Mesh Moving Techniques for Fluid-Structure Interactions With Large Displacements.pdf
-
Upload
harley-souza-alencar -
Category
Documents
-
view
225 -
download
0
Transcript of 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 -
7/27/2019 Mesh Moving Techniques for Fluid-Structure Interactions With Large Displacements.pdf
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
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.2011.02.019http://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 -
7/27/2019 Mesh Moving Techniques for Fluid-Structure Interactions With Large Displacements.pdf
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
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.2011.02.019http://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 -
7/27/2019 Mesh Moving Techniques for Fluid-Structure Interactions With Large Displacements.pdf
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.
4 T. Wick / Computers and Structures xxx (2011) xxxxxx
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.2011.02.019http://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 -
7/27/2019 Mesh Moving Techniques for Fluid-Structure Interactions With Large Displacements.pdf
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.
T. Wick / Computers and Structures xxx (2011) xxxxxx 5
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.2011.02.019http://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 -
7/27/2019 Mesh Moving Techniques for Fluid-Structure Interactions With Large Displacements.pdf
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.
6 T. Wick / Computers and Structures xxx (2011) xxxxxx
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.2011.02.019http://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 -
7/27/2019 Mesh Moving Techniques for Fluid-Structure Interactions With Large Displacements.pdf
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
T. Wick / Computers and Structures xxx (2011) xxxxxx 7
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.2011.02.019http://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 -
7/27/2019 Mesh Moving Techniques for Fluid-Structure Interactions With Large Displacements.pdf
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.
8 T. Wick / Computers and Structures xxx (2011) xxxxxx
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.2011.02.019http://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 -
7/27/2019 Mesh Moving Techniques for Fluid-Structure Interactions With Large Displacements.pdf
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.
T. Wick / Computers and Structures xxx (2011) xxxxxx 9
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.2011.02.019http://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 -
7/27/2019 Mesh Moving Techniques for Fluid-Structure Interactions With Large Displacements.pdf
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
10 T. Wick / Computers and Structures xxx (2011) xxxxxx
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.2011.02.019http://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 -
7/27/2019 Mesh Moving Techniques for Fluid-Structure Interactions With Large Displacements.pdf
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
T. Wick / Computers and Structures xxx (2011) xxxxxx 11
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.2011.02.019http://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 -
7/27/2019 Mesh Moving Techniques for Fluid-Structure Interactions With Large Displacements.pdf
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 2552 51016 linear-elastic 7.8 1.0
CSM 1 4664 93992 linear-elastic 26.0 2.6
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
12 T. Wick / Computers and Structures xxx (2011) xxxxxx
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/