Inductance Effect in Crosstalk Prediction

Grégory Servel, Denis Deschacht, Françoise Saliou, Jean-Luc Mattei, and Fabrice Huret Abstract-Rapid progress in integrated circuit technology has led to an increase in switching speeds of digital circuits. This increase is the primary reason why inductance noise causes chips to fail. As a result, there is a growing interest in the inductance associated with on-chip signal lines. In this paper, we present the electromagnetic analysis we have followed in order to determine the accurate values of the equivalent parameters from which a set of multiple coupled transmission lines could be modeled. We then determine the most critical parameters which make the inductance effect important and we propose a new analytical expression to accurately evaluate the crosstalk voltage.

Index Terms-Crosstalk, electromagnetic analysis, inductance effect, on-chip interconnections, transmission lines.

I. INTRODUCTION

M ICROELECTRONICS evolution is characterized by an important rise in integration and circuit speed running. Today the race toward integration is faced with the problem of interconnects which has become one of the blocking points in the improvement of circuit performances. Technological advances in microelectronic fabrication techniques focus on the maximum integration of devices in the same chip. This implies an increase in the role of interconnects in circuit behavior, which necessitates the analysis of interconnects in integrated circuits. Tools employed in order to get interconnect characteristics are of two kinds. The first kind makes use of tables and analytic formulations, derived from the equations of electromagnetism.

Here the number of approximations increases considerably with the problem complexity, so that the results could become unreliable even for simple structures. The second type of tool requires numerical simulation help (these utilities are often based on the finite difference time domain or on the finite element method).

Here the accuracy of the solutions obtained is better than in the first type. However these tools are mainly geared to simulate propagation phenomena in models which eventually show quite a large diversity of sizes (such as small lines imbedded in a large background), and from which the users may encounter difficulties in meshing the models. Therefore it is desirable to have numerical methods which could give confident modeling of relatively complex structures, and in particular the magnetic coupling between their constituting elements.

In deep submicron technology, the wire thickness is often greater than the wire width, and the spacing between adjacent wires is often smaller than the distance between adjacent metal layers. This makes the coupling capacitance between adjacent wires on the same layer larger than the ground capacitance to adjacent layers. The reduction of distances between wires in deep sub-micron technologies increases the coupling effects and interactions between circuit interconnection lines. Two adjacent wires form a coupling capacitor and a mutual inductor. A voltage or a current change on one wire can thus interfere with the signal on the other wire. This interference noise (crosstalk) consists of the appearance of a glitch in a line (victim line) which is in its steady state (high or low level) when a neighboring line (affecting line) makes a transition (high-low or low-high).

Noise profoundly affects the performance of a circuit and is becoming one of the most important areas for concern in the design of deep submicron integrated circuits. This problem is especially serious for designs with higher clock frequencies, lower supply voltages, and the use of dynamic logic since they have a lower noise margin. To ensure that a final layout is noise immune, accurate and efficient noise models are needed to guide interconnect optimization at various stages.

Recently, a number of simple crosstalk noise models were proposed. The effects of the coupling capacitance have been addressed by Shoji [START_REF] Shoji | Theory of CMOS Digital Circuits and Circuit Failures[END_REF] using a simple linear RC circuit and by Becer [START_REF] Becer | An analytical model for delay and crosstalk estimation with application to decoupling[END_REF] and Vittal [START_REF] Vittal | Modeling crosstalk in resistive VLSI interconnections[END_REF], using and lumped-circuit models. Recent models such as [START_REF] Kuhlmann | Efficient crosstalk estimation[END_REF], [START_REF] Kahng | Noise and delay uncertainty studies for coupled RC interconnects[END_REF] propose complex models of prediction of crosstalk requiring complex calculations, and, as a result important CPU time. Reference [START_REF] Cong | Improved crosstalk modeling for noise constrained interconnect optimization[END_REF] proposes a closed-form peak noise formula by assuming a saturated ramp input for the aggressor net. Most of these models, however, did not consider the distributed nature of RC networks, which is necessary in deep sub-micron designs. Servel [START_REF] Servel | On-chip crosstalk evaluation between adjacent interconnections[END_REF] proposes an analytical expression of crosstalk voltage for one, two and four adjacent lines, which takes into account interconnect capacitances, line resistance and their distributed nature, driver resistance and variable strengths of the buffers driving coupled lines. None of these works includes the inductive effects on crosstalk evaluation, which is becoming an important consideration regardless of the system frequency. An efficient on-chip interconnect model is critical to interconnect optimization at high-level, logic synthesis and physical design. Closed-form formulas are particularly efficient and effective for these design stages.

In this paper, we highlight the importance of on-chip inductance modeling for crosstalk voltage evaluation, and we show how the most critical parameters make the inductance effect very important in high-performance VLSI circuit design today. We also propose a new analytical expression to accurately evaluate the crosstalk voltage.

This paper is structured as follows: after the introduction, Section II presents the electromagnetic analysis we have followed in order to determine the accurate values of the RLCG equivalent parameters from which a set of multiple coupled transmission lines could be modeled. In Section III we compare the time domain response of three coupled lines for RC and RLCG models to give the respective variations of the crosstalk amplitude in order to determine the level of modeling which is necessary to represent interconnects. We investigate the most critical parameters which make the inductance effect important. Finally, a corrective term is proposed in Section IV to improve the accuracy of the simple RC model to take into account the inductive effects.

II. QUASI-STATIC CHARACTERIZATION OF INTERCONNECTS

For submicron design, where interconnect dramatically affects the performance, the broad-band transmission line behavior of interconnects have to be characterized by rigorous analytical or numerical approaches. Therefore, the simulation of high-speed analog and digital integrated circuits requires the analysis of frequency-dependent transmission lines with nonlinear devices.

The frequency-dependent behaviors of transmission lines such as losses and dispersions are accurately represented in the frequency domain, while the determination of transmission line delays and reflections requires the time-domain simulation of a nonlinear network often used as drivers and terminations. In order to resolve this dilemma, most analysis programs use numerical-transform techniques to alternate between frequency and time-domain.

Interconnect cross-sectional surfaces are inhomogeneous. According to Maxwell's equations, all of the propagating modes within an inhomogeneous structure are hybrid. Usually, to analyze such waveguides, it is necessary to perform a full-wave analysis. In such cases, the Finite Element Method, associated with the Helmholtz equation, is probably the full-wave analysis method that is the most generally applicable and the most versatile [START_REF] Legier | Coupled line modelization of lossy adjacent interconnects with a full wave tangential vector finite element method[END_REF], [START_REF] Huret | Full wave analysis of conductor and substrate losses in high speed VLSI Interconnects[END_REF]. The frequency dependent per unit length elements can be extracted from the corresponding full-wave results. However, it is advantageous to represent the broad-band interconnect characteristics in terms of equivalent circuits consisting exclusively of ideal lumped elements because, in this case, the approach is compatible with general purpose circuit simulators such as SPICE. In this paper, by considering the standard technological constraints of the 0.25 m technology, we first demonstrate that the per unit length elements of studied digital circuits on-chip interconnects can be efficiently determined by quasistatic analysis.

Since this analysis is readily extended to the multiconductor case, the discussion is focused here on two coupled lines depicted in Fig. 1.

The calculated self and mutual interconnect line parameters are shown in Figs. 234. To demonstrate the accuracy of the quasistatic calculation, these figures also include the corresponding full wave solution [START_REF] Legier | Coupled line modelization of lossy adjacent interconnects with a full wave tangential vector finite element method[END_REF].

The bandwidth used in determining relevant frequency components of a digital pulse with a transition time , is generally accepted as . We can also estimate the Useful Frequency Bandwidth from the clock frequency by clock. This leads to 5 GHz for a classical 0.25 m technology. For frequencies higher than this value, the spectral component amplitudes are very small.

In addition, the attenuation of these components increases naturally with the frequency. This phenomenon also decreases the influences of the higher frequencies' small components.

In the useful frequency bandwidth (UFB), the comparison shows good agreement between the quasistatic and the full-wave solutions, and allows us to use frequency-independent per unit length elements. We have to note that the quasistatic approximation used in this study, depends only on the frequency (UFB), and not on the wire length. Finally, the accuracy of the quasistatic calculation has been also demonstrated for on-chip interconnects on lossy silicon substrate [START_REF] Zheng | CAD-oriented equivalent-circuit modeling of on-chip interconnects on lossy silicon substrate[END_REF]. Now, this section briefly presents the principles of the analysis we have followed in order to obtain the numerical values of the ( ) equivalent parameters from which a set of coupled transmission lines could be modeled. This work has been performed by using commercial FEM packages developed for two dimensional computation of electromagnetic problems [START_REF]Vector Fields[END_REF]. These packages allow us to solve conveniently and independently both magnetic and electric fields, provided that the quasistatic condition is fulfilled. A noteworthy advantage of this static approach lies in the very reduced CPU time taken for each computation (about 4 minutes). The computational model is based on the static Maxwell's equations. This method divides the studied structure into sub-domains. It is then possible, with this tool, to fit any polygonal shape by choosing the element shapes and sizes.

We focus here on the ( ) parameters which describe one central and two adjacent parallel transmission lines. The three copper lines are imbedded in a dielectric matrix (permittivity ), with lower and upper ground planes, as sketched in Fig. 6. The electric potentials and the currents have been fixed to arbitrary values. In order to modify the coupling between a given line and the two other lines, the spacing between them (denoted ) has been changed in the range 0.6-1 m. Owing to the determination of appropriate mesh patterns, an accuracy of about 1% on the computed fields has been reached. Typical views of the field distributions are represented on Fig. 1. The arrows are the electric field [Fig. 5(a)] or the magnetic field [Fig. 5(b)]; the colors give the intensity of the electric potential or of the scalar magnetic potential (values increasing from blue to red). They give qualitative indications on the intensity of the coupling strengths.

As an example, the magnetic energy and the electric energy stored in the structures have been computed. From these the following set of parameters can be calculated: the selfs and the mutual inductances and , , the capacities and the influence coefficients and , . Their numerical values have been obtained from the following general expressions (1a) and (1b):

(1a) (1b)
where the sums are over the n conductors (here we take

). denotes the currents flowing through the th line, which is placed at the potential . Obvious symmetry considerations lead to reduce the twelve unknowns to ( , , , in (1a), and to ( , , , ) in (1b). So, in both these cases, and for each value taken by the spacing parameter , four values of the stored energy have been computed (corresponding to different values of the and of the ), and the equations solved.

Fig. 6 represents the interconnect geometry used to illustrate our study. The lower metal 3 and upper metal 5 layers are considered to be very dense and the perfect metallic walls are taken into account on both sides of the metal 4 layer. In this case, the conductance is negligible (10 mS/cm) and must not be incorporated in the model. Corresponding electrical parameters are given in Table I.

III. COMPARISON BETWEEN RC AND RLC MODELS

The first problem is to determine the level of modeling which is necessary to represent deep-submicron interconnects. With this aim, we compare the time domain response for RC and RLC models. The crosstalk amplitude is obtained by HSPICE simulations with a 0.25 m process, with a level 49 foundry specified card model, the interconnection being modeled by an RC or RLC distributed model, whose number of basic cells depend upon the considered length. The crosstalk amplitude increases with the number of adjacent lines. We only consider the case where inaccurate evaluation of the crosstalk amplitude could be at the origin of a malfunction of the circuit, so we directly study the case of the bus structure. We consider three, 6 mm length, adjacent lines in the worst-case configuration, that is to say, when and (inputs of the aggressors) change from 0 to and (input of the victim line) is at , we have the most important noise voltage induced on the victim line [START_REF] Servel | On-chip crosstalk evaluation between adjacent interconnections[END_REF] (Fig. 7). The buffer sizing is calculated to keep a constant loading factor equal to 5, and a 2 m receiver size is used. Fig. 8 shows the temporal variation of the voltage at the output of the interconnection.

This first result shows a discrepancy between the two models, and the need to evaluate in detail the inductive effects. The inductance increases the crosstalk amplitude, and a logic fault can be shielded by using a simple RC model. Fig. 9 represents the variation of the crosstalk amplitude versus the interconnection length for two different spaces, , between lines, and the corresponding discrepancy in percentage, Fig. 10, between RLC and RC models. Whatever the space between the lines, the crosstalk amplitude increases with lengths up to 6 mm with a tendance toward a constant value. Only an analytical expression can rapidly determine this maximum value. In a strategy of buffering, the knowledge of this value is fundamental. When the propagation delay poses no problem, the maximum value of interconnect length on a chip, can for some cases be equal to the half perimeter of the chip. The propagation can then be made with two or three clock cycles. So, even if the maximum number of interconnect in a bus is only a few millimeters, it is important to analyze and to validate the analytical expressions up to this maximum value.

The discrepancy increases with the interconnection length, because when the length of the wire is no longer negligible in comparison with the length of the associated wave, the quasistatic hypothesis. Current and Voltage constant in the propagation direction isn't realized, then inductance modeling becomes necessary. For long lines however, the reflection phenomena are shielded from the attenuation factor, the amplitude of the reflected wave decreases with the increase in length, so that the reflection effect becomes negligible. The inductance modeling is no longer necessary.

We now analyze this discrepancy versus all the parameters, by beginning with the sensitivity study of interconnect process parameters. Various low-materials have been introduced to reduce not only the intra-metal capacitance in the same plane, but also the inter-metal capacitance between two planes. Fig. 11 shows that we have obtained similar results with different relative permittivity, , varying between 4 and 2.5, for three coupled lines of 6 mm with three different sizing for the buffer corresponding to different loading factors . The relative permit- tivity doesn't affect significantly the discrepancy between RC and RLC models.

With the aim of higher performance, the use of some low resistivity metals has been studied to replace Al in order to minimize wire RC delays and many designers have demonstrated the success of using copper (Cu) wires. Cu is very attractive due to its low resistivity and excellent electromigration resistance. But the process integration of Cu poses many potential problems one of which is diffusion, and as a result, barrier layers need to be introduced. We compared the behavior between Al and Cu wires in Fig. 12. As was expected, the discrepancy between RLC and RC models increases when the resistivity of the lines decreases.

In Fig. 13 we show the noise amplitude and the corresponding discrepancy versus driver size for 4 mm interconnect length. Driver size ranges from 7 m to 85 m corresponding to a loading factor between 2 and 12, and two different spaces between lines. We find that a stronger driver notably increases the discrepancy. The higher the driver size, the larger the RLC/RC discrepancy. The reason for this is that strong drivers yield faster-ramping waveforms, which give higher . The following figures summarize these variations: 1) versus the variation of the inductance, written as , Fig. 14, for three different input transition times; we have verified that the mutual inductance has no significant influence on the crosstalk amplitude; 2) versus the resistivity of the metal used for the interconnection, Fig. 15, for three different input transition times.

IV. ANALYTICAL EXPRESSION

In [START_REF] Servel | On-chip crosstalk evaluation between adjacent interconnections[END_REF], the authors have presented a new simple closer form expression to calculate the crosstalk amplitude between two, three and five adjacent lines, using an RC distributed model. To take into account inductive effects, it is possible to add a corrective term to the calculation of this RC model, as with and , and are the equivalent resistances of the inverter 1, 2 and 3 respectively, is the resistance of the line 2 and the rise time at the input of the inverter 2.

To validate this expression, we have simulated more than a hundred different configurations, with a range of wire length between 0.5 mm to 20 mm. We give here in Tables II andIII, some examples of these comparisons between calculated and simulated values of the crosstalk amplitude, for the cases where m, for different configurations. For all the simulations, we considered the worst-case configuration, with . In Table II we keep the same size for the buffers of the two aggressor's lines, the size of the buffer of the victim line being different, as reported in the table. We give the simulation values of the simple RC distributed model and the values of the RLC model. These values are compared with those calculated by our expression. Finally the discrepancy between calculated and simulated values is given in the last column. We can see the good agreement and the importance of the corrective term.

In Table III, we consider different sizes for the buffers.

For these cases, we can see also a good agreement between simulated and calculated crosstalk values and the necessity of taking into account the inductance effect in a large range of length.

We now consider a new interconnect geometry. The corresponding electrical parameters obtained by electromagnetic simulations are cm fF/cm fF/cm fF/cm nH/cm nH/cm For m, and the size of the buffers calculated to keep a constant loading factor equal to 5, we have obtained the following results given in Table IV.

For this configuration, largely different from the previous one, the accuracy of our proposed model is also good, and simulations with an RLC model are essential. The inductive effect cannot be neglected.

The coefficient has been determined for the 0.25 m technology. For the passage to a new technology, this corrective term has the same form, but it will be necessary to fit the new coefficient. In each case, the error between calculated and simulated values is less than 10%.

V. CONCLUSION

It is important for designers to know if crosstalk problems may arise in their final designs. An analytical expression to calculate the crosstalk would help the circuit designers to locate the pair of lines which is most sensitive to crosstalk interference. The accuracy of RC models is no longer sufficient for deep submicron circuits. In this paper, a methodology has been proposed to determine the accurate electrical parameters . We have shown the influence of inductive effects and proposed a corrective term in order to efficiently deal with them. The accuracy and applicability of a simple closed-form model for calculated crosstalk noise was shown.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Two coupled Cu lines.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Mutual and serie inductance of the two coupled lines.

Fig. 5 .

 5 Fig. 5. (a) Electric field and (b) magnetic field.

Fig. 6 .

 6 Fig. 6. Interconnection geometry.

Fig. 7 .

 7 Fig. 7. Worst-case configuration.

Fig. 8 .

 8 Fig. 8. Signal variation at the output of the victim wire.

Fig. 9 .

 9 Fig. 9. Crosstalk amplitude variation versus interconnection length.

Fig. 10 .

 10 Fig. 10. Discrepancy between RLC and RC model versus interconnection length.

Fig. 11 .

 11 Fig. 11. Discrepancy between RLC and RC model versus permittivity.

Fig. 12 .

 12 Fig. 12. Discrepancy between RLC and RC model versus interconnection length for Al and Cu wire.

Fig. 13 .

 13 Fig. 13. RLC and RC crosstalk amplitude versus transistor size for two spacing between lines.

Fig. 14 .

 14 Fig. 14. Discrepancy between RLC and RC model versus a variation of the inductance value.

Fig. 15 .

 15 Fig. 15. Discrepancy between RLC and RC model versus the resistivity of the metal layer.

TABLE I ELECTRICAL

 I PARAMETERS USED TO MODEL THE INTERCONNECTIONS

TABLE II COMPARISON

 II OF CROSSTALK VALUES FOR W = W

TABLE III COMPARISON

 III OF CROSSTALK VALUES FOR W 6 = W

is the inductance of the line 2,

TABLE IV COMPARISON

 IV OF CROSSTALK VALUES FOR A NEW GEOMETRY

Françoise Saliou was born in Lannion, France, in 1975. She received the M.S. degree in electronics from the University of Brest, France, in 1999, where she is currently pursuing the Ph.D. degree at the Laboratory for Electronics and Communication Systems.

Her current research activities include the superconductor materials processing and its application in microwave circuits. Moreover, the theoretical approach of microwave devices and VLSI circuits, such as designing problems, are under interest.

Jean-Luc Mattei was born in


Raphaël Malak Truffet

Christian Barillot

Emmanuel Caruyer

Optimal selection of diffusion-weighting gradient waveforms using compressed sensing and dictionary learning

Acquisition sequences in diffusion MRI rely on the use time-dependent magnetic field gradients. Each gradient waveform encodes a diffusionweighted measure; a large number of such measurements are necessary for the in vivo reconstruction of microstructure parameters. We propose here a method to select only a subset of the measurements while being able to predict the unseen data using compressed sensing. We learn a dictionary using a training dataset generated with Monte-Carlo simulations; we then compare two different heuristics to select the measures to use for the prediction. We found that an undersampling strategy limiting the redundancy of the measures allows for a more accurate reconstruction when compared with random undersampling with similar sampling rate.

Purpose

Diffusion MRI enables estimation of microstructure parameters such as axon diameter and density in white matter. A diffusion-weighted measurement relies on a spin-echo sequence with a diffusion-weighting gradient, for which the waveform, intensity and direction constitute the many degrees of freedom describing the acquisition. A large number of such measurements may be needed for an accurate parameter estimation, leading to long acquisition time, often incompatible with in vivo imaging. In this work, we evaluate two strategies to reduce the number of measurements: starting from a collection of gradient waveforms, we first learn a dictionary, then we select a subset of gradients (i) giving non-redundant signal response or (ii) optimizing the sensing matrix properties for sparse reconstruction. Both approaches are evaluated and compared to random subsampling.

Data generation

Microstructure configurations

We generated synthetic signals using Monte-Carlo simulation as implemented in Camino . Microstructure configurations follow the irregularly packed, gamma-distributed radius cylinders model, with an average radius in the range [0.5μm, 3μm], a shape parameter (for the gamma distribution) in the range [1.5, 9] and an intracellular volume fraction in the range [0.015, 0.8] for a total of 180 different sets of microstructure parameters. Last, the data is augmented using spherical harmonics representation up to rank L=6 to rotate the microstructure into 100 various directions.

Gradient waveforms

We generated a set of 65 piecewise constant gradient waveforms with fixed orientations. Some examples of generated waveforms are shown on figure 4. Then, the waveforms are rotated into 40 directions, selected uniformly on the unit sphere , which gives a total of 2600 potential gradient waveforms.

Methods

Dictionary learning

The reconstruction method is based on compressed sensing; we learned a dictionary using a training set made up of signals corresponding to 20% of the generated microstructure configurations. The learning is performed with SPAMS and aims at solving where is the number of signals vectors in the learning set, is the signal for microstructure and is a regularization weight. For the learning phase, the regularization parameter was fixed to 0.15, and the size (number of atoms) of the dictionary was set to 200. In what follows, we propose two methods using this dictionary to select a subset, , of the original measurements while being able to predict the unseen data.

Minimizing redundancy of gradient response

The first strategy consists in minimizing a correlation score for a subset Ω of gradients where is the dictionary where the lines are centered and reduced. We perform a local and discrete optimization strategy: choose an initial subset Ω. while the correlation score decreases: find the line in Ω the most correlated with the others in Ω, and replace it with the gradient not in Ω the less correlated with the remaining gradients. return the subset Ω. Note that this discrete optimization problem can be relaxed, where instead of binary selecting lines to define the set , we can attribute a positive weight to each line. The proposed minimization algorithm gives a result close to the theoretical minimum (see Fig. 1).

6 || -D | + λ|| | min D,x i 1 n ∑ i=1 n 1 2 y i x i | 2 2 x i | 1 n y i i λ λ Ω f (Ω) = ∑ i,j∈Ω ( ) ∑ k D ~i,k D ~j 2

Minimizing the coherence of the dictionary

In compressed sensing, it is known that a sensing matrix with a low coherence gives a sparser representation . We propose to choose minimizing where is the column of , the extracted dictionary containing lines whose indices are in . We approximately solve this problem with a local search similar as the one described in the previous section.

Performance evaluation

Using the testing dataset (80% of the original data not used for training), we first compute the sparse representation with the LARS-Lasso algorithm using , the subsampled data, and , the dictionary with selected lines. We then reconstruct the full signal and compute the root-meansquare deviation.

Results and discussion

We compare the two subsampling strategies to random undersampling for different subsampling factors (number of measurements) and regularization weights ( ), which balances the sparsity of the representation and data fidelity. In figures 2 and 3, we plot the root-mean-square deviation for signals in the testing set. As expected, selecting the gradients to miminize the coherence leads to sparser representation; however, selecting gradients by minimizing the line correlation of the dictionary enables better prediction for a small number of samples.

We proposed a novel, data-driven method to address the problem of optimally selecting gradient waveforms for experiment design in microstructureenabled diffusion MRI. We will extend this work to evaluate the impact of the proposed sampling schemes on the accuracy of rotation-invariant microstructure estimated parameters. The number of gradients used for the reconstruction is fixed to 30. This figure shows that we have to choose carefully. If it's too small, there is over-fitting because the LARS-Lasso algorithm gives too much importance to the data fidelity and is not able to generalize for gradient waveforms that are not used to find the representation. On the contrary, if it's too high, the algorithm finds a representation that is too sparse, then the representation does not contains enough information. ). A symmetry is used to replicate the gradient waveform after the RF180 pulse. Among the 3⁴=81 such waveforms, we only keep 65 by removing waveforms that don't reach +g .

1 Ω g(Ω) = max i,j |⟨ , ⟩| D Ω,i D Ω,j || | || | D Ω,i | 2 D Ω,j | 2 D Ω,i i D Ω Ω x

FiguresFigure 1 :

 1 Figures

Figure 2 :

 2 Figure 2: Fidelity of the reconstruction depending on .The number of gradients used for the reconstruction is fixed to 30. This figure shows that we have to choose carefully. If it's too small, there is over-fitting because the LARS-Lasso algorithm gives too much importance to the data fidelity and is not able to generalize for gradient waveforms that are not used to find the representation. On the contrary, if it's too high, the algorithm finds a representation that is too sparse, then the representation does not contains enough information.

Figure 3 :

 3 Figure3: Fidelity of the reconstruction depending on the number of measures used. λ is set to 5E-5; the 3 undersampling strategies give a similar score when the number of gradients is higher than 25; for fewer measurements, it is found that the best technique consists in minimizing , the correlation of the lines of the restricted dictionary.

Figure 4 :

 4 Figure 4: Example of gradient waveforms used in the simulation. The gradient amplitude is defined on 4 time steps and 3 possible values (-g , 0, +g). A symmetry is used to replicate the gradient waveform after the RF180 pulse. Among the 3⁴=81 such waveforms, we only keep 65 by removing waveforms that don't reach +g .

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)

Acknowledgements

No acknowledgement found.


Marta Perez

Anaïs Barasinski

Benoît Courtemanche

Chady Ghnatios

Anais Barasinski
email: anais.barasinski@ec-nantes.fr.

Benoit Courtemanche

Francisco Chinesta

Sensitivity thermal analysis in the laser-assisted tape placement process

Keywords: composites, automated tape placement, thermal modeling, Proper Generalized Decomposition, statistics analysis, uncertainty propagation

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

The context of this study is the LATP process, that consists in an automated filament winding process designed to produce composite parts with a thermoplastic matrix. A fiber-reinforced composite is continuously laid-down onto the previously laid tapes around a mandrel, while the tape is heated up by the laser in order to bond it to the substrate.

The main difficulty when using this technology lies in the heating control of the tape and the variability of the material, which affects the quality and properties of the part. The heating and cooling cycles generate very high temperature gradients (between 300 and 1400 • C/s), applied to a thin prepreg material (about 150 µm). These particular conditions hinder the use, in LATP, of autoclave conventional control and measurements methods. In addition, the tape may also exhibit irregularities in terms of fiber volume fraction, porosity, roughness, and thickness which may vary along the length of the prepreg tape. In this case, the tape may not heat uniformly, therefore won't weld as expected with the consequent impact on the part properties and performances.

Many recents works addressed modeling and simulation issues, as well as experimental characterization and validation [START_REF] Kollmannsberger | Numerical analysis of the temperature profile during the laser-assisted automated fiber placement of CFRP tapes with thermoplastic matrix[END_REF][START_REF] Schaefer | Analysis and improved process response prediction of laser-assisted automated tape placement with PA-6/carbon tapes using design of experiments and numerical simulations[END_REF][START_REF] Yassin | Processing of thermoplastic matrix composites through automated fiber placement and tape laying methods: A review[END_REF][START_REF] Schledjewski | Processing of unidirectional fiber reinforced tapesfundamentals on the way to a process simulation tool (ProSimFRT)[END_REF][START_REF] Grouve | Weld strengh of laser-assisted tape-placed thermoplastic composites[END_REF][START_REF] Stokes-Griffin | Thermal modelling of the laser-assisted thermoplastic tape placement process[END_REF]. The main difficulties from the modeling point of view come from the multiple scales and physics encountered in LATP processes. First, in the laser/composite interaction the penetration depth is extremely small, two or three fiber rows, creating a high thermal localization difficult to model and to resolve numerically.

The heat partition between the incoming tape and the substrate represents another issue, because there are multiple reflections between them, and absorption depends strongly on the microstructure (fibers distribution, exhibiting large variability at that scale), on the fiber orientation and also on the electromagnetic material properties. Simulation is usually performed by using ray-tracing (geometric optics) [START_REF] Reichardt | New analytical and numerical optical model for the laser assisted tape winding process[END_REF][START_REF] Stokes-Griffin | A laser tape placement process for selective reinforcement of steel with CF/PA6 composites: Effects of surface preparation and laser angle[END_REF], even if the relation between the wavelength and the fiber diameter compromises the validity is many cases. Other alternative approches using more general electromagnetic models were addressed in [START_REF] Dedieu | Towards mastering the filament winding process with in situ consolidation for launcher application[END_REF][START_REF] Dedieu | Model of laser/composite interaction based on scattering by multiple cylinders[END_REF].

Then, consolidation is facilitated by the compression provided by a roller. The thermo-mechanical coupling is of major relevance, because the squeeze flow at the tape/substrate interface depends on the roller stiffness, its thermal properties and the applied pressure. The roller/composite interaction is difficult to simulate, due to the small tape and substrate thicknesses require levels of resolution difficult to attain by using standard mesh-based discretization techniques. Here, in-plane-out-of-plane decompositions involved in the PGD-based solvers were successfully employed in our former works [START_REF] Dedieu | On the prediction of residual stresses in automated tape placement[END_REF][START_REF] Dedieu | About the origins of residual stresses in in situ consolidated thermoplastic composite rings[END_REF].

The thermal model, and more concretely the thermo-rheological coupling, is the key factor when modeling and simulating consolidation at the tape/substrate interface. First, the description of interface roughness remains an open issue [START_REF] Saoudi | On the interfacial thermal properties of two rough surfaces in contact in preimpregnated composites consolidation[END_REF][START_REF] Leon | Wavelet-based multiscale proper generalized decomposition[END_REF][START_REF] Leon | High-resolution thermal analysis at thermoplastic pre-impregnated composite interfaces[END_REF]. Then the squeeze flow at the interface scale, where asperities are composed of resin (whose viscosity is largely affected by the temperature) that can flow under the applied compression [START_REF] Deignan | Insights into complex rheological behaviour of carbon fibre/PEEK from a novel numerical methodology incorporating fibre friction and melt viscosity[END_REF][START_REF] Algerich | Prediction of composites behavior undergoing an ATP process through data-mining[END_REF][START_REF] Leon | Effects of material and process parameters on in-situ consolidation[END_REF], and fibers, that require local accommodation and the flow of the resin located in between, remains an open issue from the modeling and simulation viewpoints. Moreover, the connexion between interface morphology and thermal resistance is of major relevance to ensure accurate time-dependent thermal evaluations, required for evaluating the time-evolution of viscosity and also the molecular diffusion that allows recovering similar properties at the interface level that the ones existing in the bulk.

Finally, modeling and simulating the cooling process allows predicting the residual stresses installation, that are responsible of the final part distortions [START_REF] Dedieu | On the prediction of residual stresses in automated tape placement[END_REF][START_REF] Dedieu | About the origins of residual stresses in in situ consolidated thermoplastic composite rings[END_REF][START_REF] Van Hoa | Manufacturing procedure to make flat thermoplastic composite laminates by automated fibre placement and their mechanical properties[END_REF]. The main issue again in the very degenerated geometries, where the thickness is orders of magnitude smaller that the other characteristic dimensions, making difficult the use of accurate 3D finite element simulations.

As just described LATP involves many scales, and many physics. The issue related to the multiple scales can be efficiently addressed using adapted numerical techniques able to resolve all the involved scales. The issue related to the multi-physics nature can also be accurately accounted, however, the existence of many models with their own parameters makes difficult to extract the most critical, explicative or significant parameters, and more precisely, where and when each one expresses its influence.

One could think that addressing this issue only requires performing a rich enough DoE (design of experiments), and from it making a parametric study for extracting all the desired information. However, when the number of parameters increases the accuracy provided by a DoE is not enough for performing precise predictions.

The main aim of the present work is to solve an almost state of the art thermal model by using an efficient numerical technique that considers parameters (geometrical and material) as model extracoordinates, the so-called Proper Generalized Decomposition (PGD). Within the PGD rationale the parametric temperature field is expressed in a separated form, as a finite sum of functional products, where each term depends on a single coordinate (space, time or each one of the parameters considered as extra-coordinates). Such a separated representation allows the explicit expression of the sensitivity fields, from the temperature derivative with respect to each parameter. These sensitivity fields represent a very valuable methodology to analyze and establish the influence of the critical input parameters on the thermal response, and therefore, for performing process optimization and control.

Moreover, having such a parametric solution allows quantifying the effects of materials and/or process parameters variability, because as highlighted latter, real properties are statistically distributed. Again, the availability of a parametric solution allows propagating uncertainty or variability in almost real-time.

The present work does not propose neither models nor simulation tools, it considers state of the art models, applies an advanced simulation tool, the Proper Generalized Decomposition (PGD) for calculating a parametric temperature, and then it proposes a new methodology for providing an exhaustive analysis on the critical parameters influencing the process (sensitivity analysis) as well as a very efficient uncertainty propagator, able to address material and process variability, and evaluating their impact on the process. In this work, we study the influence of material parameters variability into the thermal behaviour during the LATP process from the data reported in [START_REF] Fouyer | Etude de la variabilité des caractéristiques thermiques des tapes de pré-imprégnés dans le cadre de l'enroulement filamentaire thermoplastique[END_REF].

The experimental validation of the main conclusions of the present works constitutes a work in progress. As just commented, here we are considering almost state of the art models, and consequently we are confident on the pertinence of the main tendencies derived from our numerical analysis, as soon as models are expected being pertinent at least at the first order.

The present article is structured in three main parts. The first concerns the experimental characterization of the tapes as well as of its variability. Then, the second introduces parameters in the representative thermal model and computes the parametric temperature field. Finally, the third part analyses the influence of the input material parameters on the thermal response and perform a statistical analysis related to the material and process variability.

Characterization of the material and its variability

In this study, two different materials were chosen: material A (55% AS4/PEEK) and material B (50% HTS45/PEEK). Rings were manufactured consecutively by using the CETIM deposition machine, located in the Technocampus EMC2 at Nantes (France). Each ring needed approximately 4.4 m of tape, which correspond about 6 tape layers (Figure 1). Two main data were collected: temperature evolution, measured with a thermal camera (FLIR T660) and microscopic analysis of the cross-section of tapes before processing them. The process parameters were set to: circumferential velocity of 7 m/min, laser power of 1000 W and the pressure applied to the compression roller was set to 2 bars. The image analysis is used to observe the microstructure of the material to eventually define the actual properties based on the observation of different phases. The analysis of these properties is then used to simulate the composite manufacturing processes. This section addresses the characterization, as accurate as possible, of the material parameters from the microcut images.The aim is obtaining information such as, fiber volume fraction (FVF), thickness and homogenized thermal conductivity tensor.

An active contour model (also called snake in computer vision) was used in order to find the contour of the pre-impregnated (prepreg) composite tape. A snake consists of a deformable spline minimizing energy and on which image forces apply pushing it towards the object contour, in the present case the tape contour. This method looks for minimizing the energy and requires the knowledge of the desired contour shape beforehand. This allowed to obtain a mask to detect the prepreg in the image. To detect the fibers in the image, the color image is converted to a gray-scale image. The gray scale image is then converted to a binary image based on a set threshold object detection method. The thresholding method replaces each pixel in an image with a black pixel if the image intensity is lower than a given value.

In [START_REF] Trias | Determination of the critical size of a statistical representative volume element (SRVE) for carbon reinforced polymers[END_REF], the size of a RVE (representative volume element) for a typical carbon fiber polymer was established and it was concluded that the minimum size is δ = L R = 50, being L the side of the element and R the fiber radius. This leads in our case, with a fiber diameter of around 7 µm, a RVE whose size is equal to the thickness of the prepreg. This issue is discussed later when calculating effective (homogenized) thermal properties.

Fiber volume fraction

The fiber volume fraction (FVF) is the percentage of fiber volume in the entire volume of the fiberreinforced composite material. For each microscopic cross-section, FVF was measured for 25 RVE in each cross-section of material A and material B. For one RVE, the pixels whose value was equal to 1, that indicates a part of a fiber, were summed up and divided by all the pixels contained in the RVE leading to the fiber volume fraction. Figure 2 plots the distribution of FVF for material A and material B and gives an average value of 46% in the case of material B and 49% for material A.

Thickness

The thickness was measured through the z-axis (vertical direction in Figure 1) and along the prepreg's width (represented by coordinate y). The results show an important variation in the thickness in the case of prepreg B, while for material A the thickness is more homogeneous. Figure 3 confirms this observation as the distribution of the thickness for all the material A samples are around 140 µm while for material B, the data is more dispersed.

Homogenized thermal conductivity

Due to the noticed microscopic heterogeneity, the macroscopic thermal modeling requires a homogenized thermal conductivity that depends on the microscopic details. As considered in our former work [START_REF] Lopez | Advanced thermal simulation of processes involving materials exhibiting fina-scale microstructures[END_REF], we can define the macroscopic temperature gradient ∇T at the elementary volume

∇T = g(x) = 1 |ω| ω g(x) dx, (1)
where g(x) denotes the microscopic temperature gradient. Following usual computational homogenization, we assume the existence of a localization tensor L(x) relating the macroscopic temperature gradient ∇T with the microscopic local temperature gradient g(x), such that

g(x) = L(x) • ∇T. (2)
As previously indicated, due to the small tape thickness the elementary volume considered for calculating the effective thermal conductivity involves the entire tape thickness, with a width and length large enough for ensuring that the effective thermal properties remain almost stable with respect to the variation of these dimensions. Now, we consider the microscopic heat flux q(x) according to the Fourier's law

q(x) = -k(x) • g(x), (3)
and its macroscopic counterpart Q that reads

Q = q(x) = -k(x) • g(x) = -k(x) • L(x) • ∇T, (4)
from which the homogenized thermal conductivity can be defined from

K = k(x) • L(x) . (5)
As k(x) is perfectly known everywhere in the RVE ω, the definition of the homogenized thermal conductivity tensor only requires the computation of the localization tensor L(x). We consider the general 2D case that involves the solution of two boundary value problems related to the steady state heat transfer model in the microscopic domain ω for two different boundary conditions on ∂ω:

     ∇ • k(x) • ∇Θ 1 (x) = 0 Θ 1 (x ∈ ∂ω) = y (6)
and

     ∇ • k(x) • ∇Θ 2 (x) = 0 Θ 2 (x ∈ ∂ω) = z . (7)
Thus, the localization tensor results finally:

L(x) = ∇Θ 1 (x) ∇Θ 2 (x) . (8)
To obtain the macroscopic homogenized thermal conductivity tensor we analyzed 150 microstructures in the case of material A and 196 for the material B. The carbon fiber transverse conductivity was set to k f = 0.8 W/mK [START_REF] Hatta | The measurements of thermal conductivity of carbon fibers[END_REF][START_REF] Tian | Anisotropic thermal property measurements of carbon-fiber/epoxy composite materials[END_REF][START_REF] Zhang | Measurements of thermal conductivity and electrical conductivity of a single carbon fiber[END_REF][START_REF] Grujicic | Computational analysis of the thermal conductivity of the carbon-carbon composite materials[END_REF] and the matrix conductivity k m = 0.2 W/mK. The homogenized thermal conductivity distribution along the width and the thickness are depicted in Figures 4 and5 respectively.

Variability of the material

The main information extracted from the aforementioned image analysis is listed in Tables 1 and2. This data is the input parameters for the thermal model and the stochastic analysis developed in Sections 3 and 4.

Deterministic thermal model

The objective of this section is to obtain the steady state temperature in a coordinate system attached to the placement head, which is assumed to move with a constant velocity. For a given number of plies, this temperature field can be used to reconstruct the thermal history in any material point. In these conditions each material point experiences the same thermal history during the process. It is progressively heated when approaching the laser, it reaches its maximum temperature when the laser applies directly on it and it cools down relatively fast when getting far from the heat source, reaching the ambient temperature before the next pass, when placing the next layer. Therefore, the laser and the roller are kept fixed and the material is assumed moving with a speed v in the opposite direction to the one of the laser and compression head. On the other hand, before experiencing the bonding the interface between the incoming layer and the substrate is modeled as a perfect insulation, to express the lack of contact between the incoming tape and the substrate. In the material domain, consisting in the substrate (plies already placed) and the incoming layer, defining the domain Ω = [0, L x ] × [0, L y ] × [0, L z ] as depicted in Figure 6, we solve the following heat problem widely considered when modelling ATP processes:

ρC p (v • ∇T ) = ∇ • (K • ∇T ) ( 9
)
where T is the temperature, ρ is the density, C p is the heat capacity, K is the homogenized anisotropic conductivity tensor and v is the velocity of the laminate with respect to the laser. The heating source is modeled as a surface flux. The convective boundary conditions with the air, roller and the mold, are expressed from q(x, t)

• n = h(T amb -T (x, t)) ( 10
)
where q is the heat flux, h is the heat transfer coefficient, T amb is the ambient temperature, T (x, t) is the temperature on the boundary when the convective exchange operates between composite and roller, composite and air or composite and working plane. Convective exchanges scale with coefficient h that at its turn depends on the existing conditions in the systems exchanging heat. It is also assumed that at the boundary representing the incoming material the temperature corresponds to the ambient one, whereas in the opposite boundary, where the material leaves the domain, the temperature is almost stabilized and consequently the heat flux vanishes.

It is important to note that numerical techniques employed in this paper perfectly work when addressing nonlinear models. An accurate solution of the ATP thermal model requires considering the temperature dependence of thermal parameters, however, as indicated in the introduction, here we are not interested in evaluating accurately the temperature field, but only the sensitivity of that field to the thermal parameters as well as analyzing the impact of variability. Thus, fine tuning and nonlinear modeling remain of second order as it was proved in our preliminary numerical tests carried out before considering the linear model.

Proper Generalized Decomposition

In order to solve a single multidimensional problem instead of running one simulation for each choice of the analyzed material parameters, we are going to introduce the material parameters as extra-coordinates in the thermal model. The difficulties related to the increase in the model's multidimensionality can be circumvented by using a separated representation [START_REF] Ghnatios | Modeling soft, permeable matter with the Proper Generalized Decomposition (PGD) approach, and verification by means of nanoindentation[END_REF][START_REF] Ghnatios | On the space separated representation when addressing the solution of PDE in complex domains[END_REF]. The Proper Generalized Decomposition (PGD) [START_REF] Chinesta | Recent advances and new challenges in the use of the Proper Generalized Decomposition for solving multidimensional models[END_REF][START_REF] Chinesta | The Proper Generalized Decomposition for advanced numerical simulations: A primer[END_REF] is an "a priori" model order reduction technique based on the use of separated representations in order to ensure that the complexity scales linearly with the model dimensionality. It consists of expressing the unknown field as a finite sum of functional products, i.e., expressing a generic multidimensional function u

(x 1 , • • • , x d ) as, u(x 1 , • • • , x d ) ≈ i=N i=1 X 1 i (x 1 ) • • • X d i (x d ). (11)
In the case of the thermal model presented above it involves as coordinates the space x, the homogenized components of the conductivity tensor K (assumed a diagonal tensor since we are using only unidirectional prepregs), the tape thickness e and the specific heat ρC p (the last is included as extra-coordinate because the noticed fibers weight fraction variability depending on the material supplier). By denoting by x the in plane coordinates (x,y), the temperature separated representation reads

T (x, z, k yy , k zz , e, ρC p ) ≈ i=N i=1 X i (x) • Z i (z) • K yy i (k yy ) • K zz i (k zz ) • E i (e) • R i (ρC p ) (12)
that results in a problem defined in a high-dimensional space, which in the more general 3D case involves 7 dimensions but whose computational complexity is associated with the solution of some 2D problems related to the calculation of the function X i (x) and a series of 1D problems for calculating the remaining functions involved in Eq 12. An alternate directions fixed point algorithm, with rankone updates, that is, computing a term of the sum (Eq 12) at each iteration, is used to construct the temperature separated representation. Interested readers can refer to [START_REF] Chinesta | The Proper Generalized Decomposition for advanced numerical simulations: A primer[END_REF] and the references therein. The simulation configuration was the same as the one extensively considered in our former works [START_REF] Chinesta | First steps towards an advanced simulation of composites manufacturing by automated tape placement[END_REF].

Model calibration

According to [START_REF] Courtemanche | Influence des propriétés thermiques des pré-impregnés composites thermoplastiques pour l'enroulement filamentaire laser[END_REF], we first consider L x = 1 m, L y = 6 mm, the velocity of the laminate is v = 0.12 m/s and the power source P = 1000 W. The point of contact between the roller and the mandrel is placed at x r = 0.8 m (with x = 1 the incoming boundary). The conductivity along the fiber direction is set to K xx = 5 W/mK and the ambient temperature is considered

T amb = 22 • C.
The composite heat capacity is calculated from the fiber volume fraction using the rule of mixtures as follows,

ρC p = w f ρ f C p f + (1 -w f )ρ m C pm ( 13
)
where w f is the weight fraction of the fiber obtained from the image analysis. C p f = 1129 J/kgK is the fiber specific heat capacity whereas C pm = 2200 J/kgK is the specific heat capacity of the matrix. These values were collected from the suppliers specifications. On the other hand ρ m = 1300 kg/m 3 is the matrix density, ρ f = 1790 kg/m 3 and ρ f = 1700 kg/m 3 are the fiber density for material A and material B, respectively. Thus, a parametric solution is obtained with the material parameters (e, k yy , k zz and ρC p ) as extra-coordinates. Considering the mean values of the material parameters, the values of all transfers coefficients were fitted from the experimental measurements, i.e., for BT46, see Figure 7 [20]. This allowed to calibrate the model, in order to introduce the materials variability and study their influence on the thermal response as described in the next section.

Stochastic framework

The aim of this section is to quantify the material uncertainty and its propogation. The stochastic simulation approach considered in this work is based in most standard formulation of the Monte Carlo (MC) method [START_REF] O'hagan | Polynomial Chaos: A Tutorial and Critique from a Statistician's Perspective[END_REF]. Monte Carlo simulation is a method for iteratively evaluating a deterministic model using sets of random numbers as inputs. Then the statistical characteristics of the model outputs are obtained, and their impact in the process is evaluated.

In each numerical experiment, the possible values of the input random variables X(X 1 , X 2 , ..., X n ) are generated according to their distributions shown in Section 2.

The Monte Carlo method presents a robust but usually computational expensive method as N simulations must be performed. However, in our case, the parametric solution already computed can be particularized for any choice of parameters without requiring further calculations, therefore Monte Carlo simulations become almost a real-time postprocessing.

Thus, N = 10000 realizations of the stochastic process were considered in the Monte Carlo analysis. The region of the parametric domain analyzed is considered as the surroundings the measurements taken in [START_REF] Fouyer | Etude de la variabilité des caractéristiques thermiques des tapes de pré-imprégnés dans le cadre de l'enroulement filamentaire thermoplastique[END_REF].

In the numerical simulation it corresponds to the volume x ∈ [0.835, 0.855], y ∈ [0.0024, 0.0036] and z = 1 4 e. The temperature is averaged in that region and each value is plotted in Figure 8. The input random variables (k yy , k zz , e and ρC p ) are generated according to the distributions shown in Tables 1 and2 for material A and material B, respectively.

Monte Carlo simulations are carried out for material A and material B. In Table 3, different statistical results are shown: sample mean, standard deviation, kurtosis and skewness. Figure 8 shows more dispersion in the temperature results for material B than for material A. Thus, for the same process conditions as material B reaches higher temperature values it could thermally degraded. This situation does not occur for material A, although the mean values for both materials are similar as shown in Table 3. The histogram, as shown in Figure 9, is a graphical representation of the distribution of numerical data and is an estimation of the probability distribution of a continuous variable (quantitative variable). Considering that, in the numerical simulation, x = 0.8 m is the contact point between the roller and the mandrel, the values from x = 0.6 to x = 0.8 m correspond to the consolidated tape and from x = 0.8 to 0.87 m the zone heated by the laser. Figures 10 and11 show the temperature field from the MC simulation with respect to the tape placement direction, from x = 0.6 to x = 0.87 m for materials A and B respectively. The averaged temperature experienced by the material at the 1st interface is shown with a solid line, the maximum temperature with a dashed line and the minimum with a dash-dotted line.

Experiments were carried out for both materials A and B. The tapes of about 12 mm width were heated by using a laser of 200 W (wavelength 980 nm). The temperature on the incoming tape surface was measured by using a IR camera FLIR A655, 7-14 µm spectral range, 640 × 480 pixels resolution (1 pixel = 17 µm), 50Hz sampling frequency and 100-650 • C temperature range. Thus, the temperature along the incoming tape width at the hottest location was measured. Now, considering the numerical predictions for both materials referred to the incoming tape, Figure 13 depicts the temperature evolution along the tape length, being the hottest location the one associated with the coordinate x = 0.85 m, where the laser directly applies. At that location and due to the properties variability, temperature exhibits a variability in the intervals of length ∆T Fluct A,x=0.85 for material A and ∆T Fluct B,x=0.85 for material B, of magnitude 50 • C and 100 • C respectively (again the larger variability of material B properties justifies its larger interval of possible temperatures).

Experimental and numerical intervals of temperature fluctuation shown in Figures 12 and13 respectively are in quite good agreement and could constitute a first qualitative proof of the modeling framework here proposed. As just indicated these results are very preliminary and consequently a deep experimental campaign is in progress to better quantify these tendencies.

In all the cases, if we compare material A with material B, there is more variability in the thermal response of material B, because of its heterogeneity. It would be interesting in further analyses to fix the desired temperature gap in a control zone ensuring an optimal deposition. This could allow us to give, for a fixed set of process parameters, a target variability for the different material parameters in order to ensure a right processability. Moreover, all the previous information doesn't allow to say which is the material parameter that has more influence on the temperature values. For addressing this question, a sensitivity analysis is carried out. However, addressing sensitivities experimentally is difficult because varying a given parameter without affecting all the other remains reasonably impossible. Numerical simulation offers such a possibility for performing sensitivity analyses, because modifying a parameter without modifying the others becomes strait-forward.

Sensitivity analysis

The sensitivity analysis performed in this section expresses how much would a parameter variation in the input of the model affects the thermal field output. This analysis identifies the most important parameters affecting the output. The sensitivity of the output, here the temperature T with respect to any of the input parameter, here the material parameters, is defined as the derivative of T with respect to the parameter. One may note that by using the PGD and thanks to the separated form of the temperature expressed by Eq 12, the sensibilities calculation becomes straightforward. For example, if one wishes taking the derivative of the temperature T with respect to the tape thickness e, it suffices to take the derivative of functions E i with respect to e, i.e.:

∂T ∂e ≈ i=N i=1 X i (x) • Z i (z) • K yy i (k yy ) • K zz i (k zz ) • ∂E i (e) ∂e • R i (ρC p ). ( 14
)
The temperature separated representation in Eq 14 involves few tens of modes and meshes are fine enough for ensuring the solution convergence.

Because the derivatives with respect to the different parameters involve different units, their comparison becomes delicate. For that reason the derivatives are normalized by giving the percentage on the temperature change with respect to a percentage of change of the involved parameters, allowing a proper comparison.

The sensibilities in this section are computed for K yy = 0.465 W/mK, K zz = 0.51 W/mK, e p = 0.1352 mm, and ρC p = 2.56 × 10 6 J/m 3 K. The sensitivity with respect to the tape thickness is illustrated in Figure 14. In it, we can notice a negative sensitivity just above the heating zone, in the new added tape and below the heating region in the already consolidated layers. Therefore, at these locations and for those reference parameters values, increasing the tape thickness would decrease the temperature. Otherwise, the sensitivity is positive elsewhere in the part, and the effect is more pronounced on the steady state value of the temperature that corresponds to the left domain boundary. As discussed later, the tape thickness sensitivity is the highest with respect to all the sensibilities. Thus, 1% of change of it produced changes up to 1% in the temperature. Because the tape thickness can exhibit variations up to 20%, the variations in temperature will reach similar orders of magnitude.

The sensitivity to ρC p is depicted in Figure 15. Tendencies are similar to the ones related to the tape thickness. However, the percentage of variation of the specific heat being much smaller than the one related to the thickness, its final effect will be less significant.

For the same parameters values, the sensitivity with respect to the in plane conductivity (in the direction orthogonal to the fibers) K yy and the out-of-plane one K zz are depicted in Figures 16 and17 respectively. We can note that the K yy variation has a negligible sensitivity in almost all the domain, as a 1% change in this parameter input only affects the temperature by 10 -3 %. That result is not surprising as the process, due to the geometry of the prepreg tape (very thin layers compared to the in-plane dimensions) is mostly driven by out-of-plane thermal exchanges. The sensitivity to the out-of-plane conductivity is as expected higher, as Figure 17 reveals. Thus, an increase of K zz of 1% will induce a slight positive variation on the temperature (10 -1 %) just above the heating zone in the new added tape and below the heating region in the already consolidated layers. However, in the already consolidated layers, increasing K zz will slightly decrease the temperature in the rest of the domain. Because the noticed variability of the out-of-plane conductivity of about 10%, more significant variations are expected in the temperature. In summary, the parameter having the highest impact on the temperature field are the thickness and ρC p parameters, with the thickness having a much more pronounced effect as this parameter presents a high variability. Then the through-the-thickness thermal conductivity parameter comes in second order, as either its variation and also its impact are lower, and finally the in-plane thermal conductivity (in the direction orthogonal to the fibers) is totally negligible.

Concluding remarks

The main aim of this work was to study the influence of material parameters variability in the thermal behaviour during the LATP process. The material parameters analyzed were the thickness, the fiber volume fraction (directly related to the parameter ρC p in the numerical model) and the homogenized conductivity tensor in the cross-section of the tape (k yy and k zz ). Microscopic cross-sections were analyzed, allowing to characterize the material variability from two different suppliers. This characterization showed a higher variability for the material B supplier than for the material A, which might have an influence on the thermal response.

On the other hand, temperature measurements were carried out with a thermal camera during deposition, allowing to extract different informations (either by inverse analyses, the values of the thermal exchanges parameters, and also to identify the thermal variation of the temperature during the deposition on the tape). For that purpose most of experimental measurements were performed along the width of the tape in the hottest zone. It might be interesting to extend this measurements at different locations of the process, e.g., near the contact between roller and mandrel, in order to improve the confidence in the proposed simulation tool.

Moreover, the simulation tool allowed to conduct very efficiently a stochastic analysis based on the Monte Carlo method (only by post-treatment of the data with a PGD-based parametric solution) in order to quantify the material variability impact on the thermal response. By comparing the two materials, we concluded that the higher dispersion in the material and process parameters, the more dispersion in the temperature values experienced by the material during its forming stage. Finally, a sensitivity analysis was carried out in order to identify the most critical parameters on the thermal response. As expected the material thickness appears as very significant, as well as the through-thethickness thermal conductivity k zz , and the ρC p parameter. The last one having an impact localizing in the cooling domain, whereas k zz present a more local impact under the heating zone, whereas the material thickness impact appears either in the heating zone and the cooling regions. Moreover, it constitutes a work in progress. can reasonably stressed that the in-plane thermal conductivity k yy has no significant impact on the temperature field.

In summary, the results presented here show that material uncertainty can introduce significant variability in the thermal response with considerable impact in industrial practice. Consequently, variability effects should be incorporated in process design/optimization to ensure robustness.

A deeper sensitivity analysis coupled with temperature range recommandation in order to ensure a good deposition quality could provide precious indications for material parameters specifications and

Figure 1 .

 1 Figure 1. Ring fabrication scheme.

Figure 2 .

 2 Figure 2. Fiber volume fraction distribution for material A (green) and material B (blue).

Figure 3 .

 3 Figure 3. Thickness distribution for material A (green) and material B (blue).

Figure 4 .

 4 Figure 4. Through-width thermal conductivity distribution for material A (green) and material B (blue).

Figure 5 .

 5 Figure 5. Through-thickness thermal conductivity distribution for material A (green) and material B (blue).

Figure 6 .

 6 Figure 6. Simulation geometry and boundary conditions.

Figure 7 .

 7 Figure 7. Comparison between the experiment thermal measurements (blue) and the numerical model (red) for sample BT46.

Figure 8 .

 8 Figure 8. Monte Carlo thermal simulations (Averaged temperature in the zone of interest).

Figure 9 .

 9 Figure 9. Histograms from simulations.

Figure 10 .

 10 Figure 10. Averaged temperature along deposition direction (from right to left) of the incoming tape and the substrate for material A. We consider the frame attached to the laying head, and consequently the right boundary corresponds to the material inlet and the left one to the outlet.

Figure 11 .

 11 Figure 11. Averaged temperature along deposition direction (from right to left) of the incoming tape and the substrate for material B. We consider the frame attached to the laying head, and consequently the right boundary corresponds to the material inlet and the left one to the outlet.

Figure 12

 12 Figure 12 depicts the incoming tape for both materials A and B, the heated region by the laser and the line along which temperatures were measured (Line 2). As it can be noticed in that figure, there is a certain temperature fluctuation across the tapes width, larger for material B. Such a fluctuation could evidence the intrinsic spatial variability of the parameters previously mentioned (thickness, thermal conductivity, local fiber concentration, etc ...), justifying that as discussed, material B having larger properties variability exhibits larger spatial fluctuations. The fluctuation interval for material B is of about 80 • C, i.e., ∆T Exp B

Figure 12 .

 12 Figure 12. Temperature measurement along the tapes width for both materials at the hottest location (Line 2).

Figure 13 .

 13 Figure 13. Temperature prediction along the tapes length and evaluation of the intervals of hottest temperatures at location x = 0.85 m.

Figure 14 .

 14 Figure14. Temperature sensitivity with respect to the thickness of the tape. Variation of Temperature (in %) for a variation of 1% of the thickness value. We consider the frame attached to the laying head, and consequently the right boundary corresponds to the material inlet and the left one to the outlet.

Figure 15 .

 15 Figure[START_REF] Leon | High-resolution thermal analysis at thermoplastic pre-impregnated composite interfaces[END_REF]. Temperature sensitivity with respect to ρC p . Variation of Temperature (in %) for a variation of 1% of the ρC p value. We consider the frame attached to the laying head, and consequently the right boundary corresponds to the material inlet and the left one to the outlet.

Figure 16 .

 16 Figure[START_REF] Deignan | Insights into complex rheological behaviour of carbon fibre/PEEK from a novel numerical methodology incorporating fibre friction and melt viscosity[END_REF]. Temperature sensitivity with respect to K yy . Variation of Temperature (in %) for a variation of 1% of the K yy value. We consider the frame attached to the laying head, and consequently the right boundary corresponds to the material inlet and the left one to the outlet.

Figure 17 .

 17 Figure17. Temperature sensitivity with respect to K zz . Variation of Temperature (in %) for a variation of 1% of the K zz value. We consider the frame attached to the laying head, and consequently the right boundary corresponds to the material inlet and the left one to the outlet.

Table 1 .

 1 Parameters variability for material A.

	Parameter	Mean, µ	Std dev, σ	µ/σ (%)
	FVF (%)	49.2	3.79	7.7
	Thickness (µm)	135.2	12.06	8.9
	K yy (W/mK)	0.465	0.029	6.2
	K zz (W/mK)	0.510	0.028	5.4

Table 2 .

 2 Parameters variability for material B.

	Parameter	Mean, µ	Std dev, σ	µ/σ (%)
	FVF (%)	46.4	5.64	12.2
	Thickness (µm)	154.6	31.58	20.4
	K yy (W/mK)	0.442	0.042	9.5
	K zz (W/mK)	0.476	0.038	7.9

Table 3 .

 3 Monte Carlo simulation results for ring setup.

	Parameter	Temperature ( • C)	Temperature ( • C)
		for material A	for material B
	Mean, µ	403.19	402.43
	Std dev, σ	9.98	21.15
	Kurtosis	3.13	3.12
	Skewness	0.25	0.48

Conflict of interest

The authors declare that they have no conflict of interest.