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Computers and Structures xxx (2011) xxx–xxx

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Computers and Structures

journal homepage: www.elsevier .com/locate /compstruc

Fluid-structure interactions using different mesh motion techniques

Thomas WickInstitute 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 2010Accepted 25 February 2011Available online xxxx

Keywords:Finite elementsFluid-structure interactionMonolithic formulationBiharmonic equation

0045-7949/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.compstruc.2011.02.019

E-mail address: [email protected]

Please cite this article in press as: Wick T. Fluj.compstruc.2011.02.019

a b s t r a c t

In this work, we compare different mesh moving techniques for monolithically-coupled fluid-structureinteractions in arbitrary Lagrangian–Eulerian coordinates. The mesh movement is realized by solvingan additional partial differential equation of harmonic, linear-elastic, or biharmonic type. We examinean implementation of time discretization that is designed with finite differences. Spatial discretizationis based on a Galerkin finite element method. To solve the resulting discrete nonlinear systems, a Newtonmethod with exact Jacobian matrix is used. Our results show that the biharmonic model produces thesmoothest 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 manyreal-life applications, such as industrial processes, aero-elasticity,and bio-mechanics. More specifically, fluid-structure interactionsare important to measuring the flow around elastic structures, theflutter analysis of airplanes [1], blood flow in the cardiovascularsystem, and the dynamics of heart valves (hemodynamics) [2,3].

Typically, fluid and structure are given in different coordinatesystems making a common solution approach challenging. Fluidflows are given in Eulerian coordinates whereas the structure istreated in a Lagrangian framework. We use a monolithic approach(Fig. 1), where all equations are solved simultaneously. Here, theinterface conditions, the continuity of velocity and the normalstresses, are automatically achieved. The coupling leads to addi-tional nonlinear behavior of the overall system.

Using a monolithic formulation is motivated by upcominginvestigations of gradient based optimization methods [4], andfor 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 ‘arbitraryLagrangian–Eulerian’ framework (ALE), the choice of appropriatefluid mesh movement is important. In general, an additionalelasticity equation is solved. For moderate deformations, one canpose an auxiliary Laplace problem that is known as harmonic meshmotion [6,7]. More advanced equations from linear elasticity arealso available [8,9]. For a partitioned fluid-structure interactionscheme, a comparison was made between different models [10].

ll rights reserved.

e

id-structure interactions using

The pseudo-material parameters in both approaches were usedto control the mesh deformation. If the parameters do not dependon mesh position and geometrical information, both approachescan 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 theinterface [8]. There are several techniques for choosing theseparameters to retain an optimal mesh, such as a Jacobian-basedstiffening power [11] that is eventually governed by appropriatere-meshing techniques. We use an ad hoc approach for theseparameters, measuring the distance to the elastic structure andadapting the parameters to prevent mesh cell distortion as longas possible.

Here, we also use (for mesh moving) the biharmonic equationthat others have studied for fluid flows in ALE coordinates [12]. Itwas also shown there, that using the biharmonic model providesgreater freedom in the choice of boundary and interface conditions.In general, the biharmonic mesh motion model leads to a smoothermesh (and larger deformations of the structure) compared to themesh motion models based on second order partial differentialequations. Larger deformations and structure touching the wallare only possible with a fully Eulerian approach [6,7,13] or in theALE 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 setof nodes.

Although, the mesh behavior of the harmonic and the bihar-monic mesh motion models were analyzed in [12] for differentapplications, we upgrade these concepts to fluid-structure interac-tion problems. Moreover, we provide quantitative comparisons ofthe three mesh motion models.

In the discretization section, we address aspects of the imple-mentation of a temporal discretization, that is based on finite

different mesh motion techniques. Comput Struct (2011), doi:10.1016/

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

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

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

differences. In particular, we present the one step-h schemes [14]and the Fractional step-h scheme [15] in ALE fashion for themonolithic problem. Space discretization is done using a standardGalerkin finite element approach. The solution of the discretizedsystem can be achieved with a Newton method, which is veryattractive because it provides robust and rapid convergence. TheJacobian matrix is derived by exact linearization which is demon-strated by an example. Because the development of iterative linearsolvers 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, thefluid equations in artificial coordinates, and structure equations fortwo different material models, are introduced. After, the mixed for-mulation of the biharmonic equation is introduced for two kinds ofboundary conditions. Finally, fluid-structure interaction based on aclosed 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 anexact computation of the Jacobian matrix. The computation ofthe directional derivatives is shown. In Section 4, numerical testsfor four problems (in both two and three dimensions) are per-formed, showing the advantages and the differences between thethree 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 theequations 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 coupledproblem.

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 ¼ bC ¼ bCD [ bCN where bCDandbCNdenote Dirichlet and Neumann boundaries, respectively. Weadopt standard notation for the usual Lebesgue and Soboley spacesand their extensions by means of the Bochner integral for timedependent 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 :¼ L2ðI; XÞ. Concretely,we use L :¼ L2ðI; L2ðXÞÞ and V :¼ H1ðI; H1ðXÞÞ ¼ fv 2 L2

ðI; H1ðXÞÞ : @tv 2 L2ðI; H1ðXÞÞg.

Please cite this article in press as: Wick T. Fluid-structure interactions usingj.compstruc.2011.02.019

2.2. Fluid in artificial coordinates

Let bAf ðx; tÞ : Xf � It ! Xf ðtÞ be a piecewise continuously differ-entiable invertible mapping. We define the physical unknowns v f

and pf in bXf by

v f ðx; tÞ ¼ v f ðx; tÞ ¼ v f ðbAf ðx; tÞ; tÞ;pf ðx; tÞ ¼ pf ðx; tÞ ¼ pf ð bAf ðx; tÞ; tÞ:

Then, with

bF f :¼ rbAf ; bJ f :¼ det bF f ;

we get the relations [20]:

rv f ¼ rv fbF�1

f ; @tv f ¼ @tv f � ðbF�1f @t

bAf � rÞv f ;ZXf

f ðxÞdx ¼ZbX f

f ðxÞbJdx:

With help of these relations, we can formulate the Navier–Stokesequations in artificial coordinates:

Problem 2.1. (Variational fluid problem, ALE framework) Findfv f ; pf g 2 fvD

f þ bVg � bLf , such that v f ð0Þ ¼ v0f , for almost all time

steps t, and

ðbJ f qf ð@tv f þ ðbF�1f ðv f � @t

bAf Þ � rÞv f Þ; wvÞbX f

þ ðbJ f rfbF�T

f ; rwvÞbX f

� hgf ; wvibC i[bCN

¼ 0 8wv 2 bV f ;

ðddivðbJ fbF�1

f v f Þ; wpÞbX f

¼ 0 8wp 2 bLf ;

with the transformed Cauchy stress tensor

rf :¼ �pf I þ qf mf ðrv fbF�1 þ bF�TrvT

f Þ:

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 thatthe 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 domainXs(t) to the reference domain bXs. The transformation bAsðtÞ :bXs � It ! XsðtÞ is naturally given by the deformation itself:bAsðx; tÞ ¼ xþ usðx; tÞ; bF s :¼ rbAs ¼ I þ rus; bJs :¼ detðbF sÞ:

ð1Þ




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

We observe two material models. First, the elastic compressible(geometrically) nonlinear Saint Venant–Kirchhoff material (STVK).It is well suited for (relatively) large displacements with thelimitation of small strains. The strain is defined by bE :¼12 ðbF TbF � IÞ. Second, we employ the Mooney–Rivlin model (IMR)

that is useful in the description of incompressible-isotropicrubber-like materials. It is also an adequate model for deformationswith large strains. The sought physical unknowns are the displace-ment û, the velocity v , and a pressure ps (in case of the IMRmaterial).

Problem 2.2. (Structure problems, Lagrangian framework) Findfvs; us; psg 2 fvD

s þ bV0s g � fuD

s þ bV0s g � bLs, such that usð0Þ ¼ u0

s , foralmost all time steps t, and

ðqs@tv s; wvÞbXs

þ ðbJsrsbF�T

s ; rwvÞbXs

� hbJ srsbF�T

s ns; wvibC i[bCN

¼ ðqsf s; wvÞbXs

8wv 2 bV s

ð@t us � v s; wuÞbXs

¼ 0 8wu 2 bV s;

ð2Þ

where qs is the structure density, ns the outer normal vector on bCi

and bCN , respectively. The Cauchy stress tensors for STVK materialand the IMR material, respectively, are given by

rs :¼ bJ�1bF ðksðtrbEÞI þ 2lsbEÞbF T ; ð3Þ

rs :¼ �psI þ lsbFbF�T þ l2

bF�TbF�1 ð4Þ

with the Lamé coefficients ls, ks, and l2. For the STVK material, thecompressibility is related to the Poisson ratio ms (ms <

12). External

volume forces are described by the term f s.

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 ¼ bC1 [ bC2.

In the following, we investigate finite element approximationsof the biharmonic equationbD2u ¼ f in bX; ð5Þ

with boundary conditions

u ¼ @nu ¼ 0 on bC1;bDu ¼ @nbDu ¼ 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], weintroduce an auxiliary variable w ¼ �bDu obtaining two differentialequations:

w ¼ �bDu in bX;� bDw ¼ f in bX; ð6Þ

with boundary conditions

u ¼ @nu ¼ 0 on bC1;

w ¼ @nw ¼ 0 on bC2:

In order to discretize (5) with a conforming Galerkin finite elementscheme, we derive a variational formulation with standard argu-ments [21,22]:

Problem 2.3. Find fu; wg 2 bV 0 � bV such that


� ðw; wwÞ þ ðru; rwwÞ ¼ 0 8ww 2 bV ;ðrw; rwuÞ ¼ ðf ;wuÞ 8wu 2 bV 0:

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 forspatial discretization. When working with a variational formulationof 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. Forthis purpose, we need to specify the transformation bAf in thefluid-domain. On the interface bCi, this transformation is given byfollowing the structure displacement:bAf ðx; tÞjbC i

¼ xþ usðx; tÞjbC i: ð7Þ

On the outer boundary of the fluid domain, @ bXf n bCi there holdsbAf ¼ 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 thesimplest route, but it only works for small mesh deformations ifa constant number is chosen for the material parameter. Largerdeformations [11] are realized by solving a linear elasticity prob-lem. As a third approach, we use the biharmonic operator fordeforming the mesh with two types of boundary conditions [12].

In the following section, we explain how to apply the differentmesh moving techniques and how to pose the boundary and inter-face conditions. To extend usjbXs

to the fluid domain bXf , themapping bAf :¼ idþ 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 :¼ uð1Þf ; uð2Þf

� �:¼ uðxÞf ; uðyÞf

� �:

We need the single components of ûf below to apply different types ofboundary conditions to the biharmonic mesh motion model. In thefollowing, the formal description of the first two mesh motion mod-els coincides and only differ in the definition of the ‘stress’ tensors rg .

2.5.1. Mesh motion with harmonic modelThe simplest model is based on the harmonic equation, which

reads in strong formulation:

�ddivðrgÞ ¼ 0; uf ¼ us on bCi; uf ¼ 0 on @ bXf n bCi; ð8Þ

with rg ¼ auruf , A detailed explication of the artificial parameterau :¼ auðxÞ is given in Section 3.6.

2.5.2. Mesh motion with linear elastic modelThe 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 obtainthe following static-equilibrium equation:

�ddivðrgÞ ¼ 0; uf ¼ us on bCi; uf ¼ 0 on @ bXf n bCi:

where rg is formally equivalent to the STVK material in Eq. 3. It isgiven by:

rg :¼ bFðakðtr�ÞI þ 2al�Þ: ð9Þ

The pseudo-material parameters ak :¼ akðxÞ and al :¼ alðxÞ areexplained in Section 3.6. Further, � ¼ 1

2 ðruf þ ruTf Þ is the linearized

version of the strain tensor bE.





2.5.3. Mesh motion with biharmonic modelIn this work, solving the biharmonic equation is introduced as a

third possible fluid mesh deformation. It is based on the alreadyintroduced mixed model in the strong formulation Eq. 6. As before,artificial material parameters are used to control the mesh motion.Then

wf ¼ �aubDuf and � aw

bDwf ¼ 0: ð10Þ

To simplify notation, we assume au ¼ aw ¼ 1 in this section.It is more convenient to consider the single component func-

tions uð1Þf and uð2Þf ,

wð1Þf ¼ �bDuð1Þf and � bDwð1Þf ¼ 0;

wð2Þf ¼ �bDuð2Þf and � bDwð2Þf ¼ 0:

We focus on two types of boundary conditions. First, we pose thefirst type of boundary conditions

uðkÞf ¼ @nuðkÞf ¼ 0 on bC n bCi for k ¼ 1;2: ð11Þ

Second, we are concerned with a mixture of boundary conditions(see Fig. 2)

uð1Þf ¼ @nuð1Þf ¼ 0 and wð1Þf ¼ @nwð1Þf ¼ 0 on bCin [ bCout;

uð2Þf ¼ @nuð2Þf ¼ 0 and wð2Þf ¼ @nwð2Þf ¼ 0 on bCwall; ð12Þ

which we call second type of boundary conditions. The interface con-ditions for ûf are given as usual, uf ¼ us on bCi:

Remark 2.1. Using the second type of boundary conditions in arectangular domain where the coordinate axes match the Cartesiancoordinate system, as shown in Fig. 2, leads to mesh movementonly in the tangential direction. This effect reduces mesh celldistortion because only the perpendicular directions of ûf and wf

are 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 monolithicsolution algorithms. In the following, we focus on a monolithicdescription of the coupled problem. We define a continuous vari-able û for all bX defining the deformation in bXs and supportingthe transformation in bXf . Thus, we skip the subscripts, and becausethe definition of bAf coincides with the previous definition of bAs,we define in bX:bA :¼ idþ u; bF :¼ I þ ru; bJ :¼ detðbFÞ: ð13Þ

Furthermore, the velocity v is a common continuous function forboth subproblems, whereas the pressure p is discontinuous. For theconvenience 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 ). Findfv ; u; pg 2 fvD þ bV0g � fuD þ bV0g � bL, such that vð0Þ ¼ v0 andû(0) = û0, for almost all time steps t, and

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


ðbJqf @tv; wvÞbX f

þ ðqfbJðbF�1ðv � @t uÞ � rÞvÞ; wvÞbX f

þ ðbJrfbF�T ; rwvÞbX f

þ ðqs@tv; wvÞbXsþ ðbJrs

bF�T ; rwvÞbXs� hg; wvibCN

� ðqfbJ f f ; w

vÞbX f� ðqs f s; w

vÞbXs¼ 0 8wv 2 bV 0;

ð@t u� v ; wuÞbXsþ ðrg ; rwuÞbX f

� hrgnf ; wuibC i¼ 0 8wu 2 bV 0;

ðddivðbJbF�1v f Þ; wpÞbX f

þ ðps; wpÞbXs¼ 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 twocoupling conditions on the interface. The continuity of velocityacross bCi is strongly enforced by requiring one common continu-ous velocity field on the whole domain bX. The continuity of normalstresses is given by

ðbJrsbF�T ns;w

vÞbC i¼ ðbJrf

bF�T nf ;wvÞbC i

: ð14Þ

By 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 arenecessary to prevent spurious feedback of the displacementvariables û and w. For more details on this, we refer to [7].

3. Discretization

In this section, we focus on the discretization in time and spaceof the fluid-structure interaction Problem 2.4. Our method ofchoice are finite differences for time discretization and a Galerkinfinite element method for spatial treatment.

3.1. Variational formulation in an abstract setting

In the domain bX and the time interval I ¼ ½0; T�, we consider thefluid-structure interaction Problem 2.4 with harmonic or lin-ear-elastic mesh motion in an abstract setting (the biharmonicproblem is straightforward): Find bU ¼ fv ; u; pg 2 X, wherebX0 :¼ fvD þ bV0g � fuD þ bV0g � bL, such thatZ T

0

bAðbUÞð bWÞdt ¼Z T

0

bF ð bWÞdt 8 bW 2 bX0: ð15Þ

The linear form bFð bWÞ and the semi-linear form bAðbUÞð bWÞ are definedbybF ð bWÞ ¼ ðqs f s; w

vÞbXs; ð16Þ

andbAðbUÞð bWÞ ¼ ðbJqf@tv ; wvÞbX f

þ ðqfbJðbF�1v � rÞvÞ; wvÞbX f

� ðqfbJðbF�1@t u � rÞvÞ; wvÞbX f

þ hg; wvibCN

� ðqfbJ f f ; w

vÞbX f

þ ðbJrfbF�T ; rwvÞbX f

þ ðqs@tv ; wvÞbXsþ ðbJrs

bF�T ; rwvÞbXsþ ð@t u; wuÞbXs

� ðv; wuÞbXsþ ðauru; rwuÞbX f

� haunf ru; wuibC i

þ ðddivðbJbF�1v f Þ; wpÞbX fþ ðps; w

pÞbXs: ð17Þ

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




;


3.2. Time discretization

The abstract problem Eq. 15 can either treated by a full time–space Galerkin formulation, which has been investigated for fluidproblems [25]. Alternatively, the Rothe method can be used incases where the time discretization is based on finite differenceschemes. A classical scheme for problems with a stationary limitis the the (implicit) backward Euler scheme (BE), which is stronglyA-stable, but only from first order, and dissipative. It is later used inthe 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 itis therefore well-suited for computing solutions with rough dataand 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. Thesekinds of problems are now formulated as One-step h scheme [14].This design has the advantage that it can easily be extended tothe Fractional-Step-h scheme.

We (formally) define the following semi-linear forms and groupthem into four categories: time equation terms (including the timederivatives), implicit terms (e.g., the incompressibility of fluid),pressure terms, and all remaining terms (stress terms, convection,etc.):

bATðbUÞð bWÞ ¼ ðbJqf @tv ; wvÞbX f

� ðqfbJðbF�1@t u � rÞvÞ; wvÞbX f

þ ðqs@tv ; wvÞbXsþ ð@t u; wuÞbXs

;bAIðbUÞð bWÞ ¼ ðauru; rwuÞbX f� haunf ru; wuibC i

þ ðddivðbJbF�1v f Þ; wpÞbX f

þ ðps; wpÞbXs

;

bAEðbUÞð bWÞ ¼ ðqfbJðbF�1v � rÞvÞ; wvÞbX f

þ ðbJrf ;vubF�T ; rwvÞbX f

þ ðbJrs;vubF�T ; rwvÞbXs

� ðv ; wuÞbXsbAPðbUÞð bWÞ ¼ ðbJrf ;pbF�T ; rwvÞbX f

;

ð18Þ

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

rf ;vu ¼ qf mf ðrvbF�1 þ bF�TrvTÞ;

rs;vu ¼ bJ�1bF ðksðtrbEÞI þ 2lsbEÞbF�T ;

rf ;p ¼ �bJpf IbF�T :

The time derivative in bATðbUÞð bWÞ is approximated by a backwarddifference quotient. For the time step tm 2 Iðm ¼ 1;2; . . .Þ, we com-pute v :¼ vm; u :¼ um via

bATðbUm;kÞð bWÞ ¼ qfbJm�1

2v � vm�1

k; wv

� �bX f

� qf ðbJbF�1 u� um�1

k� rÞv ; wv

� �bX f

þ qsv � vm�1

k; wv

� �bXs

þ u� um�1

k; wu

� �bXs

;

where bJm�12 ¼bJm�bJm�1

2 , ûm :¼ û(tm), vm :¼ vðtmÞ, and bJ :¼ bJm :¼bJðtmÞ. The former time step is given by vm�1, etc.

3.2.1. Basic-h schemeLet the previous solution bUm�1 ¼ fvm�1; um�1; pm�1g and the

time step k :¼ km = tm � tm�1 be given.


Find bUm ¼ fvm; um; pmg such that

bATðbUm;kÞð bWÞ þ hbAEðbUmÞð bWÞ þ bAPðbUmÞð bWÞ þ bAIðbUmÞð bWÞ¼ �ð1� hÞbAEðbUm�1Þð bWÞ þ hbFðbUmÞð bWÞ þ ð1� hÞbF ðbUm�1Þð bWÞ:

The concrete scheme depends on the choice for the parameter h. Inparticular, we get the backward Euler scheme for h = 1, the Crank–Nicolson scheme for h ¼ 1

2, and the shifted Crank–Nicolson forh ¼ 1

2þ km [24].

3.2.2. Fractional-step-h schemeWe choose h ¼ 1�

ffiffi2p

2 ; h0 ¼ 1� 2h, and a ¼ 1�2h1�h ; b ¼ 1� a. The

time step is split into three consecutive sub-time steps. LetUm�1 ¼ fvm�1; um�1; pm�1g and the time step k :¼ km = tm � tm�1 begiven.

Find bUm ¼ fvm; um; pmg such thatbATðbUm�1þh;kÞð bWÞ þ ahbAEðbUm�1þhÞð bWÞ þ hbAPðbUm�1þhÞð bWÞþ bAIðbUm�1þhÞð bWÞ¼ �bhbAEðbUm�1Þð bWÞ þ hbF ðbUm�1Þð bWÞ;

bATðbUm�h;kÞð bWÞ þ ahbAEðbUm�hÞð bWÞ þ h0bAPðbUm�hÞð bWÞþ bAIðbUm�hÞð bWÞ ¼ �ah0bAEðbUm�1þhÞðWÞ þ h0bF ðbUm�hÞð bWÞ;bATðbUm;kÞð bWÞ þ ahbAEðbUmÞð bWÞ þ hbAPðbUmÞð bWÞþ bAIðbUmÞð bWÞ ¼ �bhbAEðbUm�1Þð bWÞ þ hbFðbUm�hÞð bWÞ: ð19Þ

3.3. Spatial discretization

The time discretize equations are the starting point for theGalerkin discretization in space. To this end, we construct finitedimensional subspaces bX0

h � 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. Amesh consists of quadrilateral or hexahedron cells bK . They performa non-overlapping cover of the computation domain bX � Rd, d = 2,3. The corresponding mesh is given by bT h ¼ fbKg. The discretizationparameter in the reference configuration is denoted by h andis a cell-wise constant that is given by the diameter hbK of thecell bK .

On bT h, conforming finite element spaces for vh; uh; ph, and wh

are denoted by the space bV h � bV . We prefer the biquadratic, dis-continuous-linear Qc

2=Pdc1 element. The definitions of the spaces

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

bV h :¼ vh 2 ½CðbXhÞ�d; vhjbK 2 ½Q 2ðbK Þ�d 8bK 2 bT h; vhjbCnbC i¼ 0

� �;

bPh :¼ ph 2 ½bL2ðbXhÞ�; phjbK 2 ½P1ðbK Þ� 8bK 2 bT h

n o:

We consider for each bK 2 bT h the bilinear transformationrK : bK unit ! K , where bK unit denotes the unit square. Then, the Qc

2

element is defined by

Qc2ðbK Þ ¼ q � r�1

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

with dim Qc2 ¼ 9, which means nine local degrees of freedom. The

Pdc1 element consists of linear functions defined by

Pdc1 ðbK Þ ¼ q � r�1

K : q 2 span < 1; x; y >

with dim Pdc1 ðKÞ ¼ 3.





Defining the displacement variables ûh and wh is straightfor-ward. The property of the Qc

2=Pdc1 element is continuity of

the velocity values across different mesh cells. However, thepressure is defined by discontinuous test functions. In addition,this element preserves local mass conservation, is of loworder, gains the inf-sup stability, and therefore is an optimalchoice for both fluid problems and fluid-structure interactionproblems.

Remark 3.1. Computation of fluid-structure interaction withbiharmonic mesh motion has more computational cost at eachtime step than just using a linear elasticity problem. This is becausean additional equation is added to the problem. Because we use adirect solver for the linear sub-problems, the condition numberdoes not play a role. In the context of a Galerkin finite elementscheme, 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 Qc

2elements for ûh and wh.

3.4. Linearization

Time and spatial discretization results for each single time stepin a nonlinear quasi-stationary problem

bAðbUmÞð bWÞ ¼ bF ð bWÞ 8 bW 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-correctionproblem

bA 0ðbUjmÞðdbUm; bWÞ ¼ �bAðbUj

mÞð bWÞ þ bF ð bWÞ;Ujþ1

m ¼ Ujm þ kdbUm: ð20Þ

Here k 2 (0,1] is used as damping parameter for line searchtechniques. The directional derivative bA0ðbUÞðdbU ; bWÞ; is definedby

bA 0ðbUÞðdbU ; bWÞ :¼ lime!0

1ebAðbU þ edbUÞð bWÞ � bAðbUÞð bWÞn o

¼ ddebAhðbU þ edbUÞð bWÞ��

e¼0:

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 becumbersome. Nevertheless, in this context, we use the exactJacobian matrix to identify the optimal convergence properties ofthe Newton method.

3.4.1. Implementation aspectsIn this section, we present an example of one specific direc-

tional derivative that includes all of the necessary steps. Derivationof 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 formbATðbUÞð bWÞ, Eq. 18, that is part of the fluid convection term in ALEcoordinates. It holds

bAconvðbUÞð bWÞ ¼ ðqfbJðbF�1@t u � rÞvÞ; wvÞbX f

¼ ðqf rvbJbF�1@tu; wvÞbX f:

In this case, the directional derivative bA0convðbUÞðdbU ; bWÞ in the direc-tion dbU ¼ fdv ; du; dpg is given by


bA0convðbUÞðdbU ; bWÞ ¼ rdvbJbF�1 u� um�1

k; wv

� �þ rvðbJbF�1Þ0ðduÞ u� um�1

k; wv

� �þ rvbJbF�1 du

k; wv

� �: ð21Þ

In two dimensions the deformation matrix reads in explicit form:

bF ¼ I þ ru ¼ 1þ @1u1 @2u1

@1u2 1þ @2u2

!;

which brings us to

bJbF�1 ¼ 1þ @2u2 �@2u1

�@1u2 1þ @2u2

!and its directional derivative in direction du ¼ ðdu1; du2Þ:

ðbJbF�1Þ0ðduÞ ¼ @2du2 �@2du1

�@1du2 @2du2

!:

This expression is part of the second term shown in Eq. 21. Theremaining expressions for directional derivatives can be derivedin an analogous way. For more details on computation of the direc-tional derivatives on the interface, please refer to [6,7]. Accuratedetermination of the directional derivatives is also indispensablefor optimization problems in which the performance of the Newtonalgorithms heavily depend on [26].

3.5. Mesh refinement

The computations are performed on globally-refined meshesand heuristically-refined meshes. We use two kinds of heuristicmesh refinement. The first step is geometric refinement aroundthe interface. The second step is measurement of the smoothnessof 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 controllingthe influence of these parameters. In one technique selective meshdeformation is used that is based on the shape and volume changesof the cells [8]. Another method is augmented by a stiffening powerthat determines the rate by which smaller elements are stiffenedmore than larger ones [11].

Mesh cells touching the interface are critical with respect tomesh degeneration. Therefore, the aim of these parameters shouldbe to maintain the shape of the fluid mesh cells, close to the inter-face, by controlling the determinant bJ of the transformation bF . Theparameters must be adjusted in a certain way for different testsconfigurations which is problematic because the exact parameterchoice is a priori unknown. This problem does not occur whenusing the biharmonic mesh motion model. An optimal, smoothmesh 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 motionmodel, we use au :¼ h2bK auðxÞ, with small parameter auðxÞ > 0.The parameters ak and al for the linear-elasticity mesh modelcan be defined in a similar way.




-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 biharmonicmesh motion models for the CSM 1 test. Degeneration of the mesh cells correspondsto negative values of bJ , for the case using the harmonic mesh motion model withconstant parameter.


4. Numerical tests

In this section, we compare the different mesh motion modelsusing 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]. Wecompare our results to the results given in these articles and ex-tend the CSM test to a new configuration to show the improvedperformance of the biharmonic model with regard to the meshmotion.

4.1. CSM tests

In these test cases, the fluid is set to be initially at rest in bXf . Anexternal gravitational force f s is applied only to the elastic beam,producing a visible deformation. The tests are performed astime-dependent problems (backward Euler), leading to a steadystate solution. For the harmonic and linear-elastic model, we usethe time step size k = 0.02 s; for the biharmonic model we usek = 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 onthe harmonic mesh motion model without a mesh-position depen-dent material parameter. It turns out that the harmonic modeldoes not hold any more. The reference values are taken from[27]. In the second example CSM 4, only the gravitational force isincreased causing the elastic beam to become much moredeformed.

Table 1

4.1.1. ConfigurationThe 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 lengthl = 0.35 m and height h = 0.02 m. The right lower end is positionedat (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 trailingedge of the structure, measuring x- and y-deflections of the beam.

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

DoF ux(A)[ � 10�3 m] uy(A)[ � 10�3 m]

27744 �25.2199 �121.97142024 �25.2805 �122.13272696 �25.3101 �122.214133992 �25.3268 �122.259

4.1.2. Boundary conditionsFor 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 thebiharmonic 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 imposedleading to a zero mean value of the pressure at this part of theboundary.

4.1.3. ParametersWe choose for our computation the following parameters. For

the (resting) fluid we use .f = 103 kg m�3, mf = 10�3 m2 s�1. Theelastic structure is characterized by .s = 103 kg m�3, ms = 0.4,ls = 5⁄105 kg m�1 s�2. The vertical force is chosen as f s ¼ 2 m s�2.

4.1.4. Discussion of the CSM 1 testWe 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 canbe seen in Fig. 3, where the minimal values, min (bJ), of the ALE-transformation determinant bJ are sketched as function plots. Ourresults indicate that using the harmonic approach (which is thesimplest one) is sufficient for this numerical test.

4.1.5. Discussion of the CSM 4 testDue to the higher gravitational force f s ¼ 4 m s�2 applied to the

structure, the beam is deformed to a greater extent than in the pre-viously described test.

For this test case, only the biharmonic mesh motion modelequipped with the second type of boundary conditions leads to re-sults. This effect occurs because the outermost mesh layer is notdeformed when using the first type of boundary conditions. How-ever, the second type can deal with this factor because the mesh isallowed to move in a tangential direction along the outer boundaryand 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 minðJÞ can be studied in Fig. 6.

We observe that the biharmonic mesh motion model leads to asmoother fluid mesh compared to the other two mesh motionmodels. The function plots of the min(bJ) in Figs. 3 and 6 indicatethat the global minimum of the biharmonic models is further awayfrom zero compared to the global minimums of the harmonic andlinear-elasticity approaches. In other words, the mesh distortion issmaller 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 largestructural deformations [28]. We modify the given configurationby enlarging the height of the membrane. We use the incompress-ible Mooney–Rivlin model (which is capable to deal with largedeformations and large strains) to characterize the structure. Thetest is driven by a pressure difference between bCin and bCout . Wechoose the time step size k = 0.01 and the implicit Euler time step-ping scheme.

4.2.1. Configuration and ParametersThe configuration is sketched in Fig. 7. We use the following

parameters to run the simulation: .f = 1000.0 kg m�3, and




Fig. 4. CSM 4 test with the harmonic and linear-elastic mesh motion models and gravitational force f s ¼ 4ms�2. Both models lead to mesh distortion close to the lowerboundary.

Fig. 5. CSM 4 test with biharmonic mesh motion model and gravitational force f s ¼ 4ms�2. 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

Min

(J)

Time

HarmonicPseudo elasticity

Biharmonic 1st type bcBiharmonic 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 firstthree models.

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



mf = 0.004 m2 s�1 for the fluid. For the structure, we use.s = 800.0 kg m�3, ls = 2.0 ⁄ 107 Pa, l2 = 1.0 ⁄ 105 Pa.

4.2.2. Initial conditions and boundary conditionsOn 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 allremaining parts we prescribe homogeneous Dirichlet conditionsfor the velocity and the displacement:

u ¼ 0 on bCin [ bCout [ bCwall;

v ¼ 0 on bCwall;

mf@nu� pI � nf ¼ pinflow � nf on bCin;

mf@nu� pI � nf ¼ 0 on bCout:

The pressure pin is increased during the computation, i.e.,pin ¼ t � pinitial with pinitial ¼ 5:0 � 106 Pa:

4.2.3. Quantities of comparison

(1) y-deflection of the structure at the point A(t) withA(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 minðbJÞ.4.2.4. Results

The qualitative behavior of the numerical results does agreewith the findings in [28]. However, we use quadrilaterals for thediscretization, whereas the other authors use triangles. This isone reason why we get a smaller maximal deformation of themembrane (Figs. 8 and 9). Moreover, we use the same overall meshfor the fluid and the structure domains, which leads to high




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

Min

(J)

Time

Harmonic with constant parameterHarmonic

Linear elasticityBiharmonic 1st type bc

Fig. 8. Function plots of min (bJ) for the mesh motion models of the membrane onfluid test. Degeneration of mesh cells corresponds to negative values of bJ , arising inthe first three models.


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 areapplied to an unsteady fluid-structure interaction problem. Weconsider the numerical benchmark test FSI 2, which was proposedin [27]. The configuration is the same as for the CSM tests, sketchedin 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 propermesh 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 discussedpreviously.

4.3.1. Boundary conditionsA parabolic inflow velocity profile is given on bCin by

v f ð0; yÞ ¼ 1:5U4yðH � yÞ

H2 ; U ¼ 1:0 m s�1:

At the outlet bCout the ‘do-nothing’ outflow condition is imposedwhich lead to zero mean value of the pressure at this part of theboundary. The remaining boundary conditions are chosen as inthe CSM test cases.

-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-di

s

Time

global 1global 2global 3

Fig. 9. Large deformation membrane fluid test with the biharmonic mesh model for thrstretch of the cell under the membrane.


4.3.2. Initial conditionsFor the non-steady tests one should start with a smooth in-

crease of the velocity profile in time. We use

v f ðt; 0; yÞ ¼ v f ð0; yÞ1�cos p

2tð Þ2 if t < 2:0 s

v f ð0; yÞ 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 theelastic beam. They form a closed path in which the forcescan be computed with the help of line integration. Theformula is evaluated on the fixed reference domain bX andreads:

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0

y-st

retc

h

ee differ

differe

ðFD; FLÞ ¼ZbS bJrall

bF�T � nds

¼ZbSð circleÞ

bJrfbF�T � nf dsþ

ZbSðbeamÞ

bJrfbF�T � nf ds:

ð22Þ

The quantities of interest for this time dependent test caseare represented by the mean value, amplitudes, and frequency ofx- and y-deflections of the beam in one time period T ofoscillations.

4.3.4. ParametersWe choose for our computation the following parameters. For

the fluid we use .f = 103 kg m�3, mf = 10�3 m2 s�1. The elastic struc-ture is characterized by .s = 104 kg m�3, ms = 0.4, ls = 5⁄105

kg m�1 s�2.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 2–4. The refer-ence values are taken from [30]. In general, to verify convergencewith respect to space and time, at least three different meshlevels and time step sizes should be presented. Three differentmesh levels are not possible when working with the simplestapproach: 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 onthe 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.

0.1 0.2 0.3 0.4 0.5 0.6 0.7

Time

global 1global 2global 3

ent mesh levels. Left: vertical displacement of the point (0.0,0.005). Right:

nt mesh motion techniques. Comput Struct (2011), doi:10.1016/



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 2Results 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]. Thefrequencies f1[s�1] and f2[s�1] of ux 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)[ � 10�3] uy(A)[ � 10�3] FD FL

5032 3.0e�3 �14.62 ± 13.17 1.06 ± 79.87 210.78 ± 73.97 �1.83 ± 295.85032 2.0e�3 �14.66 ± 13.19 1.02 ± 78.30 211.83 ± 73.72 �1.83 ± 295.85032 1.0e�3 �14.70 ± 13.20 0.94 ± 80.39 210.17 ± 75.34 �0.40 ± 298.45032 0.5e�3 �14.63 ± 13.17 1.08 ± 80.34 212.61 ± 74.31 �0.84 ± 297.419488 3.0e�3 �13.73 ± 11.79 1.20 ± 78.20 207.72 ± 72.63 �0.21 ± 227.119488 2.0e�3 �13.59 ± 11.79 1.25 ± 77.96 207.52 ± 72.07 �2.03 ± 226.519488 1.0e�3 �13.63 ± 11.77 1.24 ± 78.11 201.96 ± 73.15 �1.86 ± 231.219488 0.5e�3 �13.59 ± 11.77 1.23 ± 78.06 203.59 ± 70.37 �1.25 ± 221.5(Ref.) 0.5e�3 �14.85 ± 12.70 1.30 ± 81.70 215.06 ± 77.65 0.61 ± 237.8


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 andthe related benchmark computations. These values also varies inthe literature [27,30,31]. Nevertheless, on the finest meshes ofthe linear-elastic and biharmonic mesh motion models, all of thevalues 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 biharmonicmodel for moderate deformations.

4.4.1. Configuration and ParametersThe 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 m�3, and mf = 0.01 m2 s�1 for the fluid. For the structure,we use .s = 1.0 kg m�3, ms = 0.4, and ls = 500.0 kg m�1 s�2.

4.4.2. Initial conditions and boundary conditionsA constant parabolic inflow velocity profile is given on bCin by

v f ðt;0; yÞ ¼ 16:0UyzðH � yÞðH � zÞ

H4 ; U ¼ 0:45 m s�1:




Table 3Results 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]. Thefrequencies f1[s�1] and f2[s�1] of ux and uy vary in a range of 3.83 � 3.90 (Ref.3.86) and 1.91 � 1.95 (Ref. 1.93), respectively.


5032 3.0e�3 �13.93 ± 12.48 1.20 ± 78.02 205.11 ± 69.01 0.21 ± 284.25032 2.0e�3 �13.88 ± 12.55 1.21 ± 77.72 204.63 ± 68.06 0.39 ± 277.45032 1.0e�3 �13.99 ± 12.62 1.23 ± 78.03 201.65 ± 70.81 �0.15 ± 277.919488 3.0e�3 �13.47 ± 11.70 1.28 ± 77.52 205.86 ± 70.05 0.34 ± 225.319488 2.0e�3 �13.54 ± 11.71 1.29 ± 77.77 206.71 ± 70.02 �0.31 ± 226.519488 1.0e�3 �13.60 ± 11.77 1.28 ± 77.99 205.49 ± 70.46 �0.29 ± 228.029512 3.0e�3 �13.00 ± 11.33 1.26 ± 76.09 202.92 ± 67.09 0.20 ± 216.029512 2.0e�3 �13.06 ± 11.37 1.28 ± 76.29 203.74 ± 67.17 0.48 ± 216.629512 1.0e�3 �13.11 ± 11.42 1.26 ± 76.50 203.28 ± 67.69 0.54 ± 217.7(Ref.) 0.5e�3 �14.85 ± 12.70 1.30 ± 81.70 215.06 ± 77.65 0.61 ± 237.8

Table 4Results 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 quantitiesof interest: ux, uy [m], FD, FL[N]. The frequencies f1[s�1] and f2 [s�1] of ux and uy vary in a range of 3.83 � 3.88 (Ref.3.86) and 1.92 � 1.94 (Ref.1.93), respectively.


27744 3.0e�3 �13.63 ± 11.80 1.27 ± 78.72 207.22 ± 71.13 �0.57 ± 230.627744 2.0e�3 �13.72 ± 11.84 1.26 ± 78.38 208.12 ± 71.18 �0.30 ± 232.627744 1.0e�3 �13.74 ± 11.85 1.28 ± 78.48 209.46 ± 71.43 �0.06 ± 231.727744 0.5e�3 �13.66 ± 11.81 1.28 ± 78.32 208.96 ± 71.60 �0.06 ± 238.242024 3.0e�3 �13.34 ± 11.57 1.40 ± 77.08 204.81 ± 68.54 0.79 ± 221.542024 2.0e�3 �13.36 ± 11.55 1.28 ± 77.18 205.61 ± 68.67 0.51 ± 223.042024 1.0e�3 �13.38 ± 11.58 1.31 ± 77.44 206.11 ± 68.26 0.62 ± 221.242024 0.5e�3 �13.27 ± 11.52 1.23 ± 77.25 207.05 ± 68.87 0.30 ± 230.672696 3.0e�3 �14.43 ± 12.46 1.35 ± 80.71 212.50 ± 76.40 0.18 ± 234.672696 2.0e�3 �14.49 ± 12.44 1.19 ± 80.66 213.49 ± 76.39 0.13 ± 235.772696 1.0e�3 �14.49 ± 12.46 1.16 ± 80.63 213.39 ± 75.25 0.23 ± 234.272696 0.5e�3 �14.40 ± 12.39 1.25 ± 80.55 213.55 ± 76.06 0.30 ± 240.2(Ref.) 0.5e�3 �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 m 1 m

1.37 m

1.5 m

A(t)

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

Table 5Results 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]); eachscaled by 10�6. 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 – – – – – –


At the outlet bCout the ‘do-nothing’ outflow condition is imposed,leading to zero mean value of the pressure on this part of theboundary.

4.4.3. Quantities of comparison

(1) x-, y-, and z-deflection of the beam at A(t) withA(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. ResultsThe results for the different quantities of interest are in agree-

ment between both of the mesh motion models, as illustrated inTable 5.

4.4.5. Computational cost for the numerical tests. Finally, we summa-rize our observations with regard to the computational cost perNewton 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 biharmonicequation is much more expensive in each Newton step. Concretely,the cost in two dimensions is five times higher for the biharmonicmesh motion model compared to the other two models. In threedimensions the factor for low amount of degrees of freedom(DoF) is again five. Whereas for 624 cells in three dimensions thefactor becomes 70. It seems to be the linear solver, but it is stillan open question. A detailed study is given in Table 6. This resultindicate, using the biharmonic model with UMFPACK in threedimensions becomes prohibitive in a sequential solution process.




Table 6CPU Times per Newton step for solving the linear equations on a Intel Xeon machinewith a 2.40 GHz processor and sequential programming.

Test case Cells DoF Mesh motion model CPU time (in s)

CSM 1 992 19488 linear-elastic 2.2 ± 0.2CSM 1 2552 51016 linear-elastic 7.8 ± 1.0CSM 1 4664 93992 linear-elastic 26.0 ± 2.6CSM 1 992 27744 biharmonic 6.2 ± 0.5CSM 1 2552 72696 biharmonic 67.5 ± 13.5CSM 1 4664 133992 biharmonic 206.8 ± 37.63D FSI 78 5856 harmonic 0.4 ± 0.023D FSI 281 19694 harmonic 6.0 ± 0.23D FSI 624 39312 harmonic 25.2 ± 2.83D FSI 78 8628 biharmonic 10.8 ± 0.33D FSI 281 28979 biharmonic 206.0 ± 6.63D FSI 624 57720 biharmonic 2475 ± 495.0


Due to enormous memory usage for direct solvers, one shoulduse iterative solvers [16,17]. Further, adaptive mesh refinementis 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 showthat 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 thecontrary, our results suggest that the biharmonic approach is moreexpensive, 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 touse economic local mesh-refinement because of the prohibitivecomputational cost of using global mesh refinement. Therefore,we propose to use discretization in a closed variational setting thatcan be extended to a full time–space Galerkin discretization for thewhole problem. This setting is the basis for an automatic meshadaption with the ‘dual weighted residual’ (DWR) method, whichalso allows for a ‘goal-oriented’ a posteriori error estimation. Here,the adjoint solutions will have to be derived; for this task, a closedsemilinear 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.

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