Page 279 - Numerical Analysis and Modelling in Geomechanics

260 THE 2-D CHEBYSHEV SPECTRAL ELEMENT METHOD
for the P-SV (plain strain) case. Here, u (x, z, t) and u(x, z, t)=(u , u ), f (x, z, t)
x
y
z
y
and f(x, z, t)=(f , f ), define the (horizontal) out-of-plane, and in-plane
x
z
components of the displacement and exciting force, respectively; ρ(x, z) is the
density, and λ(x, z) and µ(x, z) are Lamé’s constants of the medium. Equations (9.
1) and (9.2) are defined for (x, z)≥ Ω and t≥ [0, T], where Ω is a two-
dimensional, bounded, inhomogeneous medium, and [0, T] is a bounded time
interval.
The 2-D wave propagation problem is completed by the acoustic equation,
which describes the propagation of a 2-D pressure field. This is a scalar equation
which is formally similar to equation (9.1), and is written as:
(9.3)

where p(x, z, t) is the pressure field, p(x, z) is the density, c(x, z) is the wave
velocity, and f(x, z, t) is the source forcing term, which equals the divergence of
the body force divided by the density.
Equations (9.1–9.3) are completed with suitable boundary and initial
conditions for the unknown fields.
The SPEM starts writing equations (9.1–9.3) in an equivalent variational
formulation (Marfurt, 1984; Priolo et al., 1994). In all cases, the equivalent
problem is to find Ξ such that:

(9.4)

where is the space of all functions that vanish on the boundaries, and which,
together with their first derivatives, are square integrable over Ω. The functions w
(x, z) are called weight (or test) functions. The symbols a(•,•) and (•,•) denote
Ω
Ω
symmetric, bilinear forms, and are specified for each case.
In the P-SV case (equation (9.2)), Ξ=u, and
(9.5)

(9.6)

(9.7)

D is the differential operator, given by