Water wave problems using null-field boundary integral equations: ill-posedness and remedies

Abstract

In this article, we focus on the hydrodynamic scattering of water wave problems containing circular and/or elliptical cylinders. Regarding water wave problems, the phenomena of numerical instability due to fictitious frequencies may appear when the boundary element method (BEM) is used. We examine the occurring mechanism of fictitious frequency in the BEM through a water wave problem containing an elliptical cylinder. In order to study the fictitious frequency analytically, the null-field boundary integral equation method in conjunction with degenerate kernels is employed to derive the analytical solution. The modal participation factor for the numerical instability of zero divided by zero can be exactly determined in a continuous system even though the circulant matrix cannot be obtained in a discrete system for the elliptical case. It is interesting to find that irregular values depend on the geometry of boundaries as well as integral representations and happen to be zeros of the mth-order (even or odd) modified Mathieu functions of the first kind or their derivatives. To avoid using the addition theorem to translate the Bessel functions to the Mathieu functions, the present approach can solve for the water wave problem containing circular and/or elliptical cylinders at the same time in a semi-analytical manner by using the adaptive observer system. The closed-form fundamental solution is expressed in terms of the degenerate kernel in the polar and elliptic coordinates for circular and elliptical cylinders, respectively. Three examples are considered to demonstrate the validity of the present approach, including an elliptical cylinder, two parallel identical elliptical cylinders and one circular cylinder and one elliptical cylinder. Finally, two regularization techniques, the combined Helmholtz interior integral equation formulation method and the Burton and Miller approach, are adopted to alleviate the numerical resonance due to fictitious frequency.