We intend to numerically simulate and visualize, in 2-D and 3-D dimensions, the characteristics of the deviation of energy waves from the wave normal in an anisotropic media. Theoretically, the direction of energy wave propagating in anisotropic media is the eigen-vector solution of the Christoffel's equation. And the corresponding eigen-value is the wave phase speed. In a general 3-D space, the energy wave propagates in one of three directions. These three directions (also call the principal directions) correspond to three types of waves. They are quasi-longitudinal (QL) waves, quasi-transverse (QT) waves and transverse (T) waves. While in 2-D space, QL and QT waves propagate. Therefore in 2-D and 3-D problem, there are two and three pairs of eigen-solutions respectively. The phase velocity, by definition, is the propagating wave speed in the wave normal direction. While the group velocity is the wave speed in the direction in which planar wave energy propagates in the media. All three waves types have its corresponding wave phase velocity. Generally the magnitude of wave phase velocity is smaller than the wave group velocity. The fact that the wave energy tend to propagate in the eigen-vector direction in the anisotropic media are itself an intrinsic property of anisotropic material.

Here we simulate the numerical version of these same characteristics by employing FEM wave propagation models in an anisotropic material. The FEM program we employ has been proven its correctness in an isotropic material by testing the transmitting and reflecting of elastic waves using Newmark method. Both 2-D and 3-D versions are tested and proven correct by comparing the FEM numerical results with those obtained by solving the 1-D wave equation (partial differential equation) using Laplace transformation. And both force and displacement boundary condition on the wave launching elements are employed. The simulations are carried out on Convex C3880 in NCSA supercomputing facility. Visualizations of the FEM numerical results are performed on PV-wave Commercial program and then converted to quick time movie in Macintosh platform. The three simulations followed can be regarded as the highlight of energy wave deviation properties in anisotropic materials.

1. Simulation of 2-D elastic waves propagation and visualization (QT / MPG ) in a deformed grid mesh.

A 2-D FEM mesh is built for an anisotropic material domain. The mesh consists of 8100 (45*180) 4-node linear quad-elements. The boundary conditions are such that only one side of the mesh is fixed. A 9-elements transducer region is chosen on the middle bottom of the mesh. A displacement boundary condition is prescribed on the transducer elements and 40 time steps are used to simulate a sine wave on the transducer. The wave normal is the positive y-direction. We intend to generate a single sine pulse and visualize the deviation, from wave normal, of QL and QT waves. In addition, Rayleigh waves (surface waves), reflecting waves from the fixed wall, wave diffraction phenomenon are also expected to be present in the simulation.

As the visualization shows, the energy wave deviation are clearly seen along the principal directions, though the QL energy wave is more obvious than that of the QT wave. Along the edge of the QL waves, there are diffraction phenomenon occurring 'in the wake'. Further, the Rayleigh surface waves are clearly seen on the top side of the rectangle, and a reflecting wave is seen interacting on the left side of QL waves.

2. Simulation of 2-D energy waves propagation and visualization in an intensity varying animation.

Spherical wave models have been studied less intensively than planar waves models. Here we attempt to simulate spherical waves properties in anisotropic media. A 2-D spherical media has been constructed by 10800 linear quadrilateral elements. To avoid employing triangular elements, the center of the media is left empty. There are 60 "shells", circular ring of elements, from the inner most layer to the outer most boundary, where the displacement boundary condition is fixed. The isotropic region is composed of 20 shells in the inner part of media, the others are anisotropic.

There are two reasons for employing different regions in the media. One of them is that we can observe and study the characteristic discrepancy of waves propagating in isotropic and anisotropic region. The other reason is that of avoiding the complexity of generating an initial uniform disturbance near the center of the anisotropic region. By generating the disturbances from the inner shell (isotropic region) of the media, we can easily employ the sine waves by numerical design. Besides, the energy waves can be carried through the isotropic region to anisotropic region retaining the energy uniformity on the first shell of anisotropic region.

We can observe the uniform propagation of energy in the isotropic region as expected (energy propagation animation QT / MPG). As these waves flow across the boundary into the anisotropic region, the "focusing " phenomenon becomes pronounced. Similar results have been experimentally observed [Duke, 1989]. Note the wave pattern is unmistakably symmetric which corresponds to the symmetric stiffness property in the 4 quadrants. Note also that the secondary waves reflecting back from the center is again focusing symmetrically in the anisotropic . This numerical experiment, in addition to the demonstration of the symmetric anisotropic stiffness property can lead us to a better understanding of spherical waves characteristics in anisotropic material. Also, we can draw the conclusion that, in anisotropic material, the energy flow tend to focus along the direction corresponding to the highest stiffness.

3. Simulation of 3-D energy waves propagation and visualization (QT / MPG ) in an intensity varying animation.

The 3-D mesh is consisted of 8000 (20*20*20) 8-node linear brick elements. As in 2-D case, only one side of the mesh is fixed. The remaining sides (including the top and bottom sides) are all freely displaced in x, y, and z direction. The transducer elements are chosen as a 10*10 area on the center bottom of the mesh. The prescribed sine displacement wave are such that the wave normal is along the positive z-axis direction and the displacement direction of the transducer elements are also along the z-axis. We only simulate a half pulse sine wave to limit, visually, the energy pack spreading in the 3-D domain. Since the computational cost for 3-D problems is expected to be high, obtaining the deviation phenomenon from the visualization will satisfy our purpose. Since we expect the difficulties of visualizing a 3-D domain, a rotating coordinate system is displaced in the animation to better distinguish the bifurcated beam of energy waves.

The animation is successful enough to display the deviation of QL and QT energy wave pack in the 3-D domain. Though the bifurcation of the wave pack is only observed on the last stage of the quick time movie. The pure T (transverse) wave can also been seen whose propagation path also deviates from the wave normal and bifurcates from the QL and QT waves.

1. M. J. P. Musgrave, Crystal Acoustics, Holden-Day, San Francisco, 1970.

2. J. C. Duke, Jr. , E. G. Henneke II, M. T. Kiernan, and P. P. Grosskopf, A Study of the Stress Wave Factor Technique for Evaluation of Composite Material, NASA Contractor Report 4195.