JOSS manuscript comments

[jorgensd]

Only minor comments wrt to the manuscript from https://github.com/openjournals/joss-reviews/issues/6897

Summary

• I would remove the last sentence of the summary

Statement of need

• Do you have a reference for the statement

  The current literature in computational fracture mechanics predominantly focuses on unilateral first-order criteria, systematically neglecting the exploration of higher-order information for critical points.

Three solvers

• I would rename this to Functionality
  • As a non-specialist, it would beneficial to me introduce the problem you would like to solve ([1]) prior to introducing the solvers.
  • Additionally, $V_0$ is not defined (I guess a suitable Sobolev space).
  • I guess it would be good to refer to some papers that have examples of a formulation similar to eq [1]. For instance the aforementioned paper (https://doi.org/10.1016/j.jmps.2021.104424, section 2.4?). Maybe there are some other references for non-experts that you know?

Software The interface is well described in this section, only question from reading the paper and not testing the implementation is:

• How does state # A dictionary of fields describing the system link to A list of upper bounds (upper and lower) for the state? As a dictionary is not ordered, and one has multiple states, it is not clear which bound is applied to which state.
• numerical_parameters is not described by example, would be useful to see an example in the paper.

Verification I guess you should formulate $z=(v, \beta)$ as $z$ is not explicitly defined in the definition of the Rayleigh ratio.

To me it would be worthwhile rehashing what the exact solutions from the appendix of Pham, K., Marigo, J.-J., & Maurini, C. (2011) is as I am not a domain expert. It would make figure 2-5 easier to interpret. Especially figure 3-5 contains a lot of information that I do not how to easily digest as a non-specialist.

State of the field: Do the authors describe how this software compares to other commonly-used packages? Some references of other software solving similar problems with be useful (maybe linked to my comment in Statement of need.

[kumiori]

Thank you for your comments, all addressed below.

1. we removed the last sentence as recommended

2. In so-called "phase field" fracture, questions of stability are not common practice. We have expanded our consideration in the text. As a reference, you can take the recent book "MEALOR II Damage Mechanics and Local Approach to Fracture" (doi: 10.5281/zenodo.10125170)

"the easiest to implement is the phase-field approach due to its variational nature. This explains its current success. A time-discretization is first performed which gives an incremental objective function. This function is then minimized using an alternate minimization. For instance, for elasto-damage models, a basic finite element solver is used to update the displacement field while another basic finite element solver updates the damage. These two solvers alternate until convergence. Then the next time-step is considered."

For other references of peer-reviewed papers, see also: https://doi.org/10.1007/s10704-024-00763-w https://doi.org/10.1016/j.cma.2010.04.011 https://doi.org/10.1016/j.cma.2015.06.009 https://doi.org/10.1016/j.jmps.2008.10.012 https://doi.org/10.1007/s00466-014-1109-y

3. the problem specification has been moved prior to introducing the solvers, as suggested

4. the section has been renamed as suggested

5. thank you for pointing out the missing definition of $V_0$, it has been specified in the text

6. we have expanded our references to stability problems driven by unilateral principles, as requested, to elucidate the context for non-specialists

7. we have clarified the use of the bounds which apply to the internal (irreversible) order parameter. Currently, the ordering is implicit, namely, the first element of the list bounds is the lower bound, the second is the upper bound.

8. we have added, in the appendix, an example of numerical parameters referring to the computation shown in the manuscript

9. indeed, thank you for pointing out, we have specified the argument of the operator.

10. we have added the explicit solution of the Rayleigh ratio minimisation problems in the appendix, for reference

11. we have expanded our comment in relation to other existing numerical software in the Statement of Need. To the best of our knowledge, there are no commonly-used packages for the solution of the general problem we address.

[kumiori]

Changes are indicated in bold font in the manuscript file.

[jorgensd]

@kumiori The changes look really good. Some minor typos:

of the system as a dictionary (u, alpha), whre the first elemnt is associated to the reversible field and the second to the irreversible component, its the associated constraints on the latter, and the solver’s parameters (see an example in the Appendix).

Should be

of the system as a dictionary (u, alpha), where the first element is associated to the reversible field and the second to the irreversible component, its the associated constraints on the latter, and the solver’s parameters (see an example in the Appendix).

[jorgensd]

In the appendix, could one format the three different conditions (1., 2., 3.) in a slightly more readable way? I get that there are a lot of conditionals for each $\beta$ solution, but maybe using sub-equations (4.1, 4.2 and 4.3 maybe)?

Additionally, eq [2.1] and [2.2] is not explicitly defined, only through the [2] with an implicit assumption that the "and-statement" gives the sub-numbering. I am not super fond of this, but it shouldn't be a blocker to proceed further with the paper..

[kumiori]

Thank you, we took your suggestions and indicated explicitly the sub-labeling of equation 2, for ease and clarity of reference and reformatted the definition of the minimiser, for improved clarity. Additionally, we fixed typos.