SUMMARYWe are concerned with the numerical simulation of wave motion in arbitrarily heterogeneous, elastic, perfectly‐matched‐layer‐(PML)‐truncated media. We extend in three dimensions a recently developed two‐dimensional formulation, by treating the PML via an unsplit‐field, but mixed‐field, displacement‐stress formulation, which is then coupled to a standard displacement‐only formulation for the interior domain, thus leading to a computationally cost‐efficient hybrid scheme. The hybrid treatment leads to, at most, third‐order in time semi‐discrete forms. The formulation is flexible enough to accommodate the standard PML, as well as the multi‐axial PML.We discuss several time‐marching schemes, which can be used à la carte, depending on the application: (a) an extended Newmark scheme for third‐order in time, either unsymmetric or fully symmetric semi‐discrete forms; (b) a standard implicit Newmark for the second‐order, unsymmetric semi‐discrete forms; and (c) an explicit Runge–Kutta scheme for a first‐order in time unsymmetric system. The latter is well‐suited for large‐scale problems on parallel architectures, while the second‐order treatment is particularly attractive for ready incorporation in existing codes written originally for finite domains.We compare the schemes and report numerical results demonstrating stability and efficacy of the proposed formulations. Copyright © 2014 John Wiley & Sons, Ltd.
We are concerned with high-fidelity subsurface imaging of the soil, which commonly arises in geotechnical site characterization and geophysical explorations. Specifically, we attempt to image the spatial distribution of the Lamé parameters in semi-infinite, threedimensional, arbitrarily heterogeneous formations, using surficial measurements of the soil's response to probing elastic waves. We use the complete waveform response of the medium to derive the inverse problem, by using a partial-differential-equation (PDE)-constrained optimization approach, directly in the time-domain, to minimize the misfit between the observed response of the medium at select measurement locations, and a computed response corresponding to a trial distribution of the Lamé parameters. We discuss strategies that lend algorithmic robustness to our proposed inversion scheme. To limit the computational domain to the size of interest, we employ perfectly-matched-layers (PMLs).In order to resolve the forward problem, we use a recently developed hybrid finite element approach, where a displacement-stress formulation for the PML is coupled to a standard displacement-only formulation for the interior domain, thus leading to a computationally cost-efficient scheme. Time-integration is accomplished by using an explicit Runge-Kutta scheme, which is well-suited for large-scale problems on parallel computers.We verify the accuracy of the material gradients obtained via our proposed scheme, and report numerical results demonstrating successful reconstruction of the two Lamé parameters for both smooth and sharp profiles.
Despite the ever increasing adoption of wave motion simulations for assessing seismic hazard, most assessment/simulations are still based on a flat surface earth model. The purpose of this paper is to quantify the effect of topographic irregularities on the ground motion and local site response by means of parametric investigations in the frequency-domain of typical two-dimensional features.To this end, we deploy best-practice tools for simulating seismic events in arbitrarily heterogeneous formations; these include: a forward wave simulator based on a hybrid formulation encompassing perfectly-matched-layers (PMLs); unstructured spectral elements for spatial discretization; and the Domain-Reduction-Method that permits placement of the seismic source within the computational domain, thus allowing consideration of realistic seismic scenarios.Of particular interest to this development is the study of the effects that various idealized topographic features have on the surface motion when compared against the response that is based on a flat-surface assumption. We report the results of parametric studies for various parameters, which show motion amplification that depends, as expected, on the relation between the topographic feature's characteristics and the dominant wavelength. More interestingly, we also report motion de-amplification patterns.
We discuss the application of a recently developed full-waveform-inversion-based technique to the imaging of geotechnical sites using field-collected data. Specifically, we address the profiling of arbitrarily heterogeneous sites in terms of P-and S-wave velocities in three dimensions, using elastic waves as probing agents. We cast the problem of finding the spacial distribution of the elastic soil properties as an inverse medium problem, directly in the time domain, and use perfectly-matched-layers (PMLs) to account for the semi-infinite extent of the site under investigation. After briefly reviewing the theoretical and computational aspects of the employed technique, we focus on the characterization of the George E. Brown Jr. Network for Earthquake Engineering Simulation site in Garner Valley, California (NEES@UCSB). We compare the profiles obtained from our full-waveform-inversion-based methodology against the profiling obtained from the Spectral-Analysis-of-Surface-Waves (SASW) method, and report agreement. In an attempt to validate our methodology, we also compare the recorded field data at select control sensors that were not used for the full-waveform inversion, against the response at the same sensors, computed based on the full-waveform-inverted profiles. We report very good agreement at the control sensors, which is a strong indicator of the correctness of the inverted profiles. Overall, the systematic framework discussed herein seems robust, general, practical, and promising for three-dimensional site characterization purposes.
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