M T López-López

P Kuzhir

A Meunier

G Bossis
email: bossis@unice.fr

Synthesis and magnetorheology of suspensions of cobalt particles with tunable particle size

Different samples of cobalt powder were synthesized. Particle size and shape were characterized by electron microscopy and light scattering. These measures showed that the synthesized powders consisted of monodisperse spheres with average diameters ranging between 63 and 760 nm. These powders were used for the preparation of magnetorheological (MR) fluids by dispersing them in silicone oil. It was found that particle size did not have much influence on the MR response of MR fluids for average particle diameters larger than 100 nm. On the other hand, the MR response decreased appreciably when average particle diameter was diminished below 100 nm.

Introduction

Suspensions of colloidal magnetic particles are complex fluids that exhibit magnetic field-dependent rheological (flow) properties. Upon magnetic field application, these particles experience attractive magnetostatic forces, which lead to the formation of particle structures aligned with the field direction. The formation of these structures, which strengthen the suspension and oppose to the flow, is the basic phenomenon underlying the magnetorheological (MR) effect [START_REF] Odenbach | Ferrofluids[END_REF][START_REF] Rosensweig | Ferrohydrodynamics[END_REF][START_REF] Bossis | Magnetorheology: Fluids, Structures and Rheology[END_REF][START_REF] Winslow | [END_REF]. The formation of field-induced particle aggregates, i.e. the intensity of the MR effect, will depend on the ratio of the magnetic interaction energy to the thermal energy. For two magnetic dipoles this ratio is given by [START_REF] Bossis | Magnetorheology: Fluids, Structures and Rheology[END_REF]:

kT H a f 2 2 0 3 2 0 β µ πµ λ = . ( 1
)
Here, a is the particle radius, µ 0 the permeability of vacuum, µ f the relative permeability of the carrier liquid, H 0 the external magnetic field strength, k the Boltzmann constant, T the absolute temperature, and β the magnetic contrast factor. Generally, the particles have a large permeability and, therefore, β~1. Thus, taken T = 300 K and µ f ~1, for MR fluids (

1 ã µm) 1 =
λ is obtained for H = 0.046 kA/m, whereas for FF ( 5 ã nm) the same value is obtained for H = 368 kA/m. Therefore, the Brownian forces normally dominate the magnetic forces in FF and, consequently, they only exhibit weak MR effect [5][6]. On the other hand, even at low magnetic field, the magnetic forces are much higher than the Brownian forces in MR fluids and, consequently, they exhibit strong MR effect [7][8].

From the experimental viewpoint, little is known about the effect of particle size on the MR properties of suspensions intermediate between FF and MR fluids. Lemaire et al. [9] measured the shear stress as a function of the shear rate for three different suspensions, made of polystyrene particles containing inclusions of magnetite, under the presence of an applied magnetic field of approximately 10 kA/m. The only difference between the suspensions was the average particle diameter: 0.5, 0.8 and 1.0 microns. They found that, for a given shear rate, the apparent shear stress slightly increased with the diameter of the particles. Rosenfeld et al. [10] investigated the MR properties of two suspensions of iron particles, the average particle size being the only difference between both of them: 30 microns and 26 nm. They found that, even at high magnetic field strength, the suspension of microparticles exhibited yield stresses up to 2 times higher than the suspension of nanoparticles. According to Equation 1, even for a particle diameter of 25 nm the parameter λ is still larger than unity for magnetic fields of interest (λ ~ 10 for H = 100 kA/m). Thus, from the theoretical viewpoint, the MR effect should not show any size dependence for diameters larger than 25 nm.

The difficultness of performing a rigorous experimental study about the influence of particle size on the MR properties, lies in obtaining monodisperse and well dispersed spherical particles with different average particle diameters. For instance, Kormann et al. [11] obtained a quite large yield stress (3 kPa at 0.3 Tesla) with ferrite nanoparticles of 30 nm diameter likely because of the presence of aggregates of nanoparticles. The aim of the present work is to rigorously analyze this dependence for particle diameters in the range 50 nm to 1 µm. With this objective, we synthesized different samples of cobalt powder with controlled diameter, by reduction of cobalt ions in polyols. The sosynthesized powders were used for the preparation of MR suspensions. Finally, the MR properties of these suspensions were investigated as a function of the applied magnetic field.

Experimental methods

Reduction of metallic ions in a liquid polyol has been extensively used for the preparation of spherical particles of magnetic and non-magnetic materials [8,[START_REF] Fiévet | Fine Particles[END_REF][START_REF] Fiévet | [END_REF][14][15][16]. This method was used in this work to synthesize spherical cobalt particles with different average diameter. Particles size and shape were characterized by means of scanning electron microscopy (SEM) and transmission electron microscopy (TEM). In addition, the hydrodynamic diameter of the particles was obtained by light scattering measurements using a Mastersizer 2000 instrument (Malvern Instruments, UK). The magnetization curves measured on the powders with a vibrating sample magnetometer show no noticeable difference between the different sizes and we find Ms between 13900 and 14000 Gauss whatever the size. Nevertheless it is worth noting that this value is significantly lower than the one of bulk cobalt (Ms= 17900G) which means that some part of the catalyst has been included into the cobalt nanoparticles.

MR suspensions (φ = 0.05) were prepared by dispersing the synthesized cobalt powders in silicone oil. Based on discussions reported in previous papers [17][18], aluminum stearate was used as dispersant. The MR properties of the suspensions were measured using a Haake RS150 controlled stress rheometer. The measuring system geometry was a plate-plate configuration of diameter 20 mm and gap 0.300 mm with a controlled temperature T=20°C. The magnetic field was applied in the vertical direction using a home-made electromagnet similar to the one sold by Anton Paar for the rheometer MCR301 , which allows reaching fields up to 580 kA/m in the measuring gap.

Results and discussion

The synthesized powders consisted of relatively monodisperse spheres of submicrometer size. Table I shows the average particle size obtained by electron microscopy and light scattering. As observed, there is a quite good agreement between the data obtained by electron microscopy and by light scattering which confirms the absence of aggregation in liquid phase. . I).

The steady-shear flow of suspensions containing 5 vol.% of cobalt powders of samples S1-S4 dispersed in a silicone oil of viscosity: 0.479Pa.s and density = 0.97g/cm 3 , was investigated as a function of the applied magnetic field. The values of the shear stress were obtained as a function of the shear rate, for the different applied magnetic field strengths (up to 431 kA/m). From extrapolation at zero shear rate, of the shear stress vs. shear rate curves in the range 0.01 s -1 -0.1 s -1 on a logarithmic scale, the values of the static yield stress were obtained. These values are plotted as a function of the magnetic field strength in Figure 1. As expected for MR fluids, the values of the yield stress shown by samples S1-S4 increase with magnetic field strength. Let us focus on the comparison of the values of the yield stress for the different samples. As observed, the yield stress decreases as the average particle diameter is diminished (see particle diameter in Table I). This decrement is negligible between S1(760nm) and S2 (330nm)except at the highest fields, but becomes quite significant for average diameter smaller than 100 nm. For example, taken as the reference, the value of sample S1 at H = 297 kA/m, the yield stress is respectively: 94 % for sample S2; 82 % for sample S3; and 65 % for sample S4. The reason for this decrement could be the Brownian motion, which is able to generate fluctuations on a large scale and promote the presence of gaps between the particles, which will decrease the strength of the columnar structures under strain and, consequently, the strength of the MR effect [9].Nevertheless, considering that, even for the smallest diameter, we have λ > 1000 for H = 297 kA/m it is not obvious that such a low Brownian motion can explain this size dependence; furthermore, this decrease should be less pronounced at the largest fields, which corresponds to the highest values of λ, and actually we observe the contrary. Another explanation stands on the surface energy of the aggregates, which determines their size and the condition of their rupture under strain. The energy difference produced by the breakdown of an aggregate in two smaller ones will be a decreasing function of the size of the particles as it is the case for the surface energy. The consequence is that aggregates constituted of smaller particles will break at lower strain and consequently at lower stress. This approach developed by Zubarev et al. [19] for ferrofluids could likely explain the observed behavior and will be developed in a future paper.

Conclusions

Different samples of spherical cobalt particles with average diameter in the range 60 nm -800 nm, have been synthesized by reduction of ions in liquid polyols. Being dispersed in silicone oil, these particles allowed us to carry out the first systematic study on the effect of particle size on the yield stress of MR suspensions. This effect has been shown to be almost negligible for average particle diameter higher than 100 nm. For smaller diameters, a significant decrement of the MR effect is observed. Brownian motion is probably not responsible for the observed behavior but rather the change of surface energy when an aggregate is broken and form two half aggregates of the same length. Besides the academic interest of this study, it has also been evidenced that nanometric particles have large potential applications in smart MR devices, especially those having narrow flow channels, such as active hydrostatic bearings for precision machinery [START_REF] Ochonski | The attraction of ferrofluids bearings[END_REF].

11thFigure 1 .

 1 Figure 1. Yield stress, τ y , plotted as a function of the magnetic field strength, H, for suspensions containing 5 vol.% of cobalt particles (samples S1-S4; see TableI).

11th

  Conference on Electrorheological Fluids and Magnetorheological Suspensions IOP Publishing Journal of Physics: Conference Series 149 (2009) 012073 doi:10.1088/1742-6596/149/1/012073

Table I .

 I Average diameter (nm), measured by different techniques, of the synthesized cobalt particles.

	Sample name	S1	S2	S3	S4
	Electron Microscopy	760	330	83	63.6
	Light Scattering	629	498	126	69

Acknowledgements

Eureka E! 3733 Hydrosmart project and "Conseil Régional PACA" (Biomag project) are acknowledged for their financial support. One of the authors (M.T.L.-L.) also acknowledges financial support by Secretaría de Estado de Universidades e Investigatigación (MEC, Spain) through its Postdoctoral Fellowship Program. We are also grateful to A. Zubarev for helpful discussions and V.Cagan from Versailles University for magnetization measurements.


Alexandre Imperiale
email: alexandre.imperiale@cea.fr

INCORPORATING EFFECTIVE TRANSMISSION CONDITIONS BETWEEN FLUID AND SOLID DOMAINS IN TRANSIENT WAVE PROPAGATION PROBLEMS USING THE MORTAR ELEMENT METHOD

Keywords: Transient Wave Propagation, Acoustic -Elastic waves interactions, Domain Decomposition Methods, Time Discretization, Ultrasonic Testing Modeling

We derive an efficient numerical scheme for transient wave propagation in configurations where a fluid domain and a solid domain are separated by a thin coating material. These type of configurations are numerically challenging for multiple factors: managing fluid -solid coupling, enabling non-conform space discretizations, and rendering robust time discrete algorithm w.r.t the thin layer thickness. By combining the mortar element method with effective transmission conditions we are able to address these challenges. We illustrate our approach by proposing relevant 2D numerical illustrations inspired from simple ultrasonic testing experiments.

INTRODUCTION

In the present communication, we consider the problem of efficient numerical modeling of high-frequency transient wave propagation in a configuration where a solid material is surrounded by a fluid, and at the fluid -solid interface a thin layer of material is present. This configuration typically occurs when representing Ultrasonic Testing (UT) of immersed coated specimens. The computational challenges associated to this type of configurations are threefold:

(1) The thickness of the coating material, denoted by η in the following, is significantly smaller than the wavelength of interest, thus degrading the efficiency of standard approaches with explicit time scheme, as they suffer from a stability condition on the time step in O(η).

(2) In UT applications, the wavelength in the fluid are usually smaller than in the solid (due to lower wave velocities). Hence traditional conform approaches will impose an over-refined mesh in the solid domain, leading to a potentially significant cost overhead.

(3) As any fluid -structure interaction problem, one has to operate a specific treatment in order to incorporate the normal stress and normal velocity continuity conditions within their numerical scheme.

Various tools and techniques can be found in the literature to overcome some of these computational bottlenecks. For instance, authors in [START_REF] Chabassier | Fourth-order energy-preserving locally implicit time discretization for linear wave equations[END_REF] use an implicit scheme (only) in the thin layer. Their approach alleviates the dependency of the time step w.r.t. the layer thickness.

It is based on the mortar element method [START_REF] Ben Belgacem | The mortar element method for three dimensional finite elements[END_REF] allowing for non-conform coupling between the domains. However it is done at the price of solving at each iteration a linear system of size proportionate to the number of degrees of freedom in the layer and should require some adjustment to incorporate fluid -solid transmission conditions. In [START_REF] Diaz | Energy conserving explicit local time stepping for second-order wave equations[END_REF] a local time stepping method is proposed which effectively reduces the influence of refined or complex regions of the mesh onto the stability condition of the explicit time scheme. Essentially, this is performed by adding subiterations between coarse time steps. Their approach imposes the use of discontinuous Galerkin methods [START_REF] Grote | Discontinuous Galerkin finite element method for the wave equation[END_REF] in order to address non-conform meshes, and cannot support as-is coupling between fluid and solid propagators. Other approaches rely on a simplification of the wave propagator in the coating layer based upon asymptotic developments -the asymptotic parameter being proportionate to the thickness η. A typical example in the literature is [START_REF] Rokhlin | Analysis of boundary conditions for elastic wave interaction with an interface between two solids[END_REF]. More recently authors in [START_REF] Bonnet | Effective transmission conditions for thinlayer transmission problems in elastodynamics. The case of a planar layer model[END_REF] derived Effective Transmission Conditions (ETC) between two solid domains surrounding the layer while insuring convergence to the exact solution as the thickness tends to zero. Their effective numerical procedure leads to a time step independent of the layer thickness, however supporting acoustic -elastic waves coupling with non-conforming meshes is not addressed in their approach. Inspired from this seminal body of work, our approach combine the use of ETC -as in [START_REF] Rokhlin | Analysis of boundary conditions for elastic wave interaction with an interface between two solids[END_REF] with an adequate form of the mortar element method -as in [START_REF] Chabassier | Fourth-order energy-preserving locally implicit time discretization for linear wave equations[END_REF]. It is in essence the extension of [START_REF] Imperiale | Numerical modeling of wave propagation in anisotropic viscoelastic laminated materials in transient regime: Application to modeling ultrasonic testing of composite structures[END_REF] to configurations with fluid and solid propagators. The mortar unknowns are treated with an implicit time-scheme, from which we can expect the stability condition of the complete numerical procedure to be independent of the layer thickness or effective parameters. This method supports (by construction) non-conform meshes, and we are able to adapt it to support fluid -solid coupling. Combining these elements we are able to address the various computational challenges associated to the configurations of interest.

The outline of our paper is as follows. In Section 2 we detail the modeling setting used to represent UT experiments of an immersed and coated solid material. In Section 3 we present the numerical procedure leading to a fully discrete scheme. This procedure combine spectral finite elements [START_REF] Cohen | Higher-order numerical methods for transient wave equations[END_REF] for the space discretization and second order schemes [START_REF] Joly | Variational methods for time-dependent wave propagation problems[END_REF] for the time discretization. We illustrate our approach in Section 4 with some relevant numerical results, and provide some conclusion and perspectives in the last Section.

WAVE PROPAGATION THROUGH AN IMMERSED AND COATED SOLID

Geometrical considerations

Let N = 2 or N = 3 denotes the space dimension. As depicted in Figure 1, we consider the open domain Ω s ⊂ R N occupied by a solid material subject to an UT experiment. This solid material is partially adjacent to a fluid (typically water) occupying the fluid domain Ω f . Between the solid and fluid domains, we consider a thin coating material occupying the volume Ω η , where η represents an upper bound of the (local) thickness of the coating domain. As a basic assumption of the following developments, we assume that η is significantly smaller that the wavelengths of interest, hence justifying the label of "coating" domain.

Additionally, we consider an extruded structure of Ω η that we detail in this subsection in a 2D context for simplicity -the extension for N = 3 being conceptually straightforward. Let Γ represents the "middle" line of the domain characterized by its parametric coordinate transformation ψ : [0; 1] → x ∈ Γ, assumed to be a C 1 -diffeomorphism. Let Γ ± be the "upper" and "lower" boundary lines of the domain. They are obtained from the middle line through the transformations ϕ η ± : x → x ′ ∈ Γ ± , i.e.

Γ ± = x ′ x ′ = ϕ η ± ψ(u) , ∀u ∈ [0; 1] .
These transformations depend on the upper bound of the thickness η and are also assumed to be C 1 -diffeomorphisms. The remaining "left" and "right" boundary lines of the domain denoted by Γ 0 and Γ 1 are segments defined as

∀β ∈ {0, 1}, Γ β = x ′ x ′ = vϕ η + ψ(β) + (1 -v)ϕ η -ψ(β) , ∀v ∈ [0; 1] ,
and we define the domain Ω η as the open set with boundary 1 for a representation of such configurations.

∂Ω η = Γ + ∪ Γ -∪ Γ 0 ∪ Γ 1 -see Figure

Elastic and acoustic linear wave propagation models

Within the solid domain, we consider u the space and time dependent material displacement generated by the high-frequency and low amplitude waves during their course. Internal mechanical stresses σ are associated to this displacement. As standard in ultrasonic wave propagation modeling, we assume a fully linear constitutive law, and in the particular case of this communication we consider -without loss of generality -an isotropic material. In this context we recover Hooke's law

σ(u) = λtr(ε(u))I + 2µε(u), ε(u) = 1 2 ∇u + ∇u ⊺ ,
where ε(u) is the linearized Green -Lagrange tensor associated to the displacement u, tr(•) is the trace of a second order tensor, I is the identity second order tensor, and {λ, µ} are the Lamé's coefficients. In addition to this constitutive law, we recall the standard linear elasto-dynamic equation

ρ s ∂ 2 tt u -∇ • σ(u) = 0, in Ω s , (1)
where ∇• is the divergence operator applied to second order tensors, and ρ s is the mass density of the material. Within the fluid domain, we consider v the velocity field and p the pressure field associated to the acoustic wave propagation. They are linked through the standard wave equations

1 ρ f c 2 ∂ t p + ∇ • v + f = 0, ρ f ∂ t v + ∇p = 0, in Ω f , (2)
where f is a time and space dependent source term representing some known acoustic emitter, ∇• is the divergence operator applied to vector fields, c is the speed of waves in the fluid domain, and ρ f is the mass density of the fluid. Defining ϕ the acoustic velocity potential such that

v = 1 ρ f ∇ϕ, p = -∂ t ϕ, (3)
where the second relation is obtained using (2), we recover the traditional second order wave equation

1 ρ f c 2 ∂ 2 tt ϕ - 1 ρ f ∆ϕ = f, in Ω f . (4)

Effective transmission conditions to represent the coating material

To represent the propagation within the coating material, we resort to effective transmission conditions (ETC) as in [START_REF] Rokhlin | Analysis of boundary conditions for elastic wave interaction with an interface between two solids[END_REF] that we adapt to fluid -solid coupling. To simplify the presentation, we will use extensively the following notation

∀d ∈ {1, N }, ∀f ∈ [L 2 (Γ ± )] d , f ± = f • ϕ η ± ∈ [L 2 (Γ)] d .
In particular, we define the mechanical fluxes associated to the fluid and solid fluxes as

τ + = -p + n, τ -= σ -• n, on Γ.
The mean value and jump of the fluxes are denoted by

⟨τ ⟩ = τ + + τ - 2 , [τ ] = τ + -τ -.
In the same manner, we define the mean value and jump of velocities on Γ as

⟨v⟩ = v + + ∂ t u - 2 , [v] = v + -∂ t u -.
Without the presence of the coating layer, implying in particular that ϕ η ± = I, satisfying the standard fluid -solid transmission conditions implies

[v] • n = 0, [τ ] = 0, on Γ,
which is essentially the continuity of the normal velocities and the mechanical fluxes. Note that, because of the form of the fluid flux, this is equivalent to satisfying

[v] • n = 0, [τ ] • n = 0, τ -• n ⊥ = 0, on Γ,
where n ⊥ is any tangent vector of Γ. The underlying idea behind ETC, as in [START_REF] Rokhlin | Analysis of boundary conditions for elastic wave interaction with an interface between two solids[END_REF][START_REF] Lombard | How to incorporate the spring-mass conditions in finite difference schemes[END_REF], is to perturb the two jump relations with spring -mass terms using the mean fluxes and velocities, namely

[v] • n = ηs ∂ t ⟨τ ⟩ • n, [τ ] • n = ηm ∂ t ⟨v⟩ • n, on Γ. ( 5
)
Here s and m are the effective normal compliance and mass density of the coating layer. As an illustrative example, in the case of an isotropic coating material in 2D admitting {λ ′ , µ ′ } as Lamé's parameters and ρ ′ as mass density, a typical choice of effective parameters is

s = 1 λ ′ + 2µ ′ , m = ρ ′ . (6)
In the following, without loss of generality and to simplify the presentation, we will assume that both the thickness η and the effective parameters {s, m} are constant w.r.t. space variables.

Mortar elements and coupled weak formulation

Incorporating such transmission conditions (especially in a potentially non-conform context) is not self-evident. To address this challenge we resort to the mortar element method [START_REF] Ben Belgacem | The mortar element method for three dimensional finite elements[END_REF]. It is originally a domain decomposition approach enabling coupling of non-conforming space discretizations. Recently [START_REF] Imperiale | Numerical modeling of wave propagation in anisotropic viscoelastic laminated materials in transient regime: Application to modeling ultrasonic testing of composite structures[END_REF] it has been used to incorporate ETC between two solids and we extend this approach to fluid -solid coupling. To start with, we define the following unknowns

∂ t λ f = v + • n, ∂ t λ s = τ -• n, (7)
which are, for any fixed time value, elements of L 2 (Γ). Using this definition and the jump relations (5) we have

∂ t λ f -∂ t u -• n = ηs ∂ 2 tt ϕ + + ∂ 2 tt λ s 2 , ∂ t ϕ + -∂ t λ s = ηm ∂ 2 tt λ f + ∂ 2 tt u -• n 2 ,
which can be integrated over time to reduce the order of time derivation. Satisfying these conditions weakly, and combining with the weak form of formulations ( 1) and (4) yields the following fully coupled weak formulation:

Find {ϕ(t), u(t), λ f (t), λ s (t)} ∈ H 1 (Ω f ) × [H 1 (Ω s )] N × L 2 (Γ) × L 2 (Γ) such that d 2 dt 2 Ω f 1 ρ f c 2 ϕ ϕ ⋆ dΩ + Ω f 1 ρ f ∇ϕ • ∇ϕ ⋆ dΩ + d dt Γ λ f ϕ ⋆ + dΓ = Ω f f ϕ ⋆ dΩ, d 2 dt 2 Ωs ρ s u • u ⋆ dΩ + Ωs σ(u) : ε(u ⋆ ) dΩ - d dt Γ λ s u ⋆ -• n dΓ = 0,
ETC between fluid and solid domains using mortar elements -Alexandre Imperiale

FULLY DISCRETE SCHEME

Space discretization using spectral finite elements

We discretize in space the weak formulation (8) using spectral finite elements [START_REF] Cohen | Higher-order numerical methods for transient wave equations[END_REF]. In its original form, it is a H 1 -conform, high order finite element method, based on quadrangular or hexahedral meshes, with the Gauss -Lobatto distribution of nodes in the reference element. Let us denote by T h (Ω f ) and T H (Ω s ) conform meshes of the fluid and solid domains with characteristic mesh sizes h and H. In most cases, the gap between wave speeds in the two domains may require two different mesh sizes to obtain better performances. Therefore, in the following we assume that h ≤ H, potentially entailing non-conformities at the coupling interface. For each element K in a mesh, we denote by F K the associated transformation of the reference quadrangle or hexahedron K. On the reference element, we consider Q α ( K) the space of polynomials of maximal order α ∈ N. Let p and q be two given natural integers, we define the following finite element approximation spaces

V h = v h ∈ C 0 (Ω f ) ∀K ∈ T h (Ω f ), ∃! v ∈ Q p ( K), v h | K = v • F -1 K ⊂ H 1 (Ω f ), W H = w H ∈ C 0 (Ω s ) ∀K ∈ T H (Ω s ), ∃! w ∈ Q q ( K), w H | K = w • F -1 K ⊂ H 1 (Ω s ),
and the vectorial counterpart

W H = [W H ] N ⊂ [H 1 (Ω s )] N .
Let N f and N s be the dimension of the approximation spaces V h and W H respectively, we consider

V h = span {φ I } N f I=1 , W H = span {ψ I } Ns /N I=1 , W H = span {ψ I } Ns I=1 ,
the associated Lagrange basis functions. Using these notations, we introduce the mass and stiffness matrices

M f = Ω f 1 ρ f c 2 φ I φ J dΩ N f I,J=1 , M s = Ωs ρ s ψ I • ψ J dΩ Ns I,J=1
,

K f = Ω f 1 ρ f ∇φ I • ∇φ J dΩ N f I,J=1 , K s = Ωs σ(ψ I ) : ε(ψ J ) dΩ Ns I,J=1
.

For the mortar unknowns, we consider the approximation space built from the trace of the scalar volume space W H . More precisely, we define

X H = span {ξ ℓ } N Γ ℓ=1 ⊂ L 2 (Γ), ∀ℓ = 1, • • • , N Γ , ∃! I ∈ {1, • • • , Ns /N}, ξ ℓ = ψ I | Γ -• ϕ η -,
and N Γ the number of nodes on Γ -. With these basis functions, we define the coupling matrices (from mortar to volume unknowns) and the surface mass matrix

C f = Γ φ I+ ξ ℓ dΓ N f ,N Γ I,ℓ=1 , C s = Γ ψ I-• n ξ ℓ dΓ Ns,N Γ I,ℓ=1 , M Γ = Γ ξ ℓ ξ m dΓ N Γ ℓ,m=1 .

Time discretization and stability condition

Let us consider ∆t > 0 a given time step -satisfying a specific stability condition discussed later on -and n ∈ N any step index. We consider -→ ϕ n ∈ R N f the vector of components of the discrete velocity potential in the basis of V h at time t n = n∆t. In the same manner we define

- → u n ∈ R Ns , - → λ n f ∈ R N Γ and - → λ n s ∈ R N Γ the
vector of degrees of freedom in the basis of the relevant approximation space. With these notations, for the time discretization, we apply an explicit second order time-scheme centered at t n for the volume propagators, and a second order time-scheme centered at t n+ 1 /2 for the transmission conditions, namely

1 ∆t 2 M f - → ϕ n+1 -2 - → ϕ n + - → ϕ n-1 + K f - → ϕ n + 1 2∆t C f - → λ n+1 f - - → λ n-1 f = - → f , 1 ∆t 2 M s - → u n+1 -2 - → u n + - → u n-1 + K s - → u n -1 2∆t C s - → λ n+1 s - - → λ n-1 s = 0, ηm 2∆t M Γ - → λ n+1 f - - → λ n f + ηm 2∆t C ⊺ s - → u n+1 -- → u n = 1 2 C ⊺ f - → ϕ n+1 + - → ϕ n -1 2 M Γ - → λ n+1 s + - → λ n s , ηs 2∆t C ⊺ f - → ϕ n+1 - - → ϕ n + ηs 2∆t M Γ - → λ n+1 s - - → λ n s = 1 2 M Γ - → λ n+1 f + - → λ n f -1 2 C ⊺ s - → u n+1 + - → u n . (9)
In the first line of ( 9), the values of the right-hand are simply given by the L 2 -inner product of the source function f with the basis functions of V h . Without carrying a complete analysis of the stability for ( 9), one can expect -since the dynamics satisfied by the mortar unknowns is discretized by an implicit scheme -to be able to derive the following stability condition ∆t ≤ min

   2 r(M -1 f K f ) , 2 r(M -1 s K s )    , (10)
where r(•) is the spectral radius of a matrix. Note that the two terms in the right hand side of the previous inequality are the stability conditions of the fully discrete fluid and solid propagators taken independently. In other words, if this condition would be sufficient to insure stability, it would mean that the time step is independent on both the coating layer thickness and the corresponding effective parameters.

NUMERICAL ILLUSTRATIONS

To illustrate the efficiency of our approach we consider a simple 2D case, where the fluid domain is composed of water and the solid domain is composed of a steel material, i.e.

ρ f = 1.0 g • cm -3 , c = 1.5 mm • µs -1 , ρ s = 8.0 g • cm -3 , λ = 144 GPa, µ = 72 GPa.
The source term is a line source in the fluid with a central frequency of 1.0 MHz. The size of the domains are chosen to be proportionate to the pressure wavelengths, in particular the fluid domain is the square 25λ f P × 25λ f P , while the solid domain is the rectangle 25λ f P × 25λ s P . The thickness of the coating layer is set to be η = λ f P/10 and the material of the layer is twice as stiff as the solid domain, i.e.

ρ ′ = ρ s , λ ′ = 2λ, µ ′ = 2µ, from which we can derive the effective transmission coefficients using [START_REF] Bonnet | Effective transmission conditions for thinlayer transmission problems in elastodynamics. The case of a planar layer model[END_REF]. Concerning the discretization parameter we use spectral elements of order five with three elements per λ f P in the fluid domain. For the solid domain we consider both cases of conform discretization (where we impose the same discretization as the fluid domain) and non-conform discretization where the order is still five but we set three elements per λ s P . We carry out numerical simulations by computing the upper bound on the time step provided in [START_REF] Lombard | How to incorporate the spring-mass conditions in finite difference schemes[END_REF].

In Figure 2 we represent this configuration in addition with two snapshots where we compare: the reference simulation with conform interface and a meshed coating layer, a simulation using ETC with a conform interface and finally a simulation using ETC with a non-conform interface. In these snapshots we can see a very good agreement when comparing reflected pressure wave in the fluid and bulk waves in the solid (both pressure and transverse waves). We observe some discrepancies for the surface waves. These discrepancies appear even in the conform case, implying that the ETC cannot adequately capture these contributions.

To better validate the proposed numerical scheme we propose in Figure 3 to extract over time the solution at two observation points. As depicted in Figure 2, we position a first point near the interface and a second point far from the interface, in the solid domain. Even though the time step for each discretization is different, we observe a very good agreement between the three numerical solutions, with once again some discrepancies at the observation point located near the interface.

In Figure 4 we propose a graphic representation of the performances of the three different schemes. Comparing the reference with the simulation using ETC with a conform interface we observe an identical cost per iteration, as expected, but a significant improvement on the time step, also expected since the thin coating layer is not meshed. Finally, comparing the reference with the simulation using ETC with a non-conform interface we observe an important gain in both cost per iteration and time step, leading to an overall speedup of approximately ×15 for a fixed global time window.

CONCLUSIONS AND PERSPECTIVES

In this communication we have presented a way to incorporate effective transmission conditions between a fluid wave propagator and a solid wave propagator using the mortar element method. This can typically be used to efficiently represent a thin layer between the fluid and solid domains. By using an implicit time scheme, the numerical illustrations suggest that the CFL condition of the presented numerical scheme becomes (compared to standard explicit discretization) independent of the thin layer thickness and the effective parameters. Additionnaly, relying on the flexibility of the mortar element method we are able to incorporate non-conform space discretizations between the fluid mesh and the solid mesh. Combining these two assets, we are able to obtain significant performance gains on relevant numerical simulations compared to a reference approach based on a standard fully explicit scheme with a meshed coating layer.

From a theoretical point of view, the natural perspective of this work is to be able to prove that the CFL condition is independent of the effective parameters. From a more numerical angle, we should investigate how the proposed numerical scheme performs on 3D cases or cases with curved interfaces.

Figure 1 :

 1 Figure 1: Representation of the type of geometrical configuration of interest for N = 2. The hatched area, above the solid domain Ω s and below the fluid domain Ω f , represents the coating layer Ω η .

Figure 2 :

 2 Figure 2: Illustration and snapshots of the solutions obtained by three different approaches: (i) meshed coating layer with conform interface (i.e. reference); (ii) ETC with conform interface; (iii) ETC with non-conform interface. Red dots represent the position of the observation points.

Figure 3 :

 3 Figure 3: Comparing the time variation of the solution of the reference, ETC with conform interface and ETC with non-conform interface at the observation points P 0 (left) and P 1 (right).

Figure 4 :

 4 Figure 4: Comparing performances of the reference, ETC with conform interface and ETC with non-conform interface. The cost per iteration is divided by 1.52 and the time step is multiplied by 9.96, leading to an overall speedup of ×15.

d dt Γ ηm λ f + u -• n 2 λ ⋆ f dΓ = Γ (ϕ + -λ s )λ ⋆ f dΓ, d dt Γ ηs ϕ + + λ s 2 λ ⋆ s dΓ = Γ (λ f -u -• n)λ ⋆ s dΓ,(8)for any set of test functions {ϕ⋆ , u ⋆ , λ ⋆ f , λ ⋆ s } ∈ H 1 (Ω f ) × [H 1 (Ω s )] N × L 2 (Γ) × L 2 (Γ).