Page 285 - Numerical Analysis and Modelling in Geomechanics

266 THE 2-D CHEBYSHEV SPECTRAL ELEMENT METHOD

Figure 9.1 Example of a quadrangular mesh taken from a study performed near the
village of Mels (Udine, Italy). The picture clearly displays how the grid adapts both in
shape, to fit the near surface structure and surface topography, and size, to fit the shear
wave velocity locally (from Siro (2001)). The small cross marks within each element
show the internal nodes (a total number of 25, corresponding to a polynomial order N=6).

triangles are eliminated by splitting triangles into (three) quadrangles, and
quadrangles into (four) quadrangles; finally, the mesh is regularised.
The main advantages of using the SPEM are (i) the flexibility of the
unstructured grids in describing realistic geometries; and (ii) the high
computational accuracy, which derives from the use of high-order Chebyshev
polynomials. In inhomogeneous media, SPEM has proved to be more accurate
than other grid methods (Seriani and Priolo, 1994), especially in the case of
inclined and curved interfaces, since grid lines can be exactly aligned to material
interfaces (e.g. Figure 9.1). Moreover, the total number of grid nodes is strongly
reduced, compared with methods based on structured grids. These properties of
SPEM make it particularly suitable to compute numerically accurate solutions of
the full wave equations in complex media.
More details about the numerical solution, implementation, and computational
efficiency and accuracy can be found in Padovani et al. (1994). A discussion
about the use of the method in real applications, and in particular for engineering
seismology purposes (e.g. quadrangular mesh generation, source definition and
scaling, numerical accuracy and computational efficiency, and limitations and
advantages of using a 2-D approach), can be found in Priolo (2001).