and highly forward-biased angular distribution of each collision, electron transport processes are more complex than those of neutrons and photons. For example, an electron with an incident energy of 0.5 MeV slowed down to 0.25 MeV requires approximately 2.9 × 10⁻⁶ volts in an aluminum plate. 4 In contrast, electrons undergo only a few collisions, whereas under the same conditions, neutrons or photons require only a dozen or even ten collisions. Accurately and efficiently simulating electron transport processes has always been a key challenge in radiative transport calculations.
[0003] Because of the short mean free path and highly forward-peaked scattering of electronic reactions, the Boltzmann equation, while accurately characterizing electron transport, is computationally extremely expensive to solve directly. Monte Carlo methods require extensive particle history simulations; deterministic methods, to accurately describe electron energy loss, necessitate very narrow group widths; characterizing the highly forward-peaked angular distribution may require Legendre expansions of hundreds of orders; and the short mean free path also demands extremely fine spatial grids. Therefore, in engineering practice, an approximate form of the Boltzmann equation is typically used to characterize electron transport.
[0004] In existing technologies, Monte Carlo methods generally employ condensed history methods to reduce computational load. Monte Carlo methods can realistically describe the microscopic interaction processes of electrons and are widely used for electron transport calculations. However, this method typically requires a large amount of particle history to reduce statistical noise, and its computational cost remains high in complex geometric or deep-penetration problems, making it difficult to meet the requirements of computational efficiency and result determinism in engineering analysis.
[0005] For deterministic solutions to electron transport, researchers have proposed approximate forms based on the Boltzmann transport equations, with the Boltzmann–Fokker–Planck (BFP) equation being a typical example. This equation reduces the computational complexity of the problem by dividing inelastic electron collisions into soft and hard collisions based on the magnitude of energy loss, and then using a continuous slowing-down approximation for the soft collision part. This approach maintains physical plausibility while reducing the computational complexity of the problem.
[0006] In terms of numerical solutions, existing research has discretized the continuous slowing approximation operator in the BFP equations and reconstructed the electron transport equations into the form of multi-group Boltzmann transport equations by constructing equivalent energy group shift-out terms and downscattering source terms. This allows for solution using existing multi-group discrete ordinate procedures for neutral particles. However, the discrete ordinate method is not suitable for complex geometries and is only applicable to regular and simple geometries, limiting its applicability to practical engineering problems.
[0007] In contrast, the method of characteristics, by integrating the transport equations along characteristic lines when dealing with neutron and photon transport problems, exhibits good adaptability to complex geometries and has been widely used in engineering calculations. However, for electron transport problems, existing deterministic research mainly focuses on solution frameworks based on the discrete ordinate method. There is still a lack of systematic and mature technical solutions on how to effectively incorporate electron transport equations containing continuous slowing approximations into the method of characteristics solution framework and achieve direct reuse of existing neutral particle transport procedures.
[0008] Therefore, there is an urgent need for a method that can transform the electron transport problem into a form suitable for solving using the method of characteristics while maintaining the physical rationality of the continuous slowing approximation, so as to achieve efficient and stable deterministic calculation of the electron transport process and meet the engineering application requirements under complex geometric conditions. Summary of the Invention
[0009] To address the problems existing in the prior art, this invention provides a deterministic electron transport method applicable to unstructured geometry solutions. The objective of this invention is to provide a characteristic line deterministic electron transport calculation method based on the continuous slowing approximation. By discretizing the energy of the continuous slowing approximation operator and constructing multi-group pseudo-section data, the electron transport problem is formally equivalent to a multi-group neutral particle transport problem. This allows for deterministic calculation of electron transport without modifying the core algorithm of existing neutral particle characteristic line method solvers, avoiding statistical noise in Monte Carlo methods, improving the stability and engineering applicability of the calculation results, and making it suitable for electron transport analysis and multi-particle coupled transport calculations under complex geometric conditions.
[0010] To achieve the above objectives, the present invention adopts the following technical solution: A deterministic method for calculating electron transport based on the characteristic line of the continuous slowing approximation includes the following steps: Step 1: Establishment of the electron transport model: The electron transport equation containing the continuous moderation approximation operator, namely the BFP equation, is used to describe the transport process of electrons in the medium. The continuous moderation approximation operator is used to characterize the continuous energy loss effect of electrons in the inelastic soft collision process. Step 2: Energy discretization of the continuous slowing approximation operator: In the energy dimension, the continuous slowing approximation operator is discretized, and the continuous slowing approximation operator is equivalently transformed into the particle removal term of the corresponding energy group and the downscattering source term from the high energy group to the corresponding energy group, thereby reconstructing the multi-group electron transport equation containing the continuous slowing approximation operator into the form of the standard multi-group Boltzmann transport equation; Step 3: Construction of multi-group electron pseudo-sections: Based on the energy discretization results in Step 2, construct multi-group pseudo-section data to describe the electron transport process, so that the electron transport process is formally equivalent to the multi-group neutral particle transport process. Step 4: Deterministic solution of the characteristic line: The multi-group pseudo-section data is introduced into the existing neutral particle characteristic line method transport solution framework. Without modifying the core algorithm of the transport solution framework, the reconstructed multi-group electron transport equation is solved by integral along the characteristic line to realize the deterministic calculation of the electron transport process.
[0011] Preferably, the establishment of the electron transport model in step 1 specifically involves: using an electron transport equation containing a continuous slowing approximation operator to describe the electron transport process in the medium; for inelastic collisions in the interaction between electrons and matter, classification is based on the magnitude of electron energy loss: inelastic collisions in which the scattered electron energy still falls into the adjacent energy group are defined as soft collisions, and inelastic collisions in which the scattered electron energy crosses the adjacent energy group and falls into a lower energy group are defined as hard collisions; wherein, the soft collisions are processed using a continuous slowing approximation operator, and the hard collisions are processed using the Boltzmann scattering operator, thereby establishing an electron transport model that can reflect the continuous energy loss characteristics and angular distribution characteristics of electrons.
[0012] Preferably, the energy discretization process of the continuous slowing approximation operator in step 2 is specifically as follows: the energy discretization process of the continuous slowing approximation operator adopts the diamond difference approximation, assuming that the flux in each energy group changes linearly with respect to energy, and gradually eliminates the flux at the energy group boundary in the continuous slowing term by using the flux at the highest energy point as 0 (in order to ensure particle conservation), and uses the group flux to represent the continuous slowing approximation operator. The continuous slowing approximation operator is represented by a particle removal term and a downscattering source term.
[0013] Preferably, the construction of the multi-group electron pseudo-cross section in step 3 specifically involves: constructing multi-group pseudo-cross section data based on the energy discretization results in step 2 using the constrained blocking ability at the energy group boundary. The multi-group pseudo-cross section data includes the equivalent removal cross section of electrons in each energy group and the downscattering cross section matrix describing the migration of electrons from high-energy groups to low-energy groups. The multi-group pseudo-cross section data is used to equivalently represent the energy loss effect of the continuous slowing approximation operator during the transport solution process.
[0014] Preferably, the deterministic solution of the characteristic line in step 4 specifically involves: introducing the multi-group pseudo-section data into the existing neutral particle characteristic line method transport solution framework for electron transport calculation. The continuous slowing approximation operator only introduces particle removal and downscattering source terms in the energy dimension, without introducing angle change terms. The electron transport equation is consistent with the corresponding form in the neutral particle characteristic line method transport solution framework in spatial discretization, angle discretization, and integral form along the characteristic line. Thus, a deterministic solution of the electron transport process is achieved without changing the numerical solution structure and geometric processing method of the characteristic line method transport solution framework.
[0015] Compared with the prior art, the present invention has the following outstanding advantages: 1. This invention models the electron transport process based on the continuous slowing approximation. By combining multi-group discretization with the method of characteristics, it avoids the unavoidable statistical noise problem in the Monte Carlo method and can provide stable and repeatable electron transport calculation results, meeting the deterministic requirements in engineering analysis.
[0016] 2. By transforming the continuous slowing approximation operator into an energy group shift-out term and a downscattering source term, and embedding it into a multi-group Boltzmann framework in the form of a pseudo-section, this invention can realize electron transport calculation without modifying the core algorithm of the existing neutral particle transport solver, which significantly reduces the engineering implementation and code development costs.
[0017] 3. The method of this invention employs the method of characteristics for spatial and angular solutions. In terms of spatial discretization, angular discretization, and integration along characteristic lines, it fully inherits the algorithmic structure and geometric processing capabilities of existing neutral particle characteristic line method transport solvers. Theoretically, it is applicable to any complex geometric model supported by the method of characteristics. Compared to traditional deterministic methods for electron transport based on the discrete ordinate method, the method of this invention has better adaptability to complex geometric structures, is suitable for complex model analysis at the engineering scale, and is easily integrated with existing neutron and photon transport analysis workflows. Attached Figure Description
[0018] Figure 1 This is the overall flowchart of the method of the present invention.
[0019] Figure 2 This is a schematic diagram of the electron swarm flux distribution calculated by the method of the present invention in a two-dimensional homogeneous medium model. Detailed Implementation
[0020] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.
[0021] This invention proposes a characteristic line deterministic electronic transport calculation method based on the continuous slowing-down approximation, the flowchart of which is shown below. Figure 1 The specific steps are as follows: Step 1: Considering the characteristics of electrons in matter—short mean free path and a high forward peak in angular distribution—an electron transport equation incorporating a continuous moderation approximation operator is used to describe the electron transport process in the medium. Inelastic electron collisions are classified according to the magnitude of energy loss: soft collisions are defined as those where the scattered electron energy still falls into an adjacent energy group, while hard collisions are defined as those where the scattered electron energy crosses an adjacent energy group and falls into a lower energy group. Soft collisions are handled using the continuous moderation approximation operator, and hard collisions are handled using the Boltzmann collision operator, thus establishing an electron transport model that reflects the continuous energy loss and angular distribution characteristics of electrons.
[0022] Step 2: Discretize the continuous moderating approximation operator contained in the electron transport equation of Step 1 in the energy dimension. Using the rhombus difference approximation method, approximate the continuous moderating approximation operator within each energy group. Assuming that the flux within each energy group changes linearly with energy, gradually eliminate the flux at the energy group boundary in the continuous moderating term by using the flux at the highest energy point as 0 (to ensure particle conservation). Represent the continuous moderating approximation operator using the group flux, thus transforming the continuous moderating approximation operator into an equivalent particle removal term for the corresponding energy group and a single downward scattering source term from the high-energy group. Through this discretization, the electron transport equation containing the continuous moderating approximation operator is reconstructed into the standard multi-group Boltzmann transport equation form, with only a unidirectional coupling relationship from the high-energy group to the low-energy group in the energy dimension.
[0023] Step 3: Based on the energy discretization results of the continuous slowing approximation operator in Step 2, and using the constrained stopping power data at the energy group boundaries calculated by the multi-group electron cross-section generation program, construct multi-group pseudo-cross-section data to describe the soft collision effect. The pseudo-cross-sections include the equivalent electron exit cross-sections in each energy group and the downscattering cross-section matrix describing the electron migration from high-energy groups to low-energy groups, used to equivalently characterize the physical behavior of the continuous slowing process within the multi-group framework. The pseudo-cross-sections are then superimposed with the total cross-section and scattering matrix calculated by the multi-group electron cross-section generation program to obtain an equivalent multi-group cross-section database for subsequent transport calculations.
[0024] Step 4: Introduce the multi-group pseudo-section data constructed in Step 3 into an existing neutral particle characteristic line method transport solution framework (e.g., a photon transport solver based on the characteristic line method) for electron transport calculation. The continuous slowing approximation operator does not introduce an angle change term, and the angle integral form of the electron along each characteristic line remains unchanged. Without altering the core algorithm structure of this solution framework, integrate the reconstructed multi-group electron transport equations along the characteristic lines, and scan each energy group in descending energy order to achieve a deterministic solution for the electron transport process.
[0025] It should be noted that this invention does not depend on a specific software implementation. Any neutral particle transport program with the ability to solve transport problems using the method of characteristics can be used to implement the method of this invention.
[0026] Figure 2 This diagram illustrates the results of calculating the electron group flux distribution in a two-dimensional homogeneous medium model using the characteristic line method and the discrete ordinate method of this invention. This example verifies the feasibility and stability of the proposed method for deterministic calculations of electron transport under the continuous slowing approximation. The results show that, under fully symmetric boundary conditions, the proposed method can stably provide a reasonable electron group flux distribution, and its spatial distribution trend is consistent with the calculation results of the reference deterministic method. The calculation model is a two-dimensional homogeneous nitrogen element with geometric dimensions of 1 cm × 1 cm and a medium element density of 7.034639 × 10⁻⁶. - ² n / barn / cm, the electron source is set as a monoenergetic electron source.
[0027] In this embodiment, multi-group pseudo-section data based on continuous slowing approximate discrete construction is introduced into the existing neutral particle characteristic line method transport solution framework, so as to realize deterministic calculation of the electron transport process without changing the core algorithm structure of the original solver.
[0028] Figure 2 The results verify the feasibility of constructing multiple pseudo-sections and combining them with the method of characteristics to solve the problem, and to reuse the existing neutral particle method of characteristics transport procedure to carry out deterministic calculations of electron transport under the continuous slowing approximation condition. This provides an effective technical approach for realizing engineering calculations of electron transport under complex geometric conditions.
[0029] It should be noted that although this embodiment uses a two-dimensional uniform flat plate model to verify the method of the present invention, since the method of the present invention only introduces the discrete processing of the continuous slowing approximation operator in the energy dimension and does not make any modification to the solution process of the method of characteristics in the spatial and angular dimensions, the method of the present invention is not sensitive to geometric complexity. Its applicability to complex geometric models is directly inherited from the reused neutral particle method of characteristics transport solver. The method of the present invention is also applicable to geometric models containing multi-material regions, irregular interfaces, or complex engineering structures.
Claims
1. A method for calculating characteristic lines of deterministic electron transport based on the continuous slowing-down approximation, characterized in that, Includes the following steps: Step 1: Establishment of the electron transport model: The electron transport equation containing the continuous moderation approximation operator, namely the BFP equation, is used to describe the transport process of electrons in the medium. The continuous moderation approximation operator is used to characterize the continuous energy loss effect of electrons in the inelastic soft collision process. Step 2: Energy discretization of the continuous slowing approximation operator: In the energy dimension, the continuous slowing approximation operator is discretized, and the continuous slowing approximation operator is equivalently transformed into the particle removal term of the corresponding energy group and the downscattering source term from the high energy group to the corresponding energy group, thereby reconstructing the multi-group electron transport equation containing the continuous slowing approximation operator into the form of the standard multi-group Boltzmann transport equation; Step 3: Construction of multi-group electron pseudo-sections: Based on the energy discretization results in Step 2, multi-group pseudo-section data are constructed to describe the electron transport process, so that the electron transport process is formally equivalent to the multi-group neutral particle transport process. Step 4: Deterministic solution of the characteristic line: The multi-group pseudo-section data is introduced into the existing neutral particle characteristic line method transport solution framework. Without modifying the core algorithm of the transport solution framework, the reconstructed multi-group electron transport equation is solved by integral along the characteristic line to realize the deterministic calculation of the electron transport process.
2. The method according to claim 1, characterized in that, The establishment of the electron transport model in step 1 is specifically as follows: an electron transport equation containing a continuous slowing approximation operator is used to describe the electron transport process in the medium. For inelastic collisions in the interaction between electrons and matter, they are classified according to the magnitude of electron energy loss: inelastic collisions in which the scattered electron energy still falls into the adjacent energy group are defined as soft collisions, and inelastic collisions in which the scattered electron energy crosses the adjacent energy group and falls into a lower energy group are defined as hard collisions. The soft collisions are processed using the continuous slowing approximation operator, and the hard collisions are processed using the Boltzmann scattering operator, thereby establishing an electron transport model that can reflect the continuous energy loss characteristics and angular distribution characteristics of electrons.
3. The method according to claim 1, characterized in that, The energy discretization process of the continuous slowing approximation operator in step 2 is as follows: the energy discretization process of the continuous slowing approximation operator adopts the diamond difference approximation. It is assumed that the flux in each energy group changes linearly with respect to energy. The flux at the energy group boundary in the continuous slowing term is gradually eliminated by using the flux at the highest energy point as 0. The continuous slowing approximation operator is represented by the group flux. The continuous slowing approximation operator is represented by a particle removal term and a downscattering source term.
4. The method according to claim 1, characterized in that, The construction of the multi-group electron pseudo-cross section in step 3 is as follows: Based on the energy discretization results in step 2, multi-group pseudo-cross section data is constructed using the restricted blocking ability at the energy group boundary. The multi-group pseudo-cross section data includes the equivalent removal cross section of electrons in each energy group and the downscattering cross section matrix describing the migration of electrons from high energy groups to low energy groups. The multi-group pseudo-cross section data is used to equivalently represent the energy loss effect of the continuous slowing approximation operator in the transport solution process.
5. The method according to claim 1, characterized in that, In step 4, the deterministic solution of the characteristic line is specifically as follows: the multi-group pseudo-section data is introduced into the existing neutral particle characteristic line method transport solution framework for electron transport calculation. The continuous slowing approximation operator only introduces the particle removal term and the downscattering source term in the energy dimension, without introducing the angle change term. The electron transport equation is consistent with the corresponding form in the neutral particle characteristic line method transport solution framework in spatial discretization, angle discretization, and integral form along the characteristic line. Thus, the deterministic solution of the electron transport process is achieved without changing the numerical solution structure and geometric processing method of the characteristic line method transport solution framework.