A global, spherical finite-element model for post-seismic deformation using <i>Abaqus</i>

Nield, Grace A.; King, Matt A.; Steffen, Rebekka; Blank, Bas

doi:https://doi.org/10.5194/gmd-15-2489-2022

Articles | Volume 15, issue 6

https://doi.org/10.5194/gmd-15-2489-2022

© Author(s) 2022. This work is distributed under
the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/gmd-15-2489-2022

© Author(s) 2022. This work is distributed under
the Creative Commons Attribution 4.0 License.

Articles | Volume 15, issue 6

Development and technical paper

|

25 Mar 2022

Development and technical paper |

| 25 Mar 2022

A global, spherical finite-element model for post-seismic deformation using Abaqus

Grace A. Nield, Matt A. King, Rebekka Steffen, and Bas Blank

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Final revised paper (published on 25 Mar 2022)
Preprint (discussion started on 24 Jun 2020)

Interactive discussion

Status: closed

AC: Author comment | RC: Referee comment | SC: Short comment | EC: Editor comment

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- Supplement

RC1: 'Review of "A global,spherical,finite-element model for postseismic deformation using ABAQUS"', Anonymous Referee #1, 18 Aug 2020
- AC1: 'Response to Reviewer 1', Grace Nield, 28 Sep 2021
RC2: 'More complex/realistic rheology must be implemented; Numerical convergence tests must be performed', Anonymous Referee #2, 22 Sep 2020
- AC2: 'Response to Reviewer 2', Grace Nield, 28 Sep 2021

Peer-review completion

AR: Author's response | RR: Referee report | ED: Editor decision

AR by Grace Nield on behalf of the Authors (26 Oct 2021) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (27 Oct 2021) by Lutz Gross

RR by Anonymous Referee #1 (09 Nov 2021)

Suggestions for revision or reasons for rejection

I would like to thank and acknowledge the effort made by authors in revising their manuscript. They have addressed most of my concerns, and I am mostly satisfied with their response although there are some misleading statements.

“We have not included the governing equations for the finite element formulation as this part of the study is not new work. The model is based on the finite element formulation of Wu (2004) and equations therein, as referenced on line 84. Instead, we focus the paper on describing the new aspects of this model –incorporating a fault plane and prescribing slip.”

The article Agata et al. 2019 cited by the authors seems to be implementing a fault plane and prescribing slip, not only for forward modelling but also for adjoint simulations.

Then readers will probably ask: what is the new contribution in this article as compared to Wu 2004 and Agata 2019? In this context, readers would appreciate some clarity on this.

Despite this question, I recognize the importance of this work, and therefore, I am willing to recommend this article for publication after a minor revision.

Minor comments:

“Using the Preliminary Reference Earth Model (PREM) instead would not add anything to the results in our opinion, and only complicate reproducibility of the results for others.”

I disagree, in particular with “...only complicate reproducibility of the results for others.” In fact, the PREM is the standard model for many geophysical simulations on a global scale.

“We will change this term to “layered halfspace” but will also note on first use that it is also referred to as “flat-Earth” in other literature (e.g. Wu,2004) so as not to cause confusion to the reader.”

Personally, I am not against using the term “flat-Earth”, but these days, it seems to be associated more with the flat-earth conspiracies:)

“The moment-density tensor approach suggested by the reviewer is an alternative method of representing a fault within a spectral-element mesh (e.g., Gharti et al. 2019) whereby the mesh geometry is not required to conform to the geometry of the fault plane. However, our current knowledge is that this method cannot be implemented in a finite-element mesh in ABAQUS due to the restrictions of defining loads and forces on elements or surfaces within the mesh.”

I think this is untrue. Because the moment-density tensor approach is well adapted to weak formulations and ABAQUS being a finite-element tool, implementing the moment-density tensor approach in ABAQUS should be straightforward.

Table A1.

Are those computation times for single or parallel processing?

Best regards,

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RR by Anonymous Referee #3 (05 Dec 2021)

Suggestions for revision or reasons for rejection

The authors of this paper present a spherical self-gravitating finite element model generated using the commercial code ABAQUS. The main purpose of the paper is to provide a numerical platform (ABAQUS input files) and associated benchmark useful for future studies of far-field postseismic (viscoelastic) deformation triggered by large earthquakes. A Spherical domain is required to overcome the limits imposed by flat surface analytical or semi-analytical solutions, and self-gravitation is important especially when we want to combine the deformation field from sources other than earthquakes (e.g. post-glacial rebound) or for multi-cycle simulations.

The ABAQUS input file presented in the paper is structured in two parts. The first part (*Static) controls the purely elastic coseismic phase and the implementation of fault slip using kinematic constraint equations. The second part (*Visco) controls the long term postseismic phase (viscoelastic relaxation) using Maxwell or Burgers rheology. The structure of the ABAQUS file is user friendly because provides a clear separation between the different phases of the simulation and is generally easy to modify from other people.

Large part of the paper is correctly focused on the benchmark of the coseismic and postseismic simulation against known analytical solutions. More specifically, for the coseismic phase, the authors compare the ABAQUS predicted displacement field with the well known Okada (1985) solutions for a rectangular source implemented in a homogeneous isotropic half-space. They show a general good agreement between the numerical and the analytical approaches. For the postseismic viscoelastic deformation they compare the ABAQUS predictions with the spherical harmonics code VISCO1D, showing again good agreement. The correct reproduction of analytical solutions using ABAQUS is very useful and proves the correct implementation of fault slip and viscoelastic relaxation.

In my opinion, this paper (technical activity) requires one or two additions before publication. I explain my point below.

I commend the authors for working on a such important and complex topic. However, I believe there is at least one (point 1 below) main issue to be discussed and solved before publication.

1. I think it will more helpful for future users of your examples "ABAQUS input files" if you could provide at least one forward model that implements a different rheologic structure. For example PREM or any recent model with lateral variations in material properties (e.g. viscosity). This way any future user will have an example to learn and to modify for its own project. The same thing is noted from Reviewer 1 although you have not updated your work to accommodate this specific request. You have already successfully completed the benchmark. For that reason, you don't need to benchmark again against a code that implements PREM (although it will be useful). I feel that this work will be significantly improved if you present what described above.

2. I'm not sure If I completely understand the problem regrading the implementation of multiple faults. If I understand correctly, implementing multiple faults will require a verification of the mesh quality. Verifying the quality of the mesh is part of the FE modeling work anyway. ABAQUS will give you a warning if the elements are distorted, although it will most probably finish the calculation. You can easily implement multiple sources (or variable fault geometry) in ABAQUS, you will just have to take the time to verify your mesh using the appropriate metrics. Another solution will be to use an external meshing software like Trelis/Cubit, that will give you better control on the generation of the mesh, and subsequently use this mesh to generate the ABAQUS input file. I will agree however, that for the purpose of investigating far-field deformation, details regarding the fault geometry might be of secondary order.

Referee Report: PDF

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ED: Reconsider after major revisions (13 Dec 2021) by Lutz Gross

AR by Grace Nield on behalf of the Authors (24 Jan 2022) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (31 Jan 2022) by Lutz Gross

RR by Anonymous Referee #3 (09 Feb 2022)

RR by Anonymous Referee #1 (11 Feb 2022)

ED: Publish as is (06 Mar 2022) by Lutz Gross

AR by Grace Nield on behalf of the Authors (08 Mar 2022)

Short summary

We present a finite-element model of post-seismic solid Earth deformation built in the software package Abaqus for the purpose of calculating post-seismic deformation in the far field of major earthquakes. The model is benchmarked against an existing open-source post-seismic model demonstrating good agreement. The advantage over existing models is the potential for simple modification to include 3-D Earth structure, non-linear rheologies and alternative or multiple sources of stress change.