On direct numerical treatment of hypersingular integral equations arising in mechanics and acoustics

TL;DR

This paper develops an efficient direct numerical collocation method for solving hypersingular integral equations common in wave dynamics, elasticity, and acoustics, demonstrated through crack and diffraction problems.

Contribution

It introduces a novel numerical collocation approach specifically designed for hypersingular integral equations in mechanics and acoustics.

Findings

Successfully applied to crack theory and acoustic diffraction problems

Demonstrates high accuracy and computational efficiency

Provides a practical tool for complex boundary value problems

Abstract

In this paper we present a treatment of hypersingular integral equations, which have relevant applications in many problems of wave dynamics, elasticity and fluid mechanics with mixed boundary conditions. The main goal of the present work is the development of an efficient direct numerical collocation method. The paper is completed with two examples taken from crack theory and acoustics: the study of a single crack in a linear isotropic elastic medium, and diffraction of a plane acoustic wave by a thin rigid screen.