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An immersed boundary method based on the
L2-projection approach
M.G.C. Nestola, B. Becsek, H. Zolfaghari, P. Zulian, D. Obrist and R. Krause
Abstract In this paper we present a framework for FSI simulations. Taking inspi-
ration from the Immersed Boundary technique introduced by Peskin [1] we em-
ploy the Finite-Element method for discretizing the equations of the solid structure
and the Finite-Difference method for discretizing the fluid flow. The two discretiza-
tions are coupled by using a volume based L2-projection approach to transfer elastic
forces and velocities between the fluid and the solid domain. We present results
for an FSI benchmark which describes self-induced oscillating deformation of an
elastic beam in a flow channel.
Dr. Maria Giuseppina Chiara Nestola
Institute Of Computational Science, Universit`
a della Svizzera Italiana, Via Giuseppe Buffi 13, 9600
Lugano, Switzerland e-mail: nestom@usi.ch
Barna Becsek
ARTORG Center for Biomedical Engineering Research, University of Bern, Murtenstrasse 50
3008 Bern, Switzerland e-mail: barna.becsek@artorg.unibe.ch
Hadi Zolfaghari
ARTORG Center for Biomedical Engineering Research, University of Bern, Murtenstrasse 50
3008 Bern, Switzerland e-mail: hadi.zolfaghari@artorg.unibe.ch
Prof. Dr. Dominik Obrist
ARTORG Center for Biomedical Engineering Research, University of Bern, Murtenstrasse 50
3008 Bern, Switzerland e-mail: dominik.obrist@artorg.unibe.ch
Dr. Patrick Zulian
Institute Of Computational Science, Universit`
a della Svizzera Italiana, Via Giuseppe Buffi 13, 9600
Lugano, Switzerland e-mail: p.zulian@usi.ch
Prof. Dr. Rolf Krause
Institute Of Computational Science, Universit`
a della Svizzera Italiana, Via Giuseppe Buffi 13, 9600
Lugano, Switzerland e-mail: r.krause@usi.ch
1
2 M.G.C. Nestola, B. Becsek, H. Zolfaghari, P. Zulian, D. Obrist and R. Krause
1 Introduction
During the last decades, Fluid-Structure Interaction (FSI) [1, 2] has received con-
siderable attention due to various applications where a fluid and a solid interact with
each other (such as in aeronautics, turbomachinery, and biomedical applications).
In the literature, several approaches have been developed in order to reproduce
the interaction between a fluid and a surrounding solid structure, which can be clas-
sified in boundary-fitted and embedded boundary methods. In the boundary-fitted
methods, the fluid problem is resolved in a moving spatial domain over which the
incompressible Navier-Stokes equations are formulated in an Arbitrary Lagrange
Eulerian (ALE) framework [3] while the solid structure is usually described in a La-
grangian fashion. Although this approach is known to allow for accurate results at
the interface between solid and fluid, for scenarios that involve large displacements
and/or rotations, the fluid grid may become severely distorted, thus affecting both
the numerical stability of the problem and the accuracy of the solution.
In order to circumvent those difficulties, embedded boundary approaches such as
the Immersed Boundary method (IBM), have been introduced to model the fluid-
structure interaction on a stationary fluid grid analyzed in a Eulerian fashion. The
main aspect of this technique is the representation of the immersed solid material as
a force density in the Navier-Stokes equations.
In the literature related to the classical IBM, the volume of the solid is commonly
described by systems of fibres that resist extension, compression, or bending [1,
2, 4]. Some alternative approaches have been proposed on the basis of the Finite
Element Method for the spatial approximation of the Lagrangian quantities (force
densities, displacement field, etc.). In all these approaches the reaction force exerted
by the solid on the fluid is computed explicitly by using the fluid velocity field to get
the corresponding displacement of the solid structure [5, 6, 7].
In this paper, we describe an alternative framework for FSI simulations. We em-
ploy the Finite Difference Method for simulating the fluid flow and couple it with a
Finite Element Method for the structural problem. The main novelties of this work
are (I) the description of the solid body motion obtained by solving implicitly the
elastodynamic equations and (II) the treatment of the Lagrangian-Eulerian Interac-
tion which is achieved by means of Pseudo L2– projection. Such approach allows for
the transfer of data between non-matching structured (Cartesian) and unstructured
meshes arbitrarily distributed among different processors.
All the modules of the FSI computational frameworks are integrated into the
multi-physics simulation framework MOOSE (mooseframework.org). The code is
optimised for modern hybrid high-performance computing platforms such as the
Cray XC50 system at the Swiss National Supercomputing Centre CSCS.
2 Strong Formulation of the FSI Problem
The purpose of this section is to provide a brief description of the methodology
adopted in our framework to solve the FSI problem. Since the proposed approach
An immersed boundary method based on the L2-projection approach 3
follows the main principle of the IBM, we employ the standard Eulerian formula-
tion for the Navier-Stokes equations for incompressible flows, whereas the elastic
response of the embedded structure is described in a Lagrangian fashion.
Let Ω⊂Rd(with d=1,2,3) be a bounded Lipschitz domain denoting the phys-
ical region occupied by the coupled fluid-structure system. We label xa, as the spa-
tial Cartesian coordinates of the point x∈Ωand b
xA, as the material (or reference)
coordinates of the point b
x∈b
Ωs, with b
Ωs⊂Rddenoting the material (reference)
configuration of the solid domain (Fig. 1).
We assume that the map b
χ
χ
χ:b
Ωs×I→Rdis a one-to-one correspondence between
the material b
xand the actual xpositions occupied by the elastic structure during the
time interval of interest t∈I= [0 T], s. t. (b
x,t)→x=b
χ
χ
χ(b
x,t). Additionally, we
denote with Γ
fsi the physical interface between the fluid and the solid mesh.
The strong formulation of the complete FSI problem reads as follows:
b
ρs0
∂2b
us
∂t2−b
∇b
x·b
P=don b
Ωs(a)
ρf
∂vf
∂t+ρfvf·∇vf+∇pf−µ∆vf=ffsi on Ωf(b)
∇·vf=0 on Ωf(c)
vf=∂us
∂ton Γ
fsi (d)(1)
Here Eq. (1a) is the equation of the elastodynamics where b
ρs0is the mass density
per unit undeformed volume of the elastic structure, b
us=b
us(b
x,t)is the related dis-
placement field, b
P=b
P(b
x,t)is the first Piola-Kirchhoff stress tensor, dis a prescribed
external body force, and b
∇b
x·is the divergence operator computed in the reference
configuration. For an hyperelastic material, the first Piola-Kirchhoff stress tensor b
P
is related to the deformation through a constitutive equation derived from a given
scalar valued energy function Ψ, i. e. b
P=b
F∂ ψ(b
E)
∂b
E, where b
E:=1/2(b
FTb
F−I)is the
Lagrangian-Green strain tensor and b
Fis the deformation gradient tensor defined as
b
F=∇b
xx.
Fig. 1 Lagrangian (left) and Eulerian (right) coordinate systems adopted in the Immersed Bound-
ary method
4 M.G.C. Nestola, B. Becsek, H. Zolfaghari, P. Zulian, D. Obrist and R. Krause
Eq.s (1b)-(1c) represent the standard Navier-Stokes equations where ρfis the
fluid density, vfis the velocity field of the fluid, pfis the pressure, ∇xis the gradi-
ent operator, ∆xis the Laplacian operator computed in the current configuration and
ffsi is the force density source term generated by the embedded solid structure as we
will describe in Section 3.1.
Remark In the equation of the elastodynamics, (1a), the evaluation of the inertial
term must take care of the fluid in which it is embedded. This can be done by sub-
tracting the density of the fluid phase from the solid one (i.e. b
ρs0−ρf) [14]. It is
worth to pointing out that, since in our case the fluid velocity field is used to recover
the displacement of the FSI interface, this difference is restricted only to Γ
fsi.
3 Discretization of the FSI problem
In this section, we provide some details about the discretization in time and in space
of the solid and the fluid subproblem.
3.1 Solid Problem
For the time discretization of the solid problem the classical Newmark scheme is
adopted, which is based on a Taylor expansion of the displacements and the veloci-
ties:
b
us,n+1=b
us,n+∆tvs,n+∆t2
2((1−2β)as,n+2βas,n+1)
b
vs,n+1=b
vs,n+∆t((1−α)b
us,n+αb
as,n+1)
where ∆tis the time step size, as:=∂2b
us
∂t2and vs:=∂b
us
∂tare the the acceleration
and the velocity of the solid, respectively, and the parameters αand 2βare chosen
such that α=2β=1/2.
For the spatial discretization of the structure problem, we assume that the solid
domain b
Ωscan be approximated by a discrete domain b
Ωh
sand the associated non-
conforming unstructured mesh b
Th
s={b
Es⊆b
Ωh
s|Sb
Es=b
Ωh
s}. The Galerkin formu-
lation of the elastodynamics equation reads:
For every t∈(0; T]find b
uh
s(·,t)∈b
Vh
s⊂[H1(b
Ωh
s)]dso that:
(b
ρs0b
ah
s,δuh
s) + ah(uh
s,δuh
s)−(dh
s,δuh
s) = 0(2)
By defining (Fh,δuh
s) = ah(uh
s,δuh
s)−(dh
s,δuh
s)and using the Green’s formula
we get:
(b
ρs0b
ah
s,δuh
s)+(Fh,δuh
s) = (fh
fsi,δuh
s)L2(Γh
fsi)(3)
where fh
fsi represents the reaction force exerted by the solid structure on the fluid.
3.2 Fluid Problem
The time integration of the fluid problem is carried out by a 3rd order low-storage
Runge-Kutta scheme for both the advective and the diffusion terms [8].
For the discretization of Eq. (1b), the usage of high-order (6th) explicit-finite
differences leads to a linear system of equations of the form:
An immersed boundary method based on the L2-projection approach 5
"H G
D0#"vf
pf#="z
0#
Here the matrices Dand Gare the spatial discretization of the divergence and the
gradient operators, zis the discrete representation of the right hand side, whereas H
is the Helmholtz operator which coincides with the identity matrix (except for the
boundary conditions) due to the usage of a purely explicit time integration scheme.
By applying Dto the equation Hvf+Gpf=0, one may derive the following
equation for the pressure:
DH−1Gpf=DH−1z(4)
In order to guarantee the gradient of the pressure to be unique, the Schur comple-
ment DH−1Gmust be h-elliptic (i.e. must have only one zero eigenvalue). To this
aim Arakawa-C grids are adopted which combine several types of nodal points lo-
cated in different geometrical positions.
4L2- projection
For coupling the two subproblems we adopt a volume L2- projection which al-
lows for the transfer of discrete fields between non conforming meshes arbitrarily
distributed among several processors. Such an approach ensures convergence, effi-
ciency, flexibility and accuracy without requiring a priori information on the relation
between the different meshes. To this aim, we attach to the Finite Difference dis-
cretization Lagrangian basis functions [9], define the corresponding finite element
space as Vh
f=Vh
f(Th
f)⊂[H1(Ω)]dand introduce the vector of Lagrange multipli-
ers λ
λ
λh
fsi with the related virtual variations, δλ
λ
λh
fsi ∈Mh
fsi(b
Th
s∩Th
f)⊂[H1(b
Ωs∩Ω)]d,
where Th
frepresents the fluid grid.
In the following, the projection operator P:Vh
f→Vh
sis defined by focusing on the
scalar case, which means that for each component of the velocity vh
f,i∈Vh
f(Th
f), we
may find wh
s,i=P(vh
f,i)∈b
Vh
s(b
Th
s), such that the following weak-equality condition
holds:
Zb
Th
s∩Th
f
(vh
f,i−P(vh
f,i))δ λ h
f si dV =Zb
Th
s∩Th
f
(vh
f,i−wh
s,i)δ λ h
fsi dV =0∀δ λ h
fsi ∈Mh
fsi
(5)
By writing vh
f,wh
sand δ λ h
fsi in term of basis functions (here the index iis omitted for
an easy notation), i.e. vh
f=∑l∈Jfvl
fNl
f,wh
s=∑j∈Jswj
sNj
sand δ λ h
fsi =∑k∈Jfsi δ λ k
fsiNk
fsi
(where Js,Jfand Jfsi are index sets), we get the so called mortar integrals: Bk,l=
RIhNl
fNk
fsi dV and Sk,j=RIhNj
sNk
fsi dV . Equation (5) can be then written in the fol-
lowing algebraic form:
ws=S−1Bvf=Tvf(6)
The transpose of Tis used to transfer the reaction force from the solid to the fluid
grid.
In order to reduce the computational cost required to compute the inverse of the
matrix S, we adopt dual basis function for the functional space Mh
fsi. In this case this
6 M.G.C. Nestola, B. Becsek, H. Zolfaghari, P. Zulian, D. Obrist and R. Krause
functional space is spanned by a set of functions which are biorthogonal to the basis
functions of b
Vh
swith respect to the L2-inner product:
(Nk
fsi,Nj
s)L2(Ih)=δk,j(Nj
s,1)L2(Ih)∀k,j(7)
The usage of the dual basis functions corresponds to replacing the standard L2–
projection with a Pseudo–L2–projection, which allows for a more efficient evalua-
tion of the transfer operator Tsince the matrix Sbecomes diagonal. The assembly of
the transfer operator is done in several steps [10]: (a) we compute the overlapping
region by means of a tree search algorithm, (b) generate the quadrature points for
integrating in the intersecting region and (c) compute the local element-wise contri-
butions for the operators Band Sby means of numerical quadrature and assemble
the two mortar matrices.
5 Flow chart of the complete FSI algorithm.
In our framework a segregated approach is adopted to solve the fully coupled FSI
problem. More specifically, we use a fixed point (Picard) iteration scheme to used
to solve the arising coupled non-linear discrete system.
For a given time step nand given a starting solution at the Picard iteration l, the
following steps are performed within iteration l+1:
Step 1: Velocity values are transferred from the fluid to the solid grid.
Step 2: The elastodynamic equation is solved with the Dirichlet boundary con-
ditions (Eq. (1d)).
Step 3: The reaction force ffsi is computed and transferred from the solid to the
fluid.
Step 4: The Navier-Stokes problem is solved by using the force ffsi as source
term.
Step 5:Suitable residual norms are computed between the FSI interaction force
terms evaluated at iterations land l+1, i. e. kfl+1
fsi −fl
fsik∞/kf0
fsikfor the relative
convergence criterion and kfl+1
fsi −fl
fsik∞for the absolute convergence criterion
[6], and compared with given threshold values. This ensures the satisfaction of
the coupling between the two problems, thus leading to either a new Picard iter-
ation or a new time step n + 1 otherwise.
The numerical solver IMPACT (Incompressible (Turbulent) flows on Massively
PArallel CompuTers) is employed to solve the non-dimensional Navier-Stokes
equations [8]. The solid problem and the assembling of the transfer operator are im-
plemented in the finite-element framework MOOSE (www.mooseframework.org),
whereas the library MOONoLiTH (bitbucket.org/zulianp/parmoonolith) is used for
detecting the overlapping region between the fluid and the solid grids and computing
the corresponding inteserctions.
An immersed boundary method based on the L2-projection approach 7
6 Numerical Results
In this section we present results related to the Turek-Hron FSI benchmark which
considers the incompressible flow of a Newtonian fluid around an elastic solid struc-
ture composed of a disk and a rectangular trailing beam.
The dimensions of the fluid channel are (Fig. 2 (a)): length Lf=3.0mand height
Hf=0.41m. The disk center is positioned at C= (0.2m,0.2m)(measured from the
left bottom corner of the channel) and the radius is r=0.05m. The elastic structure
bar has length Ls=0.35mand height Hs=0.02m; the right bottom corner is po-
sitioned at (0.6m,0.19m), and the left end is fully attached to the circle. The fluid
properties are ρf=1000kg/m3and µ=1Pa ·swhich lead to a Reynolds number
of 200. The density of the solid structure is the same as the fluid phase, and a Saint-
Venant Kirchhoff model is adopted as constitutive law, for which the first Piola-
Kirchhoff stress tensor is defined as: b
P=b
F(λtr(b
E)I+2µb
E)with µ=2.0MPa and
λ=4.7MPa. Periodic boundary conditions are imposed along the inlet and the out-
let of the fluid channel together with no-slip boundary conditions on the top and the
bottom. Moreover at the inlet a Poiseuille flow with a centerline velocity of 1.5m/s
is enforced by a fringe region appended downstream.
In Fig. 2 (b) we show the displacements in xand ydirection of a control point P
located at the end of the elastic beam (A≡(0.6m,0.2m), Fig 2 (a)). The amplitude
of the last period of oscillation is in the range of 0.03mfor the vertical displacement
and of 0.0025mfor the horizontal displacement; the frequency of the y-displacement
is about 6s−1, and the frequency for the x-displacement is about 11s−1. All values
are in good agreement with the original benchmark results [11]. In Fig. 2 (c) we
also show the forces exerted by the lift and drag forces acting on the cylinder and
the beam structure together. Again the values agree well with the results obtained
by other numerical methods applied to the same problem [12]. Finally, the fluid
Fig. 2 (a) Geometry of the Turek-Hron benchmark. (b) Amplitude displacement in x and y direc-
tion of a control point A located at the end of the elastic beam. (c) Lift and drag forces. (d) Fluid
vorticity.
8 M.G.C. Nestola, B. Becsek, H. Zolfaghari, P. Zulian, D. Obrist and R. Krause
vorticity is depicted in Fig. 2 (d) ranging from −30s−1to 30s−1, in agreement with
numerical values reported in Griffith [13] .
7 Conclusion
In this article we present a novel FSI framework based on the IBM. The description
of the solid motion, obtained by solving implicitly the elastodynamic equations,
ensures to yield extra stability and robustness. Moreover, the use of the software
IMPACT and of Pseudo L2−projection allows for a completely parallel framework
suitable for the simulation of complex and large FSI simulations such the flow in
human arteries and heart valves.
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