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

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, 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.

Axisymmetric semi-analytical finite elements for modelling waves in buried/submerged fluid-filled waveguides

“…where the zand θ -dependence of the displacement has been dropped. Recalling that u r (ξ) = N(ξ)u ri we can write the derivative of the displacement as du r dξ = JN ,r u ri (47) which finally yields u r r ξ=−1 = N ,r ξ=−1 u ri (48) An analogous derivation applies to the where the FE formulation for strain was derived, we redefine strain for ξ = −1 the node in the core element located at the axis of symmetry as…”

“…[46] proposed using unbounded absorbing functions for acoustic problems and developed optimal profile parameters ensuring perfect absorption. Follow-up considerations for a parameter-free formulation were also presented in [47]. Angle of incidence does not pose an issue for the problem considered in this paper.…”

“…Angle of incidence does not pose an issue for the problem considered in this paper. However, a comparison between the presented approach and rigorously optimised techniques from [46,47] is an interesting topic for future work on SAFE-PML modelling.…”

Perfect absorbing layers for wave scattering problems and related topics

“…Compared with the PML n , the PML ∞ renders the solution decay at an infinite rate near the outer boundary r = R 2 . It is therefore not surprising it is parameter-free [24]. However, from the above analysis, we infer that the PML-equation (2…”

Simulación numérica de la transmisión indirecta estructural y de la transmisión del ruido de impactos en edificios

“…La explicación detallada de estas tareas no entra en los objetivos de esta tesis, pero hay que recordar su posible influencia en los resultados obtenidos.No hay que olvidar que se están desarrollando muchos métodos basados en los elementos finitos, con planteamientos matemáticos diferentes al introducido aquí. Por ejemplo,Cimpeanu et al (2015) presentan un método basado en elementos finitos para la solución de la ecuación de Helmholtz.…”