Browsing by Subject "Finite element"
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Item Devising superconvergent HDG methods for partial differential equations(2012-08) Shi, KeThe DG methods are ideally suited for numerically solving hyperbolic problems. However this is not the case for diffusion problems,even though they are ideally suited for hp-adaptivity. Indeed, when compared with the classical continuous Galerkin methods on the same mesh, they have many more global degrees of freedom and they are not easy to implement. When compared with the mixed methods, they do not provide optimally convergent approximations to the flux and do not display superconvergence properties of the scalar variable. As a response to these disadvantages, the HDG methods were introduced in [6]. Therein, it was shown that HDG methods can be implemented as efficiently as the mixed methods. Later in [7] it was proven that the HDG methods do share with mixed methods their superior convergence properties while retaining the advantages typical of the DG methods. Inspired by these results, in this Thesis we are trying to explore HDG methods in a wider circumstance.Item Iris biomechanics in health and disease.(2010-05) Amini, RouzbehComputational models of the eye have been studied by various investigators. The main purpose of developing a computational model is to provide a better understanding of the normal function of the eye as well as the abnormalities causing ocular diseases. For instance, by using computational methods, new insights have been brought to the pathophysiology and anatomical risk factors of angle-closure glaucoma, a mysterious eye disease closely related to the mechanics of the iris. Unlike the clinical research, computational studies are neither hindered by experimental difficulties nor by patient health risks. We developed computational models of the ocular tissues at three different levels to understand the mechanisms by which ocular globe deformation, iris-aqueous-humor interaction, and detailed iris structure affect the iris configuration. These models include: a finite-element model of the whole ocular globe consisting of the iris, cornea, sclera, and limbus, a finite-element model of iris-aqueous-humor interaction in the anterior eye, and a finite element model of the iris with its active and passive constituent tissues. Our whole-globe simulation showed that corneoscleral indentation, a diagnostic and/or treatment method in various glaucoma-related complications, would lead to changes in the anterior chamber angle. Our model showed that the limbus, due to its unique mechanical properties, plays an important role in the deformation of the whole ocular globe. Simulations performed using our anterior-segment model showed that the rapid changes (~sec) in the iris-aqueous-humor system due to corneoscleral indentation may lead to long recovery times (~min). We showed that a similar long recovery mechanism prevents the iris from drifting forward during normal blinking. Finally, simulations based on the detailed iris anatomy showed that the posterior location of the dilator muscle could contribute to the iris anterior bowing following dilation even in the absence of the aqueous humor pressure difference. Clinical studies have emphasized the key role of the iris shape and configuration in physiology and pathophysiology of the eye. In the course of our research, we showed that iris configuration is ultimately affected by many parameters including deformation of the whole ocular globe, interaction with aqueous humor flow, and activation of its constituent muscles.Item A jumping multigrid method via finite element extrapolation(University of Minnesota. Institute for Mathematics and Its Applications, 2010-12) Chen, Chuanmiao; Hu, Hongling; Xie, Ziqing; Zhang, ShangyouItem Modeling of continuum transport and meso-scale kinetics during solution crystal growth(2014-05) Wang, WeiSolution crystal growth is widely applied in many industries and fundamental research, and it is employed to crystallize materials ranging from inorganic molecules, small organic molecules, to large organic molecules. However, despite the broad application, fundamental factors regarding this crystal growth process are not well understood. In this thesis, numerical models are developed to study the influences of macro-scale mass transfer limitations and meso-scale growth kinetics on solution crystal growth. A parallel, finite element model is implemented to compute three-dimensional fluid flow and mass transfer during crystal growth and is especially applied to the growth systems in Atomic Force Microscopy fluid cells. This work assesses the parametric sensitivity of growth conditions to factors such as the strength of flow, the frequency of scanning motion, the size of the crystal, and the kinetics of the growing surface. Accounting for such effects will be very important to understand solution crystal growth and to interpret AFM measurements of growth dynamics. Additionally, a simplified two-dimensional numerical model focused on the region near the growing crystal surface and the AFM cantilever was developed based on the calculated results of the three-dimensional model. With this two-dimensional model, we provide basic understanding of the fluid flow and mass transfer where the AFM measurements were made, and simplified the revision of AFM measurements interpretation.A fundamental theoretical model based on the phase-field approach is developed to simulate nano-scale island growth and spiral step growth on crystal surfaces in a supersaturated liquid and is validated by comparison to zinc oxide nanowires synthesis experiments. Results obtained by this work help to explain how experimental factors affect the crystal growth and crystal microstructures and the correlation between island growth and spiral growth mechanisms.