The spectral element method: An efficient tool to simulate the seismic response of 2D and 3D geological structures

Abstract

We present the spectral element method to simulate elastic-wave propagation in realistic geological structures involving complieated free-surface topography and material interfaces for two- and three-dimensional geometries. The spectral element method introduced here is a high-order variational method for the spatial approximation of elastic-wave equations. The mass matrix is diagonal by construction in this method, which drastically reduces the computational cost and allows an efficient parallel implementation. Absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains. The time discretization is based on an energy-momentum conserving scheme that can be put into a classical explicit-implicit predictor/multi-corrector format. Long-term energy conservation and stability properties are illustrated as well as the efficiency of the absorbing conditions. The associated Courant condition behaves as ΔtC < O (nel−1/ndN−2), with nel the number of elements, nd the spatial dimension, and N the polynomial order. In practice, a spatial sampling of approximately 5 points per wavelength is found to be very accurate when working with a polynomial degree of N = 8. The accuracy of the method is shown by comparing the spectral element solution to analytical solutions of the classical two-dimensional (2D) problems of Lamb and Garvin. The flexibility of the method is then illustrated by studying more realistic 2D models involving realistic geometries and complex free-boundary conditions. Very accurate modeling of Rayleigh-wave propagation, surface diffraction, and Rayleigh-to-body-wave mode conversion associated with the free-surface curvature are obtained at low computational cost. The method is shown to provide an efficient tool to study the diffraction of elastic waves by three-dimensional (3D) surface topographies and the associated local effects on strong ground motion. Complex amplification patterns, both in space and time, are shown to occur even for a gentle hill topography. Extension to a heterogeneous hill structure is considered. The efficient implementation on parallel distributed memory architectures will allow to perform real-time visualization and interactive physical investigations of 3D amplification phenomena for seismic risk assessment.

Research on basin effect based on 2-D geophysical prospecting profile

“…In this paper, we use the SEM (Komatitsch and Vilotte, 1998), which is popular for ground motion simulations. The SEM combines the advantages of the pseudo-spectral method and the FE method.…”

Boosting memory access locality of the Spectral Element Method with Hilbert space-filling curves

“…One such technique is the spectral element method (SEM) (Patera 1984, Pozrikidis 2014, a variant of the finite element method (FEM) (Logan 2017). The SEM has been applied to a number of wave-related problems (Seriani 1997, Komatitsch & Vilotte 1998, Mercerat et al 2006, with favorable results. In addition to being readily adaptable to various kinds of partial differential equations (PDEs), it is also flexible when it comes to the geometric details of the underlying domain, even when the domain contains complex media interfaces (Komatitsch & Vilotte 1998).…”

“…(2.29) in general does not produce exact results (Komatitsch et al 2000). However, a diagonal mass matrix allows for a significant reduction of the complexity of the solution algorithm and of the related processing time (Komatitsch & Vilotte 1998), which justifies the loss of accuracy.…”

“…The second work is a seminal paper by Komatitsch & Vilotte (1998) of elastic waves in geological structures over unstructured meshes of quadrilaterals in 2D and hexahedra in 3D. This study handles time integration through an explicit Newmark scheme.…”

Spectral element method for 3-D controlled-source electromagnetic forward modelling using unstructured hexahedral meshes

“…First applications to seismic wave propagation appeared in the nineties (Priolo et al 1994;Seriani et al 1994;Faccioli et al 1996) and used Chebyshev polynomials to approximate the unknown fields. A major improvement arrived with the work of Komatitsch & Vilotte (1998), which proposed the combination of Lagrange polynomials as interpolants and a Gaussian quadrature scheme defined on GLL points for the elastic wave equation. This particular choice enabled constructing a diagonal mass matrix in seismological applications, which in turn led to a simpler explicit time stepping scheme and allows an effective parallel implementation (Schuberth 2003).…”