Page 286 - Numerical Analysis and Modelling in Geomechanics

ENRICO PRIOLO 267
Recent developments
The kernel of the 2-D SPEM, as described in this paper, goes back to the early
1990s. Afterwards, however, the approach underwent some noteworthy
improvements. Following a similar approach, that is a high-order finite element
formulation, Komatitsch and Vilotte (1998) developed a method in both 2-D and
3-D, using Legendre polynomials as interpolants.
Faccioli et al. (1996) followed a somewhat different approach, and developed
a code that is more suitable for engineering applications. Here, the spectral
elements are very large and are connected by a domain decomposition technique
based on a variational principle. In each element, the wave equation is solved by
a global pseudo-spectral Fourier or Fourier-Legendre method. The method
solves both 2-D and 3-D cases, can also handle non-linear soil behaviour, and
allows for the simultaneous use of spectral and classical low-order finite
elements.
The Chebyshev SPEM itself is still being developed. Seriani (1997, 1998)
implemented the element-by-element technique into the Chebyshev SPEM for
the acoustic equation. Here, the matrix coefficients are computed on the fly at
each time step while solving the linear system, thus avoiding global matrix
assembly. Seriani and Priolo (2000) introduced new heterogeneous elements,
which account for medium variations inside the element itself. The advantages
are that a finely heterogeneous medium can be described in a finer way than the
problem solution, and the medium heterogeneity can be represented numerically
by the most appropriate shape function and polynomial order. Laurenzano and
Priolo (2001) are working on a simplified construction of the computational
mesh for a complex structure, based on few control points and geometrical
constraints, and an optimal adaptation of the mesh size to the medium properties.
As a consequence, the applicability of the 2-D Chebyshev spectral element
method to geo-problems is improved.
All the above methods are also implemented for parallel computers.

Method demonstration: two case histories
This section illustrates two real applications that have been tackled by the 2-D
SPEM. The first concerns the construction of a detailed scenario of ground
motion in Catania (Sicily, Italy) for a catastrophic earthquake. The second aims
at estimating, through a parametric approach, how and how much the presence of
a massive structure built at ground surface (e.g. an embankment or an earth dam)
may influence the ground motion.

Ground shaking scenario in Catania (Sicily, Italy)
This study was developed within The Catania Project, a three-year national
research programme funded by the National Research Council-National Group