Linearly Preconditioned Nonlinear Solvers for Phase Field Equations Involving p-Laplacian Terms

Phase field models are usually constructed to model certain interfacial dynamics. Numerical simulations of phase-field models require long time accuracy, stability and therefore it is necessary to develop efficient and highly accurate numerical methods. In particular, the unconditionally energy stable , unconditionally solvable, and accurate schemes and fast solvers are desirable. In this thesis, We describe and analyze preconditioned steepest descent (PSD) solvers for fourth and sixth-order nonlinear elliptic equations that include p-Laplacian terms on periodic domains in 2 and 3 dimensions. Such nonlinear elliptic equations often arise from time discretization of parabolic equations that model various biological and physical phenomena, in particular, liquid crystals, thin film epitaxial growth and phase transformations. The analyses of the schemes involve the characterization of the strictly convex energies associated with the equations. We first give a general framework for PSD in Hilbert spaces. Based on certain reasonable assumptions of the linear pre-conditioner, a geometric convergence rate is shown for the nonlinear PSD iteration. We then apply the general theory to the fourth and sixth-order problems of interest, making use of Sobolev embedding and regularity results to confirm the appropriateness of our pre-conditioners for the regularized p-Lapacian problems. The results include a sharper theoretical convergence result for p-Laplacian systems compared to what may be found in existing works. We demonstrate rigorously how to apply the theory in the finite dimensional setting using finite difference discretization methods. Based on the PSD framework, we also proposed two efficient and practical Preconditioned Nonlinear Conjugate Gradient (PNCG) solvers. The main idea of the preconditioned solvers is to use a linearized version of the nonlinear operator as a metric for choosing the initial search direction. And the hybrid conjugate directions as the following search direction. In order to make the proposed solvers and scheme much more practical, we also investigate an adaptive time stepping strategy for time dependent problems. Numerical simulations for some important physical application problems – including thin film epitaxy with slope selection, the square phase field crystal model and functionalized Cahn-Hilliard equation – are carried out to verify the efficiency of the schemes and solvers

Immersed finite element method for interface problems with algebraic multigrid solver

This thesis is to discuss the bilinear and 2D linear immersed finite element (IFE) solutions generated from the algebraic multigrid solver for both stationary and moving interface problems. In contrast to the body-fitting mesh restriction of the traditional finite element methods or finite difference methods for interface problems, a number of numerical methods based on structured meshes independent of the interface have been developed. When these methods are applied to the real world applications, we often need to solve the corresponding large scale linear systems many times, which demands efficient solvers. The algebraic multigrid (AMG) method is a natural choice since it is independent of the geometry, which may be very complicated in interface problems. However, for those methods based on finite difference formulation and a structured mesh independent of the interface, the stiffness matrix of the linear system is usually not symmetric positive-definite, which demands extra efforts to design efficient multigrid methods. On the other hand, the stiffness matrix arising from the IFE methods are naturally symmetric positive-definite. Hence the IFE-AMG algorithm is proposed to solve the linear systems of the bilinear and 2D linear IFE methods for both stationary and moving interface problems after the IFE and multi-grid methods are reviewed respectively. The numerical examples demonstrate the features of the proposed algorithm, including the optimal convergence in both Ł² and semi-H¹ norms of the IFE-AMG solutions, the high efficiency with proper choice of the components and parameters of AMG, the influence of the tolerance and the smoother type of AMG on the convergence of the IFE solutions for the interface problems, and the relationship between the cost and the moving interface location --Abstract, page iii

Numerical simulation on the aerodynamic effects of blade icing on small scale Straight-bladed VAWT

AbstractTo invest the effects of blade surface icing on the aerodynamics performance of the straight-bladed vertical-axis wind turbine (SB-VAWT), wind tunnel tests were carried out on a static straight blade using a simple icing wind tunnel. Firstly, the icing situations on blade surface at some kinds of typical attack angle were observed and recorded under different cold water flow fluxes. Then the iced blade airfoils were combined into a SB-VAWT model with two blades. Numerical simulations were carried out on this model, and the static and dynamic torque coefficients of the model with and without icing were computed. Both the static and dynamic torque coefficients were decreased for the icing effects

Numerical Simulation Based Targeting of the Magushan Skarn Cu-Mo Deposit, Middle-Lower Yangtze Metallogenic Belt, China

The Magushan Cu–Mo deposit is a skarn deposit within the Nanling–Xuancheng mining district of the Middle-Lower Yangtze River Metallogenic Belt (MLYRMB), China. This study presents the results of a new numerical simulation that models the ore-forming processes that generated the Magushan deposit and enables the identification of unexplored areas that have significant exploration potential under areas covered by thick sedimentary sequences that cannot be easily explored using traditional methods. This study outlines the practical value of numerical simulation in determining the processes that operate during mineral deposit formation and how this knowledge can be used to enhance exploration targeting in areas of known mineralization. Our simulation also links multiple subdisciplines such as heat transfer, pressure, fluid flow, chemical reactions, and material migration. Our simulation allows the modeling of the formation and distribution of garnet, a gangue mineral commonly found within skarn deposits (including within the Magushan deposit). The modeled distribution of garnet matches the distribution of known mineralization as well as delineating areas that may well contain high garnet abundances within and around a concealed intrusion, indicating this area should be considered a prospective target during future mineral exploration. Overall, our study indicates that this type of numerical simulation-based approach to prospectivity modeling is both effective and economical and should be considered an additional tool for future mineral exploration to reduce exploration risks when targeting mineralization in areas with thick and unprospective sedimentary cover sequences