Computation of a quasi-static field induced by two long straight parallel wires in a conductor with a rough surface

Prémel, Denis

doi:10.1088/0022-3727/41/24/245305

Cited by 6 publications

(8 citation statements)

References 36 publications

(50 reference statements)

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“…In practice, the Fourier space must be discretized and truncated. To do that, we exploit the same explanations as already presented in two previous papers except it must be generalized to the 2D Fourier space. In summary, let us assume a finite number of harmonics to represent the potential, hence

\begin{matrix} right φ (x^{1}, x^{2}, x^{3}) & left = T {scriptF}^{- 1} [\hat{φ} (α, β, x^{3})] & right \\ right & left \approx \sum_{m = - M}^{m = + M} \sum_{n = - N}^{n = + N} {\hat{φ}}_{m n} e^{- i α_{m} x^{1} - i β_{n} x^{2}} . \end{matrix}

The convolution product

((), ĝ * \overset{φ}{true}) false(α, β, x^{3} false)

results in the discrete Fourier space by a matrix product of the form

([], g) \overset{φ}{true}

where

\overset{φ}{true}

is a column vector, resulting from the concatenation of all the discrete values

{\overset{φ}{true}}_{m n}

.…”

Section: Physical Formalism Of the Problemmentioning

confidence: 99%

“…Although this method is very popular, in the optical domain, for the computation of the fields scattered by 2D diffraction gratings or by perfectly conducting random surfaces enlightened by a plane wave it is less known in the low frequency community. To the best of our knowledge, the CCM was used for the first time in 2008 by D. Prémel in a preliminary study for a 2D EC problem . Then, it has been used for simulating a first 2.5D eddy current nondestructive testing configuration before developing more practical cases of interest: a 3D EC probe scanning a 2D multilayered conductor with parallel or nonparallel interfaces .…”

Section: Introductionmentioning

confidence: 99%

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Development of the curvilinear coordinate method for the computation of quasi‐static fields induced by an eddy current probe scanning a 3D conductor of complex shape characterized by an arbitrary 2D surface

Prémel

Granet

2017

Int J Numerical Modelling

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This paper deals with the development of a semi-analytical model for the fast computation of the quasi-static field induced by any eddy current probe in a 3D conductor of complex shape. The workpiece is considered as a planar half-space mock-up with an interface air conductor characterized by an arbitrary 2D surface a(x, y). The curvilinear coordinate method consists in introducing a change of coordinates to be able to write analytically and easily the boundary conditions at the interface air-metal. Due to the novel generalized metric space, the covariant form of Maxwell equations must be considered. The curvilinear coordinate method is widely used in the optical community for the analysis the diffraction phenomenon on gratings. The method has been applied recently for eddy current calculations in the planar case for 2.5D configurations characterized by a 3D eddy current probe and a 2D layered stratified conducting media. By an extension to 3D problems, this work constitutes the preliminary task for the development of a complete numerical model, which has the capability to address very complex eddy current nondestructive testing configurations such as a stratified layered media constituted by a set rough interfaces. For that purpose, the modal formalism is described for the first time in the 3D case, and several numerical experiments show the validity and the efficiency of the fast numerical model in comparison to the finite element method. KEYWORDS computational electromagnetics, covariant form of Maxwell equations, eddy current nondestructive testing, rough surfaces 1 Int J Numer Model. 2018;31:e2219.wileyonlinelibrary.com/journal/jnm

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\begin{matrix} right φ (x^{1}, x^{2}, x^{3}) & left = T {scriptF}^{- 1} [\hat{φ} (α, β, x^{3})] & right \\ right & left \approx \sum_{m = - M}^{m = + M} \sum_{n = - N}^{n = + N} {\hat{φ}}_{m n} e^{- i α_{m} x^{1} - i β_{n} x^{2}} . \end{matrix}

The convolution product

((), ĝ * \overset{φ}{true}) false(α, β, x^{3} false)

results in the discrete Fourier space by a matrix product of the form

([], g) \overset{φ}{true}

where

\overset{φ}{true}

is a column vector, resulting from the concatenation of all the discrete values

{\overset{φ}{true}}_{m n}

.…”

Section: Physical Formalism Of the Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Development of the curvilinear coordinate method for the computation of quasi‐static fields induced by an eddy current probe scanning a 3D conductor of complex shape characterized by an arbitrary 2D surface

Prémel

Granet

2017

Int J Numerical Modelling

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“…Hence, the function J is still piecewise constant along the r direction since pulse functions are still used along that direction. B-splines of higher degree have been used in a first step along the z and w directions because of the spectral expression of G ee with respect to these variables, see equation (2). Examples of the new 2D expansion functions obtained along the w and z directions are shown in Figure 3.…”

Section: Choice Of Basis and Test Functionsmentioning

confidence: 99%

New discretisation scheme based on splines for volume integral method

Reboud

Prémel²,

Lesselier³

et al. 2008

COMPEL - The International Journal for Computation and Mathematics in Electrical and Electronic Engineering

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PurposeA numerical model dedicated to external eddy current inspection of tubes has been developed using the volume integral method (VIM). The purpose of this paper is to suggest new discretization schemes based on non‐uniform B‐splines for the solution of the state equation with the method of moments (MoM).Design/methodology/approachVIM is a semi‐analytical approach providing fast and accurate results for the simulation of eddy current testing (ECT) of pieces with canonical geometries. The state equation derived with this formalism is solved using the Galerkin variant of the well‐known MoM.FindingsThis paper shows that an accuracy improvement is achieved in MoM by using B‐splines with degree 1 or 2 as projection functions in MoM instead of pulse functions. Moreover, comparisons between simulation results show that, for all ECT configurations tested, the use of degree 1 B‐splines is sufficient to get this improvement.Originality/valueThe use of B‐splines functions has already been proposed for MoM in the literature, but not in the framework of the Galerkin variant of MoM. This paper also shows quantitative comparisons between experiment and simulation as well as a study of the minimal degree required to get an accuracy improvement in MoM.

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“…This approach is an extension of the curvilinear coordinate method (CCM), which remains as one of the most efficient method in the optical domain for the computation of the fields scattered by 2D diffraction gratings enlightened by a plane wave or perfectly conductive random surfaces . Indeed, this method has been transposed into the low‐frequency domain for ECNDT 2D configurations . Then, a second‐order vector potential formulation has been introduced in order to be able to tackle the case of 3D eddy current probes scanning aperiodic 2D shapes.…”

Section: Introductionmentioning

confidence: 99%

Semi‐analytical computation of a quasi‐static field induced by a 3D eddy current probe scanning a 2D layered conductor with parallel rough interfaces

Caire

Prémel

Granet

2014

Int J Numerical Modelling

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This paper deals with the fast computation of the quasi‐static fields induced in a conductor with a locally perturbed shape along the longitudinal axis and extruded along the transversal one. Besides, the material is composed of several homogeneous layers with parallel faces. The method used here is based on writing Maxwell's equations in a curvilinear system, which leads to a more simple expression of boundary conditions. The validity of this method for the computation of quasi‐static fields induced by an eddy current probe in a conductive half‐space with a rough surface has been proved recently, and we propose here to complete this innovative formalism in order to solve an eddy current non‐destructive testing (ECNDT) configuration with a rough surface conductive medium composed of several homogeneous layers with parallel faces. This extension allows to tackle new ECNDT configurations such as homogeneous media constituted by rough interfaces. Moreover, it provides sufficient solutions for non‐homogeneous conductive parts presenting a depth‐varying (i.e., along the direction of the normal local vector with respect to the surface) material property, which can be approximated by a piecewise constant function by introducing multiple thin homogeneous layers. Copyright © 2014 John Wiley & Sons, Ltd.

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Computation of a quasi-static field induced by two long straight parallel wires in a conductor with a rough surface

Cited by 6 publications

References 36 publications

Development of the curvilinear coordinate method for the computation of quasi‐static fields induced by an eddy current probe scanning a 3D conductor of complex shape characterized by an arbitrary 2D surface

Development of the curvilinear coordinate method for the computation of quasi‐static fields induced by an eddy current probe scanning a 3D conductor of complex shape characterized by an arbitrary 2D surface

New discretisation scheme based on splines for volume integral method

Semi‐analytical computation of a quasi‐static field induced by a 3D eddy current probe scanning a 2D layered conductor with parallel rough interfaces

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