Frédéric Alauzet

Benoit Fabrèges

Miguel A Fernández

Mikel Landajuela

Nitsche-XFEM for the coupling of an incompressible fluid with immersed thin-walled structures

Keywords: fluid-structure interaction, incompressible fluid, thin-walled structure, unfitted meshes, fictitious domain method, XFEM, Nitsche's method, cupling schemes interaction fluide-structure, fluide incompressible, structure mince, maillages non compatibles, méthode de domaine fictif, XFEM, méthode de Nitsche, schémas de couplage

In this paper we introduce a Nitsche-XFEM method for fluid-structure interaction problems involving a thin-walled elastic structure (Lagrangian formalism) immersed in an incompressible viscous fluid (Eulerian formalism). The fluid domain is discretized with an unstructured mesh not fitted to the solid mid-surface mesh. Weak and strong discontinuities across the interface are allowed for the velocity and pressure, respectively. The fluid-solid coupling is enforced consistently using a variant of Nitsche's method with cut-elements. Robustness with respect to arbitrary interface intersections is guaranteed through suitable stabilization. Several coupling schemes with different degrees of fluid-solid time splitting (implicit, semi-implicit and explicit) are investigated. A series of numerical test in 2D, involving static and moving interfaces, illustrates the performance of the different methods proposed.

Introduction

Mathematical models describing the mechanical interaction of an incompressible viscous fluid with an immersed thin-walled flexible structure appear in a wide variety of engineering fields: from micro-encapsulation to the aeroelasticity of parachutes and sailing boats (see, e.g., [START_REF] Pozrikidis | Computational hydrodynamics of capsules and biological cells[END_REF][START_REF] Takizawa | Computational methods for parachute fluid-structure interactions[END_REF][START_REF] Lombardi | Numerical simulation of sailing boats: Dynamics, FSI, and shape optimization[END_REF]). Such multi-physics systems are also particularly ubiquitous in nature. One can think, for instance, of the wings of a bird interacting with the air, the fins of a fish moving through the water, or the opening/closing dynamics of heart valves when blood is propelled into the arteries (see, e.g., [START_REF] Tian | Fluid-structure interaction involving large deformations: 3D simulations and applications to biological systems[END_REF][START_REF] Müller | Fish and Flag -Exploring Fluid-Structure Interaction during Undulatory Swimming in Fish[END_REF]4]). The solid is deformed under the action of the fluid and the fluid flow is disturbed by the moving solid.

These problems are generally modeled by heterogeneous (parabolic/hyperbolic) systems of equations with different types of constitutive and geometrical non-linearities. This complicates the analysis both from the mathematical and numerical standpoint. In addition, the thin-walled nature of the immersed solid introduces jumps on the fluid stresses which, respectively, results in weak and strong discontinuities of the velocity and pressure fields. Standard finite element approximations, not allowing for such discontinuities, are known to deliver suboptimal convergence behavior and spurious numerical oscillations in the vicinity of the immersed solid (see, e.g., [START_REF] Diniz Dos Santos | A partitioned fluid-structure algorithm for elastic thin valves with contact[END_REF][START_REF] Kamensky | An immersogeometric variational framework for fluid-structure interaction: Application to bioprosthetic heart valves[END_REF][START_REF] Boffi | The finite element immersed boundary method with distributed Lagrange multiplier[END_REF])

The discontinuous features of the fluid solution can be straightforwardly incorporated within a standard finite element approximation by considering fitted fluid-solid meshes. It is well known, however, that maintaining fitted meshes may be cumbersome or unfeasible in presence of large interface deflections and topological changes (e.g., due to contacting solids). Though a number of advanced mesh update techniques have been reported in the literature (see, e.g., [START_REF] Stein | Automatic mesh update with the solid-extension mesh moving technique[END_REF][START_REF] Yigit | Grid movement techniques and their influence on laminar fluid-structure interaction computations[END_REF][START_REF] Wick | Fluid-structure interactions using different mesh motion techniques[END_REF][START_REF] Alauzet | A changing-topology moving mesh technique for large displacements[END_REF][START_REF] Takizawa | Space-time interface-tracking with topology change (ST-TC)[END_REF]), the favoured alternative is to consider an unfitted mesh formulation, in which the fluidstructure interface moves independently of a background fluid mesh. Among these approaches, we can mention the Immersed Boundary/Fictitious Domain methods (e.g., [START_REF] Peskin | The immersed boundary method[END_REF][START_REF] Glowinski | A distributed Lagrange mutiplier/fictitious domain method for particulate flows[END_REF][START_REF] Baaijens | A fictitious domain/mortar element method for fluid-structure interaction[END_REF][START_REF] Zhang | Immersed finite element method[END_REF][START_REF] Diniz Dos Santos | A partitioned fluid-structure algorithm for elastic thin valves with contact[END_REF]4,[START_REF] Boffi | Finite element approach to immersed boundary method with different fluid and solid densities[END_REF][START_REF] Kamensky | An immersogeometric variational framework for fluid-structure interaction: Application to bioprosthetic heart valves[END_REF][START_REF] Boffi | The finite element immersed boundary method with distributed Lagrange multiplier[END_REF]) and the methodologies based on a fully Eulerian description of the problem (e.g., [START_REF] Cottet | Eulerian formulation and level set models for incompressible fluid-structure interaction[END_REF][START_REF] Richter | A fully Eulerian formulation for fluid-structure-interaction problems[END_REF]). In general, these methods are known to be inaccurate in space due to the continuous nature of the fluid approximations across the interface or to the discrete treatment of the interface conditions. The current trend to overcome these consistency issues is to combine a local XFEM enrichment with a cut-FEM methodology and a Lagrange multiplier treatment of the interface coupling (see, e.g., [START_REF] Legay | An Eulerian-Lagrangian method for fluid-structure interaction based on level sets[END_REF][START_REF] Zilian | The enriched space-time finite element method (EST) for simultaneous solution of fluid-structure interaction[END_REF][START_REF] Gerstenberger | An extended finite element method/Lagrange multiplier based approach for fluid-structure interaction[END_REF][START_REF] Sawada | LLM and X-FEM based interface modeling of fluid-thin structure interactions on a non-interface-fitted mesh[END_REF]). The price to pay, with respect to the original IB and FD methods, is the need of a specific tracking of the interface intersections (see, e.g., [START_REF] Mayer | Interface handling for three-dimensional higher-order XFEM-computations in fluid-structure interaction[END_REF][START_REF] Wang | Computational algorithms for tracking dynamic fluid-structure interfaces in embedded boundary methods[END_REF][START_REF] Massing | Efficient implementation of finite element methods on nonmatching and overlapping meshes in three dimensions[END_REF]) and a loss of robustness with respect to how the interface intersects the background fluid mesh (see, e.g., [START_REF] Fries | The extended/generalized finite element method: an overview of the method and its applications[END_REF][START_REF] Burman | CutFEM: discretizing geometry and partial differential equations[END_REF]).

A well-known alternative to the discrete treatment of the interface conditions via Lagrange multipliers is Nitsche's method (see, e.g., [START_REF] Nitsche | Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind[END_REF][START_REF] Stenberg | On some techniques for approximating boundary conditions in the finite element method[END_REF][START_REF] Hansbo | Nitsche's method for interface problems in computational mechanics[END_REF]). Because of its flexibility and mathematical soundness, the Nitsche mortaring has been applied to the design of numerical methods for a number of interface problems, including XFEM for elasticity [START_REF] Hansbo | An unfitted finite element method, based on Nitsche's method, for elliptic interface problems[END_REF][START_REF] Hansbo | A finite element method for the simulation of strong and weak discontinuities in solid mechanics[END_REF][START_REF] Becker | A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity[END_REF], XFEM for two-phase transport problems [START_REF] Lehrenfeld | Nitsche-XFEM with streamline diffusion stabilization for a two-phase mass transport problem[END_REF][START_REF] Lehrenfeld | Analysis of a Nitsche XFEM-DG discretization for a class of two-phase mass transport problems[END_REF], XFEM for incompressible flow [START_REF] Schott | A new face-oriented stabilized XFEM approach for 2D and 3D incompressible Navier-Stokes equations[END_REF] and robust and accurate FD methods for elliptic and mixed problems [START_REF] Burman | Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method[END_REF][START_REF] Massing | A stabilized Nitsche fictitious domain method for the Stokes problem[END_REF][START_REF] Burman | Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes' problem[END_REF]]. Nitsche's method was first applied to fluid-structure interaction problems with fitted meshes in [START_REF] Hansbo | Nitsche's method combined with space-time finite elements for ALE fluid-structure interaction problems[END_REF] and used to design stable explicit coupling (or loosely coupled) schemes in [START_REF] Burman | Stabilization of explicit coupling in fluid-structure interaction involving fluid incompressibility[END_REF][START_REF] Burman | Explicit strategies for incompressible fluid-structure interaction problems: Nitsche type mortaring versus Robin-Robin coupling[END_REF]. It has recently been extended to fluid-structure interaction problems with unfitted meshes in [START_REF] Burman | An unfitted Nitsche method for incompressible fluidstructure interaction using overlapping meshes[END_REF], yielding robust and optimal a priori error estimates (fixed interface). In [START_REF] Burman | An unfitted Nitsche method for incompressible fluidstructure interaction using overlapping meshes[END_REF], the case of the coupling with thin-walled solids is limited to structures surrounding the fluid domain (i.e., not immersed).

The first contribution of this paper consists in the introduction of a robust and accurate Nitsche-XFEM method for fluid-structure interaction problems involving a thin-walled elastic structure immersed in an incompressible viscous fluid. We consider an Eulerian description for the fluid and a Lagrangian formulation for the solid. The fluid domain is discretized with an unstructured mesh not fitted to the solid mid-surface deformed mesh. In this unfitted mesh framework, the (strong) consistency of the proposed fluid-solid coupling builds on the following two ingredients:

• across the interface, locally enriched piecewise affine fluid velocity and pressure approximations respectively allow for weak and strong discontinuities (using the XFEM approach of [START_REF] Hansbo | A finite element method for the simulation of strong and weak discontinuities in solid mechanics[END_REF][START_REF] Becker | A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity[END_REF]);

• the kinematic/dynamic fluid-solid coupling is enforced through a fluid-sided Nitsche's mortaring (based on [START_REF] Burman | An unfitted Nitsche method for incompressible fluidstructure interaction using overlapping meshes[END_REF]).

Besides, consistent symmetric stabilization operators are added to guarantee robustness with respect to arbitrary interface/element intersections (see, e.g., [START_REF] Burman | Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes' problem[END_REF]) and to circumvent the classical inf-sup and convective related instabilities (see, e.g., [START_REF] Brezzi | On the stabilization of finite element approximations of the Stokes equations[END_REF][START_REF] Burman | Continuous interior penalty finite element method for Oseen's equations[END_REF][START_REF] Burman | Continuous interior penalty finite element method for the time-dependent Navier-Stokes equations: space discretization and convergence[END_REF][START_REF] Schott | A new face-oriented stabilized XFEM approach for 2D and 3D incompressible Navier-Stokes equations[END_REF]). In this regard, it is worth noting that for robustness these operators act on the fictitious region of the computational domain, without compromising the overall optimal accuracy of the method (in the energy norm).

The second contribution has to do with the time-discretization. Several coupling schemes with different levels of fluid-solid splitting are proposed: implicit, explicit and semi-implicit. The stability and convergence properties of the resulting fully discrete methods are analyzed within a representative linear setting (static interfaces). The salient features of the semi-implicit schemes introduced in this paper are twofold: (i) they deliver superior stability and accuracy with respect to alternative methods of explicit nature (see, e.g., [START_REF] Boffi | Finite element approach to immersed boundary method with different fluid and solid densities[END_REF]); (ii) they avoid the strong coupling of alternative semi-implicit coupling schemes (see, e.g., [START_REF] Newren | Unconditionally stable discretizations of the immersed boundary equations[END_REF][START_REF] Boffi | The finite element immersed boundary method with distributed Lagrange multiplier[END_REF]) without compromising stability and accuracy.

Finally, the theoretical findings are substantiated by a series of numerical examples in 2D, involving static and moving interfaces, which illustrate the performance of the methods proposed by comparing with analytic solutions and fitted mesh approaches.

The rest of the paper is organized as follows. Section 2 is devoted to the derivation and the analysis of the methods within a linear setting (fixed interface). The space semi-discrete Nitsche-XFEM formulation is introduced in Section 2.1, Section 2.2 presents the time discretization and the different coupling schemes. In Section 3, the numerical methods are formulated within a non-linear setting involving moving interfaces. Numerical evidence illustrating the performance of the methods proposed is reported and discussed in Section 4. Finally, a summary of the conclusions and some directions of further investigation are given in Section 5.

Linear model problem: static interface

We consider a fluid-structure interaction problem in which the fluid is described by the Stokes equations and the structure by a linear thin membrane or shell model. The fluid domain is denoted by Ω ⊂ R d (d = 2, 3). The structure is assumed to be immersed within Ω, with its mid-surface represented by the oriented manifold Σ ⊂ Ω of codimension 1 and unitary normal vector n. For the time being, we assume that Σ divides Ω into two open domains Ω 1 and Ω 2 (see Figure 1). In Section 2.1.3 we address the general case in which Ω is partially intersected by Σ. We denote the outward unit normal to Ω i on Σ by n i , i = 1, 2. Note that we choose Ω 1 and Ω 2 so that n 1 = n and n 2 = -n. We set

Γ i = ∂Ω i \Σ, i = 1, 2.
In the following we will make extensive use of the following definition. For a given continuous scalar or tensorial field f defined in Ω (possibly discontinuous across Σ) we define its sided- restrictions to Σ, noted by f 1 and f 2 , as

Inria

f 1 (x) def = lim ξ→0 + f (x + ξn 1 ), f 2 (x) def = lim ξ→0 + f (x + ξn 2 ) ∀x ∈ Σ.
We also define the following jumps and average across the interface Σ:

f def = f 1 -f 2 , f n def = f 1 n 1 + f 2 n 2 , { {f } } def = 1 2 (f 1 + f 2 ) .
The coupled problem reads as follows: find the fluid velocity and pressure u :

Ω × R + → R d , p : Ω × R + → R, the solid displacement and velocity d : Σ × R + → R d , . d : Σ × R + → R d such that      ρ f ∂ t u -divσ(u, p) = 0 in Ω i × R + , i = 1, 2 divu = 0 in Ω i × R + , i = 1, 2 u = 0 on Γ i × R + , i = 1, 2 (1)
           u 1 = u 2 = . d on Σ × R + , ρ s ∂ t . d + Ld = -σ(u, p)n in Σ × R + , . d = ∂ t d in Σ × R + , d = 0 on ∂Σ × R + , (2)
complemented with standard initial conditions u(0

) = u 0 , d(0) = d 0 and . d(0) = . d 0 .
Here, the constants ρ f and ρ s stand for the fluid and solid densities, respectively, while denotes the solid thickness. The fluid Cauchy-stress tensor is given by σ(u, p) 2) the solid is geometrically modeled as a surface of co-dimension one, it does not displace any fluid. In presence of gravitational forcing, the relation (2) 1 should hence be replaced by

ρ s ∂ t . d + Ld = -σ(u, p)n + (ρ s -ρ f ) g in Σ × R + ,
where g denotes the gravitational acceleration field and the term -ρ f g models the buoyant force acting on the solid (i.e, the displaced fluid).

We will make use of the standard Sobolev spaces H m (ω) (m ≥ 0), with norm • m,ω . The closed subspaces H 1 Γ (ω), of functions in H 1 (ω) with zero trace on Γ, and L 2 0 (ω), of functions in L 2 (ω) with zero mean in ω, will also be used. The scalar product in L 2 (ω) is denoted by (•, •) ω and its norm by • 0,ω . We consider

V def = [H 1 Γ (Ω)] d and Q def = L 2 0
(Ω) as the fluid velocity and pressure functional spaces, respectively. The space W ⊂ [H 1 0 (Σ)] d denotes the space of solid admissible displacements.

The standard Stokes bi-linear form

a f Ω (u, p), (v, q) def = 2µ (u), (v) Ω -(p, divv) Ω + (q, divu) Ω ,
will be used. The elastic bi-linear form a s : W × W → R will represent the weak form of the (unbounded linear) surface differential operator L :

D(L) ⊂ [L 2 (Σ)] d → [L 2 (Σ)] d , namely, a s (d, w) = (Ld, w) Σ
for all d ∈ D(L) and w ∈ W . We assume a s to be symmetric, coercive and continuous on W with associated norm w s def = (a s (w, w))

1 2
. The weak form of the linear coupled problem (1)-( 2) reads as follows: for t > 0, find (u(t), p(t), d(t),

.

d(t)) ∈ V × Q × W × W such that u 1 | Σ = u 2 | Σ = . d, . d = ∂ t d, ρ f ∂ t u, v Ω + a f Ω (u, p), (v, q) + ρ s ∂ t . d, w Σ + a s (d, w) = 0, (3)
for all (v, q, w) 3) we get, for t > 0, the following standard energy identity

∈ V × Q × W with v| Σ = w. Taking (v, q, w) = (u(t), p(t), . d(t)) in (
ρ f 2 u(t) 2 0,Ω + ρ s 2 . d(t) 2 0,Σ + 1 2 d(t) 2 s + 2µ t 0 (u(s)) 2 0,Ω ds = ρ f 2 u 0 2 0,Ω + ρ s 2 . d 0 2 0,Σ + 1 2 d 0 2 s . (4)

Space semi-discretization

The dynamic relation (2) 2 introduces jumps in the fluid pressure and in the velocity gradient across the interface Σ. We propose to approximate the fluid velocity and pressure on triangulations of Ω which are independent of the interface Σ (see Figure 2(a)). In order to guarantee the optimality of the approximations, we allow the discrete fluid solution to be discontinuous inside the elements which are intersected by the interface. In this unfitted framework, the interface coupling conditions (2) 1,2 will be enforced through a Nitsche's type mortaring.

Nitsche-XFEM semi-discrete formulation

For the sake of simplicity, we assume that Ω and Σ are polyhedral. For the construction of the discrete approximation spaces in the fluid, we follow the unfitted approach reported in [START_REF] Hansbo | A finite element method for the simulation of strong and weak discontinuities in solid mechanics[END_REF][START_REF] Becker | A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity[END_REF].

To this purpose, we consider two family of meshes {T f i,h } 0<h<1 , i = 1, 2, where each T f i,h covers the fluid region Ω i . Each mesh T f i,h is fitted to the exterior boundary Γ i but, in general, not to Σ (see Figure 2(b)). Moreover, we assume that for every element

K ∈ T f 1,h ∩ T f 2,h we have K ∩ Σ = ∅. Note that T f 1,h ∪ T f 2,h
gives a conforming triangulation of the whole fluid domain Ω. We denote by Ω i,h the domain covered by T f i,h , viz.,

Ω i,h def = int ∪ K∈T f i,h K .

Inria

Finally, for the solid, we consider a family of triangulations {T s h } 0<h<1 of Σ. For the sake of simplicity, we assume that the three families of triangulations are quasi-uniform. We introduce the following standard spaces of continuous piecewise affine functions:

X f i,h def = v h ∈ C 0 (Ω i,h ) v h|K ∈ P 1 (K) ∀K ∈ T f i,h , X s h def = v h ∈ C 0 (Σ) v h|K ∈ P 1 (K) ∀K ∈ T s h . (5)
Associated with X f i,h , we define the spaces

V i,h def = v i,h ∈ [X f i,h ] d v i,h | Γi = 0 , Q i,h def = X f i,h , i = 1, 2. (6)
For the approximation of the fluid velocity and pressure we will consider the product spaces

V h def = V 1,h × V 2,h and Q h def = Q 1,h × Q 2,h
, respectively. The solid displacement and velocity

are approximated in W h def = [X s h ] d ∩ W . Remark 2.2.
Note that the discrete velocity and pressure are two-valued in the overlap interfacial region. Hence, for

f h = (f 1,h , f 2,h ) ∈ X f 1,h × X f 2,h , the associated function f Ω,h def = f 1,h | Ω1 in Ω 1 , f 2,h | Ω2 in Ω 2 ,
may develop discontinuities across the interface Σ. This feature is illustrated in Figures 3 and5. This is one of the fundamental ingredients of the present XFEM approach to approximate the solution of (1)-(2).

Since the velocity/pressure discrete spaces V h /Q h fail to satisfy the inf-sup condition, we consider a symmetric pressure stabilization operator s h : Q h ×Q h → R, for instance, the classical Brezzi-Pitkäranta method (see, e.g.,. [START_REF] Brezzi | On the stabilization of finite element approximations of the Stokes equations[END_REF])

s h (p h , q h ) def = γ p h 2 µ i=1,2 ∇p i,h , ∇q i,h Ω i,h , (
f h ∈ X f 1,h × X f 2,h .
with γ p > 0 a user-defined parameter. Note that the stabilization acts on the whole computational domain Ω 1,h × Ω 2,h . In order to guarantee robustness of the method with respect to the way the fluid mesh T f 1,h ∪T f 2,h is intersected by the solid mesh T s h , we consider the ghost-penalty stabilization operator

g h : V h × V h → R defined by g h (u h , v h ) = γ g µh 2 i=1 F ∈F Σ i,h ∇u i,h F , ∇v i,h F F ,
where γ g > 0 is a user-defined parameter and the symbol F denotes the jump across the edge or face F belonging to F Σ i,h , the set of interior edges or faces of the elements of T f i,h intersected by Σ, i.e., F ∈ F Σ i,h if F is a (d -1)-manifold and there exist [START_REF] Burman | Ghost penalty[END_REF], it follows that this operator extends the natural H 1 -coercivity in the physical domain to the whole Ω 1,h × Ω 2,h , viz.,

K 1 , K 2 ∈ T f i,h , with K 1 ∩ Σ = ∅ or K 2 ∩ Σ = ∅, such that F = K 1 ∩ K 2 . From
cs 2 i=1 µ (v i,h ) 2 0,Ω i,h + g h (v h , v h ) ≤ µ 2 i=1 (v i,h ) 2 0,Ωi + g h (v h , v h ), (7)
for all v h ∈ V h , with cs > 0 depending on γ g . The total stabilization operator is hence given by

S h (u h , p h ), (v h , q h ) def = s h (q h , q h ) + g h (u h , v h ), with the associated semi-norm |(v h , q h )| S def = S h (v h , p h ), (v h , q h ) .
We also introduce the fluid discrete bi-linear form

a f h (u h , p h ), (v h , q h ) def = a f Ω1∪Ω2 (u h , p h ), (v h , q h ) + S h (u h , p h ), (v h , q h ) . (8)
The proposed space semi-discrete approximation of (1)-(2) reads as follows: for t > 0, find

u h (t), p h (t), . d h (t), d h (t) ∈ V h × Q h × W h × W h , Inria such that . d h (t) = ∂ t d h (t) and ρ f ∂ t u h , v h Ω + a f h (u h , p h ), (v h , q h ) + ρ s ∂ t . d h , w h Σ + a s (d h , w h ) - 2 i=1 σ(u i,h , p i,h )n i , v i,h -w h Σ - 2 i=1 u i,h - . d h , σ(v i,h , -q i,h )n i Σ + γµ h 2 i=1 u i,h - . d h , v i,h -w h Σ = 0 (9)
for all

(v h , q h , w h ) ∈ V h × Q h × W h .
Here, γ > 0 is a positive parameter given by Lemma 2.1 below (see also Remark 2.5).

Remark 2.3. It should be noted that, unlike [47, Section 4.1], the interface fluid tractions from opposite sides do not cancel in [START_REF] Braack | Stabilized finite element methods for the generalized Oseen problem[END_REF]. This is a direct consequence of the XFEM nature of the velocity/pressure space V h × Q h , which guarantees the strong consistency of (9) with (1)-(2).

Remark 2.4. In the case of a non-polyhedral interface (see Figure 1), Σ and a s (•, •) in (9) have to be replaced by their corresponding h-dependent approximations Σ h and a s h (•, •), respectively.

The above space semi-discretized formulation can be viewed as an extension of the unfitted method recently introduced in [START_REF] Burman | An unfitted Nitsche method for incompressible fluidstructure interaction using overlapping meshes[END_REF] to the case of coupling with immersed thin-walled structures. In the next section, we will build on this relation to briefly discuss the stability and convergence properties of (9).

Stability and convergence

The next lemma exploits property [START_REF] Boffi | Finite element approach to immersed boundary method with different fluid and solid densities[END_REF] to guarantee the coercivity of the Stokes-Nitsche's operator in [START_REF] Braack | Stabilized finite element methods for the generalized Oseen problem[END_REF]. Lemma 2.1. For γ > 0 sufficiently large, there exists a constant c s > 0 such that

c s µ 2 i=1 ∇v i,h 2 0,Ω i,h + γµ h 2 i=1 v i,h -w h 2 0,Σ + |(v h , q h )| 2 S ≤ a f h (v h , q h ), (v h , q h ) - 2 i=1 σ(v i,h , q i,h )n i , v i,h -w h Σ - 2 i=1 v i,h -w h , σ(v i,h , -q i,h )n i Σ + γµ h 2 i=1 v i,h -w h 2 0,Σ + |(v h , q h )| 2 S for all (v h , q h , w h ) ∈ V h × Q h × W h .

RR n°8723

Proof. The proof follows the arguments reported in [START_REF] Burman | An unfitted Nitsche method for incompressible fluidstructure interaction using overlapping meshes[END_REF]Lemma 3.1]. First, we have

a f h (v h , q h ), (v h , q h ) - 2 i=1 σ(v i,h , q i,h )n i , v i,h -w h Σ - 2 i=1 v i,h -w h , σ(v i,h , -q i,h )n i Σ + γµ h 2 i=1 v i,h -w h 2 0,Σ + |(v h , q h )| 2 S = 2µ 2 i=1 (v i,h ) 2 0,Ωi -2 2 i=1 σ(v i,h , 0)n, v i,h -w h Σ + 2 i=1 γµ h v i,h -w h 2 0,Σ + |(v h , q h )| 2 S .
Combining the Cauchy-Schwarz inequality with the following (robust) trace inequality

h (v i,h )n 2 0,Σ ≤ C T (v i,h ) 2 0,Ω i,h ∀v i,h ∈ V i,h , i = 1, 2, (10)
with C T > 0, we have

2 i=1 2σ(v i,h , 0)n, v i,h -w h Σ ≤ 8C T γ µ 2 i=1 (v i,h ) 2 0,Ω i,h + 1 2 2 i=1 γµ h v i,h -w h 2 0,Σ .
We conclude by using the strengthened stability [START_REF] Boffi | Finite element approach to immersed boundary method with different fluid and solid densities[END_REF] provided by the ghost-penalty operator , taking

γ > 8C T cs (11)
and using Korn's inequality.

Remark 2.5. The proof of Lemma 2.1 provides a lower bound for the parameter γ > 0 through the relation [START_REF] Brooks | Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations[END_REF]. The constants therein can be estimated automatically from the solution of generalized eigenvalue problems associated with the inequalities [START_REF] Boffi | Finite element approach to immersed boundary method with different fluid and solid densities[END_REF] and [START_REF] Brezzi | On the stabilization of finite element approximations of the Stokes equations[END_REF]. We refer to [START_REF] Harari | What are C and h?: inequalities for the analysis and design of finite element methods[END_REF]Section 3] for an overview of this approach, and to [START_REF] Ruess | Weak coupling for isogeometric analysis of non-matching and trimmed multi-patch geometries[END_REF][START_REF] Embar | Imposing Dirichlet boundary conditions with Nitsche's method and spline-based finite elements[END_REF] for applications in the context of Nitsche's method.

If we take v h = u h (t), q h = p h (t) and w h = . d h (t) in ( 9) and apply the result of Lemma 2.1 we get the following discrete counterpart of (4):

ρ f 2 u h (t) 2 0,Ω + ρ s 2 . d h (t) 2 0,Σ + 1 2 d h (t) 2 s + c s t 0 2 i=1 µ ∇u i,h (s) 2 0,Ω i,h + 2 i=1 γµ h u i,h (s) - . d h (s) 2 0,Σ + u h (s), p h (s) 2 S ds ≤ ρ f 2 u h (0) 2 0,Ω + ρ s 2 . d h (0) 2 0,Σ + 1 2 d h (0) 2 s , (12)
which guarantees the energy stability of (9). The following result states the optimal accuracy of (9) in the energy norm, under regularity assumptions on the solution of (1)- (2). The symbol denotes inequality up to a multiplicative constant independent of h.

Inria

Theorem 2.1. Let (u, p, . d, d) be the solution of (1)-( 2) and (u h , p h , . d h , d h ) be given by [START_REF] Braack | Stabilized finite element methods for the generalized Oseen problem[END_REF]. We assume that the interface Σ is flat and that

u ∈ [H 1 (R + ; H 2 (Ω 1 ∪ Ω 2 ))] d , p ∈ C 0 (R + ; H 1 (Ω 1 ∪ Ω 2 )), d, . d ∈ [H 1 (R + ; H 2 (Σ))] d .
For t > 0, there holds:

ρ f 2 (u h -u)(t) 2 0,Ω + ρ s 2 ( . d h - . d)(t) 2 0,Σ + 1 2 (d h -d)(t) 2 s + c s t 0 2 i=1 µ ∇(u i,h -u)(s) 2 0,Ωi + γµ h (u i,h - . d i,h )(s) 2 0,Σ ds h 2 ,
where γ > 0 is given by Lemma 2.1.

Proof. The result follows by applying the arguments presented in the proof of [START_REF] Burman | An unfitted Nitsche method for incompressible fluidstructure interaction using overlapping meshes[END_REF]Theorem 3.1] to each Ω i,h , i = 1, 2.

Partially intersected fluid domain

In this section we discuss how the space semi-discrete formulation (9) can be generalized to the case in which the interface Σ only partially intersects the domain Ω (see Figure 4(a)). In order to set up the new discrete spaces V h and Q h for the fluid, we consider a fictitious prolongation of Σ, denoted by Σ tip h , so that Σ ∪ Σ tip h divides Ω into two open domains (see Figure 4(a)). The fluid-fluid fictitious interface is defined in terms of the partition Σ tip h = Σ h ∪ Σ h where:

• The part Σ h is included in the set of elements containing the tip of Σ. Moreover, within each of these elements, it is defined as the prolongation of the interface up to the point which is opposite to the edge or face intersected by Σ.

• The part Σ h is aligned with the edges or faces of the mesh.

f h ∈ X f 1,h × X f 2,h .
We now proceed, as in Section 2.1.1, by introducing two overlapping triangulations T f i,h , i = 1, 2 (see Figure 4(b)). Note that the overlap region reduces to the set of elements intersected by Σ. The associated discrete spaces X f i,h , i = 1, 2, are then defined as in [START_REF] Baaijens | A fictitious domain/mortar element method for fluid-structure interaction[END_REF]. At last, the fluid velocity and pressure spaces are derived from ( 6) by strongly enforcing the continuity of the velocity and pressure across Σ h (see Figure 5), viz.:

V h def = v h = (v 1,h , v 2,h ) ∈ V 1,h × V 2,h v 1,h = v 2,h on Σ h , Q h def = q h = (q 1,h , q 2,h ) ∈ Q 1,h × Q 2,h q 1,h = q 2,h on Σ h . (13)
The functions of these spaces are continuous in the domain

Ω h def = Ω\(Σ ∪ Σ h ),
but discontinuous across the interface Σ ∪ Σ h . The fluid discrete bi-linear form ( 8) is hence redefined as

a f h (u h , p h ), (v h , q h ) def = a f Ω h (u h , p h ), (v h , q h ) + S h (u h , p h ), (v h , q h ) .
The continuity of velocity and stress across the fictitious fluid-fluid interface Σ h will be enforced in a consistent weak fashion, using Nitsche's method (see, e.g., [START_REF] Di Pietro | Mathematical aspects of discontinuous Galerkin methods[END_REF]Section 6.1.2]). In summary, the resulting semi-discrete approximation of ( 1)-( 2) reads as follows: for t > 0, find

u h (t), p h (t), . d h (t), d h (t) ∈ V h × Q h × W h × W h , such that . d h (t) = ∂ t d h (t) and ρ f ∂ t u h , v h Ω + a f h (u h , p h ), (v h , q h ) + ρ s ∂ t . d h , w h Σ + a s (d h , w h ) - 2 i=1 σ(u i,h , p i,h )n i , v i,h -w h Σ - 2 i=1 u i,h - . d h , σ(v i,h , -q i,h )n i Σ + γµ h 2 i=1 u i,h - . d h , v i,h -w h Σ -{ {σ(u h , p h )} }n, v h Σ h -{ {σ(v h , -q h )} }n, u h Σ h + γµ h u h , v h Σ h = 0 (14) Inria for all (v h , q h , w h ) ∈ V h × Q h × W h .
Remark 2.6. It should be noted that the sole differences between (9) and ( 14) are the definition of the discrete fluid space V h × Q h and the three additional terms acting on the fictitious fluidfluid interface Σ h .

Standard arguments show that the energy stability ( 12) also holds for [START_REF] Burman | Continuous interior penalty finite element method for the time-dependent Navier-Stokes equations: space discretization and convergence[END_REF]. The extension of Theorem 2.1 is more delicate due to the lack of regularity in the vicinity of the interface tip (see, e.g. [START_REF] Dimitrov | On singularities in the solution of three-dimensional Stokes flow and incompressible elasticity problems with corners[END_REF]).

Mesh intersection and integrals over cut-elements

The bilinear forms of the space semi-discrete formulations ( 9) and ( 14) require the evaluation of integrals over cut-elements. This is a consequence of the fact that, for consistency, the fluid equations are integrated only in the physical zone of Ω i,h , i = 1, 2. This is a non-standard implementation problem which demands a specific track of the interface intersections (see, e.g., [START_REF] Massing | Efficient implementation of finite element methods on nonmatching and overlapping meshes in three dimensions[END_REF]), namely:

• evaluation of the intersections between the unfitted fluid and solid meshes, i.e., the computation of the cut-elements;

• evaluation of the integrals over the cut-elements.

This can be challenging in practice, particularly if (as in the present framework) the unfitted fluid and solid meshes are unstructured. Regarding the first point, a few algorithms have been recently reported in the literature (see, e.g., [START_REF] Mayer | Interface handling for three-dimensional higher-order XFEM-computations in fluid-structure interaction[END_REF][START_REF] Wang | Computational algorithms for tracking dynamic fluid-structure interfaces in embedded boundary methods[END_REF][START_REF] Massing | Efficient implementation of finite element methods on nonmatching and overlapping meshes in three dimensions[END_REF]). The second is usually faced by sub-dividing the cut-elements into sub-elements for the purpose of the numerical integration (see, e.g., [START_REF] Fries | The extended/generalized finite element method: an overview of the method and its applications[END_REF][START_REF] Mayer | Interface handling for three-dimensional higher-order XFEM-computations in fluid-structure interaction[END_REF]). Such subdivision can however be involved in 3D (particularly for general unstructured meshes).

An alternative is the use of the divergence theorem to obtain a boundary representation of the integrals (see, e.g., [START_REF] Massing | Efficient implementation of finite element methods on nonmatching and overlapping meshes in three dimensions[END_REF][START_REF] Sudhakar | An accurate, robust, and easyto-implement method for integration over arbitrary polyhedra: Application to embedded interface methods[END_REF]). where the solid mesh edges have been enforced inside the fluid mesh by a partitioning method. The resulting mesh is the intersected mesh where both meshes are fitted.

In the 2D numerical examples of Section 4, the above operations have been performed with a customized 2D algorithm which simultaneously yields the mesh intersections and the cut-elements subdivisions (see Figure 6). Its main steps are the following:

1. We first localize all the solid mesh vertices inside the fluid mesh. This is carried out using a barycentric coordinates based algorithm, which efficiently identifies the element of the fluid mesh containing a given point (see, e.g., [START_REF] Frey | Mesh generation. Application to finite elements[END_REF]Section 2.10] or [2, Section 3] for details). Once localized, the solid vertices are inserted into the fluid mesh. For efficiency, simple insertion patterns are used, instead of complex vertex insertion operators (such as the Delaunay kernel [START_REF] Frey | Mesh generation. Application to finite elements[END_REF]). In this regard, it is worth recalling that the quality of the intersected mesh is definitely not a concern here, whose sole purpose is numerical quadrature in cut-elements (not interpolation). Therefore, simple mesh validity suffices. When a point is inserted into a 2D mesh composed of triangles, three cases may arise (two of them being degenerated):

• the point falls inside a triangle, then the triangle is split into three triangles;

• the point is on a mesh edge, then the edge is split into two and the two triangles sharing this edge are split into two triangles; • the point coincides with an existing mesh vertex, then nothing is done. This step is illustrated in Figures 6(a)-6(b).

2. We consider a partitioning algorithm (see, e.g., [START_REF] George | Ultimate' robustness in meshing an arbitrary polyhedron[END_REF]) to insert the structure mesh edges into the mesh resulting from step 1. Each edge is treated one at a time. Its endpoints are vertices of the current (intersected) mesh thanks to step 1. Let AB be an edge of the solid mesh. Starting from one of its endpoints, say A, we seek for the first current mesh edge intersected by AB. The intersection point P 1 is computed (see [2, Section 5] for details) and inserted into the current mesh. The two triangles sharing the intersected edge containing P 1 are split into two triangles. At this stage, part of edge AB (i.e., the segment AP 1 ) has been added to the current mesh. Then, the process is pursued by seeking for the intersection between sub-edge P 1 B and the current mesh edges, which will give a new intersection point P 2 and so on. If after n intersection steps the sub-edge P n B belongs to the current mesh, the whole solid edge AB has been inserted into this mesh. It should be noted that, for efficiency, this algorithm works only locally and progresses only through neighboring elements. This step is illustrated in Figures 6(b)-6(c).

Once the mesh intersection has been resolved via the above two steps, the data structures needed to evaluate the terms of the Nitsche-XFEM formulation ( 9) and ( 14) are updated accordingly.

Time discretization: fully discrete schemes

This section is devoted to the time discretization of the unfitted spatial semi-discrete formulations ( 9) and [START_REF] Burman | Continuous interior penalty finite element method for the time-dependent Navier-Stokes equations: space discretization and convergence[END_REF]. In order to simplify the presentation, we mainly consider the case of ( 9). The extension to [START_REF] Burman | Continuous interior penalty finite element method for the time-dependent Navier-Stokes equations: space discretization and convergence[END_REF] simply follows by adding suitable fluid terms (see Remark 2.6). Several coupling strategies will be discussed, which differ in their degree of fluid-solid splitting: implicit, explicit and semi-implicit nature (Algorithms 1-3 below). The theoretical discussion on the stability and accuracy of the methods introduced will be substantiated by numerical experiments in Section 4.1.

In the succeeding text, τ > 0 denotes the time-step length, t n def = nτ for n ∈ N, and ∂ τ x n def = 1 τ x nx n-1 stands for the first-order backward difference. The superscript n, will denote explicit extrapolations of order r ∈ {0, 1, 2} to x n , namely,

x n, def =      0 if r = 0, x n-1 if r = 1, 2x n-1 -x n-2 if r = 2.
Inria

Implicit coupling scheme

An overall backward Euler time-stepping of (9) yields the implicit coupling scheme reported in Algorithm 1. This guarantees unconditional stability and optimal accuracy. In particular, by denoting

E n def = ρ f 2 u n h 2 0,Ω + ρ s 2 . d n h 2 0,Σ + 1 2 d n h 2
s , the total energy of the discrete system at time t n , and by

E n def = ρ f 2 u n h -u(t n ) 2 0,Ω + ρ s 2 . d n h - . d(t n ) 2 0,Σ + 1 2 d n h -d(t n ) 2 s 1 2
, the energy norm of the approximation error, the following theorem holds. We recall that , denote inequalities up to multiplicative constants independent of h and τ .

Theorem 2.2. Let {(u n h , p n h , . d n h , d n h )} n≥1
be the sequence given by Algorithm 1. Then, for γ > 0 given by Lemma 2.1 and n ≥ 1, there holds

E n ≤ E 0 .
Moreover, under for smooth enough solutions, the following a priori error estimate holds [START_REF] Burman | Stabilization of explicit coupling in fluid-structure interaction involving fluid incompressibility[END_REF]. The a priori error estimate follows by combining the arguments of the proof of Theorem 2.1 with Taylor expansions in time.

E n h + τ. Proof. The first result is a straightforward consequence of Lemma 2.1, after taking (v h , q h , w h ) = (u n h , p n h , . d n h ) in
Algorithm 1 Implicit coupling scheme For n ≥ 1, find u n h , p n h , . d n h , d n h ∈ V h × Q h × W h × W h , such that . d h = ∂ τ d n h and ρ f ∂ τ u n h , v h Ω + a f h (u n h , p n h ), (v h , q h ) + ρ s ∂ τ . d n h , w h Σ + a s (d n h , w h ) - 2 i=1 σ(u n i,h , p n i,h )n i , v i,h -w h Σ - 2 i=1 u n i,h - . d n i,h , σ(v i,h , -q i,h )n Σ + γµ h 2 i=1 u n i,h - . d n h , v i,h -w h Σ = 0 (15)
for all

(v h , q h , w h ) ∈ V h × Q h × W h .
Therefore, due to Theorem 2.2, Algorithm 1 exhibits optimal first-order accuracy (in the energy norm) both in time and space.

Remark 2.7. Second-order accuracy in time can be achieved, for instance, by considering a Crank-Nicolson time-stepping of (9).

The price to pay for the above unconditional stability and accuracy properties of Algorithm 1, is that, at each time level, the fluid (u n h , p n h ) and solid ( .

d n h , d n h
) states are fully coupled, which can be computationally demanding in practice. Indeed, besides the hybrid nature of the coupled problem, general thin-walled solid models discretized by finite elements are known to yield very ill-conditioned stiffness matrices requiring specific solvers (see, e.g., [START_REF] Gee | Parallel multilevel solution of nonlinear shell structures[END_REF]). In the next two sections, we introduce two alternative time discretizations of (9) with a certain degree of splitting between the time-stepping of the fluid and of the solid.

Stabilized explicit schemes

We consider the stabilized explicit coupling paradigm originally introduced in [START_REF] Burman | Stabilization of explicit coupling in fluid-structure interaction involving fluid incompressibility[END_REF][START_REF] Burman | Explicit strategies for incompressible fluid-structure interaction problems: Nitsche type mortaring versus Robin-Robin coupling[END_REF] for fitted meshes, and extended in [START_REF] Burman | An unfitted Nitsche method for incompressible fluidstructure interaction using overlapping meshes[END_REF] to the unfitted framework. We apply here those ideas to the Nitsche-XFEM spatial semi-discrete formulation [START_REF] Braack | Stabilized finite element methods for the generalized Oseen problem[END_REF]. To this purpose, we first formulate [START_REF] Braack | Stabilized finite element methods for the generalized Oseen problem[END_REF] in terms of two coupled sub-problems by successively taking (v h , q h ) = (0, 0) and w h = 0 in (9). This yields:

• Solid sub-problem:

ρ s ∂ t . d h , w h Σ + a s (d h , w h ) + σ(u h , p h )n , w h Σ + 2γµ h . d h -{ {u h } }, w h Σ = 0 (16)
for all w h ∈ W h .

• Fluid sub-problem:

ρ f ∂ t u h , v h Ω + a f h (u h , p h ), (v h , q h ) - 2 i=1 σ(u i,h , p i,h )n i , v i,h Σ - 2 i=1 u i,h - . d h , σ(v i,h , -q i,h )n i Σ + γµ h 2 i=1 u i,h - . d h , v i,h Σ = 0 (17) for all (v h , q h ) ∈ V h × Q h .
We combine a backward Euler time-stepping of the fluid and solid bulk terms in ( 16)-( 17) with an explicit treatment of the interface coupling terms. More precisely, for n ≥ 1, we have:

1. Solid sub-step: find . d n h , d n h ∈ W h × W h with . d n h = ∂ τ d n h and such that ρ s ∂ τ . d n h , w h Σ + a s (d n h , w h ) + 2γµ h . d n h , w h Σ = 2γµ h { {u n-1 h } }, w h Σ -σ(u n-1 h , p n-1 h )n , w h Σ (18) for all w h ∈ W h . 2. Fluid sub-step: find (u n h , p n h ) ∈ V h × Q h such that ρ f ∂ τ u n h , v h Ω + a f h (u n h , p n h ), (v h , q h ) + γµ h 2 i=1 u n i,h , v i,h Σ - 2 i=1 u n i,h - . d n h , σ(v i,h , -q i,h )n i Σ + γ 0 h γµ 2 i=1 p n i,h -p n-1 i,h , q i,h Σ = γµ h 2 i=1 . d n h , v i,h Σ + 2 i=1 σ(u n-1 i,h , p n-1 i,h )n i , v i,h Σ (19) for all (v h , q h ) ∈ V h × Q h .

Inria

A salient feature of this approach is that explicit treatment of the interface coupling in [START_REF] Burman | An unfitted Nitsche method for incompressible fluidstructure interaction using overlapping meshes[END_REF] uncouples the computation of ( . d n h , d n h ) and (u n h , p n h ). The fundamental ingredient for the stabitity of the scheme is the weakly consistent term

γ 0 h γµ 2 i=1 p n i,h -p n-1 i,h , q i,h Σ , γ 0 > 0,
which controls the temporal interface pressure fluctuations induced by the fluid-solid splitting in time and, hence, avoids added-mass stability issues.

The next result establishes the conditional stability of the fully discrete method ( 18)-( 19).

Theorem 2.3. Let {(u n h , p n h , . d n h , d n h )} n≥1
be the sequence given by ( 18)- [START_REF] Burman | Continuous interior penalty finite element method for Oseen's equations[END_REF]. Then, under conditions γ C T /c s , γτ h and γ 0 1, there holds

E n E 0 + 2 i=1 µ u 0 i,h 2 0,Σ + µ (u 0 i,h ) 2 0,Ω i,h + γ 0 h γµ τ p 0 i,h 2 0,Σ for n ≥ 1.
Proof. The result follows by combining Lemma 2.1 with the arguments reported in [15, Section 5.1].

The main drawback of the scheme ( 18)-( 19) is that it delivers poor accuracy in practice. More precisely, the explicit treatment of the penalty term in the solid sub-step [START_REF] Burman | An unfitted Nitsche method for incompressible fluidstructure interaction using overlapping meshes[END_REF], i.e,

2 γµ h . d n h -{ {u n-1 h } }, w h Σ , (20)
induces an splitting error O (τ /h) which is not uniform in h. In the spirit of [START_REF] Burman | Explicit strategies for incompressible fluid-structure interaction problems: Nitsche type mortaring versus Robin-Robin coupling[END_REF][START_REF] Burman | An unfitted Nitsche method for incompressible fluidstructure interaction using overlapping meshes[END_REF], we propose to circumvent this issue by combining ( 18)-( 19) with a predictor-corrector strategy (with K ≥ 0 corrections). The resulting solution procedure is detailed in Algorithm 2, where we have used the notation

∂ τ x n,k def = 1 τ (x n,k -x n-1 ).
The key idea is that if, instead of the first-order extrapolation [START_REF] Burman | Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method[END_REF], we consider a second-order extrapolation of the fluid velocity (r = 2), after K correction iterations, the error induced by the explicit treatment of the penalty term becomes

O τ 2 /h K+1 (23)
As a result, the contribution of the penalty term becomes O τ 2 /h) with K = 0 and, hence, τ = O(h) suffices to achieve overall first-order accuracy. Nevertheless, numerical evidence indicates that K ≥ 1 is mandatory for stability (see, e.g., the discussion of Section 4.2.2).

Remark 2.8. The original stabilized explicit coupling scheme (18)-( 19) can be retrieved from Algorithm 2 by taking K = 0 (no correction) and with first-order extrapolation for the initial guess of the fluid velocity (r = 1). On the other hand, if we let K → ∞ the splitting error [START_REF] Codina | Stabilized finite element approximation of transient incompressible flows using orthogonal subscales[END_REF] tends to zero and we retrieve the implicit coupling solution provided by Algorithm 1. In other words, Algorithm 2 with enough correction iterations (i.e., until convergence) provides a partitioned iterative solution procedure for Algorithm 1.

Algorithm 2 Stabilized explicit coupling scheme with K ≥ 0 corrections For n ≥ 1:

1. Set initial guess for predictor:

u n,0 h = u n, h , p n,0 h = p n-1 h , d n,0 h = d n-1 h , . d n,0 h = . d n-1 h .
2. Correction iterations. For k = 1, . . . , K + 1:

(a) Find . d n,k h , d n,k h ∈ W h × W h with . d n,k h = ∂ τ d n,k h
and such that

ρ s ∂ τ . d n,k h , w h Σ + a s (d n,k h , w h ) + 2γµ h . d n,k h , w h Σ = 2γµ h { {u n,k-1 h } }, w h Σ -σ(u n,k-1 h , p n,k-1 h )n , w h Σ (21) for all w h ∈ W h . (b) Find (u n,k h , p n,k h ) ∈ V h × Q h such that ρ f ∂ τ u n,k h , v h Ω + a f h (u n,k h , p n,k h ), (v h , q h ) + γµ h 2 i=1 u n,k i,h , v i,h Σ - 2 i=1 u n,k i,h - . d n,k h , σ(v i,h , -q i,h )n i Σ + γ 0 h γµ 2 i=1 p n,k i,h -p n,k-1 i,h , q i,h Σ = γµ h 2 i=1 . d n,k h , v i,h Σ + 2 i=1 σ(u n,k-1 i,h , p n,k-1 i,h )n i , v i,h Σ (22) for all (v h , q h ) ∈ V h × Q h . 3. Set u n h = u n,K+1 h , p n h = p n,K+1 h , d n h = d n,K+1 h , . d n h = . d n,K+1 h .

Semi-implicit schemes

The main drawback of the explicit coupling schemes introduced in the previous section is that the splitting error is not uniform in h. Enough correction iterations with suitable predictions (Algorithm 2) are hence needed to enhance accuracy. In this section, we propose to overcome these issues through an operator splitting approach (see [START_REF] Fernández | Splitting schemes for incompressible fluid/thin-walled structure interaction with unfitted meshes[END_REF]Section 2]). The resulting schemes deliver stability and overall first-order accuracy (uniform in h) while keeping a certain degree of fluid-solid splitting (semi-implicit coupling schemes). The schemes herein presented extend the explicit Robin-Neumann paradigm introduced in [START_REF] Fernández | Incremental displacement-correction schemes for incompressible fluidstructure interaction: stability and convergence analysis[END_REF][START_REF] Fernández | Explicit Robin-Neumann schemes for the coupling of incompressible fluids with thin-walled structures[END_REF] to the case of immersed structures with unfitted meshes.

To this purpose, we consider the following fractional-step time-marching of (9): for n > 1,

Inria 1. Find u n h , p n h , . d n-1 2 h ∈ V h × Q h × W h such that ρ f ∂ τ u n h , v h Ω + a f h (u n h , p n h ), (v h , q h ) + ρ s τ . d n-1 2 h - . d n-1 h , w h Σ - 2 i=1 σ(u n i,h , p n i,h )n i , v i,h -w h Σ - 2 i=1 u n i,h - . d n-1 2 h , σ(v i,h , -q i,h )n i Σ + 2 i=1 γµ h u n i,h - . d n-1 2 h , v i,h -w h Σ = -a s d n, h , w h . (24)
for all

(v h , q h , w h ) ∈ V h × Q h × W h . 2. Find . d n h , d n h ∈ W h × W h with . d n h = ∂ τ d n h and such that ρ s τ . d n h - . d n-1 2 h , w h Σ + a s d n h , w h = a s d n, h , w h (25)
for all w h ∈ W h .

The introduction in step [START_REF] Cottet | Eulerian formulation and level set models for incompressible fluid-structure interaction[END_REF] of the intermediate solid-velocity .

d n-1 2 h
enables the implicit treatment of the solid inertia within the fluid. This is enough to guarantee added-mass free stability. The remaining solid elastic contributions are treated explicitly (or ignored) via extrapolation. cannot be eliminated in [START_REF] Cottet | Eulerian formulation and level set models for incompressible fluid-structure interaction[END_REF] and, hence, the coupling scheme is not explicit. This is a major difference with respect to the case of fitted meshes and conformal discretizations considered in [START_REF] Fernández | Incremental displacement-correction schemes for incompressible fluidstructure interaction: stability and convergence analysis[END_REF][START_REF] Fernández | Explicit Robin-Neumann schemes for the coupling of incompressible fluids with thin-walled structures[END_REF]. In that case, we can take

. [START_REF] Cottet | Eulerian formulation and level set models for incompressible fluid-structure interaction[END_REF], which yields a standard fluid problem with an explicit Robin condition on the interface Σ.

d n-1 2 h = u n h | Σ and w h = v h | Σ in
Remark 2.10. The relation [START_REF] Cottet | Eulerian formulation and level set models for incompressible fluid-structure interaction[END_REF] has somme similarities with the explicit time-splitting procedures commonly used in the immersed boundary (IB) method (see, e.g., [START_REF] Peskin | The immersed boundary method[END_REF][START_REF] Newren | Unconditionally stable discretizations of the immersed boundary equations[END_REF][START_REF] Boffi | Finite element approach to immersed boundary method with different fluid and solid densities[END_REF]). Indeed, sub-step (24) simultaneously includes the fluid and solid inertia whereas the solid elastic contributions are treated explicitly. The key difference concerns the solid sub-step [START_REF] Di Pietro | Mathematical aspects of discontinuous Galerkin methods[END_REF], which in the IB method consists of a simple displacement-velocity relation (i.e., the structure solver is never called), which in practice enforces restrictive CFL conditions for stability. Theorem 2.4 below shows that (24)-( 25) circumvents this issue.

It is worth noting that the semi-implicit coupling scheme provided by ( 24)-( 25) has a reduced computational complexity with respect to Algorithm 1. Indeed, the solid contribution to [START_REF] Cottet | Eulerian formulation and level set models for incompressible fluid-structure interaction[END_REF] reduces to a simple interface mass-matrix, which does not degrade the conditioning of the system matrix. This reduction in the coupling complexity is particularly important when considering general shell models (see, e.g., [START_REF] Chapelle | The Finite Element Analysis of Shells -Fundamentals[END_REF]), whose elastic contributions incorporate additional unknowns (e.g., rotations). Moreover, unlike Algorithm 2, the scheme ( 24)-( 25) does not involve any correction iteration.

In the spirit of [START_REF] Fernández | Explicit Robin-Neumann schemes for the coupling of incompressible fluids with thin-walled structures[END_REF], the solid step (25) can be reformulated as a standard solid problem. Indeed, by taking v h = 0 and q h = 0 in [START_REF] Cottet | Eulerian formulation and level set models for incompressible fluid-structure interaction[END_REF] and adding the resulting expression to [START_REF] Di Pietro | Mathematical aspects of discontinuous Galerkin methods[END_REF], we get

ρ s ∂ τ . d n h , w h Σ + a s (d n h , w h ) = -σ(u n h , p n h )n , w h Σ + 2γµ h { {u n h } } - . d n-1 2 h , w h Σ ( 26
)
for all w h ∈ W h . We can also avoid the extrapolations of the solid elastic term in [START_REF] Cottet | Eulerian formulation and level set models for incompressible fluid-structure interaction[END_REF] (which can be cumbersome in practice), by reformulating step [START_REF] Cottet | Eulerian formulation and level set models for incompressible fluid-structure interaction[END_REF] in a more intrinsic fashion. To this purpose, we note that from [START_REF] Dimitrov | On singularities in the solution of three-dimensional Stokes flow and incompressible elasticity problems with corners[END_REF], we have

a s (d n, h , w h ) = -ρ s ∂ τ . d n, h , w h Σ -σ(u n, h , p n, h )n , w h Σ + 2γµ h { {u n, h } } - . d n-1 2 , h , w h Σ ( 27
)
for all w h ∈ W h and n > r. Owing to ( 26) and ( 27), the semi-implicit scheme ( 24)-( 25) can be reformulated as shown in Algorithm 3.

Algorithm 3 Semi-implicit coupling scheme For n > r:

1. Fluid-with-solid-inertia sub-step: find

u n h , p n h , . d n-1 2 h ∈ V h × Q h × W h such that ρ f ∂ τ u n h , v h Ω + a f h (u n h , p n h ), (v h , q h ) + ρ s τ . d n-1 2 h , w h Σ - 2 i=1 σ(u n i,h , p n i,h )n i , v i,h -w h Σ - 2 i=1 u n i,h - . d n-1 2 h , σ(v i,h , -q i,h )n i Σ + 2 i=1 γµ h u n i,h - . d n-1 2 h , v i,h -w h Σ = ρ s τ . d n-1 h + τ ∂ τ . d n, h , w h Σ + σ(u n, h , p n, h )n , w h Σ - 2γµ h { {u n, h } } - . d n-1 2 , h , w h Σ (28) for all (v h , q h , w h ) ∈ V h × Q h × W h . 2. Solid sub-step: find . d n h , d n h ∈ W h × W h with . d n h = ∂ τ d n h and such that ρ s ∂ τ . d n h , w h Σ + a s (d n h , w h ) = -σ(u n h , p n h )n , w h Σ + 2γµ h { {u n h } } - . d n-1 2 h , w h Σ for all w h ∈ W h .
Remark 2.11. Algorithm 3 with r = 1, 2 is a multi-step method on the interface. Thus, additional data is needed to start the time-marching. In practice, this data can be obtained by performing one step of the scheme with r = 0 and then one step of the scheme with r = 1.

The following result states the stability and convergence properties of the semi-implicit schemes reported in Algorithm 3.

Theorem 2.4. Let {(u n h , p n h , . d n-1 2 h , d n h , . d n h
)} n>r be the sequence given by Algorithm 3, initialized as indicated in Remark 2.11 for r ≥ 1. Let γ > 0 be given by Lemma 2.1. There holds

E n ≤ E 0
for all n > r, unconditionally for r ∈ {0, 1} and under the CFL-like condition τ αh 6 5 , τ α < 1 and α > 0, for r = 2. In addition, for smooth enough solutions, there holds

E n h + τ + τ 2 r-1 .

Inria

Proof. The result follows from a straightforward adaptation of the arguments used in [31, Proposition 2.1].

The contribution of the splitting error in the a priori energy estimate of Theorem 2.4 is given by the O(τ 2 r-1 ) term. Note that this guarantees the h-uniformity of the error, which is a major advantage with respect to Algorithm 2. Moreover, Algorithm 3 with r = 1 simultaneously yields unconditional stability and overall first-order accuracy, without resorting to any correction iteration.

Remark 2.12. Algorithms 1-3 can be adapted to the time discretisation of (14) with minor modifications. Indeed, it suffices to add the corresponding fictitious fluid-fluid interface terms

-{ {σ(u n h , p n h )} }n, v h Σ h -{ {σ(v h , -q h )} }n, u n h Σ h + γµ h u n h , v h Σ h
to the discrete problems (15), ( 22) and (28), respectively.

3 The non-linear case: dynamic interfaces

In this section we extend the numerical methods of Section 2.2 to the case of a non-linear fluid-structure interaction problem, involving an incompressible viscous fluid and an immersed thin-walled structure. The fluid is described by the incompressible Navier-Stokes equations and the structure by a (possibly) non-linear membrane or shell model.

Problem setting

Let Σ ⊂ R d be the reference configuration of the solid mid-surface. The current position of the interface, denoted by Σ(t), is parametrized by the one-to-one deformation map φ : Σ×R + -→ R d as Σ(t) = φ(Σ, t), with φ def = I Σ×R + + d and where d denotes the displacement of the solid. In order to ease the presentation, we introduce the notation φ t def = φ(•, t), so that we also have Σ(t) = φ t (Σ). The structure is supposed to move within a domain Ω ⊂ R d with boundary Γ def = ∂Ω (see Figure 7). For simplicity and without loss of generality, Ω is assumed to be fixed. The fluid is described in the time-dependent control volume

Ω(t) def = Ω\Σ(t) ⊂ R d ,
with its boundary partitioned as ∂Ω(t) = Σ(t) ∪ Γ.

The considered non-linear coupled problem reads as follow: find the fluid velocity and pressure u : Ω × R + → R d , p : Ω × R + → R and the solid displacement and velocity d :

Σ × R + → R d , . d : Σ × R + → R d such that            Ω(t) def = Ω\Σ(t), Σ(t) = φ t (Σ), φ t def = I Σ + d(t), ρ f ∂ t u + (u • ∇)u -divσ(u, p) = 0 in Ω(t), divu = 0 in Ω(t), u = 0 on Γ, (29)
         u 1 = u 2 = . d • φ -1 t on Σ(t), d(t) ∈ W , . d = ∂ t d, ρ s ∂ t . d, w Σ + a s d, w) = - Σ(t) σ(u, p)n • w • φ -1 t ∀w ∈ W , (30)
RR n°8723 30) is emphasized by the presence of the map φ -1 t , which transforms the solid quantities from the reference to the current configuration. This introduces geometrical non-linearities.

For the fluid, we consider the same velocity and pressure functional spaces as in Section 2 and we introduce the convective tri-linear form

c Ω (z, u, v) def = ρ f z • ∇u, v Ω . (31)
We recall that, if z ∈ {v ∈ V / divv = 0 in Ω}, using integration by parts we have

c Ω (z, v, v) = 0 ∀v ∈ V . ( 32
)
The weak form of the linear coupled problem ( 29)-( 30) reads as follows: for t > 0, find (u(t), p(t), d(t), .

d(t)) ∈ V × Q × W × W such that            φ t = I Σ + d(t), Σ(t) = φ t (Σ), u 1 = u 2 = . d • φ -1 t on Σ(t), . d = ∂ t d, ρ f (∂ t u, v) Ω + c Ω (u, u, v) + a f Ω (u, p), (v, q) + ρ s ∂ t . d, w Σ + a s (d, w) = 0 (33)
for all (v, q, w) [START_REF] Frey | Mesh generation. Application to finite elements[END_REF] and using [START_REF] Fernández | Explicit Robin-Neumann schemes for the coupling of incompressible fluids with thin-walled structures[END_REF] we retrieve an energy equality similar to (4).

∈ V × Q × W with v| Σ(t) = w • φ -1 t . Assuming that a s (d, ∂ t d) = 1 2 ∂ t a s (d, d), taking (v, q, w) = (u(t), p(t), . d(t)) in

Numerical methods

The next section presents the formulation of the numerical methods introduced in Section 2.2 within the framework of the non-linear coupled problem (29)- [START_REF] Fernández | Incremental displacement-correction schemes for incompressible fluidstructure interaction: stability and convergence analysis[END_REF]. The basic idea consists in combining the different coupling paradigms with an explicit treatment of the geometrical compatibility (29) 1 . Finally, in Section 3.2.2, we briefly comment on how to handle the integration of quantities associated with different time levels.

Inria

Nitsche-XFEM formulation and coupling schemes

For simplicity, we assume that Ω and Σ are polyhedral. We consider the general case in which the interface partially intersects the fluid domain (see Figure 7). The approximation space for the solid, W h , is the same as in Section 2.1. For a given discrete displacement d n-1 h ∈ W h at time level n -1, we introduce its corresponding deformation map φ n-1

h def = I Σ + d n-1 h
. The current configuration (i.e., at time level n) of the discrete interface is defined as

Σ n h def = φ n-1 h (Σ). (34)
In other words, the geometric compatibility (29) 1 is treated in an explicit fashion.

For the construction of the fluid discrete spaces, we proceed as in Section 2.1.3 with the sole difference that, in the present framework, the fluid-solid interface Σ n h depends on the discrete displacement d n-1 h . As a result, the fictitious fluid-fluid interface Σ n h ∪ Σ n h and the fluid spaces, V n h and Q n h given by ( 13), depend both on the mesh step h and on the time level n. We recall that the functions of these spaces are continuous in the fluid domain

Ω n h def = Ω\ Σ n h ∪ Σ n h ,
but discontinuous across the moving interface Σ n h ∪ Σ n h . It should be noted that the discrete fluid velocities do not satisfy the assumptions of (32), namely, they are not divergence free and V n h ⊂ V . Therefore, we need to modify the trilinear form [START_REF] Fernández | Splitting schemes for incompressible fluid/thin-walled structure interaction with unfitted meshes[END_REF] in order to retrieve a suitable discrete counterpart of [START_REF] Fernández | Explicit Robin-Neumann schemes for the coupling of incompressible fluids with thin-walled structures[END_REF]. This is a well-known issue when dealing with discontinuous Galerkin approximations of the Navier-Stokes equations (see, e.g., [25, Section 6.2.2]). The key idea consists in combining the so-called Temam's trick with the fact that, owing to (30) 1 , the velocity field u is continuous across Σ n h ∪ Σ n h . To this purpose, we introduce the discrete tri-linear form

c n h (z h , u h , v h ) def = c Ω n h (z h , u h , v h ) + ρ f 2 (divz h )u h , v h Ω n h -ρ f { {z h } } • n u h , { {v h } } Σ n h ∪ Σ n h - ρ f 2 z h • n , { {u h • v h } } Σ n h ∪ Σ n h .
Note that the last three terms are strongly consistent. Moreover, using integration by parts in Ω n h we can infer that, if z h ∈ V n h , there holds

c n h (z h , v h , v h ) = 0 ∀v h ∈ V n h . (35)
To cope with the numerical instabilities related to the inf-sup incompatibility of the discrete spaces and to large local Reynolds numbers, we need to resort to a stabilisation method (see, e.g., [START_REF] Brooks | Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations[END_REF][START_REF] Hughes | A new finite element formulation for computational fluid dynamics. V. Circumventing the Babuška-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations[END_REF][START_REF] Tezduyar | Stabilized finite element formulations for incompressible flow computations[END_REF][START_REF] Codina | Stabilized finite element approximation of transient incompressible flows using orthogonal subscales[END_REF][START_REF] Burman | Continuous interior penalty finite element method for Oseen's equations[END_REF][START_REF] Braack | Stabilized finite element methods for the generalized Oseen problem[END_REF][START_REF] Burman | Continuous interior penalty finite element method for the time-dependent Navier-Stokes equations: space discretization and convergence[END_REF][START_REF] Ganesan | Stabilization by local projection for convection-diffusion and incompressible flow problems[END_REF]3] and the references therein). The objective of such a procedure is basically twofold: guarantee the well-posedness of the discrete problem and improve the convergence of the approximations while limiting the propagation of spurious oscillations. A successful approach is the so-called SUPG/PSPG method (see, e.g., [START_REF] Brooks | Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations[END_REF][START_REF] Hughes | A new finite element formulation for computational fluid dynamics. V. Circumventing the Babuška-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations[END_REF][START_REF] Tezduyar | Stabilized finite element formulations for incompressible flow computations[END_REF]), which offers an unified treatment of the velocity and pressure stabilization by adding to the Galerkin formulation an element-wise weighted residual of the equation. The residual based nature of the stabilization operator guarantees the consistency of the method.

Within the present unfitted framework this last property is more delicate. Indeed, as discussed in Section 2.1.1 (see also [START_REF] Burman | Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes' problem[END_REF][START_REF] Burman | An unfitted Nitsche method for incompressible fluidstructure interaction using overlapping meshes[END_REF]), the theoretical analysis indicates that the stabilization operator must act on the whole computational domain, that is, including the fictitious zone of

RR n°8723

Algorithm 4 Non-linear version of Algorithm 1. For n ≥ 1,

1. Interface update: Ω n h = Ω\ Σ n h ∪ Σ n h , Σ n h = φ n-1 h (Σ), φ n-1 h = I Σ + d n-1 h . 2. Find u n h , p n h , . d n h , d n h ∈ V n h × Q n h × W h × W h , such that . d h = ∂ τ d n h and                          ρ f ∂ τ u n h , v h Ω + a f,n h u n h ; (u n h , p n h ), (v h , q h ) + ρ s ∂ τ . d n h , w h Σ + a s (d n h , w h ) - 2 i=1 σ(u n i,h , p n i,h )n i , v i,h -w h Σ n h - 2 i=1 u n i,h - . d n h , σ(v i,h , -q i,h )n i Σ n h -{ {σ(u n h , p n h )} }n, v h Σ n h -{ {σ(v h , -q h )} }n, u n h Σ n h + γµ h 2 i=1 u n i,h - . d n h , v i,h -w h Σ n h + γµ h u n h , v h Σ n h = 0 for all (v h , q h , w h ) ∈ V n h × Q n h × W h .
the overlapping region. However, in this zone, we cannot guarantee that the residual of a smooth extension of the solution vanishes. An alternative to circumvent this issue (see also [START_REF] Schott | A new face-oriented stabilized XFEM approach for 2D and 3D incompressible Navier-Stokes equations[END_REF]) is to use symmetric stabilization methods whose consistency does not rely on the residual (see, e.g., [START_REF] Codina | Stabilized finite element approximation of transient incompressible flows using orthogonal subscales[END_REF][START_REF] Burman | Continuous interior penalty finite element method for Oseen's equations[END_REF][START_REF] Burman | Continuous interior penalty finite element method for the time-dependent Navier-Stokes equations: space discretization and convergence[END_REF][START_REF] Ganesan | Stabilization by local projection for convection-diffusion and incompressible flow problems[END_REF]3]). As an example, we consider here the continuous interior penalty (CIP) stabilization method of [START_REF] Burman | Continuous interior penalty finite element method for Oseen's equations[END_REF][START_REF] Burman | Continuous interior penalty finite element method for the time-dependent Navier-Stokes equations: space discretization and convergence[END_REF]. To this purpose, we first introduce the set F n i,h of interior edges or faces of T n,f i,h . The corresponding velocity and pressure stabilization operators are, respectively, given by the relations

s n v,h (z h ; u h , v h ) def = γ v,1 h 2 2 i=1 F ∈F n i,h ξ Re F (z h ) z h • n L ∞ (F ) ∇u h F , ∇v h F F + γ v,2 h 2 2 i=1 F ∈F n i,h z h L ∞ (F ) divu h F , divv h F F , s n p,h (z h ; p h , q h ) def = γ p h 2 2 i=1 F ∈F n i,h ξ Re F (z h ) z h L ∞ (F ) ∇p h F , ∇q h F F , (36)
where

Re F (z h ) def = ρ f z h L ∞ (F ) hµ -1 denotes the local Reynolds number, ξ(x) def = min{1
, x} is a cut-off function and γ p , γ v,i > 0, i = 1, 2, are user-defined parameters. At last, we collect all the above fluid contributions in a single term

a f,n h z h ; (u h , p h ), (v h , q h ) def = c n h (z h , u h , v h ) + a f Ω n h (u h , p h ), (v h , q h ) + s n v,h (z h ; u h , v h ) + s n p,h (z h ; p h , q h ) + g n h (u h , v h ), (37)
with the time-dependent ghost-penalty operator now given by

g n h (u h , v h ) def = γ g µh 2 i=1 F ∈F n,Σ i,h ∇u i,h F , ∇v i,h F F ( 38
)
Inria Algorithm 5 Non-linear version of Algorithm 2.

For n ≥ 1:

1. Interface update: φ n-1 h = I Σ + d n-1 h , Σ n h = φ n-1 h (Σ), Ω n h = Ω\ Σ n h ∪ Σ n h . 2.
Set initial guess for predictor:

u n,0 h = u n, h , p n,0 h = p n-1 h , d n,0 h = d n-1 h , . d n,0 h = . d n-1 h .
3. Correction iterations. For k = 1, . . . , K + 1:

(a) Find . d n,k h , d n,k h ∈ W h × W h with . d n,k h = ∂ τ d n,k h
and such that

ρ s ∂ τ . d n,k h , w h Σ + a s (d n,k h , w h ) + 2γµ h . d n,k h , w h Σ n h = 2γµ h { {u n,k-1 h } }, w h Σ n h -σ(u n,k-1 h , p n,k-1 h )n , w h Σ n h for all w h ∈ W h . (b) Find (u n,k h , p n,k h ) ∈ V n h × Q n h such that ρ f ∂ τ u n,k h , v h Ω + a f,n h u n,k h ; (u n,k h , p n,k h ), (v h , q h ) + γµ h 2 i=1 u n,k i,h , v i,h Σ n h - 2 i=1 u n,k i,h - . d n,k h , σ(v i,h , -q i,h )n i Σ n h + γ 0 h γµ 2 i=1 p n,k i,h -p n,k-1 i,h , q i,h Σ n h -{ {σ(u n,k h , p n,k h )} }n, v h Σ n h -{ {σ(v h , -q h )} }n, u n,k h Σ n h + γµ h u n,k h , v h Σ n h = γµ h 2 i=1 . d n,k h , v i,h Σ n h + 2 i=1 σ(u n,k-1 i,h , p n,k-1 i,h )n i , v i,h Σ n h for all (v h , q h ) ∈ V n h × Q n h . 4. Set u n h = u n,K+1 h , p n h = p n,K+1 h , d n h = d n,K+1 h , . d n h = . d n,K+1 h .
and where F n,Σ i,h denotes the set of interior edges or faces of the elements intersected by Σ n h . We now have all the ingredients to extend the numerical methods of Section 2.2 to the approximation of the non-linear coupled problem ( 29)- [START_REF] Fernández | Incremental displacement-correction schemes for incompressible fluidstructure interaction: stability and convergence analysis[END_REF]. By combining the explicit treatment of the geometric compatibility [START_REF] Fries | The extended/generalized finite element method: an overview of the method and its applications[END_REF] with the strongly coupled paradigm of Section 2.2.1 we get the solution procedure given in Algorithm 4.

Remark 3.2. Note that all the appearances of . d n h and w h in the interface terms of Σ n h must be understood as .

d n h • (φ n h ) -1 and w h • (φ n h ) -1
, respectively. The purpose of this little abuse of notation is simply to ease the presentation.

In Algorithm 5 we have reported the non-linear counterpart of the stabilized explicit coupling paradigm of Section 2.2.2. It should be noted that the K ≥ 0 corrections are performed with the same configuration of the interface Σ n h . At last, Algorithm 6 presents the extension of the semi-implicit scheme introduced in Section 2.2.3 to the present non-linear framework.

RR n°8723

Algorithm 6 Non-linear version of Algorithm 3 For n > r:

1. Interface update: φ n-1 h = I Σ + d n-1 h , Σ n h = φ n-1 h (Σ), Ω n h = Ω\ Σ n h ∪ Σ n h . 2. Find u n h , p n h , . d n-1 2 h ∈ V h × Q h × W h such that ρ f ∂ τ u n h , v h Ω + a f,n h u n h ; (u n h , p n h ), (v h , q h ) + ρ s τ . d n-1 2 h , w h Σ - 2 i=1 σ(u n i,h , p n i,h )n i , v i,h -w h Σ n h - 2 i=1 u n i,h - . d n-1 2 h , σ(v i,h , -q i,h )n i Σ n h + 2 i=1 γµ h u n i,h - . d n-1 2 h , v i,h -w h Σ n h -{ {σ(u n h , p n h )} }n, v h Σ n h -{ {σ(v h , -q h )} }n, u n h Σ n h + γµ h u n h , v h Σ n h = ρ s τ . d n-1 h + τ ∂ τ . d n, h , w h Σ + σ(u n, h , p n, h )n , w h Σ n h - 2γµ h { {u n, h } } - . d n-1 2 , h , w h Σ n h for all (v h , q h , w h ) ∈ V n h × Q n h × W h . 3. Find . d n h , d n h ∈ W h × W h with . d n h = ∂ τ d n h and such that ρ s ∂ τ . d n h , w h Σ + a s (d n h , w h ) = -σ(u n h , p n h )n , w h Σ n h + 2γµ h { {u n h } } - . d n-1 2 h , w h Σ n h for all w h ∈ W h .
The stability results of Theorems 2.2-2.4 remain valid for Algorithm 4, Algorithm 5 with K = 0 and r = 1, and Algorithm 6, respectively. The proofs follow by combining the identity [START_REF] Fries | On time integration in the XFEM[END_REF] with the result of Lemma 2.1 and the arguments used in the proofs of the static case (Section 2.2). This shows that the explicit treatment of the interface location in the fluid, [START_REF] Fries | The extended/generalized finite element method: an overview of the method and its applications[END_REF], does not compromise the energy stability of the methods. Similar conclusions are known for semi-implicit time discretizations of the immersed boundary method (see, e.g., [START_REF] Newren | Unconditionally stable discretizations of the immersed boundary equations[END_REF][START_REF] Boffi | The finite element immersed boundary method with distributed Lagrange multiplier[END_REF]).

It should be noted, however, that the semi-implicit schemes reported in Algorithm 6 have a reduced computational complexity with respect to alternative semi-implicit schemes used in the immersed boundary method (see, e.g., [START_REF] Newren | Unconditionally stable discretizations of the immersed boundary equations[END_REF][START_REF] Boffi | The finite element immersed boundary method with distributed Lagrange multiplier[END_REF]), which treat the fluid-solid kinematic-dynamic coupling (30) 1,3 in a fully implicit fashion (strong coupling), as Algorithm 4.

Integrals with discontinuous functions at different time levels

At each time level, the first step of Algorithms 4-6 involves the computation of the new interface intersections and the new sub-divisions of the cut-elements. In other words, we perform the intersection algorithm of Section 2.1.4 at each time-step. Some of the fluid integrals in Algorithms 4-6, however, involve fluid discrete functions associated with different time levels, namely, Inria the bulk term related to the time-stepping scheme in the fluid,

ρ f τ (u n-1 h , v h ) Ω ,
and the interface terms

σ(u n-1 i,h , p n-1 i,h )n i , v i,h Σ n h , γ 0 h γµ p n-1 i,h , q i,h Σ n h
, associated with the fluid-solid splitting of Algorithm 5. A practical difficulty arises whenever

(u n-1 h , p n-1 h ) ∈ V n-1 h × Q n-1 h and (v h , q h ) ∈ V n h × Q n
h are discontinuous at different locations within the same element. This issue is illustrated in 1D in Figure 8(a), which shows that the evaluation of the bulk terms is much more involved than in the static framework of Section 2, since the interface locations at t n-1 and t n have to be considered in the evaluation of the intersections and sub-divisions of the same element. Instead, we propose to approximate the quantities defined at t n-1 by shifting the discontinuity to the location of the interface at t n (see Figure 8(b)). Basically, this amounts to evaluate part of the functions defined at t n-1 in the fictitious zone of the overlapping region. In this sense, it can be viewed as a class of ghost fluid method (see, e.g., [START_REF] Fedkiw | Coupling an Eulerian fluid calculation to a Lagrangian solid calculation with the ghost fluid method[END_REF]). A similar approach is discussed in [START_REF] Fries | On time integration in the XFEM[END_REF] (therein called Alternative 2) in the framework of time-stepping schemes for XFEM methods with dynamic interfaces.

Numerical examples

The purpose of this section is to illustrate, via a series of 2D numerical examples, the stability and accuracy of the methods introduced in Sections 2.2 and 3.2. We consider a simple string model for the solid, so that in (2) and [START_REF] Fernández | Incremental displacement-correction schemes for incompressible fluidstructure interaction: stability and convergence analysis[END_REF] we have

d = ηn, Ld = -λ 1 ∂ 2
s η + λ 0 η n, where η : Σ × R + → R denotes the normal displacement and λ 1 , λ 0 > 0 are given coefficients.

Static interface

We first consider the case of the linear model problem of Section 2 (static interface). The next two paragraphs illustrate numerically the accuracy properties of Algorithms 1-3 in two different configurations. We retrieve, in particular, the convergence rates reported in Section 2.2. Along this section, the user-defined parameters of Algorithms 1-3 are fixed to γ p = 10 -3 , γ g = 1, γ = 10 3 and (in Algorithm 2) γ 0 = 1.

Idealized closed valve

The purpose of this first example is to mimic the behavior of a closed valve under a given pressure drop. We consider a rectangular fluid domain with the thin-walled solid immersed along its middle cross section (see Figure 9). We take Ω = (0, 4) × (0, 1) and Σ = {2} × (0, 1) in ( 1)- (2). All the units are given in the CGS system. The fluid physical parameters are ρ f = 1 and µ = 0.035. the solid we have ρ s = 1.1, = 0.1, λ 1 = E /(2(1 + ν)) and λ 0 = 0, with Young's modulus E = 0.75 • 10 6 and Poisson's ratio ν = 0.5. The external boundary conditions for the fluid problem are shown in Figure 9. A steady pressure drop of magnitude P 0 = 20000 is enforced between the fluid inlet and outlet boundaries. The structure is fixed on its extremities. After a brief transition phase, the system reaches a steady state with a pressure jump across the interface. For this simple configuration, the exact solution can be computed analytically. The fluid and solid velocities vanish and the pressure is a piecewise constant function taking the value P 0 on the left side and zero on the right one. Hence, the solid equation reduces to the single 1D boundary value problem

-λ 1 ∂ 2 y η = p = P 0 in (0, 1), η(0) = η(1) = 0, (39)
whose solution is given by

η(y) = P 0 2λ 1 y(1 -y). (40)
This analytic solution is used to evaluate the spatial accuracy of the methods. Because of the time independent nature of the problem, we limit the discussion to the implicit scheme given by Algorithm 1. We consider four pairs of unfitted fluid-solid meshes with increasing degree of refinement, namely, h ∈ 0.1/2 i 3 i=0 . For illustration purposes, we have displayed in Figure 10 the steady state displacement provided by Algorithm 1 for h = 0.1, together with the analytical solution [START_REF] Glowinski | A distributed Lagrange mutiplier/fictitious domain method for particulate flows[END_REF] and the approximation obtained with an implicit fitted method. Note that both the fitted and the unfitted approximations perfectly match the analytical solution. The corresponding pressure fields are reported in Figure 11. The unfitted method is able to perfectly capture the pressure drop without spurious oscillations. The overlap of the approximation at the interfacial zone is also clearly visible. Finally, Figure 12 presents the convergence history of the displacement approximations in the energy norm, for the fitted and the unfitted methods. As expected, we retrieve the optimal first-order convergence rate predicted by Theorem 2.2 for Algorithm 1.

Pressure-waves

This example is a variant of the well-known fluid-structure benchmark describing the propagation of a pressure-wave within an elastic tube. The fluid domain is given by the rectangle Ω = (0, 6) × (0, 0.8) and the interface by the segment Σ = (0, 6) × {0.54} (see Figure 13). As in the previous example, all the units are given in the CGS system. The fluid physical parameters are ρ f = 1.0, µ = 0.035. For the solid we have ρ s = 1.1, = 0.1, λ 1 = E /(2(1 + ν)) and λ 0 = E /(0.25(1ν 2 )), with Young's modulus E = 0.75 • 10 6 and Poisson's ratio ν = 0.5. The boundary conditions for the fluid are reported in Figure 13. On the inlet boundary {0} × [0, 0.54] we impose a sinusoidal pressure p in (t), of maximal amplitude 2 • 10 4 , during 5 • 10 -3 seconds (half a period). The solid is clamped on its extremities. Figure 14 shows some snapshots of the elevated pressure field at three different time instants obtained with an implicit fitted method and Algorithm 1. The zooms included in Figures 14(a) and 14(b) illustrate the fitted and unfitted nature of the meshes. Algorithm 1 gives practically the same accuracy as the fitted method, predicting the propagation of a pressure jump along the tube. uniformly refined both in time and in space according to the parameters

⌃ u y = 0 u y = 0 u y = 0 u y = 0 u y = 0 ( (u, p)n) x = p in (t) ( (u, p)n) x = 0 ( (u, p)n) x = 0
(τ, h) ∈ (2 • 10 -4 /2 i , 0.1/2 i ) 3 i=0 . (41)
Note that τ = O(h). A reference solution has been generated with the fitted implicit method using τ = 6.25 • 10 -6 and h = 3.125 • 10 -3 (i.e., i = 5 in ( 41)). Figure 15 reports the convergence histories of the displacement error in the elastic energy norm at t = 0.015 obtained with Algorithms 1-3. For comparison purposes, the convergence history of the fitted implicit approximations is also displayed. The corresponding displacements, for each level of space-time refinement, are reported in Figures 16 and17.

Figure 15(a) retrieves the first-order optimal convergence rate predicted by Theorem 2.2 for Algorithm 1. This convergent behavior is also clearly visible in Figure 16, which points out the good agreement with the approximations provided by the implicit fitted method. As regards Algorithm 2, Figure 15(a) indicates that the variant without extrapolation (K = 0, r = 1) fails to converge under τ = O(h). On the contrary, stable and first-order accurate approximations are obtained with a second-order prediction and one correction iteration (K = 1, r = 2). This behavior is also clearly visible in Figure 16, hence confirming the theoretical discussion on the accuracy of the methods of Section 2.2.2.

Finally, we comment on the results obtained with Algorithm 3. Figure 15(b) shows convergence for the three variants. Note, however, that sub-optimal accuracy is obtained with the variant without extrapolation (r = 0). This behavior is also striking in Figure 17, which points out the extremely poor accuracy of the displacement approximations for all the space-time refinement levels. On the contrary, the variants with r = 1 and r = 2 retrieve the optimal firstorder accuracy of the fitted implicit scheme, hence confirming the convergence rates reported in Theorem 2.4.

Dynamic interface

In this section we provide numerical evidence on the stability and the accuracy properties of Algorithms 4-6 in different examples. For comparison purposes, we consider as reference solution an implicit fitted-ALE approximation of problem [START_REF] Frey | Mesh generation. Application to finite elements[END_REF], where the discrete ALE mapping is built from a simple harmonic lifting of the solid displacement (no advanced mesh update strategy is hence used). Along this section, the user-defined parameters in Algorithms 4-6 are always chosen to be γ p = 10 -2 , γ v,1 = 10 -2 , γ v,2 = 0, γ g = 1, γ = 10 2 and (in Algorithm 5) γ 0 = 1. Furthermore, for simplicity, a standard semi-implicit treatment of the fluid convective term, a f,n h u n-1 h ; (u n h , p n h ), (v h , q h ) , is adopted in Algorithms 4-6.

Idealized closed valve

We consider the idealized closed valve test of Section 4.1.1 within the non-linear framework of Section 3. The geometrical configuration, physical parameters and boundary conditions are those of Section 4.1.1. A remarkable feature of this setting is that both the static and the dynamic interface cases share the same steady state displacement solution. Indeed, using the parameterization of the interface configuration Σ(t), in terms of the steady state displacement, given by r(y) = η(y), y , y ∈ [0, 1], we have for all w ∈ W . We hence recover [START_REF] Gerstenberger | An extended finite element method/Lagrange multiplier based approach for fluid-structure interaction[END_REF], as the strong form of (30), and the analytical expression for the displacement given by ( 40). Due to the stationary nature of the problem, we limit the discussion to the implicit scheme given by Algorithm 4, and compare its accuracy with the implicit ALE-fitted method. Prototypical approximations obtained with both approaches are displayed in Figure 18. The mesh size for the fitted (structured) and the unfitted (unstructured) meshes is approximately h ≈ 0.05. Note the distortion of the fluid mesh in Figure 18(a), prescribed by the ALE fitted approach, in order to fit the solid mesh. In Figure 18(b), on the contrary, the structure mesh moves independently of the fluid background mesh. This is more visible in Figure 19, which presents the mesh intersection and the cut-elements subdivisions resulting from the algorithm described in Section 2.1.4. In order to show the capabilities of the proposed unfitted methods to handle situations with large interface displacements, we consider a series of increasing inlet pressures P 0 ∈ {10 4 +12•10 3 • i} 7 i=0 . In Figure 20, we have reported the displacement approximations obtained with the fitted and the unfitted methods. The analytical solutions given by ( 40) are also displayed. We observe that both approaches are in excellent agreement with the analytical solution until i = 5. From this point on, the considered ALE mesh update fails. On the contrary, the unfitted approach is able to continue for i ≥ 6 while keeping a perfect match with the analytic solution.

- Σ(t) σ(u, p)n • w • φ -1 t = 1 0 P 0 n r(y) • w φ -1 t (r (

Double cavity

We consider a variant of the classical lid-driven cavity problem in which a second cavity is attached below the lower flexible wall. The fluid domain is given by the rectangle Ω = (-0.5, 0.5) × (-0.5, 1.5) and the interface by the segment Σ = (-0.5, 0.5) × {0.5} (see Figure 21). All units are given in the SI system.

The fluid physical parameters are given by ρ f = 1.0 and µ = 0.2. For the solid we have ρ s = 250, = 0.01, λ 1 = E /(2(1 + ν)) and λ 0 = 0, with ν = 0 and we consider three different Figure 22 presents some snapshots of the fluid velocity magnitude and of the elevated fluid pressure obtained with the ALE-fitted method and Algorithm 4 for a value of the Young modulus, E = 62500, which gives relatively large interface displacements. The time-step length is τ = 0.005 and the mesh step is approximately h ≈ 0.025 in both simulations. The good agreement between the fitted and unfitted approaches is noticeable.

Inria (u, p)n = 0 (u, p)n = 0 (u, p)n = 0 (u, p)n = 0 u = 0 u y = 0 ⌃ u = 0 u = 0 u x = U (t)
Figures 23 and24 present a closer view of the discrete solutions at time t = 7.5, showing the elevated pressure and velocity components. The comparison of these figures demonstrates the capability of the present unfitted approach to accurately capture the weak and strong discontinuities of the velocity and pressure fields across the interface. No spurious oscillations appear. Note that the unfitted approximation is two-valued in the set of elements intersected by the interface, only its value in the physical zone of the overlapping region must be compared with the results of Figures 23.

In For comparison purposes, the corresponding results for the ALE-fitted method are also plotted. Figure 25 shows that Algorithm 4 gives similar results as the ALE-fitted method in all the regimes. As expected, the higher the Young modulus, the lower the displacement amplitude. Note that for E = 31250, the considered ALE-fitted method fails at a certain level of interface deformation (due to the breakdown of the mesh update technique), while Algorithm 4 does not show any lack of robustness.

The situation is more delicate for the explicit coupling schemes given by Algorithm 5. The two explicit variants do deliver stable approximations, but some spurious time oscillations appear in the case of the largest interface deflections. This issue is related to the non-uniformity in space of the splitting error discussed in Section 2.2.2. Indeed, as shown in Figure 27, reducing the timestep length τ (while keeping h fixed) improves the quality of the approximations. The enhanced accuracy of the variant with second-order prediction and one correction iteration (K = 1, r = 2) is also noticeable. Numerical evidence, not reported here, shows however that this variant may lack stability if γτ /h is not sufficiently small. As an example, with γ = 1000, as in Section 4.1, stability requires more than one correction iteration (K > 1). The results reported in Figure 26 indicate that the semi-implicit schemes given by Algorithm 6 with r = 1 and r = 2 do not suffer from the above issues. These variants deliver comparable accuracy to the implicit ALE-fitted method. On the contrary, the variant with r = 0 provides poor approximations in all the regimes. We thus retrieve, also in the dynamic interface framework, the essential ingredients of the accuracy result given by Theorem 2.4. Figure 29 presents some snapshots of the fluid velocity magnitude and of the elevated fluid pressure obtained with the ALE-fitted method and Algorithm 4. The time step is τ = 0.0025 and the step parameter of the fluid and solid meshes is approximately h ≈ 0.05. The unfitted method is able to capture the dynamics of the pressure jump across the interface delivered by the fitted approximation without any spurious oscillation. This feature is even more striking in Figures 30 and31, where we present a closer view of the elevated pressure and velocity Section 3.2. To this purpose, we have reported in Figure 32 the time history of the solid displacement at point (1.9, 0) (tip of the left leaflet) obtained with Algorithms 4-6 and the implicit ALE-fitted method. Figure 32(a) confirms the results of the above qualitative discussion on the accuracy of Algorithm 4 with respect to the reference ALE-fitted method. As regards the time splitting schemes given by Algorithms 5 and Algorithm 6, Figure 32 indicates that the best performance is obtained with the variants (K = 1, r = 2) and r = 1, 2, respectively. Once again, this numerical evidence is consistent with the theoretical discussion of Section 2.2. Finally, in order to illustrate the capability of the proposed unfitted approach to handle very large interface deflections we have performed the simulation with an increased maximum pressure drop of P max = 400. Since the present computer implementation is not able to handle contact, the top leaflet has been removed. Some snapshots of the discrete solution obtained with Algorithm 4 are displayed in Figure 33. Figure 34 shows the time history of the tip displacement provided by Algorithm 4 and the ALE-fitted algorithm. Both approaches give similar results up to a certain degree of interface deformation, above which the ALE-fitted method fails while the Inria unfitted method still delivers a stable numerical approximation.

Idealized open valve

Conclusion

We have introduced a Nitsche-XFEM method for incompressible fluid-structure interaction problems involving immersed thin-walled structures. Eulerian and Lagrangian formalisms are, respectively, considered for the fluid and the solid. The key features of the spatial discretization proposed are:

• unfitted (unstructured) fluid and solid meshes;

• affine finite element approximations including weak and strong discontinuities for the velocity and the pressure, respectively (based on the XFEM method of [START_REF] Hansbo | A finite element method for the simulation of strong and weak discontinuities in solid mechanics[END_REF][START_REF] Becker | A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity[END_REF]);

• integration of the fluid equations only in the physical domain (cut-elements);

• consistent treatment of the kinematic/dynamic fluid-solid coupling via Nitsche's method;

• symmetric velocity/pressure and ghost-penalty stabilization to guarantee robustness without compromising accuracy.

In the case of static interfaces, a priori error estimates (Theorem 2.1) guaranteeing optimal convergence (in the energy norm) towards non-singular solutions have been derived using the arguments reported in [START_REF] Burman | An unfitted Nitsche method for incompressible fluidstructure interaction using overlapping meshes[END_REF]. Several splitting schemes (implicit, explicit, semi-implicit) have been proposed, including their formulation with moving interfaces. The fundamental ingredients for the stability and accuracy of the resulting fully discrete methods have been discussed. These theoretical stability and convergence results have then been confirmed via numerical evidence in a series of 2D examples involving static and moving interfaces. The comparison of the different methods indicates that the best performance (in terms of accuracy and computational complexity) is obtained with Algorithms 3 and 6 with r = 1. The salient features of this semi-implicit method are: (i) it avoids strong coupling; (ii) it simultaneously yields unconditional stability and optimal convergence in the energy norm.

Extensions of this work can explore several directions. From the modelling point of view, we plan to incorporate more complex thin-walled solid models, including contact and fracture mechanics. The extension of all these problems to 3D presents some challenges (see, e.g., the discussion of Section 2.1.4). Another interesting point, not addressed in this paper, is the convergence analysis of the methods with curved and dynamic interfaces. The latter is a particularly delicate problem which has received little consideration in the literature (see, e.g., [START_REF] Zunino | Analysis of backward Euler/extended finite element discretization of parabolic problems with moving interfaces[END_REF] for an analysis in the parabolic case).

Figure 1 :

 1 Figure 1: Geometric description.

def= 1 2Remark 2 . 1 .

 121 -pI + 2µ (u) with (u) def = ∇u + ∇u T and µ denoting the fluid dynamic viscosity. The abstract surface differential operator L describes the solid elastic effects. Since in (1)-(

Figure 2 :

 2 Figure 2: Unfitted fluid-solid meshes. In grey the elements intersected by Σ (a). The triangulations T f 1,h and T f 2,h , with the overlapping region in grey (b).

  a) Unfitted meshes (b) Weak discontinuity (c) Strong discontinuity

Figure 3 :

 3 Figure 3: Idealized fully intersected configuration. Example of functionsf h ∈ X f 1,h × X f 2,h .

Figure 4 :

 4 Figure 4: The fluid-solid interface Σ and the fluid-fluid fictitious interface Σh in red (a). The two new triangulations with the overlapping region in grey (b).

Figure 5 :

 5 Figure 5: Idealized partially intersected configuration (the red dotted line represents the fictitious fluid-fluid interface). Example of functions fh ∈ X f 1,h × X f 2,h .

Figure 6 :

 6 Figure6: Main steps of the mesh intersection algorithm: (a) initial state where the solid mesh (in red) and the fluid mesh are not fitted; (b) end of step 1 where the solid mesh vertices have been inserted inside the fluid mesh; (c) end of step 2 where the solid mesh edges have been enforced inside the fluid mesh by a partitioning method. The resulting mesh is the intersected mesh where both meshes are fitted.

Figure 7 :Remark 3 . 1 .

 731 Figure 7: Geometric description. complemented with standard initial conditions u(0) = u 0 , d(0) = d 0 and . d(0) = . d 0 . We recall that W ⊂ [H 1 (Σ)] d denotes the space of solid admissible displacements and a s : W × W → R describes the (possibly non-linear) elastic behavior of the structure. Remark 3.1. The mixed Eulerian-Lagrangian nature of the problem (29)-(30) is emphasized by the presence of the map φ -1 t , which transforms the solid quantities from the reference to the current configuration. This introduces geometrical non-linearities.

Figure 8 : he previous solution u n- 1 h∈ V n- 1 h

 811 Figure 8: he previous solution u n-1 h ∈ V n-1 h and the test function v h ∈ V n h are discontinuous at different locations, Σ n-1 h and Σ n h respectively, within the same element. The dotted lines represent the value of the functions in the fictitious zone of the overlapping region and the circles the ghost values (a). Approximation of the function u n-1 h with the discontinuity at Σ n h (b).

0 ⌃Figure 9 :

 09 Figure 9: Geometric configuration and external boundary conditions.

Inria

Figure 10 :

 10 Figure 10: Steady state displacement.

Figure 11 :Figure 12 :

 1112 Figure 11: Steady state pressure approximations.

Figure 13 :

 13 Figure 13: Geometric configuration and external boundary conditions.

14 :

 14 Snapshots of the fluid velocity magnitude and elevated pressure at time t = 0.005, 0.01, 0.015 (from left to right) obtained with h = 0.05 and τ = 10

- 4 .

 4 In order to provide numerical evidence on the convergence rates of Algorithms 1-3, we have Inria

Figure 15 :

 15 Figure 15: Time-convergence history of the displacement at t = 0.015 obtained with h = O(τ ).

Figure 16 :

 16 Figure 16: Algorithms 1 and 2. Comparison of the solid displacements at t = 0.015 for different levels of (τ, h)-refinement (41).

1 0P 0

 10 y)) |r (y)|dy = |r (y)| -1 (1, -η (y)) • (w(y), 0) |r (y)|dy =

Figure 17 :

 17 Figure 17: Algorithm 3. Comparison of the solid displacements at t = 0.015 for different levels of (τ, h)-refinement (41).

Figure 18 :

 18 Figure 18: Steady state pressure approximation and deformed solid configuration.

Figure 19 :

 19 Figure 19: Mesh intersection and the cut-elements subdivisions.

Figure 20 :

 20 Figure 20: Steady state displacements for increasing inlet pressures.

Figure 21 :

 21 Figure 21: Geometric configuration and external boundary conditions.

Figure 22 :

 22 Figure 22: Snapshots of the fluid velocity magnitude and elevated pressure at time t = 6.25, 7.5 and 8.75 (from left to right) obtained with E = 62500, τ = 5 • 10 -3 and h ≈ 0.025.

Figure 23 :

 23 Figure 23: ALE-fitted method. Strong (pressure) and weak (velocity) discontinuities of the approximated solution at time t = 7.5 with E = 62500, τ = 5 • 10 -3 and h ≈ 0.025.

Figure 24 :

 24 Figure 24: Algorithm 4. Strong (pressure) and weak (velocity) discontinuities of the approximated solution at time t = 7.5 with E = 62500, τ = 5 • 10 -3 and h ≈ 0.025. values for the Young modulus, E = {125000/2 i } 2 i=0 . The fluid external boundary conditions are given in Figure 21, with U (t) def = 100[1cos(0.4πt)].The solid is clamped on its extremities. Figure22presents some snapshots of the fluid velocity magnitude and of the elevated fluid pressure obtained with the ALE-fitted method and Algorithm 4 for a value of the Young modulus, E = 62500, which gives relatively large interface displacements. The time-step length is τ = 0.005 and the mesh step is approximately h ≈ 0.025 in both simulations. The good agreement between the fitted and unfitted approaches is noticeable.Figures23 and 24present a closer view of the discrete solutions at time t = 7.5, showing the elevated pressure and velocity components. The comparison of these figures demonstrates the capability of the present unfitted approach to accurately capture the weak and strong discontinuities of the velocity and pressure fields across the interface. No spurious oscillations appear. Note that the unfitted approximation is two-valued in the set of elements intersected by the interface, only its value in the physical zone of the overlapping region must be compared with the results of Figures 23.In order to investigate the accuracy of the different coupling schemes, we have reported in Figures 25 and 26 the time history of the solid displacement at point (-0.2, 0.5) obtained with

Figure 25 :Figure 26 :

 2526 Figure 25: Time history of the solid displacement at point (-0.2, 0.5) obtained with Algorithms 4 and 5, τ = 5 • 10 -3 .

3 Figure 27 :

 327 Figure 27: Time history of the solid displacement at point (-0.2, 0.5) obtained with Algorithms 4 and 5, E = 31250.

Figure 28 :

 28 Figure 28: Geometric configuration and external boundary conditions.structure we have ρ s = 1.2, = 0.065, λ 1 = E /(2(1 + ν)) and λ 0 = 0, with Young's modulus E = 10000 and Poisson's ratio ν = 0.5. The external boundary conditions for the fluid are detailed in Figure28, with the inlet pressure given by p in (t) def = P max sin(2πt) with P max = 150. The solid is clamped at its bottom and top extremities, i.e., at (1.9, 0.6) and (2.1, 1).Figure29presents some snapshots of the fluid velocity magnitude and of the elevated fluid pressure obtained with the ALE-fitted method and Algorithm 4. The time step is τ = 0.0025 and the step parameter of the fluid and solid meshes is approximately h ≈ 0.05. The unfitted method is able to capture the dynamics of the pressure jump across the interface delivered by the fitted approximation without any spurious oscillation. This feature is even more striking in Figures30 and 31, where we present a closer view of the elevated pressure and velocity

Inria

Figure 29 :

 29 Figure 29: Snapshots of the fluid velocity magnitude and elevated pressure at time t = 0.25, 0.5 and 0.75 (from left to right) obtained with P max = 150, τ = 2.5 • 10 -3 and h ≈ 0.05.

Figure 30 :

 30 Figure 30: ALE-fitted method. Strong (pressure) and weak (velocity) discontinuities of the approximated solution at time t = 0.25 obtained with P max = 150, τ = 2.5 • 10 -3 and h ≈ 0.05.We now turn our attention to the accuracy of the different coupling schemes proposed in

Figure 31 :

 31 Figure 31: Algorithm 4. Strong (pressure) and weak (velocity) discontinuities of the approximated solution fields at time t = 0.25 obtained with P max = 150, τ = 2.5 • 10 -3 and h ≈ 0.05.

Figure 32 :

 32 Figure 32: Time history of the solid displacement at point (1.9, 0.6) obtained with P max = 150, τ = 2.5 • 10 -3 and h ≈ 0.05.

Figure 33 :Figure 34 :

 3334 Figure 33: Snapshots of the fluid velocity magnitude and elevated pressure at time t = 0.25, 0.5 and 0.75 (from left to right) obtained with Algorithm 4, P max = 400, τ = 2.5 • 10 -3 and h ≈ 0.05.

RR n°8723

InriaRESEARCH CENTRE PARIS -ROCQUENCOURTDomaine de Voluceau, -Rocquencourt B.P. 105 -78153 Le Chesnay Cedex Publisher Inria Domaine de Voluceau -Rocquencourt BP 105 -78153 Le Chesnay Cedex inria.fr ISSN 0249-6399

Acknowledgements

Work funded by the French National Research Agency (ANR) through the EXIFSI project (ANR-12-JS01-0004). The authors are grateful to Prof. Erik Burman (University College London, UK) for all the fruitful discussions. They also thank an anonymous reviewer of this paper for his/her insightful remarks which helped to improve the manuscript.