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Review and Recent Developments on the Perfectly Matched Layer (PML) Method for the Numerical Modeling and Simulation of Elastic Wave Propagation in Unbounded Domains

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Abstract

This review article revisits and outlines the perfectly matched layer (PML) method and its various formulations developed over the past 25 years for the numerical modeling and simulation of wave propagation in unbounded media. Based on the concept of complex coordinate stretching, an efficient mixed displacement-strain unsplit-field PML formulation for second-order (displacement-based) linear elastodynamic equations is then proposed for simulating the propagation and absorption of elastic waves in unbounded (infinite or semi-infinite) domains. Both time-harmonic (frequency-domain) and time-dependent (time-domain) PML formulations are derived for two- and three-dimensional linear elastodynamic problems. Through the introduction of only a few additional variables governed by low-order auxiliary differential equations, the resulting mixed time-domain PML formulation is second-order in time, thereby allowing the use of standard time integration schemes commonly employed in computational structural dynamics and thus facilitating the incorporation into existing displacement-based finite element codes. For computational efficiency, the proposed time-domain PML formulation is implemented using a hybrid approach that couples a mixed (displacement-strain) formulation for the PML region with a classical (displacement-based) formulation for the physical domain of interest, using a standard Galerkin finite element method (FEM) for spatial discretization and a Newmark time scheme coupled with a finite difference (Crank-Nicolson) time scheme for time sampling. Numerical experiments show the performances of the PML method in terms of accuracy, efficiency and stability for two-dimensional linear elastodynamic problems in single- and multi-layer isotropic homogeneous elastic media.

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Notes

  1. The interested reader can refer to [17, 18, 33, 42, 196, 197] for the definition of well-posedness, hyperbolicity and stability of initial boundary value problems for hyperbolic systems.

  2. As already mentioned in [59, 161, 239] and unlike stated in [216, 240], the fourth-order tensor-valued field \(\widetilde{\varvec{C}}\) does have the major symmetry property, i.e. \(\widetilde{C}_{ijkl} = \widetilde{C}_{klij}\), but both minor ones are lost i.e. \(\widetilde{C}_{ijkl} \ne \widetilde{C}_{jikl}\) and \(\widetilde{C}_{ijkl} \ne \widetilde{C}_{ijlk}\).

  3. The Ricker wavelet r(t) defined in (54) is such that \(r(t) = \dfrac{d^2p}{d t^2}(t)\), where \(p(t) = \dfrac{1}{2(\pi f_d)^2} \exp (-(\pi f_d)^2(t - t_d)^2)\) corresponds to a modified second-order derivative of the Gaussian probability density function \(t\mapsto \dfrac{1}{\sigma \sqrt{2\pi }} \exp \left( -\dfrac{(t - t_d)^2}{2\sigma ^2}\right)\) with mean value \(t_d\) and standard deviation \(\sigma =\sqrt{2}/\omega _d\) up to a multiplicative constant \(4\sqrt{\pi }/\omega _d^3\) with \(\omega _d=2\pi f_d\) the central (or dominant) angular frequency.

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Acknowledgements

The authors gratefully acknowledge and thank Christian Soize, Professor at Université Gustave Eiffel, Laboratoire MSME, for very helpful discussions, constructive remarks and valuable comments.

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    Florent Pled & Christophe Desceliers

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Pled, F., Desceliers, C. Review and Recent Developments on the Perfectly Matched Layer (PML) Method for the Numerical Modeling and Simulation of Elastic Wave Propagation in Unbounded Domains. Arch Computat Methods Eng 29, 471–518 (2022). https://doi.org/10.1007/s11831-021-09581-y

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