The spectral element method: large scale applications to earthquake and vibration engineering

The spectral element method is introduced as an effective tool to perform large-scale numerical simulations of elastic wave propagation and dynamic soil-structure interaction. The theoretical formulation was first proposed in the late 90s (Faccioli et al., 1997), where the Galerkin method applied to the dynamic linear equilibrium problem was tailored in terms of spectral approximation of the spatial domain. Specifically, the computational domain is split into hexahedral elements where the unknown displacement field is approximated by a Lagrange polynomial of degree N, and each volume integral is evaluated numerically by the Legendre-Gauss-Lobatto (LGL) quadrature formula. This way, on one hand the exponential accuracy of the spectral method is preserved and, on the other hand, the computational effort in evaluating the integrals is minimized, since the resulting mass matrix is diagonal.
More recently, the research carried out at Department of Structural Engineering, Politecnico di Milano, and at CRS4, Sardinia, lead to major improvements of the numerical code in terms of pre- and post-processing tools, and of the implementation in parallel architectures to achieve high-performance computing. The resulting numerical code GeoELSE (GeoELastodynamics by Spectral Elements, see web site geoelse.stru.polimi.it) presently enables one to carry out numerical simulations of elastic wave propagation in 3D media under various types of excitation, including an earthquake source, moving loads, vibrating machines etc. Among its specific features, GeoELSE presently includes paraxial boundary conditions, linear visco-elastic and non-linear elastic materials, domain reduction method.
Several examples of application are provided, including: (i) the seismic response of the Grenoble valley in the near-field of a moderate earthquake (Stupazzini et al., 2009); (ii) the seismic response of a railway viaduct; (iii) the analysis of ground vibrations induced by high-speed trains (Paolucci and Spinelli, 2006).


References
- Faccioli E., F. Maggio, R. Paolucci, A. Quarteroni (1997). “2D and 3D elastic wave propagation by a pseudo-spectral domain decomposition method”, Journal of Seismology, Vol. 1, 237-251.
- Paolucci R., D. Spinelli (2006). “Ground motion induced by train passage”, ASCE Journal of Engineering Mechanics, Vol. 132, 201-210.
- Stupazzini M., R. Paolucci, H. Igel (2009). “Near-fault earthquake ground motion simulation in the Grenoble Valley by a high-performance spectral element code”. Bulletin of the Seismological Society of America, Vol. 99(1), 286-301.