Immersed boundary methods for fluid-structure interaction with topological changes

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interaction with topological changes

Fannie Maria Gerosa

To cite this version:

Fannie Maria Gerosa. Immersed boundary methods for fluid-structure interaction with topological

changes. Numerical Analysis [math.NA]. Sorbonne Université, 2021. English. �tel-03240631�

Immersed boundary methods for

fluid-structure interaction with

topological changes

THÈSE DE DOCTORAT

présentée par

Fannie M. Gerosa

pour obtenir le grade de

DOCTEUR DE

SORBONNE UNIVERSITÉ

Spécialité : Mathématiques Appliquées

Soutenue publiquement le 13 Avril 2021 devant le jury composé de :

Miguel Ángel Fernández

Directeur de thèse

Luca Formaggia

Rapporteur

Lucia Gastaldi

Examinatrice

Céline Grandmont

Examinatrice

Laura Grigori

Examinatrice

Aline Lefebvre

Examinatrice

Acknowledgements

I would like to express my sincere gratitude to my advisor Miguel Á. Fernández, for

all the support and advices. Always available to generously transmitting your enormous

knowledge and passion for research. Thank you for your guidance and encouragement

during these years. You have been a wonderful scientific father.

Special thank also to the REO-COMMEDIA members Muriel Boulakia, Céline

Grand-mont, Damiano Lombardi and Olga Mula. A particular thank to Marina Vidrascu, the

quintessence of this team. I also wish to thank Erik Burman for his collaboration during

my visiting period at the University College London.

I thank Luca Formaggia and Patrick Le Tallec for having accepted to review this

manuscript and for all the constructive suggestions and remarks. I also thank Lucia

Gastaldi, Céline Grandmont, Laura Grigori and Aline Lefebvre for agreeing to be part of

the jury.

Thank to my Commedia’s friends Justine, Colette, Valeria, Sara, Fabien, Haibo, Oscar,

Mihai and Marguerite. To my office mates Mocia, Vicente and Alexandre. A special

thank to María (nuestra profesora de salsa), Ludovic (l’aspirant mathématicien cuisinier,

je n’oublierai jamais nos bavardages pendant la pause café), Felipe (músico, filósofo que

compartió conmigo todas las fases del doctorado de principio a fin) and finally Daniele

(mio testimone di nozze e miglior collaboratore sin dai lontani tempi del magico Poli).

Un grazie speciale alla mia famiglia, ai miei genitori, Bruna e Federico, ed al mio

fratellino Alessio, per il loro costante sostegno. Vi voglio bene e non vedo l’ora di

fes-teggiare insieme a voi (ovviamente anche con Giorgia).

Il mio ringraziamento più grande va a mio Marito Florent. Senza di te mi sentirei persa

ovunque nella vita. Il tuo sostegno e la tua pazienza sono stati fondamentali durante questi

ultimi mesi. Merci.

Paris, France

Fannie Gerosa

Méthodes de la frontière immergée pour l’interactions

fluide-structure avec des changements de topologie

Resumé:

Cette thèse est dédiée à la modélisation, l’analyse numérique et à la simulation

des problèmes d’interactions fluide-structure. Nous considérons des structures déformable

à parois minces et immergées dans un fluide visqueux incompressible. La motivation

sous-jacente de ce travail est la simulation des valves cardiaques.

Ce travail aborde des questions fondamentales qui vont du fractionnement efficace du

temps avec des maillages non compatibles, à la modélisation de contact dans l’interaction

fluide-structure et son approximation. Pour des raisons de robustesse et de solidité

math-ématique, l’approximation spatiale est basée sur le cadre de maillage non compatibles

Nitsche-XFEM.

Dans la première partie, nous présentons et analysons un nouveau schéma

semi-implicite, qui évite un couplage fort, sans compromettre la stabilité et la précision. Dans

la deuxième partie, nous considérons la situation dans laquelle le contact se produit. Dans

le contexte de Nitsche-XFEM, un modèle d’interaction fluide-structure-contact est

égale-ment étendu dans le cas de contact avec plusieurs structures. Une procédure de duplication

spécifique permet de préserver la consistance de la méthode également dans le cas du

con-tact. Les inconsistances mécaniques traditionnelles de la formulation du contact relaxé

dans l’interaction fluide-structure sont contournées en introduisant un modèle poreux de

surface dans la paroi de contact. Cette couche décrit la rugosité de la surface, donnant

un sens physique aux régions fluides infinitésimales, qui restent entre le solide et la

sur-face au contact. Dans la dernière partie, nous développons l’extension 3D de la méthode

de maillage Nitsche-XFEM, dans le cas de domaines fluides entièrement et partiellement

intersectés.

Mots-clés:

Interactions fluide-structure, Méthodes de maillages non compatibles,

Sché-mas de couplage, Méthode de Nitsche, XFEM, Structures minces immergés, Contact.

Immersed boundary methods for fluid-structure interaction with

topological changes

Abstract:

This thesis is dedicated to the modeling, numerical analysis and simulation

of fluid-structure interaction problems, involving thin-walled structures immersed in an

incompressible viscous fluid. The underlying motivation of this work is the simulation of

heart valves.

This work addresses fundamental issues which go from efficient time-splitting with

unfitted meshes, to contact modeling in fluid-structure interaction and its approximation.

For the sake of robustness and mathematical soundness, the spatial approximation is based

on the Nitsche-XFEM unfitted mesh framework.

In the first part, we present and analyse a new semi-implicit scheme for

Nitsche-XFEM, which avoids strong coupling, without compromising stability and accuracy. In

the second part, we consider the situation in which contact occurs. In the context of

Nitsche-XFEM, a fluid-structure-contact interaction model is extended also to the case of

contact with multiple structures. A specific duplication procedure allows to extend the

consistency of the method to the case of contact. Traditional mechanical inconsistencies of

relaxed contact formulation in fluid-structure interaction are circumvented by introducing

a surface porous model in the contact wall. This layer describes surface roughness, giving

physical meaning to the infinitesimal fluid regions between the solid and the surface at

contact. In the last part, we develop the 3D extension of the unfitted mesh Nitsche-XFEM

method in case of fully and partially intersected fluid domains.

Keywords:

Fluid-structure interaction, Unfitted mesh methods, Coupling schemes,

Nitsche’s method, XFEM, Immersed thin-walled structures, Contact.

Contents

Introduction

1

Thesis general context

3

Position of the thesis

3

Thesis outline and main contributions

4

Author’s bibliography

5

1

Numerical methods for fluid-structure interaction

7

1.1 Introduction

7

1.2 Fluid-structure interaction

8

1.3 Numerical methods

18

I

_{Time-splitting schemes for unfitted mesh approximations}

of FSI

23

2

An unfitted mesh semi-implicit coupling scheme for fluid-structure

in-teraction with immersed solids

25

2.1 Introduction

25

2.2 Linear model problem: static interfaces

27

2.3 Non-linear model: moving interface

40

2.4 Numerical experiments

43

2.5 Conclusion

56

3

Error analysis of an unfitted mesh semi-implicit coupling scheme for

fluid-structure interaction

57

3.1 Introduction

57

3.2 Problem Setting

58

3.3 Numerical methods

60

3.4 Numerical analysis

65

3.5 Numerical experiments

83

3.6 Conclusion

87

II

_{Modeling and approximation of fluid-structure-contact}

4.1 Introduction

91

4.2 Problem setting

92

4.3 Numerical methods

95

4.4 Numerical experiments

108

4.5 Conclusion

119

5

A mechanically consistent fluid-structure-contact interaction model

121

5.1 Introduction

122

5.2 Mathematical models

123

5.3 Numerical experiments

136

5.4 Conclusions

150

III

_{3D numerical simulations}

151

6

A 3D Nitsche-XFEM method for FSI with immersed thin-walled

struc-tures

153

6.1 Introduction

154

6.2 The linear model

155

6.3 The non-linear model

175

6.4 Numerical experiments

176

6.5 Conclusion

194

General conclusion and perspectives

197

Thesis general context

The mechanical interaction of an incompressible viscous fluid with an immersed

struc-ture appears in a wide variety of engineering fields and is particularly ubiquitous in nastruc-ture.

The applications span from micro-encapsulation, biomechanics of cells deformation, birds

flight, physiological flows, such as, heart dynamics and cilliary beating, to aeroelasticity

of parachutes and sailing boats (see, e.g.,

Liu and Liu

(

2006

);

Van Loon et al.

(

2005

);

Han and Peskin

(

2018

);

Nakata and Liu

(

2012

);

Weymouth et al.

(

2006

);

Takizawa and

Tezduyar

(

2012

)).

Since the beginning of this century, impressive progress has been made in building

efficient and accurate numerical methods, able to represent the physiological and

patho-logical functionality of blood dynamics. Numerical simulations are effective tools, useful

to physicians, as, e.g., support for design new devices, better understanding diseases or

malfunctioning and they can become tools for quantity estimations, otherwise concretely

inaccessible.

Position of the thesis

This thesis is devoted to the modeling, analysis and numerical simulations of

fluid-structure interaction problems with immersed solids and contact. The work is mainly

motivated by the numerical simulation of blood flow interacting with heart valves (see,

e.g.,

Kamensky et al.

(

2015

);

Lau et al.

(

2010

)), with particular focus on stability, accuracy

and robustness. Given the ratio thickness/size of the leaflets, a common assumption is to

model the heart valves as structures of co-dimension one (see, e.g.,

Diniz dos Santos et al.

(

2008

);

Astorino et al.

(

2009b

)). This simplified, but realistic, reduced order problem is

a fundamental ingredient of this thesis and we refer to as thin-walled solid model. With

regard to efficiency, one possible way to gain in performance is to introduce a certain

degrees of time-splitting between the fluid and solid problems, thus, avoid strong coupling.

Standard loosely coupled schemes are known to exhibit stability and accuracy issues, since

the interface coupling in incompressible fluid-structure interaction is extremely stiff.

In-deed, they often require severe time-step restrictions and stability conditions, which are

linked to the amount of added-mass effect (see Section

1.3.2

).

The development of fluid-structure interaction numerical methods has been extensively

investigated within the fitted and unfitted mesh frameworks. In the first strategy the fluid

and solid meshes are fitted at their interface (see Section

1.3.1

), thus, it is ideal for problems

with moderate displacement and are usually treated with moving mesh techniques and an

Arbitrary Lagrangian Eulerian description of the fluid problem. However, they becomes

cumbersome within problems featuring large interface displacement and potential contact

between solids.

In such situations, a favored numerical approach is the unfitted mesh based

formula-tions, in which an Eulerian description of the fluid problem is combined with solid meshes

which are freely to move independently of a background fluid mesh. However, the design

and analysis of splitting schemes, which avoid strong coupling, in the unfitted framework,

have been rarely addressed in the literature so far. In the first part of this thesis (Part

I), we investigate the stability and accuracy of an unfitted mesh semi-implicit scheme,

which avoids strong coupling. Therein, we will consider a consistent spatial discretization

based on Nitsche mortaring, with unfitted mesh and cut-elements, named Nitsche-XFEM

method.

Moreover, in applications where contact occurs, such as in heart valves dynamics,

topo-logical changes appear in the fluid domain. Modeling contact, in particular coupled with

fluid-structure interaction, rises many issues, from the modeling and numerical points of

view. The computational complexity is also an issue, in particular when a realistic contact

(i.e, without relaxation of the contact conditions) is considered. Real topology changes

can be avoided by relaxing the contact conditions (see Section

1.2.4

). This approximation

suffers, however, of mechanical consistency loss. In the second part of this thesis (Part

II), we address some of these issues (see Section

1.2.4

).

Furthermore, the development of efficient, accurate and robust methods is fundamental

for 3D numerical simulations. This raises some issues that are addressed in Part III by

extending the unfitted mesh, Nitsche’s and cut-elements based method, to the 3D case.

Indeed, the considered unfitted mesh method requires a specific track of the interface

intersections which becomes cumbersome in 3D.

Thesis outline and main contributions

The main contributions of this work are listed here, chapter by chapter. For the sake

of clarity, they are discussed at the beginning of each chapter.

Chapter 1.

This is an introductory chapter. We present the essential models involved

in fluid-structure interaction problems with thin-walled immersed solids. A review of the

state-of-the-art on numerical methods is provided, by discussing the different available

methods for the space and time discretization.

Part I: Time-splitting schemes for unfitted mesh approximations of FSI

Chapter 2.

We introduce a new semi-implicit coupling scheme for the numerical

approximation of incompressible fluid-structure interaction problems, involving

thin-walled immersed solids. The method combines a Nitsche based unfitted mesh spatial

approximation with a fractional-step time-marching in the fluid. The viscous part of the

coupling is treated in an explicit fashion, while the remaining fluid pressure and solid

con-tributions are treated implicitly. The presented scheme efficiently avoid strong coupling,

without compromising stability and accuracy. A stability analysis is conducted in the

chapter and the efficiency of the numerical scheme is illustrated via numerical experiments.

Author’s bibliography

5

Chapter 3.

This chapter is devoted to the error analysis of the numerical method

presented in Chapter

2

, for a linear fluid-structure coupled system, involving the transient

Stokes equations (in a fixed domain) and a thin-walled solid elastodynamics model. Robust

a priori error estimates are derived for two extrapolated variants of the solid velocity.

Further, numerical evidences on the convergence properties of the methods is provided.

Part II: Modeling and approximation of fluid-structure-contact interaction

Chapter 4.

We address the issues raised by the approximation of a basic

fluid-structure-contact interaction model using the Nitsche-XFEM method. We illustrate

that, for consistency reasons, further element duplication is needed in the fluid elements

where contact between the structures occurs. The proposed method, for fluid-structure

interaction with contact, is hence compared with the ALE and FD/Lagrange multipliers

methods, exploiting the advantages and limitations of these strategies.

Chapter 5.

We introduce a mechanically consistent mixed dimensional

fluid-structure-contact interaction model. The fluid-fluid-structure-contact interaction problem

is coupled to a thin-walled Darcy model on the contacting wall. The model gives a

mechanical justification for the fluid-structure-contact interaction with a relaxed contact

condition. The thin-walled porous layer introduces tangential creeping flow along the

boundary and allows for the modelling of boundary flow due to surface roughness.

Numerical examples are reported for both Stokes’-Darcy coupling alone, as well as

fluid-structure-Darcy-contact at the porous boundary layer.

Part III: 3D numerical simulations

Chapter 6.

In this chapter, we discuss the formulation and implementation aspects

of the Nitsche-XFEM discretization method, to the three-dimensional case. A particular

focus is made on the efficiency and robustness of the intersection and sub-triangulation

algorithms without resorting to black-box meshing software. The performance and

robustness of the presented method are explored via a series of numerical examples,

involving moving interfaces, with partially and fully intersected fluid domains.

Author’s bibliography

• Papers in peer reviewed journals:

1. M.A. Fernández and F.M. Gerosa, An unfitted mesh semi-implicit

cou-pling scheme for fluid-structure interaction with immersed solids.

International Journal for Numerical Methods in Engineering, 2019, DOI:

10.1002/nme.6449. Available online:

https://hal.inria.fr/hal-02288723

.

• Papers in conference proceedings:

2. E. Burman, M. A. Fernández, S. Frei, and F. M. Gerosa, 3D-2D

Stokes-Darcy coupling for the modelling of seepage with an application

to fluid-structure interaction with contact.

Chapter 20 of F. J.

Ver-molen, C. Vuik (eds.), Numerical Mathematics and Advanced Applications

ENUMATH 2019, Lecture Notes in Computational Science and Engineering

139. DOI: 10.1007/978-3-030-55874-1_20. Available online:

https://hal.

inria.fr/hal-02417042

.

• Submitted papers:

3. E. Burman, M.A. Fernández and F.M. Gerosa, Error analysis of an unfitted

mesh semi-implicit coupling scheme for fluid-structure interaction.

Available online:

https://hal.inria.fr/hal-03065803

.

4. S. Frei, and F.M. Gerosa, E. Burman, M.A. Fernández, A mechanically

consistent model for fluid-structure interactions with contact

in-cluding seepage.

Available online:

https://hal.archives-ouvertes.fr/

hal-03174087

• Papers in preparation:

5. F. Alauzet, D. Corti, M.A. Fernández and F.M. Gerosa, A comparison of

dif-ferent strategies for fluid-structure-contact approximation with

mul-tiple thin-walled immersed solids.

6. F. Alauzet, D. Corti, M.A. Fernández and F.M. Gerosa, Three-dimensional

Nitsche-XFEM method for the coupling of an incompressible fluid

with immersed thin-walled structures.

Chapter 1

Numerical methods for fluid-structure

interaction

In this chapter, we briefly review the basic models and numerical methods for

fluid-structure interaction with immersed thin-walled solids. In particular, we present a review

of the existing numerical methods, focusing, in particular, on the spatial discretization,

the time splitting between the fluid and structure solvers and contact formulations.

Contents

1.1

Introduction

7

1.2

Fluid-structure interaction

8

1.2.1

Fluid models

8

1.2.2

Thin-walled solid model

10

1.2.3

Fluid-structure coupled problems

11

1.2.4

Contact modeling

14

1.3

Numerical methods

18

1.3.1

Spatial discretization

18

1.3.2

Time-splitting

19

1.1

Introduction

The content of this chapter serves as basic background of the work presented in the

rest of this thesis. The reader interested in fundamentals of general continuum

mechan-ics, is referred to

Gurtin

(

1982

);

Lai et al.

(

2009

). Material regarding fundamentals of

solid mechanics can be found in

Ciarlet

(

1988

);

Chapelle and Bathe

(

2011

). The latter

reference is related, in particular, to the mathematical models and their finite element

approximation, in the context of thin-walled structures. The models commonly used for

the mathematical modeling of fluid-structure interaction (FSI) problems are presented in

the following sections. Starting from the description of the fluid equations, solid equations

and, finally, considering the full FSI problem, adding the necessary coupling conditions.

The fluid formalisms considered will be the Eulerian and Arbitrary Lagrangian Eulerian

(ALE), while for the solid equation we restrict the discussion only to the Lagrangian

de-scription. As regards additional materials about basis and introduction to fluid-structure

interaction problems, we refer to

Formaggia et al.

(

2009

) and the references therein.

Af-terward, we describe the modeling in contact. In particular, for discussion on dry contact

mechanics, we refer to

Wriggers and Zavarise

(

2004

), for contact treated via penalization

to

Chouly and Hild

(

2012

), and via Augmented Lagrangian/Nitsche’s approach to

Burman

et al.

(

2018

,

2019

). The latter approach can be seen as a consistent penalization method.

For contact considered in the context of fluid-structure interaction, we refer to

Diniz dos

Santos et al.

(

2008

);

Astorino et al.

(

2009b

);

Kamensky et al.

(

2015

);

Zonca et al.

(

2020

)

and

Burman et al.

(

2020a

);

Mayer et al.

(

2009

,

2010

);

Chouly et al.

(

2017

) for further

examples of contact for FSI treated via Nitsche’s approach.

Regarding the numerical approximation, we present a review of the existing numerical

methods, discussing the possible spatial discretization strategies (distinguish fitted mesh

from the unfitted mesh based approximations) and the degree of time splitting between

the fluid and structure sub-problems (introducing the concept of strongly coupled and

weakly-coupled schemes). Reviews on numerical methods for FSI can be found in

Hou

et al.

(

2012

);

Formaggia et al.

(

2009

);

Fernández

(

2011

).

The rest of the chapter is organized as follows. In Section

1.2

we describe the general

geometrical setting and we introduce the fundamental models involved in FSI problem. In

particular, we present the fluid problems in Section

1.2.1

, the solid problem in Section

1.2.2

and the coupled problems in Section

1.2.3

. In Section

1.2.4

, the main issues encountered

considering contact modeling are described. Section

1.3

presents a review on the existing

numerical methods regarding the space discretization (Section

1.3.1

) and the splitting

between the fluid and solid (Section

1.3.2

).

1.2

Fluid-structure interaction

In this section, we consider the mechanical interaction between a deformable

thin-walled structure and an incompressible viscous fluid. In the following, the structure domain

and the fluid-structure coupling interface are identified by the solid mid-surface.

In continuum mechanics, the Lagrangian formalism is typically used when the interest

is on following the material particles, while the Eulerian point of view describes the state

of the system in a given control volume in the physical space. Depending on the context

one formulation is preferred to the other (see

Formaggia et al.

(

2009

)). The classical choice

is to consider the Eulerian representation for the fluid and the Lagrangian for the solid.

Another possible formulation is the intermediate formalism, called Arbitrary Lagrangian

Eulerian (ALE) and is often considered in hemodynamics simulations (see

Nobile

(

2001

);

Formaggia et al.

(

2009

)).

In the following paragraphs, we will first present the fluid equations in their Eulerian

and ALE formalism, afterwards, the solid Lagrangian equations and finally the coupled

FSI problem.

1.2.1

Fluid models

Blood is a complex non-Newtonian fluid characterized by a suspension of cells in a

liquid (the plasma) made of water for its 90%. The constitutive particles are mainly

1.2.

Fluid-structure interaction

9

red blood cell, white blood cell and platelets. The red blood cell are responsible for

the Newtonian blood behavior, due to their highly deformable structure. The

Non-Newtonian effects become significant when the vessel size is small (see, e.g.,

Fåhræus-Lindqvist effect in

Possenti et al.

(

2019

);

Pries et al.

(

1994

)), while they can be neglected for

medium and large size arteries. In this work we will consider the numerical approximation

of blood flow in large/medium size vessels, hence, it is reasonable to assume the blood

as an homogeneous, incompressible and Newtonian fluid, governed by the Navier-Stokes

equations. Let now introduce the Navier-Stokes equations in the different frameworks.

1.2.1.1

Eulerian Incompressible Navier-Stokes equations

In this paragraph we provide the necessary notations to describe the Navier-Stokes

equations in theirs Eulerian framework. Let Ω

f

_(t)

_{∈ R}

d

_{, d = 2, 3, be a bounded}

time-dependent domain, with a Lipschitz boundary ∂Ω

f

_{(t). We denote with n the unit outward}

normal on ∂Ω

f

_(t)

_{. The previously discussed fluid assumptions translate to constant (in}

space and time) fluid density ρ

f

_{and dynamic viscosity µ. The Incompressible}

Navier-Stokes equations in Eulerian framework read as follow: find the fluid velocity u : Ω

f

_(t)

_×

R

+

→ R

d

and pressure p : Ω

f

(t)

× R

+

→ R, such that for all t ∈ R

+

we have

(

ρ

f

_∂

t

u + (u

· ∇)u

− div σ(u, p) = 0 in Ω

f

_(t),

d

iv u = 0 in Ω

f

(t),

(1.1)

where σ(u, p)

def

= 2µ(u)

_{− pI is the Cauchy stress tensor and (u)}

def

=

1

2

∇u + ∇u

T

₎

_the

strain rate tensor, in which I denotes the identity tensor.

Problem (

1.1

) needs to be completed with proper initial condition u(0) = u0

and

boundary conditions on ∂Ω

f

_{(t), namely}

u = u

D

on ΓD

(t),

σ(u, p)n = g

_N

on ΓN(t),

where uD

and gN

denote, respectively, a velocity and a pressure profile. Finally, ΓD

and

ΓN, are such that ∂Ω

f

_{(t) = ΓD(t)}

_{∪ Γ}

_N(t)

_.

1.2.1.2

Arbitrary-Lagrangian-Eulerian Incompressible Navier-Stokes

equa-tions

In this paragraph we reformulate the Eulerian Navier-Stokes equation (

1.1

) in the ALE

framework. Only the essential notions are presented. A more extensive presentation can

be found in

Formaggia et al.

(

2009

) and

Nobile

(

2001

). Let bΩ

f

_{= Ω}

f

₍₀₎

_{be the reference}

fluid domain. The Arbitrary-Lagrangian-Eulerian description is based on the introduction

of an appropriate one-to-one mapping A : bΩ

f

_{× R}

+

_{→ R}

d

_{, defined in terms of the fluid}

domain displacement d

f

_{: b}

_Ω

f

_{× R}

+

_{→ R}

d

_{, given by the following expression}

The purpose of the ALE mapping A is to parametrize the motion of fluid domain and

facilitate the time discretization, when working with an evolving computational domain.

The fluid domain velocity is hence defined as w

def

= ∂

t

d

f

and the initial configuration is

such that bΩ

f

₌

_A(b

_Ω

f

_{, 0). It should be noted that, the initial configuration coincides with}

the reference fluid domain. We will also introduce the notation At

(

_bx)

def

=

_{A(bx, t), by fixing}

t > 0

and with bx ∈ bΩ

f

_.

We recall that for a given functional defined on the ALE reference domain b

f : b

Ω

f

_×R

+

_,

we can define its Eulerian counterpart, by

f (x, t) = b

f (x, t)

_{◦ A}

−1

_t

= b

f (

_A

−1

_t

(x), t),

_{∀ (x, t) ∈ Ω}

f

(t)

_{× R}

+

,

and conversely

b

f (

x, t) = f (

_b

_At

(

x), t),

_b

_{∀ (b}

x, t)

_{∈ b}

Ω

f

_{× R}

+

.

For instance, at each point of the current configuration, the Eulerian domain velocity is

such that w(x, t) = b

w(

x, t).

b

Notice that in general the fluid velocity and the domain

velocity are different (if w = 0 we retrieve the Eulerian formulation, while if w = u the

Lagrangian description).

The last necessary ingredient is the so-called ALE time-derivative. For a given Eulerian

field q, we define the ALE time-derivative as follows:

∂q

∂t









_A

def

= ∂

t|

A

q = w

· ∇q +

∂q

∂t

.

Hence, the Arbitrary-Lagrangian-Eulerian description of the incompressible Navier-Stokes

equations is obtained by introducing the ALE time derivative inside (

1.1

). The problem

reads as follow:

find the velocity bu = bu(bx, t) : bΩ

f

_{× R}

+

_{→ R}

d

_{and pressure bp = bp(bx, t) : bΩ}

f

_{× R}

+

_{→ R,}

such that

(

ρ

f

∂t|A

u + u

− w · ∇

u

− div σ(u, p) = f in Ω

f

_(t),

div u = 0 in Ω

f

_(t),

(1.2)

As for the Eulerian description, problem (

1.2

) is similarly completed with initial condition

on Ω

f

₍₀₎

_{and boundary conditions on ΓD}

_{and ΓN.}

1.2.2

Thin-walled solid model

For a full presentation of the theory of shell models we refer to

Chapelle and Bathe

(

2011

);

Bischoff et al.

(

2018

). Given the ratio thickness/size of the heart valve leaflets, a

common assumption is to model the valves as co-dimensional one structures (i.e., (d −

1)-dimensional models where d is the dimension of the problem under analysis, see, e.g.,

Diniz dos Santos et al.

(

2008

);

Astorino et al.

(

2009b

)). These simplified (but still

real-istic) dimensionally reduced problems, that we refer to as thin-walled solid models, are

those considered in the models presented in this work. Starting form a thick-walled solid

description, with the assumption of ration thickness/size of the solid structure small, we

1.2.

Fluid-structure interaction

11

define the solid equations on the solid mid-surface Σ and we refer to its thickness as ε. Note

that Σ is at the same time the fluid-structure interface as well as the solid reference

config-uration. We will consider the Reissner-Mindlin kinematic assumption (see

Chapelle and

Bathe

(

2011

)), which states that, a material line orthogonal to the reference mid-surface

is assumed to remain straight and unstretched during deformation. The displacement is,

hence, given in terms of a global displacement and a rotation vector around the normal to

the mid-surface. Additionally we well consider a shear-membrane-bending model within a

non-linear framework. This will be the model considered in the numerical examples of the

next chapters, unless specified otherwise.

For sake of simplicity, the shear-membrane-bending model that we consider in the

description of the methods reads as follow: find the solid displacement d : Σ × R

+

_{→ R}

d

and velocity

d : Σ

.

× R

+

_{→ R}

d

_{, such that}

(

ρ

s

ε∂

t

.

d + L(d) = T

on Σ,

.

d = ∂

t

d

on Σ,

(1.3)

where ρ

s

_{represents the solid density and ε its thickness. Additionally, T denotes a given}

source term, hence a force per unit area, and the surface operator L represents the strong

formulation of the thin-walled solid elastic contributions.

Finally, problem (

1.3

) must be completed with initial conditions, namely,

(

d(0) = d

0

on Σ,

.

d(0) =

d

.

0

on Σ,

(1.4)

as well as boundary conditions on ∂Σ.

1.2.3

Fluid-structure coupled problems

Considering the models presented previously, we can now describe the full

fluid-structure interaction problem including the coupling/transmission conditions. Thus, we

will couple the fluid equations introduced in Section

1.2.1

(considering both Eulerian and

ALE formalism) with the thin-walled solid model presented in Section

1.2.2

in Lagrangian

formalism.

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⌃(t)

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⌃

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t

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⌦

f

2

(t)

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n

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n

2

Figure 1.1: Geometrical configuration.

We consider Σ ⊂ R

d

_{, with d = 2, 3, the solid mid-surface and let Σ(t) be the current}

position of the interface, given in terms of a deformation map φ : Σ × R

+

_{→ R}

d

_and

the solid displacement d, namely φ

def

= I

_Σ×R

+

+ d

and such that Σ(t) = φ(Σ, t). The

geometric configuration is shown in Figure

1.1

. The structure is allowed to move within a

fixed domain Ω

f

_{⊂ R}

d

_{, with boundary Γ}

def

_{= ∂Ω}

f

_{. The fluid time-dependent domain is then}

defined as Ω

f

_(t)

def

_{= Ω}

f

_{\ Σ(t) and with boundary ∂Ω}

f

_{= Γ}

_{∪ Σ(t). We also introduce the}

following notation φ

t

def

= φ(

·, t). The immersed interface Σ(t) is oriented with unit normal

n

Σ

and it divides the fluid domain Ω

f

(t)

into two subdomains, with normals respectively

n1

def

= nΣ

and n2

def

=

−nΣ.

For a given field f in Ω

f

_(t)

_{(possibly discontinuous on Σ(t)), we define its}

sided-restrictions, denoted by f1

and f2, as

f1(x)

def

= lim

ξ→0

−

f (x + ξn1),

f2(x)

def

= lim

ξ→0

−

f (x + ξn2),

for all x ∈ Σ(t), and the following jump and average operators across Σ(t):

Jf K

def

= f1

_{− f}

2

Jf nK

def

= f1n1

+ f2n2,

_{{f}}

def

=

1

2

f1

+ f2

.

In the coupling with a thin-walled solid the transmission condition are applied directly

to the solid mid-surface Σ, this means that the solid thickness effects are neglected in the

interface coupling. This is a common assumption in the coupling of thin-walled solids with

general 3D media (see, e.g.,

Chapelle and Ferent

(

2003

)). The fluid and solid problems are

coupled via the so-called kinematic and dynamic coupling conditions. The first is a no-slip

condition, representing the fact that, due to its viscosity, the fluid sticks perfectly to the

fluid-structure interface. The second accounts for the Newton’s third law, hence, for the

balance of stresses at the interface. Additionally a geometrical compatibility condition

need to be fulfilled between the fluid and solid domains Ω

f

_(t)

_{and Σ(t).}

In the sequel, we introduce the Eulerian-Lagrangian and the ALE-Lagrangian

descrip-tion of the coupled FSI problem.

1.2.3.1

Eulerian-Lagrangian formalism

We can define the first coupled fluid-structure problem by considering the Eulerian

Navier-Stokes equations of Section

1.2.1.1

and the membrane model of Section

1.2.2

. The

coupled problem read as follow: find the fluid velocity u : Ω

f

_{× R}

+

_{→ R}

d

_{and pressure}

p : Ω

f

_{× R}

+

_{→ R, the solid displacement d : Σ × R}

+

_{→ R}

d

_{and velocity}

_{d : Σ}

.

_{× R}

+

_{→ R}

d

_,

such that











ρ

f

_∂

t

u + u

· ∇u

− div σ(u, p) = 0 in Ω

f

_(t),

d

iv u = 0 in Ω

f

(t),

u = 0

on Γ,

(1.5)

(

ρ

s

ε∂t

d + L(d) = T

.

on Σ,

.

d = ∂td

on Σ,

(1.6)