|M.Sc Student||Hanan Amar|
|Subject||Coupling Methods for Dynamic Models of One-and|
|Department||Department of Aerospace Engineering||Supervisor||Full Professor Givoli Dan|
Lately various methods have been proposed for solving problems that extend over domains that include high- and low-dimensional regions. These methods involve a mechanism for coupling the high- and low-dimensional models, to produce a single hybrid model. This thesis describes the implementation and typical results of such hybrid methods. The focus of the work is on the coupling of two-dimensional (2D) and one-dimensional (1D) models in time-dependent linear wave problems. This work is an extension to a recent research that focused on the use of hybrid methods for coupling of 2D and 1D models in time-harmonic wave problems. The resulting hybrid 2D-1D model is solved using standard FEM procedures and standard time stepping methods of the Newmark family. The main benefit of using a well designed hybrid model is in its efficiency in terms of computational resources required, when compared to the standard computation of a full 2D model taken for the entire domain.
Two important issues related to such hybrid 2D-1D models are: (a) the design of the hybrid model and its validation (with respect to the original problem), and (b) the way the 2D-1D coupling is done, and the coupling error generated. This research focuses on the second issue.
In the first part of the document, two numerical methods are adapted to the 2D-1D coupling scenario, for the time-dependent wave problem describing the lateral motion of an elastic membrane. These methods are the Panasenko method and the Nitsche method. Both are existing methods that deal with interfaces; however none of them has previously been adopted and applied to the type of problem under study here. We present the derivation and FE formulation of each method in detail. The second part of the document describes the extensive numerical experimentation done through the course of the research. The accuracy of the 2D-1D coupling by both methods is compared numerically for a specially designed benchmark problem, and conclusions are drawn on their relative performances. Then the methods are applied to a couple of more complicated problems, and once again compared.
The conclusion we draw from the results of this work is that both methods are viable candidates for coupling 2D and 1D regions, but that the Nitsche method may generate smaller errors in certain scenarios.