The aim of this work is to demonstrate a Helmholtz potential based approach for the development of the constitutive equations for a shape memory polymer undergoing a thermomechanical cycle. The approach is motivated by the use of a simple spring-dashpot type analogy and the resulting equations are classified as state-equations and suitable kinetic equations for the recoverable-energy elements and the dissipative elements in the model respectively. These elements have mechanical properties which vary with temperature. The governing equations of the model are developed starting from the basic conservation laws together with the laws of thermodynamics. The entire set of equations are written in a state-evolution form as a set of ordinary differential equations to be solved using Matlab. It is shown that the results of the simulation in Matlab are in qualitative and quantitative agreement with experiments performed on polyurethane. Subsequently, we study the dependence of the yield-stress on temperature to be similar and different functions of heating or cooling processes.

In this dissertation we describe a coordinate scaling technique for the numerical approximation of solutions to certain problems posed on unbounded domains in two and three dimensions. This technique amounts to introducing variable coefficients into the problem, which results in defining a solution coinciding with the solution to the original problem inside a bounded domain of interest and rapidly decaying outside of it. The decay of the solution to the modified problem allows us to truncate the problem to a bounded domain and subsequently solve the finite element approximation problem on a finite domain. The particular problems that we consider are exterior problems for the Laplace equation and the time-harmonic acoustic and elastic wave scattering problems. We introduce a real scaling change of variables for the Laplace equation and experimentally compare its performance to the performance of the existing alternative approaches for the numerical approximation of this problem. Proceeding from the real scaling transformation, we introduce a version of the perfectly matched layer (PML) absorbing boundary as a complex coordinate shift and apply it to the exterior Helmholtz (acoustic scattering) equation. We outline the analysis of the continuous PML problem, discuss the implementation of a numerical method for its approximation and present computational results illustrating its efficiency. We then discuss in detail the analysis of the elastic wave PML problem and its numerical discretiazation. We show that the continuous problem is well-posed for sufficiently large truncation domain, and the discrete problem is well-posed on the truncated domain for a sufficiently small PML damping parameter. We discuss ways of avoiding the latter restriction. Finally, we consider a new non-spherical scaling for the Laplace and Helmholtz equation. We present computational results with such scalings and conduct numerical experiments coupling real scaling with PML as means to increase the efficiency of the PML techniques, even if the damping parameters are small.

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Cette thèse se rapporte à certaines de nos publications (avant 1995) concernant des solutions explicites exactes ou asymptotiques pour des problèmes variés de diffraction par un dièdre imparfaitement conducteur.

No description available.

In this thesis I present a new numerical scheme for 2.5-D/3-D direct current resistivity modelling in heterogeneous, anisotropic media. This method, named the ‘Gaussian quadrature grid’ (GQG) method, co-operatively combines the solution of the Variational Principle of the partial differential equation, Gaussian quadrature abscissae and local cardinal functions so that it has the main advantages of the spectral element method. The formulation shows that the GQG method is a modification of the spectral element method and does not employ the constant elements and require the mesh generator to match the earth’s surface. This makes it much easier to deal with geological models having a 2-D/3-D complex topography than using traditional numerical methods. The GQG technique can achieve a similar convergence rate to the spectral element method. It is shown that it transforms the 2.5-D/3-D resistivity modelling problem into a sparse and symmetric linear equation system, which can be solved by an iterative or matrix inversion method. Comparison with analytic solutions for homogeneous isotropic and anisotropic models shows that the error depends on the Gaussian quadrature order (abscissae number) and the sub-domain size. The higher order or smaller the subdomain size employed, the more accurate the solution. Several other synthetic examples, both homogeneous and inhomogeneous, incorporating sloping, undulating and severe topography are presented and found to yield results comparable to finite element solutions involving a dense mesh. The thesis also presents for the first time explicit expressions for the Fréchet derivatives or sensitivity functions in resistivity imaging of a heterogeneous and fully anisotropic earth. The formulation involves the Green’s functions and their gradients, and is developed both from a formal perturbation analysis and by means of a numerical (finite element) method. A critical factor in the equations is the derivative of the electrical conductivity tensor with respect to the principal conductivity values and the angles defining the axes of symmetry; these are given analytically. The Fréchet derivative expressions are given for both the 2.5-D and the 3-D problem using both constant point and constant block model parameterisations. Special cases like the isotropic earth and tilted transversely isotropic (TTI) media are shown to emerge from the general solutions. Numerical examples are presented for the various sensitivities as functions of the dip angle and strike of the plane of stratification in uniform TTI media. In addition, analytic solutions are derived for the electric potential, current density and Fréchet derivatives at any interior point within a 3-D transversely isotropic homogeneous medium having a tilted axis of symmetry. The current electrode is assumed to be on the surface of the Earth and the plane of stratification given arbitrary strike and dip. Profiles can be computed for any azimuth. The equipotentials exhibit an elliptical pattern and are not orthogonal to the current density vectors, which are strongly angle dependent. Current density reaches its maximum value in a direction parallel to the longitudinal conductivity direction. Illustrative examples of the Fréchet derivatives are given for the 2.5-D problem, in which the profile is taken perpendicular to strike. All three derivatives of the Green’s function with respect to longitudinal conductivity, transverse resistivity and dip angle of the symmetry axis (dG/dσ₁,dG/dσ₁,dG/dθ₀ ) show a strongly asymmetric pattern compared to the isotropic case. The patterns are aligned in the direction of the tilt angle. Such sensitivity patterns are useful in real time experimental design as well as in the fast inversion of resistivity data collected over an anisotropic earth. / Thesis (Ph.D.) -- University of Adelaide, School of Chemistry and Physics, 2009

Thesis (Ph. D.)--Brigham Young University. Dept. of Chemical Engineering, 2005. / Includes bibliographical references (p. 145-156).

Zugl.: Aachen, Techn. Hochsch., Diss., 2007.

Thesis (M.S.)--West Virginia University, 2010. / Title from document title page. Document formatted into pages; contains viii, 75 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 72-75).

Techn. Universiẗat, Diss., 2000--Berlin.