Thesis (Ph. D.)--University of Iowa, 2007. / Supervisor: Shaoping Xiao.. Includes bibliographical references (leaves 174-182).

Radial Basis function (RBF) method for solving partial differential equation (PDE) has a lot of applications in many areas. One of the advantages of RBF method is meshless. The cost of mesh generation can be reduced by playing with scattered data. It can also allow adaptivity to solve some problems with special feature. In this thesis, RBF method will be considered to solve several problems. Firstly, we solve the PDEs on surface with singularity (folded surface) by a localized method. The localized method is a generalization of finite difference method. A priori error estimate for the discreitzation of Laplace operator is given for points selection. A stable solver (RBF-QR) is used to avoid ill-conditioning for the numerical simulation. Secondly, a {dollar}H^2{dollar} convergence study for the least-squares kernel collocation method, a.k.a. least-square Kansa's method will be discussed. This chapter can be separated into two main parts: constraint least-square method and weighted least-square method. For both methods, stability and consistency analysis are considered. Error estimate for both methods are also provided. For the case of weighted least-square Kansa's method, we figured out a suitable weighting for optimal error estimation. In Chapter two, we solve partial differential equation on smooth surface by an embedding method in the embedding space {dollar}\R^d{dollar}. Therefore, one can apply any numerical method in {dollar}\R^d{dollar} to solve the embedding problem. Thus, as an application of previous result, we solve embedding problem by least-squares kernel collocation. Moreover, we propose a new embedding condition in this chapter which has high order of convergence. As a result, we solve partial differential equation on smooth surface with a high order kernel collocation method. Similar to chapter two, we also provide error estimate for the numerical solution. Some applications such as pattern formation in the Brusselator system and excitable media in FitzHughNagumo model are also studied.

This study presents a numerical technique that enables exact treatment of all boundary conditions including those that are given on the interface boundary of two distinct media. This interface boundary conditions for Poisson equation are formulated as equality of the physical field and fluxes across the interface boundary. In this work first, the range of physical and geometric parameters which allow the applicability of the meshfree method with distance fields are tested and compared with analytical solution. Second, it investigates how the solution error depends on the ratio of B-spline support and thickness of the interface layer. Further, this study also concentrates on developing improved computational tools like 1D integration and modification of distance fields for analysis of diffusion concentration in heterogeneous material with high contrast of physical and geometrical properties. These improved computational tools for meshfree method with distance fields improve the accuracy of solution and decreases the computational time. Finally, these improved tools are used to solve a 2D problem for analysis of diffusion concentration and the results are compared to FEM solution to show that the improved tools yield computationally better results.

Porosity is known to be one of the primary detrimental factors controlling fatigue life and total elongation of several cast alloy components. The two main aims of this work are to examine pore nucleation and growth effects for predicting gas microporosity and to study the physics of bifilm dynamics to gain understanding in the role of bifilms in producing defects and the mechanisms of defect creation. In the second chapter of this thesis, an innovative technique, based on the combination of a set of conservation equations that solves the transport phenomena during solidification at the macro-scale and the hydrogen diffusion into the pores at the micro-scale, was used to quantify the amount of gas microporosity in A356 alloy castings. The results were compared with published experimental data. In the reminder of this work, the Immersed Element-Free Galerkin method (IEFGM) is presented and it was used to study the physics of bifilm dynamics. The IEFGM is an extension of the Immersed Finite Element method (IFEM) developed by Zhang et al. [50] and it is an attractive technique for simulating FSI problems involving highly deformable bifilm-like solids.

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This work presents a partitioned approach to simulating free-surface flow interaction with hyper-elastic structures in which a smoothed particle hydrodynamics (SPH) solver is coupled with a finite-element (FEM) solver. SPH is a mesh-free, Lagrangian numerical technique frequently employed to study physical phenomena involving large deformations, such as fragmentation or breaking waves. As a mesh-free Lagrangian method, SPH makes an attractive alternative to traditional grid-based methods for modeling free-surface flows and/or problems with rapid deformations where frequent re-meshing and additional free-surface tracking algorithms are non-trivial. This work continues and extends the earlier coupled 2D SPH-FEM approach of Yang et al. [1,2] by linking a double-precision GPU implementation of a 3D weakly compressible SPH formulation [3] with the open source finite element software Code_Aster [4]. Using this approach, the fluid domain is evolved on the GPU, while the CPU updates the structural domain. Finally, the partitioned solutions are coupled using a traditional staggered algorithm. / Ph. D.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / O aumento da capacidade dos computadores nessas trÃs ultimas dÃcadas tem tornado possÃvel a soluÃÃo de problemas de engenharia cada vez mais complexos. Essa ampliaÃÃo na possibilidade de soluÃÃo de tais problemas à resultado do avanÃo nos mÃtodos numÃricos e do desenvolvimento de algoritmos eficientes. Entretanto, estes mÃtodos numÃricos sÃo baseados na construÃÃo de malhas de discretizaÃÃo e a geraÃÃo de malhas ainda representa o maior desafio desses mÃtodos. Por esse motivo, nos Ãltimos anos o foco dos estudos de modelagem de problemas relacionados Ãs Ãguas subterrÃneas està voltado para o desenvolvimento de âmÃtodos livres de malhasâ ( meslhess ou meshfree methods) que tÃm como objetivo eliminar ou, pelo menos aliviar os problemas associados à construÃÃo e/ou reconstruÃÃo de malhas. Em problemas transientes, nos mÃtodos numÃricos tradicionais, o espaÃo à discretizado e em seguida à feita uma nova discretizaÃÃo para o tempo que requer a escolha de uma relaÃÃo Ãtima entre o intervalo de tempo escolhido e a discretizaÃÃo do espaÃo. Esta tese propÃe o desenvolvimento de um modelo numÃrico-analÃtico para problemas transientes de fluxo e contaminaÃÃo de Ãgua subterrÃnea. à um mÃtodo numÃrico com relaÃÃo à descriÃÃo do espaÃo, onde serà utilizado o mÃtodo RBF livre de malha e à analÃtico com relaÃÃo ao tempo, onde serÃo geradas expressÃes matemÃticas para a parte transiente. TrÃs configuraÃÃes de problemas unidimensionais de Ãgua subterrÃnea foram modeladas pelo mÃtodo RBF livre de malha e pelo mÃtodo numÃrico-analÃtico (MNA), utilizando o MAPLE e ( versÃo 10.0) como linguagem de programaÃÃo. Nos trÃs casos estudados, o meio poroso à homogÃneo e saturado. Os resultados apresentados mostram a validaÃÃo de fÃsica do MNA, mas possuem algumas restriÃÃes em sua aplicaÃÃo, tais como domÃnios poucos discretizados e a escolha de um fator de forma Ãtimo. O presente trabalho mostra tambÃm que o modelo proposto acomoda condiÃÃes de contorno que variam com o tempo. / Improvements in computer capabilities in the last three decades make it possible to solve more and more complex engineering problems. The increase in possibilities for solving such problems has been due to advances in numerical methods and development of efficient algorithms. Nevertheless, these numerical methods are based on mesh discretization and it is widely acknowledged that mesh generation remains one of the biggest challenges in mesh-based methods. During recent years, groundwater problems modeling studies are focused on the development of meshless or mesh-free methods. The aim of the so-called mesh-free methods is to eliminate or at least minimize the problems associated with meshing and/or remeshing. Traditional numerical methods discretize space and then discretize time in order to solve transient problems. This procedure requires na optimal relationship between space and time discretizations. This work proposes the development of a numerical-analytical model for flow and contaminant transport groundwater transient problems. It is a numerical method with respect to space, with RBF meshfree method and it is analytical with respect to time, with mathematical expressions. Three onedimensional problems configurations were teste using RBF meshless method and numerical-analytical (MNA) method, in MAPLE program. In all three cases, porous media is homogeneous and saturated. Results show MNAâs physical validation, but there are some restrictions to its use, such as domain discretization, PÃclet number and optimal shape parameter. The present work also shows that MNA accommodates well varying boundary conditions.

In this thesis, we develop a temperature-dependent homogenization technique and implement it into the meshfree particle method for nanoscale continuum simulations. As a hierarchical multiscale method, the nanoscale meshfree particle method is employed to model and simulate nanostructured materials and devices. Recently developed multiscale methods can overcome the limitations of both length and time scales that molecular dynamics has. However, multiscale methods have difficulties in investigating temperature-dependent physical phenomena since most homogenization techniques employed in continuum models have an assumption of zero temperature. A new homogenization technique, the temperature-related Cauchy-Born (TCB) rule, is proposed with the consideration of the free energy instead of the potential energy in this thesis. This technique is verified via stress analyses of several crystalline solids. The studies of material stability demonstrate the significance of temperature effects on nanostructured material stability. Since meshfree particle methods have advantages on simulating the problems involving extremely large deformations and moving boundaries, they become attractive options to be used in the hierarchical multiscale modeling to approximate a large number of atoms. In this thesis, a nanoscale meshfree particle method with the implementation of the developed homogenization technique, i.e. the TCB rule, is proposed. It is shown that numerical simulations in nanotechnology can be beneficial from this technique by saving a great amount of computer time. The nanoscale meshfree particle method is employed to investigate the crack propagation in a nanoplate with the development of cohesive zone model and a thermal-mechanical coupling model. In addition, the nanoscale meshfree particle method is simplified to successfully study mechanisms of nanotube-based memory cells.

This thesis presents the Sequentially Optimized Meshfree Approximation (SOMA) method, a new and powerful Computational Fluid Dynamics (CFD) solver. While standard computational methods can be faster and cheaper that physical experimentation, both in cost and work time, these methods do have some time and user interaction overhead which SOMA eliminates. As a meshfree method which could use adaptive domain refinement methods, SOMA avoids the need for user generated and/or analyzed grids, volumes, and meshes. Incremental building of a feed-forward artificial neural network through machine learning to solve the flow problem significantly reduces user interaction and reduces computational cost. This is done by avoiding the creation and inversion of possibly dense block diagonal matrices and by focusing computational work on regions where the flow changes and ignoring regions where no changes occur.

A meshfree approximation scheme based on the radial basis function methods is presented for the numerical solution of the options pricing model. This thesis deals with the valuation of the European, Barrier, Asian, American options of a single asset and American options of multi assets. The option prices are modeled by the Black-Scholes equation. The θ-method is used to discretize the equation with respect to time. By the next step, the option price is approximated in space with radial basis functions (RBF) with unknown parameters, in particular, we con- sider multiquadric radial basis functions (MQ-RBF). In case of Ameri- can options a penalty method is used, i.e. removing the free boundary is achieved by adding a small and continuous penalty term to the Black- Scholes equation. Finally, a comparison of analytical and finite difference solutions and numerical results from the literature is included.