fsi

Numerical Methods for Fluid-Structure InteractionProblems

Thomas [email protected]

Heidelberg

July 23, 2010(in process)

Contents

Bibliography 4

1 Introduction 71.1 Fluid-Structure Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Fundamentals of Continuum Mechanics 132.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 The strain tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1.2 The rate-of-strain tensor . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.1 The Conservation Equations in Fluid Dynamics . . . . . . . . . . . . . 192.2.2 Material Laws in Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . 222.2.3 Incompressible Flows, the Navier-Stokes Equations . . . . . . . . . . . 232.2.4 Variational formulation of the incompressible Navier-Stokes Equations 24

2.3 Structure Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3.1 The Conservation Equations in Structure Dynamics . . . . . . . . . . 252.3.2 The Piola Transformation . . . . . . . . . . . . . . . . . . . . . . . . . 272.3.3 Material Laws in Structure Dynamics . . . . . . . . . . . . . . . . . . 292.3.4 Linear Models of Elasticity . . . . . . . . . . . . . . . . . . . . . . . . 302.3.5 Variational formulation of the elastic structure equations . . . . . . . 32

3 The Fluid-Structure Interaction Problem 333.1 Fluid Flows in ALE Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 353.2 Construction of the ALE-mapping Tf . . . . . . . . . . . . . . . . . . . . . . 383.3 Fluid-structure interaction in ALE coordinates . . . . . . . . . . . . . . . . . 41

4 The Finite Element Method for Continuum Mechanics 434.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.1.1 Finite Elements for elliptic partial differential equations . . . . . . . . 444.1.2 Discretization of time-depending partial differential equations . . . . . 454.1.3 Discretization of nonlinear problems . . . . . . . . . . . . . . . . . . . 464.1.4 Discretization of transport-dominant problems . . . . . . . . . . . . . 48

4.2 Finite Elements for the incompressible Navier-Stokes Equations . . . . . . . . 494.2.1 Finite Element discretization of the Stokes Equations . . . . . . . . . 494.2.2 Finite Element discretization of the Navier-Stokes Equations . . . . . 514.2.3 Finite Elements for Navier-Stokes in ALE-formulation . . . . . . . . . 53

4.3 Finite Elements for elastic structure equations . . . . . . . . . . . . . . . . . . 554.4 Finite Elements for the fluid-structure interaction problem . . . . . . . . . . . 56

3

Contents

4.5 Some mathematical theory for fluid-structure interaction problems (noch einwenig experimentell und mit Vorsicht zu geniessen) . . . . . . . . . . . . . . . 57

4.6 Solution of the monolithic FSI formulation . . . . . . . . . . . . . . . . . . . . 62

5 Partitioned Approaches 655.1 Coupling of the subproblems . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.2 Weakly coupled approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.3 Strongly coupled partitioned approaches . . . . . . . . . . . . . . . . . . . . . 715.4 Acceleration schemes for partitioned fsi-solvers . . . . . . . . . . . . . . . . . 72

5.4.1 Aitken relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.4.2 Steepest descent relaxation . . . . . . . . . . . . . . . . . . . . . . . . 74

5.5 Stability analysis and the added mass effect . . . . . . . . . . . . . . . . . . . 75

6 Alternative Approaches for FSI-problems 816.1 Fully Eulerian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.2 A Fixed Mesh Euler-Lagrange Approach . . . . . . . . . . . . . . . . . . . . . 86

6.2.1 The Flow Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.2.2 The Interface Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.2.3 The Structure Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 896.2.4 The coupled problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4

Bibliography

[Ran08] R. Rannacher. Numerische Methoden für Probleme der Kontinuumsmechanik.Universität Heidelberg, http://numerik.uni-hd.de/˜lehre/notes/, 2008. Vor-lesungsskriptum.

[Joh87] C. Johnson: Numerical Solution of Partial Differential Equations by the FiniteElement Method. Cambridge University Press, 1987

[RST08] H.-G. Roos, M. Stynes, L. Tobiska. Robust Numerical Methods for Singularly Per-turbed Differential Equations. Springer Series in Computational Mathematics 24,Second Edition, Springer Verlag 2008.

[Gal1] G.P. Galdi. An Introduction to the Mathematical Theory of the Navier-Stokes Equa-tions. Volume I, Linearised Steady Problems. Springer, 1994.

[Gal2] G.P. Galdi. An Introduction to the Mathematical Theory of the Navier-Stokes Equa-tions. Volume II, Nonlinear Steady Problems. Springer, 1994.

[GR86] V. Girault and P.-A. Raviart. Finite Elements for the Navier Stokes Equations.Springer, Berlin, 1986.

[SG97] H. Schlichting and K. Gersten. Grenzschicht-Theorie. Springer Verlag, 1997.

[Tem84] R. Temam. Navier-Stokes Equations. North-Holland, 1984.

Grundlagen der Funktionalanalysis

[Alt08] H.W. Alt. Lineare Funktionalanalysis. Eine anwendungsorientierte Einführung.Springer Verlag, Berlin, 5. Auflage, 2008.

[Wer07] D. Werner. Funktionalanalysis. Springer, Berlin, 6. Auflange, 2007.

Methods for Fluid Structure Interaction

[GeWa08a] A. Gerstenberger, W.A. Wall. Enhancement of fixed-grid methods towars com-plex fluid-structure interaction applications. International Journal for NumericalMethods in Fluids, 57(9), pp 1227-1248 (2008)

[GeWa08b] A. Gerstenberger, W.A. Wall. An extended finite element method / Lagrangemultiplier based approach for fluid-structure interaction. Computer Methods inApplied MEchanics and Engineering, 197(19-20), pp 1699-1714 (2008)

5

Bibliography

[GeWa10] A. Gerstenberger, W.A. Wall. An embedded Dirichlet formulation for 3d continua.International Journal for Numerical Methods in Engineering, 82 (5), pp 537-563(2010)

6

1 Introduction

Fluid-structure interaction problems describe the coupled dynamics of fluid mechanics andstructure mechanics. They are classical multi-physics problems. Examples for fluid-structureinteraction exist in many engineering applications:

aerodynamics The simulation of the aerodynamical properties of planes is the classical com-putational fluid dynamics (CFD) problem. To estimate the forces, drag- or lift-values,of a given configuration, simulations are inevitable. In particular for new and largeconstructions, like the Airbus 380, experimental data from wind tunnel tests are notavailable.

For an accurate simulation however, the deformation of the plane under the aerody-namical forces has to be taken into account, since this deformation alters the shape ofthe plane and thus changes the aerodynamical behavior.

bio-engineering For a better understanding of the flow of blood (hemodynamics) in vesselsor in the heart, simulations are the only way to obtain data. The walls of the bloodvessels are highly elastic and deformed under the pulsating blood flow. Further, theflow itself is driven by the contraction of the heart muscles, that is, by the deformationof the structure surrounding the flow domain.

The described problems have in common, that the evolving flow acts on the surface of thestructure leading to its deformation. This deformation changes the flow domain itself. Weare looking at two-way coupled systems, where each part influences the other.

Definition 1 (Coupled Systems). A coupled system SSS is one in which physically or compu-tationally heterogeneous mechanical components interact dynamically.[?, ?, ?, ?, ?, ?]. BySSS1 and SSS2 we denote the subsystems. The coupled system SSS is called one-way, if there is nofeedback between the subsystems and two-way, if there is feedback between the subsystems.

The fluid-structure interaction examples described above are two-way coupled systems. Anexample for a one-way coupled system would be the interaction of a walking human with thesurrounding air: The movement of the human will give rise to an airflow around it, this slowairflow will however not impose significant forces on the human. The analysis of coupledsystems and especially the numerical methods will depend on the type of coupling betweenthe subsystems.

The concept of coupled systems can be generalized to multi-coupled systems with subsystemsSSS1, . . . ,SSSN . This can be multi-structure-fluid interaction, fluid-structure-fluid-interaction(e.g. water-boat-air). We then call a fully coupled system multi-way coupled.

7

1 Introduction

In most problems it cannot be decided if the problem is one-way or two-way. Regarding theinteraction of a very slow driving car with the surrounding air, the influence of the air onthe car can be neglected. At a certain speed however, the aerodynamic resistance plays animportant role.

1.1 Fluid-Structure Interaction

For the following, we introduce a prototypical fluid-structure interaction problem: Figure 1.1shows a flow domain with an obstacle. We call the common domain Ω, the flow domain Ωf

and the structure domain Ωs:

Ωf

Ωs

Figure 1.1: Fluid-structure interaction domain.

Now assume, that the fluid domain Ωf is filled with air, and Ωs is a rigid moving body ofsteel. This movement will set the fluid into motion. The air however will not significantlyact on the obstacle:

Ωs

Ωf

Ωs

Ωf

Ωf

Ωs

Figure 1.2: Fluid motion imposed by moving structure.

This problem is a one-way fluid-structure interaction problem. The movement of the struc-ture controls the motion of the fluid but the fluid’s motion does not impair the movementof the structure.

Next assume, that the flow is driven by an inflow condition and the obstacle is an elas-tic structure. The evolving flow will act on the surface of the structure and will cause adeformation. This deformation changes the flow domain:

8


Ωs

Ωf

ΩsΩs

Ωf Ωf

Figure 1.3: Fluid-structure interaction.

Due to the deformation of the obstacle, the flow domain is altered. Here, there is a realfeedback between both subsystems and the coupling is two-way.

Both coupled problems in Figure 1.2 and 1.3 have in common, that the problems are formu-lated on moving domains. Here, the common domain Ω keeps the same, but the subdomainsof the fluid Ωf and the solid Ωs problem change with time: Ω = Ωf (t) ∪ Ωs(t). This is oneof the main difficulties connected with the modeling of fluid-structure interaction problemsas well the design of numerical methods for their solution.

The different degree in coupling and interaction is important for the treatment of the prob-lems. We will consider all fluid-structure interaction problems as time-dependent problemsSSS(t). The solution is approximated at time-steps t1, t2, . . . :

tn n+1S( ) S(t )

Figure 1.4: Time-approximation of the coupled problem.

In every time-step tn → tn+1 both problems need to be solved. The solution SSSn+1 dependson the state of both subproblems at time tn as well as on the interaction between bothsubproblems. The straightforward way for simulation the coupled problem is the monolithicapproach: we simultaneously solve the fluid and the structure problem at the same time:

tnS( )

tnS( )

f

s

n+1S(t )

n+1S(t )s

f

Figure 1.5: Monolithic solution of the coupled system.

9

1 Introduction

For this approach, we need to formulate both subproblems, fluid and structure, as onecombined problem. Sometimes however the coupling between both problems suggests astaggering of the solutions. Considering the problem described in Figure 1.2, the flow fieldhas no influence on the body, which is moved by some external mechanism. Here, it may beadvisable to first solve for the new shape of the structure and flow domain and then for theflow field:

s

tnS( )

tn

n+1S(t )

n+1S(t )

f

S( )s

f

Figure 1.6: Staggered solution of the coupled system.

This configuration can now be treated with standard methods: the structure deformationis computable with a structure solver, the flow problem can be computed separately witha fluid dynamics code. Unfortunately, interesting fluid-structure interaction problems aremostly two-way coupled.

Still, it is possible to numerically decouple the interaction problem. These methods arecalled partitioned approaches:

tnS( )

tn

n+1S(t )

n+1S(t )

f

S( )s

f

s

Figure 1.7: Partitioned solution of the coupled system.

In every time-step tn → tn+s both problems are solved separately. The flow problem SSSf attime tn+1 depends on the flow and on the structure problem at time tn, but the interactionat time tn+1 is not taken into account for. For the structure problem the same approachis used. In terms of time-stepping methods this approach can be called an semi-explicitapproach: while the fluid and structure dynamics itself is considered in an implicit fashion,the interaction between both problems is included in an explicit way giving rise to stabilityproblems and asking for small time steps. An advantage of the partitioned approach is thatdifferent solvers can be used for the different subproblems. The coupling between fluid and

10


structure comes into the problem by means of boundary conditions on the interface betweenboth subdomains. Aside, this decoupling allows for a parallel solution of the fluid and thestructure problem. However, in most applications the coupling between both problems is tostrong for partitioned approaches.

A further development of the partitioned approaches are the strongly coupled partitionedapproaches. The two subproblems are solved independently in a decoupled way. Every time-step tn → tn+1 is however iterated yielding approximate solutions SSS(tn+1)if and SSS(tn+1)is.To compute the i-th iterative, the solution at time tn and the last approximations SSS(tn+1)i−1

f

and SSS(tn+1)i−1s are considered:

s

tnS( )

tn

n+1S(t )

n+1S(t )

f

S( )s

f

Figure 1.8: Strongly coupled partitioned approach for the coupled system.

The strongly coupled partitioned approach still allows for the use of different solvers for thefluid and structure dynamics subsystem. Due to the outer iteration, stability problems aredamped, even though not removed. If a very large number of sub-iterations is necessary tofind the stable state, the strongly coupled partitioned approach can be less efficient than themonolithic solution of the coupled problem.

11

1 Introduction

12

2 Fundamentals of Continuum Mechanics

In this chapter we develop the fundamental equations in continuum mechanics.

2.1 Kinematics

V V (t)

x

u

x

u

Figure 2.1: Deformation of a material particle x in a material body V .

Let V be amaterial body in the reference configuration. Then, let x ∈ V be amaterial particlein the material body V . After some time t, the particle x is moved to a new position x. Byu = x− x we denote the deformation of the particle x. We assume, that the deformation isa continuous (and perhaps differentiable) function defining the displacement of all particlesx ∈ V :

u = x− x ⇒ x = x(x, t) = x+ u(x, t). (2.1)

Regarding the displacement of all particles x ∈ V , the reference body V is deformed to V (t),see Figure 2.1. By x(x, t) ∈ V (t) for t ≥ 0 we denote the trajectory of the particle x. Physicalevidence tells us that at time t, no two different particles x ∈ V and x′ ∈ V will be at thesame position position x ∈ V , thus the relation (2.1) is invertible and we define on V (t)

u = x− x ⇒ x = x(x, t) = x− u(x, t). (2.2)

By summing up (2.1) and (2.2) we get

u(x, t) = u(x, t).

If we follow the trajectory of a particle x(x, t) we can define its velocity v(x, t) by

v(x, t) = dtx(x, t) = ∂tx · ∇xu+ ∂tu(x, t) = ∂tu(x, t).

We distinguish two coordinate frameworks when observing moving bodies and particles. TheLagrangian framework viewpoint is material-centered: the focus follows a specify particle x.Deformation u(x, t) and velocity v(x, t) denote the deformation and velocity of the particle

13


x at time t. The Eulerian framework is space-centered: the focus is on a fixed point x ∈ V (t)in space. Deformation u(x, t) and velocity v(x, t) denote the deformation and velocity of theparticle x = x(x, t) which is in location x at time t.

The deformation u(x, t) in the Eulerian framework is defined by (2.2), the velocity v(x, t) inthe Eulerian system is the velocity of the particle x which is at location x at time t:

v(x, t) := ∂tx(x, t) = ∂t(x+ u(x, t)) = ∂tu(x, t).

The Lagrangian framework is always described on the reference system V . Throughout thiswork, variables in Lagrangian coordinates are always marked by a “hat”. The Lagrangianframework is the natural way of describing problems of structure dynamics, where the de-formation of a material body V into a new equilibrium V is modeled. In the Eulerianframework, variables and properties are described on the actual deformed (and possiblymoving) body V . Trajectories of single particles are not followed, instead, the point of focusis on a fixed location in space. Variables in Eulerian coordinates are always given as Romanletters without further marking.

V V (t)u

u

δx δx

Figure 2.2: Deformation of differential line-segment δx from the reference system V to δx inthe deformed system V .

Kinematics is the study of motion. Motion includes the movement (velocity and accelera-tion) of a particle (or body) and also the change of the position relative to the referenceconfiguration. Apart from the deformation of single particles, for structural mechanics weneed to observe the change in the relation between particles. Let x and xδ be two materialpoints with

xδ = x+ δx.

By x ∈ V and xδ ∈ V we denote the deformed points given by a differentiable deformationfield u in V via

x = x+ u(x), xδ = xδ + u(xδ) = x+ δx+ u(x) + ∇u(x)δx+O(|δx|2).

See Figure 2.2. Thus, we calculate the deformed line-segment δx in first order by

δx := xδ − x ≈ δx+ ∇u(x)δx = (I + ∇u(x))︸︷︷︸=:F (x)

δx.

The second-order tensor F is called the deformation gradient and is one of the measures forthe deformation. We will use F to describe the change of bodies, in terms of volume, strain,strain-rates. F is a function of x in the reference framework. By

F := F−1 = (I −∇u),

14

2.1 Kinematics

we denote the inverse deformation gradient:

δx = F−1δx = Fδx.

To distinguish these two tensors, we also call F the Lagrangian deformation gradient and Fthe Eulerian deformation gradient.

2.1.1 The strain tensor

In continuum mechanics the strain tensor E is one of the key quantities. It is used todescribe the relative length-change of a differential line-segment δx under deformation. Let‖e‖ = 1 be a unit vector and with δx := se for s > 0 we define the two material points xand xδ = x+ δx = x+ se. For, the deformed points x and xδ, by

de(x) := lims→0

‖x− xδ‖ − ‖x− xδ‖‖x− xδ‖

,

we denote the relative length-change in direction e for the material point x. With thedeformation gradient F = I + ∇u we obtain for small s→ 0

de(x) = ‖F e‖ − ‖e‖‖e‖

= ‖F e‖ − 1 =(F e, F e

) 12 − 1 =

(e+ ∇u+ ∇uT + ∇uT ∇ue, e

) 12 − 1

We define the strain tensor E(x) by

E(x) = 12∇u(x) + ∇u(x)T + ∇u(x)T ∇u(x) = 1

2FT F − I,

to obtain for every direction e

de(x) =(1 + 2(E(x)e, e)

) 12 − 1.

We call F T F the right Cauchy Green tensor and E = 12(F T F − I) the Green strain tensor

or Lagrangian strain tensor .

If the gradient of the deformation ∇u is small, the quadratic term ∇uT ∇u is negligible andwe define the linear strain tensor ε by

ε := 12(∇u(x) + ∇u(x)T

).

For the Cartesian unit vectors ei, the principal elongations in direction of the coordinateaxes are

d(i) =(1 + 2(Eiiei, ei)

) 12 − 1 = (1 + 2Eii)

12 − 1.

For small displacements ‖∇u‖ 1 this is in first order

d(i) = Eii +O(E2ii)

15


Examples Let u be a homogenous translation u(x, t) = u0. Then, the strain tensor is zeroE = 0. The length of line-segment keeps the same.

Next, let u be a scaling with a scaling parameter s ∈ R+

u(x, t) = sx− x, ∇u =(s− 1 0

0 s− 1

)

Strain tensor and linear strain tensor (in two dimensions) are given by

E =(

(s− 1)(1 + s−1

2)

00 (s− 1)

(1 + s−1

2)) , ε =

(s− 1 0

0 s− 1

)

For small scalings (s close to 1), the linear tensor is a good approximation. In direction e,the relative length-change is

de =(1 + 2(εe, e)

) 12 − 1 =

(1 + 2(s− 1)

) 12 − 1 = (2s− 1)

12 − 1 = s− 1 +O(|s− 1|2)

For large deformations, e.g. s = 1/10 or s = 10 the linear approximation is not accurate.For the full tensor E we have

de =(

1 + 2(s− 1)(

1 + s− 12

)) 12− 1 = s− 1

Next, let u be a shearing:

u(x) =(sy0

), E = 1

2

(0 ss s2

), ε = 1

2

(0 ss 0

),

with a shearing parameter s ∈ R. Again, for small values of s the gradient ‖∇u‖ 1 is verysmall and the linear tensor is a good approximation. In the general case, with a unit vectore = (e1, e2), ‖e‖ = 1 we get

de =(1 + 2se1e2 + s2e2

2

) 12 − 1

For the Cartesian unit-vectors e = (1, 0) and e = (0, 1) we have d(1,0) =√

1 − 1 = 0 andd(0,1) =

√1 + s2 − 1:

u u

√1 + s2

s

Figure 2.3: Deformation of a line-segment under a shearing.

16

2.1 Kinematics

The shearing can also be used to explain the off-diagonal entries in the strain tensor. Lete1 and e2 be two unit-vectors in a right angle, e.g. (e1, e2) = 0. Assume, that the shearingparameter s is small again. Then, the angle ω between the two vectors in the deformedsystem with ei = F ei can be estimated as

cos(ω) = (F e1, F e2)‖F e1‖ ‖F e2‖

=((I + ∇u)e1, (I + ∇u)e2)

(1 + d(1))(1 + d(2))︸︷︷︸≈1

≈(∇u+ ∇uT + ∇uT ∇u︸︷︷︸

=2E

e1, e2) = 2Eij

u

Figure 2.4: Angular deformation under a shearing. Deformation of a line-segment under ashearing for the three directions e1 = (0, 1), e2 = (1, 0) and e3 = (1, 1)/

√2.

For angles ω close to π/2 we use the expansion:

cos (ω) ≈ π

2 − ω, ω ≈ π

2 − 2Eij .

2.1.2 The rate-of-strain tensor

In fluid-dynamics even very small forces can lead to a arbitrary large ‘deformation’ of fluid-particles, that is, to a ‘flow’ of the fluid. Thus, not the strain itself, but the rate of changein strain is of importance. The derivation is similar to the previous section: we again regarda differential line-element with the two end-points x and xδ = x+ sδx, where ‖δx‖ = 1. Wefollow the two points over a time-span t 7→ t+ k and approximate

x(k) = x+ u(x, t+ k) = x+ u(x, t)︸︷︷︸=0

+k ∂tu(x, t)︸︷︷︸=v(x,t)

+ k2

2 ∂ttu(x, t) +O(k3)︸︷︷︸=:R(x,k)

.

Then,x(k) = x+ kv(x) +R(x, k), xδ(k) = xδ + kv(xδ) +R(xδ, k).

By δx(k) := xδ(k)− x(k) we denote the differential line-segment after he time-span k. Thelength-change dδx is given by

dδx(x, k) := lims→0

‖xδ(k)− x(k)‖ − ‖xδ − x‖‖xδ − x‖

= lim ‖δx(k)‖‖sδx‖

− 1

= lim ‖sδx+ k(v(xδ)− v(x)) +R(xδ, k)−R(x, k)‖s

− 1

17


and the rate of length-change by

dδx(x, k) := limk→0

dδx(x, k)k

.

Assuming differentiability of v we see

v(xδ)− v(x) = s∇v(x)δx+O(s2), R(xδ, k)−R(x, k) = sO(k2),

to get in first order in s:

‖δx(k)‖ = ‖δx+ k∇vδx+O(k2)‖ =(1 + (k∇v +∇vT δx+O(k2)δx, δx)

) 12.

By (1 + z)12 = 1 + z/2 +O(z2) we have:

‖δx(k)‖ = 1 + k

2(∇v +∇vT δx, δx

)+O(k2).

And finally, for the rate of length-change

dδx(x, k) = limk→0

k2 (∇v +∇vT δx, δx) +O(k2)

k=( 1

2∇v +∇vT ︸︷︷︸=:ε

δx, δx)

=(εδx, δx

).

We call ε the rate-of-strain tensor . It is the fundamental measure to describe strains influid-dynamics. The rate-of-strain tensor is linear and symmetric with the component-wiserepresentation

ε(x) = 12

2∂1v1(x) ∂1v2(x) + ∂2v1(x) ∂1v3(x) + ∂3v1(x)∂2v1(x) + ∂1v2(x) 2∂2v2(x) ∂2v3(x) + ∂3v2(x)∂3v1(x) + ∂1v3(x) ∂3v2(x) + ∂2v3(x) 2∂3v3(x)

.The diagonal entries εii describe the relative rate of strain in direction of the coordinate axesxi, and the trace of the matrix ε describes the relative speed of change in volume

tr (ε) =∑

∂ivi = div v.

A single entry εij can be interpreted as the velocity of angular change in the xi − xj plane.

2.2 Fluid Dynamics

We describe the motion of a fluid (that is a liquid or a gas). The fluid flow is modeled bydescribing its properties in points x in a volume V . This space-centered focus is the classicalEulerian view-point. In every point x we model the velocity v and density ρ. Further, by pwe denote the pressure. These set of variables are called the primitive variable in contrastto the conservative variables ρ the mass-density and ρv the momentum.

18

2.2 Fluid Dynamics

2.2.1 The Conservation Equations in Fluid Dynamics

In continuum mechanics we do not describe the physical properties of single particles or theinteraction of one particle with another, instead, we describe the behavior of densities ina continuum. This way we assume, that the observed space is completely filled with thesubstance. We ignore the fact, that matter is made of discrete atoms on a very fine scale.On length scales much greater than the atom-distant, continuum models are very accurate.

Let V (t) ⊂ R3 be a (moving) volume. The material in V (t) has certain distributed propertieslike the mass-density ρ, the momentum ρv and the energy-density ρe. The basic equationsof continuum mechanics are derived from the following physical conservation principles

conservation of mass mass is not added nor removed

conservation of momentum change in momentum is equivalent to the acting force

conservation of energy energy is not added nor removed

For instance the first principle means, that the mass of a volume

m(V (t)) =∫V (t)

ρ dx,

will not change in timedtm(V (t)) = 0.

Let Φ be some physical value (e.g. density) given for every mass point ξ and for every t.Then, by

Φ(x, t) = Φ(x(ξ, t), t) := Φ(ξ, t),

we can access this value also in Eulerian coordinates (which is necessary to describe it onthe moving volume V (t)). The ‘local derivative’

∂tΦ = ∂tΦ(x, t),

indicates the local change in a fixed point in space x (that is the Eulerian view-point of afixed focus). The ‘material derivative’ (or total derivative)

dtΦ = dtΦ(x(ξ, t), t) = ∂tΦ(x, t) + v · ∇xΦ(x, t),

indicates the change of Φ for a certain particle ξ which is at x = x(ξ, t) at time t. Byv(x, t) = ∂tx(ξ, t) we denote the velocity of the particle.

Theorem 1 (Reynolds Transport Theorem). Let Ψ = Ψ(x, t) be a smooth scalar function.Then, for every volume V (t) it holds

d

dt

∫V (t)

Φ dx =∫V (t)

∂tΦ + div (Φv)

dx.

19


For a value Φ withd

dt

∫V (t)

Φ dx = 0,

we get with Reynolds transport theorem the following relation:

d

dt

∫V (t)

Φdx =∫V (t)

∂tΦ + div (Φv) dx ⇒∫V (t)

∂tΦ dx = −∫∂V (t)

n · vΦdo.

Thus, the change of a value Φ in V (t) over time equals the negative flux over V (t)’s boundary.

Conservation of Mass Physical evidence tells us, that mass is neither created nor destroyed.We thus assume, that for each volume V it holds

dtm(V ) = dt

∫Vρdx = 0.

Using Φ = ρ in Reynolds transport theorem, we obtain∫V∂tρ+∇ · (ρv) dx = 0.

This holds for every volume V . We assume that the integrand is continuous and concludein every spatial point x the equation of mass conservation:

∂tρ+∇ · (ρv) = 0. (2.3)

Conservation of Momentum (Impuls) Let V be a volume. The forces acting on V canbe split into volume forces in V and surface forces on ∂V . Volume forces will be denotedby a distributed function f . An example for a volume force is the gravity, which acts onevery material point in the volume. Surface forces only act with contact on the surface ofthe body. Examples are pressure or frictional forces. They are described by a tensor σ. Theforce acting on a surface with normal n is given by n · σ. Together, the total force F (V )acting on V is

F (V ) =∫Vρf dx+

∫∂Vn · σ do =

∫Vρf +∇ · σdx.

This stress-tensor σ includes the material properties of the fluid and will be developed later.The momentum M(V ) of V is given by

M(V ) =∫Vρv dx.

Newton’s law tells that the change in momentum is equal to the acting forces

dtM(V ) = F (V ) ⇒ dt

∫Vρv dx =

∫Vρf +∇ · σdx.

We apply Reynolds transport theorem for the scalar values Φ = ρvi for i = 1, 2, 3 to get

dt

∫Vρvi dx =

∫V∂t(ρvi) +∇ · (ρviv) dx

20

2.2 Fluid Dynamics

which yields the point-wise equation

∂t(ρvi) +∇ · (ρviv) = ρfi + (∇ · σ)i

In a compact vectorial formulation, we have

∂t(ρv) +∇ · (ρv ⊗ v) = ρf +∇ · σ,

where v ⊗ v := (vivj)3i,j=1 is the dyadic product. With the conservation of mass (2.3) we get

∂t(ρv) = ∂tρv + ρ∂tv = ρ∂tv − v∇ · (ρv).

Further, with∇ · (ρv ⊗ v) = v∇ · (ρv) + ρv · ∇v,

we get the equation of momentum conservation

ρ∂tv + ρv · ∇v = ρf +∇ · σ.

Conservation of Angular Momentum (Drehimpuls) The angular momentum (or momentof momentum) of a volume with regard to the origin is given by

L(V ) =∫Vx× (ρv)dx,

and the torque (or moment or moment of force, “Drehmoment”):

T (V ) =∫Vx× (ρf) dx+

∫∂Vx× (n · σ) dx.

Torque and angular momentum are the rotational analogies of force and momentum. Con-servation of angular momentum says:

dtL(V ) = T (V ).

Using Reynolds transport theorem and after lengthy computation we obtain for the changeof angular momentum:

dtL(V ) =∫Vx×

(ρ∂tv + ρv · ∇v

)dx.

Likewise, the torque can be rewritten as

T (V ) =∫Vx× (ρf + divσ) dx.

Here, by ε = (ε)ijk=1,2,3 we denote the permutation tensor or Levi-Civita-Tensor , which isgiven by

εijk =

1 if (i, j, k) is an even permutation of (1, 2, 3)−1 if (i, j, k) is an odd permutation of (1, 2, 3)0 if (i, j, k) if any symbol is repeated, i = j, i = k, j = k.

21


We conclude with the point-wise equation for conservation of angular momentum

x× (ρ∂tv + ρv · ∇v − divσ − ρf) = ε : σ.

The first part is given by the conservation of momentum. Thus, conservation of angularmomentum implies

ε : σ = 0 ⇔ σjk = σkj ⇔ σ = σT .

Conservation of angular momentum is hence an additional condition requiring the symmetryof the stress-tensor σ.

The set of conservation equations is now given by:

∂tρ+∇ · (ρv) = 0,ρ∂tv + ρv · ∇v = ρf +∇ · σ.

(2.4)

2.2.2 Material Laws in Fluid Dynamics

The conservation equations are derived from very basic principles and basically hold forall different materials. The set of equations (2.4) includes 4 equations for the 10 physicalquantities density (1), velocity (3) and symmetric tensor σ (6). To close this gap, we willderive constitutive laws for the dependence of the stress tensor σ on the other variables.These laws have to be understood as modeling.

A so called Stokes fluid has the property, that the stress tensor σ is spherically symmetric ifthe fluid is at rest v = 0. That is, for every two normal unit-vectors n1 and n2 we have

n1 · σ|v=0n1 = n2 · σ|v=0n

2.

With the symmetry σ = σT it follows, that the stress-tensor for a fluid at rest is diagonal

σ|v=0 = −pI,

where p is the scalar hydrostatic pressure. For a general moving fluid, we introduce the shearstress tensor τ to capture the remaining parts:

σ = −pI + τ.

τ = τT must be symmetric. Further, the trace of τ must be zero

tr (τ) = 0,

otherwise it would be “hidden” in the pressure-part “−pI”. We link the shear tensor τ tothe strain rate tensor ε via a general material law

τ = F (ε).

For a Stokes flow we assume the following properties

symmetry if T is a symmetric tensor, then F (T ) is also a symmetric tensor. Further,F (0) = 0.

22

2.2 Fluid Dynamics

isotropy the material law F in invariant with regard to volume preserving, orthogonal co-ordinate transformation, i.e.

F (QTQT ) = QF (T )QT ,

for every symmetric tensor T and every orthogonal transformation Q = (Qij)i,j=1,2,3with QQT = I and det (Q) = 1.

If we further assume, that the material law is linear, we call the fluid a Newtonian fluid andthe stress tensor has to have the form:

σ = −pI + F (ε) = −pI + 2µε+ λtr (ε)I,

with the two material constants µ (shear viscosity) and λ (volume viscosity). These twoparameters usually depend on the temperature and the density of the fluid. For isothermalfluids, the shear and volume viscosity are constant in space and time, thus

divσ = −∇p+ div τ, τ := µ∇v +∇vT + λdiv vI.

From physical observation we use the relation

3λ+ 2µ = 0, µ ≥ 0 ⇒ λ = −23µ,

between the parameters. This relation holds for one-atom-gases. The stress tensor for aStokes fluid is:

σ = −pI + µ∇v +∇vT − 23µdiv vI (2.5)

Finally, the conservation equations for a Newtonian Stokes fluid are given by

∂tρ+∇ · (ρv) = 0,

ρ∂tv + ρv · ∇v − µ∇v +∇vT + 23µdiv vI +∇p = ρf.

2.2.3 Incompressible Flows, the Navier-Stokes Equations

Even with large forces, fluids are very difficult to compress. Thus, the volume V (t) nearlydoes not change with time. If the volume V (t) is constant in time, the flow is called anincompressible flow:

d

dt

∫V

dx = 0.

Reynolds transport theorem, used on the constant Φ = 1 yields∫V∇ · v dx = 0 ⇒ ∇ · v = 0.

Conservation of mass then simplifies to

∂tρ+ v · ∇ρ = 0.

23


For the divergence of the stress tensor (2.5) it holds

−divσ = ∇p− µ∇ · ∇v︸︷︷︸=∆v

−µ ∇ · ∇vT︸︷︷︸=∇div v=0

+23∇ · divv︸︷︷︸

=0

I = −µ∆v +∇p.

If the fluid is homogeneous, the density does not vary in space and time ρ(x, t) = ρ0 and theincompressible Navier-Stokes equations are given by

∂tv + v · ∇v − ν∆v + 1ρ0∇p = f,

∇ · v = 0.(2.6)

The value ν := µ/ρ0 is called the kinematic viscosity opposed to the dynamic viscosity µ.

This system is closed by boundary and initial values. Let Ω ⊂ Rd be the domain withboundary Γ = ∂Ω. We split this boundary into Γ = ΓD ∪ ΓN , where ΓD represents parts ofthe boundary with Dirichlet condition and ΓN with Neumann or Robin boundary conditions.Then, we impose the boundary and initial values:

v(x, 0) = v0 in Ω, v = gD on ΓD, n · σ = gσ on ΓN ,

where gD and gσ are appropriate functions.

2.2.4 Variational formulation of the incompressible Navier-Stokes Equations

We derive the weak formulation of Equation (2.6) by multiplying with suitable test-functions.We introduce the following function spaces:

H10 (Ω; ΓD) = φ ∈ H1(Ω)d : v = 0 on ΓD,L2

0(Ω) = ξ ∈ L2(Ω) : (ξ, 1)Ω = 0.

The variational formulation of problem (2.6) is

Problem 1 (Incompressible Navier-Stokes equations). On Ωf (t) find v ∈ vD + Vf , Vf :=H1

0 (Ω; ΓD) and p ∈ Qf , Qf := L20(Ω), such that(

ρ0(∂tv + (v · ∇)v), φ) + (µ∇v,∇φ) = (ρ0f, φ) + 〈gσ, φ〉ΓN ∀φ ∈ Vf(∇ · v, ξ) = 0 ∀ξ ∈ Qf ,

(2.7)

whereσf := −pI + ρfνf

∇v +∇vT

.

While the Dirichlet-condition v = gD is embedded into the function space: the functionvD ∈ H1(Ω)d is a continuation of gD into the domain Ω. The Neumann-condition on thestresses n · σ on ΓN is ensured due to integration by parts:

(µ∇v,∇φ)Ω −1ρ0 (p, ·∇φ)Ω = (−µ∆v +∇pI, φ)Ω + 〈µn · ∇v − p · nI, φ〉ΓN .

24

2.3 Structure Mechanics


In structure mechanics, we describe the deformation and movement of a volume V filled withan elastic material. By V we will denote the volume in the undeformed reference state. Dueto forces acting in the volume or on the surface of the volume, the structure will experiencesome deformation u(x, t). This deformation can either be a real dynamic process yieldinga time-depending deformed volume V (t) or it can result in a new equilibrium V after someinitial dynamic deformation. In structure dynamics we are interested in the deformationfield u(x, t) for every material-point x of the reference volume V . This is the classical,material-centered Lagrangian viewpoint.

2.3.1 The Conservation Equations in Structure Dynamics

In structure dynamics, the velocity of particles is of lesser interest. Instead, we regard thetransition of a volume V under strains from a reference state to a new equilibrium V (t). Wesearch for the deformation vector u(x, t)

V 3 x = x(x, 0) → x(x, t) = x+ u(x, t) ∈ V (t)

For structure mechanics, the Lagrangian viewpoint is natural. We denote all variables andentities given in the reference framework V by a “hat” and all variables on the deformeddomain V (t) or V without it. By v(x, t) we denote the velocity of particle x at time t(wherever this particle may be in V (t)):

v(x, t) := dtx(x, t) = dtu(x, t) = ∂tu(x, t).

The deformation in the deformed volume V is defined by u(x, t) := u(x(x, t), t) or likewise(since x(x, t) is invertible) u(x(x, t), t) := u(x, t). This is the Eulerian formulation of thedeformation. Then, for the Eulerian velocity we get

v(x, t) := v(x(x, t), t) = dtx(x, t) = ∂tu(x, t).

Thus, by this consideration, velocity and displacement can be given in both frameworks viathe easy relation v(x, t) = v(x, t) and u(x, t) = u(x, t) for the couple x = x(x, t). Howeverwe note, that the spatial derivatives are not immediately available. In particular, we have∇u 6= ∇u!

We assume the following material properties. The deformation is isothermal: no mechanicaleffects are caused by temperature gradients. In the reference configuration V , all physicalparameters (that is the density ρ(x, 0) = ρ0(x), initial deformation u(x, 0) = u0(x) andinitial velocity v(x, 0) = v0(x) are given.

The behavior of the structure is then determined by the two conservation principles of massand momentum valid on the deformed domain V (t). Conservation of mass reads

d

dt

∫V (t)

ρ(x, t) dx = 0, (2.8)

25


and momentum:d

dt

∫V (t)

ρ(x, t)v(x, t) dx =∫V (t)

ρ(x, t)f(x, t)dx−∫∂V (t)

n(x) · σ(x, t)dox, (2.9)

where ρf is a prescribed volume force acting on V (t) and σ is the stress tensor describingthe surface forces. The conservation principles are first derived on the (moving) deformeddomain V (t).

Here, the difficulty of structural modeling is obvious (this will also be the underlying difficultyof fluid-structure interaction): we intend to formulate the structure equations in Lagrangiancoordinates on the reference system V . The governing principles however are determinedon the deformed equilibrium system V (t). To derive the set of equations we thus need totransform the conservation principles to the reference system V . In principle, it would bepossible to formulate the structure equations in the Eulerian framework, then however thecoordinates itself x = x+ u are unknowns in the system.

Conservation of Mass The density distribution ρ0(x) is known at time t = 0. Conservationof mass (2.8) means for all t ≥ 0∫

Vρ0(x) dx =

∫V (t)

ρ(x, t)dx.

Transformation via T : V → V (t) yields∫V (t)

ρ(x, t) dx =∫Vρ0(x)dx =

∫V (t)

ρ0(x(x, t))J−1 dx.

Conservation of mass gives us the constituting law for the density ρ(x, t) in the deformedframework:

ρ(x, t) = J−1ρ0(x), (2.10)where J = det(I + ∇u). In Eulerian formulation we can express the density by

ρ(x, t) = J(x, t)ρ0(x(x, t))

Conservation of Momentum We transform the left side of (2.9) and use the conservationof mass (2.10):

d

dt

∫V (t)

ρv dx = d

dt

∫Vρ(x(x, t))J︸︷︷︸

ρ0(x)

v dx =∫Vρ0∂tv dx =

∫V (t)

ρ0(x(x, t))J−1∂tv dx =∫V (t)

ρ∂tv dx.

Together with the right side of (2.9) conservation of momentum gets (written in the Eulerianframework)

ρ∂tv = ρf + divσ in V (t).With v = ∂tu we can write the conservation equations in structure mechanics as

ρ = Jρ0,

ρ∂ttu = ρf + divσ.(2.11)

26


Like in the fluid case, we close this general system with initial and boundary values. Theboundary Γ = ∂Ω is again split into a Dirichlet part ΓD and a Neumann part ΓN . Since thestructure problem is described by a wave-equation, we have to specify initial values for thedeformation and its derivative (the velocity):

u(x, 0) = u0(x) in Ω, v(x, 0) = v0(x) in Ω, u = gD on ΓD, n · σ = gσ on ΓN .

These equations are written on the deformed domain V (t) which is determined by the un-known deformation u(x, t) itself. To close this gap, it is necessary to express the balanceequations (2.11) in the Lagrangian reference system.

2.3.2 The Piola Transformation

We need to build a transformation between the Eulerian space V (t) and the Lagrangian Vwhich maps forces (vectors) in the right way. The transformation of the coordinate system(e.g. of normal vectors) needs to be considered.

Definition 2 (Piola Transform). Let T : V → V be a diffeomorphism with F := ∇T ,J := det F . Further, let v ∈ H1(Ω) be a differentiable vector field. Then, the Piola transformof v is given by

J F−1v.

First, we derive the following Lemma

Lemma 1 (Divergence of the Piola transform). Let T : V → V with T (x) = x + u(x) be aC2-diffeomorphism and v : V → Rd with d = 2, 3 be a differentiable vector field in V . Then,for the divergence of the Piola transform J F−1v it holds

Jdiv v = div (J F−1v), (2.12)

where v : V → Rd is given by v(x) = v(T (x)).

Proof: We show the proof in the two-dimensional case. With x(x) = T (x), Fij = (∇T )ij =∂jxi and J := det F , we write

J F−1 =(∂2x2 −∂2x1−∂1x2 ∂1x1

), J F−1v =

(∂2x2v1 − ∂2x1v2−∂1x2v1 + ∂1x1v2

).

The divergence on the right side of (2.12) can be written as

div (J F−1v) = ∂1(∂2x2v1 − ∂2x1v2) + ∂2(−∂1x2v1 + ∂1x1v2)= v1(∂12x2 − ∂21x2) + v2(∂21x1 − ∂12x1)

+ ∂2x2∂1v1 − ∂2x1∂1v2 − ∂1x2∂2v1 + ∂1x1∂2v2.

Since T is two times differentiable, the partial derivatives commute ∂i∂jx = ∂j ∂ix) and thefirst row in the last term dissappears. Transforming to the Eulerian system using

∂ivj = ∂1vj ∂ix1 + ∂2vj ∂ix2,

27


yields (if we collect the terms with regard to vj)

div (J F−1v) = (∂1v1 + ∂2v2)(∂2x2∂1x1 − ∂1x2∂2x1)+ ∂2v1(∂2x2∂1x2 − ∂1x2∂2x2) + ∂1v2(∂1x1∂2x1 − ∂2x1∂1x1).

By noting J = ∂1x1∂2x2 − ∂1x2∂2x1 the result follows.

This lemma helps us to get a formula for the transformation of surface integrals:Lemma 2 (Transformation of Surface Integrals). Let T : V → V be the transformation asabove. Further, let v ∈ H1(V ) be a differentiable vector-field. Then,∫

∂Vv · n do =

∫∂V

(J F−1v) · n do.

Proof: With Gauss’ theorem followed by transformation to V∫∂Vv · n do =

∫Vdiv v dx =

∫VJdiv v dx.

Using Lemma 1 and with Gauss’ theorem∫VJdiv v dx =

∫Vdiv(J F−1v) dx =

∫∂V

(J F−1v) · n do,

we complete the proof.

The transformation process is necessary to correctly express the surface forces n · σ in thereference system. We need to find a corresponding tensor P to write the normal stresses inthe Lagrangian system satisfying∫

∂Vn · σ do =

∫∂Vn · P do.

Lemma 3 (First Piola-Kirchhoff stress-tensor). Let T : V → V be defined as above. Then,the surface force on ∂V is transformed to the reference system by∫

∂Vn · σ do =

∫∂Vn · (J σF−T ) do.

The tensorP := J σF−T ,

is called the first Piola-Kirchhoff stress-tensor.

Proof: By Fi we denote the i-th component of the surface force:

Fi :=∫∂V

(n · σ)ido =∑k

∫∂Vnkσki,do =

∫∂Vn · σi do,

where σi is the i-th column-vector of σ. Since σ = σT is symmetric, σi can also be understoodas the i-th row-vector. With Lemma 2 we get

Fi =∫∂Vn · (J F−1σi)do.

Using the symmetry of the stress-tensor σ we complete the proof.

We further define

28


Definition 3 (Second Piola-Kirchhoff stress-tensor). The symmetric tensor

S = F−1P = J F−1σF−T ,

is the second Piola-Kirchhoff stress-tensor.

We emphasize the differences between the three stress-tensors σ, P and S. The Cauchystress-tensor σ comprises the “real stress” in the current configuration at a point x. Thefirst Piola-Kirchhoff tensor indicates the stress expressed in the current configuration for amaterial point x in reference configuration. The second Piola-Kirchhoff tensor comprises thestress with regard to the reference configuration.

Finally, we can transform the balance equations in structure dynamics (2.11) to the La-grangian reference system:∫

Vρ0∂tv dx =

∫Vρ0f + div (J σF−T ) dx.

Likewise, if ρ0, v and σ are smooth:

ρ0∂ttu = ρ0f + div (J σF−T ). (2.13)

2.3.3 Material Laws in Structure Dynamics

The material law describes the dependence of the stress tensor σ on the strain-tensor E

P = C(E), E = 12∇u+ ∇uT + ∇uT ∇u,

via the elasticity law C. We are interested in elastic materials. Elasticity means, than ifa forces are applied on a body V in reference configuration it will be deformed V (t). Ifthe forces are driven back to zero, the deformed body will turn back into the referenceconfiguration V (t) → V . We note that real elasticity is an idealization. In contrast, amaterial called plastically, if the deformation (or parts of the deformation) are irreversible.Finally, a material is called viscoelastic, if it has properties of a viscous fluid as well as of anelastic structure.

Further we distinguish between isotropic and anisotropic material behavior. Common ex-amples for an anisotropic materials are blood vessels. They are by more easy to deform inradial than in axial direction.

Finally, a body is said to be materially homogeneous, if its response to applied loads isindependent of the position within the body.

We call a material incompressible, if it preserves volume:

t ≥ 0 :∫Vdx =

∫V (t)

dx =∫VJ dx,

thus if J = 1. In that case, the density is given by ρ(x, t) = ρ0(x(x, t)).

29


2.3.4 Linear Models of Elasticity

We only consider materials, where the stress to strain relation is linear (in Lagrangian andEulerian coordinates):

P = CE, E = 12(F T F − I).

For a linear, isotropic and symmetric material, the following law can be derived:

Lemma 4 (Hooke’s Law). Under the assumption of a linear, homogenous and isotropicmaterial law, the stress strain relation must be of the type

S = 2µE + λtr (E)I,

with two material parameters µ and λ.

With Definition 3 we can calculate the Cauchy stress-tensor σ by

σ = 1JF (2µE + λtr(E)I)F T .

The first Piola-Kirchhoff tensor is given by

P = F (2µE + λtr(E)I).

If µ 6= 0 and 3λ+ 2µ 6= 0 Hooke’s law is invertible:

E = 12µS −

λ

2µ(3µ+ 3λ)tr(S)I.

We introduce the Poisson’s ratio ν and the Young’s modulus E by

ν := λ

2(λ+ µ) , E := µ(3λ+ 2µ)λ+ µ

,

to write in compact formE = 1 + ν

ES − ν

Etr(S)I.

We try to identify reasonable condition on the elasticity parameters λ, µ, ν and E. Thedeformation is assumed to be small, so that ε is a good approximation of E. We analyze thefollowing cases:

1. Uniform pressure: Sij = −pδij . Hooke’s law yields the corresponding strains by

εij = 1 + ν

ESij −

ν

Etr(ε)δij = −p1− 2ν

Eδij = − p

3λ+ 2µδij .

The diagonal entries εii indicate the length-change in directions of the coordinate axes.The trace tr(ε) the relative volume change. A positive pressure p must go along witha decrease of volume, thus we note

3λ+ 2µ > 0.

30


2. Pressure in x1-direction:

S =

−p 0 00 0 00 0 0

The strain is estimates as

ε = p

E

−1 0 00 ν 00 0 ν

A pressure in x1-direction (in negative normal direction) should result in a contractionin x1 direction and an expansion in x2 and x3 direction, thus ε11 < 0 and ε22, ε33 > 0.We note

E > 0, ν > 0.

3. Finally, we apply a shearing in the x1 − x2 plane:

S =

0 1 01 0 00 0 0

, ε = 12µ

0 1 01 0 00 0 0

.A shearing leads to an angular change. For small deformations, we have found therelation εij = 1

2(12π − ωij), where ωij is the angular change between in the xi − xj

plane. We thus conclude withµ > 0.

We combine the previous results to find the conditions

3λ+ 2µ > 0, µ > 0, ν > 0, E > 0.

The material governed by this linear material law is called the St. Venant Kirchhoff-material.It considers the full nonlinear strain tensor and full geometric nonlinearities. Just the ma-terial is assumed to have a linear stress-strain relationship. The problem is to find u, suchthat

ρ0∂ttu− div(F (2µE + λtr(E)I)

)= ρ0f .

Further simplification can be made by assuming that the deformation is very small ‖∇u‖ 1. Then, the strain tensor can be approximated by the linear E ≈ ε and we use the simpli-fication F ≈ I. We derive the set of equations

ρ0∂ttu− div (2µε+ λtr(ε)) = ρ0f .

This equation can be transformed into

∂ttu−µ

ρ0 ∆u+ λ+ µ

ρ0 ∇div u = f .

This fully linear equation is called the Navier-Lame problem.

31


The equations are closed by boundary and initial values. These have to be transformed ontothe reference coordinates:

u(x, 0) = u0(x), v(x, 0) = v0(x) in Ω, u = gD on ΓD, n · (J σF−T ) = J gσF−T on ΓN .

2.3.5 Variational formulation of the elastic structure equations

In the context of fluid-structure interaction, we write the structure-equations as a set offirst-order equations in time, introducing the velocity v = ∂tu as an additional variable.Then, by

H10 (Ω; ΓD)d = φ ∈ H1(Ω)d : φ = 0 on ΓD,

we denote the test-space. Find u ∈ u0 +H10 (Ω; ΓD)d, v ∈ v0 +H1

0 (Ω; ΓD)d, such that

(ρ0∂tv, φ) + (J σF−T , ∇φ) = (ρ0f , φ) + 〈J gσF−T , φ〉ΓN ∀φ ∈ H10 (Ω; ΓD),

(dtu, ψ) = (v, ψ) ∀ψ ∈ H10 (Ω; ΓD).

(2.14)

The Dirichlet boundary conditions are built into the function space and the Neumann con-dition on the stresses is given due to integration by parts.

32

3 The Fluid-Structure Interaction Problem

In this chapter we describe the coupled fluid-structure interaction problem. As a prototypicalapplication we have the configuration from the introduction in mind:

Γi

Ωf (t1)

Γi(t)

Ωf (t2)Γout

Γin

Γi(t)Ωs(t1) Ωs(t2)

Ωf Γwall

Ωs Γbase

Figure 3.1: Prototypical FSI configuration.

A domain Ω ∈ R2 in reference configuration at time t = 0 is split into a fluid and solid part

Ω = Ωf ∪ Ωs.

An incompressible fluid is flowing in Ωf and an elastic structure is given in Ωs which issticked to at the boundary of the domain us = 0. Due to the coupled dynamics of fluid andstructure, the solid domain will be deformed Ωs → Ωs(t) which also imposes a deformationof the flow domain Ωf → Ωf (t). We split the boundary of Ω = Ωf ∪ Ωs = Ωf (t) ∪ Ωs(t)into the Dirichlet part ΓD = Γin ∪ Γwall ∪ Γbase as noted in Figure 3.1, and into the outflowboundary Γout:

∂Ω = Γin ∪ Γwall ∪ Γbase ∪ Γout.

Since these boundaries do not move with time, Lagrangian and Eulerian coordinates coincidehere. We further denote by Γi the interface between fluid and structure and we define inEulerian and Lagrangian coordinates

Γi = Ωf ∪ Ωs, Γi(t) = Ωf (t) ∪ Ωs(t).

This internal boundary moves in time.

On the moving domain Ωf (t) the incompressible Navier-Stokes equations are valid: findvf ∈ vD +H1

0 (Ωf (t); ΓD)d and pf ∈ L20(Ωf (t)) such that

(ρf (∂tvf + vf · ∇vf ), φ)Ωf (t) + (σf ,∇φ)Ωf (t) = (ρfff , φ)Ωf (t) + 〈gσf , φ〉Γi(t) + 〈goutf , φ〉Γout ,

(∇ · vf , ξ)Ωf (t) = 0,(3.1)

33


where σf = νf (∇vf +∇vTf )− pfI. On the outflow boundary Γout, we impose the do nothingboundary condition νf∂nv−pn = 0 by choosing an appropriate gout

f to remove the transposedpart in the stress tensor σf . The boundary function gσf on the interface Γi(t) will cope withthe stresses from the solid-part.

In the fixed solid reference domain, the elastic structure equation is given. Find u ∈ uD +H1

0 (Ωs; Γbase)d and v ∈ vD +H10 (Ωs; Γbase)d such that

(ρs∂tvs, φ)Ωs + (J σsF−T , ∇φ)Ωs = (ρsfs, φ)Ωf + 〈J gσs F−t, φ〉Γi ,

(∂tu, ψ)Ωs = (v, ψ)Ωs ,(3.2)

where σs is the Cauchy stress tensor for a St. Venant-Kirchhoff material (formulated in theLagrangian coordinate system)

σs = J−1s Fs(2µEs + λstr(Es)I)F Ts , Fs := I + ∇us, Js := det(Fs), Es := 1

2(F Ts Fs − I).

On the interface Γi, the function gσs copes with the stresses from the fluid domain.

The interaction of the fluid with the solid is given by the continuity of the velocity and bya balance of forces:

vf = vs on Γi(t), n · σf = n · σs on Γi(t). (3.3)

With these notations, we can formulate a first version of the fluid-structure interactionproblem

Problem 2 (First Fluid-Structure Interaction Formulation). Find vf ∈ v0f + Vf , pf ∈ Lf ,

and us ∈ u0s + Vs and vs ∈ v0

s ∈ Vs, such that (3.1, 3.2, 3.3) are fulfilled for all φ ∈ Vf ,ξ ∈ Lf and φ, ψ ∈ Vs.

The formulation of the coupled system is not satisfying:

1. Both systems (3.1) and (3.2) are formulated in different coordinate systems. Ωf (t) ismoving, while Ωs is fixed. We will need different triangulations for both subdomains.

2. The coupling conditions (3.3) are formulated on the common interface. However, sinceusing different coordinate systems, we do not have a common interface. Instead weneed to express both conditions in one of the two coordinate systems, e.g. by using (inLagrangian formulation)

x ∈ Γi : vs(x) = vf (x) = vf (x+ us).

The transformed coordinate x = x + us will most certainly not be a mesh-point inthe fluid domain. The interface conditions thus cannot be fulfilled exactly, insteadwe need some interpolation operators. For the normal-stresses condition we furtherneed to transform the tensors with the Piola transformation onto the same coordinatesystem.

34

3.1 Fluid Flows in ALE Formulation

3. Once the solid is deformed, the domain partitioning will change

Ωf ∪ Ωs → Ωf (t) ∪ Ωs(t),

thus, the Navier-Stokes equations are formulated on a moving domain. In addition,the interface-coupling condition is given on the moving interface

Γi(t) = Ωf (t) ∩ Ωs(t).

And since in one time-step tn → tn+1 the domain partitioning is changing, we havedifferent domains and we thus need different triangulations at the beginning and atthe end of the time-step. How should an appropriate function space look like?

The clue to treat these problems is to formulate both systems in one common coordinatesystem. In the following section we introduce the Arbitrary Lagrangian Eulerian Coordi-nates (ALE) for the Navier-Stokes equations. We express the fluid problem on the fixedcomputational domain Ωf which matches the solid domain for all times.


In fluid-structure interaction, the fluid problem is given on the moving domain Ωf (t). Asdiscussed above, this gives rise to various problems, most severe, the domain and with it thecomputation mesh changes from one time-step to the other tn → tn+1. To work around thisproblem we want to formulate the flow problem on a fixed domain Ωf . Let Tf : Ωf → Ωf (t)be a mapping between this fixed domain Ωf and the moving domain Ωf (t). We will usethis mapping to transform the Navier-Stokes equations onto the fixed domain. While thisis not necessarily required, we think of this fixed domain Ωf as Ωf , the counterpart to theLagrangian structure domain Ωs. Later on, we will discuss cases, where we will allow forgeneral reference domains of the flow problem, where Ωf does not have to coincide withΩf . Because the fluid’s reference domain is arbitrary, and the structure will be formulatedin Lagrangian coordinates, the resulting fluid-structure formulation is called the ArbitraryLagrangian Eulerian (ALE) formulation.

Let Tf : Ωf → Ωf (t) be a C2-diffeomorphism. We call Tf the ALE mapping. We define thegradient Ff and its determinant Jf by

Tf : Ωf → Ωf (t), Ff := ∇Tf , Jf := det(Ff ).

We define the principle variables velocity v and pressure p on the reference domain Ωf viathis transformation:

v(x, t) = v(Tf (x, t), t), p(x, t) = p(Tf (x, t), t).

The spatial and temporal derivatives of the variables are given by the following Lemma:

35


Lemma 5 (Derivatives of the ALE mapping). Let Tf : Ωf → Ωf (t) be a C1-diffeomorphism,f ∈ H1(Ω(t)) be a differentiable function and v ∈ H1(Ω(t))d a differentiable vector-field.Then it holds for f(x, t) := f(x, t) and v(x, t) = v(x, t)

∂tf = ∂tf − (F−1f ∂tTf · ∇)f , (3.4)

∇f = F−Tf ∇f , (3.5)

∇v = ∇vF−1f , (3.6)

(v · ∇)f = (F−1f v · ∇)f . (3.7)

Proof: For the partial temporal derivative ∂tf(x, t) we get with the chain rule

∂tf = ∂tf(x, t) = ∂tf(T−1f (x, t), t) = ∂tf(x, t) + ∂tT

−1f · ∇f . (3.8)

For the inverse of the ALE mapping it holds by differentiating

Tf T−1f = id ⇒ 0 = ∂tTf (T−1

f (x, t), t) = ∂tTf + ∇Tf∂tT−1f (x, t) ⇒ ∂tT

−1f = −F−1

f ∂tTf .(3.9)

Then, combining (3.8) and (3.9) we derive the first statement (3.4).

For the spatial derivative we get by component-wise calculation

∂if(x, t) = ∂if(T−1f (x, t), t) =

d∑j=1

∂j f∂i[T−1f ]j =

d∑j=1

∂j f [F−1f ]ji ⇒ ∇f = F−Tf ∇f ,

which yields (3.5). Relation (3.6) follows by applying (3.5) to the components of v and bynoting that the Jacobian ∇v is given by the row-vectors ∇vTi . For the convective term (3.7)we get

(v · ∇)f =d∑i=1

vi∂if =d∑

i,j=1vi∂j f [F−1

f ]ji =d∑j=1

[F−1f v]j ∂j f = (F−1

f v · ∇)f .

Then, the variational formulation of the Navier-Stokes equations (2.7) on the moving domainΩf (t) can be mapped onto the reference domain.

We treat the different terms separately, starting with the divergence condition:

(∇ · v, ξ)Ωf (t) =∫

Ωf (t)div(v)ξdx =

∫ΩfJfdiv(v)ξdx.

We assume that Tf is a C2-diffeomorphism. Then, with Lemma 1, we conclude

(div(v), ξ)Ωf (t) = (div(Jf F−1v), ξ)Ωf .

Using (3.4) and (3.7) to transform the momentum equations, we get

(ρf∂tv, φ)Ωf (t) =(Jf ρf (∂tv − (F−1

f ∂tTf · ∇)v), φ)Ωf,(

ρf (v · ∇v), φ)Ωf (t) =

(Jf ρf (F−1

f v · ∇)v, φ)Ωf,

36


where ρf (x, t) = ρf (x, t) is the fluid’s density expressed in the artificial coordinates on Ωf .Combined, we get(

ρf (∂tv + (v · ∇)v), φ)Ωf (t) =

(Jf ρf (∂tv + (F−1

f (v − ∂tTf ) · ∇)v), φ)Ωf.

Finally, the viscous term in the momentum equation gets

(σf ,∇φ)Ωf (t) = (J σf , ∇φF−1f )Ωf = (J σf F−Tf , ∇φ)Ωf ,

where σf (x) = σf (Tf (x, t)) is the fluid’s stress tensor in the artificial coordinate system onΩf . It holds with (3.6)

σ(x) = −p(Tf (x))I + ρfνf∇v(Tf (x)) +∇v(Tf (x))T

= −pI + ρfνf

∇vF−1

f + F−Tf ∇vT.

Finally, the right hand side of the equation is easily transformed by

(ρff, φ)Ωf (t) = (Jf ρf f , φ)Ωf .

Then, the complete set of equations in ALE coordinates is

Problem 3 (Navier-Stokes equations in ALE coordinates). Let Tf : Ωf → Ωf (t) be a C2

diffeomorphism. Then, the Navier-Stokes equations in artificial coordinates are given on Ωf

by finding v ∈ v0 + Vf , Vf := H10 (Ωf ; ΓD)d and p ∈ Qf , Qf := L2

0(Ωf ) such that(Jf ρf (∂tv + (F−1

f (v − ∂tTf ) · ∇)v), φ)Ωf

+ (Jf σf F−Tf , ∇φ)Ωf = (Jf ρf f , φ)Ωf

(div(Jf F−1v), ξ)Ωf = 0, ∀φ ∈ Vf , ξ ∈ Qf ,(3.10)

where ρf is the density given on Ωf and

σf = −pI + ρfνf∇vF−1

f + F−Tf ∇vT.

If Ω = Ωf ∪ Ωs, we can formulate the fluid-structure interaction problem on one commondomain. The interface Γi = Ωf ∩ Ωs is just the interface between both computationaldomains. Further, by solving the Navier-Stokes equations in ALE coordinates, the domainis fixed in time. Discretization is straightforward: we generate a triangulation Ωf,h of Ωf

and can discretize with time-stepping schemes in time and finite elements in space. Theinterface condition is given by

vf (x, t) = vs(x, t) and nf · (Jf σf F−Tf ) + ns · (JsσsF−Ts ) = 0 on Γi.

We note, that even though the formulation looks just like the Lagrangian structure formu-lation, we stress the fact, that the transformation Tf is arbitrary and in general not thetransformation to the Lagrangian coordinate system. Comparing the flow equations in ALEcoordinates (Problem 3) with the Eulerian formulation in Problem 1, we observe strongnonlinearities introduced by the transformation. In particular, its gradient Ff enters intothe equation in various ways. For the formulation to be well-posed, it has to be at leastinvertible. Since this transformation will not be given analytically (because it will have todepend on the unknown solution itself to fit the structure’s domain), this will give rise toserious problems.

37


3.2 Construction of the ALE-mapping Tf

The ALE formulation of the Navier-Stokes equations is useful whenever the problem is givenon a moving domain. Thinking of fluid-structure interaction problems, the domain movementis stated by the solution of the coupled problem. We will construct the transformation Tfsimilar to the transformation of the structure domain. This is given by:

Ts(x) = x+ us(x) in Ωs,

where the deformation us can be regarded as the deformation of the domain. In the fluiddomain, we will follow the same approach. Let Tf : Ωf → Ωf (t) be given by

Tf (x) = x+ uf (x) in Ωf ,

where we denote by uf the “deformation of the fluid domain”.

A natural choice for the ALE-mapping would be the transformation to material-centeredLagrangian coordinates. The deformation uf follows the trajectories of the fluid and isnaturally given by

uf = ∂tvf . (3.11)Then, with

Ff = I + ∇uf , ∂tTf = ∂tuf = vf ,

the acceleration terms in the transformed momentum equation (3.10) get

(Jf ρf∂tvf , φ)Ωf + (Jf σf F−Tf , ∇φ)Ωf = (J ρf f , φ)Ωf ,

and the equations comes without the convective term. That is natural for the Lagrangianviewpoint. Since the convective term gives rise to stability problems when dealing withhigher Reynolds numbers, this choice of Tf looks beneficial.

Using (3.11) however does not yield a mapping between the domains of interest

Tf : Ωf 6→ Ωf (t),

since particles x ∈ Ωf can be transported out of the area of interest, see Figure 3.2 for an easyexample, where a Poiseuille flow is modeled using ALE transformation to the Lagrangiancoordinate system. The domain of interest is the fixed rectangle Ωf (t) = [0, X] × [0, Y ] forall times. The ALE-domain however will get more and more distorted and does not reflectΩf (t).

Ωf = Tf (t0)Ωf Tf (t1)Ωf Tf (t2)Ωf

Figure 3.2: Using the “natural deformation” to construct the ALE-mapping.

The ALE-mapping should have the following properties:

38

3.2 Construction of the ALE-mapping Tf

1. It has to be a bijective mapping of the reference domain Ωf onto the domain of interest:

Tf (t) : Ωf → Ωf (t), T−1f (t) : Ωf (t)→ Ωf .

2. Along the interface, it should coincide with the Lagrange-Euler transformation of thestructure domain:

Tf (x, t) = Ts(x, t) ∀x ∈ Γi.

Introducing a flow-deformation field uf this means

uf (x, t) = us(x, t) ∀x ∈ Γi.

3. The transformation should be regular

Jf = det(Ff ) > 0,

or most preferable Jf ∼ 1.

We construct uf as the solution of an artificial partial differential equation to be solved inΩf :

L(uf ) = 0 in Ωf , uf = us on Γi, uf = 0 on Γ/Γi.

The differential-operator L should be such way, that the resulting solution uf is as smoothas possible. The first choice is to consider the Laplace-equation to get a harmonic extensionof us to uf into the fluid domain

∆uf = 0, uf = us on Γi, uf = 0 on Γ/Γi.

This additional equation can be formulated and discretized along with the Navier-Stokesequations. Then, the ALE-mapping is given as a further solution variable in the system. Wenote, that this variable uf is not a deformation in the sense that it fits to the velocity, ingeneral ∂tuf 6= vf .

One problem of using the harmonic extension to define the ALE mapping is the lack ofregularity. Often, the structure Ωs has sharp edges (like in our example, Figure 3.1). Theseedges introduce reentrant corners in the flow domain Ωf . Here, the solution of the Laplace-equation locally behaves like

|u(r)| ∼ rπω ,

where r is the distance to the corner and ω the outer angle. For ω > π (i.e. for reentrant cor-ners), u 6∈ H2(Ωf ) and the derivatives tend to infinity. In ALE-context, the transformationgradient Ff looses its regularity.

Jf →∞

Nevertheless, using the harmonic extension for defining the ALE mapping is the method ofchoice. If the deformation of the structure (and the flow domain) is small, this method worksvery reliable.

39


For larger deformations, especially close to reentrant edges, using the biharmonic equationis a good alternative, which leads to more regular solutions:

∆2uf = 0 in Ωf , uf = us, ∂nuf = 0 on Γ/Γi.

Here, the corner singularity is usually about one level better, yielding H2 solutions even forthe limit case ω = 2π.

While the bi-harmonic equation yields excellent ALE mappings, it is very costly to solve.One either has to use C1-conforming finite elements, or one has to use mixed methods forits solution introducing an additional set:

∆uf = wf , ∆wf = 0.

For very large deformations, using the bi-harmonic equation is one of the view possibilitiesto generate a robust ALE method.

A further possibility is to define the ALE-mapping using a elastic-structure equations

Ff (2µf Ef + λf tr(Ef )I) = 0 in Ωf , Ff := I + ∇uf , Ef = 12(F Tf Ff − I).

One problem of using the structure equation for generating the ALE mapping is the tendencytowards incompressibility. This can lead to a degeneration of elements close to the interface:a force in x-direction will lead to a compression in this direction which will be balancedby a stretching in y-direction. This way, the transformed elements can get very anisotropic.Using a negative Poisson ratio ν < 0 (which is not physical for most materials), this behaviorcan be reversed. Compression in x-direction will lead to compression in y-direction and thetransformed elements stay isotropic, see Figure 3.3.

ν > 0 ν < 0

Figure 3.3: Material behavior with positive and negative Poisson ratio ν.

By choosing the “structure parameters” µf and λf large close to the interface, the “material”gets stiffer here, yielding smaller transformation gradients Ff and Jf ∼ 1.

40

3.3 Fluid-structure interaction in ALE coordinates

3.3 Fluid-structure interaction in ALE coordinates

In this section we can finally state the monolithic formulation of the fluid-structure inter-action problem in ALE-coordinates. The structure equation is formulated in Lagrangiancoordinates on Ωs, the Navier-Stokes equations in ALE coordinates on the domain Ωf . Inaddition we here solve the Laplace equation for the definition of the ALE mapping. Thecontinuity conditions vs = vf and uf = us on Γi = Ωf ∩Ωs are strongly enforced by includingthem in the function space. We search for u ∈ H1(Ω; ΓD)d and v ∈ H1(Ω; ΓD)d, where thelocal quantities are defined by restriction vf := v|Ωf , vs := v|Ωs , us := u|Ωs and so on.

Since there is no pressure variable given in the solid domain, we harmonically extend thefluid-pressure pf to Ωs. On all Ω we denote the pressure field by p.

We note, that this extension of the pressure is an inconsistency. While the fluid’s pressureis of low regularity pf ∈ L2

0(Ωf ), the Laplace equation yields ps ∈ H1(Ωs). This additionalregularity will feed back into the fluid domain, if the extension is not properly decoupled.

Since the ALE-mapping is defined in accordance to the Lagrange-Euler structure mappingvia Tf := x+ uf , we can define on all Ω:

T := x+ u, F := I + ∇u, J := det(F ).

In the structure domain, T takes the place of the Lagrangian-Eulerian coordinate transfor-mation, while in the fluid domain, T has no physical meaning but serves as ALE mapping.

Problem 4 (ALE formulation of the Fluid Structure Interaction problem). Find v ∈ vD +H1

0 (Ω; ΓD), u ∈ uD +H10 (Ω; ΓD) and p ∈ L2

0(Ω), such that

(J ρf (∂tv + (F−1(v − ∂tT ) · ∇)v), φ)Ωf + (ρ∂tv, φ)Ωs

+(J σsF−T , ∇φ)Ωs + (J σf F−T , ∇φ)Ωf = (J ρf ff , φ)Ωf + (ρsfs, φ)Ωf

(div(J F−1v), ξ)Ωf + (αp∇p, ∇ξ)Ωs = 0,

(αu∇u, ∇ψ)Ωf + (∂tu− v, ψ)Ωs = 0,

where the stress tensors are given as

σf = −pI + ρfνf∇vF−1 + F−T ∇vT ,σs = J−1F (2µsE + λstr(E)I)F T .

In Problem 4, both momentum equations are multiplied by the same continuous test-functionφ ∈ H1

0 (Ω; ΓD). Thus, integration by parts in both subdomains yields the boundary termon Γi:

(nf · (J σsF−T ), φ)Γi + (ns · (J σf F−T ), φ)Γi = 0.

The interface condition is hence implicitly included in the formulation.

41


Integration by parts also yields artificial boundary terms in the equations describing theextension of the deformation and pressure:

(αu∇u, ∇ψ)Ωf = (−αu∆u, ψ)Ωf + 〈n · ∇u, ψ〉Γi = 0,

and(αp∇p, ∇ψ)Ωs = (−αp∆p, ψ)Ωs + 〈n · ∇p, ψ〉Γi = 0,

On the interface, both equations include homogeneous Neumann boundary values. However,we want to impose Dirichlet-boundary values to guarantee continuity uf = us on Γi. Thisway, the equations are overdetermined and the extension will feedback to the other domaindistorting the results. In the discretization and solution schemes we carefully have to treatthis additional boundary terms.

42

4 The Finite Element Method for ContinuumMechanics

In this chapter we describe the discretization of the equations with finite elements.

4.1 Basics

Let Ω ∈ Rd with d = 2, 3 be an open domain. The boundary Γ := ∂Ω of the domain issplit into Γ := ΓD ∪ ΓN , a Dirichlet- and Neumann part. We describe the finite elementdiscretization of an elliptic partial differential equation

L(u) = f in Ω, u = gD on ΓD, B(u) = gN on ΓN . (4.1)

Here, L is the elliptic differential operator, e.g. L := −∆. By gD we denote the prescribedDirichlet values on ΓD. B is some boundary operator and gN is the prescribed Neumann-data. E.g., by

B(u) = ∂nu+ αu, gN = 0,

a homogenous Robin boundary condition is indicated.

With the bilinear form a(·, ·) we write Equation (4.1) in the weak formulation

a(u, φ) = (f, φ) ∀φ ∈ V,

derived by multiplying (4.1) with a suitable test-function φ, by integrating over the domainΩ and applying integration by parts to all second order terms. For the Laplace-equationwith Robin-boundary values on ΓR this bilinear-form is given by

a(u, φ) = (∇u,∇φ) + 〈αu, φ〉ΓN .

The weak formulation of (4.1) is to find u ∈ uD +H10 (Ω; ΓD), such that

a(u, φ) = (f, φ) + 〈gD, φ〉ΓN ∀φ ∈ H10 (Ω; ΓD). (4.2)

Here, we understand uD ∈ H1(Ω) as an extension of the Dirichlet boundary data into thedomain.

43

4 The Finite Element Method for Continuum Mechanics

4.1.1 Finite Elements for elliptic partial differential equations

The finite element discretization aims at finding a discrete solution uh ∈ Vh of (4.2). If thediscrete function space Vh is a subspace Vh ⊂ H1

0 (Ω; ΓD) we call the the resulting method aconforming finite element method, otherwise a nonconforming finite element method .

For the definition of the discrete space Vh, we introduce:

Definition 4 (Finite Element Mesh). By Ωh = K we denote a triangulation of the domainΩ into open elements K (quadrilaterals in two and hexahedrals in three dimensions):

Ω =⋃

K∈Ωh

K.

We require the mesh to be structurally regular

∀K,K ′ ∈ Ωh : K 6= K‘ ⇒ K ∩K ′ = ∅, K ∩ K ′ =

∅ orxi ∈ Ωh a corner nodee ∈ Ωh a common edgef ∈ Ωh a common face

and shape regular

∀K ∈ Ωh : π

2 − cα < αK,i <π

2 + cα,dKρK≤ cρ,

where αK,i are the interior angles of K, dK the diameter of K and ρK the radius of thelargest inscribed circle. cα < π/2 and cρ ≥ 2 are constants.

By K = (0, 1)d we denote the reference element. On K we define the space of polynomialsup to degree r:

Q(r) = span(xγ11 · · ·x

γdd ), 0 ≤ γ1, . . . , γd ≤ r, P (r) = span(xγ1

1 · · ·xγdd ), 0 ≤

d∑i=1

γi ≤ r.

The first one is the space of polynomials up to degree r in every direction, the second spacehas maximum degree r. Every element K can now be written as mapping of a simplytransformation TK : K 7→ K.

Remark 1. The mapping TK is affine, if K is a parallelogram or parallelepiped (in threedimensions). For general quadrilaterals and hexahedrals this mapping is nonlinear. If trian-gular and tetrahedral meshes are used, every element is result of an affine mapping.

Next, we can define the polynomial spaces on every element K by transformation:

Q(r)(K) = φ : φ T−1K ∈ Q(r).

If TK ∈ Q(r) we call the finite element space Q(r)(K) isoparametric. For higher order finiteelements r > 1 using isoparametric finite elements is essential to approximate domains withcurved boundaries. Then, the edges and faces of the elements are not necessarily straight.

44

4.1 Basics

On the domain Ω we can finally define continuous and discontinuous finite element spacesV

(r)h and V (r),dc

h as

V(r)h = φ : Ω→ R : φ ∈ C(Ω), φ|K ∈ Q(r)(K),

V(r),dch = φ : Ω→ R : φ|K ∈ Q(r)(K) or V (r),dc

h = φ : Ω→ R : φ|K ∈ P (r)(K)

We introduce the Lagrangian nodal basis of V (r)h in a similar fashion. By xi ∈ K, i =

1, . . . , (r + 1)d we denote equally spaced nodal points of the reference element. Then, byφi, i = 1, . . . , (r + 1)d we denote a basis of Q(r) on K with the property

φi(xj) = δij , i, j = 1, . . . , (r + 1)d.

This basis is mapped ontoQ(r)(K) and combined to a continuous basis of V (r)h = spanφih, i =

1, . . . , N. For vh ∈ V(r)h we write

vh =N∑i=1

vvvihφih.

Then, the finite element solution of (4.2) is given by:

vvvh ∈ RN : a( N∑j=1

vvvjhφjh

)(φih)

= (f, φih), i = 1, . . . , N.

By introducing the system matrix Ah ∈ RN×N and the load vector bbbh ∈ RN :

(Ah)ij = a(φjh, φih), i, j = 1, . . . , N, bbbih = (f, φih), i = 1, . . . , N.

The solution vvvh is hence given as the solution of a system of linear equations

Ahvvvh = bbbh.

4.1.2 Discretization of time-depending partial differential equations

In this section we describe the discretization of non-stationary partial differential equations.First, we have parabolic initial value problems in mind. Let I := [0, T ] we a time-interval.In weak formulation we search u ∈ uD +X0

u(0) = u0, (ut, φ) + a(u, φ) = (f, φ) ∀φ ∈ H10 (Ω; ΓD), u(0) = u0,

where uD ∈ X are extensions of the Dirichlet values into the domain and u0 ∈ uD(0)+H10 (Ω)

is the initial value.

We discuss the discretization with Rothe’s method. First, we discretize in time by introducingdiscrete time-steps

0 = t0 < t1 < . . . , TM = T, Im = (tm−1, tm], km := tm − tm−1,

45


and by setting for m = 0, . . . ,M : um := u(tm). For approximation of the time-derivative weuse the θ-method with a parameter θ ∈ [0, 1]:

(um − um−1, φ) + kmθa(um, φ) + km(1− θ)a(um−1, φ) = km(Fmθ , φ), m = 1, . . . ,M,

where u0 is the given initial value and

Fmθ := θf(tm) + (1− θ)f(tm−1).

For θ = 1 this method is the implicit backward Euler method, for θ = 0 the fully explicitforward Euler method. These two methods are first order accurate in time. The explicitforward Euler method is unstable and not usable for stiff problems, e.g. if a(·, ·) representsthe Laplace-equation. The backward Euler method is unconditionally stable. For θ = 1

2 theresulting method is the second order accurate Crank-Nicolson scheme:

(um, φ) + km2 a(um, φ) = km(Fm1/2, φ) + (um−1, φ)− km

2 a(um−1, φ)

m = 1, . . . ,M. (4.3)

Every time-step can then be solved separately and is regarded as a stationary partial differ-ential equation:

(u, φ) + θmkma(u, φ) = (F, φ),

where the right hand side and all the explicit parts of the time-discretization are gatheredin (F, φ). These “stationary” equations can be solved as indicated in the previous section.It is possible to use different finite element meshes Ωm

h for different time steps tm−1 → tm.

The elastic structure equations are wave-equations with second order temporal derivatives:

u(0) = u0, ∂tu(0) = v(0) = v0, (utt, φ) + a(u, φ) = (f, φ) ∀φ ∈ H10 (Ω; ΓD).

Instead of using special time-discretization schemes like the Newmark Scheme, we alwayswrite these equations as a set of two equations:

u(0) = u0, v(0) = v0,

(vt, φ) + a(u, φ) = (f, φ) ∀φ ∈ H1

0 (Ω; ΓD)(ut, φ) = (v, φ) ∀ψ ∈ H1

0 (Ω; ΓD) .

Then, we can apply simple time-stepping methods like the θ-scheme on this system of equa-tions:

(vm, φ) + θkma(vm, φ) = (vm−1, φ) + (Fmθ , φ)− (1− θ)kma(vm−1, φ),(um, ψ)− θkm(vm, ψ) = (um−1, ψ) + (1− θ)km(vm−1, ψ).

4.1.3 Discretization of nonlinear problems

Next, we assume that the differential operator contains nonlinearities. We write the weakformulation of the equations with a bilinear-form a(φ)(φ), which is linear in the second andpossibly nonlinear in the first argument:

u ∈ uD +H10 (Ω; ΓD) : a(u)(φ) = (f, φ) ∀φ ∈ H1

0 (Ω).

46

4.1 Basics

1

1

ε

Figure 4.1: Exact and numerical (dashed lines) solution to the convection-diffusion modelproblem.

For discretization, we first linearize this equation by using a Newton-iteration to find thesolution u(i) i→∞−−−→ u ∈ H1

0 (Ω; ΓD):

Algorithm 1 (Newton-Iteration). Let u(0) ∈ uD + H10 (Ω; ΓD) be an initial guess of the

solution, e.g. u(0) = uD. For i ≥ 1 iterate

1. Assemble Newton residual

(r(i), φ) = (f, φ)− a(u(i))(φ) ∀φ ∈ H10 (Ω).

2. Check residual‖r(i)‖ ≤ TOL ⇒ STOP!

3. Solve for w ∈ H10 (Ω; ΓD):

a′(u(i−1))(w, φ) = (r(i), φ) ∀φ ∈ H10 (Ω; ΓD).

4. Updateu(i) := u(i−1) + w(i).

In Step 3 of the algorithm a linear system has to be solved. By a′(u)(w, φ) we denote thedirectional derivative of a(·)(·) at point u in direction of w. It is defined by

a′(u)(w, φ) := d

dsa(u+ sw)(φ),

and it is linear in the second and third argument. These linear Newton update equationscan be solved as described the two previous sections. Quadratic convergence of the Newtonmethod is guaranteed, if a(·)(·) is two times differentiable and if the initial guess is goodenough.

47


4.1.4 Discretization of transport-dominant problems

In this section we discuss the finite element discretization of transport diffusion equations ofthe type

a(u, φ) = ε(∇u,∇φ) + (β · ∇u, φ), (4.4)

where ε > 0 is the diffusion constant and β : Ω → Rd is a given transport field. For smalldiffusion values ε 1, the finite element method is known to yield numerically unstablesolution. This is easily seen by analyzing the 1d-example

−εu′′ + u′ = 0 in (0, 1), u(0) = 1, u(1) = 0, (4.5)

with the exact solution

u(x) = exε − e

1ε

1− e1ε

.

This solution behaves like u(x) ≈ 1 for x < 1− ε and quickly turns to zero in a small layerof size O(ε). A finite element solution of this problem will yield over- and undershoots if themesh-size is too large (to be precise, for h > 2ε), see Figure 4.1.

A stable solution of transport-diffusion equations can either be reached by using discontinu-ous finite elements for discretization or by adding stabilization terms to Equations 4.4. Theaim of these additional stabilization terms is to gain control over the derivative in transportdirection ‖β · ∇u‖. The most common stabilization technique is the streamline diffusionmethod. Here, the weak formulation is build with the special test-function:

(−ε∆u+β·∇u, φ+δhβ·∇φ) = (ε∇u,∇φ)+(β·∇u, φ)+∑K∈Ωh

(−ε∆u, δhβ·∇φ)K+∑K∈Ωh

(β·∇u, δhβ·∇φ)K .

For obtaining a consistent discretization, the right hand side of the equation is tested withthe same adapted test-function (f, φ + δhβ · ∇φ). The stabilization parameter δh is definedelement-wise by

δh∣∣K

= δ0min(h2

K

ε,hK|β|∞

),

with a constant δ0 ≈ 12 . This stabilization scheme is able to find stable solutions for arbi-

trary ε > 0. One problem of the streamline diffusion method is that the scheme has to beimplemented in a consistent fashion, multiplying every term in the equation with δhβ · ∇φ.This way, not-natural terms like (∆u, δhβ · ∇φ) (which is of third order) appear. We willrefer to this problem when discussing the stabilization of the Navier-Stokes equations andthe full fluid-structure interaction problem.

One alternative stabilization technique is the Local Projection Method (LPS). Again, stabilityis gained by adding stabilization terms to the bilinear-form:

(ε∇u,∇φ) + (β · ∇u, φ) +∑K∈Ωh

(δhβ · ∇πhu, β · ∇πhφ)K = (f, φ),

where the element-wise parameter δh is defined as above. The operator πh : Vh → Vh isthe so called fluctuation operator. The method aims at penalizing only the oscillations (see

48

4.2 Finite Elements for the incompressible Navier-Stokes Equations

Figure 4.1) on the finest mesh level, i.e. we only want to penalize the derivatives in β-direction on the finest mesh scale. For this, let Vh be a coarser finite element space thanV

(r)h , e.g. the space V (r)

2h of same degree but on a coarser mesh. Then, the fluctuationoperator πh is defined by

πhφh := φh − i2hφh,

as the difference between the function φh itself and its interpolation to the next coarsemesh. This LPS-method yields optimal convergence results (like the streamline diffusionmethod). It however is not able to stabilize the limit-case ε → 0. The great benefit ofthis method is its diagonal character : the stabilization term only acts on the value to bestabilized (the transport-derivative) and it does not introduce additional artificial terms tothe equation. Especially for treating complex problems like fluid-structure interaction, thisis a very valuable property.

4.2 Finite Elements for the incompressible Navier-StokesEquations

4.2.1 Finite Element discretization of the Stokes Equations

We first discuss the finite element discretization of the stationary Stokes equation in a domainΩ ⊂ Rd: find v ∈ v0 + V0 and p ∈ L, such that

(∇v,∇φ)− (p,∇ · φ) = (f, φ), (∇ · v, ξ) = 0, ∀φ ∈ V0, ξ ∈ L.

The function spaces are

V0 := H10 (Ω; ΓD)d, L := L2

0(Ω) := ξ ∈ L2(Ω) :∫

Ωξdx = 0.

Since the Stokes equations determine the pressure only up to a constant, these are removedfrom the space L. If the boundary ∂Ω contains a Neumann part ΓN , integration by partsyields:

(∇v,∇φ)− (p,∇ · φ) = (−∆v +∇p, φ) + 〈∂nv − n · p, ξ〉ΓN

This boundary condition〈∂nv − n · p, ξ〉ΓN = 0,

is called the do nothing or outflow boundary condition. The name “do nothing” arises fromthe property, that some prototypical flow fields are not tempered with by this condition. Thisholds e.g. for the Poiseuille flow or the Couette flow. The do nothing condition includesa further “hidden boundary condition”. Let ΓN be a connected Neumann boundary withnormal vector n and tangential vector τ . Since v is divergence-free, it holds with vn := n · vand vτ := τ · v

∇ · v = ∂nvn + ∂τvτ ⇒ ∂nvn = −∂τvτ .

Then, with the test function ξ = 1 on ΓN :∫ΓN

n · p ds =∫

ΓN∂nv ds =

∫ΓN

∂nvn ds = −∫

ΓN∂τvτ ds = 0,

49


since vτ = 0 on the adjacent Dirichlet boundaries. On every connected Neumann boundarysegment, the pressure has zero average value. If Neumann boundary is present, we can thussearch the pressure in the full space p ∈ L := L2(Ω).

Stable Finite Element Pairs for the Stokes Equations

For a consistent discretization of the Stokes equations we need discrete subspaces

Vh ⊂ V0, Lh ⊂ L.

To yield a stable discretization this finite element pair Vh×Lh has to fulfill the BB-condition:

Lemma 6 (BB-condition). Let Vh ⊂ V0 and Lh ⊂ L be finite element spaces fulfilling thecondition

γ‖ξh‖ ≤ maxφh∈Vh

(ξh,∇ · φh)‖∇φh‖

, ∀ξh ∈ Lh, (4.6)

with a constant γ > 0 (independent of h). Then, there exists a unique solution vh ∈ Vh andph ∈ Lh of

(∇vh,∇φh)− (ph,∇ · φh) = (f, φh), (∇ · vh, ξh) = 0 ∀φh ∈ V0, ξh ∈ Lh.

This condition (4.6) is also called the discrete inf-sup condition.

The most simple equal order finite element pairs, e.g. Vh = [V (r)h ]d for the velocity Lh = V

(r)h

for the pressure are known not to fulfill the BB-condition. A common inf-sup-stable Stokes-elements is the Taylor-Hood Element, which is given by a piecewise bi-quadratic velocity anda bi-linear pressure:

Vh := V(2)h , Lh := V

(1)h .

On quadrilateral (and hexahedral) meshes, the finite elements pairs V (r+1)h − V (r)

h are allstable for r ≥ 1.

Remark 2. If the right hand side is f ∈ H1(Ω) and the boundary of the domain is “smooth”,the following a priori estimate holds for the approximation with the Taylor-Hood element:

‖∇(v − vh)‖+ ‖p− ph‖ ≤ Ch2‖∇3u‖+ ‖∇2p‖.

Stabilized finite elements for the Stokes equations

Sometimes it is preferable to use simple finite element pairs that are not inf-sup-stable asthe equal-order elements Q1 − Q1 or Q2 − Q2. This can be for ease of implementation.Stability is reached by adding certain stabilization terms to the equation. The most well-known technique is the Pressure Stabilized Petrov Galerkin method (PSPG), similar to the

50


streamline upwind method. The weak formulation of the Stokes equations is derived byusing a modified test-function pair (φh + αh∇ξh, ξh):

(∇vh,∇φh)− (ph,∇ · φh) = (f, φh) ∀φh ∈ Vh.

(∇ · vh, ξh) +∑K∈Ωh

(αh∇ph,∇ξh)K +∑K∈Ωh

(αh(f −∆vh),∇ξh)K = 0 ∀ξh ∈ Lh.

The stabilization parameter αh is chosen element-wise as

αh = α0h2K , α0 ≈

12 .

The drawbacks of this stabilization technique are similar to those of the streamline diffusionmethod. Additional coupling are introduced to the system. In the error analysis the term‖f −∆vh‖ appears whose implication is not completely understood.

As an alternative technique we again introduce the Local Projection Stabilization, whichreaches stability by adding local fluctuations of the pressure-gradient:

(∇vh,∇φh)− (ph,∇ · φh) = (f, φh) ∀φh ∈ Vh.

(∇ · vh, ξh) +∑K∈Ωh

(αh∇πhph,∇πhξh)K = 0 ∀ξh ∈ Lh.

The stabilization parameter αh is defined as for the PSPG method, the operator πh againmeasures the fluctuation with respect to a coarse functions space Vh. A common projectionspace for the LPS-method is a finite element space of lower degree on the same mesh:

Vh = V(r)h , Vh = V

(r−1)h : πh := id− ir−1,

which yields optimal approximation results (since the natural regularity of the pressure isone order lower than the velocities regularity).

Remark 3. The function space pairing Qr − Qr−1 yields a stable method for transport-stabilization. We however do not reach optimal order of convergence. For transport stabi-lization, we need to project to an equal-order space on a coarse mesh Qr −Qr2h.

4.2.2 Finite Element discretization of the Navier-Stokes Equations

The incompressible Navier-Stokes equations on the time-interval I = [0, T ] in weak formu-lation are given by

(ρ∂t(v + v · ∇v), φ) + (ρν∇v +∇vT , φ)− (p,∇ · φ) = (ρf, φ),(∇ · v, ξ) = 0.

For time discretization difference schemes like presented in Section 4.1.2 are used. Whentreating flow problems, further demands are made on the time-stepping scheme:

Stability The difference scheme should at least be A-stable. Stronger stability results likeStrong A-Stability are desirable so that oscillations in the data or numerical errors aredamped.

51


Dissipation The scheme should be energy preserving: physical oscillation may not be damped.

First order schemes like backward Euler are not competitive. Further, the backward Eulerscheme is very dissipative, physical oscillations are damped too much. The easy and well-established Crank-Nicolson scheme is second order, A-stable and exactly energy-preserving.It however lacks the ability to damp unwanted oscillations. A remedy is given by the im-plicitly shifted Crank-Nicolson scheme, a θ-scheme with the value θ = 1

2 + k. This methodhas slightly better stability properties and still has low dissipation.

A further method, which combines most preferable properties is the Fractional-Step-Theta(FST) scheme. Here, one time-step tn−1 → tn is assembled by three sub-steps of the θ-scheme, using different θ-values and different time-steps for every sub-step:

tn−1θ−→ tn−1+k′

1−θ−−→ tn−k′θ−→ tn,

The step-size is k′ for the first and the last and k − 2k′ for the middle step. This method isof second order for

k′ :=(

1− 1√2

)k,

and strongly A-stable for all values θ ∈ (12 , 1]. Further, the FST-scheme has very low

dissipation, making it well-suited for flow problems.

In the incompressible Navier-Stokes equations, the pressure is regarded as a point-wise (intime) Lagrange-multiplier to guarantee the incompressibility. We thus always treat thepressure-part fully implicit and one step of the θ-scheme has the form (where v is the velocityin the old time-step):

(ρv, φ) + kθ(ρv · ∇v, φ) + kθ(ρν∇v +∇vT ,∇φ)− k(p,∇ · φ) = F (k, θ, v)(∇ · v, ξ) = 0,

(4.7)

where by F (k, θ, v) we denote the right hand side containing the data and all explicit termsevaluated at the old time-step:

F (k, θ, v) := (ρv, φ)+k(θf+(1−θ)f , φ)−k(1−θ)(ρv ·∇v, φ)−k(1−θ)(ρν∇v+∇vT ,∇φ).

After time-discretization, the resulting nonlinear equation is solved with Newton’s method.Let u = (v, p) be the approximation from the last Newton step. Then, the update (w, q) isgiven by the linear problem:

(ρw, φ) + kθ(ρv · ∇w + w · ∇v, φ) + kθ(ρν∇w +∇wT ,∇φ)− k(q,∇ · φ) = F (k, θ, v)−kθ(ρv · ∇v, φ)− kθ(ρν∇v +∇vT ,∇φ) + k(p,∇ · φ)

(∇ · w, ξ) = −(∇ · v, ξ),(4.8)

We write this equation in a compact notation by

A′(v, p)(w, q, φ, ξ) = F (k, θ, v, v, q). (4.9)

52


When using higher equal order finite elements, and for problems with dominant convection(i.e. for high Reynolds numbers), the Galerkin discretization of this equation is not stableand further stabilization terms are necessary. Let Qh × Vh be the finite element pair. Then,stability could by reached by combining the PSPG and the SUPG-method with the test-function:

φ := φ+ δw · ∇φ+ α∇ξ.

To get a consistent scheme, this test-function is used on both sides of (4.8) and thus alsoacts on the explicit terms and in such a way is coupling the old with the new time velocity:

1k

(v − v, φ+ δw · ∇φ+ αξ) . . .

As an alternative, we use LPS-stabilization for both, convective terms and inf-sup condition.Instead of (4.9) we solve

A′(v, p)(w, q, φ, ξ) + Slps(v, p)(w, q, φ, ξ) = F (k, θ, v, v, q),

where the stabilization form is defined by

Slps(v, p)(w, q, φ, ξ) =∑K∈Ωh

δK(ρv · ∇πw, v · ∇πφ)K + αK(1/ρ∇πq,∇πξ)K

,

with parameters defined locally on every element K as:

δK = αK = α0

(ν

h2K

+ |v|K;∞hK

)−1

.

4.2.3 Finite Elements for Navier-Stokes in ALE-formulation

In ALE formulation, the Navier-Stokes equations are strongly nonlinear. We repeat Formu-lation (3.10) using Tf := x+ u as used in FSI context:(

Jf ρf (∂tv + (F−1f (v − ∂tu) · ∇)v), φ

)Ωf

+ (Jf σf F−Tf , ∇φ)Ωf = (Jf ρf f , φ)Ωf

(div(Jf F−1v), ξ)Ωf = 0.

For discretization we have to follow the presented course: discretize in time, linearize, dis-cretize in space. Here we have to face an additional curiosity since the time-derivativeappears in a nonlinear form. Further, when the deformation u is defined implicitly, a secondtime derivative appears. It is not obvious, how to derive a proper, in particular a secondorder and stable, time-stepping formulation.

A rigorous derivation is done, if Galerkin methods are used for time-discretization. Weanalyze the following model-problem:

(∂tu, φ) + (∇u,∇φ) = (f, φ) on [0, T ]× Ω, u(u) = u0.

53


We can write this equation in a space-time Galerkin formulation by integrating over I = [0, T ]and by multiplying with suitable test-functions:∫

I

(∂tu, φ) + (∇u,∇φ)

dt =

∫I(f, φ)dt ∀φ ∈ X.

The solution u is found in the space:

u ∈ u0 +X, X = v : v ∈ L2(I;H1(Ω)), ∂tv ∈ L2(I;H1(Ω)∗).

For this space it holds X → C(I;H1(Ω)), thus, these functions are continuous in time.

We discretize this equation in time by introducing discrete-time steps ti, i = 0, . . . ,M andby defining sub-intervals

0 = t0 < t1 < · · · < tM = T, Im := [tm−1 − tm], km := tm − tm−1.

We use a Petrov-Galerkin approach with continuous trial and discontinuous test-functions:

Xtrialk = v ∈ C(I;H1(Ω)) : v|Im ∈ P

m(Im;H1(Ω)), Xtestk = v ∈ L(I;H1(Ω)) : v|Im ∈ H

1(Ω),

where Pm(Im) is a polynomial space with dimension 2. We write uk ∈ Xtrialk as:

uk =M∑m=0

umΦm, um ∈ H1(Ω).

Then, the time-discretized model-problem is given by

M∑m=1

∫Im

(∂tuk, φm) + (∇uk,∇φm)

dt =

M∑m=1

(f, φm)dt, ∀φ1, . . . , φM ∈ H1(Ω).

Since the test-functions are discontinuous, the system is decoupled and can be written as atime-stepping scheme, with global coupling only due to the continuity of the trial-space Xk.

We now define the space Pm(Im) and indicate the two basis functions for a parameterθ ∈ (0, 1):

Φ0m(t) := tm − t

km−(θ−1

2)φθ(t), Φ1

m(t) := t− tm−1km

+(θ−1

2)φθ(t), φθ(t) := 6(t− tm−1)(tm − t)

k2m

.

For Φ0m and Φ1

m it holds:∫Im

Φ0mdt = (1− θ)km,

∫Im

Φ1mdt = (θ)km,

∫Im∂tΦ1

mdt = −1,∫Im∂tΦ1

mdt = 1.

In every time-step tm−1 → tm, the solution is known at the left boundary uk(tm−1) = um−1

and we need to solve:

(−um−1 + um, φ) + (1− θ)km(∇um−1,∇φ) + θkm(∇um,∇φ) =∫Im

(f, φ)dt ∀φ ∈ H1(Ω).

Up to an integration error in the right hand side, this scheme is exactly the θ-scheme.

54

4.3 Finite Elements for elastic structure equations

We can now use this approach to derive a time-stepping scheme for more complex situationsas arising in fluid-structure interaction problems: we discuss the model-problem

(h∂tu, φ) + (∇u,∇φ) = (f, φ) on [0, T ]× Ω, u(u) = u0,

where h(t, x) is some function, also to be discretized via

hk =M∑m=0

hmΦm, hm ∈ H1(Ω).

The Galerkin discretization of the first term with the time derivative now gets:∫Im

(hk∂tuk, φ)dt =∫Im

((hm−1Φm−1 + hmΦm)∂t(um−1Φm−1 + umΦm), φ

)dt

=(hm−1 + hm

2 (um − um−1), φ).

The full time-stepping scheme for the model problem then reads:

(hm−1 + hm

2 (um−um−1), φ)+(1−θ)km(∇um−1,∇φ)+θkm(∇um,∇φ) =

∫Im

(f, φ)dt ∀φ ∈ H1(Ω).

Here, the solution at old and new time-step are always coupled. In the full Navier-Stokesequations in ALE formulation, the second time-dependent term is discretized likewise.

4.3 Finite Elements for elastic structure equations

In weak formulation, the equations describing the deformation of an elastic structure aregiven by

(ρ∂tv, φ) + (J σsF−T , ∇φ) = (ρfs, φ)(∂tu− v, ψ) = 0,

with the stress-tensor

σs = J−1F (2µE + λstr(E)I)F T , E := 12(F T F − I).

Even tho this equation is very nonlinear, the discretization is straight-forward and followsthe presented route: discretize in time (by a Galerkin or time-stepping method), linearizeand discretize in space by finite elements. In Lagrangian formulation there are no additionalstabilization terms necessary, since the equation does not include transport-terms. Whenusing incompressible material laws however, the function spaces need to fulfill an inf-supcondition and “pressure-stabilization” may be required for equal-order finite elements.

55


4.4 Finite Elements for the fluid-structure interaction problem

The coupled fluid-structure interaction problem as derived in Chapter 3 reads (compare toProblem 4):

(J ρf (∂tv + (F−1(v − ∂tT ) · ∇)v), φ)Ωf + (ρ∂tv, φ)Ωs

+(J σsF−T , ∇φ)Ωs + (J σf F−T , ∇φ)Ωf = (J ρf ff , φ)Ωf + (ρsfs, φ)Ωf


(αu∇u, ∇ψ)Ωf + (∂tu− v, ψ)Ωs = 0,

where the stress tensors are given as

σf = −pI + ρfνf∇vF−1 + F−T ∇vT ,σs = J−1F (2µsE + λstr(E)I)F T .

The finite element discretization has to take care of several difficulties:

• Strong nonlinearities

• Time-dependent problem including “non-standard terms”

• Stabilization is necessary

• Different type of equation in different subdomains

• Artificial extension of variables (“uf” and “ps”)

We already have dealt with the first three issues. In every subdomain Ωf and Ωs can beassumed to be regular, if the data is regular and if the boundary of the domain (and theshape of the interface Γi) is regular. Across the interface however, velocity v and deformationu are continuous, but higher regularity cannot be postulated. If the interface Γi = ∂Ωf ∩∂Ωs

cuts through a mesh-element K ∈ Ωh, we cannot expect a good approximation property:

‖∇(u− ihu)‖K ≤ ch?K‖∇1+?u‖K .

This additional approximation error is only local along the interface. This region howeverplays a decisive role for the overall dynamics of the fluid-structure system. Whenever possi-ble, the mesh should thus be aligned with the interface Γi!

The coupled formulation includes two parameter αp and αu controlling the harmonic exten-sion of deformation and pressure to the other domain. The choice of these parameter willinfluence the approximation of the discrete schemes as well as the properties of the algebraicsystem and thus the convergence of the linear solvers. This will be discussed at some furtherpoint.

Both values, pressure and deformation are extended with the Laplace-equation. On the in-terface Γi continuity is postulated. Integration by parts yields (exemplarily for the pressure):

0 = (div(J F−1v), ξ)Ωf +(αp∇p, ∇ψ)Ωs = (div(J F−1v), ξ)Ωf +(−αp∆p, ψ)Ωs +〈ns · ∇p, ψ〉Γi .

56

4.5 Some mathematical theory for fluid-structure interaction problems (noch ein wenig experimentell und mit Vorsicht zu geniessen)

On the Interface, this formulation includes a Neumann boundary condition:

〈ns · ∇p, ψ〉Γi = 0.

This boundary condition is in addition to the already present Dirichlet condition givenby the continuity postulate “ps = pf” and will lead to a spurious feedback to the flow-domain. We can remove this feedback by adding this unwanted interface-boundary term tothe formulation. A better way to work around this feedback is to decouple both problemsand to enforce the Dirichlet-values strongly by removing the additional basis-functions:

ΩsΩfΓi

Figure 4.2: Delete basis functions on interface seen from the solid-side

The extension of the pressure to the structure is only a numerical necessity since we do notneed a pressure in the structure domain at all. It would be more appropriate to find thepressure in the space pf ∈ L2

0(Ωf ) and to write the divergence equation

(div(J F−1), ξ)Ωf = 0 ∀ξ ∈ L20(Ωf ),

and not to have an equation in the structure domain at all. Further, it would be moreappropriate to not combine the extension of the deformation to the flow domain and thecondition ∂tu = v in one equation since they are independent. From a theoretical point ofview one should use two different test-functions, one defined only on Ωf and the other onlyon Ωs:

(αu∇u, ∇ψf )Ωf + (∂tu− v, ψs)Ωs ∀ψf ∈ H10 (Ωf ; ΓD), ∀ψs ∈ H1

0 (Ωs; ΓD).

4.5 Some mathematical theory for fluid-structure interactionproblems (noch ein wenig experimentell und mit Vorsicht zugeniessen)

In this section we will analyze a simplified model for fluid-structure interaction:

57


• We only consider the stationary case:

∂tv = ∂tu = 0, v∣∣∣Γs

= 0.

• The elasticity model is supposed to be linear:

E ≈ 12∇u+∇uT ,

• We do not consider geometric nonlinearities Ts = I and we assume the ALE-transformationto be the identity Tf = I:

F = I, J = 0.

• We neglect the transposed entries in the stress-tensors, and we neglect the divergence-condition of the fluid problem:

−∇ · σf = ν∆v, −∇ · σs = µ∆u.

What follows is a linear set of partial differential equations:

−ν∆v = f−α∆u = 0

in Ωf ,

−µ∆u = fv = 0

in Ωs, ν∂nv = µ∂nu on Γi := ∂Ωf ∩ ∂Ωs.

Since we omit the divergence condition and the transposed parts in the tensors, no couplingsbetween the different velocity and deformation components remain. We can thus considervelocity and deformation to be scalar.

With U := (v, u) and Φ := (φ, ψs, ψv) we introduce the bilinear-form

a(U,Φ) := ν(∇v,∇φ)f + µ(∇u,∇φ)s + α(∇u,∇ψf )f − α〈∂nu, ψf 〉s − (v, ψs)s,

where

U = (v, u) ∈ H10 (Ω; ΓD)×H1

0 (Ω; ΓD), Φ = (φ, ψs, ψv) ∈ H10 (Ω; ΓD)×H1

0 (Ωf ; ΓD)×H10 (Ωs; ΓD).

The following remark is easily seen:

Lemma 7 (Continuity). With the triple-norm

|||Φ||| :=(‖∇v‖2f + ‖∇u‖2s + ‖∇u‖2f + ‖v‖2s

) 12 ,

the bilinear-form a(·, ·) is continuous:

|a(U)(Φ)| ≤ cc|||U ||| |||Φ||| ∀U,Φ ∈ H10 (Ω; ΓD)×H1

0 (Ω; ΓD).

58


Let Ψ = (ψ,−ψ) with ψ = 0 on Ωf . Then, with Poincaré inequality

a(Ψ,Ψ) = −µ‖∇ψ‖2s + ‖ψ‖2 ≤ (c2p − µ)‖∇v‖ss,

which is negative for large values of µ. We thus cannot get ellipticity of the bilinear-form.We can however formulate the following stability result:Lemma 8 (Stability of the model equation). The following inf-sup condition holds withsome γ > 0:

supU∈X

a(U,Φ)− µ(∇v,∇ψ)s|||Φ||| ≥ γ (|||U |||+ ‖∇v‖s) ∀U ∈ X.

Proof: Let U = (v, u) be arbitrary. Diagonal testing with Φ1 := (v, u) yields:

a(U,Φ1) = ν‖∇v‖2f + µ(∇u,∇v)s + α‖∇u‖2f − (v, u)s − µ(∇v,∇u)s≥ ν‖∇v‖2f + α‖∇u‖2f −

ε

2‖v‖2s −

cp2ε‖∇u‖

2s,

where ε > 0 is arbitrary and cp is the constant of the Poincaré inequality. Then, withΦ2 := (u, 0)

a(U,Φ2) = ν(∇v,∇u)f + µ‖∇u‖2s,and Φ3 := (0,−v)

a(U,Φ3) = −α(∇u,∇v)f + ‖v‖2s + ‖∇v‖2s,we can test with the combined test-function

Φ4 := Φ1 + Φ2 + ν

αΦ3 =

(v + uu− ν

αv

)to get

a(U,Φ4) = ν‖∇v‖2f +(µ− cp

2ε

)‖∇u‖2s + α‖∇u‖2f + ‖v‖2s

(ν

α− ε

2

)+ ν

α‖∇v‖2s.

For ε = cp/ν we have

a(U,Φ4)− µ(∇v,∇ψ)s = ν‖∇v‖2f + µ

2 ‖∇u‖2s + α‖∇u‖2f + ‖v‖2s

(2µν − αcp2µα

)+ ν

α‖∇v‖2s.

For α < 2µνcp

we get

a(U,Φ4)− µ(∇v,∇ψ)s ≥(ν + µ

2 + cp2µν + cp

2µ

)(|||U |||2 + ‖∇v‖2s

). (4.10)

Finally, we need to estimate the triple-norm of this test-function Φ4:

|||Φ4|||2 = ν‖∇(v + u)‖2f + ‖v + u‖2s + µ‖∇(u− ν

αv)‖2s + α‖∇(u− ν

αv)‖2f

≤ 2ν‖∇v‖2f + 2ν‖∇u‖2f + 2‖v‖2s + 2‖u‖2f + 2µ‖∇u‖2s + 2να‖∇v‖2s + 2α‖∇u‖2f + 2ν

α‖∇v‖2f

≤(

2ν + 2να

)‖∇v‖2f + (2ν + 2cp + 2α) ‖∇u‖2f + 2‖∇u‖2f + 2µ‖∇u‖2s

≤(

4ν + 2µ+ 2cp + 4µνcp

+ 2 + cpµ

)(|||U |||2 + ‖∇v‖2s

)(4.11)

59


Thus, combining the constants c1 of (4.10) and c2 of (4.11)

supΦ

a(U,Φ)− µ(∇v,∇ψ)s|||Φ||| ≥ a(U,Φ4)− µ(∇v,∇ψ4)s

|||Φ4|||≥ c1

|||U |||2 + ‖∇v‖2s|||Φ4|||

≥ c1c2︸︷︷︸=:γ

|||U |||,

we conclude the proof.

With this stability result, we can proof the existence of a unique solution:

Lemma 9 (Existence and Uniqueness). For every f ∈ H−1(Ω), there exists a unique solutionU ∈ H1

0 (Ω; ΓD)×H10 (Ω; ΓD) of

A(U,Φ)− µ(∇v,∇ψ)s = (f, φ) ∀Φ = (φ, ψ) ∈ H10 (Ω; ΓD)×H1

0 (Ω; ΓD).

For this solution U the following stability estimate holds

|||U |||+ ‖∇v‖s ≤1γ‖f‖−1,

where γ > 0 is the constant from Lemma 8 and cp the Poincaré constant.

Proof: (i) Let X := H10 (Ω; ΓD)×H1

0 (Ω; ΓD) and X ′ be its dual space. Then, we introducethe linear operator L : X → X ′ by

〈LU,Φ〉 = a(U,Φ)− µ(∇v,∇ψ)s ∀Φ ∈ X.

This operator is continuous:‖LU‖X′ ≤ c‖U‖X ,

due to the continuity of the bilinear form:

‖LU‖X′ := sup|||W |||=1

|〈LU,W 〉| = sup|||W |||=1

|a(U,W )− µ(∇v,∇ψ)s| ≤ cc (|||U |||+ ‖∇v‖s) .

(ii) Next, we show that this operator is injective. U1, U2 be given with a(U1,Φ)−µ(v1, ψ)s =a(U2,Φ) − µ(v2, ψ)s for all Φ. E.g., U1 and U2 can be two solutions of the model problem.Then, by Lemma 8

|||U1 − U2||| ≤1γ

sup a(U1 − U2,Φ)− µ(∇(v1 − v2),∇ψ)s|||Φ||| = 0.

Thus, for every f ∈ L(V ) there exists a unique inverse U = L−1f which is solution of themodel-problem.

(iii) It holds:

|||U |||+‖∇v‖s ≤1γ

sup|||Φ|||=1

(a(U)(Φ)− (∇v,∇ψ)s) = 1γ

sup|||Φ|||=1

(f, φ) ≤ 1γ

sup|||Φ|||=1

||f ||−1‖φ‖1 = 1γ||f ||−1

Thus, the operator L and its inverse L−1 are continuous on the range of L.

60


(iv) Since L and L−1 are continuous, it follows that L(X) is closed. With the closed rangetheorem we conclude that L is an isomorphism on

L : X → Φ ∈ X : a(U,Φ)− (∇v,∇ψ) = 0, ∀U ∈ X0 ⊂ X ′.

The space X0 := l ∈ V ′ : 〈l, φ〉 = 0, ∀φ ∈ X ⊂ X ′ is the polar of X.

(v) We finally get surjectivity by showing that

Φ ∈ X : a(U,Φ)− µ(∇v,∇ψ)s = 0, ∀U ∈ X = 0,

e.g. by showing that for every Φ ∈ X, there exists a U ∈ X, such that

a(U,Φ)− µ(∇v,∇ψ)s 6= 0.

This is easily seen by constructing special trial-functions. E.g. if the given Φ = (φ, ψ) fulfills‖∇φ‖s > 0, we choose v = 0 and u = δφ, where δ ≥ 0 is zero in Ωf and ‖δ∇φ‖s > 0. Then,a(U,Φ) = µ‖δ∇u‖2s. This approach works likewise for the other terms.

Next, let Vh × Vh = Xh ⊂ X be a conform discretization of the model problem. Then, wecan proof the following a best-approximation result:

Lemma 10 (Best approximation). Let Xh := Vh × Vh ⊂ H10 (Ω; ΓD)2 be a conform dis-

cretization fulfilling the inf-sup condition

supUh∈Xh

a(Uh,Φh)− µ(∇vh,∇ψh)s|||Φh|||

≥ γh (|||Uh|||+ ‖∇vh‖s) ∀Uh ∈ Xh,

with γh ≥ γ0 > 0. Further,

∀Φh ∈ Vh ∃Uh ∈ Vh : a(Uh,Φh)− (∇vh,∇ψh)s 6= 0.

Then, for the solution Uh ∈ Vh × Vh of the discretized model-problem it holds:

|||U − Uh|||+ ‖∇(v − vh)‖s ≤ccγh

infWh∈Xh

(|||U −Wh|||+ ‖∇(v − wh)‖s) .

Proof: Since Vh is a conform approximation space we have Galerkin orthogonality. Thusfor arbitrary Wh ∈ Vh:

a(Uh −Wh,Φh)− µ(∇(vh − wh,∇ψh)s = a(U −Wh,Φh)− µ(∇(v − w), ψh)s ∀Φh.

With the stability result

|||Uh −Wh|||+ ‖∇(uh − wh)‖s ≤1γ

sup a(Uh −Wh,Φh)− µ(∇(vh − wh), ψh)s|||Φh|||

= 1γ

sup a(U −Wh,Φh)− µ(∇(v − wh), ψh)s|||Φh|||

≤ ccγh

(|||U −Wh|||+ ‖∇(v − wh)‖s) ,

and the triangle inequality the result follows:

|||U − Uh|||+ ‖∇(v − vh)‖s ≤ |||U −Wh|||+ ‖∇(v − wh)‖s + |||Uh −Wh|||+ ‖∇(vh − wh)‖s

≤(

1 + ccγh

)(|||U −Wh|||+ ‖∇(v − wh)‖s)

61


4.6 Solution of the monolithic FSI formulation

As in the previous section, we consider first consider the simplified set of equations:

ν(∇v,∇φ)f + µ(∇u,∇φ)s = (f, φ)α(∇u,∇ψ)f − (v, ψ)s = 0.

This corresponds to a linear system of equations given by

Axxx = bbb,

where xxx = (v, u) and bbb = (f, 0). We split the matrix into a fluid-part and into a structure-partwritten as:

A = Af +As =(νL 00 αL

)f

+(

0 µL−M 0

)s

,

where Lij = (∇φj ,∇φi) is the discretized Laplace-operator (on regular meshes the usual9-point-stencil) and Mij = (φj , φi) the mass-matrix. Integration of the matrices is done onlyin the corresponding subdomain. Then, the linear system of equations gets:

νLfvvvf + µLsuuus = fff

αLfuuuf −Msvvvf = 0.

We solve this linear system with iterative methods. These methods use the norm of thedefect as stopping criteria. For the second equation we have due to scaling reasons

di = ‖αLfuuuif‖+ ‖Msvvvif‖ ∼ αf‖uuuif‖+ h2‖vvvif‖,

and on finer meshes we lack control over the velocity. vh = 0 will not be satisfied. By choosingα = h2 we balance the two parts in the norms. Further, by scaling the lower equation by 1/α,we reach that all norms in the entire set of equations have the same asymptotic behaviorwith respect to the mesh size h.

Hence, the set of equations is given by the matrices

A = Af +As =(νL 00 L

)f

+(

0 µL−h−2M 0

)s

.

First, we want to estimate the condition number of the two matrices. We note, that for Land M we have

λmin(M) ∼ chd, λmax(M) ∼ chd, cond2(M) ∼ 1λmin(L) ∼ chd, λmax(L) ∼ chd−2, cond2(M) ∼ h−2,

where d is the spacial dimension. The matrix Af is block-diagonal and symmetric. We canestimate the largest and smallest eigenvalue as:

λmax(Af ) = maxνλmax(L), λmax(L) = λmax(L) maxν, 1,λmin(Af ) = minνλmin(L), λmin(L) = λmin(L) minν, 1.

62

4.6 Solution of the monolithic FSI formulation

Since the viscosity is usually small, we derive

cond2(Af ) = ν−1cond2(L) ∼ ν−1h−2

The matrix As is not symmetric. Here, the condition number is defined as:

cond2(As) = ‖As‖2‖A−1s ‖2 = λmax(ATs As)

12λmax(A−Ts A−1

s )12 = λmax(ATs As)

12

λmin(AsATs )12

For ATs As and AsATs holds due to symmetry of M and L:

ATs As = AsATs =

(ν2LL 0

0 1α2MM

)Hence,

λmax(ATs AS)12 = λmax(As) = maxµλmax(L), h−2λmax(M),

λmin(ATs AS)12 = λmin(As) = minµλmin(L), h−2λmin(M),

and finally for the condition number of the matrix As:

cond2(As) = maxµλmax(L), h−2λmax(M)minµλmin(L), h−2λmin(M) ∼

maxµhd−2, hd−2minµhd, hd−2

= maxµ, 1minµh2, 1

For small mesh sizes, the condition number goes like

cond2(As) = h−2.

Considered separately, both subproblems have a condition number as expected. It is howeveressential to scale the second equation by h−2, otherwise, the condition number would behavelike cond2(Af ) = h−4. Due to the coupling over the interface, where both subproblemsfall together, the condition number of the overall system is difficult to estimate. Numericalestimations of the eigenvalues however show that the overall condition still goes like h−2.

A geometric multigrid method is thus optimal for solving the system Axxx = bbb to deal withthe bad conditioning. It is however very difficult to obtain robust multigrid smoothers forthe coupled problem. Methods that are available for either flow- or structure problems willin general not work on the coupled system. The multigrid smoother however only needs tosolve high frequent error parts which are local. Thus it is advisable to split the system intothe two subproblems for smoothing. On mesh-level l, we write the smoother in general formas a preconditioned Richardson iteration:

xxxi = xxxi−1 + P−1(bbb−Axxxi−1),

where P ∼ A is the preconditioner. For the inverse of P , use an approach from domaindecomposition:

P−1 := A−1f +A−1

s ,

with the goal to split the system into the both subproblems. Then, each step of the iterationreads:

xxxi = xxxi−1 + (A−1f +A−1

s )(bbb−Axxxi−1),and can be written in two sub-steps:

63


1. rrri−1 := bbb−Axxxi−1

2. Afwwwif = rrri−1

3. Aswwwis = rrri−1

4. xxxi := xxxi−1 + ω(wwwif +wwwis)

The two systems in Step 2 and 3 of this algorithms can by smoothed using available stan-dard techniques. For the flow-problem a smoother of Vanka-type or an incomplete LU-decomposition of the matrix Af is robust. Due to the nonlinear transformations, and ILU-type smoother is also necessary for the structure equation.

This linear solution process can be directly transferred to the full set of equations.

Finally, we shortly refer to the full set of (stationary) fluid-structure interaction equations:

(J ρf F−1(v · ∇)v, φ)Ωf + (J σsF−T , ∇φ)Ωs + (J σf F−T , ∇φ)Ωf = (J ρf ff , φ)Ωf + (ρsfs, φ)Ωf


(αu∇u, ∇ψ)Ωf + (−v, ψ)Ωs = 0,

The system is highly nonlinear and we a Newton’s method is used for solving. In everyNewton-Update step the linear system can again be split into the two subdomains and wesolve with a matrix:

A = Af +As =

K(vvv,uuu) + Σf (uuu) B(uuu) FK,Σf (uuu,vvv)C(uuu,vvv) 0 FC(uuu,vvv)

0 0 αuL

f

+

0 0 Σs(uuu)0 αpL 0−M 0 0

s

.

Here, by K we denote the matrix belonging to the convection, Σf and Σs are the stress-tensors, B is the pressure-gradient, C the divergence and L the Laplace-matrix. The matricesF denote the derivatives with regard to the ALE-mapping. These terms cannot easily belinked to a specific differential operator. The dependence on the solution uuu and vvv is due tothe nonlinearity.

For the solution process we again split the system with a domain decomposition approachusing as preconditioner

P−1 = A−1f +A−1

s .

For the approximation of both these subproblems, the matrix-structure can be used to furthersimplify, e.g. the solution of the structure-block can be split into three subproblems:

Asxxxδ = rrr ⇔ −Mvvvδ = rrrv, αpLppp

δ = rrrp, Σs(uuu)uuuδ = rrru.

The fluid-problem is more difficult, but it is still possible to split this system into a Laplaceequation and a saddle-point system:

Afxxxδ = rrr ⇔ αuLuuu

δ = rrru,

(K(vvv,uuu) + Σf (uuu) B(uuu)

C(uuu,vvv) 0

)(vvvδ

pppδ

)=(rrrv − FK,Σf (uuu,vvv)uuuδrrrp − FC(uuu,vvv)uuuδ

).

64

5 Partitioned Approaches

In this chapter we discuss partitioned approaches for the solution of fluid-structure interactionproblems. Opposed to the previously treated monolithic approach, the system is decoupledand fluid- and solid-problem are solved independently. One big advantage of a decouplingis that existing and efficient solution methods can be used for both subproblems. Theequilibrium condition on the interface is no longer implicit part of the formulation, insteadit has to be coupled into the subproblems via boundary values or loads.

In every time-step tn−1 → tn we split the system into the two subproblems, namely the fluidproblem F and the solid problem S:

F F F

tn tn+1tn−1

S S S

Opposed to monolithic approaches, we aim at solving the subproblems independently of eachother. Thus, the equilibrium condition given on the interface (in Eulerian formulation)

vf = vs, n · σs = n · σf on Γi(t),

is not implicit part of the problem formulation. Instead it must be contributed in form ofboundary conditions and loads. In the following, we describe the two subproblems.

Solution of the fluid problem F Within one time-step tn−1 → tn the fluid domain Ωf (t) canchange. We have already discussed the problem of solving the Navier-Stokes equations on thetime depending domain Ωn−1

f := Ωf (tn−1) → Ωf (tn) =: Ωnf . Key is the ALE-Formulation

of the Navier-Stokes equations. Here we describe a more general approach than used formonolithic formulations. By Ωn

f we denote the reference-domain to be used for the time-steptn−1 → tn and by Tn−1

f : Ωnf → Ωn−1

f and Tnf : Ωnf → Ωn

f we denote the ALE-mappings atthe beginning and at the end of the time-interval [tn−1, tn]. We note that Ωn

f is an unknownin this time-step. In general, we thus cannot assume that Tnf is known exactly! At the oldtime-step tn−1, the domain partitioning is given by the structure’s deformation and Tn−1

f isknown. Without further information, the approximation Tnf ≈ T

n−1f is the only possibility.

One choice for the ALE-mapping would be to use the reference domain Ωf := Ωf = Ωf (0)for all time-steps. This would be the ALE-mapping as used in the monolithic approach

65


Ωf

Ωs

Figure 5.1: Non-matching meshes for the fluid and structure problem.

described before. One can however also use different reference domains in every time step, e.g.Ωf := Ωn−1

f which is the domain at the old time-step. Then, Tn−1f = id is the identity. Using

different reference domains in every time-step will ask for a remeshing of the computationaldomain after every step. Then, the ALE-mapping only refers to the difference between theold and the new time-step. We will stress the benefit of – at first glance very expensive –technique for problems involving large deformations later on.

The coupling to the structure problems is embedded by prescribing Dirichlet conditions forthe fluid’s velocity vf on the interface Γi. This velocity has to match the velocity vs of thestructure:

vf = vs on Γi.

Here, we have to face the problem, that the structural velocity is given in Lagrangian co-ordinates vs(x) on Ωs, while the fluid’s velocity is given in arbitrary ALE coordinate vf (x),where the fluid’s reference domain Ωf does not need to match the Lagrangian referencedomain Ωn

f if different ALE-mappings are used in different time-steps. This makes the ac-curate evaluation of the Dirichlet data difficult. Furthermore, even if one ALE mappingTf : Ωf → Ωf (t) is used for all time-steps, it is possible (and most often advisable) to usedifferent triangulations Ωf,h and Ωs,h. Then, if these triangulations do not match on thecommon interface Γi as shown in Figure 5.1, the interpolation of the Dirichlet values is notstraightforward.

We formulate fluid-problem ALE-formulation using the θ-scheme for time-discretization:

Problem 5 (Fluid Problem in ALE coordinates F−ALE). Given vn−1, pn−1, Tn−1 andTn find vn ∈ vD,i +H1

0 (Ωf ; ΓD,i) and pn ∈ L20(Ωf ), such that

(ρfJ(vn − vn−1), φ

)Ωf−(ρf J F−1(Tn − Tn−1) · ∇v, φ

)Ωf

+θkn(F−1vn · ∇vn, φ)Ωf +θkn(Jnσnf F−T , ∇φ)Ωf = θkn(ρf Jn−1fn−1f , φ)Ωf +(1−θ)kn(ρf Jnfnf , φ)Ωf

− (1−θ)kn([Fn−1]−1vn−1 · ∇vn−1, φ)Ωf − (1−θ)kn(Jn−1σn−1f [Fn−1]−T , ∇φ)Ωf ∀φ ∈ H1

0 (Ωf ; ΓD,i),

(div(Jn[Fn]−1vn), ξ)Ωf = 0 ∀ξ ∈ L20(Ωf ). (5.1)

Here, with the bar we denote the average values over the time-interval:

v := vn−1 + vn

2 , J := Jn−1 + Jn

2 , JF−1 := Jn−1[Fn−1]−1 + Jn[Fn]−1

2 .

66

Further, for the extension of the boundary values into the domain it holds

vD,i = vD on ∂Ωf/Γi, vD,i = vns on Γi.

The structure’s velocity on the interface is a further prerequisite (that also cannot be givenexactly since it is unknown of the coupled system). Finally, Fn := ∇Tn and Jn := det(Fn).

We have already stated, that the ALE-mapping Tn : Ωnf → Ωn

f is not known a prioriand needs to be determined in an additional problem, which will be call M, the mappingproblem. The solution of this problem will be described separately. Here it gets obvious, thata partitioned approach cannot readily result in the correct solution of the coupled problem,since the mapping Tn must be provided beforehand. The correct mapping however dependson the domain layout at old and new time-step. By approximating Tn ≈ Tn−1 the correctdomain is not considered.

Solution of the solid problem S The solid problem is formulated and will be solved inLagrangian coordinates on the reference domain Ωs. The coupling to the fluid-problem isgiven by means of Neumann boundary values on the interface:

n · (J σsF−T ) = n · (J σf F−T ) on Γi,

where σf is the fluid’s stress tensor on the interface evaluated with the best known approx-imation of velocity vf and pressure pf . Again, the proper evaluation of the interface forceis difficult, since it is required in Lagrangian coordinates which do not have to match thefluid’s coordinate system Ωf . We write the structure problem discretized in time with theθ-scheme:

Problem 6 (Solid Problem S). Given un−1 and vn−1 find un ∈ uD + H10 (Ωs; ΓD) and

vn ∈ H10 (Ωs; ΓD) such that

(ρs(vn − vn−1), φ)Ωs + θkn(Jnσns [Fn]−T , ∇φ)Ωs = θkn(ρsfns , φ)Ωs

+ (1− θ)kn(ρsfn−1s , φ)Ωs − (1− θ)kn(Jn−1σn−1

s [Fn−1]−T , ∇φ)Ωs

+ θkn〈ns · gnf , φ〉Γi + (1− θ)kn〈ns · gn−1f , φ〉Γi ∀φ ∈ H

10 (Ωs; ΓD)

(un − un−1, ψ)Ωs − θkn(vn, ψ)Ωs = (1− θ)kn(vn−1, ψ)Ωs ∀ψ ∈ H10 (Ωs; ΓD) (5.2)

where uD is some extension of the Dirichlet data on ΓD into the domain. The interface loadis given by

gn−1f = Jn−1σn−1

f [Fn−1]−T , gnf = Jnσnf [Fn]−T .

The fluid’s stresses (again unknown at tn) need to be given for the solution of the time-step.

Solution of the mapping problem M The third subproblem consists of finding the newdomain partitioning Ω := Ωn

f∪Ωns and of finding an appropriate ALE-mapping Tf : Ωf → Ωn

f .

In the structure domain, the deformation describes the layout of the domain via Tns = id+un.This mapping also indicates the position of the interface Γni and thus describes the boundary

67


of the fluid domain Ωnf . To define the mapping Tf : Ωn

f → Ωnf we need to provide a fluid-

deformation unf . This can be done by either interpolating the boundary values into thedomain or by solving an partial differential equation. Here, we describe the easy method ofusing a harmonic extension:

Problem 7 (Mapping Problem M). On Ωf with given uns on Γi, find uf , such that

−∆unf = 0, unf = uns on Γi, unf = 0 on ∂Ωf/Γi.

Again, due to different coordinate systems Ωnf and Ωs the boundary values need to be

interpolated.

5.1 Coupling of the subproblems

If these three problems where all solved simultaneously, vnf , pnf , uns , vs and unf would solvethe coupled fluid-structure interaction problem as given in the monolithic formulation.

Every time-step of a partitioned solution approach will consist of the following three sub-steps. These can be combined in various order or solved parallel.

F Solve the flow problem

vn−1f , pn−1

f , vn−1s , vns , T

n−1f , Tnf ⇒ vnf , p

nf

S Solve the structure problem

un−1s , vn−1

s , σn−1f , σnf ⇒ uns , v

ns

M Solve the mapping problemun−1f , uns ⇒ unf

In every subproblem some input-values are not known exactly but part of the problem to besolved. We will distinguish between two different kinds of coupling algorithms: staggered orweakly coupled on the one and iterative staggered or strongly coupled on the other side.

Weakly coupled approaches solve every subproblem only once per time-step. These ap-proaches are generic explicit approaches. The interface condition will not be fulfilled exactly.In strongly coupled approaches, the three subproblems are solved in an iterative manner.This way, the unknown data (like the interface velocity uns in the fluid problem) will beapproximated in an outer iteration.

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5.2 Weakly coupled approaches

5.2 Weakly coupled approaches

Here, we describe so called weakly coupled or staggered solution approaches. We supposethe general case, where a different reference domain Ωn

f := Ωn−1f is used in every time-step

tn−1 → tn. The following sub-tasks need to be solved:

1. Create the new fluid-domain and mesh for tn−1 → tn:

Ωn−1f

un−1f−−−→ Ωn

f , Ωn−1f,h

un−1f−−−→ Ωn

f,h.

2. Transfer the fluid variables onto the new mesh

vn−1f

∣∣Ωn−1f

un−1f−−−→ vn−1

f

∣∣Ωnf, pn−1

f

∣∣Ωn−1f

un−1f−−−→ pn−1

f

∣∣Ωnf

3. Transfer the structure deformation and velocity on the interface to the new fluid-coordinates:

un−1s

∣∣∣Γi

un−1f−−−→ un−1

s

∣∣∣Γi, vn−1

s

∣∣∣Γi

un−1f−−−→ vn−1

s

∣∣∣Γi.

4. Solve the mapping problem to match the last structure-velocity

−∆unf = 0, unf = un−1s on Γi.

5. Solve the fluid problem

vn−1f , pn−1

f

unf−→ vnf , pnf .

6. Transfer the interface-load to the structure domain

σnfunf−→ σnf

7. Solve the structure problem

un−1s , vn−1

s

σnf−→ uns , vns .

If the standard ALE approach with just one reference domain Ωnf = Ωf is used for all

time-steps n, the first two steps of the algorithm do not need to be considered, since thecomputational mesh stays the same. Along the interface however, the structure’s velocityand deformation needs to be transferred since possibly non-matching meshes are used.

In the following, we describe the different solution-steps of this basic partitioned algorithmin detail:

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Step 1: mesh-movement We discuss the case, where the reference domain Ωnf for the

time-step tn−1 → tn is given by the deformed domain of the previous time-step, e.g. by thetransformation

Ωnf := Tn−1

[tn−2,tn−1](Ωn−1f ).

There are basically two possibilities for generating the new mesh Ωnf,h. First, based on the

boundary Ωnf a new mesh can be generated with a mesh-generator. The advantage of this

method is that in every step, a good mesh quality fulfilling shape regularity can be assured.As a drawback, by generating a new mesh there would be no topological connection betweenthe old and new meshes. This would complicate the transfer of the solution variables fromΩn−1f,h to Ωn

f,h in Step 2 of the algorithm.

Alternatively, the domain deformation un−1f can be used to move the mesh nodes to the new

mesh:Ωn−1f,h 3 x

n−1i ⇒ xn−1

i + un−1f (xn−1

i ) ∈ Ωnf,h.

This way, the new mesh is generated by moving the nodes (and elements) of the old one.As indicated in Figure ??, moving the mesh nodes can lead to strongly distorted elements.Thus, in a second step, the new mesh needs to be locally remeshed by adding or removingelements to produce a valid, shape-regular mesh. In the following, we assume, that themeshes are modified by this second approach.

Step 2: solution transfer Velocity and pressure need to be transferred to the new meshΩn−1f,h → Ωn

f,h. If the mesh is generated by moving with un−1f , one could simply interpolate

viavn−1f

∣∣Ωnf,h

(xni ) := vn−1f

∣∣Ωn−1f,h

(xn−1i ), where xni = xn−1

i + un−1f (xn−1

i ).

This however is not conforming in a Galerkin-sense. Instead we need to L2-project thesolution onto the new mesh. Let vn−1

f,h be the discrete solution from the last time-step inALE-coordinates and vn−1

f,h the old solution on the new mesh Ωnf,h to be given in Eulerian

coordinates. Then with the basis-functions φi on the new mesh it holds by transformationwith T[tn−2,tn−1]:

(vn−1, φi)Ωnf

= (J vn−1f,h , φi)Ωn−1

f,

where φi are the corresponding test-function on Ωn−1f,h given in ALE-coordinates. To properly

project the old solution onto the new mesh, we first need to evaluate the right hand side onthe old mesh, followed by solving the mass-matrix on the new mesh:

i = 1, . . . , N : ri := (J vn−1f,h , φi)Ωn−1

f, (vn−1

f,h , φi) = ri ∀i = 1, . . . , N.

This L2-projection vn−1f,h is to be used as vn−1

f,h

∣∣Ωnf,h

on the new mesh as initial value.

If the new mesh would be generated by a complete remeshing operation, there would be nolink between the old and new test-functions. In this case, the evaluation of the right handside ri is very expensive.

70

5.3 Strongly coupled partitioned approaches

Step 4: solution transfer In this step, using the Laplace equation is only one alterna-tive. For problems with large deformation, an elastic structure equation or the bi-harmonicequation yields better results.

For partitioned approaches, it is even possible to use a simple interpolation for constructingthe deformation of the flow-domain. In contrast to monolithic models, this mesh-deformationequation is solved separately and does not introduce couplings in the flow-systems systemmatrix.

Step 5, Step 6: the fluid and structure problem Here, the two subproblems need to besolved. This is done separately and standard solvers can be used.

In weakly coupled solution approaches, the interface conditions vs = vf and n · (σf −σs) = 0are not fulfilled exactly. These approaches are only successful if the coupling between thetwo problems is very weak.

5.3 Strongly coupled partitioned approaches

Strongly coupled partitioned approaches try to satisfy the interface conditions by iteratingin every time-step until the solution does not change any more. For simplicity, we introducesome notation for the different sub-steps in the algorithm:

Mesh Solve for fluid-deformation(uf,(i)) = M(uΓ,(i)),

Fluid Solve the flow problem

(vf,(i), pf,(i)) = F (vn−1f , uf,(i), vΓ,(i))

Solid Solve the solid problem

(vs,(i), us,(i)) = S(vn−1s , un−1

s , σΓ,(i))

Fluid-Transfer Transfer the solid’s velocity to the flow domain

(vΓ,(i)) = TF (vs,(i−1))

Solid-Transfer Transfer the fluid-forces to the solid

(σΓ,(i)) = TS(vf,(i), pf,(i))

Interface-Restrict Restrict solid’s deformation to the interface

(uΓ,(i)) = TR(us,(i))

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Then, the basic iterative algorithm for every time-step is as follows: Get an initial guess ofthe interface displacement uΓ,(0) and iterate for i ≥ 1

1. Solve mesh equation (uf,(t)) = M(uΓ,(i−1))

2. Transfer interface-velocity (vΓ,(i)) = TF (vs,(i−1))

3. Solve flow problem (vf,(i), pf,(i)) = F (uf,(i), vΓ,(i))

4. Transfer interface loads (σΓ,(i)) = TS(vf,(i), pf,(i))

5. Solve structure problem (vs,(i), us,(i)) = S(σΓ,(i))

6. Restrict new interface deformation uΓ,(i) = TR(us,(i))

7. Check residual |uΓ,(i) − uΓ,(i−1)| < tol

The fix-point iteration can be written in a compact form:

uΓ,(i) = TR S TS F TF M(uΓ,(i−1)).

And if we combine the mesh problem M and the flow problem F together with the fluid-transfer TF into one combined problem, denoted by F , and if we include TS , TR and thestructure problem S into S, the fix-point iteration gets:

uΓ,(i) = S F (uΓ,(i−1)). (5.3)

For convergence control we introduce the interface residual as

rΓ,(i) = uΓ,(i) − uΓ,(i−1), (5.4)

which is used in Step 7 of the algorithm as stopping criteria.

The Dirichlet-Neumann setup of the iteration assumes, that the coupling from the solid tothe fluid problem is stronger than the other way around. If the structure has a nearly rigidwall – like the surface of an airplane – the fluid’s forces have little influence on the structure’sdisplacement, whereas a deformation of the structure immedeatly alters the flow-problem.Then, the proposed iteration seems reasonable and it can be assumed that the predictedstructure displacement will be accurate. If on the other hand the structure is very flexibleand the fluid’s density is large compared to the structure’s density, the influence of the flowfield can be significant. Then, extensive iteration is necessary, and for some problems thescheme will not converge at all.

5.4 Acceleration schemes for partitioned fsi-solvers

Written as (5.3), the partitioned fluid-structure interaction scheme is given as a fix-pointiteration on the position of the interface. For acceleration of this iteration we use relaxationmethods by first estimating a prediction of the new displacement and then adding only ashare of it

uΓ,(i) = S F (uΓ,(i−1)), rΓ,(i) := uΓ,(i) − uΓ,(i−1), uΓ,(i) := uΓ,(i−1) + ωirΓ,(i),

72

5.4 Acceleration schemes for partitioned fsi-solvers

(uΓ,(i−1), rΓ,(i))

(uΓ,(i), rΓ,(i+1))

(uΓ,(i+1), 0)

Figure 5.2: Secant method for convergence acceleration with Aitken relaxation.

with a variable relaxation parameter ωi ∈ R. The simple choice of one fixed relaxationparameter ωi := ω for all steps leads to a robust convergent scheme, if ω 1 is smallenough. The convergence rate however will be very low. In the following we describedifferent methods for choosing a dynamic relaxation parameter.

5.4.1 Aitken relaxation

Aitken relaxation is best explained by assuming, that uΓ,(i) is a scalar value. Using thelast to iterates, the new deformation is approximated. Let uΓ,(i−1), uΓ,(i) = S F (uΓ,(i−1))and uΓ,(i), uΓ,(i+1) = S F (uΓ,(i)) be given. Then, the residual rΓ,(i) = uΓ,(i) − uΓ,(i−1) andrΓ,(i+1) = uΓ,(i+1) − uΓ,(i) are known.

Then, we can figure rΓ,(i) as function of uΓ,(i−1) and using the last given values we try tofind uΓ,(i+1) such that rΓ,(i+2)(uΓ,(i+1)) = 0. For finding this root we use secants method,see Figure 5.2, describing uΓ,(i+1) as:

0 = rΓ,(i+1) +rΓ,(i+1) − rΓ,(i)uΓ,(i) − uΓ,(i−1)

(uΓ,(i+1) − uΓ,(i))

⇔ uΓ,(i+1) = uΓ,(i) − rΓ,(i+1)uΓ,(i) − uΓ,(i−1)rΓ,(i+1) − rΓ,(i)

= uΓ,(i) +uΓ,(i−1) − uΓ,(i)

uΓ,(i−1) − uΓ,(i) + uΓ,(i+1) − uΓ,(i)︸︷︷︸ωi+1

(uΓ,(i+1) − uΓ,(i))︸︷︷︸rΓ,(i+1)

.

We can derive a recursion formula for the relaxation parameter ωi+1:

ωi+1 =uΓ,(i−1) − uΓ,(i)

uΓ,(i−1) − uΓ,(i) + uΓ,(i+1) − uΓ,(i)

=uΓ,(i−1) − (uΓ,(i−1) + ωirΓ,(i))

rΓ,(i+1) − rΓ,(i)

= −ωirΓ,(i)

rΓ,(i+1) − rΓ,(i).

73


This result cannot directly be transferred to the vector-valued case, since dividing by rΓ,(i+1)−rΓ,(i) is not possible. Hence, to transform the vector entities into scalars, we project all valuesin the rΓ,(i+1) − rΓ,(i)-direction to get:

ωi+1 = −ωi(rΓ,(i), rΓ,(i+1) − rΓ,(i))‖rΓ,(i+1) − rΓ,(i)‖2

.

Aitken’s method works very well for a large range of fluid structure interaction problems. Itis very cheap in a sense, that no additional information than the last two approximationsare necessary. In particular, we do not need derivatives or sensitivities from the solution. Tostart the iteration, ω0 must be suitably chosen, either by using ωni from the last time-stepor by taking a small enough value.

5.4.2 Steepest descent relaxation

We try to find the best relaxation parameter ωi in the given search direction rΓ,(i). Forthis purpose, we introduce a merit function φ which is minimal at the searched interfacedisplacement uΓ. The relaxation parameter is given by

ωi := arg minωi

φ(uΓ,(i−1) + ωirΓ,(i)).

Assuming that φ is differentiable, the condition to determine ωi is given by

0 = dφ

dωi=∂φ(uΓ,(i−1) + ωirΓ,(i))

∂uΓrΓ,(i).

Thus, the condition to specify ωi is:

φ′(uΓ,(i−1) + ωirΓ,(i)) = 0.

We Taylor expand φ around the last iteration uΓ,(i−1) to get

φ(uΓ,(i−1) + ωirΓ,(i)) ≈ φ(uΓ,(i−1)) + ωi(φ′(uΓ,(i−1))

)TrΓ,(i) + ω2

i

2 rTΓ,(i)φ′′(uΓ,(i−1))rΓ,(i).

And for determining ωi we derive the relation

φ′(uΓ,(i−1) + ωirΓ,(i)) ≈φ(uΓ,(i−1) + ωirΓ,(i))− φ(uΓ,(i−1))

ωi

= φ′(uΓ,(i−1))T rΓ,(i) + ωi2 r

TΓ,(i)φ

′′(uΓ,(i−1))rΓ,(i) = 0,

⇔ ωi = −2φ′(uΓ,(i−1))T rΓ,(i)

rTΓ,(i)φ′′(uΓ,(i−1))rΓ,(i)

.

At this point, we need to specify the merit function φ. We assume, that

φ′(uΓ,(i−1)) = rΓ,(i),

74

5.5 Stability analysis and the added mass effect

which stands for the assumption, that the residual rΓ,(i) is the direction of the steepestdescent. Further, since φ is a scalar function, this assumption implies, that the residual canonly be the gradient of a scalar function. This assumption is not fulfilled, however it leadsto a symmetric Jacobian:

JΓ = φ′′(uΓ,(i−1)) =∂rΓ(uΓ,(i−1))

∂uΓ= φ′′(uΓ,(i−1)).

Finally, the best search step ωi is given by

ωi = −2rTΓ,(i)rΓ,(i)

rTΓ,(i)JΓrΓ,(i).

The evaluation of this relation requires the evaluation of the matrix vector product with theJacobian, JΓrΓ,(i). This Jacobian is not available, however the product can be approximated.A simple way to approximate the Jacobian is by finite differences:

JΓrΓ,(i) = d

dsrΓ(dΓ,(i−1) + srΓ,(i))

∣∣∣s=0≈rΓ(dΓ,(i−1) + δrΓ,(i))− rΓ,(i)

δ

To evaluate the residual at dΓ,(i−1) + δrΓ,(i) one step of the fsi-iteration (5.3) needs to beadded:

JΓrΓ,(i) ≈1δ

(S F (uΓ,(i−1) + δrΓ,(i))− uΓ,(i−1) − δrΓ,(i) − rΓ,(i)

).

The computational cost of this relaxation method is one additional cycle of the fsi iteration.Further, due to the finite differences, this method is very sensitive.

Other ways of expressing the Jacobian JΓ are based on approximating the derivative of (5.3):

JΓrΓ,(i) = S′ F (uΓ,(i−1))F ′(uΓ,(i−1)).

The derivative of the structure operator S is available, if a Newton’s method is used forsolving the structure problem. The derivative of the fluid field however is not given, sincewe here need the derivative with respect to the interface derivation uΓ,(i−1). This needs tobe approximated. We do not give details on this methods and refer to the literature.


Here, we discuss stability issues of partitioned approaches for fluid-structure interaction. Ananalysis of the full set of equations including all nonlinearities is very difficult. Hence, westrongly simplify the set of equations by assuming:

• There are no external forces or Dirichlet conditions

• The deformation is small and we neglect all geometric nonlinearities in the ALE map-ping and in the structural displacement

• The convective term in the Navier-Stokes equations is neglected

75


• Viscid effects are neglected

The set of equations is then simplified to

(ρf∂tv, φ)f − (p,∇ · φ)f = 〈ffΓi , φ〉Γi , (∇ · v, ξ)f = 0

for the flow system and

(ρs∂ttu, φ)s + (ε(u), φ)s = 〈−f sΓi , φ〉Γi ,

for the solid part. By ffΓi and fsΓi we denote the interface forces. In a iterative partitioned

scheme, the equilibrium shows ffΓi − fsΓi → 0.

Semi-discrete System We first discretize the system only in time. With the usual notationof Mf for the fluid’s and Ms solid’s mass matrix scaled with the densities, and K for thesolid’s diffusive matrix, B for the gradient operator on the pressure and the divergencematrix BT , we write the system as:

ρfMfv′ + Bp = ffΓi ,

BT v = 0,

andρsMsu′′ + Ku = fsΓi .

By neglecting the ALE-mapping and assuming linearity, all coefficients matrices do notchange in time. Thus, for the divergence we get:

d

dtBT v = BT v′ = 0,

and the flow system is given as [ρfMf B

BT 0

] [v′

p

]=[ffΓi0

].

Next, we split the systems for flow and structure into the interior part and into the interfacepart. Then, with v = (vΩ, vΓ) and u = (uΩ, uΓ) (similar for the matrices) haveρfMf

ΩΩ ρfMfΩΓ BΩ

ρfMfΓΩ ρfMf

ΓΓ BΓBT

Ω BTΓ 0

v′Ωv′Γp

=

0ffΓi0

.

In a final simplification, we use lumped versions of the mass-matrices. MMMfl . We assume, that

the mesh is regular and that a linear discretization is used. Then, we can write the lumpedmass matrix as

MMMfl = |K|If ,

where If is the identity and |K| the volume of the support of a fluid node.

76


Then, the off-diagonal entries MfΓΩ and Mf

ΩΓ are zero and the overall flow system getsρf |K|IfΩΩ 0 BΩ

0 ρf |K|IfΓΓ BΓBT

Ω BTΓ 0

v′Ωv′Γp

=

0ffΓi0

. (5.5)

In a partitioned scheme, the acceleration on the interface v′Γ is given by the structuralprediction. Assuming that v′Γ is known, we can express the remaining fluid variables as

p = ρf |K|(BTΩBΩ)−1BT

Γv′Γ

v′Ω = −ρ−1f |K|

−1BΩp = −BΩ(BTΩBΩ)−1BT

Γv′Γ.

Next, we can use the second line of system (5.5) to express resulting boundary force as

ffΓi = ρf |K|IfΓΓv′Γ + BΓp = ρf |K|

IfΓΓ + BΓ(BT

ΩBΩ)−1BTΓ

︸︷︷︸

=:MA

v′Γ

We callMA the added mass operator . It maps the interface acceleration onto a force-vectoron the interface. MA is a dimensionless and symmetric operator. If the flow system has aunique solution, the matrix BT

ΩBΩ is positive. Thus, the eigenvalues ofMA are larger thanone.

The added mass operator describes, how the prediction of the interface acceleration v′Γ relatesto the new interface forces for the structure problem:

fsΓi = ρf |K|MAv′Γ. (5.6)

Hence, we can write the next structure-step with the estimated interface force as:

ρs

[Ms

ΩΩ MsΩΓ

MsΓΩ Ms

ΓΓ

] [u′′Ωu′′Γ

]+[Ks

ΩΩ KsΩΓ

KsΓΩ Ks

ΓΓ

] [uΩuΓ

]=[

0−ρf |K|MAv

′Γ

]. (5.7)

On the interface we can identify the fluid’s acceleration v′Γ with the second derivative of thesolid’s displacement u′′Γ. This points up the relation

ρsMsΓΓu

′′Γ ∼ −ρf |K|MAv

′Γ,

and by understanding ρf |K| as a unit-mass of the fluid in the proximity of a fluid node, theadded mass operator acts as additional mass on the degrees of freedom on the interface.

Time discretization schemes For a stability analysis of the partitioned scheme, we applytime-discretization of the two sub-problems. For simplicity, we use lowest order, fully implicittime stepping schemes for the step tn → tn+1, ∆t = tn+1 − tn.

77


In the flow problem, the interface acceleration v′Γ is given as a predictor and needs to beestimated using the solid’s deformation. With u′′Γ = v′Γ we approximate using the last knowntime-steps:

v′Γn+1 = u′′Γ

n+1 = 1∆t2 (unΓ − 2un−1

Γ + un−2Γ ).

Then, the right hand side fsΓi of the solid system is given by (5.6) as

fsΓin+1 = 1

∆t2 ρf |K|MA(unΓ − 2un−1Γ + un−2

Γ ).

We further simplify the structure system (5.7) by assuming small time-steps. Then, thestiffness terms can be neglected and we use KKKs = 0. Further, with the lumped mass matrixMMM s = |S|Is the system is reduced onto the interface:

ρs|S|Isu′′Γ = −ρf |K|MAv′Γ.

By discretizing with simple backward second order differences, and by inserting the righthand side approximated above, we get:

ρs|S|1

∆t2 (un+1Γ − 2unΓ + un−1

Γ ) = − 1∆t2 ρf |K|MA(unΓ − 2un−1

Γ + un−2Γ ).

For a closer analysis, we note that the matrix MA is positive and symmetric. We thushave an orthogonal system of eigenvectors vi ofMA with corresponding eigenvalues µi. Thedeformation on the interface unΓ can be developed in this eigenvector system by unΓ =

∑i u

ni vi.

For the coefficients it must hold

(un+1i − 2uni + un−1

i ) + ρf |K|ρs|S|

µi(uni − 2un−1i + un−2

i ) = 0. (5.8)

With the ansatz un+1i = λiu

ni the amplification factors λi need to be roots of the characteristic

polynomial:(λ3i − 2λ2

i + λi) + ρf |K|ρs|S|

µi(λ2i − 2λi + 1) = 0,

which has a double root at λ1 = 1 and by

(λi − 1)2(λi + ρf |K|

ρs|S|µi)

= 0,

we find the third root atλ3 = −ρf |K|

ρs|S|µi.

For a scheme to be stable, these amplification factors all need to be smaller than one. Thus,for

ρf |K|ρs|S|

µi > 1,

the partitioned scheme is unstable, since the added mass effect on the interface is dominant.Since µi ≥ 1, note that stability of the scheme is governed by the ratio fluid-mass to solid-mass.

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Here, the great success of partitioned scheme in aerodynamics is obvious, where the densityof the air ρf ≈ 1kgm−3 is small in proportion to the density of e.g. aluminum of ρs ≈3000kgm−3. This ratio allows for a stable solution with even very simple iterative schemes.For problems like hemodynamics, where ρf ∼ ρs ∼ 103kgm−3, partitioned schemes fail toconverge. Here, monolithic approaches are more efficient.

lat

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80

6 Alternative Approaches for FSI-problems

Partitioned approaches are successful, if the coupling between fluid and structure is weak.When combined with remeshing, partitioned approaches are very efficient and can handleproblems with large deformation. If the coupling between the subproblems is strong, parti-tioned approaches fail to converge.

The monolithic ALE formulation on the other hand is very robust and also suitable forproblems, where partitioned approaches break down. They however have the problem ofmodeling large deformation since then, the ALE-mapping Tf : Ωf → Ωf (t) will loose itsregularity. In some applications, even the contact of the elastic structure with the boundaryof the domain or with another structural part can be given. Then, the topology of the domainΩf (t) changes, see Figure 6.1. Here, it is not possible to indicate an invertible ALE-mapping.

In this chapter we discuss alternative approaches for fsi-problems. First, we consider theFully Eulerian formulation, where both flow-problem and structure-problem are given inEulerian formulation.

6.1 Fully Eulerian Coordinates

For the ALE-formulation we need a mapping Tf : Ωf → Ωf (t) of the flow reference domainto the actual flow domain Ωf (t). This mapping is used to transform the Navier-Stokesequations onto the fixed reference domain. For the ALE formulation to work, the mappingneeds to be invertible. If this property is lost, the ALE-formulation breaks down.

Here we try to use the opposite approach: the flow problem is left in the natural Euleriancoordinates and the structure problem is transformed onto the moving domain Ωs(t). Here,this transformation is given by the structure’s deformation itself:

Ts : Ωs → Ωs(t), Ts(x, t) = x+ us(x, t).

Ωf

Ωs

Ωf (t)

Ωs(t)

Figure 6.1: Fluid-structure interaction with contact of structure with another structure do-main leading to a topology change of the domain.

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The difference between the Fully Eulerian and the ALE approach is best demonstratedwith Figure 6.1: for ALE, the computational domain is the left sketch in the figure, forthe Eulerian approach, the computation is done on the domain on the right half. In bothformulations, one of the two systems needs to be mapped. In ALE, this mapping Tf isarbitrary in a sense, that it has no physical relevance. This is the reason, why this mappingcan deteriorate, det(∇Tf ) → ∞. In the Fully Eulerian approach, the mapping is given bythe structural displacement itself and it holds:

det(J) := det(∇Ts) = det(I + ∇us).

For incompressible materials it holds det(J) = 1 and for compressible materials, the limitsdet(J) → 0 or det(J) → ∞ are prohibited due to physical reasons. They would imply thateither different material points are at the same location or that holes would appear.

In Eulerian coordinates we define on Ωs(t)

u(x, t) = u(x, t), v(x, t) = v(x, t), x = x+ u(x, t) ⇔ x = x− u(x, t).

With this definition, the inverse mapping Ts is given by

Ts : Ωs(t)→ Ωs, Ts(x, t) = x− u(x, t).

For simplicity, we write:

F := ∇Ts = (I −∇u), J := det(F ) = det(I −∇u).

While velocity and deformation are easily available in Eulerian coordinates, their gradient∇u and ∇v is not. By chain rule, we get:

x = Ts(Ts(x, t), t) ⇒ I = (I −∇u)(I + ∇u).

And thus∇u = (I −∇u)−1 − I. (6.1)

For the deformation gradient it holds

F = F−1, J = J−1 (6.2)

For the temporal derivative we have:

dtv(x, t) = ∂tv(x, t) = ∂tv(x, t) + v · ∇v. (6.3)

Then, the equations for an elastic material in Lagrangian formulation (see (2.14)) are givenby:

(ρ0∂tv, φ)Ωs + (J σF−T , ∇φ)Ωs = (ρ0f , φ)Ωs

(dtu, ψ)Ωs = (v, ψ)Ωs

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We remember, see (2.10) that for the density we have:

ρ(x, t) = J−1ρ0(x) ⇔ ρ0(x) = J−1ρ(x, t).

Transformation to Ωs(t) and using (6.1), (6.2) and (6.3) yields

(Jρ0(∂tv + v · ∇v), φ)Ωs(t) + (σs,∇φ)Ωs(t) = (Jρ0f, φ)Ωs(t)

(J(∂tu+ v · ∇u), ψ)Ωs(t) = (Jv, ψ)Ωs(t).

For St. Venant-Kirchhoff materials, the stress-tensor is

σs = J−1F (2µE + λtr(E)I)F T , E = 12(F T F − I),

In Eulerian coordinates, this transforms to

σs = JF−1(2µE + λtr(E)I)F−T , E = 12(F−TF−1 − I).

For fluid-structure interaction, the interface condition is again continuity of the velocity andthe stress condition:

n · σf = n · σs.

In Fully Eulerian coordinates, we do not need a transformation of the flow domain, thus thefluid’s deformation uf is not required. In this sense, the Eulerian formulation appears tobe easier than the ALE formulation. The big drawback of the Eulerian approach howeveris that the domain partitioned into Ωf (t) and Ωs(t) is not known a priori but is itself anunknown in the system. By writing the full set of fsi-equations the gap gets obvious

(ρf (∂tv + v · ∇v), φ)Ωf (t) + (σf ,∇φ)Ωf (t) = (ρff, φ)Ωf (t)

(div v, ξ)Ωf (t) = 0n · σs = n · σf , vf = vs on Γi(t)

(Jρ0s(∂tv + v · ∇v), φ)Ωs(t) + (σs,∇φ)Ωs(t) = (Jρ0

sf, φ)Ωs(t)

(J(∂tu+ v · ∇u), ψ)Ωs(t) = (Jv, ψ)Ωs(t).

As in ALE, on different parts of the domain Ω different equations are valid. Opposed to theALE formulation, this partitioning is however not known. A certain point x ∈ Ω can eitherbe in the fluid domain x ∈ Ωf (t) or in the solid domain x ∈ Ωs(t). For the assembly of thesystem it is of course required to know which is the domain of influence. This informationhowever is given by the solution itself: if x− u(x, t) ∈ Ωs then x ∈ Ωs(t).

To obtain a simular relation in the flow problem, we need to find a displacement uf of thefluid-domain. If we denote by u the displacement variable on all Ω, with u = us on Ωs(t)and u = uf on Ωf (t), we can decide:

x ∈ Ω ⇒x ∈ Ωs(t) x− u ∈ Ωs

x ∈ Ωf (t) x− u 6∈ Ωs.

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Figure 6.2: Non-matching mapping of the flow domain with artificial deformation uf .

We have chosen to check x − u 6∈ Ωs instead of x − u ∈ Ωf on purpose. This way, we havemore freedom with defining the extension of u to the flow-domain. x− u|Ωf (t) does not needto assemble a mapping Ωf (t)→ Ωf . See Figure 6.2 for an example of such a non-matchingdeformation. Is is only important that x− u is not mapping into Ωs for x ∈ Ωf (t).

For the extension of u to the fluid domain Ωf (t) the only important condition is, that afluid-particle x ∈ Ωf (t) will be transported within the fluid domain x u−→ x′ ∈ Ωf (t′). Here,the obvious choice is to use the deformation onto Lagrangian coordinates and to extend uby:

(J(∂tu+ v · ∇u), ψ)Ωf (t) = (Jv, ψ)Ωf (t).

This extension was discussed and rejected in the context of ALE coordinates: first, fluidparticles can be transported out of the computational domain Ω, whenever there are outflowor inflow conditions. Second, the deformation u can awkward, e.g. if a rotational flow is pre-scribed. Here however, this deformation is not used to define a transformation, derivatives,or the inverse is not required. u is only needed for lookup of the domain of influence. Wecan write the implicit system for the Fully Eulerian fsi-problem:

Problem 8. Fluid Structure Interaction in Fully Eulerian Coordinates Find v ∈ vD +H1

0 (Ω; ΓD)d, p ∈ L20(Ω) and u ∈ H1

0 (Ω; ΓD)d such that

(ρf (∂tv + v · ∇v), φ)Ωf (t) + (Jρ0s(∂tv + v · ∇v), φ)Ωs(t)

+(σf ,∇φ)Ωf (t) + (σs,∇φ)Ωs(t) = (ρff, φ)Ωf (t) + (Jρ0sf, φ)Ωs(t),

(∇ · v, ξ)Ωf (t) + (∇p,∇ξ)Ωs(t) − 〈ns · ∇p, ξ〉Γi(t) = 0,(∂tu+ v · ∇u, ψ)Ω = (v, ψ)Ω,

where ρf is the fluid’s density and ρ0s(x) is the solid’s density in reference state and with the

stress-tensors as mentioned above. For a point x ∈ Ω we decide:

x ∈ Ω ⇒x ∈ Ωs(t) ⇔ x− u ∈ Ωs

x ∈ Ωf (t) ⇔ x− u 6∈ Ωs

This formulation goes without arbitrary transformation. In ALE coordinates, this trans-formation was the reason for most numerical problems. In particular the flow problems isalways well defined in Fully Eulerian coordinates.

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The big disadvantage of the Eulerian scheme is the implicit dependence on the domain. Evenif the material laws would be considered to be linear, the overall problem is nonlinear due tothis dependence. For solving the weak formulation with a Newton’s method, the Jacobianhas to be computed. Derivatives with respect to the deformation include derivatives withrespect to the domain-partitioning. We discuss a simplified fluid tensor given by σf = ∇v.Then, the viscous term gets:

(σf ,∇φ)Ωf (t) = (∇v,∇φ)Ωf (t).

For evaluating the Jacobian, we need to consider derivatives in direction of v and u. Thev-derivative is straightforward:

d

ds(∇(v + sη),∇φ)Ωf (t)

∣∣s=0 = (∇η,∇φ)Ωf (t),

whereas for the u-derivative the deformation dependence is within the integral boundaries.We write Ωf (t) = Ωf (u) to get:

d

ds(∇v,∇φ)Ωf (u+sη)

∣∣∣s=0

= d

ds

∫Ωf (u+sη)

∇v : ∇φdx∣∣∣s=0

.

These derivatives appear in shape-optimization problems and are called shape gradients. Fortheir evaluation we introduce the characteristic function:

χ(x) =

1 x ∈ Ωf

0 x 6∈ Ωf

,

and derive the characteristic function of the moving Eulerian domain by:

χ(x, u) = χ(x− u).

Then, we can write ∫Ωf (u)

f(x)dx =∫

Ωχ(x− u)f(x)dx,

to evaluate the derivative as:

d

ds

∫Ωf (u+sη)

f(x)dx∣∣∣s=0

= d

ds

∫Ωχ(x, u+ sη)f(x)dx

∣∣∣s=0

.

The characteristic function χ is not differentiable. By Hadamard structure theorem we canwrite this derivative as:

d

ds

∫Ωχ(x, u+sη)f(x)dx

∣∣∣s=0

= limε→0

1ε

( ∫Ωχf (x, u+εη)f(x)dx−

∫Ωχf (x, u)f(x)dx

)=∫

Ωi(u)nf ·ηf(x)−dx,

where f(x)− is the value of f(x) on the interface Γi(u) as seen from the inside of Ωf (t).

Some terms in the Jacobian of the system include integrals over the interface of the domain.This interface is moving and crossing mesh elements, see Figure 6.3.

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Figure 6.3: Left: Mesh partitioning in the Fully Eulerian formulation. The interface Γi(t)crosses the mesh elements. Right: implicit transformation of the solid domain.This transformation is well-defined.

A careful integration of these terms is essential. Since the interface crosses through elementsK of the mesh, in different parts of the elements, different equations are valid. Usual Gaussrules are not appropriate for evaluating the integrals at the interface, since the bi-linear formto be integrated is not differentiable across the interface. Instead special integration ruledneed to be applied. In the vicinity of the interface, we split the quads into triangles andperform the integration separately on every triangle, which is now either part of the fluid orof the structure domain. For further increasing the accuracy, we allow these triangles to becurved by introducing additional degrees of freedom on the interface. See Figure 6.4 for asketch.

In theory, the Fully Eulerian approach is able to handle very large deformation, even contactof the structure, and this in a fully implicit monolithic formulation. The dependence of thecomputational domain partitioning on the solution itself however leads to big problems whensolving the algebraic systems.

Ks

Ks

Ks

Ks

KfKfKfKf

Figure 6.4: Integration close to the interface by using specially adapted integration rules.Marked points: degrees of freedom eliminated from system.

6.2 A Fixed Mesh Euler-Lagrange Approach

Finally, we present a (possibly monolithic) approach, where both subsystems fluid and struc-ture are left in their natural coordinate systems. The idea is to formulate the fluid problemon moving domain in an Eulerian mesh and the structure problem on a fixed Lagrangianmesh. The coupling on the interface is realized with the help of Lagrange-multipliers. This

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Ωf (t) Γi Ωs

Figure 6.5: Three subdomains for fluid Ωf , interface Γi and the structure Ωs.

approach has recently been described by Gerstenberger and Wall ([GeWa08a], [GeWa08b],[GeWa10]).

We split the problem into three sub-problems. On the (moving) fluid domain Ωf (t) weformulate the Navier-Stokes equations:

∂tvt + v · ∇v −∇ · σf = f∇ · v = 0

in Ωf (t)

v = vΓi on Γi(t),(6.4)

where vΓi is the velocity of the interface.

The solid equation is first formulated on the moving Eulerian domain Ωs(t). Here, we findthe deformation us as:

ρsdttus −∇ · σs = f in Ωs(t)n · σs = n · σf on Γi(t).

(6.5)

Later on, this equation will be solved in Lagrangian coordinates on a fixed mesh. Finally,on the interface Γi(t) we have an equation for the velocity of the interface vΓi :

vΓi = dtus on Γi(t). (6.6)

By treating Equations (6.4), (6.5) and (6.6) together, the coupled fsi-problem is solved.The interface condition is realized as a Dirichlet-condition for the fluid-field, a Neumann-condition for the interface stresses in the solid equation and lower-dimensional equation forthe interface velocity.

For discretization, the flow problem is discretized in Eulerian formulation on a fixed meshΩh with Ωf (t) ⊂ Ωh. This means, that the interface is moving and is not resolved by themesh elements. The structure equation will be transformed to Lagrangian coordinates andthen solved on a fixed Lagrangian mesh Ωs,h. The equation on the interface is discretizedon an own mesh Γi,h. Altogether, three different meshes and discretizations are used. Thecoupling between the different fields is realized by Lagrange-multipliers in an implicit way.

6.2.1 The Flow Problem

It is necessary to solve the Navier-Stokes equations on a moving domain Ωf (t). Therefore,the fluid domain is embedded in a larger surrounding, e.g. Ωf (t) ⊂ Ω, the complete com-putational domain. This domain is triangulated with Ωh. The Navier-Stokes equations are

87


Figure 6.6: Navier-Stokes equations on a moving domain. Shaded part: ficticious fluid do-main, where solid equations are given. Crossed degrees of freedom: interfacenodes with special test-functions. Boxed degrees of freedom: removed fromsystem.

only valid on those degrees of freedom inside Ωf (t). See Figure 6.6. The remainder Ωf (t)/Ωis called the ficticious flow domain. While the next degrees of freedom behind the interfaceare included in the computation, all the boxed degrees of freedom (in the figure) are removedfrom the computation and velocity and pressure are extended with zero.

On the interface, the velocity is required to fulfill the condition

v = vΓi on Γi(t),

and in the ficticious domain, we extend v = 0. To allow for this discontinuity in a finiteelement simulation we add additional degrees of freedom in the interface nodes (the onescrossed in Figure 6.6):

vh =Nf∑i=1

vvviφfi +

NΓi∑i=1

vvvΓii φ

Γii ,

where the special test-functions φΓii are discontinuous across the interface and given by

φΓii

∣∣∣K3xi

=

1 in K ∩ Ωf (t)0 in K ∩ Ω/Ωf (t).

This concept, of adding specially adapted basis functions to the computation is called XFEMfor eXXXtended FFFinite EEElement MMMethod.

The velocity condition on the interface is realized by means of an Lagrange multiplier λ. Weassume that the interface velocity vn+1

Γi is known at time tn+1. Then, discretized in timewith the one-step θ-scheme, the new flow field is given by

(ρfv, φ)Ωf + ∆tθ

(ρfv · ∇v, φ)Ωf + (σf ,∇φ)Ωf − 〈λ, φ〉Γi

= (ρfvn, φ)Ωf + ∆t(θ − 1)An

(∇ · v, ξ) = 0〈v, χλ〉Γi = 〈vn+1

Γi , χλ〉Γi .

where for simplicity by An we describe all the explicit parts evaluated at the old time-step tn.The Lagrange multiplier λ and the corresponding test-function χλ are living on the interfaceΓi only and will be discretized on a lower-dimensional mesh. To understand the meaning of

88


the Lagrange multiplier, we need to integrate the term including the stress-tensor by partsto get:

· · · − (divσf , φ)Ωf + 〈n · σf , φ〉Γi = 〈λ, φ〉Γi .

The value of λ is exactly the interface-stress.

The introduction of the Lagrange multiplier is best understood with an easy model problem:assume, homogenous Dirichlet conditions for the Laplace equation are to be included bymeans of a Lagrange multiplier. Then, we aim at minimizing:

J(u) = 12‖∇u‖

2 where u = 0 on Γ.

The Lagrange functional is given by

L(u, λ) = J(u) + (λ, u),

and the optimality condition says:

0 = ∇L ⇒

(∇u,∇φu) + (λ, φu) = 0,(u, φλ) = 0.

6.2.2 The Interface Problem

The interface velocity is given by the simple equation

vΓi = dtuΓi ,

as the temporal derivative of the interfaces deformation. We discretize by the one-stepθ-method to get

un+1Γi − u

nΓi

∆t = θvn+1Γi + (1− θ)vnΓi .

6.2.3 The Structure Problem

The structure problem is likewise coupled to the interface with an additional Lagrangemultiplier µ:

(ρsdttus, ψ)Ωs + (σs,∇ψ)− 〈µ, ψ〉Γi = 0,〈us, χµ〉Γi = 〈uΓi , χ

µ〉Γi .

As mentioned before, the value of the Lagrange multiplier µ is the interface stress and thebalance of forces condition for the coupled problems states

µ = λ.

This system can be discretized in time by splitting into a first-order system and applyingthe θ-scheme or by Newmark’s method.

89


6.2.4 The coupled problem

Combining the three subproblems, the system can be solved in a monolithic approach by aNewton’s method. In every Newton step a linear system of the following type is to be solved:

Fvv Fvp Xvλ 0 0 0Fpv 0 0 0 0 0Xλv 0 0 Xλui 0 0

0 0 Xuiλ 0 Xuiµ 00 0 0 Xµui 0 Xµus

0 0 0 0 Xusµ Susus

vpλuΓiµus

=

fvfpfλfuΓifµfus

Here, F describes the Navier-Stokes system with couplings between velocity and pressureunknown, S the structure system. The different X-terms describe the couplings to and fromthe Lagrange multipliers on the interface. The equation using the χui test-function of theinterface displacement is used to assure the balance of forces by requiring that the Lagrangemultipliers coincide.

This system can either be solved in a monolithic way or partitioned by splitting into aDirichlet-Neumann system. The solution of the coupled system is difficult since it is asaddle-point system in terms of the flow system (pressure) and in terms of the interfacecouplings λ and µ. The resulting method however is able to handle very large deformation,and has the benefit of formulating the subproblems in their natural coordinates without theneed of transformation.

For the numerical solution the construction of appropriate spaces for the Lagrange multipliersis a difficult task. As with the inf-sup condition for the Navier-Stokes equations certainconstraints must be fulfilled.

90

Index

added mass operator, 77ALE, 35ALE mapping, 35angular momentum, 21Arbitrary Lagrangian Eulerian, 35

backward Euler, 46

conservative variables, 18

deformation, 13deformation gradient, 14do nothing boundary condition, 34, 49dyadic product, 21

elasticity law, 29Eulerian framework, 14

finite element methodconforming, 44nonconforming, 44

forward Euler, 46fractional-step-theta scheme, 52Fully Eulerian, 81

Green strain tensor, 15

homogeneous fluid, 24hydrostatic pressure, 22

incompressible flow, 23isoparametric, 44isothermal fluid, 23

Lagrangian framework, 13Lagrangian strain tensor, 15Levi-Civita-Tensor, 21Local Projection Method, 48LPS, 48

material body, 13

material particles, 13

Navier-Stokes equationsincompressible, 24

Newtonian fluid, 23

outflow boundary condition, 49

partitionediterative staggered, 68staggered, 68strongly coupled, 68weakly coupled, 68

permutation tensor, 21poisson’s ratio, 30primitive variables, 18PSPG, 50

rate-of-strain tensor, 18right Cauchy Green tensor, 15

shear stress tensor, 22St. Venant Kirchoff, 31stabilization parameter, 48Stokes Flow, 22strain tensor, 15

linear, 15streamline diffusion stabilization, 48surface forces, 20system matrix, 45

Taylor-Hood Element, 50torque, 21trajectory, 13

velocity, 13viscosity

dynamic, 24kinematic, 24

volume forces, 20

91

Index

XFEM, 88

92

fsi

Documents

structure dynamics292

structure dynamics252

structure problem896

fluid dynamics

finite elements

conservation equations

elastic structure equations554

finite element discretization