Page 278 - Numerical Analysis and Modelling in Geomechanics

ENRICO PRIOLO 259
formulation of the seismic wave propagation equations. The computational
domain is discretised into an unstructured grid composed of irregular
quadrilateral elements. This property makes the SPEM particularly suitable to
compute numerically accurate solutions of the full wave equations in complex
media, which can be taken into account to the finest detail. The earthquake is
simulated following an approach that can be considered “global”, that is, all the
factors influencing the wave propagation—source, crustal heterogeneity, fine
details of the near-surface structure, and topography—are taken into account and
solved simultaneously.
In this chapter, the author reorganises and synthesises material from his last
ten years’ work. The chapter is organised into four main sections. First, the
author summarises the basic theory, the spectral element numerical solution, and
focuses on some implementation issues. The method’s effectiveness in dealing
with real applications is illustrated through the description of two case studies:
the strong ground motion estimation in Catania (Sicily, Italy) for a catastrophic
earthquake, and the study of the influence of a massive structure on the nearby
ground motion.

Mathematical formulation of the SPEM
The Chebyshev spectral element method (SPEM) is a high-order finite element
technique, which solves the variational formulation of the equation. The
computational domain is decomposed into non-overlapping quadrilateral
subdomains. In each subdomain, the solution of the variational problem is
expressed as a truncated expansion of Chebyshev orthogonal polynomials, as in
the spectral methods. This section describes the mathematical formulation and
the modelling algorithm.

Equations of motion
The equations of the linear elastodynamics, which govern the wave propagation,
split in 2-D into two uncoupled equations (Eringen and Suhubi, 1975), which
describe the in-plane (P-SV vector equation) and antiplane (SH scalar equation)
particle motion, respectively. In the differential formulation they are written in
the well-known form as (Marfurt, 1984):

(9.1)

for the SH case, and

(9.2)