Clément Touzeau

Benoit Magnain

Gilles Lubineau

Éric Florentin

Benoît Magnain

Eric Florentin
email: eric.florentin@insa-cvl.fr

Robust method for identifying material parameters based on virtual fields in elastodynamics

Keywords: Robust identification, time-dependent virtual fields, transient dynamics, linear elasticity

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Progress in mechanical engineering and image processing has led to the continuous development of contact-less full-field measurement techniques such as the digital image correlation (DIC) [START_REF] Sutton | Determination of displacements using an improved digital correlation method[END_REF][START_REF] Chu | Applications of digital-image-correlation techniques to experimental mechanics[END_REF]. Inverse methods using information provided by the DIC technique have been created to characterize materials by identifying the parameters governing the constitutive law of a given material. The identification process of material parameters can be especially challenging in transient dynamics, due to the important flow of registered data. In this paper, we propose a new, multi-step, identification method for extracting elastic properties of materials in a transient state.

This new method, based on a variational formulation in transient dynamics, introduces a particular set of virtual fields that leads to a linear system of equations. Solving the system allow us to directly extract the material elastic properties. This last step, as well as the absence of any finite element resolution during the extraction of the parameters, are two key characteristics of our proposed method. The use of virtual fields for identification purposes has been investigated by Grédiac et al., by applying the virtual field method (VFM) to numerous configurations [START_REF] Grédiac | Special virtual fields for the direct determination of material parameters with the virtual fields method. 1-principle and definition[END_REF][START_REF] Grédiac | The virtual fields method for extracting constitutive parameters from full-field measurements: a review[END_REF]. The FEMU method [START_REF] Kavanagh | Finite element applications in the characterization of elastic solids[END_REF] is more intuitive than the VFM but requires an iterative procedure involving costly finite element resolution to extract its constitutive parameters. FEMU is based on the minimisation of a cost function measuring the gap between one measure (usually obtain by DIC) and several finite element resolutions. It is common to create the cost function by using the least square error between a measured displacement field and the displacement calculated by finite element simulations [START_REF] Rosenzveig | A fe model updating method for the simulation of the assembly process of large and lightweight aeronautical structures[END_REF]. Another approach for constructing the cost function with geometric quantities was recently presented by Touzeau et al., in [START_REF] Touzeau | Identification in transient dynamics using a geometry-based cost function in finite element model updating method[END_REF]. In their study, the authors report interesting results in the framework of large transformations.

The constitutive equation gap method (CEGM) [START_REF] Feissel | Modified constitutive relation error identification strategy for transient dynamics with corrupted data: The elastic case[END_REF][START_REF] Florentin | Identification of the parameters of an elastic material model using the constitutive equation gap method[END_REF][START_REF] Faverjon | Identification of boundary conditions by solving cauchy problem in linear elasticity with material uncertainties[END_REF] and the equilibrium gap method (EGM) [START_REF] Claire | A finite element formulation to identify damage fields: The equilibrium gap method[END_REF] are two examples of other methods based on the minimization of a cost function. Various popular identification strategies can be found in the scientific literature, including review articles by Bonnet and Constantinescu [START_REF] Bonnet | Inverse problems in elasticity[END_REF] or Avril et al. [START_REF] Avril | Overview of identification methods of mechanical parameters based on full-field measurements[END_REF] for the case of linear elasticity. Other method, also based on the choice of special fields, such as the reciprocity gap method (RGM) can be used for material parameters identification purposes. An example for elastodynamics studies is given in [START_REF] Bui | Numerical identification of linear cracks in 2d elastodynamics using the instantaneous reciprocity gap[END_REF].

Our proposed method is assessed by comparison with the classical finite element model updating (FEMU) method [START_REF] Kavanagh | Finite element applications in the characterization of elastic solids[END_REF]. Our data are synthetic, and built using a refined finite element simulation. In order to take into account imperfections inherent to the practical use of the method, perturbations are introduced. We generate the synthetic data starting from the exact value of material parameters which are known, this makes it possible to measure the identification error at the end of the identification process. A probabilistic study is performed by operating a Monte Carlo simulation on several test case configurations. Both noise on measurements and uncertainties on the loading conditions are considered, as done in many other studies [START_REF] Erdogan | Structural identification (st-id) using finite element models for optimum sensor configuration and uncertainty quantification[END_REF][START_REF] Toussaint | The virtual fields method with piecewise virtual fields[END_REF][START_REF] Passieux | Multiscale displacement field measurement using digital image correlation: Application to the identification of elastic properties[END_REF][START_REF] Alfano | Global sensitivity analysis in the identification of cohesive models using full-field kinematic data[END_REF]. During our numerical investigations, methods are evaluated in accuracy, robustness and CPU time consuming.

Problem framework

In this section, we present the notations and the equations of the direct problem used in the identification procedure and provide a concise description of the finite element resolution. We also describe the build of the synthetic displacement field used for the numerical investigations. Finally, we give an overview of the FEMU method used for comparison of accuracy and robustness.

Continuous problem in transient dynamics

The direct problem deals with the theory of continuum mechanics in transient dynamics under the assumption of small perturbations and plane stress state in linear elasticity. Here, we present the general statement of the direct problem in 2D. We consider a solid as a closed subspace of R 2 , denoted Ω, as shown in Figure 1. We study the mechanical behaviour of this solid for the time interval τ = [0,

t f ]. Ω Γ σ T d Γ u u d Figure 1: Continuous problem
The partition of its boundary ∂Ω can be defined as follows:

Γ σ ∪ Γ u = ∂Ω, Γ σ ∩ Γ u = ∅. ( 1
)
where the tractions T d are defined on Γ σ and the displacements u d are imposed on Γ u . The body forces are neglected and no contact occurs during the study. The problem consists of finding the displacement u(x, t) by knowing the geometry, the boundary conditions and the material constitutive law. The solution is unique and the governing equations are:

div [2µǫ + λtr(ǫ)I] = ρ ü in Ω × τ (2)
[2µǫ + λtr(ǫ)I] . n = T d on Γ σ × τ (3)
u d = 0 on Γ u × τ (4) ǫ = 1 2 (∇ u + ∇ T u) in Ω × τ (5)
u| t=0 = 0 and u| t=0 = 0 in Ω

In this equation system, all the fields depend on space and time, but for the sake of clarity, we do not make this appear in the notations. We define θ = (λ, µ) T as the set of Lamé parameters representing the elastic properties of the constitutive material. ρ represent the density of the constitutive material, ǫ the linearised strain tensor, ü the acceleration term of the solution, I the matrix identity and n the normal vector to Γ σ . Finally, the initial conditions are defined by equation (6).

Finite element analysis

In order to resolve the system of equations introduced in Section 2.1, we perform a finite element analysis and use a classical space discretization:

u(x, t) = NF E i=1 u i (t)N i (x) (7)
where u i (t) are the vector of nodal displacements and N i (x), i = 1, 2, . . . , N F E are the finite elements shape functions associated to the mesh of Ω.

The global vector of unknowns

U (t) = [u 1 (t) T u 2 (t) T . . . u NF E (t) T ]
T is the solution to the equation:

M Ü (t) + K(θ)U (t) = F (t) ∀ t ∈ τ ( 8
)
where M is the mass matrix, K(θ) the rigidity matrix and F (t) the external forces vector.

In order to solve equation ( 8), we perform a time discretization of the time interval τ :

t n = n∆t with n ∈ [0, N meas ] and N meas = t f ∆t (9)
with N meas the number of records made during the experimental procedure. In the following, the term

U (t n ) is simply denoted U n .
We then solve the equation ( 8) using the well-known Newmark scheme by making the following assumptions:

Un+1 = Un + ∆t (1 -γ) Ün + γ Ün+1 (10)
U n+1 = U n + ∆t Un + (∆t) 2 ( 1 2 -β) Ün + β Ün+1 (11)
where β and γ are the parameters of the scheme. By choosing the classical set of values 0.25 for β and 0.5

for γ, we obtain an implicit scheme called the average acceleration method. In the case of linear elasticity, this scheme is of the second order of accuracy and is unconditionally stable.

Using equation ( 8) at the time step n + 1 with ( 10) and [START_REF] Feissel | Modified constitutive relation error identification strategy for transient dynamics with corrupted data: The elastic case[END_REF], one can write:

K(θ)U n+1 = Fn+1 (12)
with:

K(θ) = K(θ) + 1 β∆t 2 M (13)
and:

Fn+1 = F n+1 + M 1 β∆t 2 U n + 1 β∆t Un + ( 1 2β -1) Ün (14)
Starting from the initial conditions at t = 0 (equation ( 6)), the linear system in equation ( 12) can be solved iteratively to obtain the displacement for all time steps.

Building the synthetic data

In order to validate the proposed identification method, the measured displacement field U (t) is replaced by synthetic data. First, a refined solution to the direct problem is computed with the exact values of Lamé parameters (θ ex ). Indeed, the Newmark scheme is performed using a very small time steps ∆t combined with a very fine mesh. For each time steps, the solution is stored on a measurement grid, denoted U ref n . Thus, U ref n (for all n in [1, N meas ]) constitutes the reference for our measurements. These synthetic data are interesting because they make it possible to estimate the quality of the identification procedure.

The displacement vectors U ref n of size N eq (the number of equations), ∀ n ∈ [1, N meas ] are gathered into a matrix U ref :

U ref = U ref 1 , ..., U ref Nmeas ( 15
)
We consider the displacement matrix U ref to be exact, and the error due to the finite element is neglected.

In reality, experimental set-ups and image processing always provide a measured displacement field with perturbations. Noise can be created because of the quality of the CCD sensors, the experimental benchmark and/or the user (see [START_REF] Blaysat | On the propagation of camera sensor noise to displacement maps obtained by dic -an experimental study[END_REF] for more details). In a small perturbation framework, the signal-to-noise ratio is low and the noise perturbation cannot be ignored in the study.

In our study, we choose to add noise synthetically by considering a random perturbation (i.e. uniform white noise) and use a Monte-Carlo process to study the robustness of the proposed method. The randomness is characterized by the parameter ω, with ω p , p ∈ {1, 2, ..., N MC } the N MC Monte Carlo realizations. The number of Monte-Carlo realizations must be large to obtain a good accuracy of the Monte-Carlo analysis; therefore, we use N MC = 10 4 in our study. The new displacement field U meas , considered as more realistic, is build from U ref as follows:

U meas (ω) = U ref + δU(ω) (16)
The perturbation matrix δU(ω) is a proportional white noise perturbation. Each component δU ij of matrix δU(ω) is expressed as follows:

δU ij (ω) = αψ ij (ω)U ref ij ( 17
)
where ψ ij (ω) is a N meas × N eq uniform random variable-centered on zero with values within the range

[-1, 1]
. α is a scalar parameter characterizing the magnitude of the white noise.

N MC Monte Carlo realizations U meas (ω p ) (p ∈ {1, 2, ..., N MC }) of the random matrix U meas (ω) can easily be constructed for the Monte-Carlo analysis.

Reference identification method

The accuracy and the robustness of our proposed method, is assessed by comparison with the FEMU method [START_REF] Kavanagh | Finite element applications in the characterization of elastic solids[END_REF], [START_REF] Rosenzveig | A fe model updating method for the simulation of the assembly process of large and lightweight aeronautical structures[END_REF], a popular, established method for extracting constitutive parameters that can easily be extended to transient dynamics. This method aims at minimizing a cost function that describes the distance between the measured field and a calculated field obtained by using the finite element method:

θ id (ω) = argmin θ C(θ, ω) (18)
In equation [START_REF] Sutton | Determination of displacements using an improved digital correlation method[END_REF], θ id (ω) is a random vector due to the random perturbation δU(ω) associated to the measured displacements. The displacement vectors U calc n of size N eq (the number of equations), ∀ n ∈ [1, N meas ] are gathered into a matrix U calc . We introduce the cost function C u (θ, ω) as follows:

C u (θ, ω) = Nmeas i Neq j U meas ij (ω) -U calc ij (θ) 2 (19)
where U calc ij are the components of matrix U calc .

Because of the large number of simulations needed to minimize each realizations

C u (θ, ω p ) (p ∈ {1, 2, ..., N MC })
of the random variable C u (θ, ω), we use a centered difference scheme with a lumped mass matrix, instead of the Newmark scheme, to calculate U calc n . Similarly, a coarser mesh is built for the reference solution.

Finally, the time step ∆t is chosen with respect to the Courant-Friedricks-Levy condition:

∆t ≤ ∆t cr = l c ( 20
)
where c is the wave celerity in the material and l is the critic length of the smallest element of the mesh.

This scheme is selected because it does not need a system resolution since the mass matrices are lumped.

Developed identification strategy

Here, we propose an inverse method for extracting the constitutive parameters of the material directly, without any finite element resolutions. This makes it different from the FEMU method, which needs iterative resolutions during the minimization process. In the following equations, the random displacement

U meas (t, ω) is simply denoted U .

General principle for extracting two parameters

The linear elasticity of the material is expressed using a stiffness matrix K(θ) and the Lamé coefficients λ and µ:

K(θ) = λK λ + µK µ (21)
where K λ and K µ are symmetric definite positive matrices.

Using equation ( 21), one can rewrite equation ( 8) as follows:

M Ü + (λK λ + µK µ )U = F ∀ t ∈ [0, t f ] (22)
The first idea of the method consists in writing a variational formulation of the problem defined in equation ( 22). U * (t) (denoted U * in the following) is introduced as a virtual field defined over [0, t f ].

U * is selected according to the Dirichlet conditions; it is continuous for the whole domain Ω (kinematic admissibility). Therefore:

t f 0 M Ü + (λK λ + µK µ )U -F T U * dt = 0 (23)
or, after some simple operations:

t f 0 Ü T MU * dt + t f 0 U T λK λ + µK µ U * dt - t f 0 F T U * dt = 0 (24)
In equation ( 24), the first term is of great importance as it provides an expression of the acceleration field, Ü , in terms of the measured data. Ü is computed from the measured displacement field using two temporal derivations. This step can lead to major errors due to noise in the measured data. To address this issue, we perform two successive partial integrations, which leads to the following equations:

t f 0 U T M Ü * dt + t f 0 U T λK λ + µK µ U * dt - t f 0 F T U * dt + U T MU * t f 0 -U T M U * t f 0 = 0 (25)
The main advantage of this new formulation is that it only requires knowledge of the measured displacement field over the time and the measured velocity field at t 0 and at t f . Ü * must be chosen derivable twice over time. In this paper, we propose to study the following set of virtual fields:

   U * 1 = U ∀t ∈ [0, t f ] U * i+2 = t i 1 ∀t ∈ [0, t f ] i ∈ N (26)
where vector 1 represents a vector filled with ones (1, 1) T .

U * 1 , corresponds to the measured displacement field. The second order differentiation Ü * = Ü * 1 cannot be performed using a basic differentiation. Indeed, due to noise propagation, it is very challenging to estimate it properly. A specific algorithm has been used and is presented in section 3.3.3. For U * i+2 , (i ∈ N) the dependency with t is analytical, which conduce to an analytical expression of the second order derivative :

Ü * = Ü * 2+i .
In order to identify the two Lamé coefficients, we choose two virtual fields and build the linear system with two equations and two unknowns. We propose to use the first two virtual fields of the set: U * 1 and U * 2 .

Thus, we obtain:

   λ t f 0 U T K λ U dt + µ t f 0 U T K µ U dt = t f 0 F T U dt -U T MU t f 0 + U T M U t f 0 - t f 0 U T M Ü dt λ t f 0 U T K λ t 0 1dt + µ t f 0 U T K µ t 0 1dt = t f 0 F T t 0 1dt -U T Mt 0 1 t f 0 (27)
The system (27) can be rewritten as follows:

A ij θ j = b i (28)
with:

A 11 = t f 0 U T K λ U dt (29)
A 12 = t f 0 U T K µ U dt (30)
A 21 = t f 0 U T K λ t 0 1 dt (31)
A 22 = t f 0 U T K µ t 0 1 dt (32) b 1 = t f 0 F T U dt + U T M U t f 0 -U T MU t f 0 - t f 0 U T M Ü dt (33) b 2 = t f 0 F T t 0 1dt -U T Mt 0 1 t f 0 (34)
Because the value of the determinant of A is not equal to zero, a unique solution exists for each realization of ω. The solution θ id can be expressed as follows:

λ id = b 1 A 22 -b 2 A 12 det A and µ id = A 11 b 2 -A 21 b 1 det A (35)
The unfortunate case, det(A) = 0, is likely to occur if we introduce a virtual field that conduces to a linear combination of previous equations. A simple manner to avoid this problem is to skip this field and take the next one in the proposed family.

Extension of the method

Here, we present an extension of the proposed method for an arbitrary number of parameters (e.g. orthotropic or heterogeneous material). If n is the amount of material parameters to extract so that θ = (θ 1 , θ 2 , ..., θ n ) T , the stiffness matrix K(θ) must be expressed as:

K(θ) = θ 1 K θ1 + θ 2 K θ2 + ... + θ n K θn (36)
By using as much virtual fields as needed (n), we can establish a linear system of virtual fields as follows:

                           θ 1 t f 0 U T K θ1 U dt + ... + θ n t f 0 U T K θn U dt = t f 0 F T U dt -U T MU t f 0 + U T M U t f 0 - t f 0 U T M Ü dt θ 1 t f 0 U T K θ1 t 0 1dt + ... + θ n t f 0 U T K θn t 0 1dt = t f 0 F T t 0 1dt -U T Mt 0 1 t f 0 θ 1 t f 0 U T K θ1 t1 T dt + ... + θ n t f 0 U T K θn t1 T dt = t f 0 F T t1 T dt -U T Mt1 T t f 0 + U T M11 T t f 0 ... θ 1 t f 0 U T K θ1 t n 1 T dt + ... + θ n t f 0 U T K θn t n 1 T dt = t f 0 F T t n 1 T dt -U T Mt n 1 T t f 0 + U T Mnt n-1 1 T t f 0 - t f 0 U T Mn(n -1)t n-2 1 T dt (37)
or, in matrix form:

A ij θ j = b i (38)
A is a n by n square matrix. An analytical resolution can be performed to find the n parameters θ = (θ 1 , θ 2 , ..., θ n ) T as far as n is generally small.

The decomposition described by equation ( 36) is a limitation. Nevertheless, different extensions remains possible such as orthotropic materials (linear elasticity) or heterogeneous materials like multi-material domain for instance. Such a case will be treated numerically in the section 4.

Numerical approximations

In our proposed method, we introduce a set of variational formulations of the direct problem. After performing two successive partial integrations, we extract material parameters directly, using a linear system of material parameters. Here, we present the mathematical tools used to calculate the different terms of the matrix A (see sections 3.1 and 3.2).

Numerical integration

Time integration is performed numerically by using the Simpson scheme. Based on equation ( 9), we define f as a continuous function on [0, t f ] and {y 0 , • • • , y Nmeas } as the discrete values of the function f . We approximate t f 0 f (t) dt by:

t f 0 f (t) dt = ∆t 3 y 0 + 4 Nmeas/2 i=0 y 2i+1 + 2 Nmeas/2 i=1 y 2i + y Nmeas ( 39
)
Where N meas must be an even number. The performed numerical tests shows that the integration error can be neglected in front of the identification error (see equation ( 43)).

Smoothing of U * 1

In the case of small perturbations, the signal-to-noise ratios are low and the accuracy of the identification method can be limited by measurement uncertainties. Indeed, the time derivation of measured data can lead to major oscillations in calculated terms, which may happen when using U * 1 = U as virtual field. To address this, we propose a numerical approximation that consists of smoothing the original data. Using the same notations as in the previous part, the smoothed value, ỹi , is calculated by: ỹi = y i-2∆t + 2y i-∆t + 4y i + 2y i+∆t + y i+2∆t 10 (40)

One of the advantages of this method is a reduction of the random variation of U * 1 (t) and its derivatives, which means that the signal-to-noise ratios are increased. Our work aims at improving this method in terms of accuracy and robustness, without costing severe additional CPU time computations.

Numerical derivation for U * 1

The second order differentiation for the displacement U (t) is also a delicate part of the procedure. In order to reduce the loss of information due to a bad differentiation approximation, here, we use a numerical model based on a five-point-centered difference scheme to provide an approximation ḟ (t) on the time interval [0, t f ]. The differentiation term can then be expressed as:

∂f (t) ∂t t=ti = y i-2∆t -8y i-∆t + 8y i+∆t -y i+2∆t 12∆t = ẏi (41)

Evaluation of the identification strategy

Using the previously established values of θ ex which are used to generate the synthetic measurements, we evaluate the identification error, ǫ(ω), a random vector, as the difference between the identified parameters θ id (ω) and the exact parameters θ ex :

ǫ(ω) = θ id (ω) -θ ex (42)
Each component of the vector ǫ(ω) corresponds to an error on each physical parameters (relative to the corresponding component of θ). Practically, we also define a relative error e(ω), for each component as:

e i (ω) = ǫ i θ i,ex i = λ or µ (43)
The different statistics of e can be obtained by using Monte Carlo simulations. In order to evaluate accuracy and robustness of our proposed method, we define the mean of the relative identification error e(ω)

with ē expressed as:

ē ≈ 1 N MC NMC p=1 e(ω p ) (44)
Finally, we define the standard deviation of the relative identification error e(ω) denoted ê as follows:

ê ≈ 1 N MC NMC p=1 e(ω p ) -ē (45)
Using Monte-Carlo simulations, we determinate the discrete Probability Density Function (PDF) and the discrete Cumulative Density Function (CDF). In our work, we focus on the error value e 90% i that leads to 90 % CDF CDF (e 90% i ) = 90%, meaning that 90% of the realizations e i (ω p ) (p ∈ {1, 2, ..., N MC }) are below

e 90% i .

Numerical investigations

Our method is tested for four different systems: one very simple mass-spring system, two systems consisting of 2D squares (with and without a hole) and one system describing a multi-materials domain.

First case study: mass-spring system

We consider a mass-spring system with a single degree of freedom (see Figure 2). The goal is to identify the stiffness k of the spring using the method presented in section 3. This model provide a numerical benchmark for which an analytical solution is available.

Statement of the problem and data construction

The measured positions are built from the vertical position of the mass m subject to the transient loading, as shown in Figures 2 and3. The measured position of m is denoted x meas (t, ω p ) and can be expressed as the sum of two terms: the exact part (e.g. the analytical solution), denoted x ex (t) and the random perturbation (e.g. the measurement error), denoted δx(ω):

k m T d (t)
T d (t) t t f 2t 1 t 1 P max
x meas (t, ω) = x ex (t) + α ψ(ω) x ex (t) ∀t ∈ [0, t f ] (46)
where ψ(ω) is an uniform random variable centered on zero within [-1, 1].

The first term is a solution of the equilibrium equation:

ẍex (t) + k m x ex (t) = 1 m T d (t) ∀t ∈ [0, t f ] (47)
where ẍex (t) is the exact acceleration term. The analytical solution for equation (47) is:

x ex (t) = P max kt 1 t - sin( k m t) k m ∀t ∈ [0, t 1 ] (48)
x ex (t

) = P max kt 1 t -2t 1 + 2 sin k m (t -t 1 ) -sin( k m t) k m ∀t ∈ [t 1 , 2t 1 ] (49)
x ex (t) =

P max kt 1 2 sin k m (t -t 1 ) + sin k m (2t 1 -t) -1 k m ∀t ∈ [2t 1 , t f ] (50)

Reference identification method

The minimization problem equation ( 18) can be simplified for the mass-spring system. It then becomes:

k id (ω) = argmin k C x (k, ω) (51)
where k id (ω) is the identified parameter. The cost function is based on the position evaluation and can be defined by:

C x (k, ω) = Nmeas i=0 x meas i (ω) -x calc i (k) 2 (52)
where x meas i (ω) are the measured positions of the mass and x calc i (k) is the calculated position obtained by using the equation (47) and a Newmark scheme.

Proposed identification method

In this case, the method presented in section 3 can be simplified. The variational formulation of the problem is then written using equation (47) applied to x meas (t, ω p ):

t f 0 ẍmeas (t, ω p ) + k m x meas (t, ω p ) - 1 m T d (t) x * dt = 0 (53)
Where x * is a virtual position field defined on [0, t f ]. The only parameter to extract is then identified by solving one equation:

k id (ω) = t f 0 T d (t)x * dt - t f 0 mx meas (t, ω)ẍ * dt + [m ẋ * x meas (t, ω)] t f 0 -[m ẋmeas (t, ω)x * ] t f 0 t f 0 x meas (t, ω)x * dt (54)
The same choices can be made for virtual fields:

   x * 1 = x meas (t, ω) ∀t ∈ [0, t f ] x * i+2 = t i ∀t ∈ [0, t f ] i ∈ N (55)
A comparison between several virtual fields leads to the resolution of several equations. Some of them are represented in the following development:

for x * 1 : k id (ω) = t f 0 T d (t)x meas dt - t f 0 mx meas ẍmeas dt + [m ẋmeas x meas ] t f 0 -[m ẋmeas x meas ] t f 0 t f 0 (x meas ) 2 dt (56) for x * i+2 : k id (ω) = t f 0 T d (t)t i dt - t f 0 mx meas i(i -1)t i-2 dt + [mit i-1 x meas ] t f 0 -[m ẋmeas t i ] t f 0 t f 0 x meas t i dt (57)
The numerical approximations are performed with the mathematical tools defined in the section 3. The identification methods used in our study are listed in Table 1.

Results and discussion

The parameters of the loading evolution are: t 1 = 1.0 s; t f = 4.0 s and P max = 10 N . The mass is equal to 30 kg and the stiffness to 745 N.m -1 . The time step is ∆t = 0.1 s so the number of records is N meas = 40. The number of Monte Carlo simulations is N MC = 10 4 . The noise magnitude α varies from 0% to 8% which is considered as coherent for the case of linear elasticity (see [START_REF] Azzouna | Identification of elastic properties from full-field measurements: a numerical study of the effect of filtering on the identification results[END_REF]).

Table 1: Names of the identification methods used in our study

Name identification method m1

x * 2 = 1 is used, m2

x * 3 = t is used, m3

x * 4 = t 2 is used, m4

x * 1 = x meas (t, ω) is used, m5

x * 1 = x meas (t, ω) is used with a smoothing (equation (40

)), femu C x (k, ω) = Nmeas i=0 x meas i (ω) -x calc i (k) 2 .
The notation femu, used in this part, stand only for the 0D case study. In the other studies (2D studies), the notation FEMU will be used.

As a result of the Monte-Carlo study, Figure 4 for the most part, virtual fields lead to better results than the femu method for reasonable values of noise amplitude. They also show that although the smoothing technique is not absolutely necessary for such a simple study, it does slightly improve the identification process without inducing any loss of mechanical information. An illustration of this is given in Figure 5 which shows the discrete PDF of the realisation e k (ω p ) (p ∈ {1, 2, ..., N MC }) associated to Monte-Carlo simulations for α = 8% and Figure 6, which shows the discrete CDF for the exact same conditions, and so compares the best choice (m5) of the proposed method with femu. The matrix of noisy displacement U meas is built as described in Section 2. A refined FE resolution is performed using a 1024 TRI3 element mesh and a fine time step (∆t = 10 -8 s). Measurements are performed using exact values of parameters : λ ex = 40.4 GP a and µ ex = 26.9 GP a.

For calculations made during the identification process, U calc is built using a coarse FE mesh with 16

TRI3 elements and a centered difference scheme with time step equal to 10 -7 s in accordance with CFL conditions (see equation ( 20)). The number of recordings for the identification is N meas = 250.

For illustration purpose, Figure 8 show respectively the fine mesh, used to construct the synthetic measurements, and the coarse mesh, used for the FEMU calculations. Identification methods are listed in Table 3. The Lamé coefficients are simultaneously identified by the FEMU method and our proposed method.

We first analyse the influence of the mesh used for the calculations made during the identification process with FEMU. To that end, we evaluate the behavior of the FEMU method without adding noise (α = 0%). For comparison purpose, the identification errors obtained with M1 are also given in the Figures 9 and10.

One can observe that better results are obtained with the proposed method denoted M1. The residual error observed when FEMU is performed with a very fine mesh (N F E = 1024) is due to the difference between the integration schemes used for the exact measurement and the FEMU calculations. In the following we chose N F E = 16 for FEMU, wich is seems a good balance between time consumption reasons and quality of the results (around 1%error).

In a second place, we analyse the behavior of the methods when noise on measurements is considered (α = 0). Figures 11 (resp. 12) shows the error values e 90% λ (resp. e 90% µ ) at CDF (e 90% λ ) = 90% (resp. CDF (e 90% µ ) = 90% ) for different values of α. The mean variations (ē λ , ēµ ) and the standard deviations (ê λ , êµ ) of the errors e λ and e µ are given in Table 4 for λ and Table 5 for µ. Based on Figures 13, 14, 15 and 16, we conclude that our proposed technique improves robustness and accuracy significantly. In Table 6, the CPU time for one identification is presented for the developed methods (M1 and M2) and is compared to the duration of one unique FE calculation. For accuracy purposes, all CPU times in Table 6 are based on 10 000 simulations. It is important to note that the smoothing technique is not time consuming. For example, one identification with M2 is 24 times less time-consuming than one FE simulation. In fact, the complete identification process is less time-consuming with our proposed methods than with the FEMU method. The absence of finite element iterations in our method is also an asset in transient dynamics. Indeed, FEMU identification requires at least one FE calculation.

Third case study: 2D square with a hole

The third case study differs from the previous one by its square hole at the center of the domain as shown in Figure 17. The loading evolution shape remains unchanged from previous studies. As for the previous case study, the boundary conditions are due to the symmetries of the considered problem. In this case, P max = 0.4 MPa, t 1 = 1.0 ms and 2t 1 = t f = 2.0 ms and the lengths are L = 0.1 m and l = 0.04 m. The calculations made during the identification process, U calc are built using a coarse FE model with 29

T d (t) L l
TRI3 elements and a centered difference scheme time step equal to 10 -7 s. This value is still chosen with respect to the CFL condition (see [START_REF] Touzeau | Identification in transient dynamics using a geometry-based cost function in finite element model updating method[END_REF]). The number of recordings for the identification is N meas = 200.

For illustration purpose, Figure 18 show respectively the fine mesh (used for generate synthetic data) and the coarse mesh (for FEMU identification) used in the following numerical examples. The mean variations (ē λ , ēµ ) and standard deviations (ê λ , êµ ) of the errors e λ and e µ are given in Table 7 for λ and Table 8 for µ. Results are consistent with those obtained with a no-hole 2D square and show that our proposed method increases robustness. The smoothing technique also appears to be very beneficial.

Influence of the material

In this third case study (2D square with a hole) we identify the Lamé coefficients of an aluminum-like material. We now test the robustness of the proposed method, for the same case study, but with a different type of material. We perform numerical investigations to illustrate the influence of the material properties on our proposed method. Table 9 gives an overview of the standard elastic properties for different types of material. Tests are performed similarly to those conducted for an aluminum-type material (section 4.3) with α = 0%. In doing so, we only consider the deterministic part of the study.

Table 10 shows the relative identification errors e λ and e µ obtained for each material listed in Table 9. We observe that, when a smoothing technique is used, the proposed method leads to better results for both parameters.

Loading uncertainties consideration

Previous studies only take into consideration the noise on measurements. However, during the measuring process of the displacement field (for instance by a DIC during a tensile test), the loading sensor can also lead to uncertainties as well as the CCD sensor. Here, we assess the behavior of the proposed method when noise on measurements and uncertainties on the loading evolution are taken into account simultaneously. To this end, we consider a random perturbation on the loading duration t 1 and on the loading intensity P max .

The evolution of the loading does not change (see Figure 3) but the loading duration becomes a random variable t1 (ω):

t1 (ω) = 1.0 + β ψ 1 (ω) t 1 (58)
where ψ 1 is a centered uniform random variable with values within [-1,1] and β is a coefficient with values within [0%; 2%; 4%; 6%; 8%]. By doing so, the mean of t1 is t 1 .

The loading intensity also becomes a random variable Pmax :

Pmax (ω) = 1.0 + β ψ 2 (ω) P max (59)
where ψ 2 is also a centered uniform random variable with values within [-1,1].

Based on these considerations, new numerical investigations, illustrated on the third case study, are carried out to provide an assessment of the method (M2 We show that the uncertainties on the loading are more likely to deteriorate the behaviour performance of the proposed method (M2) than noise does on measurements. The robustness of the FEMU method is still observed but our new strategy leads to better results until (α, β) = 6% which is an acceptable noise level for linear elasticity [START_REF] Azzouna | Identification of elastic properties from full-field measurements: a numerical study of the effect of filtering on the identification results[END_REF].

Similarly to previous observations made in the third case study, results are improved with the use of a smoothing technique on the virtual fields, without any significant loss of mechanical information.

Illustration of extension to multi-material

In this last case study, we illustrate one of the possible extensions of the proposed method (such as described in 3.2). Let us consider a multi-materials body as show in Figure 23. The domain initially used for the second case study is now split into two equal parts. The first half is represented by an isotropic elastic material with λ 1 ex = 40 GP a and µ 1 ex = 26 GP a. The second half is represented by an isotropic elastic material with λ 2 ex = 80 GP a and µ 2 ex = 52 GP a. Both parts are perfectly assembled.

(λ 2 , µ 2 ) (λ 1 , µ 1 ) L 2 L 2 T d (t)
The loading remains unchanged from the previous studies, the numerical values are: P max = 16.0 MPa, t 1 = 0.1 ms and t f = 0.25 ms. Other boundary conditions are symmetries. The length is L = 0.01m and the number of recording for identification is N mes = 250.

In order to have reliable synthetic data, we construct the reference data with a N F E = 2048 mesh and the calculated data (for FEMU) with a N F E = 32 mesh.

In this final case, we are interested in the identification of the 4 parameters: θ = (λ 1 , µ 1 , λ 2 , µ 2 ).

According to the developed strategy, one must chose 4 virtual fields in order to construct a small linear system. The following virtual fields are selected:

U * 1 = U (t, ω) , U * 2,3 = t 0 1 and U * 4 = t 1 1 (60)
The choice U * 2,3 = t 0 1 can be divided into two virtual fields as follow: Table 11 show the relative identification errors (e λ 1 , e µ 1 , e λ 2 and e µ 2 ) for α = 0% and β = 0% (none of the introduced perturbation are considered). One can observe that the proposed method lead to better results. Furthermore, the smoothing technique do not deteriorate the quality of the identification as previously observed for the previous cases.

U * 2 =    t 0 1 if y < L 2 0 else U * 3 =    t 0 1 if y ≥ L
In the following, we assess the robustness of the methods M1 and M2 when noise on measurements is considered. Table 12 show the mean variations (ē λ 1 ,ē µ 1 ,ē λ 2 ,ē µ 2 ), the standard deviations (ê λ 1 ,ê µ 1 ,ê λ 2 ,ê µ 2 ) and the errors (e 90% λ 1 ,e 90% µ 1 ,e 90% λ 2 ,e 90% µ 2 ) for α = 8% and β = 0%. The comparison with the FEMU method is not available for this case study. Indeed, the identification of the four parameters is too long, due to the choice of a simple minimization based on dichotomy algorithm.

The observed conclusions are in agreement with the previous cases. Indeed, a reasonable noise level (8%) conduce to an error (e 90% θi ) lower than 4% when M2 is used. The use of a smoothing technique slightly improve (or does not change) the robustness of the method. Nevertheless, the mean variations and the errors (e 90% θi ) are significantly improved.

Conclusions

In our study, we propose a new identification method for extracting elastic properties of materials in transient dynamics. The method is based on the discrete finite element formulation from which a linear system is created by using a set of virtual fields in the time dimension. Our new method is less time-consuming than the FEMU method which is used as a reference for evaluation purposes. In fact, a significant advantage of the developed method is the direct identification process that allows to extract the parameters directly.

For instance, current methods such as FEMU or CEGM are indirect and require an iterative minimization procedure.

The noise is one of the problems that can corrupt the data. This article is only based on numerical synthetic data. Therefore, the robustness of the developed approach has been studied numerically regarding noise on measurements and bad knowledge of the loading conditions.

Monte-Carlo simulations performed during numerical investigations, using synthetic data, demonstrate the accuracy and the robustness of this new method. A smoothing technique is also proposed in order to lower noise propagation; it appears to greatly improve the quality of the identification without causing any significant loss of mechanical information.

Other different problems can occur when using real data, for example misalignment between specimens and the acquisition system. In forthcoming work the method will be applied to real measurements in order to identify in a more realistic situation. This work is a first step of the development of the method. The proposed method seems adapted to the transient dynamics framework in linear elasticity. The proposed virtual fields are satisfying when a quasi-perfect framework is considered. The next step is to confront the developed method to a more complex situation. In the future, it would be also interesting to investigate new identification methods for large transformations and other non-linear real cases.

Figure 2 :

 2 Figure 2: Test case configuration

Figure 3 :

 3 Figure 3: Loading evolution

Figure 5 :Figure 6 :

 56 Figure 5: PDF of e k using m5 (blue) and femu (black)

Figure 7 :

 7 Figure 7: Test case 2 configuration

Figure 8 :

 8 Figure 8: Illustration of the fine mesh (N F E = 1024) and the coarse mesh (N F E = 16)

Figures 9 Figure 9 :Figure 10 :

 9910 Figures 9 and 10 show respectively the error e λ (%) and e µ (%) for different values of N F E .

Figure 11 Figure 12

 1112 Figure 11: e 90% λ s.t CDF (e 90% λ ) = 90% vs α

Figure 13 :Figure 14 :Figure 15 :Figure 16 :

 13141516 Figure 13: PDF of e λ (ω) using M1 (red) and M2 (blue)

Figure 17 :

 17 Figure 17: Test case 3 configuration

Figure 18 :Figure 19

 1819 Figure 18: Illustration of the fine mesh (N F E = 1856) and the coarse mesh (N F E = 29)

Figure 20

 20 Figure 20: e 90% µ s.t CDF (e 90% µ ) = 90%vs α

Figure

  Figure 21: e 90% λ s.t CDF (e 90% λ ) = 90% vs α and β

Figure 22

 22 Figure 22: e 90% µ s.t CDF (e 90% µ ) = 90% vs α and β

Figure 23 :

 23 Figure 23: Forth case study

  is the vertical component of the position.

Table 2 :

 2 ēk and êk for different values of α Table shows the mean ē and the standard deviation ê of e k (ω) as a function of α. Results show that,

	shows e 90% k	versus α. Level e 90% k	is defined by CDF (e 90% k	) =

k (ω p ) (p ∈ {1, 2, ..., N MC }) are below e 90% k .

Table 3 :

 3 Name of the identification methods used in our study

	Name	Identification method
	M1	U * 2 and U * 1 = U (t, ω) are used,
	M2	U * 2 and U * 1 = Ũ (t, ω)(smoothing displacement) are used,
	FEMU	the cost function C u is used.

Table 4 :

 4 ēλ and êλ vs α

	α	M1	M2	FEMU
	0% 0.042 0.042	1.55
	2% 0.20	0.14	1.55
	ēλ (%) 4% 0.59	0.31	1.55
	6% 1.27	0.56	1.55
	8% 2.20	0.90	1.55
	0% 0.00	0.00	0.00
	2% 0.13	0.10	0.049
	êλ (%) 4% 0.29	0.22	0.098
	6% 0.45	0.37	0.15
	8% 0.60	0.54	0.20

Table 5 :

 5 ēµ and êµ vs α Results show that the FEMU method is a robust identification procedure, similarly to what was observed in the previous case study (see section 4.1). In the present case study, the smoothing technique appears to be beneficial; it significantly improves our method without causing any important loss of mechanical

	α	M1	M2	FEMU
	0% 0.069 0.069	0.93
	2% 0.27	0.14	0.93
	ēµ (%) 4% 0.89	0.37	0.93
	6% 1.90	0.76	0.93
	8% 3.31	1.30	0.93
	0% 0.00	0.00	0.00
	2% 0.090 0.081	0.026
	êµ (%) 4% 0.18	0.17	0.053
	6% 0.27	0.27	0.079
	8% 0.36	0.36	0.10

information. The benefits of the smoothing technique for α = 8% are shown in Figures

13 and 14

(for λ) and Figures 15 and 16

(for µ).

Table 6 :

 6 CPU time calculations for comparison

	methods time (ms)
	M1	4.25
	M2	4.32
	FE	102.5

Table 7 :

 7 ēλ and êλ vs α

	α	M1	M2	FEMU
	0% 0.043 0.043	6.63
	2% 0.33	0.17	6.63
	ēλ (%) 4% 1.19	0.50	6.63
	6% 2.62	1.05	6.63
	8% 4.58	1.82	6.64
	0% 0.00	0.00	0.00
	2% 0.15	0.12	0.07
	êλ (%) 4% 0.30	0.27	0.13
	6% 0.45	0.44	0.20
	8% 0.61	0.60	0.25

Table 8 :

 8 ēµ and êµ vs α

	α	M1	M2	FEMU
	0% 0.052 0.052	5.10
	2% 0.43	0.19	5.10
	ēµ (%) 4% 1.57	0.63	5.10
	6% 3.45	1.37	5.10
	8% 6.02	2.40	5.10
	0% 0.00	0.00	0.00
	2% 0.072 0.072	0.01
	êµ (%) 4% 0.14	0.14	0.02
	6% 0.22	0.22	0.049
	8% 0.31	0.29	0.065

Table 9 :

 9 Standard values of parameters reflecting elastic properties of materials

		Steel Aluminum Glass Wood Concrete Diamond
	E(GPa)	210	70	60	7	35	1000
	ν	0.285	0.3	0.25	0.2	0.2	0.1
	ρ(kg.m -3 ) 7800	2700	2800	400	2400	3517
	λ(GPa)	108.3	40.4	24	1.9	9.7	113.6
	µ (GPa)	81.7	26.9	24	2.9	14.6	454.5

Table 10 :

 10 Quality of identification vs. material type, for both M2 and FEMU methods, with and without smoothing technique

			Steel Aluminum Glass Wood Concrete Diamond
	e λ (%)	M2	0.032	0.043	0.11	0.11	0.22	0.12
		FEMU 5.09	6.63	2.56	1.65	1.72	9.15
	e µ (%)	M2	0.028	0.052	0.056 0.047	0.077	0.027
		FEMU 6.12	5.10	6.62	7.23	3.91	5.61

Table 11 :

 11 Quality of the identification for FEMU, M1 and M2 methods

	Material parameters:		λ 1	µ 1	λ 2	µ 2
		FEMU 6.11 1.88 2.52	1.25
	e θi (%)	M1	0.43 0.18 0.049 0.084
		M2	0.43 0.18 0.051 0.085

Table 12 :

 12 Accuracy and robustness of the proposed method (M1 and M2)

	Material parameters:	λ 1	µ 1	λ 2	µ 2
	ēθi (%)	M1 3.55 3.91 4.75 3.23
		M2 1.45 1.60 1.94 1.32
	êθi (%)	M1 2.06 2.26 2.75 1.86
		M2 2.01 2.24 2.73 1.86
	e 90% θi (%)	M1 6.20 6.85 8.30 5.60
		M2 2.54 2.80 3.80 2.29


Lise Dauban

Emilie Cerezo

Anthony Henras

Olivier Gadal

David Engelke

Joachim Griesenbeck

Ross Hannan

Deborah Johnson

Richard Maraia

Christoph Müller

Craig Pikaard

David Schneider

Meeting report from the first European OddPols meeting: Toulouse 2018

The organizers are grateful to the "Centre de Biologie Intégrative de Toulouse", for support in the organization of the event. Participants from 13 different countries presented their newest exciting results during the scientific sessions and during extended conversation hours in the cosy atmosphere of the botanic garden of the Museum. Here

Introduction:

The 2018 OddPols meeting was held for the first time in Europe, at the Museum of Natural History of Toulouse, France, under the supervision of a giraffe proposed to become the meeting mascot. The meeting gathered 85 participants who shared their science through 37 oral presentations, divided in 8 general topics and 27 posters. The scientific committee awarded prizes for the best student talk and poster to encourage the upcoming generation of OddPols specialists. Congratulations to Aishwarya Iyer-Bierhoff for her talk during the session on regulation of transcription (see below). Congratulations to Marcel Böhrer for his poster dealing with the characterization of genetic and epigenetic determinants of RNA Pol IV activity in Arabidopsis thaliana). limiting its flexibility, while the Rpa43 C-terminus remains disordered.

Herbert Tschochner (Universität Regensburg, Germany) focused on Pol I-specific heterodimer Rpa49/Rpa34.5 activity using an in vitro approach. The study uncoupled the functions of Rpa49 C-terminal domain (A49-Ct), which shares homology with RNA polymerase II (Pol II) transcription factor TFIIE, and of the complex dimerization module composed of Rpa34.5 and A49 N-terminus (A34.5/A49-Nt), related to TFIIF. When present in the polymerase, A49-Ct enhances DNA binding and promotes transcription speed but leads to introduction of more errors. In contrast, an A34/A49-Nt containing Pol I is slowed down in elongation, but enhances fidelity. Moreover A49-Ct supports the polymerase to transcribe through a nucleosomal template. Interestingly, both Pol I and Pol III are able to transcribe through nucleosomal templates in vitro contrary to Pol II. This property is correlated with the presence of Pol I A349/A34.5 heterodimer or with the Pol III equivalent Rpc37/Rpc53 complex, which also shares homology with TFIIE and TFIIF.

Another Pol I specific feature, compared to Pol II and III, is its relatively high sensitivity to DNA-intercalating agents. The rRNA gene promoter sequences are not conserved across species and Konstantin I. Panov (Queen's University Belfast, United Kingdom) proposed that their structures and physical properties, such as bendability and meltability are conserved. He further exposed that Pol I inhibitors all share a flat structural region that can disrupt DNA 3D structure, increase DNA melting temperature and thus alter polymerase recruitment to the promoter and/or initiation events.

Christoph W. Müller (European Molecular Biology Laboratory, Germany) compared Pol I and Pol II mechanistic, based on the analysis of their respective atomic structures. One striking difference is the path the DNA adopts when engaged with the RNA polymerase. While the DNA fibre remains almost straight when transcribed by Pol I, it is bent when engaged within Pol II. In Pol I, this is explained by the presence of the Pol I transcription factor Rrn7 cyclin folds which bind further upstream on the DNA compared to those of TFIIB and by the absence of TBP bound to the Pol I core promoter.

As Pol I and Pol II are quite divergent, Pol III is closer to Pol II. Indeed Matthias K. Vorländer (from Christoph Muller's lab, EMBL, Germany) presented 3.7 Å to 5.5 Å resolution structures obtained by cryo-electron microscopy of yeast Pol III Pre-Initiation Complex (PIC) comprising Pol III bound to U6 promoter DNA and TFIIIB (composed of Brf1, Bdp1 and TBP). The group observed that TFIIIB binds to DNA on top of the polymerase cleft, with Brf1 contacting directly the polymerase as TFIIB does in Pol II. They could also determine that TFIIIB is tightly bound to DNA, forming a cage around the promoter, thus creating a stable platform capable of sustaining Pol III high initiation rate. Within the preinitiation complex, TFIIIB contacts transcription factor-like polymerase subunits Rpc37 and Rpc34 and stabilizes the transcription bubble.

Type 1 and type 2 Pol III promoters are peculiar in that they contain intragenic promoter elements. Type 3 Pol III promoter architecture is similar to snRNA gene promoters transcribed by Pol II. Jerome Gouge (The Institute of Cancer Research, United Kingdom) presented a study focused on Brf2, a transcription factor specific for these type III promoters. Very interestingly, they showed that Brf2 can be oxidised, and in turn modulates Pol III recruitment in a redox sensitive manner [START_REF] Gouge | Redox Signaling by the RNA Polymerase III TFIIB-Related Factor Brf2[END_REF]. Under prolonged oxidative conditions, Brf2 impedes transcription of seleno-cysteine tRNAs required for translation of detoxifying enzymes. Accumulation of reactive oxygen species then leads to cell apoptosis. Alessandro Vannini's fellow proposed that Brf2 activation in cancer cells lead to sustained detoxification and apoptosis evasion.

Seok-Kooi Khoo (Institute of Molecular Biology, Taiwan) focused on the functional characterization of Rpc82/Rpc34/Rpc31 Pol III subcomplex. Rpc82 subunit, which is Pol II TFIIE paralogue, contains a structurally disordered insertion in the third Winged-Helix domain, conserved in fungi and required for optimal cell growth and transcription initiation in vitro. Using site-specific photo-crosslinking analysis to explore Rpc82 function, the group showed that it interacts with both TFIIB-related Brf1 protein in the PIC and DNA upstream of the transcription start site. Rpc82 also contacts Pol III Rpc31 and Rpc34 subunits on the protrusion and wall domains of the cleft.

Session 3. Regulation of transcription: activators and repressors

Aishwarya Iyer-Bierhoff (German Cancer Research, Germany), from Ingrid Grummt's group, presented data on the regulation of fibrillarin (FBL), an enzyme that catalyzes methylation of both rRNA and nucleolar histone H2A. Glutamine methylation of histone H2A is exclusively present at the rDNA locus and correlates with Pol I transcription [START_REF] Tessarz | Glutamine methylation in histone H2A is an RNA-polymerase-I-dedicated modification[END_REF]. The group identified FBL as a direct target of the NAD + dependent deacetylase, SIRT7. They showed that acetylation and deacetylation of FBL by CBP and SIRT7 respectively, impact on H2A methylation and Pol I transcription without altering pre-rRNA methylation or processing. The study reveals that modulation of FBL acetylation and histone H2A glutamine methylation contributes to cell cycle-dependent fluctuations in Pol I transcription. In interphase, SIRT7 dependent deacetylation of FBL promotes H2A methylation and high levels of Pol I transcription, whereas hyperacetylation of FBL attenuates H2A methylation during mitosis and correlates with repression of Pol I transcription.

Transcription initiation by Pol III requires the activity of TFIIIB transcription factor. Guillermo Abascal-Palacios from the group of Alessandro Vannini (The Institute of Cancer Research, United Kingdom) reported the cryo-EM structure of Pol III pre-initiation complex at 3.4-4.0 Ǻ and apo-Pol III at 3.1-3.4 Ǻ resolution. These structures revealed a function of TFIIIB in the rearrangement of Rpc34 and Rpc37 Pol III specific subunits to facilitate DNA opening during Pol III transcription initiation.

Magdalena Boguta (Institute of Biochemistry and Biophysics, Poland) highlighted a dual role of a second major Pol III transcription factor, TFIIIC. This complex activates Pol III transcription by binding to the intragenic promoter elements of tRNA genes and allowing TFIIIB recruitment. The group reported that, depending on growth conditions, TFIIIC also functions as a tDNA transcription repressor. Under repressive conditions, interaction between TFIIIC subunit Tfc4 and TFIIIB subunits Brf1 and Bdp1 is strengthened. This leads to TFIIIB sequestration away from the tDNA transcription start site (TSSs), failure in Pol III recruitment and impairment in transcription initiation.

Session 4. Regulation of transcription: elongation and termination

Richard Maraia (National Institutes of Health, MD, USA) presented a new activity of Rpc37/Rpc53 Pol III subunits during termination which occurs at an oligo(T) tract on the nontemplate strand; oligo(rU:dA) hybrid on the template strand. The Rpc37/Rpc53 heterodimer promotes Pol III elongation in the proximal part of this tract by maintaining the RNA 3' end in the active site, ensuring processivity until sufficient instability of the (rU:dA) RNA:DNA hybrid results in transcript release [START_REF] Mishra | RNA polymerase III subunits C37/53 modulate rU:dA hybrid 3' end dynamics during transcription termination[END_REF].

David A. Schneider (University of Alabama, AL, USA) presented a study focused on Pol I subunit Rpa12 required for hydrolytic clivage of the nascent rRNA within backtracked or arrested transcription complexes. The results show that Rpa12 also participates in transcription elongation. Using in vitro transient-state kinetic analyses, they showed that Pol I lacking Rpa12 displays a dramatic defect in nucleotide addition during transcription elongation, suggesting that Rpa12 takes part directly in nucleotide addition mechanism. Authors also found that polymerases lacking Rpa12 are more stably associated with DNA template in vitro and are more resistant to high salt concentrations, indicating that Rpa12 influences the structure of the elongating polymerase. This new destabilization role of Rpa12 is also in agreement with data suggesting its involvement in transcription termination.

Work from Tommy Darriere from Olivier Gadal's group (Centre de Biologie Intégrative de Toulouse, France) demonstrated that point mutations in the two largest Pol I subunits can bypass the need for Rpa49 specific subunit in living yeast cells. Moreover, these mutations were all localized in a very specific area, near Rpa12 subunit, at the interface between the jaw and the lobe. Authors also showed that one of the mutations, in an otherwise WT context, allows overproduction of rRNAs, leading to the formation of a putative super-Pol I.

Tom Moss (Laval University, Quebec, Canada) reported a combination of ChIP-seq data using Mouse Embryonic Fibroblast (MEFs) and Embryonic Stem Cells (ESCs) conditional for the transcription factors UBF, RRN3 and TAF1B. Analysis of these data suggests that UBF, in collaboration with SL1, is a key factor in establishing and maintaining rDNA repeat activity. RRN3 release from elongating transcription complex is stochastic and depends on cell type and possibly initiation rate. Furthermore, a peak of arrested Pol I is observed at the "spacer promoter" immediately upstream of the enhancer repeats and the gene body. This peak is positioned just downstream of a unique chromatin boundary consisting of CTCF and cohesin complexes. Active chromatin marks and phased nucleosomes are detected adjacent to this boundary, but not elsewhere in the rDNA repeat.

Session 5. Chromatin effect on transcription Konstantin I. Panov (Queen's University, Belfast, United Kingdom) presented a study focused on the epigenetic regulation of rDNA chromatin in response to stress in human cells. The group discovered a new link between stress and chromatin epigenetic changes at the rDNA locus involving histone demethylase KDM4A. Upon stress, such as starvation, DNA damage or heat shock, KDM4A dissociates from rDNA. This correlates with downregulation of Pol I transcription and closing of rRNA genes as visualized by ATAC-see method. Conversely, KDM4A is required for recovery after stress since upon its relocalization to nucleoli, rRNA genes adopt an open conformation and rRNA synthesis resumes. Association of KDM4A with rDNA in starved-refed cells depends on PI3K/SGK1 signaling pathway but other non-identified stress-dependent signalling pathways are involved in the cases of other stresses.

A complementary study presented by Ann-Kristin Östlund Farrants (Stockholm University, Sweden) highlighted the crucial role of B-WICH complex (SNF2h, WSTF, NM1) in Pol I transcription. Following glucose stimulation, B-WICH is recruited at rDNA locus where it is implicated in chromatin opening. Indeed, in absence of B-WICH subunit WSTF, rDNA remains in a repressive chromatin state despite glucose stimulation. This prevents association of c-Myc and histone deacetylase SIRT7. Furthermore dissociation from the promoter of the NuRD silencing complex component CDH4, is impaired in absence of WSTF. Therefore, upon glucose stimulation, B-WICH is required to counteract the repressive poised state induced by NuRD complex.

To understand whether rRNA gene expression is differentially regulated depending on growth, Jean-Clément Mars (Moss group, Laval University, Quebec, Canada) studied rDNA activity in mouse cell lines displaying different proliferation rates. In contrast to MEFs, fast growing ESCs have almost all their rRNA genes transcriptionally active. The authors characterized the distribution of histone modifications across the rDNA as well as the presence of a CTCF boundary adjacent to the spacer promoter [START_REF] Herdman | A unique enhancer boundary complex on the mouse ribosomal RNA genes persists after loss of Rrn3 or UBF and the inactivation of RNA polymerase I transcription[END_REF]. Histone modifications associated with gene activity were all detected at this boundary, but not elsewhere. Different distributions of RRN3 and Pol I/RPI indicated a distinct regulation of transcription in ESCs as compared to MEFs. Moreover, ESCs displayed a lower per gene density of engaged Pol I as compared to MEFs, suggesting that in MEFs the reduced number of active genes is compensated by an increase in polymerase density.

Martina Dvorackova (Masaryk University, Czech Republic) uses Arabidopsis thaliana to better decipher the role of CAF-1 complex, responsible for depositing histone variant H3.1 during replication. The group showed that depleting CAF-1 subunits does not impact replication timing but results in increase in DNA damages and chromosome instability. Indeed, both telomeric and rDNA repeats are progressively lost in these mutants. In reverted plant lines telomeric and rDNA repeat number are recovered in a stochastic manner.

Session 6. Interplay between transcription, genome organization and nuclear structures

The group of Brian McStay (Centre for Chromosome Biology, Ireland) investigates the genomic architecture of nucleolar organizer regions (NORs) to address the conundrum on what dictates nucleolar architecture. On the short arms of acrocentric chromosomes, ribosomal gene arrays are flanked by proximal and distal sequences (relative to centromeres). The distal regions contain a large inverted repeat transcribed by Pol II. Sequencing of acrocentric chromosome distal junctions by PacBio revealed that the precise barrier between rDNA array and distal region on each acrocentric chromosome displays strong sequence identity. Recombination events occur not only in the ribosomal gene array but also in this region.

Interestingly a study presented by Aline Marnef (Legube group, Centre de Biologie Inte ́grative de Toulouse, France) showed that Pol I transcription inhibition and nucleolar cap formation can be uncoupled. They observed that induction of DSBs in nucleoli using the AsiSI restriction endonuclease affects Pol I transcription in all cell cycle phases (G1, S, G2). However, Pol I transcription inhibition upon DSB induction in G1 phase does not induce nucleolar cap formation, indicating that transcription inhibition is not sufficient for nucleolar cap formation. Nucleolar DSBs can be repaired by homologous recombination at the nucleolar periphery. Moreover, CTCF and cohesin complex are required for cap formation and recombination events.

Nutrient deprivation also leads to major nucleolar rearrangements in yeast such as massive rDNA condensation. This is caused by inhibition of Pol I and Pol III through inactivation of protein kinase TOR and loading of condensin complex on rDNA, as explained by X.F. Steven Zheng (Rutgers Cancer Institute of New Jersey, NJ, USA). Indeed, rapamycin treatment leads to a rapid loading of condensin complex onto rDNA. Inhibition of Pol I transcription in absence of condensin results in hyper recombination at the rDNA locus correlated with increased association of Rad52. High rDNA condensation is therefore essential to prevent deleterious recombination between rDNA units and cancer cell transformation. mTOR activation leads to nucleolar hypertrophy due to an increase in both Pol I and Pol III activities. This hypertrophy is associated with elevated growth rate and genomic instability, two major actors in tumour progression.

During his plenary lecture, Craig Pikaard (Howard Hughes Medical Institute at Indiana University, IN USA) gave a retrospective account of the 20-year history of multisubunit RNA Polymerases IV and V, beginning with his identification of their atypical catalytic subunits in preliminary sequence data for the first plant genome (that of Arabidopsis thaliana). He highlighted key discoveries made in his and other laboratories that revealed distinct roles for Pols IV and V in siRNA-directed DNA methylation and transposon silencing and that identified their transcripts, subunit compositions and interacting partners, leading to current models for the RNA-directed DNA methylation pathway. The biochemical activities of Pol IV and Pol V are still not well defined, such that more studies are needed to understand their mechanisms of action to the same extent as for Pols I, II and III.

Session 7. Mechanisms of gene silencing

Later on during the meeting, Craig Pikaard exposed his work based on the development of in vitro systems to study the first steps of RNA-directed DNA methylation and gene silencing in plants. They recapitulated the synthesis of 24 nt siRNAs from template DNA using purified Pol IV, RNA-dependent RNA polymerase 2 (RDR2) and the dicer endonuclease, DCL3. Pol IV transcribes the single-stranded DNA template and terminates soon after it encounters the non-template strand of double-stranded DNA. The resulting termination complex enables RDR2, which needs a free RNA 3' end to function, to initiate its transcription, converting the Pol IV RNA strand into a dsRNA which is then processed by DCL3 into 24 nt siRNAs.

Todd Blevins (Institut de Biologie Mole ́culaire des Plantes, France) presented a talk about the mechanism of Pol IV silencing of transposable elements (TE) in plants. Using partial loss-of-function alleles of Pol IV subunits in Arabidopsis, his lab discovered that the Pol IV enzyme itself could trigger an initial wave of TE silencing without generating high levels of siRNAs. A second wave, involving amplification of the siRNAs, is then needed to complete transposable element silencing.

Gene silencing in plants involves another DNA-dependent RNA polymerase, namely Pol V. Studies in the group of Andrzej T. Wierzbicki (University of Michigan, MI, USA) aim at understanding the role of this polymerase. Loci to be silenced need to be transcribed by Pol V, this enzyme is therefore an important decision maker in the silencing process. While transcribing, Pol V recruits chromatin remodelers such as SWI/SNF complex. Nucleosome positioning and epigenetic landscape of target genes are thus affected, enhancing their silencing. Fine-tuning this mechanism is fundamental to prevent heterochromatin spreading to neighbouring genes. However, Pol V transcripts are hardly detectable. Pol V transcribes loci harbour CG methylations but this mark is not the only determinant in the recruitment of the polymerase, CHH methylation also contributes to some extent. Using a RIP-seq approach, they captured and sequenced Pol V transcripts and detected transcription start sites (TSSs) all over the transcribed regions. The presence of CHG methylation around these TSSs suggests that this mark could be involved in Pol V initiation.

Apart from gene silencing mechanism, Charlotte Montacié (Sáez-Vásquez group, Laboratoire Génome et Développement des Plantes, France) presentation dealt with the influence of iron level on rDNA organisation and ribosome biogenesis in plants. Plant nucleoli contain high iron concentrations. To assess the impact of iron on ribosome biogenesis they studied nicotinamide synthase (nas) mutant plants characterized by reduced iron levels in nucleoli. In these plants, an increased number of fibrillar centres is observed but no evidence for changes in ribosome synthesis levels could be obtained. Remarkably, iron reduction led to nucleolar rearrangement with reactivation of the normally silenced NOR2, now present in the nucleolus of plant cells and supporting expression of rDNA variant 1. This reactivation may be caused by epigenetic modifications since changes in CHG and CHH methylation are observed. These data are in agreement with those showing that high iron amounts in animal cells cause rDNA hyper methylation and oxidation.

Session 8,9. OddPols involvement in development, genetic disorders, cancer and diseases Nadine Hein and Ross Hannan (John Curtin School of Medical Research, Australia) presented studies on the development of drugs targeting Pol I to treat cancer. Ross Hannan presented advanced studies on CX-5461, the first Pol I inhibitor to enter clinical trials in Australia and Canada. Investigations to understand the mechanism of action of CX-5461 are crucial to reveal additional activities, which can contribute to efficacy, toxicity and mechanisms of acquired resistance. The gene encoding TOP2α is frequently mutated in cells with acquired resistance to CX-5461. Extracts from resistant cells show a decrease in TOP2α activity in vitro and TOP2α knock down is sufficient to cause resistance to CX-5461. As CX-5461, TOP2α inhibits PIC formation and transcription initiation but not elongation. CX-5461 induces a DNA damage response restricted to nucleoli characterized by a strong increase in DSBs at rDNA promoters. In addition to Pol I, CX-5461 also inhibits TOP2α, which may impair the DNA Damage Repair response in nucleoli and increase rDNA instability.

In parallel, Nadine Hein is working with Pimera, Inc (San Diego, CA) on the development of a second generation Pol I inhibitors that could be used in combination therapies to avoid acquired resistances. Authors showed preliminary studies on a new selective inhibitor of Pol I, PMR-116. Cell treatment with PMR-116 impairs Pol I recruitment to the rDNA in a similar fashion as CX-5461 and does not affect Pol II transcription. PMR-116 inhibits Pol I transcription both in cancer and primary cells but these latter do not display a cytotoxic response. The main cellular response to PMR-116 is cell death by apoptosis. PMR-116 induces phosphorylation and accumulation of p53 but does not activate CHK1 or CHK2 as CX-5461. In clinical terms, PMR-116 is a promising drug since it increases survival compared to CX-5461, outperforms CX-5461 therapeutic window in Acute Myeloid Leukaemia (AML) and prostate cancers and displays good pharmacological properties.

Marikki Laiho (Johns Hopkins University School of Medicine, MD, USA) presented the newest results on another Pol I inhibitor, BMH-21, a potential systemic cancer therapeutic. BMH-21 is a selective inhibitor of Pol I that impacts Pol I transcription cycle by causing transcription blocks. BMH-21 treatment induces specific degradation of Pol I subunit RPA194. This degradation requires the presence of transcribing Pol I on rDNA as shown by the requirement of UBF, TAF1C, and RRN3. The design and characterization of BMH-21 derivatives allowed to show that DNA intercalation per se is not the only determinant of RPA194 degradation. The ongoing studies in the Laiho group also aim at developing patient selection strategies by analyzing the oncogenic drivers or genetic alterations that upregulate Pol I transcription in cancers. A new rRNA probe detecting the 45S rRNA precursor has also been designed to study Pol I activity in cancer specimens.

Bruce A. Knutson (SUNY Upstate Medical University, NY, USA) talked about Treacher Collins Syndrome (TCS). This craniofacial disorder is characterized by facial bone malformations resulting from defects in neural crest cell migration during pharyngeal arches development. POLR1C or POLR1D genes encode subunits conserved in Pol I and Pol III (Rpac40 and Rpac19 in yeast, respectively) and are often found mutated in affected patients, as well as TCOF1, encoding a factor involved in rRNA synthesis and processing. As Treacher Collins mutations target highly conserved residues within yeast and human proteins, authors explored the molecular basis leading to the disease in Saccharomyces cerevisiae. Most of the TCS mutations are believed to affect dimerization of these subunits. One particular mutation (T71I) does not affect dimerization. It induces a cryo-sensitive phenotype associated with a defect in the production of the 18S and 25S rRNAs. Impaired recruitment of both Pol I and Pol III to rDNA is observed in this mutant and co-IP experiments showed that Rpac19 and Rpac40 no longer interact with other Pol I or Pol III subunits. Altogether, their results show that Treacher Collins mutations affect both Pol I and Pol III machineries [START_REF] Walker-Kopp | Treacher Collins syndrome mutations in Saccharomyces cerevisiae destabilize RNA polymerase I and III complex integrity[END_REF].

The presentation by Martin Teichmann (University of Bordeaux, France) dealt with the regulation of Pol III transcription during tumorigenesis. The group is studying the relationship between tumor promoting factors and Pol III transcription regulation during human IMR90 fibroblast transformation. This model system allows to recapitulate full tumoral transformation in vitro. Their data revealed that among Pol III components, only a strong increase of POLR3G and SNAPC1 expression at the mRNA levels was observed with progression of transformation. No deregulation of Pol I and Pol II components could be observed. Strikingly, several Pol III subunits and Pol III transcription factors (TFIIIB and TFIIIC) are up-regulated at the protein level. This increase is accompanied by a 2-fold increase of a subset of Pol III-synthesized RNAs such as 7SK, MRP or U6 snRNA. Collectively, these results indicate that overexpression of only a subset of the Pol III machinery components is sufficient to sustain transformation and tumor formation.

Shuping Zhong (University of Southern California, CA, USA) is interested in pathways leading to alcohol-associated cancer development. The group reported previously that alcohol stimulates JNK1 signaling, which in turn activates Estrogen Receptor α (ERα) and results in increase in Brf1 expression and activation of Pol III transcription. In the present study, they investigate the role of Runx2 (Runt-related transcription factor 2), a downstream target of the ER pathway, in Brf1 expression and Pol III gene activation in the development of breast cancer. Runx2 is overexpressed in human biopsies of breast cancer suggesting a role in tumorigenesis. Alcohol induces Runx2 overexpression in an ER positive breast cancer cell line (MCF-7) and this activation is mediated by ERα. RNAi-mediated knock down of Runx2 following alcohol treatment prevents Brf1 expression and Pol III gene activation. The group showed that the anti-tumor compound Tamoxifen reduces alcohol-mediated increase in Brf1 expression of breast cancer cells.

Deborah L. Johnson (Baylor College of Medicine, Texas, USA) described the regulation of the key repressor of Pol III transcription, MAF1, by proteasomal degradation. MAF1 protein has a short half-life and it is regulated through proteosome-mediated degradation. This occurs through both ubiquitin-dependent and independent pathways. mTOR-induced phosphorylation of MAF1 promotes its ubiquitination by the Cullin2-RING E3 ligase and this enhances its degradation. The deubiquitination enzyme, USP10, deubiquitinates MAF1 to increase its stability. These results are consistent with the previously observed decrease in USP10 expression in liver cancers and the loss of MAF1 protein with enhanced Pol III transcription.

Session 10. Novel links with Physiology and immunity

When the MAF1 gene is knocked out in mice (MAF1 KO mice), the resulting phenotype is quite unexpected: mice are lean and remain so even on a high fat diet. They display a reduced metabolic efficiency and have increased energy expenditure but no change in locomotor activity. Ian Willis (Albert Einstein College of Medicine, NY, USA) reported that in these mice, an energy sink is provided by constantly active Pol III transcription in all tested tissues leading to an energy-consuming, futile RNA cycle [START_REF] Willis | Metabolic programming a lean phenotype by deregulation of RNA polymerase III[END_REF]. To feed the energy demand, mice exhibit high levels of lipid and protein catabolism. Metabolite profiling revealed that the MAF1 KO impacts central metabolism in the mouse liver. They display an increased activity of two pathways linked to the TCA cycle: the urea cycle and the purine nucleotide cycle. The metabolomic data also point to increased nucleotide synthesis and turnover. These data suggest that constitutively high Pol III transcription in MAF1 KO mice reprograms central metabolic pathways and wastes metabolic energy through a futile RNA cycle. SINE elements such as Alu or MIR are transposable elements, the transcription of which is assumed by RNA Pol III: their left arm encompasses A and B boxes driving Pol III transcription initiation. To ensure genome stability, these sequences have been domesticated by epigenetic silencing mechanisms involving DNA and histone methylation. However, they are found expressed at higher levels when cells are exposed to stress and notably in cancer cells. Davide Carnevali from the group of Giorgio Dieci (University of Parma, Italy) developed a bioinformatic pipeline [START_REF] Carnevali | Identification of RNA Polymerase III-Transcribed SINEs at Single-Locus Resolution from RNA Sequencing Data[END_REF] to identify genuine Pol III-derived SINE RNAs expressed at single-locus resolution and distinguish them from those that are merely 'passengers' of longer Pol II-transcribed genes. Using RNA-seq libraries derived from normal or Acute Myeloid Leukemia (AML) affected patient cell lines, they determined whether Alu expression is deregulated in AML. The group observed a very significant increase in the expression of many Alu elements in these cancer cells whose expression pattern seems to correlate with tumour subtype, opening the possibility to use them as a molecular biomarker.

Acknowledgements:

This meeting was organized with the help of Célia Plisson-Chastang, Marta Kwapisz, Christophe Dez and Julio Saez-Vasquez. For financial support, the organizers are grateful to the Centre National de la Recherche Scientifique, Centre de Biologie Intégrative de Toulouse, Université de Toulouse, Région Occitanie, and Promega company. The conference was organized in the context of the Euro Science Open Forum 2018. Special thanks to Christine Baillat and Manon Blanc (CBI) for financial administration; the Museum external conference manager, Josiane Beltran for her devoted implication; David Villa (CBI) for art work and "Mirabili'Art" association for giving a flavour of Toulouse culture.