published_or_final_version / abstract / toc / Physics / Master / Master of Philosophy

In this research, we study two different geometric approaches, namely, the discrete exterior calculus and differential complexes, for developing numerical schemes for linear and nonlinear elasticity. Using some ideas from discrete exterior calculus (DEC), we present a geometric discretization scheme for incompressible linearized elasticity. After characterizing the configuration manifold of volume- preserving discrete deformations, we use Hamilton’s principle on this configuration manifold. The discrete Euler-Lagrange equations are obtained without using Lagrange multipliers. The main difference between our approach and the mixed finite element formulations is that we simultaneously use three different discrete spaces for the displacement field. We test the efficiency and robustness of this geometric scheme using some numerical examples. In particular, we do not see any volume locking and/or checkerboarding of pressure in our numerical examples. This suggests that our choice of discrete solution spaces is compatible. On the other hand, it has been observed that the linear elastostatics complex can be used to find very efficient numerical schemes. We use some geometric techniques to obtain differential complexes for nonlinear elastostatics. In particular, by introducing stress functions for the Cauchy and the second Piola-Kirchhoff stress tensors, we show that 2D and 3D nonlinear elastostatics admit separate kinematic and kinetic complexes. We show that stress functions corresponding to the first Piola-Kirchhoff stress tensor allow us to write a complex for 3D nonlinear elastostatics that similar to the complex of 3D linear elastostatics contains both the kinematics an kinetics of motion. We study linear and nonlinear compatibility equations for curved ambient spaces and motions of surfaces in R3. We also study the relationship between the linear elastostatics complex and the de Rham complex. The geometric approach presented in this research is crucial for understanding connections between linear and nonlinear elastostatics and the Hodge Laplacian, which can enable one to convert numerical schemes of the Hodge Laplacian to those for linear and possibly nonlinear elastostatics.

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This thesis is devoted to two aspects of nonlinear PDEs which are fundamental for the understanding of the order and coherence observed in the underlying physical systems. These are symmetry properties and soliton solutions. We analyse these fundamental aspects for a number of models arising in various branches of theoretical physics and appli ed mathematics. We start with a fluid model of a plasma in the case of a general polytropic process. We propose a method of the analysis of unmagnetized travelling structures, alternative to the conventional formalism of Sagdeev 's pseudopotential. This method is then utilized to obtain the existence domain for compressive solitons and to establish the absence of rarefactive solitons and monotonic double layers in a two-component plasma. The second class of models under consideration arises in (2+1)-dimensional condensed matter physics. These are the Abelian gauge theories with Chern-Simons term, which are currently considered as candidates for the description of high-Te superconductivity and fra ctional quantum Hall effect. The emphasis here is on nonrelativistic theories. The standard model of a self-gravitating gas of nonrelativistic bosons coupled to the Chern-Simons gauge field is capable of describing asymptotically vanishing field configurations , such as lump-like solitons. We formulate an alternative model, which describes systems of repulsive particles with a background electric charge and allows to incorporate asymptotically nonvanishing configurations, such as condensate and its topological excitations. We demonstrate the absence of the condensate state in the standard nonrelativistic gauge theory and relate this fact to the inadequate Lagrangian formulation of its nongauged precursor. Using an appropriate modification of this Lagrangian as a basis for the gauge theory naturally leads to the new model. Reformulating it as a constrained Hamiltonian system allows us to find two self-duality limit s and construct a large variety of self-dual solutions. We demonstrate the equivalence of the model with the background charge and the standard model in the external magnetic field. Finally we discuss nontopological bubble solutions in Chem-Simons-Maxwell theories and demonstrate their absence in nonrelativistic theories. Finally, we consider a model of a nonhomogeneous nonlinear string. We continue the group theoretical classification of the string equations initiated by Ibragimov et al. and present their preliminary group classification with respect to a countable dimensional subalgebra of their equivalence algebra. This subalgebra is an extension of the 10-dimensional subalgebra considered by Ibragimov et al. Our main result here is a table of non-equivalent equations possessing an additional symmetry. / Thesis (Ph.D.)-University of Natal, 1994.

This thesis is divided into five chapters. The scope of problems considered is defined in chapter I. Next, chapter II provides background material on image processing with partial differential equations and a review of prior work in the field. Chapter III covers the medical imaging portion of the research; the key contribution is a control-based algorithm for interactive image segmentation. Applications of the feedback-augmented level set method to fracture reconstruction and surgical planning are shown. Problems in vision-based control are considered in Chapters IV and V. A method of improving performance in closed-loop target tracking using level set segmentation is developed, with unmanned aerial vehicle or next-generation missile guidance being the primary applications of interest. Throughout this thesis, the two application types are connected into a unified viewpoint of open-loop systems that are augmented by exogenous data.

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Typescript. / Thesis (Ph. D.)--University of Hawaii at Manoa, 1990. / Includes bibliographical references. / Microfiche. / vi, 96 leaves, bound ill. 29 cm

Diss. (sammanfattning) Uppsala : Univ., 2004. / Härtill 8 uppsatser.

Diss. (sammanfattning) Uppsala : Univ., 2004. / Härtill 8 uppsatser.

Diss. (sammanfattning) Uppsala : Uppsala universitet, 2005. / Härtill 5 uppsatser.