Numerical modeling and simulation for ultrasonic inspection of anisotropic austenitic welds using the mass-spring lattice model

NDT&E International 44 (2011) 571–582

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Numerical modeling and simulation for ultrasonic inspection of anisotropic austenitic welds using the mass-spring lattice model Eunsol Baek, Hyunjune Yim n Department of Mechanical Engineering, Hongik University, 72-1 Sangsu-dong, Mapo-gu, 121-791, Seoul, Republic of Korea

a r t i c l e i n f o

abstract

Article history: Received 24 May 2010 Received in revised form 19 May 2011 Accepted 27 May 2011 Available online 13 June 2011

This paper deals with two-dimensional numerical modeling and simulation of ultrasonic nondestructive testing of austenitic welds, which are assumed to be homogeneous and transversely isotropic. The numerical computer model used in this work is the rectangular mass-spring lattice model (RMSLM). The model’s capability has first been investigated for accurately simulating ultrasonic waves in the austenitic weld medium. Compatible numerical models for transmitting probes ‘tailored’ for the austenitic weld have also been developed so that incident waves can be sent precisely in desired directions. A two-dimensional numerical simulator, consisting of the probe models and the RMSLM, has been applied to realistic, typical ultrasonic testing problems of austenitic welds, and effective test setups for them have been found. This demonstrates the capability and usefulness of the simulator. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Anisotropic austenitic weld Ultrasonic testing Numerical simulation Mass-spring lattice model Effective test setup

1. Introduction Ultrasonic inspection of an austenitic weld is generally regarded difficult because of the material anisotropy, in addition to the high attenuation of waves due to wave scattering from coarse grain structure and wave absorption by the material. While high attenuation reduces the probability of detecting defects, anisotropy gives rise to beam skew, which makes it difficult to transmit ultrasonic waves in a desired direction, or to correctly interpret the paths of received waves and thus reliably estimate the location and size of a defect. Furthermore, the anisotropy of an austenitic weld often exhibits inhomogeneity, formed during gradual cooling process after welding. Such inhomogeneity of the weld may vary from specimen to specimen, and thus may not precisely be known until its cross sections are viewed. Fundamental characteristics of elastic wave propagation in anisotropic media have been studied analytically by many researchers, e.g. Fedorov [1] and Musgrave [2]. Recent analytical studies with a focus on nondestructive testing of anisotropic materials include a traction model proposed by Niklasson [3], a Fouriertransform-based model derived by Eriksson et al. [4], and various Gaussian beam models developed by Spies [5]. Of particular relevance here are those studies reported by Ogilvy [6–9], on ultrasonic testing of anisotropic welds. In order to find the optimal test configurations, she used the ray method to

n

Corresponding author. Tel.: þ822 320 1489; fax.: þ822 322 7003. E-mail addresses: [email protected] (E. Baek), [email protected] (H. Yim). 0963-8695/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ndteint.2011.05.011

model the wave phenomena in two dimensions. More recently, Schmitz et al. [10] developed the synthetic aperture focusing technique (SAFT), by considering the varying grain directions, to realize three-dimensional ray tracing. Shlivinski and Langenberg compared the SAFT results with those from an elastodynamic finite integral technique (EFIT) [11]. Also to solve ultrasonic testing problems of austenitic welds, Liu and Wirdelius [12] used a traction probe model with twodimensional ray tracing. In order to consider the varying grain directions of the weld, they divided the weld into small regions and assigned to each region a tilted coordinate system according to the rotated axes of material symmetry in that region. However, studies to develop complete numerical simulators for ¨ the entire ultrasonic testing are scarce. Bostrom et al. [13] completed a numerical simulator by integrating a mathematical model for a transmitting probe based on the traction continuity condition, a model for a receiving probe based on the electromechanical reciprocity relation, and wave scattering formulation using Green’s tensor. They demonstrated their model by considering an example problem of detecting a crack, yet its application to real problems seems limited because reflections from isotropic–anisotropic material interfaces, such as weld–parent medium interfaces, or from a side traction-free boundary were not addressed. A complete ultrasonic testing simulator, called CIVATM, is commercially available [14], which uses the ‘pencil method’ and the geometrical theory of diffraction (GTD) to simulate ultrasonic wave propagation in three dimensions. In addition to the largely analytical methods reviewed above, purely numerical methods have also been developed. Examples include Langenberg et al. [15], who used the elastodynamic finite

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integral technique (EFIT) to predict the output rf signals, and Yamawaki and Saito [16], who modeled various wave phenomena, including refraction at the austenitic cladding interface, using the finite difference method (FDM). Also, a numerical model for the austenitic weld itself, called the ‘modeling anisotropy from notebook of arc welding (MINA)’, was developed, and was used in conjunction with a finite-element method to simulate ultrasonic wave propagation [17]. In spite of the significant number of studies reviewed above, most of them are not readily applicable to real problems because of their complexity or excessive computational load. Also, those approaches based on the ray method per se are not capable of simulating wave diffraction, and those purely analytical methods cannot deal with the complex geometry of real test objects. Thus, the present paper uses a much simpler numerical model, called the rectangular mass-spring lattice model (RMSLM), which is both computationally efficient and flexible enough to deal with real problems. That is, it can deal with test objects that have finite dimensions and complex geometry, consist of different materials (i.e. isotropic parent medium and anisotropic weld medium), and include various anomalies such as voids and cracks that give rise to complex wave phenomena such as diffraction and scattering in addition to reflection and refraction. Yet, all problems considered in this paper are two-dimensional because a three-dimensional RMSLM has not yet been developed (see Section 2). The RMSLM was developed for transversely isotropic media [18,19] through modifications of the (square) mass-spring lattice model (MSLM), which in turn had been originally developed by a research group led by Harumi et al. [20–22] based on the mass point model [23]. Harumi and the present authors [20–22,24–28] have studied and demonstrated the capability of the MSLM and the RMSLM to predict ultrasonic wave behaviors such as propagation, reflection, refraction, and diffraction. In contrast with the qualitative nature of the previous demonstration (i.e. concerned only with the capability of predicting the presence of reflected and refracted waves that should exist), the present paper first aims at closely investigating the RMSLM’s quantitative accuracy (e.g. orientations of phase normals, directions of wave propagation, and displacement amplitudes of reflected waves) in simulating fundamental wave phenomena in a typical austenitic weld (Section 3). The paper is then concerned with developing numerical models for probes that are exclusively effective for the austenitic weld (Section 4). Finally, using the complete ultrasonic testing simulator consisting of the RMSLM and the probe models, various realistic testing problems are simulated to illustrate the usefulness of the method (Section 5).

2. The numerical model—RMSLM It seems orderly to briefly introduce here the numerical model of RMSLM before its capability is investigated. A representative cell of the two-dimensional lattice of the RMSLM is depicted in Fig. 1, where nine lumped masses are shown as solid circles, and only those (both linear and rotational) springs that exert forces on the center mass are shown for simplicity. The coordinate axes (x2 and x3) are set in Fig. 1 such that the x3-axis is the axis of material symmetry for a transversely isotropic medium. It is assumed in this study that the austenitic weld is homogeneous with transverse isotropy because modeling its inhomogeneity (caused by gradual cooling of the weld) will require further development of the RMSLM as discussed in Section 6 of this paper. This assumption is made at the present stage of study even though it will render the model unable to precisely predict the (slightly) curved wave paths in real austenitic weld. Also, signal attenuation and noises due to wave absorption and scattering by the material are ignored in this study though wave absorption may be modeled by adding dashpots to the

Fig. 1. Representative cell of two-dimensional rectangular mass-spring lattice model (RMSLM).

model in Fig. 1 [29]. The elastic constants and mass density of the homogeneous, lossless austenitic weld in this work are assumed as follows [6]: C2222 ¼263 GPa; C1122 ¼98.0 GPa; C2233 ¼145 GPa; C3333 ¼ 216 GPa; C2323 ¼129 GPa; C1212 ¼82.5 GPa; and, r ¼7900 kg/m3. Recall that only two-dimensional motions (in the x2–x3 plane) of masses can be simulated using this two-dimensional model. This is equivalent to assuming plane–strain states and thus precludes all SH waves that by definition involve out-of-plane displacements. Then, the equations of motion for the center mass in Fig. 1 may be derived by summing all the spring forces, acting on it, which can be expressed in terms of the relative displacements of the masses. The resulting equations of motion, which are similar in form to the difference equations of a typical FDM, are used for the simulation computation. These RMSLM difference equations contain the time step Dt and lattice spacing h1 and h2 (see Fig. 1), which are related to one another for the numerical stability and optimal accuracy of the computation [25]. Note that the spring constants (kis and ais in Fig. 1) are expressed in terms of the material’s elastic constants [25].

3. Numerical simulation of wave phenomena using the RMSLM In this section, various fundamental phenomena of ultrasonic waves are numerically simulated using the RMSLM, and the results are compared with the corresponding analytical results. These comparisons are made quantitatively where possible to confirm the capability of the RMSLM to accurately simulate waves in the austenitic weld. 3.1. Generation of single-mode plane waves by line loads Unlike in isotropic media, line loads with loading directions are either perpendicular or parallel to the line of loading do not generate in anisotropic media plane waves of a single mode, but plane waves of two different modes: quasi-longitudinal (qP) and quasi-shear (qSV) modes. In this study, as an attempt to generate plane waves purely of a single mode, the loading direction of the line load has been deliberately aligned along the direction of the particle displacement (or, equivalently, that of the traction) associated with the desired plane wave. Relationships among these different directions may be obtained from the analytical wave physics [30]. Before discussing the numerical results, it is useful to define sign conventions for three important directions associated with a plane wave (qP or qSV) in anisotropic media. These three directions are all measured counterclockwise from the x3-axis as

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defined in Fig. 2: the phase normal direction ( ,901r y r þ901) perpendicular to the wave front, the displacement polarization direction (2, 901r j r þ 901), and the energy propagation direction ( , 1801r g r þ1801). These conventions will be used throughout this work. A number of numerical simulations was conducted to generate single-mode plane waves with varying wave normal directions y at an increment of 51, for both qP and qSV modes. They all produced numerical results exhibiting single-mode plane waves, and the angles of j and g measured from them are tabulated in Table 1 along with the corresponding analytical angles [30] and the error magnitudes. It may be observed from the table that the RMSLM is capable of producing single-mode plane waves with errors less than 0.401 in the displacement polarization direction, and less than 1.01 in the energy propagation direction, except for three cases. 3.2. Reflection of plane waves at a free boundary

is no critical angle for qP incidence as in all isotropic cases, whereas there are three critical angles for qSV incidence: 28.81, 37.81, and 53.61. These three critical angles divide the entire range of the angle of qSV incidence into four regimes, each of which exhibits different numbers and modes of reflected waves as listed in Table 2. In this paper, cases of numerical simulation are presented as shown in Fig. 3. Fig. 3(a) and (c) schematically shows the line loads that generate incident plane waves with angles of incidence Table 2 Modes of reflected waves, according to the angle of incidence in austenitic weld. Mode of incident wave

Angle of incidence (yI) (deg.)

Modes of reflected waves

qP qSV

0–90 0–28.8 28.8–37.8 37.8–53.6 53.6–90

RqP, RqSV RqP, RqSV RqSV RqSV1, RqSV2 RqSV

This subsection investigates the capability of the RMSLM to accurately simulate the reflection of plane waves at a planar traction-free boundary of the austenitic weld medium. To this end, numerical simulation has been conducted for various angles of incidence (at 51 intervals) and for both incident modes of qP and qSV, and the numerical results have been compared with the corresponding analytical results, with regard to the modes, reflection angles, and displacement reflection coefficients of the reflected waves. Before the numerical results are presented, the critical angles of incidence for the austenitic weld, analytically found [30] as shown in Table 2, need to be discussed. Note in Table 2 that there

Fig. 2. Definitions of directions of phase normal, displacement polarization, and energy propagation.

573

Fig. 3. Reflection of plane qSV wave incident to the traction-free bottom boundary: (a) schematic diagram with 401 angle of incidence; (b) snapshot of simulated wavefield showing two reflected qSV waves corresponding to (a); (c) schematic diagram with 701 angle of incidence; and (d) snapshot of simulated wavefield showing only one reflected qSV wave corresponding to (c). (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Table 1 Comparison of numerically measured displacement polarization directions and energy propagation directions with analytically obtained directions, for plane qP and qSV waves having various wave normal directions.

y [deg.] Displacement polarization direction (j) qP

Energy propagation direction (g)

qSV

qP

qSV

Anal. [deg.] Num. [deg.] Error [deg.] Anal. [deg.] Num. [deg.] Error [deg.] Anal. [deg.] Num. [deg.] Error [deg.] Anal. [deg.] Num. [deg.] Error [deg.] 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

14.57 24.70 31.11 35.43 38.62 41.19 43.41 45.46 47.45 49.49 51.70 54.18 57.12 60.78 65.51 71.82 80.11

14.77 24.74 31.08 35.58 38.55 41.23 43.43 45.50 47.42 49.53 51.72 54.21 57.17 60.85 65.57 71.88 80.15

0.20 0.04 0.03 0.15 0.07 0.04 0.02 0.04 0.03 0.04 0.02 0.03 0.05 0.07 0.06 0.06 0.04

 75.43  65.30  58.89  54.57  51.38  48.81  46.59  44.54  42.55  40.51  38.30  35.82  32.88  29.22  24.49  18.16  9.89

 75.07  65.59  59.14  54.84  51.39  49.01  46.64  44.50  42.60  40.45  38.28  35.85  32.78  29.06  24.37  18.09  9.50

0.36 0.29 0.25 0.27 0.01 0.20 0.05 0.04 0.05 0.06 0.02 0.03 0.10 0.16 0.12 0.07 0.39

20.00 30.85 36.28 39.50 41.75 43.53 45.08 46.52 47.93 49.39 50.98 52.81 55.04 57.95 62.07 68.25 77.60

19.80 30.37 35.66 38.74 41.20 42.86 44.60 46.29 47.75 50.54 52.10 53.76 56.00 58.98 63.05 68.76 77.69

0.20 0.48 0.62 0.76 0.55 0.67 0.48 0.23 0.18 1.15 1.12 0.95 0.96 1.03 0.98 0.51 0.09

 19.06  26.14  26.73  24.26  19.35  11.21 2.35 24.53 53.55 78.13  86.43  77.29  71.85  68.91  68.30  70.75  77.88

 19.14  26.26  26.67  24.31  19.35  11.13 2.38 24.40 53.54 78.13  86.90  77.84  72.54  68.70  68.38  70.75  77.92

0.08 0.12 0.06 0.05 0.00 0.08 0.03 0.13 0.01 0.00 0.47 0.55 0.69 0.21 0.08 0.00 0.04

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number and modes of reflected waves at a traction-free boundary of the austenitic weld medium. For quantitative validation, the angle of reflection (yR) and phase speeds of reflected waves (cR) were obtained from the numerical results, as tabulated in Tables 3 and 4 for incident qP and qSV waves, respectively. In both cases, the numerically found angles of reflection and speeds of reflected waves show excellent agreements with the corresponding analytical results [30], exhibiting less than 0.251 error in angles and less than 3% error in wavespeeds, except for a few cases. It may also be noted in Table 4, the number and modes of reflected waves change in different regimes of the angle of incidence, in accordance with Table 2. Finally, displacement reflection coefficients, defined as the displacement amplitude of a reflected wave divided by that of the incident wave, have been computed from the simulation results and plotted in Fig. 4(a) and (b) for incident qP and qSV waves, respectively, versus the angle of incidence. Solid squares

yI ¼401 and 701, respectively, where the forcing direction of single-cycle sinusoidal loading is indicated by double-headed arrows. Fig. 3(b) and (d) shows example snapshots of the numerically computed wavefield, during the process of reflection at the bottom boundary, corresponding to the cases of Fig. 3(a) and (c), respectively. In Fig. 3(b) and (d), wave labels beginning with ‘I’ and ‘R’ denote the incident and reflected waves, respectively, and the dotted and dashed arrows indicate the phase normal and energy propagating directions, respectively, as defined in Fig. 2. Note that in all wavefield images presented throughout this paper (such as Fig. 3(b) and (d)), the brightness of each pixel shows the magnitude of strain at that location whereas the color indicates the wave mode, with red and blue for purely dilatational and purely shear strains, respectively—thus, purple meaning a mixture of the two strain components. Examining Fig. 3(b) and (d) and comparing them with Table 2 confirms that the RMSLM is capable of predicting the correct

Table 3 Analytical and numerical values of reflection angles and phase speeds of reflected waves for incident plane qP waves.

yIqP [deg.] Angle of reflection

Phase speed of reflected wave

yRP (RqP wave)

cRP (RqP wave)

yRS (RqSV wave)

cRS (RqSV wave)

Anal. [deg.] Num. [deg.] Error [deg.] Anal. [deg.] Num. [deg.] Error [deg.] Anal. [m/s] Num. [m/s] Error [%] Anal. [m/s] Num. [m/s] Error [%] 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00 55.00 60.00 65.00 70.00 75.00 80.00 85.00

5.02 10.00 14.97 19.98 24.99 29.96 35.04 39.98 45.78 50.74 54.81 59.75 65.92 70.38 74.87 80.04 85.06

0.02 0.00 0.03 0.02 0.01 0.04 0.04 0.02 0.78 0.74 0.19 0.25 0.92 0.38 0.13 0.04 0.06

3.77 7.11 9.93 12.33 14.40 16.22 17.84 19.31 20.67 21.94 23.13 24.25 25.32 26.33 27.24 28.03 28.57

3.76 7.12 9.98 12.28 14.40 16.29 17.87 19.29 20.67 21.89 23.30 24.37 25.36 26.37 27.32 28.01 28.55

0.01 0.01 0.05 0.05 0.00 0.07 0.03 0.02 0.00 0.04 0.18 0.12 0.04 0.04 0.08 0.01 0.02

5295.38 5453.21 5642.33 5829.51 5998.97 6142.59 6255.81 6336.01 6381.87 6393.10 6370.46 6315.88 6232.85 6127.23 6008.68 5892.96 5804.07

5341.07 5427.66 5683.79 5810.48 6029.79 6121.91 6272.82 6322.32 6600.43 6552.64 6445.74 6424.88 6416.44 6270.62 6099.69 5930.49 5810.02

0.86 0.47 0.73 0.33 0.51 0.34 0.27 0.22 3.42 2.50 1.18 1.73 2.95 2.34 1.51 0.64 0.10

3993.30 3884.44 3759.97 3640.01 3530.18 3430.98 3341.40 3260.08 3185.76 3117.38 3054.13 2995.48 2941.24 2891.67 2847.74 2811.56 2786.73

4001.29 3852.69 3752.14 3617.20 3500.71 3408.50 3334.45 3241.63 3176.85 3120.22 3006.26 2965.31 2921.20 2874.13 2824.12 2789.39 2750.36

0.20 0.82 0.21 0.63 0.83 0.66 0.21 0.57 0.28 0.09 1.57 1.01 0.68 0.61 0.83 0.79 1.31

Table 4 Analytical and numerical values of reflection angles and phase speeds of reflected waves for incident plane qSV waves.

yIqSV [deg.] Angle of reflection

Phase speed of reflected wave

yRP (RqP wave)

cRP (RqP wave)

yRS (RqSV wave)

cRS (RqSV wave)

Anal. [deg.] Num. [deg.] Error [deg.] Anal. [deg.] Num. [deg.deg.] Error [deg.] Anal. [m/s] Num. [m/s] Error [%] Anal. [m/s] Num. [m/s] Error [%] 5 10 15 20 25 30 35 40

6.76 15.13 26.58 42.48 63.47 – – –

6.70 15.00 26.78 42.46 63.44 – – –

0.06 0.13 0.20 0.02 0.03 – – –

45

–

–

–

50

–

–

–

55 60 65 70 75 80 85

– – – – – – –

– – – – – – –

– – – – – – –

5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00  76.83 45.00  65.07 50.00  57.60 55.00 60.00 65.00 70.00 75.00 80.00 85.00

4.92 9.87 15.18 20.16 25.65 30.81 34.95 39.99  76.78 45.00  66.92 49.95  58.24 53.76 60.01 65.02 69.98 75.04 79.95 85.05

0.08 0.13 0.18 0.16 0.65 0.81 0.05 0.01 0.05 0.00 1.85 0.05 0.64 1.24 0.01 0.02 0.02 0.04 0.05 0.05

5343.98 5647.39 6047.52 6363.12 6260.94 – – –

5276.33 5557.93 5908.98 6311.07 6234.77 – – –

1.27 1.58 2.29 0.82 0.42 – – –

–

–

–

–

–

–

– – – – – – –

– – – – – – –

– – – – – – –

3959.17 3756.70 3497.65 3222.36 2957.34 2725.23 2548.15 2446.23 3705.55 2432.56 3119.70 2508.06 2764.30 2660.86 2870.99 3115.96 3373.62 3621.81 3835.87 3985.95

3875.52 3708.15 3427.16 3166.83 2843.83 2622.47 2510.05 2424.32 3707.94 2432.51 3279.72 2501.55 2752.31 2747.72 2888.04 3121.12 3377.58 3623.68 3842.06 3981.05

2.11 1.29 2.02 1.72 3.84 3.77 1.50 0.90 0.06 0.00 5.13 0.26 0.43 3.26 0.59 0.17 0.12 0.05 0.16 0.12

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Fig. 4. Comparison of numerical and analytical displacement reflection coefficients for reflected waves, as functions of the angle of incidence: (a) for incident qP waves, and (b) for incident qSV waves.

(’), asterisks (%), and solid circles (K) in Fig. 4 are for the reflected qP wave, the (first) reflected qSV wave and, if any, the second reflected qSV wave, respectively. Note that it was not possible to obtain numerical values for the case of qSV incidence with y ¼551 (see Fig. 4(b)). This is because the energy propagating direction of this incident wave turns out to be g ¼  86.431, meaning that it travels almost parallel to the free boundary and thus yields no measurable reflection. Comparisons of the numerical results with the corresponding analytical values [30], shown by curves in Fig. 4, exhibit good agreements in both cases of incident qP and qSV waves.

3.3. Diffraction and scattering of plane waves from cracks and circular voids Physics underlying wave diffraction and scattering from obstacles in anisotropic media is the same as that in isotropic media [31]. Various such problems have been simulated using the RMSLM, and only two cases are presented in this subsection. Fig. 5(a) schematically shows the case where a horizontally traveling incident plane P wave encounters a horizontal ‘crack’. Note that ‘cracks’ are modeled in the RMSLM by disconnecting the springs (see Fig. 1) that link the two crack faces [25,26]. Because of the lattice structure of the model, the thus obtained defect model should appropriately be called a rectangular slit that has a very small width. Yet, wave scattering behavior predicted from this defect model turned out to reasonably agree with the correct crack-tip scattering behavior in its essential nature [27]. Thus, rectangular slits modeled in this manner are called cracks in this paper, with a justifiable loss of rigor. Fig. 5(b) shows a snapshot of the numerically simulated wavefield for the case of Fig. 5(a). Diffracted qP (labeled ‘DqP’) and diffracted qSV (labeled ‘DqSV’) waves, both centered at the crack tip, as well as the head waves (labeled ‘H’) are observed in Fig. 5(b), which agree with the results in case of isotropic media [32]. Similarly, Fig. 5(c) and (d) is for the case of an SV incident plane wave that encounters a circular void. The diffracted waves (‘DqP’ and ‘DqSV’ in Fig. 5(d)) again exhibit wave surfaces similar to those generated by a point source, and are centered at a point inside the void, not at the front end of the void. This also agrees with other numerical results in the relevant literature [16].

Fig. 5. Diffraction and scattering of plane waves: (a) schematic diagram for P incident plane wave encountering a horizontal crack; (b) simulated wavefield corresponding to (a); (c) schematic diagram for SV incident pane wave encountering a circular void; and (d) simulated wavefield corresponding to (c). (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

4. Numerical models of ultrasonic probes for austenitic weld Because of beam skew, ultrasonic beams emitted into an anisotropic medium by probes that are designed for isotropic media will propagate in different directions from those in the

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isotropic case. Further, considering different characteristics of beam skew in different anisotropic media, an exclusively customized probe design is needed for each anisotropic medium in order to transmit ultrasonic wave beams in a desired direction for that medium. In this work, numerical models are developed for transmitting probes ‘tailored’ for the austenitic weld material at hand. Also, numerical models for receiving probes previously developed for isotropic media are slightly modified to match the wave behavior in the austenitic weld. 4.1. Models of transmitting probes for austenitic weld The transmitting probe model developed in this subsection is similar to others in the literature, e.g. Ref. [3]. The most significant difference between the model developed here and the one in Ref. [3] is that the former directly imposes a traction distribution on the test object, which would produce only the desired wave of a given mode and propagation direction, whereas the latter begins with the excitation of an isotropic wedge that is coupled with the anisotropic test object. The model in Ref. [3] produces complex wavefield of multiple incident waves with various modes and propagation directions, therefore is not convenient for the purpose of this study. A typical transmitting ultrasonic probe exerts a distribution of force on a finite region of the test object’s boundary where it contacts the object. The corresponding numerical model to accompany the RMSLM will thus be a set of forces applied to the boundary masses of the model, located in the excited region (say, AB in Fig. 6). The way in which these forces are modeled in this study is based on the idea schematically shown in Fig. 6. First, imagine as though the medium were extended above the upper boundary of the test object, where a plane wave of the desired mode propagated in the desired direction g towards AB. Then it may be expected that the forces (or tractions) exerted to the l masses in AB in Fig. 6, as the imaginary wave passes through AB, will give rise to the same desired plane wave into the test object. The mathematical formulation for this model is derived below. Though the real ultrasonic waves emitted from the probe are in the form of pulses, equations are first derived for a continuous, harmonic plane wave. For the desired wave mode and desired energy propagating direction g, the associated phase normal direction y and displacement polarization direction j may be found from the analytical wave mechanics. Then, the displacement of the imaginary plane harmonic wave, explained above, may be expressed as ui ðx,tÞ ¼ Adi exp½Iðk dxotÞ

ð1Þ

where uis (i¼2, 3) are the displacement components in the x2–x3 plane; A is the amplitude of the wave, arbitrarily set to unity here; dis (i¼ 2, 3) are the components ofpaffiffiffiffiffiffiffi unit vector in the displacement polarization direction j; I ¼ 1; k¼ kn¼ {k2, k3}T, where k and n are the wavenumber and phase normal unit vector,

Fig. 6. Schematic diagram illustrating the idea for modeling a transmitting probe.

respectively; x ¼{x2, x3}T is the position vector of the point of interest; and, o is the angular frequency. The phase normal unit vector for the case shown in Fig. 6 is n¼ {sin y, cos y}T. Recall that stress components sij in any homogeneous linearlyelastic medium may be expressed in terms of the displacement components as [1]   1 @um @ul sij ¼ Cijlm þ ð2Þ 2 @xl @xm where the summation convention has been used, and Cijlms are the elastic constants of the material. Substituting Eq. (1) into Eq. (2) and retaining only the real part will yield the stress components of the plane harmonic wave. The resulting expression may be substituted into the formula, ti ¼ sijmj (i, j¼2, 3), to yield the traction vector t on AB having a unit (inward) normal vector m¼{0, 1}T and x3 ¼ a (see Fig. 6). The thus obtained traction distribution in the probe’s contact area AB (b arx2 rbþ a, x3 ¼ a) may be written, after some modifications explained below, as tðx2 ,tÞ ¼ Agðx2 Þ½dC2323 ðd3 k2 þ d2 k3 Þe2 þ ðC3322 d2 k2 þC3333 d3 k3 Þe3  sinðotk2 x2 k3 aÞ,

ba rx2 r bþ a

ð3Þ

where e2 and e3 are the unit vectors in the x2- and x3-directions, respectively, and only the nonzero elastic constants for transverse isotropy have been retained. In writing the final expression of Eq. (3), two new factors have been incorporated. The first factor is a spatial function g(x2) that models a non-uniform distribution of traction on the probe’s face. In this work, g(x2) is assumed to be a parabolic function, with its maximum at the center, x2 ¼b. The second factor incorporated is d, named here ‘shear transmission factor’, in the e2-component. Its value ranges from 0 to 1, depending upon the capability of the couplant to transmit shear traction. For example, d ¼0 corresponds to an inviscid couplant whereas d ¼ 1 to a perfect coupling. Throughout this study, d ¼1 is assumed such that all waves that can exist may be generated to their full strength and thus easily be studied. As mentioned above, to model the real pulse-type waves, Eq. (3) is first fast-Fourier-transformed (FFT), and the transform is multiplied by the probe’s frequency characteristic function, F(o), defined ¨ and Wirdelius [34] and by Baek [33] in accordance with Bostrom Silk [35]. Furthermore, because of the nonzero angle y in Fig. 6, the forcing on the l masses does not occur at the same time; the excitation begins with the mass at point A and continues sequentially toward point B with a time delay of (h sin y)/c between two adjacent masses, where h is the inter-mass spacing along AB and c is the phase speed of the plane wave. The numerical probe model is now completed, and is ready to produce numerical results. Before the numerical simulation results are presented, analytical results to be compared with them must be discussed. Fig. 7 shows the analytically obtained wave surfaces in the x2–x3 plane as generated, in the austenitic weld medium, by a point source located at the origin of the coordinate system shown. Note that only those surfaces in the fourth quadrant, except some parts of qSV cusps, are shown in Fig. 7 because they are sufficient for the present study. The four energy propagating directions – g ¼0, 301, 451, and 601 – to be studied in the next subsection are indicated in dashed lines through the origin. For each of these four ‘ray’ directions, segments of the corresponding plane waves are shown by short thick lines tangent to the wave surfaces at the intersections between the ray and wave surfaces [30]. Nine such plane wave segments shown in Fig. 7 correspond to the desired plane waves to be generated through the numerical simulation in the following subsection, and are labeled with the mode and ray direction, e.g. qSV301 and qP301 for the rays in the direction of g ¼301. Note that the direction of g ¼0 is special because the material is isotropic in this direction; therefore, purely

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longitudinal and purely shear plane waves labeled P0 and SV0 exist in this direction, in addition to the quasi-shear qSV0 wave. 4.2. Simulated wavefields emitted from transmitting probe models As mentioned above, using the transmitting probe model developed here, wavefields emitted from nine transmitting ‘probes’ have been simulated for four desired directions of energy propagation—g ¼0, 301, 451, and 601. The frequency characteristics of the probes have been assumed as follows: for all P and qP probes, both the center frequency and the 6 dB-bandwidth [33]

Fig. 7. Analytically obtained wave surfaces in austenitic weld (only in the fourth quadrant) as generated by a point source at the origin.

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are equal to 5.0 MHz; and, for all qSV probes, both 2.5 MHz whereas for the SV0 probe, both 3.5 MHz. Fig. 8 shows a snapshot of the simulated wavefield for each of the nine probes considered, where the location and extent of a probe are indicated by a short white strip in the upper boundary of the medium. The extent or width of all probes in this work is set equal to 6.35 mm. The white straight line emitting from the center of the probe in each image indicates the desired direction of wave propagation. Fig. 8(a) shows a simulated wavefield for the normal (g ¼0) transmitting P probe, where a red (i.e. purely longitudinal) horizontal planar wave, corresponding to the wave labeled P0 in Fig. 7, is clearly observed. Fig. 8(b) and (c) shows simulated wavefields for the normal SV and the normal qSV probe models, respectively. Recall that these two cases (SV0 and qSV0), in spite of the same direction of energy propagation (i.e. g ¼0), exhibit different phase normal directions y as shown in Fig. 7, and also different k and d vectors in Eq. (3). The wavefield snapshots obtained for the 301, 451, and 601 angle-beam qP probe models are shown in Fig. 8(d) through (f), respectively. In all these cases, planar qP waves are observed, which correspond to the plane wave segments (qP301, qP451, and qP601) in Fig. 7. In Fig. 8(g), (h), and (i) are observed similar results for the 301, 451, and 601 angle-beam qSV probe models, respectively. Though not shown here, analyzing a few consecutive snapshots of wavefield enables the quantitative assessment of the energy propagating directions. Table 5 shows a summary of the directions of energy propagation and phase normal measured from the numerically simulated wavefields in comparison with the corresponding analytical values. The comparisons confirm that the probe model developed here is capable of accurately simulating the transmitting probes, with a maximum error of 3.81 observed in the case of qSV601.

Fig. 8. Simulated wavefields for nine transmitting probes: (a) normal P probe; (b) normal SV probe; (c) normal qSV probe; (d) 301 angle-beam qP probe; (e) 451 angle-beam qP probe; (f) 601 angle-beam qP probe; (g) 301 angle-beam qSV probe; (h) 451 angle-beam qSV probe; and (i) 601 angle-beam qSV probe. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

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4.3. Models of receiving probes The receiving probe model that has been used for isotropic media [24] is used here without much modification. This model produces a voltage output signal in proportion to the sum of the normal tractions on the RMSLM’s boundary masses, which are in contact with the receiving probe or with an angled wedge, exerted by concurrently arriving waves. In computing the output signal, the weighting function g(x2) used in Eq. (3) is taken into account as well as the probe’s frequency characteristics. Details may be found in Ref. [24]. The receiving probe model for anisotropic media is different from the one for isotropic media in that it takes into account the effects of the three mutually independent directions of wave energy propagation, phase normal, and displacement polarization. By combining the RMSLM and the probe models developed here for the austenitic weld, a complete numerical simulator has been established for ultrasonic testing of the austenitic weld. The following section illustrates the applications of the simulator.

5. Simulation of ultrasonic testing for typical defects in austenitic welds In order to demonstrate the capability and usefulness of the numerical simulator developed above, it is used in the present section to simulate various ultrasonic testing problems that contain typically encountered defects in austenitic welds. Fig. 9 shows schematic diagrams of four such ultrasonic testing problems, containing an Table 5 Comparison of energy propagation and phase normal directions of numerically simulated waves generated by transmitting probe models with the corresponding analytical results. Label in Fig. 7

P0 SV0 qSV0 qP301 qP451 qP601 qSV301 qSV451 qSV601

Simulated wavefield in Energy propagation Phase normal (y) Fig. 8 (g)

(a) (b) (c) (d) (e) (f) (g) (h) (i)

Anal. [deg.]

Num. [deg.]

Anal. [deg.]

Num. [deg.]

0 0 0 30 45 60 30 45 60

0.0 0.0 0.5 28.6 44.9 59.9 29.3 44.4 56.2

0 0 34.3 9.4 34.7 72.7 41.0 43.5 46.2

0.0 0.0 34.2 9.6 34.4 71.0 40.5 42.4 43.7

inclined crack in (a), circular voids in (b), a sidewall fusion lack in (c) or a centerline crack in (d). Each part of Fig. 9 shows the geometry of the austenitic weld—filled with light vertical lines. Recall here that the austenitic steel in the weld is assumed to be homogeneous and transversely isotropic with the x3-axis being the axis of material symmetry, and its elastic constants were given in Section 2. The isotropic material properties of the parent steel medium in all parts of Fig. 9 are assumed to be those of carbon steel: Young’s modulus, E¼211 GPa; Poisson’s ratio, n ¼0.286; and, density, r ¼7860 kg/m3 [36]. To the masses and springs of the RMSLM (see Fig. 1) along the interfaces between the weld and the parent medium are assigned the averages of the masses and spring constants for the two adjoining media [26]. Note that there is a numerical modeling conflict between the isotropic parent medium and the transversely isotropic weld because they are best modeled using the (square) MSLM and the rectangular MSLM (or RMSLM), respectively, but an identical type of lattice must be used throughout the entire modeling space to avoid mismatch of grids at the material interfaces. In the present work, the square lattice has been selected for both media, yet the spring configuration of the RMSLM is retained because this ‘RMSLM’ with a square lattice can model both isotropic and transversely isotropic media, with some adjustments of a few spring constants. Further, numerical error in wavespeeds, caused by the use of a square lattice instead of the optimal rectangular lattice having an aspect ratio of 1.103 for the austenitic weld, has been found to be smaller than 0.25%. This justifies the use of a square lattice in both media. The dimensions of the test object for each problem are given in Fig. 9 in terms of the square lattice spacing, h¼0.072 mm, which is set in conjunction with the numerical integration time step Dt¼ 0.0122 ms such that numerical stability and accuracy are guaranteed within the frequency range considered here [25,26]. Therefore, the dimensions of test objects in Fig. 9(a) through (c) are 43.2 mm  14.4 mm, and those in Fig. 9(d), 43.2 mm  10.8 mm. These dimensions are too small to be practical, yet are used here, with the matching (small) size of probes, to save computation time. For each of the four problems studied, an effective, if not optimal, test setup has been found. An effective test setup consists of the specification of the probe(s) selected from a list of those assumed available, and their positions. In this work, it is assumed that only those probes listed in Table 5 are available, and that only the top boundary of the test object is accessible to the probes. Finding an effective test setup usually requires many trials and errors [37].

Fig. 9. Schematic diagrams of typical ultrasonic testing problems for austenitic welds: (a) 301-inclined embedded crack in weld; (b) two circular voids in weld; (c) sidewall fusion lack; and (d) centerline crack in bottom weld.

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5.1. Detection of an inclined embedded crack in weld The first problem shown in Fig. 9(a) is to detect a 301-inclined crack embedded in the austenitic weld. The effective test setup found here for this problem is schematically shown in Fig. 10(a). Note here that in Fig. 10(a) and all similar schematic diagrams (part (a) of Figs. 10 through 13) showing effective test setups for the four problems being studied, the transmitting and receiving probes are shown as rectangular boxes labeled ‘T’ and ‘R’, respectively, and the transmitter-receiver in a pulse-echo scheme is denoted as ‘T/R’. In all these diagrams, three wave rays emitted from the transmitting probe are traced and displayed, where the center ray is highlighted with thick lines while two side rays are shown in thin lines; and the qP (or P) rays are in dashed lines whereas the qSV (or SV) rays in solid lines.

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The effective test setup shown in Fig. 10(a) is a pulse-echo setup using the normal qSV probe (see Fig. 8(c)), whose center is positioned at the center of the top surface, i.e. at (x2, x3)¼(300h, 200h). This setup is effective because the incident wave is inclined y ¼34.31 (see Fig. 8(c)), which will yield a strong reflected wave from the 301-inclined crack because of its small angle of incidence, i.e. merely 4.31¼34.31 301. Three snapshots of the resulting wavefield are shown in Fig. 10(b) through (d). Note that these and similar images in Figs. 10 through 13, beginning in part (b), show several important snapshots of simulated wavefields for the selected test setup. In all these snapshots, the ‘T’ and ‘T/R’ probes are shown as white horizontal strips in the upper boundary of the test object while the ‘R’ probe (see Figs. 12 and 13) as a set of white ‘scale’ markings.

Fig. 10. Simulated results for problem in Fig. 9(a): (a) effective test setup and ray paths; (b)–(d) simulated wavefields, at successive instants; and (e) simulated output voltage signal. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Fig. 11. Simulated results for problem in Fig. 9(b): (a) effective test setup and ray paths; (b)–(d) simulated wavefields, at successive instants; and (e) simulated output voltage signal. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

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Fig. 12. Simulated results for problem in Fig. 9(c): (a) effective pitch-catch test setup and ray path; (b)–(d) corresponding simulated wavefields, at successive instants; (e) corresponding output voltage signal; and (f) simulated output voltage signal for similar pulse–echo setup. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Fig. 13. Simulated results for problem in Fig. 9(d): (a) effective test setup and ray paths; (b)–(e) simulated wavefields, at successive instants; and (f) simulated output voltage signal. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

It may be observed from the wavefield snapshots in Fig. 10 that a qSV wave with a 34.31-inclined phase plane has been produced by the probe and propagates downward toward the crack (Fig. 10(b)), reflects from the crack and propagates back upward (Fig. 10(c)), and is finally received by the probe (Fig. 10(d)). In Fig. 10(c) and (d), respectively, diffracted qP waves from the crack tip and mode-converted reflected qP waves from the probe face, both reddish, are also observed. The simulated output voltage signal predicted to be picked up by the receiving probe is shown in Fig. 10(e). The signal exhibits a

clear, isolated flaw signal. In this particular example, the magnitude of the flaw signal is approximately 1/4 that of the input signal, often called the main bang. It may thus be stated that the test setup depicted in Fig. 10(a) is effective in detecting the crack. 5.2. Detection of two circular voids in weld The second problem shown in Fig. 9(b) is to detect two circular voids of an identical diameter of 10h¼0.72 mm. One of the most effective ultrasonic testing plans for this particular problem was

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found to use diffracted waves from the voids. To this end, a normal P probe is considered as a T/R probe for a pulse–echo setup. The corresponding ray paths are schematically shown in Fig. 11(a), where the T/R probe is placed at the center of the weld’s top boundary or at (x2, x3) ¼(300h, 200h). The corresponding simulated wavefields are shown in Fig. 11(b) through (d) at successive instants: the reddish P diffracted waves from the upper void propagate upward while diffraction from the lower void occurs (Fig. 11(b)); both the first and second diffracted waves propagate upward, followed by reflected waves from the bottom boundary (Fig. 11(c)); and the first diffracted wave is received by the probe (Fig. 11(d)). Fig. 11(e) shows the corresponding simulated output signal. A simple time-of-flight analysis has confirmed that the two flaw signals, labeled ‘A’ and ‘B’, correspond to the diffracted waves from the upper and lower voids, respectively. Much stronger waves observed after 5.7 ms in Fig. 11(e) represent the back-wall echoes from the bottom boundary of the test object. The existence of flaw signals ‘A’ and ‘B’, though they are approximately an order of magnitude smaller than the input signal, demonstrates that the pulse–echo test setup in Fig. 11(a) is effective in detecting and distinguishing the two circular voids. 5.3. Detection of sidewall fusion lack This problem is to detect a sidewall fusion lack situated along the right-hand boundary of the weld, as shown in Fig. 9(c). Among several test setups considered, the pitch-catch setup whose ray paths are shown in Fig. 12(a) has been found to be most effective. In this setup, the transmitting probe is a 601 angle-beam qP probe positioned at (x2, x3)¼ (208h, 200h) whereas the receiving probe is a 451 angle-beam qP probe at (x2, x3)¼(300h, 200h). The corresponding, simulated wavefields are shown in Fig. 12(b) through (d) for three important instants: a plane qP wave with a phase normal inclined by y ¼72.71 has been emitted from the transmitting probe and propagates in the direction of g ¼ 601 (Fig. 12(b)); the incident wave has been reflected from the fusion lack, and propagates back toward the receiving probe (Fig. 12(c)); and, the reflected qP wave reaches the receiving probe (Fig. 12(d)). The numerically simulated output voltage signal is shown in Fig. 12(e), which clearly exhibits an isolated flaw signal. In this particular problem, another test setup is considered to emphasize the beam skew characteristic of the anisotropic medium. This alternative setup is a pulse–echo setup with its T/R probe at the location of the transmitter (T) in Fig. 12(a). If the medium were isotropic, this pulse–echo setup would be the most effective because the angle of incidence to the anomaly would be zero (i.e. normal incidence). However, this is not the case in the anisotropic austenitic weld, as confirmed by the much smaller magnitude of the corresponding flaw signal shown in Fig. 12(f) than that in Fig. 12(e)—both plots are drawn in the same scale for direct comparison. 5.4. Detection of a centerline crack in weld The final problem studied in this paper is to detect a crack that vertically breaks the center of the weld’s bottom boundary, as shown in Fig. 9(d). The test setup that was found most effective of all considered is depicted in Fig. 13(a). Because it is not possible to impinge an incident beam onto the crack face with a sufficiently small angle of incidence using a transmitting probe located within the top boundary of the weld, a 451 angle-beam P probe (T) is located in the parent medium region as shown in Fig. 13(a), i.e. at (x2, x3)¼ (360h, 150h), for a minimum angle of incidence. In this setup, as shown by the thick dashed-line center ray in Fig. 13(a),

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the ultrasonic P wave emitted from the transmitting probe goes through a complex process before received by a 601 angle-beam SV probe (R) located at (150h, 150h). Fig. 13(b) through (e) shows the corresponding, simulated wavefields at four important sequential instants. In Fig. 13(b), the incident P wave in red transmits into the weld region while being refracted; in Fig. 13(c), the refracted qP wave is being reflected from the crack and, as a result, a mode-converted (purple) qSV wave is generated; in Fig. 13(d), the reflected qSV wave is about to encounter the left boundary of the weld; and, in Fig. 13(e), the transmitted and refracted SV wave is reflected from the bottom boundary of the test object, headed toward the receiving probe. The corresponding, simulated output voltage signal is shown in Fig. 13(f), where the flaw signal is clearly seen. Thus, it is confirmed that the ultrasonic test setup shown in Fig. 13(a) can effectively detect the centerline crack.

6. Conclusions In this work, a numerical simulator has been developed based on the rectangular mass-spring lattice model (RMSLM), for the purpose of simulating ultrasonic tests of austenitic welds that are assumed to be homogeneous and transversely isotropic. To this end, the capability of the RMSLM has first been confirmed to accurately simulate fundamental behaviors of ultrasonic waves, e.g. generation, reflection, and diffraction of planar waves, in such a homogeneous and anisotropic austenitic weld. Then, numerical models have been developed for transmitting probes that are tailored to the elastic properties of the austenitic weld such that incident waves of desired modes can be sent in desired directions. By combining the RMSLM and the probe models, a complete numerical simulator has been developed. The simulator then was used to simulate four ultrasonic testing problems involving frequently encountered anomalies in austenitic welds. As a result of the numerical simulations, effective test setups were found for each of the simulated problems, and the capability and usefulness of the numerical simulator developed have been confirmed. Limitations of the model developed in this study result from the assumptions made, such as material inhomogeneity of the austenitic weld medium and the absence of wave absorption and scattering by the weld material. Consequently, the model cannot precisely simulate the curvature of wave paths inside the inhomogeneous austenitic weld and the material-originated wave attenuation and noises. Also, the two-dimensionality of the model poses another limitation on its applicability. Thus, one immediately obvious future work is to develop a numerical model that can model the inhomogeneity in the austenitic weld, i.e. the gradually changing orientation of the axis of material symmetry. Therefore, it is of significant interest whether the RMSLM used in this work may be extended or modified to model such inhomogeneity. This issue may be addressed by first dividing the austenitic weld into small regions, which are each homogeneous but differ from one another in the orientation of material symmetry axis, as in Ref. [12]. Then, the question of applicability of the RMSLM becomes whether the spring constants of the model can be adjusted such that it can simulate a transversely isotropic medium with an arbitrary orientation of the axis of material symmetry. When the axis of material symmetry for a transversely isotropic material is rotated, the associated elastic constants Cijlm change accordingly, and the number of nonzero, independent Cijlms exceeds five, beyond the present modeling capability of the RMSLM. In order to simulate such an elastic solid, the RMSLM must be modified by adding more springs of independent spring constants to match the increased number of independent elastic constants. A preliminary

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study has revealed that for a special case of the 451-rotated axis of material symmetry, the eight independent elastic constants for the medium may be simulated by adding four more rotational springs to the RMSLM. Cases of arbitrary orientations of the material symmetry axis require further study. Incorporation of dashpots [29] into the numerical simulator will be another promising topic for future work, which will enable the consideration of the effects of wave absorption by the material. Another topic for future work is the seemingly straightforward development of a three-dimensional version of the massspring lattice model. This development is more viable now than before because the ever-increasing capability of computers can more easily deal with the huge increase of computational load needed for three-dimensional simulation. In spite of all the limitations addressed above, the numerical simulator developed in this work is easy to learn and implement and, furthermore, is highly efficient in computation time. Therefore, it is expected to help render the ultrasonic testing of austenitic welds, considered as a difficult problem for long, much more tractable.

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