Long-time stable high-order absorbing boundary conditions for elastodynamics

Baffet, Daniel H.; Bielak, Jacobo; Givoli, Dan; Hagstrom, Thomas; Rabinovich, Daniel

doi:10.1016/j.cma.2012.05.007

Cited by 42 publications

(32 citation statements)

References 41 publications

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“…In the stable high-order ABC for elastodynamics, all the recursive relations except the last one are scalar in nature (as in the H-W ABC), while the last one is the vectorial LK condition. In [20], we have proved this combination to be stable and converging, at the continuous level.…”

Section: Introductionmentioning

confidence: 99%

“…At the continuous level, DAB for elastodynamics is believed to be stable. In [14] a well-posedness proof was provided for the DAB scheme, albeit for the acoustic problem, and in [20] stability was proved for an elastodynamics formulation using the same high-order ABC that the DAB in the present paper is based on. Since, as noted above, the semi-discrete problem is found to be unstable, we conclude that the FE formulation destabilizes the DAB scheme.…”

Section: Stabilitymentioning

confidence: 99%

“…We define the boundary conditions on both boundaries recursively. The recursive boundary conditions are defined in a manner that is in many respects similar to the ABC for elasticity defined by equation (28) in [20], with added low-order terms: Also, in (15) and (16), a m , b m ,ã m andb m are the parameters of the boundary conditions as defined in the "Computational aspects" section below.…”

Section: The Problem In the Computational Domain And The Double Absormentioning

confidence: 99%

“…Recently, an H-W type ABC was applied to problems in elastodynamics [20,21]. This is the first known high-order ABC for elastodynamics that is long-time stable.…”

Section: Introductionmentioning

confidence: 99%

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The double absorbing boundary method for elastodynamics in homogeneous and layered media

Rabinovich

Givoli

Bielak

et al. 2015

Adv. Model. and Simul. in Eng. Sci.

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Background:Recently the Double Absorbing Boundary (DAB) method was introduced as a new approach for solving wave problems in unbounded domains. It has common features to each of two types of existing techniques: local high-order Absorbing Boundary Conditions (ABC) and Perfectly Matched Layers (PML). However, it is different from both and enjoys relative advantages with respect to both. Methods: The DAB method is based on truncating the unbounded domain to produce a finite computational domain, and on applying a local high-order ABC on two parallel artificial boundaries, which are a small distance apart, and thus form a thin non-reflecting layer. Auxiliary variables are defined on the two boundaries and within the layer, and participate in the numerical scheme. In previous studies DAB was developed for acoustic waves which are solutions to the scalar wave equation. Here the approach is extended to time-dependent elastic waves in homogeneous and layered media. The equations are written in second-order form in space and time. Standard Finite Elements (FE) are used for space discretization and the damped Newmark scheme is used for time discretization. Results: The performance of the scheme is demonstrated via numerical examples. The DAB was applied to elastodynamics problems in conjunction with the FE method to demonstrate the performance of the method. Conclusions: DAB is a viable method for solving wave problems in unbounded domains.

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Section: Introductionmentioning

confidence: 99%

Section: Stabilitymentioning

confidence: 99%

Section: The Problem In the Computational Domain And The Double Absormentioning

confidence: 99%

“…Recently, an H-W type ABC was applied to problems in elastodynamics [20,21]. This is the first known high-order ABC for elastodynamics that is long-time stable.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The double absorbing boundary method for elastodynamics in homogeneous and layered media

Rabinovich

Givoli

Bielak

et al. 2015

Adv. Model. and Simul. in Eng. Sci.

Self Cite

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show abstract

“…What makes BIE formulations well suited to configurations involving infinite media is the fact that integral representation formulae automatically verify decay or radiation conditions. BEMs thus constitute an appealing alternative to domain discretization methods for wave propagation problems, as artificial boundary conditions [7] are not required for dealing with the radiation conditions, and grid dispersion cumulative effects are absent [45].…”

Section: Introductionmentioning

confidence: 99%

Recent advances on the fast multipole accelerated boundary element method for 3D time-harmonic elastodynamics

Chaillat

Bonnet

2013

Wave Motion

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To cite this version:Stéphanie Chaillat, Marc Bonnet. Recent advances on the fast multipole accelerated boundary element method for 3D time-harmonic elastodynamics. Wave Motion, Elsevier, 2013Elsevier, , 50, pp.1090Elsevier, -1104 Abstract This article is mainly devoted to a review on fast BEMs for elastodynamics, with particular attention on time-harmonic fast multipole methods (FMMs). It also includes original results that complete a very recent study on the FMM for elastodynamic problems in semi-infinite media. The main concepts underlying fast elastodynamic BEMs and the kernel-dependent elastodynamic FM-BEM based on the diagonal-form kernel decomposition are reviewed. An elastodynamic FM-BEM based on the half-space Green's tensor suitable for semi-infinite media, and in particular on the fast evaluation of the corresponding governing double-layer integral operator involved in the BIE formulation of wave scattering by underground cavities, is then presented. Results on numerical tests for the multipole evaluation of the half-space traction Green's tensor and the FMM treatment of a sample 3D problem involving wave scattering by an underground cavity demonstrate the accuracy of the proposed approach. The article concludes with a discussion of several topics open to further investigation, with relevant published work surveyed in the process.

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Space–time finite element method with domain reduction techniques for dynamic soil–structure interaction problems

Sharma,

Shimizu,

Fujisawa

2024

Int Journal of Mech Sys Dyn

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Design of earth structures, such as dams, tunnels, and embankments, against the vibrational loading caused by high‐speed trains, road traffic, underground explosions, and, more importantly, earthquake motion, demands solutions of the dynamic soil–structure Interaction (SSI) problem. This paper presents a velocity‐based space–time finite element procedure, v‐ST/finite element method (FEM), to solve dynamic SSI problems. The main goal of this study is to present the computation details of implementing viscous boundary conditions of Lysmer–Kuhlemeyer to truncate the unbounded soil domain. Furthermore, additional time‐dependent boundary conditions, in terms of the free‐field response, are included to facilitate energy flow from the far field to the computation domain at the vertical truncated boundaries. In the FEM, seismic input motion is applied to an effective nodal force vector, which can be obtained explicitly in the numerical simulations. Finally, the response of a concrete gravity dam resting on an elastic half‐space to the horizontal component of earthquake motion is computed and successfully compared with the results of semidiscrete FEM using the Newmark‐ method.

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Long-time stable high-order absorbing boundary conditions for elastodynamics

Cited by 42 publications

References 41 publications

The double absorbing boundary method for elastodynamics in homogeneous and layered media

The double absorbing boundary method for elastodynamics in homogeneous and layered media

Recent advances on the fast multipole accelerated boundary element method for 3D time-harmonic elastodynamics

Space–time finite element method with domain reduction techniques for dynamic soil–structure interaction problems

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