- Email: [email protected]
Location of continuous AE sources by sensory neural networks
Ultrasonics 36 ( 1998) 525-~530
Location of continuous AE sources by sensory neural networks Igor Grabec *, Tadej Kosel, Peter Mui? FLICIII~~~of’M~~chmicul
En@wering,
Universit~~ of’i$ddjunu.
PO Bo.\- 394. S-1 01)O L..uh@mtr. Slovct~icr
Abstract A brief classification of location problems which appear in acoustic emission (AE) analysis is given. Empirical treatment of corresponding inverse problems is explained and applied to location of sources which generate continuous AE signals. A continuous AE phenomenon is treated as a stochastic process which is represented by the source coordinates and the correlation function of the emitted sound. The empirical model of AE phenomenon is formed based on a set of samples. The model includes a network of AE sensors and a neural network (NN ). During formation of the model, the AE signals are generated by sources at typical positions on a specimen. Recorded ultrasonic signals are transmitted to the NN together with the source coordinates. The first layer of NN determines the cross-correlation functions of signals and forms from them and source coordinates the data vectors. In the second layer, a set of prototype vectors is formed from the data vectors by a self-organized learning. After learning. the network is capable to locate the source based on detected sound. For this purpose, the sensors provide AE signals. while the NN determines the corresponding correlation function and associates to it the source coordinates. The association is performed by a non-parametric regression which is implemented in the third layer of NN. (C) 1998 Elsevier Science B.V. Kqxwrtk
Acoustic
emission:
Empirical
modelling:
Locator;
1. Introduction The basic task of acoustic emission (AE) analysis in non-destructive testing of materials is to find the position of the source [ 11. Location problem is most simple in the case of discrete acoustic emission when the AE signals are comprised of distinct bursts. For the solution of the corresponding problem. a method of associative mapping of acoustic bursts to source properties has been applied recently [2-41. In this article, we try to develop this method still further so that it could also be applied for a location of continuous AE sources. With this aim, the fundamentals of the physical description of AE phenomena and the basic problems of AE location are analysed. The analysis leads to the empirical modelling of continuous AE phenomena and an associative estimation of source position by a self-organized information processing system [2]. An AE phenomenon is physically described by a relationship between the force density .f’(r, t) and the displacement field u(u. t), which represent the source and the emitted sound, respectively [ 11. In the theory of elastodynamics, this relationship is described by a linear
* Corresponding
author.
Fax: + 386-61
-I 253-315:
e-mail: igor._cr;tbeciccfsuni-lj.si
004l-624X,“98 PI/
$19.00 ‘v 1998
SO041-624X(97)00056-5
Elsevier
Science B.V.
All rights reserved.
Neural network
partial differential equation. Its solution, includes the boundary and initial conditions. described by the convolution:
u(r, r)’
s
G(V,f”; t-r’)f(r,
f’)dvdt’,
which also is generally
(1)
Green’s dyadic G(.) describes a sound wave generated by a unit impulse in a point-like source [ 11. The fundamental problem of acoustic emission analysis is to estimate the source fieldf(r, t) from the acoustic signals detected at various positions {r,; r/= I.. .Di on a specimen [ 1.41. This inverse problem is ill conditioned because the signals from a finite set of points do not describe the complete displacement field [5]. Therefore, some method of regularization is needed by which this incomplete information could be transformed into applicable information about the source. Beside this, Green’s dyadic can be determined analytically only for very simple configurations of samples, while for most cases of practical interest it should be determined experimentally. This is usually too demanding and we are therefore usually forced to avoid a rigorous physical description of source properties and turn to an approximate treatment of the problem by a priori assuming a model of sources under consideration. The most simple models can be described as follows.
1.1. Locul source oj’discretr
Its force density
AE
is described
[2,4]. In this case, the database {x,, =(I: u,,(t)),,. d= l...D,
by:
f(r, r)=a6(r-r,~)6(u-Vs),
(2)
where u, i, and Y, denote the force magnitude, the time and the position of excitation, respectively. The sound field is in this case expressed by: U(U,l)=G@,
Ys; r-t,)a.
(3)
For the sake of simplicity, we further assume that the direction of the source force is given and that only one component of the displacement is detected. Eq. (3) is then simplified to: U(P,
t) = G(u,
P,;
t - t,y)a.
by:
IZ= l...Ni.
(6)
A location of continuous AE sources cannot directly be performed by any of the above-mentioned methods because acoustic bursts cannot be distinguished. However. it has already been shown that one can still apply the triangulation method if the time delays between continuous acoustic signals from various positions can be estimated from their cross-correlation functions [7]. Our purpose is to show that the correlation function could also be utilized in the comparison method as the input signal from which the neural network could associatively estimate the position of a continuous AE source. With this aim, we turn to the statistical description of continuous AE phenomena.
(4)
Green’s function in this equation generally represents a wave-burst. It is comprised of various wave modes which are propagating with different velocities. For the purpose of source location a wave mode is usually selected which could be recognized in a burst. An AE source can be then located by the following procedure: ( 1 ) From the burst signals detected at various positions on a specimen the differences of burst arrival times, Ar,, are estimated using a threshold element; and (2) from the differences of arrival times and the known path and velocity of the selected mode, the source position is estimated using triangulation techniques. A weak point of the triangulation method is that it is based on an analysis of ray paths which cannot always be simply carried out. Beside this, the non-linear equations from which the source coordinates are calculated are sensitive to experimental errors in determination of burst arrival times and often yield non-unique solutions. In order to avoid these deficiencies an associative method has been developed recently which is based on comparison of a detected burst with a set of prototype bursts [3,6]. With this aim, a set of N bursts is first excited by sources at various positions of the sample. From the source coordinates and the recorded time delays a set of prototype data vectors is then formed: (x, =(v, Atd)n, d= l...D,
is described
tz= l...N).
(5)
At the estimation of the source position, the differences of burst arrival times are first measured and then the corresponding vector rs is associatively determined based on comparison of the measured At, with those stored in the database. The corresponding estimation procedure is described in Section 1.2. For the purpose of associative estimation of the source position, the empirical information about the detected burst need not be given in terms of differences of arrival times At,, but can also be presented by the burst waveforms ud(t), which are detected by D detectors
1.2. Locul SOM~~~J of’continuous The force expression
density
AE
is in this case described
by the
.f(V, t)=a(t)ci(r-v,).
(7)
in which u(t) denotes a time-dependent force amplitude. The sound field is in then expressed by the convolution: u(r, t)=
G(u, u,%;t-
t’)u(t’) dt’.
(8)
s
We assume that the force amplitude represents a stationary stochastic process with a zero mean value E[u(v, t)] =O. Its basic characteristic is described by the correlation matrix: R,(t,
(9)
-rl)=Eta(t,)Oa(t,)l.
Here, 0 denotes the outer or matrix product of vectors. When the process is ergodic. the statistical average E[...] may be substituted by the average with respect to time t. One of the most applicable models is a white noise. or uncorrelated stationary source, which is described by the correlation matrix: R,,(t2-f,)=ROS(t2-tl).
(10)
The constant R, denotes the source intensity. correlation matrices of the source and the emitted are related by the equation: R,(v,,
~2, f,, tz)=E[u(v, =
The wave
t,)Orc(v, r,)]
ssss
G(r, , r’; t, - t’)G(u,,
x E[u(t’)@u(t”)]
r’; tz -t’)
d V’ dt’ d I”’ dt”
G(v,, U,$iI, - t’)G(r2, v,; f, -t”) =is x R,(t’-
t”) dt’dt”.
(11)
In the case when the source process is in time uncorrelated the displacement wave correlation matrix is described by the expression:
Mu,.
~2. t,
-tz)=Ro
s
G(r,. u,,; t)G(r2. r,; f+t,
=R,R,;(v,.rz,r,;fz-ft,).
-r,)
dr (12)
This clearly demonstrates that the wave correlation matrix depends on source position Y,, but the problem is to express the coordinates explicitly. Let us for this purpose consider the basic properties of Green’s function in an infinite medium [ 1,5]. It generally represents the excitation of the medium by an impulse in a point source at r,,. The excited disturbance propagates as a wave burst away from the source and generally arrives to the points of detection at rl and Ye at different times. Consider a case when the source and points of observation are distributed on a line. If there is just one wave propagation mode with the group velocity, c, then the propagation time from source to the ith detector is Ati= ]lri -ry,//lc.. One could expect that the correlation matrix R,(r,, rz, ry, t,- I~) should have a maximum when the time delay tz - t, compensates the difference between times of propagation At2 - Ar,. In a one-dimensional case this condition yields a linear relationship between the position of the source and the time delay t2- t, at which the wave correlation function has a maximum [7]. However, in practice the problem is usually much more complicated. First of all real specimens represent bounded media for which various modes of wave propagation with different velocities are characteristic. The structure of the wave burst is therefore generally very complex and does not exhibit a single, well-defined maximum from which the time delay could be reliably determined [ 11. In order to proceed to an exact relationship between source coordinates and the wave correlation matrix, one should specify the Green’s dyadic in detail, which is out of the scope for most applications [ 1.51. Besides this, the assumption that the source of continuous AE is in time uncorrelated may be false and there may also be several sources active simultaneously. Based on the relationship between the force and the wave correlation matrix, we can thus only conjecture that the wave correlation matrix contains an information about the source position, while the corresponding function cannot be specified in detail [7]. However. we can still conclude that this function can be modelled based on empirical prototype samples obtained by exciting the medium by a set of prototype sources. The corresponding data base can then be represented by N data vectors which are comprised of the source coordinates and the correlation matrices of the force amplitude and the
signals: (x,=[I:
R,(t),
Ru(r,,r2.
t)],;~=l~.~/‘v’I.
(13)
Such a general database still appears too extensive for practical applications. The treatment can be simplified by considering only one component of the source force and displacement field, which are represented by a correlation functions. Based on the linear relationship between source and the wave correlation matrix one can conclude that the magnitude of source amplitude is irrelevant for the estimation of its position, and utilize only normalized correlation functions in the database. If the mechanism of the prototype sources and the analysed source is always the same. then the source correlation matrix is also superfluous in the database and we can represent it in an abbreviated form as: (x,, =[I; @Jr,, I’,. r)],; II = 1 ..,N].
( 14)
Here e,, denotes a normalized cross-correlation function of AE signal which is detected at two different points on the specimen: (15) When there are more than two points of AE detection. the corresponding correlation functions can be concatenated in the data vector. The remaining problem is then to develop an information processing system by which the sound correlation function could be mapped to the source coordinates. For this purpose we have developed a radial basis function neural network by following the non-parametric approach to empirical modelling of natural phenomena which is briefly explained in the next section [ 2.41.
2. Non-parametric
modelling of AE phenomena
The empirical description of natural phenomena deals with the presentation of physical laws in terms of probability distributions. It has been explained in detail elsewhere and therefore we present here just the most important concepts of the formulation [2.4.8]. The basic object of the empirical description of the natural phenomena is a set of physical variables which are simultaneously measured by some array of sensors. In our example, this set represents a data vector comprised of source coordinates and AE signal characteristics. Let this vector be represented by A4 components: x=(&...&). In an empirical description of the AE phenomenon, we repeat the basic experiment N times to obtain the database (x,..,x,~] of N samples. Instead of a physical law that describes the relationship between the components of vector X, we further treat this vector as a random variable and characterize it by the empirical
528
joint
I. Gruhrc et al. / Liltrusonit:v 36 (19%‘) 525-530
probability
density
function:
(16) When the number of samples N increases without limit during the observation of the phenomenon we prefer to use a fixed number K of prototype vectors {ql...qk) to represent the properties of the random vector x. The empirical probability density function is then substituted by the representative one:
L(x)= ;
kil&--4k).
(17)
At N = K, we initialise the prototypes by setting qk = xk, while for N>K, we introduce an adaptation. With this aim, we first approximate the singular delta function in Eqs. ( 16) and (17) by a smooth Gaussian window or radial basis function:
-“ij”“],
lV=(x-q)=exp[
in which the parameter (T corresponds to an average distance between the prototype vectors. The representative probability density function is then adapted to the empirical one by minimizing the discrepancy between both distributions. An optimal adaptation is approximately described by an iteration formula for changes of prototype vectors [2,8]: Aq,z
K N+l
i
(XIV+1-4rMx,+1
-4,)
(19) The prototype vectors can be interpreted as the memorized contents of neurons in a radial basis function NN. Their adaptation is influenced by the input samples x, as well as by the interaction between the neurons that represents a kind of self-organization which maximally preserves the empirical information [2,8]. In the case of a low number N, all the samples can be treated as initialized prototype ones and an additional adaptation is not needed, which is convenient for applications. The prototype vectors represent an empirical model of the phenomenon under consideration. Their initialization and adaptation characterize the learning phase of the empirical modelling. The next theoretical problem related to this modelling is to show how the prototypes can be further applied [2,4]. Let us for this purpose assume that the observation of the phenomenon provides only partial information that is given by a truncated vector: g(gY1...4,; N),
(20)
in which 0 denotes the missing part. The question is then how to estimate in an optimal way the complementary or hidden truncated vector comprised of missing components:
h=(8; &+* ..&,f),
(21)
such that the complete concatenation
x=gOh=(&
data
vector
is obtained
...5,7,
by the
(22)
A statistically optimal solution of this problem is given by the conditional average estimator L =A( g). It is expressed in terms of truncated prototype vectors as [2,8]: ii= 5
B,@)h,,
where Bk(g)=
tv@-gk:k)
K
(23)
c \“k-K1)
I=1
The basis functions B,(g) in this expansion represent the measure of similarity between the truncated vector g given by measurement and stored truncated prototypes g,; therefore, the estimation resembles an associative recall which is characteristic of intelligence. The estimation of the hidden vector 4 is the main task of the application of the empirical model represented by prototypes. It corresponds to a non-parametric regression [2]. For the location of continuous acoustic emission sources, the corresponding locator must be able to detect AE signals and source coordinates, to determine the correlation function, to create the prototype vectors, and to estimate the source position from the detected AE signals. Each of these operations can be performed in a separate unit that can be interpreted as one layer of a sensory network.
3. Experiments The intelligent AE source locator, which was developed based on the empirical modelling of AE phenomena, is shown in Fig. 1. It is implemented on an automatic data-acquisition system controlled by a computer and supplemented by a network of sensors and preamplifiers for detecting AE signals and measuring source position during calibration procedure. As the AE detectors, piezoelectric transducers (pinducers) with broadband characteristic in the ultrasonic region between 0.1 and 1 MHz are applied. The diameter of the transducer active area is 1 mm and it can be therefore considered a point-like detector. The locator operates in two different modes. (1) In the learning mode, a set of N experiments is performed in which a complete information about the AE phenomenon is acquired. From the source coordinates and preprocessed AE signals. the sample vectors are created and supplied to the
prototype source 0 actual source n estimated source
0
Fig.
I,
Scheme of the intelligent
neural network in the PC where the prototype vectors are adapted to them. In the case of discrete AE phenomenon. either the AE signal wave forms or the time delays are used to describe the bursts. In the case of continuous acoustic emission, the cross-correlation function between signals from sensors is first calculated and then concatenated with source coordinates. (2) In the application mode. only acoustic data are provided which are then completed by the estimated source coordinates. The performance of the locator is demonstrated here by an example of continuous AE generated by grinding on a steel plate of diameter 250 mm and thickness 20 mm. Its applicability to associative location of discrete AE sources has been demonstrated elsewhere [3.6]. The purpose of the experiment reported here was to examine the accuracy of continuous AE source location. For the sake of simplicity, only linear location was considered. Fig. 2 shows a typical example of detected stochastic signals and their cross-correlation function. During training, eight prototypes were created in the memory. In the analysis mode of operation the performance was tested by 15 sources. Eight of them were generated approximately on the same basic spots as during training. while the remaining seven were placed betwecn basic ones. Fig. 3 shows the comparison of the true and the estimated coordinate. The experimental points are scattered around the prime line which corresponds to the exact solution of location problem. The errors mainly appear when sources that lie between prototype ones are located. The location of the sources at the positions of the prototype sources is uncertain mainly at terminal points. Presumably the reason for this uncertainty are the reflections of the acoustic waves on the specimen boundary. On average. the error of estimated position is less than the distance between two nearby prototype sources. This means that the locator properly estimates the zone from which the AE signal arrives, while it is less successful in exact interpolating. The reason for the
locator of ,AE sourcc~.
II’1
AE signal
I
s2
j ‘I
I
‘I
.
weak ability of generalization is presumably in the very complex structure of the correlation function of this stochastic AE phenomenon, which is a consequence of the wave propagation in the plate and the SOL~I-cc mechanism.
4. Conclusions
Our examination shows that the empirical treatment of the location problem leads to the development of an intelligent information processing system with a structure of sensory neural network that yields location results acceptable for NDT. Its intelligence is a conse-
530
I. Grabec et al. / Ultrasonics 36 (I 998) 525-530
Achl
Sowa
E~Soww Pnwvw SDUCS
Fig. 3. Results of location experiment. Horizontal coordinate; vertical axis, estimated coordinate.
axis,
true
quence of the adaptation procedure. For this purpose, the conventional calibration procedure is generalized such that a database of a sufficient number of prototype vectors is obtained. With an increasing number of prototypes the estimation error generally decreases. However, as with the calibration of conventional source locators, there is a problem with a proper simulation of AE sources. This is not an inherent problem of the intelligent method utilized, but rather a problem of the experimental reproducibility of experimental observation of the phenomenon. A similar conclusion is also valid for the problem of non-unique mapping which can be, caused by reflections of the sonic waves on the specimen boundaries. The greatest advantage of the empirical approach stems from the fact that no specific analysis of ray paths or adaptation of the corresponding location program is needed when the structure of the sensory network that forms the AE antenna is changed. The same locator is thus very flexible and applicable to different configurations. Even the variation of the number of sensors needs nothing more than a new setting of the number of components used in the description of the data vector. For the modelling, various paradigms developed by neural network research are applicable. The most well known among them is a multi-layer perceptron that learns to perform a mapping by back-propagation of errors. As it has been shown elsewhere the performance of such a network is equivalent to the system proposed here, although the physical interpretation of the parameters in terms of prototype vectors is more simple in our
case [2]. This also makes feasible a simple inscription of data without any training, when the number of statistical samples is lesser then the number of memory cells. In this case, the learning time is negligible in comparison with the time needed for execution of the prototype experiments. Moreover, the splitting of the data vectors into given and hidden components can be performed arbitrarily after the training procedure. The basic idea for the intelligent locator stems from the non-parametric modelling of empirical relations between multicomponent variables [2,4]. One could expect that the same method should also be applicable for a solution of initially mentioned general source identification problem in which the force field must be determined from the detected AE signals. For instance, if a set of joint records of both fields: {[NC t),f@, t)L; n= 1...Ni,
(24)
were obtained, one could train a neural network to map a detected AE signal to the corresponding source field. Some attempts to prove this possibility have been carried out recently [2,4], but the experimental specification of vector fields represents great problems. Therefore, this general method could hardly be applied to nondestructive testing in an industrial environment outside a laboratory.
References [I ] K. Miller, P. Mclntire P] [3 ]
[4] [5]
[6]
[7] [S]
(Eds.), Nondestructive Testing Handbook, vol. 5: Acoustic Emission, Am. Sot. NDT, Philadelphia, PA, 1987. I. Grabec, W. Sachse, Synergetics of Measurements, Prediction and Control, Springer, Heidelberg, 1997. I. Grabec, B. Antolovic, Intelligent Locator of AE Sources, Progress in Acoustic Emission, The Japanese Society for NDI, Tokyo, 1994, pp. 565-570. 1. Grabec, W. Sachse, Automatic modeling of physical phenomena: application to ultrasonic data, J. Appl. Phys. 69 ( 1991) 623336244. Y.H. Pao, L.E. Payne, W.W. Symes, F. Santosa, Inverse Problems of Acoustic Waves in Fluids and Solids, Report ONRjSRO III, TAM, Cornell University, 1986. T. Kosel, 1. Grabec, B. Antolovii-, P. MuiiE, Intelligent locator of acoustic emission sources in composite materials. in: V. Krstelj (Ed.), Proceedings of Inservice Inspection ‘95, Croatian Society NDT, Zagreb, 1995, pp. 235-243. I. Grabec. Application of correlation techniques for localization of acoustic emission sources, Ultrasonics I6 ( 1978 ) I I l-1 15. I. Grabec, Self-organization of neurons described by the maximumentropy principle, Biol. Cybernet. 63 (1990) 4033409.
Location of continuous AE sources by sensory neural networks Igor Grabec *, Tadej Kosel, Peter Mui? FLICIII~~~of’M~~chmicul
En@wering,
Universit~~ of’i$ddjunu.
PO Bo.\- 394. S-1 01)O L..uh@mtr. Slovct~icr
Abstract A brief classification of location problems which appear in acoustic emission (AE) analysis is given. Empirical treatment of corresponding inverse problems is explained and applied to location of sources which generate continuous AE signals. A continuous AE phenomenon is treated as a stochastic process which is represented by the source coordinates and the correlation function of the emitted sound. The empirical model of AE phenomenon is formed based on a set of samples. The model includes a network of AE sensors and a neural network (NN ). During formation of the model, the AE signals are generated by sources at typical positions on a specimen. Recorded ultrasonic signals are transmitted to the NN together with the source coordinates. The first layer of NN determines the cross-correlation functions of signals and forms from them and source coordinates the data vectors. In the second layer, a set of prototype vectors is formed from the data vectors by a self-organized learning. After learning. the network is capable to locate the source based on detected sound. For this purpose, the sensors provide AE signals. while the NN determines the corresponding correlation function and associates to it the source coordinates. The association is performed by a non-parametric regression which is implemented in the third layer of NN. (C) 1998 Elsevier Science B.V. Kqxwrtk
Acoustic
emission:
Empirical
modelling:
Locator;
1. Introduction The basic task of acoustic emission (AE) analysis in non-destructive testing of materials is to find the position of the source [ 11. Location problem is most simple in the case of discrete acoustic emission when the AE signals are comprised of distinct bursts. For the solution of the corresponding problem. a method of associative mapping of acoustic bursts to source properties has been applied recently [2-41. In this article, we try to develop this method still further so that it could also be applied for a location of continuous AE sources. With this aim, the fundamentals of the physical description of AE phenomena and the basic problems of AE location are analysed. The analysis leads to the empirical modelling of continuous AE phenomena and an associative estimation of source position by a self-organized information processing system [2]. An AE phenomenon is physically described by a relationship between the force density .f’(r, t) and the displacement field u(u. t), which represent the source and the emitted sound, respectively [ 11. In the theory of elastodynamics, this relationship is described by a linear
* Corresponding
author.
Fax: + 386-61
-I 253-315:
e-mail: igor._cr;tbeciccfsuni-lj.si
004l-624X,“98 PI/
$19.00 ‘v 1998
SO041-624X(97)00056-5
Elsevier
Science B.V.
All rights reserved.
Neural network
partial differential equation. Its solution, includes the boundary and initial conditions. described by the convolution:
u(r, r)’
s
G(V,f”; t-r’)f(r,
f’)dvdt’,
which also is generally
(1)
Green’s dyadic G(.) describes a sound wave generated by a unit impulse in a point-like source [ 11. The fundamental problem of acoustic emission analysis is to estimate the source fieldf(r, t) from the acoustic signals detected at various positions {r,; r/= I.. .Di on a specimen [ 1.41. This inverse problem is ill conditioned because the signals from a finite set of points do not describe the complete displacement field [5]. Therefore, some method of regularization is needed by which this incomplete information could be transformed into applicable information about the source. Beside this, Green’s dyadic can be determined analytically only for very simple configurations of samples, while for most cases of practical interest it should be determined experimentally. This is usually too demanding and we are therefore usually forced to avoid a rigorous physical description of source properties and turn to an approximate treatment of the problem by a priori assuming a model of sources under consideration. The most simple models can be described as follows.
1.1. Locul source oj’discretr
Its force density
AE
is described
[2,4]. In this case, the database {x,, =(I: u,,(t)),,. d= l...D,
by:
f(r, r)=a6(r-r,~)6(u-Vs),
(2)
where u, i, and Y, denote the force magnitude, the time and the position of excitation, respectively. The sound field is in this case expressed by: U(U,l)=G@,
Ys; r-t,)a.
(3)
For the sake of simplicity, we further assume that the direction of the source force is given and that only one component of the displacement is detected. Eq. (3) is then simplified to: U(P,
t) = G(u,
P,;
t - t,y)a.
by:
IZ= l...Ni.
(6)
A location of continuous AE sources cannot directly be performed by any of the above-mentioned methods because acoustic bursts cannot be distinguished. However. it has already been shown that one can still apply the triangulation method if the time delays between continuous acoustic signals from various positions can be estimated from their cross-correlation functions [7]. Our purpose is to show that the correlation function could also be utilized in the comparison method as the input signal from which the neural network could associatively estimate the position of a continuous AE source. With this aim, we turn to the statistical description of continuous AE phenomena.
(4)
Green’s function in this equation generally represents a wave-burst. It is comprised of various wave modes which are propagating with different velocities. For the purpose of source location a wave mode is usually selected which could be recognized in a burst. An AE source can be then located by the following procedure: ( 1 ) From the burst signals detected at various positions on a specimen the differences of burst arrival times, Ar,, are estimated using a threshold element; and (2) from the differences of arrival times and the known path and velocity of the selected mode, the source position is estimated using triangulation techniques. A weak point of the triangulation method is that it is based on an analysis of ray paths which cannot always be simply carried out. Beside this, the non-linear equations from which the source coordinates are calculated are sensitive to experimental errors in determination of burst arrival times and often yield non-unique solutions. In order to avoid these deficiencies an associative method has been developed recently which is based on comparison of a detected burst with a set of prototype bursts [3,6]. With this aim, a set of N bursts is first excited by sources at various positions of the sample. From the source coordinates and the recorded time delays a set of prototype data vectors is then formed: (x, =(v, Atd)n, d= l...D,
is described
tz= l...N).
(5)
At the estimation of the source position, the differences of burst arrival times are first measured and then the corresponding vector rs is associatively determined based on comparison of the measured At, with those stored in the database. The corresponding estimation procedure is described in Section 1.2. For the purpose of associative estimation of the source position, the empirical information about the detected burst need not be given in terms of differences of arrival times At,, but can also be presented by the burst waveforms ud(t), which are detected by D detectors
1.2. Locul SOM~~~J of’continuous The force expression
density
AE
is in this case described
by the
.f(V, t)=a(t)ci(r-v,).
(7)
in which u(t) denotes a time-dependent force amplitude. The sound field is in then expressed by the convolution: u(r, t)=
G(u, u,%;t-
t’)u(t’) dt’.
(8)
s
We assume that the force amplitude represents a stationary stochastic process with a zero mean value E[u(v, t)] =O. Its basic characteristic is described by the correlation matrix: R,(t,
(9)
-rl)=Eta(t,)Oa(t,)l.
Here, 0 denotes the outer or matrix product of vectors. When the process is ergodic. the statistical average E[...] may be substituted by the average with respect to time t. One of the most applicable models is a white noise. or uncorrelated stationary source, which is described by the correlation matrix: R,,(t2-f,)=ROS(t2-tl).
(10)
The constant R, denotes the source intensity. correlation matrices of the source and the emitted are related by the equation: R,(v,,
~2, f,, tz)=E[u(v, =
The wave
t,)Orc(v, r,)]
ssss
G(r, , r’; t, - t’)G(u,,
x E[u(t’)@u(t”)]
r’; tz -t’)
d V’ dt’ d I”’ dt”
G(v,, U,$iI, - t’)G(r2, v,; f, -t”) =is x R,(t’-
t”) dt’dt”.
(11)
In the case when the source process is in time uncorrelated the displacement wave correlation matrix is described by the expression:
Mu,.
~2. t,
-tz)=Ro
s
G(r,. u,,; t)G(r2. r,; f+t,
=R,R,;(v,.rz,r,;fz-ft,).
-r,)
dr (12)
This clearly demonstrates that the wave correlation matrix depends on source position Y,, but the problem is to express the coordinates explicitly. Let us for this purpose consider the basic properties of Green’s function in an infinite medium [ 1,5]. It generally represents the excitation of the medium by an impulse in a point source at r,,. The excited disturbance propagates as a wave burst away from the source and generally arrives to the points of detection at rl and Ye at different times. Consider a case when the source and points of observation are distributed on a line. If there is just one wave propagation mode with the group velocity, c, then the propagation time from source to the ith detector is Ati= ]lri -ry,//lc.. One could expect that the correlation matrix R,(r,, rz, ry, t,- I~) should have a maximum when the time delay tz - t, compensates the difference between times of propagation At2 - Ar,. In a one-dimensional case this condition yields a linear relationship between the position of the source and the time delay t2- t, at which the wave correlation function has a maximum [7]. However, in practice the problem is usually much more complicated. First of all real specimens represent bounded media for which various modes of wave propagation with different velocities are characteristic. The structure of the wave burst is therefore generally very complex and does not exhibit a single, well-defined maximum from which the time delay could be reliably determined [ 11. In order to proceed to an exact relationship between source coordinates and the wave correlation matrix, one should specify the Green’s dyadic in detail, which is out of the scope for most applications [ 1.51. Besides this, the assumption that the source of continuous AE is in time uncorrelated may be false and there may also be several sources active simultaneously. Based on the relationship between the force and the wave correlation matrix, we can thus only conjecture that the wave correlation matrix contains an information about the source position, while the corresponding function cannot be specified in detail [7]. However. we can still conclude that this function can be modelled based on empirical prototype samples obtained by exciting the medium by a set of prototype sources. The corresponding data base can then be represented by N data vectors which are comprised of the source coordinates and the correlation matrices of the force amplitude and the
signals: (x,=[I:
R,(t),
Ru(r,,r2.
t)],;~=l~.~/‘v’I.
(13)
Such a general database still appears too extensive for practical applications. The treatment can be simplified by considering only one component of the source force and displacement field, which are represented by a correlation functions. Based on the linear relationship between source and the wave correlation matrix one can conclude that the magnitude of source amplitude is irrelevant for the estimation of its position, and utilize only normalized correlation functions in the database. If the mechanism of the prototype sources and the analysed source is always the same. then the source correlation matrix is also superfluous in the database and we can represent it in an abbreviated form as: (x,, =[I; @Jr,, I’,. r)],; II = 1 ..,N].
( 14)
Here e,, denotes a normalized cross-correlation function of AE signal which is detected at two different points on the specimen: (15) When there are more than two points of AE detection. the corresponding correlation functions can be concatenated in the data vector. The remaining problem is then to develop an information processing system by which the sound correlation function could be mapped to the source coordinates. For this purpose we have developed a radial basis function neural network by following the non-parametric approach to empirical modelling of natural phenomena which is briefly explained in the next section [ 2.41.
2. Non-parametric
modelling of AE phenomena
The empirical description of natural phenomena deals with the presentation of physical laws in terms of probability distributions. It has been explained in detail elsewhere and therefore we present here just the most important concepts of the formulation [2.4.8]. The basic object of the empirical description of the natural phenomena is a set of physical variables which are simultaneously measured by some array of sensors. In our example, this set represents a data vector comprised of source coordinates and AE signal characteristics. Let this vector be represented by A4 components: x=(&...&). In an empirical description of the AE phenomenon, we repeat the basic experiment N times to obtain the database (x,..,x,~] of N samples. Instead of a physical law that describes the relationship between the components of vector X, we further treat this vector as a random variable and characterize it by the empirical
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joint
I. Gruhrc et al. / Liltrusonit:v 36 (19%‘) 525-530
probability
density
function:
(16) When the number of samples N increases without limit during the observation of the phenomenon we prefer to use a fixed number K of prototype vectors {ql...qk) to represent the properties of the random vector x. The empirical probability density function is then substituted by the representative one:
L(x)= ;
kil&--4k).
(17)
At N = K, we initialise the prototypes by setting qk = xk, while for N>K, we introduce an adaptation. With this aim, we first approximate the singular delta function in Eqs. ( 16) and (17) by a smooth Gaussian window or radial basis function:
-“ij”“],
lV=(x-q)=exp[
in which the parameter (T corresponds to an average distance between the prototype vectors. The representative probability density function is then adapted to the empirical one by minimizing the discrepancy between both distributions. An optimal adaptation is approximately described by an iteration formula for changes of prototype vectors [2,8]: Aq,z
K N+l
i
(XIV+1-4rMx,+1
-4,)
(19) The prototype vectors can be interpreted as the memorized contents of neurons in a radial basis function NN. Their adaptation is influenced by the input samples x, as well as by the interaction between the neurons that represents a kind of self-organization which maximally preserves the empirical information [2,8]. In the case of a low number N, all the samples can be treated as initialized prototype ones and an additional adaptation is not needed, which is convenient for applications. The prototype vectors represent an empirical model of the phenomenon under consideration. Their initialization and adaptation characterize the learning phase of the empirical modelling. The next theoretical problem related to this modelling is to show how the prototypes can be further applied [2,4]. Let us for this purpose assume that the observation of the phenomenon provides only partial information that is given by a truncated vector: g(gY1...4,; N),
(20)
in which 0 denotes the missing part. The question is then how to estimate in an optimal way the complementary or hidden truncated vector comprised of missing components:
h=(8; &+* ..&,f),
(21)
such that the complete concatenation
x=gOh=(&
data
vector
is obtained
...5,7,
by the
(22)
A statistically optimal solution of this problem is given by the conditional average estimator L =A( g). It is expressed in terms of truncated prototype vectors as [2,8]: ii= 5
B,@)h,,
where Bk(g)=
tv@-gk:k)
K
(23)
c \“k-K1)
I=1
The basis functions B,(g) in this expansion represent the measure of similarity between the truncated vector g given by measurement and stored truncated prototypes g,; therefore, the estimation resembles an associative recall which is characteristic of intelligence. The estimation of the hidden vector 4 is the main task of the application of the empirical model represented by prototypes. It corresponds to a non-parametric regression [2]. For the location of continuous acoustic emission sources, the corresponding locator must be able to detect AE signals and source coordinates, to determine the correlation function, to create the prototype vectors, and to estimate the source position from the detected AE signals. Each of these operations can be performed in a separate unit that can be interpreted as one layer of a sensory network.
3. Experiments The intelligent AE source locator, which was developed based on the empirical modelling of AE phenomena, is shown in Fig. 1. It is implemented on an automatic data-acquisition system controlled by a computer and supplemented by a network of sensors and preamplifiers for detecting AE signals and measuring source position during calibration procedure. As the AE detectors, piezoelectric transducers (pinducers) with broadband characteristic in the ultrasonic region between 0.1 and 1 MHz are applied. The diameter of the transducer active area is 1 mm and it can be therefore considered a point-like detector. The locator operates in two different modes. (1) In the learning mode, a set of N experiments is performed in which a complete information about the AE phenomenon is acquired. From the source coordinates and preprocessed AE signals. the sample vectors are created and supplied to the
prototype source 0 actual source n estimated source
0
Fig.
I,
Scheme of the intelligent
neural network in the PC where the prototype vectors are adapted to them. In the case of discrete AE phenomenon. either the AE signal wave forms or the time delays are used to describe the bursts. In the case of continuous acoustic emission, the cross-correlation function between signals from sensors is first calculated and then concatenated with source coordinates. (2) In the application mode. only acoustic data are provided which are then completed by the estimated source coordinates. The performance of the locator is demonstrated here by an example of continuous AE generated by grinding on a steel plate of diameter 250 mm and thickness 20 mm. Its applicability to associative location of discrete AE sources has been demonstrated elsewhere [3.6]. The purpose of the experiment reported here was to examine the accuracy of continuous AE source location. For the sake of simplicity, only linear location was considered. Fig. 2 shows a typical example of detected stochastic signals and their cross-correlation function. During training, eight prototypes were created in the memory. In the analysis mode of operation the performance was tested by 15 sources. Eight of them were generated approximately on the same basic spots as during training. while the remaining seven were placed betwecn basic ones. Fig. 3 shows the comparison of the true and the estimated coordinate. The experimental points are scattered around the prime line which corresponds to the exact solution of location problem. The errors mainly appear when sources that lie between prototype ones are located. The location of the sources at the positions of the prototype sources is uncertain mainly at terminal points. Presumably the reason for this uncertainty are the reflections of the acoustic waves on the specimen boundary. On average. the error of estimated position is less than the distance between two nearby prototype sources. This means that the locator properly estimates the zone from which the AE signal arrives, while it is less successful in exact interpolating. The reason for the
locator of ,AE sourcc~.
II’1
AE signal
I
s2
j ‘I
I
‘I
.
weak ability of generalization is presumably in the very complex structure of the correlation function of this stochastic AE phenomenon, which is a consequence of the wave propagation in the plate and the SOL~I-cc mechanism.
4. Conclusions
Our examination shows that the empirical treatment of the location problem leads to the development of an intelligent information processing system with a structure of sensory neural network that yields location results acceptable for NDT. Its intelligence is a conse-
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I. Grabec et al. / Ultrasonics 36 (I 998) 525-530
Achl
Sowa
E~Soww Pnwvw SDUCS
Fig. 3. Results of location experiment. Horizontal coordinate; vertical axis, estimated coordinate.
axis,
true
quence of the adaptation procedure. For this purpose, the conventional calibration procedure is generalized such that a database of a sufficient number of prototype vectors is obtained. With an increasing number of prototypes the estimation error generally decreases. However, as with the calibration of conventional source locators, there is a problem with a proper simulation of AE sources. This is not an inherent problem of the intelligent method utilized, but rather a problem of the experimental reproducibility of experimental observation of the phenomenon. A similar conclusion is also valid for the problem of non-unique mapping which can be, caused by reflections of the sonic waves on the specimen boundaries. The greatest advantage of the empirical approach stems from the fact that no specific analysis of ray paths or adaptation of the corresponding location program is needed when the structure of the sensory network that forms the AE antenna is changed. The same locator is thus very flexible and applicable to different configurations. Even the variation of the number of sensors needs nothing more than a new setting of the number of components used in the description of the data vector. For the modelling, various paradigms developed by neural network research are applicable. The most well known among them is a multi-layer perceptron that learns to perform a mapping by back-propagation of errors. As it has been shown elsewhere the performance of such a network is equivalent to the system proposed here, although the physical interpretation of the parameters in terms of prototype vectors is more simple in our
case [2]. This also makes feasible a simple inscription of data without any training, when the number of statistical samples is lesser then the number of memory cells. In this case, the learning time is negligible in comparison with the time needed for execution of the prototype experiments. Moreover, the splitting of the data vectors into given and hidden components can be performed arbitrarily after the training procedure. The basic idea for the intelligent locator stems from the non-parametric modelling of empirical relations between multicomponent variables [2,4]. One could expect that the same method should also be applicable for a solution of initially mentioned general source identification problem in which the force field must be determined from the detected AE signals. For instance, if a set of joint records of both fields: {[NC t),f@, t)L; n= 1...Ni,
(24)
were obtained, one could train a neural network to map a detected AE signal to the corresponding source field. Some attempts to prove this possibility have been carried out recently [2,4], but the experimental specification of vector fields represents great problems. Therefore, this general method could hardly be applied to nondestructive testing in an industrial environment outside a laboratory.
References [I ] K. Miller, P. Mclntire P] [3 ]
[4] [5]
[6]
[7] [S]
(Eds.), Nondestructive Testing Handbook, vol. 5: Acoustic Emission, Am. Sot. NDT, Philadelphia, PA, 1987. I. Grabec, W. Sachse, Synergetics of Measurements, Prediction and Control, Springer, Heidelberg, 1997. I. Grabec, B. Antolovic, Intelligent Locator of AE Sources, Progress in Acoustic Emission, The Japanese Society for NDI, Tokyo, 1994, pp. 565-570. 1. Grabec, W. Sachse, Automatic modeling of physical phenomena: application to ultrasonic data, J. Appl. Phys. 69 ( 1991) 623336244. Y.H. Pao, L.E. Payne, W.W. Symes, F. Santosa, Inverse Problems of Acoustic Waves in Fluids and Solids, Report ONRjSRO III, TAM, Cornell University, 1986. T. Kosel, 1. Grabec, B. Antolovii-, P. MuiiE, Intelligent locator of acoustic emission sources in composite materials. in: V. Krstelj (Ed.), Proceedings of Inservice Inspection ‘95, Croatian Society NDT, Zagreb, 1995, pp. 235-243. I. Grabec. Application of correlation techniques for localization of acoustic emission sources, Ultrasonics I6 ( 1978 ) I I l-1 15. I. Grabec, Self-organization of neurons described by the maximumentropy principle, Biol. Cybernet. 63 (1990) 4033409.