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SIMULATIONS OF PULSED

EDDY CURRENT TESTING

VIA SURROGATE MODELS

R. MIORELLI1, C. REBOUD1, A. DUBOIS1

T. THEODOULIDIS2, S. BILICZ3

1 CEA, LIST, Centre de Saclay, Gif-sur-Yvette, France

2 Department of Mechanical Engineering, University of Western

Macedonia, Greece

3 Budapest University of Technology and Economics, Budapest,

Hungary

OUTLINE

Overview on Pulsed Eddy Current (PEC) Time-stepping vs. Frequency Domain Summation (FDS) and inverse Fourier transform Semi-analytical models in harmonic regime, based on integral equations Standard spectrum interpolation Robust spectrum interpolation with metamodels Numerical validations and discussion Conclusion and perspectives

3

OVERVIEW ON PULSED EDDY CURRENT TESTING (PECT)

PECT generalities:

Pros:

• Deep penetration of eddy current inside the tested medium

• Relatively simple electronic devices (compared to multi-frequency ECT)

Cons:

• Lack of phase data

PECT applicative domain:

• Subsurface corrosion inspections

• Test of joints between layers

• Measurements of layers thickness

• Cracks detection in planar structures

• Tube wall thinning measurements (Remote Field-PECT)

• Cracks detection in tube (Remote Field-PECT)

4

SEMI-ANALYTICAL INTEGRAL METHODS FOR ECT

SIMULATION IN HARMONIC REGIME

• Volume Integral Method: very suitable for volumetric cracks (high generality vs. low CPU time efficiency)

• Boundary Element Method: dedicated to narrow & ideal crack (very high CPU time efficiency vs. less generality)

Volumetric (or surface) dipole density

Narrow or surface (ideal) crack

BEM model

• BEM assumptions: [Bow94], [The10], [Mio13]

1. crack gap ≤ d/2

2. Lateral faces lie parallel to each other

3. Crack is a barrier to the incident EC

• Green dyad Gnn adapted to narrow cracks (2D-mesh required)

variation of electromotive force (V)

1. Calculation of primary field

2. Calculation of total field (state equation)

Volumetric crack

VIM model

Volume integral

Total Electric Field Primary field

3. Calculation of coil response via reciprocity theorem

variation of electromotive force (V)

[Bow94] Bowler, App. Phys. (75) 1994

[The10] Theodoulidis, NdT&E (43) 2010

[Mio13] Miorelli, Reboud, Theodoulidis, and Lesselier, IEEE Trans. On Mag. (49) 2013

5

TIME-STEPPING AND FREQUENCY DOMAIN SUMMATION WITH INVERSE-FOURIER TRANSFORM

PECT and RF-PECT modeling

Time domain approach

Direct calculation of PECT signals

Integral Equation Methods (IEMs): use of time-

domain Dyadic Green Function (DGF) [Fu08]

Finite Element Method (FEM): resolution of a

system of differential equations in time domain

(time-step or Time Domain Integration (TDI))

Pros.:

• Precision in obtained results (no

approximations)

Cons.:

• It may be a very time consuming approach

(FEM)

Frequency domain approach

1. Calculation of harmonic solution at different

(wisely chosen) frequencies

2. Interpolation of results on a given range of

frequencies

3. Time-domain signal obtained via Inverse

Fourier Transform (IFFT)

Pros.:

• Use of “classic” harmonic solvers based on

IEMs and FEM

• Normally faster than time-step techniques

Cons.:

• Accuracy depends on the interpolation

(interpolation functions, how many

frequencies and which ones)

[Fu08] Fu and Bowler, IEEE Trans on Mag. (8) 2006

6 FREQUENCY DOMAIN SUMMATION

Classical spectrum interpolation

• Interpolation of the frequency response H(f) of the system, from a set of evaluations at log-

spaced frequencies: precision and/or efficiency depend on the excitation signal (i.e. H(f) is

not accurately interpolated everywhere), number of frequencies chosen manually

Metamodel H*(f) built from a database of frequency response evaluations H(fi)

• Radial Basis Function (RBF) built from sequentially designed database [Dou11]

• Kriging interpolator built from a database verifying an “Output Space Filling” criterion [Bil10]

Pros.:

• Possibility to generate a robust database (H(f) is accurately interpolated everywhere, i.e. the

accuracy obtained is independent from the excitation signals)

• Possibility to generate multi-dimensional (2D, 3D, …) database (ex. coil lift-off, crack

dimension(s) and position, specimen characteristic(s), etc.)

Cons.:

• more complicated to develop than the classical signal interpolation

Frequency Domain Summation (FDS) and interpolation schemes [Xie11] [The12]

[Xie11] Xie, Chen, Takagi, and Uchimoto, IEEE Trans. On. Mag. (47) 2011

[The12] Theodoulidis, Wang, and Tian, NdT&E (47) 2012

[Bil10] Bilicz, Lambert, and Gyimóthy Inv. Prob. (26) 2010

[Dou11] Douvenot, Lambert, and Lesselier, IEEE Trans. On. Mag. (47) 2011

7

SPECTRUM INTERPOLATION AND DATABASE GENERATION APPROACHES

Overview on FDS with logarithmic interpolation

Time step: 5 e-4 s

Output data

[H(f1), H(f2)…, H(f20)]

Calling Forward

Solver (20 times)

Inverse Fourier Transform

Interpolation (ex. cubic splines)

[H*(f1), H*(f2), …, H*(fN)]

Input Signal

I(f) x Output Signal y*(t)

How many and which frequencies

8


Initialization [H(fmin), H(fmean), H(fmax)]

Interpolated spectrum (RBF or kriging) [H*(f1), H*(f2),…, H*(fN)]

Database generation and exploitation via

sequential design and Radial Basis Function

(RBF) or Output Space Filling (OSF) and kriging

Inverse Fourier Transform Output Signal y*(t)

FDS with database and metamodel interpolation

Input Signal I(f) x

9


Error criteria:

Constrained mesh refinement: dichotomy principle

it may be expensive from the computational point of view

Overview on SD and RBF basic concepts in 2D [Dou11]


Mesh refinement process: evaluation at

every center of mesh edges

Each of them involve a

forward solver call

Maximum error

next point

ECT signals Frequencies Forward solver

0

1

0

1 Param. 1

Para

m. 2

10


Overview on SD and RBF basic concepts [Dou11]

Database in frequency domain:

Pros.:

• Guaranteed precision of the metamodel

• TPS: no kernel parameter

Cons.:

• The database algorithm is quite expensive in terms of number of evaluations of the

direct model (every interval is tested in its central point)


Computation of weights wi

TPS on the database points

Evaluated

signals Weights

RBF metamodel interpolation:

Thin Plate Spline (TPS) kernel

ECT signals Frequencies

x

y

x1 x4 x2 x5 x3

Forward solver

11


Overview on OSF and functional kriging basic concepts [Bil10]

Functional kriging Best Linear Unbiased Predictor (BLUP)

Database in frequency domain:

[Bil10] BIlicz, Vazquez, Gyimóthy, Pávó, and Lambert, IEEE Trans. On. Mag. (46) 2010

Covariance (based on Matérn function hp.

stationary process)

Kriging coefficients

Calculated statistical

behavior

Estimated random

process

ECT signals

Distance function Associated stochastic process

Frequencies Forward solver

Unknown (new)

frequency Already calculated

ECT signals

Distance function

12


Linear system to solve in order to find the coefficients

Pros.:

• Description of approximated values in stochastic form (increases interpolation flexibility)

• At least as good as RBF in terms of calculation load

Cons.:

• Tuning of some statistical parameters in the Matérn covariance matrix

Interpolated values via kriging interpolation

Overview on OSF and functional kriging basic concepts [Bil10]

[Bil10] BIlicz, Vazquez, Gyimóthy, Pávó, and Lambert, IEEE Trans. On. Mag. (46) 2010

Covariance matrix Evaluated signals

Lagrange multiplicator

Kriging coefficients

13

NUMERICAL VALIDATION: COMPARISON VS. TIME-STEPPING FINITE ELEMENT METHOD CODE (PART 1/3)

13

Test case #1: planar multilayered structure with narrow crack

Map size: (x,y)= 60x80 samples BEM unknowns: Nx

x Ny x Nz = 1 x 20 x 10

First layer thickness: 2.0 mm

Conductivity: 15.0 MS/m

Second layer thickness: 1.5 mm


Third layer thickness: 2.5 mm


Last plate crack (l x w x d)= 20.0 x 0.2 x 2.5 mm

Excitation (pulse) signal:

Rise time: t= 250 us

i(t)= I0[1-exp(-t/t)]

Signal period: 10 ms

GMR sensor (receiver)

Driving coil (emitter)

Buried crack

[Mio13] Miorelli, Reboud, Theodoulidis, and Lesselier, IEEE Trans. On Mag. (49) 2013

• Harmonic signals have been model by using the BEM [Mio13]

Inspected area

14


Test case#1: planar multilayered structure with crack (comparison with time-step FEM)

Good agreement with respect to FEM data (Comsol time-step results): • RBF interpolated values obtained with 35 forward solver

• Kriging interpolated values obtained with 15 forward solver

High efficiency: one forward solver calculation is performed in about 30 sec

(on standard PC for the complete map calculated over 60 x 80 scan positions)

Peak time 0.5 ms

15


Test case#1: planar multilayered structure with crack 2D-database

Study of frequency and crack length variations (L = [10,20] mm)

Results for OSF-kriging database built with 200 samples: results displayed for coil centered

over the crack

20

increasing

crack length

increasing

crack length

Time signal computed for all 60 x 80 scan positions

16

Test case#2: Remote-Field PECT in homogeneous tube [Yan13]

Pick-up coil

Tube inner radius 120 mm

Wall thickness: 5.0 mm


Relative permeability: [1]

Transversal crack (l x w x d): 15 x 5 x 2 mm



i(t)= I0[1-exp(-t/t)]


Driven coil

Transversal crack

TUBE CASE: OBTAINED RESULTS WITH CIVA WITH SPLINE INTERPOLATION (LOGSPACED FREQUENCIES) (PART 1/3)

• Time-harmonic signals have been model by using the VIM [Ska08]

[Ska09] Skarlatos, Pichenot, Lesselier, Lambert, and Duchêne, IEEE Trans. On Mag. (40) 2008

[Yan13] Yang and Li, NdT&E (53) 2013

VIM unknowns: Nx

x Ny x Nz = 10 x 10 x 5

17


• Time-harmonic signals have been model by using the VIM [Ska08]

[Ska09] Skarlatos, Pichenot, Lesselier, Lambert and, Duchêne, IEEE Trans. On Mag. (40) 2008

[Yan13] Yang and Li, NdT&E (53) 2013

Pick-up coil

Driven coil

Axial crack

Tube inner radius 120 mm

Wall thickness: 5.0 mm


Relative permeability: [1]

Axial crack (l x w x d): 15 x 5 x 2 mm



i(t)= I0[1-exp(-t/t)]


Test case#2: Remote-Field PECT in homogeneous tube [Yan13]

VIM unknowns: Nx

x Ny x Nz = 10 x 10 x 5

18

Test case#2: 100 frequencies (all the values of FFT of i(t)) vs. kriging


Kriging interpolated values obtained with 25 forward solver for all 351 (linear scan)

samples

Results displayed for the pickup coil centered under the crack

19 CONCLUSION AND PERSPECTIVES

Conclusion: • Overall good agreement between the frequency summation approach via integral equation

based methods (BEM or VIM) and time-step FEM simulations (planar case)

• Speed-up in CPU time obtained by the presented work vs. time-step FEM (BEM or VIM are

almost independent from the number of coil positions)

• Reliability of database approach

• Unsupervised choice of computed frequencies (even if sub-optimum number of frequencies

is chosen by RBF)

• Possibility to model any output signal (within the database frequency range) independently

from the applied excitation (different pulse shapes, linear ramp, etc.) via an already built

database and an “real-time” interpolation

Further perspectives: • Further numerical (time-step FEM) comparison and experimental validations of the model

• Study of database generation with larger number of dimensions (i.e. coil lift-off, different

crack parameters, different specimen characteristics variation of conductivity,

permeability, driven-pickup coil distance in tubes, etc.)

http://www-civa.cea.fr/

Direction DRT

Département DISC

Laboratoire LSME

Commissariat à l’énergie atomique et aux énergies alternatives

Institut Carnot CEA LIST

Centre de Saclay | 91191 Gif-sur-Yvette Cedex

T. +33 (0)1 69 08 58 28 | F. +33 (0)1 69 08 70 08

Etablissement public à caractère industriel et commercial | RCS Paris B 775 685 019

Thank you




22


Sequential Design (SD) and RBF with Thin Plate Spline (TPS) kernel [Dou11]

Database generation (SD)

Initialization

[X(fmin,x,y), X(fmean,x,y), X(fmax,x,y)]

Database evaluation

Stop Criterion

Calling Forward

Problem

NO

YES Interpolated spectrum

[X(f1,x,y), X(f2,x,y), X(fN,x,y)]

TPS RBF Interpolation

[X*(f1,x,y), X*(f2,x,y), X*(fM,x,y)]

IFFT y*(t,x,y)

Database exploitation via

RBF

Evaluation of

interpolation error

[Dou11] Douvenot, Lambert and Lesselier, IEEE Trans. On. Mag. (47) 2011

23


Outline in Output Space Filling (OSF) and kriging in database generation and

exploitation [Bil10]

Initialization

[X(fmin,x,y), X(fmean,x,y), X(fmax,x,y)]

Kriging Prediction

New Observation

Remove samples Stop Criteria Output Database

[X(f1,x,y), X(f2,x,y), X(fN,x,y)]

Kriging Interpolation

[X*(f1,x,y), X*(f2,x,y), X*(fM,x,y)]

IFFT y*(t,x,y)

NO NO YES YES

Database exploitation

via kriging

Database generation with OSF

Calling Forward

Problem

[Bil10] BIlicz, Vazquez, Gyimóthy, Pávó and Lambert, IEEE Trans. On. Mag. (46) 2010

24


Interpolation problem via interpolation

Overview on RBF basic concepts [Dou11]

Thin Plate Spline (TPS) kernel

Forward problem

Pros.:

• Possibility guarantee a required error among the interpolated data

• No estimation of kernel parameters

Cons.:

• It may be very expensive from the computational point of view ( )

[Dou11] Douvenot, Lambert and Lesselier, IEEE Trans. On. Mag. (47) 2011

Refined mesh of the database

Evaluated

signals

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