Browsing by Author "Xiao-Biao Lin, Committee Member"

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Finite Element Methods for Interface Problems with Locally Modified Triangulations
(2009-08-04) Xie, Hui; Kazufumi Ito, Committee Member; Xiao-Biao Lin, Committee Member; Sharon Lubkin, Committee Member; Zhilin Li, Committee Chair
Interface problems arise in many applications such as heat conduction in different materials. The partial differential equations (PDEs) that describe these applications have domains that consist of different subdomains. The different subdomains can have complicated shapes or can have different properties. For instance, different subdomains can represent different phases of the same material, such as water and ice. The coefficients of the PDEs can be discontinuous across the interfaces of the subdomains, and the source terms can be singular. Due to these irregularities, the solutions to the PDEs can be nonsmooth or even discontinuous. Here we restrict ourselves to interface problems that do not depend on time and can be expressed in terms of elliptic or elasticity PDEs. We present finite element methods (FEMs) for elliptic and elasticity problems with interfaces. The FEMs are based on body-fitted meshes with a locally modified triangulation. A FEM based on a body-fitted mesh uses a triangulation that is aligned with the interfaces. However, for complicated interfaces it can be difficult and expensive to generate such triangulations. That is why we use a locally modified triangulation based on Cartesian meshes. We first form a Cartesian mesh, then move the grid points near the interfaces to the interfaces. This leads to a locally modified triangulation. We use the standard FEM with the locally modified triangulation to solve the elliptic and elasticity problems with interfaces. By FEM theory, the method is second order accurate in the infinity norm for piecewise smooth solutions. We present some numerical examples to show the second order accuracy of the method. We also present a new second order finite difference method that does not require to compute the curvature. At points away from the interface we can approximate the PDE by using the standard 5-point scheme. At points where the interface crosses the 5-point scheme, we still use the 5-point scheme by introducing some ghost values for the grid points on the other side of interface. The price is that we need to find an equation for each ghost value. We will use the interface conditions, either the jump in Dirichlet or Neumann boundary conditions, to form the equations for the ghost values to complete the linear system. We also present some numerical examples to show the second order accuracy of the method.
The Fixed Points of a Seasonal Model of Population Infectives
(2007-04-30) Gaither, Jeffrey Beau Sellers; John Franke, Committee Chair; Xiao-Biao Lin, Committee Member; James Selgrade, Committee Member
We model the spread of epidemics among insect populations. The mapping Ft = (1−e−INt )(Nt −I) + I iterates on the current number of infectants to produce the number of infectants in the next time-period. The value Nt is the current population, and it is known that population follows a globally attracting cycle N1 . . .Np, which represents the population at various times of the year. Thus, the function F = Fp ο ... οF1 maps infectants to infecants on a month-to-month or seasonto- season basis. We show that for p = 2, F has only one attractor. We also show that for any F there is 0 such that for any > 0, F has only one attractor. We give an example of multiple attractors in the p = 4 case, and provide a means by which the composition F can be represented as a composition of functions which are all scalar multiples of F1.
Immersed-Interface Finite-Element Methods for Elliptic and Elasticity Interface Problems
(2007-07-31) Gong, Yan; Jason Osborne, Committee Member; Sharon Lubkin, Committee Member; Zhilin Li, Committee Chair; Xiao-Biao Lin, Committee Member
The purpose of the research has been to develop a class of new finite-element methods, called immersed-interface finite-element methods, to solve elliptic and elasticity interface problems with homogeneous and non-homogeneous jump conditions. Simple non-body-fitted meshes are used. Single functions that satisfy the same non-homogeneous jump conditions are constructed using a level-set representation of the interface. With such functions, the discontinuities across the interface in the solution and flux are removed; and equivalent elliptic and elasticity interface problems with homogeneous jump conditions are formulated. Special finite-element basis functions are constructed for nodal points near the interface to satisfy the homogeneous jump conditions. Error analysis and numerical tests are presented to demonstrate that such methods have an optimal convergence rate. These methods are designed as an efficient component of the finite-element level-set methodology for fast simulation of interface dynamics that does not require re-meshing. Such simulation has been a powerful numerical approach in understanding material properties, biological processes, and many other important phenomena in science and engineering.
An Investigation of Aerosol Filtration via Fibrous Filters
(2008-11-06) Wang, Qiqi; Behnam Pourdeyhimi, Committee Co-Chair; Hooman Vahedi Tafreshi, Committee Co-Chair; Timothy Clapp, Committee Member; Xiao-Biao Lin, Committee Member
The most common method of removing particles from a gas stream is via fibrous filters. However, most of the previous studies have been limited to systems consisting of rows of fibers (often in two-dimensional geometries) perpendicular to the flow direction. The current work is aiming to develop an understanding of the role of filter?s microstructure and manufacturing process. In the first part of this study, pressure drop and nanoparticle collection efficiency of lightweight spun-bonded media are simulated by solving the Navier-Stokes equations inside three-dimensional geometries resembling the microstructure of such media. These pressure drop and collection efficiencies showed a perfect agreement with experimental data. In the second part of this work, the influences of fiber length and compaction ratio of filter media on the pressure drop are discussed. Simulation data of staple fiber media have shown good agreement with Davies? empirical equation. Such an agreement indicates that, within the range of dimensions considered, the fiber length has no significant influence on the materials? through-plane permeability as long as the SVF remains constant. Our simulation results for nonwovens with different compaction ratios, together with our experimental data, indicate that pressure drop of the porous media increases with increasing the compaction ratio or temperature of the calender rolls. In the third part of this work, we presented our approach for modeling permeability of fibrous filters with bimodal fiber size distributions (referred to as bimodal filters in this context). The three-dimensional microstructures resembling bimodal filter media with random in-plane fiber orientation distribution were generated to compute their permeability constants. These results were compared with the previous analytical and numerical models as well as our experimental data. Here we concluded that there exists an area-weighted equivalent average diameter for each bimodal filter that can be used in the existing expressions for calculating the permeability of unimodal filters. The last part of this thesis is dedicated to studying the permeability woven fabrics. Concerned with the accuracy of the homogeneous anisotropic lumped model of Gebart (1992) for predicting the permeability of multifilament fabrics, we devised a series of numerical simulations conducted in full three-dimensional geometry of idealized multifilament woven fabrics wherein the filaments were packed in Hexagonal arrangements. While a relatively good agreement was obtained, our results indicate that Gebart?s model underestimates the permeability of multifilament fabrics at high yarn?s solid volume fractions. We also simulated the pressure drop of monofilament woven fabrics under tension where we observed a logarithmic relationship between the discharge coefficient and the Reynolds number of the flow.