published_or_final_version / abstract / Civil Engineering / Doctoral / Doctor of Philosophy

published_or_final_version / Mechanical Engineering / Doctoral / Doctor of Philosophy

Three-dimensional acoustic propagation in shallow water waveguides is studied using the longitudinally invariant finite element method. This technique is appropriate for environments with lateral variations that occur in only one dimension. In this method, a transform is applied to the three-dimensional Helmholtz equation to remove the range-independent dimension. The finite element method is employed to solve the transformed Helmholtz equation for each out-of-plane wavenumber. Finally, the inverse transform is used to transform the pressure field back to three-dimensional spatial coordinates. Due to the oscillatory nature of the inverse transform, two integration techniques are developed. The first is a Riemann sum combined with a wavenumber sampling method that efficiently captures the essential components of the integrand. The other is a modified adaptive Clenshaw-Curtis quadrature. Three-dimensional transmission loss is computed for a Pekeris waveguide, underwater wedge, and Gaussian canyon. For each waveguide, the two integration schemes are compared in terms of accuracy and efficiency. / text

Advection-dispersion is generally solved numerically with methods that treat the problem from one of three perspectives. These are described as the Eulerian reference, the Lagrangian reference or a combination of the two that will be referred to as Eulerian-Lagrangian. Methods that use the Eulerian-Lagrangian approach incorporate the computational power of the Lagrangian treatment of advection with the simplicity of the fixed Eulerian grid. A modified version of a relatively new adaptive Eulerian-Lagrangian finite element method is presented for the simulation of advection-dispersion. Advection is solved by an adaptive technique that automatically chooses a local solution technique based upon a criterion involving the spatial variation of the gradient of the concentration. Moving particles (the method of characteristics; MOC) are used to define the concentration field in areas with significant variation of the concentration gradient. A modified method of characteristics (MMOC) called single-step reverse particle tracking is used to treat advection in areas with fairly uniform concentration gradients. As the simulation proceeds, the adaptive technique, as needed to maintain solution accuracy and optimal simulation efficiency, adjusts the advection solution process by inserting and deleting moving particles to shift between MMOC and MOC. Dispersion is simulated by a finite element formulation that involves only symmetric and diagonal matrices. Despite evidence from other investigators that diagonalization of the mass matrix may lead to poor solutions to advection-dispersion problems, this method seems to allow "lumping" of the mass matrix by essentially decoupling advection and dispersion. Based on tests of problems with analytical solutions, the method seems capable of reliably simulating the entire range of Peclet numbers with Courant numbers that range to 15.

This research is an investigation into two different aspects of the faulting process. The first part of the study focuses on the initial stages of fault formation, while the second analyzes the deformation produced by an existing fault. The section on fault formation is an attempt to determine whether slip on an existing fault has a significant effect on the formation of subsequent faults. A two-dimensional elastic finite element technique is used to examine the system of stresses produced by slip on an initial fault, assuming that deformation occurs either elastically or by brittle failure. A Mohr-Coulomb failure criterion is used to determine the most likely region of secondary fault initiation. A strain energy criterion is then used to find the preferred direction of fault propagation. The study on fault formation is subdivided into two sections representing two idealized tectonic environments: purely extensional and purely compressional. The section on extensional fault formation explains the prevalence of grabens in extensional tectonic regimes as a consequence of the stress perturbations due to slip on an initial normal fault. Slip on the initial fault produces a region of high proximity to failure at the surface of the downthrown block. A secondary fault would be expected to initiate in this region. The direction of propagation of this fault that most effectively relieves the shear stress (and therefore minimizes the total strain energy) is toward the initial fault, resulting in an antithetic orientation, or graben. The width of the graben is found to be controlled by the depth of the initial normal fault, rather than the depth to a change in material properties. The study of compressional fault formation indicates that, except for steeply-dipping faults, the presence of an initial thrust fault tends to suppress the formation of other faults in its vicinity. However, if a secondary fault initiates near an initial thrust fault, the direction in which it propagates will be influenced by the presence of the initial fault. The way in which it is influenced is dependent on the fault dip. The final part of this study examines the deformation produced by repeated earthquake cycles on the San Andreas fault in southern California. A three-dimensional, time-dependent kinematic finite element model is used to investigate the influence of slip distribution and rheological parameters on the predicted horizontal and vertical deformation. The models include depth-varying rheological properties and power-law viscoelastic behavior. The predicted deformation patterns are fairly sensitive to the parameters used in this study. Of particular importance is the calculation of vertical uplift rate since, in many cases, models that cannot be distinguished from each other on the basis of horizontal deformation may produce distinctive vertical uplift patterns.

This project investigates the heat flow from the rock into ventilating airways by studying various parameters. Two approaches have been used: laboratory measurement of thermal properties to study their variation, and analytic and numerical models to study the effect of these variations on the heat flow. Access to a heat-flux system and special treatment of contact resistance has provided the opportunity to study thermal conductivity as a function of moisture contained in rock specimens. For porous sandstone, tuff, and concretes, thermal conductivity can double when the specimens are soaked; the functional dependence of conductivity on moisture for the first two cases is definitely non-linear. Five previous models for conductivity as a function of porosity are shown not to explain this new phenomenon. A preliminary finite element model is proposed which explains the key features. Other variations of conductivity with applied pressure, location, constituents, weathering or other damage, and anisotropy have been measured. In the second phase of the research, analytical and numerical methods have been employed to consider the effects of the variation in the thermal properties plus the use of insulation on the heat flow from the rock into the ventilated and cooled airways. Temperature measurements taken in drill holes at a local mine provide confirmation for some of the models. Results have been provided in a sensitivity analysis mode so that engineers working on other projects can see which parameters would require more detailed consideration. The thermal conductivity of the rock close to the airways is a key factor in affecting heat loads. Dewatering and the use of insulation, such as lightweight foamed shotcretes, are recommended.

A finite element formulation, based on assumed stress functions, is developed for the linear elastic analysis of the stresses in two-dimensional elasticity problems, including multiply-connected regions and flat plate bending. For planar analysis the Airy stress function is utilized. The physical significance of the Airy stress function and its normal derivatives are brought out. A new technique is introduced to account for traction type boundary conditions. A family of rectangular finite elements, which enables the direct insertion of stress type boundary conditions, and two higher-order rectangular elements which enable continuous stress variations along the interelement boundaries are constructed using blending function interpolants. In addition, a C° continuous triangular plate bending element is adapted for use as a plate stretching element. The Southwell stress function is employed for the analysis of flat plates in bending. A computer program is developed to substantiate the proposed methodology. The formulations are evaluated through the comparison of solutions obtained from the proposed method with classical solutions and solutions obtained from the assumed displacement finite element method. The elements are evaluated by solving the same example problem with different element types. Extensions of the proposed method to account for body forces, initial stresses, material nonlinearities, and shells are briefly discussed. It is demonstrated that the proposed method can directly be integrated with minimal modifications into existing general purpose finite element programs.

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In the research undertaken a finite element formulation has been developed for an elastodynamic problem using a least squares approach. The special requirements of the problem demanded a study of suitability of various elements. The emergence of the final element is a result of both theoretical and numerical study of three different elements. The approximation function is assumed on the basis of the order of the governing differential equations. Then the square of the error resulting from the approximate solution is minimized over the entire domain as well as the boundaries in the same functional. The element equation emerging from the formulation does not yield a singular stiffness matrix, since the boundary conditions are already taken into account in the element equation. The formulation presented in this thesis is only for the normal propagation of phi-wave. A finite element code has been developed based on the new formulation.

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