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A Nonconvolutional, Split-Field, Perfectly Matched Layer for Wave Propagation in Isotropic and Anisotropic Elastic Media: Stability Analysis

European School for Advanced Studies in Reduction of Seismic Risk (ROSE School), University of Pavia, Pavia 27100, Italy

Apostolos S. Papageorgiou

Department of Civil Engineering, University of Patras, GR–26500 Patras, Greece

A nonconvolutional, split-field, perfectly matched layer, referred to as the multiaxial perfectly matched layer (M-PML), is proposed and implemented. The new formulation is obtained by generalizing the classical perfectly matched layer (PML; as originally proposed by Bérenger, [1994]) to a medium in which damping profiles are specified in more than one direction. Under the hypothesis of small damping and using an eigenvalue sensitivity analysis based on first derivatives, we propose a method to study the stability of the M-PML. With this method we demonstrate that the stability of the M-PML is related to the ratios of the specified damping profiles. Recognition of this fact leads to a general procedure for constructing robust, stable M-PML models for anisotropic media. It is also demonstrated that for any anisotropic medium the classical PML exhibits instabilities related to an eigenvalue with zero real part of multiplicity higher than one. Furthermore, we show that exponential growth due to eigenvalues with positive real part can be present in the classical PML for some orthotropic media. The effectiveness of the proposed M-PML and its advantages relative to the classical PML are demonstrated by constructing stable terminations for the aforementioned anisotropic media. The method of stability analysis is developed and demonstrated for two-dimensional elastodynamics problems, but its extension to three-dimensional configurations is straightforward.