Weixuan Xiao

Peng Du

Dong Ding

Abdellatif Ouahsine
email: ouahsine@utc.fr

CFD/DEM coupled approach for the stability of caisson-type breakwater subjected to violent wave impact

Keywords: Breakwater stability, Violent wave impacts, Fluid-Porous model, CFD/DEM coupling

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Introduction

Breakwaters are used for the protection of harbors and beaches against wave action. Their failure may be caused by the motion of the caissons and the global instability of the rubble mound [START_REF] Oumeraci | Review and analysis of vertical breakwater failures-lessons learned[END_REF]; [START_REF] Takahashi | Typical failures of composite breakwaters in Japan[END_REF]). For the motion of caissons, the most common forms are sliding, subsidence, and overturning [START_REF] Takahashi | Stability of caisson-type breakwater foundation under tsunami-induced seepage[END_REF]), which may induce the movement of armour units to increase the failure 5 of the breakwater.

Elsewhere, breakwater failures have been investigated by several empirical studies [START_REF] Cuomo | Breaking wave loads at vertical seawalls and breakwaters[END_REF]; [START_REF] Doan | Probabilistic risk evaluation for overall stability of composite caisson breakwaters in Korea[END_REF]) and by some numerical and experimental methods [START_REF] Elchahal | The effects of reflection coefficient of the harbour sidewall on the performance of floating breakwaters[END_REF]; [START_REF] Hofland | Large scale wave impacts on a vertical wall[END_REF]). For the computational techniques, the computational fluid dynamics (CFD) method is the most common tool used to describe the wave impacts. [START_REF] Kocaman | Investigation of dam-break induced shock waves impact on a vertical wall[END_REF] discussed the impact 10 of dam-break induced shock waves on a vertical wall by a CFD RANS-VOF solution. The simulation results

show that the impact of dam-break flood waves on the vertical wall causes wave reflection against the wall and the occurrence of a negative wave. The impacts of waves on the vertical wall were also investigated by [START_REF] Liu | CFD simulations of violent breaking wave impacts on a vertical wall using a two-phase compressible solver[END_REF] using a two-phase compressible CFD solver. Recently, many researchers made a series of attempts to couple the fluid and solid models. Discrete Element Method (DEM) is usually used to calculate the movement of armour units. A wave-structure interaction method was proposed by [START_REF] Latham | Coupled FEM DEM/Fluids for coastal engineers with special reference to armour stability and breakage[END_REF].

In this method, the forces and the volume fraction from the CFD model are mapped onto the DEM structure. [START_REF] Ren | SPH-DEM modeling of the hydraulic stability of 2D blocks on a slope[END_REF] used a coupled SPH-DEM method to investigate the stability of armour units in rubblemound breakwaters. An interfacial force-balance condition achieved the coupling between the fluid particle (SPH) and the solid spheres (DEM).

Due to the complexity of the breakwater structure, the porous medium should also be considered. Traditionally, Darcy's law and Forchheimer law were used to investigate linear and non-linear structures for porous armour layer [START_REF] Whitaker | The Forchheimer equation: a theoretical development[END_REF]). [START_REF] Hsu | A numerical model for wave motions and turbulence flows in front of a composite breakwater[END_REF] proposed a mathematical model based on the Volume-Averaged Reynolds-Averaged Navier-Stokes (RANS) equations that coupled the fluid and the porous medium to describe surface wave motions in the vicinity of a coastal structure. In this model, the Forchheimer law was adds by the volume-averaged process. Additionally, this fluid and porous medium coupled model are also developed in OpenFoam [START_REF] Higuera | Three-dimensional interaction of waves and porous coastal structures using OpenFOAM®. part i: Formulation and validation[END_REF]; Alcérreca-Huerta and [START_REF] Alcérreca-Huerta | Wave-induced pressures in porous bonded revetments. part i: Pressures on the revetment[END_REF]; [START_REF] Liang | A three-dimensional model for the seabed response induced by waves in conjunction with currents in the vicinity of an offshore pipeline using openfoam[END_REF]; [START_REF] Guler | Numerical assessment of tsunami attack on a rubble mound breakwater using openfoam®[END_REF]), an extended Forchheimer law which adds a mass term that accelerates a certain amount of water considered.

In this paper, we present a triple-coupled Fluid-Porous-Solid model. The fluid model is described by the Volume-Averaged Reynolds-Averaged Navier-Stokes (VARANS) equations in which the extended Forchheimer law used to calculate the porous medium flow is added to the inertia terms [START_REF] Hsu | A numerical model for wave motions and turbulence flows in front of a composite breakwater[END_REF]).

The solid model, which is based on the Discontinuous Deformation Analysis (DDA) method-a kind of DEM method, is used to compute the movement of the caisson and armour units [START_REF] Shi | Discontinuous deformation analysis: a new numerical model for the statics and dynamics of deformable block structures[END_REF]). This method enables to take into account the shapes of armour units, as well as the contact between blocks. The coupling between the fluid and the solid is carried out by a transmission strategy of the fluid mesh nodes' pressure towards the solid polygon vertices, while the coupling between the fluid and the porous medium consists of the equations of the porous model in the terms of the inertia of the fluid model. The structure of this paper is as follows: In section 2, equations of motion for the fluid and for the solid are introduced. The governing equations of the fluid-porous coupled model and of the DDA method are detailed. A transmission strategy is also given to couple the fluid and solid model. Section 3 is devoted to the validation of fluid and solid models. Simulation results of the flow patterns around the breakwater and the stability of armour units are presented in section 4. Section 5 concludes the paper.

Mathematical formulation of the problem

A schematic illustration of the studied problem is shown in Fig. 1. In order to investigate the problem, a numerical approach that couples the Fluid-porous model and the Solid-DDA model is proposed, wherein the fluid model is based on the Volume-Averaged RANS equations, the solid model is based on the DDA method, and the porous model is described by the non-linear Forchheimer model. The governing equations of the models are shown below:

∇ U = 0
(1)

∂ρU ∂t + ∇ (ρU U ) = -∇P + g X∇ρ + ∇µ ef f ∇U + σκ∇α (2)
where U is the velocity vector, X is the Cartesian position vector, g denotes the gravitational acceleration vector, and ρ represents the weighted averaged density. The term µ ef f = µ + µ t , where µ is the weighted average dynamic viscosity and the µ t is the dynamic turbulence viscosity calculated by k -model. σκ∇α signifies the surface tension effects, where σ is the surface tension, α is the fluid volume fraction, and

κ = ∇ α |α| .

Extended Forchheimer equations for porous medium

Darcy's law has been traditionally used for describing the transport properties of porous media; however, as the flow velocity increases, Darcy's law became inapplicable as the relationship between pressure and velocity becomes non-linear. A correction term (see the second term in Eq.( 3)), based on a quadratic velocity, was added by Forchheimer [START_REF] Whitaker | The Forchheimer equation: a theoretical development[END_REF]) to take this non-linearity into account. Furthermore, in the present study, an added mass term C A [START_REF] Higuera | Three-dimensional interaction of waves and porous coastal structures using OpenFOAM®. part i: Formulation and validation[END_REF]) was considered. To accelerate the same volume of water in a porous medium, additional momentum is required [START_REF] Mèndez | Mean magnitudes induced by regular waves in permeable submerged breakwaters[END_REF]). The extended Forchheimer equation can be written as:

P porous = AU + BU |U | + ρ n C A ∂U ∂t (3)
where P porous is the porous medium pressure, and the coefficients A and B are [START_REF] Van Gent | Wave interaction with permeable coastal structures[END_REF][START_REF] Higuera | Three-dimensional interaction of waves and porous coastal structures using OpenFOAM®. part i: Formulation and validation[END_REF]) :

A = µ 1 (1 -n) 3 n 2 µ D 2 50 (4) B = µ 2 (1 + 7.5 K C ) 1 -n n 2 ρ D 50 (5)
where D 50 is the mean diameter of the porous material. µ 1 and µ 2 are empirical coefficients related to the linear and nonlinear drag force, respectively. K C = ToU M nD50 is the Keulegan-Carpenter number where U M is the maximum oscillatory velocity. T o is the period of the oscillation [START_REF] Van Gent | Wave interaction with permeable coastal structures[END_REF]).

Volume-averaged RANS equation for coupled fluid-porous

In order to correspond the extended Forchheimer Eq.( 3) with the RANS equations, we introduce the volume-averaged velocity U that exists in the interstices of the solid framework of the porous medium, given by:

U = 1 V V f U dV (6)
where U is the hydrodynamic velocity with respect to the fluid, V is the total volume, and V f is the part of V which is occupied by the fluid. In what follows, we set the porosity n = V f V . Then, by substituting volume-averaged velocity Eq.( 6) in the RANS Eqs. (1-2), we obtain the following VARANS equations:

∇ U = 0 (7) (1 + C A ) ∂ ∂t ρ U n + ρ n 2 U ∇ U = -∇P + gX ∇ρ + 1 n ∇µ ef f ∇ U + σκ∇α -A U n -B| U n | U n (8)
where C A = 0.34 denotes the added mass coefficient, which is kept constant in the present study [START_REF] Higuera | Three-dimensional interaction of waves and porous coastal structures using OpenFOAM®. part i: Formulation and validation[END_REF]).

VOF method and numerical procedure

The computational model is built with a two-phase model (Fluid-air) and the free surface that separates these two phases can cut a cell of the computational mesh into two unequal parts. Each part contains a quantity of each phase. In what follows, the Volume-of-Fluid (VOF) method is used to describe the volume fraction of the fluid inside each computational cell which will be transmitted to VARANS. It is based on the following transport equation [START_REF] Hirt | Volume of fluid (VOF) method for the dynamics of free boundaries[END_REF]):

∂α ∂t + ∇ • αU + ∇ • U c α(1 -α) = 0 (9)
where α is the fluid phase fraction laying between 0 and 1, where α=0 corresponds to full of air and α=1 corresponds to full of fluid. However, in order to obtain physical results, there must be some limitations:

a clear interface must be maintained, and α must be limited between 0 and 1. [START_REF] Weller | Derivation, modelling and solution of the conditionally averaged two-phase flow equations[END_REF] added an artificial compression term ∇ • α(1 -α). This method only uses non-zero values at the interface. In addition,

|U c | = [min(C α |U |), max(|U |)],
where the factor C α can be specified. If U c is normal to the interface, the fluid will not be compressed, which points to a larger value of α and therefore from the air to the water phase.

The boundedness of this equation is achieved by the specially designed solver MULES (multi-dimensional universal limiter for explicit solution).

In the present study, the VARANS models are solved by using the PIMPLE algorithm (pressure implicit with the splitting of operators) to compute the pressure and SIMPLE (semi-implicit method for pressurelinked equations) algorithm to obtain the velocity fields. Its main structure is inherited from the original PISO but allows equations to be relaxed to ensure the convergence of all equations at each time step [START_REF] Higuera | Three-dimensional interaction of waves and porous coastal structures using OpenFOAM®. part i: Formulation and validation[END_REF]; [START_REF] Cai | Moving immersed boundary method[END_REF]).

Governing equations of the solid model

In order to investigate the movement and the stability of the caisson and armour units, the Discontinuous Deformation Analysis (DDA) method is used. In this method, the displacement at any point (x, y) of a block i can be described by its first-order approximation [START_REF] Shi | Discontinuous deformation analysis: a new numerical model for the statics and dynamics of deformable block structures[END_REF]):

χ i =   u i v i   = T i D i ( 10
)
where

T i =   1 0 -(y -y 0 ) (x -x 0 ) 0 (y -y 0 )/2 0 1 (x -x 0 0 (y -y 0 ) (x -x 0 )/2
  is the transformation matrix, and D i = (u 0 v 0 γ 0 ε xx ε yy ε xy ) T is the displacement matrix which is associated with six variables. u 0 and v 0 are the translations of the block gravity center in x and y directions, γ 0 is the rotation around (x 0 ,y 0 ), and ε xx , ε yy and ε xy are the normal and shear strains.

Based on the relationship between the strain and the displacement for block i, we have:

ε =      ∂ ∂x 0 0 ∂ ∂y 1 2 ∂ ∂y 1 2 ∂ ∂x      u v = Lχ ( 11
)
where L is the differential operator matrix for a 2D problem.

Here, we assume that the deformation is elastic and linear. Therefore, we get:

ε i = LT i D i = BD i ( 12
)
σ i = Eε i = EBD i ( 13
)
where E is the elastic matrix [START_REF] Shi | Discontinuous deformation analysis: a new numerical model for the statics and dynamics of deformable block structures[END_REF]) and

B =      0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1      .
The total potential energy Π p of the block i which is the summation of all the potential energy sources, which includes the elastic strain energy, initial stress potential energy, body force potential energy, and inertial energy, is given by:

Π p = Ai 1 2 ε T i σ i dA i + Ai ε T i σ 0 dA i - Ai χ T i f b dA i + Ai χ T i mT i Di dA i (14)
where f b is the body forces applied on a block i, m is the block mass per unit area and σ 0 is the initial stress.

Substituting Eq. ( 10),( 12) and ( 13) into Eq. ( 14), the total potential energy of a system of N blocks is expressed as:

Π p = N i=1 D T i M Di + 1 2 D T i KD i -D T i F (15)
where M = Ai mT T i T i dA i is the mass matrix, K= Ai B T EBdA i is the stiffness matrix, and F = Ai (T T i f b -B T σ 0 )dA i is the external forces matrix. By minimizing the potential energy, we obtain:

∂Π p ∂D T i = 0 ⇒ M D + KD = F (16)
Then, the displacement and the velocity in Eq. ( 16) can be approximated by the Newmark-β method:

D n+1 = D n + ∆t Ḋn + ∆t 2 2 (1 -2β 1 ) Dn + 2β 1 Dn+1 Ḋn+1 = Ḋn + ∆t (1 -β 2 ) Dn + β 2 Dn+1 (17)
where D and Ḋ are the acceleration and velocity matrices, respectively, and β 1 = 1/2 and β 2 = 1 for the implicit scheme [START_REF] Ding | Numerical study of ballast-flight caused by dropping snow/ice blocks in high-speed railways using Discontinuous Deformation Analysis (DDA)[END_REF]. Substituting Eq. ( 17) into Eq. ( 16)), we obtain:

(K + 2M ∆t 2 )D n+1 = F + 2M ∆t Ḋn (18)
And then the compact form is given by:

KD = F (19)
Consequently, we get the global matrix:

           K11 K12 K13 . . . K1n K21 K22 K23 . . . K2n K31 K32 K33 . . . K3n . . . . . . . . . . . . . . . Kn1 Kn2 Kn3 . . . Knn                       D 1 D 2 D 3 . . . D n            =            F1 F2 F3 . . . Fn            (20)
where Fi and D i are the sub-matrices of force and displacement, which are 6 × 1 sub-matrices. The Kij are 6 × 6 sub-matrices. Sub-matrices Kij (i = j) are determined by the block material properties, whereas Kij (i = j) are related to the contacts between blocks.

It should be noted that when the contact between the blocks takes place, the associated potential energy must be added to the global equilibrium equation (Eq. ( 15)), and then the associated contact sub-matrices have to be added to the global matrix Eq. ( 20). In the present study, the surface contact constraints of blocks are enforced by the penalty method. Taking the contact of two blocks i and j as an example, where the penetration distance of point P 1 into edge P 2 P 3 of block j is δ. Using the penalty method is equivalent to placing a spring between the two blocks, as shown in Fig. 2. The strain energy of the contact spring reads:

Π c = 1 2 P δ 2 ( 21
)
where P is the coefficient of penalty which can vary between 10E and 1000E, where E denotes Young's modulus [START_REF] Shi | Discontinuous deformation analysis: a new numerical model for the statics and dynamics of deformable block structures[END_REF]). The block kinematics of the DDA method are obtained by the open-close iteration which is adding or removing a stiff spring in each time step at the contact position where tension and interpenetration do not exist.

Coupling procedure of Fluid-Solid interaction

The coupling between the fluid and the solid models requires a transfer of data from one model to another.

In our case, the fluid model uses meshes while the solid model is meshless (see Fig. 3 23)).

It is specified that in this fluid-solid coupling, the transfer is retroactive, where the fluid transfers to the solid the pressure force, a force considered external to the solid.

f i = p i d 2 ; f j = p j d 2 ( 22
)
where p i and p j are the fluid pressures at nodes i and j, and d is the distance of the finite element edge from node i to node j. Finally, the global forces (F 1 , F 2 ) acting on the DDA block can be written as:

F 1x = n i=1 f i n x ( d 2 d 1 + d 2 ), F 1y = n i=1 f i n z ( d 2 d 1 + d 2 ) F 2x = n i=1 f i n x ( d 1 d 1 + d 2 ), F 2y = n i=1 f i n z ( d 1 d 1 + d 2 ) ( 23
)
where n is the total number of nodes in the edge, d 1 and d 2 are the distances from the given node to vertices 1 and 2, and n x and n y represent the directions along x and y respectively.

A flowchart of procedures and connections between the coupled fluid-porous-solid models, including boundaries, inputs and outputs of each model, has been created as previously described, and is shown in Fig. 4.

Numerical model and validation

The numerical simulations were performed to analyze the flow evolution and to characterize the forces on the breakwater subjected to violent wave impacts. The breakwater is composed of a vertical caisson with shoreward armour units and a porous seaward armour layer (see Fig. 5). The hydrodynamic impact is generated by the solitary wave, with a wave height (h w =6 m) and a water depth d w =10 m. The caisson position x/h c = 15.96 (where h c =13 m is the height of the caisson) [START_REF] Fang | A central numerical scheme to 1D Green-Naghdi wave equations[END_REF]; Martin-Medina et al.

(2018)). It is worth noting that the armour layer protects the structure from the attacks of incident waves by dissipating their energy. These armour layers are often built of large armour units and can be considered as a porous medium. Indeed, many recent scientific studies have considered the porosity parameter to represent different types of armour units in the CFD simulations. [START_REF] Van Gent | Wave interaction with permeable coastal structures[END_REF] proposed a value laying between 0.25 to 0.4 for cube-shaped units and suggested a porosity value of 0.49 for tetrapod-shaped units. An armour layer thickness of 2 m and a porosity value of 0.49 are used in the following simulations. The location of the gauges is also identified in Fig. 5, numbered from 1 to 5, to monitor the surface and the velocity of waves.

Boundary conditions

• In the input computational domain, the free surface elevation η is given by:

η = h w sech 2 3h w 4d 3 w X + ψ (24)
where X = (x -ct), h w is the wave height, d w is the water depth, ψ is the wave phase shift, and the wave celerity c is g(h w + d w ). The horizontal and vertical velocity components U h and U v verify [START_REF] Lee | Measurmment of velocities in solitary waves[END_REF]):

U h = η √ gd w d w 1 - η 4d w + d 2 w 3η 1 - 3y 2 2d 2 w d 2 η dX 2 (25)
and

U v = -y √ gd w d w 1 - η 2d w dη dX + 1 3 d 2 w 1 - y 2 2d 2 w d 3 η dX 3 (26)
where x and y are the horizontal and the vertical space variables. Here, we assume that the boundary is at x= 0, and the free surface will start from the highest point of the solitary wave.

• At the top boundary, we assume p = p atmospheric ; at the bottom boundary, we set a slip condition,

where the roughness can be assumed to be negligible.

Mesh and time step convergence

The validation of the fluid model is carried out by comparing the calculated horizontal velocity and water height with the theoretical free surface elevation η at transect P1 (see Fig. 5) deduced from Eq. ( 24). Three types of meshes are used for mesh convergence validation, which contain 24510, 48000 and 94080 tetrahedral mesh elements, respectively. A refinement ratio √ 2 is used for the mesh validation (Ji et al. (2014b)). The grid spacing of the coarser meshes (∆x c , ∆y c ) to grid spacing of the finer meshes (∆x f , ∆y f ) is given as:

r g = ∆x c ∆x f = ∆y c ∆y f = √ 2 (27)
Fig. 6(1) shows that the fine and medium meshes have similar results and give better than coarse mesh.

Considering both the accuracy and the simulation time, the medium mesh is used for the following studies.

Three time steps are designed. Fig. 6 (2) shows that the results of the three time step cases are all reasonable.

The accuracy of the case (∆t=0.1 s) is clearly enough; therefore, we use time steps of 0.1 s in this work.

The refinement of the space or time steps can improve the quality of the results; however this is not enough.

Solid model validation

The validation of the solid model is carried out by comparing the DDA numerical results with both the experimental and the numerical results given by [START_REF] Komodromos | Simulation of the response of ancient columns under harmonic and earthquake excitations[END_REF]. Thus, the behavior of seven 48 mm × 48 mm × 29 mm blocks with a mass of 135.5 g under harmonic excitations (at the base) is simulated by DDA wherein we assume that the contact stiffness is 10 7 N/m and friction angle φ is 34 o , and the peak ground accelerations is 2.15 m/s 2 . For four different frequencies, the computed results are compared with the experimental observations. The DDA simulations show similar responses to experimental observations, as shown in Fig. 7. Furthermore, compared to the DEM results, the DDA results have higher accuracy.

the DDA model. The results from the experimental have a great agreement with the numerical simulations, as shown in Fig. 8. Therefore, the DDA can be used to investigate the displacement of the vertical caisson and the discrete rear structures.

Simulation results

Flow patterns around the breakwater

The simulation of the flow patterns around the breakwater was done by considering the fluid-porous coupling. The wave evolution around the breakwater presented in Fig. 9 shows that as the wave approaches the vertical caisson grows considerably in amplitude until the breaking process occurs (Fig. 9(b)). This then leads to an up-moving jet due to the wave-squeezing processes (Fig. 9(c)).

The distribution of the wave pressure in front of the vertical caisson is shown in Fig. 10. The normal pressure due to solitary waves is mainly located between 20 s and 25 s. The results show that the fluid pressure is relatively high in the lower half of the caisson, which can induce a risk of sliding of the structure or even give rise to a liftforce which could accentuate the overturning process since the maximum pressure is located at the caisson's toe (see point A, Fig. 10).

At the vicinity of the breakwater, the height of the waves (expressed as H=η + d w ) increases considerably as they approach the breakwater until the breaking process occurs, as shown in Fig. 11. The wave-breaking process happens after the impact between the wave and the caisson [START_REF] Ouahsine | Numerical study of coastal sandbar migration, by hydro-morphodynamical coupling[END_REF]). Three main reasons cause this wave-breaking process:(1) non-linear terms and secondary waves, (2) reflection on the porous medium and the caisson, (3) wave-wave interactions. We then define the maximum impacting wave height (IWH max ) as the peak of the incoming wave height after it impacts the caisson, as shown in Fig. 12. Due to the squeezing process of the breakwater, we observe that a larger S induces a bigger (IWH max ) but less water is ejected. Hence, the Impacting Wave Height (IWH max ) can be fitted through the following formula:

IW H max = α 1 + α 2 e α3S (28)
where α 1 =0.41, α 2 =1.45 and α 3 =0.5. Fitting results are shown in Fig. 13.

The seaward porous medium affects the wave. The turbulent kinetic energy K for various thickness of the porous layer is shown in Fig. 14. The large porous layer has less turbulent kinetic energy due to a large dissipation. Several turbulences can be found on the upper side of the caisson on the shoreward side, in the middle of caisson on the seaward side, as well as at the top and bottom of the porous medium. The flow in the porous medium initially coincides with the direction of the wave and then opposes it (Van der Meer et al. ( 2016)). The porous medium significantly reduces the effects of the breaking waves.

Solution behavior with the shape of shoreward armour units

In order to analyze the stability of the shoreward armour units and the influence of their shape, three typically shaped armour [START_REF] Kaidi | Stability of breakwaters under hydrodynamic loading using a coupled DDA/FEM approach[END_REF])(see Fig. 15) are modeled and placed on the shoreward side of the breakwater. The armour units are optimally arranged to ensure the initial position stability. Six units, numbered from 1 to 6, are shown in Fig. 16. The breakwater is then subjected to a violent solitary wave impact whose input external force is calculated by Eq. ( 23) ((see the flow in Figs. 9 and 10). The material parameters used for the Solid-DDA model are shown in Table 1. As for the displacement, the accropod and the tetrapod units followed the cubic units in that there is no significant drop; however, the rotation of the units can be seen. Because the two shaped units are under force, the armour units move to rearrange and the structure becomes more stable, with the tetrapod units.

The standard deviation (SD) of the displacement or the rotation of the armour units is used to describe the stability of the breakwater, which is calculated by:

SD = N i=1 (φ i -φ) 2 N ( 29
)
where φ i denotes the displacement or rotation value of units i at the final position, φ is the average displacement or rotation value, and N = 6. The standard deviations of the horizontal displacement, the vertical displacement, and the rotation are shown in Fig. 17. For the displacement of the armour units, we have obtained SD cube > SD accropod > SD tetrapod . For the rotation, however, the accropod units are greater than the cubic and tetrapod units. Furthermore, the cubic units have larger displacements, most of which are in the form of sliding along the slope.

Based on the simulation results, the tetrapod units are the most stable, followed by the accropods, and then by the cubic armour units. The reason is that cubic armour units bring resistance to the breakwater through mass whereas tetrapod and accropod blocks bring resistance through mass and interlock forces among units.

Effect of Cohesion

In this simulation, a cohesion C f =2 M P a and a tensile strength σ t = 0.7M P a are imposed, which may actually represent concrete placed on the surfaces and contact points. [START_REF] Kaidi | Stability of breakwaters under hydrodynamic loading using a coupled DDA/FEM approach[END_REF]). These values are obtained by adopting the extended Mohr-Coulomb failure criterion based on the tension cut-off. It consists in reducing the tensile strength of the material by imposing a value of σ t while ensuring the following condition:

σ t ≤ Ccotanφ. Thus, by fixing σ t = 0.7M P a, we decrease the value of C to C f , which corresponds to the movement of one block of the caisson. C f is considered optimal.

Fig. 18 shows that the displacement of the armour units without cohesion is bigger than the units with cohesion. All the units behind the caisson have been moved but the units No. formula has been established for this purpose. Moreover, the results also show that the shape of the armour units is a major factor to be taken into account in the study of the stability of the structure. In particular, they show that tetrapod-shaped units are the most stable, followed by acropod-shaped units and then by cubic-shaped units, and that cohesion enforces the stability of the breakwater.

Figure 1 :

 1 Figure 1: Schematic illustration of the studied problem

Figure 2 :

 2 Figure 2: Interaction between two contacting blocks

  (a)) since it is based on the DDA method. Therefore, an adequate data transfer strategy is required in order to faithfully reproduce the studied physical phenomena (Kaidi et al. (2012)). It consists in converting the nodal pressure forces of the fluid (f i , f j ) into equivalent forces F i at the vertex points of the solid (see Fig. 3 (b)-(c) and Eq. (

Figure 3 :

 3 Figure 3: Fluid/solid interface:(a) fluid finite element mesh and solid; (b) fluid pressures in the mesh points (c) conversion of fluid pressures into equivalent vertex forces

Figure 5 :

 5 Figure 5: Schematic illustration of the computational domain. Wave type: solitary wave; wave height hw=6 m; water depth dw = 10m; armour layer thickness Ta=2 m; breakwater caisson height hc=13 m; caisson position: x/hc=15.96. The precise location of gauges P1-P5 is 2hc, 11hc, 13hc, 14hc and 15.5hc (m), respectively.

  It is therefore necessary to use digital diagrams or models, such as Serre (Do Carmo et al. (2018, 2019)) or Extended Boussinesq-like models (Ouahsine et al. (2013)), which have good dispersive capacities for accurate prediction of wave activity.

Figure 6 :

 6 Figure 6: (a) Horizontal velocity and free surface elevation of mesh, and (b) time step convergence at gauge P1 (see Fig.Fig. 5 for the wave parameters and gauges location).

Figure 7 :

 7 Figure 7: Comparison between the experimental results (Komodromos et al. (2008)) and the numerical results computed by DEM and DDA. The dimension of one block is 48 mm ×48 mm ×29 mm, the mass is 135.5 g, the contact stiffness is 10 7 N/m, the friction angle is φ = 34 o , and the peak ground accelerations is 2.15 m/s 2 .

Figure 8 :

 8 Figure 8: Comparison between the experimental values of a initiate (Komodromos et al. (2008)) and DDA results.

Figure 9 :

 9 Figure 9: Pattern changes: (a) t=19.5 s; (b) t=20.5 s; (c) t=21.5 s. The impacting wave is a solitary wave with a wave height 6 m and a water height 10 m. The caisson height is 13 m. The thickness of the porous medium (porosity = 0.49) is 2 m. The slope is 1:2.

Figure 10 :

 10 Figure 10: Pressure distribution along the vertical caisson. The impacting wave is a solitary wave with a wave height 6 m and a water height 10 m. Point A is located at the bottom of the caisson. Points B-D are 1 3 hc, 2 3 hc, and hc away from point A.

Figure 11 :

 11 Figure 11: Water height evaluation at the vicinity of breakwater (see Fig.5)

Figure 12 :

 12 Figure 12: Surface wave deviation with various seaward slopes (1:S). For the wave parameters, see Fig. 5. The IWHmax of the four slopes are: (a) 1.86 m,(b) 3.71 m,(c) 5.57 m and (d) 9.28 m, respectively.

Figure 13 :

 13 Figure 13: Variation of IWHmax (see Fig. 12) for various seaward slopes.

Figure 14 :

 14 Figure 14: Turbulent kinetic energy K for various porous layer thickness in t=24.0 s. (see Fig. 5 for the wave parameters).

Figure 15 :

 15 Figure 15: Shapes of armour units used in the simulation.

Fig. 16

 16 Fig.16shows the initial and the final position of the three various shapes of armour units. The displacement of the caisson hindered by armour units is almost zero and stable while varying degrees of sliding and rotating happen on the armour units. The cubic units are more unstable than the other two; in fact, cubic unit No.2 experienced a significant fall. We found that units No.2 and No.3 have larger vertical displacement while units No.3 and No.6 have large horizontal displacement. Therefore, a falling process always happens on the rear side of the caisson and a sliding process happens on the toe of the breakwater structure. As for the displacement, the accropod and the tetrapod units followed the cubic units in that

Figure 16 :

 16 Figure 16: Simulated movement for various shapes of armour units: (a) Cube, (b) Accropod, (c) Tetrapod. The breakwater was subjected to solitary wave impacts whose input external force is calculated by Eq. (23). (See Fig. 5 for the wave parameters).

Figure 17 :

 17 Figure 17: Variation of standard deviations for three shapes of armour units.

  Fig. 19. For unit No.2, when cohesion forces work, the steady-state can be reached in about 22.0 s and the maximum displacement is 1.2 m which is far less than 1.90 m without cohesion. For unit No.5, it reachessteady-state at 22.0 s (with cohesion) and 24.0 s (without cohesion). In total, the displacement of the two units was lower by 36.8% and 23.8% respectively, as compared to when cohesion is absent. Therefore, we conclude that cohesion enforces the stability of the breakwater.

Figure 18 :

 18 Figure 18: Comparison of the motion of cubic units at final position: (a) cohesion=0 Pa, tensile strength σt = 0.7M P a; (b) cohesion= 2 MPa, tensile strength σt = 0.7M P a; (c) Comparison of the effect of cohesion. The breakwater was subjected to solitary wave impacts whose external force is calculated by Eq. (23) (See Fig. 5 for the wave parameters).

  This paper presents a fluid-porous-solid triple coupled model to simulate the stability of a caissontype breakwater. The fluid model is described by the Volume-Averaged Reynolds-Averaged Navier-Stokes (VARANS) equations in which the nonlinear Forchheimer equations for the porous medium are implemented as the terms of inertia. The solid model is based on the DDA method to take the discrete behavior of armor units into account. The coupling between the fluid and the solid is carried out by a transmission strategy of the fluid mesh nodes' pressure towards the solid polygon vertices. The results of the numerical simulation show that the porosity and the thickness of the porous layer have a significant influence on the distribution of the kinetic energy of turbulence (TKE) around the structure of the breakwater. The greater the thickness, the lower the intensity of the TKE. Indeed, the porous layer, located just in front of the caisson, acts as a support structure that effectively dissipates and absorbs the turbulent kinetic energy of the impacting waves. It reduces the growth of wave crests and the overtopping

Figure 19 :

 19 Figure 19: Displacement of cubic armour units 2 and 5 with and without cohesion

Table 1 :

 1 Material parameters used for simulations

	50 GP a	0.30	2400 kg/m 3	2×10 8 N/m	0.6

Young's modulus E Poisson's ratio ρ unit weight m Penalty spring constant P Coefficient of friction µ

Acknowledgments

The authors wish to thank the financial support from the China Scholarship Council (CSC).


Xavier Ohl
email: xohl@chu-reims.fr

Renaud Siboni

Xavier Pr

Ohl

Surgical treatment of terrible triad of the elbow

Keywords: elbow, dislocation, terrible triad, radial head, coronoid process

Terrible triad of the elbow associates posterior dislocation, radial head fracture and coronoid process fracture. It is a complex trauma, associating severe bone and ligament lesions, with high more-or-less long-term risk of residual instability, stiffness, pain and osteoarthritis. During the last 20 years, pathologic, biomechanical and clinical knowledge has greatly progressed. Prevention of these severe complications requires initial understanding of the lesion mechanism and precise analysis of all lesions. Surgery aims to restore perfect stability by sequential anatomic repair, enabling early mobilization to prevent onset of stiffness.

Introduction

Terrible triad (TT) is a frequent and highly unstable form of fracture-dislocation of the elbow associating posterior dislocation of the elbow, radial head fracture and coronoid process fracture. Management needs to be well codified to avoid definitive complications, the most frequent of which are stiffness, especially in extension, and residual instability, especially posterolateral.

Treatment aims to restore perfect stability to allow rapid mobilization of the joint.

The present paper seeks to answer the following questions:

-Why is this triad so terrible, and how should lesions be assessed? -How should the radial head be managed? -Should the coronoid process be repaired, and if so how? -In what order should ligaments be repaired? -What is the role of external fixation? -What are the results of the various treatment options?

1. Why is this triad so terrible, and how should lesions be assessed?

Lesion mechanism

TT is a form of elbow dislocation associating posterior or posterolateral humeroulnar dislocation with radial head fracture and coronoid process fracture due to posterolateral trauma with the hand in pressure and the forearm in compression under the humerus. The dislocating force combines forearm rotation (supination) and valgus. Dislocation is simple if the rotation is enough for the radial head and coronoid process to be brought under the distal humerus; otherwise, impact on the distal humerus fractures the radial head and coronoid process [START_REF] Sanchez-Sotelo | Complex elbow instability: surgical management of elbow fracture dislocations[END_REF]. The first compartment to be affected is lateral, with lateral collateral ligament (LCL) lesion. Forces are then transmitted to the anterior and posterior capsule and then the medial collateral ligament (MCL) [START_REF] O'driscoll | Elbow subluxation and dislocation. A spectrum of instability[END_REF] (Figure 1).

The association impacts all elbow stabilizers: primary (LCL, MCL) and secondary (radial head and humeroradial joint, and anterior capsule fixed on the coronoid process).

Initial treatment

Complete clinical examination screens for cutaneous and neurovascular complications. Standard AP and lateral X-ray confirms diagnosis and detects any bone fragments from the radial head, coronoid process or medial or lateral epicondyle. Elbow CT should be performed after reduction to visualize fragments and plan treatment. The wrist should be systematically examined to screen for pain and/or distal radioulnar joint deformity, and comparative bilateral radiographs should screen for Essex-Lopresti syndrome.

Reduction uses external maneuvers under fluoroscopy, with the patient under anesthesia in the operating room. General anesthesia is to be preferred, to obtain complete muscle relaxation and as operative time can be hard to predict, and allows postoperative neurologic surveillance, unlike peripheral nerve block.

When reduction has been achieved, elbow stability can be assessed, notably in extension. In some rare cases of TT with non-displaced radial head fracture and fracture of the tip of the coronoid, non-operative treatment may be indicated if the elbow is stable on testing [START_REF] Mazhar | Evaluation of functional outcome after nonsurgical management of terrible triad injuries of the elbow[END_REF]. If on the other hand the joint dislocates in 45° extension, it needs to be stabilized. "Too good" a joint line on post-reduction radiographs also suggests possible residual instability.

Surgical strategy is codified. It should begin with a lateral approach to expose and detail lesions "from outside inward" (LCL, radial head, coronoid process), then repair them "from inside outward". Some authors recommend a single posterior approach, giving simultaneous access to both compartments.

Lateral approach

The patient is positioned supine, with the operated limb on a table and a pneumatic tourniquet. We use the lateral Kocher approach rather than Cadenat posterolateral approaches, as there is generally no need for downward extension, which would risk damaging the deep branch of the radial nerve. The lateral approach provides better access to the coronoid process than do posterolateral approaches. The skin landmarks comprise the lateral epicondyle and the radial head (or neck), located under pronation-supination. All lateral approaches come close to the radial nerve, which has to be borne in mind throughout surgery; pronation tends to move the nerve and its branches out of the way.

Skin incision is vertical, about 5 cm above the lateral epicondyle, then curving at the lateral epicondyle and following the lateral side of the forearm down to the radial neck. The muscle fascia incision is between the carpal ulnar extensor and the anconeus muscle. In practice, the fascial tear induced by the trauma is often used (Figure 2). At this point, the hemarthrosis evacuates through the lateral capsule-ligament detachment, which should be extended down toward the annular ligament, which is then sectioned and located, to be repaired at end of surgery. Proximally, the proximal LCL is usually found to be detached and the lateral epicondyle to be bare (Figure 3). Intra-ligament tear and distal detachment are rarer.

After washing out the hemarthrosis, the humeroradial and humeroulnar joints can be explored for lesion assessment: although X-ray or CT investigate fractures, cartilage lesions, which are just as important, often need screening at this point, and are detailed in the surgery report.

Single posterior approach

This is an extensive approach, although medial compartment access is in fact seldom needed. We therefore do not use it in first line. It requires lateral decubitus positioning with arm in 90° flexion on a support.

To sum up:

-TT is a severe and complex elbow trauma involving both bone and ligament elements.

-Reduction is performed in theater, and post-reduction CT is necessary to analyze bone lesions. After reduction, surgery is indicated in case of instability in extension. -A single lateral approach is often sufficient.

How should the radial head be managed?

In non-displaced radial head fracture, non-operative treatment is indicated [START_REF] Mazhar | Evaluation of functional outcome after nonsurgical management of terrible triad injuries of the elbow[END_REF]. In displaced or comminuted fracture, treatment is surgical. Radial head resection is to be avoided, and the choice is between internal fixation or prosthesis. Comminution, patient age and bone quality are determining factors in this choice.

Internal fixation

Internal fixation is feasible if there are no more than 3 fragments and if there is bone continuity between part of the radial head and the neck. Simple screwing is the most common technique.

All radial head fragments must be located; if one is missing, it needs to be found under fluoroscopy. Once all have been found, temporary K-wire fixation can be performed. Definitive fixation uses intra-osseous screws and/or screws in compression (Figure 4). If fixation is insufficiently stable, low-profile anatomic plates specific to the radial head can be used, requiring a more extensive approach to the neck, with risk of injuring the posterior interosseous nerve. This plate needs to be positioned in the safe zone of the radial head and neck surface seen on the lateral approach with the forearm in neutral pronation (Figure 5).

In case of radial neck fracture and intact but impacted radial head, the technique of choice is retrograde pinning to raise the head.

Prosthesis

When internal fixation is not possible, resection is to be avoided due to risk of Essex-Lopresti syndrome and residual posterolateral instability [START_REF] Bonnevialle | Radial head replacement in adults with recent fractures[END_REF]. Radial head replacement is then needed, to reconstruct the humeroradial column. Various parameters have to be taken into account in choosing the type and positioning of the implant.

Type of implant

Choice is based on intra-prosthetic mobility, type of stem fixation, and implant modularity. a) In traumatology, a fixed head is preferred, restoring better stability than bipolar implants with intra-prosthetic mobility, as shown in-vitro [START_REF] Chanlalit | The biomechanical effect of prosthetic design on radiocapitellar stability in a terrible triad model[END_REF] and in-vivo [START_REF] Allavena | Outcomes of bipolar radial head prosthesis to treat complex radial head fractures in 22 patients with a mean follow-up of 50 months[END_REF]. b) There are several types of fixation: cementless, cemented, and "mobile" within the medullary canal. Cemented implants are interesting, but there is "no going back" in case of error in height. With uncemented implants, it is difficult to use a press-fit technique in small-diameter bone, due to risk of fracture. Some manufacturers provide a compression screw to stabilize the stem in the medullary canal. An unfixed stem "floating" in the canal theoretically allows implant alignment to be adapted in movement, but with risk of residual elbow pain [START_REF] Sanchez-Sotelo | Complex elbow instability: surgical management of elbow fracture dislocations[END_REF].

c) The main argument concerns implant modularity, enabling in-situ choice of the best combination of head, neck and stem to optimize stability, avoiding impingement and allowing optimal implant positioning.

Positioning

Good implant positioning determines efficacy and joint congruence. The relevant factors are head diameter, implant height and congruence on fluoroscopy. a) Head diameter is selected on a simple rule. Head fragments are collected and reassembled on the table to get a precise idea of native head diameter. Too small a prosthetic head gives too narrow a contact area with the capitulum and incomplete LCL tensioning. Too large a head incurs risks of "overstuffing" and poor congruence between the implant and the ulnar sigmoid cavity, excessive LCL tension, and postoperative stiffness. In case of hesitation between two sizes, the smaller should be chosen. b) Most present-day implants allow height to be adjusted independently of head diameter. Thus, dynamic tests and fluoroscopic control on a trial implant are essential. Two radiographic criteria help optimize positioning: implant head height should not exceed the inferior edge of the ulnar groove on AP view in extension; and the humeroulnar space should be medially and laterally symmetrical [START_REF] Bonnevialle | Radial head replacement in adults with recent fractures[END_REF]. Fluoroscopic views of the wrist should also be taken in case of suspected Essex-Lopresti syndrome. Recent but contradictory meta-analyses compared internal fixation versus head replacement in TT [START_REF] Chen | Replacement or repair of terrible triad of the elbow: A systematic review and meta-analysis[END_REF][START_REF] Kyriacou | Radial head replacement versus reconstruction for the treatment of the terrible triad injury of the elbow: a systematic review and meta-analysis[END_REF][START_REF] Sun | Comparison between radial head arthroplasty and open reduction and internal fixation in patients with radial head fractures (modified Mason type III and IV): a meta-analysis[END_REF]. At all events, all radial head internal fixation methods and a total radial head prosthesis need to be available in theater.

To sum up:

-Radial head resection is never indicated. Internal fixation or replacement is the rule, as it is essential to reconstruct the humeroradial column. -The gold standard in internal fixation is compression screwing.

-Radial head replacement has to be rigorous to avoid residual instability or overstuffing. -Wrist X-ray screening for Essex-Lopresti syndrome should be performed in case of the slightest doubt.

3. Should the coronoid process be repaired, and if so how?

Pathophysiology and classifications of coronoid process fracture

Coronoid process repair is presently the center of controversy. The coronoid process contributes to stability by contributing to bone surface congruence; anatomic studies showed that the elbow cannot be stable in case of fracture detaching >50% of the coronoid process surface. It is also the insertion site for the main (anterior) MCL bundle and also for the anterior capsule and the anterior brachial muscle, which also contribute to stability.

Coronoid process fracture classification progresses regularly. Regan and Morrey's was the first classification, based on standard AP views, in 3 grades: grade 1, coronoid tip; grade 2, <50%; and grade 3, >50% of the coronoid process [START_REF] Morrey | Fractures of the coronoid process of the ulna[END_REF] (Figure 6). O'Driscoll et al. proposed a new classification, based on 3D and CT analysis of the fracture line [START_REF] O'driscoll | Difficult elbow fractures: pearls and pitfalls[END_REF], distinguishing tip (type 1), anteromedial (type 2) and base (type 3) fracture (Figure 6).

New anatomic studies [START_REF] Jeon | The contribution of the coronoid and radial head to the stability of the elbow[END_REF] analyzed the contributions of the coronoid process and the radial head to stability. All agreed that base fracture (type 3) should be stabilized. In other cases, stabilization is necessary only if stability is not restored after radial head reconstruction and fixation of the lateral capsule-ligament complex. There are various means of stabilization, choice depending on type of coronoid fracture, approach, and type of radial head fracture.

When the radial head cannot be conserved

In this case, access to the coronoid process is facilitated by a lateral approach after resecting the radial neck. We often perform coronoid bone suture in type 1 or 2 fracture, using a metal anchor in the fracture site or, preferably, using posteroanterior bone tunnels with posterior lacing. In type 3 fracture with a large fragment, we prefer a cannulated screw under visual and fluoroscopic control.

Bone suture via bone tunnels can easily be performed on a lateral approach when the radial head is resected. The first step consists in lacing the coronoid tip with non-absorbable suture, taking care to have a sufficient length of each thread to be able to pass them through the tunnels and tie them behind. The tunnels can be performed with the help of a knee ligament reconstruction visor to control the exit into the fracture site. We prefer to drill two parallel 2-mm tunnels, then using straight suture-passers to bring out the two strands at the ulnar crest. The two strands are then held by forceps on stand-by, as definitive fixation of the coronoid process has to await radial head implantation. Coronoid process suturing is performed on the reduced elbow in 90° flexion.

Internal fixation by cannulated screw, posterior to anterior, is also quite simple. The screws have to be long enough and 3-4 mm in diameter. Reduction often uses a spatula or small pointed forceps while positioning 1 or 2 K-wires from posterior to anterior under fluoroscopic control. If K-wire positioning is correct, drilling and screw positioning can proceed. Using 2 cannulated screws is mechanically preferable, but seldom possible given coronoid fragment size.

When the radial head is conserved

After internal fixation of the radial head, the remaining unfractured part restricts coronoid process access. In that case, we do not systematically fix the coronoid process in Regan-Morrey types 1 and 2. Elbow testing after head fixation and lateral capsule-ligament suture guides any complementary coronoid process fixation, which in fact we find rarely necessary. In type 3, the coronoid process can be reduced on a lateral approach even when the radial head has been conserved, by screwing under fluoroscopy.

A second approach (pure anterior or medial) can be necessary for coronoid process fixation by direct screwing or using a dedicated anatomic plate. A medial approach [START_REF] Lor | Buttress plate fixation of coronoid process fractures via a medial approach[END_REF] involves prior location of the ulnar nerve and medial ligament repair in addition to coronoid process fixation. The anterior approach is more delicate technically, and allows only fixation of the coronoid process [START_REF] Reichel | Anterior approach for operative fixation of coronoid fractures in complex elbow instability[END_REF][START_REF] Yang | A novel anterior approach for the fixation of ulnar coronoid process fractures[END_REF] (Figures 7A and7B). Systematic fixation of the coronoid process is controversial [START_REF] Dyer | Terrible Triad Fracture-Dislocations of the Elbow[END_REF][START_REF] Giannicola | Terrible triad of the elbow: is it still a troublesome injury?[END_REF]. Papatheodorou et al. [START_REF] Papatheodorou | Terrible triad injuries of the elbow: does the coronoid always need to be fixed?[END_REF] reported excellent clinical and radiological results in 14 TTs treated by isolated fixation or replacement of the radial head and lateral suturing. The coronoid fractures were type 1 or 2, with no type 3s, and were never fixed. This report confirmed the in-vitro findings reported by Jeon et al. [START_REF] Jeon | The contribution of the coronoid and radial head to the stability of the elbow[END_REF], who recommended fixation essentially of type 3 fracture.

To sum up:

-Type 3 coronoid process fracture should be systematically fixed.

-In types 1 and 2, the choice depends on whether the radial head is conserved and on elbow stability after reconstructing the humeroradial column and lateral ligament.

In what order should ligaments be repaired?

In TT, the lateral ligament is always involved, whereas the medial ligament may be intact [START_REF] Sanchez-Sotelo | Complex elbow instability: surgical management of elbow fracture dislocations[END_REF]. Lateral repair is systematic and constitutes the final step in the lateral approach; medial repair is rarely necessary.

LCL repair

The LCL is usually detached from the lateral epicondyle, which is "bare", due to this proximal detachment. Sometimes the ligament is torn in its body, or, more rarely, detached distally, in which case direct suture or distal reinsertion on the ulna is required. The lateral ligament complex ensures stability in varus and posterolateral stability. For recent lesions, LCL repair is usually enough; reconstruction can be considered in case of posterolateral instability persisting some time after trauma.

The classic technique consists in bone suture or, more often, anchors fixed on the lateral epicondyle. This allows permanent control of suture tension and the elbow to be cycled ahead of definitive suture [START_REF] Mcguire | Medial and Lateral Collateral Ligament Repair or Reconstruction of the Elbow[END_REF]. We usually use 2 metal anchors impacted on the epicondylar crest (Figure 8). Mason-Allen sutures are then performed to fix the capsule-ligament and tendon on the lateral epicondyle (Figure 9). After complete reduction, the elbow is put in 90° flexion in neutral pronation-supination, and the knots are tightened. The annular ligament is then sutured with absorbable sutures.

MCL repair

In our experience, this step is rarely necessary, but it is important to know when and how it should be undertaken. Like for the LCL, the MCL is usually detached from its humeral insertion. In TT, it is by definition associated with coronoid process fracture.

If the elbow is still unstable in extension after coronoid fixation, head fixation or replacement and lateral ligament repair, the medial ligament has to be repaired. Isolated frontal instability in valgus, on the other hand, without instability in flexion-extension does not require MCL repair [START_REF] Chemama | Terrible triade du coude : description et prise en charge[END_REF].

The medial approach is centered on the medial epicondyle, to explore and release the ulnar nerve. The medial epicondyle is found to be bare, and MCL repair is performed like for the LCL, with non-absorbable anchors. If the coronoid process needs to be accessed, it is sufficient to release and retract the medial epicondylar and anterior brachial muscles forward to give access to the anteromedial capsule and the coronoid process.

A recent retrospective study comparing a single lateral approach and a double lateral plus medial approach [START_REF] Zhou | Comparison of a single approach versus double approaches for the treatment of terrible triad of elbow-A retrospective study[END_REF] found much better clinical results and fewer complications (notably concerning the ulnar nerve) in the former.

Postoperative course

Once stability has been restored, the usually recommended 6 weeks' cast immobilization is superfluous. We prescribe 10 days' long-arm contention (resin or made-to-measure rigid splint) then, after a radiological and clinical check-up on day 10, an articulated elbow orthosis allowing free pronation-supination, set for 2 weeks in 120° flexion and 45° extension; at 1 month, ranges of motion are increased. After 6 weeks, the orthosis is discarded after a radiographic check. The patients then perform self-rehabilitation associating flexion-supination and extension-pronation.

To sum up:

-Lateral ligament repair is fundamental, and should be systematic on a lateral approach. -Medial ligament repair should not be systematic, but reserved to cases of instability in extension persisting after lateral repair.

What is the role of external fixation?

If, after all lesions have been repaired, the elbow is still unstable in flexion-extension, an articulated (dynamic) external fixator may be considered [START_REF] Duparc | Prevention and management of early treatment failures in elbow injuries[END_REF]. In our experience, this is exceptional, and repair quality should be checked first. Some authors advocate first-line articulated external fixation ahead of medial repair [START_REF] Dyer | Terrible Triad Fracture-Dislocations of the Elbow[END_REF]. Whatever the case, transarticular ulnohumeral pinning is not indicated.

A recent multicenter prospective study specified the role of external fixation and the expected results [START_REF] Iordens | Good functional recovery of complex elbow dislocations treated with hinged external fixation: a multicenter prospective study[END_REF], in 27 patients (9 TTs) with external fixation following elbow reduction and lesion repair. Results were excellent, but the authors stressed the difficulty of achieving perfect joint congruence, essential for rapid mobilization.

To sum up:

-If, after repairing the various bone and ligament structures, the elbow is still unstable in extension, an articulated external fixator may be considered. -The technique has to be rigorous, to achieve good joint congruence.

-We propose a decision-tree for TT treatment after lesion assessment (Figure 10).

What are the results of the various treatment options?

6.1 What results can be expected?

The first clinical series reported catastrophic results, greatly contributing to the notoriety of TT [START_REF] Egol | Fracture-dislocation of the elbow functional outcome following treatment with a standardized protocol[END_REF][START_REF] Pugh | Standard surgical protocol to treat elbow dislocations with radial head and coronoid fractures[END_REF][START_REF] Ring | Posterior dislocation of the elbow with fractures of the radial head and coronoid[END_REF][START_REF] Winter | Surgical management of elbow dislocation associated with non-reparable fractures of the radial head[END_REF]. Associated complications include residual instability, stiffness, persistent pain and humeroulnar and/or humeroradial osteoarthritis [START_REF] Chen | Complications of treating terrible triad injury of the elbow: a systematic review[END_REF].

These poor early results were due to lack of knowledge of the pathology and biomechanical consequences of the lesions. Many biomechanical studies have since been conducted and standardized protocols have been designed [START_REF] O'driscoll | Difficult elbow fractures: pearls and pitfalls[END_REF][START_REF] Papatheodorou | Terrible triad injuries of the elbow: does the coronoid always need to be fixed?[END_REF][START_REF] Egol | Fracture-dislocation of the elbow functional outcome following treatment with a standardized protocol[END_REF][START_REF] Pugh | Standard surgical protocol to treat elbow dislocations with radial head and coronoid fractures[END_REF][START_REF] Ring | Posterior dislocation of the elbow with fractures of the radial head and coronoid[END_REF][START_REF] Winter | Surgical management of elbow dislocation associated with non-reparable fractures of the radial head[END_REF][START_REF] Forthman | Elbow Dislocation With Intra-Articular Fracture: The Results of Operative Treatment Without Repair of the Medial Collateral Ligament[END_REF][START_REF] Garrigues | Fixation of the coronoid process in elbow fracture-dislocations[END_REF][START_REF] Kani | Terrible triad injuries of the elbow[END_REF]. The radial head is now systematically reconstructed or replaced and the lateral ligaments are reinserted, both being requirements for good outcome (Figure 11A-G). It is always essential to restore elbow stability at end of procedure, allowing very early mobilization to prevent rapid severe stiffening.

In a recent review of the literature, Chen et al. [START_REF] Chen | Complications of treating terrible triad injury of the elbow: a systematic review[END_REF] found satisfactory clinical scores and MEPS (Mayo Elbow Performance Score) 78-95/100. However, the revision surgery rate was 22%, due to hardware-related complications, stiffness, instability, and ulnar neuropathy. A recent study reported long-term results at a mean 9 years in 12 cases treated by the standardized procedure described here [START_REF] Zaidenberg | Treatment of terrible triad injuries at a mean follow-up of nine years[END_REF]. Clinical results were very encouraging, with a mean 151° flexion-extension and 162° pronation-supination, but with osteoarthritic remodeling in 66% of cases.

What factors affect results?

Zhou et al. [START_REF] Zhou | Does Timing of Surgery Affect Treatment of the Terrible Triad of the Elbow?[END_REF] analyzed the impact of time to treatment in a retrospective study, with 3 groups: treatment within 24 hours, at 2-14 days, and >14 days. Complications and pain did not differ between groups, except for greater stiffness in the last. It emerged that, after initial reduction, specialized treatment could be within 14 days without negative consequences.

According to other authors [START_REF] Iordens | Good functional recovery of complex elbow dislocations treated with hinged external fixation: a multicenter prospective study[END_REF][START_REF] Sorensen | Treatment of persistent instability after posterior fracture-dislocation of the elbow: restoring stability and mobility by internal fixation and hinged external fixation[END_REF], when external fixation is required, it should be implemented rapidly. Giannicola et al. [START_REF] Giannicola | Terrible triad of the elbow: is it still a troublesome injury?[END_REF] identified certain factors for poor prognosis: obesity, poor treatment adherence, soft-tissue lesions, and lack of surgeon experience.

Antoni [START_REF] Antoni | Anterior capsule re-attachment in terrible triad elbow injury with coronoid tip fracture[END_REF] recently analyzed the contribution of anterior capsule bone suture onto the tip of the coronoid process, and found no clinical or radiographic benefit.

To sum up:

-At end of surgery, the elbow should be stable, to allow rapid mobilization to avoid stiffening. -After initial reduction, specialized treatment is recommended, to optimize results and avoid revision procedures. -Osteoarthritic remodeling is usual in the medium and long term.

Conclusion

-Terrible triad of the elbow is a particular form of fracture-dislocation associating severe bone and soft-tissue lesions jeopardizing elbow stability. -Management is now well codified: o Emergency reduction. o Specialized surgery, imperatively including lateral compartment reconstruction associating radial head reconstruction by internal fixation or arthroplasty and lateral repair. The coronoid process should be systematically fixed in type-3 fracture and considered on a case-by-case basis in types 1 and 2. Medial repair and external fixation are indicated only if instability persists after all repair procedures.

Figure 1 :Figure 2 :Figure 3 :Figure 4 :Figure 5 :Figure 6 :Figure 8 :Figure 9 :Figure 10 :

 1234568910 Figure 1: Elbow dislocation mechanisms. From Mansat P, Bonnevialle N. Luxations du coude. EMC (Elsevier Masson SAS Paris), Appareil Locomoteur, 14-042-A-10, 2009. axial compression / supination / valgus

Acknowledgments

The authors thank Dr Cécile Nérot for advice and attentive re-editing.

Disclosure of interest:

The authors have no conflicts of interest to disclose in relation to the present review Funding: none Author contributions: Xavier Ohl: literature analysis, article writing. Renaud Siboni: data collection, article re-editing.


M Boxberger

A Antezack

S Magnien

N Cassir

B La Scola

Draft genome and description of Mixta mediterraneensis strain Marseille-Q2057 T sp.nov., a new bacterium isolated from human healthy skin

Keywords: Bacteria, culturomics, genome, Mixta mediterraneensis, species, sp. nov., taxonogenomics

In 2019, by culturing a skin swab from the hand of a 30-year-old healthy woman using the culturomic method, we isolated the new bacterial strain Marseille-Q2057 T (= CSUR-Q2057).

Matrix-assisted desorption-ionization time-of-flight mass spectrometry (MALDI-TOF MS) failed to identify this isolate. Analysis of the 16S rRNA gene and Genome-to-Genome comparison suggested that this taxon belongs to a novel bacterial species within the family Erwiniaceae, phylum Proteobacteria. We describe here its main phenotypic characteristics, genome sequence and annotation of Mixta mediterraneensis strain Marseille-Q2057 T , a new member of the Mixta genus, that we propose as type strain.

Introduction

The genus Mixta was created in 2018 to resolve certain approximations in the taxonomy of the Erwiniaceae family, in the light of recent advances in combined genomic and phylogeny approaches. [START_REF] Palmer | Mixta gen. nov., a new genus in the Erwiniaceae[END_REF] Mixta mediterraneensis strain Marseille-Q2057 T was isolated using the culturomics approach, based on the use of a large panel of culture conditions to describe the microbial composition of a sample by high-throughput culture [START_REF] Lagier | Culturing the human microbiota and culturomics[END_REF][START_REF] Lagier | Microbial culturomics: paradigm shift in the human gut microbiome study[END_REF][START_REF] Lagier | Culture of previously uncultured members of the human gut microbiota by culturomics[END_REF] A taxonogenomics approach, including matrix-assisted laser desorption-ionization time-of-flight mass spectrometry (MALDI-TOF MS), phylogenetic analysis, main phenotypic description and genome sequencing, was used to describe this species [START_REF] Lagier | Culturing the human microbiota and culturomics[END_REF][START_REF] Ramasamy | A polyphasic strategy incorporating genomic data for the taxonomic description of novel bacterial species[END_REF].

Materials and methods

Strain isolation and phenotypic tests

Mixta mediterraneensis strain Marseille-Q2057 T was initially isolated by direct seeding of 50 μL of sample on an Acinetobacter-specific medium [START_REF] Zebral | A selective medium for the isolation of the Acinetobacter genus bacteria[END_REF] incubated in aerobiosis at 31°C, MALDI-TOF MS protein analysis was carried out using a Microflex spectrometer (Bruker Daltonics, Bremen, Germany) [START_REF] Tatusova | NCBI prokaryotic genome annotation pipeline[END_REF]. Spectra from strain Marseille-Q2057T were imported into the MALDI BIOTYPER software (version 3.0, Bruker) and analysed by standard pattern matching (with default parameter settings). The study was validated by the ethics committee Sud-Est IV under number ID-RCB: 2019-A01508-49. Different growth temperatures (30°C, 37°C, 45°C and 56°C), atmospheric conditions-anaerobic, aerobic and microaerophilic (CampyGEN, Oxoid, USA) and pH (5, 6.5, 7.5, 8.5) were tested. API ZYM, API 20E and API 50 CH strips (BioMérieux, Marcy L'Étoile, France) were used to evaluate the biochemical properties of the strain according to the manufacturer's instructions. For scanning electronic microscopy, a colony was collected from agar and immersed into a 2.5% glutaraldehyde fixative solution. The slide was gently washed in water; air-dried and examined to evaluate bacterial structure on a TM4000 microscope approximately 60 cm in height and 33 cm in width. The standard disc method was applied for antimicrobial susceptibility testing according to the French Microbiology Society. Motility test was performed using the semi-solid TCC media as described by Tittsler and Sandholzer [START_REF] Tittsler | The use of semi-solid agar for the detection of bacterial motility[END_REF].

Genome sequencing

Genomic DNA (gDNA) of M. mediterraneensis strain Marseille-Q2057 T was extracted in two steps: a mechanical treatment was first performed with glass beads acid-washed (G4649-500g; Sigma, St Louis, MO, USA) using a FastPrep-24™ 5G Grinder (mpBio, Irvine, CA, USA) at maximum speed (6.5) for 90 seconds. Then after 30 minutes of lysozyme incubation at 37°C, DNA was extracted on the EZ1 biorobot (Qiagen, Hilden, Germany) with an EZ1 DNA tissues kit. The elution volume was 50 µL. The gDNA of M. mediterraneensis strain Marseille-Q2057 T was quantified by a Qubit assay with the high sensitivity kit (Life Technologies, Carlsbad, CA, USA) to 0.2 ng/µL. Genomic DNA was next sequenced on the MiSeq Technology (Illumina Inc., San Diego, CA, USA) with the paired end strategy and was barcoded in order to be mixed respectively with 21 other genomic projects prepared with the Nextera XT DNA sample prep kit (Illumina). To prepare the paired end library, dilution was performed to require 1ng of each genome as input to prepare the paired end library. The 'tagmentation' step fragmented and tagged the DNA. Then limited cycle PCR amplification (12 cycles) completed the tag adapters and introduced dual-index barcodes. After purification on AMPure XP beads (Beckman Coulter Inc., Fullerton, CA, USA), the libraries were then normalized on specific beads according to the Nextera XT protocol (Illumina). Normalized libraries were pooled into a single library for sequencing on the MiSeq. The pooled single strand library was loaded onto the reagent cartridge and then onto the instrument along with the flow cell. Automated cluster generation and paired end sequencing with dual index reads were performed in a single 39-hour run in 2 × 250 bp. Total information of 4.5 Gb was obtained from a 462 K/mm 2 cluster density with a cluster passing quality control filters of 93.9%. Within this run, the index representation for M. mediterraneensis strain Marseille-Q2057 T was determined to index 3.06%. The 9 045 583 paired end reads were filtered according to the read qualities. To improve the quality of the assembly, an Oxford Nanopore approach was performed on 1D gDNA sequencing for the MinIon device using an SQK-LSK109 kit. The library was constructed from 1-µg gDNA without fragmentation and end repair. Adapters were ligated to both ends of gDNA. After purification on AMPure XP beads (Beckman Coulter), the library was quantified by a Qubit assay with the high sensitivity kit (Life Technologies, Carlsbad, CA, USA). A total of 1376 active pores were detected for the sequencing and the workflow WIMP was chosen for bioinformatic analysis in live. After 2 hours of run time and the end of life of the flow cell, 32 541 000 reads as raw data were generated.

Genome annotation and genome comparison

Genome annotation was obtain through the NCBI Prokaryotic Genome Annotation Pipeline [START_REF] Tatusova | NCBI prokaryotic genome annotation pipeline[END_REF]. The genome sequence data were uploaded to the Type (Strain) Genome Server (TYGS), a free bioinformatics platform available under https://tygs.dsmz.de, for a whole genome-based taxonomic analysis [START_REF] Meier-Kolthoff | TYGS is an automated high-throughput platform for state-of-the-art genome-based taxonomy[END_REF]. Determination of closest type strain genomes was performed in two complementary ways: first, all user genomes were compared against all type strain genomes available in the TYGS database via the MASH algorithm, a fast approximation of intergenomic relatedness [START_REF] Ondov | Mash: fast genome and metagenome distance estimation using MinHash[END_REF], and, the ten type strains with the smallest MASH distances chosen per user genome. Second, an additional set of ten closely related type strains was determined via the 16S rDNA gene sequences. These were extracted from the user genomes using RNAMMER [START_REF] Lagesen | RNAmmer: consistent and rapid annotation of ribosomal RNA genes[END_REF] and each sequence was subsequently BLASTED [START_REF] Camacho | BLAST+: architecture and applications[END_REF] against the 16S rDNA gene sequence of each of the currently 12 983 type strains available in the TYGS database. This was used as a proxy to find the best 50 matching type strains (according to the bitscore) for each user genome and to subsequently calculate precise distances using the Genome BLAST Distance Phylogeny approach (GBDP) under the algorithm 'coverage' and distance formula d5 [START_REF] Meier-Kolthoff | Genome sequence-based species delimitation with confidence intervals and improved distance functions[END_REF]. These distances were finally used to determine the ten closest type strain genomes for each of the user genomes. All pairwise comparisons among the set of genomes were conducted using GBDP and accurate intergenomic distances inferred under the algorithm 'trimming' and distance formula d5. One hundred distance replicates were calculated each. Digital DNA-DNA hybridization values and confidence intervals were calculated using the recommended settings of the GGDC2. Complementarily, the degree of genomic similarity of strain Marseille-Q2057 with closely related species was estimated using ORTHOANI software with default parameters [START_REF] Lee | OrthoANI: An improved algorithm and software for calculating average nucleotide identity[END_REF], the nine closest species were determined on a DNA-DNA hybridization basis. Antibiotic-resistance genes and presence of pathogenesis-related proteins was investigated using the ABRICATE TOOLS v1.0.1 against ARG-ANNOT [START_REF] Gupta | ARG-ANNOT, a new bioinformatic tool to discover antibiotic resistance genes in bacterial genomes[END_REF], EcOH [START_REF] Ingle | EcOH: In silico serotyping of E. coli from short read data[END_REF], NCBI Bacterial Antimicrobial Resistance reference Gene Database [START_REF] Feldgarden | Validating the AMRFinder Tool and resistance gene database by using antimicrobial resistance genotype-phenotype correlations in a collection of isolates[END_REF], PLASMIDFINDER [START_REF] Carattoli | In silico detection and typing of plasmids using PlasmidFinder and plasmid multilocus sequence typing[END_REF], RESFINDER [START_REF] Bortolaia | ResFinder 4.0 for predictions of phenotypes from genotypes[END_REF], CARD [START_REF] Mcarthur | The Comprehensive Antibiotic Resistance Database[END_REF] and VFDB [START_REF] Chen | VFDB: a reference database for bacterial virulence factors[END_REF] using the Online Galaxy platform [START_REF] Afgan | The Galaxy platform for accessible, reproducible and collaborative biomedical analyses: 2016 update[END_REF].

Results

Strain identification and classification

Mixta mediterraneensis strain Marseille-Q2057 T was isolated from the hand skin swab of a 30 -year-old healthy woman. Mixta mediterraneensis strain Marseille-Q2057 T was not identified by our systematic MALDI-TOF MS screening, suggesting that the corresponding species was not in the database (https://www.mediterranee-infection.com/acces-ressources/base-de-donnees/urms-data-base/) (Fig. 1); it analysed within the closest members of Erwiniaceae on the IHU databases available spectra and did not belong to any known cluster (Fig. 2).

Moreover, strain Marseille-Q2057 T exhibited 97.66% 16S rRNA sequence similarity with Mixta gaviniae strain DSM 22758 (extracted from the genome accessible CP026377.1), the phylogenetically closest bacterium with standing in nomenclature (Fig. 3a) Furthermore, digital DNA-DNA hybridization revealed a maximum identity similarity of only 23.6% (Fig. 3b and Table 1) and an ORTHOANI parameter provided a value of 80.76% (Fig. 4) between the novel organism and Pantoea conspicua LMG 24534 (GCA_002095315). Taken altogether these results confirm the status of this strain as a new member of the Mixta genus for which the name Mixta mediterraneensis strain Marseille-Q2057 T is proposed.

Phenotypic characteristics

Growth of M. mediterraneensis strain Marseille-Q2057 T was initially isolated by direct seeding of 50 μL of sample on Acinetobacter-specific medium [START_REF] Zebral | A selective medium for the isolation of the Acinetobacter genus bacteria[END_REF] incubated in aerobiosis at 31°C. Colonies from strain Marseille-Q2057 T showed a beige pigmentation and no hemolysis.

Bacterial cells were Gram-negative, motile bacilli with a length of about 3.8 µm and a width of about 0.8 µm determined by electronic scanning microscopy (Fig. 3). Strain Marseille-Q2057 T is aerobic, anaerobic and microaerophilic. Optimum pH of this bacterium is comprised between pH 5 and pH 7.5. The sporulation test (20 minutes at 80°C) was negative.

Using API strips, positive reactions were shown for alkaline phosphatase, leucine arylamidase, acid phosphatase, naphthol-AS-BI-phosphohydrolase, β-glucuronidase, sodium pyruvate, carbon substrate, D-glucose, D-fructose, D-mannose, N-acetyl glucosamine, esculin, D-trehalose. All other reactions tested were negative. In addition, this bacterium was catalase positive and oxidase negative. These results are summarized in Table .2

Genome properties

The genome of strain Marseille-Q2057 was 4 532 310 bp long with a 51.76% G+C content.

The genome assembly of this strain was achieved on 34 contigs. Of the 4537 predicted genes, 4067 were protein-coding genes and 108 were RNAs (7 16S rRNA, 8 additional 5S rRNAs, 7 additional 23S rRNAs, 77 tRNAs and 9 ncRNAs) (Fig. 6). The distribution of genes into clusters of orthologous groups (COGs) functional categories for strain Marseille-Q2057 T and other closely related bacterial taxa is detailed in Table 3. Analysis of the COGs categories shows that the mobilome, amino acid transport and metabolism elements of the strain Marseille-Q2057 appear to be the more numerous putative functions (by COGs) (412 in category X, 343 in category E, respectively). Through this analysis, we can see that the repartition of all COG categories is similar across these species. (Fig. 7) The in silico resistome of the strain Marseille-Q2057 T and the search for virulence factors [START_REF] Chen | VFDB: a reference database for bacterial virulence factors[END_REF] of this strain showed on the 7 contig an 85.77% identity gene with Crp gene that could be implied in fluoroquinolone, macrolide and penam resistance (using CARD). Two IncFII plasmids were detected on the 18 and 19 contigs.

Discussion and conclusion

In the past 8 years, a culturomic approach has led to the discovery of more than 500 bacterial species [START_REF] Lagier | Culturing the human microbiota and culturomics[END_REF]. Using the taxonogenomics concept, i.e. the combination of the genomic and phenotypic properties of a putative new taxon [START_REF] Abdallah | The impact of culturomics on taxonomy in clinical microbiology[END_REF], we have characterized a new bacterial species representing a new species within the family Erwiniaceae found on human hand skin.

It was named as M. mediterraneensis strain Marseille-Q2057 T . Members of Erwiniaceae are commonly found associated with plants, so it is reasonable to think that our species, found on skin, is part of the transient cutaneous microbiota.

Mix'ta N.L. fem. n. Mixta, the mixed one, referring to the mixed lifestyles of species in the genus. Me.di.ter.ra.ne.en′sis, L. masc. adj., mediterraneensis, 'of Mediterraneum,' the Latin name of the Mediterranean Sea by which Marseille is located and the bacteria isolated.

Deposit in culture collections and Sequences Database

Mixta mediterraneensis strain Marseille-Q2057 T , was deposited in CSUR collections under accession CSUR-Q2057. The 16S rRNA and genome sequences are available under accession numbers MW177953 and JACFXY000000000.1, respectively. relative to other closely related bacterial taxa. Trees were generated with FASTME 2.1.6.1 [START_REF] Lefort | FastME 2.0: A Comprehensive, Accurate, and Fast Distance-Based Phylogeny Inference Program: Table 1[END_REF] from Genome BLAST Distance Phylogeny (GBDP) distances calculated from genome sequences our 16S sequences. The branch lengths are scaled in terms of GBDP distance formula d5. The numbers above branches are GBDP pseudo-bootstrap support values >60% from 100 replications, with an average branch support of 66.6%. The tree was rooted at the midpoint [START_REF] Farris | Estimating phylogenetic trees from distance matrices[END_REF]. Information storage and processing

FIG 1 .FIG. 2 .

 12 FIG 1. MALDI-TOF MS reference mass spectrum. Spectra from 12 individual colonies of strain Marseille-Q2057 T were compared and a reference spectrum was generated.

FIG. 3 .

 3 FIG. 3. (a) 16s rRNA-based phylogenetic tree, and (b) whole-genome-based phylogenetic tree highlighting the position of Mixta mediterraneensis sp. nov., strain Marseille-Q2057 T

FIG. 4 .FIG. 5 .FIG. 6 .FIG. 7 .

 4567 FIG. 4.Scanning electron microscopy of Mixta mediterraneensis sp. nov., strain Marseille-Q2057 T using a Tabletop microscope TM 4000plus (Hitachi, Tokyo, Japan). The scale bar represents 5 µm.

TABLE 1 . Digital DNA-DNA hybridization values obtained by sequence comparison of all studied genomes using TYGS comparison server using the second formula

 1

	Subject strain	dDDH (%) with Mixta mediterraneensis	95% CI	G+C content difference (in %)
	Pantoea conspicua LMG 24534	23.6	21.3-26.0	3.8
	Pantoea brenneri LMG 5343	23.6	21.3-26.0	3.99
	Pantoea vagans LMG 24199	23.2	20.9-25.6	3.58
	[Curtobacterium] plantarum LMG 16222	23	20.7-25.4	3.3
	Pantoea agglomerans NBRC 102470	23	20.7-25.4	3.36
	Pantoea eucalypti LMG 24197	22.8	20.5-25.2	2.51
	Pantoea stewartii CCUG 26359	22.7	20.4-25.2	1.83
	Pantoea anthophila LMG 2558	22.6	20.4-25.1	5.01
	Pantoea ananatis LMG 2665	21.9	19.6-24.3	1.64
	Pectobacterium betavasculorum NCPPB 2795	21.1	18.9-23.5	0.6
	Pantoea rwandensis LMG 26275	20.8	18.6-23.3	0.85
	Mixta theicola DSM 29212	20.7	18.5-23.1	2.23
	Pantoea eucrina LMG 5346	20.7	18.4-23.1	4.5

TABLE 2 . Differential characteristics of Mixta mediterraneensis strain Marseille-Q2057 and its most closely related species with standing in nomenclature Mixta mediterraneensis Pantoea conspicua Pantoea brenneri Pantoea vagans [Curtobacterium] plantarum Pantoea agglomerans

 2

	Properties	Marseille-Q2057	LMG 24534	LMG 5343	LMG 24199	LMG 16222	NBRC 102470
	Cell size	0.8 × 3.8 µm	0.9 × 1.5-3.0 µm	0.9 × 1.5-3.0 µm 0.9 × 1.5-3.0 µm	0.3-0.5 × 0.6-3.0 µm	NA
	Oxygen requirement	facultative	facultative	facultative	facultative	+	facultative
	Gram stain	-	-	-	-	+	-
	Motility	-	+	+	+	+	+
	Endospore formation	NA	-	-	-	-	-
	Optimum temperature for growth (°C)	31°C	28°C-30°C	28°C-30°C	NA	28°C-30°C	30°C
	Production of :
	Alkaline phosphatase	+	NA	NA	NA	NA	NA
	Catalase	+	NA	NA	NA	NA	NA
	Oxidase	-	-	-	-	NA	-
	α-Glucosidase	-	NA	NA	NA	NA	NA
	β-Galactosidase	+	NA	NA	+	NA	+
	Acid from:
	N-Acetylglucosamine	+	+	+	+	+	+
	L-arabinose	-	+	+	+	+	+
	D-ribose	-	+	+	+	+	+
	D-mannose	+	+	+	+	+	+
	D-mannitol	-	+	+	+	+	+
	D-glucose	+	+	+	+	+	NA
	D-fructose	+	+	+	+	+	+
	D-maltose	-	+	+	+	+	+
	D-lactose	-	+	+	+/-	-	-

TABLE 3 . Detailed functional classes of predicted genes according to the clusters of orthologous groups of proteins of Mixta mediterraneensis sp. nov. other closely related bacterial taxa

 3

Acknowledgements

The authors are indebted to Ludivine Brechard for sequencing the genome and the platform of electron microscopy of IHU-MI for the electron micrographs.

Funding sources MB is PhD granted by the collaboration between M&L Laboratories and Aix Marseille University referenced PVM:2018-200. This study was supported by the French State managed by the National Research Agency under the Investissements d'avenir (Investments for the Future) programme under the reference ANR-10-IAHU-03 (Méditerranée Infection) and by the Région Provence-Alpes-Côte-d'Azur and the European funding FEDER PRIMI.

Conflicts of interest

The authors have no conflicts of interest to declare.