An hp-adaptive discontinuous Galerkin finite-element method for 3-D elastic wave modelling

Abstract

International audienceWe present a discontinuous Galerkin finite-element method (DG-FEM) formulation with Convolutional Perfectly Matched Layer (CPML) absorbing boundary condition for 3-D elastic seismic wave modelling. This method makes use of unstructured tetrahedral meshes locally refined according to the medium properties (h-adaptivity), and of approximation orders that can change from one element to another according to an adequate criterion (p-adaptivity). These two features allow us to significantly reduce the computational cost of the simulations. Moreover, we have designed an efficient CPML absorbing boundary condition, both in terms of absorption and computational cost, by combining approximation orders in the numerical domain. A quadratic interpolation is typically used in the medium to obtain the required accuracy, while lower approximation orders are used in the CPMLs to reduce the total computational cost and to obtain a well-balanced workload over the processors. While the efficiency of DG-FEMs have been largely demonstrated for high approximation orders, we favour the use of low approximation orders as they are more appropriate to the applications we are interested in. In particular, we address the issues of seismic modelling and seismic imaging in cases of complex geological structures that require a fine discretization of the medium. We illustrate the efficiency of our approach within the framework of the EUROSEISTEST verification and validation project, which is designed to compare high-frequency (up to 4 Hz) numerical predictions of ground motion in the Volvi basin (Greece). Through the tetrahedral meshing, we have achieved fine discretization of the basin, which appears to be a sine qua non condition for accurate computation of surface waves diffracted at the basin edges. We compare our results with predictions computed with the spectral element method (SEM), and demonstrate that our method yields the same level of accuracy with computation times of the same order of magnitude

Frequency‐Scaled Curvature as a Proxy for Topographic Site‐Effect Amplification and Ground‐Motion Variability

“…Khors-Sansorny, et al, 2007) with a 10 m spatial discretization (i.e., more than 70 numerical cells per minimum wavelength; Bohlen and Saenger, 2006). Verification tests of our finite-difference method including topography were done by Maufroy (2010) through quantitative comparisons with the discontinuous Galerkin method introduced by Etienne et al (2010). The tests revealed solution discrepancies smaller than 25% for frequencies below 4 Hz in most of the studied area (see fig.…”

Seismic Waves - Research and Analysis

“…However, there is no mathematical proof for unstructured meshes that guarantees numerical stability. (Hammer & Stroud, 1958) and explicit forms of these matrices could be found in Etienne et al (2010) for P 0 , P 1 and P 2 orders.…”

“…These authors proposed the application of an additional damping in the PML, onto the directions parallel to the layer, leading to a multiaxial PML (M-PML) which does not follow strictly the matching property of PML in the continuum and which has a less efficient absorption power. Through various numerical tests, Etienne et al (2010) has shown that instabilities could be delayed outside the time window of simulation when considering extended M-PML from CPML. Table 2 gives the computation times for updating the velocity and stress wavefields in one element for one time step, for different approximation orders, without or with the update of the CPML memory variables (i.e.…”

Accurate seabed modeling using finite difference methods

“…For example, the finite-element method [8,30,35] can approximate complex geometries and topographies by discretizing the models using triangular or tetrahedral cells. Other similar methods, for example, the discontinuous Galerkin [6,7,11,15,21], the finite-volume method [10,57,59], and the boundaryelement method [31,32], can also provide a measure of flexibility towards rugged interfaces by employing irregular grids. However, compared with finite-difference methods, these alternatives are almost always more computationally expensive and generally more complicated to implement.…”

“…Due to the heterogeneity of the Earth, the simulation of seismic wavefields is achieved most commonly using numerical methods, such as the finite-element method [8,30,35], the finite-difference method [1,25,44,45], spectral-element method [23,24], or discontinuous Galerkin [6,7,11,15,21]. The most widely used is the finitedifference method, which uses finite differences to calculate the partial derivatives of wave equations numerically.…”