Ph.D Thesis

Ph.D StudentOfir Yoav
SubjectLowD-HighD Hybrid Models in Wave Problems
DepartmentDepartment of Applied Mathematics
Supervisor PROF. Dan Givoli


This thesis focuses on the implementation of hybrid models in wave problems. The coupling of two-dimensional (2D) and one-dimensional (1D) models in time-harmonic elasticity stands in the majority of the work, where the hybrid 2D-1D model is justified by assuming some regions in the 2D computational domain behave approximately in a 1D way. The 2D and 1D structural regions are discretized by using 2D and 1D Finite Element (FE) formulations.

This hybrid model, if designed properly, is much more efficient than the standard 2D model taken for the entire problem. 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 phase of the research, three numerical methods are adapted to the 2D-1D

coupling scenario, for elastic time-harmonic waves: the Panasenko method, the Dirichlet-to-Neumann (DtN) method and the Nitsche method. All three 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. The accuracy of the 2D-1D coupling by the three methods is compared numerically for a specially designed benchmark problem, and conclusions are drawn on their relative performances. Additionally, the Nitsche method is implemented in more complicated problems, and compared to the Penalty method.

In the second phase of the research, three closely-related coupling methods are considered. They are all based on the DtN map associated with the 1D problem, on the interface: (i) The DtN method in its usual form, where the DtN map is calculated numerically, and where the 1D and 2D problems are solved separately; (ii) The CML method, devised by Carka, Mear and Landis, which is equivalent to the former one, but in which the 2D and 1D interface solutions are solved simultaneously; (iii) An iterative DtN map method. Direct application of these three methods results in low accuracy. Therefore, these methods are used here in conjunction with a Boundary Stress Recovery (BSR) technique, originally proposed by Mizukami, which provides the same order of accuracy for the stress as for the primary variable. The resulted improvement is demonstrated through numerical examples.

An extended DtN method, dealing with a more general class of problems, is considered in the last phase of the research. The computational domain still includes a slender region, but the solution in that region does not necessarily behave in a 1D way. Such a persistent 2D behavior occurs for sufficiently large wave numbers. The problem in the slender part is reduced to a sequence of 1D problems. The coupling is done through the numerically calculated DtN maps associated with these 1D problems. Even for this class of problems, the hybrid model is more efficient than the standard 2D model taken for the entire problem, yet its accuracy is not significantly lower.