V European Conference on Computational Fluid DynamicsECCOMAS CFD 2010

J. C. F. Pereira and A. Sequeira (Eds)Lisbon, Portugal,14-17 June 2010

THE GERMAN NATIONAL JOINT PROJECT MUNA: MANAGEMENTAND MINIMIZATION OF UNCERTAINTIES AND ERRORS IN

NUMERICAL AERODYNAMICS

Bernhard Eisfeld∗

∗German Aerospace Center (DLR)Institute of Aerodynamics and Flow Technology

Lilienthalplatz 7, D-38108 Braunschweig, Germanye-mail: bernhard.eisfeld@dlr.de

Key words: Uncertainties, Errors, Numerical Simulation, Aerodynamics

Abstract. Results of the German national joint project MUNA – Management andMinimization of Uncertainties and Errors in Numerical Aerodynamics – are presented.From January 2007 to March 2010 altogether 12 partners from research and industryhave been developing and applying methods and tools indicating and reducing inaccuraciesin numerical flow simulations associated with the turbulence model, numerical methods,geometry and deformation and the numerical grid.

Sensors have been developed, indicating errors and limits of the validity of RANS tur-bulence models. Numerical methods have been developed, reducing the discretization errorand providing stochastic information on the solution, including shape optimization undergeometrical uncertainties. Uncertainties associated with the structural model, the inter-polation algorithm as well as the required grid deformation within coupled fluid/structuresimulations have been investigated, leading to more reliable tools. A dissipation based sen-sor for insufficiently resolved grid regions, an improved grid adaptation based e. g. onadjoint information and an a posteriori grid manipulation tool have been developed, allreducing the influence of the computational grid on the predictions.

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1 INTRODUCTION

Numerical flow simulation has become a key technology in aeronautics. In particular,the design of future aircraft will mainly rely on numerical simulation results, thus de-manding for safe estimates of validity and accuracy of the underlying results in the wholeflight regime, including its borders. Within the German national joint project MUNA –Management and Minimization of Uncertainties and Errors in Numerical Aerodynamics– , running from January 2007 until March 2010, the following 12 partners have beencooperating on various aspects of this subject:

• German Aerospace Center (DLR), Institute of Aerodynamics and Flow Technology

• Airbus Germany

• EADS Military Air Systems

• Eurocopter

• RWTH Aachen University, Institute of Aerodynamics (RWTH-AIA)

• RWTH Aachen University, Computational Analysis of Technical Systems (RWTH-CATS)

• TU Berlin, Institute of Fluid Mechanics and Technical Acoustics (TUB-ISTA)

• TU Braunschweig, Institute of Aircraft Design and Lightweight Structures (TUBS-IFL)

• TU Braunschweig, Institute of Fluid Mechanics (TUBS-ISM)

• TU Braunschweig, Institute of Scientific Computing (TUBS-WiRe)

• Stuttgart University, Institute of Aero- and Gasdynamics

• Trier University, Department of Mathematics

The focus of the activities was on all kinds of errors and uncertainties associated with thecomputational mesh, the turbulence modelling, the numerical methods and the geometryrepresentation, including items of fluid-structure coupling and structure mechanics mod-elling. These fields have been considered the most critical with respect to the reliabilityof numerical simulation results.

Various techniques for analysing and minimizing errors and uncertainties have beendeveloped throughout the project, where the DLR TAU code has been the central simu-lation tool used by the majority of the partners. Furthermore the participation of Airbus,EADS-MAS and Eurocopter ensured the industrial relevance and applicability of majordevelopments.

In the following the most relevant results obtained in MUNA will be presented.

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2 TURBULENCE MODELLING

2.1 Sensors for RANS Turbulence Models

Turbulence modelling is known to be crucial in complex flow situations. Wheneverdeviations of simulation results from corresponding experiments are observed, the under-lying turbulence model is likely to be suspected for being at least one of the reasons.In general this refers to statistical turbulence models based on the Reynolds averagedNavier-Stokes (RANS) equations.

The task of RANS turbulence modelling has been beyond the scope of MUNA. Never-theless the influence of non-ideal operation conditions on the predictions by various modelshas been investigated by TUB-ISTA. In particular the interaction with the computationalmesh in terms of e. g., the wall normal first grid spacing, the cell growth ratio or the gridskewness in boundary layers, has been systematically tested for various models. Fromthese results error estimators, e. g. for the local skin friction coefficient have been devel-oped, allowing correcting the numerical result with respect to the investigated non-idealgrid characteristics. Additionally, sensors have been developed, indicating critical entitieslike shocks or separation and re-attachment lines, where the predictions of RANS modelsare known to deviate.

Fig. 1 shows the drag coefficient predicted on different grids, using different turbulencemodels for the transonic flow around the RAE 2822 airfoil. As one can see, the developederror estimators with respect to wall-normal grid spacing and cell growth ratio allowfor correcting the respective coarse grid results, so that the fine grid results are nearlyrecovered.

Figure 1: RAE 2822. Prediction of drag on different grids by different turbulence models and corre-sponding error corrrection (yellow). Influence of first grid spacing (left) and wall-normal cell growthratio. Result by TUB-ISTA.

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2.2 Sensor for Zonal RANS/LES

Since the predictions with standard RANS models deteriorate in case of separation,at RWTH-AIA a zonal Large Eddy Simulation (LES) is favoured, where the much moreaccurate, but also more expensive LES is embedded into a RANS simulation in criticalregions where the RANS predictions cannot be trusted. In order to detect such regionsof high RANS uncertainty, a sensor has been developed, based on flow characteristics likethe pressure gradient, the Reynolds shear stress and the wall shear stress. The sensor isconstructed in such a way that the required LES region is minimized, in order to keep thenumerical effort as low as possible. The procedure has been validated for the flow over aflat plate, a flat-plate case with shock-boundary layer interaction and a transonic airfoil.

Fig. 2 shows a snapshot of a zonal RANS/LES simulation for an oblique shock, im-pinging on a flat plate, leading to a separation of the turbulent boundary layer. Syntheticeddies are generated at the LES inlet where controlled forcing is applied in four planesfurther downstream for a smooth transition from RANS to LES. At the LES outlet theRANS model’s eddy viscosity is reconstructed.

Figure 2: Zonal RANS/LES for shock-boundary layer interaction on a flat plate. Iso-Mach contours inthe RANS region and λ2-isolines in the LES region. Result by RWTH-AIA.

3 NUMERICAL METHODS

3.1 Iterated Defect Correction Method (IDeC)

The accuracy of any flow simulation depends on the numerical error of the appliedmethod. Traditionally this is reduced by improving the grid resolution, i. e. by increasingthe number of cells used for discretizing the domain. For a given grid however, one would

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instead have to increase the order of accuracy of the numerical scheme which is usuallynot forseen with standard methods.

A possible remedy could be the so-called iterated defect correction method, developedat Stuttgart University within MUNA. The idea is to feed a low order solution into ahigher order discretization for determining the corresponding defect. This defect is thenfed back into the original method as a source term. As has been demonstrated withinMUNA, the coupled approach leads to an order of accuracy corresponding to the higherorder scheme used for computing the defect.

Fig. 3 shows the convergence of lift and drag for the transonic flow around the RAE2822 airfoil, computed with a first order scheme coupled to a third order defect correctionmethod. The result is compared to the second order TAU solution. Clearly the thirdorder defect correction pushes the first order result closer to the reference solution.

Figure 3: RAE 2822. Convergence of lift (left) and drag (right), using a 3rd order iterated defectcorrection method coupled to a 1st order baseline method. Comparison with 2nd order accurate TAUsolution. Result of Stuttgart University.

3.2 Taguchi Method

Given a highly accurate numerical solution of highly reliable physical model equa-tions, the result nevertheless is uncertain because of uncertainties in the input e. g., dueto boundary or initial conditions. While any individual numerical simulation result isdeterministic, it can be considered one single realisation of a stochastic distribution.

Whereas determining the complete probability density function of the solution in case ofseveral uncertain parameters is very expensive, the Taguchi method applied at StuttgartUniversity allows efficiently determining the relative sensitivities of the solution. Onlyfew samples are needed because different parameters are varied simultaneously. Further-more the method can be used to provide information on interactions between the testedparameters.

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Fig. 4 shows the result of a Taguchi analysis applied to the transonic flow around theRAE 2822 airfoil, where the flow conditions in terms of Reynolds number (parameter A),Mach number (parameter B) and incidence (parameter C) have been varied. As one cansee, Reynolds number and incidence mainly influence the lift coefficient Cl, whereas theMach number has a major impact on the pressure drag Cdp, dominating also the totaldrag coefficient Cd. In contrast, the viscous contribution to the drag coefficient, Cdv ismainly influenced by the Reynolds number.

Figure 4: RAE 2822. Relative sensitivities of lift and drag coefficients with respect to flow conditions.Parameter A: Reynolds number. Parameter B: Mach number. Parameter C: Incidence. Result ofStuttgart University.

3.3 Stochastic Analysis of Uncertainties

A true non-deterministic treatment of numerical results requires an assumption on therespective probability density functions of uncertain parameters. In a so-called Monte-Carlo simulation a very large number of samples, each referring to an individual numericalsimulation, is taken then for determining the corresponding stochastic characteristics ofthe solution. Typical of a non-linear system response is that the mean of the stochasticsolutions will not be identical with the solution of the mean of the input parametersconsidered.

Within MUNA TUBS-WiRe has developed approximation methods for significantlyreducing the required number of samples for establishing the stochastics of the non-deterministic solution. By a suitable representation of the probability sparse-grid tech-niques can be applied reducing the required number of numerical simulations fromO(105) . . .O(106) (Monte-Carlo) to O(102).

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Fig. 5 shows the error bars on the pressure and skin friction distributions for thetransonic flow around the RAE 2822 airfoil associated with the combined uncertainties inMach number and incidence. As one can see, the largest uncertainties occur close to theshock, reflecting the sensitivity of the shock position to the tested parameters.

Figure 5: RAE 2822. Error bars in terms of standard deviations on pressure distribution (left) andskin friction distribution (right) associated with combined uncertainties of Mach number and incidence.Result of TUBS-WiRe.

3.4 Robust Design with Geometric Uncertainties

A major field of interest in non-deterministic methods is the design of geometricalshapes under consideration of manufacturing imperfections, where the latter representdistributed uncertainties of the geometry. Within MUNA Trier University has developeda robust design method, allowing for optimising e. g., an airfoil geometry, while takinginto account such geometric uncertainties. By suitable representation of the uncertaintiesin terms of eigenvectors, the number of required samples has been reduced to as little asO(20), allowing for practical applications. The method is robust as it allows for additionalconstraints to be fulfilled.

Fig. 6 shows the result of two optimisations of the RAE 2822 airfoil with respect todrag, one obtained with a standard single set point optimisation and a robust optimi-sation, including geometrical uncertainties. In both cases a minimum lift constraint hasbeen imposed. Comparing the performance of the respective geometries unter the samegeometrical uncertainties, the robust design result yields by 10 drag counts lower drag interms of the expected value than the single set point result. Moreover the robust designresult fulfills the minimum lift constraint for all disturbed samples, whereas the single setpoint result violates the constraint for the majority of disturbed geometries.

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Figure 6: Drag optimisation of RAE 2822 airfoil with respect to drag with constraint on minimum lift.Comparison of drag (top) and lift results (bottom) for deterministic single set point optimisation androbust design, considering 21 geometrically disturbed samples. Result of Trier University.

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4 GEOMETRY AND DEFORMATION

4.1 Uncertainties in Structural Modelling

Traditionally uncertainty analysis is more common to computational strucure mechan-ics than to computational fluid dynamics due to the non-linear effects dominating in thelatter. Besides analysing the influence of different structural modelling approaches inMUNA, TUBS-IFL has applied a stochastic First Order Reliability Method (FORM) toa coupled fluid/structure simulation for analysing the probability of failure in terms ofdeviation of the angle of attack αg at the bended wing tip from the undeformed referenceαg,ref . The uncertainty in Young’s modulus has been used as parameter in the simulations,assuming a Gaussian distribution, where the variance was varied between σ = 0.02 andσ = 0.05. The lift at aeroelastic equilibrium has been kept constant for all simulations.

Fig. 7 shows the result of the FORM analysis for different margins of allowed deflec-tion. Clearly, the smaller the allowed deflection and the wider the probability densitydistribution (variance) is, the higher is the risk of structural failure.

Figure 7: HIRENASD wing. FORM analysis of structural failure in terms of wing tip incidence forvarying variance in uncertain Young’s modulus and different margins of allowed deflection. Result ofTUBS-IFL.

4.2 Influence of Coupling Strategy on Coupled CFD/CSM Solutions

Besides the uncertainties associated with computational fluid dynamics and computa-tional structure mechanics, the coupling between the disciplines, i. e. the interpolationof forces and deformations, introduces an additional uncertainty to the simulation re-sult. Special care has to be taken, if multi-component configurations, consisting e. g. ofa multi-element wing attached to a fuselage, are to be handled. In particular gaps or

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overlaps in the deformed geometry as well as mappings to wrong structural componentsmust be safely avoided. In MUNA different interpolation schemes have been investigatedby RWTH-CATS, where a moving least squares interpolation appeared the most suitableone

Fig. 8 shows the results of a sensitivity study for the HIRENASD wing in transonicflow. As one can see, in contrast to the global spline-based interpolation the moving leastsquares interpolation exhibits virtually no influence of the radial basis function supportradius on the predicted surface deflection. Moreover the root mean square of normalizeddifferences with respect to a reference finite interpolation element method is much smaller.

Figure 8: HIRENASD wing. Influence of radial basis function support radius on global spline-basedinterpolation (left) and moving least squares interpolation (right) on root mean square of normalizeddifferences in surface deflections compared to finite interpolation element method. Result of RWTH-CATS.

4.3 Unified Grid Deformation Tool for Coupled Simulations

In geometrical shape design and optimisation the re-generation of the grid for each indi-vidual configuration introduces an additional uncertainty, preventing a reliable assessmentof the results, in particular with unstructured meshes. Since geometry changes are usuallysmall in this process, deforming the initial grid is considered an appropriate technologyfor eliminating most of the uncertainties associated with a complete re-generation of thegrid.

Within MUNA Airbus has developed a unified grid deformation tool based on radialbasis functions that is integrated into the FlowSimulator environment and thus can beused with any of their flow solvers. In order to keep the numerical effort feasible, differentstrategies have been developed for reducing the total number of required deformationbasis vectors.

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Fig. 9 shows the deformation error on a wing obtained with the different strategies.As one can see, simple equidistant reduction of basis vectors leads to a large deformationerror at the wing tip. This error is significantly reduced, when correcting the interpolationcoefficients based on the associated error. Further improvement is achieved when selectingthe basis vectors based on the associated error. However the second strategy yields almostthe same quality of results at lower cost.

Figure 9: Interpolation error with different strategies of basis vector reduction for grid deformation.Left: Equidistant reduction. Middle: Error correction by local interpolation of coefficients. Right: Errorweighted selection of basis vectors. Result of Airbus.

5 COMPUTATIONAL MESH

5.1 Grid Generation for Helicopters

Within MUNA Eurocopter has tested different strategies for efficiently generating gridsaround helicopter configurations, allowing for reliable predictions with limited resources.Different unstructured grid generator software has been applied to a helicopter fuselageproblem with different total number of points and varying resolution of the boundarylayers. Furthermore different turbulence models have been applied, in order to checktheir performance.

Fig. 10 shows the large scatter between the observed results in the force and momentcoefficients. Based on the findings, Eurocopter recommends using a medium sized gridwith approximately 3 · 106 nodes and a resolution of the boundary layer with 24 layersof prisms. The Wilcox k-ω turbulence model has been found the most robust one, wheresome improvements can be achieved with the Menter SST model.

5.2 Discretization of Wakes

Grid imperfections in the wake of airfoils and wings have been found by TUBS-ISM tohave a sensible effect on the predicted drag. In particular O-type grids around blunt trail-

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Figure 10: Helicopter fuselage. Variation of time averaged force and moment coefficients obtained withdifferent turbulence models on various grids. Result of Eurocopter.

ing egdes, as they appear with standard unstructured grid generators, show an increaseddrag caused by an insufficient resolution of the developing wake behind the airfoil.

Within MUNA TUBS-ISM has developed a sensor, based on the artificial dissipationof a central discretization scheme, indicating regions of insufficient mesh resolution. Fur-thermore a particular refinement strategy for structured parts of an unstructured grid hasbeen developed, allowing for an improved wake resolution.

Fig. 11 shows on the left hand side the sensor in the vicinity of the trailing edge of anairfoil, indicating high numerical error in the developing wake. As one can see, the rapidlateral widening of the cells, associated with the O-topology is responsible for the toorapid downstream decay of velocity gradients in the wake. On the right hand side of Fig.11 the capability of the developed structured wake refinement technology is demonstrated.

5.3 Adjoint Based Grid Adaptation

In case of insufficient grid resolution, local adaptation can be used for remedy, inparticular with unstructured meshes. However adaptation sensors based on local flowfeatures, like gradients, appear to be unsuitable for systematically reducing grid inducederrors. The reason is that feature based indicators detect only the location of erroroccurrence, but not the location of error origin and transport. This problem is solvedwhen using indicators based on the adjoint flow problem with respect to a characteristictarget functional.

Based on the idea that the artificial dissipation of a central numerical discretizationscheme is sensitive to grid imperfections, DLR has developed an adaptation indicatorwithin MUNA that is based on the adjoint with respect to the corresponding dissipationcoefficients. Furthermore, integrating this indicator yields estimates of the error in lift

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Figure 11: NLF-0416 airfoil. O-type grid, velocity profile in the wake and dissipation based error indicator(left). Initial and refined structured wake mesh (right). Result of TUBS-ISM.

and drag associated with a given grid, including its sign, thus reducing the uncertainty ofintegral coefficients.

Additional effort has been spent by DLR in MUNA on improving the adaptation strat-egy itself. Besides evaluating the adjoint based indicator, the refinement strategy has beenimproved based on considerations on element de-composability and geometrical elementquality.

Fig. 12 shows regions in the flow field around a generic aircraft configuration wherethe adjoint based error indicator is high. As one can see, the main contribution to thenumerical error is generated some distance away from the geometry where it may hardlybe assumed when considering gradients only.

5.4 Application of Adjoint Based Adaptation to Complex Flow Fields

The solution of the adjoint problem assumes zero residual of the primal problem, i. e. afully converged flow solution. Unfortunately such highly converged solutions are difficultto obtain in complex applications, where unsteady phenomena may occur locally. In thiscase time-accurate computations for a limited period may be required for convergencetogether with engineering judgement for interpreting the result, in order to get meaningfulinformation on possible grid improvements.

Within MUNA EADS-MAS has applied the above adjoint method to various flowproblems ranging from a highly stretched wing configuration to a complete fighter aircraftin different operation conditions. The resulting adjoint sensor has then been used formanually adapting the grid in the detected critical regions.

Fig. 13 shows the adjoint error indicator in the field around a complete poweredfighter aircraft before and after manual grid adapation. Clearly the error is reduced byadapting the grid, indicating the suitability of the adjoint approach. Based on these

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Figure 12: NASA Common Research Model (AIAA Drag Prediction Workshop IV). Adjoint sensor,indicating locations of numerical error generation. Result of DLR.

results, EADS-MAS has been able to distinguish between influences of the computationalmesh and effects of the turbulence model on the prediction of forces and moments in arange of incidences for this configuration.

5.5 A Posteriori Grid Improvement

In case of complex aircraft configurations, it is sometimes observed that computationsfail due to degenerated cells occurring somewhere in the flow field. Typical situations aregaps between fixed and movable parts of a wing, like deployed high-lift devices, spoilersor rudders. Furthermore grid deformation, e. g. when deflecting the horizontal tail fin, orgrid adaptation may lead to badly shaped cells that prevent a computation to converge.Unfortunately, due to the automatism of the involved process chains the occurrence ofsuch situations is almost unpredictable.

In order to be able to fully exploit e. g. the potential of grid deformation and adaptationshown above, Airbus has developed a tool within MUNA, allowing repairing badly shapedcells of a given grid. The modification algorithm is based on element quality measures,including anisotropic metrics. Additional restrictions prevent degrading the modified gridwith respect to the initial one.

Fig. 14 shows a wing with deployed movables where 26 cells with negative volumeoccurred at the gaps between fixed wing and flap, preventing a computation. Theseelements have been automatically detected and corrected with the developed tool.

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Figure 13: Eurofighter at Ma = 0.4. Adjoint error indicator before (top) and after manual adaptation(bottom). Result of EADS-MAS.

6 CONCLUSIONS

In the German national joint project MUNA – Management and Minimzation of Uncer-tainties and Errors in Numerical Aerodynamics – 12 partners from research and industryhave developed and applied methods increasing the reliability of numerical flow simula-tions.

With respect to turbulence modelling sensors have been developed for checking thecompatibility of RANS models with a given grid, delivering error estimates. Furthermorea sensor has been developed, indicating regions where the RANS approach is no longervalid and a more accurate LES should be performed.

With respect to numerics, an iterated defect method has been developed, increasingthe order of accuracy of a given numerical scheme of lower accuracy order. The Taguchimethod has been used for efficiently determining the relative influence of multiple param-eters like flow conditions. Efficient stochastic methods based on sparse-grid technologyhave been developed for accounting for multiple and distributed stochastic uncertainties.In particular an efficient robust design technique has been developed allowing for shapeoptimization with geometrical uncertainties.

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Figure 14: Complex wing geometry with deployed movables. Red crosses indicate 26 elements withnegative volume, preventing a computation. The a posteriori grid manipulation tool successfully detectedand repaired the elements. Result of Airbus.

Uncertainties of geometry have also been treated in the context of deforming surfaces.The influence of structural model uncertainties on the probability of failure has beenstudied as well as the characteristics of interpolation algorithms on coupled fluid/structuresimulations for multi-component configurations. Furthermore a uniform grid deformationmodule has been developed that can in principle be coupled to any numerical flow solver,involving algorithms for reducing the deformation related error to a minimum.

Finally several methods and tools for improving the computational grid have beendeveloped, including dissipation and adjoint based error sensors and a posteriori gridmanipulation. In particular the applicability of the adjoint based grid adaptation and ofthe a posteriori grid manipulation tool to complex configurations have been demonstratedby the industrial partners.

ACKNOWLEDGEMENTS

The German national joint research project MUNA – Management and Minimisationof Uncertainties and Errors in Numerical Aerodynamics – has been funded by the Ger-man Ministry of Economics within the Luftfahrtforschungsprogramm IV under contractnumber 20A0604A.

The author is indebted to contributions by the follwing colleagues: Holger Barnewitz(Airbus), Dr. Stephan Albensoeder (Voith), Willy Fritz (EADS-MAS), Dr. HerbertRieger (EADS-MAS), Dr. Frederic Le Chuiton (Eurcopter), Alessandro d’Alascio (Eu-rocopter), Sascha Schneider (Eurocopter), Prod. Dr. Nicolas Gauger (DLR), Prof. Dr.

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Norbert Kroll (DLR), Dr. Matthias Orlt (DLR), Dr. Richard Dwight (DLR), Dr. Si-mone Crippa (DLR), Prof. Dr. Wolfgang Schroder (RWTH-AIA), Dr. Matthias Meinke(RWTH-AIA), Benedikt Roidl (RWTH-AIA), Prof. Dr. Marek Behr (RWTH-CATS),Prof. Dr. Josef Ballmann (RWTH-LFM/CATS), Georg Wellmer (RWTH-CATS), LarsReimer (RWTH-CATS), Prof. Dr. Frank Thiele (TUB-ISTA), Dr. Charles Mock-ett (TUB-ISTA), Tobias Schmidt (TUB-ISTA), Prof. Dr. Peter Horst (TUBS-IFL),Dr. Matthias Haupt (TUBS-IFL), Paul Reich (TUBS-IFL), Andreas Reim (TUBS-IFL),Prof. Dr. Rolf Radespiel (TUBS-ISM), Ekrem Mazlum (TUBS-ISM), Prof. Dr. Her-mann G. Matthies (TUBS-WiRe), Dr. Alexander Litvinenko (TUBS-WiRe), Elmar Zan-der (TUBS-WiRe), Prof. Dr. Claus-Dieter Munz (Stuttgart University), Dr. ThorstenLutz (Stuttgart University), Alexander Filimon (Stuttgart University), Alexander Wolf(Stuttgart University), Prof. Dr. Volker Schulz (Trier University) and Claudia Schillings(Trier University).

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