# On the use of model order reduction for simulating automated fibre placement processes

- Nicolas Bur
^{1}, - Pierre Joyot
^{1}, - Chady Ghnatios
^{2, 3}, - Pierre Villon
^{4}, - Elías Cueto
^{5}and - Francisco Chinesta
^{3}Email author

**3**:4

**DOI: **10.1186/s40323-016-0056-x

© Bur et al. 2016

**Received: **14 August 2015

**Accepted: **28 January 2016

**Published: **5 March 2016

## Abstract

Automated fibre placement (AFP) is an incipient manufacturing process for composite structures. Despite its conceptual simplicity it involves many complexities related to the necessity of melting the thermoplastic at the interface tape-substrate, ensuring the consolidation that needs the diffusion of molecules and control the residual stresses installation responsible of the residual deformations of the formed parts. The optimisation of the process and the determination of the process window requires a plethora of simulations because there are many parameters involved in the characterization of the material and the process. The exploration of the design space cannot be envisaged by using standard simulation techniques. In this paper we propose the off-line calculation of rich parametric solutions that can be then explored on-line in real time in order to perform inverse analysis, process optimisation or on-line simulation-based control. In particular, in the present work, and in continuity with our former works, we consider two main extra-parameters, the first related to the line acceleration and the second to the number of plies laid-up.

### Keywords

Composites Automated fibre placement Numerical simulation Model order reduction PGD Simulation based control## Background

The numerical simulation of such a process is the subject on an intensive research work. Indeed, because of the successive heating and cooling of the structure during the addition of new tapes, residual stresses appear in the formed part. The evaluation of these residual stresses is crucial because they have a significant impact on both the mechanical properties and the geometry of the manufactured plate or shell due to the spring-back. In order to evaluate and control the evolution of such residual stresses an accurate evaluation of the thermal history is required.

Several models were proposed since the early 90’s. We can mention in particular the numerical analysis made by Sonnez et al. [19] and the work by Pitchumani et al. [15] interested in the study of interfacial bonding. In the latter, the domain considered was only 2d and strong assumptions were introduced in the thermal model, in particular concerning the boundary conditions. Moreover, in order to simplify the geometry of the domain, an incoming tow was assumed instantaneously laid down all along the substrate, which is far from being the case in the real process. Finally, the thermal/mechanical contact was assumed to be perfect at the inter-ply interfaces, which again seems to be a crude assumption. First attempts of the modelling and simulations of this process can be found in [14, 18].

In [8] we proposed some improvements to existing models. First of all, the domain was considered 3d and the material anisotropic. In order to take into account the imperfect adhesion at the inter-ply interface, thermal contact resistances were introduced. Regarding the mechanical problem, the incoming tow was progressively laid down on the substrate and was subjected to a tension force in order to reproduce the pre-tension applied in the real process. But actually, beyond the model itself, the numerical method employed for the solution of the thermal and mechanical problems associated to the AFP process was novel. That work represented a first step towards a global thermo-mechanical process modelling using robust and efficient numerical tools. The numerical strategy we proposed was based on the proper generalized decomposition (PGD) [1, 2]. This method uses a separated representation of the unknown field, in that case temperature or displacements, and results in a tremendous reduction of the computational complexity of the model solution. Moreover, it entails the ability to introduce any type of parameters (geometrical, material, etc.) as extra-coordinates into the model, to obtain, by solving only once the resulting multidimensional model, the whole envelope containing all possible solutions, a sort of computational vademecum that can be then exploited on-line even on light, deployed, computing platforms like smartphones or tablets [4, 9, 11, 12, 16].

However, in those simulations the laying velocity was considered constant. Thus, transient regimes were not taken into account, and these regimes are of special interest for controlling processes that usually involve repeated accelerations and decelerations. When accelerating, the heating power should increase to ensure melting and molecular diffusion, and when decelerating the heating power must decrease in order to prevent thermal degradation. Because the process control must operate in real time, parametric solutions should be computed off-line in order to be used on-line for process control or process optimisation Moreover, in our previous works we considered fixed the number of plies involved in the laminate, and then many parametric solutions were needed, one for each number of plies. The present work represents a step forward and considers the number of plies as a new model extra-parameter. Thermal contact resistances were successfully addressed in [8] and for the sake of simplicity, and even if they are discussed in next sections, they are not be considered in the numerical examples addressed in the last section of the present work.

The main aim of the present work is the proposal of some advanced tools for the efficient simulation of a complex composites manufacturing process, and their numerical analysis. Their consideration into an integrated simulation platform of the real industrial process is beyond the objective of the present work.

In what follows we revisit in “PGD at a glance” section the PGD discretisation technique and in “Parameters becoming coordinates” section its application for computing parametric solutions involving material parameters, initial and boundary conditions and parameters defining the domain in which the problem is defined. Modelling of the AFP manufacturing process is addressed in “Process modelling” section, with special emphasis in the consideration of the number of plies as model parameter. “From steady-state to transient parametric solutions” section focusses on transient regimes and the use of the resulting parametric solutions for process control purposes. Finally “Conclusion” section addresses few conclusions and perspectives.

## PGD at a glance

Consider a problem defined in a space of dimension *d* for the unknown field \(u(x_1,\ldots ,x_d)\). Here, the coordinates \(x_i\) denote any usual coordinate (scalar or vectorial) related to physical space, time, or conformation space in microscopic descriptions [1], for example, but they could also include, as we illustrate later, model parameters such as boundary conditions or material parameters.

*N*functional products involving each a number

*d*of functions \(X_i^j(x_j)\) that are unknown

*a priori*. It is constructed by successive enrichment, whereby each functional product is determined in sequence. At a particular enrichment step \(n+1\), the functions \(X_i^j(x_j)\) are known for \(i \le n\) from the previous steps, and one must compute the new product involving the

*d*unknown functions \(X_{n+1}^j(x_j)\). This is achieved by invoking the weak form of the problem under consideration. The resulting problem is non-linear, which implies that iterations are needed at each enrichment step. A low-dimensional problem can thus be defined in \( \varOmega _j \) for each of the

*d*functions \(X_{n+1}^j(x_j)\).

If \(\mathcal M\) nodes are used to discretise each coordinate, the total number of PGD unknowns is \(N \times \mathcal M \times d\) instead of the \(\mathcal M^d\) degrees of freedom involved in standard mesh-based discretisations.

*Q*parameters \(p^1, \ldots , p^Q\), where \(p^j \in \varOmega _{p^j}\), with \(j=1, \ldots , Q\), the solution is sought under the separated form

PGD solution procedures have been extensively described in our former works and successfully applied in a plethora of applications. The interested reader can refer to the reviews [5, 6, 7] as well as to the primer [10] that describes the practical issues related to its computational implementation. For this reason in “Process modelling” section we will focus in some novel aspects that AFP processes involve. Among them we are considering two issues: (i) the consideration of the number of plies as a model parameter, allowing the solution of the thermal model for any number of plies; and (ii) the consideration of the heating cycle in a parametric way, leading to a transient parametric solution to be applied for control purposes.

## Parameters becoming coordinates

In this section we summarize the developments described in [9] in order to illustrate how parameters of different nature become coordinates. In particular we consider three types of parameters: (i) parameters related to the model; (ii) parameters related to initial and boundary conditions; and (iii) geometrical parameters defining the space-time domain in which the model is defined.

### Model parameters as extra-coordinates

*k*is here viewed as a new coordinate defined in the interval \(\mathcal I_k \). Thus, instead of solving the thermal model for different discrete values of the conductivity parameter, we wish to solve only once a more general problem. For that purpose we consider the weighted residual form related to Eq. (3)

- 1.
With \(T_{n+1}^{(r-1)}(t)\) and \(K_{n+1}^{(r-1)}\) given at the previous iteration of the non linear solver \((r-1)\) (arbitrarily initialized at the first iteration: \(T_{n+1}^{(0)}(t)\) and \(K_{n+1}^{(0)}(k)\)), all the integrals in \(\mathcal I_t \times \mathcal I_k\) are performed, leading to a boundary value problem involving \(X_{n+1}^{( r )}(\mathbf x)\).

- 2.
With \(X_{n+1}^{( r )}(\mathbf x)\) just calculated and \(K_{n+1}^{(r-1)}\) given at the previous iteration of the non linear solver \((r-1)\), all the integrals in \(\varOmega \times \mathcal I_k\) are performed, leading to an one-dimensional initial value problem involving \(T_{n+1}^{( r )}(t)\).

- 3.
With \(X_{n+1}^{( r )}(\mathbf x)\) and \(T_{n+1}^{(r)}\) just updated, all the integrals in \(\varOmega \times \mathcal I_t\) are performed, leading to an algebraic problem involving \(K_{n+1}^{( r )}(k)\).

- 4.The convergence is checked by calculatingWhen \(\mathcal E^{r}\) becomes small enough the just computed functions are incorporated into the approximation of the solution:$$\begin{aligned} \mathcal E^{r}&= \Vert X_{n+1}^{( r )}(\mathbf x) - X_{n+1}^{( r-1 )}(\mathbf x) \Vert \nonumber \\&\quad \, +\Vert T_{n+1}^{( r )}(t) - T_{n+1}^{( r-1 )}(t) \Vert + \Vert K_{n+1}^{( r )}(k) - K_{n+1}^{( r-1 )}(k) \Vert . \end{aligned}$$(9)$$\begin{aligned} \left\{ \begin{array}{l} X_{n+1}(\mathbf x)=X_{n+1}^{( r )}(\mathbf x); \\ T_{n+1}(t)=T_{n+1}^{( r )}(t); \\ K_{n+1}(k)=K_{n+1}^{( r )}(k). \end{array} \right. \end{aligned}$$(10)

### Boundary and initial conditions as extra-coordinates

In what follows we address the simplest scenarios consisting in constant Neumann, Dirichlet and initial boundary conditions. More complex and general situations were addressed in [9].

#### Neumann boundary condition as extra-coordinate

#### Dirichlet boundary condition as extra-coordinate

### Initial conditions as extra-coordinates

### Parametric domains

*u*(

*x*,

*t*) in many domains of length \(L \in [L^-, L^+]\) and for many time intervals of length \(\Theta \in [\Theta ^-, \Theta ^+]\), more than solving the model for many possible choice in order to define a meta-model, it is preferable to compute the parametric solution \(u(x,t;L,\Theta )\).

*L*and \(\Theta \), both defining the domain of integration. In order to explicit this dependence, we consider the coordinate transformation

## Process modelling

In the AFP process many parametric solutions are of interest. In [8] the authors focused on the solution of the steady-state thermal problem where the thermal contact resistances (it was proved that their consideration is a key point for modelling appropriately the thermal process), the laser power and the line velocity were considered as parameters and then included into the PGD parametric solution as extra-coordinates.

Here we extend these results by addressing two major issues: (i) the consideration of the number of plies composing the laminate as a model parameter; and (ii) the consideration of transient solutions induced by non constant laying velocities, both of major interest for controlling the process.

The numerical approaches developed in this section can be applied for any material (from the consideration of its thermal properties) and any tape dimension, with the only constraint of having a radius of curvature of the part much larger than the length of the analyzed region, such that the plane configuration analyzed remains representative. For smaller radius of curvature, it should be taken into account. Moreover, for varying tape directions thermal properties vary from one layer to other and consequently the ply orientation should be introduced as extra-parameter. This situation was successfully addressed in our former works (e.g. [4]). For the sake of clarity tape orientation is not considered in the modelling that follows.

### Number of plies as parameter

*p*plies with equal thickness \(e_p\), such that \(H=p \cdot e_p\).

*p*the number of plies and \(h_i\) the interface thermal resistance.

#### Domain transformation

*H*depends on the considered number of plies as depicted in Fig. 3. In order to define the problem in a reference domain we consider the coordinate transformation

*p*it suffices considering \(p \in \mathcal I_p = [1, 2, \cdots , P_M]\), \(P_M \in \mathbb N\). The fact of having a discrete nature is not an issue because the model does not imply derivatives with respect to the coordinate

*p*.

#### Interface treatment

However, when interfacial thermal resistances must be considered an important issue appears suddenly. First we must take into account the temperature discontinuity across the plies interfaces. The simplest possibility consists in duplicating the nodes at those interfaces. However the interfaces positions depend on the number of plies considered. For example when considering two plies (\(p=2\)) the interface is located at \(\lambda = 1/2\). When considering three plies (\(p=3\)) the two interfaces are located at \(\lambda =1/3\) and \(\lambda =2/3\).

*p*the consideration of both continuity and discontinuity transmission conditions, we propose enforcing continuity by applying the Nitsche’s method. Imagine for a while that we are solving

*h*is the cell-size.

If now we come back to the enforcement of temperature continuity across the interface located at \(\lambda _k\) for a certain *p* for which \(\lambda _k\) is not a real interface (it will be for another *p*), the transmission condition writes from one side of the interface where the temperature is denoted by \(u^-\) assuming that \(u^-=u^+\), and on the other side by enforcing \(u^+=u^-\), both written by using the Nitsche’s formulation (47).

#### Numerical results

## From steady-state to transient parametric solutions

### Steady-state parametric solution

In order to control the process, other parameters should be introduced as extra-coordinates, in particular the laser power *q* and the line velocity *V*. Thus the parametric solution involves the space coordinates, the number of plies, the laser power and the line velocity, i.e. \(u(x,y,\lambda ,p,q,V)\).

*n*solution, the approximation \(u^h_{n+1}\) results from

#### Heating law determination from the steady-sate parametric solution

### Transient parametric solution

#### Heating law determination from the transient parametric solution

*c*in order to minimize the gap with respect to the target temperature. The minimisation process is illustrated in Fig. 10.

#### Other possible computational vademecums

In order to simulate more complex scenarios involving an acceleration phase, followed by a plateau, to finish with a decelerating regime, we decided to create a parametric solution with the space coordinates, the time, three characteristic thermal resistances (one representative of the interfaces within the substrate, another representing the ply-substrate interface and lastly the one existing with the environment on the upper boundary) and the process parameters as coordinates of the system.

The process parameters concern 3 times, the plateau velocity and the plateau heating power. Thus, in the interval \([t_0=0,t_1]\) both velocity and laser increase linearly (with respect to time) to reach at time \(t_1\) both target values: the plateau velocity and heating power. Then the system evolves with constant velocity and power within the interval \([t_1,t_2]\). If the length of this interval is large enough the steady-state conditions predicted by the model [8] are attained. This check served to validate transient model. Finally, within the interval \([t_2,t_3]\) both the velocity and the power decrease linearly to vanish at the terminal time \(t_3\). The parametric solution contains in this case 12 coordinates.

## Conclusion

This paper proposes an original approach to simulate AFP composites manufacturing processes. First, using a spatial transformation to match a reference domain, the number of plies composing the laminate was considered as model parameter and then as problem extra-coordinate within the PGD framework. Different parametric solutions (computational vademecums) were defined by incorporating model parameters, boundary conditions and geometrical parameters. These parametric solutions were then used in order to define the heating laws in a very simple and efficient manner. The online use of all these offline pre-computed solutions allows for real time simulation, optimization and simulation based control of heating in AFP processes. Even if the offline calculations could be expensive from the computational point of view, they are performed offline and only once. Then, further online calculation are accomplished under the real-time constraint, opening a plethora of unimaginable and appealing possibilities.

## Declarations

### Acknowledgements

This research is part of the Impala project, which is a FUI 11 project, funded by OSEO, Conseil régional d’Aquitaine and Conseil général des Pyrénées Atlantiques. Francisco Chinesta thanks the support of the *Institute Universitaire de France – IUF –* and the financial support of ESI group within the ESI-ECN Chair. Elías Cueto acknowledges the financial support of the Spanish Ministry of Economy and Competitiveness through grants number CICYT-DPI2011-27778-C02-01/02 and DPI2014-51844-C2-1-R.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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