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The purpose of this paper is to develop a general method for using Finite Elements in the Steepest Descent Method. The main application is to a partial differential equation for a Transonic Flow Problem. It is also applied to Burger's equation, Laplace's equation and the minimal surface equation. The entire method is tested by computer runs which give satisfactory results. The validity of certain of the procedures used are proved theoretically. The way that the writer handles finite elements is quite different from traditional finite element methods. The variational principle is not needed. The theory is based upon the calculation of a matrix representation of operators in the gradient of a certain functional. Systematic use is made of local interpolation functions.

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This research investigation explores the feasibility of implementing a laser welded sandwich steel panel bridge deck system as a viable alternative to standardized reinforced concrete bridge decks. Generally used in naval ship building applications, steel sandwich panels possess attractive characteristics towards the integration with bridge infrastructure such as service life in excess of 100 plus years, dead load reduction, rapid construction, decreased closure time, and automated mass production. The lack of fatigue data for the laser "stake" welds used to create the enclosed sandwich panel geometry raised concerns with respect to fatigue life. The primary focus of this study was to determine whether or not infinite fatigue life was possible. Two different laser welding technologies were investigated, Laser Beam Welding (LBW) and Hybrid-Laser Arc Welding (HLAW). Test specimens were fabricated and tested in order to examine fatigue resistance based on a localized load effect between adjacent core stiffeners. Finite element models were used to obtain the stress range for each individual test due to complex geometry and partially restrained boundary conditions. In order to assess the fatigue performance of the overall deck system, additional finite element models were created to study the local and global behavior of different sandwich panel configurations. As a whole the investigation yielded promising results. Infinite fatigue life is achievable due to outstanding fatigue performance. The HLAW stake welds demonstrated superior fatigue resistance in comparison to the LBW process. Localized load effects can be minimized through the modification of different panel parameters. Pushing forward, full scale testing is essential to the future employment of this innovative bridge deck system. / Master of Science

Thesis: B.S., Massachusetts Institute of Technology, Department of Mechanical Engineering, 1978 / Includes bibliographical references. / by Arthur P. Cimento. / B.S. / B.S. Massachusetts Institute of Technology, Department of Mechanical Engineering

In this scientific work, we use two effective methods : Lie groups theory and the finite element method, to explain why the transition from laminar flow to turbulence flow depends on the variation of the Reynolds number. We restrict ourselves to the case of incompressible viscous Newtonian fluid flows. Their governing equations, i.e. the continuity and Navier-Stokes equations are established and investigated. Their solutions are expressed explicitly thanks to Lie's theory. The stability theory, which leads to an eigenvalue problem is used together with the finite element method, showing a way to compute the critical Reynolds number, for which the transition to turbulence occurs. The stationary flow is also studied and a finite element method, the Newton method, is used to prove the stability of its convergence, which is guaranteed for small variations of the Reynolds number. / Mathematical Sciences / M.Sc. (Applied Mathematics)

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published_or_final_version / Mechanical Engineering / Doctoral / Doctor of Philosophy