臺灣博碩士論文加值系統

分子動力學模擬近年來已廣泛應用於計算微觀尺度材料性質。然而受限於建 模時的真實尺度，使用分子動力學進行跨越到介觀尺度的非平衡態模擬，其計算 量相當高。即使發展多尺度方法的需求迫在眉睫，但在過往的研究中，多尺度方 法的發展仍受限於原子尺度到連體之間失真性波反射的問題。
本研究發展適用於任意晶格排列的多尺度分子動力學半解析法。本方法將原 子系統分為實體域與虛擬域的耦合區域架構，並拆解出虛擬域在頻域的貢獻，推 導出隱含虛擬域效應的時間歷時卷積核函數。再依照介面原子的自由度關係配 對，即可透過此數值方法控制介面虛擬原子，使得動力模擬在不考慮虛擬域的原 子下，仍能等價於原生的全原子系統。
本研究發展的原子系統多尺度介面，廣義上不受限於實體域的數量與耦合方 式，為一跨實體域的雙向隱含介面。而在單實體域的特例下，多尺度介面則簡化 為吸收性邊界條件。這兩種應用在數值方法上有些關鍵上的不同。吸收性邊界條 件將資訊捨去到虛擬域，因此可以透過卷積的時間截斷來降低運算量以及使系統 維持邊界無反射的特性。另一方面，多尺度介面必須使不同實體域的資訊能跨虛 擬域傳遞，這段交互作用的關係若透過時間截斷干涉會導致模擬失真。除此之外， 透過本方法解析出的時間歷時卷積核函數於具有強烈的實體空間物理性質，也因 此能保有尺寸效應。有了此特性，多尺度方法在進行跨尺度分析時，方能使模擬 結果符合模擬材料的尺寸。
本研究最後基於時間歷時卷積核函數的物理性質，發展機器學習應用在吸收 性邊界條件的可能性，並且進行了初步驗證。機器學習吸收性邊界條件能大幅簡 化拆解虛擬域頻域貢獻的過程。未來若能完全實作泛用的機器學習多尺度介面， 將可突破目前使用分子動力學進行跨越到介觀尺度非平衡態模擬的瓶頸。

Molecular dynamics (MD) simulation has been widely applied in studying materials behavior in the past decades. However, owning to its great demands on the modeling fidelity, MD is often limited by the length scale it can handle. Although it is much desired to develop multiscale modeling to overcome this limitation, the development of dynamic multiscale modeling is limited by the spurious wave-reflection problems on the interface between atoms and continuum.
This study has developed a multiscale semi-analytical formulation for arbitrary lattices in MD. This method decomposes MD system into a coupled-domain of real and virtual domains, dissembles the contribution from the virtual domain in frequency, and derives the time-history kernel function (THKF) accounting for the interaction of virtual domain implicitly. The virtual atoms on the interface can further be controlled by this numerical method after pairing of degree-of-freedom of THKF and atoms on the interface. By doing so, the dynamics simulation without considering virtual atoms will be equivalent to the original full MD simulation.
In a broad sense, multiscale interface (MI) from this study is a two-way implicit interface across real domains, which is not limited to the number of real domains and the way of coupling. However, in the special case of single real domain, multiscale interface reduces to the classical absorbing boundary condition (ABC). There is some key difference between these two applications. ABC totally ignores the information of the virtual domain, thus the time-cutoff of convolution can reduce the computational cost and maintain the bound of the non-reflecting system. On the other hand, due to the transition of the information between different real domains, the simulation of MI will result in spurious wave reflection if THKF are interfered by the time-cutoff. In addition, THKF analyzed from the MI has strong physical meanings of real space that contains the size effect, which can be further developed to resolve length scale limitation in MD.
In this thesis, we also explore the possibility of applying machine learning (ML) in ABC based on the function form in THKF. The preliminary verification has been completed and ML for ABC can reduce the computational efforts substantially. This poses a good direction to develop ML for MI in the future.

誌謝 I
摘要 II
Abstract III
目錄 IV
圖目錄 VI
第一章 緒論 1
1.1 研究背景 1
1.2 分子動力學的多尺度方法回顧與挑戰 5
1.3 研究目的 7
1.4 論文架構 8
第二章 理論與方法 9
2.1 多尺度分子動力學半解析法 9
2.2 於 MD 中建立多尺度介面與吸收性邊界條件的方法 12
2.3 於 LAMMPS 的實作流程 14
2.4 小結 15
第三章 多尺度介面 16
3.1 一維原子鏈 17
3.2 三維面心立方晶格 22
3.3 THKF 的貢獻──以一維原子鏈為例 26
3.4 小結 28
第四章 吸收性邊界條件 29
4.1 一維原子鏈 30
4.2 二維三角晶格 35
4.3 虛擬域第一模態貢獻時間點與時間截斷點 tc 41
4.4 小結 55
第五章 多尺度分子動力學半解析法的延伸討論 56
5.1 THKF 隱含的 Real Space 尺寸效應 56
5.2 多尺度介面演算法的運算量與 Real Space 的關聯 60
5.3 小結 61
第六章 機器學習吸收性邊界條件 62
6.1 多尺度分子動力學半解析法的限制 62
6.2 機器學習簡介 63
6.3 THKF 結合多元線性迴歸模型 63
6.4 機器學習吸收性邊界條件訓練成果 64
6.5 小結 67
第七章 結論與展望 68
參考文獻 70

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