mesh moving techniques for fluid-structure interactions with large displacements.pdf

7/27/2019 Mesh Moving Techniques for Fluid-Structure Interactions With Large Displacements.pdf

1/12

Fluid-structure interactions using different mesh motion techniques

Thomas Wick

Institute of Applied Mathematics, University of Heidelberg, INF 293/294 69120 Heidelberg, Germany

a r t i c l e i n f o

Article history:

Received 17 November 2010

Accepted 25 February 2011Available online xxxx

Keywords:

Finite elements

Fluid-structure interaction

Monolithic formulation

Biharmonic equation

a b s t r a c t

In this work, we compare different mesh moving techniques for monolithically-coupled fluid-structure

interactions in arbitrary LagrangianEulerian coordinates. The mesh movement is realized by solving

an additional partial differential equation of harmonic, linear-elastic, or biharmonic type. We examinean implementation of time discretization that is designed with finite differences. Spatial discretization

is based on a Galerkin finite element method. To solve the resulting discrete nonlinear systems, a Newton

method with exact Jacobian matrix is used. Our results show that the biharmonic model produces the

smoothest meshes but has increased computational cost compared to the other two approaches.

2011 Elsevier Ltd. All rights reserved.

1. Introduction

Fluid-structure interactions are of great importance in many

real-life applications, such as industrial processes, aero-elasticity,

and bio-mechanics. More specifically, fluid-structure interactionsare important to measuring the flow around elastic structures, the

flutter analysis of airplanes [1], blood flow in the cardiovascular

system, and the dynamics of heart valves (hemodynamics) [2,3].

Typically, fluid and structure are given in different coordinate

systems making a common solution approach challenging. Fluid

flows are given in Eulerian coordinates whereas the structure is

treated in a Lagrangian framework. We use a monolithic approach

(Fig. 1), where all equations are solved simultaneously. Here, the

interface conditions, the continuity of velocity and the normal

stresses, are automatically achieved. The coupling leads to addi-

tional nonlinear behavior of the overall system.

Using a monolithic formulation is motivated by upcoming

investigations of gradient based optimization methods [4], and

for rigorous goal oriented error estimation and mesh adaptation

[5], where a coupled monolithic variational formulation is an inev-

itable prerequisite.

For fluid-structure interaction based on the arbitrary

LagrangianEulerian framework (ALE), the choice of appropriate

fluid mesh movement is important. In general, an additional

elasticity equation is solved. For moderate deformations, one can

pose an auxiliary Laplace problem that is known as harmonic mesh

motion [6,7]. More advanced equations from linear elasticity are

also available [8,9]. For a partitioned fluid-structure interaction

scheme, a comparison was made between different models [10].

The pseudo-material parameters in both approaches were used

to control the mesh deformation. If the parameters do not depend

on mesh position and geometrical information, both approaches

can only deal with moderate fluid mesh deformations. This prob-

lem is resolved by using mesh-position dependent material param-eters that are used to increase the stiffness of cells near the

interface [8]. There are several techniques for choosing these

parameters to retain an optimal mesh, such as a Jacobian-based

stiffening power [11] that is eventually governed by appropriate

re-meshing techniques. We use an ad hoc approach for these

parameters, measuring the distance to the elastic structure and

adapting the parameters to prevent mesh cell distortion as long

as possible.

Here, we also use (for mesh moving) the biharmonic equation

that others have studied for fluid flows in ALE coordinates [12]. It

was also shown there, that using the biharmonic model provides

greater freedom in the choice of boundary and interface conditions.

In general, the biharmonic mesh motion model leads to a smoother

mesh (and larger deformations of the structure) compared to the

mesh motion models based on second order partial differential

equations. Larger deformations and structure touching the wall

are only possible with a fully Eulerian approach [6,7,13] or in the

ALE framework with a full or partial re-meshing of the mesh, i.e.,

generating a new set of mesh cells and sometimes also a new set

of nodes.

Although, the mesh behavior of the harmonic and the bihar-

monic mesh motion models were analyzed in [12] for different

applications, we upgrade these concepts to fluid-structure interac-

tion problems. Moreover, we provide quantitative comparisons of

the three mesh motion models.

In the discretization section, we address aspects of the imple-

mentation of a temporal discretization, that is based on finite

0045-7949/$ - see front matter 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.compstruc.2011.02.019

E-mail address: [email protected]

Computers and Structures xxx (2011) xxxxxx

Contents lists available at ScienceDirect

Computers and Structures

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p s t r u c

Please cite this article in press as: Wick T. Fluid-structure interactions using different mesh motion techniques. Comput Struct (2011), doi: 10.1016/

j.compstruc.2011.02.019
http://dx.doi.org/10.1016/j.compstruc.2011.02.019mailto:[email protected]://dx.doi.org/10.1016/j.compstruc.2011.02.019http://www.sciencedirect.com/science/journal/00457949http://www.elsevier.com/locate/compstruchttp://dx.doi.org/10.1016/j.compstruc.2011.02.019http://dx.doi.org/10.1016/j.compstruc.2011.02.019http://dx.doi.org/10.1016/j.compstruc.2011.02.019http://dx.doi.org/10.1016/j.compstruc.2011.02.019http://www.elsevier.com/locate/compstruchttp://www.sciencedirect.com/science/journal/00457949http://dx.doi.org/10.1016/j.compstruc.2011.02.019mailto:[email protected]://dx.doi.org/10.1016/j.compstruc.2011.02.019


2/12

differences. In particular, we present the one step-h schemes [14]

and the Fractional step-h scheme [15] in ALE fashion for the

monolithic problem. Space discretization is done using a standard

Galerkin finite element approach. The solution of the discretized

system can be achieved with a Newton method, which is very

attractive because it provides robust and rapid convergence. The

Jacobian matrix is derived by exact linearization which is demon-

strated by an example. Because the development of iterative linear

solvers is difficult for fully coupled problems (however, sugges-

tions have been made [16,17]), and we are only interested in solv-

ing problems for a low amount of unknowns, we use a direct solverto solve the linear systems.

The outline of this paper is as follows. In the second section, the

fluid equations in artificial coordinates, and structure equations for

two different material models, are introduced. After, the mixed for-

mulation of the biharmonic equation is introduced for two kinds of

boundary conditions. Finally, fluid-structure interaction based on a

closed variational setting is proposed. Section 3 presents discreti-

zation in time and space of the fluid-structure interaction prob-

lems. Moreover, the nonlinear problem is examined through an

exact computation of the Jacobian matrix. The computation of

the directional derivatives is shown. In Section 4, numerical tests

for four problems (in both two and three dimensions) are per-

formed, showing the advantages and the differences between the

three mesh motion models. The computations are performed usingthe finite element software package deal.II [18].

2. Equations

In this section, we briefly introduce the basic notation and the

equations describing both the fluid (in the ALE-transformed coor-

dinate system) and structure (in its natural Lagrangian coordi-

nates). Then, we present the monolithic setting for the coupled

problem.

2.1. Notation

We denote by X & Rd, d = 2, 3, the domain of the fluid-structureinteraction problem. This domain is supposed to be time indepen-

dent but consists of two time dependent subdomains Xf(t) andXs(t). The interface between both domain is denoted by Ci(t) =

oXf(t) \ oXs(t). The initial (or later reference) domains are denotedby bXf and bXs, respectively, with the interface bCi. Further, we de-

note the outer boundary with @bX bC bCD [ bCN where bCDandbCNdenote Dirichlet and Neumann boundaries, respectively. We

adopt standard notation for the usual Lebesgue and Soboley spaces

and their extensions by means of the Bochner integral for time

dependent problems [19]. We use the notation (, )X for a scalarproduct on a Hilbert space X and h, i@X for the scalar product onthe boundary oX. For the time dependent functions on a time inter-

val I, the Sobolev spaces are defined by X : L2I;X. Concretely,we use L : L

2

I; L2

X and V : H1

I; H1

X fv 2 L2

I; H1X : @tv 2 L2I; H1Xg.

2.2. Fluid in artificial coordinates

Let bAf^x; t : Xf It ! Xft be a piecewise continuously differ-entiable invertible mapping. We define the physical unknowns vfand ^pf in bXf by

vf^x; t vfx; t vfbAf^x; t; t;^pf^x; t pfx; t pf

bAf^x; t; t:Then, with

bFf : rbAf; bJf : detbFf;we get the relations [20]:

rvf rvfbF1f ; @tvf @tvf bF1f @tbAf rvf;ZXf

fxdx

ZbXf ^f^xbJd^x:

With help of these relations, we can formulate the NavierStokes

equations in artificial coordinates:

Problem 2.1. (Variational fluid problem, ALE framework) Findfvf; ^pfg 2 fvDf bVg bLf, such that vf0 v0f , for almost all timesteps t, and

bJfqf@tvf bF1f vf @tbAf rvf; wvbXf bJfrfbFTf ; rwvbXf h^gf; w

vibC i[bCN 0 8wv 2 bVf;ddivbJfbF1f vf; wpbXf 0 8wp 2 bLf;with the transformed Cauchy stress tensor

rf : ^pfI qfmfrvfbF1 bFTrvTf:The viscosity and the density of the fluid are denoted by mf and qf,respectively. The function gf represents Neumann boundary condi-

tions for both physical boundaries (e.g., stress zero at outflowboundary), and normal stresses on bCi. Later, this boundary repre-

sents the interface between the fluid and structure. We note that

the specific choice of the transformation bAf is up to now arbitraryand left open.

2.3. Structure in Lagrangian coordinates

Usually, structural problems are formulated in Lagrangian coor-

dinates, which means to find a mapping from the physical domain

Xs(t) to the reference domain bXs. The transformation bAst :bXs It ! Xst is naturally given by the deformation itself:

bAs^x; t ^x us^x; t; bFs : rbAs I rus; bJs : detbFs: 1

Fig. 1. The monolithic solution approach for fluid-structure interaction.

2 T. Wick / Computers and Structures xxx (2011) xxxxxx


http://dx.doi.org/10.1016/j.compstruc.2011.02.019http://dx.doi.org/10.1016/j.compstruc.2011.02.019http://dx.doi.org/10.1016/j.compstruc.2011.02.019http://dx.doi.org/10.1016/j.compstruc.2011.02.019


3/12

We observe two material models. First, the elastic compressible

(geometrically) nonlinear Saint VenantKirchhoff material (STVK).

It is well suited for (relatively) large displacements with the

limitation of small strains. The strain is defined by bE :12bFTbF I. Second, we employ the MooneyRivlin model (IMR)

that is useful in the description of incompressible-isotropic

rubber-like materials. It is also an adequate model for deformations

with large strains. The sought physical unknowns are the displace-

ment , the velocity v, and a pressure ^ps (in case of the IMR

material).

Problem 2.2. (Structure problems, Lagrangian framework) Find

fvs; us; ^psg 2 fvDs bV0s g fuDs bV0s g bLs, such that us0 u0s , foralmost all time steps t, and

qs@tvs; wvbXs bJsrsbFTs ; rwvbXs

hbJsrsbFTs ns; wvibC i[bCN qs^fs; wvbXs 8wv 2 bVs@tus vs; w

ubXs 0 8wu 2 bVs;2

where qs is the structure density, ns the outer normal vector on bCiand bCN, respectively. The Cauchy stress tensors for STVK material

and the IMR material, respectively, are given by

rs : bJ1bFkstrbEI 2lsbEbFT; 3rs : ^psI ls

bFbFT l2bFTbF1 4with the Lam coefficients ls, ks, and l2. For the STVK material, thecompressibility is related to the Poisson ratio ms (ms < 12). Externalvolume forces are described by the term fs.

2.4. The mixed formulation of the biharmonic equation

In this section, we focus on a mixed formulation of the bihar-

monic equation. To be convenient for later purposes we use the

hat notation as introduced before. Let bX & Rd be a polygonal do-main with boundary bC bC1 [ bC2.

In the following, we investigate finite element approximations

of the biharmonic equationbD2u ^f in bX; 5with boundary conditions

u @nu 0 on bC1;bDu @nbDu 0 on bC2:This equation is well-known from structure mechanics where u de-

scribes the deflection of a clamped plate under the vertical force f.

To derive a mixed formulation in the sense of Ciarlet [21], we

introduce an auxiliary variable w bD

u obtaining two differentialequations:

w bDu in bX; bDw ^f in bX; 6

with boundary conditions

u @nu 0 on bC1;w @nw 0 on bC2:In order to discretize (5) with a conforming Galerkin finite element

scheme, we derive a variational formulation with standard argu-

ments [21,22]:

Problem 2.3. Find fu; wg 2 bV0 bV such that

w; ww ru; rww 0 8ww 2 bV;rw; rwu f;wu 8wu 2 bV0:

Problem 2.3 has computational advantages compared to other var-

iational formulations of the biharmonic equation. This mixed for-

mulation avoids the use of H2-conforming finite elements for

spatial discretization. When working with a variational formulation

of the original Eq. 5, higher order finite elements are indispensable.

2.5. The coupled problem in ALE coordinates

Combining the reference domains bXf and bXs leads to the well-

established ALE formulation for fluid-structure interactions. For

this purpose, we need to specify the transformation bAf in thefluid-domain. On the interface bCi, this transformation is given by

following the structure displacement:bAf^x; tjbC i ^x us^x; tjbCi : 7On the outer boundary of the fluid domain, @bXf n bCi there holdsbAf id. Inside bXf, the transformation should be as smooth and reg-

ular as possible, it is otherwise arbitrary.There are several possible ways to pose the artificial problem.

Often, the fluid mesh movement is resolved by solving a (linear)

elasticity equation [6,8,23]. Solving the Laplace equation is the

simplest route, but it only works for small mesh deformations if

a constant number is chosen for the material parameter. Larger

deformations [11] are realized by solving a linear elasticity prob-

lem. As a third approach, we use the biharmonic operator for

deforming the mesh with two types of boundary conditions [12].

In the following section, we explain how to apply the different

mesh moving techniques and how to pose the boundary and inter-

face conditions. To extend usjbXs to the fluid domainbXf, the

mapping bAf : id u in bXf is defined. In two dimensional config-urations, the mesh moves in x- and y-direction, which allows find-

ing a vector-valued artificial displacement variable

uf : u1f ; u

2f

: u

xf ; u

yf

:

Weneed thesingle componentsof fbelow to apply differenttypes of

boundary conditions to the biharmonic mesh motion model. In the

following, the formal description of the first two mesh motion mod-

elscoincides andonly differin thedefinition of thestresstensors rg.

2.5.1. Mesh motion with harmonic model

The simplest model is based on the harmonic equation, which

reads in strong formulation:

ddivrg 0; uf us on bCi; uf 0 on @bXf n bCi; 8with rg auruf, A detailed explication of the artificial parameter^au :

^au

^x is given in Section 3.6.

2.5.2. Mesh motion with linear elastic model

The linear-elasticity equation is formally based on the well-

known momentum equations from structure mechanics. If we as-

sume a steady state process and neglect the body forces, we obtain

the following static-equilibrium equation:

ddivrg 0; uf us on bCi; uf 0 on @bXf n bCi:where rg is formally equivalent to the STVK material in Eq. 3. It isgiven by:

rg : bFaktrI 2al: 9The pseudo-material parameters ak : ak^x and al : al^x areexplained in Section 3.6. Further,

1

2 ^ruf

^ru

T

f is the linearizedversion of the strain tensor bE.

T. Wick / Computers and Structures xxx (2011) xxxxxx 3




4/12

2.5.3. Mesh motion with biharmonic model

In this work, solving the biharmonic equation is introduced as a

third possible fluid mesh deformation. It is based on the already

introduced mixed model in the strong formulation Eq. 6. As before,

artificial material parameters are used to control the mesh motion.

Then

wf aubDuf and awbDwf 0: 10To simplify notation, we assume au aw 1 in this section.

It is more convenient to consider the single component func-

tions u1f

and u2f

,

w1f

bDu1f and bDw1f 0;w

2f

bDu2f and bDw2f 0:We focus on two types of boundary conditions. First, we pose the

first type of boundary conditions

uk

f @nu

k

f 0 on bC n bCi for k 1; 2: 11

Second, we are concerned with a mixture of boundary conditions

(see Fig. 2)

u1f

@nu1f

0 and w1f

@nw1f

0 on bCin [ bCout;u

2f

@nu2f

0 and w2f

@nw2f

0 on bCwall; 12which we call second type of boundary conditions. The interface con-

ditions for f are given as usual, uf us on bCi:

Remark 2.1. Using the second type of boundary conditions in a

rectangular domain where the coordinate axes match the Cartesian

coordinate system, as shown in Fig. 2, leads to mesh movement

only in the tangential direction. This effect reduces mesh cell

distortion because only the perpendicular directions of f and wfare constrained to zero at the different parts of bC.

Up to now, the description of the problems has been derived in

a general manner that serves for both partitioned and monolithic

solution algorithms. In the following, we focus on a monolithic

description of the coupled problem. We define a continuous vari-

able for all bX defining the deformation in bXs and supporting

the transformation in bXf. Thus, we skip the subscripts, and because

the definition of bAf coincides with the previous definition of bAs,we define in bX:bA : id u; bF : I ru; bJ : detbF: 13Furthermore, the velocity v is a common continuous function for

both subproblems, whereas the pressure ^p is discontinuous. For the

convenience for the reader, we only state the full variational formu-

lation of the harmonic and the linear-elastic mesh motion models.

Problem 2.4 (Variational fluid-structure interaction framework ). Find

fv; u; ^pg 2 fvD bV0g fuD bV0g bL, such that v0 v0 and(0) = 0, for almost all time steps t, and

bJqf@tv; wvbXf qfbJbF1v @tu rv; wvbXf bJrfbFT; rwvbXf qs@tv; w

vbXs bJrsbFT; rwvbXs h^g; wvibCN qfbJff; wvbXf qs^fs; wvbXs 0 8wv 2 bV0;

@tu v; wu

bXs

rg; rwu

bXf

hrgnf; wui

bCi

0 8wu 2

bV0;

ddivbJbF1vf; wpbXf ^ps; wpbXs 0 8wp 2 bL;with qf, qs, mf, ls, ks, bF, and bJ. The stress tensors rf, rs, and rg aredefined in Problems 2.1, 2.2, and the Eqs. 8 and 9, respectively.

The Problem 2.4 is completed by appropriate choice of the two

coupling conditions on the interface. The continuity of velocity

across bCi is strongly enforced by requiring one common continu-

ous velocity field on the whole domain bX. The continuity of normal

stresses is given by

bJrsbFTns;wvbC i bJrfbFTnf;wvbC i : 14By omitting this boundary integral jump over bCi the weak continu-

ity of the normal stresses becomes an implicit condition of the fluid-

structure interaction problem.

Remark 2.2. The boundary terms on bCi in Problem 2.4 are

necessary to prevent spurious feedback of the displacement

variables and w. For more details on this, we refer to [7].

3. Discretization

In this section, we focus on the discretization in time and space

of the fluid-structure interaction Problem 2.4. Our method of

choice are finite differences for time discretization and a Galerkin

finite element method for spatial treatment.

3.1. Variational formulation in an abstract setting

In the domain bX and the time interval I 0; T, we consider thefluid-structure interaction Problem 2.4 with harmonic or lin-

ear-elastic mesh motion in an abstract setting (the biharmonic

problem is straightforward): Find bU fv; u; ^pg 2 ^X, wherebX0 : fvD bV0g fuD bV0g bL, such thatZT

0

bAbUbWdt ZT0

bFbWdt 8bW 2 bX0: 15The linear form bF bW and the semi-linear form bAbU bW are definedby

bFbW qs^fs; w

vbXs ; 16andbAbUbW bJqf@tv; wvbXf qfbJbF1v rv; wvbXf

qfbJbF1@tu rv; wvbXf h^g; wvibCN qfbJff; wvbXf bJrfbFT; rwvbXf qs@tv; w

vbXs bJrsbFT; rwvbXs @tu; wubXs v; wubXs auru; rwubXf haunfru; wuibCi

ddiv

bJ

bF1vf; w

p

bXf

^ps; wp

bXs

: 17

The fluid convection term in Eq. 17 is decomposed into two parts forlater purposes.

Fig. 2. Flow around cylinder with elastic beam with circle-center C= (0.2,0.2) andradius r= 0.05.





5/12

3.2. Time discretization

The abstract problem Eq. 15 can either treated by a full time

space Galerkin formulation, which has been investigated for fluid

problems [25]. Alternatively, the Rothe method can be used in

cases where the time discretization is based on finite difference

schemes. A classical scheme for problems with a stationary limit

is the the (implicit) backward Euler scheme (BE), which is stronglyA-stable, but only from first order, and dissipative. It is later used in

the numerical Examples 1,2 and 4.

The Fractional-step-h scheme is used for unsteady simulations

[15]. It has second-order accuracy and is strongly A-stable, and it

is therefore well-suited for computing solutions with rough data

and computations over long time intervals.

After semi-discretization in time, we obtain a sequence of gen-

eralized steady fluid-structure interaction problems that are com-

pleted by appropriate boundary values for every time step. These

kinds of problems are now formulated as One-step h scheme [14].

This design has the advantage that it can easily be extended to

the Fractional-Step-h scheme.

We (formally) define the following semi-linear forms and group

them into four categories: time equation terms (including the time

derivatives), implicit terms (e.g., the incompressibility of fluid),

pressure terms, and all remaining terms (stress terms, convection,

etc.):

bATbUbW bJqf@tv; wvbXf qfbJbF1@tu rv; wvbXf qs@tv; w

vbXs @tu; wubXs ;bAIbUbW auru; rwubXf haunfru; wuibC i ddivbJbF1vf; wpbXf ^ps; wpbXs ;bAEbUbW qfbJbF1v rv; wv

bXf

bJrf;vubFT; rwvbXf bJrs;vubF

T; rwvbXs v; wubXs ;bAPbUbW bJrf;pbFT; rwvbXf ;

18

where the reduced tensors rf,vu, rs,vu, and rf,p, are defined as:

rf;vu qfmfrvbF1 bFTrvT;

rs;vu bJ1bFkstrbEI 2lsbEbFT;rf;p bJpfIbFT:The time derivative in bATbU bW is approximated by a backwarddifference quotient. For the time step tm

2I

m

1;2; . . .

, we com-

pute v : vm; u : um via

bATbUm;kbW qfbJm12 v vm1k

; wv

bXf qfbJbF1 u um1

k rv; wv

bXf

qsv vm1

k; wv

bXs u u

m1

k; wu

bXs ;

where bJm12 bJmbJm1

2, m : (tm), vm : vtm, and bJ : bJm :

bJtm. The former time step is given by vm1, etc.

3.2.1. Basic-h scheme

Let the previous solutionb

Um

1

fvm

1

; um

1

; ^pm

1

g and thetime step k : km = tm tm1 be given.

Find bUm fvm; um; ^pmg such that

bATbUm;kbW hbAEbUmbW bAPbUmbW bAIbUmbW 1 hbAEbUm1bW hbFbUmbW 1 hbFbUm1bW:

The concrete scheme depends on the choice for the parameter h. In

particular, we get the backward Euler scheme for h = 1, the Crank

Nicolson scheme for h 12, and the shifted CrankNicolson forh 1

2 km [24].

3.2.2. Fractional-step-h scheme

We choose h 1 ffiffi2

p2 ; h

0 1 2h, and a 12h1h ; b 1 a. Thetime step is split into three consecutive sub-time steps. Let

Um1 fvm1; um1; ^pm1g and the time step k : km = tm tm1 begiven.

Find bUm fvm; um; ^pmg such thatbATbUm1h;kbW ahbAEbUm1hbW hbAPbUm1hbW bAIbUm1hbW bh

bAE

bUm1

bW h

bF

bUm1

bW;

bATbUmh;kbW ahbAEbUmhbW h0bAPbUmhbW bAIbUmhbW ah0bAEbUm1hW h0bFbUmhbW;

bATbUm;kbW ahbAEbUmbW hbAPbUmbW bAIbUmbW bhbAEbUm1bW hbFbUmhbW: 19

3.3. Spatial discretization

The time discretize equations are the starting point for the

Galerkin discretization in space. To this end, we construct finite

dimensional subspaces bX0h & bX0 to find an approximate solutionto the continuous problem. In the context of monolithic ALE for-mulations, the computations are done on the reference configura-

tion bX. We use two or three dimensional shape-regular meshes. A

mesh consists of quadrilateral or hexahedron cells bK. They perform

a non-overlapping cover of the computation domain bX & Rd, d = 2,3. The corresponding mesh is given by bTh fbKg. The discretizationparameter in the reference configuration is denoted by h and

is a cell-wise constant that is given by the diameter hbK

of the

cell bK.

On bTh, conforming finite element spaces for vh; uh; ^ph, and whare denoted by the space bVh & bV. We prefer the biquadratic, dis-continuous-linear Qc2=P

dc1 element. The definitions of the spaces

for the unknowns vh and ^ph on a time interval Im read:

bVh : vh 2 CbXhd; vhjbK 2 Q2bKd 8bK 2 bTh; vhjbCnbC i 0& ';bPh : ^ph 2 bL2bXh; ^phjbK 2 P1bK 8bK 2 bThn o:We consider for each bK 2 bTh the bilinear transformationrK : bKunit ! K, where bKunit denotes the unit square. Then, the Qc2element is defined by

Qc2bK q r1K : q 2 span < 1;x;y;xy;x2;y2;x2y;y2x;x2y2 >

with dim Qc2 9, which means nine local degrees of freedom. ThePdc1 element consists of linear functions defined by

Pdc1 bK q r1K : q 2 span < 1;x;y >

with dim Pdc1 K 3.





6/12

Defining the displacement variables h and wh is straightfor-

ward. The property of the Qc2=Pdc1 element is continuity of

the velocity values across different mesh cells. However, the

pressure is defined by discontinuous test functions. In addition,

this element preserves local mass conservation, is of low

order, gains the inf-sup stability, and therefore is an optimal

choice for both fluid problems and fluid-structure interaction

problems.

Remark 3.1. Computation of fluid-structure interaction with

biharmonic mesh motion has more computational cost at each

time step than just using a linear elasticity problem. This is because

an additional equation is added to the problem. Because we use a

direct solver for the linear sub-problems, the condition number

does not play a role. In the context of a Galerkin finite element

scheme, the spatial discretization of the mixed biharmonic equa-

tion is stable for equal-order discretization on polygonal domains,

which was part of our assumptions. Here, we work with Qc2elements for h and wh.

3.4. Linearization

Time and spatial discretization results for each single time step

in a nonlinear quasi-stationary problem

bAbUmbW bFbW 8bW 2 bX0h;which is solved by a Newton-like method. Given an initial guess U0m ,

find for j = 0,1,2, . . . the update dbUm of the linear defect-correction

problem

bA 0bUjmdbUm; bW bAbUjmbW bFbW;Uj1m U

jm kd

bUm: 20

Here k 2 (0,1] is used as damping parameter for line searchtechniques. The directional derivative bA0bUdbU; bW; is definedby

bA 0bUdbU; bW : lime!0

1

ebAbU edbUbW bAbUbWn o

d

debAhbU edbUbW

e0:

Due to the large size of the Jacobian matrix and the strongly nonlin-

ear behavior of fluid-structure interaction problems in the mono-

lithic ALE framework, calculating the Jacobian matrix can be

cumbersome. Nevertheless, in this context, we use the exact

Jacobian matrix to identify the optimal convergence properties of

the Newton method.

3.4.1. Implementation aspects

In this section, we present an example of one specific direc-

tional derivative that includes all of the necessary steps. Derivation

of the other expressions is straight forward, but for the conve-

nience of the reader, it is not shown here.

Let us consider the second term of the semi-linear formbATbU bW, Eq. 18, that is part of the fluid convection term in ALEcoordinates. It holds

bAconvbUbW qfbJbF1@tu rv; wvbXf

qfrvbJbF1@tu; wvb

Xf:

In this case, the directional derivativeb

A0convb

U^db

U;b

W in the direc-tion dbU fdv; du; d^pg is given by

bA 0convbUdbU; bW rdvbJbF1 u um1k ; wv rvbJbF10du u um1

k; wv

rv

bJ

bF1

du

k; wv

: 21

In two dimensions the deformation matrix reads in explicit form:

bF I ru 1 @1u1 @2u1@1u2 1 @2u2

!;

which brings us to

bJbF1 1 @2u2 @2u1@1u2 1 @2u2

!and its directional derivative in direction du du1; du2:

bJbF10du @2du2 @2du1

@1du2 @2du2 !:This expression is part of the second term shown in Eq. 21. The

remaining expressions for directional derivatives can be derived

in an analogous way. For more details on computation of the direc-

tional derivatives on the interface, please refer to [6,7]. Accurate

determination of the directional derivatives is also indispensable

for optimization problems in which the performance of the Newton

algorithms heavily depend on [26].

3.5. Mesh refinement

The computations are performed on globally-refined meshes

and heuristically-refined meshes. We use two kinds of heuristic

mesh refinement. The first step is geometric refinement aroundthe interface. The second step is measurement of the smoothness

of the discrete solutions that also lead to local refinement in the re-

gions around the interface.

3.6. Influence of the artificial parameters

We use an ad hoc method to define the artificial (material-)

parameters: au, aw, ak , and al. They are used to control the meshmotion of the fluid mesh. There are several choices for controlling

the influence of these parameters. In one technique selective mesh

deformation is used that is based on the shape and volume changes

of the cells [8]. Another method is augmented by a stiffening power

that determines the rate by which smaller elements are stiffenedmore than larger ones [11].

Mesh cells touching the interface are critical with respect to

mesh degeneration. Therefore, the aim of these parameters should

be to maintain the shape of the fluid mesh cells, close to the inter-

face, by controlling the determinant bJ of the transformation bF. The

parameters must be adjusted in a certain way for different tests

configurations which is problematic because the exact parameter

choice is a priori unknown. This problem does not occur when

using the biharmonic mesh motion model. An optimal, smooth

mesh is automatically achieved using this mesh motion model,

see Fig. 3 and [12]. Therefore, the material parameters au, aw donot depend on the mesh-position. For the harmonic mesh motion

model, we use au : h2bKau^x, with small parameter au^x > 0.The parameters ak and al for the linear-elasticity mesh modelcan be defined in a similar way.





7/12

4. Numerical tests

In this section, we compare the different mesh motion models

using numerical tests. The first three tests are two dimensional,based on the Computational Structure Mechanics (CSM) test [27],

the large deformation membrane on fluid test [28], and Fluid Struc-

ture Interaction (FSI) benchmark configurations [27,23,29]. We

compare our results to the results given in these articles and ex-

tend the CSM test to a new configuration to show the improved

performance of the biharmonic model with regard to the mesh

motion.

4.1. CSM tests

In these test cases, the fluid is set to be initially at rest in bXf. An

external gravitational force ^fs is applied only to the elastic beam,

producing a visible deformation. The tests are performed as

time-dependent problems (backward Euler), leading to a steady

state solution. For the harmonic and linear-elastic model, we use

the time step size k = 0.02 s; for the biharmonic model we use

k = 0.1 s.

In the first test case CSM 1, the same parameters used by [27]

validate the code and are used to compare the different mesh mo-

tion approaches. In particular, we run one computation based on

the harmonic mesh motion model without a mesh-position depen-

dent material parameter. It turns out that the harmonic model

does not hold any more. The reference values are taken from

[27]. In the second example CSM 4, only the gravitational force is

increased causing the elastic beam to become much more

deformed.

4.1.1. Configuration

The computational domain (Fig. 2) has length L = 2.5 m and

height H= 0.41 m. The circle center is positioned at C=

(0.2 m,0.2 m) with radius r= 0.05 m. The elastic beam has length

l = 0.35 m and height h = 0.02 m. The right lower end is positioned

at (0.6 m,0.19 m), and the left end is attached to the circle.

Control points A(t) (with A(0) = (0.6, 0.2)) are fixed at the trailing

edge of the structure, measuring x- and y-deflections of the beam.

4.1.2. Boundary conditions

For the upper, lower, and left boundaries, the no-slip condi-

tions for velocity and no zero displacement for structure are given.

When using the second type of boundary conditions with the

biharmonic mesh motion model, the displacement should be zeroin normal direction and free in the tangential direction. This allows

the fluid mesh the freedom to move along the boundary and re-

sults in a better partition of the fluid mesh.

At the outlet bCout, the do-nothing outflow condition is imposed

leading to a zero mean value of the pressure at this part of the

boundary.

4.1.3. Parameters

We choose for our computation the following parameters. Forthe (resting) fluid we use .f = 103 kg m3, mf = 10

3 m2 s1. Theelastic structure is characterized by .s = 10

3 kg m3, ms = 0.4,ls = 510

5 kg m1 s2. The vertical force is chosen as ^fs 2 m s2.

4.1.4. Discussion of the CSM 1 test

We observe, that the harmonic mesh motion without the mesh-

position dependent parameter leads to mesh degeneration and,

therefore, does not hold in this example. A quantitative study can

be seen in Fig. 3, where the minimal values, min (bJ), of the ALE-

transformation determinant bJ are sketched as function plots. Our

results indicate that using the harmonic approach (which is the

simplest one) is sufficient for this numerical test.

4.1.5. Discussion of the CSM 4 test

Due to the higher gravitational force fs 4 m s2 applied to thestructure, the beam is deformed to a greater extent than in the pre-

viously described test.

For this test case, only the biharmonic mesh motion model

equipped with the second type of boundary conditions leads to re-

sults. This effect occurs because the outermost mesh layer is not

deformed when using the first type of boundary conditions. How-

ever, the second type can deal with this factor because the mesh is

allowed to move in a tangential direction along the outer boundary

and prevent mesh degeneration. The measurements can be ob-

served in Table 1. Screenshots of the meshes are given in the Figs.

4 and 5. A quantitative study of the minJ can be studied in Fig. 6.We observe that the biharmonic mesh motion model leads to a

smoother fluid mesh compared to the other two mesh motion

models. The function plots of the min(bJ) in Figs. 3 and 6 indicatethat the global minimum of the biharmonic models is further away

from zero compared to the global minimums of the harmonic and

linear-elasticity approaches. In other words, the mesh distortion is

smaller when using the biharmonic mesh motion model.

4.2. Large deformation membrane on fluid test

The purpose of this example is to test our framework for large

structural deformations [28]. We modify the given configuration

by enlarging the height of the membrane. We use the incompress-

ible MooneyRivlin model (which is capable to deal with large

deformations and large strains) to characterize the structure. The

test is driven by a pressure difference between bCin and bCout. We

choose the time step size k = 0.01 and the implicit Euler time step-ping scheme.

4.2.1. Configuration and Parameters

The configuration is sketched in Fig. 7. We use the following

parameters to run the simulation: .f = 1000.0 kg m3, and

-0.2

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14

Min(J)

Time

Harmonic with constant parameterHarmonic

Linear elasticityBiharmonic 1st type bc

Biharmonic 2nd type bc

Fig. 3. Comparison of the min (bJ) for the harmonic, linear-elastic, and biharmonic

mesh motion models for the CSM 1 test. Degeneration of the mesh cells corresponds

to negative values ofbJ, for the case using the harmonic mesh motion model with

constant parameter.

Table 1

Results for CSM 4 with biharmonic mesh motion and second type of boundary

conditions.

DoF ux(A)[ 103 m] uy(A)[ 103 m]27744 25.2199 121.97142024 25.2805 122.13272696

25.3101

122.214

133992 25.3268 122.259





8/12

mf = 0.004 m2 s1 for the fluid. For the structure, we use

.s = 800.0 kg m3, ls = 2.0 10

7 Pa, l2 = 1.0 105 Pa.

4.2.2. Initial conditions and boundary conditions

On the lower boundary bCin and upper boundary bCout we pre-

scribe Robin-type boundary condition for the velocity and pressureand homogeneous Dirichlet condition for the displacement. On all

remaining parts we prescribe homogeneous Dirichlet conditions

for the velocity and the displacement:

u 0 on bCin [ bCout [ bCwall ;v 0 on bCwall;mf@nu ^pI nf ^pinflow nf on bCin;mf@nu ^pI nf 0 on bCout:The pressure ^pin is increased during the computation, i.e.,^pin t ^pinitial with ^pinitial 5:0 106 Pa:

4.2.3. Quantities of comparison

(1) y-deflection of the structure at the point A(t) with

A(0) = (0.0,0.005) [m].

(2) Principal stretch of the fluid cells under the membrane, i.e.

the stretch between the points (0.0,0.005) [m] and

(0.0,0.0025) [m].

(3) Measuring minbJ.

4.2.4. Results

The qualitative behavior of the numerical results does agree

with the findings in [28]. However, we use quadrilaterals for the

discretization, whereas the other authors use triangles. This is

one reason why we get a smaller maximal deformation of the

membrane (Figs. 8 and 9). Moreover, we use the same overall meshfor the fluid and the structure domains, which leads to high

Fig. 4. CSM 4 test with the harmonic and linear-elastic mesh motion models and gravitational force ^fs 4ms2. Both models lead to mesh distortion close to the lowerboundary.

Fig. 5. CSM 4 test with biharmonic mesh motion model and gravitational force ^fs 4ms2 . In the left picture the mesh cells distort using the first set of boundary conditions.In the right picture the second kind of boundary conditions are used.

-0.2

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14

M

in(J)

Time

Harmonic

Pseudo elasticity

Biharmonic 1st type bc

Biharmonic 2nd type bc

Fig. 6. Function plots of min (bJ) for the mesh motion models of the CSM 4 test.

Degeneration of mesh cells corresponds to negative values of bJ, arising in the first

three models.

Fig. 7. Configuration: large deformation membrane on fluid test.





9/12

anisotropies in the structure when working with a very thin mem-brane (Fig. 10). For that reason, we enlarged the membrane to pre-

vent difficulties due to the anisotropies.

4.3. Flexible beam in 2D

In this example, the three proposed mesh motion models are

applied to an unsteady fluid-structure interaction problem. We

consider the numerical benchmark test FSI 2, which was proposed

in [27]. The configuration is the same as for the CSM tests, sketched

in Fig. 2. New results can be found in [23,30,31]. The Fractional-

Step-h scheme, as presented in Eq. 19, was used for time discreti-

zation with different time step sizes k.

Due to large deformations of the elastic beam, using the proper

mesh motion model becomes crucial (Fig. 11). The mesh-dependent parameters used for the harmonic and linear-elastic ap-

proaches are the same as were used for the CSM tests discussed

previously.

4.3.1. Boundary conditions

A parabolic inflow velocity profile is given on bCin by

vf0;y 1:5U4yH y

H2; U 1:0 m s 1:

At the outlet bCout the do-nothing outflow condition is imposed

which lead to zero mean value of the pressure at this part of the

boundary. The remaining boundary conditions are chosen as in

the CSM test cases.

4.3.2. Initial conditions

For the non-steady tests one should start with a smooth in-

crease of the velocity profile in time. We use

vft; 0;y vf0;y

1cos p2t

2if t< 2:0 s

vf0;y otherwise:

(The term vf(0,y) is already explained above.

4.3.3. Quantities of comparison and their evaluation

(1) x- and y-deflection of the beam at A(t).

(2) The forces exerted by the fluid on the whole body, i.e.,

drag force FD and lift force FL on the rigid cylinder and the

elastic beam. They form a closed path in which the forces

can be computed with the help of line integration. The

formula is evaluated on the fixed reference domain bX and

reads:

FD; FL

Z

bS

bJrallbFT nds ZbS circlebJrfbFT nfds ZbSbeambJrfbFT nfds:

22

The quantities of interest for this time dependent test case

are represented by the mean value, amplitudes, and frequency of

x- and y-deflections of the beam in one time period T of

oscillations.

4.3.4. Parameters

We choose for our computation the following parameters. For

the fluid we use .f = 103 kg m3, mf = 10

3 m2 s1. The elastic struc-ture is characterized by .s = 10

4 kg m3, ms = 0.4, ls = 5105

kg m1 s2.

We observe the same qualitative behavior in each of our ap-proaches for the quantities of interest (ux(A),uy(A), drag, and lift);

these results are in agreement with [30].

The computed values are summarized in Tables 24. The refer-

ence values are taken from [30]. In general, to verify convergence

with respect to space and time, at least three different mesh

levels and time step sizes should be presented. Three different

mesh levels are not possible when working with the simplest

approach: harmonic mesh motion. For the third mesh level,

the min (J) becomes negative, and the ALE-mapping bursts off.

The x-displacements show the same behavior for all configura-

tions. For the y-displacements, we observe the same behavior on

the coarse mesh as we do for the harmonic and biharmonic ap-

proaches. However, the elastic approach yields nearly the same re-

sults on the different mesh levels.

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

M

in(J)

Time

Harmonic with constant parameter

Harmonic

Linear elasticity

Biharmonic 1st type bc

Fig. 8. Function plots of min (bJ) for the mesh motion models of the membrane on

fluid test. Degeneration of mesh cells corresponds to negative values ofbJ, arising in

the first three models.

-0.001

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

y-dis

Time

global 1global 2global 3

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

y-stretch

Time

global 1global 2global 3

Fig. 9. Large deformation membrane fluid test with the biharmonic mesh model for three different mesh levels. Left: vertical displacement of the point (0.0,0.005). Right:stretch of the cell under the membrane.





10/12

The drag values are similar for the first two mesh levels for eachmesh motion model. The results on the finest mesh for the bihar-

monic approach match the reference values.

The most difficult task is to compute the lift values. These diffi-

culties are a well-known phenomenon from fluid mechanics and

the related benchmark computations. These values also varies in

the literature [27,30,31]. Nevertheless, on the finest meshes of

the linear-elastic and biharmonic mesh motion models, all of the

values have the same sign and come relatively close to the refer-

ence values.

4.4. 3D bar behind a square cross section

In the last example, we consider a configuration in three dimen-sions. The steady state is derived in a similar fashion to the first

example, using the backward Euler time stepping scheme. Wecompare the harmonic mesh motion model with the biharmonic

model for moderate deformations.

4.4.1. Configuration and Parameters

The configuration (Fig. 12) is based on the fluid benchmark

example proposed in [32].

We use the following parameters to drive the simulation:

.f = 1.0 kg m3, and mf = 0.01 m

2 s1 for the fluid. For the structure,we use .s = 1.0 kg m

3, ms = 0.4, and ls = 500.0 kg m1 s2.

4.4.2. Initial conditions and boundary conditions

A constant parabolic inflow velocity profile is given on bCin by

vft; 0;y 16:0UyzH yH z

H4; U 0:45 m s1:

Fig. 10. Large deformation membrane on fluid test. The mesh deformation using the biharmonic model at the times t= 0.12 (left) and t= 0.7 (right) are displayed.

Fig. 11. FSI 2 test case: mesh (left) and velocity profile in vertical direction (right) at time t= 16.14 s.

Table 2

Results for the FSI 2 benchmark with the harmonic mesh motion model. The mean value and amplitude are given for the four quantities of interest: ux, uy[m], FD, FL[N]. The

frequencies f1[s1] and f2[s

1] ofux and uy vary in a range of 3.83 3.87 (Ref. 3.86) and 1.91 1.94 (Ref. 1.93), respectively.

DoF k[s] ux(A)[ 103] uy(A)[ 103] FD FL5032 3.0e3 14.62 13.17 1.06 79.87 210.78 73.97 1.83 295.85032 2.0e3 14.66 13.19 1.02 78.30 211.83 73.72 1.83 295.85032 1.0e3 14.70 13.20 0.94 80.39 210.17 75.34 0.40 298.45032 0.5e3 14.63 13.17 1.08 80.34 212.61 74.31 0.84 297.419488 3.0e3 13.73 11.79 1.20 78.20 207.72 72.63 0.21 227.119488 2.0e3 13.59 11.79 1.25 77.96 207.52 72.07 2.03 226.519488 1.0e3 13.63 11.77 1.24 78.11 201.96 73.15 1.86 231.219488 0.5e3 13.59 11.77 1.23 78.06 203.59 70.37 1.25 221.5(Ref.) 0.5e3 14.85 12.70 1.30 81.70 215.06 77.65 0.61 237.8





11/12

At the outlet bCout the do-nothing outflow condition is imposed,

leading to zero mean value of the pressure on this part of the

boundary.

4.4.3. Quantities of comparison

(1) x-, y-, and z-deflection of the beam at A(t) with

A(0) = (8.5,2.5,2.73) [m].

(2) Drag and lift around square cross section and elastic beam,with help of Eq. 22.

4.4.4. Results

The results for the different quantities of interest are in agree-

ment between both of the mesh motion models, as illustrated in

Table 5.

4.4.5. Computational cost for the numerical tests. Finally, we summa-

rize our observations with regard to the computational cost per

Newton step. In each nonlinear step (see Eq. 20), the Jacobian ma-

trix and the residual are evaluated and then solved by a direct sol-

ver (UMFPACK). Our results indicate that using the biharmonic

equation is much more expensive in each Newton step. Concretely,

the cost in two dimensions is five times higher for the biharmonic

mesh motion model compared to the other two models. In three

dimensions the factor for low amount of degrees of freedom

(DoF) is again five. Whereas for 624 cells in three dimensions the

factor becomes 70. It seems to be the linear solver, but it is still

an open question. A detailed study is given in Table 6. This result

indicate, using the biharmonic model with UMFPACK in threedimensions becomes prohibitive in a sequential solution process.

Table 3

Results for the FSI 2 benchmark with the linear-elastic mesh motion model. The mean value and amplitude are given for the four quantities of interest: ux, uy[m], FD, FL[N]. The

frequencies f1[s1] and f2[s

1] ofux and uy vary in a range of 3.83 3.90 (Ref.3.86) and 1.91 1.95 (Ref. 1.93), respectively.

DoF k[s] ux(A)[ 103] uy(A)[ 103] FD FL5032 3.0e3 13.93 12.48 1.20 78.02 205.11 69.01 0.21 284.25032 2.0e3 13.88 12.55 1.21 77.72 204.63 68.06 0.39 277.45032 1.0e3 13.99 12.62 1.23 78.03 201.65 70.81 0.15 277.919488 3.0e3 13.47 11.70 1.28 77.52 205.86 70.05 0.34 225.319488 2.0e3 13.54 11.71 1.29 77.77 206.71 70.02 0.31 226.519488 1.0e3 13.60 11.77 1.28 77.99 205.49 70.46 0.29 228.029512 3.0e3 13.00 11.33 1.26 76.09 202.92 67.09 0.20 216.029512 2.0e3 13.06 11.37 1.28 76.29 203.74 67.17 0.48 216.629512 1.0e3 13.11 11.42 1.26 76.50 203.28 67.69 0.54 217.7(Ref.) 0.5e3 14.85 12.70 1.30 81.70 215.06 77.65 0.61 237.8

Table 4

Results for the FSI 2 benchmark with the biharmonic mesh motion model and second type of boundary conditions. The mean value and amplitude are given for the four quantities

of interest: ux, uy [m], FD, FL[N]. The frequencies f1[s1] and f2 [s

1] ofux and uy vary in a range of 3.83 3.88 (Ref.3.86) and 1.92 1.94 (Ref.1.93), respectively.

DoF k[s] ux(A)[ 103] uy(A)[ 103] FD FL27744 3.0e3 13.63 11.80 1.27 78.72 207.22 71.13 0.57 230.627744 2.0e3 13.72 11.84 1.26 78.38 208.12 71.18 0.30 232.627744 1.0e3 13.74 11.85 1.28 78.48 209.46 71.43 0.06 231.727744 0.5e3 13.66 11.81 1.28 78.32 208.96 71.60 0.06 238.242024 3.0e3 13.34 11.57 1.40 77.08 204.81 68.54 0.79 221.542024 2.0e3 13.36 11.55 1.28 77.18 205.61 68.67 0.51 223.042024 1.0e3 13.38 11.58 1.31 77.44 206.11 68.26 0.62 221.242024 0.5e3 13.27 11.52 1.23 77.25 207.05 68.87 0.30 230.672696 3.0e3 14.43 12.46 1.35 80.71 212.50 76.40 0.18 234.672696 2.0e3 14.49 12.44 1.19 80.66 213.49 76.39 0.13 235.772696 1.0e3 14.49 12.46 1.16 80.63 213.39 75.25 0.23 234.272696 0.5e3 14.40 12.39 1.25 80.55 213.55 76.06 0.30 240.2(Ref.) 0.5e3 14.85 12.70 1.30 81.70 215.06 77.65 0.61 237.8

inflow bc

outflow bc

x

z

y4.1 m

25 m

4.1 m

4.5 m

1 m

3 m1 m

1.37 m

1.5 m

A(t)

Fig. 12. Configuration: flow around square cross section with elastic beam.

Table 5

Results for steady 3D FSI test case with harmonic (four upper rows) and biharmonic

(four lower rows) mesh motion. Evaluation of x-, y-, and z-deflections (in [m]); each

scaled by 106. In the last two columns drag and lift forces are displayed (in [N]).

Cells DoF ux (A) uy(A) uz(A) FD FL

78 5856 9.5106 32.7193 4.0278 0. 6633 0 .0502281 19694 23.8909 17.7207 2.9588 0.7647 0.1996624 39312 17.1212 0.4168 2.7161 0. 7753 0 .01034992 286368 18.6647 0.1522 3.0243 0. 7556 0 .011378 8628 9.5115 32.7149 4.0277 0. 6632 0 .0502281 28979 23.794 17.2999 2.9692 0.7671 0.1964624 57720 17.123 0.41921 2.7155 0. 7753 0 .01034992





12/12

Due to enormous memory usage for direct solvers, one should

use iterative solvers [16,17]. Further, adaptive mesh refinement

is an efficient tool to reduce the computational cost [6,13,33].

5. Conclusions

In this work, three different types of fluid mesh movement forfluid-structure problems are used and compared: harmonic, lin-

ear-elastic, and biharmonic structure extension. Our results show

that the biharmonic mesh model works fine for large displace-

ments of the elastic structure and leads to a smoother fluid mesh.

Compared to the harmonic and linear-elastic mesh motion models,

the biharmonic equation is easier to use. This ease of use is the re-

sult of the artificial parameters that do not depend on the mesh po-

sition for the biharmonic model in our proposed method. On the

contrary, our results suggest that the biharmonic approach is more

expensive, because of the second displacement variable. In upcom-

ing works, we will study different mesh motion models for unstea-

dy three dimensional configurations. Here, it is indispensable to

use economic local mesh-refinement because of the prohibitive

computational cost of using global mesh refinement. Therefore,we propose to use discretization in a closed variational setting that

can be extended to a full timespace Galerkin discretization for the

whole problem. This setting is the basis for an automatic mesh

adaption with the dual weighted residual (DWR) method, which

also allows for a goal-oriented a posteriori error estimation. Here,

the adjoint solutions will have to be derived; for this task, a closed

semilinear form is indispensable.

Acknowledgement

The financial support by the DFG (Deutsche Forschungsgeme-

inschaft) and the IGK 710 is gratefully acknowledged. Further,

the author thanks Dr. Th. Richter and Dr. M. Besier for discussions.

References

[1] Piperno S, Farhat C. Paritioned procedures for the transient solution of coupledaeroelastic problems Part II: Energy transfer analysis and three-dimensionalapplications. Comput Methods Appl Mech Eng 2001;190:314770.

[2] Figueroa CA, Vignon-Clementel IE, Jansen KE, Hughes TJR, Taylor CA. A coupledmomentum method for modeling blood ow in three-dimensional deformablearteries. Comput Methods Appl Mech Eng 2006;195:5685706.

[3] Nobile F, Vergara C. An effective fluid-structure interaction formulation forvascular dynamics by generalized Robin conditions. SIAM J Sci Comput2008;30(2):73163.

[4] Becker R, Kapp H, Rannacher R. Adaptive finite element methods for optimalcontrol of partial differential equations: basic concepts. SIAM J Optim Control2000;39:11332.

[5] Becker R, Rannacher R. An optimal control approach to error control and meshadaptation in finite element methods. In: Iserles A, editor. ActaNumerica. Cambridge University Press; 2001.

[6] Th. Dunne, Adaptive finite element approximation of fluid-structure inter-action based on eulerian and arbitrary LagrangianEulerian variational for-mulations, Dissertation, University of Heidelberg; 2007.

[7] Richter T, Wick T. Finite elements for fluid-structure interaction in ALE andfully Eulerian coordinates. Comput Meth Appl Mech Eng 2010;199(4144):263342.

[8] Tezduyar TE, Behr M, Mittal S, Johnson AA. Computation of unsteadyincompressible flows with the finite element methodsspace- timeformulations. In: Smolinski P, Liu WK, Hulbert G, Tamma K, editors. Iterativestrategies and massively parallel implementations, new methods in transient

analysis, vol. AMD-143. New York: ASME; 1992. p. 724.[9] Sackinger PA, Schunk PR, Rao RR. A NewtonRaphson pseudo-solid domain

mapping technique for free and moving boundary problems: a finite elementimplementation. J Comput Phys 1996;125(1):83103.

[10] Yigit S, Schfer M, Heck M. Grid movement techniques and their influence onlaminar fluid-structure interaction problems. J Fluids Struct 2008;24(6):81932.

[11] Stein K, Tezduyar T, Benney R. Mesh moving techniques for fluid-structureinteractions with large displacements. J Appl Math 2003;70:5863.

[12] Helenbrook BT. Mesh deformation using the biharmonic operator. Int J NumerMeth Eng 2001:130.

[13] Dunne Th, Rannacher R, Richter T. Numerical simulation of fluid-structureinteraction based on monolithic variational formulations. In: Galdi GP,Rannacher R, et al., editors. Numerical fluid structure interaction. Springer;2010.

[14] Turek S. Efficient solvers for incompressible flow problems. Springer-Verlag;1999.

[15] R. Glowinski, J. Periaux. Numerical methods for nonlinear problems in fluiddynamics. Proc Int Semin Sci Supercomput. Paris, Feb. 26, North Holland;1987.

[16] Heil M. An efficient solver for the fully coupled solution of large-displacementfluid-structure interaction problems. Comput Methods Appl Mech 2004;193:123.

[17] Badia S, Quaini A, Quarteroni A. Splitting methods based on algebraicfactorization for fluid-structure interaction. SIAM J Sci Comput 2008;30(4):1778805.

[18] W. Bangerth, R. Hartmann, G. Kanschat. Differential equations analysis library.Technical Reference; 2010. .

[19] Wloka J. Partielle differentialgleichungen. Stuttgart: B.G. Teubner; 1987.[20] Quarteroni A, Formaggia L. Mathematical modeling and numerical simulation

of the cardiovascular system. In: Ayache N, editor. Modelling of living systems.Ciarlet PG, Lions JL, editors. Handbook of numerical analysisseries. Amsterdam: Elsevier; 2002.

[21] Ciarlet PG, Raviart P-A. A mixed finite element method for thebiharmonic equation. In: de Boor C, editor. Mathematical aspects of finiteelements in partial differential equations. New York: Academic Press; 1974. p.12545.

[22] Ciarlet PG. The finite element method for elliptic problems. North-Holland;1987.

[23] In: Bungartz H-J, Schfer M, editors. Fluid-structure interaction: modelling,simulation, optimization, Springer series: Lecture Notes in ComputationalScience and Engineering 2006; 53(VIII).

[24] Heywood JG, Rannacher R. Finite-element approximation of the nonstationaryNavierStokes problem. Part IV: Error analysis for second order timediscretization. SIAM J Numer Anal 1990;27(2):35384.

[25] Schmich M, Vexler B. Adaptivity with dynamic meshes for space-time finiteelement discretizations of parabolic equations. SIAM J Sci Comput 2008;30(1):36993.

[26] Becker R, Meidner D, Vexler B. Efficient numerical solution of parabolicoptimization problems by finite element methods. Optim Methods Softw2007;22(5):81333.

[27] Hron J, Turek S. Proposal for numerical benchmarking of fluid-structureinteraction between an elastic object and laminar incompressible flow. In:Fluid-structure interaction: modeling, simulation, optimization. In: Bungartz,Hans-Joachim, Schafer, Michael, editors. Lecture notes in computationalscience and engineering, vol. 53. Springer; 2006. p. 14670.

[28] Bathe K-J, Ledezma GA. Benchmark problems for incompressible fluid flowswith structural interactions. Comput Struct 2007;85:62844.

[29] Bungartz H-J, Schfer M, editors. Fluid-structure interaction II: modelling,simulation, optimization. Springer series: Lecture Notes in ComputationalScience and Engineering 2010.

[30] Turek S, Hron J, Madlik M, Razzaq M, Wobker H, Acker JF. Numericalsimulation and benchmarking of a monolithic multigrid solver for fluid-structure interaction problems with application to hemodynamics.Ergebnisberichte des Instituts fr Angewandte Mathematik 403, Fakultt frMathematik, TU Dortmund 2010.

[31] Degroote J, Haelterman R, Annerel S, Bruggeman P, Vierendeels J. Performanceof partitioned procedures in fluid-structure interaction. Comput Struct2010;88:44657.

[32] Schaefer M, Turek S. Benchmark computations of laminar flow around acylinder. In: Hirschel EH, editor. Notes on numerical fluid mechanics, vol.52. Vieweg; 1996. p. 54766.

[33] Bathe K-J, Zhang H. A mesh adaptivity procedure for CFD and fluid-structureinteractions. Comput Struct 2009;87:60417.

Table 6

CPU Times per Newton step for solving the linear equations on a Intel Xeon machine

with a 2.40 GHz processor and sequential programming.

Tes t case Cells DoF Mes h motion model CPU time (in s )

CSM 1 992 19488 linear-elastic 2.2 0.2



CSM 1 992 27744 biharmonic 6.2 0.5

CSM 1 2552 72696 biharmonic 67.5 13.5CSM 1 4664 133992 biharmonic 206.8 37.6

3D FSI 78 5856 harmonic 0.4 0.02

3D FSI 281 19694 harmonic 6.0 0.2

3D FSI 624 39312 harmonic 25.2 2.8

3D FSI 78 8628 biharmonic 10.8 0.3

3D FSI 281 28979 biharmonic 206.0 6.6

3D FSI 624 57720 biharmonic 2475 495.0


Pl it thi ti l i Wi k T Fl id t t i t ti i diff t h ti t h i C t St t (2011) d i 10 1016/
http://www.dealii.org/http://www.dealii.org/http://dx.doi.org/10.1016/j.compstruc.2011.02.019http://dx.doi.org/10.1016/j.compstruc.2011.02.019http://www.dealii.org/

mesh moving techniques for fluid-structure interactions with large displacements.pdf

Documents