M Hofmann

B Stoffel

O Coutier-Delgosha

R Fortes-Patella

J L Reboud

EXPERIMENTAL AND NUMERICAL STUDIES ON A CENTRIFUGAL PUMP WITH 2D-CURVED BLADES IN CAVITATING CONDITION

In the presented study a special test-pump with 2D curvature blade geometry in cavitating and non-cavitating conditions was investigated using different experimental techniques and a 3D numerical model of cavitating flows. Experimental and numerical results concerning pump characteristics and performance breakdown were compared at different flow conditions. Appearing types of cavitation and the spatial distribution of vapour structures within the runner were also analysed.

Introduction

In the scope of the European Research Program PROCOPE, researchers of the TUD (Laboratory for Turbomachinery and Fluid Power at Darmstadt) and of the LEGI (Laboratoire des Ecoulements Géophysiques et Industriels de Grenoble) work together in order to improve the understanding of the unsteady behaviour of cavitating flows and the related erosive aggressiveness. In order to extend previous analyses concerning the cavitating flows around a 2D-hydrofoil [START_REF] Hofmann | Numerical and experimental investigations on the self-oscillating behavior of cloud cavitation: part 1 visualisation / part 2 dynamic pressures[END_REF] and in a cascade of three hydrofoils [START_REF] Lohrberg | Numerical and Experimental Investigations on the Cavitating Flow in a Cascade of Hydrofoils[END_REF], the present study consists of investigations by experimental means and numerical simulation concerning a special centrifugal test-pump in cavitating and noncavitating conditions. Experiments were carried out at Darmstadt University of Technology using different experimental techniques. The measurement of classical pump characteristics and performance breakdown at different flow conditions were associated with flows visualisations. Appearing types of cavitation and the spatial distribution of vapour structures within the runner were analysed. Three-dimensional Navier-Stokes codes taking into account the cavitation process have been developed during the last years [START_REF] Takasugi | Study on cavitating flow around a finite span hydrofoil[END_REF], Alajbegovic et al., 1999[START_REF] Kunz | Multi-phase CFD analysis of natural and ventilated cavitation about submerged bodies[END_REF], Bunnel and Heister, 2000] based on different multi-phase flow approaches [START_REF] Delannoy | Two phase flow approach in unsteady cavitation modelling[END_REF]Kueny, 1990, Kubota et al. 1992]. Industrial CFD codes are now starting to take into account cavitation models, allowing first applications to pump geometries [START_REF] Combes | Etude de la cavitation dans la pompe SHF[END_REF]. In this context, a numerical model for three-dimensional cavitating flows is developed at LEGI, based on the 3D code FINE/TURBO TM , developed by NUMECA International. That work is performed in cooperation with SNECMA Moteurs and the French space agency CNES, with the final objective to provide accurate simulations of unsteady cavitating flows in the inducers of rocket engine turbo-pumps [Coutier-Delgosha et al. 2001a,b]. The application to the centrifugal pump represents a first step of validation of the model on steady-state cavitating flow in turbomachinery.

Geometry

A special runner geometry has been chosen to easily adapt existing measuring techniques for a single hydrofoil in a test section [START_REF] Hofmann | Numerical and experimental investigations on the self-oscillating behavior of cloud cavitation: part 1 visualisation / part 2 dynamic pressures[END_REF][START_REF] Hofmann | Ein Beitrag zur Verminderung des erosiven Potentials kavitierender Strömungen[END_REF] to a pump test-rig. Optical access in 2 planes was made possible to enable a view perpendicular to the blade surface on both suction and pressure side by Plexiglas windows in the housing and a Plexiglas shroud. The runner has 5 single-curved blades with two different radii at inlet and outlet. The part of the blade with the larger radius (the second part in flow direction) was also made of Plexiglas to obtain optical access to the pressure side of the following blade and the entire channel respectively. Figure 1a shows the intersection of the 2 blade radii. Another characteristic is the parallel hub and shroud to get an almost 2-dimensional blade-to-blade channel with constant width. A radial-symmetric housing (Figure 1b) is used to obtain almost constant conditions on the runner outlet (if we neglect gravitational forces which are small compared to the performance of the runner). With these preconditions we obtain comparable cavitation conditions in each channel without the influence of a volute casing.

Nominal conditions are at a rotational speed of 36 Hz and a flow-rate q = 210 m 3 /h. Specific speed of the runner is n s = n Q 1/2 / H 3/4 = 20 (europ. value: n in rpm, Q in m 3 /s and H in m), the outer runner diameter equals 278 mm. Cavitation conditions are defined by NPSH value based on the upstream total pressure, water vapour pressure and density at ambient temperature.

Visualisations

Besides the measurements of the characteristics of the pump at different flow-rates and cavitation conditions, various visualisation techniques were used. All images shown in this paper were done at nominal flow rate with various values of NPSH, where both cavitation on suction and pressure side of the blades occurs. Stroboscopic light was used on one hand for standard imaging and High-Speed-Video with the Light-Sheet-Illumination (Figure 2) was applied to observe self-oscillating states of the cavitation on the blade pressure side or other unsteady effects [START_REF] Hofmann | Similarities an geometrical effects on rotating cavitation in 2 scaled centrifugal pumps[END_REF]. The investigated flow conditions during the experiments show always unsteady behaviour of the leading edge cavitation, not only concerning the unsteadiness of the closure region of the attached cavitation but also the shedding of vapour structures in the channel. Besides the well known attached cavitation on pressure and suction side, another type of cavitation occurs on the inlet of the runner caused by the strong curvature of the streamlines. It is visible either as attached cavitation caused by the depression at the radius (regions B and C in Figure 3), or as shear cavitation because the flow was separating at the inlet radius of the shroud at lower values of NPSH (region B in figure 4). Dependent on the length of the attached part of the cavitation (region A in all figures) the extent of this shear layer cavitation leads into the channel as it is shown in Figure 4 at two different stages of a cavitation cycle. In Figure 4a the larger extent of the shear cavitation (region B) on top of the attached part (see region A, the illuminated interface of the cavitation sheet on pressure side) is visible. In Figure 4b (with a smaller cavitation sheet) no shear cavitation can be seen. The contour of the interface of the attached part of the cavitation (region A) already indicates a different state of the typical self oscillating cloud cavitation condition. With the aid of Laser-Light-Sheet-Illumination of the vapour-fluid-interface, an analysis of the unsteadiness of the attached part of the cavitation in the runner could be done in the middle of the channel. The images were taken with a rate of 2 Hz but triggered by an angular decoder within the motor and a special conditioning which allowed to take the image at every phase angle compared to a reference angle with a step size of 1 degree. Hence, the illuminated blade had 18 revolutions between every image. Based on 500 images, illuminated with 50 µs, a mean grey value distribution was calculated to identify the mean cavity on the leading edge of the blade.

The standard deviation of the grey values or its variance can serve as a value for the unsteadiness of the cavitation. In regions of the image with higher values of the variance, the fluctuations of the grey values, and therefore the fluctuations of the reflecting vapour structures, are larger than in regions with a smaller variance. Those regions are constantly filled with either water or vapour. The result of such a treatment is shown in Figures 5 and6. Mean distribution (Figure 5a and 6a): The attached part of the cavitation can only be identified by its 2-phaseinterface, because the light sheet is mainly reflected. But the mean region with a cloud shedding is also indicated by a higher mean grey value just after the closure region of the cavitation sheet. For the numerical comparison we just identify the attached part. As expected, the extent of both regions enlarges with decreasing NPSH-value. As already mentioned, the flow has the tendency to separate on the leading edge of the blade, which is supported (Figure 6a) at lower pressure conditions. [START_REF] Delannoy | Two phase flow approach in unsteady cavitation modelling[END_REF][START_REF] Reboud | Two-phase flow structure of cavitation: experiment and modelling of unsteady effects[END_REF]]. This fluid is characterised by a density ρ that varies in the computational domain: when the density in a cell equals the liquid one (ρ l ), the whole cell is occupied by liquid, and if it equals the vapour one (ρ v ), the cell is full of vapour. Between these two extreme values, a liquid/vapour mixture, still considered as one single fluid, occupies the cell. The void fraction α = (ρ -ρ l ) / (ρ v -ρ l ) can thus be defined as the local ratio of vapour contained in this homogeneous mixture.

Velocities are assumed to be locally the same for liquid and for vapour. An empirical state law is used to manage the mass fluxes resulting from vaporisation and condensation processes. That barotropic law links the density to the local static pressure ρ(P). When the pressure is higher or lower than vapour pressure, the fluid is supposed to be purely liquid or purely vapour, according to the Tait equation or to the perfect gas law respectively. The two fluid states are joined smoothly in the vapour-pressure neighbourhood. It results in the evolution law presented in Figure 7, characterized mainly by its maximum slope 1/A min 2 , where A min 2 = ∂P/∂ρ. A min can thus be interpreted as the minimum speed of sound in the mixture. The numerical model of cavitating flows based on that physical description is developed from the commercial code FINE/TURBO TM developed by NUMECA International. FINE/TURBO TM is a three-dimensional structured mesh code that solves the time dependant Reynolds-Averaged Navier-Stokes equations. Time accurate resolutions of the equations use the dual time stepping approach. Pseudo-time derivative terms are added to march the solution towards convergence at each physical time step. The range of application is extended to low-compressible or incompressible flows by introducing a preconditioning matrix [START_REF] Hakimi | Preconditioning methods for time dependent Navier-Stokes equations[END_REF]. The discretization is based on a finite volume approach. Convection terms are treated by a second order central scheme associated with artificial dissipation terms. The pseudo-time integration is made by a four-step Runge-Kutta procedure. The physical time-derivative terms are discretized with a second order backward difference scheme. The code resorts to a multigrid strategy to accelerate the convergence, associated with a local time stepping and an implicit residual smoothing. The model was applied to the presented centrifugal pump geometry. 300.000 cells multi-blocks mesh of one single blade-to-blade channel was used (Figure 8). Conditions applied for the simulations are the following: -Turbulence model: we use for the simulations presented in this paper a Baldwin-Lomax turbulence model. -Boundary conditions: Velocity is imposed at the inlet of the suction pipe. Laws of the wall are imposed along solid boundaries. The relative motion between the inlet pipe walls and the runner is taken into account. On the other hand, the outlet housing shape is not described and the parallel walls are treated as hub and shroud extensions up to the outlet, at 1.5 times the runner outer radius, where a uniform static pressure is imposed. -Initial transient treatment: First of all, a steady step is carried out, with a pseudo vapour pressure low enough to ensure non cavitating conditions in the whole computational domain. Then, the NPSH is slowly lowered by increasing smoothly the pseudo vapour pressure at each new time step up to the physical value. Vapour structures spontaneously appear and grow during that process, in the regions of low static pressure. The final NPSH value, depending on the outlet static pressure imposed, is then kept constant throughout the computation.

Figure 8. View of the mesh on shroud side of the pump. (the entire pump geometry is reconstructed by rotation of the single blade-to-blade channel)

Non-cavitating characteristics

Experimental tests and numerical calculations were performed considering a large range of flow-rates, in noncavitating conditions. Figure 9a illustrates the computational result at nominal flow rate: the total pressure elevation is represented in the middle of the pump channels. Figure 9b presents a comparison between the numerical and experimental performance charts. Directly based on the measurements performed, the head of the pump is defined as the difference between downstream static pressure and upstream total pressure. We observe a reliable agreement between the pump characteristics given by measurement and computation in the whole range of flow rates investigated experimentally. The model gives a better prediction when the flow rate is over 50% of the nominal value (Qn = 210m 3 /h), and the numerical simulation becomes unstable at very low partial flow rate. As a matter of fact, the numerical simulation slightly overestimates the pump head. This is an expected result, since the flow through the side chamber is not taken into account in numerical simulations. This gap flow of the pump impeller has to be added to the flow rate actually passing through the blade-to-blade channels and is therefore slightly higher in the experiments than in the model. At nominal flow rate, the head is overestimated of about 5% (40.5m instead of 38.5m). The uncertainty on the measured value was estimated to max. 80 cm and corresponds to 2% of the value, which also has to be taken into account. We summarize in Figure 12 the whole comparison, by drawing the head drop curves for three different flow rates: namely 0.8Qn, Qn and 1.08Qn. From the numerical point of view, while decreasing the NPSH, the performance drop appears first as a smooth decrease of the pump head. The final head-drop is only partially simulated because the computation rapidly becomes unstable and stops (this is more particularly the case at 1.08Qn). Our upstream boundary condition consists in imposing in a strict manner the mass flow rate passing in the pump. Because the coupling between the pump and the hydraulic loop is not taken into account, the effects of the cavitation blockage on the flow rate are neglected and the head-drop is less progressive than in the experiments.

Results obtained from first simulation of the pump cavitation behaviour are promising: the head drop is predicted with a good homogeneity with respect to the three flow rates. The NPSH values obtained for the 3% and 10% head drop are globally overestimated with respect to experimental values (of about 1m for the 10% head drop, and 1.5 to 3 m for the 3% head drop, see Figure 13). These results correspond to our first try of predicting the cavitation characteristic of a pump, and a study of the effect of the model parameters (e.g. the A min value in the state law is set to 2 m/s in that computation or the turbulence model) might probably improve the quantitative agreement.

Conclusion

Numerical and experimental results were presented in this study, concerning a test runner with 2D curvature blade geometry. From the experimental point of view, besides the measurements of the cavitation characteristics of the pump in various conditions, a special visualisation set-up was developed to investigate the unsteady behaviour of leading edge cavitation. Image processing and statistical treatment of the photographs taken at given runner position allowed to quantify the attached and cloud cavitation extent. A numerical model of 3D cavitating flows, based on the 3D code FINE/TURBO TM , has been developed to predict the cavitation behaviour in turbomachinery [Coutier-Delgosha et al. 2001a,b]. This model was applied to the centrifugal pump geometry. Non-cavitating and cavitating conditions were investigated. Calculations were found to be in good agreement with experimental measurements and visualisations. Experimental and numerical results concerning the pump characteristics and performance breakdown were drawn at different flow conditions and the mean spatial distributions of vapour structures within the runner were compared. The results obtained show the ability of the model to simulate the main features of 3D cavitating flows in rotating machinery. However, the fluctuating twophase areas are not simulated and work is in progress to improve the numerical model in that way.

Figure

  Figure 1a. runner geometry ; 1b. housing

Figure 2 .

 2 Figure 2. Visualisation set-up

Figure 3 .

 3 Figure 3. Unsteady state of blade cavitation on suction side, NPSH = 8m. (stroboscopic light illumination) (A scaling bar is added to each image, representing a length of 10 mm)

Fig. 4 :

 4 Fig. 4: unsteady state of blade cavitation on pressure side at two different time and NPSH = 7 m (Laser light sheet)

A

  Standard deviation (Figure5band 6b): The fluctuation of the shedding becomes larger with decreasing NPSH and its extension fills almost half of the height of the channel. This is a result of the higher production of transient vapour structures from the leading edge cavitation. The interface of the attached cavitation seems in contrast to be rather stable. Probably the flow is clearly separated from the leading edge, the separation zone fills almost steady with vapour, and only in the closure region vapour is shed into the channel.

Figure 5 .

 5 Figure 5. Mean vapour distribution and standard deviation Qn , NPSH = 7 m

Figure 7 :

 7 Figure 7: The barotropic state law ρ(P) for water.

  Figure 9a. Total pressure field in the pump (nominal flow rate)

Figure 10 .

 10 Figure 10. Vapour structures on suction side (experiment NPSH=8m, computation NPSH = 7m). Calculation: iso-density contour (ρ ≈ 0.95ρ l : void ratio >5%) drawn in yellow, shroud in blue, blade in grey 10 mm

Figure 11

 11 Figure 11: a) Numerical density distribution ; b) pressure side cavity, comparison with experiment (NPSH = 7m).

  Figure 12. Head-drop curves: comparison at 0.8Qn, Qn and 1.08Qn

Figure 13

 13 Figure 13: NPSH values for 3% and 10% head drop.

Acknowledgements

This work is part of a European exchange program PROCOPE with the research teams of Grenoble and Darmstadt as members. The numerical model has been developed with the support of SNECMA Moteurs (Rocket Engine Division), and the French space agency CNES.


Camille Sonneville

Dominique Planson

Viêt Luong

Pascal Phung

Besar Bevilacqua

Asllani

Interest of using a micro-meter spatial resolution to study SiC semi-conductor devices by Optical Beam Induced Current (OBIC)

Interest of using a micro-meter spatial resolution to study SiC semiconductor devices by Optical Induced Current (OBIC)

C. Sonneville 1) , D. Planson 1) , L. V. Phung 1) , P. Bevilacqua 1) , B. Asllani 1) dominique.planson@insa-lyon.fr 1) Univ Lyon, INSA Lyon, Univ. Claude Bernard Lyon 1, Ecole Centrale Lyon, CNRS, Ampère, F-69621, France In order to fully benefit of the advantages provided by SiC and to avoid premature breakdown of the high voltage devices, it is mandatory to have efficient peripheral protections such as MESA and/or JTE.

The OBIC characterization can help the technology computer-aided design (TCAD) and the device process to optimize the efficiency of the periphery protection by analyzing the electric field distribution in the structure and especially at the junction periphery. In this talk, we will present an in-house testbench called micro-OBIC which will allow us to characterize SiC high voltage (HV) PiN diodes (10 kV class) with a micro-meter spatial resolution.

A laser emitting UV at 349 nm is used to generate electron-hole pairs into SiC epilayer close to the top surface. In order to reduce the laser beam spot size, we developed a testbench represented in figure 1.a)

The laser beam intensity is first controlled by a set of neutral density filter, before passing through a beam expander. Finally, the laser spot is focused by a microscope objective. The spot size is estimated around 1-2μm. The HV device under test (DUT) is fixed on a X,Y,Z motorized stage and the DUT can be moved under the laser beam focus point. Induced current is then measured with a Source-Measure Unit (SMU). The DUT can be reverse biased by applying voltage thanks to the SMU. 2D (X,Y) micro-OBIC scans were performed on HV PiN diode. The PiN diode was fabricated on a 4H-SiC wafer using a 110 μm thick epilayer with a doping concentration of 7.10 14 cm -3 . These diodes are protected by a MESA and a JTE of 400 𝜇m length as shown in Figure 1.b) Figure 2b and2c. compared the optical image and the OBIC signal (at 0V) of the same part of the diode. The OBIC signal is null in the electrode, reached a peak (around 120nA) in the MESA region, then decreases to lower value to stay almost constant (around 30nA) in the whole JTE region. Figure 3 represents a 3-D view of the OBIC signal with a reverse biased at 100V. The reverse biased induced an increase of the OBIC peak to 200nA and of the signal in the JTE region to 130nA. Moreover the microscopic spatial resolution of the micro-OBIC testbench highlights very small features in the MESA region, which were not observable with previous lower spatially resolved OBIC testbench [START_REF] Planson | Experimental and simulation results of optical beam induced current technique applied to wide bandgap semiconductors[END_REF]. Some defaults in the JTE region are visible too. To better understand the physical origin of these fine structures and defaults, these experimental results will be compared to OBIC simulations and coupled with other characterization technics such as scanning electron microscopy (SEM) or micro-Raman spectroscopy.

b)

Fig. 1a). Schematic representation of the micro-OBIC testbench.

  Fig.1a). Schematic representation of the micro-OBIC testbench. Fig.1b) schematic representation of the SiC HV PiN diode.

Fig. 2a )Fig. 3 .

 2a3 Fig. 2a) Optical image of the diode (top view), microscope objective x50. Fig. 2b) micro-OBIC 2D-scan of the SiC HV PiN diode at 0V.

The OBIC, is a non-destructive characterization technique, which has been previously successfully used to characterize High Voltage (HV) SiC devices [1,2,3,4].

  UV Laser can be used to generate electron-hole pairs into Wide Band-Gap (WBG) semiconductors. In the space charge region, the electric field drives the collected carriers and a current, so-called Optical Beam induced Current (OBIC), can be measured. The induced current is directly related to the electrical field in the device.

Mo-P-23 ICSCRM2019 © ICSCRM2019


Alexis Joly

Henning Muller

Yannis Kompatsiaris

Andreas Rauber

Henri Gouraud

Pieter Van Der Linden

Joost Geurst

Comments: First version based on think-tank5 and consortium discussions 0 2

Introduction

The main objective of this report is to provide recommendations to the European Commission concerning the funding and the steering of public evaluation campaigns as a tool to foster multimedia search technology transfer at the European level. These recommendations rely on the results of 3 complementary preliminary actions conducted within CHORUS+ coordination action: 1-Report on Multimedia Retrieval evaluation dimensions: Based on the literature and CHORUS+ consortium experience in the evaluation of multimedia search technologies, a report synthesizing the role of evaluation at different levels of the innovation workflow was published in November 2011 (deliverable D3.3). International benchmarks and evaluation campaigns were identified as a key component that might fill the gap between scientific criteria used by the researchers (to measure fundamental progress) and industrial & business-oriented criteria used by companies to identify relevant technologies and build innovative products.

2-Survey on the evaluation of Multimedia Search technologies:

To better understand the suitability of benchmarking as a tool to foster exchange between academia and industry, a survey was set up and delivered to both communities. The content of this survey was already published within D3.3 deliverable (November 2011). It was subsequently distributed in December 2011 to several communities including ImageCLEF, TRECVID and MIREX benchmarking campaigns, ACM Multimedia 2011 conference, EU projects of the Media Search cluster and a LinkedIn group on enterprise search. The 20 participants to the 5th Think Tank on Multimedia search technology transfer driven by benchmarking were finally asked to fill the survey. The result's analysis of the survey was used as a support of discussion during the ThinkTank and is provided in the annex of this report.

3-Thinktank on Multimedia Search technology transfer driven by benchmarking: A

Thinktank was organized by CHORUS+ on April 19th 2012 during the international conference WWW 2012 in Lyon. This event brought together experts and stakeholders of multimedia search related benchmarking efforts in order to exchange on lessons learned and to assess suitability of benchmarking to foster technology transfer. The results of the above mentioned survey were presented and discussed during the meeting. The question of whether EU should play a role in further strengthening and supporting these efforts was also addressed as an important issue. Around 20 people including leading industrials, expert SMEs, EC representatives and highly known researchers in the multimedia search technology field gathered in Lyon to discuss these subjects. The notes synthesising the whole discussions of this Thinktank are available as one CHORUS+ deliverable (D5.2.5, Think-tank 5 Meeting Notes).

The content of this report is organized in two main sections. The first part (section 1) synthesized the main conclusions and lessons learned from the three actions described above.

The second part provides the recommendations of CHORUS+ consortium towards sustaining and/or improving European practices concerning public benchmarking.

Main conclusions and lessons learned about benchmarking campaigns

Benchmarking campaigns are suitable to foster exchange between academia and industry. Challenges measured in benchmarking campaigns are overall judged as relevant by both academia and industry. The motivations differ from an actor to another (e.g. between SME's, big companies and research institutes) but each of them finds an interest to participate in or to follow the results of benchmarking campaigns. Identified benefits for these actors include: (i) measuring and boosting global research progress (ii) increasing the visibility of good research (iii) facilitating access to evaluation data (iv) facilitating the emergence and the sustainability of research communities (v) fostering the convergence of evaluation methodologies (vi) fostering the emergence of private benchmarks modeled on public ones but using business-specific data

Benchmarking campaigns have a positive scientific, technical and economic impact. NIST (one of the largest evaluation campaign organized by the US National Institute of Standards and Technology) has measured significant technical, industrial and scientific impact of its campaign. In particular the TRECVID campaign has allowed to double performances of systems over 3 to 10 year span (depending on the topic). According to a study of RTI International return on investment reached a factor 3 to 51 . And finally TRECVID has generated more than 2000 publications. Similarly significant technical and scientific impact has also been described in publications about other campaigns.

The results of CHORUS+ survey as well as the discussions during the Thinktank confirmed that benchmarking campaigns became an important tool for companies to identify relevant research progress and select new technologies for their products. An increasing interest for participating in, and, organizing benchmarking campaigns in the future was measured in both academia and industry.

Benchmarking campaigns are criticized in some points. An important criticism is the implicit cost to participate in an evaluation campaign. Up to 10 additional man months over usual R&D costs are required to participate in an evaluation campaign for the first time.

Even if this cost decreases for further participations, this expensive entry price has a negative impact on the participation of SME's as well as many research groups world wide. Another frequently mentioned shortcoming is related to the scale and scope of data used for benchmarking. Shipping real-world and big data is indeed logistically very difficult and limited by access rights. The consequence is that systems might converge to ad-hoc solutions and therefore generalize poorly when transferred to real-world content. A last criticism concerns the way technologies are evaluated in benchmarking campaigns, and notably the controversial question of user-centered vs. system-oriented evaluation. Some actors from both academia and industry complain that end-users of the technologies are not involved enough in the evaluation process. The large companies who participated in our survey particularly identified this point as critical. On the other side, user-centered evaluations strongly increase the evaluation cost and are suspect of being more subjective.

There is a lack of support to the organization of benchmarking campaigns in Europe.

There is no dedicated funding in Europe to sustain the organization of public benchmarking campaigns at the international level. Large initiatives such as CLEF or MediaEval typically live through heterogeneous and opportunistic research funds including national and European projects, and volunteer resources from research institutes. In this context it appears particularly difficult to assess the impact of campaigns over longer periods (5-10 year area). On the other side, the American National Institute of Standards and Technology is in charge of organizing most benchmarking campaigns in US with significant permanent resources (complemented by contributing external researchers). There was a consensus during the Thinktank on that Europe should not simply leave the floor to NIST (for several reasons related to scientific, cultural and social diversity as well as economic strategy). As a result of its central role on stimulating research and innovation in Europe, the EU commission appears as a highly recognized candidate to efficiently set up and support a sustainable and efficient way to fund and synchronize benchmarking campaigns in Europe.

Steering of benchmarking campaigns is controversial. Selecting and synchronizing scientific challenges measured in public benchmarking campaigns is a complex process sensitive to impartiality and biases. In the US, NIST employs several mechanisms: sometimes the challenge is defined by an agency, and in other cases the challenge is defined collectively by the research community. Most European benchmarking campaigns such as CLEF and MediaEval are based on a bottom-up mechanism. New challenges are proposed by individual research groups or research projects, and the organizers of previous campaigns decide collectively whether this new task should be integrated. There does not exist a specific mechanism to synchronize the campaigns between each other's. Some participants to the Thinktank rather suggested a top-down approach where the challenges would be defined by public agencies. A EU based effort should concentrate on evaluating results that are funded by EU funds. Some other participants tempered this approach to avoid adding a layer of bureaucracy and to avoid fragmentation of research evaluation.

Recommendations

According to the conclusions and lessons learned about multimedia search technology benchmarking, CHORUS+ consortium believes that a more sustainable and efficient way to fund and synchronize benchmarking campaigns in Europe is required. Here is a list of recommendations that go in that direction:

Ensuring transparency, sustainability and efficiency of benchmarking campaigns funding. As long as EU benchmarking campaigns rely on opportunistic and unaccountable funds, efficiency won't be measurable (by both the funders and the organizers of these campaigns). EU funding for the organization of benchmarking campaigns should therefore be more centralized and conditioned to a clear budget and work plan (for instance through specific calls for projects or through a dedicated EIT service similarly to NIST). The additional cost will be compensated by the reduction of the current costs (that are split over several projects) and by an overall efficiency gain.

Ensuring that benchmarking campaigns steering is balanced. The definition of the challenges measured in benchmarking campaigns has to be done collectively by the research community, the industry and the authorities. It is in particular crucial to keep an important place for innovation diversity by ensuring that new task proposals come from both the research community and the industry. Acceptance mechanisms should also rely on a balanced pool of experts.

Ensuring that the costs to participate in benchmarking campaigns are eligible for funding in EU research projects. This would help covering the additional engineering costs required to participate in benchmarking campaigns and foster the participation of small organizations (PME's, small research groups, etc.).

Encouraging participation in benchmarking campaigns.

Benchmarking campaigns are a tool to boost technological progress and foster exchanges between industry and academia. They should not be considered as a way to rate companies or research groups.

The technical performances measured in these challenges are actually reflecting only partially the scientific excellence of the underlying works or the quality of the tested products.

Conditioning funding to benchmarking results should in particular be avoided. The simple fact that an organization participates to a campaign will stimulate results and motivation to go ahead. Successful organizations are free to communicate on their results as an argument of scientific excellence or for advertising their products. Allowing anonymous submissions could be a rather good means to increase participation of companies. Companies have been reported to defer participation because they fear bad publicity in case of poor results.

Opening up the number and reach of the participants will mechanically foster technology transfer (e.g. : Companies may take over ideas and algorithms from academic teams ranked better than them).

Besides these structural recommendations, CHORUS+ consortium also would like to highlight two key objectives towards improving current practices in benchmarking:

Moving to larger and real-world data. The consequence of too small or too narrow data is that technologies generalize poorly when transferred to real-world content. This gap between the performances measured in benchmarking campaigns and what can be expected at scale-one is weakening technology transfer. Integrating new technologies in large infrastructures without enough guaranties on performances is actually too risky for many industrials.

Allowing user-centric and external evaluations. System-oriented evaluation metrics used in current benchmarks are essential but not sufficient to cover a vast range of usage of the evaluated technologies. Furthermore, evaluation methodologies are often not scalable because of the huge human work required to build appropriate evaluation data.

Complementary to current practices, a good evaluation framework should allow other research groups, companies or even end-users to evaluate a technology with their own criteria or in the context of their own workflow.

These two objectives are actually conditioned to more general concerns in the multimedia research community: data openness, availability of large-scale infrastructures and technology sustainability. CHORUS+ recommendations towards achieving these objectives therefore go beyond benchmarking issues but we believe making such recommendations can help converging to solutions:

Ensuring data openness in EU projects. Companies often refuse to share the large data they are using in their scientific publications, sometimes for competitive reasons and sometimes to protect customers' privacy. On the other side, as big data is becoming an important research area, this practice is criticized by many researchers for its secrecy and the risks of bad science, potential frauds, etc. The problem occurs as well within EU funded projects. Our recommendation is therefore to condition EU funding to some guaranties on data openness, at least for the project's consortium, and possibly to the research community (typically through benchmarking campaigns).

Funding large-scale infrastructures. Besides privacy and copyright issues, hardware resources and data management problems prevent many research groups from working on real-world and big data. We advocate for setting up a shared infrastructure at the European level adapted to research on information retrieval and data mining. Such infrastructure should allow hosting large-scale multimedia data as well as services developed by research projects (such as the services that could be evaluated in benchmarking campaigns). This could be done in collaboration with major content providers and owners of big infrastructures in Europe.

Ensuring sustainability of technologies built within EU funded projects.

When not exploited commercially, many relevant technologies built within EU projects are lost. New projects often re-develop the same piece of work and this results in a large waste of time and money. Ensuring the sustainability of the technical components developed in EU projects is therefore crucial. Our recommendation is that the developed components should be either commercially exploited or shared (with new EU projects and/or with the research community). A moratorium could be applied for making thinks easier, notably for industrial partners: any results may be locked up for one or two years, but then should be shared if no commercial exploitation occurred. An open infrastructure such as the one discussed previously could make such sharing easier.

Annex 1 -Results of CHORUS+ survey on the evaluation of multimedia retrieval technologies

CHORUS+ survey on the evaluation of multimedia retrieval technologies, described in

Respondent's profiles Organizations

A large fraction of the respondents come from academia (75%). In the detailed result's analysis we refer to this group as academics. The remaining 25% coming from industry still represent a tolerable number of respondents (20 companies). A great majority of respondents from academia are benchmarking aware. Less than half of the respondents from industry are aware about public benchmarking campaigns.

Companie's profiles

Companies Size Companies activity

Small, intermediate and big companies are well represented. In the detailed result's analysis we refer to small companies as the one having up to 50 salaries and to big companies as the one having more than 200 salaries.

People's profiles

Are you? Your main activity?

The main activity of 86% of the respondents is research but at different levels of responsibility (62% are researchers or engineers). In the detailed result's analysis we refer to the 9% of respondents whose main activity is development as developers and to the 14% of

Q1.2 How do you identify new technical components that you would like to experiment and/or benchmark?

Overall results The results show that according to all groups of respondents, scientific articles are the best way to identify new technologies, followed by benchmarking campaigns.

Detailed results

Big comp

Interestingly, watching competitors is a good source of information according to big companies whereas small companies and academics ranked it the worst criterion.

Q1.3 What criteria do you use for selecting technical components (for experimentation, proof of concept or integration in products) ?

Overall results

Detailed results

Big comp

Benchmarking results

Adequacy user needs

Benchmarking results

Benchmarking results

Benchmarking results

Worst

Novelty

Security Purchase price

Security Security Security

Overall results show that benchmarking campaigns are on average the best criteria to select new technical components for integration or deeper testing (whereas in previous question scientific articles were judged as the best way to identify/discover new components). But it is important to notice that benchmarking campaigns are ranked as the second best criteria by almost all groups when looking at the detailed result's table. Top-1 criteria for academics is actually scientific impact whereas the top-1 criteria for companies is technical skills (e.g. scalability, response times, portability, etc.). In between, benchmarking appears as the best compromise between research (technology suppliers) and exploitation (technology integrators). For both academics and companies, TRECVID is far away the most suited evaluation campaigns for their business or research activity, followed by ImageCLEF, MIREX and MediaEval. Notice that these results might be biased by the proportion of respondent's coming from the TRECVID community. But still, according to other statistics on the number of participants to these different campaigns (provided in D3.3) it is highly believable that TRECVID is the most popular one. In 2011 for instance, the number of participants was 73 at TRECVID, 43 at ImageCLEF, 40 at MIREX, 39 at MediaEval, 25 at PASCAL VOC, 15 at SHREC.

PART 1 Synthesis

Q2.2 To your opinion, the challenges measured in public evaluation campaigns are

Q2.3 To your opinion, the evaluation criteria used in public evaluation campaigns are

Both academics and companies consider that challenges measured in public evaluation campaign as well the used evaluation criteria are reasonably relevant and very relevant for about 20% of them.

Q2.3 In the future do you plan to

An important conclusion of this graphic is that almost 60% of the companies who responded to the questionnaire plan to use technologies selected as the best ones within benchmarking campaigns.

Future intentions of respondents about their participation in benchmarking campaigns show a stable interest compared to actual participation (45% of respondent's companies and 70% of respondent's academics did participate in a campaign in a the past).

PART 2 Synthesis

-There is an agreement on the relevance of existing benchmarks -60% of companies plan to use technologies selected by benchmarks -Attractiveness to participate in and organize public benchmarks is still there PART3 of the questionnaire Scientific Evaluation Criteria Q3.1 What are the best criteria that you think should be taken into account when benchmarking multimedia IR components ? The results show that according to all groups of respondents effectiveness is the best criteria to evaluate multimedia IR components, followed by scalability and efficiency. Ergonomy of the Graphical User Interface is not considered an important criterion in such evaluations.

Overall results

Detailed results

Big comp

Looking at the user satisfaction criterion alone, we see that a majority of big companies would like to see some user trials in benchmarking campaigns.

Q3.2 What criteria do you use to judge that a scientific article is an important contribution ?

Overall results

Detailed results

Big comp

Discussions in cinferences

Scientific excellence and biblio-metrics (number of citations, H-index, etc.) are ranked first by managers and big companies. On the other side, experimental results are ranked first by small companies, developers and academics. Finally, claims of the authors of a paper are ranked first by students whereas they are among the worst criterion for big companies and managers. Overall, we can remark that:

-Confidence in claims decreases with financial impact of the respondents -Confidence in research community increases with with financial impact of the respondents -Relevance of experimental results is quite stable over the different groups and on the average the best criterion Q3.3 What are the greatest difficulties in the scientific evaluation of multimedia retrieval ? The results clearly show that according to all groups of respondents data availability is most critical issue in evaluating multimedia retrieval technologies. Human resources appear as the second main limitation whereas hardware resources do not appear as a problem. This last point has to be mitigated by the fact that the scale of currently available data is relatively small compared to real-world data. So that if the main limitation (data availability) was solved, it is probable that hardware limitations would become more critical to process (to process very large amount of data).

Overall results

Detailed results

Big comp

PART 3 Synthesis

-Effectiveness is considered as the top-1 evaluation criterion and this validates the approach of current benchmarking campaigns. It is followed by scalability and Efficiency concerns. Only big companies are convinced by human-centered evaluation as a complementary criterion to be used in benchmarking campaigns.

-Criteria used to evaluate scientific publications are diverse and evolve with the financial impact of the underlying decisions to be taken. Experimental results are the most consensual criteria.

-Data availability is most critical issue in evaluating multimedia retrieval technologies

  D3.3 deliverable (November 2011), was published as an online questionnaire 2 in December 2011. It was then advertised across several communities including ImageCLEF, TRECVID and MIREX benchmarking campaigns, ACM Multimedia 2011 conference, EU projects of the Media Search cluster and a LinkedIn group on enterprise search. The 20 participants to the 5th Think Tank on Multimedia search technology transfer driven by benchmarking were finally asked to fill the survey as well. At the time of compiling the results (mid-April 2011), 80 respondents had filled the online questionnaire.

-

  Scientific literature is the best way to prospect and discover

new technologies -Technical performances are the best key of commercial success whereas

	scientific
	excellence is judged as the worst one
	-Academics & Companies differ on how they select technologies in practice (for
	integration or testing): scholarly impact vs. technical skills
	-But Academics & Companies agree on that Benchmarking is a good way to select
	technologies in practice. So that benchmarking appears as the best compromise between
	research and exploitation. This central position makes it a powerful tool for boosting
	technology transfer.

PART2 of the questionnaire Public Evaluation campaigns Q2.1 Which evaluation campaign is the most suitable for your business or research activity ?

GUI/ergonomy GUI/ergonomy GUI/ergonomy User satisfaction criteria alone

			Small comp	Develop	Accadem	Manager	Student
	Top-1 Effectiveness	Effectiveness Effectiveness Effectiveness	Effectiveness	Effectiveness
	Top-2 Scalability	Efficiency	Scalability	Scalability	Scalability	Efficiency
	Worst GUI/ergonomy User	Diversity,
			satisfaction	exploration
	In	0,56	0,20	0,43	0,36	0,33	0,37
	top-2

Small comp Develop Accadem Manager Student Top-1 Scientific excellence, citations Experiments Experiments Experiments Scientific excellence, citations Claims Top-2 Third parties expes Claims Claims Claims Experiments Experiments Worst Claims Theoretical statements Discussions in cinferences Theoretical statements Theoretical statements

Please refer to: http://trec.nist.gov/pubs/2010.economic.impact.pdf

https://docs.google.com/spreadsheet/viewform?formkey=dEJodDJPRG1HcW94NHNfOUgx cXBBYWc6MQ#gid=0

respondents who are either manager or director as managers. The group referred as student in the detailed results is simply composed by the 24% of the first diagram.

PART1 of the questionnaire

Using benchmarking for technology transfer Q1.1 Which aspects are likely to contribute to the commercial success of a technical component ?

Overall results

Detailed results

Big comp

Ease of integration

Ease of integration

Ease of use Ease of use Ease of use The results show that according to all groups of respondents, technical performances is the most important aspect that contributes to the commercial success of a technical component. Ease of integration is an important aspect for small companies and developers in general.

Worst

Interestingly commercial terms appear to be the less contributive aspect for big companies whereas reciprocally, scientific excellence is considered as the less contributive aspect by academics.

Scientific excellence is overall the less contributive factor to commercial success.


Nicolas Di
email: nicolas.di-palma@ec-lyon.fr

Adrien Martin

Fabrice Thouverez

Vivien Courtier

Nonlinear Harmonic Analysis of a Blade Model Subjected to Large Geometrical Deflection And Internal Resonance

This paper is devoted to the study of the nonlinear harmonic response of an industrial blade model subjected to large geometrical deflection. A reduction procedure is performed on the blade model using the linear normal modes of the structure. Geometrical nonlinear effects are taken into account by considering cubic and quadratic stiffnesses in the dynamical reduced model. Reduced nonlinear stiffness coefficients are computed with the STiffness Evaluation Procedure (STEP) and periodic solutions are sought using the Harmonic Balance Method (HBM) coupled to a pseudo-arclength continuation. Along with the harmonic response, a bifurcation analysis is performed to compute both turning and branching points. Specific attention is paid to the internal resonance phenomenon. 2 to 1 internal resonance occurred during the frequency response analysis close to the first and second modes of the reduced model. Mode coupling phenomena occurred during the harmonic analysis and secondary branches of solutions were obtained from branching point bifurcations.

NOMENCLATURE x(t)

Vector of unknows in the physical basis M, C, K Mass, damping and linear stiffness matrices f nl , f ext External and nonlinear forces q Vector of unknowns in the modal basis Φ Truncated modal basis • Variables in the modal basis K c , K q Quadratic and cubic reduced stiffness tensors • Vector of Fourier coefficients N h Number of harmonics Z Dynamic stiffness matrix F, F Direct and inverse discrete Fourier transform matrices s Arc-length parameter HBM Harmonic Balance Method STEP STiffness Evaluation Procedure DOF Degree Of Freedom AFT Alternating Frequency-Time BP, TP Branching and Turning Points

INTRODUCTION

The aeronautic industry is constantly confronted by economic and environmental issues, such as gas emission and noise level reduction, and the need to decrease fuel consumption. The development of high-bypass ratio machines is a current trend in engine design intended to reach such objectives. This has led to the conception of slender components including fan blades which may undergo large deflections. In such conditions, the response of the blades to harmonic excitation can involve mode couplings leading to so-called internal resonances.

Internal resonances are a feature of nonlinear systems. Having no counterpart in linear systems, they correspond to an exchange of energy between several modes of a structure. They occur when two or more frequencies of the structure are commensurable or nearly commensurable [START_REF] Nayfeh | Nonlinear oscillations[END_REF], which can be summarized by the following condition,

k m k ω k ≈ 0 m k ∈ Z, k ≥ 2 (1)
where the frequencies ω k need not be the linear frequencies of the structure [START_REF] Kerschen | Nonlinear normal modes, part i: A useful framework for the structural dynamicist[END_REF].

Understanding internal resonance phenomena is essential in the design process of the components mentioned previously. As such phenomena are usually encountered during experimental trials, being able to predict and simulate them may lead to considerable savings of money and time. Several approaches to study internal resonance phenomenon can be found in the literature. The multiple scales analysis [START_REF] Nayfeh | Nonlinear oscillations[END_REF] is commonly used and furnishes pertinent results for small nonlinear systems [START_REF] Nayfeh | Nonlinear oscillations[END_REF][START_REF] Nayfeh | Nonlinear coupling of pitch and roll modes in ship motions[END_REF][START_REF] Haddow | Theoretical and experimental study of modal interaction in a two-degree-of-freedom structure[END_REF]. However, its application is limited to weak nonlinear systems. Furthermore, multiple scales analysis is an analytical approach, then, its use for large systems is not feasible. Boivin et al. proposed in [START_REF] Boivin | Non-linear modal analysis of structural systems featuring internal resonances[END_REF] an approach based on multi-mode invariant manifolds to treat interactions between modes [START_REF] Jiang | The construction of non-linear normal modes for systems with internal resonance[END_REF][START_REF] Pesheck | Nonlinear modal analysis of structural systems using multi-mode invariant manifolds[END_REF]. In [START_REF] Kerschen | Nonlinear normal modes, part i: A useful framework for the structural dynamicist[END_REF][START_REF] Peeters | Nonlinear normal modes, part ii: Toward a practical computation using numerical continuation techniques[END_REF], internal resonances were obtained for nonlinear normal modes (NNM) calculation by a shooting method. In addition, King and Vakakis proposed in [START_REF] King | An energy-based approach to computing resonant nonlinear normal modes[END_REF] an energy-based approach to compute resonant NNM. Moreover, frequency-based methods like the Harmonic Balance Method (HBM), have shown promising results concerning internal resonances [START_REF] Ribeiro | Non-linear vibration of beams with internal resonance by the hierarchical finite-element method[END_REF][START_REF] Ribeiro | Non-linear free vibration of isotropic plates with internal resonance[END_REF]. Though these methods have been able to predict modal interactions through internal resonance with good precision, there is little in the literature on their application to industrial cases.

In practice, industrial models are reduced to smaller ones with reduction methods. In these cases, the evaluation of nonlinear reduced terms can prove difficult. Indeed, the direct application of the projection basis onto the full numerical model to extract the nonlinear terms may be time-consuming. One solution is to estimate a priori the nonlinear reduced forces associated with the reduction basis. That is the purpose of the STiffness Evaluation Procedure (STEP) [START_REF] Muravyov | Determination of nonlinear stiffness with application to random vibration of geometrically nonlinear structures[END_REF]. Moreover, obtaining the harmonic response of dynamical systems is of particular interest and the Harmonic Balance Method (HBM) is a relevant tool for achieving this goal. Avoiding transient phase calculation, it is used to find the periodic solution of an ordinary differential equation that guarantees significant time-cost reductions in comparison with classical direct time integration methods. HBM is often completed by a continuation technique. These approaches enhance the following solution branches and optimize time-cost reduction.

In this paper, we propose a study of the internal resonances of a reduced industrial blade model subjected to large geometrical deflection. The reduction is performed by modal projection. The nonlinear stiffnesses are determined by a STiffness Evaluation Procedure (STEP) and the simulations are performed using HBM coupled with a pseudo arc-length continuation procedure.

NUMERICAL APPROACH

In the following, the equations of motion of an n-degrees-of-freedom (DOFs) system subjected to geometrical nonlinearities are written in the form:

Mẍ(t) + C ẋ(t) + Kx(t) + f nl (x(t)) = f ext (t) (2)
where x(t) is the vector of unknown displacements, M, C, K are the usual structural matrices, i.e. mass, damping and stiffness, respectively. The f nl (x(t)) vector refers to the geometrical nonlinear terms. Finally f ext (t) is the vector of external forces acting on the system.

Reduction

For industrial models, the size of the system (2) is generally large and calculation costs quickly reach prohibitive levels. To overcome this difficulty, reduction methods are generally employed to restrict the number of unknowns and lead to appropriately sized systems. In this paper, the reduction is carried out by classical modal projection using the linear normal modes (LNM) Φ i of the underlying undamped linear system of Eqn. [START_REF] Kerschen | Nonlinear normal modes, part i: A useful framework for the structural dynamicist[END_REF]. To this end, matrix Φ including a subset of r eigenvectors is introduced to expand the displacement vector as follows:

x(t) = Φq(t) (3)
Substituting this expression in Eqn. (2) leads to,

Mq(t) + C q(t) + Kq(t) + Φ f nl (Φq(t)) = f ext (t) (4)
where

M = Φ MΦ C = Φ CΦ K = Φ KΦ f ext (t) = Φ f ext (t) (5)
The size of the reduced system (4) is r with r n. Generally, the expressions of the nonlinear terms in Eqn. (2) are not available since they originate from a finite element program. Their expression in Eqn. ( 4) is therefore unknown. The stiffness evaluation procedure (STEP) allows overcoming this difficulty.

STiffness Evaluation Procedure (STEP)

The STiffness Evaluation Procedure is a method for determining the nonlinear modal stiffness coefficients from an arbitrary finite element model. The principle is to assume the expression of the nonlinear reduced modal forces in the following form [START_REF] Muravyov | Determination of nonlinear stiffness with application to random vibration of geometrically nonlinear structures[END_REF],

Φ f nl (Φq(t)) = r j=1 r k=j K q jk q j q k + r j=1 r k=j r l=k K c jkl q j q k q l ( 6
)
where K q and K c are second and third order tensors whose components are quadratic and cubic nonlinear stiffness coefficients. Then, the method proposed in [START_REF] Muravyov | Determination of nonlinear stiffness with application to random vibration of geometrically nonlinear structures[END_REF] suggests determining the unknown coefficients K q jk and K c jkl using a series of prescribed nodal displacements x p . The procedure is summarized as follows.

By assuming the nodal displacement along one of the linear eigenvectors Φ j , x p = Φ j q j , coefficients K q jj and K c jjj associated with the nonlinear modal restoring force can be isolated in Eqn. [START_REF] Pesheck | Nonlinear modal analysis of structural systems using multi-mode invariant manifolds[END_REF], Φ f nl (Φ j q j ) = K q jj q j q j + K c jjj q j q j q j [START_REF] Pesheck | Nonlinear modal analysis of structural systems using multi-mode invariant manifolds[END_REF] Then, by employing another displacement expression, x p = Φ j q j + Φ k q k , a new relationship can be obtained between the coefficients K q jk q j q k , K c jjk q j q j q k and K c jkk q j q k q k and the nonlinear modal force, Φ f nl (Φ j q j + Φ k q k ) = K q jj q j q j + K c jjj q j q j q j + K q kk q k q k + K c kkk q k q k q k + K q jk q j q k + K c jjk q j q j q k + K c jkk q j q k q k (8)

Finally, coefficients K c jkl are obtained by enforcing the displacement x p to be equal to Φ j q j + Φ k q k + Φ l q l , leading to the following equation, Φ f nl (Φ j q j + Φ k q k + Φ l q l ) = K q jj q j q j + K q kk q k q k + K q ll q l q l + K q jk q j q k + K q jl q j q l + K q kl q k q l + K c jjj q j q j q j + K c kkk q k q k q k + K c lll q l q l q l + +K c jjk q j q j q k + K c kjj q k q j q j + K c jjl q j q j q l + K c llj q l q l q j + K c kkl q k q k q l + K c llk q l q l q k + K c jkl q j q k q l (9)

Coefficients K q jk and K c jkl are determined by combining Eqns.( 7), ( 8) and [START_REF] King | An energy-based approach to computing resonant nonlinear normal modes[END_REF]. For instance, the unknowns K q jj can be obtained by summing Φ f nl (Φ j q j ) and Φ f nl (-Φ j q j ). In the same manner, subtracting Φ f nl (Φ j q j ) and Φ f nl (-Φ j q j ) enables computing coefficients K c jjj .

Harmonic Balance Method

A harmonic balance strategy was developed [START_REF] Krylov | Introduction to non-linear mechanics (am-11)[END_REF] to estimate the nonlinear response of the nonlinear dynamical equations. The principle of the Harmonic Balance Method (HBM) is to approximate the periodic solution of an ordinary differential equation using a truncated Fourier series:

x(t) = a 0 + N h k=1 a k cos(kωt) + b k sin(kωt) (10)
Substituting Eqn. [START_REF] Ribeiro | Non-linear vibration of beams with internal resonance by the hierarchical finite-element method[END_REF] and its derivatives in the equations of motion and projecting the resulting expression, using the Galerkin method [START_REF] Urabe | Galerkin's procedure for nonlinear periodic systems[END_REF], onto the harmonic function basis B = {1, cos(kωt), sin(kωt), k = 1 . . . N h } by way of the following scalar product,

f (t), g(t) = ω π 2π ω 0 f (t)g(t)dt (11)
yields the nonlinear algebraic system,

Z(ω)x + f nl (x) = f ext ( 12
)
where x is the vector of unknowns [a 0 , a 1 , b 1 , . . . ] , f nl and f ext are the Fourier coefficients of f nl and f ext . Z is the so-called dynamic stiffness matrix defined by,

Z =      2K Z 1 . . . Z N h      , Z k = K -(kω) 2 M kωC -kωC K -(kω) 2 M (13)
In spite of its limitation to harmonic solutions, the harmonic balance method is a useful method since no hypotheses are made concerning the transient phase of the response. Indeed, the latter is ignored during the procedure. Furthermore, despite the increase of unknowns, the resolution of an algebraic system offers a significant reduction in computational costs compared to traditional time integration methods.

Alternating Frequency Time (AFT)

Determining the Fourier coefficients f nl (x) in Eqn. ( 10) is generally not straightforward except in some rare cases where analytical expressions can be derived. Even in these cases, the Alternating Frequency/Time (AFT) procedure is generally preferred. The AFT method allows computing the coefficients required by transposing frequency-domain terms in the time-domain terms [START_REF] Cameron | An alternating frequency/time domain method for calculating the steady-state response of nonlinear dynamic systems[END_REF]. The general procedure, recalled by Eqn. [START_REF] Urabe | Galerkin's procedure for nonlinear periodic systems[END_REF], is summarized as follows. By expressing x in time space using inverse discrete Fourier transform (iDFT), allows evaluating the nonlinear forces, after which the Fourier coefficients of the nonlinear forces in frequency space are derived using the direct Fourier transform (DFT).

x iDFT ---→ x(t) ----→ f nl (x(t)) DFT ---→ f nl (x) (14)
As an algebraic nonlinear system, Eqn. ( 12) can be solved using usual Newton-like methods for which an analytical expression of the Jacobian is required. Using both the AFT procedure and the composed function derivation rule, the computation of the first derivatives of the nonlinear terms can be done easily by applying Eqn. [START_REF] Cameron | An alternating frequency/time domain method for calculating the steady-state response of nonlinear dynamic systems[END_REF],

∂ f nl ∂ x = F ∂f nl ∂x F ( 15
)
where F (resp. F) stands for the DFT (resp. iDFT). Note that nonlinear terms are independent of ω since the nonlinearity is purely displacement dependent.

Most of the time, system ( 12) is extended so as to include a parametrization equation to ensure the continuation of the solution branch.

Continuation

Continuation or path following methods consist in finding a set of solutions of a system by varying a parameter called continuation parameter [START_REF] Allgower | Numerical continuation methods: an introduction[END_REF]. In frequency response analysis, this parameter often corresponds to the system excitation frequency Ω. The most common continuation methods rely on a predictor-corrector scheme.

Geometrical nonlinear systems often exhibit multiple solution branches featuring bifurcation points such as turning points (TP) and branching points (BP) (Fig. 1). For such dynamical systems, predictor/corrector approaches such as sequential continuation generally fail to track solution branches. To overcome turning point failures, branches of solutions are parametrized by the arc-length parameter s, that is, (x, Ω) = (x(s), Ω(s)). A wide range of predictor/corrector methods using arc-length parametrization are available in the literature [START_REF] Nayfeh | Applied nonlinear dynamics: analytical, computational and experimental methods[END_REF]. In this paper, the estimated solutions are computed with tangent prediction while the correction uses the pseudo arc-length method [START_REF] Keller | Numerical solution of bifurcation and nonlinear eigenvalue problems[END_REF].

The pseudo arc-length method forces the corrected point to belong to a hyperplane orthogonal to the tangent prediction vector (Fig. 1). Assuming that the equations of the predicted point are:

x pre i+1 = x i + ∆x ω pre i+1 = ω i + ∆ω ( 16
)
with t = [∆x , ∆ω] the tangent to the branch at previous solution point [x i , ω i ] and [x pre i+1 , ω pre i+1 ] the prediction of the next solution point, pseudo arc-length is reduced to:

x i+1 -x pre i+1 ω i+1 -ω pre i+1 .t = 0 ( 17
)
or equivalently,

(x i+1 -x i ) ∆x + (ω i+1 -ω i )∆ω -∆s 2 = 0 ∆ω 2 + ∆x ∆x = ∆s 2 (18)
with ∆s = ||t||.

Pseudo arc-length continuation is carried on by appending Eqn. [START_REF] Keller | Numerical solution of bifurcation and nonlinear eigenvalue problems[END_REF] to Eqn. [START_REF] Muravyov | Determination of nonlinear stiffness with application to random vibration of geometrically nonlinear structures[END_REF], leading to the so-called extended system.

Bifurcation

As mentioned earlier, nonlinear systems can exhibit nonlinear phenomena such as bifurcation points. They appear at points for which the Jacobian of the system becomes singular [START_REF] Seydel | Practical bifurcation and stability analysis[END_REF]. Although turning points (TP) are automatically computed thanks to arc-length parametrization, branching points (BPs) are tricky to handle [START_REF] Grolet | Vibration analysis of a nonlinear system with cyclic symmetry[END_REF]. At the same time, since internal resonances originate from BP [START_REF] Kerschen | Nonlinear normal modes, part i: A useful framework for the structural dynamicist[END_REF], being able to predict them and follow bifurcated branches is of particular interest for our study. In the following we denote the nonlinear system as f (x, ω) and assume that its size is equal to n. BP analysis can be summarized in 3 steps:

1. Localization of the BP 2. Resolution of the branching system; 3. Computation of the bifurcated branches. The first step relies on monitoring the determinant of the Jacobian f x (= ∂f /∂x) of the system. A change of sign of this determinant for two consecutive solution points indicates the presence of a bifurcation point. This step is similar for both the turning and the branching points. The type of bifurcation is determined along with the next step. The second step is aimed at finding the exact position of the previously identified bifurcation point. This procedure is performed using the so-called branching system [START_REF] Moore | The calculation of turning points of nonlinear equations[END_REF],

(x pre i+1 , Ω pre i+1 ) (x pre i+2 , Ω pre i+2 ) (x i , Ω i ) (x i+1 , Ω i+1 ) (x i+2 , Ω i+2 ) (
  f (x, ω) f x (x, ω)h ||h|| -1   = 0 ( 19
)
where h is a nontrivial eigenvector associated with the null eigenvalue of f x (x, ω). The type of bifurcation is determined by the rank of the rectangular matrix [f x f ω ] (= [∂f /∂x, ∂f /∂ω]). For a full rank n the bifurcation is a turning point, while for a rank at most equal to n -1 the bifurcation is a branching point and it is necessary to go on with step 3. An alternative to the system [START_REF] Seydel | Practical bifurcation and stability analysis[END_REF], is described in [START_REF] Xie | Bifurcation tracking by harmonic balance method for performance tuning of nonlinear dynamical systems[END_REF]. It prevents the singularity of the Jacobian of the system [START_REF] Seydel | Practical bifurcation and stability analysis[END_REF] near BPs. This approach was used in this document. The last step consists in determining the branches emanating from the identified BP. This can be done by applying a branch switching algorithm whose procedure is fully described by Kuznetsov in [START_REF] Kuznetsov | Elements of applied bifurcation theory[END_REF]. Here, we briefly present the main idea of the branch switching algorithm.

The purpose of the method is to determine the tangents to the branches at the branching point. For the sake of simplicity, we here consider the case of only two emanating branches. In the following, we define y := (x, ω) and denote by superscript '0' variables at the branching points. A BP will hence be noted y 0 . Kuznetsov showed that the tangents can be written in the form,

t = β 1 q 1 + β 2 q 2 (20)
with (β 1 , β 2 ) ∈ R 2 and {q 1 , q 2 } = span{Ker f y y 0 }. Supposing the null-space of f y at the BP already identified, the definition of the tangents resumes to determine the values of the coefficients β 1 and β 2 in Eqn. [START_REF] Grolet | Vibration analysis of a nonlinear system with cyclic symmetry[END_REF]. As demonstrated in [START_REF] Kuznetsov | Elements of applied bifurcation theory[END_REF], they are the solution of the quadratic form system,

b 11 β 2 1 + 2b 12 β 1 β 2 + b 22 β 2 2 = 0 with b ij = ϕ, B(q 1 , q 2 ) , i, j = 1, 2 (21)
with ϕ belonging to the of f y y 0 and B being the second order derivative of f (i.e. the Hessian). The scalar product in Eqn. ( 21) is defined as follows,

ϕ, B(q, q) = n i=1 n+1 j,k=1 ϕ i ∂ 2 f i (y) ∂y j ∂y k y=y 0 q j q k ( 22
)
The second order system (21) possesses 2 independent nontrivial solutions if b 11 b 22 -b 2 12 < 0. These two solutions correspond to the two desired tangents. By diagonalizing the matrix associated with the quadratic form of Eqn. ( 21), it is possible to express β 1 in terms of β 2 . The values of these two coefficients will be estimated by enforcing the norm of the tangent vectors.

APPLICATION FOR AN INDUSTRIAL MODEL 2.1 Presentation of the industrial blade model

The model considered for the application of the numerical methods described previously is a finite element industrial blade model. The blade, composed of 59765 DOFs, is shown in Fig. 2.

Fig. 2: Finite element model of the industrial blade

As recalled in the introduction of this document, slender components such as engine blades can undergo large displacements leading to nonlinear behavior. Here, we focus on a particular feature of nonlinear systems, namely internal resonance. In particular we focus on the first torsional mode and the second bending mode whose linear eigenfrequencies are nearly commensurable, so that ω 2 2ω 1 .

Model reduction and hypothesis

As we are dealing with an industrial blade model, the number of DOFs is too large to carry out the nonlinear analysis planed in this paper. Consequently, the blade model must be reduced. This reduction was performed using the procedure presented in the first part of the present document. Since we are interested in the interaction between the first torsional mode and the second bending mode of the blade, the reduction basis is composed only of these two modes. The resulting reduced model therefore depends on two generalized coordinates denoted q 1 and q 2 .

To complete the reduced model, the following hypotheses are made. First, we assume an external force in the form of a punctual cosine excitation with pulsation Ω and amplitude F 0 acting on the blade tip. The damping matrix is assumed to be diagonal with a damping ratio ζ 1 equal to 0.02% for the first mode and ζ 2 equal to 0.016% for the second mode. Finally, the reduced stiffnesses are computed using the STEP method with prescribed displacements q j = 1×10 -2 and q k = 5×10 -2 in Eqns. ( 7) and ( 8). The nonlinear stiffness coefficients obtained using the STEP method are given in Tab. 1. It is important to point out that a set of values ranging from 1×10 -2 to 1×10 1 was used for the prescribed modal displacements q j and q k . For each choice, the STEP simulations provided very similar results. Besides, these choices of modal displacements correspond to a deflection magnitude at blade tip ranging from 0.1 mm to around 5 cm. These remarks highlight the great stability of the STEP method and its potential for handling large nonlinearities. Tab. 1, clearly shows that reduced cubic stiffness coefficients are predominant in comparison with the quadratic coefficients. The reduced model has the form of Eqn. (4), and in the rest of this article we consider the two pulsations ω 1 and ω 2 such that ω 2 2ω 1 .

In the following, the frequency response of the model is analyzed near the two pulsations of the reduced system using the numerical methods presented in the first part. Results in physical coordinates are retrieved using the modal projection (3) and vibration motion is studied at blade tip. Particular attention is given to the computation of bifurcated branches. All the pulsations will be made dimensionless with respect to the first pulsation of the system and the dimensionless variables will be denoted by superscript '*'.

Numerical results

Excitation near ω 1

The frequency response at blade tip corresponding to an excitation of amplitude F 0 = 0.015 N around the first pulsation ω 1 of the reduced model is shown in Fig. 3. The simulation was carried out with N h = 5 harmonics in the Fourier series decomposition and 128 time-steps for the AFT procedure. Due to the predominance of reduced cubic stiffness coefficients, the frequency response exhibits a significant stiffening effect.

The evolution of the Fourier coefficients of the response at blade tip for points (a) to (d) (Fig. 3) is depicted in Fig. 4. Starting form point (a), the response is linear with an harmonic 1 dominant compared to the others. For point (b) near Ω * = 1 the second harmonic begins to participate in the response. The first effects of the 2 to 1 internal resonance occur. From (b) to (c) the relative level of the second harmonic growths with the excitation pulsation. Finally from (c) to (d) the presence of the harmonic 2 stops i.e. the effects of the internal resonance cease and only the fundamental harmonic responds to the excitation.

The internal resonance occurrence can be further characterized by observing the temporal signal of the two generalized coordinates q 1 (t) and q 2 (t) for points (a) to (d) (Fig. 5). The signals were obtained by applying the inverse discrete Fourier transform on the Fourier coefficients. From (b) to (c) the nonlinear coupling between the modal coordinates allows q 2 to vibrate with a pulsation equal to the double of the excitation pulsation. Harmonic 2 in q 2 becomes dominant over harmonic 1. This situation clearly shows that the internal resonance acts on the second mode of the reduced system. From (c) to (d) the signals get back to an harmonic 1 response.

Hence, the characteristics of the nonlinear system permits the occurrence of a 2 to 1 internal resonance. In other words, when the excitation is close to ω 1 the intrinsic nonlinear behavior of the reduced system activates the internal resonance and harmonic 2 of the excitation appears in the response.

Excitation near ω 2

Let us now focus on the frequency response at blade tip for an excitation around the second pulsation of the reduced system. The response was obtained with a force amplitude F 0 = 0.10 N and the results are shown in Fig. 6. It is important to note that we here expect a response with a pulsation equal to half the pulsation of the excitation. Hence, we consider a pulsation of excitation in the form 2Ω with Ω close to ω 2 /2. In this manner, in absence of internal resonance, only harmonic 2 responds to the excitation. As in the previous case, the curve still exhibits Temporal integration [ ] a stiffening behavior. However, contrary to the previous case, two bifurcated branches of solutions can be seen near the high amplitudes. These branches, referred to as secondary branches, were computed using the algorithm presented in section 1.6. Branches with commensurable shapes were observed in [START_REF] Grolet | On a new harmonic selection technique for harmonic balance method[END_REF]. It is noteworthy that the branches reach high amplitudes for a very small range of pulsations. A first conclusion can be drawn; in conception phase, missing the secondary branches can lead to underestimate the level of vibration.

Because of the atypical shapes of the response branches, we wished to validate our results by a temporal method. Thus, a temporal integration method using an explicit Runge-Kutta formula was performed on the reduced model and initial conditions was chosen close to the harmonic balance results. Results for several excitation pulsations are superimposed to the frequency response in Fig. 6. Results present a very good correlation for the principal branch of solution as well as the secondary branches. Then, the comparison enables to validate the results obtained by HBM. Note that owing to probable instability purposes, only a few solution points were obtained on the secondary branches.

The evolution of the Fourier coefficients at blade tip for four chosen solution points was determined. It is illustrated in Fig. 7(a) to (d). On the principal branch of solutions, only harmonic 2 participates in the response (harmonic 0 also participates but at a much smaller level). For this branch, internal resonance occurs (Fig. 7 (a) and (b)). However, on the first and second bifurcated branches (points (c) and (d)), the first harmonic participates in the frequency response (Fig. 7(c) and (d)). Thus, a 2 to 1 internal resonance occurs for secondary branches. In addition, the amplitude of harmonic 1 growths along the secondary branches and can be considerably superior than the amplitude of the second harmonic (Fig. 7(d)). Thus, in contrast with the first case of excitation, the internal resonance only occurs for bifurcated branches.

Again, an overview of the temporal signals of the modal coordinates q 1 and q 2 helps understanding of the transition from classical nonlinear interaction to the occurrence of internal resonance (Fig. 8). For points belonging to the principal branch of solutions, both q 1 (t) and q 2 (t) respond with the harmonic of the excitation, corresponding to harmonic 2 (Fig. 8(a) and (b)). By examining the signals in Fig. 8(c) and (d), it is clear that q 1 (t) responds with the first harmonic while q 2 (t) responds with the second harmonic. Thus, internal resonance impacts the dynamic of the first modal coordinate. In addition, Fig. 8(d) shows that for an excitation case close to ω 2 , the response amplitude of q 1 (t) can be higher than the amplitude of q 2 (t).

0 1 2 3 -4 -2 0 2 4 t* Amplitude [×10 -4 ] (a) 0 1 2 3 -1 0 1 t* Amplitude [×10 -2 ] (b) 0 1 2 3 -2 0 2 t* Amplitude [×10 -2 ] (c) 0 1 2 3 -2 0 2 t* Amplitude [×10 -2 ] (d)
Fig. 8: Temporal signals of the modal coordinates 1 (t) [ ] and q 2 (t) [ ]

CONCLUSION

Large engine components such as fan blades may undergo large geometrical deflection due to their slender shape. This behavior implies serious nonlinear effects whose comprehension is essential for designers. Indeed, these effects may significantly alter the linear behavior of the structures. One feature of nonlinear effects is the internal resonance phenomenon which, involving modes coupling, can increase vibratory levels. Thus, their prediction and identification in conception phase represents an important challenge for industrial designers.

In this paper, the nonlinear harmonic response of an industrial blade model undergoing large deflection has been studied. For computational purposes, the model has been reduced using the linear normal modes of the structure. Nonlinear effects have been taken into account through polynomial stiffnesses and reduced nonlinear stiffness coefficients were obtained by a stiffness evaluation procedure. The harmonic balance method coupled to a pseudo arc-length continuation approach has been used to determine the periodic solutions of the nonlinear dynamic system. During the simulations, attention has been paid to internal resonance risks.

Excitations close to the first and second mode of the reduced model have been studied. In both cases, frequency response at blade tip has been obtained by modal projection. In the first case, the internal resonance was activated as soon as the excitation pulsation reached the pulsation of the first mode. In this case the response of the second mode initiated with a pulsation equal to two times the excitation pulsation. In the second case, internal resonance occurred through secondary branches of solutions computed by a branch switching method. On these branches, participation of harmonic 1 corresponding to half the harmonic of the excitation has been observed. The atypical behavior of the frequency response has been verified thanks to temporal integration. The comparison has permitted to validate the results obtained by the harmonic balance method.

The study presented in this document has been conducted on a significantly reduced model, however this paper highlights the potential of the described methods to predict internal resonance phenomena for an industrial model with the harmonic balance method. In further works, it would be interesting to complete the reduction basis using, for instance, modal derivatives [START_REF] Martin | Dynamic analysis and reduction of a cyclic symmetric system subjected to geometric nonlinearities[END_REF].

Fig. 1 :

 1 Fig. 1: Nonlinear frequency response. (a) Bifurcation points: turning points [ ], branching points [ ]. (b) Pseudo arc-length continuation: Solution points [ ], Iteration points [ ]

  From (a) to (b) the signals are in phase with a

Fig. 3 : 1 0Fig. 4 :coefficients a 2 k + b 2 k

 31422 Fig.3: Frequency response at blade tip near ω 1

Fig. 5 :

 5 Fig. 5: Temporal signals of the modal coordinates q 1 (t) [ ] and q 2 (t) [ ]

Fig. 6 :

 6 Fig. 6: Frequency response at blade tip near ω 2 . Principal branch [ ], Secondary branch [ ]. Temporal integration [ ]

Fig. 7 :

 7 Fig. 7: Fourier coefficients a 2 k + b 2 k of the response at blade tip for an excitation close to ω 2

Table 1 :

 1 Modal quadratic and cubic stiffness coefficients of the reduced model

	Modal Equation	1	2
	K q 11	8.79 × 10 3 -9.04 × 10 3
	K q 12	-1.81 × 10 4 3.83 × 10 4
	K q 22	1.91 × 10 4 -1.97 × 10 4
	K c 111	5.85 × 10 6 -3.24 × 10 6
	K c 112	-9.73 × 10 6 1.66 × 10 7
	K c 122	1.66 × 10 7 -1.42 × 10 7
	K c 222	-4.72 × 10 6 1.51 × 10 7

Acknowledgements

The authors would like to thanks Safran Aircraft Engines for providing the financial support for this project, and for giving permission to publish this work.