V Cagli

J.-F Jousseaume

B Nodin

Marc De Salvo

Christophe Bocquet

Muller

T Cabout

J Buckley

C Cagli

V Jousseaume

J-F Nodin

B De Salvo

M Bocquet

Ch Muller

Role of Ti and Pt electrodes on resistance switching variability of HfO 2 -based RRAMs

Role of Ti and Pt electrodes on resistance switching variability of HfO_2-based Resistive Random Access Memory

Introduction

Regardless their NAND (data memorydense and cheap) or NOR (code memoryfast) architecture, Flash technology is still today the undisputed reference for the non-volatile memory market and related applications. However, as downscaling and cost reduction trend for conventional floating gate memories appear ever more complicated below the 20 nm technological node, opportunities are opened for alternative concepts. Among emerging technologies, Oxide Resistive Random Access Memory (OxRRAM) appears as one of the most promising solutions for the next non-volatile memory generation. Relying on a reversible voltage-controlled resistance change between high resistive state (HRS) and low resistive state (LRS), the simple capacitor-like Metal/Insulator/Metal (MIM) structure of OxRRAM offers suitable CMOS compatibility with possible integration between two metal interconnect lines into Back-End of Line (BEOL). Besides, this technology attracts an actual interest for very high-density storage and paves the way towards multilevel tri-dimensional architectures [START_REF] Ch | Emerging Memory Concepts: Materials, Modeling and Design[END_REF]. Excellent electrical performances were already demonstrated in terms of low voltage operations [START_REF] Govoreanu | [END_REF], high programming speed [3] and long retention [4]. However, OxRRAM concept is still in its infancy compared to other technologies and the device-todevice variability of switching parameters is still a key issue for future industrialization [START_REF] Chen | Proceedings of International Reliability Physics Symposium[END_REF].

Regarding the switchable material, several metal oxides were reported to exhibit voltagecontrolled resistance change [START_REF] Kim | Proceedings of European Solid State Device Research Conference[END_REF] and recently a growing interest was turned towards hafnium oxide HfO2 that undergoes a bipolar resistance switching. Beside the nature of switchable metal oxide, a particular attention must be paid to electrodes that may deeply modify the electrical behavior of MIM capacitors. In this context, this paper deals with the electrical characteristics of two sets of capacitor-like structures integrating either Pt or Ti-TiN electrodes and it brings to light the role of electrodes on the switching parameters and the intrinsic capacitor-to-capacitor variability.

Experimental

Fabrication of capacitor-like structures

Capacitor-like MIM structures were fabricated on top of a tungsten pillar bottom plug (600 nm in diameter) connecting metal lines M1 and M2 as shown in Fig. 1. MIM capacitors consisted in a 10 nm thick HfO2 oxide layer grown by Atomic Layer Deposition (ALD) at 350°C and sandwiched between two sputtered metallic electrodes. To apprehend the influence of electrode materials on switching characteristics, two sets of devices were fabricated with either Pt (25 nm thick) as top and bottom electrodes (namely Pt/HfO2/Pt structures) or Ti-TiN electrodes corresponding to Ti(10 nm)/HfO2/TiN(25 nm) structures. Memory stacks were then etched to form MIM capacitors with a typical diameter of 1 µm. Finally a TiN-covering layer was deposited over the top electrode to connect the second metal line M2. It has to be mentioned that the device architecture is called "1R" (one Resistor), since it does not integrate an access metal-oxide-semiconductor transistor. X-ray diffraction patterns recorded on HfO2 blanket film showed that the polycrystalline oxide layer crystallizes in a monoclinic symmetry when sandwiched with Pt electrodes while orthorhombic symmetry is favored by using Ti-TiN electrodes. Besides, composition profiles measured by either Energy Dispersive X-ray spectroscopy (EDX) [START_REF] Cagli | [END_REF] or X-ray Photoelectron Spectroscopy (XPS) [8] revealed the formation of an additional layer TiOx/HfOx at the interface Ti/HfO2 due to oxygen-getter nature of titanium. This latter interface layer that was shown to be sub-stoichiometric in oxygen may subsequently enhance the defect-assisted conduction by acting as a reservoir of oxygen vacancies.

Electrical characterization

Quasi-static measurements were performed on capacitors to assess their electrical behavior and apprehend their switching characteristics at low voltage sweeps. Current-voltage (I-V) characteristics were measured with a standard semiconductor parameter analyzer HP 4155 (Agilent Company) with a current compliance fixed at 1 mA for forming and set operations.

In complement, dynamic measurements were performed to evaluate the electrical performances of MIM structures submitted to voltage pulses. The dependence of switching parameters on pulse rise time was carefully studied for both set and reset operations with a pulse generator Agilent 81110A that enabled varying voltage sweeps from 10 -2 up to 10 7 V/s.

Quasi-static measurements

Polarity assessment

Several set and reset cycles were applied to both sets of MIM structures to assess their electrical behavior, i.e. unipolar, bipolar or non-polar. Voltage was considered as positive when applied on top electrode. In agreement with results previously published by Goux et al.

[9], the capacitors with Pt top and bottom electrodes showed a non-polar behavior: as shown on I-V characteristics in Fig. 2a, set operation (HRS-to-LRS) was achieved for both voltage polarities and reset (LRS-to-HRS) succeeded whatever the polarity (and independently of the polarity used in preceding set operation). In contrast, MIM capacitors with Ti-TiN electrodes systematically exhibited a bipolar behavior: as depicted in Fig. 2b, set operation was done only as a positive voltage was applied on top electrode and subsequent reset operation required the opposite polarity to switch back the memory element into a high resistance state. Whatever the electrodes, set operation was achieved with a current of about 10 µA. In contrast, non-polar and bipolar behaviors were clearly distinguished during the reset operation. Non-polar capacitors with Pt electrodes exhibited a reset with a sudden and sharp current drop towards a very high resistance state (average resistance of 410 6 ), whereas bipolar capacitors with Ti-TiN electrodes showed a gradual reset with successive current jumps leading to smaller resistance in HRS (410 5 ). The abrupt reset transition for Pt electrodes is rather in favor of a thermal dissolution of conductive filaments proposed for non-polar/unipolar devices [10][11][12][13][14] (requiring a high reset current IReset as shown in Table 1), while "multi-steps" reset for Ti-TiN electrodes is consistent with a field-driven migration of oxygen ions [15] with possibly a gradual variation of the local density of vacancy defects as proposed by Goux et al. [16]. This latter mechanism is surely strengthened by the oxygen vacancies reservoir existing within a native TiOx/HfOx interfacial layer [START_REF] Cagli | [END_REF]8] (or insert layer as shown in ref. [9]). This layer acts as respectable oxygen buffer layer during electroforming and exhibits low release energy of oxygen species required for reset operation. Note that a positive voltage applied on Ti top electrode is required to achieve set operation, in agreement with the fielddriven movement of oxygen-vacancies ions towards TiN Ti top bottom electrode that leads to the formation of conductive filaments of positively charged defects (i.e. oxygen vacancies) within HfO2 layer [17,[START_REF] Huang | Proceedings of International Reliability Physics Symposium[END_REF]. A similar scenario was also proposed for Cr2O3-based memories [START_REF] Chen | [END_REF]20].

In the following, for clarity and for a better comparison of switching parameters, it was chosen to switch both sets of capacitors in a bipolar mode, forming and set operations being achieved with positive voltage on top electrode while reset state was reached by applying negative voltage.

Capacitor-to-capacitor variability for set and reset operations

Capacitor-to-capacitor variability was studied from a systematic analysis of more than 250 capacitors for each split of electrodes. Before being cycled (20 set and 20 reset operations), electroforming operation was done on pristine capacitors by applying a positive voltage sweep on top electrode with a current compliance fixed to 1 mA. Nevertheless, due to the very fast switching the current compliance does not avoid current overshoot occurring during electroforming stage. Tirano et al. [21] attributed this phenomenon to the discharge of a parasitic capacitance in parallel to the resistive device in this simple 1R architecture (one resistor without dedicated select device). These local transient voltage, current and temperature [22,23] boost the motion of oxygen anions O 2-[24] [START_REF] Sowinska | Hard x-ray photoelectron spectroscopy study of the electroforming in Ti/HfO2-based resistive switching structures[END_REF] and in turn favor the generation of oxygen-vacancy defects that subsequently form conductive filaments [START_REF] Cagli | [END_REF]. Table 1 summarizes electroforming voltages VForming (and their error bars) measured on both sets of capacitors. The decrease of forming voltage for Ti/HfO2/TiN capacitors (2 V instead of 4.3 V for Pt electrodes) was previously explained by the formation of an oxygen-vacancies-rich TiOx/HfOx interfacial layer at the top interface [START_REF] Cagli | [END_REF]. This oxygen sub-stoichiometric layer surely enhances the defect-assisted conduction and contributes to the decrease of the effective thickness of switchable oxide layer.

Beside electroforming stage, capacitor-to-capacitor variability was also studied for set and reset operations. A summary of corresponding switching parameters is given in Table 1. First, regarding LRS-to-HRS reset operations, Fig. 3a shows the cumulative capacitor-to-capacitor distribution functions of resistance in LRS (so-called set resistance Rset) for both sets of electrodes: a low variability was observed whatever the nature of electrodes. Nevertheless, capacitors with Ti-TiN electrodes exhibited a larger average resistance in set state of about 480  instead of 120  for Pt/HfO2/Pt capacitors. This latter feature might be related to the current overshoot that occurred during set operation. Indeed, Ti/HfO2/TiN capacitors exhibited a lower average set voltage (cf. Table 1) and consequently generated a lower current overshoot that led to a higher resistance in LRS [21]. In complement, Fig. 3b shows the variability of reset voltage VReset: unsurprisingly, since the average set resistances were found similar for both sets of MIM structures, reset voltages are subsequently close with a comparable variability as the one observed on RSet.

HRS-to-LRS set operations were also carefully analyzed. The cumulative distribution functions of set voltages VSet presented in Fig. 4a clearly show that Ti/HfO2/TiN capacitors switched at a significantly lower voltage, typically 0.6 V instead of 2.5 V for Pt/HfO2/Pt capacitors (cf. Table 1). Besides, it appears that the capacitor-to-capacitor variability was substantially narrower for capacitors with Ti-TiN electrodes. This latter characteristic may be linked to the capacitor-to-capacitor variability observed on RReset resistance in HRS (Fig. 4b). Indeed, one can observe a clear narrowing of variability for Ti/HfO2/TiN capacitors while Pt electrodes evidently led to a higher reset resistance with a significantly broader distribution. This feature matches well the two types of reset operations observed for either nonpolar/unipolar or bipolar capacitors. As previously mentioned in section 3.1, non-polar Pt/HfO2/Pt capacitors exhibited a very sharp current drop during reset operation leading to a very high resistance state. In contrast, bipolar Ti/HfO2/TiN capacitors showed a softer and gradual reset operation with several current jumps (cf. Fig. 2b) and a resulting lower resistance in HRS. Moreover, the lower variability of RReset evidently demonstrated a better repeatability of set operations. Nevertheless, it should be noted that the lower resistance in HRS led to a closing of the programming window (i.e. RReset/RSet ratio), decreasing from 810 5 to 710 2 for Pt and Ti-TiN electrodes, respectively.

To summarize, bipolar asymmetric Ti/HfO2/TiN capacitors evidently exhibited a better repeatability revealed by a narrower capacitor-to-capacitor variability of lower set voltages VSet and lower reset resistances RReset. In contrast, in agreement with previous published data [9], non-polar Pt/HfO2/Pt capacitors showed a worse reliability evidenced by significantly broader capacitor-to-capacitor distribution functions of larger set voltages and higher reset resistances. As suggested by Goux et al. [9], the permeability of Pt electrode to oxygen certainly limits the switching yield and explains why asymmetric Ti/HfO2/Ti stack exhibits better reliability performances as compared to Pt/HfO2/Pt stack.

Data retention

Data retention was estimated by measuring the time-to-failure from either LRS or HRS state at different temperatures. Whatever the electrodes and in agreement with results published by Huang et al. [START_REF] Huang | Proceedings of International Reliability Physics Symposium[END_REF], HRS state was found to be highly stable in time since no failure was observed after 10 hours at 300°C. In contrast, LRS appeared unstable with failures detected at high temperature. To give a better insight on LRS retention capability, the time-to-failure was measured as MIM capacitors were heated at different temperatures. First successive electroforming, reset and set operations were performed at the testing temperature to check the capacitors functionality. Capacitors were then programmed in LRS state and their resistance was subsequently measured in time. As shown in inset of Fig. 5, the time-to-failure was determined when resistance reached 10 times the initial LRS resistance. Results for different testing temperatures are displayed in the form of an Arrhenius plot shown in Fig. 5. This latter plot is in good agreement with behavior generally observed on OxRRAM memory elements [26] and clearly demonstrates better retention capabilities for Ti/HfO2/TiN capacitors. The extrapolation to 10 years enabled estimating 65°C as retention temperature for Ti-TiN electrodes against 50°C for Pt electrodes. Hence, Ti and TiN electrodes sandwiching HfO2 oxide layer enable improving data retention capabilities. This improved characteristic may be related to a higher set resistance for Ti/HfO2/TiN capacitors and to a smaller larger amount of oxygen vacancy defects that decelerates the time-dependent dissolution of conductive filaments.

Pulsed measurements

As shown in inset of Fig. 6a Ramp speed dependence of set and reset voltages was measured on both sets of MIM structures (Fig. 6b). In agreement with quasi-static measurements previously described, set and reset voltages and their associated variability were shown to be significantly lower for Ti/HfO2/TiN capacitors, with VSet and VReset below 1 V even at 10 7 V/s (i.e. 0.7 V at 810 6 V/s). In contrast, Pt/HfO2/Pt capacitors exhibited higher voltages with significantly larger dispersions. Consequently, Ti-TiN electrodes enabled improving dynamic performances with set and reset operations achieved below 1 V (in absolute values). Once again, the existence of TiOx/HfOx interface layer may be invoked to explain the better dynamic electrical characteristics since this layer may represent an optimized trade-off between efficient oxygen confinement and low energy of oxygen release.

Conclusions

Figure captions

Capacitors

VForming

[V] VSet [V] VReset [V] RSet [Ω] RReset [Ω] IReset [mA]
Pt/HfO2/Pt 4.3 ± 0.6 2.5 ± 0.9 -0.38 ± 0.09 120 ~ 410 6 4.2 Ti/HfO2/TiN 2.0 ± 0.7 0.6 ± 0.4 -0.42 ± 0.06 480 ~ 410 5 1.2

  , the experimental setup associated a load resistance in series with the device under test. Applied voltage VAppl and voltage across capacitor VCell were monitored during set and reset operations to determine the switching voltages VSet and VReset in pulsed mode. Fig. 6a presents set operation achieved as Ti/HfO2/TiN capacitor was submitted to a triangular voltage pulse. Switching was detected as a sudden variation of VCell induced by the resistance change that modifies the ratio of the voltage divider formed by load resistance and capacitor resistance.

  This paper compared switching characteristics and capacitor-to-capacitor variability of MIM structures relying on HfO2 oxide layer sandwiched by either platinum or titanium-titanium nitride electrodes. Capacitor-like structures were deposited on top of a tungsten pillar and integrated in-between two interconnect metal lines. While Pt/HfO2/Pt capacitors exhibited a non-polar behavior, Ti/HfO2/TiN capacitors unambiguously showed bipolar switching. Quasistatic and dynamic measurements clearly demonstrated better performances for Ti/HfO2/TiN capacitors: (i) lower electroforming and set voltages with significantly narrower capacitor-tocapacitor distribution functions; (ii) a improved data retention (10 years at 65°C, instead of 50°C for Pt electrodes); (iii) better dynamic electrical characteristics with lower set and reset voltages for ramp speed up to 10 7 V/s. The better performances of Ti/HfO2/TiN capacitors are surely related to switching mechanisms dominated by thermal-driven oxygen movement during electroforming/set operations and electrochemical re-oxidation during reset operation. The native TiOx/HfOx interface layer certainly plays a predominant role in the switching mechanism by providing a reservoir of oxygen vacancies. It acts as an oxygen buffer layer during electroforming/set operations and exhibits low release energy of oxygen species required for reset operation.

Fig. 1 .

 1 Fig. 1. Schematic representation of HfO2-based memory elements in the form of capacitorlike MIM structures integrated between two interconnect metal lines noted M1 and M2.Memory stack is deposited on top of a tungsten pillar bottom electrode.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Depending on the chemical nature of electrodes sandwiching HfO2 layer, the switching behavior is either non-polar with Pt top and bottom electrodes (a) or bipolar with Ti-TiN electrodes.

Fig. 4 .

 4 Fig. 4. Cumulative capacitor-to-capacitor distribution functions for set voltage (a) and resulting resistance in reset state (b). Distributions are given for MIM capacitors integrating either Pt (blue) or Ti-TiN (red) electrodes.

Fig. 5 .

 5 Fig. 5. Temperature-dependent Arrhenius plot of time-to-failure of LRS state for both sets of

Fig. 6 .

 6 Fig. 6. (a) Triangular waveform of applied voltage pulse VAppl for dynamically measuring set

TableTable 1 .

 1 Summary of switching parameters extracted on the two sets of capacitors (operations achieved in bipolar mode).


Nadhir Lebaal
email: nadhir.lebaal@insic.fr

Fabrice Schmidt
email: fabrice.schmidt@enstimac.fr

Stephan Puissant
email: stephan.puissant@insic.fr

Optimisation of extrusion flat die design and die wall temperature distribution, using Kriging and response surface method

Keywords: polymer extrusion, thermal regulation, response surface method, optimisation, Kriging interpolation, finite element analysis finite element method, optimisation, intelligence die design

HAL is

Introduction

Extrusion defects such as weld-lines, non-uniform exit velocity distribution across the extrusion die and problems of stagnation zones are influenced by the geometry of the extrusion die and by the operating conditions such as temperature of regulation, flow rate and the rheological parameters of the melt.

A coat hanger die (Figure 1) is commonly used to extrude thermoplastic thin sheets [START_REF] Puissant | Two-dimensional multilayer coextrusion flow in a flat coat-hanger die. Part I: modelling[END_REF][START_REF] Michaeli | Approach of automatic extrusion die optimisation[END_REF][START_REF] Reifschneider | Automated sheet die design[END_REF][START_REF] Gifford | The effect of wall slip on the performance of flat extrusion dies[END_REF][START_REF] Kaymaz | Application of Kriging method to structural reliability problems[END_REF]. However, if the melt distribution channel is not designed properly, the velocity at the exit may not be uniform [START_REF] Chen | Optimization of the coat hanger manifold via computer simulation and orthogonal array method[END_REF][START_REF] Wang | The flow distribution of molten polymers in slit dies and coat hanger die through three-dimensional finite element analysis[END_REF]. This leads to a variation in the sheet thickness across the width of the die. Since a tight tolerance on thickness is required for a high-quality plastic sheet, the ultimate goal of this work is to minimise the velocity variations at the die exit.

The design of dies for polymer extrusion is complicated by the non-linear relationship between the resin viscosity, shear rate and temperature dependence; this often involves trial and error corrections to the die geometry to achieve uniform velocities distribution at the exit. Manual correction of the die geometry is a time consuming and costly procedure. The most challenging and demanding work is to find methods of how to reduce or even eliminate die correction. If this is achieved, productivity and cost would be improved and the product will be more competitive.

Regarding the polymer extrusion process, [START_REF] Winter | Design of dies fort the extrusion of sheets and annular parisons: the distribution problem[END_REF] have used the one-dimensional flow analysis, to design coat-hanger dies, which present a uniform exit velocity distribution; these designs are verified experimentally. [START_REF] Michaeli | Approach of automatic extrusion die optimisation[END_REF] have used a combination of finite-element-analysis and flow analysis network with isothermal flow to optimise the die geometry using an evolution strategy algorithm.

Broyden Fletcher Goldfarb Shanno (BFGS) algorithm is commonly used for the resolution of non-linear optimisation problems. We can mention [START_REF] Sun | Optimization of a flat die geometry[END_REF] who have optimised a flat die to obtain a homogeneous velocity distribution using BFGS algorithm. A penalty function was introduced to enforce a limit on the maximum allowable pressure drop in the die. Smith et al. (1998a) optimised the coat hanger die to minimise simultaneously the velocity and residence time variations using BFGS algorithm and penalty function. Smith et al. (1998a) modelled Newtonian and non-Newtonian isothermal flow in a coat hanger die using a generalised Hele-Shaw (HS) approximation, and optimised the die by minimising the pressure drop under the constraint of exit flow uniformity within a pre-set tolerance. The results of the sensitivity analysis needed for the Sequential Quadratic Programming (SQP) algorithm were compared then using direct differentiation and adjoint method. The same author [START_REF] Smith | An optimisation-based approach to compute sheeting die designs for multiple operating conditions[END_REF] makes a comparison between two optimisation algorithms with constraint based on SQP and Sequential Linear Programming (SLP) to optimise the geometry for two different materials at various temperatures. The optimisation problem consists in minimising the pressure loss in the die, with an imposed constraint such that a homogeneous velocity distribution within an imposed tolerance is obtained on the outlet side of the die.

All calculation methods (analytical or computational), except the three-dimensional flow model, are unable to simultaneously compute the velocity across the gap of the die land and across the width of the land. However, for complex geometries, the computational resources and time required for the analysis of extrusion dies are considerable. Consequently, the use of evolution strategy algorithm [START_REF] Michaeli | Computer aided optimisation of extrusion dies[END_REF][START_REF] Sienz | Flow restrictor design for extrusion slit dies for a range of materials: simulation and comparison of optimization techniques[END_REF] to optimise the extrusion die in three-dimensional analysis is less attractive for designers because of the unacceptable turnaround time for results. The gradient methods (Smith et al., 1998b;[START_REF] Smith | An optimisation-based approach to compute sheeting die designs for multiple operating conditions[END_REF][START_REF] Sun | Optimization of a flat die geometry[END_REF][START_REF] Xiaorong | Optimal design for polymer sheeting dies[END_REF][START_REF] Ettinger | Parameterization and optimisation strategies for the automated design of u PVC profile extrusion dies[END_REF][START_REF] Hua | Development of a novel meshless local Kriging (LoKriging) method for structural dynamic analysis[END_REF] require the computations of the gradients of the functions, and the computation of gradients by finite difference is time consuming.

The majority of the used optimisation algorithms require a great number of simulations, increasing by this fact the computing time. For the above reasons, in this work we have developed an automatic optimisation algorithm based on a response surface method together with Kriging interpolation and SQP algorithm. This optimisation algorithm is applied to solve two different optimisation problems (optimise the geometrical design and the wall temperature distribution in the coat hanger die). In the first example, the objective function is minimised by varying the geometrical design of the melt distribution channel and in the second example we vary the wall temperature settings.

We used REM3D ® software [START_REF] Pichelin | A Taylor discontinuous Galerkin method for the thermal solution in 3D mold filling[END_REF][START_REF] Gruau | 3D tetrahedral, unstructured and anisotropic mesh generation with adaptation to natural and multidomain metric[END_REF][START_REF] Silva | Viscoelastic Compressible Flow and Applications in 3D Injection Molding Simulation[END_REF] to compute a 3D flow in extrusion dies. This software takes into account strain rate and temperature dependence of the material extruded.

The optimisation benchmark

In this paper, the coat hanger die is optimised for an Acrylonitrile Butadiene Styrene (ABS, Astalac EPC 10000). The polymer flow is supposed to be purely viscous (the viscoelastic behaviour is not taken into account). This assumption is in agreement with [START_REF] Sun | Effect of elongational viscosity on the flow in flat die[END_REF] who show that the elongation effect does not influence the exit velocity distribution at the die exit. The constitutive equations used in REM3D ® give an expression of the viscosity in function of the shear rate and temperature.

The simulations are performed using a flow rate of 50,000 mm 3 /s with a melt temperature at the die entrance of 240°C.

The rheological parameters of the ABS are referred in Table 1. A Carreau Yasuda/WLF viscosity model [START_REF] Balasubrahmanyan | Thermal control of melt flow in cylindrical geometries[END_REF] is used to characterise the temperature and shear rate dependence of viscosity:

1 1r e f 1 0r e f 0 2r e f 2 ()( ) () e x p 1 . ()( ) m ss ss s AT T AT T T ATT AT T α α γ ηη η τ - -- =- + +- + - (1)
The thermal conductivity (k), density ( ) and heat capacity (C p ) of the polymer used for the flow simulation were assumed to be constant throughout the range of temperature in the flat die and are referred in Table 2. Owing to the symmetry of the die, only one-half of the passage is modelled as shown in Figure 1. The boundary condition of no slip at the die wall is assumed. Because of the complexity of the geometry and the important aspect ratio, it is necessary to use an anisotropic mesh with elements of minimum size across the thickness of the die, and much larger along the length and width.

Automation of the finite element model

To save time at the optimisation process and to control efficiently and easily the optimisation strategy, it is necessary to use all the possibilities of automation offered by this code (via Matlab ® ). Figure 2 shows a diagram of the principle of operation and possible interactions of optimisation routine by Matlab ® with REM3D ® and design routine. It is also possible to automate all the tasks usually carried out through graphic interface from the creation of FEM until the recovery of the results. In Figure 3 and in the next section, we will see the optimisation procedure that we develop to obtain a good homogenisation of velocity at the die exit.

Design variables and objective functions

As a first example, the extrusion die is optimised with the distributor channel height H as a variable. The initial height is fixed at 36.5 mm and allowed to vary between 30 mm and 110 mm during the optimisation process (Figure 1). The wall die temperature is fixed at 230°C. As a second example, we optimise the die wall temperature. To this end, die wall temperatures are fixed at three locations. During the optimisation process, the three temperatures can vary between 200°C and 280°C, starting from a uniform temperature of 230°C. The maximum temperature of 280°C is chosen so that there is no polymer degradation.

It was mentioned earlier that the goal in flat die design is to minimise the velocity variation across the die exit. Thus, the objective function is defined as follows:

2 1 1 N i i vv J Nv = - = (2)
where N is the total number of nodes at the die exit in the middle plane, v i is the velocity at the mid-plan of the gap (i.e., maximum velocity of the Poiseuille profile calculated by the finite element programme) and v is the average exit velocity.

Optimisation procedure

The response surface method [START_REF] Myers | Response Surface Methodology, Process and Product Optimization Using Designed Experiments[END_REF][START_REF] Lebaal | Rheological parameters identification using in situ experimental data of a flat die extrusion[END_REF] consists in the construction of an approximate expression of the objective function starting from a limited number of evaluations of the real function. To obtain a good approximation, we used a Kriging interpolation described in the next section.

In this method, the approximation is computed by using the evaluation points by Composite Design of Experiments (CDEs) [START_REF] Montgomery | Design and Analysis of Experiments[END_REF]. After the interpolation of the objective function, we minimise it using an SQP method (see Figure 3). In addition, to avoid local minima, a successive minimisation is carried out by changing the initial points of the SQP algorithm starting from each point of the DOE. The best solution among those obtained in various iterations is then considered.

After we have obtained the best minimum, a weight function of Gaussian type is used to slightly vary the approximate problem and make the response surface more accurate locally. The iterative procedure stops when the successive points are superposed with a tolerance = 10 -3 . In the first example (geometrical optimisation), with one geometric variable, we use five evaluation points to obtain a good interpolation.

In the second example (die wall temperature optimisation), with three variables, we use 35 evaluation points to have a good interpolation.

Kriging interpolation

Kriging was introduced to meet the needs for minerals prospection. It was used to reconstitute the position of an ore seam starting from concentrations obtained by drilling. We limit ourselves to present the Kriging as an interpolation technique [START_REF] Kaymaz | Application of Kriging method to structural reliability problems[END_REF][START_REF] Hua | Development of a novel meshless local Kriging (LoKriging) method for structural dynamic analysis[END_REF], which allows to model very efficiently complex surfaces of arbitrary shapes. It is applied here to represent the objective and constraint functions in an explicit form, according to the optimisation variables. The construction of a Kriging model can be explained as follows:

() () () T J xp x a Z x =+ (3)
with p(x)= [p 1 (x), …, p m (x)] T , where m denotes the number of the basis function in regression model, a = [a 1 , …, a m ] T , is the coefficient vector, x is the design variables, () J x is the unknown objective or constraint interpolation function and Z(x) is the random fluctuation. The term p T (x)a in equation ( 3) indicates a global model of the design space, which is similar to the polynomial model in a Moving Least Squares (MLS) approximation. The second part in equation ( 3) is a correction of the global model. It is used to model the deviation from p T (x)a so that the whole model interpolates response data from the function.

The output responses from the function are given as follows:

{ } 12 () () , () , , ().

n F xf x f x f x = (4)
From these outputs, the unknown parameters a can be estimated:

11 1 () TT aP R P P R F -- - = ( 5
)
where P is a vector including the value of p(x) evaluated at each of the design variables and R is the correlation matrix, which is composed of the correlation function evaluated at each possible combination of the points of design:

1 11 1 n 2 n1 n () 0 0 (, x ) (, x) 0( ) 0 (, x ) (, x ) 00 ( ) n n wx x Rx Rx wx x R Rx Rx wx x - - =+ - (6) 3 (| |) (Geometrical optimisation) . (| |) (Die wall temperature regulation) ij ij ij xx R xx - = - (7)
A weight function of Gaussian type with a circular support is adopted for the Kriging interpolation because its derivatives with respect to the coordinates exist to any desired order. It takes the following relationship:

22 2 e x p ( ( /)) e x p ( ( /)) 1 () 1e x p ((/ ) )

iw iw i w iw dc rc dr Wx rc dr χ χ -- - -× ≤ = -- ≥ (8)
where 2 1 ()

n JJ ii J dx x = =-
is the distance from a discrete node x i to a sampling point x in the domain of support with radius r w , and c is the dilation parameter. c = r w /4 is used in computation. 0 ≤ x ≤ 1 represents the degree of the importance of the weight function.

The second part in equation ( 3) is in fact an interpolation of the residuals of the regression model p T (x)a. Thus, all response data will be exactly predicted and are given as follows:

() () T Z xr x β = (9)
where r T (x) = {R(x, x 1 ), …, R(x, x n )}

The parameters β are defined as follows:

1 () . RFP a

β - =- (10)
3 Results and discussion

Geometrical optimisation

The objective of this technique is to reach the global minimum on the field domain.

The optimisation consists in adjusting the height of the melt distribution channel (H) (see Figure 1) to minimise the velocity variation at die exit. The optimisation algorithm, described in the previous section, is used to optimise a flat die for ABS. The flat die based on geometric parameters shown in Figure 1 was employed as the initial design for the optimisation scheme.

A representative convergence history during an optimisation run is represented in Figure 4. In this case, the initial value of the melt distribution channel depth (variable H) is 36.5 mm. After two iterations, the superposition of the successive points fell below the cutoff of = 10 -3 and the simulation stopped. This optimisation run represents 11 h 23 mm of CPU time on a computer Pentium IV, 3 GHz, 1 Go RAM. Under the constraints of die geometry, melt flow rheology and operating conditions, the optimal value of the melt distribution channel depth was found at H = 94.9 mm. The same figure illustrates the evolution of the normalised objective function according to the number of iteration. We can see that it is varying quite fast at the beginning of the optimisation until the second step. Its final value is about 5.3 × 10 -2 , which represents a total variation of about 94.7%.

Qualitatively, velocity distributions at the exit of the optimised die (Figure 5(B)) are more uniform than the exit velocity in the initial die design (Figure 5(A)). It is obvious from Figure 5(A) that in the initial die design the exit velocity at the border of the die is significantly higher than at the centre of the die. After the optimisation, we observe in Figure 6 that the exit velocity distributions at the mid-plane in the gap for the optimised die design is much more uniform across the entire width of the half die. The exit velocity drops to zero in the small regions near the border of this die to satisfy the no slip condition at the walls. The pressure drop in the initial die design is 4.423 MPa (Figure 7(A)). It should be noted that the pressure drop decreases during the optimisation iteration. The total pressure drop in the optimised die was decreased by 49.9% or 2.215 MPa (Figure 7(B)).

Compared with the initial geometry, the flow rate can be increased by 140% at the same pressure drop (Figure 8). Non-uniform exit temperature distribution is observed in the initial die (Figure 9(A)). However, it is to be noted that more polymer is flowing towards the edge because of the shear heating. Hence, the temperature is lower near the middle of the die and higher near the edge. By comparison, the shear heating in the optimal die and the temperature distribution are more uniform (Figure 9(B)) because of the homogeneous distribution of the exit velocity.

Optimisation of the die wall temperature

The convergence history during an optimisation run is represented (Figure 10). After three iterations, the simulation stopped. This optimisation run represents Five days and 9 h of CPU time in a computer Pentium IV, 3 GHz, 1 Go RAM. We notice that the normalised objective function is varying quite fast at the beginning of the optimisation until the third step. Its final value is about 6.88 × 10 -2 . The initial values of the die wall temperatures are T 1 = T 2 = T 3 = 230°C, where T 1 represents the temperature in the centre of die, T 3 the temperature near the border of the die and T 2 the temperature between T 1 and T 3 (Figure 11).

To improve the uniformity of velocity at the die exit, the optimal values for the temperatures profile of the die wall in the three zones are varied from T in = 230°C to T 1 = 276.4°C, T 2 = 257.02°C and T 3 = 223.88°C (Figure 12), respectively, in zones one, two and three. We find that the temperature of the initial regulation (Figure 13(A)) gives a non-uniform exit velocity distribution. This velocity distribution in the initial die is obtained with a uniform die wall temperature T = 230°C. We notice in Figure 14, that in the centre of the die, the velocities are lower; on the other hand, in the two parts close to the end of the die we observe that the velocities increase until 50%. The optimal solution shows higher wall temperatures in the centre of the die. On the other hand, the temperatures are lower on the edge to increase the viscosity of polymer, which generates a reduction in the flow towards the edge. With these optimal temperatures of regulation, we could obtain a balance, which provides homogeneous exit velocities distribution according to all the width of the die (Figures 13(B) and 14).

For the initial temperature of regulation, the pressure drop is about 4.42 MPa (Figure 15(A)). It should be noted that the pressure drop decreases during the optimisation iteration. The total pressure drop in the die for the optimal temperature of regulation was decreased by 26.88%, which represents 3.23 MPa (Figure 15(B)).

We note a higher exit temperature for the optimal regulation profile because the wall temperature has increased in the centre of the die by 46°C and 27°C, respectively, in T 1 and T 2 . In the small regions near the border of this die, the temperature decreases by only 7°C (T 3 ) (Figure 12). We note a difference in melt temperature along the die width for the optimal die temperature regulation (Figure 16(B)). But, the great difference of regulation temperature (about 50°C) on the wall die leads to a variation of melt temperature of only 10°C across the die exit width. This variation could, however, generate difficulties during drawing and cooling of the sheet obtained on the outlet side of the die.

Conclusion and prospects

We have obtained good optimisation results using a response surface method with a very fast convergence (two iterations). This is due to the good interpolation obtained using Kriging interpolation. The weight function and the various evaluations of the interpolation function (requiring only small CPU time) allow to determine an accurate global minimum. An optimisation methodology for the die design for polymer sheet extrusion has been developed. This optimisation methodology may be applicable to optimise both operating conditions (in our case the die wall temperature) or the geometrical design.

To save time and control easily the optimisation procedure, we have linked this programme with REM3D ® software and a separate design routine. The optimisation programme uses the REM3D ® software to simulate the polymeric flow in the flat die.

This programme successfully optimised in the first approach the geometrical design of a flat die for a uniform velocity and temperature distribution at the die exit and decreasing in the same time the pressure drop in the die.

In the second approach, to preserve the existing geometry and avoid to design and manufacture a new die, the wall temperature distribution is optimised in heterogeneous way (i.e., the wall temperatures are different in the various zones of the die). This same approach permits to optimise the velocity distribution in the existing die.

For our future work, we plan to optimise more complex geometries of extrusion die for various polymers and for multiple operating conditions.

Figure 1

 1 Figure 1 Geometry of the flat die

Figure 2 Figure 3

 23 Figure 2 Optimisation methodology

Figure 4

 4 Figure 4 Fast convergence of the objective function 'design optimisation'

Figure 5 Figure 6

 56 Figure 5 (A) Velocity distributions of the initial and (B) optimised designs of the flat die (see online version for colours)

Figure 7

 7 Figure 7 (A) Pressure distributions in initial and (B) optimised designs of the flat die (see online version for colours)

Figure 9

 9 Figure 9 (A) Temperature distributions in the mid-plane of the initial and (B) optimised flat die (see online version for colours)

Figure 10 Figure 11 Figure 12

 101112 Figure 10 Fast convergence of the objective function 'regulation of temperature'

Figure 13 (Figure 14

 1314 Figure 13 (A) Velocity distributions of the initial and (B) optimised die temperature distribution (see online version for colours)

Figure 16 (

 16 Figure 16 (A) Temperature distributions in the mid-plane of the initial and (B) optimised die wall temperature distribution (see online version for colours)

Table 1

 1 Parameter of the behaviour laws of Carreau Yasuda, WLF

	η 0 (Pa.s)	M (no unit)	α 0 (no unit)	τ s (Pa)
	716.997	0.15862	1	224063
	A 1 ( no unit)	A 2 (K)	T s (K)	T ref (K)
	20.4	101.6	397.7	524.7

Table 2

 2 Properties of the polymer under study

	Specific heat, Cp (J/Kg/°K)	1900
	Conductivity, k (W/C/ m)	0.115
	(Kg/m 3 )	969

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Pollen analysis of the coprolite of Mammoth from Chukotka Jacqueline Argant

1-Introduction

It is well known that animals, either herbivores or carnivores, swallow pollen with their food. The analysis of the gut of herbivores shows that it contains many pollen grains which are getting more and more numerous along the intestine and, after, in the feces. This content varies with the season but it is not always easy to precise this one because remanent pollen accumulated on the ground, or in dry flowers, can be absorbed together with flowering plants and produce a mix of pollen from different periods in the year [START_REF] Argant | Climat et environnement au Quaternaire dans le Bassin du Rhône d'après les données palynologiques[END_REF]. But it reflects well the vegetation (trees as well as herbaceous plants) in the area where the animal was feeding. Many studies of fossilized feces of mammoth were still carried on and provided interesting data about the diet of these animals as well as about their environment and climate [START_REF] Mol | Results of the CERPOLEX/Mammuthus Expeditions on the Taimyr Peninsula, Arctic Siberia, Russian Federation[END_REF][START_REF] Van Geel | The ecological implications of a Yakutian mammoth's last mealQuaternary[END_REF]. The discovery, in 2010, of coprolites of Mammoth in a collapse of the bank of the river Small Anjuj in the far East of Russia was particularly interesting as no other remains of the animal were found. The coprolites were consequently the only way to get information on this Mammoth and its environment. Irina Kirillova took a fragment in the heart of one coprolite, therefore protected against pollution by modern pollen rain, to perform its pollen analysis.

2-Preparation of the coprolite

The preparation of the coprolite comprises the following principal stages: -Decarbonatation with hydrochloric acid, desilicification with hydrofluoric acid (concentration 40 %, cold test).

-Removal of the organic matter by heating in potassium hydroxide solution for 10 minutes.

-Concentration in a dense solution (d = 2) -Washing out with dimineralised water after each operation.

-Mounting in glycerin.

3-Pollen content of the coprolite

This coprolite is made of a dark grey, very compact powder, enclosing a lot of plant fibres. As could be expected, the coprolite showed many microscopic remains of plant tissues. Well preserved pollen grains were dispersed among them. Twenty-one taxa could be observed. The table 1 provides the number of pollen grains of each taxon and its percentages calculated in relation to the total pollen sum: arboreal pollen (AP) + non arboreal pollen (NAP) + spores. The figure 1 is a simplified representation of the pollen spectra.

Pollen grains found in the coprolite are mainly from herbs, defining an open landscape. Among these, pollen grains of grass largely dominate with several species of Poaceae, which can be distinguished essentially by their various sizes. Unfortunately, based on this only characteristic, it is impossible to identify them precisely to species. The second more abundant taxon is Artemisia (15,81 %), represented by at least two species. We find also Caryophyllaceae (4,99 %), Rosaceae (Potentilla type and the genus Dryas, respectively 2,83 and 0,33 %), Fabaceae (2,33 %), Cyperaceae (1%). The other herbaceous plants are all in low percentages, less than 1%. Only one spore of fern was observed. Some taxa provide the indication of wet habitats, but they are few: Polygonum amphibium type, Ranunculus aquatilis type, Thalictrum, and two Rosaceae, Sanguisorba officinalis and Filipendula, the first growing well on grassy banks, the second in damp meadows. Cyperaceae is probably, as most often, also associated with wetlands.

As for the trees, they seems almost inexistant : a single pollen grain of Alnus, one of a Rosaceae Sorbus type, and one ballonnet of Pinus. It was likely that these trees were not present in the close vicinity of the site of Chukotka, as their pollen can fly a long distance, especially in the case of Pine. Some spores of coprophilous fungi have also been observed. The pollen spectra of the coprolite of Chukotka Mammoth is dominated by Poaceae, Artemisia and some other light-demanding taxa, without any tree pollen. The pollen assemblage is close for instance to those recorded from the Taimyr peninsula, especially the Jarkov Mammoth [START_REF] Mol | Results of the CERPOLEX/Mammuthus Expeditions on the Taimyr Peninsula, Arctic Siberia, Russian Federation[END_REF]. It reflects a steppic, treeless environment, corresponding to the so-called "Mammoth Steppe", under cold and dry climatic conditions. After Van Geel et al. ( 2008) "coprophagy which has been repeatedly reported from herbivores [...] is suggested as a purposeful behavior [...] for mammoths", as spores of coprophilous fungi were sometimes found inside the gut of these animals. The presence of similar spores inside the heart of the Chukotka coprolite may verify this assertion.

Fig. 1 :

 1 Fig. 1 : Simplified pollen spectra of the coprolite of Mammoth from Chukotka

Table 1 :

 1 Pollen analysis of a coprolite of mammoth from Chukotka


Maxime Bordival

Yannick Le Maoult

Fabrice Schmidt

Optimisation of Preform Temperature Distribution For the

published or not. The documents may come

INTRODUCTION

Among the techniques devoted to the manufacture of PolyEthylene Terephtalate (PET) bottles, the twostage Stretch-Blow-Moulding (SBM) process is probably the most popular. This process involves the manufacture of structurally amorphous semiproducts, called preforms, made by injection moulding of PET resin. A reheating step is necessary to heat the preforms to the appropriate temperature distribution above the glass transition, which is typically around 80°C for PET. This stage is generally performed using infrared (IR) heaters, taking advantage of the semitransparent behaviour of PET submitted to IR radiation. In a second stage, the preforms are stretched using a cylindrical rod, and blown using two levels of air pressure. Then, the bottles are cooled down by a mould whose temperature is regulated using cooling channels.

The heating conditions, that control the preform temperature distribution, strongly affect the blowing kinematics (stretching and inflation), and consequently the thickness distribution of the bottle. Temperature also affects the orientation induced by biaxial stretching, which in turn, affects mechanical, optical and barrier properties of bottles [1].

Regarding to BM, Lee and Soh [2] presented a FE optimization method to determine the optimal thickness profile of a preform, given the required wall thickness distribution for the blow-moulded part. More recently, Thibault et al. [3] proposed an automatic optimization of the preform geometry (initial shape and thickness) and operating conditions, using the nonlinear constrained algorithm Sequential Quadratic Programming (SQP). The robustness of the method was discussed through a comparison with experiments performed within industrial conditions. SQP was also used in order to optimize heating system parameters [START_REF]Proc. of the Int. Conf. ESAFORM[END_REF]. The ABSTRACT: This study presents an optimization strategy developed for the stretch-blow moulding process. The method is based on a coupling between the Nelder-Mead optimization algorithm, and Finite Element (FE) simulations of the forming process developed using ABAQUS®. FE simulations were validated using in situ tests and measurements performed on 18.5g -50cl PET bottles. To achieve that, the boundary conditions were carefully measured for both the infrared heating and the blowing stages. The temperature distribution of the perform was predicted using a 3D finite-volume software, and then applied as an initial condition into FE simulations. Additionally, a thermodynamic model was used to predict the air pressure applied inside the preform, taking into account the relationship between the internal air pressure and the enclosed volume of the preform, i.e. the fluid-structure interaction. It was shown that the model adequately predicts both the blowing kinematics and the thickness distributions of the bottle. In a second step, this model was combined to an optimization loop to automatically compute the best perform temperature distribution, providing a uniform thickness for the bottle. Only the last part will be fully detailed in this paper. In this work, we propose a numerical optimization strategy for SBM. For that, we developed an iterative procedure allowing to automatically compute the best temperature distribution along the preform length, providing a uniform thickness for the bottle. We solve the optimization problem by coupling FE simulations to the Nelder-Mead optimization algorithm (nonlinear simplex). Results were validated by careful in situ tests and measurements performed on 18.5g -50cl PET bottles. To achieve that, special attention was given to the measurement of boundary conditions required for both the infrared heating stage, and the blowing stage.

OPTIMIZATION OF PREFORM TEMPERATURE

The performance of a bottle manufactured by SBM is drastically affected by its thickness distribution. In order to achieve bottles with appropriated thickness distributions, it is more desirable to adjust the process conditions, and to use the same design of preform for making different shapes of bottles. This approach aims to minimize the cost associated with the design of a new perform (especially the manufacture of a new injection mould). Determining adequate operating conditions remains nevertheless costly and time consuming. Different approaches are possible, such as trial-and error methods, or design of experiments. Both of them require a large number of experiments (or simulations), especially when the parameters are strongly interdependent. As a consequence, they become inadequate and impracticable for complex problems. In contrast, the optimization algorithms make the optimization process fully automatic, and from this point of view, yield a significant assist in the development cycle.

In this section, we propose to couple an optimization algorithm to FE simulations in order to optimize the temperature distribution along the preform length. The goal will be to provide an homogeneous thickness for the bottle. Infrared heating and blow moulding numerical models have been fully detailed in a previous paper [START_REF]Proc. of the Int. Conf NUMIFORM[END_REF]. A typical blow moulding simulation is presented in figure 1 where we compare the thickness distribution computed, and the thickness profile measured. Measurements were averaged on a set of three trials. We observe a good agreement along most part of the bottle (less than 15 % error on the mean thickness).

Parameterisation and constraints

In order to describe the temperature distribution along the preform length, we consider three optimization variables. They correspond to three temperatures located at different heights of the preform, as illustrated by figure 2. The whole temperature distribution is then deduced using the Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) method [START_REF] Fritsch | [END_REF]. To provide an accurate interpolation, an additional temperature is added on the preform neck. This fourth temperature is not optimized, but fixed to 80°C, which corresponds approximately to the glass transition of PET. Indeed, throughout the reheating stage, the preform neck is generally protected from IR radiation in order to prevent its temperature from exceeding the PET glass transition. This approach aims to prevent any deformation of the bottle neck during the forming process. Finally, to simplify the problem, the temperature is assumed to be uniform through the preform thickness. The optimization variables are constrained using lower and upper bounds, corresponding respectively to the PET glass transition temperature, and to the PET crystallization temperature. These two physical limits have been naturally chosen to prevent serious strengthening of the structure from appearing, in which case, any deformation would be proscribed during the forming stage. Let us note that neither linear nor nonlinear constraint is required.

Objective function

In this application, we attempt to provide a uniform thickness for the bottle. This objective must be mathematically formulated by an appropriate costfunction. A simple way to proceed is to define the objective function F as the standard deviation of the computed thicknesses, as following:

( )

∑ = - - = n 1 i 2 i th th 1 n 1 ) x ( F r (1)
where x r represents the set of optimization variables, n is the number of nodes along the bottle height, i th is the thickness at the node " i ", and th is the mean thickness. The nodal thicknesses are computed using a Python script that we have developed into ABAQUS® FEM software. Such a function is null for a bottle with perfectly uniform thickness.

Choice of an Algorithm

The choice of the optimization algorithm is closely related to the type of cost function. In our application, we attempt to minimize a nonlinear realvalued function, subject to bound constraints. In addition, strong mechanical and geometric nonlinearities could induce significant numerical instabilities, making the objective-function noisy, and therefore non-differentiable. As a consequence, the gradient-based algorithms might not be adapted to this type of problem. In contrast, the direct search methods (which do not require the computation of the cost-function gradient) remain particularly adapted to the non-derivative optimization. Among this family of methods, the Nelder-Mead simplex algorithm is probably one of the most popular. However, this local method provides relatively slow convergence rates [7]. Nevertheless, when the derivatives can not be explicitly written, this method can save a significant amount of computation time compared to gradient-based methods. Indeed, the computation of the cost-function gradients can become strongly time consuming when they are approximated using the finite-difference method. This is particularly true when the number of optimization variable is large. On the other hand, the Nelder-Mead simplex algorithm is restricted to unconstrained problems. In this work, we used the method proposed by Luersen et al. [START_REF] Luersen | Proc. of the Int. Conf. on Eng. Comp. Tech[END_REF] in order to add bound-constraints into the Nelder-Mead simplex algorithm available in Matlab® .

Results and discussion

All numerical results reported in the sequel were obtained on a 2.8 GHz-512 Mo Pentium 4. Figure 3 displays the decrease of the objective-function value in terms of the number of optimization iterations. We observe that the objective function is reduced by 60% of its initial value after the first iteration, and by more than 80% at the end of the optimization process. Consequently, the thickness distribution of the formed bottle is 80% more uniformed after optimization. The algorithm converges after 5 iterations, which involves only 10 objective-function evaluations (that is to say, 10 FE simulations).

On average, one cost-function evaluation requires 26 min CPU. Thus, the total CPU time required for the optimization is approximately 3h20min. Figure 4 illustrates the temperature distribution along the preform length before and after optimization. Initial conditions were chosen in order to apply a uniform temperature (100°C) on the preform. Such temperature distribution leads to a strongly nonuniform thickness distribution for the bottle, as illustrated by Fig. 5. After optimization, there is a temperature gradient along the perform length, which provides a more uniform thickness and a full blowing of the bottle. Fig. 4 also illustrates the optimal temperature distribution determined by Logoplast Company using an experimental trial-and-error method. This result has been obtained using the same preform, but with a different shape of mould. However, we can notice that there is a good agreement in the trends between the temperature profile experimentally determined within industrial conditions, and the temperature distribution computed using our optimization method.

CONCLUSION

For SBM optimization, we have proposed a practical methodology to numerically optimize the temperature distribution of a PET preform, in order to provide a uniform thickness for the bottle.

Encouraging preliminary results have shown the viability of our approach. However, it would probably be more desirable to directly optimize the process parameters of the heating systems. But to do so, both the infrared-heating simulation and the blowing simulation would need to be included into the optimization loop, resulting in further complications essentially due to long computation times.

Nevertheless, this approach would implicitly account for the influence of the temperature distribution through the preform thickness, which is of prime interest.

  Key words: Stretch-blow moulding, PET bottles, heat transfer, finite element method, optimization. Optimisation of Preform Temperature Distribution For the Stretch-Blow Moulding of PET Bottles M. Bordival, Y. Le Maoult, F.M. Schmidt CROMeP, Ecole des Mines Albi, Campus Jarlard, 81000 Albi, France objective was to homogenize the temperature along the preform length, by modifying the process parameters related to the IR oven. It is interesting to point out that authors questioned the relevance of the objective chosen for the optimization.

Fig. 1 .

 1 Fig. 1. Wall thickness distribution of the bottle. The error bars show ± 1 standard deviation for a set of 3 trials.

Fig. 2 .

 2 Fig. 2. Temperature distribution along the preform length -Optimization variables.

Fig. 3 .

 3 Fig. 3. Objective-function value versus iterations.

Fig. 4 .

 4 Fig. 4. Initial and optimized temperature distributions along the preform length.

Fig. 5 .

 5 Fig. 5. Thickness distribution of the bottle before and after optimization.

ACKNOWLEDGEMENTS

This study was conducted within the frame of 6th EEC framework. STREP project APT_pack; NMP -PRIORITY 3. www.apt-pack.com. Special thanks to Logoplast Technology for manufacturing the preforms, and QUB for their collaboration.