Journal ArticleDOI

# A Reduced Basis Model with Parametric Coupling for Fluid-Structure Interaction Problems

01 Apr 2012-SIAM Journal on Scientific Computing (Society for Industrial and Applied Mathematics)-Vol. 34, Iss: 2, pp 1187-1213
TL;DR: A reduced order model with reliable a posteriori error bounds is obtained for steady fluid-structure interaction problems and rapid convergence of the reduced solution of the parametrically coupled problem as the number of geometric parameters is increased.
Abstract: We present a new model reduction technique for steady fluid-structure interaction problems. When the fluid domain deformation is suitably parametrized, the coupling conditions between the fluid and the structure can be formulated in the low-dimensional space of geometric parameters. Moreover, we apply the reduced basis method to reduce the cost of repeated fluid solutions necessary to achieve convergence of fluid-structure iterations. In this way a reduced order model with reliable a posteriori error bounds is obtained. The proposed method is validated with an example of steady Stokes flow in an axisymmetric channel, where the structure is described by a simple one-dimensional generalized string model. We demonstrate rapid convergence of the reduced solution of the parametrically coupled problem as the number of geometric parameters is increased.

### Schematic of the control points and resulting free-form parametric deformation

• This allows the user to keep the number of FFD parameters to a desired low level (in their case roughly 5-10 parameters).
• Parametric coupling of fluid and structure.
• 1. Formulation of the coupled problem in the parameter space.
• This requires showing that the nearest point projection is continuous in the strong H 2 -norm topology.

### 5.2. Empirical interpolation method for nonaffine problems.

• Terms, and similarly for the other forms.
• In practice the EIM has been quite useful for solving nonaffinely parametrized PDEs with the reduced basis method [20, 36, 47] .
• For the free-form deformation detailed in Sect. 3.2 in fact the forms B and F are affine due to the fact that the map T FFD is polynomial.
• For generic nonpolynomial shape parametrizations the situation remains more challenging.

### 6.3. Convergence and accuracy of the coupling algorithm.

• By introducing a parametric free-form deformation of the flow geometry the fluid equations can be written as parametric partial differential equations on a fixed domain.
• The authors then applied the reduced basis method to the fluid equations to obtain an efficient reduced model with certified error bounds.
• The geometric deformation parameters were also used to couple the fluid domain to a 1-d wall equation, where the parameters acted as the coupling variables.
• The authors demonstrated that for a modest number of free-form deformation parameters an approximate coupling between fluid and structure can be achieved.
• Numerical simulations were based on the rbMIT toolkit [23] developed by the group of Anthony Patera as well as the MLife fluid mechanics solvers originally authored by Fausto Saleri.

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fluid-structure interation problems.
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Version: Submitted Version
Article:
Lassila, T., Quarteroni, A. and Rozza, G. (2012) A reduced basis model with parametric
coupling for fluid-structure interation problems. SIAM Journal on Scientific Computing , 34
(2). A1187 - A1213. ISSN 1064-8275
https://doi.org/10.1137/110819950
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A REDUCED BASIS MODEL WITH PARAMETRIC COUPLING
FOR FLUID-STRUCTURE INTERACTION PROBLEMS
TONI LASSILA
, ALFIO QUARTERONI
§
, AND GIANLUIGI ROZZA
Abstract. We present a new model reduction technique for steady ﬂuid-structure interaction
problems. When the ﬂuid domain deformation is suitably parametrized, the coupling conditions
between the ﬂuid and structure can be formulated in the low-dimensional space of geometric p aram -
eters. Moreover, we apply the reduced basis method to reduce the cost of repeated ﬂuid solutions
necessary to achieve convergence of ﬂuid-structure iterations. In this way a reduced order m odel
with reliable a posteriori error bounds is obtained. The proposed method is validated with an exam-
ple of steady Stokes ﬂow in an axisymmetric channel, where the structure is described by a simple
1-d generalized string model. We demonstrate rapid convergence of the reduced solution of the
parametrically coupled problem as the number of geometric parameters is increased.
AMS subject classiﬁcations. 65N30, 74F10, 76D07
Key words. ﬂuid-structure interaction, mo del reduction, incompressible Stokes equations, re-
duced basis method, free-form deformation
1. Introduction. The numerical simulation of Fluid-Structure Interaction (FSI)
problems is an important topic in wide areas of engineering and medical research.
Concerning the latter, of great importance is the modelling of blood ﬂow in the large
arteries of the human cardiovascular system, where pulsatile ﬂows combined with a
high degree of deformability of the arterial walls togeth er cause large displacement
eﬀects that cannot be neglected when attempting to accurately model the ﬂow dy-
namics of the system. High ﬁdelity computational ﬂuid dynamics and structural
mechanics solvers b ased on, for example, th e Finite Element Method (FEM) need to
be combined in a framework that is challenging both from a mathematical as well as
implementation viewpoint. For an ove rv i ew of cardiovascular modelling techniques we
refer to [42, 44] and the bo ok [14]. The comple xi ty and nonlinearity of FSI problems
has u ntil rece ntly lim i te d the scope of physically meaningful simulations to just small
and isolated sections of arteries. When at t em pt i n g to consider the entire cardiovas-
cular sy st e m as a complex network of diﬀerent time and spatial scales, Model Order
Reduction (MOR) techniques can accurately and reliably re du ce the nonlinear FSI
models to computationally more cost-eﬃcient ones.
In the geom e tr i c multiscale ap pr oach to MOR [13] the ﬂow network is decomposed
to smaller parts that are joined t oget h er using physical coupling conditions, and each
part of which is model l ed at a level necessary to capture the relevant local dynamics
of the system. The target for our proposed reduced model is those parts of the car-
diovascular network, where a full ﬁdeli ty 3-d Navier-Stokes solution is not nece ss ary,
but whe re ﬂuid-stru ct u re interaction eﬀects are still important. The reduced model
should fulﬁll two conditions: (i) it should have certiﬁed a posteriori error bou nd s that
can be tuned to the user’s requirements, and (ii) it should have suﬃciently low online
computational memory require ments to ﬁt on one parallel node of a supercomputer.
Mode lling and Scientiﬁc Computing (CMCS),
´
Ecole Polytechnique ed´erale de Lausanne, Lau-
sanne, Switzerland (toni.lassila@epﬂ.ch, gi anlu igi.ro zza@ epﬂ. ch, alﬁo.quarteroni@epﬂ.ch). Support
provided by ERC-Mathcard Project (ERC-2008-AdG 227058).
Department of Mathematics and Systems Analysis, Aalto University, Helsinki, Finland. Sup-
ported by the Emil Aaltonen Foun datio n.
§
Mode lling and Scientiﬁc Computing (MOX), Politecnico di Milano, Milan, Italy.
1

2 T. LASSILA, A. QUARTERONI AND G. ROZZA
An important aspect of any large-displacement FSI problem is ﬁnding the conﬁg-
uration of the interface between ﬂuid and structure. The pro c ess is typically iterative:
a trial conﬁgu rat i on of the geometry is used to solve the ﬂuid and structure subprob-
lems, the coupling conditions are test ed , and if th ey are not satisﬁed within a desired
degree of accuracy then the trial conﬁguration is updated and the step is repeated. A
traditional approach to FSI is that the discrete mesh is updated on each iteration by
moving the boundary nodes and adjusting the interior mesh points to ensure mesh
quality. This approach lead s to a large number of coupling variables (the total number
of mesh points on the fr ee boundary). An external parametrization of the geometr y
can be used to drive down the number of coupling variables. When considering sim-
ple ﬂow geometries the shape of the deformable wall can be direct ly parametrized
e.g. with splines. For realistic geometries it might be necessary to parametrize the
geometry in a way that is relatively indepen de nt of its description .
There are many shape parametrization methods to choose from. Comparisons of
diﬀerent shape parametrization techniques from a ﬂuid dynam ic s viewpoint can b e
found in [52], and from a model reduction viewpoi nt in [33]. We propose to descri be
the deformat ion s of the ﬂuid channel with Fre e -Form Deformations (FFDs) [53]. They
are a technique for smooth parametric deformations of arbitr ar y shapes embedded in
the grid of control points. FFDs can be used to give a ﬂexible and global parametric
deformation of a ﬁxed reference domain that is comp l et e ly independent of the shape
and its computational mesh. Model reduction for FFD-based shape p aram et ri z at ion s
has been previously considered for the shape design of airfoils in potential [27] and
thermal ﬂows [50]. In cardiovascular applications, FFDs have been used to track the
motion of the left ventricle (see [34] for a review), and to solve an optimal shape
design problem of an aorto-coronaric bypass anastomoses [32].
After parametrizing the geometry with a FFDs we need to address the coupling
between ﬂuid and structure. We use the deformation parameters of the FFD as
coupling variables. A ﬁxed point coupling algorithm can be written in the parameter
space rather than the displacement space. Again an iterat i ve procedure is needed to
ensure the coupling conditions are satisﬁed to a desired tolerance. Thus a pot e ntially
large number of parametric PDE solutions for the ﬂuid equations need t o be performed
in diﬀerent parametric conﬁgurations.
To reduce the memory requirements and the online computat i onal cost of solving
the ﬂuid system, we apply the Reduced Basis (RB) method (originally proposed and
analyzed in [1, 11, 37 , 41]). It is a rel iabl e MOR method for paramet r ic PDEs. An
overview can be found in [49] and a more detailed exposition in [38]. The attractive-
ness of these methods is based on their ability to give certiﬁed a posteriori bounds
on the error of the ﬁeld solutions and their outputs when compared to the underlying
FEM solution. We use the reduced basis method to reduce the computational cost of
the steady Stokes equations in diﬀerent conﬁgurations of the geometry.
The s t ru ct u r e is t he following: in Sect. 2 we introduce the steady FSI problem of
incompressible Stokes e q uat i ons cou pl e d to an elli p t ic 1-d gen er ali z ed st ri n g equ ati on .
This is a benchmark probl em for which the existence of solutions has been demon-
strated in [18, 19] and whose numerical solution has been previously considered e.g.
in [29, 35, 54]. In Sect. 3 we discuss the geometric parametrization and introduce the
free-form deformations. In Sect. 4 we couple the ﬂuid and structure in the space of
parametric deformations. In Sect. 5 the reduced basis method for the ﬂuid equations
is detailed, and we discuss a posteriori error bounds of the solutions. In Sect. 6 we
present numerical results valid ati n g our approach. Sect. 7 contains some conclusions.

REDUCED BASIS WITH PARAMETRIC COUPLING FOR FSI 3
2. Th e steady ﬂuid-stru ctu r e interaction model. We use the following
standard notations: R
d
, d = 1, 2, 3, is a bounded open set, H
k
(Ω) is the
Sobolev space of functions with weak derivatives up t o order k square-integrable on
X, H
k1/2
(Ω) i s the space of functions that are traces of H
k
(Ω) on the boundary
Ω, H
k
0
(Ω) is the subspace of functions whose trace vanishes on Ω; C
k,α
(Ω) is the
space of functions with derivatives up t o order k being older-continuous with expo-
nent 0 < α 1 (if α = 1 these are the Lipschitz-continuous functions); L
2
(Ω) is the
space of square-integrable functions, and L
(Ω) is the space of essentially bounded
functions on Ω.
o
Γ
w
Γ
in
x
2
x
1
φ
η(x
1
)
R(x
1
)
Γ
out
Fig. 2. 1. Axisymmetric ﬂow geometry for the ﬂuid-structure intera c tio n model problem
2.1. Fluid model: the steady incompressible Stokes equations. We as-
sume the ﬂow geometry represented in Fig. 2.1 that is axisymmetric with cylindrical
coordinates (x, φ) = (x
1
, x
2
, φ)
o
× [0, 2π). The lengthwise cross-section of the
domain
o
:= (0, L) × (0, R) depend s on the unknown radius R(x
1
) of the channel,
which satisﬁes R(x
1
) := R
0
+ η(x
1
) > 0, where η H
2
0
(0, L) is a function describing
the smooth displacement of the outer wall from its reference conﬁgurati on ( a cyli n -
0
> 0). We assume also axis ym met r i c forces, f = f (x) and
g = g(x
2
). Owing to the axial symmetry we can consider the steady Stokes equa-
tions for incompressible ﬂuid ﬂow i n the two-dimensional d omai n
o
(η) with m i xe d
boundary conditions on its boundary Γ(η) = Γ
in
Γ
out
Γ
w
(η), that is
· σ + f = 0 in
o
(η)
· u = 0 in
o
(η)
u = 0 on Γ
w
, u = g on Γ
in
, σ · n = 0 on Γ
out
, (2.1)
where u is the ﬂuid velocity ﬁeld, and σ is the symmetric Cauchy stress tensor.
The data are assumed to have the following regularity: f [L
2
(Ω
o
)]
2
and g
H
1/2
(Γ), where the space [H
1/2
(Γ)]
2
= γ
Γ
([H
1
(Ω
o
)]
2
) is deﬁned as usual wit h the
continuous trace operator γ
Γ
on Γ. We denote by
b
g [H
1
0
(Ω
o
)]
2
any continuous
extension of the Dirichlet data to the ﬂuid domain. Assuming a Newtonian ﬂuid,
the stress-s t rain relationship is given by σ = pI + ν (u + u
t
) , where ν denotes
the dynamic viscosity and p is t he pressure ﬁeld. After choosin g the velocity space
V := [H
1
Γ
d
(Ω
o
(η))]
2
of functions that vanish on Γ
d
= Γ
in
Γ
w
and the pressure space
Q := L
2
(Ω
o
(η)), a mixed weak formulation of the eq u at ion s is to ﬁnd u V and

4 T. LASSILA, A. QUARTERONI AND G. ROZZA
p Q s.t.
Z
o
[νu : v p · v] d =
Z
o
f · v d
Z
o
ν
b
g · v d for all v V
Z
o
q · u d =
Z
o
q ·
b
g d for all q Q
.
(2.2)
The treatment of the inhomogeneous Dirichlet condition is done by lifting this is the
standard way when ai mi n g at reduced basis approximations in parameter-dependent
domains
1
. For notational b r ev i ty we deﬁne the bilinear forms
A(u, v) := ν
Z
o
u : v d , B(q, v) :=
Z
o
q · v d (2.3)
and the linear f orm
F(v) :=
Z
o
f · v d. (2.4)
Then (2.2) can be compactly written as
(
A(u, v) + B(p, v) = F(v) A(
b
g, v) for all v V
B(q, u) = −B(q,
b
g) for all q Q
. (2.5)
With our assumptions on the displacement function η the domain
o
is of class C
0,1
and the Stokes equations have a unique solution (u, p) V × Q [16].
2.2. Structural model: the 1-d generalized string equation. Next we give
the equ at i ons for the structural displacement function η. These equations are in the
Lagrangian form on the undeformed conﬁguration of the wall, which we identify as
the interval (0, L) in our simpliﬁ ed 1-d case. We assume the displacements are small
and always in the normal direction of Γ
w
, the tangential displacement being equal
to zero. The equilibrium equation for the structural displacement is chosen as the
second order equation with a fourth orde r perturbation (with ε > 0 small)
ε
4
η
x
4
1
kGh
2
η
x
2
1
+
Eh
1 ν
2
P
η
R
0
(x
1
)
2
= τ
Γ
w
, x
1
(0, L) (2.6)
where h is the wall thickness, k is the Timoshenko shear cor re ct i on factor, G the
shear modulus, E the Young modulus, ν
P
the Poisson ratio, R
0
reference conﬁguration, and τ
Γ
w
denotes the applied traction. This is a simpliﬁed
1-d equation for the structure that is often used in haemodynamic ﬂuid-struct u r e
interaction problems as a “ﬁrst approximation” [44]. We h ave added a fourth order
term in order to have added regularity for the displacement. The weak form of (2.6)
is to ﬁnd the structural displacement in the normal dire ct i on η D s.t .
τ
Γ
w
(φ) = ε
R
L
0
2
η
x
2
1
2
φ
x
2
1
dx
1
+kGh
R
L
0
η
x
1
φ
x
1
dx
1
+
Eh
1ν
2
P
R
L
0
η φ
R
0
(x
1
)
2
dx
1
:= C(η, φ) (2.7)
for all φ in the space D := H
2
0
(0, L) of kinematically admissible displacements.
1
Other approaches, such as Lagrange multipliers or Nitsche’s method, that might seem more
attractive from a mathematical viewpoint may run into pro blems when dealing with the reduced
basis method.

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### "A Reduced Basis Model with Parametr..." refers background or methods in this paper

• ...discretized problem, and thus a sufficient condition for stability is that the finite element velocity and pressure spaces Vh and Qh should be chosen such that they satisfy the discrete Ladyzhenskaya-Babuška-Brezzi (LBB) condition [16]...

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