跳到主要內容

臺灣博碩士論文加值系統

(44.220.44.148) 您好!臺灣時間:2024/06/21 15:54
字體大小: 字級放大   字級縮小   預設字形  
回查詢結果 :::

詳目顯示

我願授權國圖
: 
twitterline
研究生:李銘育
研究生(外文):Li, Ming-Yu
論文名稱:移動邊界截斷格點法對於波茲曼方程式之應用
論文名稱(外文):Moving Boundary Truncated Grid Method: Application to the Boltzmann–BGK Equation
指導教授:周佳駿
指導教授(外文):Chou, Chia-Chun
口試委員:蕭百沂蘇蓉容
口試委員(外文):Hsiao, Pai-YiSu, Jung-Jung
口試日期:2023-07-27
學位類別:碩士
校院名稱:國立清華大學
系所名稱:化學系
學門:自然科學學門
學類:化學學類
論文種類:學術論文
論文出版年:2023
畢業學年度:111
語文別:中文
論文頁數:171
中文關鍵詞:有限差分法移動邊界截斷格點法統計力學波茲曼–BGK方程式擴散克拉默斯逃逸問題計算流體力學混合體系問題半導體物理元件模擬
外文關鍵詞:Finite-Difference MethodMoving Boundary Truncated Grid MethodStatistical MechanicsBoltzmann–BGK EquationDiffusionKramers Escape ProblemComputational Fluid DynamicsMixed-Regime ProblemSemiconductor PhysicsDevice Simulation
相關次數:
  • 被引用被引用:0
  • 點閱點閱:77
  • 評分評分:
  • 下載下載:0
  • 收藏至我的研究室書目清單書目收藏:0
移動邊界截斷格點法是一種能自適決定計算區域邊界,並顯著降低格點使用率的固定網格有限差分算則。隨著格點截斷和邊界外插,相空間格點被動態啟用、關閉,以高效並準確積分動力學方程式。在本研究中,截斷格點法被拓展到求解二維相空間中的波茲曼–BGK方程式,並應用於三個領域⸺化學物理、計算流體力學以及半導體物理。在化學物理案例上,吾人首先檢驗截斷格點法,能在斯莫魯霍夫斯基極限下重現擴散動力學,印證其滿足擴散漸進解;隨後,計算欠阻尼條件下、雙阱位能中的逃逸問題,截斷格點法將系統演化至波茲曼分布,並給出和全格點法一致的穿透機率動力學。在計算流體力學案例上,截斷格點法成功描述局域平衡起始態的弛豫過程,與文獻中援引高階傅立葉擬譜法所建構之基準解諳合,並將系統演化至正確的全域平衡態;此外,截斷格點法被進一步應用於更複雜的遠平衡起始態之混合體系問題,計算出與階層式高階不連續伽遼金隱式-顯式算法相吻合的流體變數模式,並能得出流體力學變數均勻分布之平衡態解。在半導體物理案例上,透過半導體波茲曼–泊松系統,以截斷格點法研究砷化鎵二極體,對於常數弛豫時間模型,截斷格點法精確描述準彈道電子之輸送現象,並給出與五階加權本質無振盪算則一致的穩態解;吾人隨之採用更加真實的極性光學聲子模型,得出與常數弛豫時間模型有定量差異之結果,而截斷格點法之通道模擬結果再次與全格點法相吻合,且其精度足以區分出不同模型之區別。在所有數值案例中,截斷格點法皆成功透過減少格點的使用,降低數值計算之實時成本,展現截斷比例越高則省時益多,截斷比例越低則準度越精之趨勢;綜合考量,移動邊界截斷格點法可作為求解波茲曼–BGK方程式之高效能全動力學算則。
The moving boundary truncated grid (TG) method is a fixed-mesh finite-difference scheme that self-adaptively determines the boundary of the computational domain, leading to a substantial reduction in grid-point usage. By employing grid truncation and boundary extrapolation, grid points are dynamically activated and deactivated, allowing for the integration of dynamical equations with remarkable accuracy and high efficiency. In this study, the TG method is extended to solve the Boltzmann–BGK equation in a two-dimensional phase space, and we explore its applications in chemical physics, computational fluid dynamics (CFD), and semiconductor physics. In chemical physics, we validate the TG method by reproducing diffusive dynamics in the Smoluchowski limit. The method is confirmed to satisfy the asymptotic solution derived from the diffusion equation. Furthermore, the TG method is applied to investigate the Kramers escape problem in the underdamping regime. It evolves the system toward the Boltzmann distribution as an equilibrium solution. The transmission probability obtained from the TG method aligns well with that provided by the conventional full-grid (FG) method. In CFD, the TG method successfully depicts the relaxation process of a local-equilibrium initial state, in accordance with the benchmark solution constructed from a high-order Fourier pseudo-spectral method in the literature. It reliably evolves the system toward the expected global-equilibrium state. The method is further applied to a more complex mixed-regime problem with a far-from-equilibrium initial state. It captures the pattern of hydrodynamic variables comparable to that obtained from a hierarchical high-order discontinuous Galerkin implicit-explicit scheme. The method successfully predicts a global-equilibrium solution characterized by spatial homogeneity across all hydrodynamic variables. In semiconductor physics, the TG method is utilized to study a gallium-arsenide diode, modelled by a Boltzmann-Poisson system. For the constant relaxation-time models, it accurately describes the transport phenomena of quasi-ballistic electrons and provides a steady-state solution consistent with that furnished by the fifth-order weighted essentially non-oscillatory scheme. Nevertheless, we adopt a more realistic polar-optical-phonon model, which yields results with quantitative differences from that of the constant relaxation-time models. The TG solver demonstrates excellent agreement with the FG solver in channel simulations, effectively distinguishing between the two models. In all numerical examples, the TG method manages to reduce the computational costs. The TG parameters directly influence the trade-off between accuracy and efficiency. Heavier truncation results in greater time savings, whereas lighter truncation produces more accurate results. In summary, the TG method serves as an efficient full-kinetic solver for the Boltzmann–BGK equation.
第一章 理論基礎 1
1-1. 理論力學概論 1
1-1-1. 哈密頓力學 1
1-1-2. 劉維爾定理 3
1-2. 氣體動力理論 6
1-2-1. BBGKY層系 6
1-2-2. 物理圖像及分子混沌假設 8
1-2-3. 波茲曼輸送方程式 11
1-3. 波茲曼方程式之性質和模型方程式 13
1-3-1. 碰撞不變量、動差方程式和守恆律 13
1-3-2. 馬克士威分布和熱平衡 18
1-3-3. 線性化碰撞算符之數學性質 21
1-3-4. 弛豫時間模型與BGK方程式 25
第二章 研究方法 27
2-1. 數值微分方程 27
2-1-1. 數值微分:有限差分近似 27
2-1-2. 數值積分:牛頓-寇次公式 32
2-1-3. 起始值問題:龍格-庫塔法 37
2-1-4. 起始‐邊界值問題:有限差分算則 41
2-2. 移動邊界截斷格點法 46
2-2-1. 計算流程 47
2-2-2. 對於相空間分佈函數時間演化之應用 49
第三章 研究成果 60
3-1. 化學物理學上的應用 60
3-1-1. 斯莫魯霍夫斯基極限 60
3-1-2. 克拉默斯之逃逸問題 100
3-2. 計算流體力學上的應用 114
3-2-1. 局域平衡態之弛豫動力學 115
3-2-2. 遠平衡態之混合體系問題 125
3-3. 半導體物理學上的應用 135
3-3-1. 常數散射速率模型 147
3-3-2. 極性光學聲子模型 159
第四章 結論與未來研究方向 165
參考文獻 167
1. Thornton, S. T.; Marion, J. B. Classical Dynamics of Particles and Systems; Brooks/Cole: Belmont, 2004.
‌2. Tuckerman, M. E. Statistical Mechanics: Theory and Molecular Simulation; Oxford University Press: Oxford England; New York, 2010.
‌3. Pauli, W. E.; Enz, C. P. Pauli Lectures on Physics. 4, Statistical Mechanics; Dover Publications; Mineola, New York, 2000.
‌4. Kardar, M. Statistical Physics of Particles; Cambridge University Press, 2007.
‌5. Harris, S. An Introduction to the Theory of the Boltzmann Equation; Courier Corporation, 2012.
6. Guo, Z.; Liu, H.; Luo, L.-S.; Xu, K. A Comparative Study of the LBE and GKS Methods for 2D near Incompressible Laminar Flows. J. Comput. Phys. 2008, 227 (10), 4955–4976.
‌7. Cercignani, C. Mathematical Methods in Kinetic Theory; Springer Science & Business Media, 2013.
8. Gross, E. P.; Jackson, E. A. Kinetic Models and the Linearized Boltzmann Equation. Phys. Fluids 1959, 2 (4), 432–432.
9. Burden, R. L.; Faires, J. D. Numerical Analysis; Brooks/Cole, Cengage Learning, 2010.
‌10. Cheney, E. W.; Kincaid, D. Numerical Mathematics and Computing; Thomson Brooks/Cole, 2007.
‌11. Fornberg, B.; Flyer, N. A Primer on Radial Basis Functions with Applications to the Geosciences; SIAM, 2015.
‌12. Pettey, L.; Wyatt, R. E. Wave Packet Dynamics with Adaptive Grids: The Moving Boundary Truncation Method. Chem. Phys. Lett. 2006, 424 (4-6), 443–448.
‌13. Pettey, L. R.; Wyatt, R. E. Application of the Moving Boundary Truncation Method to Reactive Scattering: H + H2, O + H2, O + HD. J. Phys. Chem. A 2008, 112 (51), 13335–13342.
14. Pettey, L.; Wyatt, R. E. Quantum Wave Packet Dynamics on Multidimensional Adaptive Grids: Applications of the Moving Boundary Truncation Method. Int. J. Quantum Chem. 2007, 107 (7), 1566–1573.
15. Lee, T.-Y.; Chou, C.-C. Moving Boundary Truncated Grid Method for Wave Packet Dynamics. J. Phys. Chem. A 2018, 122 (5), 1451–1463.
16. Lu, C.-Y.; Lee, T.-Y.; Chou, C.-C. Moving Boundary Truncated Grid Method: Multidimensional Quantum Dynamics. Int. J. Quantum Chem. 2020, 120, No. e26055.
‌17. Lee, T.-Y.; Lu, C.-Y.; Chou, C.-C. Moving Boundary Truncated Grid Method: Application to the Time Evolution of Distribution Functions in Phase Space. J. Phys. Chem. A 2020, 125 (1), 476–491.
‌18. Lu, C.-Y.; Lee, T.-Y.; Chou, C.-C. Moving boundary truncated grid method for electronic nonadiabatic dynamics. J. Chem. Phys. 2022, 156 (4), 044107.
19. Hänggi, P.; Talkner, P.; Borkovec, M. Reaction-Rate Theory: Fifty Years after Kramers. Rev. Mod. Phys. 1990, 62 (2), 251–341.
‌20. Wyatt, R. E. Quantum Dynamics with Trajectories: Introduction to Quantum Hydrodynamics; Springer New York: New York, Ny, 2005.
‌21. Zachos, C.; Fairlie, D.; Curtright, T. Quantum Mechanics in Phase Space; World Scientific, 2005.
22. Di\acute{o}si, L. Caldeira-Leggett master equation and medium temperatures. Physica A 1993, 199, 517−526.
23. Di\acute{o}si, L. On High-Temperature Markovian Equation for Quantum Brownian Mo-tion. Europhys. Lett. 1993, 22, 1−3.
24. Skinner, J. L.; Wolynes, P. G. Derivation of Smoluchowski Equations with Corrections for Fokker-Planck and BGK Collision Models. Phys. A: Stat. Mech. 1979, 96 (3), 561–572.
25. Skinner, J.; Wolynes, P. G. Relaxation Processes and Chemical Kinetics. J. Chem. Phys. 1978, 69 (5), 2143–2150.
‌26. Skinner, J.; Wolynes, P. G. General Kinetic Models of Activated Processes in Condensed Phases. J. Chem. Phys. 1980, 72 (9), 4913–4927.
‌27. Kuharski, R. A.; Chandler, D.; Montgomery, J. A.; Rabii, F.; Singer, S. J. Stochastic Molecular Dynamics Study of Cyclohexane Isomerization. J. Phys. Chem. 1988, 92 (11), 3261–3267.
‌28. Donoso, A.; Martens, C. C. Solution of phase space diffusion equations using interacting trajectory ensembles. J. Chem. Phys. 2002, 116 (24), 10598−10605.
29. Boelens, A. M. P.; Venturi, D.; Tartakovsky, D. M. Tensor Methods for the Boltzmann-BGK Equation. J. Comput. Phys. 2020, 421, 109744.
30. Filbet, F.; Jin, S. A Class of Asymptotic-Preserving Schemes for Kinetic Equations and Related Problems with Stiff Sources. J. Comput. Phys. 2010, 229 (20), 7625–7648.
‌ 31. Filbet, F.; Rey, T. A Hierarchy of Hybrid Numerical Methods for Multiscale Kinetic Equations. SIAM J. Sci. Comput. 2015, 37 (3), A1218–A1247.
32. Xiong, T.; Jang, J.; Li, F.; Qiu, J.-M. High Order Asymptotic Preserving Nodal Discontinuous Galerkin IMEX Schemes for the BGK Equation. J. Comput. Phys. 2015, 284, 70–94.
‌33. Xiong, T.; Qiu, J.-M. A Hierarchical Uniformly High Order DG-IMEX Scheme for the 1D BGK Equation. J. Comput. Phys. 2017, 336, 164–191.
34. Ansgar Jüngel. Transport Equations for Semiconductors; Springer Science & Business Media, 2009.
‌35. Poupaud, F.; Ringhofer, C. Semi-Classical Limits in a Crystal with Exterior Potentials and Effective Mass Theorems. Commun. Partial. Differ. Equ. 1996, 21 (11-12), 1897–1918.
36. Kittel, C.; Kroemer, H. Thermal Physics; W. H. Freeman and Company. New York, 1980.
37. Lundstrom, M. Fundamentals of Carrier Transport; Cambridge University Press: Cambridge, U.K. ; New York, 2000.
‌38. Fischetti, M. V. Master-Equation Approach to the Study of Electronic Transport in Small Semiconductor Devices. Phys. Rev. B 1999, 59 (7), 4901–4917.
‌39. Cercignani, C.; Gamba, I. M.; Jerome, J. W.; Shu, C.-W. Device Benchmark Comparisons via Kinetic, Hydrodynamic, and High-Field Models. Comput. Methods Appl. Mech. Eng. 2000, 181 (4), 381–392.
‌40. Baranger, H. U.; Wilkins, J. W. Ballistic Electrons in an Inhomogeneous Submicron Structure: Thermal and Contact Effects. Phys. Rev. B 1984, 30 (12), 7349–7351.
41. Anile, A. M.; Romano, V.; Russo, G. Extended Hydrodynamical Model of Carrier Transport in Semiconductors. SIAM J. Appl. Math. 2000, 61 (1), 74–101.
42. Majorana, A.; Pidatella, R. M. A Finite Difference Scheme Solving the Boltzmann–Poisson System for Semiconductor Devices. J. Comput. Phys. 2001, 174 (2), 649–668.
43. Csontos, D.; Ulloa, S. E. Quasiballistic, Nonequilibrium Electron Distribution in Inhomogeneous Semiconductor Structures. Appl. Phys. Lett. 2005, 86 (25), 253103.
44. Frensley, W. R. Boundary Conditions for Open Quantum Systems Driven far from Equilibrium. Rev. Mod. Phys. 1990, 62 (3), 745–791.
45. Fatemi, E.; Odeh, F. Upwind Finite Difference Solution of Boltzmann Equation Applied to Electron Transport in Semiconductor Devices. J. Comput. Phys. 1993, 108 (2), 209–217.
46. Csontos, D.; Ulloa, S. E. Modeling of Transport through Submicron Semiconductor Structures: A Direct Solution to the Coupled Poisson-Boltzmann Equations. J. Comput. Electron. 2004, 3 (3-4), 215–219.
47. Hasha, D. L.; Eguchi, T.; Jonas, J. J. Dynamical Effects on Conformational Isomerization of Cyclohexane. J. Chem. Phys. 1981, 75 (3), 1571–1573.
‌48. Hasha, D. L.; Eguchi, T.; Jonas, J. J. High-Pressure NMR Study of Dynamical Effects on Conformational Isomerization of Cyclohexane. J. Am. Chem. Soc. 1982, 104 (8), 2290–2296.
‌49. Garrity, D. K.; Skinner, J. L. Effect of Potential Shape on Isomerization Rate Constants for the BGK Model. Chem. Phys. Lett. 1983, 95 (1), 46–51.
50. Su, W.; Alexeenko, A. A.; Cai, G. A Parallel Runge–Kutta Discontinuous Galerkin Solver for Rarefied Gas Flows Based on 2D Boltzmann Kinetic Equations. Comput. Fluids 2015, 109, 123–136.
51. Meng, J.; Wu, L.; Reese, J. M.; Zhang, Y. Assessment of the Ellipsoidal-Statistical Bhatnagar–Gross–Krook Model for Force-Driven Poiseuille Flows. J. Comput. Phys. 2013, 251, 383–395.
‌52. Frensley, W. R. Wigner-Function Model of a Resonant-Tunneling Semiconductor Device. Phys. Rev. B 1987, 36 (3), 1570–1580.
53. Frensley, W. R. Quantum Transport Modeling of Resonant-Tunneling Devices. Solid-State Electron. 1988, 31 (3-4), 739–742.
54. Antonius Dorda; Schürrer, F. A WENO-Solver Combined with Adaptive Momentum Discretization for the Wigner Transport Equation and Its Application to Resonant Tunneling Diodes. J. Comput. Phys. 2015, 284, 95–116.
‌55. Yamada, Y.; Tsuchiya, H.; Ogawa, M. Quantum Transport Simulation of Silicon-Nanowire Transistors Based on Direct Solution Approach of the Wigner Transport Equation. IEEE Trans. Electron Devices 2009, 56 (7), 1396–1401.
電子全文 電子全文(網際網路公開日期:20280809)
連結至畢業學校之論文網頁點我開啟連結
註: 此連結為研究生畢業學校所提供,不一定有電子全文可供下載,若連結有誤,請點選上方之〝勘誤回報〞功能,我們會盡快修正,謝謝!
QRCODE
 
 
 
 
 
                                                                                                                                                                                                                                                                                                                                                                                                               
第一頁 上一頁 下一頁 最後一頁 top
無相關期刊