the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 12 Aug 2025
Overcoming the numerical challenges owing to rapid ductile localization
Abstract. Strain localization is among the most challenging mechanical phenomena for computational Earth sciences. Accurately capturing it is difficult because strain localization initiates spontaneously, is self-accelerating, and its characteristic length and time scales are typically significantly smaller than the spatial and temporal resolutions of the model. This results in an undesirable dependence of the model behavior on numerical parameters and a large computational cost. Strain localization is most commonly associated with brittle failure, but ductile processes such as thermal runaway can also result in rapid ductile localization. Here, we present a numerical model to investigate thermal runaway, and further propose strategies to overcome the challenges associated with resolving rapid localization: (i) adaptive time stepping; (ii) adaptive rescaling; and (iii) two types of regularization. We demonstrate the effect of these strategies in one- and two-dimensional models. We rely on the accelerated pseudo-transient method to solve the governing equations and use graphics processing units to accelerate two-dimensional computations. Our adaptive time stepping strategy allows us to accurately capture spontaneous and rapid stress release during thermal runaway while reducing time steps by more than ten orders of magnitude. Adaptive rescaling further reduces rounding errors and the number of required iterations by two orders of magnitude. Viscosity regularization and gradient regularization enable us to mitigate resolution dependencies but may differently impact the physical response of the model.
Competing interests: At least one of the (co-)authors is a member of the editorial board of Geoscientific Model Development.
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Status: open (until 08 Oct 2025)
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CEC1: 'Comment on egusphere-2025-2417', Astrid Kerkweg, 10 Sep 2025
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AC1: 'Reply on CEC1', Arne Spang, 19 Sep 2025
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Dear Astrid Kerkweg,
thank you for bringing this oversight to our attention. We will change the title of the manuscript to
“Overcoming the numerical challenges owing to rapid ductile localization with DEDLoc (version 1.0.0)”,
according to the journal guidelines.Best regards,
Arne SpangCitation: https://doi.org/10.5194/egusphere-2025-2417-AC1
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AC1: 'Reply on CEC1', Arne Spang, 19 Sep 2025
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RC1: 'Comment on egusphere-2025-2417', Laetitia Le Pourhiet, 27 Sep 2025
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General Assessment
The paper by Spange et al. compares different methods to regularize ductile localization associated with thermal runaway in the ductile regime. This is a challenging and important issue for the community, since the length scales of shear zones produced by this process in the Earth’s mantle may be on the order of nanometers to millimeters and evolve over seconds, while models typically operate at scales of hundreds of kilometers and millions of years. Regularization is therefore essential.
The paper is well written and the results are presented clearly. The main weakness, in my opinion, is the lack of separation between the physical aspects and the numerical methodology in Sections 2 and 3. I believe this could be addressed by a modest reorganization of the text.
Broader Comments
- The paper is primarily methodological, but a stronger discussion of the underlying physics would be very valuable. In particular, it would help to reflect on how the approximations in the heat equation, mass conservation, boundary conditions, and the assumption of a single shear band simplify or complicate the problem compared to more Earth-like conditions.
- It would also be interesting to discuss whether the length scales or gradients introduced by the regularization might have a physical meaning. Ideally, the problem should be governed by material parameters rather than numerical parameters. While the proposed regularizations alleviate mesh dependence, they introduce sensitivities to new parameters that are equally numerical. Introducing physically motivated mechanisms that could act as natural regularization would strengthen the study.
Suggestions for Reorganization
- Introduce the governing equations (Sec. 2.2.1), rheology (Sec. 3.2), and density (Sec. 3.3) before describing the model setup (Sec. 2.1) and the 1D simplifications (Sec. 2.2.2).
- In Section 3, begin with the spatial discretization, then proceed through the pseudo-transient scheme, nonlinear viscosity iterations, and finally regularization.
- Please add the thermal boundary conditions to the description of the model setup.
Notation and Tables
- Since you use λ as a regularization length scale, I suggest using κ for heat diffusivity (instead of λ). Consequently, use k for heat conductivity in Eq. 17 and in the text (L.122), and update the last line of Table 1. Currently, you list conductivity but label it as diffusivity.
- In Table 1, group the elastic bulk modulus K with the shear modulus G. It does not need the subscript “b”. You could compute K using Poisson’s ratio of 0.25 and then remove Poisson’s ratio from the table, along with Eq. 28. For an isotropic elastic material, only K and G are needed; Poisson’s ratio can simply be mentioned as an explanatory note.
Technical and Line-by-Line Comments
- L.23: While plate-scale models may overestimate shear zone width due to coarse resolution, could you add a reference or explanation for grain-scale models potentially underestimating them? If Braeck et al. are correct, shear zones could be as thin as nanometers—smaller than grain-scale models can capture.
- Eq. 27: Please provide a brief explanation of how it follows from the mass conservation equation. I suggest moving this equation (with the explanation) next to Eqs. 2 and 4, without devoting a full subsection to density.
- Eq. 3: Explicitly note that τᵢⱼ eᵢⱼᵛⁱ is the shear heating term, and that all dissipated work is assumed to convert into heat (i.e., no grain size reduction).
- Consider placing remarks on neglected terms close to the relevant equations:
- inertial and gravity terms after Eq. 1,
- adiabatic and radiogenic heating after Eq. 3,
- thermal expansion after Eq. 4.
- Use e instead of ε˙\dot{\varepsilon} for the deviatoric strain rate, consistent with your notation for deviatoric stress (τ vs. σ). Also, define τ explicitly, just as you do for the strain rate tensor.
- L.84: effective shear viscosity (clarify wording).
- L.89: Gravity does not need to be listed here, as it does not affect Eqs. 7–8 if initial stress conditions already include gravity and they should.
- L.97: Since you previously called them “deviatoric,” maintain that wording consistently.
- Eq. 17: Replace λ with κ
- L.123: Use k=κ/(ρCp).
- L.145–150: Please separate physics from numerics; for instance, move stabilization viscosity details to a new subsection of Section 3.
- L.203: Clarify why τ is used instead of change in strain rate? If this comes from the cited reference, a short explanation in the text would help.
- L.260–265: A short note on how elasticity may improve conditioning would be useful.
- L.355/L.375: You mention using the same parameters in 1D and 2D, but then describe differences in how the anomaly is defined. Please clarify.
- L.370: A comment on the effect of periodic boundary conditions would be helpful. In principle, once the shear band spans the entire domain, 1D and 2D solutions should converge and not diverge as 1D is an infinite shear band…. I don’t really understand.
- L.382: When the timestep drops to seconds, and given the large stress drops you show, is it still valid to neglect inertial terms? Could you add a small comment on that ?
Recommendation
I recommend minor revisions, focused mainly on clarifying the separation between physical and numerical aspects, improving the discussion of the physical meaning of regularization, and addressing the minor notational and organizational issues listed above.
Laetitia Le Pourhiet
Citation: https://doi.org/10.5194/egusphere-2025-2417-RC1
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Dear authors,
in my role as Executive editor of GMD, I would like to bring to your attention our Editorial version 1.2: https://www.geosci-model-dev.net/12/2215/2019/
This highlights some requirements of papers published in GMD, which is also available on the GMD website in the ‘Manuscript Types’ section: http://www.geoscientific-model-development.net/submission/manuscript_types.html
In particular, please note that for your paper, the following requirements have not been met in the Discussions paper:
Following the GMD publication criteria, in order to simplify reference to your developments, please add the model name DEDloc and a version number in the title of your article in your revised submission to GMD.
Yours, Astrid Kerkweg