Inverse Shape Reconstruction of Inner Cavities Using Guided SH-Waves in a Plate

The paper proposes an inverse reconstructionmethod for inner cavities in a 2D plate using guided SH-waves.When a pure incident wave mode is sent toward the flaw area, reflected waves are observed at the far field due to mode conversion. From wave scattering theory, the reflected wave field is expressed by an integral over the unknown flaw surface concerning the total wave field. By the introduction of Born approximation and far-field expressions of Green’s function, it is found that the reflection coefficient for the mode of the same order as that of the incident mode is related to the unknown shape of the cavity by Fourier transform relations. By mathematical deduction, we show that if the basic (0th-order) symmetric and the 1st-order antisymmetric modes are used as incident waves, the locations and shapes of upper and lower boundaries of the cavity can be reconstructed. Numerical examples are illustrated in the paper, and a proper condition for applying this method is discussed. The research can act as a basis of nondestructive inspection for latent flaws within more complicated structures.


Introduction
Ultrasonic guided waves have been widely used in nondestructive testing (NDT) for pipes, plate structures, and composite materials.Compared with bulk waves, they can travel over long distances and interrogate the whole cross section, which allow us to inspect a large range from a single access point [1].The nondestructive testing methods using guided waves are based on the principle that when the incident wave is scattered from a flaw, a part of the energy is reflected back and the information of reflected wave is related to the location and size of the flaw.
Most researches on NDT concerning guided waves make use of signal-processing technology to analyze the timeof-flight (TOF) of reflected waves, in order to locate flaws within a plate, mostly surface defects or through holes.For example, Fromme et al. [2] and Rajagopalan et al. [3] used a single-transmitter-multireceiver configuration to inspect over a large span of the plate for defect locations and directivities using Lamb modes.Masserey and Fromme [4] used a simple pitch-catch device to detect the surface notch on a ribbed plate by Rayleigh-like waves.Hayashi and Murase [5] and Nishino et al. [6] also used torsional guided wave modes to detect flaws on a thin-wall pipe.Recently, Lee and Park [7] used pulsed laser to generate Lamb wave and a piezoelectric sensor to monitor the structure response and adopted wavenumber filtering to extract damage-sensitive features.Other researchers also tried tomographic approaches with the use of guided waves, such as [8][9][10][11][12].Due to the complexity, fewer researches concern inspection of inner flaws using guided waves.Recently, Lee et al. [13] suggest a wavelet-transformed ultrasonic propagation imaging method to detect debondings in a sandwich plate.Watkins and Jha [14] proposed a modified time-reversal approach for detection of both inner and surface flaws in a plate using Lamb waves.Generally, the TOF-based methods can well detect the defects' positions and ranges; however, there are fewer researches available about reconstruction of flaws' exact shapes.
Recently, some researchers begin to make use of more information of reflected waves such as the amplitude and phase for detailed flaw sizing and reconstruction, which is based on a better understanding of interaction between flaw and guided waves.A notable case is made by Singh et al. [15] and Castaings et al. [16] who studied flaw sizing in a 2D plate model.They used a finite-element-based model to calculate reflection coefficients for various values of flaw's geometric parameter, followed by an optimization process to match these values to coefficients induced by actual flaws.However, this method requires a basic knowledge of the flaw's location as a priori condition and needs excessive FEM modeling as a database.Wang et al. [17] studied the interaction between guided Lamb waves and internal rectangular defects by hybrid BEM formulation and pointed out that the variation of reflection or transmission coefficients with the defect's parameter can provide information for inverse reconstruction.Wang and Hirose also developed an inversion method based on wave scattering theory, using the reflection coefficients of guided SH-waves [18] and Lamb waves [19] to reconstruct the locations and shapes of surface thinning flaws.
In this paper, we propose an inverse reconstruction method for the location and shape of an inner cavity in a 2D isotropic plate, using ultrasonic guided SH-waves.This study is subsequent to the previous research on inverse analysis of shape reconstruction of surface flaws [18].When a pure SHwave mode is incident into the flaw zone, various reflected wave modes are generated and observed at the far field due to mode conversion.The reflection coefficients, defined by ratios between the amplitudes of the reflected waves and the incident wave, depend on the position and shape of the cavity.The cavity's position and shape can be defined by two functions, representing upper and lower boundaries of the cavity.On the basis of scattering theory, it is found that the two functions can be determined by the inverse Fourier transform of the reflection coefficients of 0th-order symmetric and 1st-order antisymmetric SH-wave modes, with the introduction of Born and far-field approximations.We will describe the problem statement in Section 2 and explain the formulation of the inverse reconstruction method in Section 3. Numerical examples are shown in Section 4 to illustrate validity of the inverse reconstruction method and discuss the proper application condition.This research provides a fast detection and reconstruction method for latent flaws inside plates using guided waves, which is especially useful when conventional point-to-point ultrasonic nondestructive inspection is unavailable.The current study also serves as a basis for theoretical research of NDT using more complex guided wave modes, such as Lamb waves, or its application in more complicated plate structures.

Problem Statement
We consider a 2D isotropic elastic plate with an inner elliptic cavity, as shown in Figure 1.The thickness of plate is 2b.Two functions are defined to depict the inner cavity: the depth of the cavity's upper and lower boundaries from the upper surface, termed as ℎ( 1 ) and ℎ  ( 1 ), respectively, and we define ( 1 ) = ℎ  ( 1 ) − ℎ( 1 ).
Ultrasonic guided SH-waves, which have antiplane displacements in  3 -direction, are used for reconstruction of the inner cavity's location and shape.As shown in Figure 1, the incident guided SH-wave propagates from right to left along  1 -direction and is scattered by the cavity.The reflected wave is observed at the far field on the right side of the flaw area.
Suppose the incident wave is a pure guided SH-wave mode of th-order with frequency , as shown by Here,   = /2,   = √ 2 − For convenience, the time factor  + in wave fields will be omitted in the following context.
Due to mode conversion, various SH-waves, including propagating and nonpropagating modes, can be excited by the scatterer as the incident wave hits the cavity.However, as the observation position is far away from the flaw area, nonpropagating modes decay and the reflected wave  ref , which is received from the incident side, is composed of only propagating modes, as illustrated by the following equation: where  ref  is the amplitude of the th wave mode and summation  is taken for all the propagating modes travelling from left to right.
These modes are orthogonal with each other on a cross section of a plate, which was proved by Auld [20].If there are two modes termed  and , propagating with wavenumbers   and   , respectively, for the case of SHwave, the orthogonality has the following formation: where   and   are the modal displacement and shear stress  13 of the mode .The asterisk symbol means complex conjugate.  is taken as unit when  =  and zero otherwise, and () is the integral value when  = .If the mode is normalized, () = .Therefore, using the orthogonality, we can obtain the amplitude  ref  by the following equation: where  ref and  ref represent overall displacement and stress of the reflected wave field.In numerical calculations, ( 4) is adopted to separate each mode from one another.While in experimental studies, various separation methods are used, for example, 2D Fourier transform in [21], application of Bragg grating sensor in [22], and mode compensation method in [23].
The aim of inverse reconstruction is to find the location and shape of the inner cavity.Mathematically, it is the process of reconstructing functions ( 1 ) and ℎ( 1 ), using the reflection coefficients  ref  as input data.

Inverse Reconstruction Procedure
The formulation of inverse reconstruction method is based on the integral expression for the scattered wave by the inner cavity.The scattered wave field at the observation point X can be expressed by an integral over the surface of cavity S, as shown by (5).If X is located at the incident side (as the situation discussed here), the scattered wave observed is also the reflected wave.Consider where  sca ( ref ) is the displacement of scattered (reflected) wave at observation point X,  tot =  inc +  sca is the total displacement on the cavity surface , and   is the unit normal vector pointing outward.In (5), (x − X) is Green's function for SH-wave in a plate waveguide, which has been obtained by Hirose [24] in an integral form.If the observation point is far away from the flaw area, the far-field expression of Green's function can be deduced as follows: where   is a trigonometric function defined by (1b).
As shown by (6), it is seen that far-field Green's function can be expressed as the combination of different modes of guided SH-waves.The first term on the right-hand side of ( 6) is the fundamental (0th symmetric) mode, and the summation index  is taken for all the propagating symmetric and antisymmetric modes.
If the wavelength of an incident wave is much larger than the size of the cavity, the magnitude of scattered displacement is much less than that of the incident wave, due to weak scattering by a cavity.Then, the total wave field  tot is replaced by incident wave field  inc , which is known as Born approximation.Applying Gauss's theory, we can transfer the surface integral into the volume integral: As mentioned in the previous section, we firstly assume that the incident wave is a single fundamental (0th symmetric) SH-mode, propagating from right to left side in Figure 1, and the reflected wave is observed at the far field on the right side of the cavity.Also, by mode separation, the portion of the same mode as the incident wave is obtained.
In this situation, the displacement field of incident wave is written as (8), and Green's function corresponding to the fundamental SH-mode is expressed as (9).Consider Substituting ( 8) and ( 9) into (7) and taking the limit of  1 → ∞, the reflected wave of the 0th symmetric mode  ref 0 is written as The volume integral in (10a) becomes Thus, we have the following Fourier transform relation: where the reflection coefficient  ref 0 is defined as the quotient of amplitudes for reflected wave over the incident wave, which is a function of wavenumber .From (11b), the function ( 1 ) is obtained.
Secondly, the vertical location ℎ( 1 ) of the cavity is determined by using the reflection coefficient of higher-order SH-wave mode.If the incident wave and the reflected wave are single th-order ( ≥ 1) SH-wave mode (symmetric or antisymmetric), the displacement of incident wave and corresponding Green's function are expressed by the following equations: where guided wavenumber for the th-order SH-mode is Substituting ( 13) into (7), we have (14) which resembles (10a) and (10b): where () is the trigonometric function defined by (1b).By defining  ref  (  ) =  ref  / inc  and rewriting the volume integral as a double integral, we have It is noticed that the integral of the first item in the square bracket over  1 , that is, as seen in (11a).Thus, we can simplify (15) into The integrand ( 1 ) ≡ sin(/)(ℎ( 1 ) −  + ( 1 )) − sin(/)(ℎ( 1 ) − ) in ( 16) can be obtained by performing the inverse Fourier transform of the coefficient The function ℎ( 1 ) is obtained by solving the transcendental equation as follows: However, it should be noted that, within the definition range of − ≤ ℎ( 1 ) ≤ , (18) has multiple solutions.The larger  is, the more roots it has.To avoid too many "virtual images" of the cavity, we choose  = 1 (1st-order antisymmetric mode).In this case, ℎ( 1 ) has two solutions between − and , which means a "real image" and a "virtual image" of the cavity are formed by the inversion process.

Numerical Examples
For numerical verifications of the proposed inverse method, the reflection coefficients of 0th-order symmetric and 1storder antisymmetric guided SH-wave modes due to various elliptic cavities are calculated by the means of mode-exciting method [25] and are used as input data in the inverse analysis.
We also let  ref  (  ) =  ref *  (−  ), (,  = 0, 1), where the asterisk symbol means complex conjugate.Figure 2 shows the absolute values of the reflection coefficients | ref  | versus the normalized frequency /  of th-order guided SH-wave modes ( = 0 or 1).As a reference, the reflection coefficients estimated by Born approximation given by ( 15) are also drawn in Figure 2.
Table 1 illustrates the validation chart of the obtained reflection and transmission coefficients at different frequency points.The coefficients   are defined as the quotient of amplitudes of reflected/transmitted waves over the incident wave (see Table 1).Two relations are used for validation.Firstly, due to energy conservation, the sum of the squares of reflection coefficients plus that of transmitted coefficients should be 1.Secondly, because the flaw is symmetric, for incident waves excited from the right side and the left side, their reflection/transmission coefficients should have the same absolute value.
Table 1 lists the situation where 1 or 2 modes exist in the waveguide at the given frequency.The cases where higherorder modes exist are also checked.From "energy check" in Table 1, it is seen that the energy conservation is satisfied.From the "relative error" column, we can notice that most of relative errors do not exceed 5%, while the biggest one occurs where nominator is very small, so the absolute error is also not large.
Substituting these coefficients into (11b) and ( 17), the function ( 1 ) and the intermediate function concerning flaw position information ( 1 ) are obtained, as shown by Figure 3.The function of position ℎ( 1 ) can be determined by solving (18), after determining the functions ( 1 ) and ( 1 ).Ideally, for intact parts with no flaw, ( 1 ) is zero, and to obtain ℎ( 1 ) is meaningless.However, as seen in Figure 3, the function ( 1 ) is not zero even in the nonflaw region because of errors in inversion.Therefore, in this study, a flaw region is defined as the area where ( 1 ) shows values larger than 25% of the maximum value of ( 1 ), and the upper boundary coordinate ℎ( 1 ) is calculated only within the flaw region.By determining the functions of ( 1 ) and ℎ( 1 ), the upper and lower boundaries of the cavity can be obtained and the image of the cavity is reconstructed.
In Figure 4, the reconstructed upper and lower boundaries are shown with thick curves and the shape of the original circular flaw is shown with thin curves.The "actual" reconstructed image is shown in solid curves, and the "virtual" one is illustrated in dashed curves, which appears at the mirror image position against the "actual" image.The inverse method is furthermore applied to other cavities at different locations and with different sizes.Figures 5 and 6 illustrate the reconstructed boundaries of the circular cavities with the radius of  1 =  2 = 0.2 and center locations (0, −0.7) and (0, 0), respectively.Figures 7-11 show the results for elliptic cavities with different combinations of the radii  1 and  2 , while the center location ( x1 , x2 ) of elliptic cavities is kept at (0, −0.5) for all the cases.
From Figures 4-6, it can be seen that the shapes of cavities are fairly well reconstructed, regardless of different vertical positions of a flaw.In Figure 6, we can see that as the cavity's center is on the  1 -axis, the "actual" and "virtual" images overlap each other.
The results of reconstructed cavities with different elliptic radiuses are twofold.In the cases of  1 = 0.1, 0.05 Definition of coefficients is as follows.Suppose the mth mode is used as the incident wave; then, we introduce the direction numbers, 1: the left direction of the flaw, 2: the right direction, and the reflection/transmission coefficients   .Meanings of subindices:   : A-incident direction ( = 1,2); a-incident mode order; -scattered direction ( = 1,2); -scattered mode order ( and  the same: reflection coefficient;  and  different: transmission coefficient).Wave field:  inc =  inc    (   2 ) ±   1 (+ for incident wave from 2; − from 1); and  2 = 0.2, as seen in Figures 7 and 8, respectively, the vertical positions of upper and lower boundaries are well found, but the flaw ranges in  1 -direction are enlarged compared to reality.In general, for a flaw shape with rapid change in  1 -direction, high-frequency components of reflection coefficients are needed to obtain good accuracy in shape reconstruction.However, in our approach, only relatively low-frequency components are available because Born approximation is appropriate in the low-frequency zone.That is why we cannot reconstruct images for narrow flaws as shown by Figures 7 and 8 with fine resolution.On the other hand, in the cases of  1 = 0.3 and 0.5, as shown by Figures 9-11, respectively, the ranges of flaw area in  1 -direction agree well with the actual locations, whereas the positions of upper and lower boundaries are deviated from the reality, especially for the case of  1 = 0.5 and  2 = 0.2.The reasons may be stated as follows.
In Figure 12, the absolute values of reflection coefficients in the case of  1 = 0.5 and  2 = 0.2 are illustrated.We can see that, for the 0th SH-wave mode, the reflection coefficients given by Born approximation are close to the numerical results, whereas, for the 1st-order SH-wave mode, the differences between numerical reflection coefficients and Born approximation values become very large, especially within some frequency bands; for example, /  = 1.5∼2.0 and 2.5∼3.0.The large reflection coefficients are due to structural resonance.If the cavity is relatively long in  1direction and close to the surface, the region between the cavity boundary and the plate surface behaves like a thin "plate" supported by elastic ends, which can vibrate in large amplitude near its resonance frequencies.In this case, the cavity can be seen as a strong scatterer and Born approximation is not valid anymore.On the other hand, the reflection coefficients for the 0th-order SH-wave mode still match Born approximation values.Consequently, the function ( 1 ) is still well reconstructed, but the reconstructed ℎ( 1 ) deviates from reality.As a comparison, for the case of  1 =  2 = 0.2, as shown in Figure 2, it is seen that the reflection coefficients given by Born approximation are close to numerical values for both 0th-and 1st-order wave modes, except at some limited points which are cutoff frequencies of other wave modes.Thus, both functions ( 1 ) and ℎ( 1 ) are well reconstructed.
In a summary, the applicability of Born approximation, which is accurate for low-frequency components and for weak scatterers, is the key factor for the accuracy of the proposed reconstruction method.The accuracy of the inversion method depends on the size and location of the cavity.If the cavity's dimension in  1 -direction  1 is relatively large and the cavity is close to the surface of plate, due to structural resonance, the reflection coefficients for 1st-order SH-wave mode cannot be well estimated by Born approximation, which makes the function ℎ( 1 ) lose accuracy.On the other hand, if  1 is relatively small, ( 1 ) changes rapidly on  1 -direction, and after Fourier transform it contains high-frequency components that cannot be covered by lowfrequency Born approximation.As a result, the boundaries in  1 -direction tend to be vaguer.

Conclusion
In this paper, a new inverse reconstruction method for inner cavities in a 2D plate using guided SH-wave has been proposed.The location and shape of unknown cavities have been reproduced by performing inverse Fourier transforms of reflection coefficients of both 0th-order symmetric and 1storder antisymmetric SH-wave modes in frequency domain.
The validity of this method was illustrated by numerical examples.If the dimensions of cavity in  1 -and  2 -directions are within an appropriate range, the image of the cavity can be well reconstructed.Because of the applicable condition of Born approximation, the accuracy of reconstruction depends on horizontal size and location of cavity.If the cavity's dimension is relatively wide in  1 -direction and close to the plate surface, Born approximation no longer holds for 1storder SH-wave mode because of structural resonance; thus, the reconstructed position in  2 -direction loses accuracy.If the cavity is narrow in  1 -direction, the low-frequency limitation of Born Approximation makes the boundary of flaw image vaguer.For further research, we hope to improve the method and apply it for reconstruction of inner holes in a realistic 3D plate model.Experimental studies, followed by field applications, will also be investigated in the future.

Figure 1 :
Figure 1: Schematic of the inverse problem.
2  is the longitudinal wavenumber, and  = /  .  = √/ is the -wave velocity of material, with  being the shear stiffness and  the density of a plate. inc  is the amplitude of the incident wave.  (   2 ) is a trigonometric function defined as   (   2 )

Table 1 :
An illustration of validation chart for reflection and transmission coefficients.