Validity of some asymptotic models for eddy current inspection of highly conducting thin deposits

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inspection of highly conducting thin deposits

Houssem Haddar, Zixian Jiang

To cite this version:

Houssem Haddar, Zixian Jiang. Validity of some asymptotic models for eddy current inspection of highly conducting thin deposits. [Research Report] RR-8556, INRIA. 2014. �hal-01016568v2�

0249-6399ISRNINRIA/RR--8556--FR+ENG

RESEARCH REPORT N° 8556

July 2014

Validity of some

asymptotic models for eddy current inspection of highly conducting thin deposits

Houssem Haddar , Zixian Jiang

RESEARCH CENTRE SACLAY – ÎLE-DE-FRANCE 1 rue Honoré d’Estienne d’Orves Bâtiment Alan Turing

deposits

Houssem Haddar

∗

, ZixianJiang

†

Projet-TeamDÉFI

ResearhReport n°8556July201430pages

Abstrat: Highly onduting thin deposits mayblind eddyurrentprobesin non-destrutive

testingofsteamgeneratortubesandthusshould beidentied. Inthisreport,various asymptoti

models are studied to model the axisymmetri thin onduting layers by eetive transmission

onditionsdependingonre-salingparameterandasymptotiexpansionorder,soastoavoidthe

highomputationalostin afullmodeldue tothosethinlayers. Wealsoseletthemostadapted

modelsforpratialongurationsvianumerialomparisonsinasimpliedase.

Key-words: axisymmetri eddy urrentmodel, asymptoti models, eetivetransmissionon-

ditions.

∗

houssem.haddarinria.fr,INRIASalayCMAPEolePolytehnique,91120Palaiseau,Frane

†

zixian.jiangpolytehnique.edu,INRIASalayCMAPEolePolytehnique,91120Palaiseau,Frane

fortement onduteurs

Résumé : Des ouhes mines hautement ondutries peuvent masquer des défauts prob-

lématiques lors d'un ontrle non-destrutif des tubes dans un générateur de vapeur via des

sondesourantdeFouault. Ainsiil estessentieldepouvoirentenirompte. Dans erapport,

onétudiedesmodèlesasymptotiquesavediérentesonditionsdetransmissioneetivesayant

pour objetif de modéliserune ouhemine axis-symétrique an de réduire le oût de alul

numérique. Parmi des onditions de transmission qui dépendent d'un paramètre de redimen-

sionnementet del'ordredudéveloppementasymptotique,onenséletionnelespluspertinentes

pourlesappliationspratiquesviadestestsnumériquesdansdesongurationssimpliées.

Mots-lés: modèledeourantdeFouaultaxis-symétrique,modèlesasymptotiques,onditions

detransmissioneetives.

1 Introdution

Non-destrutive eddy urrent testing of the steam generatortubes is an essentialtask for the

safety andfailure-free operatingof nulearpowerplants. This testing generallyaims todetet

harmfuldefetssuhasraksoftubesandloggingdepositsinoolingiruitbetweentubesand

supporting plates(see forexample[14,8,9,1315℄). These problemati faultsannevertheless

be masked by some thin layers of opper overing the shell side of the tube due to its high

ondutivity. This iswhyitisimportantto beableto evaluatetheirinuene oneddyurrent

testing.

The opper layers are haraterized by a very high ondutivity (as ompared to steam

generatortubes) and averysmall thikness, seeTable1. A majornumerialhallenge to deal

withthisproblemwiththefulleddyurrentmodelistheexpensiveomputationalostresulting

from the fat that the domain disretization should use a very ne mesh of the same sale

to the thikness of the thin layer. To redue the numerial ost, we replae the thin layer

by someeetive transmission onditionsusing the asymptoti expansionof the solutionwith

respet to the thikness of the deposit, whih yields the so-alled asymptoti models. A rih

literatureonasymptotimodelshasbeendevelopedandwemayiteamongothersTordeux[12℄,

Claeys [5℄, Delourme[6℄, Poignard [10℄ and thereferenes thereinfor dierentapproahesand

variousappliations.

tubewall opperlayer

ondutivity(inS·m⁻¹⁾ σt= 0.97×10⁶ σc = 58.0×10⁶

thikness(inmm⁾ rt2−rt1 = 1.27 0.005∼0.1

Table1: Condutivityand saledierenesbetweentubewallandopperlayer.

In this report, we onsider the ase where a opper layerof onstant thikness overs ax-

isymmetrially theshell side of thetube. Aording to the hoie of aresaling parameterm

for theondutivityand theasymptoti expansionordern^{, weanobtainafamilyofeetive}

transmission onditions Z^{m,n linking up the solutions at the two sides of the thin layer. We}

aimtohoosetheappropriateeetivetransmissiononditions,i.e. theparameters(m, n)^,with

whihthediretasymptotimodelnotonlygivesagoodapproximationofthefullmodelinreal

onguration,butalsoallowstoestablishquikinversionmethods.

Although mainly onsidering here the aseof thin layerswith onstant thikness, weshall

introdue the asymptoti method for general thin deposit layers, i.e. with variable thikness

(Setion2). Withthatmethod,someeetivetransmissiononditionswithdierentparameters

(m, n)^{arealulatedforthin layerswithonstantthikness(Setion3). Itisworthmentioning}

thatsimilarstudieshavebeendevelopedin[11℄forthinondutingsheetwithonstantthikness

withadierentgeometrial setting. Finally inSetion 4wegivesome1-Dnumerialexamples

whih allow to verify and ompare the asymptoti models with these dierent transmission

onditions,andthendisussthemostpertinentmodelinviewofdiretandinversesimulations.

2 Asymptoti approximation of axisymmetri eddy urrent

model

This setion onentrateson theonstrution ofasymptoti models for eddyurrentproblems

withthepreseneofhighlyondutingthinlayers. Theobjetiveistogettheeetivetransmis-

siononditionsontheinterfaebetweenthethin layerand thetubewithwhihthevariational

asymptotimodelhasnolongerthevolumeintegralonthethin layerdomain.

Let us briey introdue the axisymmetri eddy urrent problem. Formore details readers

may refer to [7℄. In the ylindrial oordinates, a vetor eld a an be deomposed into the meridian parta_m =are_r+aze_z andthe azimuthal parta_θ =aθe_θ. a isaxisymmetri if ∂θa

vanishes. Under the assumption of axisymmetry and the high ondutivity / low frequeny

regime(ωǫ≪σ^),the3-Dtime-harmoniMaxwellequationsfortheeletriandmagnetields

(E,H)

(curlH+ (iωǫ−σ)E=J inR³, curlE−iωµH= 0 ⁱⁿR³,

(1)

with adivergene-free axisymmetri applied soure J (divJ = 0^{) anberedued to a seond}

order equation on a 2-D domain R²

+ := {(r, z) : r ≥ 0, z ∈ R} ^{for the azimuthal part of the}

eletrield Eθ^{that wedenoteinthesequelby}u=Eθ^:

−div 1

µr∇(ru)

−iωσu= iωJθ= iωJ ⁱⁿ R²₊, ⁽²⁾

where ∇ := (∂r, ∂z)^{t and} div := ∇· ^{are gradient and divergene operators in 2-D Cartesian}

oordinates. Assume that J ∈L²(R²₊) ^{hasompatsupport, andthat} µ ^andσ^{are in} L^∞(R²₊)

suhthat µ≥µv>0^,σ≥0^andthat µ=µv^,σ= 0 ^forr≥r0 ^{suientlylarge. Thenproblem}

(2)withadeayondition(u→0^asr²+z²→0^{)hasauniquesolutionin}H(R²₊)^whereforany Ω⊂R²₊ wedenote

H(Ω) :=n

v:r^1/2(1 +r²)^−λ/2v∈L²(Ω), r^−1/2∇(rv)∈L²(Ω)o

withλ^anyreal>1^{(see [7℄). Letusindiate thatif}Ω^{isbounded inthe}r^{-diretion then}H(Ω)

isequivalenttothefollowingspaeforwhihweshallusethesamenotation

H(Ω) :=n

v :r^1/2v∈L²(Ω), r^−1/2∇(rv)∈L²(Ω)o .

Hene,theeddyurrentequation(2)writesin thevariationalform

a(u, v) :=

Z

Ω

1

µr∇(ru)· ∇(r¯v)−iωσrw¯v

drdz= Z

Ω

iωJvr¯ drdz ∀v∈H(Ω). ⁽³⁾

For numerial reasons, the omputational domain will be trunated in radial diretion at

r=r∗ ^where r∗> r0 ^{issuientlylargeandimposeaNeumannboundaryonditionon}r=r∗^.

Then the solution for the trunated problem would satisfy (2) on Ω = Br∗ := {(r, z) ∈ R² : 0 ≤ r ≤ r∗}^{. This is why we shall use in the sequel the generi notation for the} variational spae H(Ω) ^with Ω ^denotingR²₊ orBr∗^{. We also reallthat theproblem on} Ω =Br∗ ^anbe

equivalently trunated to a bounded domain Br_∗,z_∗ = {(r, z) ∈ R² : 0 ≤r ≤ r∗,|z| < z∗} ^by

introduingappropriateDirihlet-to-Neumannoperatorsonz=±z∗^{. Thiswould beonvenient}

foraeleratingnumerialevaluationofthesolution(see[7℄). Asaorollaryofthewell-posedness

oftheproblem foru^weanstate:

Corollary 2.1. Assumethatthe soureJ ∈L²(Ω) ^{with ompat support. Then the} variational formulation (3)has auniquesolution uⁱⁿH(Ω)^.

Tube Deposit uδ

− uδ

uδ + z

r

δd(z) Γc Γt₁

rt₁ rt₂ Γt₂

Figure 1: Representationofathin layerdeposit.

2.1 Resaled in-layer eddy urrent equation

We onsider a thin layer of deposit with high ondutivity (in our ase, a layer of opper)

overingaxisymmetrially(a partof)theshell sideofthetube(seeFigure1foraradialutof

the settingin theylindrialoordinatesystem). Therst stepis to rewritethe in-layereddy

urrent equation by resaling the oordinate in the transverse diretion and the ondutivity

with respet to the layerthikness. Theanalytialsolution of thisresaled equation allowsto

get therelationship betweenthe boundaryvaluesof bothDirihlet and Neumanntype onthe

twolongitudinalboundariesofthethin layer.

Onthedomainofproblem Ω^,weset

Ω± :={(r, z)∈Ω :r≷rt2}

The thin layer is depited by the domain Ω^δ_c ⊂ Ω+^{. We denote by} u^δ_± ^{the elds outsidethe}

depositlayer,withu^δ₋ ⁱⁿ Ω− ^andu^δ₊ ⁱⁿ Ω+\Ω^δ_c ^{(at theshellsideof thedeposit layer),andby} u^{δ thein-layereld,i.e. in} Ω^δ_c ^{(see Figure1). Assume that thethikness}fδ(z) ^{at thevertial}

positionz ^isoftheorder δ

fδ(z) =δd(z),

whereδ^{isasmallparameterand}d(z)^isindependentofδ^{. Assumingthatthedepositondutivity}

writes

σc= σm

δ^m, ⁽⁴⁾

where σm ^{is an} appropriately re-saledondutivity and m ^{the re-saling parameter. We will}

partiularly interestin the ases where m = 0,1,2^{. So in the deposit layer the eddy urrent}

equation(2)writes

−div 1

r∇(ru^δ)

−k²_m

δ^mu^δ= 0 ^onΩ^δ_c, ⁽⁵⁾

wherek_m² := iωµcσm^{. Weonsiderthevariable}substitution

ρ=r−rt2

δ , ρ∈[0, d(z)],

andwedenotebyu˜= ˜u(ρ, z) :=u^δ(rt2+δρ, z)^{there-saledin-layersolution. From(5),onegets} r²

δ²∂_ρ²u˜+r

δ∂ρu˜−u˜+r²

∂_z²u˜+k_m² δ^mu˜

= 0.

Bysubstitutingr ^withrt2+ρδ^,wegetform= 0,1^,

∂_ρ²˜u=−δB¹mu˜−δ²B²mu˜−δ³Bm³u˜−δ⁴Bm⁴u,˜ ⁽⁶⁾

with





























B¹0= 2ρ rt2

∂ρ²+ 1 rt2

∂ρ, B²0= ρ²

r_t²₂∂_ρ²+ ρ

r_t²₂∂ρ− 1

r²_t2 +∂_z²+k²₀, B³0= 2ρ

rt2

∂_z²+k²₀ , B⁴0= ρ²

r_t²₂ ∂_z²+k²₀ ,





























B¹1= 2ρ rt2

∂_ρ²+ 1 rt2

∂ρ+k₁², B²1= ρ²

r_t²₂∂_ρ²+ ρ

r²_t2∂ρ− 1

r²_t2 +∂_z²+ 2ρ rt2

k₁², B³1= 2ρ

rt2

∂_z²+ ρ² r²_t2k₁², B⁴1= ρ²

r_t²₂∂_z².

(7)

Form= 2^{,weonsideraweightedin-layereld} w^δ(r, z) :=√

ru^δ(r, z) r∈[rt2, rt2+fδ(z)],

andafterre-salingonehas

˜

w(ρ, z) :=w^δ(rt2+δρ, z)

Sotheeddyurrentequationforthein-layereld u^{δ (5)beomeshere} 1

δ²∂_ρ²w˜− 3

4r²w˜+k₂²

δ²w˜+∂_z²w˜= 0.

Bysubstitutingr ^withrt2+ρδ^,weobtain

∂_ρ²+k²₂

˜

w=−δB2¹w˜−δ²B2²w˜−δ³B³2w˜−δ⁴B⁴2w,˜ ⁽⁸⁾

with





























B2¹= 2ρ rt2

∂_ρ²+k₂² , B2²= ρ²

r_t²₂ ∂_ρ²+k₂²

− 3 4r²_t2 +∂²_z, B2³= 2ρ

rt2

∂_z², B2⁴= ρ²

r_t²₂∂z².

(9)

2.2 Taylor developments for u^δ₊

Wewould liketoextendthesolutionoutsidethedeposit layeru^δ₊ ^{throughthelayerdomaintill}

theinterfaeΓt2^{,i.e. from}Ω+\Ω^δ_c ^toΩ+^{,suhthat the}transmissiononditionsonΓc ^between

u^and u^δ₊ ^{ouldbeexpressedin termsof} u^δ₊ ^onΓt2^{. As}u^δ₊ ^{satises theeddy urrentequation}

withoeientsµ=µv ^andσ=σv= 0ⁱⁿΩ+\Ω^δ_c^{,itisnaturaltoassumethatitsextensionon} Ω^δ_c ^{satises thesameequation}

−div 1

µvr∇(ru^δ₊)

= 0 ⁱⁿΩ+.

Usingthevariablesubstitutionν =r−rt2^{,onerewritestheaboveequationinthefollowingform}

X4 j=0

ν^jAj(ν∂ν, ∂z)u^δ₊= 0, ⁽¹⁰⁾

where

A⁰(ν∂ν, ∂z) = (ν∂ν)²−ν∂ν, A1(ν∂ν, ∂z) = 2

rt2

(ν∂ν)²− 1 rt2

ν∂ν, A²(ν∂ν, ∂z) = 1

r_t²₂ (ν∂ν)²− 1 r_t²₂ +∂_z², A³(ν∂ν, ∂z) = 2

rt2

∂_z², A⁴(ν∂ν, ∂z) = 1

r_t²₂∂_z².

Iftheasymptotiexpansionofu^δ₊ ^withrespettoδ^writes u^δ₊(r, z) =

X∞ n=0

δⁿuⁿ₊(r, z),

theneahtermuⁿ₊(r, z)^{veriesthesameequation(10) . WithTaylorseriesexpansion,onehas} uⁿ₊(rt2+ν, z) =

X∞ k=0

ν^ku^n,k₊ (z) ^where u^n,k₊ (z) = 1

k! ∂_ν^kuⁿ₊ (rt2, z).

Sine

ν∂ν

ν^ku^n,k₊ (z)

=k

ν^ku^n,k₊ (z) ,

weanindeedwriteAi(ν∂ν, ∂z)^asAi(k, ∂z)^{whileitisappliedto}

ν^ku^n,k₊ (z)

. Thus,from(10)

X4 j=0

X∞ k=0

Aj(k, ∂z)(ν^k+ju^n,k₊ ) = 0.

Theequalityat orderO(ν^k)^yields

A0(k, ∂z)u^n,k₊ =− X4 j=1

Aj(k−j, ∂z)u^n,k−j₊ ,

with u^n,−1₊ = u^n,−2₊ = u^n,−3₊ = u^n,−4₊ = 0^{. Now we onsider} A⁰(k, ∂z) = k²−k^{. For}k ≥2^, A⁰(k, ∂z)6= 0^{,thusinvertiblewithitsinverse}A⁻¹0 (k, ∂z) = _k2¹−k^{. Sowehave}

u^n,k₊ =−A⁻¹0 (k, ∂z)



 X4 j=1

Aj(k−j, ∂z)u^n,k−j₊



, k≥2. ⁽¹¹⁾

Nowwedenereurrentlytwofamiliesofoperators{Sk⁰(∂z),Sk¹(∂z)}^: S0⁰:= Id, S0¹:= 0, S1⁰:= 0, S1¹:= Id,

k≥2

















Sk⁰:=−A⁻¹0 (k, ∂z)



 X4 j=1

A^j(k−j, ∂z)Sk−j⁰ (∂z)



,

Sk¹:=−A⁻¹0 (k, ∂z)



 X4 j=1

Aj(k−j, ∂z)Sk−j¹ (∂z)



.

(12)

Fromthereurrentrelation(11),oneobserves

u^n,k₊ (z) =Sk⁰(∂z)uⁿ₊(rt2, z) +Sk¹(∂z)∂ruⁿ₊(rt2, z).

Thereforewehavethefollowingdevelopments













uⁿ₊(rt2+ν, z) = X∞ k=0

ν^k

Sk⁰(∂z)uⁿ₊+Sk¹(∂z)∂ruⁿ₊

(rt2, z),

∂ruⁿ₊(rt2+ν, z) = X∞ k=0

ν^k(k+ 1)

Sk+1⁰ (∂z)uⁿ₊+Sk+1¹ (∂z)∂ruⁿ₊

(rt2, z).

Wealsodene theoperators

Sek⁰:=Sk⁰− 1 rt2

Sk¹, Sek¹:= 1 rt2

Sk¹. ⁽¹³⁾

ThentheTaylorseriesexpansionswrite













uⁿ₊(rt2+ν, z) = X∞ k=0

ν^k

Sek⁰(∂z)uⁿ₊+Sek¹∂r(ruⁿ₊)

(rt2, z),

∂r(ruⁿ₊)(rt2+ν, z) = X∞ k=0

ν^k(k+ 1)

(rt2Sek+1⁰ +Sek⁰)uⁿ₊+ (rt2Sek+1¹ +Sek¹)∂r(ruⁿ₊) (rt2, z).

(14)