An Improved Quasicontinuum Method for Coupling Atomistic Modeling and Continuum Formulation

許多工程材料的變形與破壞在本質上都是跨尺度的行為，亦即巨觀實驗所觀察到的現象實際上係由不同尺度的行為所操控。故在計算材料力學方面，長久以來始終希望能透過瞭解物質微觀缺陷的組成，合理且正確地預測物質在巨觀下所觀察到的力學行為。本研究最主要的挑戰在於建構一個跨尺度的理論，一方面保留連體力學架構，以處理較大的區域範圍；另一方面，同時加入原子尺度的微觀資料以處理我們所感興趣的局部區域。 擬連體法（quasicontinuum method）在介觀尺度上為分子動力學與連體力學提供了一個跨尺度的架構。該法假設系統只需要少數代表性原子，而其他原子可透過有限元素方法中形狀函數的概念內差計算其相對位置。本研究透過回顧文獻上的相關研究，特別針對能量與力的計算方式加以改良。其次，進一步將擬連體法擴充應用至三維問題，結合高效率的Delaunay網格自動生成軟體，產生三維問題所需之四面體元素網格。 在能量計算方面，依據變形程度可將系統區分為原子區與有限元素區。在有限元素區，將對於每一元素儲存其節點原子之晶格向量座標，如此即可以計算於目前卡式座標下之元素晶格向量。將其代入Cauchy-Born法則後，即可計算單位原子所含能量。再透過嚴謹的方式計算元素中所含原子數目，便可正確地求得元素所含之總勢能。而原子區則以傳統分子動力學之方法，透過古典勢能計算總能量，加上有限元素區之總勢能，即可得目前系統所含之總勢能。在計算力方面，本研究藉由束縛原子（slaved atom），以古典勢能方式計算每一代表原子所受的作用力，消除傳統耦合法中常見的不平衡力。最後，透過以上法則，將可利用共軛坡降法疊代求解系統之平衡態。 本研究亦開發一物件導向的軟體套件來實作此擬連體法。此套件架構於Digital Material之上，大部份的類別繼承自Molecular Dynamics套件並加以延伸。由於多尺度模擬在軟體實作上具有高度的複雜性，是故如何保留軟體設計上的彈性又能兼顧計算效能亦是本研究於開發軟體時的重點。Digital Material運用了許多新的軟體技術，例如design patterns使物件之間的互動能更具備彈性、延伸性及可再使用性。因此本研究可以從既有的架構上輕易延伸新的功能，使新的擬連體法套件能與現有的分子動力模擬套件於同一個軟體平台上密切地結合。 為了檢核本研究所提出的方法確實能有效應用於介觀尺度的材料模擬，藉由模擬完美晶體於不考慮表面能效應（於周期性邊界條件）、考慮表面能效應之二維與三維的例子來探討計算能量及力的正確性。更進一步地藉由模擬原子堆置錯誤、Lomer差排、刃差排之拆解與演進等例子來探討所觀察到的現象是否與分子動力計算或實驗觀察的結果具有一致性。前者數值結果顯示本研究所提之方法能利用少數的代表原子即可得到與分子動力計算完全相同的結果，後者更顯示所得之數值結果與實驗所觀察到的物理現象及物理量相符合。因此，本文所提之改良式擬連體法確實能兼顧跨尺度計算所需的效率與正確性。Mechanical deformation and failure of many engineering materials are inherently multi-scale. Observed macroscopic material behavior is often governed by processes that occur at many different length and time scales. One of the challenging themes faced today is to develop a robust methodology which captures atomistic phenomena while using continuum descriptions to reduce redundant degrees of freedom. The quasicontinuum method provides an appealing framework to adaptively blend atomistic realism with continuum at mesoscale. The method assumes that only a few atoms are represented in a solid, whereas the other atoms are kinematically constrained. In this study, we revisit the formulation with the aims to improve its energy and force calculations. In addition, application of the method is extended to three dimensions, in which the representative atoms are triangulated with a set of tetrahedra using an efficient Delaunay-based mesh generator. We keep track of primitive lattice vectors in an elementwise fashion to establish positions of constrained atoms in finite elements. For each element, an interatomic potential energy density is computed from a periodic arrangement of atoms using its elementwise lattice vectors. An efficient yet rigorous counting algorithm is developed to compute the total number of the atoms and the total potential energy accurately. Interatomic forces are computed for all the representative atoms by an atomistic-based formulation, whereas the conjugated gradient method is used to resolve the equilibrium configuration. An object-oriented software package is developed to implement our improved quasicontinuum. This package extends the Molecular Dynamics package in the Digital Material framework. Modern software techniques, such as design patterns and generic programming, are employed for extensibility and flexibility. The major responsibilities, design concerns and specifications for the package are described. Verification and validation examples at mesoscale scales are used to demonstrate the capabilities of the proposed methodology. Test examples include perfect crystals and crystals with microscopic defects. Prefect crystal examples are used to verify the methodology implemented herein, particularly for energy and force calculations. Examples with microscopic defects are used to demonstrate the feasibility of the method to capture the requisite atomic resolution. The defects and their related mechanisms been studied include the calculation of a stacking fault energy, the Lomer lock phenomenon and the dissociation of an edge dislocation into two Shockley partial dislocations. All the computed results using the improved quasicontinuum method agree excellently with those from the molecular dynamic analysis, and are comparable with experimental measurements. We conclude that the method can accurately compute energies and forces acting on the representative atoms while properly maintain desirable computation efficiency.Table of Contents 誌謝 i Abstract iii 摘要 v Table of Contents vii List of Figures xi List of Tables xvii 1 Introduction 1 1.1 Background 1 1.2 Literature Review 3 1.3 Organization of the Thesis 6 2 Improved Quasicontinuum Method 13 2.1 Atomistic Modeling 15 2.2 Crystalline Solids, Kinematic Constraints and Representative Atoms 17 2.3 Cauchy-Born Rule 20 2.3.1 Continuum-based Local Energy 21 2.3.2 Atomistic-based Local Energy 22 2.4 Coupling Atoms with Continua 26 2.4.1 Effects of the Local/Nonlocal Interface 26 2.4.2 Slaved Atoms for Energy coupling 28 2.5 Forces Calculation 31 2.5.1 Energy-based formulation 31 2.5.2 Force-based formulation 32 2.5.3 Atomistic -based formulation 34 2.6 Mesh Generation 35 2.7 Summary 36 3. Software Development 45 3.1 Overview 45 3.2 Components of Molecular Dynamics 46 3.2.1 ListOfAtoms 46 3.2.2 Potential 47 3.2.3 AtomsMover 47 3.2.4 Transformer 49 3.2.5 NeighborLocator 49 3.2.6 BoundaryCondition 50 3.2.7 Constraint 51 3.2.8 AtomsInitializer 51 3.2.9 ListOfAtomsObserver 53 3.3 Components of Quasicontinuum 55 3.3.1 QCListOfAtoms and ListOfSlavedAtoms 55 3.3.2 QCSlaveConstraint 57 3.3.3 QCPotential 57 3.3.4 Element 58 3.3.5 MeshData 59 3.3.6 Mesher2D 60 3.3.7 Mesher3D 61 3.4 Putting It All Together 62 4. Numerical Examples 69 4.1 Perfect Crystals 70 4.1.1 Perfect Crystals with Periodic Boundary Conditions 70 4.1.2 Perfect Crystals with Surface Effects 72 4.2 Stacking Fault 73 4.3 Lomer Dislocation 75 4.4 Edge Dislocation Dissociation 77 5 Conclusions and Future Work 105 5.1 Conclusions 105 5.2 Recommendation for Future Work 106 Reference 109 Appendix A EAM Potential 11