Hypersingular Boundary Integral Equations: Some Applications in Acoustic and Elastic Wave Scattering

Abstract

The properties of hypersingular integrals, which arise when the gradient of conventional boundary integrals is taken, are discussed. Interpretation in terms of Hadamard finite-part integrals, even for integrals in three dimensions, is given, and this concept is compared with the Cauchy Principal Value, which, by itself, is insufficient to render meaning to the hypersingular integrals. It is shown that the finite-part integrals may be avoided, if desired, by conversion to regular line and surface integrals through a novel use of Stokes’ theorem. Motivation for this work is given in the context of scattering of time-harmonic waves by cracks. Static crack analysis of linear elastic fracture mechanics is included as an important special case in the zero-frequency limit. A numerical example is given for the problem of acoustic scattering by a rigid screen in three spatial dimensions.

A transformation approach for efficient evaluation of oscillatory surface integrals arising in three‐dimensional boundary element methods

“…Comparing this to the approach in (8), the main change is that @o j ı @n will now be used to capture the oscillation of @ˆ=@n, in place of o j . This is justifiable because most common choices for o j (ours in (9) included) are solutions of the Helmholtz equation themselves; hence, they are well suited to represent bothˆand @ˆ=@n on S .…”

“…Indeed, most of the contour integral transforms published in the BEM literature are solely focussed on this case (e.g. [7][8][9]). …”

“…The focus here has particularly been on integrals arising from the hyper-singular integral operator in collocation schemes (e.g. [7,8]). An early and influential exposition of this approach was given by Terai [9], based on conversion to polar coordinates; this included support for higher-order polynomial basis functions, but was restricted to planar elements.…”

“…It is interesting to note that it was the solid angle S j .x/ that was removed to give this numerically-robust edge kernel, because that is also what is subtracted in many regularisation schemes for the hyper-singular operator (e.g. [7,8]]. …”

Real and Complex Hypersingular Integrals and Integral Equations in Computational Mechanics*

“…Consider the case of p isotropic elastic blocks. For the i-th block with the boundary B 1 we have a hypersingular equation (see, e.g., [10], [11] where a x = 0.5(1 -i/)(-l//i"), a 2 = 0.5(1 -v)(l/n + + 1 //*"), ct(x) is a traction in the point x, Au = u + -u~ is a displacement discontinuity, the normal n is fixed on the contacting surfaces of the blocks, the sign "plus" ("minus") corresponds to the block for which the normal n is outward (inward), for external boundaries the normal n is assumed to be outward and l/fi~ = 0, u~ = 0. The derived hypersingular equation contains only tractions and displacement discontinuities on contacts of blocks.…”