Browsing by Subject "Applied Mathematics"

Now showing 1 - 2 of 2
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    Development and Analysis of Computationally Efficient Methods for Analyzing Surface Effects
    (2017-01) Binder, Andrew
    Experiments have shown that materials at the nanoscale exhibit new material prop- erties compared to their macro-counterparts as a result of surface effects. We rigorously examine an atomistic model that exhibits surface effects and estimate the rate of de- cay of this influence. Despite the highly localized nature of surface effects, the regular Cauchy-Born method is shown to be incapable of capturing the surface physics in these systems. Two methods that seek to accurately model the influence of surfaces in a molecular statics problem are examined. First, the surface Cauchy-Born method is examined. An asymptotic analysis is performed to investigate the behavior of this method, and it is shown that the method does represent an improvement over the regular Cauchy-Born method. However, it does not fully capture the surface behavior. Next, a novel predictor-corrector method is introduced to better capture these effects. Using the regular Cauchy-Born solution as a predictor for material behavior, the solution is corrected over a small boundary layer at the surface of a 1D material. The decomposition of the approximation into a bulk and surface component is justified in the analysis, and the convergence of the approximation to the atomistic solution is shown. The analysis for both methods is then verified numerically.
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    A new approach to finite element simulations of general relativity
    (2015-06) Quenneville-Belair, Vincent
    In order to study gravitational waves, we introduce a new approach to finite element simulation of general relativity. This approach is based on approximating the Weyl curvature directly through new stable mixed finite elements for the Einstein-Bianchi system. We design and analyze these novel finite elements by adapting the recently developed Finite Element Exterior Calculus (FEEC) framework to abstract Hodge wave equations. This framework enables us to borrow key ideas from Reissner-Mindlin plate bending and elasticity with weakly imposed symmetries to maintain stability of the method. The stability of a discretization often relies on deep connections between fundamental branches of mathematics: the FEEC mimics these connections for the numerical method to achieve similar stability to that of the original equations. The recent development of FEEC has had a transformative impact on electromagnetism and related computational problems, and we are expanding it to general relativity.