Continuity requirements for density functions in the boundary integral equation method

Radiation of water waves by a heaving submerged horizontal disc

“…Both of these definitions have analogues for one-dimensional finite-part integrals. For more information on hypersingular integrals over surfaces, see Martin & Rizzo (1989, 1996, Krishnasamy et al (1990) and Krishnasamy, Rizzo & Rudolphi (1992).…”

Evaluation of non-singular BEM algorithms for potential problems

“…In spite of the very successful numerical results reported by Richardson et al (1997), Richardson and Cruse (1999), Chien et al (1991), Huang and Cruse (1994) using various forms of these relaxed algorithms combined with piece-wise C 1,α interpolations, Martin and Rizzo (1996), Krishnasamy et al (1992) have concluded that these algorithms could not be theoretically justified. This means that, from a strictly mathematical point of view, only boundary element implementations that ensure C 0,α or C 1,α continuity at each collocation point can be applied in the discretizations of the standard, or the hypersingular boundary integral equations, respectively.…”

“…Nevertheless, the higherorder of the singularities in the kernels requires more smoothness of the densities for the given integral to be finite, just like on the standard hypersingular formulation. According to Krishnasamy et al (1992), a sufficient condition for the existence of the hypersingular integral is the C 1,α continuity of the density function at the source point. Standard isoparametric boundary elements do not satisfy this requirement and for this reason, approximate solution techniques for solving hypersingular integral equations by means of the BEM require special consideration.…”

On the prediction of self-noise due to supercavitating underwater vehicle moving at subsonic speed using boundary element approach

“…The final form of the modified boundary integral equation based on Chien's method of hypersingular regularization [5] as: is termed as the density function. For the finite part (FP) of the hypersingular integral in Term A to exist, it is important that the density function is C 1 continuous at the interelement edges [7]. It should be noted that when using constant elements to describe the pressure function variation, Term A in (Eq.12) is zero since the field point is collocated at the same point where the source point lies (i.e.…”