Comparison of high‐order absorbing boundary conditions and perfectly matched layers in the frequency domain

Rabinovich, Daniel; Givoli, Dan; Bécache, Éliane

doi:10.1002/cnm.1394

Cited by 87 publications

(57 citation statements)

References 39 publications

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“…[37,38]) have also considered this class of functions and have conducted studies on the variation of the various parameters in their respective formulations (e.g. [26,39]). It is therefore important to stress that the damping function (9) used in the present paper is the only function that has the key property that the complex coordinate mapping ( and (d) shows a zoom into the region near the interface between the computational domain and the PML, to illustrate the extremely rapid variation of f for small PML thicknesses.…”

Section: Discussionmentioning

confidence: 99%

“…Numerical experiments for a variety of 2D acoustic scattering problems demonstrated that the method works extremely well (and much better than many previously used absorbing functions) for problems involving non-planar waves too. Rabinovich et al [26] subsequently also showed that the method outperforms the commonly used polynomial absorbing functions and presents significant advantages over ABC-based discretisations. A key feature of Bermúdez et al 's approach is that the only parameter to be adjusted in order to achieve the optimal numerical solution for a given computational effort is the thickness of the PML region.…”

Section: Introductionmentioning

confidence: 99%

“…[22,28]. Rabinovich et al [26] analysed the effect of variations in δ PML and N PML on the accuracy of the solution, but restricted themselves to PML thicknesses within that range. Popular choices for σ i are powers of the normal distance from ∂D c , i.e.…”

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confidence: 99%

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A parameter-free perfectly matched layer formulation for the finite-element-based solution of the Helmholtz equation

Cimpeanu

Martinsson

Heil

2015

Journal of Computational Physics

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This paper presents a parameter-free perfectly matched layer (PML) method for the finite-element-based solution of the Helmholtz equation. We employ one of Bermúdez et al.'s unbounded absorbing functions for the complex coordinate mapping underlying the PML. With this choice, the only free parameter that controls the accuracy of the numerical solution for a fixed numerical cost (characterised by the number of elements in the bulk and the PML regions) is the thickness of the perfectly matched layer, δ PML . We show that, for the case of planar waves, the absorbing function performs best for PMLs whose thickness is much smaller than the wavelength. We then perform extensive numerical experiments to explore its performance for non-planar waves, considering domain shapes with smooth and polygonal boundaries, different solution types (smooth and singular), and a wide range of wavenumbers, k, to identify an optimal range for the normalised PML thickness, kδ PML , such that, within this range, the error introduced by the presence of the PML is consistently small and insensitive to change. This implies that if the PML thickness is chosen from within this range * Corresponding author URL: http://www3.imperial.ac.uk/people/radu.cimpeanu11 (Radu Cimpeanu), http://www.maths.manchester.ac.uk/~mheil/ (Matthias Heil) Preprint submitted to Journal of Computational PhysicsMay 4, 2015 no further PML optimisation is required, i.e. the method is parameter-free. We characterise the dependence of the error on the discretisation parameters and establish the conditions under which the convergence of the solution under mesh refinement is controlled exclusively by the discretisation of the bulk mesh.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A parameter-free perfectly matched layer formulation for the finite-element-based solution of the Helmholtz equation

Cimpeanu

Martinsson

Heil

2015

Journal of Computational Physics

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“…Complex coordinate stretching is well known for viscoelastic media to understand the nature of the intrinsic attenuation for wave propagation; thus, the PML presents a nice rule in designing the optimal attenuation coefficients by tuning the attenuation factor. Rabinovich et al (2010) compared high-order ABC with the PML in the frequency domain.…”

Section: Introductionmentioning

confidence: 99%

Comparison of artificial absorbing boundaries for acoustic wave equation modelling

Gao

Song

Zhang

et al. 2017

Exploration Geophysics

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Abstract. Absorbing boundary conditions are necessary in numerical simulation for reducing the artificial reflections from model boundaries. In this paper, we overview the most important and typical absorbing boundary conditions developed throughout history. We first derive the wave equations of similar methods in unified forms; then, we compare their absorbing performance via theoretical analyses and numerical experiments. The Higdon boundary condition is shown to be the best one among the three main absorbing boundary conditions that are based on a one-way wave equation. The Clayton and Engquist boundary is a special case of the Higdon boundary but has difficulty in dealing with the corner points in implementaion. The Reynolds boundary does not have this problem but its absorbing performance is the poorest among these three methods. The sponge boundary has difficulties in determining the optimal parameters in advance and too many layers are required to achieve a good enough absorbing performance. The hybrid absorbing boundary condition (hybrid ABC) has a better absorbing performance than the Higdon boundary does; however, it is still less efficient for absorbing nearly grazing waves since it is based on the one-way wave equation. In contrast, the perfectly matched layer (PML) can perform much better using a few layers. For example, the 10-layer PML would perform well for absorbing most reflected waves except the nearly grazing incident waves. The 20-layer PML is suggested for most practical applications. For nearly grazing incident waves, convolutional PML shows superiority over the PML when the source is close to the boundary for large-scale models. The Higdon boundary and hybrid ABC are preferred when the computational cost is high and high-level absorbing performance is not required, such as migration and migration velocity analyses, since they are not as sensitive to the amplitude errors as the full waveform inversion.

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“…Moreover, they are relatively easy to implement in conjunction with virtually any conventional approximation method, like the finite difference, finite element or spectral methods, and can be adapted to solve problems originating from electromagnetism, acoustics, or elasticity, to name a few. They are also generally very efficient and often compare favorably with other existing techniques for artificially handling unbounded domains (see for instance the reference [2], in which a comparison of the performances of high-order absorbing boundary conditions and several types of perfectly matched layers in two dimensions, for problems governed by the Helmholtz equation, is offered).…”

Section: Introductionmentioning

confidence: 99%

On the use of perfectly matched layers in the presence of long or backward propagating guided elastic waves

Dhia¹,

Chambeyron²,

Legendre³

2014

Wave Motion

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An efficient method to compute the scattering of a guided wave by a localized defect, in an elastic waveguide of infinite extent and bounded cross section, is considered. It relies on the use of perfectly matched layers (PML) to reduce the problem to a bounded portion of the guide, allowing for a classical finite element discretization. The difficulty here comes from the existence of backward propagating modes, which are not correctly handled by the PML. We propose a simple strategy, based on finite-dimensional linear algebra arguments and using the knowledge of the modes, to recover a correct approximation to the solution with a low additional cost compared to the standard PML approach. Numerical experiments are presented in the two-dimensional case involving Rayleigh-Lamb modes.

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Comparison of high‐order absorbing boundary conditions and perfectly matched layers in the frequency domain

Cited by 87 publications

References 39 publications

A parameter-free perfectly matched layer formulation for the finite-element-based solution of the Helmholtz equation

A parameter-free perfectly matched layer formulation for the finite-element-based solution of the Helmholtz equation

Comparison of artificial absorbing boundaries for acoustic wave equation modelling

On the use of perfectly matched layers in the presence of long or backward propagating guided elastic waves

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