Thesis (M.S.)--University of Wisconsin--Madison, 1969. / eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

This work is concerned with the use of isogeometric analysis based on Non- Uniform Rational B-Splines (NURBS) to develop efficient and robust numerical techniques to deal with the problems of incompressibility in the fields of solid and fluid mechanics. Towards this, two types of formulations, mixed Galerkin and least-squares, are studied. During the first phase of this work, mixed Galerkin formulations, in the context of isogeometric analysis, are presented. Two-field and three-field mixed variational formulations - in both small and large strains - are presented to obtain accurate numerical solutions for the problems modelled with nearly incompressible and elasto-plastic materials. The equivalence of the two mixed formulations, for the considered material models, is derived; and the computational advantages of using two-field formulations are illustrated. Performance of these formulations is assessed by studying several benchmark examples. The ability of the mixed methods, to accurately compute limit loads for problems involving elastoplastic material models; and to deal with volumetric locking, shear locking and severe mesh distortions in finite strains, is illustrated. Later, finite element formulations are developed by combining least-squares and isogeometric analysis in order to extract the best of both. Least-squares finite element methods (LSFEMs) based on the use of governing differential equations directly - without the need to reduce them to equivalent lower-order systems - are developed for compressible and nearly incompressible elasticity in both the small and finite strain regimes; and incompressible Navier-Stokes. The merits of using Gauss-Newton scheme instead of Newton-Raphson method to solve the underlying nonlinear equations are presented. The performance of the proposed LSFEMs is demonstrated with several benchmark examples from the literature. Advantages of using higher-order NURBS in obtaining optimal convergence rates for non-norm-equivalent LSFEMs; and the robustness of LSFEMs, for Navier-Stokes, in obtaining accurate numerical solutions without the need to incorporate any artificial stabilisation techniques, are demonstrated.

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Partial Least Squares (PLS) is an important statistical technique with multipleand diverse applications, used as an effective regression method for correlated orcollinear datasets or for datasets that are not full rank for other reasons. A shorthistory of PLS is followed by a review of the publications where the issues with theapplication PLS that have been discussed. The theoretical basis of PLS is developedfrom the single value decomposition of the covariance, so that the strong links between principal components analysis and within the various PLS algorithms appear as a natural consequence. Latent variable selection by crossvalidation, permutation and information criteriaare examined. A method for plotting crossvalidation results is proposed that makeslatent variable selection less ambiguous than conventional plots. Novel and practicalmethods are proposed to extend published methods for latent variable selection byboth permutation and information criteria from univariate PLS1 models to PLS2 multivariate cases. The numerical method proposed for information criteria is also more general than the algebraic methods for PLS1 that have been recently published as it does not assume any particular form for the PLS regression coefficients. All of these methods have been critically assessed using a number of datasets, selected specifically to represent a diverse set of dimensions and covariance structures. Methods for simulating multivariate datasets were developed that allow controlof correlation and collinearity in both regressors and responses independently. Thisdevelopment also allows control over the variate distributions. Statistical design ofexperiments was used to generate plans for the simulation that allowed the factorsthat infuence PLS model fit and latent variable selection. It was found that all thelatent variable selection methods in the simulation tend to overfit and the feature inthe simulation that causes overfitting has been identified.

This thesis designs new methods for Q tomography and Q-compensated prestack depth migration when the recorded seismic data suffer from strong attenuation. A motivation of this work is that the presence of gas clouds or mud channels in overburden structures leads to the distortion of amplitudes and phases in seismic waves propagating inside the earth. If the attenuation parameter Q is very strong, i.e., Q<30, ignoring the anelastic effects in imaging can lead to dimming of migration amplitudes and loss of resolution. This, in turn, adversely affects the ability to accurately predict reservoir properties below such layers. To mitigate this problem, I first develop an anelastic least-squares reverse time migration (Q-LSRTM) technique. I reformulate the conventional acoustic least-squares migration problem as a viscoacoustic linearized inversion problem. Using linearized viscoacoustic modeling and adjoint operators during the least-squares iterations, I show with numerical tests that Q-LSRTM can compensate for the amplitude loss and produce images with better balanced amplitudes than conventional migration. To estimate the background Q model that can be used for any Q-compensating migration algorithm, I then develop a wave-equation based optimization method that inverts for the subsurface Q distribution by minimizing a skeletonized misfit function ε. Here, ε is the sum of the squared differences between the observed and the predicted peak/centroid-frequency shifts of the early-arrivals. Through numerical tests on synthetic and field data, I show that noticeable improvements in the migration image quality can be obtained from Q models inverted using wave-equation Q tomography. A key feature of skeletonized inversion is that it is much less likely to get stuck in a local minimum than a standard waveform inversion method. Finally, I develop a preconditioning technique for least-squares migration using a directional Gabor-based preconditioning approach for isotropic, anisotropic or anelastic least-squares migration. During the least-squares iterations, I impose sparsity constraints on the inverted reflectivity model in the local Radon domain. The forward and the inverse mapping of the reflectivity to the local Radon domain is done through 3D Fourier-based discrete Radon transform operators. Using numerical tests on synthetic and 3D field data, I demonstrate that the proposed preconditioning approach can discriminate against artifacts in the image resulting from irregular or insufficient acquisition and can produce images with improved signal-to-noise ratio when compared with standard migration.

The purpose of this paper is to examine the interactions between two plant species endemic to Florida and develop a model for the growth of one of the plant species. An equation for the growth of Hypericum cumulicola is developed through analyzing how the distance to and the height of the nearest Ceratiola ericoides (Florida rosemary) affects the growth of Hypericum cumulicola. The hypericums were separated into five separate regions according to the distance to the nearest rosemary plant. The parameters for a basic growth equation were obtained in each of the five regions and compared to each other along with the average deviations in each of the five regions. Analysis of the five separate regions aided in the creation of different growth equations that each encompassed all of the regions together. Four different growth equations are developed and then compared and analyzed for their accuracy.

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