FINITE ELEMENT MODELING OF TRANSIENT WAVE PHENOMENA AT

FINITE ELEMENT MODELING OF TRANSIENT WAVE PHENOMENA AT SOLIDIFLUID INTERFACES T. Xue, W. Lord, S. Udpa, L. Udpa and M. Mina Department of Electrical and Computer Engineering Iowa State University Ames, IA 500 11 INTRODUCTION The solid/fluid interface appears in many ultrasonic measurement systems. Models for the system must take account of the interface. Analytical models for wave phenomena at the interface (especially curved interfaces) are either difficult or subject to severe approximation. The finite element method is ideal for this especially when the problem domain is bounded. A survey of this subject has been given by Kalinowski [1]. In this paper, an axisymmetric finite element model is developed for a solid medium and a fluid medium in contact. Displacement is used as the primary variable in the solid media and pressure in the fluid. The scalar pressure in the fluid medium makes the total degrees of freedom less than if displacement is used. The global mass matrix and stiffness matrix are rendered symmetric by introducing a potential variable for the fluid medium [2]. The final finite element equations are solved by the explicit integration approach. FINITE ELEMENT MODELING Solid Medium The governing equation for an isotropic medium with neglected body force and viscous damping is as follows Upon discretization of the problem domain to quadrilateral isoparametric elements, the following matrix equation is derived using either the weighted residual or energy functional approach [3-6] (2) where K is the global stiffness matrix, M is the global mass matrix, F incorporates the traction boundary condition, and U is the displacement vector. (1) Review o/progress in Quantitative Nondestructive Evaluation, Vol. 15 Edited by D.O, Thompson and D.E. Chimenti, Plenum Press, New York, 1996 299

Fluid Medium The governing wave equation for pressure is as follows: n2 1.. v P=-P c 2 (3) where p=pressure and c=l wave velocity. Eq. (3) can be derived from Eq. (1) provided that A. =-11 (4) with a further assumption that U r = O. Then Uz stands for the pressure field. It can be shown that in the pressure analogy, the z component of surface traction becomes the product of the fluid bulk modulus and the normal derivative of the pressure. The finite element equations assume the following form M/p+KfP=f where f = Apc 2 ap / an. Note that from now on p stands for the fluid density. (5) SolidlFluid Coupling At the interface, the force applied to the solid by the fluid is - pa, while the force applied to the fluid by the solid is A(pC)2 Un' then the couple equation is as follows (6) Where A is an area matrix associated with the interfacing nodes. Let p = -q, we have t where G = -J fdt o In a more compact form, (8) Direct Time Integration Using a central difference approximation for the second derivatives and a backward difference approximation for the first derivatives, we have 1 q ( J...D-~M)Uq (J... D 1 M)Uq (9)!l.t2 MUt+flt - F, K + /).t!l.t2 t +!l.t /).t2 t-flt 300

O.3cm Front surface interface I --r z c=1500 mj 0=1000 k2/m V1=6300 mj V.=3100 mj O.7cm p=2700 kglm 1 Back surface 2cm Fig. 1. Axiymmetric geometry containing a solid/fluid interface. The mass matrix is diagonalized through a mass lumping technique, then Stability Conditions The explicit central difference scheme is conditionally stable. Then the integration time step is restricted for numerical stability by the following critical value Mer =2/wmax, r" "'max < - \l:g I 1/2 (11) where g is a geometric parameter. For a square element with sides h, the condition is which is valid for the lumped mass matrix. h M::;;- V; (12) RESULTS AND DISCUSSION The basic axisymmetric geometry shown in Fig. 1 is modeled using the finite element procedure presented above. The fluid and solid media are taken as water and aluminum respectively and their mechanical properties are indicated in the figure. A raised cosine function [3-6] with a center frequency of 5 MHz is used as the driving signal, while the source location and structure are varied for a few cases. In each case the source is also symmetric about the z-axis. As the first case, we assume a point source inside the solid medium 0.2 cm from the interface. The driving force of the source is along the axial direction. The wave profiles at different time instants are shown in Fig. 2. Note from now on that in each profile plot the solid side is represented by the z-displacement while the fluid side by the potential which is normalized such that the magnitude is on the same order as the z-displacement in the solid. Fig. 2(a) indicates that the reflection and refraction of the longitudinal wave front have occurred at the interface and the shear wave front from the source has reached the interface. In Fig. 2(b) the reflection and refraction of the shear wave as well as the reflection of the 301

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(a) (b) /.Ls, and (c)!::4 /.Ls. F;g. 3. Wave profile, fo, a POint -Jil<" 'ou"", on the flu;d,"nace, (aj M. 75 ~', (b J /;3.25 (C) 303

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(a) (b) Fig. 5. Wave Profiles fo, a 'ph",;oa/ly foou,," wave from 'he fluid me<fiwn '0 the 'Olid medium, 9_>9" (a) "'3.5 ~', (b) "'4 ~', and (0) "'5 ~,. (c) 305

longitudinal wave from the back surface are shown. Each reflection associated with the solid medium is accompanied by a mode conversion. Fig. 2(c) records the various wave fronts at a later time. As a second case, we consider a point-like source, i.e., a small finite aperture, on the front surface of the fluid medium. The aperture diameter is 2a=0.01 cm, which corresponds to ka=1t/3. The wave profiles at three different time instants are shown in Fig. 3. It can been seen that after the longitudinal wave front reaches the interface, both reflection and refraction occur. Among the refracted waves, we can identify the longitudinal, shear, head and interface Rayleigh waves. A further point to note is that as the interface Rayleigh wave propagates, it radiates (or leaks) waves (at the Rayleigh angle) back to the fluid medium. By examining the time dependent signals at points along the interface, the interface Rayleigh velocity is estimated to be VR=2940 mis, which is slightly greater than the corresponding Rayleigh wave velocity on a traction free surface. As a third case, assume a spherically focused wave from an aperture with diameter of 1.5 cm and focal length of 2 cm. The wave profiles are shown in Fig. 4. From Fig. 4(a) we see the reflected longitudinal wave and the refracted shear wave. At that time instant a longitudinal wave has not been refracted into the solid medium. This is expected as the maximum angle of incidence is between the critical angles for longitudinal and shear waves (9max =20.56,91=13.77, 9s=28.94 ). An interface Rayleigh wave does not exist as the incident angle is always smaller than the critical Rayleigh angle which is estimated to be 9 R=30.66. As the incident wave proceeds as shown in Figs. 4(b) and (c), the incident angle becomes smaller and then a focused longitudinal wave is transmitted into the solid medium. It is defocused after passing the focal point. As a final case, consider a spherically focused wave similar to the previous wave except that the focal length is 1 cm. The corresponding wave profiles are shown in Fig. 5. An essential difference is that the maximum angle of incidence (9max=36.87 ) is greater than all the three critical angles. As the incident wave proceeds, the interface Rayleigh wave, shear wave, longitudinal wave and head wave gradually appear. The head wave on the fluid side radiated from the interface Rayleigh wave is clearly indicated especially when the interface wave passes through the axis of symmetry. CONCLUSIONS Transient wave propagation phenomena associated with solid/fluid interfaces have been presented using the finite element method. Various wave profiles are predicted based on different source conditions. In this study, the longitudinal wave velocity in the fluid medium is smaller than the Rayleigh wave velocity for the solid medium. The results indicate that an interface Rayleigh wave is generated when an incident wave from the fluid medium matches the critical Rayleigh angle, and a propagating interface Rayleigh wave radiates waves back into the fluid medium. Although only planar interfaces are considered here, the extension to arbitrarily curved interfaces is trivial for the finite element method with the expense of some extra computer resources. REFERENCES 1. A. 1. Kalinowski, in Shock and Vibration Computer Programs, Reviews and Summaries, Edited by W. Pikley and B. Pikley, (1975), pp 405-452. 2 G. C. Everstine, J. Sound and Vibration, 79, (1981), pp 157-160. 3. R. Ludwig. and W. Lord, IEEE Trans. UFFC, 35, (1988), pp 809-20. 4. R. Ludwig. and W. Lord, IEEE Trans. UFFC, 36, (1989), pp 342-50. 5. W. Lord, R. Ludwig and Z. You, J. NDE, 9, (1990), pp 129-43. 6. Z. You, M. Lust, R. Ludwig and W. Lord, IEEE Trans. UFFC 38, (1991), pp 436-45. 306