To reduce computational complexity and memory requirement for 3-D elastodynamics using the boundary element method (BEM), a multi-level fast multipole BEM (FM-BEM) is proposed. The diagonal form for the expansion of the elastodynamic fundamental solution is used, with a truncation parameter adjusted to the subdivision level, a feature necessary for achieving optimal computational efficiency. Both the single-level and multi-level forms of the elastodynamic FM-BEM are considered, with emphasis on the latter. Crucial implementation issues, including the truncation of the multipole expansion, the optimal number of levels, the direct and inverse extrapolation steps are examined in detail with the backing of numerical experiments. A complexity analysis for both the single-level and multi-level versions is conducted. The correctness and computational performances of the proposed elastodynamic FMM are demonstrated on numerical examples, featuring up to O(10 6 ) DOFs run on a single-processor PC and including the diffraction of an incident P plane wave by a semi-spherical or semi-ellipsoidal canyon, representative of topographic site effects.Keywords: Fast multipole method; Boundary element method; 3-D elastodynamics; Topographic site effects IntroductionThe boundary element method (BEM), pioneered in the sixties [6,41], is a mesh reduction method, subject to restrictive constitutive assumptions but yielding highly accurate solutions. It is in particular well suited to deal with unbounded-domain idealizations commonly used in e.g. acoustics [50], electromagnetics [35,39] or seismology [7,22]. In contrast with domain discretization methods, artificial boundary conditions [18] are not needed for dealing with the radiation conditions, and grid dispersion cumulative effects are absent [24,51].However, in traditional boundary element (BE) implementations, the dimensional advantage with respect to domain discretization methods is offset by the fully-populated nature of the BEM coefficient matrix, with set-up and solution times rapidly increasing with the problem size N . It is thus essential to develop alternative, faster strategies that allow to still exploit the known advantages of BEMs when large N prohibit the use of traditional implementations. Fast BEMs, i.e. BEMs of complexity lower than that of traditional BEMs, appeared around 1985 with an iterative integral-equation [19,20], in the context of many-particle simulations. The FMM then naturally led to fast multipole boundary element methods (FM-BEMs), whose scope and capabilities have rapidly progressed, especially in connection with application in electromagnetics [21,31,32,53], but also in other fields including acoustics [14,36,48] and computational mechanics [30]. Many of these investigations are summarized in a review article by Nishimura [37]. The FMM, as well as other fast BEM approaches [23,27,55,56], intrinsically relies upon an iterative solution approach for the linear system of discretized BEM equations, with solution times typically of order O(N lo...
International audienceThis work focuses on the analysis of the multiple interactions between soil layers and civil-engineering structures in dense urban areas submitted to a seismic wave. To investigate such phenomena, called site-city interaction (SCI) herein, two simplified site-city configurations are considered: a homogeneous, periodically spaced city and a heterogeneous, nonperiodically spaced city, both on a constant- depth basin model. These 2D boundary-element method models are subjected to a vertically incident plane SH Ricker wavelet. A parametric study of the city parameters (density of buildings and their natural frequencies) and the thickness of the basin is carried out to characterize the SCI and to investigate its sensitivity to some governing parameters. The following parameters are analyzed: building vibrations, induced ground motion, ground-motion perturbations inside and outside the city, spatial coherency, and kinetic energy of the "urban wave field." A so-called site-city resonance is reached when the soil fundamental frequency and structure eigenfrequencies coincide; building vibrations and ground motion are then significantly decreased and the spatial coherency of the urban field is also strongly modified. Building density and city configuration play a crucial role in the energy distribution inside the city
International audienceThe analysis of site effects is very important since the amplification of seismic motion in some specific areas can be very strong. In this paper, the site considered is located in the centre of Nice on the French Riviera. Site effects are investigated considering a numerical approach (Boundary Element Method) and are compared to experimental results (weak motion and microtremors). The investigation of seismic site effects through numerical approaches is interesting because it shows the dependency of the amplification level on such parameters as wave velocity in surface soil layers, velocity contrast with deep layers, seismic wave type, incidence and damping. In this specific area of Nice, a one-dimensional (1D) analytical analysis of amplification does not give a satisfactory estimation of the maximum reached levels. A boundary element model is then proposed considering different wave types (SH, P, SV) as the seismic loading. The alluvial basin is successively assumed as an isotropic linear elastic medium and an isotropic linear viscoelastic solid (standard solid). The thickness of the surface layer, its mechanical properties, its general shape as well as the seismic wave type involved have a great influence on the maximum amplification and the frequency for which it occurs. For real earthquakes, the numerical results are in very good agreement with experimental measurements for each motion component. Two-dimensional basin effects are found to be very strong and are well reproduced numerically
S U M M A R YThe analysis of seismic wave propagation and amplification in complex geological structures raises the need for efficient and accurate numerical methods. The solution of the elastodynamic equations using traditional boundary element methods (BEMs) is greatly hindered by the fully-populated nature of the matrix equations arising from the discretization. In a previous study limited to homogeneous media, the present authors have established that the fast multipole method (FMM) reduces the complexity of a 3-D elastodynamic BEM to N log N per GMRES iteration and demonstrated its effectiveness on 3-D canyon configurations. In this paper, the frequency-domain FM-BEM methodology is extented to 3-D elastic wave propagation in piecewise homogeneous domains in the form of a FM-accelerated multi-region BE-BE coupling approach. This new method considerably enhances the capability of the BEM for studying the propagation of seismic waves in 3-D alluvial basins of arbitrary geometry embedded in semi-infinite media. Several fully 3-D examples (oblique SV -waves) representative of such configurations validate and demonstrate the capabilities of the multi-domain FM approach. They include comparisons with available (low-frequency) results for various types of incident wavefields and time-domain results obtained by means of Fourier synthesis.
This article investigates the effects of the nonlinear behavior of soils on site response, through various earthquake recordings from the KiK-net database in Japan. This network is composed of more than 688 surface–borehole instruments, from which a characterization of the shear- and compressive-wave velocity profiles down to the borehole depth is available. We selected events with a peak ground acceleration (PGA) at the downhole station of <10 cm=s2 in order to characterize the linear soil behavior by computing the surface to downhole spectral ratios at each site. Modifications of site-response curves computed with strong events (PGA >50 cm=s2) compared to the linear characterization are supposed to be caused by nonlinear soil behavior. To describe the effects of soil nonlinear behavior on site response per event, we propose the percentage of modification (either amplification or deamplification) of the site-response curve compared to the linear evaluation (PNLev, percentage of nonlinearity) and the associated shift frequency (Shev). These parameters are used to estimate the probability that nonlinear site response is significantly different that the linear counterpart. We find that, regardless the site, this probability is important even for low input-motion PGA (values equal to or larger than 30 cm=s2 at downhole sensor). This indicates that nonlinear soil behavior must be taken into account in site-response evaluation for moderate to strong motion. In addition, for 54 sites of the KiK-net database that have recorded at least two strong events (PGA at the downhole station >50 cm=s2), we define four additional parameters that characterize the effects of soil nonlinear behavior on site responses for each site: (1) a PGA threshold (PGAth), defined as the PGA value for which PNLev is higher than 10%; (2) a site-specific PNL for a PGA of 50 cm=s2 (PNLsite); (3) a site-specific shift of the predominant frequency for a PGA of 50 cm=s2 (Shsite); and (4) a frequency from which we observe deamplification between nonlinear and linear site responses (fNL). We observe that nonlinear soil behavior can increase the amplification at frequencies below fNL.We find that fNL lies in between the fundamental and the predominant resonance frequencies of the site response and that sites having VS contrast close to the surface trigger nonlinear behavior at a lower input-motion PGA threshold. These results suggest that nonlinear behavior occurs mostly in the superficial soil layers. Furthermore, by investigating the nonlinear behavior of soils on earthquake horizontal-to-vertical spectral ratios at the surface, we find that they can give satisfactory results (equivalent to the analysis of borehole site responses) for the evaluation of the fNL frequency and shift frequency (Shsite), which indicates that part of the results obtained in this study can be extended to other databases without downhole sensors
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