南方科技大学知识苑(SUSTech KC): Numerical Simulations of Neutron Transport Equation and Its Parameter Estimations

题名	Numerical Simulations of Neutron Transport Equation and Its Parameter Estimations
其他题名	中子输运方程的数值模拟以及参数估计
姓名	昝博千
姓名拼音	ZAN Boqian
学号	12032858
学位类型	硕士
学位专业	0701 数学
学科门类/专业学位类别	07 理学
导师	张振
导师单位	数学系
论文答辩日期	2024-05-13
论文提交日期	2024-07-01
学位授予单位	南方科技大学
学位授予地点	深圳
摘要	The neutron transport equation (NTE) plays a critical role in various engineering applications, such as nuclear reactors and CT imaging. With the rapid development of science and computing, numerical simulation and parameter estimations of NTE are becoming increasingly important in engineering and medical fields. However, solving the neutron transport equation faces two challenges: Firstly, the NTE lacks an analytical solution, necessitating the use of numerical methods. Secondly, direct computation requires solving a complex matrix equation, making iteration methods preferable. So far, many acceleration methods based on the source iteration method have been used to solve the NTE more quickly, especially in complicated and high-dimensional cases. With the development of deep learning, using a neural network instead of a mesh-based method is preferred due to shorter computation times and higher accuracy. In this thesis, we have completed two main tasks: Firstly, we do a method review. We review and investigate the source iteration method and several acceleration methods based on it, including the nonlinear diffusion method (NDA), diffusion synthetic method (DSA), quasi diffusion method (QD), S2SA, and the KP method. Additionally, we introduce the Andersen Acceleration method, a universal acceleration method in iteration methods. Then we implement these methods in 1-D isotropic and anisotropic cases and compare the acceleration effects of these methods combined with Andersen acceleration. Besides traditional numerical methods, we also utilize an auxiliary physics-informed neural network (A-PINN), which is a specialized framework for solving integral-differential equations (IDE), to solve the NTE in 1-D and 2-D cases. Based on PINNs, A-PINN changes the single-output neural network into a multi-output neural network. The multi output includes the approximation of the solution and an auxiliary output as the integral of the solution. Secondly, we use A-PINN to do some parameter estimations and find that it produced good results even with noisy data. As far as I know, this is the first attempt to use such a framework to wwwwsolve inverse problems of NTE.
关键词	NTE Source Iteration Method Acceleration Methods A-PINN Inverse Problems
语种	英语
培养类别	独立培养
入学年份	2020
学位授予年份	2024-06
参考文献列表	[1] HE X, LUO L S. Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation[J]. Physical Review, 1997, 56: 6811. [2] BIAGI S F. Accurate solution of the Boltzmann transport equation[J]. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 1988, 273: 533-535. [3] MASLOWSKI A, WANG A, SUN M, et al. Acuros CTS: A fast, linear Boltzmann transport equation solver for computed tomography scatter-Part I: Core algorithms and validation[J]. American Association of Physicists in Medicine, 2018, 45(5): 1899-1913. [4] WANG A, MASLOWSKI A, MESSMER P, et al. Acuros CTS: A fast, linear Boltzmann transport equation solver for computed tomography scatter-Part II: System modeling, scatter correction, and optimization[J]. American Association of Physicists in Medicine, 2018, 45(5): 1914-1925. [5] LEWIS E E, MILLER W F. Computational methods of neutron transport[M]. 1984. [6] LOUIS A H, DAVID M Y. Applied iterative methods[M]. Academic Press, 1981. [7] VARGA R S. Iterative solution of elliptic systems and applications to the neutron diffusion equations of reactor Physics[J]. SIAM Review, 1967, 9: 756-758. [8] LYUSTERNIK L A. Remarks on numerical Solution of the boundary value problems for Laplace equation and calculation of eigenvalues by the mesh method[J]. Trudy Matem. Inst. An. USSR, 1964, 20: 49-64. [9] LEBEDEV V. The iterative KP method for the kinetic equation[J]. Mathematical Methods for Solution of Nuclear Physics Problems, 1964. [10] ALCOUFFE R E. Diffusion synthetic acceleration method for diamond differenced discrete ordinates equation[J]. Nuclear Science and Engineering, 1977, 64(2): 344-355. [11] ALCOUFFE R E. A stable diffusion synthetic acceleration method for neutron transport iterations[J]. Transactions of the American Nuclear Society, 1976, 23: 203. [12] GOL’DIN V. A Quasi-Diffusion method of solving the kinetic equation[J]. USSR Computational Mathematics and Mathematical Physics, 1964, 4(6): 136-149. [13] WILLERT J, PARK H. Applying nonlinear diffusion acceleration to the neutron transport k-Eigenvalue problem with anisotropic scattering[J]. Nuclear Science and Engineering, 2015, 181: 351-360. [14] JEFFREY W, PARK H, KNOLL D. A comparison of acceleration methods for solving the neutron transport k-eigenvalue problem[J]. Journal of Computational Physics, 2014, 274: 681-694. [15] PARK H, KNOLL D A, NEWMAN C K. Nonlinear acceleration of transport criticality problems[J]. Nuclear Science and Engineering, 2012, 172: 52-65. [16] BALSARA D. Fast and accurate discrete ordinates methods for multidimensional radiative transfer. Part I, basic methods[J]. Nuclear Science and Engineering, 2001, 69: 671-707. [17] WILLERT J A, KELLEY C. Efficient solutions to the NDA-NCA low-order eigenvalue problem[J]. American Nuclear Society, 2012. [18] WALKER H, PENG N. Andersen Acceleration for fixed-point iterations[J]. SIAM Journal on Numerical Analysis, 2011, 49: 1715-1735. [19] WILLERT J, PARK H, TATTANO W. Using Anderson acceleration to accelerate the convergence of neutron transport calculations with anisotropic scattering[J]. Nuclear Science and Engineering, 2015, 181(3): 342-350. [20] RAISSI M, PERDIKARIS P, KARNIADAKIS G E. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations[J]. Journal of Computational Physics, 2018, 378(2019): 686-707. [21] LI R, LEE E, LUO T. Physics-informed neural networks for solving multiscale mode-resolved phonon Boltzmann transport equation[J]. Materials Today Physics, 2021, 19: 100429. [22] XIE Y, WANG Y, MA Y. Boundary dependent physics-informed neural network for solving neutron transport equation[J]. Annals of Nuclear Energy, 2024, 195: 110181. [23] YUAN L, NI Y Q, DENG X Y, et al. A-PINN: Auxiliary physics informed neural networks for forward and inverse problems of nonlinear integro-differential equations[J]. Journal of Computational Physics, 2022, 462(2022): 111260. [24] RIGANTI R, NEGRO L D. Auxiliary physics-informed neural networks for forward, inverse, and coupled radiative transfer problems[J]. Applied Physics Letters, 2023. [25] WILLERT J A. Hybrid deterministic/monte carlo methods for solving the neutron transport equation and k-eigenvalue problem[D]. North Carolina State University, 2013. [26] VASSILIEV O N. Monte Carlo methods for radiation transport: fundamentals and advanced topics[M]. Springer, 2017. [27] LARSEN E W. Diffusion-synthetic acceleration methods for discrete-ordinates problems[J]. Transport Theory and Statistical Physics, 1984, 13:1-2: 107-126. [28] BELL G I, GLASSTONE S. Nuclear reactor theory[M]. US Atomic Energy Commission, 1970. [29] MARCHUK G, LEBEDEV V. Numerical methods in the theory of neutron transport[M]. 1986. [30] KNOLL D, SMITH K, PARK. H. Application of the Jacobian-free Newton-Krylov method to nonlinear acceleration of transport source iteration in slab geometry[J]. Nuclear Science and Engineering, 2011, 167(2): 122-132. [31] MIFTEN M M, LARSEN E W. The Quasi-diffusion method for solving transport problems in planar and spherical geometries[J]. Transport Theory and Statistical Physics, 1993, 22(2&3): 165-186. [32] 刘东, 王雪强, 张斌, 等. 深度学习方法求解中子输运方程的微分变阶理论[J]. 原子能科学技术, 2023, 57: 946-959.
所在学位评定分委会	数学
国内图书分类号	O29
来源库	人工提交
成果类型	学位论文
条目标识符	http://sustech.caswiz.com/handle/2SGJ60CL/778782
专题	理学院_数学系
推荐引用方式 GB/T 7714	Zan BQ. Numerical Simulations of Neutron Transport Equation and Its Parameter Estimations[D]. 深圳. 南方科技大学,2024.

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