臺灣博碩士論文加值系統

本論文基於過去吸收性邊界條件理論發展，引入正交模態分析，成功將時間歷史核心函數(THKF)以解析解型式描繪出，與共振頻率資訊連結，賦予THKF更多物理意義。本論文所發展的吸收性邊界條件運用上，任意原子晶格與任意邊界拓撲的THKF，皆可透過擴展虛擬區域的概念下趨近求得，大幅的提升吸收性邊界條件的適用性。
在本論文中同時介紹吸收性邊界條件程式架構，將僅需一次性計算的THKF切割為獨立程式以避免重複求取，並透過平行計算加速計算。所撰寫的程式架構設計，提供各大離散分析軟體在增加最少的介面下，即可擴充並運用本吸收性邊界條件軟體。最後並運用本分析程式進行一系列的THKF求解、分析與離散系統模擬。
將程式計算求得的THKF進行分析，觀察到幾個THKF的特性。首先，THKF時間累加值必定會震盪收斂，該收斂值震盪偏離的幅度會直接影響吸收性邊界模擬系統的反彈因子。透過反彈因子分析，我們連結了時間截斷係數 與THKF震盪的關係，並提供了 的選擇策略，除了擴大該系數來取得較佳的分析結果外，更存在許多局部最佳的 ，同時要避免取到會使系統發散的THKF貢獻高估區段。
在多維系統中，距離越遠的內部-邊界關係，THKF會在較大的時間出現第一個峰值，且整體貢獻也較小。在二維三角晶格系統中透過平面波傳分析反彈因子與空間截斷係數 與入射波長的關係，模擬分析結果中可以發現空間截斷係數 與時間截斷係數 相依， 與 必須適當的選擇以避免遠距離的內部-邊界THKF座落於遠離THKF積分收斂處，導致吸收性邊界條件呈現較大的非物理波反彈現象。
最後，運用了Lennard–Jones勢能進行點波傳例題，在微小變形的假設下成功的呈現吸收性邊界條件在非諧和勢能作用下的模擬成果，並與週期性邊界條件模擬行為做比較，描述了吸收性邊界條件能成功地描述關切原子區域座落於無窮域中的影響，避免了波週期性的穿越邊界影響關切區域，提供分子模擬系統另一個邊界條件運用的考量。

In this dissertation, the time history kernel function (THKF) with the absorbing boundary condition was solved analytically. The THKF’s exact solution was formulated by eigenvalues (vibration frequencies) and eigenvectors, which gives the THKF with physical meaning. Any lattice arranged boundary topology can be solved by extending the virtual domain approximating process. This approximating process enables us to apply the absorbing boundary condition for arbitrary geometries and arbitrary lattices.
To implement the absorbing boundary condition process, we complete two individual programs. The first one calculates the THKFs while the second one uses it to predict the boundary atom positions. Inside these program, parallel programming was used to provide better performance. The design and implementation can be easily extended to any molecular dynamics program by completing two of its interfaces.
We further analyzed the THKF and found several convergent properties. First of all, the THKF is a vibration convergence function. The integrated THKF converged asymptotically to a value. We also found the THKF vibration property correlates well with the results from vibration reflection index. By analyzing the relation between the reflection index and the integrated THKF, we propose a method to choose time cutoff that will be suboptimal to provide a local minimum reflection index.
Secondly, farther inner-boundary correlates with a longer zero-initial THKF, and a smaller integrated THKF convergence value. The space cutoff depends on the time cutoff . This property causes the farther inner-boundary pair THKF at a fixed time cutoff with a worse reflection index condition. To the end, higher non-physical wave reflected at the boundary.
Finally, we use the Lennard–Jones potential to simulate a point wave propagation problem. Based on the small deformation assumption, we successfully use the aharmonic potential to absorb the point wave. We compare the Hamiltonian obtained from the absorbing boundary conditions with a much larger periodic boundary system so the waves have not traveled to the boundaries. Results indicate that the absorbing boundary conditions provide the target region as surrounding by an infinite homogenous atom arrangement. The absorbing boundary condition proposed herein is a promise boundary condition and it can provide a very useful boundary condition type to be used in the molecular dynamics simulation for wave transimission and non-equilibrium molecular dynamics.

口試委員會審定書 i
誌謝 ii
摘要 iii
Abstract v
目錄 vii
圖目錄 ix
一、 簡介 1
1.1 研究背景 1
1.2 研究目的 5
1.3 論文大綱 5
二、 理論分析 6
2.1 吸收性邊界條件理論推導 6
2.2 一維原子鏈時間歷史核心函數解析解 8
2.3 吸收性邊界分析模式 12
2.3.1 分子動力模擬系統分析 12
2.3.2 吸收性質分析 14
三、 吸收性邊界條件程式架構與實作策略 15
3.1 計算THKF程式 15
3.1.1 建立動態矩陣 15
3.1.2 計算特徵值與特徵向量 16
3.1.3 計算THKF 16
3.2 THKF檔案格式 17
3.3 吸收性邊界條件程式 19
3.3.1 與LAMMPS連接設計 20
3.3.2 初始化吸收性邊界條件程式本體 20
3.3.3 動態分析與管理原子位置 25
3.4 小結 26
四、 吸收性邊界條件分析與收斂性 27
4.1 多維系統時間歷史核心函數計算結果 27
4.1.1 二維正方晶格 27
4.1.2 二維三角晶格 30
4.1.3 三維系統：面心立方晶格 32
4.2 THKF收斂性與影響 33
4.2.1 THKF的收斂性 33
4.2.2 吸收性邊界條件時間截斷係數 34
4.3 二維三角晶格平面波傳分析 37
4.4 點波源波傳分析 43
4.5 小結 47
五、 總結 48
5.1 結論 48
5.2 未來展望 49
參考資料 52

1Zwanzig, R. W. (1960). Collision of a gas atom with a cold surface. The Journal of Chemical Physics, 32(4), 1173-1177.
2Adelman, S. A., & Doll, J. D. (1974). Generalized Langevin equation approach for atom/solid‐surface scattering: Collinear atom/harmonic chain model. The Journal of Chemical Physics, 61(10), 4242-4245.
3Doll, J. D., Myers, L. E., & Adelman, S. A. (1975). Generalized Langevin equation approach for atom/solid‐surface scattering: Inelastic studies. The Journal of Chemical Physics, 63(11), 4908-4914.
4Adelman, S. A., & Doll, J. D. (1976). Generalized Langevin equation approach for atom/solid‐surface scattering: General formulation for classical scattering off harmonic solids. The Journal of Chemical Physics, 64(6), 2375-2388.
5Cai, W., de Koning, M., Bulatov, V. V., & Yip, S. (2000). Minimizing boundary reflections in coupled-domain simulations. Physical Review Letters, 85(15), 3213.
6Appelö, D., & Kreiss, G. (2006). A new absorbing layer for elastic waves. Journal of Computational Physics, 215(2), 642-660.
7To, A. C., & Li, S. (2005). Perfectly matched multiscale simulations. Physical Review B, 72(3), 035414.
8Appelö, D., & Colonius, T. (2009). A high-order super-grid-scale absorbing layer and its application to linear hyperbolic systems. Journal of Computational Physics, 228(11), 4200-4217.
9Fang, M., Wang, X., Li, Z., Tang, S. (2013). Matching Boundary Conditions for Scalar Waves in Body-Centered-Cubic Lattices. Advances in Applied Mathematics and Mechanics, 5(03), 337-350.
10Wang, X., & Tang, S. (2013). Matching boundary conditions for lattice dynamics. International Journal for Numerical Methods in Engineering, 93(12), 1255-1285.
11Weinan, E., & Huang, Z. (2001). Matching conditions in atomistic-continuum modeling of materials. Physical Review Letters, 87(13), 135501.
12Li, X., & Weinan, E. (2006). Variational boundary conditions for molecular dynamics simulations of solids at low temperature. Communications in Computational Physics, 1(1), 135-175.
13Li, X., & Weinan, E. (2007). Variational boundary conditions for molecular dynamics simulations of crystalline solids at finite temperature: Treatment of the thermal bath. Physical Review B, 76(10), 104107.
14Jones, R. E., & Kimmer, C. J. (2010). Efficient non-reflecting boundary condition constructed via optimization of damped layers. Physical Review B, 81(9), 094301.
15Venturini, G., Yang, J. Z., Ortiz, M., & Marsden, J. E. (2011). Replica time integrators. International Journal for Numerical Methods in Engineering, 88(6), 586-611.
16Wagner, G. J., & Liu, W. K. (2003). Coupling of atomistic and continuum simulations using a bridging scale decomposition. Journal of Computational Physics, 190(1), 249-274.
17Wagner, G. J., Karpov, E. G., & Liu, W. K. (2004). Molecular dynamics boundary conditions for regular crystal lattices. Computer Methods in Applied Mechanics and Engineering, 193(17), 1579-1601.
18Park, H. S., Karpov, E. G., Liu, W. K., & Klein, P. A. (2005). The bridging scale for two-dimensional atomistic/continuum coupling. Philosophical Magazine, 85(1), 79-113.
19Karpov, E. G., Wagner, G. J., & Liu, W. K. (2005). A Green''s function approach to deriving non‐reflecting boundary conditions in molecular dynamics simulations. International Journal for Numerical Methods in Engineering, 62(9), 1250-1262.
20Tang, S., Hou, T. Y., & Liu, W. K. (2006). A pseudo-spectral multiscale method: interfacial conditions and coarse grid equations. Journal of Computational Physics, 213(1), 57-85.
21Medyanik, S. N., Karpov, E. G., & Liu, W. K. (2006). Domain reduction method for atomistic simulations. Journal of Computational Physics, 218(2), 836-859.
22Tang, S. (2010). A two-way interfacial condition for lattice simulations. The Advance in Applied Mathematic and Mechanics, 2, 45-55.
23Pang, G., & Tang, S. (2011). Time history kernel functions for square lattice. Computational Mechanics, 48(6), 699-711.
24Rahman, A. (1964). Correlations in the motion of atoms in liquid argon. Physical Review, 136(2A), A405.
25Tang, S., and Liu, B. (2015). Heat Jet Approach for Atomic Simulations at Finite Temperature. Communications in Computational Physics, 18(05), 1445-1460.
26Leach, A. R. (2001). Molecular modelling: principles and applications. Pearson education.
27Hu, J., Ruan, X., & Chen, Y. P. (2012). Molecular dynamics study of thermal rectification in graphene nanoribbons. International Journal of Thermophysics, 33(6), 986-991.
28Makov, G., & Payne, M. C. (1995). Periodic boundary conditions in ab initio calculations. Physical Review B, 51(7), 4014.
29Ruiz, G., & Michell, J. A. (2013). Design and Architectures for Digital Signal Processing, InTech, ISBN 978-953-51-0874-0.
30Plimpton, S. (1995). Fast parallel algorithms for short-range molecular dynamics. Journal of computational physics, 117(1), 1-19.