This project focuses on the computational difficulties associated with steep gradients in the velocity and displacement fields in structures subjected to impulsive loadings. The conventional finite element approach would suggest creating a finer element mesh around the areas with large gradients, but since the gradients move through the structure, such a procedure is not efficient. This could be solved by increasing the number of finite elements throughout the structure, but it would be computationally impractical to do so. An alternative approach would call for formulation of the problem in terms of a modal analysis. The large number of modes thereby required to model the structure implies similar computational difficulties. A third alternative, finite differences in space and time is computationally inefficient for use in design oriented analysis. The approach adopted herein, finite elements formulated in the space time domain, offers the possibility of overcoming these computational problems. Three classes of physical systems will be analyzed including a string under constant tension, a string under nonuniform tension, and a linear Timoshenko beam. For the string models, initial value problems including those for which a wave travels down the string will be solved as well as those involving moving load and impulsive load problems. For the Timoshenko beam, moving loads and moving constraint problems will be considered. The project comprises three phases including: (1) determining shape functions and convergence criteria for finite elements that cut across space and time; (2) computing solutions for string problems and study of convergence properties, and set up of the general solution for a space time finite element formulation for Timoshenko beams; and (3) determining solutions for beam problems with moving loads and moving constraints. The impact of this research will be on the methods of analysis for systems with moving loads or constraints, deployment problems, and flexible manipulator arms.
