Computing methods of hypersingular integral applied to eddy-current testing

Abstract

The detection of thin-opening cracks is an important part of the eddy-current nondestructive testing (NDT). The integral formulation is well adapted for this modelization if the geometry of the tested piece is simple. However, some integrals involved in the computation contain strong singularities. The aim of this paper is to improve the classical numerical resolution using a general computing method of hypersingular integral.

Modelisation of thin cracks with varying conductivity in eddy‐current testing

“…If r ′ is on the crack then the integral has to be interpreted as the finite part of Hadamard, noted FP. The physical meaning and the computation methods of the finite part of the integral are explained in Beltrame and Burais (2002a). In order to obtain an equation on p the observation point r ′ tends to a point on the crack surface and is projected on the crack normal n , then:Equation 4If the crack is a perfect insulating J n =0, but in the case of a local current leakage on the surface crack, the component J n has to be expressed.…”

“…The density p is the solution of an integral equation with an hypersingular kernel on the crack surface related to the incident current. The resolution with a collocation method and second order shape functions is carried out by Beltrame and Burais (2002a) and the improvement with special elements at the crack edges is described by Beltrame (2002). The variation Δ Z of the coil impedance is obtained by a regular integration on the density p of the crack surface.…”

“…It is not the case for the conductivity, first we give a criterion of the validity of the formulation presented by Beltrame and Burais (2004). Secondly, the convergence of the numerical scheme used in Beltrame and Burais (2002a, b, 2004) is discussed. We prove that it fails in the case of a discontinuity of the local conductivity.…”

Strong Singularity Reduction for Curved Patches for the Integral Equations of Electromagnetics

“…T HE accurate computation of singular potential integrals is crucial to the creation of general-purpose integral-equation-based codes for the computation of radiation and scattering and to many other branches of science and engineering [1], [2]. In computational electromagnetics, the importance of this topic springs from several sources.…”

Hybridization of volumetric and surface models for the computation of the T/R EC probe response due to a thin opening flaw

“…In the case of a transverse planar flaw located in the ( x , y ) plane, the spatial variable x along the thickness of the flaw disappears, and the flaw is represented by a current surface dipole density, referred to as p ( y , z ), which is a scalar quantity depending on two spatial variables corresponding to the dimensions of the surface of the ideal flaw. The surface dipole density is solution of an integral equation on the surface of the crack, which involves an hypersingular kernel when it is evaluated in the spatial domain (Beltrame and Burais, 2002a, b). In an alternative way, the kernel can be evaluated in the spectral domain (Pávó and Miya, 1994), and the use of a global approximation of p ( y , z ) (Pávó and Lesselier, 2006) may overcome some numerical difficulties coming from specific boundary conditions to be satisfied by p ( y , z ) (Bowler et al , 1997).…”