# Open Review of “An implicit staggered algorithm for CPFEM-based analysis of aluminum”

Pedro Areias, Charles Dos Santos, Rui Melicio, Nuno Silvestre, Frans van der Meer, Tushar Kanti Mandal, Vinh Phu Nguyen

► To cite this version:

Pedro Areias, Charles Dos Santos, Rui Melicio, Nuno Silvestre, Frans van der Meer, et al.. Open Review of “An implicit staggered algorithm for CPFEM-based analysis of aluminum”. 2024. hal-04700728

HAL Id: hal-04700728

https://hal.science/hal-04700728v1

Submitted on 23 Jan 2025

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Identifiers

Open Review

OAI hal-04700728

Reviewed Article

DOI 10.46298/jtcam.12960

Licence

CC BY 4.0

©The Authors

# Review of “An implicit staggered algorithm for Crystal Plasticity FEM-based analysis of aluminum”

Pedro AREIAS<fnref n="1" />,<fnref n="2" />, Charles DOS SANTOS<fnref n="3" />, Rui MELICIO<fnref n="2" />,<fnref n="4" />, Nuno SILVESTRE<fnref n="1" />,<fnref n="2" />,

Frans VAN DER MEER<fnref n="5" />,R, Tushar KANTI MANDAL<fnref n="6" />,R, and Vinh Phu NGUYEN<fnref n="7" />,E

## Review of version 1

Permalink: https://zenodo.org/records/10581765

### Reviewer 1 (Frans VAN DER MEER)

**Reviewer** The paper presents a crystal plasticity finite element model in which the accuracy of a staggered time stepping algorithm is improved with a substepping method. It is convincingly demonstrated that the baseline numerical implementation is sound and that the new method improves the accuracy of the method for large time steps.

**Authors** We thank the Reviewer for the review and sharpness and tried to improve all points under criticism.

**Reviewer** The paper is suitable for publication in JTCAM. At several key points, organization of the manuscript can be improved.

**Authors** We thank the Reviewer and tried to improve the organization of the manuscript. Organization of sections was modified according to the indications of the Reviewers.

**Reviewer** I advise to revise the paper on the following points:

**Reviewer** In the introduction, the problem description “Staggered solution can be a solution, but naive implementation produce drifting”. Is very minimal. It would be helpful to point out what is meant by “drifting” and what the source of the error is.

**Authors** We agree, we’ve substantially improved the introduction and toned-down the text. E.g. “Staggered algorithms are often used, but naive implementations produce drifting, since there is no measure to reduce the solution error. In that case, both time step and sequence of operations affect the solution [1]. A comprehensive description is given by Felippa, Park, and Farhat [2].”

**Reviewer** Same paragraph: “These are here adopted to improve the accuracy of the staggered algorithm”, specify “the staggered solution algorithm in crystal plasticity simulations” - Next sentence: “which ties the solutions”, this statement is unclear. What is the meaning of “tying” in this context?

**Authors** We agree, the revised text is now: “Effective implicit staggered algorithms have been developed for thermoelasticity [3] and phase-field simulations [4]. These are here adopted to improve the accuracy of the staggered algorithm. For FCC (aluminum), we propose an implicit staggered algo-

<footnote n="1">¹ IST – Departament of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal</footnote>

<footnote n="2">² IDMEC – Instituto de Engenharia Mecânica, Lisboa, Portugal</footnote>

<footnote n="3">³ INSA – Institut National des Sciences Appliquées (alumni), Rouen Normandie, France</footnote>

<footnote n="4">⁴ AEROG – Universidade da Beira Interior, Calçada Fonte do Lameiro, Covilhã, Portugal</footnote>

<footnote n="5">⁵ Delft University of Technology, Faculty of Civil Engineering and Geosciences, The Netherlands</footnote>

<footnote n="6">⁶ Royal Society Newton International Fellow, Department of Engineering Science, University of Oxford, Great Britain</footnote>

<footnote n="7">⁷ Department of Civil Engineering, Monash University, Clayton, Victoria, Australia</footnote>

<footnote>R Reviewer</footnote>

<footnote>E Editor</footnote>
rithm which strongly couples the solutions (in the sense of Matthies, Niekamp and Steindorf [1]) to avoid drifting.”

**Reviewer** In the outline it is stated that Section 4 presents a verification, but there is none in that section.

**Authors** Yes, it is there, spread in two sections. The introduction was modified and hopefully this is better for the journal.

**Reviewer** It would be more logical to move Section 5 before Section 3. (Supposing the EAS element is also used in the numerical example of Section 3).

**Authors** Yes. We changed the order of the sections. Yes the EAS element is used. It is also in Github for inspection (https://github.com/PedroAreiasIST/element).

**Reviewer** I expect in Section 3 to find coupled equations with a staggered solution scheme where one is solved after the other. It remains unclear what the staggered algorithm looks like. It is stated that Equation (18) is solved with Newton iterations. I suppose the $\xi_{s+1}^\alpha$ is computed outside those iterations and that Equation (18) and (20) are evaluated in a staggered fashion. Is that right? Which is evaluated first?

**Authors** Yes, that is exactly what is done. We thank the Reviewer for the excellent insight. We now explicitly state that in the text and follow the system in Section 5:

$$e_{\varepsilon} = \varepsilon_{s+1}^p - \varepsilon_s^p - \Delta t \sum_{\alpha=1}^{12} M_{\alpha} \dot{\gamma}_0^{\alpha} \left| \frac{P_{\alpha} : (\varepsilon - \varepsilon_{s+1}^p)}{\xi_{s+1}^{\alpha}} \right|^n \operatorname{sgn}[\tau_{\alpha}] = 0 \quad (1)$$

$$e_{\gamma} = \left( \delta_{\alpha\gamma} + h_0 h_{\alpha\gamma} \frac{|\Delta\gamma_{s+1}^{\gamma}|}{\xi_{\infty}^{\gamma}} \right) \xi_{s+1}^{\gamma} - \xi_s^{\alpha} - h_0 |\Delta\gamma_{s+1}^{\beta}| h_{\alpha\beta} = 0; \quad \gamma = 1, \dots, 12 \quad (2)$$

**Reviewer** No conclusion is drawn from Figure 8, while Figure 12 is not referred to at all in the text. It should not be left to the reader to determine the significance of these results.

**Authors** Yes, thank you. Figure 8 was improved and Figure 12 is now referenced.

**Reviewer** Equation (40) could use more explanation. Maybe after clarifying the staggered scheme in Section 3 this can be done with little effort. What do $\xi$ and $\gamma$ stand for, is it for solving Eq (20) at fixed $\gamma$ and Eq (18) at fixed $\xi$ respectively?

**Authors** Yes, we’ve done that:

$$\tilde{\xi}_{s+1}^{i+1} = \Xi_{\star}(\xi_{s+1}^{i}) = \Xi\left[\underbrace{\Gamma(\xi_{s+1}^{i}), \xi_{s+1}^{i}}_{\Delta\gamma}\right], \quad (3)$$

see also Eq (28) in [3].

**Reviewer** The opening words of the conclusions are not very informative. “We introduced an alternative method”, alternative to what? “We propose a robust and efficient method” would be better.

**Authors** We thank the Reviewer and replaced the sentence.

**Reviewer** Minor Corrections.

**Reviewer** “constitutive unknowns are at least 18”, should be “there are at least 18 constitutive unknowns”.

**Authors** We performed the correction, the sentence was reworked.

**Reviewer** In the statement after equation (17), it is confusing to say that “non-negativity of (17) is assumed” when the equation does become negative. Please rephrase.

**Authors** Yes, you’re right. The condition is $\tau_{\alpha}^* \Delta\gamma_{s+1}^{\alpha} \ge 0$.

**Reviewer** “time step $n$” above Eq (38) should be “time step $s$”.

**Authors** Yes, we apologize for this. It was corrected.
Reviewer 2 (Tushar KANTI MANDAL)

**Reviewer** The manuscript presents a dynamic relaxation based implementation of a Crystal Plasticity FEM (CPFEM) code. The implementation is applied for the stress analysis of Aluminium having 12 slip planes. There are some novelty in the contribution and certainly within the scope of the journal.

**Authors** We thank the Reviewer for the review and strong insights.

**Reviewer** However, some of results raises concerns regarding the accuracy of the implementation (in terms of discretisation converged solution or not). Therefore, the manuscript requires a major revision to address the following comments.

**Reviewer** Figure 7: Could the author explain mesh size sensitivity of the average stress-strain response? Is it possible to achieve mesh convergence for this problem?

**Authors** Yes, we've gone further in the analysis, with a finer mesh and convergence was obtained. Our code was not especially optimized at the time of submission, it was too slow to run the finer mesh, but now it is done. It's not perfect and some oscillations occur (see Figure 1 below and new Figure 7 in the manuscript)

<figure><img src="image_2.png" /><figcaption>**Figure 1** Effect of $h$ on the average stress $\bar{\sigma}_{33}$ and $\bar{E}_{33}$ for two orientations.</figcaption></figure>

**Reviewer** Also, why we observe significant softening in the same plots? Did the author incorporated in damage law or something equivalent? Kindly clarify.

**Authors** Yes, we've used $\xi_{\infty}^* = 7 \times 10^6$ Pa in that specific analysis, that was stated in the text, but perhaps it was a little hidden. Now it is more visible.

**Reviewer** Could we achieve time step size converged solution in Figure 9(b)? In the same context, how do we choose the time step size for this problem, using CFL condition or something equivalent? (Again, we are talking about the implicit algorithms here!). This needs clarification in the text.

**Authors** Viscosity is used, as in most single-crystal plasticity works, to circumvent the combinatorial nature of standard flow laws for 12 systems. We use $\bar{E}_{33} = 0.001/(12.5 \times 10^{-3} = 0.08s^{-1})$ which is sufficient to produce the required effect. Time step is $\Delta t = 1 \times 10^{-3}$ s, which results in a reasonable strain increment of $\Delta \bar{E}_{33} = 8 \times 10^{-5}$. Concerning convergence, we used more iterations in dynamic relaxation and the result is shown in Figure 2 below and in new Figure 9 in the manuscript.

**Reviewer** Figure 13: The results for different $\xi$ seems to be very similar, could the author show only a few but more legible figures.

**Authors** Yes, that was performed according to the indications. Only $\xi_1, \xi_5, \xi_6, \xi_7, \xi_9$ and $\xi_{12}$ are now shown, and are more visible.

**Reviewer** The authors should summarise the different algorithms (uncoupled, staggered, etc) compared in the manuscript before the numerical comparison. Also, the overall quality of the manuscript should be improved in terms of conciseness in writing.
<figure><img src="image_3.png" /><figcaption>**Figure 2** Effect of dynamic relaxation on the evolution of $[\epsilon_p]_{33}$.</figcaption></figure>

**Authors** We addressed that request:

* Classical full Newton iteration for the coupled constitutive system (e.g. [5]).

* Sequential solution of the constitutive equations, in a staggered form.

* Fixed point iteration with relaxation (e.g. in the fluid-structure interaction context [6, 7]).

* Block-Newton partition approach with approximate out-of-diagonal blocks (e.g. [8]).

* Block-Newton-Krylov where only multiplications are used for the out-of-diagonal blocks [9].

**Reviewer** It is mentioned that implementation of Eq. 31 requires 840 lines of Fortran code. Which seems to be practically very cumbersome and possibility of many mistakes. Could the authors comment of possible advantages in the implementation of CPFEM in recent finite element codes with automatic differentiation capabilities (e.g. FeniCS)?

**Authors** this was performed automatically with AceGen and Mathematica, which removes the presence of errors, please confirm Github [10], https://github.com/PedroAreiasIST/logs. We are aware of FeniCS but have our own code to maintain.

## Review of version 2

Permalink: https://zenodo.org/records/10581765

### Reviewer 1 (Frans VAN DER MEER)

I recommend to accept the paper in its present form.

### Reviewer 2 (Tushar KANTI MANDAL)

The authors have responded to all of the comments and suggestions. The manuscript is revised accordingly. The manuscript can be accepted in its present form.
Editor's assessments (Vinh Phu NGUYEN)

The paper presents a crystal plasticity finite element model in which the accuracy of a staggered time stepping algorithm is improved with a sub-stepping method. The results are convincing. The paper is suitable for publication in JTCAM. After two revisions, the authors have addressed all the comments raised by two reviewers, and we are happy to accept it.

References

[1] H.G. Matthies, R. Niekamp, and J. Steindorf. Algorithms for strong coupling procedures. *Comp Method Appl M*, 195:2028–2049, 2006.

[2] C.A. Felippa, K.C. Park, and C. Farhat. Partitioned analysis of coupled mechanical systems. *Comp Method Appl M*, 190:3247–3270, 2001.

[3] P. Erbts and A. Düster. Accelerated staggered coupling schemes for problems of thermoelasticity at finite strains. *Comput Math Appl*, 64(8):2408–2430, 2012.

[4] Y. Schapira, L. Radtke, S. Killmannsberger, and A. Düster. Performance of acceleration techniques for staggered phase-field solutions. *Comp Method Appl M*, 410:116029, 2023.

[5] de Souza Neto, E.A. and Perić, D. and Owen, D.R.J. *Computational Methods for Plasticity: Theory and Applications*. John Wiley and Sons, Ltd., West Sussex, England, 2008.

[6] P. Le Tallec and J. Mouro. Fluid structure interaction with large structural displacements. *Comp Method Appl M*, 190(24-25):3039–3067, 2001.

[7] U. Küttler and W.A. Wall. Fixed-point fluid-structure interaction solvers with dynamic relaxation. *Comput Mech*, 43:61–72, 2008.

[8] H.G. Matthies and J. Steindorf. Partitioned strong coupling algorithms for fluid-structure interaction. *Comput Struct*, 81:805–812, 2003.

[9] C. Michler, E. van Brummelen, and R. de Borst. Error-amplification analysis of subiteration-preconditioned GMRES for fluid-structure interaction. *Comp Method Appl M*, 195(17-18):2124–2148, 2006.

[10] P. Areias. Material logarithmic strain: transformations and tangent modulus. https://github.com/PedroAreiasIST/logs, 2023.