FIFTH CANADIAN CONFERENCE ON NONDESTRUCTIVE ... - IAEA

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At Chalk River, we have concentrated on modelling the physics of an ultrasonic
pulse inside the object being tested and not on the detailed modelling of
specific transducers. Pulses of both shear and compressional types can be
input at arbitrary angles to free surfaces and absorbing boundaries. This is
achieved at free surfaces by applying stresses and at absorbing boundaries by
specifying a combination of stress and displacement.
9. COMPUTATIONAL COSTS
Computer simulation of the Elastic Wave Equation is limited on any computing
device by two main factors: the availability of fast access storage; and the
cost of executing the program. The execution cost is simply the cost of
performing the arithmetic required to calculate successive time levels of the
solution. Past storage and retrieval of the values of the approximate solution
at the two preceding time levels is important because of the large amount of
data _nvolved and the fact that it must be done every timestep. Information
storage and retrieval becomes very expensive if it cannot be done in the
machine's central memory (or equivalent) and alternative storage, such as disk,
is used.
Execution and storage costs are related to the number of grid points used in
the construction of the finite difference approximation. To illustrate this we
consider an approximation in which the number of grid points for each of the
space dimensions is directly proportional to N (a positive integer). The total
number of space grid points is then proportional to N to the power D, where D
is the number of space dimensions. The amount of storage required for the
approximation and the number of arithmetic operations per timestep are
multiples of the number of space points. The number of timesteps is proportional
to N and so the total number of arithmetic operations is a multiple of N
to the power (D+l).
The number of space points required to approximate the solution to a given
accuracy depends on the physics of the problem. Probably the most significant
factor is the ratio of the wavelength of the ultrasonic pulse to the length
scale of the object it is traversing. The number of subdivisions in each space
dimension is inversely proportional to this ratio, making it more expensive to
model shorter wavelength pulses in the same object.
To achieve reasonable accuracy in typical problems encountered at Chalk River
it was necessary to use values of N of at least 100. To model all the problems
we have considered so far, values of N of 400 or more would be required. The
storage and execution costs of two-dimensional modelling of the easier problems
severely stretched the capabilities of our CYBER 175, and difficult
two-dimensional problems would stretch most Supercomputers. General threedimensional
modelling of most problems is certainly not possible with current
technology; the execution and storage costs are prohibitive on any existing
machine.