README_FEM_dispersion

When using finite elements to model wave propagation, dispersion 
is observed when wave velocity is a function of the number of 
elements contained within the wave length of a propagating wave. 

Here we study dispersion in unidirectional graphite/epoxy, which 
is hexagonal and highly anisotropic. Our ultimate objective is 
to study the effect of anisotropy on the flux deviation (group
velocity) propagation direction as a function of different fiber
orientations, 0 -> 90 degrees. Based on prior research, transverse
(T) and quasi-transverse (QT) wave speeds and propagation direction 
are observed to be the most sensitive to small changes in anisotropy.
In general the actual wave "group" velocity vector, energy flux, 
deviates from the wave "phase" velocity, which propagates normal to 
the plane wave front. Between 0 and 90 degrees, group velocity 
deviates from the phase velocity and the pure transverse (T) wave 
becomes a quasi-transverse (QT) wave. At 0 and 90 degree fiber 
orientations the group and phase velocities are the same. This case
is used here for simplicity to study dispersion of pure transverse
(T) waves before we study dispersion at various fiber orientations. 

In the accompanying figure, WhatToDo.jpg, six different cases are
outlined for a 30x60 FEM mesh. This small mesh size allows us to 
simulate many different cases of interest in a reasonable time 
using desktop computers. Increasing the number of elements at the
boundary transducer results in an increase in the number of 
elements contained in the propagating T waves. For 12, 16, 20, 
and 24 elements per transducer wave length, we observe 3, 4, 5, 
and 6 elements respectively contained with in the propagating T 
wave. We also observe increasing wave velocity with increasing 
number of elements per wave length contained in the propagating T 
wave. This is expected since larger number of elements contained 
in a propagating wave approach the ideal case of a wave propagating 
in a nondispersive continuum. With only 4 elements per transducer 
wave length the propagating T wave is alter by the FEM mesh, that 
is the group velocity is zero, which demonstrates significant 
dispersion. Dispersion exists even at 24 elements per transducer 
wavelength. However 24 elements per transducer wave length will 
require larger FEM meshes where QT waves are predicted to deviate 
as much as 50 degrees to the left. As in most cases when modeling
complex phenomena simplifying approximations are necessary and 
results must be interpreted within these approximations. Here 20 
elements per transducer wave length results in a trade-off between 
working with larger FEM meshes, e.g. 45x180, but with an acceptable 
degree of dispersion.  -- R.D. Kriz