To ensure safe operation, nuclear power plants must be inspected periodically. One of the most commonly used nondestructive inspection methods uses ultrasound to detect internal flaws. However, the complex structure of some joints between different components greatly complicates the ultrasonic inspection. These joints, which consist of welds with varying and sometimes anisotropic elastic properties, can distort the ultrasonic beam and produce unreliable results. To understand the propagation of elastic waves through such materials, beam models are used

Minachi, A.

Mould, J.

Thompson, R. Bruce

English

To ensure safe operation, nuclear power plants must be inspected periodically. One of the most commonly used nondestructive inspection methods uses ultrasound to detect internal flaws. However, the complex structure of some joints between different components greatly complicates the ultrasonic inspection. These joints, which consist of welds with varying and sometimes anisotropic elastic properties, can distort the ultrasonic beam and produce unreliable results. To understand the propagation of elastic waves through such materials, beam models are used.</p

Thompson, R.

Digital Repository @ Iowa State University (ISU)

ULTRASONIC BEAM PROPAGATION THROUGH A BIMETALLIC WELD - A COMPARISON OF PREDICTIONS OF THE GAUSS-HERMITE BEAM MODEL AND FINITE ELEMENT METHOD A. Minachi*, J. Mould** and R. B. Thompson* *Center for Nondestructive Evaluation Iowa State University Ames, Iowa 50011 **Weidlinger Associates Los Altos, CA 94022 INTRODUCTION To ensure safe operation, nuclear power plants must be inspected periodically. One of the most commonly used nondestructive inspection methods uses ultrasound to detect internal flaws. However, the complex structure of some joints between different compo-nents greatly complicates the ultrasonic inspection. These joints, which consist of welds with varying and sometimes anisotropic elastic properties, can distort the ultrasonic beam and produce unreliable results. To understand the propagation of elastic waves through such materials, beam models are used. Two of the existing models used for the simulation of elastic wave propagation are the finite difference and finite element methods [1-2]. These methods have good accuracy and can be applied to wide variety of geometrical and material conditions. However, they are rather computationally intense. A few years ago, an approximate Gauss-Hermite model was developed [3] to predict the propagation of ultrasonic waves through isotropic and anisotropic materials. This model has the advantage of being computationally fast and simple. Although the Gauss-Hermite model incorporates beam spreading and can predict the radiation by focused and planar transducers and subsequent propagation of the beam through both plane and curved interfaces, it has limitations due to the use of the Fresnel approximation. In previous work, predictions of the Gauss-Hermite model have been compared to those of the finite element model [4]. In that study, a rather simplistic layered medium was considered in which the beam propagated from a contacting strip transducer into a ferritic steel layer and through a planar interface at normal incidence into a columnar, stainless steel layer. The comparison between the two models showed an excellent agreement in the vicinity of the beam center with disagreements developing as one moved away. These deviations were interpreted as being a consequence of the Fresnel approximation. In this paper a more complex problem is investigated. The predictions of the Gauss-Hermite model and the finite element method are compared for ultrasonic beam propaga-tion through a bimetallic weld at an oblique angle. Both models consider the generation of the beam by a strip, piston transducer producing the same initial ultrasonic pulse shape. Review of Progress in Quantitative Nondestructive Evaluation, Vol. 12 Edited by D.O. Thompson and D.E. Chimenti. Plenum Press. New York. 1993 911 ULTRASONIC BEAM MODELS Finite Element Method The finite element method has been exhaustively described and analysed in the literature [5-6]. The basic idea behind finite elements is to spatially discretize the partial differential equations governing elastic wave propagation, reducing them to a system of ordinary differential equations in time. These are integrated with a finite difference scheme. For this study, we used FLEX, an explicit finite element code [7]. FLEX consists of distinct ID, 2D and 3D processors developed to maximize computational efficiency across a wide range of computers including work stations, vector supercomputers and massively parallel machines such as the Intel IPSC 860. An explicit, central difference time integra-tion algorithm is used in conjunction with isoparametric, constant strain elements and lumped masses. This eliminates the need for storing a global stiffness matrix and for mass matrix inversion. Instead, nodal forces are accumulated element-by-element and the nodal velocity increments are calculated from Newton's law in incremental form. Nodal forces are computed with single point integration of stresses over each element for maximum efficiency. The algorithm is stable provided the time step is less than the fastest wave's travel time across the smallest element. Transmitting boundary conditions based on the work of Lysmer are incorporated. Different time integration steps are permitted in differ-ent regions of the grid which can significantly increase the efficiency of the solution. Gauss-Hermite Model More detailed discussion on the Gauss-Hermite beam model can be found in several publications [3, 8-10]. The formulation initiates by representing the beam as an angular spectrum of the plane waves. By employing the Fresnel approximation, certain integrals over spatial frequency can be solved analytically. In this approximation one assumes that the range of radiation angles contained in the ultrasonic beam is restricted to a sufficiently narrow range. Thus the slowness surface can be represented by a Taylor series expansion in the vicinity of the propagation direction. For a two-dimensional problem, this has the form (1) where (kim) is the slowness or inverse velocity. Also, So is the slowness along the central ray, and kx is the transverse component of the wave vector in a coordinate system which has its z-axis along the central ray. A and C are the constants that are related to the deriva-tive of the slowness with respect to k . These constants (A and C) play an important role in the Gauss- Hermite formulation. t.. defines the slope of the slowness surface with respect to the central ray. As is known, the energy travels along the normal to the slowness surface; thus, A determines the beam skew in the formulation. Furthermore, C is related to the curvature of the slowness surface and simply determines the rate of beam spread in the Gauss-Hermite model. Within the Fresnel approximation, analytical forms can be derived governing the laws of propagation of certain wavepackets whose initial cross-sections can be described as products of Gaussian and Hermite polynomials depending on the transverse coordinate. Since these functions form a complete, orthogonal set, any beam can be represented as a superposition of them. Thus, the radiation of an ultrasonic source detected along z-axis can be represented in the two-dimensional case considered here by -;; (x, z) = L Cm -;;m(x, z) (2) m=<! 912 ---> where urn are the Gauss-Hennite eigenfunctions. Each of these eigenfunctions satisfies the wave equation, within the Fresnel approximation, and follows the principle of diffrac-tion. Moreover, their amplitudes, phases and widths depend only on z. The Gauss-Hennite complex constant coefficients, Cm: are computed by using the orthogonality property of the Gauss-Hennite eigenfunctions ano the initial radiation pattern of the source (where z=O). THE BIMETALLIC WELD MODEL The basic structure of the bimetallic weld was based on the paper by B. Nouailhas et al. [2]. Figure 1 shows the geometry of the bimetallic weld studied in this paper. The waves are generated by a two-dimensional (i.e strip) 2.5 MHz, 1.27 cm width transducer immersed in water and located 1 cm. below region (I) of the weld, with a 9.6 degree inci-dent angle. The transducer was assumed to act as a piston, producing a unifonn displace-ment over its surface. The input signal to the transducer was assumed to be a raised cosine. RESULTS The ultrasonic beam radiated from the transducer enters region (I), and passes through regions (II), (III) and (IV). Some observation points were selected in each region at which the predictions of the two models were compared. These observation points were selected according to the following procedure. First, the central ray (directed along the group velocity) was traced through each region. Second, at each interface, a local wavevector was constructed which left the interface at the same point as central ray. The observation points were placed on a line perpendicular to this local "phase path" extending 1 cm. on each side with a separation distance of 0.2 cm. The origin of the 1; coordinate system, measured on the perpendicular line, was taken on the local "phase path". Figure 1 shows the locations of observation points. Also, in Figure 1 the energy path of the central ray is indicated by a dashed line and the local "phase path" is shown by an arrow. The comparison at each region is as follows. Re~ion (II) In this isotropic region, the group and the phase velocities are in the same direction. Figures 2 shows the comparison of the two models. As can be seen from the comparisons, there is excellent agreement in this region, especially in the vicinity of the central ray. Although the Gauss-Hennite model predicts a symmetrical wave along the energy direc-tion, there is actually a slight asymmetry, revealed by the finite element method, due to the passage of the beam obliquely through interfaces and through region (I) (where the material is anisotropic). This is one of the limitations of the Gauss-Hennite model; it always pre-dicts a wavefonn that is symmetric about the central ray. The finite element results show later arriving signals due to the reflection of the beam from the boundaries (when ~ is positive). The Gauss-Hennite model can be adjusted to consider such reflections from planar or curved boundaries. However, this was not done in the present study. Re~ion (III) The material in this region is anisotropic, and the beam is skewed to the left of the phase direction. Figures 3 shows that the beam is peaked near ~=-0.2 cm. Although the agreement between the two models is not as good as in region (II), the general shapes of the wavefonns are very similar, with slight differences in amplitude. These appear to be associated with a slightly greater beam width predicted by the Gauss-Hennite model. One would expect the error in the Gauss-Hermite model to accumulate as the beam passes through the successive interfaces. This is mostly due to the deviation of the true transmis-sion laws from the approximate fonnula based on paraxial rays. The beams with higher incident and transmitted angles generate larger errors in the Gauss-Hennite model. 913 CD ~ y (cm) (0.0,6.0) (0.0,0.6) (0.0,0.0) (3 I (9.0,6.0) I ~ Q Cladding (Austenitic metal, anisotropic, dendrites parallel to y-axis) @ Ferritic steel (isotropic) ® Buttering (Austenitic metal, anisotropic, dendrites perpendicular to line AB) @ Ferritic steel Cll = C22 = C33 = 275.0 C12 = Cl3 = C23 = 113.2 C44 = Css = C66 = 80.9 P = 7900 Kg/m3 (20.2,6.0) ----. (20.2,0.0) x (cm) -Austenitic metal (11 = C22 = 262.7 C33= 216.0 (12 = 98. 2 Cl3 = C23 = 145.0 C44 = Css = 129.0 0;6 = 82.3 @ Weld (Austenitic metal, anisotropic, dendrites parallel to y-axis) 8 Ferritic steel (isotropic) Units for elastic constants are 1 cf N m-2 3 P = 8120 KWm Figure 1. Bimetallic Weld. ~ = 0.0 em ~ = 0.2 em 0.15,------=----------. .,.-----~---;=;F,..... "". ---.,,.,.--...,..., - mIte Element o Gauss-Hermite 0.05 -0.05 -0.15-+--........ --r--""T"'"-~--.,....--__j -t--..__--.,.---..--...,-----.---i 10 11 12 13 10 11 12 13 ~ = -0.4 em ~ = 0.6 em 0.15....------=----------, -r------"-----------. 0.05 -0.05 -0.15+---'---r--""T"'"-~--..----H--..--_.,._--.---...._-......... ---I 10 11 12 13 10 11 12 13 ~ = -0.8 em ~ = LO em 0.15....--------"---------, .,.--------::..---------, 0.05 -0.05 -0.15+---.---r--...,--~--..---_i -+--~-_.,._----...._-~---I 10 11 12 13 10 11 12 13 Time ().tsee) Figure 2. Comparison between finite element and the Gauss-Hermite models in region (II). 915 0.10-r-____ --'~'_=_0_._0 ____ ___, -r--------'-------":-:-:-----",,..---, - Finite Element 0.05 0.00 -0.05 o o o Gauss-Hermite -O.lO-+----,---"°'---r--...-----,.---.----++--..------O'""""T--...-----,.---.----l 14 15 16 17 14 15 16 17 ~ = 0.4 em ~ = -0.6 em 0.10...--____ --=-________ --, ...--_____ --=-_______ --, o 0.05 o O.OO-""_'OC>oC> -0.05 -0.10·+-_---,-__ ...--_---,-__ ...--_---,-_-1 -1-----,---.,.-----,----.------,-----1 14 15 16 17 14 15 16 17 ~ = 0.8 em ~ = -1.0 em 0.10...,-____ --='--_______ -----, ...--_____________ --, 0.05 o -0.05 CO -0.1O+-_~--_,_----_,_---___l -+-_--,-__ .,.-_---,-__ ...--_---,-_--\ 14 15 16 17 14 Time (Il-see) 15 16 17 Figure 3. Comparison between finite element and the Gauss-Hermite models in region (III). 916 ~ = 0.0 em ~ = -0.2 em 0.08....------.:.----------,.,.-------='---_--r;F::-in ..... it~e-;E,,-;I-em-e~nt 0 Gauss-Hermit 0.04 o -0.04 -0.08+_--..,.--,.----.----,,---...___--++--...___-_.--.---,_---.----t 17 18 19 2017 18 19 2 ~ = 0.4 em ~ = -0.6 em 0.08:..,.------.....:...----------,,-------'------------, -0.08+_---..--....------r--....------r-----1 +-----r---r-----.---r----..---i 17 18 19 20 17 18 19 20 ~ = 0.8 em ~ = -1.0 em 0.08:..,.-------=------------, ,-------'-------------, ~.08+_-_r--r_-~-~--...----~~--..---_.---,r--,_---.--~ 17 18 19 20 17 18 19 20 Time (Ilsee) Figure 4. Comparison between finite element and the Gauss-Hermite models in region (IV). 917 Region (IV) In this anisotropic region, the beam is skewed to the right of the phase direction. The skewing effect is not as great as region (III), and the beam is almost centered along the phase direction. Figure 4 shows the comparisons between the two models in this region. The agreement between the two models is generally good in the vicinity of the central ray, and again decreases as one moves away. It must be noted that, in the finite element predic-tions, the reflection of the waves from the top boundary of the weld have distorted the incoming wave near the edges (when ~ is negative). Due to the absence of any material beyond the top boundary of the weld, the amplitudes of these reflected waves are relatively large and can not be ignored. CONCLUSION The predictions of the Gauss-Hermite and finite element models are in good agree-ment in the vicinity of the central ray in all regions. The main disagreements are at the edges of the observation planes. This is mainly due to the paraxial approximation in the Gauss-Hermite formulation. The most advantageous feature of the Gauss-Hermite model is its computational speed. The CPU time for simulating wave propagation through the weld with the finite element model was 4.46 hours. On the other hand, the Gauss-Hermite model took about 5 seconds to predict the waves in each region. However, the finite element model has the advantage of being more complete. ACKNOWLEDGEMENT This work was sponsored by the Electric Power Research Institute under contract RP-2687-01 and was performed at the Ames Laboratory. Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under Contract No. W-7405-ENG-82. REFERENCES 1. A. H. Harker, J. A. Ogilvy and J. A. G. Temple, "Modeling Ultrasonic Inspection of Austenitic Weld," J. Nondestr. Eva!. Vol. 9 (1990): 155-165. 2. B. Nouailhas, G. V. Chi Nguyen, F. Pons and S. Vermersch, "Ultrasonic Modeling and Experiments: An Industrial Case: Bimetallic Weld in Nuclear Power Plants," L Nondestr. Eval. Vol. 9 (1990): 145-153. 3. B. P. Newberry and R. B. Thompson, "A Paraxial Theory for the Propagation of Ultrasonic Beam in Anisotropic Solid," J. Acoust. Soc. Amer. Vol. 85 (1989): 2290-2300. 4. A. Minachi, Z. You, R. B. Thompson and W. Lord, "Predictions of the Gauss-Hermite Beam Model and Finite Element Method For Ultrasonic Propagation Through Aniso-tropic Stainless Steel," to be submitted to IEEE Transactions on Ultrasonics. Ferro-electrics and Freguency Control. 5. T. J. R. Hughes, The Finite Element Method, Prentice Hall, Inc., Englewood Cliffs, N.J., (1987). 6. O. C. Zienkiewicz, The Finite Element Method, McGraw-Hill Book Co. Ltd., London, (1977). 7. D. K. Vaughan and E. Richardson, FLEX User's Manual Version I-H.l, Weidlinger Associates, Los Altos, Ca., (1990). 8. B. P. Newberry, A. Minachi and R. B. Thompson, "Ultrasonic Beam Propagation in Cast Stainless Steel," Review of Progress in ONDE Edited by D. O. Thompson and D. E. Chimenti, Plenum, New York, N.Y., Vol. 8B (1989): 2089-2096. 9. R. B. Thompson, T. A. Gray, J. H. Rose, V. G. Kogan and E. F. Lopes, "The Radiation of Elliptical and Bicylindrically Focused Piston Transducers," J. Acoust. Soc. Amer. Vol. 82 (1987): 1818-1828. 10. E. Cavanagh and B. D. Cook, "Lens in the Nearfield of a Circular Transducer: Gaussian-Laguerre Formulation," J. Acoust. Soc. Amer. Vol. 69 (1981): 345-351. 918 

Ultrasonic beam propagation through a bimetallic weld — a comparison for predictions of the gauss-hermite beam model and finite element method

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Ultrasonic beam propagation through a bimetallic weld — a comparison for predictions of the gauss-hermite beam model and finite element method

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