Hydrodynamic Response of Structures with Variable Structural Properties using Hypersingular Integral Equations Approach

My doctoral research focused on the study of linear water waves, particularly the study of fluid-structure interaction problems from the mathematical viewpoint. The study of water wave phenomena is quite an important branch of Applied Mathematics. The main reason for the growing interest in this area is due to the increase in the number of marine activities and protection of coastlines from wave attacks. Thus, I have rigorously worked on the problems of water wave scattering by various submerged structures which are either rigid, porous or elastic in nature. Apart from the usual structure orientations, i.e., horizontal or vertical, I have also considered structures with arbitrary inclinations in my study. Breaking the norms of uniformity, which is assumed in most of the works in the literature, I have also tried to incorporate the concept of non-uniform structural properties like variable porosity and non-uniform thickness, so as to relate to the problems of the real world. 

Employing a combination of rigorous mathematical analysis, innovative modeling techniques, and advanced numerical simulations utilizing MATLAB, I have delved into areas such as non-uniform plate dynamics and the development of efficient mathematical tools to tackle problems in both the frequency and time domains, with applications to harnessing renewable energy and understanding sea-ice dynamics. 

To deal with the physical problems, I have employed various mathematical tools like the integral equations technique, the perturbation method, eigenfunction expansion method, the Green's function technique, the Green's integral theorem and the Fourier transform method. These method compared to other methods used for solving fluid structure problems, are computationally efficient, simple and rapidly convergent. The boundary value problems are converted into set of integral equations with the help of Green's function and Green's integral theorem. The obtained integral equations contain hypersingular singular kernels, which are referred to as hypersingular integral equation. Semi-analytical solutions are obtained for all the problems using appropriate numerical techniques. My works are mainly related to two-dimensional thin submerged plates interacting with incoming waves, where the fluid motion is steady and irrotational, and the fluid is inviscid and incompressible. The emphasis is laid on the hydrodynamic performance of the structures/obstacles which is studied both analytically and graphically.