## Advances in Computational Methods for Water-Wave Problems

- Author(s): Hariri Nokob, Mohamed
- Advisor(s): Yeung, Ronald W.
- et al.

## Abstract

We developed three different techniques to enhance the computational methods used for time-harmonic linear water-wave theory. The first makes use of hypersingular boundary integral equations to model thin solid bodies in a wave field. The method developed allows for numerical treatment of the problem in its original hypersingular form using higher-order Overhauser elements without the need for regularization. We then use the method to study the case of a thin bottom-touching barrier in waves with particular interest in the effect of the harbor mouth (opening) over its hydrodynamic characteristics. Our results shed light on the drastic changes that occur in the presence of the opening and its size. We also point to the discovery of particular opening widths that allow for zero total loads on the harbor at particular frequencies.

\indent The second part still deals with hypersingular boundary integral equations but in the context of thin arbitrarily shaped plates. A Galerkin approach is used to treat the high-order singularity and results are used to fill some gaps in the literature of the values of added mass for these plates, particularly those plates with openings or holes.

\indent The third numerical method we present allows for the efficient treatment of general linear water-wave problems with multiple arbitrarily shaped bodies and arbitrarily shaped seabed geometry when the number of unknowns in the problem N is large. The problem domain is divided into an internal region that is modeled using a simple-source boundary integral equation allowing for arbitrarily shaped body and bottom geometry and an outer domain that is assumed free of solid bodies and of uniform water depth. The two domain solutions are then matched to complete the solution. The major contribution in this present topic is introducing the "fast-multipole method" to the solution procedure and therefore changing the complexity of the problem from O(N^3) (or O(N^2) if an iterative solver is used) to O(N) for large values of N. The memory requirements scale linearly with $N$ as well. This methodology is needed when the number of solid bodies is large or when very large floating structures with complex shapes are needed. It is also necessary when the bottom topography for a particular problem varies considerably. We use the computational code developed to study the effects of variable bottom topography over two case examples of multiple floating bodies. The first case is a set of 4 truncated floating vertical cylinders over a bottom protrusion, for which results indicate that variations in bottom topography cause slight to medium level changes on the wave loads on the floating bodies. The second case is applied to a configuration of 16 floating cylinders (with $N \sim 150,000$) and the effects of the variable ocean floor are again considered. Wave elevation details of this complex geometry is shown for illustration.