Introduction
The simulation of physical phenomena has been much simplified and extended by the use of numerical methods, which avoid limitations and
simplifying assumptions frequently inherent in analytical solutions of mathematical representations. There are many ways in which this can be done.
The equations can be solved by replacing integrals and derivatives by finite sums and finite differences. An alternative strategy involves the replacement
of the equations by analogue models, which express the same behavior, on the basis that these may be easier to solve numerically in particular circumstances. Perhaps the best-known example is the equivalent electrical network. The use of electrical network models in mechanics is well established.
There are direct analogues between springs, masses, and dampers on one side and capacitors, inductors, and resistors on the other. The solution to the
mechanical problem can then be obtained using conventional circuit analysis techniques with results in either the time or frequency domains. As will be
seen, in the case of transmission line matrix (TLM), the equivalent electrical analogue has the further major advantage that it leads directly to a simple
and natural numerical discretization scheme.
There is a relatively new time-domain modeling technique, called cellular automaton (CA) modeling. Particles, which may represent, for example,
concentration, amplitude, or population of a species are distributed on a mesh, which, in two dimensions, may be a Cartesian or hexagonal grid.
These are then subjected to the repeated application of a simple set of rules and the evolving behavior is monitored. With the right set of rules it may
be possible to define a CA system whose behavior closely parallels that of the physical problem of interest. In many instances the set of rules may
appear to have no obvious physical basis and, perhaps because of this,researchers in this area have worked hard at providing a good theoretical
foundation for their subject.
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