Optimizing perfectly matched layers in discrete contexts

Abstract

International audiencePerfectly matched layers (PMLs) are widely used for the numerical simulation of wave-like problems defined on large or infinite spatial domains. However, for both time-dependent and time-harmonic cases, their performance critically depends on the so-called absorption function. This paper deals with the choice of this function when classical numerical methods are used (based on finite differences, finite volumes, continuous finite elements and discontinuous finite elements). After reviewing the properties of the PMLs at the continuous level, we analyze how they are altered by the different spatial discretizations. In the light of these results, different shapes of absorption function are optimized and compared by means of both one-dimensional and two-dimensional representative time-dependent cases. This study highlights the advantages of the so-called shifted hyperbolic function, which is efficient in all cases and does not require the tuning of a free parameter, by contrast with the widely used polynomial functions

Perfectly Matched Layer for Galbrun׳s aeroacoustic equation in a cylindrical coordinates system with an axial and a swirling steady mean flow

“…These observations are in agreement with Modave et al [31], but contradict Bermudez et al [30], although the finite elements used are different from one another. This suggests that the stabilization technique (PSPG) employed by Modave [31] is not responsible for the differences observed by him with Bermudez [30] as he advocates. Indeed, Fig.…”

“…For each configuration, it is important to optimize the PML to make it attractive and competitive compared to other methods (DtN, BEM, IEM, etc.). This aspect has been extensively studied for classical wave problems [29,17,30,31]. The key points affecting the PML performances are the PML thickness, the mesh density and the absorption function.…”

“…The performances of three PML absorption functions proposed in the literature [29,17,30,31] are compared. The first one is a parabolic function (polynomial quadratic), noted σ BC for "bounded continuous", given by…”

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

“…[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.…”

Optimized Schwarz Domain Decomposition Methods for Scalar and Vector Helmholtz Equations

“…can be used, where δ corresponds to the thickness of the PML layer and x PML is the local coordinate inside the PML [8,38].…”