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The boundary integral technique was implemented in a computer code for the general static analysis of three dimensional elastic solids. The was based on a formulation of the problem in which the governing boundary equation is developed from the known solution to Kelvin's problem, by the application of Betti's reciprocal relationship. Modeling the boundary of the region being analyzed with plane elements and assuming the tractions and displacements constant across these elements leads to a set of simultaneous algebraic equations approximating the boundary integral equation. Numerical techniques are used in the computer code to assemble and solve this set of equations. The operation of this code was demonstrated by the solution of several example problems. The results of these problems show the code to be successful. It's practical application however is limited due to the large solution time required. This time would be significantly reduced if a more efficient equation solver were employed. The time requirement could be a severe limitation when a relatively large number of elements is needed to model displacement gradients. The development of an element based on linear or higher order variation of displacements would greatly reduce the required mesh size in this case and thus the solution time.

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Mathematical modeling of physical processes often leads to systems of differential and algebraic equations involving quantities of interest. A computer model created based on these equations can be numerically integrated to predict future states of the system and its evolution in time. This thesis investigates current methods in numerical time-stepping schemes, identifying a number of important features needed to speed up and increase the accuracy of the solutions. The focus is on developing new methods suitable for large-scale applications with multiple physical processes, potentially with significant differences in their time-scales. Various families of new methods are introduced with special attention to multirating, low computational cost implicitness, high order of convergence, and robustness. For each family, the order condition theory is discussed and a number of examples are derived. The accuracy and stability of the methods are investigated using standard analysis techniques and numerical experiments are performed to verify the abilities of the new methods. / Doctor of Philosophy / Mathematical descriptions of physical processes are often in the form of systems of differential equations describing the time-evolution of a phenomenon. Computer simulations are realizations of these equations using well-known discretization schemes. Numerical time-stepping methods allow us to advance the state of a computer model using a sequence of time-steps. This thesis investigates current methods in time-stepping schemes, identifying a number of additional features needed to improve the speed and accuracy of simulations, and devises new methods suitable for large-scale applications where multiple processes of different physical nature drive the equations, potentially with significant differences in their time-scales. Various families of new methods are introduced with proper mathematical formulations provided for creating new ones on demand. The accuracy and stability of the methods are investigated using standard analysis techniques. These methods are then used in numerical experiments to investigate their abilities.

In many physiological studies, variables of interest are not directly accessible, requiring that they be estimated indirectly from noisy measured signals. Here, we introduce two empirical methods to estimate the true physiological signals from indirectly measured, noisy data. The first method is an extension of Tikhonov regularization to large-scale problems, using a sequential update approach. In the second method, we improve the conditioning of the problem by assuming that the input is uniform over a known time interval, and then we use a least-squares method to estimate the input. These methods were validated computationally and experimentally by applying them to flow-through respirometry data. Specifically, we infused CO2 in a flow-through respirometry chamber in a known pattern, and used the methods to recover the known input from the recorded data. The results from these experiments indicate that these methods are capable of sub-second accuracy. We also applied the methods on respiratory data from a grasshopper to investigate the exact timing of abdominal pumping, spiracular opening, and CO2 emission. The methods can be used more generally for input estimation of any linear system. / Master of Science / The goal of an inverse problem is to determine some signal or parameter of interest that is not directly accessible but can be obtained from an observed effect or a processed version that is measurable. Finding the gas exchange signal in animals is an example of an inverse problem. One method to noninvasively measure the gas exchange rate of animals is to put them in a respirometry chamber, flow air through the chamber, and measure the concentration of the respiratory gasses outside the chamber. However, because the gasses mix in the chamber and gradually flow through the gas analyzer, the pattern of the measured gas concentration can be dramatically different than the true pattern of real instantaneous gas exchange of the animal. In this thesis, we present two methods to recover the true signal from the recorded data (i.e., for inverse reconstruction), and we evaluate them computationally and experimentally.

Variational inverse problems integrate computational simulations of physical phenomena with physical measurements in an informational feedback control system. Control parameters of the computational model are optimized such that the simulation results fit the physical measurements.The solution procedure is computationally expensive since it involves running the simulation computer model (the emph{forward model}) and the associated emph {adjoint model} multiple times. In practice, our knowledge of the underlying physics is incomplete and hence the associated computer model is laden with emph {model errors}. Similarly, it is not possible to measure the physical quantities exactly and hence the measurements are associated with emph {data errors}. The errors in data and model adversely affect the inference solutions. This work develops methods to address the challenges posed by the computational costs and by the impact of data and model errors in solving variational inverse problems. Variational inverse problems of interest here are formulated as optimization problems constrained by partial differential equations (PDEs). The solution process requires multiple evaluations of the constraints, therefore multiple solutions of the associated PDE. To alleviate the computational costs we develop a parallel in time discretization algorithm based on a nonlinear optimization approach. Like in the emph{parareal} approach, the time interval is partitioned into subintervals, and local time integrations are carried out in parallel. Solution continuity equations across interval boundaries are added as constraints. All the computational steps - forward solutions, gradients, and Hessian-vector products - involve only ideally parallel computations and therefore are highly scalable. This work develops a systematic mathematical framework to compute the impact of data and model errors on the solution to the variational inverse problems. The computational algorithm makes use of first and second order adjoints and provides an a-posteriori error estimate for a quantity of interest defined on the inverse solution (i.e., an aspect of the inverse solution). We illustrate the estimation algorithm on a shallow water model and on the Weather Research and Forecast model. Presence of outliers in measurement data is common, and this negatively impacts the solution to variational inverse problems. The traditional approach, where the inverse problem is formulated as a minimization problem in $L_2$ norm, is especially sensitive to large data errors. To alleviate the impact of data outliers we propose to use robust norms such as the $L_1$ and Huber norm in data assimilation. This work develops a systematic mathematical framework to perform three and four dimensional variational data assimilation using $L_1$ and Huber norms. The power of this approach is demonstrated by solving data assimilation problems where measurements contain outliers. / Ph. D.

The discussion in this paper focuses on how boundary based smooth shape design can be carried out. For this we treat surface generation as a mathematical boundary-value problem. In particular, we utilize elliptic Partial Differential Equations (PDEs) of arbitrary order. Using the methodology outlined here a designer can therefore generate the geometry of shapes satisfying an arbitrary set of boundary conditions. The boundary conditions for the chosen PDE can be specified as curves in 3-space defining the profile geometry of the shape. We show how a compact analytic solution for the chosen arbitrary order PDE can be formulated enabling complex shapes to be designed and manipulated in real time. This solution scheme, although analytic, satisfies exactly, even in the case of general boundary conditions, where the resulting surface has a closed form representation allowing real time shape manipulation. In order to enable users to appreciate the powerful shape design and manipulation capability of the method, we present a set of practical examples

This thesis is concerned with the analysis of a Rectilinear Distance Location Allocation Problem, where the costs are directly proportional to rectilinear distances and the amount shipped. The problem is formulated as a Mixed Integer Bilinear Programming Problem and as a Discrete Location Allocation Problem. Using linear programming relaxations constructed via the Reformulation-Linearization Technique (RLT), the latter formulation is shown to provide stronger lower bounds and is therefore adopted for implementation. In addition, cutting planes are developed to further strengthen the linear programming relaxation. The special structure of the resulting linear program is exploited in order to get a quick lower bound via a suitable Lagrangian dual formulation. This lower bounding scheme is embedded within a finitely convergent Branch and Bound algorithm that enumerates over the location decision variable space. An illustrative example and computational experience are provided to demonstrate the efficacy of the proposed algorithm. / Master of Science

This study draws on social problems literature about rhetoric in claims-making and social movement literature about credibility in framing to understand the construction of YouTube abuse and relationships between member role in the community and their frames/the reception of those frames. I also draw on feminist, non-feminist, and postfeminist literature to understand how YouTubers incorporate feminism into their claims about why YouTube abuse is wrong. Here feminism refers to understandings of sexual harassment as stemming from gender inequality, and non-feminist understandings of sexual harassment refer to individualized and degendered violations of rights and power imbalances. Postfeminist literature informs this study in understanding how a feminist issue has been disassociated with gender inequality and individualized. Drawing on this literature, I conducted a content analysis of YouTube videos and the comment sections on these YouTube video webpages to address how the community members responded to the sexual harassment problem. First, how do the YouTubers describe the problem? Second, what explanations for why the behavior is wrong, do the YouTubers use? Options include portraying the issue using a more feminist frame of "gender equality," a post-feminist frame of gender-neutral "consent," or a gender-neutral frame of "power imbalance." Lastly, are there relationships between the YouTubers' position in the community and/or gender, their responses, and positive and negative comments left on the videos? Analysis supports that YouTubers did not connect the issue to feminism and that YouTubers' positions in the community relate to how they politicized the abuse and how much commentator support they received. / Master of Science