Dear Hauke,

As you clearly pointed out, the main disadvantage of PMLs is the role of somewhat non-physical parameters which define the PML profile. Furthermore, some people claim that PMLs do not work well at angles close to grazing incidence, but I have not encountered this issue myself.

You may remember from the emails I exchanged with you some time ago - I used an exponential variant for the PML profile and never came into trouble with it. For the choice of the parameters, described in our paper, that type of PML is very effective. At the long wavelength limit, you may need to increase the length of the PML (expressed in the number of wavelengths) as explained in the paper and the associated code. Note, that my application was confined to guided waves modelling, and I did not use PMLs in the conventional finite element context:

• Michał K. Kalkowski, Jennifer M. Muggleton, Emiliano Rustighi, Axisymmetric semi-analytical finite elements for modelling waves in buried/submerged fluid-filled waveguides, Computers & Structures, Volume 196, 2018, Pages 327-340, ISSN 0045-7949, https://doi.org/10.1016/j.compstruc.2017.10.004.

Another, very robust and stable solution is to use the stiffness reduction method. This has been extensively tested for conventional FE in my current group. We are using it every day for our finite element simulations for elastic waves. You may find applications of this method in the guided wave context (spectral collocation method) as well.

• J.R. Pettit, A. Walker, P. Cawley, M.J.S. Lowe, A Stiffness Reduction Method for efficient absorption of waves at boundaries for use in commercial Finite Element codes, Ultrasonics, Volume 54, Issue 7, 2014, Pages 1868-1879, ISSN 0041-624X, https://doi.org/10.1016/j.ultras.2013.11.013.

Hope this helps. I'm happy to discuss any details, if you wish.

Best wishes,

Michal

Thank you very much for the detailed response Michal. My PML implementation works fine now in the frequency domain (I am still unclear about which approach I will try first for the time domain).

Thank you also for pointing out the stiffness reduction method, I was not aware of this approach.

Best,

Hauke

Hi Hauke Gravenkamp ,

I have implemented a time-domain PML formulation for modeling viscoelastic waves with Rayleigh damping in ABAQUS by merging it with a user element subroutine. To obtain the time domain formulation, the governing equation of motion needs to converted from the frequency domain into the time-domain by performing IFT. You may use Maple for this purpose. The following paper may be helpful.

Article Time-domain PML formulation for modeling viscoelastic waves ...

Best wishes,

Masoud

Hi Hauke Gravenkamp ,

The following papers have used the PML in time domain elastic wave propagation. Also, Kausel has reviewed the PML in these papers:

Article Time-domain hybrid formulations for wave simulations in thre...

Article Full-waveform inversion in three-dimensional PML-truncated e...

Article A symmetric hybrid formulation for transient wave simulation...

Article PMLs: A direct approach

Article Perfectly matched layers in the thin layer method

Implementation of the convolutional PML for Elastodynamic Finite Integration Technique (EFIT)

Phongsuk Ampha

2016 13th International Conference on Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology (ECTI-CON)