A discrete unified gas kinetic simulation method and system for particle transport

Abstract

The present invention belongs to the technical field related to cross-scale particle transport simulation. It discloses a discrete unified gas kinetic simulation method and system for particle transport, especially for simulations with complex non-conservative collision operators. The method uses a universal compact reconstruction method to achieve high-precision compact reconstruction of the distribution function at the characteristic line in the compact space of a single grid, and effectively processes the complex non-conservative collision implicit terms of particle transport under time trapezoidal integrals through a universal prediction-correction method. It overcomes the application limitations of existing discrete unified gas kinetic methods for transport models with complex non-conservative collision term operators, or the shortcomings of existing iterative methods, time-splitting-discrete unified gas kinetic methods, such as low computational efficiency, error accumulation, and numerical instability. It also overcomes the problem of relatively low computational accuracy of existing interface reconstruction methods based on multiple adjacent grids when solving particle transport problems with complex material arrangements or strong inhomogeneity.

Description

Technical Field

[0001] The present invention belongs to the technical field related to cross-scale particle transport simulation, and more specifically, relates to a discrete unified gas kinetic simulation method and system for particle transport, especially suitable for particle transport simulation with complex non-conservative collision operators. Background Art

[0002] Particle transport is widely present in many engineering application fields, including gas kinetics, microelectronics systems, and energetic particles. For example, in the aerospace field, there is cross-scale non-equilibrium gas transport in the re-entry aerodynamics of space vehicles, cross-scale neutron transport in advanced nuclear power systems, phonon heat transport in semiconductor chip heat dissipation, and thermal radiation transport in energy storage systems.

[0003] The discrete unified gas kinetic method (DUT) is a relatively successful cross-scale particle transport solution developed in recent years, featuring "asymptotic preservation" properties. It follows the discrete angle concept of discrete ordinates and employs a trapezoidal integral solution of characteristic lines to construct particle transport at the grid interface during the particle transport evolution process. It also introduces an auxiliary distribution function to simplify the calculation of collision implicit terms. Its coupling considers the free migration and collision effects of particles, so its grid size and time step are not restricted by relaxation time. Currently, the DUT, as an important method for solving multiscale particle transport problems, has garnered attention in various particle transport fields, including multiscale gas molecule transport, radiation transport, and phonon transport, and has been initially applied and developed in some simplified particle transport models.

[0004] Currently, discrete unified gas kinetic methods are primarily applied to simplified particle transport models with conservative collision operators, such as simplified single-molecule gas transport and gray-body phonon transport models. This is primarily due to the fact that, in existing discrete unified gas kinetic evolution processes, implicit collision terms are simplified by tracking auxiliary particle distribution functions under time trapezoidal integration. The true macroscopic quantities are directly obtained by momenting the auxiliary distribution functions, resulting in consistency between tracking the auxiliary distribution functions and the true particle distribution. However, for transport models with non-conservative collision operators, macroscopic quantities cannot be directly obtained from the auxiliary distribution functions.

[0005] For some specific, simple non-conservative collision terms, the true macroscopic quantities can be analytically solved by additionally considering the relationship between the auxiliary distribution function and the original distribution function. For complex non-conservative collision terms, such as those with external forces in non-equilibrium gas transport models and those with energy spectrum effects in energetic particle transport models, one approach is to use an iterative method to recover the original distribution function or true macroscopic quantity. This approach significantly increases computational cost, thereby reducing computational efficiency and increasing the implementation complexity of the discrete unified gas kinetics method. Another approach is to use a time-splitting method to split the non-conservative collision terms in particle transport models into a conservative and a non-conservative part. The conservative part still uses the standard discrete unified gas kinetics method, while the non-conservative part is usually treated using a simple time-splitting method. This time-splitting method can lead to large error accumulation and numerical instability, especially in strongly nonlinear systems, often reducing computational accuracy and convergence stability.

[0006] In general, existing discrete unified gas kinetic simulation methods for particle transport cannot be directly applied to transport models with non-conservative collision operators or suffer from shortcomings such as low computational efficiency, error accumulation, and numerical instability. In addition, for the reconstruction of interface distribution functions based on characteristic lines, existing discrete unified gas kinetic methods generally reconstruct the interface distribution function based on the distribution functions at the centers of multiple adjacent grids, such as the widely used central difference and flux limiter reconstruction methods based on the distribution functions at the centers of three adjacent grids. For particle transport problems with strong heterogeneity and inhomogeneity, this reconstruction method based on multiple adjacent grids has low computational accuracy. Summary of the Invention

[0007] In response to the above-mentioned defects or improvement needs of the prior art, the present invention provides a discrete unified gas kinetic simulation method and system for particle transport, which is used to solve the problems that the existing discrete unified gas kinetic simulation solution method for particle transport cannot be directly applied to transport models with non-conservative collision operators or has low computational efficiency, error accumulation and numerical instability.

[0008] To achieve the above objectives, according to one aspect of the present invention, a discrete unified gas kinetic simulation method for particle transport is provided, comprising:

[0009] Perform geometric modeling and meshing of the particle transport system to determine the computational domain. Establish a particle transport model for the computational domain and solve the particle transport model based on the discrete unified gas kinetics method to update the computational domain and implement particle transport simulation. This includes:

[0010] S1: Based on the finite volume method, the particle transport model is discretized in time and space to obtain the space-time discrete equation at the center of the grid. At the same time, the particle transport model is integrated along the characteristic line at half time step to obtain the characteristic line integral discrete equation at the grid interface.

[0011] S2: Reconstruct the distribution function at the characteristic line in the characteristic line integral discrete equation;

[0012] S3: Based on the characteristic line integral discrete equation and the reconstructed characteristic line distribution function, the implicit term in the characteristic line integral discrete equation is estimated by the interface distribution function at the old time step to obtain the estimated interface distribution function; then, the estimated interface distribution function is used to calculate the estimated value of the implicit term, and the estimated value is brought into the characteristic line integral discrete equation to obtain the corrected interface distribution function;

[0013] S4: Based on the space-time discrete equation and the corrected distribution function at the interface, the distribution function at the center of the grid corrected in the space-time discrete equation is obtained through the same estimation and correction calculation;

[0014] S5: Based on the corrected distribution function at the interface and the corrected distribution function at the grid center, the computational domain is updated and simulated, and it is determined whether the iteration termination condition is met. If so, the simulation is completed; otherwise, the process returns to S2.

[0015] According to the discrete unified gas kinetic simulation method for particle transport provided by the present invention, the particle transport model is:

[0016] ;

[0017] in, is the particle distribution function, is the spatial position, is the particle discrete direction, For time, is the relaxation time; is the collision term of the particle transport model; is the gradient operator; is the source term of the particle transport model, which is the macroscopic quantity transported by the particle Composition, macroscopic It is calculated by moment calculation of particle distribution function;

[0018] The space-time discrete equation is:

[0019] ;

[0020] ;

[0021] in, Grid Center The distribution function is at ; the discrete time step is , Indicates the n Iteration time steps; For Grid The grid volume, F is the grid interface microflux; is the grid unit normal vector; is the mesh interface area; is the grid interface position; for Distribution function at the interface at time; is the grid interface area differential;

[0022] The characteristic line integral discrete equation is:

[0023] ;

[0024] in, represents the distribution function at the interface; , represents half a time step; represents the distribution function at the characteristic line; represents the collision term at the interface; Represents the collision term at the feature line.

[0025] According to the discrete unified gas kinetic simulation method for particle transport provided by the present invention, the distribution function at the characteristic line in S2 The reconstruction is as follows:

[0026] ;

[0027] in, For the i The grid center of the grid Distribution function at is the corresponding grid slope; is the grid interface position; Indicates the n Iteration time steps; is the particle discrete direction; Represents half a time step.

[0028] According to the discrete unified gas kinetic simulation method for particle transport provided by the present invention, S2 specifically includes:

[0029] The left and right compact slopes are constructed based on the distribution functions of a single grid at the center and at the interface;

[0030] Based on the compact slope on the left and the compact slope on the right, the grid slope is calculated;

[0031] The distribution function at the characteristic line is reconstructed based on the grid slope.

[0032] According to the discrete unified gas kinetic simulation method for particle transport provided by the present invention, the left compact slope and the right compact slope are specifically:

[0033] ;

[0034] in, is the left-side compact slope, is the right-side compact slope, Subscript i Indicates the i grids, Indicates the i The center position of the grid; is the particle distribution function; is the grid length.

[0035] According to the discrete unified gas kinetic simulation method for particle transport provided by the present invention, S3 specifically includes:

[0036] Based on the characteristic line integral discrete equation and the reconstructed characteristic line distribution function, the implicit collision term or implicit source term at the interface in the characteristic line integral discrete equation is estimated by the interface distribution function at the old time step to obtain the estimated interface distribution function;

[0037] Obtaining an estimated collision term or an estimated source term at the interface based on the estimated distribution function at the interface;

[0038] Based on the characteristic line integral discrete equation and the reconstructed distribution function at the characteristic line, the estimated collision term or the estimated source term at the interface is substituted to obtain the corrected distribution function at the interface.

[0039] According to the discrete unified gas kinetic simulation method of particle transport provided by the present invention, the source term in the implicit collision term at the interface is estimated to obtain the estimated distribution function at the interface , as follows:

[0040] ;

[0041] Or by comprehensively estimating the implicit collision term at the interface, we can obtain the estimated distribution function at the interface. , as follows:

[0042] .

[0043] According to the discrete unified gas kinetic simulation method of particle transport provided by the present invention, based on the estimated interface distribution function Calculate the estimated collision term or estimated source term at the interface as follows:

[0044] ;

[0045] ;

[0046] ;

[0047] in, is the estimated macroscopic quantity at the interface, which is usually calculated by taking the moment of the distribution function, where is a moment vector, which can represent the higher-order moments of the distribution function and is determined by the specific transport model; is the estimated source term at the interface, which can usually be calculated based on macroscopic quantities; is the estimated collision term at the interface.

[0048] According to the discrete unified gas kinetic simulation method for particle transport provided by the present invention, S4 specifically includes:

[0049] Based on the space-time discrete equation and the corrected interface distribution function, the implicit collision term or implicit source term at the grid center in the space-time discrete equation is estimated by the grid center distribution function at the old time step to obtain the estimated grid center distribution function;

[0050] Obtaining an estimated collision term or an estimated source term at the center of the grid based on the estimated distribution function at the center of the grid;

[0051] Based on the space-time discrete equation and the corrected distribution function at the interface, the estimated collision term or the estimated source term at the grid center is substituted to obtain the corrected distribution function at the grid center.

[0052] According to another aspect of the present invention, a discrete unified gas kinetic simulation system for particle transport is provided, the system comprising a memory and a processor, the memory storing a computer program, and the processor executing any one of the above-described discrete unified gas kinetic simulation methods for particle transport when executing the computer program.

[0053] In general, compared with the prior art, the above technical solutions conceived by the present invention provide a discrete unified gas kinetic simulation method and system for particle transport:

[0054] 1. A universal prediction-correction calculation process is proposed for solving the distribution function at the interface and the distribution function at the grid center in the particle transport model. That is, the implicit terms in the model are first estimated and calculated to estimate the distribution function. The estimated distribution function is then used to calculate the estimated implicit terms, and the corrected distribution function is obtained using the estimated implicit terms. Unlike the existing discrete unified gas kinetic solution method, the present invention can effectively handle the non-conservative collision implicit terms under the time trapezoidal integral through a universal prediction-correction method, overcoming the application limitations of the existing discrete unified gas kinetic method for complex non-conservative collision term operator transport models or alleviating the shortcomings of existing iterative methods, time splitting-discrete unified gas kinetic methods, such as low computational efficiency, error accumulation, and numerical instability. It is applicable to the solution and simulation of cross-scale particle transport models with complex non-conservative collision operators, meeting the high-precision simulation requirements of complex and real cross-scale particle transport engineering problems.

[0055] 2. A general method for compact reconstruction of grid slopes is disclosed. This method achieves high-precision compact reconstruction of distribution functions at characteristic lines within the compact space of a single grid. This overcomes the problem of low computational accuracy in particle transport problems with strong heterogeneity or inhomogeneity, which is often encountered in existing discrete unified gas kinetics solutions, such as reconstruction methods based on multiple adjacent grid points, such as the central difference method based on the distribution function of the centers of three adjacent discrete grid points or various flux limiter methods.

[0056] 3. The present invention has good universality and can adapt to the simulation of various particle transport models with complex non-conservative collision term operators, such as the simulation of multi-group neutron transport models, the simulation of multi-component gas molecular transport models, the simulation of photon radiation transport, and the simulation of phonon transport. At the same time, the compact reconstruction format of the present invention has stronger grid size independence because it is only implemented in the compact space within a single grid. BRIEF DESCRIPTION OF THE DRAWINGS

[0057] Figure 1 This is a flow chart of the discrete unified gas kinetic simulation method for particle transport provided by the present invention.

[0058] Figure 2 Schematic diagram of the slope construction of the compact reconstruction method provided by the present invention.

[0059] Figure 3 This is a schematic diagram of the International Atomic Energy Agency (IAEA) benchmark calculation example of multi-group neutron transport embodiment 1 provided by the present invention.

[0060] Figure 4 This is an accuracy diagram of the compact reconstruction method of Example 1, in which the present invention is applied to multi-group neutron transport.

[0061] Figure 5It is a schematic diagram of the calculation results of Example 1 of the present invention applied to multi-group neutron transport.

[0062] Figure 6 This is a schematic diagram of a benchmark calculation example of a rectangular core (TWIGL) for multi-group neutron transport embodiment 2 provided by the present invention.

[0063] Figure 7 This is a comparison diagram of Example 2 of the present invention applied to multi-group neutron transport and the time splitting method.

[0064] Figure 8 It is a schematic diagram of the calculation results of Example 2 of the present invention applied to multi-group neutron transport. DETAILED DESCRIPTION

[0065] In order to make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely for the purpose of explaining the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.

[0066] See also Figure 1 This embodiment provides a discrete unified gas kinetic simulation method for particle transport, which includes:

[0067] Perform geometric modeling and meshing of the particle transport system to determine the computational domain; establish a particle transport model for the computational domain, and solve the particle transport model based on the discrete unified gas kinetic method disclosed in the present invention to update the computational domain to achieve particle transport simulation; specifically, the following steps are included:

[0068] S1: Based on the finite volume method, the particle transport model is discretized in time and space to obtain the space-time discrete equation at the center of the grid. At the same time, in order to effectively reconstruct the distribution function at the grid interface, the particle transport model is integrated along the characteristic line at half the time step to obtain the characteristic line integral discrete equation at the grid interface.

[0069] S2: Reconstruct the distribution function at the characteristic line in the characteristic line integral discrete equation in S1;

[0070] S3: Based on the characteristic line integral discrete equation and the characteristic line distribution function reconstructed in S2, the characteristic line integral discrete equation of the particle transport model is estimated and corrected. First, the estimation step is solved, and the implicit term in the characteristic line integral discrete equation is explicitly calculated using the interface distribution function under the old time step to obtain the estimated interface distribution function; then the correction step is solved, and the estimated interface distribution function is used to calculate the estimated value of the implicit term, and then it is again brought into the characteristic line integral discrete equation to obtain the corrected interface distribution function, completing the correction calculation step;

[0071] S4: Based on the space-time discrete equation and the distribution function at the interface corrected in S3, the space-time discrete equation of the particle transport model is also estimated and corrected to obtain the distribution function at the grid center corrected in the space-time discrete equation;

[0072] S5: Based on the corrected distribution function at the interface and the corrected distribution function at the grid center, the computational domain is updated and simulated, and it is determined whether the iteration termination condition is met. If so, the simulation is completed; otherwise, the process returns to S2.

[0073] The purpose of this embodiment is to achieve discrete unified gas kinetic evolution simulation of particle transport, especially transport models with complex non-conservative collision term operators, and to improve the accuracy of distribution function reconstruction at the interface to meet the high-precision simulation requirements of complex real engineering problems of cross-scale particle transport. The method flow chart of this embodiment is as follows: Figure 1 The specific steps and implementation process are as follows:

[0074] S1: Spatiotemporal Discretization of Particle Transport Equations: The particle transport model is discretized in spatiotemporal space based on the finite volume method to obtain the spatiotemporal discrete equations at the center of the grid. The particle transport model includes transport terms, namely the distribution function, collision terms, and source terms. The time integral of the transport terms uses the midpoint rule, and the time integral of the collision terms uses the trapezoidal rule. At the same time, in order to reconstruct the distribution function at the control volume grid interface, the particle transport equation is integrated along a specific characteristic line at the interface at half a time step to obtain the characteristic line integral discrete equation at the grid interface. The time integral of the collision term also uses the trapezoidal rule.

[0075] Specifically, the particle transport Boltzmann equation that needs to be solved is the particle transport model:

[0076] ;

[0077] in, is the particle distribution function, is the spatial position, is the particle discrete direction, For time, is the relaxation time; is the collision term of the particle transport model; is the gradient operator; is the source term of the particle transport model, which is the macroscopic quantity transported by the particle Composition, macroscopic It is obtained by calculating the moment of the particle distribution function; by modifying the specific equation model, Equation (1) can represent various Boltzmann-type particle transport models such as multi-group neutron transport model, multi-component gas molecule transport model, thermal radiation transport, and phonon transport.

[0078] The particle transport model is discretized in time and space based on the finite volume method. The centered grid from time arrive Integrate, where the time integral of the transport term uses the midpoint rule, and the time integral of the collision term uses the trapezoidal rule. The space-time discrete equation at the center of the grid is obtained as:

[0079] ;

[0080] ;

[0081] in, Grid Center The distribution function is at ; the discrete time step is , Indicates the n Iteration time steps; For Grid The grid volume, F is the grid interface microflux; is the grid unit normal vector; is the mesh interface area; is the grid interface position; for Distribution function at the interface at time; is the mesh interface area differential.

[0082] It can be seen that the space-time discrete equation of the above particle transport model includes the distribution function at the grid center, the distribution function at the interface, the implicit term of the collision term at the grid center, and the explicit term of the collision term. The implicit term is the quantity corresponding to the unknown time step, and the explicit term is the quantity corresponding to the known time step, that is, the quantity corresponding to the old time step. The key to solving equation (2) lies in the distribution function at the interface. The reconstruction of the implicit collision term on the right side of equation (2) This is also the two key points of the specific implementation of the present invention. In order to realize the distribution function at the interface The particle transport equation (1) is discretized along the characteristic line at the interface. Direction at half time step The time integral of the collision term is also based on the trapezoidal rule.

[0083] The space-time discrete equation of the characteristic line at the particle transport interface, that is, the characteristic line integral discrete equation, is:

[0084] ;

[0085] in, represents the distribution function at the interface; , represents half a time step; represents the distribution function at the characteristic line; represents the collision term at the interface; Represents the collision term at the feature line.

[0086] For transport models with non-conservative collision term operators, such as multi-group neutron transport models with complex non-conservative collision terms such as fission terms, energy group scattering terms, and absorption terms, and non-equilibrium gas transport models with external force terms, it is impossible to treat the implicit collision terms in the above-mentioned space-time discrete equations (2) and (4) by introducing auxiliary distribution functions as in the existing standard discrete unified gas kinetic method. Specifically, the real macroscopic quantity cannot be obtained directly by calculating the moment of the auxiliary distribution function, that is:

[0087] ;

[0088] However, the use of iterative methods, time splitting and other methods may have shortcomings such as low computational efficiency, error accumulation and numerical instability. Therefore, the present invention discloses a unified gas kinetic method for particle transport, which is particularly suitable for prediction-correction-compactness-discretization of particle transport models with complex non-conservative collision operators.

[0089] S2, compactly reconstructs the distribution function at the characteristic line in the characteristic line integral discrete equation in S1; the present invention discloses a universal compact reconstruction method, the slope of which is constructed based on the distribution function at the interface / center of a single grid to achieve higher-precision spatial discrete reconstruction, thereby improving the reconstruction accuracy of the distribution function at the interface.

[0090] Specifically, to solve equation (4), we first need to reconstruct the distribution function at the characteristic line , for a certain spatial discrete direction, the distribution function at the characteristic line The reconstruction is as follows:

[0091] ;

[0092] in, For the i The grid center of the grid Distribution function at is the corresponding grid slope; is the grid interface position; Indicates the n Iteration time steps; is the particle discrete direction; Represents half a time step. Subscript i Indicates the i grid, indicating the i The center position of the grid.

[0093] Among them, the grid slope is obtained by compact reconstruction based on the distribution function of a single grid at the center and at the interface. The grid slope can be specifically calculated based on the compact slope on the left and the compact slope on the right using one of the central difference method, the flux limiter method proposed by Van Leer (Van Leer), the flux limiter method proposed by Van Albada (Van Albada), the flux limiter method proposed by Waterson and Deconinck (OSPRE), the essential non-oscillation (ENO) and other flux limiter methods and the weighted essential non-oscillation (WENO) discretization method. The compact slope on the left and the compact slope on the right are constructed based on the distribution function of a single grid at the center and at the interface.

[0094] Existing discrete unified gas kinetics generally uses the distribution function based on the center of multiple adjacent grids to reconstruct the grid slope. ; Such as the central difference method of three adjacent grids, various flux limiter methods and discrete methods. Taking the three-point central difference and the flux limiter method proposed by Van Leer as examples, the existing grid slope The construction can be expressed as:

[0095] ;

[0096] ;

[0097] in 、 is the slope of the distribution function at the center of the adjacent grid, as shown in the schematic diagram Figure 2 As shown, the specific form is:

[0098] ;

[0099] The simplified form omits the time term. The above existing reconstruction scheme based on multiple adjacent grids can be regarded as the slope of the distribution function of three adjacent grids. 、 It can be seen that the above slope reconstruction spans three adjacent grids. For complex material arrangements and highly non-uniform particle transport problems, this reconstruction scheme based on multiple adjacent grids has low accuracy.

[0100] Based on this, this embodiment discloses a general compact reconstruction method. The slope of this compact reconstruction method is constructed based on the distribution function at the interface / center of a single grid to achieve higher-precision spatial discrete reconstruction, thereby improving the reconstruction accuracy of the distribution function at the interface. S2 specifically includes:

[0101] The left and right compact slopes are constructed based on the distribution functions of a single grid at the center and at the interface;

[0102] Based on the compact slope on the left and the compact slope on the right, the grid slope is calculated;

[0103] The distribution function at the characteristic line is reconstructed based on the grid slope.

[0104] Specifically, the slope selection within a single grid, that is, the left compact slope and the right compact slope are as follows:

[0105] ;

[0106] in, is the left-side compact slope, is the right-side compact slope, Subscript i Indicates the i grids, Indicates the i The center position of the grid; is the particle distribution function; is the grid length; Indicates the i Interfaces in a grid Department; Indicates the i Interfaces in a grid Place.

[0107] The above compact slope diagram is as follows Figure 2 As shown in FIG, the compact reconstruction method of this embodiment is applied to the flux limiter such as the central difference and the flux limiter method proposed by Van Leer, and finally the single grid The distribution function grid slope It can be expressed as:

[0108] ;

[0109] ;

[0110] The above tight slope can be and Constructing tight grid slopes for central difference methods, various flux limiter methods, and discrete methods , several optional compact grid slopes The expression is:

[0111] ;

[0112] ;

[0113] in, represents the compact slope gradient, represents a specific compact weight function. This embodiment can achieve high-precision reconstruction within the compact space of a single grid, overcoming the problem of low computational accuracy in particle transport problems with strong heterogeneity or inhomogeneity, which is often encountered in existing discrete unified gas kinetic methods, which rely on reconstruction methods based on multiple adjacent grid points. This can be conveniently applied to central difference methods, various flux limiter methods, and discrete methods.

[0114] Note that the above compact reconstruction format is merely a specific example of the compact reconstruction method disclosed herein applied to a central difference or flux limiter, and does not constitute a limitation of the present invention. By introducing the compact slope of the distribution function at the interface / center of a single grid, the present invention can be applied to various reconstruction formats, such as central differences, flux limiters, and discrete methods, to achieve high-precision compact reconstruction of the interface distribution function. Furthermore, since the compact reconstruction format of this embodiment is implemented only in the compact space within a single grid, it exhibits enhanced grid size independence.

[0115] S3: Estimation and correction of the characteristic line of the distribution function at the interface. S3 specifically includes:

[0116] Based on the characteristic line integral discrete equation and the reconstructed characteristic line distribution function, the implicit collision term or implicit source term at the interface in the characteristic line integral discrete equation is estimated by the interface distribution function at the old time step to obtain the estimated interface distribution function;

[0117] Obtaining an estimated collision term or an estimated source term at the interface based on the estimated distribution function at the interface;

[0118] Based on the characteristic line integral discrete equation and the reconstructed distribution function at the characteristic line, the estimated collision term or the estimated source term at the interface is substituted to obtain the corrected distribution function at the interface.

[0119] Specifically, S31: the distribution function at the interface is reconstructed based on the estimation of the distribution function at the characteristic line.

[0120] For the particle transport interface characteristic line integral discrete equation (4) that needs to be solved, the characteristic line distribution function is calculated based on the compact reconstruction of the interface distribution function in the S2 step. , first use the estimation step to solve. The implicit collision term or implicit source term at the interface caused by the trapezoidal integral is estimated by the interface distribution function under the old time step, and finally the estimated distribution function at the interface is calculated. Optionally, the estimated distribution function at the interface is obtained by estimating the source term in the implicit collision term at the interface. , as follows:

[0121] ;

[0122] Alternatively, the estimated interface distribution function can be obtained by performing an overall estimation of the implicit collision term at the interface. , as follows:

[0123] .

[0124] Among them, formula (15) is an embodiment of using estimated calculation for the source term in the implicit collision term at the interface, and formula (16) is an embodiment of using estimated calculation for the implicit collision term at the interface.

[0125] S32: Correction and reconstruction of the characteristic line of the distribution function at the interface.

[0126] For the characteristic line integral discrete equation (4) at the particle transport interface to be solved, the implicit collision term or implicit source term at the interface is the estimated distribution function at the interface calculated by the S31 estimation step Calculate and bring it into the characteristic line integral discrete equation again, and finally calculate the correction distribution function at the half time step interface, that is, the distribution function at the interface; based on the estimated distribution function at the interface Calculate the estimated collision term or estimated source term at the interface as follows:

[0127] ;

[0128] ;

[0129] ;

[0130] in, is the estimated macroscopic quantity at the interface, which is usually calculated by taking the moment of the distribution function. is a moment vector, which can represent the higher-order moments of the distribution function and is determined by the specific transport model; is the estimated source term at the interface, which can usually be calculated based on macroscopic quantities; is the estimated collision term at the interface.

[0131] The corrected interface distribution functions obtained using the estimated collision term or the estimated source term are as follows:

[0132] ;

[0133] ;

[0134] Wherein, formula (20) is an embodiment in which the source term in the implicit collision term at the interface is calculated using the estimated distribution function at the interface, and formula (21) is an embodiment in which the implicit collision term at the interface is calculated using the estimated distribution function at the interface. is the corrected distribution function at the interface.

[0135] S4: Based on the distribution function at the interface estimated and corrected in S3, perform estimation and correction calculations on the space-time discrete equation of the particle transport model to obtain the distribution function at the center of the grid. S4 specifically includes:

[0136] Based on the space-time discrete equation and the corrected interface distribution function, the implicit collision term or implicit source term at the grid center in the space-time discrete equation is estimated by the grid center distribution function at the old time step to obtain the estimated grid center distribution function;

[0137] Obtaining an estimated collision term or an estimated source term at the center of the grid based on the estimated distribution function at the center of the grid;

[0138] Based on the space-time discrete equation and the corrected distribution function at the interface, the estimated collision term or the estimated source term at the grid center is substituted to obtain the corrected distribution function at the grid center.

[0139] Specifically, S41: estimated calculation of the distribution function at the center of the grid.

[0140] For the particle transport space-time discrete equation (2) at the center of the grid to be solved, the distribution function at the interface calculated based on the S3 step is , first use the estimation step to solve. The implicit collision term or implicit source term at the grid center is estimated by the distribution function at the grid center of the old time step, thereby calculating the estimated distribution function at the grid center; the calculation process is:

[0141] ;

[0142] ;

[0143] Among them, formula (22) is an embodiment of using estimated calculation for the implicit collision term at the center of the grid, and formula (23) is an embodiment of using estimated calculation for the implicit source term in the implicit collision term at the center of the grid. is the estimated distribution function at the center of the grid. is the collision term at the center of the grid at the old time step. is the source term at the center of the grid at the old time step.

[0144] S42: Calculation of distribution function correction at the center of the grid.

[0145] For the particle transport space-time discrete equation (2) at the grid center to be solved, the implicit collision term or implicit source term at the grid center is calculated by the estimated distribution function at the grid center calculated in the S41 estimation step, and finally the corrected distribution function at the grid center of the new time step is calculated, that is, the distribution function at the grid center of the new time step; the calculation process is:

[0146] ;

[0147] ;

[0148] ;

[0149] ;

[0150] Wherein, formula (26) is an embodiment in which the implicit collision term at the grid center is calculated using the estimated distribution function at the grid center, and formula (27) is an embodiment in which the implicit source term in the implicit collision term at the grid center is calculated using the estimated distribution function at the grid center. is the estimated macroscopic quantity at the center of the grid; is the estimated source term at the grid center; is the estimated collision term at the center of the grid; is the distribution function at the center of the corrected grid.

[0151] S5: Time convergence verification.

[0152] Verify whether the time step is completed or the time advancement iteration converges, calculate the macroscopic quantity of particle transport, and judge whether the convergence condition is met based on the difference of the macroscopic quantity between adjacent time steps, or judge whether the termination condition is met based on the number of iterations. If the convergence condition is not met, continue the time advancement iteration of steps S2 to S5 until the convergence condition is met.

[0153] Furthermore, this embodiment provides a discrete unified gas kinetic simulation system for particle transport, the system comprising a memory and a processor, the memory storing a computer program, and the processor executing any one of the above-described discrete unified gas kinetic simulation methods for particle transport when executing the computer program.

[0154] In some specific embodiments, a prediction-correction-compactness-discrete unified gas kinetic simulation method applicable to a cross-scale particle transport model with complex non-conservative collision operators is provided. The particle transport simulation method specifically includes: geometric modeling and meshing of the particle transport system to determine the computational domain; setting the initial physical field and boundary conditions for the computational domain, and establishing a particle transport model of the computational domain; solving the particle transport model based on the discrete unified gas kinetic method of the above embodiment to update the physical field of the computational domain to realize the simulation of particle transport. Specifically, at each iterative time step, the particle transport model is solved by the discrete unified gas kinetic method to obtain the distribution function at the interface and the distribution function at the grid center, and then the various macroscopic quantities of the physical field are obtained according to the distribution function to realize the update of the physical field. As time progresses, the update simulation of the computational domain of the particle transport system is performed.

[0155] Specifically, the particle transport system can be a cross-scale gas transport system for a spacecraft or a cross-scale multi-group neutron transport system for an advanced nuclear power system. For example, the non-equilibrium gas transport model with external force terms and the multi-group neutron transport model, two particle transport models with non-conservative properties of collision operators, can be written as:

[0156] ;

[0157] The corresponding relationship with equation (1) is: is the specific gas molecule distribution function, , is the Maxwell equilibrium source term of the gas, which can be obtained by macroscopic calculation. is the external force term. The macroscopic quantity of the non-equilibrium gas transport model can be expressed as ,in is the fluid density, is the fluid momentum, For speed, is the fluid energy.

[0158] For the multi-group neutron transport model, it can be specifically written as:

[0159] ;

[0160] ;

[0161] ;

[0162] ;

[0163] ;

[0164] ;

[0165] The corresponding relationship with equation (1) is: is the angular flux density of the grouped neutrons, and its macroscopic quantity is , expressed as the clustered neutron flux density. is the neutron speed, is the total macroscopic cross section of the grouped neutrons, is the neutron scattering source, is a neutron fission source, is other external source of neutrons. for Can be grouped The neutron scattering cross section of the group, is the group neutron fission cross section. The delayed neutron precursor nucleus production share, is the neutron fission spectrum, is the average number of neutrons produced in nuclear fission. is the delayed neutron precursor decay constant, is the concentration of delayed neutron precursor nuclei, i = 1,2, ... , N is the group number of the delayed neutron precursor nucleus, is the delayed neutron grouping fraction. It can be seen that by modifying the specific distribution function and collision term, the above-mentioned Boltzmann equation form (1) can express a variety of cross-scale particle transport models, including the cross-scale non-equilibrium gas transport model and the cross-scale multi-group neutron transport model. Therefore, the above-mentioned embodiment does not serve as a specific limitation on the technology and applications of the present invention.

[0166] The multi-group neutron transport model has complex non-conservative collision operators and generally has strong complex heterogeneous material distribution and non-uniform characteristics in application. Therefore, this embodiment can effectively illustrate the advantages of the present invention. Figure 3 A schematic diagram of Multi-Group Neutron Transport Example 1 - International Atomic Energy Agency (IAEA) benchmark example is shown. Figure 4 The results show a comparison of the accuracy of the compact reconstruction method disclosed in this invention and existing limiters, where the existing flux limiter shown in the figure is a commonly used VanLeer flux limiter. As can be seen, the compact reconstruction format disclosed in this invention can achieve high-precision reconstruction within the compact space of a single grid, overcoming the low computational accuracy of the reconstruction method based on multiple adjacent grid points, which is widely used in existing discrete unified gas kinetic methods, when applied to particle transport problems with strong heterogeneity or inhomogeneity. Figure 5 The calculation results of Example 1 of the present invention applied to multi-group neutron transport are shown.

[0167] Figure 6A schematic diagram of the benchmark calculation example of multi-group neutron transport embodiment 2 - rectangular core (TWIGL) is shown. Figure 7 Comparisons between the present invention and an existing time-splitting method for handling complex non-conservative collision terms, namely the time-splitting-discrete unified gas kinetics method, are presented. The existing time-splitting-discrete unified gas kinetics method exhibits significant error accumulation, often requiring a smaller time step to ensure accuracy, which reduces computational efficiency. This present invention effectively addresses the limitation of existing standard discrete unified gas kinetics methods, which cannot be directly applied to non-conservative collision operators. It also effectively alleviates the shortcomings of the time-splitting-discrete unified gas kinetics method, such as low computational efficiency, error accumulation, and numerical instability. Figure 8 The calculation results of Example 2 of the present invention applied to multi-group neutron transport are shown.

[0168] It will be easily understood by those skilled in the art that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A discrete unified gas kinetic simulation method for particle transport, characterized in that: include: Perform geometric modeling and meshing of the particle transport system to determine the computational domain. Establish a particle transport model for the computational domain and solve the particle transport model based on the discrete unified gas kinetics method to update the computational domain and implement particle transport simulation. This includes: S1: Based on the finite volume method, the particle transport model is discretized in time and space to obtain the space-time discrete equation at the center of the grid. At the same time, the particle transport model is integrated along the characteristic line at half time step to obtain the characteristic line integral discrete equation at the grid interface. S2: Reconstruct the distribution function at the characteristic line in the characteristic line integral discrete equation; S3: Based on the characteristic line integral discrete equation and the reconstructed characteristic line distribution function, the implicit term in the characteristic line integral discrete equation is estimated by the interface distribution function at the old time step to obtain the estimated interface distribution function; then, the estimated interface distribution function is used to calculate the estimated value of the implicit term, and the estimated value is brought into the characteristic line integral discrete equation to obtain the corrected interface distribution function; S4: Based on the space-time discrete equation and the corrected distribution function at the interface, the distribution function at the center of the grid corrected in the space-time discrete equation is obtained through the same estimation and correction calculation; S5: Based on the corrected distribution function at the interface and the corrected distribution function at the grid center, the computational domain is updated and simulated, and it is determined whether the iteration termination condition is met. If so, the simulation is completed; otherwise, the process returns to S2.

2. The discrete unified gas kinetic simulation method for particle transport according to claim 1, characterized in that: The particle transport model is: in, is the particle distribution function, is the spatial position, is the particle discrete direction, For time, is the relaxation time; is the collision term of the particle transport model; is the gradient operator; is the source term of the particle transport model, which is the macroscopic quantity transported by the particle Composition, macroscopic It is calculated by moment calculation of particle distribution function; The space-time discrete equation is: in, Grid Center The distribution function is at ; the discrete time step is , Indicates the n Iteration time steps; For Grid The grid volume, F is the grid interface microflux; is the grid unit normal vector; is the mesh interface area; is the grid interface position; for Distribution function at the interface at time; The characteristic line integral discrete equation is: in, represents the distribution function at the interface; , represents half a time step; represents the distribution function at the characteristic line; represents the collision term at the interface; Represents the collision term at the feature line.

3. The discrete unified gas kinetic simulation method for particle transport according to claim 2, wherein: Distribution function at the characteristic line in S2 The reconstruction is as follows: in, For the i The grid center of the grid Distribution function at is the corresponding grid slope; is the grid interface position; Indicates the n Iteration time steps; is the particle discrete direction; Represents half a time step.

4. The discrete unified gas kinetic simulation method for particle transport according to any one of claims 1 to 3, characterized in that: S2 specifically includes: The left and right compact slopes are constructed based on the distribution functions of a single grid at the center and at the interface; Based on the compact slope on the left and the compact slope on the right, the grid slope is calculated; The distribution function at the characteristic line is reconstructed based on the grid slope.

5. The discrete unified gas kinetic simulation method for particle transport according to claim 4, characterized in that: The left compact slope and the right compact slope are specifically: in, is the left-side compact slope, is the right-side compact slope, Subscript i Indicates the i grids, Indicates the i The center position of the grid; is the particle distribution function; is the grid length, is the particle discrete direction.

6. The discrete unified gas kinetic simulation method for particle transport according to claim 2, wherein: S3 specifically includes: Based on the characteristic line integral discrete equation and the reconstructed characteristic line distribution function, the implicit collision term or implicit source term at the interface in the characteristic line integral discrete equation is estimated by the interface distribution function at the old time step to obtain the estimated interface distribution function; Obtaining an estimated collision term or an estimated source term at the interface based on the estimated distribution function at the interface; Based on the characteristic line integral discrete equation and the reconstructed distribution function at the characteristic line, the estimated collision term or the estimated source term at the interface is substituted to obtain the corrected distribution function at the interface.

7. The discrete unified gas kinetic simulation method for particle transport according to claim 6, characterized in that: By estimating the source term in the implicit collision term at the interface, the estimated distribution function at the interface is obtained , as follows: Or by comprehensively estimating the implicit collision term at the interface, we can obtain the estimated distribution function at the interface , as follows:

8. The discrete unified gas kinetic simulation method for particle transport according to claim 6, wherein: Based on the estimated distribution function at the interface Calculate the estimated collision term or estimated source term at the interface as follows: in, is the estimated macroscopic quantity at the interface, which is calculated by taking the moment of the distribution function, where is a moment vector, representing the higher-order moment of the distribution function, which is determined by the specific transport model; is the estimated source term at the interface; is the estimated collision term at the interface.

9. The discrete unified gas kinetic simulation method for particle transport according to claim 2, wherein: S4 specifically includes: Based on the space-time discrete equation and the corrected interface distribution function, the implicit collision term or implicit source term at the grid center in the space-time discrete equation is estimated by the grid center distribution function at the old time step to obtain the estimated grid center distribution function; Obtaining an estimated collision term or an estimated source term at the center of the grid based on the estimated distribution function at the center of the grid; Based on the space-time discrete equation and the corrected distribution function at the interface, the estimated collision term or the estimated source term at the grid center is substituted to obtain the corrected distribution function at the grid center.

10. A discrete unified gas kinetic simulation system for particle transport, characterized in that: The system includes a memory and a processor, wherein the memory stores a computer program, and when the processor executes the computer program, the discrete unified gas kinetic simulation method for particle transport according to any one of claims 1 to 9 is executed.

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