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SUMMARYOne approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outward from the bounded domain. A perfectly matched layer (PML) is an unphysical absorbing layer model for linear wave equations that absorbs, almost perfectly, outgoing waves of all non-tangential angles-ofincidence and of all non-zero frequencies. In a recent work [Computer Methods in Applied Mechanics and Engineering 2003; 192:1337-1375, the authors presented, inter alia, time-harmonic governing equations of PMLs for anti-plane and for plane-strain motion of (visco-)elastic media. This paper presents (a) corresponding time-domain, displacement-based governing equations of these PMLs and (b) displacement-based finite element implementations of these equations, suitable for direct transient analysis. The finite element implementation of the anti-plane PML is found to be symmetric, whereas that of the plane-strain PML is not. Numerical results are presented for the anti-plane motion of a semi-infinite layer on a rigid base, and for the classical soil-structure interaction problems of a rigid strip-footing on (i) a half-plane, (ii) a layer on a half-plane, and (iii) a layer on a rigid base. These results demonstrate the high accuracy achievable by PML models even with small bounded domains.

SUMMARYThe use of a perfectly matched layer (PML) model is an efficient approach toward the bounded-domain modelling of wave propagation on unbounded domains. This paper formulates a three-dimensional PML for elastic waves by building upon previous work by the author and implements it in a displacement-based finite element setting. The novel contribution of this paper over the previous work is in making this finite element implementation suitable for explicit time integration, thus making it practicable for use in large-scale three-dimensional dynamic analyses. An efficient method of calculating the strain terms in the PML is developed in order to take advantage of the lack of the overhead of solving equations at each time step. The PML formulation is studied and validated first for a semi-infinite bar and then for the classical soil-structure interaction problems of a square flexible footing on a (i) half-space, (ii) layer on a half-space and (iii) layer on a rigid base. Numerical results for these problems demonstrate that the PML models produce highly accurate results with small bounded domains and at low computational cost and that these models are long-time stable, with critical time step sizes similar to those of corresponding fully elastic models.

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SUMMARYThe damping-solvent extraction method for the analysis of unbounded visco-elastic media is evaluated numerically in the frequency domain in order to investigate the in uence of the computational parameters -domain size, amount of artiÿcial damping, and mesh density -on the accuracy of results. An analytical estimate of this in uence is presented, and speciÿc questions regarding the in uence of the parameters on the results are answered using the analytical estimate and numerical results for two classical problems: the rigid strip and rigid disc footings on a visco-elastic half-space with constant hysteretic material damping.As the domain size in increased, the results become more accurate only at lower frequencies, but are essentially una ected at higher frequencies. Choosing the domain size to ensure that the static sti ness is computed accurately leads to an unnecessarily large domain for analysis at higher frequencies. The results improve by increasing artiÿcial damping but at a slower rate as the total (material plus artiÿcial) damping ratio t gets closer to 0.866. However, the results do not deteriorate signiÿcantly for the larger amounts of artiÿcial damping, suggesting that t ≈ 0:6 is appropriate; a larger value is not likely to in uence the accuracy of results. Presented results do not support the earlier suggestion that similar accuracy can be achieved by a large bounded domain with small damping or by a small domain with larger damping.

ERRATUMErratum to 'Perfectly matched layers for transient elastodynamics of unbounded domains' (Int.

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