A mechanically consistent model for fluid-structure interactions with contact including seepage

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Preprint submitted on 18 Mar 2021

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A mechanically consistent model for fluid-structure

interactions with contact including seepage

Stefan Frei, Fannie Gerosa, Erik Burman, Miguel Angel Fernández

To cite this version:

Stefan Frei, Fannie Gerosa, Erik Burman, Miguel Angel Fernández. A mechanically consistent model

for fluid-structure interactions with contact including seepage. 2021. �hal-03174087�

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(will be inserted by the editor)

A mechanically consistent model for fluid-structure interactions with

contact including seepage

Stefan Frei, Fannie M. Gerosa, Erik Burman and

Miguel A. Fern´

andez

Received: date / Accepted: date

Abstract

We present a new approach for the mechanically consistent modelling and simulation of

fluid-structure interactions with contact. The fundamental idea consists of combining a relaxed contact

formulation with the modelling of seepage through a porous layer of co-dimension 1 during contact. For

the latter, a Darcy model is considered in a thin porous layer attached to a solid boundary in the limit

of infinitesimal thickness. In combination with a relaxation of the contact conditions the computational

model is both mechanically consistent and simple to implement. We analyse the approach in detailed

numerical studies with both thick- and thin-walled solids, within a fully Eulerian and an immersed

approach for the fluid-structure interaction and using fitted and unfitted finite element discretisations.

Keywords

Fluid-structure interaction

_{· contact mechanics · Nitsche’s method · finite elements · seepage}

1 Introduction

The design and analysis of computational methods for systems where several solids are immersed in a

fluid and that can come into contact is an outstanding problem. Already fluid-structure interaction (FSI)

without contact is challenging due to the moving geometries and the stiff coupling between the solid and

the fluid systems. If contact between solids is to be modelled as well, the complexity increases drastically.

Indeed, the addition of contact introduces several important aspects, such as:

–

Topological changes in the fluid domain;

–

Non-linearly changing interface conditions: The interface condition changes from a fluid-solid

inter-action to a solid-solid contact problem which is described by variational inequalities;

–

Important differences in the characteristic scales of the different physical phenomena: The contact

represents a singular phenomenon in time. In three space dimensions, the contact zone is a two

dimensional subset of the solid-solid interface and there is also a one dimensional subset of the

contact zone forming the solid-solid-fluid line.

Ideally, a computational method should be consistent with the physics, be amenable to mathematical

analysis and convenient to implement in a computational software.

In the case of fluid-structure interaction with contact, an additional complication is that it is unclear

what mathematical modelling will produce the best results. Indeed, it is known that a naive imposition

of no-slip conditions on one of the boundaries of a solid will prevent contact of smooth bodies in the

solution of the PDE system [Hesla, 2004, Hillairet, 2007, Gerard-Varet et al., 2015], contrary to what is

observed in experiments [Hagemeier et al., 2020].

Stefan Frei

Department of Mathematics and Statistics, University of Konstanz, Universittsstr. 10, 78457 Konstanz, Germany. E-mail:

stefan.frei@uni-konstanz.de

Miguel A. Fern´

andez & Fannie M. Gerosa

Inria, 75012 Paris, Sorbonne Universit´

e & CNRS, UMR 7598 LJLL, 75005 Paris, France. E-mail: miguel.fernandez@inria.fr,

fannie.gerosa@inria.fr

Erik Burman

Department

of

Mathematics,

University

College

London,

Gower

Street,

London

UK,

WC1E

6BT.

E-mail:

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Therefore, the design of computational methods for FSI-contact problems can not be completely

dissociated from the problem of modelling, but it is important to keep a certain flexibility concerning

the contact modelling to be able to include a wide range of physics, depending on the characteristics

of the considered system. In this work, we will build on previous work for FSI with contact [Burman

et al., 2020a]. There, recent techniques merging the ideas of weak imposition of fluid-structure interface

conditions [Hansbo et al., 2004, Burman and Fern´

andez, 2014] with a multiplier free formulation for

contact [Chouly and Hild, 2013] were developed, leading to an automatic handling of both fluid-solid and

solid-solid coupling conditions. The main idea was to merge different versions of Nitsche’s method using an

augmented Lagrangian formulation for variational inequalities dating back to Rockafellar [Rockafellar,

1973]. The resulting method is consistent and shown in numerical examples to be both accurate and

robust. It can also easily be combined with tools developed to facilitate the handling of the moving

interfaces such as cutFEM [Burman et al., 2015, Burman and Fern´

andez, 2014], XFEM [Mo¨es et al.,

1999], GFEM [Babuˇska et al., 2004] or fitted finite element approaches [Frei and Richter, 2014].

Already in [Burman et al., 2020a] some modelling aspects were developed. In order to avoid the

singularity of vanishing pressure in the contact zone the fluid was extended into the solid in a porous

medium model. Alternatively, the distance between the contacting bodies can be lower bounded by

some small value (proportional to the mesh size) representing the idea that a fluid layer always remain

between the contacting bodies. A similar approach was taken in [Ager et al., 2019], but here the physical

modelling went further, introducing porous layers on the solids and modelling the full 3D poro-elastic

fluid-structure interaction with contact. In the computational model for contact, a Lagrange multiplier

technique was applied in contrast to the Nitsche-approach in [Burman et al., 2020a]. In [Zonca et al.,

2020] a polygonal DG discretisation is used within a penalty method in combination with a switch

between Navier-slip to slip conditions to enable the transition to contact. The recent work [Ager et al.,

2020] showcases the potential of combining the Nitsche FSI-contact conditions of [Burman et al., 2020a]

with the FSI-cutFEM approach from [Burman and Fern´andez, 2014] using realistic physical models in

some impressive computational examples.

In the present work, we wish to build on the ideas of [Burman et al., 2020a] by considering a model for

contact with seepage due to microscopic roughness of the contacting bodies. Let Ω

_{⊂ R}

d

_{for d = 2, 3 be}

the overall domain consisting of fluid and solid subdomains. The seepage is modelled by the introduction

of a d

_{− 1-dimensional porous layer that adheres to a solid boundary, where contact might take place.}

This can be considered as a generalised boundary condition, or a bulk surface coupling in the spirit

of [Elliott and Ranner, 2013]. In our case, however, the free-flow Navier-Stokes’ system is coupled to a

surface Darcy equation. This model goes back to [Martin et al., 2005] and is of interest in its own right,

as discussed in the note [Burman et al., 2021]. By combining this porous layer approach for seepage with

the contact approach of [Burman et al., 2020a], herein extended to the case of thin-walled solids, we

obtain an approach that inherits the simplicity, accuracy and robustness of [Burman et al., 2020a], but

provides a mechanically consistent model for fluid-structure interaction with contact.

We implement the approach using different coordinate systems, discretisations and solid models.

Concerning coordinate systems, we consider both an Immersed approach going back to Peskin [Peskin,

1972] as well as a Fully Eulerian approach [Dunne, 2006, Cottet et al., 2008, Richter, 2013, Frei, 2016]. For

discretisation, we use the unfitted finite element method of [Hansbo et al., 2004, Burman and Fern´andez,

2014, Alauzet et al., 2016] and the two-scale interface fitting approach of [Frei and Richter, 2014]. We

illustrate the modelling capacity in a series of computational examples in two dimensions, including a

beam solid model and a thick-walled solid model.

Indeed, depending on if no-slip conditions are imposed or if the porous medium approach proposed

here is used the approximations will converge to different solutions for mesh size h

_{→ 0. If we consider the}

case of a bouncing ball the use of no-slip conditions will lead to a sequence of solutions that converge to

a ball that does not bounce, whereas the solutions obtained with the porous medium approach converge

to a certain bouncing height that depends on the parameters of the Darcy model. Recent comparisons

of computational methods with experimental studies ([Hagemeier et al., 2020], [von Wahl et al., 2020])

confirm that the second behavior is the physical one. Of course the parameters of the model need to be

fixed through experimental studies, or otherwise.

An outline of the paper is as follows. In Section 2, we introduce the Navier-Stokes-Darcy coupling

as well as the FSI-contact model. The variational formulation and the discretisation based on Nitsche’s

method is described in Section 3. In Section 4 we give detailed numerical studies both in the case of a

beam model and a thick solid. We conclude in Section 5.

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⌃

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<latexit sha1_base64="1GHsuPfCUrgrdol4hPou/mkpkwM=">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</latexit>

⌦

f

_(t)

<latexit sha1_base64="R+UjjQ4VLxd+l7JWub73HqdAWnA=">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</latexit>

⌦

s

_(t)

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⌦

s

(a) Coupling with a thick-walled solid

<latexit 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<latexit 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⌦

f

1 (t)

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⌦

f

2 (t)

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⌃

p

(b) Coupling with a closed thin-walled solid

Fig. 1 Geometric configurations of the fluid and solid domains.

2 Equations

In this section, we derive the Navier-Stokes-Darcy coupling, and subsequently the equations for

fluid-structure-porous-contact interaction. For simplicity, we will consider that contact takes place at a given

fixed plane surface. This can be either an exterior rigid wall or a symmetry boundary within the fluid

domain, which is relevant for example in the case of contact between two symmetric valves. The case of

to two-body contact is not considered here.

The fluid equations in Ω

f

_(t)

⊂ R

d

_{will be coupled to a fixed (d}

− 1)-dimensional porous layer Σ

p

on the exterior boundary, where contact might take place. The fluid is described by the Navier-Stokes

equations in Eulerian formalism and the structure by a possibly non-linear solid model. We consider

both (d

_{− 1)-dimensional thin-walled solids and d-dimensional thick-walled solids.}

2.1 Problem setting

Let Ω = Ω

s

_(t)

_∪Ω

f

_(t)

⊂ R

d

_{be a current configuration of the complete domain of interest, with boundary}

∂Ω:=Γ

_{∪ Σ}

p

, where Σ

p

denotes the part of the boundary where contact might take place (see Figure 1).

There, a thin porous fluid layer is considered. The solid domain Ω

s

_{(t) can be either a surface (actually the}

solid mid-surface) or a domain with positive volume in R

d

_{in the case of the coupling with a thick-walled}

solid. The current fluid-structure interface is denoted by Σ(t) and coincides with Ω

s

_{(t) in the case of a}

thin-walled solid. The corresponding reference configurations are denoted by Σ and Ω

s

_.

The structure is allowed to move freely within the domain Ω. The current position of the interface

Σ(t) and the solid domain Ω

s

_{(t) are described in terms of a deformation map φ : Ω}

s

× R

+

−→ R

d

_such

that Ω

s

_{(t) = φ(Ω}

s

_{, t) and Σ(t) = φ(Σ, t), with φ:=I}

Ω

s

+ d and where d denotes the solid displacement.

To simplify the notation we will refer to φ

t

:=φ(

·, t), so that we can also write Ω

s

(t) = φ

t

(Ω

s

), Σ(t) =

φ

t

(Σ). The fluid domain is time-dependent, namely Ω

f

(t):=Ω

\(Ω

s

(t)

∪ Σ(t)) ⊂ R

d

with boundary

∂Ω

f

_{(t) = Σ(t)}

_{∪ Γ ∪ Σ}

p

. In the case of a closed thin-walled structure, the solid domain Ω

s

(t) divides

Ω

f

_{(t) into two subdomains Ω}

f

_{(t) = Ω}

f

1 (t)

∪ Ω

2 f

(t), with respective unit normals n

1 :=n and n

2 :=

− n, as

shown in Figure 1(b). Similarly, in the case of a thick-walled solid, the interface Σ(t) divides Ω into a

solid part Ω

s

(t) and a fluid part Ω

f

(t). We write H

Γ

1 (Ω) for the first-order Sobolev space with vanishing

trace on Γ

_{⊂ ∂Ω.}

For a given field f defined in Ω (possibly discontinuous across the interface), we can define its

one-sided restrictions, denoted by f

1 and f

2 , as

f

1 (x):= lim

ξ

→0

−

f (x + ξn

1 ),

f

2 (x):= lim

_ξ

_→0

−

f (x + ξn

2 ),

for all x

_{∈ Σ(t), and the following jump and average operators across Σ(t):}

Jf K:=f

1 − f

2 Jf nK:=f

1 n

1 + f

2 n

2 ,

{{f}}:=

1

2 f

1 + f

2 .

(1)

In the case of a thin structure that has a boundary inside the fluid domain (for example with a tip),

these quantities can be defined similarly. For the details, we refer to [Alauzet et al., 2016] and Remark 1

below.

While the fluid and solid equations are standard and will be introduced in Section 2.3, we give some

details on the porous medium model in the following section.

(5)

2.2 Porous medium model and Navier-Stokes-Darcy coupling

We consider the configuration sketched in Figure 2, where a thin porous layer Ω

p

= Σ

p

× (−

₂

p

,

₂

p

)

∈ R

d

(d = 2, 3) with midsurface Σ

p

is coupled to a surrounding fluid in a fixed domain Ω

f

⊂ R

d

. The

surrounding fluid is governed by the Navier-Stokes equations

(

ρ

f

∂

t

u + u

· ∇u

− div σ

f

(u, p) = 0 in Ω

f

,

div u = 0 in Ω

f

.

(2)

Here, u denotes the fluid velocity and p the fluid pressure in Ω

f

_{. The Cauchy stress tensor is given by}

σ

f

:= 2µ

f

ε(u)

− pI, ε(u) =

1

2 ∇u + ∇u

T

_,

where I denotes the identity matrix.

γf

Ωp

n

γo

Ωf

Σp

p

Fig. 2

Porous medium domain Ω

p

with interface γ

f

to Ω

f

and exterior boundary γ

o

.

In the porous domain Ω

p

, we assume a Darcy law

(

u

l

+ K

∇p

l

= 0 in Ω

p

,

∇ · u

l

= 0 in Ω

p

,

(3)

where u

l

denotes the Darcy velocity, p

l

the Darcy pressure and K is a d

×d matrix such that the following

decomposition holds

K

_∇p

l

= K

τ

∇

τ

p

l

+ K

n

∂

n

p

l

,

with K

τ

, K

n

∈ R

+

. Here, n is the unit normal vector of the mid-surface Σ

p

that points towards the

exterior boundary γ

o

, ∂

n

= n∂

n

and

∇

τ

:= P

τ

∇ stands for the corresponding tangential part of the

gradient

P

τ

:= (I

− n ⊗ n).

Furthermore, we assume that the porous layer is very thin and consider the limit case

p

→ 0. Let the

outer boundary of Ω

p

be denoted by γ

o

and the interior boundary connecting to the fluid domain Ω

f

by

γ

f

, see Figure 2. We assume zero normal velocity (u

l

· n = 0) on the outer boundary γ

o

and continuity

of normal velocities and normal stresses on γ

f

. For the tangential fluid stresses, we consider the

Beavers-Joseph-Saffman coupling conditions [Saffman, 1971]. By u

τ

:= P

τ

u, we denote the tangential part of the

velocity vector and by σ

f,n

= n

T

σ

f

n and σ

f,τ

= P

τ

σ

f

n the normal and tangential part of the Cauchy

stress tensor σ

f

introduced above. The coupling conditions between porous medium Ω

p

and fluid Ω

f

read

_









σ

f,n

=

−p

l

on γ

f

,

u

_{· n = u}

l

· n

on γ

f

,

σ

f,τ

=

−

α

p

K

τ

p

u

τ

on γ

f

.

(4)

We note that the condition for the tangential stresses, in the last line, corresponds to a Navier-slip

boundary condition for the fluid. In contrast to this boundary condition for the fluid, the normal velocity

u

_{· n is not zero here, as the fluid can enter the porous layer.}

The appropriate choice of the parameter α in the last line of (4) depends on the application. In

the case that γ

f

correponds to a symmetry boundary within a larger fluid domain, where contact can

take place, for example between two contacting valves, it is appropriate to set α = 0 (pure slip). If the

porous layer is, however, placed at a rigid wall, the Beavers-Joseph-Saffman condition with α > 0 is more

(6)

porous layer. Values 0.01 < α < 5 have been suggested in [Nield et al., 2006]. We will consider both kind

of conditions in the numerical examples of Section 4.

Introducing the averaged porous pressure P

l

as

P

l

=

1

2 (p

l

|γ

f

+ p

l

|γ

o

)

in Σ

p

,

(5)

the following equations can be derived in the limit case

p

→ 0 (see [Martin et al., 2005, Burman et al.,

2021])











−∇

τ

· (

p

K

τ

∇

τ

P

l

) = u

· n on Σ

p

,

σ

f,n

=

−P

l

−

p

K

−1

n

4 u

· n on Σ

p

,

σ

f,τ

=

−

α

p

K

τ

p

u

τ

on Σ

p

.

(6)

Note that the only remaining porous medium variable is the averaged pressure P

l

. In the limit K

n

, K

τ

→

0, the coupling conditions turn into a Navier-slip boundary condition for the fluid on Σ

p

.

2.3 Fluid-structure-porous-contact interaction model

We assume that Ω

f

_{(t) is filled by an incompressible fluid governed by the Navier-Stokes equations. The}

domain Ω

s

_{(t) is occupied by a solid media described by a beam or shell solid model (given in terms of}

an abstract surface differential operator L) on a (d

− 1)-dimensional domain Σ or by the elastodynamics

equations in the case of a d-dimensional domain Ω

s

_{. The fluid and solid equations are coupled with}

no-slip interface conditions on the fluid-structure interface Σ(t). The solid is constrained to not penetrate

into the porous medium Σ

p

via the (relaxed) unilateral frictionless contact conditions

d

_{· n − g}

≤ 0, λ ≤ 0, λ(d · n − g

) = 0

on Σ,

(7)

Here, g

:= g

−

g

, where g denotes the gap function to Σ

p

and

g

> 0 is a small parameter. The symbol

λ stands for the normal component of the contact traction, which corresponds to the Lagrange multiplier

associated to the no-penetration condition.

The proposed fluid-structure-porous-contact interaction model is hence formulated as follows: Find

the fluid velocity and pressure u : Ω

f

× R

+

→ R

d

_{, p : Ω}

f

× R

+

→ R, the solid displacement and velocity

d : Ω

s

× R

+

→ R

d

_{, ˙}

_{d : Ω}

s

× R

+

→ R

d

_{, the Darcy porous pressure P}

l

: Σ

p

× R

+

→ R and the Lagrange

multiplier λ : Σ

_{× R}

+

→ R such that, for all t ∈ R

+

_{, the following relations are satisfied}

–

Fluid problem:











ρ

f

_∂

t

u + u

· ∇u

− div σ

f

(u, p) = 0 in Ω

f

(t),

div u = 0 in Ω

f

(t),

u = 0 on Γ,

(8)

–

Porous layer:

(

−∇

τ

· (

p

K

τ

∇

τ

P

l

) = u

l

· n on Σ

p

,

p

K

τ

· ∇

τ

P

l

= 0 on ∂Σ

p

,

(9)

–

Solid problem:











ρ

s

∂

t

d + L(d) = T

˙

on Ω

s

= Σ,

˙

d = ∂

t

d

on Ω

s

= Σ,

d = 0 on ∂Ω

s

∩ Γ,

(10)

in case of a thin-walled solid, or











ρ

s

∂

t

d

˙

− div σ

s

(d) = 0

on Ω

s

,

˙

d = ∂

t

d

on Ω

s

,

d = 0 on ∂Ω

s

∩ Γ,

(11)

(7)

–

Contact conditions:

d

· n − g

≤ 0, λ ≤ 0, λ(d · n − g

) = 0

on Σ.

(12)

–

Fluid-structure coupling conditions:

(

φ = I

Ω

s

+ d,

Ω

_s

(t) = φ

_t

(Ω

s

),

Ω

f

(t) = Ω

\Ω

s

(t),

u = ˙

d

_{◦ φ}

−1

_t

on Σ(t),

(13)

and

Z

Σ

(T

− λn) · w = −

Z

Σ(t)

Jσ

f

(u, p)nK · w ◦ φ

−1

t

,

(14)

or

Z

Σ

(σ

s

− λI)n · w = −

Z

Σ(t)

σ

f

(u, p)n

· w ◦ φ

−1

t

(15)

for all test functions w, respectively in the case of the coupling with a thin- or thick-walled solid.

–

Fluid-porous coupling conditions:











u

l

· n = u · n on Σ

p

,

σ

f,n

=

−P

l

−

p

K

n

−1

4 u

· n

|

{z

}

=: σ

p

on Σ

p

,

σ

f,τ

=

−

p

α

K

τ

p

u

τ

on Σ

p

.

(16)

The relations in (13)-(15) enforce the geometrical compatibility and the kinematic and the dynamic

coupling at the interface between the fluid and the solid, respectively. It should be noted that the

no-penetration condition in (12) is already imposed at an

g

-distance to the porous layer Σ

p

. This

modeling simplification circumvents most of the numerical difficulties associated with the topological

change in the fluid domain induced by the exact contact condition (i.e., with = 0), such as switching

between the contact and fluid-solid interfaces and presence of isolated small fluid regions (see [Ager et al.,

2020]). Moreover, it also facilitates the explicit treatment of the geometric condition in the fluid-structure

coupling (see Section 3).

2.3.1 Mechanical consistency

In the fluid-structure-porous-contact interaction model (8)-(16), a very thin fluid layer always remains

between the solid and porous medium during contact. Owing to the relations (16), the behavior of the

fluid confined in the contact layer is expected to be very close to the one of the porous fluid. Indeed,

this is a consequence of the kinematic-dynamics relations (16)

1,2

, which are enforced both during and

in absence of contact. If a part of Σ(t) is in contact with Σ

p

accoding to (7), the value of σ

f,n

(resp.

u

_{· n) on this part Σ(t) will be close to σ}

p

(resp. u

l

· n) on the corresponding part of Σ

p

. As a result, all

the kinematic and dynamic relations acting on the solid during contact have a physical meaning, which

guarantees the mechanical consistency of the proposed fluid-structure-contact interaction model.

More precisely, owing to (15), in the case of a thick-walled solid the Lagrange multiplier for the

no-penetration condition will formally assume the form

λ = σ

s,n

− σ

f,n

|

{z

}

=:

Jσ

n

K

≈ σ

s,n

− σ

p

◦ π on Σ,

where we write σ

s,n

:= n

T

σ

s

n for the solid normal traction and π denotes a (closest-point) projection

from Σ(t) to Σ

p

. Both, the solid stresses σ

s,n

and the ”porous stresses” σ

p

have a physical meaning

during solid-porous contact. Hence, this porous-contact approach gives a physical meaning to the stresses

generated in the infinitesimal fluid layer, in contrast to the relaxed contact formulation in [Burman

et al., 2020a], where the fluid stresses σ

f,n

did not allow for a direct physical interpretation. A similar

argumentation holds true for the thin-walled solid case, where

λ = ρ

s

∂

t

d + L(d)

˙

· n +

Jσ

f,n

K ≈

ρ

s

_∂

(8)

In the spirit of [Alart and Curnier, 1991], the Lagrange multiplier can further be eliminated, which

in the case of a thick-walled solid results in the non-linear contact condition

Jσ

n

K = −γ

C

d

_{· n − g}

− γ

C

−1

Jσ

n

K

|

{z

}

=: P

γ

C

(d

· n,

Jσ

n

K)

+

on Σ,

(17)

for γ

C

> 0. This can be embedded in an elegant way in the variational formulation using a Nitsche-based

approach, see [Burman et al., 2020a] and Section 3. For a thin solid, a similar approach is possible, with

the additional difficulty that the normal solid traction on the mid-surface is given in terms of the normal

PDE residual ρ

s

_∂

t

d + L(d)

˙

· n, which is rarely available at the discrete level. On the other hand,

it has been shown (see [Burman et al., 2017, Scholz, 1984, Chouly and Hild, 2012]) for the case of a

thin-walled solid that a pure penalty approach (i.e. neglecting the normal traction λ in the term P

γ

c

)

leads to a first-order approximation. The detailed variational formulations for both the case of a

thick-and a thin-walled solid will be given in the next section.

The main advantage of this method is its simplicity. Moreover, it is expected from a mechanical

point of view that the behavior of the fluid confined in the contact layer is close to the porous fluid, as

explained above. The porous medium and the structure are always coupled with the fluid only and never

directly to each other. This avoids switches in the variational formulation, which would be necessary

in the transition between fluid-solid and solid-porous interaction ([Burman et al., 2021]). On the other

hand, the solid perceives indirectly the presence of the porous layer through the fluid stresses and velocity

during contact. The resulting numerical approach is highly competitive in terms of computational costs

compared to approaches using Lagrange multipliers and/or active-sets.

2.3.2 Seepage

The proposed fluid-structure-porous-contact interaction model (8)-(16) allows for seepage in the sense

that fluid can flow through the porous layer Σ

p

, for example to connect a cavity in the central part of

the contact surface with the exterior fluid. These could emerge when the impact of the structure happens

in the lateral parts of the structure first or when contact of the solid is released in a central part of the

contact surface only. This is an important aspect in the modelling of fluid-structure-contact interaction,

as otherwise unphysical configurations might result. As an example consider the situation sketched in

Figure 3, where a solid body is in contact with the lower wall Σ

p

at initial time (left sketch). When a

(sufficiently strong) force f is applied in the central part of Ω

s

_{, while the body is fixed at the lateral}

parts, contact will be released in the central part only. If no seepage along Σ

p

is allowed, a vacuum would

emerge between Ω

s

_{and Σ}

p

. While one could argue that this paradox is already circumvented by using

the relaxed contact conditions (7), we note that only the porous layer gives a physical meaning to the

fluid filling the contact layer.

Ωf

Σp

Ωs

f

Ωs

Ωf

Σp

f

Fig. 3

Illustrative example to motivate the role of seepage in fluid-structure interaction with contact: Contact of a solid

body with the lower wall is released in the central part of the contact surface, when a specific force f is applied. Without

seepage on Σ

p

a vacuum would be created.

3 Numerical methods

This section is devoted to the numerical discretisation of the fluid-structure-porous-contact interaction

model (8)-(16). Two numerical approaches will be considered which basically depend on thin- or

thick-walled nature of the solid model and on the formalism used for the fluid-structure coupling (mixed

Lagrangian-Eulerian or fully Eulerian formalisms). For an accurate discretisation of the

fluid-structure-porous-contact interaction model (8)-(16), two different strategies will be considered in the numerical

examples reported in Section 4. The first strategy is based on an unfitted Nitsche-XFEM method, drawing

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on [Burman and Fern´andez, 2014, Alauzet et al., 2016]. The second strategy is a fitted finite element

method, following [Frei and Richter, 2014]. In both cases, we will use equal-order finite element methods.

To fix ideas, we will present the numerical approaches for two specific combinations of these

com-ponents (mixed Lagrangian-Eulerian vs. Fully Eulerian, fitted vs. unfitted finite elements, thick-walled

vs. thin-walled structures), namely a mixed Lagrangian-Eulerian approach with a thin-walled solid using

unfitted finite elements in Section 3.1 and a fully Eulerian approach with a thick-walled solid using fitted

finite element discretisation in Section 3.2. Different combinations are possible as well, but will not be

considered in the remainder of this article.

3.1 Lagrange-Eulerian formalism with immersed thin-walled solids

In what follows, the parameter δt > 0 stands for the time-step length and t

n

:= nδt denotes the time

instant at time level n

∈ N. The symbol x

n

_{generally denotes an approximation of x(t}

n

), for a given time

valued function x(t). We also introduced the notation

∂

δt

x

n

:=

1 δt

x

n

− x

n

−1

,

for the first-order backward difference.

We consider the fluid-structure-porous-contact interaction problem (8)-(16) in the case of the coupling

with immersed thin-walled solids. The time discretisation is performed with a backward-Euler scheme,

including a semi-implicit treatment of the the convective term in (8) and an explicit treatment of the

geometric coupling (13)

1 . As regards the spatial discretisation, an unfitted finite element approximation

with overlapping meshes is considered for the fluid-solid coupling, by drawing on the Nitsche-XFEM

method reported in [Alauzet et al., 2016, Burman and Fern´andez, 2014]. The fluid-porous system is

discretised by a standard fitted finite element approximation.

For the solid, we start by assuming that there exists a positive form a

s

_{: W}

× W −→ R, linear with

respect to the second argument, such that

a

s

_{d, w) = L(d), w}

Σ

for all w

_{∈ W := [H}

1 Γ

∩∂Σ

(Σ)]

d

, where W stands for the space of admissible displacements. Let

{T

s

h

}

0<h<1

be a family of triangulations of Σ. We consider the standard space of continuous piecewise

affine functions

X

h

s

:=

v

h

∈ C

0 (Σ)

v

h

|K

∈ P

1 (K),

∀K ∈ T

h

s

,

W

h

:= [X

h

s

]

d

∩ W .

The contact condition (12) is approximated via a penalty method (see, e.g., [Scholz, 1984]), by adding

the following penalty term into the solid discrete problem:

γ

c

E

s

h

2 d

n

h

· n − g

ε

₊

, w

h

_Σ

,

where E is the solid Young modulus and γ

c

> 0 is the (dimensionless) penalty parameter.

For a given discrete displacement approximation d

n

h

∈ W

h

at time t

n

, we define its associated

deformation map by φ

n

h

:=I

Σ

+ d

n

h

. This map characterises the current solid configuration (i.e., at time

level n), as Σ

n

_:=φ

n

h

(Σ). As indicated above, we consider an explicit update for the physical fluid domain

in (13)

1 , namely,

Ω

f,n

_:=Ω

\Σ

n−1

,

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which has the effect of removing the geometrical non-linearities in the fluid problem (8).