# Review of “Multi-material topology optimization of structural load-bearing capacity using limit analysis”

Leyla Mourad, Jeremy Bleyer, Romain Mesnil, Joanna Nseir, Karam Sab, Wassim Raphael, Fabian Duddeck, Alexander Popp

► To cite this version:

Leyla Mourad, Jeremy Bleyer, Romain Mesnil, Joanna Nseir, Karam Sab, et al. Review of “Multi-material topology optimization of structural load-bearing capacity using limit analysis”. 2024. hal-04693259

HAL Id: hal-04693259

https://hal.science/hal-04693259v1

Submitted on 14 Mar 2025

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Distributed under a Creative Commons Attribution 4.0 International License
Identifiers

Open Review

OAI hal-04693259

Reviewed Article

DOI 10.46298/jtcam.12427

Licence

CC BY 4.0

©The Authors

# Review of “Multi-material topology optimization of structural load-bearing capacity using limit analysis”

Leyla MOURAD<sup>1,2</sup>, Jeremy BLEYER<sup>1</sup>, Romain MESNIL<sup>1</sup>, Joanna NSEIR<sup>2</sup>, Karam SAB<sup>1</sup>, Wassim RAPHAEL<sup>2</sup>, Fabian DUDDECK<sup>3,R</sup>, and Alexander POPP<sup>4,E</sup>

## Review of version 1

Permalink: hal-03528107v2

Please note that the first version submitted to JTCAM is v2 of the hal-03528107 depository.

### 1 Reviewer 1 (Anonymous)

**Reviewer** In this manuscript, the work in Mourad et al. (2021) is extended to also include reinforcement and bi-materials. I only have some minor concerns that I suggest the authors consider before the manuscript is accepted.

**Reviewer** Section 2, I believe that this was introduced by Kammoun et al. (2014), please change.

**Authors** In Kammoun et al. (2014) a first formulation of limit analysis-based formulation was indeed proposed but limited only to the volume minimization formulation, von Mises plasticity and no penalization or filtering strategy. In the present paper, we build upon our previous work Mourad et al. (2021) which presented a generic formulation in terms of load capacity maximization, for any strength criterion and with penalization and filtering. In this paper, we also discussed its link to a volume minimization formulation which is not obvious at first sight. We changed the sentence accordingly but we believe that credit has been properly given here as well as in the introduction.

**Reviewer** Switch 2.1 and 2.2, and give clear credit to Kammoun et al. (2014).

**Authors** Again, our initial motivation was to maximize load-bearing capacity to produce mechanically efficient structures. Volume-minimization formulation is indeed extremely related to this problem as we have shown in Mourad et al. (2021). We prefer starting the presentation with section 2.1 as it is much closer to elastic stiffness maximization and limit analysis formulations.

**Reviewer** Please discuss the difference to 2.1 and 2.2 to a standard elastic problem with stress constraints.

**Authors** We added a sentence discussing the difference with a stress-constrained elastic problem.

<footnote n="1">¹ Laboratoire Navier, Ecole des Ponts ParisTech, Univ Gustave Eiffel, CNRS, France</footnote>

<footnote n="2">² Université Saint Joseph, Faculté des sciences, Mar Roukos-Dekwaneh, Lebanon</footnote>

<footnote n="3">³ Technical University of Munich, School of Engineering and Design, Germany</footnote>

<footnote n="4">⁴ Institute for Mathematics and Computer-Based Simulation, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 85577, Neubiberg, Germany</footnote>

<footnote>R Reviewer</footnote>

<footnote>E Editor</footnote>
**Reviewer** Why are you using (6) and not (5)? Why complicate (6) using the Minkowski sum in (4)? Section 3.1 is central, but written in an unclear manner. I don't like this. You only add two stress fields, where you use different limit criterions for each field. Or, am I missing something?

**Authors** We simplified the presentation and removed formulation (5) since we use only (6). We prefer discussing the latter to avoid any discussion regarding the influence of the reinforcement volume fraction which we assume to be small in any case.

**Reviewer** I even believe that you could see this reinforcement formulation in (6) as a special case of the bi-material formulation in Section 4. If so, please include Section 3 in Section 4 and derive (6) as a special case.

**Authors** Indeed, section 3 can be obtained as special case of the bi-material formulation when $1$ denotes the matrix phase and when fixing $\rho_1 = 1$ everywhere and ignoring the constraint $\rho_1 + \rho_2 \le 1$ which enforces in a relaxed manner that only one phase can exist simultaneously. We added a comment on this matter in section 4 and recalled it in the conclusion. However, for pedagogical reason we find it easier for the reader to start with the simpler reinforcement formulation which takes into account only one density to optimize rather than start with the more general formulation and simplify after, accounting also for the fact that we need to remove a constraint so particularization is not that immediate.

**Reviewer** Please provide plots on the convergence history.

**Authors** Plots of convergence history are already provided in Fig. 9 and 15.

## 2 Reviewer 2 (Fabian DUDDECK)

**Reviewer** Thanks for your clear and interesting paper. I have only minor comments (complemented with additional comments in the pdf)

**Authors** We took into account the comments made in the pdf.

**Reviewer** Page 3: You write "... with the material strength criterion "$\sigma \in G$"" which means that $\sigma$ is not the stress but a stress criterion (yield strength?). Then you use $\sigma$ in eq. (1) as stress. This is irritating - could you please clarify this better in the text. You also have "Here, $G$ is a convex set containing ..." in the text - for me, there is something missing.

**Authors** Indeed, this is an abuse of language. We corrected with $\sigma \in G$ being the "strength condition" and $G$ being the strength criterion, which is a convex set as usually assumed in limit analysis. We propagated the corrections at several places in the text.

**Reviewer** In standard topology optimization, you have checkerboard issues; do you have the same problems in your approach as well? I recommend adding a paragraph on this.

**Authors** Checkerboard issues indeed happen when selecting low-order spaces (e.g. piecewise constant) for the density field and when applying no spatial regularization techniques. Here we use a bound on the density gradient to limit mesh localization or instabilities. We added a paragraph in Section 5.1 on this topic.

**Reviewer** Page 11: Define in the text the $P$ in $P_1 / P_2$ - Lagrange interpolation.

**Authors** We added (quadratic/linear polynomial) in the text.

**Reviewer** I also recommend improving the replicability of the results from the examples. How many elements are used in each case? Which element types? Do you have a regular mesh? Etc.

**Authors** We added some information regarding the element types and mesh regularity in the text. The mesh sizes and geometry are available as demos of the `fenics_optim` package so that results are completely replicable.

**Reviewer** Fig. 17(b): Define in the caption why you have dotted and solid lines here.

**Authors** This has been added.
**Reviewer** I also miss some information on the numerics of the optimization process; is it reasonable to provide information on iterations, convergence, and/or locality of the optima (this is clear for the convexified problems, but can that be related to the non-convex original problems?).

**Authors** Regarding the numerics, we provided convergence history plots in Fig. 9 and 15. for the penalization continuation procedure. Each iteration requires solving a convex conic-constrained optimization problem. For this, the MOSEK solver is particularly robust and does not pose any convergence issues in general. Depending on the selected tolerance criteria, each resolution requires roughly 20-30 iterations of the primal-dual interior point solver used in MOSEK. We added a comment on this aspect.

Regarding optimum locality, we cannot risk any comment on the optimality of the computed solutions with respect to the original non-convex problem. Our point of view is to mimick what is traditionally done with the SIMP method which proved quite efficient and convincing. However, it is known for the elastic topology optimization case that optimal solutions are multi-rank laminates so that SIMP solutions are not optimal, although very close in practice. We are not aware of any similar result in the limit analysis case but it might certainly be quite similar. We added a comment on the conclusion section to question the optimality of the computed solutions.

**Reviewer** Using some other methods, we had problems with the pseudo-voids generated by the topology optimizer, i.e. the elements were not completely eliminated but only the material parameters were reduced to very low values. Then, we realized that a plasticity model with the assumption of plastic incompressibility like the von Mises model represents voids with a remaining resistance to volume change, which is not appropriate. Do you have the same issue? Would it make sense to include such a statement into your paper?

**Authors** Indeed, we agree that with the present approach where we apply a homothetic reduction, a von Mises model would induce a remaining resistance to volume change. In our approach, we advocate for the use of bounded criteria, in particular the $L_1$-Rankine criterion. In this case, in the limit of $\rho \to 0$, the latter degenerates to $\{0\}$ so that this issue does not appear. We added a small remark on this aspect in section 2.3.

## Review of version 2

Permalink: hal-03528107v3

## 3 Reviewer 1 (Anonymous)

**Reviewer** Could you please explain the convergence histories shown in Fig. 9 and Fig. 15 more? One would expect much more than just ten iterations for solving a complex topology optimization problem. It seems that these are not the overall convergence histories, but maybe just some part of the procedure? Please provide some more details.

**Authors** We have highlighted in text and captions of fig 9 and 15 that we refer to the outer iterations of the global penalization procedure. The last paragraph of sec 5.3.4 clearly states that one of such outer iteration corresponds to solving a convex optimization problem, the latter being solved iteratively (inner iterations) by the primal-dual interior point method used by Mosek. The total amount of outer*inner iterations therefore amounts to a few hundreds in practice. This might be closer to what the reviewer is expecting in terms of iteration count for a topology optimization problem.

## 4 Reviewer 2 (Fabian DUDDECK)

**Reviewer** Thanks for considering the comments in the first review. I now do not have additional comments except two spelling comments:

1. Please replace "refering" by "referring" in page 3.

2. Please replace "peanlization" by "penalization" on page 21.
# 5 Editor’s assessments (Alexander POPP)

**Editor** The present contribution extends the problem of optimizing structural load-bearing capacity to include multiple materials, focusing on reinforcement optimization within a fixed matrix and bi-material topologies. The approach employs convex optimization and limit analysis theory to maximize load-bearing capacity or minimize volume, with applications demonstrating versatility in strength criteria, orientation constraints, and comparisons to reinforced concrete models. The initial manuscript underwent a rigorous review process, with two expert referees offering their assessments. Both review reports have been very positive and only requested minor modifications. The authors provided such a minor revision and the manuscript was accepted for publication accordingly.

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