Theses and Dissertations from UMD

Permanent URI for this communityhttp://hdl.handle.net/1903/2

New submissions to the thesis/dissertation collections are added automatically as they are received from the Graduate School. Currently, the Graduate School deposits all theses and dissertations from a given semester after the official graduation date. This means that there may be up to a 4 month delay in the appearance of a give thesis/dissertation in DRUM

More information is available at Theses and Dissertations at University of Maryland Libraries.

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    Surface Tension Free Boundary Problems: Formulation, Optimal Control and Numerics
    (2013) Carlos, Patrick Sodré; Nochetto, Ricardo H; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    The goal of this work is to treat the formulation, optimal control and numerical analysis of free boundary problems with surface tension effects. From a formulation point of view, we introduce a (dimension independent) abstract framework which captures the essential behavior of free boundary problems with surface tension effects. We then apply this framework to two scenarios. The first is where the underlying bulk system is governed by the Laplacian with non-homogeneous essential boundary condition, and the second is modeled by the Stokes equations with slip and no-slip boundary conditions. We do not impose a fixed contact angle between the free surface and any fixed part of the boundary. Although the formulation and numerics involving the Laplacian was available in the literature, the Stokes free boundary problem in Rn is novel. To obtain this last result we also had to prove the existence and uniqueness in Sobolev spaces for the pure slip problem for domains of type C1,\epsilon. This is a significant improvement over the current best result involving C1,1 domains. The results from the abstract formulation also carry over to the optimal control aspect. We obtain differentiability conditions which guarantee existence and (local) uniqueness of a minimizer to well-behaved cost functions. In the Laplacian case we go beyond the theoretical results and give precise second-order sufficient conditions for the (local) uniqueness of a minimizer for cost functions of the tracking type. The contribution in this area is significant in the sense that sufficient conditions are usually only assumed to be true, while we actually show that it indeed holds for our specific problem. The last piece of this work is the numerical treatment of the free boundary optimal control problem based on the Laplace equation. We are able to prove optimal convergence results using the finite element method. Moreover, we construct experiments to study the behavior of various metrics associated with the optimization problem.
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    A Hybrid Variational-Level Set Approach to Handle Topological Changes
    (2007-05-08) Walker, Shawn; Nochetto, Ricardo H; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    We present a method for allowing explicit, Lagrangian meshes to undergo topological changes in an automatic way. We employ a method for detecting when topological changes are imminent. When a change occurs, we use a level set method to guide the change of topology of the domain mesh. This is then followed by an optimization step, using a variational formulation, that seeks to improve boundary mesh conformity to the zero level contour of the level set function. The advantage of this method is that it directly allows for using a variational formulation of the physics being modeled and simulated, including the ability to account for important geometric information in the model (such as for surface tension driven flow). Furthermore, the level set update and optimization step are only needed during a topological change. Hence, our method does not significantly affect computational cost.