# Review of "On the bifurcation analysis of thin multilayer structures by asymptotic numerical method"

Hamza Azzayani, Hamid Zahrouni, Norman Mathieu, Pascal Ventura, Michael

Brun, Michel Potier-Ferry, Olivier Thomas

► To cite this version:

Hamza Azzayani, Hamid Zahrouni, Norman Mathieu, Pascal Ventura, Michael Brun, et al. Review of "On the bifurcation analysis of thin multilayer structures by asymptotic numerical method". 2024. [hal-04819388](https://hal.science/hal-04819388v1)

HAL Id: hal-04819388

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Submitted on 14 Mar 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-04819388

Reviewed Article

DOI 10.46298/jtcam.10170

Licence

CC BY 4.0

©The Authors

# Review of “On the bifurcation analysis of thin multilayer structures by the Asymptotic Numerical Method”

©Hamza AZZAYANI<fnref n="1" />, ©Hamid ZAHROUNI<fnref n="1" />, ©Norman MATHIEU<fnref n="1" />, ©Pascal VENTURA<fnref n="1" />,

©Michael BRUN<fnref n="1" />, ©Michel POTIER-FERRY<fnref n="1" />, and ©Olivier THOMAS<fnref n="2" />,<fnref n="3" />

## Review of version 1

Permalink: hal-03817307v1

**Authors** The authors thank the reviewers for their enlightening comments and remarks. Great care has been taken to correct the manuscript accordingly.

## Reviewer 1 (Anonymous)

**Reviewer** Authors of this study have performed an investigation on the bifurcation analysis of thin multilayer structures by asymptotic numerical method. In general, this manuscript is interesting and brings novel results. It deserves to be published. However following comments should be addressed.

**Reviewer** The Abstract is not well written. There is no explanation of the necessity and innovation of this study.

**Authors** We have improved the abstract by emphasizing the important points of this paper.

**Reviewer** In Fig.3, the authors compared their results with Abaqus results, however, mesh parameter settings, and Abaqus solving steps should also be given and discussed in detail

**Authors** We gave more details about Abaqus parameters in page 17 of the paper. Reviewer is right, it is important to give the type of element used in Abaqus and the procedure of resolution. We have chosen the shell element S8R which is equivalent to the one used in the ANM algorithm. The mesh is sufficiently accurate with 36 elements. This gives the same response curve in the both cases of ANM and of Abaqus. We have used the implicit version of Abaqus for nonlinear analysis: Newton Raphson algorithm is associated to Riks technique to compute complex nonlinear responses involving limit points.

**Reviewer** Please provide the corresponding program code in the attachment for the reviewer to verify its correctness

**Authors** Reviewer could refer to the numerous publications about ANM. This technique has been developed by our team for several decades. It has shown its effectiveness on several applications and has been published in hundreds of papers see web of science

**Reviewer** The physical explanations of the results are insufficient. In Figs. 5 and 8, the authors should present essential explanations and tell the reasons in detail.

**Authors** It is true that we did not insist on the physical aspect concerning the response curves. We focused our attention on the numerical aspect and on the ability of our technique to compute the different

<footnote n="1">¹ Université de Lorraine, CNRS, Arts et Métiers, LEM3, Metz, France, Laboratory of excellence on Design of Alloy Metals for low-mAss Structures (DAMAS), Université de Lorraine, France</footnote>

<footnote n="2">² Arts et Métiers Institute of Technology, LISPEN, HESAM Université, F-59000 Lille, France</footnote>

<footnote n="3">³ Editor</footnote>
singular points on the response curve. Indeed, the physical aspect of buckling and post buckling of thin structures has been widely discussed in several publications. If we consider for example the response curve of figure 3 (see Fig 1 below), we can distinguish different parts of the solution branch. For example, from the starting point to point 20, we have a stable branch of the solution until the limit point which is followed by an unstable branch corresponding to a decrease in loading until the second limit point near the step 70. Then we have a transition zone and then an increase in loading corresponding to a branch of stable solution. In practice, when starting from zero, the structure supports loading up to the limit point (step 20 in figure 3) then we observe a jump of the solution to the stable branch as shown in the figure below (snap through phenomena). In figures 5, all the solutions exist and correspond to equilibrium states. They can be computed numerically but in practice, the structure will jump from a stable solution to a second one.

<figure><img src="image_2.png" /><figcaption>**Figure 1** Copy of figure 3 of the corrected version of the manuscript.</figcaption></figure>

**Reviewer** In addition, as the author stated that, ANM is an efficient method to solve nonlinear problems with high accuracy (in particular for instability modeling of thin structures). What are the advantages and disadvantages of the asymptotic numerical method compared with other numerical methods in the calculation of bifurcation? Is it possible to increase the comparison results of the asymptotic numerical method and other numerical methods in calculating the bifurcation of the thin multilayer structures

**Authors** Traditionally, an imperfection has to be introduced in the system to capture the bifurcated branch with a continuation method. ANM permits to detect these bifurcations with small or without imperfection. Moreover, the computation of natural frequencies permits to check the stability of the solutions along the branch. As far as the authors are aware, no other method currently exists for accurately identifying bifurcation points in the context of non linear pre buckling, which is the specific focus of this article.

## Review of version 2

Permalink: hal-03817307v2

## Reviewer 1 (Anonymous)

**Reviewer** Author proposes several calculation methods to improve the accuracy of calculating the stability of thin plates, this study is very interesting but there are some small problems that need to be solved.

**Reviewer** The quality of the figures should be improved. The numbers of the figures should appear in the text in order.

**Authors** I've improved the quality of Figures 1 and 2 (see the new version of the paper). The other figures are in good quality with eps format.
**Reviewer** Please explain what “g” represents in Equation 2.

**Authors** The vectors $g_i$ and $\bar{g}_i$ represent the covariant basis vectors in the initial and deformed configurations, respectively. See the updated version of the paper (Section 2.1).

**Reviewer** In Section 4, the authors should provide some details of the employed finite element type and the properties of the mesh used to model the structures.

**Authors** For the proposed application, we used a regular mesh with 36 eight-node shell element (S8R in Abaqus / Standard). We used Abaqus software for validation with Riks control. Both ANM and Abaqus use the same number of elements. See the updated version of the paper in Section 4, page 17: "The structure is discretized using 36 shell elements presented in section 2." and "The same problem is simulated using Abaqus with Newton Raphson algorithm and Riks control. The mesh is the same as the one used for the ANM with an eight node shell element S8R available in Abaqus / Standard."

**Reviewer** The author mentioned in Introduction that "For reasons of cost and competitiveness, manufacturers tend to use strong and light materials and complex structures." and Auxetic Mechanical Metamaterial is a lightweight structure with good performance. The author may refer to the following articles: 10.3390/ma16051808; 10.3390/ma16083262; 10.1016/j.compositesb.2023.110585; 10.1002/adem.202000312.

**Authors** We added the example of auxetic mechanical metamaterials in the first paragraph of the introduction with the references given below. See Section 1 of the updated paper. "For reasons of cost and competitiveness, manufacturers tend to use light materials and complex structures such as auxetic mechanical metamaterials [1-4]."

## Reviewer 2 (Anonymous)

**Reviewer** The paper "On the bifurcation analysis of thin multilayer structures by asymptotic numerical method" by H. Azzayani is a sound scientific report on the use of the asymptotic numerical method for the path following and bifurcation analysis of shell structures.

It deals with a difficult subject and provides convincing results based on an established methodology. The results are verified against numerical simulations obtained with commercial software, proving the correctness and the efficiency of the proposed numerical method on a very difficult nonlinear shell problem.

My main concern is about the originality, since the authors do not clearly state in the abstract, introduction, and the conclusions which are the main original contribution with respect to the existing literature on the ANM. Similar problem have been addressed in the past (see e.g. [19, 24]).

My opinion is that the paper can be published in JTCAM provided that the author improve the abstract, introduction, and conclusions to state clearly the addition with respect to the existing literature. I suggest few further revisions on the presentation.

**Reviewer** Please find below a list of specific comments:

**Reviewer** Please add at the end of the introduction, for example at line 90, a paragraph about the scope of the paper and its originality with respect to the existing literature.

**Authors** In this paper, we have reviewed various techniques for detecting bifurcation points. In addition, we focus on the effects of changing the solution presentation (polynomial or rational), the series truncation order and the accuracy parameter on the equilibrium path. These techniques have been applied for the first time to multilayer composite structures. See introduction section.

**Abstract:** "The paper evaluates these three bifurcation detection techniques for multilayer composite structures. It also analyses the impact of the solution representation by power series or Padé approximants, the truncation order, and the accuracy parameter on the solution path."

**Introduction:** "This paper examines the application of three previously discussed bifurcation detection methods to multilayer composite shells. The aim is to assess the reliability of these techniques in the context of strong geometrical nonlinearities. The results are not limited to the first bifurcation point, which is the most critical, but allow the detection of all possible bifurcation points of a shell structure, which constitutes an improvement regarding the state of the art. We
also analyze the influence of solution representation (polynomial or rational), truncation order, and accuracy parameter on the equilibrium path."

**Reviewer** English and formatting need some further revisions. For example: punctuation is not present after equations and the text after the equations is indented even it is on the same sentence of the equation (see e.g. after equation 1, 11, 12 ...)

**Authors** The above remark is taken into account in the new paper version.

**Reviewer** The formulation of the Hu-Washizu mixed formulation in section 2.3 is not clear. The authors provide the energy functional, but they not formulate the variational problem. The problem is not simply “principle of minimum potential energy” as stated before eq. (14), because they deal with a mixed method. Please formulate clearly the stationarity principle, including information on the functional space where the fields must lie.

**Authors** We have tried to rewrite the whole section "A three-field variational formulation". The starting point is the Hu-Washizu functional. Considering:

i - the orthogonality condition between the stress field in the mth layer $S^m$ and the additional strain $\tilde{\gamma}$.

ii - The variational problem must be expressed within a quadratic framework to apply the perturbation technique.

Consequently, we arrive at the following variational problem:

$$ \left\{ \begin{array}{l} \displaystyle \sum_{m=1}^{N} \int_{\Omega} \delta\gamma_u : S^m d\Omega - \lambda P_e(\delta u) = 0 \\ \\ \displaystyle \sum_{m=1}^{N} \int_{\Omega} \delta\tilde{\gamma} : S^m d\Omega = 0 \\ \\ \displaystyle S^m = D^m : (\gamma_u + \tilde{\gamma}) \end{array} \right. \qquad (1) $$

We have also defined the functional spaces for each field (see the updated paper): "Let us consider a material domain $\Omega$ which will consist in a shell structure. The shell formulation considered in the present study is based on a three-dimensional formulation. The unknowns are the displacement field $u$, the Green-Lagrange strain $\gamma$ and the second Piola-Kirchhoff stress tensor in the mth layer $S^m$. A consistent functional framework can be obtained by considering Sobolev spaces: $u \in W^{1,4}(\Omega)$, $\tilde{\gamma} \in L^2(\Omega)$ and $S^m \in L^2(\Omega)$. The equations of the problem are equivalent to the stationarity of the Hu-Washizu functional, which is written as follows:

$$ \pi_{HW}(u, \gamma, S^m) = \sum_{m=1}^{N} \int_{\Omega} \left\{ \frac{1}{2} \gamma : D^m : \gamma^{-t} S^m : (\gamma - \gamma_u) \right\} d\Omega - \lambda P_e(u) \qquad (2) $$

where N refers to the number of layers of laminated shells and $\lambda P_e(u)$ is the work of the external forces ($\lambda$ is a scalar load parameter). Considering the orthogonality condition between the stress field in the mth layer $S^m$ and the additional strain field $\tilde{\gamma}$, equation (2) becomes:

$$ \pi_{EAS}(u, \tilde{\gamma}) = \sum_{m=1}^{N} \int_{\Omega} \frac{1}{2} (\gamma_u + \tilde{\gamma}) : D^m : (\gamma_u + \tilde{\gamma}) d\Omega - \lambda P_e(u) \qquad (3) $$

Thus, the variational problem is written as follows:

$$ \delta\pi_{EAS} = \sum_{m=1}^{N} \int_{\Omega} (\delta\gamma_u + \delta\tilde{\gamma}) : D^m : (\gamma_u + \tilde{\gamma}) d\Omega - \lambda P_e(\delta u) \qquad (4) $$

This formulation is not the optimal one for applying the perturbation technique because the equilibrium equation is cubic with respect to the displacement field $u$. High efficiency can be achieved by employing a variational formulation within a quadratic framework, which is easily accomplished by reintroducing the stress field and the behavior law, as detailed in [33]. Thus,
the variational problem leads to the equilibrium equation, the compatibility equation which is equivalent to the orthogonality condition and the behavior equation:

$$ 
\left\{ 
\begin{array}{l}
\displaystyle \sum_{m=1}^{N} \int_{\Omega} \delta \boldsymbol{y}_u : S^m d\Omega - \lambda P_e(\delta u) = 0 \\[1.5em]
\displaystyle \sum_{m=1}^{N} \int_{\Omega} \delta \tilde{\boldsymbol{y}} : S^m d\Omega = 0 \\[1.5em]
S^m = D^m : (\boldsymbol{y}_u + \tilde{\boldsymbol{y}})
\end{array}
\right. 
\qquad (5) $$

Equations (5) are quadratic with respect to the unknowns and can be expressed in the general form [see Cochelin et al. 2008, https://doi.org/10.1080/17797179.2008.9737353]:

$$ R(U, \lambda) = L(U) + Q(U, U) - \lambda F = 0 \qquad (6) $$

where $U = (u, \tilde{y}, S^m)$ is an unknown vector, $L(.)$ a linear operator, $Q(., .)$ a quadratic one, $F$ the external load vector and $R$ the residual vector. Thus, the problem is to find the mixed variable $U$ and the loading parameter $\lambda$ verifying equation (6)."

**Reviewer** In line 165 the authors write “We present a new algorithm for the numerical computation of the solution path.” What is the new algorithm? In what it differs from previous methods presented in the literature (e.g. [19] and related references)?

**Authors** The new algorithm concerns the representation change of the solution (from polynomial to rational representation). It is the same algorithm used in reference [19] of the previous version of this paper.

Editor’s assessment (Olivier THOMAS)

**Editor** This article presents an application of the use of the asymptotic numerical method for the computation of the geometrically nonlinear statics of shell structures, with a focus on the comparison of three techniques to detect bifurcation points. The two reviews were accurate and deep and the authors provided an excellent job in revising the paper by addressing all the points raised by the reviewers, improving the quality of the presentation of the work.

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<figure><img src="image_3.png" /></figure>