Stéphanie Chaillat

A new Fast Multipole formulation for the elastodynamic half-space Green's tensor

In this article, a version of the frequency-domain elastodynamic Fast Multipole-Boundary Element Method (FM-BEM) for semi-infinite media, based on the half-space Green?s tensor (and hence avoiding any discretization of the planar traction-free surface), is presented. The half-space Green?s tensor is often used (in non-multipole form until now) for computing elastic wave propagation in the context of soil-structure interaction, with applications to seismology or civil engineering. However, unlike the full-space Green?s tensor, the elastodynamic half-space Green?s tensor cannot be expressed using derivatives of the Helmholtz fundamental solution. As a result, multipole expansions of that tensor cannot be obtained directly from known expansions, and are instead derived here by means of a partial Fourier transform with respect to the spatial coordinates parallel to the free surface. The obtained formulation critically requires an efficient quadrature for the Fourier integral, whose integrand is both singular and oscillatory. Under these conditions, classical Gaussian quadratures would perform poorly, fail or require a large number of points. Instead, a version custom-tailored for the present needs of a methodology proposed by Rokhlin and coauthors, which generates generalized Gaussian quadrature rules for specific types of integrals, has been implemented. The accuracy and efficiency of the proposed formulation is demonstrated through numerical experiments on single-layer elastodynamic potentials involving up to about N = 6 i? 10 5 degrees of freedom. In particular, a complexity significantly lower than that of the non-multipole version is shown to be achieved. The 3D elastodynamic half-space Green?s tensor is expressed as a 2D Fourier integral.It possesses the needed separated-variable form allowing multipole-like acceleration.The 2D Fourier integral is computed by a dedicated generalized Gaussian quadrature.Efficiency and accuracy are assessed on comprehensive numerical tests.A complexity significantly lower than using non-multipole version is achieved.

Fast iterative boundary element methods for high-frequency scattering problems in 3D elastodynamics

Abstract The fast multipole method is an efficient technique to accelerate the solution of large scale 3D scattering problems with boundary integral equations. However, the fast multipole accelerated boundary element method (FM-BEM) is intrinsically based on an iterative solver. It has been shown that the number of iterations can significantly hinder the overall efficiency of the FM-BEM. The derivation of robust preconditioners for FM-BEM is now inevitable to increase the size of the problems that can be considered. The main constraint in the context of the FM-BEM is that the complete system is not assembled to reduce computational times and memory requirements. Analytic preconditioners offer a very interesting strategy by improving the spectral properties of the boundary integral equations ahead from the discretization. The main contribution of this paper is to combine an approximate adjoint Dirichlet to Neumann (DtN) map as an analytic preconditioner with a FM-BEM solver to treat Dirichlet exterior scattering problems in 3D elasticity. The approximations of the adjoint DtN map are derived using tools proposed in [40] . The resulting boundary integral equations are preconditioned Combined Field Integral Equations (CFIEs). We provide various numerical illustrations of the efficiency of the method for different smooth and non-smooth geometries. In particular, the number of iterations is shown to be completely independent of the number of degrees of freedom and of the frequency for convex obstacles.

Seismic‐Wave Amplification in 3D Alluvial Basins: 3D/1D Amplification Ratios from Fast Multipole BEM Simulations

In this article, we study seismic‐wave amplification in alluvial basins having 3D standard geometries through the fast multipole boundary‐element method (FMBEM) in the frequency domain. We investigate how much 3D amplification differs from the 1D (horizontal layering) case. Considering incident fields of plane harmonic waves, we examine the relationships between the amplification level and the most relevant physical parameters of the problem (impedance contrast, 3D aspect ratio, and vertical and oblique incidence of plane waves). The FMBEM results show that the most important parameters for wave amplification are the impedance contrast and the so‐called equivalent shape ratio. Using these two parameters, we derive simple rules to compute the fundamental frequency for various 3D basin shapes and the corresponding 3D/1D amplification factor for 5% damping. Effects on amplification due to 3D basin asymmetry are also studied and incorporated in the derived rules.

Multi-Level Fast Multipole BEM for 3-D Elastodynamics

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) based on the diagonal form for the expansion of the elastodynamic fundamental solution is proposed and demonstrated on numerical examples involving single-region and multi-region configurations where the scattering of seismic waves by a topographical irregularity or a sediment-filled basin is examined.

A fast multipole accelerated BEM for 3-D elastic wave computation

The solution of the elastodynamic equations using boundary element methods (BEMs) gives rise to fully-populated matrix equations. Earlier investigations on the Helmholtz and Maxwell equations have established that the Fast Multipole (FM) method reduces the complexity of a BEM solution to N log2 N per GMRES iteration. The present article addresses the extension of the FM-BEM strategy to 3D elastodynamics in the frequency domain. Efficiency and accuracy are demonstrated on numerical examples involving up to N = O(106) boundary nodal unknowns.

Theory and implementation of H-matrix based iterative and direct solvers for Helmholtz and elastodynamic oscillatory kernels

In this work, we study the accuracy and efficiency of hierarchical matrix (

Metric-based anisotropic mesh adaptation for 3D acoustic boundary element methods

\mathcal{H}

A wideband Fast Multipole Method for the Helmholtz kernel

-matrix) based fast methods for solving dense linear systems arising from the discretization of the 3D elastodynamic Greens tensors. It is well known in the literature that standard

Seismic response of three-dimensional rockfill dams using the Indirect Boundary Element Method

\mathcal{H}

A multi-level fast multipole BEM for 3-D elastodynamics in the frequency domain

-matrix based methods, although very efficient tools for asymptotically smooth kernels, are not optimal for oscillatory kernels.