Why dual boundary element method is necessary?

Abstract

Dual boundary integral equation (BIE) was developed for problems containing degenerate boundaries in 1988 by Hong and Chen [Journal of Engineering Mechanics-ASCE, 114, 6, 1988] and was termed the dual boundary element method (BEM) in 1992 by Portela et al. [International Journal for Numerical Methods in Engineering, 33, 6, 1992]. After near 30 years, the dual BIE/BEM for the problem containing a zero-thickness barrier was revisited mathematically to study the rank deficiency from the viewpoint of the updating term and the updating document of singular value decomposition (SVD) [Journal of Mechanics, 31, 5, 2015]. In this paper, we revisit the dual BEM from the physical point of view. Although there is no zero-thickness barrier in the real world, it is always required to simulate a finite-thickness degenerate boundary to be zero-thickness in comparison with sea, air or earth scale. For example, a sheet pile, a screen, a crack problem, a thin airfoil and a breakwater were modeled by the geometry of zero-thickness. The role of the dual BEM is evident since Lafe et al. [Journal of the Hydraulics Division-ASCE, 106, 6, 1980] used the conventional BEM to model the finite-thickness pile wall to geometrically approximate zero-thickness barrier but numerically yielding divergent solution. On the contrary, we physically model the finite-thickness breakwater as a zero-thickness barrier. The breakwater is employed as an illustrative case to demonstrate that the dual BEM simulated by a zero-thickness barrier can yield more acceptable results to match the experiment data in comparison with those of the finite thickness using the conventional BEM. Finally, a single horizontal plate and two dual horizontal plates in vertical direction and in horizontal direction are three illustrative cases to tell you why the dual BEM is necessary not only in mathematics but also in physics.