A parameter-free perfectly matched layer formulation for the finite-element-based solution of the Helmholtz equation
File(s)
Accepted version
Author(s)
Cimpeanu, R
Martinsson, A
Heil, M
Type
Journal Article
Abstract
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, δPML. 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, kδPML, 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 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.
Date Issued
2015-05-11
Date Acceptance
2015-05-05
Citation
Journal of Computational Physics, 2015, 296, pp.329-347
ISSN
1090-2716
Publisher
Elsevier
Start Page
329
End Page
347
Journal / Book Title
Journal of Computational Physics
Volume
296
Copyright Statement
© 2015 Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/
Subjects
Science & Technology
Technology
Physical Sciences
Computer Science, Interdisciplinary Applications
Physics, Mathematical
Computer Science
Physics
Perfectly matched layers
Helmholtz equation
Acoustic scattering
Finite element method
ABSORBING BOUNDARY-CONDITIONS
SEISMIC-WAVE EQUATION
ACOUSTIC SCATTERING
UNBOUNDED-DOMAINS
FREQUENCY-DOMAIN
SIMULATION
MEDIA
PROPAGATION
Applied Mathematics
01 Mathematical Sciences
02 Physical Sciences
09 Engineering
Publication Status
Published