Modeling of ultrasonic fields and their interaction with ... - IEEE Xplore

Modeling of ultrasonic fields and their interaction with defects P. Calmon, A. Lhkmery, L. Paradis Commissariat a I’Energie AtomiqueKEREM, Saclay, France

- Ultrasonicmodeling at the FrenchAtomic is based onto continuous EnergyCommission(CEA) developments o f two approximate models covering realisticcommonly encountered NDT configurations.The first model, based on an extension of the Rayleigh integral, is dedicated to the prediction o f ultrasonic fields radiated byarbitrary transducers. Being computed the field may be inputted in a model o f the echo formation. Thistwo step modelingyields the prediction o f echostructures detected with transducer scanning. The associated software is implanted in the ClVA system for processing and imaging multipletechnique NDT data.

Abstract

INTRODUCTION This communicationreviews the modelsdeveloped for several years at the French Atomic Energy Commission (CEA) for computing ultrasonic fields radiated by NDT transducers and for predicting their interaction with defects or more generally with reflectors encountered in the inspected component [l-31. These models are not devoted only to laboratory uses, therefore, emphasis has been on finding the best compromise between as accurate as possible quantitativepredictions and ease, simplicity and speed, crucial requirements in the industrial context. Twomaincomplementarymodels have been implelmented in the data acquisition andprocessing system CIVA, covering themost commonly encountered N D T configurations. The first one is devoted to the computation o f the fields radiated by transducers into pieces under examination. I t is based on an extension o f the Rayleigh integral to account for the refraction at the coupling-piece interface. Thismodel has been recentlygeneralized to deal withanisotropic and heterogeneous materials.The second model simulates a whole testingexperiment by predicting the echo-structure for each scanning position of the probe andsynthesizing images typical o f those actually measured. I t assumesan a priori knowledge o f the incident field and therefore is complementary o f the first model. The beam-defect interaction applies K i r c h o f f S approximation adapted to elastodynamics. The theory has been established in the case o f isotropic and homogeneous material and has been recently extended in the aim o f taking into account realistic, and often complex, geometry o f inspected components. In the following we describe the theoretical basis of the two models.

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FIELDMODELING Principles Consider an arbitrarily shaped piezoelectric transducer immersed in a fluid (generally focused transducer) or in a solid wedge ( a so-called contact transducer). The tested component may be constituted o f multiple different isotropic or anisotropic materials separated with interfaces o f complexgeometry.The purpose o f this model is to calculate the field radiated by the transducer at any point intothe component. T h e model i s based on the Rayleigh integral formulation for the radiated field in the coupling medium. When the coupling medium is a solid wedge, it can be approximated for a fluid medium [4]. The effect of shear wave propagation inside and transmission from the wedge are thus assumed to be negligible onto theradiated field. Let us divide the emitting surface into small elementary ones, considered as point sources. We then apply the discrete Rayleigh integral in the coupling, that is a discrete summation of elementary fields multiplied by the related elementary surface. Each source point generates a sphericalwave in the coupling medium, distorted at each interface it crosses and possibly converted into other types o f reflected orrefracted waves. The Geometrical Optics (GO) approximation allows to consider such a wave at the observation point as a plane wave, since it is far enough from the point source (say a few wavelengths), as it can be for a spherical wave in a fluid. The direction o f this plane wave is given by the GO path, defined to respect Snell-Descartes law at each interface. Along this path, the time-of-flight from the source to the observation point is calculated. By this mean, and after the summation over all the elementary point sources, interferencesbetween them, thus the diffraction o f the whole source, are taken into account. In a transient model, this is represented mathematically by an impulse response. The GO approximation applied to each point source is accurate since a divergence factor ( D O is multiplied to the amplitude o f the plane wave, as for thespherical wave for which this coefficient is reduced to the inverse o f the distance between the observation point and the source.

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Source

4 Calculation Point Figure I . Schematic description o f a pencil

The Pencri method The divergence factor is derived with the help of the pencil method. A pencil is a collection o f rays included in a cone which vertex is the point source and which aperture has a solid angle dQ. The energy conservation principle is applied to this pencil and under the assumption that the energy, following the rays, only goes through the section surface (in gray on FigureI),DF is given by[ 5 ] :

where dS is the surface ofthe section. I n order to describe a pencil and its evolution during the propagation, one w i l l first define two rays : the axial ray is the symmetry axis of the pencil and i s nothing but the GO path. The paraxial ray belongs to the pencil envelope. Since the paraxial ray is given as a function of the axial ray, the surface dS is known and linearly depends on the solid angle. Therefore, the paraxial ray is mathematically described with respect to the axial ray. A four component is vector, w i l l be the so-called pencil vector denoted by required to describe any pencil at a given position along the considered geometrical optic path. One has :

W.

(ds..)

These four quantities are depicted on Figure 2 The gray plane is perpendicular to the axial ray and is tangent to the wavefront at a given time, and then follows the propagation. The first two quantities give the position o f the paraxial ray with respect to the axial ray in the gray plane. The last two quantities give the direction o f the slowness vector ; only two out of three components o f this latter vector are required since the third one is given by the elastic characteristics o f the medium, one type of wave being considered at once. After these settings. the calculation under GO approximationof the propagation of a wave generated by a point source w i l l consist o f the calculation of a 4x4 matrix, say, the pencil propagation matrix denoted by L, linking two pencil vectors on the same GO path taken at two different positions, namely at the source and at the observation points (the related pencil vectors are denoted by subscripts l and 2, respectively). This yields : v2 = L.\y, =

[:

3

.

(3)

Here four 2x2 matrix A, B. C and D are used in order to simplify the following notations. The calculation of DF is leaded by Equation (I)where the different quantities can be evaluated with the help of the pencil vectors. I t can be shown that D F is given by :

DF-'

-

= S:

det E .

(4)

The formulation o f the pencil propagation is reduced to the calculation ofthe pencil propagation matrixL. Let usfirstly consider the propagation in one homogeneous medium. In the isotropic case, two pencil vectors are linkedwith a homothetic transformation. In the anisotropic case, the transformation also involves the energy direction (which gives the GO path). This direction being given by the normal to the slowness surface, the partial curvatures of this surface are included in the matrix formulation. The result is :

where g ( S x , s y )= sz and the first and second subscripts denote the number o f derivations with respect to

S, and

axial ray Figure 2. Definition ofthe pencil vector.

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S,. , respectively.

The ( s x ,S,,) plane is tangent

to the wavefront at i t s intersection with the axial ray, but not necessarily orthogonal to the axial ray. Therefore a subscript k is added to the distance of propagation

rk in order to denote that it is calculated along the wave vector direction. Let us consider now the crossing o f an arbitrarily curved interface. The 2x2 curvature matrix .\ describes

this interface. One w i l l consider that the two pencil vectors are located immediately before and immediately after the interface. The matrix is then exmessed as :

point) and the imaginary parts with the Hilbert transform ofthis signal [Z]. Appiicutions

where h = k ' c o s 8 - k c o s 8 , B a n d 6' being the incident and reflection or transmission angles (evaluated between the wave vector and the normal o f the interface), and k and k ' the wave numbers.The 2x2 matrices 0 and EY represent the projection o f a vector belonging tothe plane tangent to the wave front (previously defined) on a plane tangent to the interfaceand containing its intersection with the axial ray. The direction of this projection is the energy direction, i.e. the axial ray. The formulation o f the matrix is then :

1

@ =

X

co~8-sin6(sinpg~,~ siny,cosO-sinGgo,l

[

cosy,

+co~pg~,~)

-cosy,c~se+sin@,,~ sin9

in which one recognizes the first partial derivatives o f the slowness function, then introducing the energy velocity. The angle 9 defines a rotationaround the axialrayto relate the system axes o f the pencil to those o f the interface curvatures [ 6 ] . From these two previous configurations wecan derive the formulationof a matrixdefining any complex case, by lnultiplying elementary matrices describing each elementary step (propagation. surface crossing).

Compu!urion ofrhe eiemenrav contributions To obtain the amplitude o f the elementary contribution ofone pointsource to the whole impulse response, it is necessary to !multiply the divergencefactor deduced from the pencilmethodwith thetransmission oriand reflectioncoefficients related to every interfaces. In the general case the product o f these coefficients is described by a3x3 matrix yielding the three components o f the displacementelementary field. Other elastic quantities such as displacementpotentials or stresses can bededuced from the displacement field considering the locally planar propagation assumed by GO approximation. When one or more evanescent waves are generated at the interface, the transmission coefficient owns a non zero phase proportional to sgn(o) [7]. Therefore the 3x3 coefficient matrix, andconsequently the elementaryimpulse responsehas real and imaginary parts leadingto a pulse distorsion in the transientregime.The real par! of the impulse responses have to be convolved with the input signal (the normal component o f the velocity at the source

A wide set of experimentalvalidationshave been performed in the case o f isotropic and homogeneous materials and show the high accuracy o f the predictions [ 2 ] . Also, comparisons with finite elements method [S] in the simplest cases have shown very good agreement. Confrontations with experiments results on heterogeneous and anisotropic cases are in progress. The model has been widely applied to various NDT configurations and especially for the design o f phased array transducers and the conception o f adaptive focusing inspections [9,lO]. I n the above paragraphs we have only mention the case o f one single piezoelectric element. I t is possibleto easily deal with phased array transducers by computing each element contribution and then sum up these contributions with adequate amplitude and delay laws. Similarly,to deal with transducers focused by ]means o f an acoustic lens, we model the lens as a continuous distribution o f delay and amplitude.Oneimportantapplication o f the ]model is the prediction o f echoes detected during the inspection of aspecified component containing defects. MODELING THE ECHO FORMATION Princrp1e.s

Indeed, once the field radiated into the tested component being computed with the previous model it ]nay be used as an input o f an echo formation model yielding the whole inspection simulation. This second lnodel is based on the following hypotheses and approximations. Firstly, we consider that the radiated field is invariant during the transducer scanning, and that the specimen is homogeneous and isotropic. Then we assume that echo structure arisingfrom the interaction with a reflector (defect or boundary) can be described as a suln of echoes, corresponding to the different possible ultrasonic paths, and that each o f them can be independently computed. We also make some assumptions upon the field in the inspected component. We assume thatthe incident wavefronts on the defect are plane in the case of a focused transducer and spherical in the case o f a planar transducer. Therefore the field is modeled by the product o f a spatial function q describing the amplitude distribution in the beam and a time-dependent function describing the wave propagation : Q ( M , t ) = q ( p . Q , ~ ) Q o (- A i !)

(8)

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where ( p , S, z ) are the cylindrical co-ordinates of the point M in a beam-referenced system. That is, p measures

E ( x , y , t )=

the distance to the focal axis, Q is the angle between the plane of incidence and the vector joining the point M and the focal axis, and z is the distance along the focal axis. The amplitude distribution y ( p . 8 , ~is) deduced from the

defec1

result o f field computation performed by the model described in the previous section. In the case of pressure waves the quantity @ being computed is the scalar velocity potential while in the case o f shear wave it is the comthe ponent o f the velocity vector potentialnormalto transducer plane o f incidence. The function @,,(t A t ) describes the propagation of a ~

waveform Q , ( r ) assumed to he invariant in the beam. The analytical expresssion o f the time of flight At contains the hypothesis of plane or spherical wavefronts and depends on the transducer type being considered. I n the case o f a focused transducer, one has : H

z

c

c,

Ar=-+where

H is the coupling medium

(9) height, c

and c,

res-

pectively are the wave speeds in the couplingmedium and in the specimen. I n order to predict echoes arising from defects after reflections on the backwall of the specimen we model these reflections considering a locally planar propagation and a perfect planar mirror tangent to the backwall at the intersection point with the transducer axis reflection. Thus the field after reflection is obtained by lnultiplying the right member of Eq. (8) with adequate values o f p. Q, and :by the planewave reflectioncoefficient. The beam-defect interaction is modelled using Kirchhoffs diffraction theory applied to elastodynamics. The theory gives the amplitude of the scattered wave in the f o r m o f coefficients after interaction with defects and takes account of the possible mode-conversion that may occur. They are issued from previously published work by Chapman [I I] for the scattering by cracks based upon a simplified expression for the crack-opening displacement, and by Lhemery [ 121 for the scattering byvolumetric flaws. Lastly, we assume the principle of reciprocity between radiation and reception, allowing us to associate the same ) reception as for radiation. distribution y ( p , 0 , ~for Consider as a first step the case where no modeconversionoccurs. In the framework o f the previous approximations we can write the echo E ( x , y , t ) received at time t and for the scanning position ( x . y ) (the x-axis beingin the plane o f incidence while the y-axisbeing normal to it) as

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IC.~~~~(~~(A~~+A~~))~(~~,S (10)

where @ denotes temporal convolution. Computing the echo therefore consists in summing the contributions o f all the infinitesimal surface elements ds distributed along the defect. The contribution o f a given surface element ds w i l l depend on its position relative to the emission (pc,Se,ze)and relative to reception,

(pr,Sr,zr), The positions relative to the emission and to the reception are different as long as the paths going and coming are not the same, which is the case for the corner echoes. The diffraction by the defect, as well as the different reflectionsof the beam.are taken into account by the Crt transfer function.Lastly, Eq. (IO) can he generalised to take mode conversions into account. One only replaces the function y by two different functions qr and yr describing either the L or T beam depending on the echo considered. Applicarion.r

The modelallowstopredict echoes arisingfrom planar or volumetric defects which may be located near the backwall o f the specimen In such a case. in addition to the diffracted echoes arising f r o r the defect tips, there exist supplementary echoes. the so-called corner echoes, that involvetwo successive reflections : one on the defect and the other on the backwall (or vice-verso) and the so-called (( indirect )) echoes involvingreflections onthe backwallboth before the interaction with the defect and after it. Furthermore. the model predicts all theseechoes including thepossible mode conversions that may occur along the ultrasonic path [ l ] . The model can also be applied to predict the geometrical echoes generated by reflection 011 thespecimen boundaries [3,13]. For example. irregularitieson the hackwall surface and also on the entrance surface can give rise to more or less strong echoes and lead to difficulties in the interpretation o f acquisitions. Strong echoes are expected as soon as the reflecting surface (backwall or entrance surface after a reflection on the backwall) contains areas normal to the incident wavefronts. Another configuration responsible o f noticeable echoes is the presence o f slope discontinuitiesor irregularities on the reflecting surface. These mechanisms of echo formation can he modeled applying the same approach as described above for field-defect interaction [ 3 ] . Here the echoes are obtained by a surface integration onthe area intercepted by the beam.

For several years the model has been implemented in the ClVA system and has been widely used as a tool to help expert’s interpretation of indications,especially in the context of on-siteinspection of Frenchpressured water reactors vessel [ 141. It has been also used as kernel of an inverse procedureallowingautomatic defectcharacterization [ 151. CONCLUSION

In this communication we have presentedthe ultrasonic NDT modeling activity at the French Atomic EnergyCommission.Approximatedmodelsare developed in order to simulate realistic industrial configurations. The field radiated by arbitrarypiezoelectrictransducers are computed by applying a model based on an extension of the Rayleigh integral. These fields are then inputted in a second model of echo formation to predict the inspection results. REFERENCES

[I] P. Calmon, 0. Roy, Ph. Benoist, IEEE Ultrasonic Symposium Proceedings 1994, p 1265. [2] M. El Amrani, P. Calmon, 0. Roy, D. Royer and 0. Casula, in Review of Progress in QNDE. Vol. 14,

[3]

eds. D.O. Thompson andD.E. Chimenti(Plenum, New York, 1995) p. 1075. A. Lhemery, P. Calmon, R. Raillon and L. Paradis, in Review offrogress in QNDE, Vol. 17., op. cir. (1998), p. 899.

[41 T.P. Lerch, L. Schmerr and A. Sedov, in Review of Progress in QNDE, Vol. 16, op. r i t . (1997). p. 885. [51 G.A. Deschamps, Proceedings of the I.E.E.E., 60, I022 (I972). [6] S.W. Lee, M.S. Sheshadri, V.Jamnejad, R. Mittra.

I.E.E.E. Trans.MicrowaveTheoryTech..MTT30, 12 (1982). 171 L. G. Chambers, Wove Motion. no. 2. pp. 247-253, 1980. [8] P. Calmon, A. Lhemery and J. Nadal. in Review of Progress in QNDE. Vol. 15., op. cil. (1996). p. 1019.

[9] S. Mahaut. C. Gondard, M. AI Amrani, P. Benoist and G . Cattiaux, Ultrasonics 34. p 503 (1996) [ I O ] S. Mahaut, 0. RoyandM. Serre.Ultrasonics 36. 127(1998).

[ l I] R. K. Chapman,CEGB Report, North Western Region applications Centre, NDT NWR/SSD/82/0091M (1982). 98 (1995). p [l21 A. Lhemery, J . Acoust.Soc.Am. 2197. [ l 3 1 P. Calmon, A. Lhemery, I. Lecoeur. R. Raillon and L. Paradis, Nuclear Engineering and Design, 180, (1998),p271 [l41 F. Lasserre, L. Hernandez and L. Paradis, Nuclear Engineering and Design, 180, (1998), p 37 [ l 5 1 M. Faur, 0 . Roy 0.. Ph. Benoist, J . Oksman and Ph. Morisseau, in Review of Progress in QNDE, Vol. 16. op. cif (l997), p. 67.

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