This thesis investigates some of the properties of a variational approximation to scalar field theories: a trial wavefunctional which has a gaussian form is used as a ground state ansatz for an interacting scalar field theory - the expectation value of the Hamiltonian in this state is then minimized. This we call the Gaussian Approximation; the resulting effective potential we follow others by calling the Gaussian Effective Potential (GEP). An equivalent but more general finite temperature formalism is then reviewed and used for the calculations of the GEP in this thesis. Two scalar field theories are described: ϕ⁴ theory in four dimensions (ϕ⁴₄) and ϕ⁶ theory in three dimensions (ϕ⁶₃). After showing what the Gaussian Approximation does in terms of Feynman diagrams, renormalized GEP's are calculated for both theories. Dimensional Regularization is used in the renormalization and this this is especially convenient for the GEP in ϕ⁶₃ theory because it becomes trivially renor-malizable. It is noted that ϕ⁶₃ loses its infrared asymptotic freedom in the Gaussian Approximation. Finally, it is shown how a finite temperature GEP can be calculated by finding low and high temperature expansions of the temperature terms in ϕ⁶₃ theory. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

This dissertation evaluates higher-order modal methods for predicting thermal and structural response. More accurate methods or ones which can significantly reduce the size of complex, transient thermal and structural problems are desirable for analysis and are required for synthesis of real structures subjected to thermal and mechanical loading. A unified method is presented for deriving successively higher-order modal solutions related to previously developed, lower-order methods such as the mode-displacement and mode-acceleration methods. A new method, called the force derivative method, is used to obtain higher-order modal solutions for both uncoupled (proportionally-damped) structural problems as well as thermal problems and coupled (non-proportionally damped) structural problems. The new method is called the force-derivative method because, analogous to the mode-acceleration method, it produces a term that depends on the forcing function and additional terms that depend on the time derivatives of the forcing function. The accuracy and convergence history of various modal methods are compared for several example problems, both structural and thermal. The example problems include the case of proportional damping for: a cantilevered beam subjected to a quintic time varying tip load and a unit step tip load and a muItispan beam subjected to both uniform and discrete quintic time-varying loads. Examples of non-proportional damping include a simple two-degree-of-freedom spring-mass system with discrete viscous dampers subjected to a sinusoidally varying load and a multispan beam with discrete viscous dampers subjected to a uniform, quintic time varying load. The last example studied is a transient thermal problem of a rod subjected to a linearly-varying, tip heat load. / Ph. D.

Equations of subsurface flow of water, the Richards equation and the Boussinesq equation, have no known exact analytical solutions. Approximate analytical solutions to these equations have been developed under linearizing simplifications. In the first part of the dissertation two commonly used linear methods of computation of groundwater flow are investigated. The equation considered includes a recharge term and slope of an impervious bed. A new method of computation with improved accuracy has been developed. The second part of the dissertation deals with vertical, unsaturated flow of water in a homogeneous soil column of finite length with arbitrary initial conditions. The boundary conditions considered at the soil surface correspond to pre-ponding and post-ponding infiltration. / Ph. D.

A numerical scheme has been developed to solve the quasilinear form of the transonic stream function equation. The method is applied to compute steady two-dimensional axisymmetric solar wind-type problems. A single, perfect, non-dissipative, homentropic and polytropic gas-dynamics is assumed. The four equations governing mass and momentum conservation are reduced to a single nonlinear second order partial differential equation for the stream function. Bernoulli's equation is used to obtain a nonlinear algebraic relation for the density in terms of stream function derivatives. The vorticity includes the effects of azimuthal rotation and Bernoulli's function and is determined from quantities specified on boundaries. The approach is efficient. The number of equations and independent variables has been reduced and a rapid relaxation technique developed for the transonic full potential equation is used. Second order accurate central differences are used in elliptic regions. In hyperbolic regions a dissipation term motivated by the rotated differencing scheme of Jameson is added for stability. A successive-line-overrelaxation technique also introduced by Jameson is used to solve the equations. The nonlinear equationfor the density is a double valued function of the stream function derivatives. The velocities are extrapolated from upwind points to determine the proper branch and Newton's method is used to iteratively compute the density. This allows accurate solutions with few grid points. The applications first illustrate solutins to solar wind models. The equations predict that the effects of vorticity must be confined near the surface and far away the streamlines must resemble the spherically symmetric solution. Irrotational and rotational flows show this behavior. The streamlines bend toward the rotation axis for rapidly rotating models because the coriolis force is much larger than the centrifugal force. Models of galactic winds are computed by considering the flow exterior to a surface which surrounds a uniform density oblate spheroid. Irrotational results with uniform outward mass flux show streamlines bent toward the equator and nearly spherical sonic surfaces. Rotating models for which Bernoulli's function is not constant show the sonic surface is deformed consistent with the one-dimensional theory.

Monte Carlo sampling-based methods are frequently used in stochastic programming when exact solution is not possible. In this dissertation, we develop two sets of Monte Carlo sampling-based algorithms to solve classes of two-stage stochastic programs. These algorithms follow a sequential framework such that a candidate solution is generated and evaluated at each step. If the solution is of desired quality, then the algorithm stops and outputs the candidate solution along with an approximate (1 - α) confidence interval on its optimality gap. The first set of algorithms proposed, which we refer to as the fixed-width sequential sampling methods, generate a candidate solution by solving a sampling approximation of the original problem. Using an independent sample, a confidence interval is built on the optimality gap of the candidate solution. The procedures stop when the confidence interval width plus an inflation factor falls below a pre-specified tolerance epsilon. We present two variants. The fully sequential procedures use deterministic, non-decreasing sample size schedules, whereas in another variant, the sample size at the next iteration is determined using current statistical estimates. We establish desired asymptotic properties and present computational results. In another set of sequential algorithms, we combine deterministically valid and sampling-based bounds. These algorithms, labeled sampling-based sequential approximation methods, take advantage of certain characteristics of the models such as convexity to generate candidate solutions and deterministic lower bounds through Jensen's inequality. A point estimate on the optimality gap is calculated by generating an upper bound through sampling. The procedure stops when the point estimate on the optimality gap falls below a fraction of its sample standard deviation. We show asymptotically that this algorithm finds a solution with a desired quality tolerance. We present variance reduction techniques and show their effectiveness through an empirical study.

Approved for public release, distribution unlimited / This research will be used to evaluate the feasibility of robotically, or remotecontrolled firefighting nozzles aboard air-capable ships. A numerical model was constructed and analyzed, using the program CFD-ACE, of a fire hose stream being deflected by the influence of a crosswind, tailwind, or headwind. The model is intended to predict the reach of the fire hose stream, indicate the distribution pattern, and estimate the volume of fire fighting agent available at the end of the stream. Preliminary results for a two fluid cross flow model have been obtained. / US Navy (USN) author.

Thesis (Ocean E)--Massachusetts Institute of Technology, Dept. of Ocean Engineering, 1980. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING / Includes bibliographical references. / by Pierre Francois Koch. / Ocean E

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Ocean Engineering, 1981. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING. / Bibliography: leaves 124-126. / by Paul Sclavounos. / Ph.D.

Two unique spatial discretizations employing generalized indexing strategies suitable for use with space-marching algorithms are presented for the numerical solution of the equations of fluid dynamics. Both discretizations attempt to improve geometric flexibility as compared to structured indexing strategies and have been formulated while considering the current and future availability of unstructured grid generation techniques. The first discretization employs a generalized indexing strategy utilizing triangular elements in the two dimensions normal to the streamwise direction, while maintaining structure within the streamwise direction. The second discretization subdivides the domain into a collection of computational blocks. Each block has inflow and outflow boundaries suitable for space marching. A completely generalized indexing strategy utilizing tetrahedra is used within each computational block. The solution to the flow in each block is found independently in a fashion similar to the cross-flow planes of a structured discretization. Numerical algorithms have been developed for the solution of the governing equations on each of the two proposed discretizations. These spatial discretizations are obtained by applying a characteristic-based, upwind, finite volume scheme for the solution of the Euler equations. First-order and higher spatial accuracy is achieved with these implementations. A time dependent, space-marching algorithm is employed, with explicit time integration for convergence of individual computational blocks. Grid generation techniques suitable for the proposed discretizations are discussed. Applications of these discretization techniques include the high speed flow about a 5° cone, an analytic forebody, and a model SR71 aircraft. / Ph. D.

The accurate finite element analysis of subregions of large structures is difficult to carry out because of uncertainties about how the rest of the structure influences the boundary conditions and loadings of the subregion model. This dissertation describes the theoretical development and computer implementation of a new approach to this problem of modeling subregions. This method, the specified boundary stiffness/force (SBSF) method, results in accurate displacement and stress solutions as the boundary loading and the interaction between the stiffness of the subregion and the rest of the structure are taken into account. This method is computationally efficient because each time that the subregion model is analyzed, only the equations involving the degrees of freedom within the subregion model are solved. Numerical examples are presented which compare this method to some of the existing methods for subregion analysis on the basis of both accuracy of results and computational efficiency. The SBSF method is shown to be more accurate than another approximate method, the specified boundary displacement (SBD) method and to require approximately the same number of computations for the solution. For one case, the average error in the results of the SBD method was +2.75% while for the SBSF method the average error was -0.3%. The comparisons between the SBSF method and the efficient and exact zooming methods demonstrate that the SBSF method is less accurate than these methods but is computationally more efficient. In one example, the error for the exact zooming method was -0.9% while for the SBSF method it was -3.7%. Computationally, the exact zooming method requires almost 185% more operations than the SBSF method. Similar results were obtained for the comparison of the efficient zooming method and the SBSF method. Another use of the SBSF method is in the analysis of design changes which are incorporated into the subregion model but not into the parent model. In one subregion model a circular hole was changed to an elliptical hole. The boundary forces and stiffnesses from the parent model with the circular hole were used in the analysis of the modified subregion model. The results of the analysis of the most refined mesh in this example had an error of only -0.52% when compared to the theoretical result for the modified geometry. The results of the research presented in this dissertation indicate that the SBSF method is better suited to the analysis of subregions than the other methods documented in the literature. The method is both accurate and computationally efficient as well as easy to use and implement. The SBSF method can also be extended to the accurate analysis of subregion models with design changes which are not incorporated into the parent model. / Ph. D.