FIFTH CANADIAN CONFERENCE ON NONDESTRUCTIVE ... - IAEA

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5. ADVANTAGES OF USING THE FIRST ORDER FORM
An important advantage of the first order form is that the most common boundary
conditions can be expressed without involving space derivatives of the solution
components. This makes numerical approximation at the boundaries much easier.
For example, the free surface (stress free) boundary condition involves only
two linear combinations of the three stresses. More general conditions on the
normal and tangential stress and displacement are equally easy to apply.
In general, the space domains of interest include a wide variety of shapes that
do not lend themselves to modelling on a Cartesian grid. Many cases are suited
to a polar coordinate system or require the use of a non-uniform grid. It is
generally much more difficult to modify the second order form of the equations
to use non-rectangular grids than it is to modify the first order form. The
modelling of inhomogeneous materials is also easier when the first order form
is used, and the modified set of equations which results is similar to the
modified equation set for a non-uniform grid. In both cases, the advantage of
using the first order form is due simply to the ease of manipulation of lower
order space derivatives.
Little is known about the properties of numerical approximations to the second
order form of the equations, but approximations of first order hyperbolic PDE's
have been investigated in great detail. The stability of difference
approximations to initial-boundary value problems, such as we are discussing,
is not easy to guarantee; results are given in [9] and [10] for the first order
equations. The stability of approximations to the second order form of the
Elastic Wave Equation has only been investigated experimentally and should not
be assumed to hold in all cases.
6. LIMITING THE COMPUTATIONAL AREA
In most cases, it is not necessary to model the whole object being tested. One
can safely ignore the parts that cannot contribute to the response signal at
the transducer during the time period of interest. When modelling the testing
of weld joints in pipelines, for instance, there is no need to consider more
than the small region that includes the transducer and weld. Any signal
spilling out of this region is effectively lost down the pipeline.
To limit the extent of the computational region, an artificial boundary is used
to truncate the object and absorb the energy of any disturbance reaching it.
Ideally, there should be no reflected disturbance at this boundary, and waves
should appear to pass through it undistorted. It is possible to impose
perfectly absorbing boundary conditions for hyperbolic systems in one
dimension, but it is usually impossible to do so for multi-dimensional
problems. A study of absorbing boundary conditions for wave equations can be
found in Clayton and Enquist [11].
The simplest absorbing boundary conditions have been used in our numerical
model. They are designed to absorb waves travelling perpendicularly to the
boundary perfectly, but have proved to be most satisfactory for waves of all
orientations tried so far. In fact, the energy reflected by this artificial
boundary usually amounts to only a few percent of the energy reaching it.