Ultrasound Propagation in Heterogeneous Media: Model Study

Transcription

1 Ultrasound Propagation Heterogeneous Media: Model Study R. A. Roberts Center for NDE, Iowa State University, Ames, IA Abstract. A computational approach to prediction of ultrasound backscatter from heterogeneous microstructure is presented. The problem is formulated as a volume tegral equation (VIE), and solved by an FFT-based iterative method which exploits a convolutional VIE kernel. enerally anisotropic heterogeneous microstructure is specified on the micron scale, and spatial filterg is applied to generate ultrasonically relevant heterogeneous elastic properties. The approach is used to exame mean square backscatter voltage as a function of spatially varyg gra size crystalle metals. Keywords: Ultrasound, Heterogeneous Media PACS: Yb Zc Hz INTRODUCTION This paper reports on a computational study of ultrasound propagation heterogeneous microstructures, such as granular metals and fiber reforced composites. Random spatial fluctuations elastic properties and density over a range of length scales relative to ultrasound wavelength can give rise to scatter-duced attenuation, backscatter noise, and phase front aberration. It is of terest to quantify the dependence of these phenomena on parameters describg the microstructure, for the purpose of quantifyg deleterious consequences on flaw detectability, and for the purpose of material characterization. While the equations governg the physical phenomena volved are well understood, obtag practical solutions to these equations is challengg due to the multiple length scales volved. Usg a typical aerospace metal alloy as an example, micro-structural detail appears on a scale 1 to 2 orders of magnitude smaller than the ultrasound wavelength. To fully capture the fluence of microstructure detail, computational methods commonly applied to study ultrasound scatterg must be refed accordgly, thereby rapidly exceedg computational resources. Alternatively, applyg statistical wavefield metrics to the governg equations yields scatterg formulations for quantities such as mean field amplitude and backscatter tensity.[1,2] The resultg formulations are solved approximately, yieldg results of herently limited validity. This paper reports on an alternate means of formulatg and computg ultrasound scatterg by microstructures explicitly specified arbitrarily fe detail. The approach adopted here employs a volumetric tegral equation (VIE) formulation, which scatterg with a heterogeneous volume is expressed via a kernel volvg the reen function for the mean host medium. Significantly, tegration the volumetric formulation is convolutional form, enablg efficient evaluation by Fast Fourier Transformation (FFT). To avoid explicit formation and version of a prohibitively large scatterg matrix, an iterative solution usg FFT-based convolution is employed to reduce the VIE error residual. In fortuitous cases, the error residual serves as the cremental solution improvement, resultg a rapidly convergg series. In complementary cases, gradient means of residual mimization are employed. The VIE formulation and computational approach employed here were origally explored previous work addressg propagation heterogeneous acoustic media, governed by the variable velocity scalar wave equation.[3] Ongog work is extendg this previous formulation to generally heterogeneous elastic media. This paper reports on progress thus far accommodatg general learly elastic media described by 21 spatially varyg elastic constants with constant density, as would describe most granular metal alloys. In our previous work microstructure heterogeneity was prescribed on the wavelength scale an ad hoc fashion, usg estimates of spatial variation longitudal wave velocity based on visual micrographic assessment of microstructure. The present work seeks to use as computational put explicit specification of material elastic properties on the microstructure length scale, which is then translated to a mathematically equivalent prescription on the ultrasonic length scale. Problem formulation and computational approaches have thus far been explored for one and two dimensional implementations. Results of the study to date are presented here, which demonstrate application to granular metal havg varyg gra size.

2 PROBLEM FORMULATION The time harmonic equation of motion governg propagation a learly elastic heterogeneous medium is expressed differential form as 2 ( cijkl (x) uk,l(x)), j ui(x) 0 (1) where c ijkl(x) are the spatially dependent elastic constants, is (spatially variant) mass density, and u i(x) are particle displacements.[4] The dependence of field quantities on time harmonic frequency is implicitly assumed. It is assumed that material heterogeneity is confed to a fite volume V embedded an fite homogeneous host medium havg elastic constants c 0 ijkl and like density. Integral equation formulation utilizes the reen function for the host medium, defed as the response to a directed pot force im (x-x ) actg at position x 0 2 (2) cijkl u k,lj: m (x x') u i: m (x x') im (x x') Although an tegral equation for field displacements can be formulated directly usg eqs. (1 and 2), the resultg equation contas terms volvg displacement derivatives of different order (an tegro-differential equation), resultg a problematic computational implementation. To avoid these complications, a formulation is pursued which volves only displacement gradients. To this end, eq. (2) is first differentiated with respect to x n. Equation (1) and the differentiated form of eq. (2) are then multiplied by u i:m,n(x x ) and u i (x), respectively, and the difference between the resultg equations is tegrated over all space V, excludg an fitesimal sphere V centered at x. 0 {(c (x) u (x)), u (x x') c u (x x') u (x) } dv 0 V V ijkl k,l j i:m,n ijkl k,lj:m,n i (3) Application of the product rule for differentiation yields 0 {c (x) u (x) u (x x') c u (x x') u (x) }, dv V V ijkl k,l i:m,n i j ijkl k,l:m,n (4) 0 {c (x) u (x) u (x x') c u (x x') u (x) } dv 0 V V ijkl k,l i, j:m,n ijkl k,l:m,n i, j The divergence theorem is applied to the first tegral eq. (4). Usg the elastic constant symmetry c ijkl= c klij and writg c ijkl (x) = c ijkl(x) c 0 ijkl yields 0 {c (x) u (x) u (x x') c u (x x' ) u (x) } n ds V ijkl k,l i:m,n V i j ijkl k,l:m,n (5) c (x) u (x) u (x x') dv 0 V V ijkl k,l i, j:m,n where n j(x) is the outward normal to surface V + V. In treatg the surface tegrals over V displacements are expressed as the sum of cident u i (x) and scattered u sc i (x) fields. Notg that c ijkl(x) = c 0 ijkl on V, the tegral volvg the scattered field is seen to vanish by the radiation condition, and the tegral volvg the cident field is seen to equal to the negative cident field gradient -u m,n (x ) evaluated at x. The tegral over V extracts the essential contribution from the reen function sgularity, represented by J ijmn (x') lim V 0 u V i:m, n (x Usg this notation, the governg tegral equation is expressed c (x) u (x) u (x x') dv u (x') c (x') u (x') J (x') u (x') V ijkl k,l i,j:m, n m,n ijkl k,l ijmn m, n where the identity c 0 ijkl J ijmn = km ln is employed. By contractg with c pqmn(x ), eq.(7) can also be written terms of the stress crements ij(x)=c ijmn(x) u m,n(x), ij (x)=c ijmn(x) u m,n (x) (not to be confused with scattered field stress), rather than displacement gradients (8) (x) c (x') u (x x') dv (x') (x') c (x') J (x') (x') V ij pqmn i,j:m, n pq ij pqmn ijmn pq Use of Voigt notation writg eq. (7 or 8) results a simpler expression to manipulate and code, e.g. followg evident algebraic manipulations and variable defitions eq. (7) takes the followg Voigt form terms of stra i c (x) (x) (x x') dv (x') (x') c (x') J (x') (x') V ij j i: m m k ik im m i, j,k,m 1,..., 6 (9) where the established conventions for Voigt notation are assumed. An iterative solution to eq. (9) is sought, which the error residual (the difference between the left and right sides of the equation) is crementally reduced with each successive iteration. This approach is preferred over explicit conversion to an algebraic matrix equation, due to the prohibitively large computational resources required to hold and manipulate the resultg matrices, particularly for future 3D implementations. Previous work volvg x') n j ds (6) (7)

3 the correspondg scalar wave equation showed that a straightforward implementation of a Neumann series often provides an efficiently convergg solution, due to the weak nature of microstructure scatterg. Such a solution for eq. (9) is expressed p1 p p (10) m (x') m (x') k (x') cik (x') Jim (x') c (x) (x) (x x') dv p 0,1,2,... V ij j i: m where the superscript p denotes successive estimates of the solution, begng with 0 i = i. Importantly, convergence of eq. (10) is not generally assured. However, convergence is consistently observed when c ijkl(x) is sufficiently small. Examples presented this paper are restricted to this class of problem, for which convergence is obtaed with a few iterations. When eq. (10) does not converge, other methods of residual mimization can be employed, such as conjugate gradient mimization. Interest this paper is predictg received ultrasonic backscatter signals as would be observed experimentally. For this purpose a volumetric expression of Auld s reciprocity relation [3] is formulated for time harmonic backscatter output voltage v() terms of the stra i v( ) T( ) c (11) V ij (x) j (, x) i (, x) dv where T() embodies, among other thgs, the frequency response of the ultrasonic measurement system. Computational efficiency evaluation of eq. (10) is significantly enhanced by notg that the tegral is the convolution of the reen state with the stress crement i(x)=c ij(x) j(x). By the convolution theorem for Fourier transformation, the tegral eq. (10) is evaluated as the product of the reen state and the stress crement the k-space transform doma. Fast Fourier Transformation (FFT) efficiently transforms functions between x and k domas, assumg the functions meet requisite criteria. Specifically, assumg the x-space solution doma is sampled on an equally-spaced Cartesian grid, use of FFT convolution requires 1) the bandwidth of the functions beg convolved must be less than the reciprocal of the function samplg terval, and 2) for non-periodic functions, at least one of the functions has to be zero-valued over half of its x-space doma to avoid recirculant contamation. The reen state has fite bandwidth by defition, and must therefore be filtered to the bandwidth used to support the sought solution. This can be achieved rigorously by truncatg the doma of tegration the evaluation the Fourier spatial frequency tegral representation of the reen state to the spectral bandwidth to be used the FFT. When a closed form reen state representation is available, a less rigorous but simpler approach is to sample the reen state symmetrically about the sgularity, then multiply by a phase factor k-space to shift the sampled function by a half samplg crement, so that the formerly sgular position aligns with the samplg grid. High frequency error this latter approach requires a somewhat smaller samplg terval to atta the same accuracy of problem solution, but its simpler implementation may justify this expense. The spatially filtered reen state is evaluated on an x-space samplg grid twice the extent of the volume supportg non-zero c ij(x). The bandwidth required to support i(x) is now discussed. Time harmonic wave motion is determed by the elastic properties of the medium averaged over some fraction of the ultrasonic wavelength. This tuitive notion suggests that wave motion should be unfluenced by a priori spatial filterg of the elastic properties, given a filter of sufficient bandwidth. This turn suggests that, given a microstructure specified fe detail, it should be possible to generate an ultrasonically equivalent medium through spatial filterg usg a filter of sufficient bandwidth. In problems of terest here, the bandwidth of the microstructure exceeds that of the cident field, which case the bandwidth required to support i(x) will be that required to support the filtered microstructure. The metric to determe bandwidth sufficiency is eq. (11): the desired bandwidth is the mimum observed to have negligible fluence on the computed backscattered signal. The implementation of backscatter signal prediction usg ultrasonically equivalent media is subject of the next section of this paper. (a) (b) c 11(x) Distance (mm) FIURE 1. a) c11(x) as a function of depth layered medium, b) layer thickness probability density.

4 NUMERICAL IMPLEMENTATION Example computations of backscattered signals heterogeneous media usg the approach outled the precedg section of this paper are presented. The underlyg concepts are first explored a 1D formulation, correspondg to 1D propagation a layered medium. The concepts are then applied to a 2D formulation of more pertent terest to propagation granular metals. The first example considers 1D propagation of an cident longitudal plane wave perpendicular to a layered medium consistg of parallel planar terfaces. Motion is limited to the x 1 direction, and is determed by a sgle elastic constant c 11(x). Usg random number generation, layer thickness is randomly specified with a normal probability density centered at 20 m, and havg 10% of its peak center value at thicknesses of 10 m and 30 m. With the layers, c 11(x) takes on a constant value randomly assigned with uniform probability density between specified limits, where mimum and maximum values of c 11 are those correspondg to a 4% longitudal 0 velocity variation an isotropic host medium c 11 havg longitudal velocity of c L=5820 m/s and density of =8.94 kg/m 3. A realization of a random layered medium meetg these criteria is depicted fig. (1a), which plots c 11(x) as a function of x 1, sampled at a 1 m resolution over mm. The histogram of layer thickness for the sequence from which fig. (1a) is extracted is plotted fig. (1b), showg agreement with the specified probability distribution of layer thickness. The backscatter signal received response to this 1D medium is computed for a 10 MHz center frequency cident plane wave pulse of longitudal motion. The spectral response T() eq. (11) is specified to be a Hanng wdow havg center frequency at 10 MHz, with a 100% 6dB spectral bandwidth. The stra field is evaluated through iterative evaluation of eq. (10) usg the 1D reen state 0 1 i kl x1x' (12) 11: (x) (2i c11 kl) e 1 where k L=/c L is the longitudal wave number for the host medium. An FFT sequence contag 2048 pots 1 m crements is used to support the solution. Numerical convergence of the solution is observed after a few iterations, but for good measure a total of 10 iterations were employed. Interest is examg the effect of spatial filterg of the heterogeneous medium c 11(x) on the predicted backscatter signal. To this end, a aussian spatial filter is applied to c 11(x), centered with unit amplitude at zero spatial frequency, and havg an amplitude of.01 (1% amplitude pot) at the Nyquist frequency of the spatial samplg, k Ny=210 6 m -1. The filtered medium is shown figs. (2a, b) the x and k domas, respectively. It is seen that the filterg supplies a slight smoothg of the discontuities of c 11(x). The backscattered signal computed by eqs.(10,11) for this slightly filtered medium is plotted followg Fourier transformation to the time doma fig. (2c). reater degrees of filterg are examed the remag plots of fig. (2). The result of reducg the filter bandwidth by a factor of 8 is shown figs. (2d, e, f). It is seen that the medium of fig. (2d) is substantially different than that of fig. (2a). However, it is seen that the backscatter signal of fig. (2f) is identical to that of fig. (2c), dicatg that the filtered medium of fig. (2d) is ultrasonically equivalent to that of fig. (2a) for the purposes of ultrasonic backscatter generation. The result of reducg the filter bandwidth by a factor of 16 is shown figs. (2g, h, i). The filtered medium of fig. (2g) displays a further deviation from that of fig. (2a). It is noted that at this level of filterg the backscatter signal begs to display a slight deviation from that of fig. (2c). Significantly, the stra field computed usg the filtered medium is over-sampled by a factor of 16. The same backscattered signal is obtaed by samplg the equivalent medium of fig. (2g) usg 64 pots at a 16 m samplg terval, thereby reducg the computation the iteration of eq. (10) to a 128 pot FFT. This observation demonstrates the computational savgs gaed by filterg the microstructure to an ultrasonically equivalent medium. Importantly, this process is not equivalent to simply samplg the unfiltered microstructure at 16 m, which would violate the Nyquist samplg theorem. The prciples demonstrated figs. (1-2) are now applied to scatterg by a 2D microstructure. A 2D prescription of c ij(x) is implemented to simulate a metal gra structure. A doma is filled with contactg circles havg random diameters displayg a normal probability distribution. Elastic constants and density are specified with each circle correspondg to sgle crystal nickel, for which, when material symmetry is aligned with ni ni ni ni ni ni Cartesian coordates, non-zero elastic constants are c 11 =c 22 =c 33 =2.50E11, c 12 =c 13 =c 23 =1.59E11, ni ni ni c 44 =c 55 =c 66 =1.28E11, Pascal units, and density =8.94 kg/m 3. To restra motion to two dimensions, the 2D doma is assumed perpendicular to the 001 crystallographic direction, and the elastic properties with each circle are prescribed by rotatg the crystal through some angle about the 001 direction. The angle of rotation is randomly specified, with a uniform probability density. At pots between the circles, properties are assigned to be those of the nearest circle. Figure (3) shows a 20 m gra structure generated this fashion at a 1 m resolution, usg the probability distribution of fig. (1b) to assign circle diameters. The image grey scale level dicates the crystallographic rotation (0 to 90 degrees) to be applied with the gra. Contactg circles a 256 m x 256 m

5 FIURE 2. Equivalent media x and k domas and resultg backscattered signals for three spatial filter bandwidths. (a) (b) 256 m. FIURE 3. Specification of 2D gra structure: a) contactg circles, b) gaps filled with nearest neighbor. doma are shown fig. (3a). The resultg microstructure after fillg the terveng spaces with the nearest neighbor properties is shown fig. (3b). Upon material rotation, motion which is restricted to the x 1-x 2 plane will be governed by 4 dependent elastic constants: c 11(x) = c 22(x), c 12(x), c 16(x) = -c 26(x), and c 66(x). Arrays contag these constants are filled over the 1 m resolution computational doma, where material rotation at each pot is assigned accordg to a gra structure map as exemplified by fig. (3b).

6 (a) (b) (c) FIURE 4. Focused beam propagation: a) cident stra field, b) spatially filtered c11(x), c) total stra field. Interest this paper is predictg backscatter signals for a mildly focused ultrasound beam, as would be used for NDE spection. The case chosen for study is a 10 MHz aussian beam focused to a 2.54 mm 6dB mimum beam width. The real part of the time harmonic focused beam the vicity of the focal zone is plotted fig. (4a), where it is seen to occupy a width of approximately 4 mm. uided by this observation, the procedure depicted fig. (3) is applied to prescribe elastic constants c ij(x), i,j=1,2,6, on a x m resolution grid, usg the same 20 m gra size probability distribution. Denotg this doma of prescribed heterogeneity as V eq.(10), the complementary doma outside V is assumed to be homogeneous and isotropic with non-zero elastic constants c 0 11 =c 0 22 =c 0 33 =+2, c 0 12 =c 0 13 =c 0 23 =, c 0 44 =c 0 55 =c 0 66 =. The Lame constants, are extracted as the mean value of c 12(x) and c 66(x) over the computational doma V, respectively. The crement elastic properties c ij(x) is then calculated as the difference between elastic properties with V and the homogeneous isotropic medium outside V. Specification of an isotropic host medium implies use of the correspondg reen function evaluation of eq. (10), expressed Cartesian (as opposed to Voigt) notation as i u (13) i,j:m, n (x x') {kt [H 0 (k L r) H 0 (k T r)], ijmn [H 0 (kt r)], jnim} r x x' i, j,m,n 1,2 4 where H 01 (z) is the order zero Hankel function of the first kd. Followg procedures applied to the 1D example, an equivalent medium is derived by spatially filterg c ij(x) with a 2D version of the aussian filter applied fig.(2). Figure (4b) shows the spatially filtered c 11(x), where a aussian spatial filter is employed havg its 1% amplitude pot at k NY/16, correspondg to the filterg applied figs. (2g, h, i). Followg spatial filterg, the x 4096 grid is decimated by a factor of 16, yieldg a 1024 x 256 c ij(x) array for use numerical evaluation of eq. (10). To facilitate FFT-based convolution, c ij(x) is loaded to the first quadrant of a 2048 x 512 computational array. The reen function of eq.(13) is filtered and sampled over a 2048 x 512 array, with values at x i-x i 0 occupyg the first half of array dimension i, and x i-x i <0 occupyg the second, as is customary with FFT manipulations. The iteration of eq. (10) then proceeds by formg the stress crement i(x)=c ij(x) j(x), transformg via FFT to k-space, multiplyg by the reen state, verse transformg, then summg with p x') (x') c (x') J (x') to obta the updated field stra. Convergence was observed ~3 iterations, m ( k ik im however several iterations beyond the observed pot of convergence were employed for good measure. The real part of the total computed stra field is plotted fig. (4c).

7 FIURE 5. Overlay of exact and approximate backscatter signals for 8 mm x 4 mm heterogeneity. FIURE 6. Comparison of mean square backscatter for uniform 20 m and dual 20 m/10 m microstructure. The iterative solution yieldg fig. (4c) updates the stra i throughout the entire 16 mm x 4 mm doma with each iteration. An alternative approach was explored which rearranges the computation so as to step sequentially through 4 mm x 4mm sections the direction of cidence. Assumg a section under consideration alone occupies the host medium, the stra field with the section is iterated to convergence usg the total forward field emergg from the precedg section as the cident field, then the backscattered signal contributed by that section is evaluated by eq. (11). This approximation is noted to exclude multiple scatterg teractions across the boundary between neighborg sections. To gauge the error troduced by this approximation, the time doma backscattered signal from an 8 mm x 4 mm doma is obtaed by applyg the exact iteration to the full doma, and then by applyg the approximate method sequentially to the two adjacent 4 mm x 4 mm domas. Time harmonic results are computed at 255 discreet frequencies distributed over a 20 MHz bandwidth, and time doma signals are obtaed by Fourier transformation assumg a Hanng spectral response T() eq.(11) with a 100% 6 db bandwidth. The backscattered signals resultg from the exact and approximate computations are overlaid fig. (5). The expanded view of the later arrivg multiply scattered noise tail reveals the largest error of approximation, which is seen to be negligible. The sequential approximation is advantageous because 1) smaller, manageable arrays havg a size dependent of the total doma length are beg manipulated at any given time, and 2) the number of required time harmonic frequencies depends only on the size of the smaller array, likewise dependent of the total doma length. Usg the approximate sequential method, the result of fig.(8) actually only requires 127 discrete frequencies, sce the backscattered signal generated by each 4 mm long section is approximately half the duration of that generated by the 8 mm long doma. The dividual backscatter signal contributions must, of course, be appropriately positioned time prior to summation. It is noted that each iteration of eq. (10) for the 4 mm square doma required 0.14 seconds on a modest capability laptop computer. As a fal result, time doma backscatter from a 16 mm x 4 mm doma is computed usg the sequential doma approximate method. Two cases are considered: 1) a uniformly distributed 20 m mean gra size over the length of the sample, and 2) a 20 m mean gra size over the first 8 mm, and a 10 m mean gra size over the followg 8 mm. ra diameters are assumed to have scaled versions of the normal probability distribution of fig. (1b), and gra orientation is assumed uniformly distributed. 100 member ensembles of microstructure realization

8 meetg these statistical descriptors were generated, backscatter signals were computed for each realization, and the mean value of the square of backscatter time doma voltage were summed for the two cases. 127 discreet frequencies were distributed over 20 MHz, and a Hanng spectral response was prescribed with a 100% 6 db bandwidth. The means square voltages plotted fig. (6) display the expected dependence on gra size. The uniform 20 m microstructure reveals a profile roughly proportion to the axial profile of the cident beam. The dual microstructure case reveals a significant drop backscatter amplitude at times correspondg to the scatter by the 10 m gras. Significantly, a noise tail is observed at times correspondg to depths at which the cident pulse is beyond the 16 mm heterogeneous doma, and therefore not generatg backscatter. The dicated noise tail fig. (6) can therefore be entirely attributed to multiple scatterg. This observation provides an dication of the level at which multiple scatterg is accumulated the propagation path. CONCLUSIONS The computational approach presented here provides a rigorous quantification the dependence of ultrasound propagation and scatter on microstructure parameters. The formulation accommodates a general prescription of spatially dependent anisotropic elastic properties on an arbitrarily fe length scale, and hence provides a tool to study a variety of microstructure phenomena, such as variable gra size, elongation, and preferred orientation, on multiple length scales, e.g. micro- and macro-gra descriptions. Spatial filterg of microstructure detail reduces the scope of computation to that sufficient for accurate prediction of ultrasound propagation and scatter, with post-filter microstructure detail on the order of field wavelength. Use of a volume tegral equation (VIE) problem formulation results a convolutional form efficiently evaluated usg Fast Fourier Transformation. Solutions are obtaed usg an iterative reduction of the VIE error residual, thereby circumventg explicit formation, retention, and version of scatterg matrices. Solutions are mathematically rigorous, providg means to assess the range of validity of approximate formulations such as Born and dependent scatterer theory, and to quantify consequent errors their use for prediction of backscatter noise levels and microstructure parameter estimation (e.g. gra size measurement). Importantly, the approach fully captures signal accumulation associated with multiple scatterg. As the approach computes scatterg for explicitly specified microstructure, computation of statistical field measures requires ensemble generation of random realizations, and explicit computation of scatter for each member. The approach is therefore computation tensive, and likely not a practical tool for route material characterization data version. It is anticipated, however, that the approach will contribute to the development of computationally-derived corrections to data version based on approximate scatterg theory. While the 1D and 2D formulations presented here give guidance and sight to the implementation of the computational approach, the predicted phenomena likely deviate sufficiently from that of the 3D problem to yield their use suspect for quantitative representation of 3D microstructure scatterg. The present 2D formulation will therefore be extended upcomg work to accommodate full 3D microstructure prescription. Due to reliance on FFT-based operations, this extension volves an crease computational operations from ~2N 2 log(n) to ~3N 3 log(n), thereby remag with the reach of available computation resources. ACKNOWLEDEMENT This research was funded through the Pratt & Whitney Center of Excellence at Iowa State University. REFERENCES 1. F. Margetan, R. Thompson, and I. Yalda-Mooshabad, Backscattered Microstructural Noise Ultrasonic Toneburst Inspections, Journal of Nondestructive Evaluation 13, , (1994). 2. F. Stanke and. Ko, "A Unified Theory for Elastic Wave Propagation Polycrystalle Materials," J. Acoust. Soc. Am. 75, , (1984). 3. Li, A., Yu, L., Roberts, R., Margetan, F. J., and Thompson, R. B., A 2-D Numerical Simulation Study of Microstructure- Induced Ultrasonic Beam Distortions, Review of Progress QNDE 23, edited by D.O. Thompson and D.E. Chimenti, AIP, , (2002). 4. J. Achenbach, Wave Propagation Elastic Solids, Elsevier, (1982). 5. B. Auld, eneral Electromechanical Reciprocity Relations Applied to the Calculation of Elastic Wave Scatterg Coefficients, Wave Motion 1, (1979).