Thesis (Ph. D.)--Drexel University, 2006. / Includes abstract and vita. Includes bibliographical references (leaves 66-76).

Yiu Yun Yip. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1997. / Includes bibliographical references (leaves 86-87). / Yiu Yun Yip. / Contents --- p.i / List of Figures --- p.iii / List of Tables --- p.vi / Abstract --- p.vii / Acknowledgements --- p.viii / Chapter Chapter 1. --- Introduction --- p.1 / Chapter Chapter 2. --- The Fragmentation Model --- p.4 / Chapter 2.1 --- Interaction Potential --- p.5 / Chapter 2.2 --- Initial Configuration --- p.6 / Chapter 2.2.1 --- Enforcement of momentum conservation at time zero --- p.8 / Chapter 2.3 --- Time Evolution --- p.9 / Chapter 2.4 --- Summary --- p.10 / Chapter Chapter 3. --- Results for object with an initial circular shape --- p.11 / Chapter 3.1 --- Measurement of F(m) --- p.11 / Chapter 3.2 --- Results and Analysis --- p.14 / Chapter 3.3 --- Discussion --- p.24 / Chapter Chapter 4. --- Results for object with an initial square shape --- p.35 / Chapter 4.1 --- Results and Analysis --- p.35 / Chapter 4.2 --- Discussion --- p.43 / Chapter Chapter 5. --- Comparsion with experimental observations --- p.49 / Chapter 5.1 --- The Experiment --- p.49 / Chapter 5.2 --- Comparison --- p.50 / Chapter 5.2.1 --- Relation between the power-law exponent and the falling height --- p.50 / Chapter 5.2.2 --- Relation between the falling height and the total number of fragments --- p.50 / Chapter 5.2.3 --- Relation between the power-law exponent and the total num- ber of fragments --- p.53 / Chapter 5.3 --- Summary --- p.53 / Chapter Chapter 6. --- Maximum entropy formalism for fragment distributions --- p.55 / Chapter 6.1 --- The formalism --- p.55 / Chapter 6.2 --- Average potential and kinetic energies for fragments with mass m --- p.57 / Chapter 6.3 --- Comparison with simulation results --- p.67 / Chapter 6.3.1 --- Analysis for small R --- p.67 / Chapter 6.3.2 --- Analysis for large R --- p.69 / Chapter 6.4 --- Conclusion --- p.69 / Chapter Chapter 7. --- Conclusion --- p.75 / Appendix A. Main program --- p.76 / Appendix B. Derivation of P*(nm，m) and n*(m) --- p.84 / Bibliography --- p.86

This thesis concerns the formulation of integration algorithms for non-Hamiltonian molecular dynamics simulation at constant temperature. In particular, the constant temperature dynamics of the Nosé-Hoover, Nosé-Hoover chain, and Bulgac-Kusnezov thermostats are studied. In all cases, the equilibrium statistical mechanics and the integration algorithms have been formulated using non-Hamiltonian brackets in phase space. A systematic approach has been followed in deriving numerically stable and efficient algorithms. Starting from a set of equations of motion, time-reversible algorithms have been formulated through the time-symmetric Trotter factorization of the Liouville propagator. Such a time-symmetric factorization can be combined with the underlying non- Hamiltonian bracket-structure of the Liouville operator, preserving the measure of phase space. In this latter case, algorithms that are both time-reversible and measure-preserving can be obtained. Constant temperature simulations of low-dimensional harmonic systems have been performed in order to illustrate the accuracy and the efficiency of the algorithms presented in this thesis. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2011.

Wong Ngai-ning. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 65-67). / Abstracts in English and Chinese. / Abstract --- p.ii / Content --- p.iv / List of Table --- p.v / List of Figures --- p.v / Acknowledgement --- p.vii / Dedication --- p.viii / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Related work --- p.2 / Chapter 1.2 --- Background --- p.6 / Chapter 1.3 --- Contribution --- p.23 / Chapter 1.4 --- Thesis roadmap --- p.24 / Chapter 2 --- The Capacitance method --- p.25 / Chapter 2.1 --- Theoretical Comparison --- p.29 / Chapter 3 --- Collision detection --- p.32 / Chapter 3.1 --- Searching the hierarchy --- p.33 / Chapter 3.2 --- Neighborhood algorithm --- p.35 / Chapter 3.3 --- Regional sphere tree update --- p.38 / Chapter 4 --- Implementation --- p.41 / Chapter 4.1 --- System Architecture --- p.41 / Chapter 4.2 --- Multi-contact latency --- p.45 / Chapter 5 --- Result and Analysis --- p.46 / Chapter 5.1 --- Pre-computation --- p.46 / Chapter 5.2 --- Relation matrix Establishment --- p.47 / Chapter 5.3 --- Sphere tree construction --- p.49 / Chapter 5.4 --- Regional sphere tree update --- p.50 / Chapter 5.5 --- Graphic result --- p.52 / Chapter 6 --- Conclusion and Future work --- p.62 / Chapter 6.1 --- Conclusion --- p.62 / Chapter 6.2 --- Future work --- p.64 / Reference --- p.65 / Appendix A --- p.68 / Appendix B --- p.70

Physics simulation computationally models physical phenomena. It is the bread-and-butter of modern-day scientific discoveries and engineering design: from plasma theory to digital twins. However, viable efficiency remains a long-standing challenge to physics simulation. Accurate, real-world-scale simulations are often computationally too expensive (e.g., excessive wall-clock time) to gain any practical usage. In this thesis, we explore two general solutions to tackle this problem. Our first proposed method is a multiscaling approach. Simulating physics at its fundamental discrete scale, e.g., the atomic-level, provides unmatched levels of detail and generality, but proves to be excessively costly when applied to large-scale systems. Alternatively, simulating physics at the continuum scale governed by partial differential equations (PDEs) is computationally tractable, but limited in applicability due to built-in modeling assumptions. We propose a multiscaling simulation technique that exploits the dual strengths of discrete and continuum treatments. In particular, we design a hybrid discrete-continuum framework for granular media. In this adaptive framework, we define an oracle to dynamically partition the domain into continuum regions where safe and discrete regions where necessary. We couple the dynamics of the discrete and continuum regions via overlapping transition zones to form one coherent simulation. Enrichment and homogenization operations convert between discrete and continuum representations, which allow the partitions to evolve over time. This approach saves the computation cost by partially employing continuum simulations and obtains up to 116X speedup over the discrete-only simulations while maintaining the same level of accuracy. To further accelerate PDE-governed continuum simulations, we propose a machine-learning-based reduced-order modeling (ROM) method. Whereas prior ROM approaches reduce the dimensionality of discretized vector fields, our continuous reduced-order modeling (CROM) approach builds a smooth, low-dimensional manifold of the continuous vector fields themselves, not their discretization. We represent this reduced manifold using neural fields, relying on their continuous and differentiable nature to efficiently solve the PDEs. CROM may train on any and all available numerical solutions of the continuous system, even when they are obtained using diverse methods or discretizations. Indeed, CROM is the first model reduction framework that can simultaneously handle data from voxels, meshes, and point clouds. After the low-dimensional manifolds are established, solving PDEs requires significantly less computational resources. Since CROM is discretization-agnostic, CROM-based PDE solvers may optimally adapt discretization resolution over time to economize computation. We validate our approach on an extensive range of PDEs from thermodynamics, image processing, solid mechanics, and fluid dynamics. Selected large-scale experiments demonstrate that our approach obtains speed, memory, and accuracy advantages over prior ROM approaches while gaining 109X wall-clock speedup over full-order models on CPUs and 89X speedup on GPUs.