Hirofumi Doi
email: hdoi@stanfordalumni.org

Juan J Alonso
email: jjalonso@stanford.edu

FLUID/STRUCTURE COUPLED AEROELASTIC COMPUTATIONS FOR TRANSONIC FLOWS IN TURBOMACHINERY

The present study demonstrates the capabilities of a fluid/structure coupled computational approach which consists of an unsteady three-dimensional Navier-Stokes flow solver, TFLO, and a finite element structural analysis package, MSC/NASTRAN. The parallelized flow solver relies on a multiblock cell-centered finite volume discretization and the dual time stepping time integration scheme with multigrid for convergence acceleration. High accuracy is pursued with respect to load transfer, deformation tracking and synchronization between the two disciplines. As a result, the program successfully predicts the aeroelastic responses of a high performance fan, NASA Rotor 67, over a range of operational conditions. The results show that the unsteady pressure generated at the shock may act to damp or excite the blade motion mainly depending on the inter-blade phase angle. It is concluded that the level of sophistication in the individually sophisticated disciplines together with an accurate coupling interface will allow for accurate prediction of flutter boundaries of turbomachinery components.

INTRODUCTION

The unstable, self-excited or forced vibrations of rotor blades must be avoided in the design of high performance turbomachinery components because they may induce structural fail-ures. In order to predict these instabilities, the presence of strong shocks in the flow needs to be accounted for, especially for transonic flows. It is, therefore, necessary to use the Euler equations or the Reynolds-averaged Navier-Stokes (RANS) equations to represent the unsteady flow fields. Unsteady aerodynamics around oscillating cascades has been studied using these nonlinear equations by many researchers. For these approaches, timemarching methods [START_REF] Gerolymos | Advances in the Numerical Integration of the Three-Dimensional Euler Equations in Vibrating Cascades[END_REF][START_REF] He | Three-Dimensional Time-Marching Inviscid and Viscous Solutions for Unsteady Flows Around Vibrating Blades[END_REF][START_REF] Bakhle | A 3D Euler/Navier-Stokes Aeroelastic Code for Propulsion Applications[END_REF][START_REF] Ji | Flutter Computation of Turbomachinery Cascades Using a Parallel Unsteady Navier-Stokes Code[END_REF] or time-linearized methods [START_REF] Hall | Calculation of Three Dimensional Unsteady Flows in Turbomachinery Using the Linearized Harmonic Euler Equations[END_REF][START_REF] Ning | Computation of Unsteady Flows Around Oscillating Blades Using Linear and Nonlinear Harmonic Euler Methods[END_REF] are used to calculate aerodynamic work per cycle over a period of oscilaation prescribing frequencies and mode shapes.

On the other hand, the coupling of a structural model and a fully nonlinear aerodynamic model requires a time-marching method and determines the frequency of the problem rather than specifying it as an input parameter. The most noticeable work on the coupled computations is done by Vahdati, Imregun and their colleagues [START_REF] Vahdati | Non-linear Aeroelasticity Analysis Using Unstructured Dynamics Meshes, Unsteady Aerodynamics and Aeroelasticity of Turbomachines[END_REF][START_REF] Chew | Part-Speed Flutter Analysis of a Wide-Chord Fan Blade, Unsteady Aerodynamics and Aeroelasticity of Turbomachines[END_REF]. In their work, a mode superposition of the structure is incorpolated into a finite element RANS solver. However, since the methodologies of each individual discipline have matured independently, each methodology has usually evolved to use a different type of grid generation, a different discretization method and a different time integration scheme so that high accuracy and efficiency individually can be achieved. In order to take advantage of the maturity of both, a reasonable alternative would be to construct an interface procedure between a flow solver and a structural solver for direct integrations of structural equations, in which the two solvers exchange the interface information given by updating the fluid and structural variables alternatively.

The present study explores this possibility by integrating an unsteady RANS flow solver and a finite element structural solver for aeroelastic problems in turbomachinery, using advanced fluid/structure coupling techniques. The flow solver used here is an unsteady three-dimensional RANS solver called TFLO [START_REF] Yao | Development and Validation of a Massively Parallel Flow Solver for Turbomachinery Flows[END_REF][START_REF] Yao | Unsteady Flow Investigations in an Axial Turbine Using the Massively Parallel Flow Solver TFLO[END_REF], originally developed to simulate unsteady flows due to blade row interactions in turbomachinery. The structural solver used here is one of the industrial standards, MSC/NASTRAN. Furthermore, the interface procedure is based on the approach proposed by Brown [START_REF] Brown | Displacement extrapolations for CFD+CSM aeroelastic analysis[END_REF]. Using this coupling approach, the aeroelastic responses of a compressor rotor are predicted, and the influence of the flow fields on the stabilities is observed.

DESCRIPTION OF THE METHOD

In order to predict the dynamic response of a flexible structure in a fluid flow, the equations of motion of the structure and the fluid equations must be interacted. One difficulty in handling the fluid/structure coupling numerically comes from the fact that the structural equations are usually formulated with material (Lagrangian) coordinates while the fluid equations are typically written using spatial (Eulerian) coordinates. In such an approach, the procedure is advanced in time by solving the flow field and the structural deformation alternatively by independent flow and structural solvers which exchange information on the structural body surface as illustrated in Figure 1. The flow solver provides the aerodynamic loads to the structural solver in order for the structural solver to calculate the displacement field of the structure. In return, the structural solver provides the surface deflections to the flow solver which changes the flow fields through the boundary conditions on the structural body surface.

Flow Solver: TFLO

TFLO is an unsteady, three-dimensional, turbomachinery flow solver in which the unsteady RANS equations are solved using a cell-centered discretization on multiblock meshes. The solution procedure is based on explicit Runge-Kutta methods with several convergence acceleration techniques such as multigrid, residual smoothing and local time-stepping. A parallelization strategy is adapted to TFLO based on the multiblock decomposition [START_REF] Reuther | An Efficient Multiblock Method for Aerodynamic Analysis and Design on Distributed Memory Systems[END_REF] using the Message Passage Interface (MPI) standard (www-unix.mcs.anl.gov/mpi/index.html). For turbulence models, the k-ω model [START_REF] Yao | Development and Validation of a Massively Parallel Flow Solver for Turbomachinery Flows[END_REF] is found to be the most suitable choice among the implemented models for complicated viscous transonic flows in turbomachnery with timely changing grids in this particular study.

For an aeroelastic analysis in a time-marching fashion, it is necessary to deform the fluid mesh at each physical time step so that it continuously conforms to the instantaneous shape of the aeroelastically deforming body under consideration. In the present study, a high quality mesh of the whole computational domain is divided into an appropriate multiblock system for parallelization purposes. Because the flow solver assumes a pointto-point match between adjacent blocks in the mesh, each block may be independently perturbed, provided that modified surfaces are treated continuously across block boundaries. For edge, face and block perturbation in these procedures, the arc-length attenuation algorithm [START_REF] Reuther | Aerodynamic Shape Optimization of Complex Aircraft Configurations via an Adjoint Formulation[END_REF] is used.

Fluid-Structure Interface

While the flow solver employs relatively finer multiblock structured meshes, the structural solver employs triangular elements. This mismatch of mesh points on the interface gives rise to the necessity of a numerical approximation to transfer the pressure distribution on the fluid surface mesh to a corresponding nodal load distribution on the structure and also for the transfer of the displacements of the structural nodes to corresponding perturbations of the fluid mesh.

In addition, the fluid system usually requires a finer temporal resolution than the structural vibration for most aeroelastic problems. In computations, while the flow solver employs the Runge-Kutta time integration which has a stability limit, the structural solver uses the Newmark time integration [START_REF] Sitton | MSC/NASTRAN Basic Dynamic Analysis User's Guide[END_REF] which is unconditionally stable. This difference in the stability limits need to be taken care of by a synchronization approach between the two solvers alternatively advancing in time.

Conservation of Loads and Energy

One of the principles that is needed to support the maintenance of coupling accuracy is the conservation of loads and energy. In general while the fluid system addresses the pressure field on the cell surfaces on the interface, the structural system is solved based on a set of concentrated forces at the nodes on the interface. A distributed pressure load, therefore, must be first transfered into equivalent nodal forces. Such a transformation must satisfy two requirements. One is that the nodal forces must yield the same net forces as the original distributed pressure loads do. Thus,

∑ m f (m) = Z ∂Ω pdS, ( 1
)
where f (m) is the nodal force vector at the node m in the structure, and p is the pressure distribution on the surface ∂Ω whose projected area vector is S. The second requirement states the maintenance of a proper energy balance. Equating the virtual work performed by f (m) working on a virtual nodal displacement δq (m) and that by p moving through the equivalent distributed virtual displacement δµ µ µ, the second requirement is given as follows,

∑ m f (m) δq (m) = Z ∂Ω pδµ µ µdS.
(2)

In this study, the approach by Brown [START_REF] Brown | Displacement extrapolations for CFD+CSM aeroelastic analysis[END_REF] is chosen to ensure that the transfer of the pressure fields to the nodal forces is both consistent and conservative. Brown's approach can also be used in the extrapolation of the nodal displacement in the structural system to the mesh deformation on the surface of the fluid system as explained in the following section.

Deformation Tracking System Consider one of a set of finite elements describing a structural model and a fluid mesh point on the blade surface shown in Figure 2. Let x denote the closest point on the closest element from a fluid mesh point X, it can be assumed that the vector connecting both points remain perpendicular to the element after deformation. Then, the displacements µ µ µ(X) and rotations µ µ µ θ (X) at X can be expressed as follows,

µ µ µ(X) = u(x) -(X -x) × u θ (x), µ µ µ θ (X) = u θ (x), (3)
where u(x) and u θ (x) are the displacement and rotation at x. In finite element analysis, the displacement at any point within the domain of an element in the model can be determined by the assumed interpolation functions in terms of the nodal displacements on the element. However, since MSC/NASTRAN is used as a structural solver and the interpolation function used in it is z y x µ (X) DEFORM u(x) q , q , q 1 2 3 q , q , q q , q , q 1 2 3 7 8 9 13 14 15

A A A 1 2 3 Fluid Mesh Point Finite Element X x Figure 2. DEFORMATION TRACKING SYSTEM
unknown, all of the finite elements were assumed to use the simple standard iso-parametric interpolation function based on the area coordinate as the weighting [START_REF] Reuther | A Coupled Aero-Structural Optimization Method For Complete Aircraft Configurations[END_REF]. When X is associated with x on the m-th element whose displacements are denoted as q (m) , this interpolation function η (m) and η (m) θ can be expressed as,

u(x) = η (m) (x) • q (m) , u θ (x) = η (m) θ (x) • q (m) . (4)
The displacements at any point on the fluid mesh surface can thus be written as,

µ µ µ(X) = η (m) (x) • q (m) -[X -x] • η (m) θ (x n ) • q (m) , (5)
where, the matrix [Xx] is in cross product form. Introducing the displacement extrapolation functions N(X) based on the global nodal displacements q of the structural model, Equation 5 can be rewritten using q because q (m) is a part of q.

µ µ µ(X) = [[η(x)] -[X -x] • [η θ (x)]] • q = [N(X)] • q (6)
Load Transfer System For a load transfer algorithm, it is required to satisfy the conservation of load and energy expressed in Equations 1 and 2, respectively. In Equation 2, δµ µ µ(X) can be related with δq introducing a set of assumed displacement interpolation functions N(X) given in Equation 6. Substituting Equation 6 into Equation 2 yields,

f • δq = ∑ m f (m) δq (m) = Z ∂Ω p • [N(X)] δqdS, ( 7
)
where f is the global nodal force vector for the structural model. Consider the complementary projected surface area vector S i around the fluid mesh point X i , S i is surrounded by the corner points which are given by simply averaging the corner points of four fluid mesh faces around X i as shown in Figure 3. Thus the right hand side of the Equation 7 can be discretized as follows,

f • δq = ∑ i f i • δq = ∑ i p i S i • [N(X i )] δq, (8)
where f i is the nodal force vector given by the contribution of the pressure at X i , which components are shown in Figure 3. f (m) is now obtained by simply summing the all f i whose fluid mesh point is associated with the m-th element. Equation 8 allows the fluid pressure within each face to be taken out of the summation,

f i = {p i S i } [N(X i )] , (9)
Since N(X i ) consists of the area coordinate system normalized by the entire area of the element associated with the point i, it is obvious that the norm of the function is equal to one. Therefore, the conservation of loads is validated by collecting f i .

∑ m f (m) = ∑ i f i = ∑ i {p i S i } [N(X i )] = ∑ i p i S i = Z ∂Ω pdS. (10)
Thus the algorithm satisfies the conservation of loads and energy.

Synchronization If a same time step in both fluid and structure system is used to solve the aeroelastic equations, the time step must be governed only by the critical time step of the explicit fluid solver. It is also noted that since the fluid system must be advanced in the physical time scale, the fast convergence methods such as the local time stepping, or multigrid can not be available. Thus, using the same time step is not efficient in advancing the aeroelastic system. This difference in allowable time step size is, however, overcome by the dual time stepping scheme [START_REF] Jameson | Time Dependant Calculations Using Multigrid with Applications to Unsteady Flows Past Airfoils and Wings[END_REF] in which the fluid system is updated implicitly in physical time, and the pseudo time step is introduced to integrate the fluid equations within the physical time period. In this way, a same large physical time step is used for both solvers.

In aeroelastic calculations, a steady flow is first computed around a structure in equilibrium or in a given stationary condition with a set of initial displacements. Next the structure is allowed to respond to the aerodynamic forces determined by the

FEM Elements Fluid Mesh p S f f f f f f f f f f i i i i9 i8 i1 i2 i3 i4 i5 i6 i7 i X Figure 3. LOAD TRANSFER SYSTEM
steady flow field. This should be followed by a simple and popular staggered procedure as described below.

(1) Transfer the motion of the boundary of the structure to the fluid system using the Deformation Tracking System and update the fluid mesh accordingly. At this time, the updated cell volumes, projected areas and surface normals are calculated.

(2) Advance the fluid system using the dual time stepping scheme to a steady-state and obtain new pressure fields using the solutions at the previous time levels stored in memory for the third order accurate backward time discretization.

(3) Convert the new pressure fields into a set of nodal forces for the structural model using the Load Transfer System.

(4) Advance the structural system using the Newmark scheme under the given set of nodal forces.

The procedure repeats from step (1) through (4) until the desired time-marching solutions are obtained.

RESULTS

The goal of this study is to predict the aeroelastic response of a compressor, to examine the influence of the inter-blade phase angle and to observe how the flow field affects the instability of the blade vibrations. NASA Rotor 67 has been recognized as a popular test case for three-dimensional viscous flow prediction procedures [START_REF] Chima | Viscous Three-Dimensional Calculations of Transonic Fan Performance[END_REF][START_REF] Jennions | Three-Dimensional Navier-Stokes Computations of Transonic Fan Flow Using an Explicit Flow Solver and an Implicit k-ω Solver[END_REF][START_REF] Arnone | Viscous Analysis of Three-Dimensional Rotor Flow Using a Multigrid Method[END_REF] because of its detailed experimental data using a laser anemometer [START_REF] Strazisar | Laser Anemometer Measurements in a Transonic Axial-Flow Fan Rotor[END_REF]. Although the structural properties for this configuration are not available, Rotor 67 has been also used to validate aeroelastic applications by a few researchers [START_REF] Chuang | A Numerical Simulator for Three-Dimensional Flows Through Vibrating Blade Rows[END_REF][START_REF] He | Three-Dimensional Time-Marching Inviscid and Viscous Solutions for Unsteady Flows Around Vibrating Blades[END_REF][START_REF] Vahdati | Non-linear Aeroelasticity Analysis Using Unstructured Dynamics Meshes, Unsteady Aerodynamics and Aeroelasticity of Turbomachines[END_REF]. Thus, Rotor 67 was chosen for the current investigation to demonstrate the capability of the previously described nonlinear aeroelastic prediction procedure. NASA Rotor 67 consists of 22 blades. The blade aspect ratio is 1.56. At the design point, the rotational speed of the rotor is 16,043 RPM, with a tip inlet relative Mach number of 1.38, a total pressure ratio of 1.63 and a mass flow of 33.25 Kg/s.

The grid used for the present study is shown in Figure 4 with the blade-to-blade view. The grid for the single passage consists of 137 mesh points in the stream-wise direction, 65 points in the blade-to-blade direction, and 81 points in the span-wise direction. In the stream-wise direction, the grid extends one axial chord upstream and downstream from the blade row at the hub, and has 33, 73 and 33 axial points along the upstream periodic boundary, in the blade passage and along the downstream periodic boundary, respectively. A total of 17 points are used in the span-wise direction to describe the tip-clearance.

A NASTRAN finite element model for the blade consists of 11×11 nodes forming 200 varying-thickness three-node triangular linear shell elements, TRIA3. The thickness is distributed to match the actual thickness of the blade. For a boundary condition, both translational and rotational displacements are not allowed at the root of the blade. The direct transient response solution sequences, SOL129, [START_REF] Herting | MSC/NASTRAN Advanced Dynamic Analysis User's Guide[END_REF] are used to solve the structural response of the blade with time-dependent nonlinear aerodynamic loads with initial conditions provided by the displacements and velocities of the last transient solution sequence. In modeling NASA Rotor 67, an imaginary material whose Young's modules, Poisson's ratio, and density are chosen to be 1.422e+11 Pa, 0.3 and 4539.5 kg/m 3 , respectively, such that the first natural frequency becomes around 400 Hz.

For the design rotation speed of the Rotor 67, the reduced frequency based on the tip section becomes 0.27 according to the first mode frequency calculated with the structural model by MSC/NASTRAN. Finally, for multi-passage simulations, the cascade is assumed to be well tuned, which means that all the blades have an identical structural property.

Steady Flow Calculations

Prior to performing the aeroelastic unsteady calculations, validation of the steady flow solution needs to be performed such that the proper initial conditions for unsteady calculations can be provided. Steady flow computations are performed with the measured total pressure, total temperature and absolute flow angle profiles specified at the inlet and the measured static pressure profile specified at the exit.

To show the ability of the flow solver to capture the correct performance, the numerically calculated performance map of the rotor is shown in Figure 5 along with the experimentally measured values. As shown in Figure 5, the overall predicted performance of the compressor is in good agreement with the experiment except for the obvious underpredicted absolute total pressure ratio. This underprediction is likely due to an over- prediction of shock and viscous flow losses and flow deviation. However, the steady-state solution including the structural deformation mentioned in the next section results in a higher mass flow and higher pressure ratio. Although the structural model is not based on the real structure of Rotor 67, this suggests that the underestimated pressure ratio and mass flow may be caused by the structural deformation not included in the present calculations.

Because of the availability of the measured data, detailed comparisons between the calculations and the experiment are made at the near peak efficiency point. In Figure 6, relative Mach number contours at three different span wise locations, i.e. 10, 30, and 70 percent span measured from the casing, are shown for the near peak efficiency condition. At 10 percent of span, a bow shock at the leading edge and a normal in-passage shock near the trailing edge on the suction surface can be observed. At 30 percent of span, a lower Mach number moves the in-passage shock slightly forward but the shock structure is qualitatively similar. At 70 percent of span, the inlet relative Mach number is subsonic and a supersonic bubble appear near the leading edge on the suction surface.

Aeroelastic Calculations

Next, the time dependent aeroelastic results are considered. The behavior of the blade oscillation can be determined by computing the aeroelastic response to a set of forced initial displacements. As previously shown, the steady-state solutions can be obtained for the rigid unloaded blade geometry. Prior to the aeroelastic response calculations, the blade is allowed to move and dynamically relax to a steady-state loaded condition. Then, the blade is deformed an additional amount predominantly corresponding to the first mode. Once the new steady solution for the blade with the initial deformations obtained, the blade is released A constant inter-blade phase angle σ is determined by the initial displacement and velocity of the displacement for multipassage computations. In addition, the inter-blade phase angle is forced by the periodic boundary condition. For example, consider the case for σ=180 degree. The computation needs to be performed with two passages where one blade is given an initial displacement and zero initial velocity of the displacement and the other is given the initial displacement in the opposite direction with the same amplitude and zero initial velocity of the displacement as well. In this study, cases for σ=0 (one passage) and σ=180 (two passages) degrees are examined in order to save computational costs.

Five cycles of oscillation, which is about 0.02 seconds, were found to be sufficient to distinguish whether the oscillation decays or grows. A total of 36 time steps per vibration period of the first oscillation mode was chosen to ensure the accuracy of the coupled system with the dual time stepping scheme and the Newmark scheme [START_REF] Alonso | Parallel Computations of Unsteady and Aeroelastic Flow Using an Implicit Multigrid-Driven Algorithm[END_REF]. Between 40 to 60 W-cycle multigrid cycles per unit physical time step were necessary to provide a reasonable time-accurate pressure distribution to the structure.

The flutter stability is assessed from the time history of the Figure 9 shows the mean pressure and work per cycle distributions for σ = 180 degrees. For this case, the two kinds of shocks on the suction side mentioned above do not form a continuous line. The forces generated by the in-passage shock on the suction side works to dampen the oscillation, while the peaks along the oblique shock and the lower part of the shock disappear. Is is also obvious that the in-passage shock on the pressure side produces the large damping forces toward the leading edge. Thus the physical mechanism for the decaying response of σ = 180 case is clearly explained. A similar argument about the trend with inter-blade phase angle is reported by Chuang [START_REF] Chuang | A Numerical Simulator for Three-Dimensional Flows Through Vibrating Blade Rows[END_REF] for the same configuration though the reduced frequency is different.

CONCLUSIONS

A fluid/structure coupled aeroelastic solver for turbomachinery based on a three-dimensional unsteady RANS solver, TFLO, and the finite element structural analysis package, MSC/NASTRAN, was developed for use in turbomachinery flutter simulations. The capabilities of the solver are demonstrated by applying it to investigate the aeroelastic response characteristics of a transonic fan rotor, NASA Rotor 67. The aeroelastic solver successfully produces different time history of blade displacements for turbomachinery depending on conditions. Results presented for Rotor 67 revealed that the main contribution to the stability of a transonic fan is the unsteady forces generated by the shock motions. In determining these shock motions, inter-blade phase angles play the most important role.

One of the biggest concern for these fluid/structure computations is that the coupled procedure is still computationally expensive. Advanced engines, however, will likely require aeroelastic analysis for multiple blade rows, or a whole wheel system to admit all possible frequencies and inter-blade phase angles. Fluid/structure couplings such as presented here would contribute to these kind of numerical predictions for aeroelastically severe conditions. Furthermore, one of the biggest benefit of the numerical analysis is its richness in data that the experiment has been lacking due the difficulties in installing the measurement instruments.

  Figure 1. FLUID/STRUCTURE COUPLING

Figure 4 .

 4 Figure 4. Blade-to-Blade View of Computational Grid for NASA Rotor 67

Figure 5 .

 5 Figure 5. Comparison of Rotor Performance at Design Speed

Figure 6 .Figure 7 .

 67 Figure 6. Experimental and Numerical Relative Mach Number Contour for Near Peak Efficiency

Figure 8 .

 8 Figure 8. MEAN PRESSURE AND WORK PER CYCLE DISTRIBUTION ON THE BLADE SURFACES FOR NEAR PEAK EFFICIENCY, σ = 0 deg

Figure 9 .

 9 Figure 9. MEAN PRESSURE AND WORK PER CYCLE DISTRIBU-TION ON THE BLADE SURFACES FOR NEAR PEAK EFFICIENCY, σ = 180 deg


Professor, Alain-Marc Rieu

Matoba Hiroshi

The European Union systemic crisis. From deconstruction to reconstruction

Context

This paper responds to two different contexts. The first one was the coming UK referendum and, whatever the result, the challenge for the EU to reinvent itself or dissolve if it continues failing European populations. The second context was a deep questioning in Japan, but also in South Korea, about the future of East Asia and of ASEAN: is a convergence possible beyond economic interactions? How to breach cultural divides? How to overcome the search for hegemony, deep historical hatred, in order to reach and establish a common ground?

Europe: an urgent need for new perspectives

Europe's unification process needs to be put in perspective in order to open new perspectives on present problems and to search alternative solutions leading eventually to policies 1 . People within national governments and European institutions tend to reject or marginalize such research because the institutions and their practices determine how these 1

Europe: the trap of an unbroken modern trajectory

As we all know, Europe's unification process started in the aftermath of World War 2, as part of the reconstruction of European economies, political institutions and social policies.

The project to unify nations at war with each other for centuries was and still is a historical event of worldwide significance, with major consequences for international relations and for the political, social and economic evolution of many nations and regions in this world. In fact this project concerns all nations, which have been affected in their history by the type of institutional arrangement (political, social and economic) implemented in Western Europe since the early 17 th century. This institutional arrangement is called Modernization. The main feature of this arrangement is a political structure, which concentrates in a centralized power the means to various populations by controlling the territory on which these populations live.

On this territory and under this political control, the populations have been unified and transformed in a "people" managed by a government and a State apparatus. The monopolization by a State of the means necessary to control a population on a territory made possible the construction of a strong army to defend the territory and the formation of a strong police to control the population.

Michel Foucault in the late 1970ies formulated the real competitive advantage of this political structure, of this theory and practice of power. The modern political technology to control a population on a territory was not based on State domination, repression and violence. A transition took place during the 17 th century when European governments in England and France realized that the cost of such policies was too high. It was impoverishing the state and poverty generates dissent and disorder. A different political technology emerged: instead of controlling the populations on the territory, the modern state started to design and implement policies and regulations putting the population, individuals, families and groups, in the situation of developing their own conditions of subsistence, the wealth of their family, the economic development of the whole society and, as a largely unintended consequence, the commercial power of the State. The emergence of modern capitalism in the 18 th century is in Europe the result of a political strategy.

In other words, the modern political model, the Nation State generated modern capitalism. The institutional system, which presided over Europe's history since the 17 th century is also the model through which Europeans were able to conceive their joint reconstruction and unification after 1945. What divided Europe and lead to these wars is also the institutional arrangement (political, economic and social), which rebuilt and unified Europe. European modern history remained unbroken. The contemporary European project is a reassertion of European modern ideals within the same institutional framework. This political structure has for goal to organize the next steps, which all strictly followed the model of the nation-state: to enlargement to 25 European nations having a common destiny (both obvious and unclear), to create of a common and open market, to establish the free circulation of goods, persons and capital. The creation of a common currency (the €) managed by a European central bank followed logically. As conclusion, a Constitutional treaty was negotiated, written and eventually was signed in 2004 to organize and summarize all from the principles of this gigantic construction. European citizens voted the Constitutional treaty with great difficulty. The fact that the ultimate state of the model has been criticized and even rejected proves that this construction was reaching its limits. The Constitution was summarizing the historical process. At the same time, a new step was open.

A transition was starting.

In transition

This first stage is over. The impact on the European Union of the 2007 systemic crisis proves that some steps have been missed, some presuppositions ignored. The trajectory had seemed obvious but the outcome of this construction is flawed or at least not adapted to the present world (dis)order. The rise of the EU, what it stands for, is probably one factor of this disorder. These flaws are weakening European union. The model for the construction has turned counter-productive. It cannot be a model for the European union as it has emerged, needs to be managed and is growing in a world, which has completely changed while the Europeans were edifying their unification. The outcome of this unification and construction is unique in this world but it also raises new problems, which cannot find solutions in the construction model. Real innovation is required in an inter-national environment, which is still dominated by old political models, the empire model (Russia, China), the nation-state, the federal model (the USA), with their cohorts of nationalism, national, cultural, ethnic and religious identities, which make no sense, only problems, for the EU future.

When one is European thinking about this historical achievement, there is a sense of pride and a feeling of danger. It seems too much for the rest of the world still caught in nationalist, ethnic and religious wars of the most extreme violence, but also for a significant part of the European population, which did not, and could not, follow the project and its achievements. A solution cannot be drawn from somewhere else. It does not exist another or new model waiting to be discovered. It has to be invented by Europeans but also by all those in the world, who understand the dynamics of this unification process, of this post-national construction and its historical significance. The only solution is to examine the process, uncover its flaws and presuppositions in order to reform and redesign the structure, the institutions and the culture 3 . The problem is not that the EU can break apart.

The first risk is that it becomes empty and meaningless, as the USA and Great Britain wish it to be, just a common market and nothing else.

The second risk is more serious: intense pressure, like the present systemic crisis, the massive migration crisis and the energy transition, can alter the trajectory in a direction contradicting the original purpose, the common ideals and values, to such a point that European nations would, one after the other, lose interest in the project. It is a serious risk: the EU could turn into a sort of empire like Russia or imperialism like the USA by seeking relentlessly their interest and power. The highest risk for a post-national "union" is turned into a sort of empire.

For instance, in response to present migration crisis, the EU could either close itself to refugees or let itself be submerged by immigrants for whom there is a lack of resources, even in Germany. Referring to European values is not a response to a complex geopolitical, economic and social conjuncture. The European humanist tradition leading, for instance, to opening up to migrants always had a dark side: a sharp opposition between "us", the civilized Europeans and the "others", non-Europeans, invaders and barbarians. European civilization 3 The flaws of the Union resulting from the nation-state model are the idea of enlargement and deepening based on contiguity, the method of hybridization and harmonization, which intended to design norms and regulations in all European activities and productions in order to reproduce at the level of the Union the behavior of each nation, the search for a common border. The counter-balance for these extensions of the model is the principle of subsidiarity.

always had these two sides, like most other civilizations. One side can easily turn into the other.

The real problem remains economic growth, job creation, and energy transition.

Responding to this pressure is not a matter of political institutions but a matter of political culture. It does not concern politicians and bureaucrats only; it concerns European populations, the conception and empowerment of a European common civil society. The present conjuncture is a dangerous situation because this decisive parameter is not taken into account. European populations have radically lost trust into politicians and existing political institutions. Bur the next democratic wave is still hard to design. In this respect the situation in Europe is very similar to the situation in Japan. Politicians and bureaucrats have the power to decide and they are not ready to renounce their power and its privileges in order to respond to the evolutions of society. If they had to choose between their power and public distrust, they would choose their power. From this point of view, the English Tories are right: the EU has become a Leviathan cut from European populations. The main challenge in the EU is turn post-democracy into a new democracy.

The economic paradigm within the political model

I can only sketch the problems and potential solutions. I am also standing over the cliff of present political institutions and philosophy. I don't wish to stand in this awkward position alone. As mentioned before, I see three untenable positions. It is impossible to simply respond to present pressures, one after the other, as if there was no yesterday and no tomorrow. It is also impossible to dream and imagine what should be. Finally it is impossible to refer to any historical model. In summary, the EU is « over the cliff », a dangerous and exciting moment. As I see it (I might be wrong or partial), the only path forward is to change perspective, to step out of the model and to learn to analyze the situation from different angles.

The model of European unification was the nation-state as an institutional arrangement. The role of the model was to neutralize nationalism, political and cultural competition. The common interest was reduced to economic development and its social benefits. They were supposed to bring peace and reduce violence. Therefore the economy was and still is the core of the unification project. So there was an economic paradigm 4 within the model and on the long term this paradigm determined the unification process: an open market, the free circulation of goods, individuals and capital, understood as financial capital and the human capital. This generated a strong economic interdependence: trade within the EU is larger than trade outside the EU. Each European nation depends on the growth or non-growth of the European economy, which depends on the growth of the American and East Asian economies.

This explicit consensus was considering political, cultural and social differences as the private life of nations, as differences, which were a fact and could not be overcome because they were expressing Europe's historical diversity. It was explained as respect for cultural differences. But what it really meant and still means is that a unified economic sphere was the core and goal of the project. But to neutralize political, cultural and social differences was also to underestimate the institutional system (in a broad sense) in which economic development takes place.

This institutional system is quite different from nation to nation. It explains what

German economic sociology calls « styles of capitalism ». In this European arrangement, governments had the responsibility to adapt the national economy and society to respond and take advantage of the unified market. But some governments, in Greece, France, Italy, etc., chose on the contrary to protect their institutional system. They considered the potential growth induced by the opening of unified market as a solution for financing their "national difference", which meant in fact the established social hierarchies and power networks. After thirty years, the result is clear: the nations, which chose not to adapt to the new situation created by the unification process, did not and could not profit from the unified market. They lost jobs, businesses and overall competitiveness.

One thing is clear: in the end, the institutional system makes the difference in economic performance and it also explains this difference. Economic performance cannot be reduced to management. To neutralize is not to ignore, but it is an explicit or implicit decision, a methodological statement also, of not taking into account the social, political and cultural environment and to reduce it to a parameter but not a necessary condition of possibility. Furthermore, to separate the economic sphere from its institutional system is de facto to reduce it to a market, to a general exchange of goods and services regulated by 4 The notion of paradigm is used in his initial narrow sense of Thomas Kuhn: the construction of a field of investigation and teaching based on various methods and cultural presuppositions. This decoupling between the economic sphere and its institutional environment is an historical construction, which was studied (for instance) by Karl Polanyi: it is the construction of a distinct field and of a discipline studying and managing this studying. There is no doubt that the extraction of this domain from its institutional environment participated quite positively in its development. It gives this field a degree of autonomy, which simplifies its management by reducing the number of parameters having an impact on activities developed in this field.

The same paradigm 5 was reproduced in the 1990s by the USA in order to open and organize after the first Golf war the so-called globalization process, a worldwide market, which was supposed to bring economic development and growth to the nations agreeing to reorganize their economy for gaining access to a worldwide market. Each country had to figure out how to adapt its institutional system and through this system its population. The strategy behind was that implementing the paradigm was supposed to slowly implement the political model 6 . The problem is that markets, firms and economies are fully embedded into different types of institutional systems. Firms certainly compete with each other but what really competes through firms, organizational processes, products and services are institutional systems. We all know that the success of a company or a product comes from the institutional systems expressed in a given company or products. This is why we all buy Apple, BMW, Sony or Audi, etc. I don't bring anything new. But the reason for this conceptual denial and voluntary ignorance was a deeply rooted tacit consensus for many different reasons. The deepest reason explaining this consensus is the belief or idea that any economy, any nation had and has its chance to grow and prosper if it adopts the best management technology, whatever the state of its institutional system. It is always the same idea: decoupling the economic sphere from its institutional system. Be it China, Brazil or Russia, the belief is that it is just a question of good management, strong productivity and competitiveness, i.e. a good commercial positioning, good quality, stable and adequately trained work force, cheap energy and raw materials. This is quite an important illusion because it reduces the temptation to resort to violence and war. Nations, which understand their true situation, run the risk of exiting history and to become highly dangerous. The economy paradigm is an efficient illusion. There were in Europe other reasons for this consensus. The French liked particular this consensus. They thought they could profit from European economic growth, have access to this « common market » and at the same time keep their conception of the state and their « republican » ideology. The English participated also in this consensus because they could keep their monarchy, their conception of a civil society independent from government, in tune with their liberal conception of economic activities. The Germans saw the European unification as a way of overcoming the Nazi period and to reconstruct their economy based on Ordo-liberalism, their own version of liberalism developed during the 1930s against the Nazi regime and inspiring their conception and management of Germany's economic and social system. The other members thought they could only profit from this project and we all did.

Each nation had its own agenda within the overall unification. This explains the differences between national policies within the Union. What is clear is that Great Britain and Germany were the closest from the economic paradigm at the core of the Union. It explains why in retrospect, they were and remain the two main economic powers in Europe. Institutional reforms in Germany after the war created an environment conducive to industrial and commercial growth, with great success.

« Over the cliff »

« Over the cliff » is an expression coming from American cartoons: the hero has not realized that he ran beyond the cliff. Suddenly he realizes that he hangs in the air before falling down. Cartoon characters or individuals fall. But societies do not have to fall; they can learn how to fly and land on another cliff. They just have to think fast and radically. I think Europe is over the cliff. The fall can take a long time but will at a certain point accelerate. Si it is time to think without taboo.

The systemic crisis, which erupted in 2008 (but long in the making), has deconstructed the economic paradigm within the political model. This deconstruction continues and includes the political model. Some nations in Europe are not even sure anymore if a united Europe has not turned counter-productive to such a point that it endangers their economy and welfare programs. It is obviously the case of England, of Scandinavian countries and of many people in Germany as well. But since 2015, the mass migration crisis seems to prove to all Europeans that a unified Europe can help negotiate and implement solutions that no single nation could even imagine. Even the Swiss these days are constantly referring to the EU as if they were (already) members. The second factor reinforcing the Union is the energy transition. A third factor, more controversial for the moment, is the role of the European Central Bank in strengthening the banking system and in financing investment for innovation and growth. These are reasons why the EU is not really in danger. But to respond to intense internal and external constraints, the unification process must be reformed in depth. The EU institutional system is flawed. As mentioned before, it is not sure if the EU institutional system, this new Leviathan, can be reformed 7 .

If the systemic crisis has weakened the first step of Union by revealing many of its presuppositions, design faults and defects, it can also potentially strengthen the Union. These defaults are well known: unequal levels of economic development in the EU, "democratic deficit", absence of reforms in several countries of their institutions by fear of their political consequences, a level of sovereign debt equivalent to the GDP and beyond, lack of investment in industry and innovation, high unemployment and reduced public benefits, rising inequalities and violence.

But the typical and predictable outcome of the nation-state model and its economic paradigm was the creation of a common currency. The currency is the main attribute of political sovereignty, with the constitution, the border and the army. The model is not dead and will still produce its effects for years to come. The Euro condenses all the present problems of the EU. It has far deeper consequences than a Constitution: they touch all aspects and corners of the economy and society. The present crisis questions its concept, its purpose and feasibility. I am not an economist but simply said, in a common market within a globalized economy, a common currency requires from the members of this community to balance exports and imports by constantly adapting their economy and through their economy their institutional system. This balance seems impossible to reach: the Euro is always too high and too low for some industries and some nations. The weaker members are supposed to have access to credit and invest in profitable and competitive sectors. But this does not happen:

7 An obvious solution to reform the Union would be for each government not to reduce the budget of the EU but to evaluate how budgets are allocated. The heads of government and state never trusted anyway the EU bureaucracy, the Commission in particular. investment tends to cover political status quo and social policies, to finance the weaker economic sectors and not the most competitive and profitable ones. The result is increased national debts.

Countries in the Eurozone trade in a common currency when some, in fact most of them, do not generate the revenues to do so. Greece is a typical example. A common currency cannot equally profit to all members of a union. The conditions for managing the social and economic fundamentals of a common currency do not exist. All Europeans know that if the EU has a future, these conditions need to be reached and eventually established. What does it really mean? Is it the role of the European Central Bank? Certainly not. It is the responsibility of each member state. What does this mean? It means that governments and state apparatus have not be able or willing to adapt and reform. This situation questions the institutional system organizing and driving the evolution of each nation. Are Europeans ready to transit in this new phase of the European construction? It is not a question of individual or collective will. It concerns the capacity of an institutional system to reform itself.

Once more, the institutional environment is the problem and it also makes the difference. Politicians and administrators think and act inside the box: they are a component of the institutional system. They are duty and legitimacy is to reproduce it, eventually to correct it. But they do not have the perspective to reform it. It is impossible to have an overview of the whole system, to criticize it according to external criteria. There is readymade model. Research in human and social sciences need to step it by multiplying perspectives, by "complexifying" the situation 8 . The core of the unification model is an economic paradigm having for goal to neutralize the different factors explaining not only the evolution of the economic sphere of each nation but also the political, social and cultural constraints on the economy. What is explaining differences in economic performance is well known but denied at the same time. Because of this denial, to establish a common market was certainly a long and complicated negotiation between various economic and national interests.

But it was conceptually relatively easy: the method and the goal were clearly in sight. To put into brackets the institutional framework might have seemed possible during an initial period dominated by the political model of the nation state. This period is over and the denial also. If in retrospect, to establish a common market was very complicated, to harmonize institutional systems is extremely difficult because one 8 It is a type of conceptual scaling, a version of the deconstruction paradigm.

enters the flesh of societies. To go beyond the economy is to enter real differences. To hybridize and harmonize institutional systems does not even make sense. Is it even possible to organize a convergence between different institutional systems, without enticing strong nationalist or chauvinist reactions? Nobody wants such a convergence because of the resulting standardization and loss of identity. But at the same time, differences cannot be essentialized because they do have a history, which can be drawn and which is constantly under influence and evolving. Differences cannot be denied but they can be compared and even imitated. It is all a question of perspective and knowledge. This approach opens two ideas.

The first idea is simple hint. If the institutional system makes the difference, then Europe is not characterized by its economy, by a market-based capitalism, not even by its political culture. It is characterized by the interplay of the differences and similarities of its different institutional systems. In this case, there is no convergence to be organized or hoped for. There are "family resemblances" between this differences and similarities and these resemblances constitute a common civilization for Europeans, without being solidified into a common identity or culture. These differences cannot be reduced to the study of political institutions, social policies, conceptions of society, philosophies and ideologies, the arts, etc. Family resemblances cannot be exhaustively defined. It would be an endless list of similarities and differences, of influences, exchange and transfer. This endless list would reinforce, even justify, the idea that economic phenomena at least can be measured, managed and reformed, with tangible benefits. Failed attempts to draw this list, i.e. to define an identity, would justify the idea that differences are obstacle, which in the end can be denied and even ignored.

A solution is to change perspective in order to have a different understanding of what the notion of "institutional system" stands for. Institutions and their connections are historical constructions. The real problem is not the system but the dynamics of its formation and the connections of its components. I have analyzed these last years the modernization trajectories of societies, more precisely the formation and evolution of the basic structure organizing a society and its evolution. This basic structure is the social system, i.e. society as a system of different spheres or functions 9 . The institutional environment is in fact structured by the social system. The modernization trajectory is the comparative analysis of the differentiation of the 9 There are various approaches and methods, which are not competing but reinforcing each other.

different spheres constituting a social system. These spheres are the religious, the political, society (in the sense of civil society) and the economic sphere. This differentiation is different according to societies. But all European societies have in common a long-term and advanced differentiation and autonomization of these spheres of activities. This advanced differentiation explains first the secularization process and the transformation of religion from a major social institution into a private faith and collective morality. Churches do exist but only as the institutional support of collective faith. Europe is a highly secularized social system. People who do not accept or share this advanced secularism cannot really live in a European society because they belong to a different type of social system 10 .

The initial differentiation is the decoupling during the West European Middle Age between the Church and the state. It distinguishes societies based on a transcendental ground (ethnic, religious and cosmological), from a different type of society and evolutionary trajectory. This divide was a sort of earthquake; it still produces strong after-shocks. But the second differentiation is probably the fundamental one for Europe, with deep long-term consequences. It is characterized by a decoupling between the State and society, which in England first brought about the concept of a "civil society", a society made of individuals and groups, with their own interests, values and rights. The State is not considered anymore as unifying a population and transforming this population into a society, which the state has for mission to protect, like a father and his family or a pastor and his flock. On the contrary, in the image of themselves constituting a civil society, the individuals give to themselves, choose and select a government. This second decoupling has for obvious consequence the ideals and institutions of modern democracy.

Already at this level, the differences between the modernization trajectory in England, France and Germany, or in Japan, are obvious, with deep consequences until today. The widening chasm between the State and society opened historically the formation, autonomization and identification of a new range of phenomena with their own rationality, trade and other activities constituting the economy of a nation, supplying to individuals their means of subsistence and welfare. This third decoupling is the source of modern capitalism.

Obviously the different modes of the second decoupling condition the third decoupling and generate different types of capitalism, different relations between the State, society and the economy. These unfolding scissions generate also a transformation of conception, 10 This statement does not condemn, nor reject Muslims: on the contrary it emancipates Muslim people from Islamic religious power. The idea is in European societies, no communities can exist without living according to the structure of the social system. The social system is clearly a set of disciplines, constraints and rules. European conception of freedom is based on these disciplines and rules. organization and role of knowledge activities. It gave birth in the 17° century to what is commonly called "modern science". Science was from the beginning embedded in the evolution of modern societies and it progressively provided a scientific basis to techniques and technical development. In other words, these decoupling, which are happening one inside the other explain what was called earlier the "institutional system".

Conclusion. The political economy of the Union

The overcoming of the European Union dead end can be explained by the fact that the To harmonize and construct a common market is one thing. But it does not mean that it is possible to harmonize the economies, which are supposed to exchange goods, services and competence on this market. The next step of the European unification process is to organize collaborations between social systems coming from different modernization trajectories. This is the Union present frontier. Is it even possible? I am not sure. The problem can be reformulated in a different dimension. It is a question of knowledge and the capacity to share this knowledge on a large scale, to train people to these problems and to develop research.

From the perspective of modernization theory, Europe's different societies associated in the Union have many common features.

-First, they are post-ethnic societies: the ethnic origins of the population are not relevant at this level of development. What people have in common is the modernization process, which fathoms the societies in which they were raised and educated, in which they work and have a life.

-Secondly, these societies are post-religious. Churches and religious institutions do exist but religions are reduced to private faith, common morality or radically transformed into shared legal values, like human rights, individual freedom, freedom of expression.

-Thirdly, the role of the economic sphere, of its requirements (profits, productivity, competitiveness) and organizational criteria, in the development of social systems are known, debated and accepted as common to all societies in the Union and in an increasing part of the world economy.

-Fourth, science and technology are considered the source of long-term growth. A democratic society has for duty to invest in research and to establish the proper conditions for the transfer of the knowledge produced into economic growth and social benefits.

-Fifth, the ideals of civil society, of the independence of people and groups in their common life, outside from the sphere of government and State apparatus, the economic sphere and the sphere of religion, are fully understood and a basic requirement by the vast majority of individuals.

-Finally, democracy and democratic politics are considered the interface between civil society and the economy, a field of constant debate and adjustment. The State apparatus has for duty to protect the people under the control of democratic institutions.

What is typical of this social system in its present stage is the role played by human and social sciences in order to study and explain what is the present stage. What people have in common is the knowledge they share about their society and this shared knowledge is the basis of the political debate. This general template is common to all the Union members. It is the frame of what could be called the political economy of the Union, which is quite different from the economy of each nation in the Union. A Union requires a double economy; the political economy of the union is regulating the economy of each member state according to their specificities and situation. In other words, cohesion, solidarity, security and collective growth are the values of the Union according to which the present problems of debts and deficits could find solutions. I am fully aware this is quite abstract but I intended to show how far an alternative reasoning could go.

Finally, in a period of systemic crisis and skepticism, I would like to propose an answer to the question: What is Europe? Today, my answer would be: the European union is a family resemblance between different nations in this part of the world, which have in common a modernization process, which has shaped their societies, economies and conceptions of knowledge. Through the Union, these nations have experienced that they share a common long-term trajectory. From this point of view, the European union has for enemies all those, which reject and oppose the modernization process. At the same time, the European union has no borders but it has strong limits. These limits are drawn by the different scissions, which have shaped the social system shared by these nations.

  European nations have therefore started to associate their destiny in full continuity with Europe's modernization project since the Renaissance. Politicians and administrations in charge of the unification project repeated and adapted the model. The model had a strong legitimacy: it provided a guideline and obvious steps to follow one after the other. Europe was just reproducing itself at a higher, post-national, level. Today Europe unification seems to have unfolded a historical concept. The first step had for goal to harmonize local (i.e. national) laws under a set of basic legal principles acting as a norm: the universal Declaration of human rights in 1948. The 1951 Treaty of Paris signed by initial or core members established the joint basis of industrial development: the European coal and steel community was for fifty years, until 2002, the axis of the European construction. In 1957, was created the European atomic energy community (Euratom) to produce and distributed nuclear energy to all member states. Discussions were always difficult, but the next steps followed very logically with a form of evidence within the same frame: the Treaty of Rome established in 1958 the European Economic Community. The 1965 Treaty of Brussels combined the contents of the Paris, Rome and Brussels in a political structure distinguishing a Commission (the European commission), a Council (the European council) and a Parliament. The European parliament has limited initiative because the Council, which associates elected heads of state and government, runs the European union.

  demand and offer. My intention is not to criticize the construction of a field in which the unification process and the original project could progress. My intention is to explain what happened and what are today the consequences.

  formation and the development of the European Union were based on a political model and economic paradigm, which unfolded their consequences in full legitimacy. The systemic crisis, which started in the USA and engulfed the whole Europe revealed, proved and increased what everybody knew but underestimated: inequalities of development, level of education and innovation potentials, of wealth, plus massive sovereign debts. The initial European project has not failed but the crisis brought its first stage to an end. This situation is forcing all Europeans to analyze the present situation in order to, eventually, overcome it. Economic differentials are the problem condensing all problems. The main failure of this economic paradigm is to have ignored the institutional environments, which determine the joint but unequal economic development of different nations associated in a union. A union does not behave and cannot be managed like a nation. It requires a different perspective and different methods.

See the report The EU in the world, Eurostat, 2015 edition, http://ec.europa.eu/eurostat/documents/ .

This paradigm is based on various cognitive operations: decoupling the field from its institutional environment, defining this field and its different internal or specific parameters, conception of methods to manage this field, including the external parameters needed to manage the external constraints on this field, etc.

This is what Francis Fukuyama called the « end of history » i.e. the triumph of liberal democracy based on an open economy, a type of capitalism constructed and managed since the end of the 1970s.


Frédéric Wrobel
email: frederic.wrobel@um2.fr

Antoine Touboul

Frédéric Saigné

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 1 

Keywords: Transient currents, single event upset, MC-Oracle, atmospheric neutron, double exponential law, diffusion model, Monte Carlo method I

We calculated neutron induced Single Event Upset (SEU) cross-section as well as the Soft Error Rate (SER) at ground level. For this purpose, we first used an accurate model based on simulation of atmospheric neutron-induced transient currents in a 90-nm drain electrode, through a detailed diffusion model. Then, we performed the same simulations by replacing each transient current by a simple double exponential law model, for which the parameters were set in order to keep the same total charge as for the diffusion model, as well as the same value of maximum current and its corresponding occurrence time. Our results show a little increase of the cross section while using the double exponential law and we established a correlation between the parameters characterizing the double exponential and the diffusion model curves.

device, it is mandatory to understand in detail the main properties of the transient currents (e.g. the shape) that are generated during particle incidence. In [START_REF] Heijmen | Factors that impact the critical charge of memory elements[END_REF], it has been asserted that a double exponential law can model the shape of the transient current, but it is a priori difficult to determine realistic values of the parameters of this law. Moreover, these parameters strongly depend on the technology. Some Monte Carlo tools, as in [START_REF] Hubert | A review of DASIE code family: Contribution to SEU/MBU understanding[END_REF][START_REF] Correas | Prediction of multiple cell upset induced by heavy ions in a 90 nm bulk SRAM[END_REF][START_REF] Wrobel | MC-ORACLE: A tool for predicting Soft Error Rate[END_REF], use the ambipolar diffusion law in order to calculate, event by event, the transient current at each node of the electronic circuit. Then, the tool calculates directly the effect of the transient current on the system without storing the information on the properties of the transient current pulse.

The double exponential law is often used to represent the shape of the transient current induced by particles but there was no attempt to show its validity compared to a physical model such as the diffusion model.

The aim of this paper is to compare the incidence of the choice of the model of parasitic current pulses, in particular their shape, on the SEU cross section. We compared the double exponential current shape and the diffusion model, through simulating their effects on the SRAM cell. Starting from a transient current provided by the diffusion law, we calculated the corresponding double exponential current for which we kept the total charge, the maximum current and the time of its occurrence.

The rest of the paper is structured as followed. Section II introduces the simulation tool and shows how the current shapes can be compared. It shows how the double exponential law parameters are extracted from the diffusion model, in order to have a consistent comparison. In Section III, we provide some results, for a 90 nm SRAM device, in terms of SEU cross section and SER. Results for more integrated devices will be given in the final paper. In Section IV, we show how we can link the parameters of the two models. Section V concludes the paper.

II. CURRENT PULSE CHARACTERIZATION

We used the Monte Carlo code MC-Oracle which has been presented in [START_REF] Wrobel | MC-ORACLE: A tool for predicting Soft Error Rate[END_REF] and will be described in the final paper.

In our study, the simulated structure is a bulk made of a silicon layer with a surface of 30 µm x 30 µm and a thickness of 20 µm (see fig. 2). Above this bulk, we added a layer of 10 µm of silicon dioxide to account for the Back End Of Line (BEOL). At the Si/SiO 2 interface, we simulate 20 x 20 drain The nuclear reactions are triggered in the structure by using a Monte Carlo approach. Incident neutrons have either a given energy with a given direction or an isotropic distribution and their energies are drawn in an atmospheric-like spectrum [START_REF] Jesd | Measurement and reporting of alpha particles and terrestrial cosmic ray induced Soft Error in Semiconductor devices[END_REF].

By using the MC-Oracle tool we investigated the SEU cross-section variation due to the use of a double exponential current pulses instead of those resulting from the more precise diffusion model. While the diffusion model is based on physical phenomena, the double exponential law is based on a simple analytical law, easier to use since it does not need any integration. Its common expression is given by the following equation:

[ ] eq. 1
where Q is the total collected charge (in C), t is the time (in s), and and are two features of the current peak named rising time and falling time respectively (in s).

The double exponential law is often used as an input of simulation tools which aim to evaluate the sensitivity of a device. Nevertheless, the main drawback is that neither the collected charge nor the rising and falling times are actually known. Here, we propose to use data calculated for the diffusion current pulses in order to determine these parameters for each transient phenomenon. Thus, generating the double exponential current pulses as simplified model of the diffusion model, we are able to consistently compare the effects of the two current shapes.

From the diffusion model we extract, for each current pulse, the total collected charge Q (which also appear in the double exponential law), the current maximum I max and the time at which the current reaches the peak T max . Vanishing the derivate of the double exponential law gives a first relationship between the maximum current and the rising and falling times:

( ) eq. 2
Moreover, since t=T max the current is I max we obtain a second relationship:

[ ] eq. 3
Starting from a transient current provided by the diffusion law, it is thus possible to deduce an equivalent double exponential current for which the total charge, the maximum current and the time at which the current is maximum are conserved. While Q is immediately extracted, and are obtained by solving eq. ( 3) and (4) through a numerical method which will be detailed in the final paper.

Fig. 3 gives an example of an obtained diffusion current and its equivalent double exponential law. Even if it is clear that some differences exist between the two shapes the main issue is then to know whether there is an impact of the shape on the SEU cross section or not. Notice that the log-log scale lets think that the area of the two curves are different but they are actually identical.

III. CROSS SECTIONS AND SER CALCULATIONS

We performed extensive simulations for monoenergetic neutrons and for an energy spectrum representative of the atmospheric environment at ground level in New York City.

The graph in Fig. 4. plots the SEU cross section obtained for both current shapes (diffusion and double exponential) for a typical SRAM cell array in 90 nm technology. Despite the huge differences between the shapes of the diffusion model and double exponential (Fig. 3), the deviation in terms of SEU cross-section is not very important and it is typically around 10-20%. The double exponential law provides systematically a higher cross section than the diffusion model. This increment is due to the fact that the double exponential current is higher than the diffusion one during a time that is longer than T max . This extended high intensity of the current increase the probability of cell upset. The simulations made with a ground level neutron spectrum lead to a soft error rate of 389 FIT/Mbit for the diffusion law and 456 FIT/Mbit (+17%). Both values are consistent with the data published in literature for 90 nm SRAMs. More results will be given in the final paper for more integrated technologies.

IV. CORRELATION BETWEEN THE DIFFUSION MODEL AND THE DOUBLE EXPONENTIAL LAW

Since the difference between the two investigated shapes is not significant, it is useful to establish a formal correlation between the parameters of the diffusion law and those of the double exponential law. The graphs in Figs. 5 and6 plot respectively the rising time and the falling time of the double exponential law as a function of T max (time when the current reaches its highest value) in the diffusion law. Each spot on these graphs corresponds to a transient current pulse. Both the rising and the falling times follow approximately a power law. Therefore, while using a double exponential law, instead of using three parameters (Q, and ) it is possible, in the framework of the diffusion model, to use only two parameters (Q and T max ) : [START_REF] Taber | Single event upsets in avionics[END_REF] where:

[ ] eq.
eq. 5 eq. 6 The use of these equations allows having pulses with more realistic features.

V. SUMMARY AND CONCLUSIONS

In order to study SET, it is crucial to obtain realistic information on the characteristics of the transient currents generated by ionizing particles. Monte Carlo tools are very useful for this purpose as it allows studying each event independently. For each event the diffusion process is then used to calculate the transient current at a given drain electrode.

Practically, a double exponential law is often assumed to well represent the transient current induced by an ionizing particle because such analytical law is easy to handle. However, the three involved parameters such as the total collected charge, the rising and the falling times are actually not known, even approximately.

In this work, we mainly focus on the shape of the transient pulse in order to state whether the double exponential law is realistic enough to well represent the current induced by a particle regardless the choice of the parameters. In order to make a fair comparison, we imposed that the equivalent double exponential law would have the same total collected charge, maximum current and its time occurrence as the diffusion law.

Our simulations results revealed that the current shape has a small incidence on the SEU cross-section and SER since the double exponential law exhibits an increase of only 20% for the cross section. The use of a double exponential law is thus acceptable in order to have a rough evaluation of the SEU cross-section. Furthermore, we proved that it is possible to fully determine the double exponential current with the help of only two parameters, which are the collected charge Q and the time T max , in which the current reaches its maximum.

In conclusion, the double exponential law shape is acceptable to simulate the transient currents induced by ionizing particles and it actually requires only two parameters (the collected charge and T max ) instead of the three that are used. Even if there are some differences between the shape of the double exponential law and the diffusion model, this is not significant to modify significantly le SEU cross section. However, there is still incertitude on the value of the remaining parameters which should be used and future work will give some guidelines about the distributions of these parameters.

  Comparison of the Transient Current Shapes Obtained With the Diffusion Model and the Double Exponential Law -Impact on the SER F. Wrobel, L. Dilillo, A. D. Touboul, F. Saigné S electrodes.

Fig. 3 .

 3 Fig. 3. An example of a transient pulse obtained by the diffusion model and the equivalent current obtained with a double exponential law. Notice that the log-log scale lets think that the area of the two curves are different but they are actually identical.

Fig. 4 .

 4 Fig. 4. SEU cross section obtained for both current shapes (diffusion and double exponential) for a typical 90 nm SRAM.

Fig. 5 .

 5 Fig. 5. Relationship between the rising time (from the double exponential law) and the Tmax (from the diffusion law).

Fig. 6 .

 6 Fig. 6. Relationship between the falling time (from the double exponential law) and the Tmax (from the diffusion law).

This work has been supported by the National Research Agency in the framework of project ANR-09-BLAN-0155-01.


Abstract Downscaling of devices increases the M ultiple-C ell-Upset (M C U) cross section of SR A Ms making them an important threat for the robustness of systems. In this paper we present different types of M C Us as they were recorded during atmospheric-neutron ir radiation experiments, while a commercial 90nm SR A M was tested in dynamic mode. T his study shows that when the memory is in dynamic mode, not only the typical M C U that involve a few flipped cells may appear but also, large clusters of upsets are possible to occur with hundreds of cells being affected. We identify different patterns of M C Us and categorize them according to their shapes and sizes.

Index Terms M ultiple C ell Upsets (M C U), neutron ir radiation, SR A M , dynamic mode I. INTRODUCTION HE effects of neutron induced upsets to ICs have been studied extensively for over two decades. Until recently, the main source of errors induced by neutrons has been considered Single Event Upsets (SEUs) where a single bit cell of the memory is affected. However, with the downscaling of devices, Multiple Cell Upsets (MCU) have started to appear more often, affecting significantly IC robustness. MCUs are upsets induced by a single impinging particle, when cells neighboring to the hit cell are upset. The importance of MCUs comes to the possibility of being Multiple Bit Upsets (MBU). Depending on the mapping of the memory bit cells, MCUs may include upset cells belonging to the same word, a case extremely difficult to be mitigated by Error Correction Codes (ECC) at the system level.

Many studies investigating MCUs exist in the literature. Most of them are focused on MCUs appearing on SRAMs at both the simulation and the experimental level. The failure mechanism of MCUs is described as follows: a particle strike generates a parasitic current to a memory cell, inducing an upset (SEU). If big enough, the current can propagate to neighboring cells upsetting them as well, and resulting to an MCU. When more than one of the affected cells belongs to the same word the result is an MBU. At the simulation level, [START_REF] Tipton | Bit Upset in 130 IE[END_REF] shows that the proton induced MCU cross sections is a result of the downscaling of the sensitive nodes (cells) among other factors. Also in [START_REF] Wrobel | Simulation of nucleon-induced nuclear reactions in a simplified SRAM structure: scaling effects IE EE Trans[END_REF] another study is performed in which the SEU and MCU cross sections are calculated with respect to the deposited charge on the SRAM cells. [START_REF] Gasiot | Multiple Cell Upsets as the Key Contribution to the Total SER of 65 nm CMOS SRAMs and Its IE E E Trans[END_REF] explains the role of the triple-well in the MCU frequency increase, due to the amplification of the collected charge.

Besides the work done at the simulation level, several studies have confirmed the existence of MCU at the experimental level. An extensive work where the MCUs are categorized according to their shape and size has been presented in [START_REF] Radaelli | Investigation of multibit upsets in a 150 nm technolo IE E E Trans[END_REF]. The reported MCUs have been retrieved by irradiating an SRAM with different neutron energies while the memory was in retention (static) mode. In [START_REF] Tausch | Neutron Induced Micro SEL Events in COTS SRAM Devices[END_REF] micro-Single Event Latchups (SEL) have been recorded as a form of big clusters of upsets. In [START_REF] Hands | Dyer Large-Scale Multiple Cell Upsets in 90 nm Commercial SRAMs During Neutron Irradia IE E E Trans[END_REF] MCUs are observed with the form of big clusters of upsets as well while a 90nm SRAM was irradiated with neutrons and read back during the irradiation run.

This work presents an analytical study on MCU clusters observed during neutron irradiation experiments on an SRAM, similar of which have never been observed before to the best of our knowledge. The MCUs are categorized according to their shapes, sizes and source of occurrence. The experiments have been performed under an atmospheric-like neutron beam at the ISIS facilities in UK [7]. The memories were tested in dynamic mode during the irradiation run and as it will be shown, this is the main reason for observing such big clusters of upsets.

The rest of the paper is organized as follows: in section II a brief presentation of the experimental setup and conditions is given. Section III shows the MCUs retrieved during the irradiation experiments and explains the source of these events, while section IV concludes the paper.

II. EXPERIMENTAL SETUP AND CONDITIONS

The memory used during the experiments is a commercial 8-word, 32Mbit 90nm SRAM. A Finite State Machine (FSM) has been implemented to an FPGA responsible for the execution of the different applied tests to the memory. We applied tests while the memory was in both the static and --G. Tsiligiannis, L. Dilillo, A. Bosio, P. Girard, S. Pravossoudovitch, A. Todri, A. Virazel, C. Frost, F. Wrobel and F.Saigné T dynamic mode and under different operating conditions. In dynamic mode testing different March algorithms [START_REF]Stress Test for Efficient Evaluation ofNeutron Impact on Commercial SRAMs[END_REF], [START_REF]Proc. IE E E Int. Test Conf[END_REF], [START_REF] Niggemeyer | Integration of non-classical[END_REF] have been used as already described in our previous work [START_REF] Tsiligiannis | algorithms in Proc. IE E E International On-Line Testing Symposium (IOLTS)[END_REF], while the static mode testing was performed using a checkerboard sequence. When the memories operate in dynamic mode, different sequences of read and write operations are applied repeatedly, according to each March test. During the read operations, each cell is compared with the value to be expected. In case a faulty value is detected, the word containing the corrupted cell is transmitted from the FPGA to a computer along with its corresponding address and the timestamp of the event. When the memory is in static mode, the read back is performed every 30 seconds where all the recorded upsets are transmitted. The memories were irradiated under the atmospheric-like neutron beam of the ISIS facilities in UK. Emitted neutrons where of energies in the range of 10-800MeV. Experiments were held for different devices under different operating conditions, and also under different irradiation runs, such that the results would not be related to a specific device, but rather to the technology and architecture of the SRAM. TABLE I shows the total number of single events retrieved for each applied test, their corresponding Soft Error Rates (SER) and the conditions under which each test took place. The SER was calculated by considering each MCU as a single event occurring by a single particle (i.e. recorded faulty cells that were part of MCU clusters were not counted separately). Finally, a relative error of 10% should be considered for the SER according to the log files provided by the facility.

III. MCU ANALYSIS

In order to facilitate the presentation of our results, we categorize the recorded MCUs according to their shapes and sizes. In Fig. 1 an example of the results obtained by applying the Dynamic Stress March test is depicted. Each white pixel of the picture represents a bit cell while the pixels in black color are the recorded upsets. This visual representation provides a global view of the memory, with all the accumulated upsets during the irradiation experiment.

As Fig. 1 shows, besides the SEUs, different types of clusters of upsets are recorded when the memory was tested under the Dynamic Stress March test. Similar results were obtained for all the applied tests when the memory was in dynamic mode. However, when the memory was tested in static mode, we have never observed such big clusters. From the results displayed in Fig. 1 and from several other radiation tests we distinguish four categories that will be detailed in the following sub-sections. The categorizing of the four event categories was possible with the use of the upset timestamp.

A. Typical MC U: Type A

Type A category corresponds to all the typical cases of MCUs that have been observed and presented in most of the previous studies [START_REF] Tipton | Bit Upset in 130 IE[END_REF][START_REF] Wrobel | Simulation of nucleon-induced nuclear reactions in a simplified SRAM structure: scaling effects IE EE Trans[END_REF], [START_REF] Radaelli | Investigation of multibit upsets in a 150 nm technolo IE E E Trans[END_REF]. Such MCUs have been observed through all the different tests that we applied so far, in static and dynamic mode. In Fig. 3 According to the studies [START_REF] Tipton | Bit Upset in 130 IE[END_REF], [START_REF] Wrobel | Simulation of nucleon-induced nuclear reactions in a simplified SRAM structure: scaling effects IE EE Trans[END_REF], such type of upsets occurs when a particle strike generates a transient current high and long enough to travel and upset neighboring cells, besides the cell that it initially collided. Another case also would be to have more than one generated transient currents from the impinging particle, traveling through different directions that would also induce upsets to neighboring cells. Both cases of MCUs have been verified in several studies at the simulation level. ECC can mitigate these types of MCU as long as other cells are interfering physically between the bits of the word. Memories are designed with a distance, of 8 to 16 cells between two consecutive bits of the same word for that reason. Type A events were the only type of MCUs that was observed when the memory was in static mode.

B. Rectangular horizontal MC U: Type B

Continuing our analysis, we will present a type of MCU that occurred in all the memories while tested in dynamic mode. Fig. 4 displays some random occurrences of these upsets when the memory was tested with the C-, the Mats+ and the Dynamic Stress March tests in different operating conditions. They are identified as one upset since they occur concurrently. This type of errors appears rather frequently and considering the number of cells they affect, they have to be taken into consideration. The major problem is due to the noneffectiveness of common ECC algorithms to correct more than two upset bits, belonging to the same word. From measurements performed on our experiments the number of upset bit-cells is in the order of 100-700. Often, more than two bit cells belong to the same word, making such errors difficult to mitigate.

These events cannot be characterized as MCUs with the typical definition since a single particle cannot induce a transient current that would affect such a large number of cells. Errors similar to these have been reported in [START_REF] Tausch | Neutron Induced Micro SEL Events in COTS SRAM Devices[END_REF] but in that case they formed different shapes. The difference on the shapes is probably due to the different architecture in particular the different position of the p and n well taps that stop the propagation of latchup events, reducing them to localized micro-latchups. According to the technology used for the memory under test in that study, when a block is in retention mode it is not fully powered. The levels of voltage are lowered so that the cells ensure the information and at the same time keep the power consumption low. Once an operation is applied to a cell, the block it belongs to is fully powered for the operation cycle. As reported in [START_REF] Tausch | Neutron Induced Micro SEL Events in COTS SRAM Devices[END_REF] if a microlatchup occurs in that block during an operation, the lowering of voltage levels allows limiting the micro-latchup to one operation cycle. The memories used in our experiments may empty imply a similar architectural scheme, which would explain the micro-latchups that we observe. These latchups do not expand to the rest of the device and also the occurrence of such upsets is during dynamic mode testing and not static. Although the details of the well taping scheme are not known, from the shapes of the recorded micro-latchups we presume that the well taps are located every 16 cells vertically and 64 cells horizontally, and assist to the blocking of the transmission of the latchup as explained in [START_REF] Radaelli | Investigation of multibit upsets in a 150 nm technolo IE E E Trans[END_REF].

C. Rectangular horizontal MC U: Type C

The third type of observed events, Type C affects a larger area with respect to Type B. To the best of our knowledge type C events have not been observed in the past. Fig. 4 displays the events as recorded in two different devices tested under different algorithms. Type C clusters are significantly larger than Type B, since they have an horizontal extension of upsets that cover from one edge of the memory to the other as seen in Fig 4. (c3,[START_REF] Radaelli | Investigation of multibit upsets in a 150 nm technolo IE E E Trans[END_REF], or in some cases as seen in Fig 4 (c1,2) from the middle of the memory to the one edge. In addition the vertical extension covers approximately 20-60 bit cells (i.e. vertical lines). It appears that for all the presented cases, there is a repeated sequence. Although this sequence differentiated for different applied test algorithms we do not have enough statistical data to correlate the error sequence with the test. What is important to mention is that the sequences of Fig. 4 (c1) and (c2) were recorded concurrently as well as the ones of Fig. 4 (c3) and (c4), meaning that these sequences are considered as one event. The difference between these two cases is their size, shape and placement at the memory as observed in Fig. 1. A first thing to note is the appearance of Type C events in different devices and also different irradiation experiments (one year distance between them). Another important observation is that for all the observed cases, the corrupted bits of the words were all instead of being all The cluster shapes that are formed in this particular way can be related to the addressing scheme that is followed in this memory. Combined with the incidence of a Single Event Functional Interrupt (SEFI) to the memory periphery, or to the sense amplifier can be a possible explanation. We come to this conclusion since Type C events are symmetric and cannot be correlated to the same source of occurrence with Type B events. Another possible explanation is that these upsets can be related to the address decoder (if placed in the middle of the array, as in common butterfly architecture) malfunction, since at the same cluster of upsets, a different sequence is observed between the first half (left part of the cluster) and the second half (right part of the cluster) as can be clearly seen in Fig. 4(c3) and (c4). Also in Fig 4 (c1) and (c2) the corrupted cells cover exactly half of the width of the memory. It is important to note that such events have not been observed for all the tests that we applied something that indicates that their frequency of occurrence is rather small. However they should be taken into account when it comes to the evaluation of the sensitivity of the memory since the number of cells affected is rather large (30k up to 50k bit-cells). Such MCUs can cover a large part of a memory region as can be seen in Fig. 5(d2) or smaller parts but still significant as in Fig 5(d3). Type D events involve 20.000-90.000 corrupted cells, inside an area of 64x2048 cells in a vertical positioned rectangular cluster as according to our experimental results. A possible cause for such events could be micro-latchups that are localized in the failing areas as in the Type B event, but on a larger scale, such that the anti-latchup mechanism does not restrict the event to a smaller area. Type D events cannot be correlated to Type C, since there is no specific sequence on the upset cells that was observed. It is also interesting to note, that these Type D events appear always at the vertical edges of the memory array (left and/or right). This fact may be possibly explained with the concurrence of micro-latchups in this peripheral locations and major delay of the controlling signals (such as word line selection) that are more important in positions tat are far from the address decoder. Othjer possible explaiations may come from the topology of the power grid or the lower effectiveness of the well taps in these areas in reducing the propagation of the latchups, but we do not have elements to ratify these possible causes. Finally, this type of events present regions of 64x64 cells that are not affected by errors and represent discontinuities in cluster. These regions have been observed in different devices and experiments, thus they cannot be correlated to a faulty device.

IV. CONCLUSIONS

Besides the Type A MCU in which a few cells are affected, the rest of the reviewed cases impact a big number of cells, which can be of great importance at a system level. It has to be clarified that such events have been observed only when the memories are tested in dynamic mode. It seems that the architecture of the memory, does not allow these microlatchups to propagate to the entire memory, but limits them to small areas and to one memory cycle duration. For this reason, the device does not need to be power cycled when a latchup occurs. Through this work we revealed some of the issues that may occur during the dynamic mode operation of the memory. Although usually static mode testing is considered a standard for neutron testing, for high-reliability systems, the presented events may potentially occur and should be taken into account.

Fig. 1 .

 1 Fig. 1. 32 Mbit SRAM bitmap with all the upsets accumulated during two hours of atmospheric neutron irradiation. The applied test was the Dynamic Stress and the memory was under the temperature of 50°C. Pixels in black represent the upset cells.

  a few representative examples are displayed while the memory was tested in static mode. (a1) (a2) (a3) (a4) (a5) Fig. 2. Type A MCU: typical case of MCU occurring while the memory was tested in static mode at 50°C.

Fig. 3 .

 3 Type B MCU occurring when tested with: (b1) March C-algorithm 337 cells upset (b2) Dynamic Stress algorithm at 50°C 582 cells upset (b3) Mats + algorithm -553 cells upset -and (b4) Dynamic Stress algorithm at nominal conditions 261 cells upset. The area affected is usually 16x128 cells.

Fig. 4 .

 4 Type C MCU: (c1),(c2) Dynamic Stress algorithm at 50°C 32112 cells upset (c3),(c4) March C-algorithm 46504 cells upset.

DFig. 5 .

 5 . Rectangular vertical MC U: Type D Type D cluster was observed through all the different test algorithms applied in dynamic mode. Fig. 5 indicates a few examples observed through h the different test runs as they were isolated from the rest of incident upsets. Type D MCU: (d1) Dynamic Stress at 50°C 38246 cells upset (d2) March C-algorithm 89934 cells upset, (d3) Mats+ at 110°C 20176 cells upset (d) Dynamic Stress nominal 41022 cells upset.

  TABLE II summarizes the obtained results with the device cross sections for each type of MCU.