Eduardo Kausel-Fundamental solutions in elastodynamics_ a compendium-Cambridge University Press (2006)

10 Stiffness matrix method for layered media
Closed-form solutions for layered media, or for homogeneous plates and strata with arbitrary
boundary conditions, do not exist. Indeed, even the free–free plate – the so-called
Mindlin plate – is ultimately intractable by purely analytical means. Thus, such problems
must be solved with the aid of numerical tools. Among these, a widely used scheme is
the propagator matrix or transfer matrix method of Haskell 1 and Thomson. 2 Nonetheless,
we choose instead to present herein the related stiffness or impedance matrix method,
which has the advantages over the propagator matrix method that are listed below and,
at least in our judgment, no disadvantages in comparison with the latter. Thus, readers
familiar with the propagator method are strongly encouraged to familiarize themselves
with and switch to this superior method. Among the advantages of the stiffness matrix
method are:
Stiffness matrices are symmetric, while propagator matrices are not.
Stiffness matrices involve half as many degrees of freedom as propagator matrices, so
their bandwidth is only half as large. The former involve only displacements, whereas
the state vector in propagator matrices contains both stresses and displacements.
On account of the two previous items, the stiffness matrix method is nearly an order
of magnitude faster than the propagator matrix method: The computational effort
is smaller by a factor 2 due to symmetry, and a factor of more than 4 on account of
bandwidth, which gives a total reduction of more than 8. Since the computations must
be repeated for each frequency and each wavenumber, the savings are considerable.
Stiffness matrices remain robust and stable for thick layers and/or high frequencies.
In these situations, layer interfaces decouple naturally as a result of coupling terms
tending to zero. Propagator matrices, on the other hand, contain terms of exponential
growth that require special treatment.
Stiffness matrices lead naturally to the solutions for normal modes (eigenvalue problems),
source problems, and wave amplification problems, all without the need for
1 Haskell, N. A., 1964, The dispersion of surface waves on multilayered media, Bulletin of the Seismological
SocietyofAmerica, Vol. 43, pp. 17–34.
2 Thomson, W. T., 1950, Transmission of elastic waves through a stratified soil medium, Journal of Applied
Physics, Vol. 21, pp. 98–93.
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