ChapterPDF Available

An immersed boundary method based on the L2-projection approach


Abstract and Figures

In this paper we present a framework for FSI simulations. Taking inspiration from the Immersed Boundary technique introduced by Peskin, we employ 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 L 2-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.
Content may be subject to copyright.
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:
Barna Becsek
ARTORG Center for Biomedical Engineering Research, University of Bern, Murtenstrasse 50
3008 Bern, Switzerland e-mail:
Hadi Zolfaghari
ARTORG Center for Biomedical Engineering Research, University of Bern, Murtenstrasse 50
3008 Bern, Switzerland e-mail:
Prof. Dr. Dominik Obrist
ARTORG Center for Biomedical Engineering Research, University of Bern, Murtenstrasse 50
3008 Bern, Switzerland e-mail:
Dr. Patrick Zulian
Institute Of Computational Science, Universit`
a della Svizzera Italiana, Via Giuseppe Buffi 13, 9600
Lugano, Switzerland e-mail:
Prof. Dr. Rolf Krause
Institute Of Computational Science, Universit`
a della Svizzera Italiana, Via Giuseppe Buffi 13, 9600
Lugano, Switzerland e-mail:
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 ( 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 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 xand b
xA, as the material (or reference)
coordinates of the point b
s, with b
sRddenoting the material (reference)
configuration of the solid domain (Fig. 1).
We assume that the map b
s×IRdis a one-to-one correspondence between
the material b
xand the actual xpositions occupied by the elastic structure during the
time interval of interest tI= [0 T], s. t. (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:
P=don b
t+ρfvf·vf+pfµvf=ffsi on f(b)
·vf=0 on f(c)
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
x,t)is the related dis-
placement field, b
x,t)is the first Piola-Kirchhoff stress tensor, dis a prescribed
external body force, and b
x·is the divergence operator computed in the reference
configuration. For an hyperelastic material, the first Piola-Kirchhoff stress tensor b
is related to the deformation through a constitutive equation derived from a given
scalar valued energy function Ψ, i. e. b
F∂ ψ(b
E, where b
FI)is the
Lagrangian-Green strain tensor and b
Fis the deformation gradient tensor defined as
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
ρs0ρf) [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
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-
where tis the time step size, as:=2b
t2and vs:=b
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
scan be approximated by a discrete domain b
sand the associated non-
conforming unstructured mesh b
s}. The Galerkin formu-
lation of the elastodynamics equation reads:
For every t(0; T]find b
s)]dso that:
s) + ah(uh
s) = 0(2)
By defining (Fh,δuh
s) = ah(uh
s)and using the Green’s formula
we get:
s) = (fh
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
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:
In order to guarantee the gradient of the pressure to be unique, the Schur comple-
ment DH1Gmust 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)[H1()]dand introduce the vector of Lagrange multipli-
ers λ
fsi with the related virtual variations, δλ
fsi Mh
where Th
frepresents the fluid grid.
In the following, the projection operator P:Vh
sis defined by focusing on the
scalar case, which means that for each component of the velocity vh
f), we
may find wh
s), such that the following weak-equality condition
f,i))δ λ h
f si dV =Zb
s,i)δ λ h
fsi dV =0δ λ h
fsi Mh
By writing vh
sand δ λ h
fsi in term of basis functions (here the index iis omitted for
an easy notation), i.e. vh
sand δ λ h
fsi =kJfsi δ λ k
(where Js,Jfand Jfsi are index sets), we get the so called mortar integrals: Bk,l=
fsi dV and Sk,j=RIhNj
fsi dV . Equation (5) can be then written in the fol-
lowing algebraic form:
The transpose of Tis used to transfer the reaction force from the solid to the fluid
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
swith respect to the L2-inner product:
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
Step 4: The Navier-Stokes problem is solved by using the force ffsi as source
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
fsikfor the relative
convergence criterion and kfl+1
fsi fl
fsikfor 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 (,
whereas the library MOONoLiTH ( 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
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 6s1, and the frequency for the x-displacement is about 11s1. 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
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 30s1to 30s1, 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 L2projection 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.
1. Peskin, Charles S.“Flow patterns around heart valves: a numerical method.” Journal of com-
putational physics 10.2 (1972): 252-271.
2. Liu, Wing Kam, et al.“Immersed finite element method and its applications to biological
systems.” Computer methods in applied mechanics and engineering 195.13 (2006): 1722-
3. Nestola, Maria GC, et al. “Three-band decomposition analysis in multiscale FSI models
of abdominal aortic aneurysms.” International Journal of Modern Physics C 27.02 (2016):
4. Devendran, Dharshi, and Charles S. Peskin.“An immersed boundary energy-based method for
incompressible viscoelasticity.” Journal of Computational Physics 231.14 (2012): 4613-4642.
5. Griffith, B. E., and Luo, X. (2017). “Hybrid finite difference/finite element immersed bound-
ary method”. International journal for numerical methods in biomedical engineering. 33.12
6. Gil, Antonio J., et al. The immersed structural potential method for haemodynamic applica-
tions. Journal of Computational Physics 229.22 (2010): 8613-8641.
7. Boffi, D., et al. On the hyper-elastic formulation of the immersed boundary method. Computer
Methods in Applied Mechanics and Engineering 197.25 (2008): 2210-2231.
8. Henniger, Rolf, Dominik Obrist, and Leonhard Kleiser.“High-order accurate iterative solu-
tion of the Navier-Stokes equations for incompressible flows.” PAMM 7.1 (2007): 4100009-
9. Fackeldey, K., et al. “Coupling molecular dynamics and continua with weak constraints”.
Multiscale Modeling and Simulation, 9.4 (2011) 1459-1494.
10. Krause, Rolf, and Patrick Zulian.“A parallel approach to the variational transfer of discrete
fields between arbitrarily distributed unstructured finite element meshes.” SIAM Journal on
Scientific Computing 38.3 (2016): C307-C333.
11. Turek, Stefan, and Jaroslav Hron.“Proposal for numerical benchmarking of fluid-structure
interaction between an elastic object and laminar incompressible flow.” Springer Berlin Hei-
delberg, 2006. 371-385.
12. Turek, Stefan, et al.“Numerical benchmarking of fluid-structure interaction: A comparison
of different discretization and solution approaches.” Fluid Structure Interaction II. Springer,
Berlin, Heidelberg, 2011. 413-424.
13. Griffith, B. E., and Xiaoyu Luo.“Hybrid finite difference/finite element version of the im-
mersed boundary method.” International Journal for Numererical Methods in Engineering
1-26 ( Submitted in revised form, 2012) DOI: 10.1002/nme.
14. Hesch, C., et al.“A mortar approach for fluid-structure interaction problems: Immersed strate-
gies for deformable and rigid bodies.” Computer Methods in Applied Mechanics and Engi-
neering 278 (2014): 853-882.
... Several strategies have been already developed for the two-field and mixed three-field formulations, e.g. [11][12][13][14][15][16][17][18][19][20][21][22][23]. Here, a class of block-triangular preconditioners for accelerating the iterative convergence by Krylov subspace methods is proposed for the stabilized mixed hybrid approach. ...
... where In (19), · e denotes the jump across the edge/face e, |M| is the d-measure of M, and β M is a stabilization term depending on the physical parameters. It is worth noticing that, unlike other stabilization techniques [6,8], the pressure-jump method has also a physical interpretation. ...
Full-text available
We consider a mixed hybrid finite element formulation for coupled poromechanics. A stabilization strategy based on a macro-element approach is advanced to eliminate the spurious pressure modes appearing in undrained/incompressible conditions. The efficient solution of the stabilized mixed hybrid block system is addressed by developing a class of block triangular preconditioners based on a Schur-complement approximation strategy. Robustness, computational efficiency and scalability of the proposed approach are theoretically discussed and tested using challenging benchmark problems on massively parallel architectures.
The Internodes method is a general purpose method to deal with non-conforming discretizations of partial differential equations on 2D and 3D regions partitioned into disjoint subdomains. In this paper we are interested in measuring how much the Internodes method is conservative across the interface. If hp -fem discretizations are employed, we prove that both the total force and total work generated by the numerical solution at the interface of the decomposition vanish in an optimal way when the mesh size tends to zero, i.e., like $\mathcal {O}(h^{p})$ O ( h p ) , where p is the local polynomial degree and h the mesh-size. This is the same as the error decay in the H ¹ -broken norm. We observe that the conservation properties of a method are intrinsic to the method itself because they depend on the way the interface conditions are enforced rather then on the problem we are called to approximate. For this reason, in this paper, we focus on second-order elliptic PDEs, although we use the terminology (of forces and works) proper of linear elasticity. Two and three dimensional numerical experiments corroborate the theoretical findings, also by comparing Internodes with Mortar and WACA methods.
Full-text available
Hydro-mechanical processes in rough fractures are highly non-linear and govern productivity and associated risks in a wide range of reservoir engineering problems. To enable high-resolution simulations of hydro-mechanical processes in fractures, we present an adaptation of an immersed boundary method to compute fluid flow between rough fracture surfaces. The solid domain is immersed into the fluid domain and both domains are coupled by means of variational volumetric transfer operators. The transfer operators implicitly resolve the boundary between the solid and the fluid, which simplifies the setup of fracture simulations with complex surfaces. It is possible to choose different formulations and discretization schemes for each subproblem and it is not necessary to remesh the fluid grid. We use benchmark problems and real fracture geometries to demonstrate the following capabilities of the presented approach: (1) resolving the boundary of the rough fracture surface in the fluid; (2) capturing fluid flow field changes in a fracture which closes under increasing normal load; and (3) simulating the opening of a fracture due to increased fluid pressure.
Full-text available
An essential ingredient for the discretization and numerical solution of coupled multiphysics or multiscale problems is stable and efficient techniques for the transfer of discrete fields between nonmatching volume or surface meshes. Here, we present and investigate a new and completely parallel approach. It allows for the transfer of discrete fields between unstructured volume and surface meshes, which can be arbitrarily distributed among different processors. No a priori information on the relation between the different meshes is required. Our inherently parallel approach is general in the sense that it can deal with both classical interpolation and variational transfer operators, e.g., the L2-projection and the pseudo-L2-projection. It includes a parallel search strategy, output dependent load-balancing, and the computation of element intersections, as well as the parallel assembling of the algebraic representation of the respective transfer operator. We describe our algorithmic framework and its implementation in the library MOONoLith. Furthermore, we investigate the efficiency and parallel scalability of our new approach using different examples in three dimensions. This includes the computation of a volume transfer operator between 2 meshes with 2 billion elements in total and the computation of a surface transfer operator between 14 different meshes with 5:9 billion elements in total. The experiments have been performed with up to 12;288 cores.
Full-text available
Comparative benchmark results for different solution methods for fluid-structure interaction problems are given which have been developed as collaborative project in the DFG Research Unit 493. The configuration consists of a laminar incompressible channel flow around an elastic object. Based on this benchmark configuration the numerical behavior of different approaches is analyzed exemplarily. The methods considered range from decoupled approaches which combine Lattice Boltzmann methods with hp-FEM techniques, up to strongly coupled and even fully monolithic approaches which treat the fluid and structure simultaneously.
Full-text available
We describe new benchmark settings for the rigorous evaluation of different methods for fluid-structure interaction problems. The configurations consist of laminar incompressible channel flow around an elastic object which results in self-induced oscillations of the structure. Moreover, characteristic flow quantities and corresponding plots are provided for a quantitative comparison.
Full-text available
One of the most challenging problems in dynamic concurrent multiscale simulations is the reflectionless transfer of physical quantities between the different scales. In particular, when coupling molecular dynamics and finite element discretizations in solid body mechanics, often spurious wave reflections are introduced by the applied coupling technique. The reflected waves are typically of high frequency and are arguably of little importance in the domain where the finite element discretization drives the simulation. In this work, we provide an analysis of this phenomenon. Based on the gained insight, we derive a new coupling approach, which neatly separates high and low frequency waves. Whereas low frequency waves are permitted to bridge the scales, high frequency waves can be removed by applying damping techniques without affecting the coupled share of the solution. As a consequence, our new method almost completely eliminates unphysical wave reflections and deals in a consistent way with waves of arbitrary frequencies. The separation of wavelengths is achieved by employing a discrete L2-projection, which acts as a low pass filter. Our coupling constraints enforce matching in the range of this projection. With respect to the numerical realization this approach has the advantage of a small number of constraints, which is computationally efficient. Numerical results in one and two dimensions confirm our theoretical findings and illustrate the performance of our new weak coupling approach.
Computational modeling plays an important role in biology and medicine to assess the effects of hemodynamic alterations in the onset and development of vascular pathologies. Synthetic analytic indices are of primary importance for a reliable and effective a priori identification of the risk. In this scenario, we propose a multiscale fluid-structure interaction (FSI) modeling approach of hemodynamic flows, extending the recently introduced three-band decomposition (TBD) analysis for moving domains. A quantitative comparison is performed with respect to the most common hemodynamic risk indicators in a systematic manner. We demonstrate the reliability of the TBD methodology also for deformable domains by assuming a hyperelastic formulation of the arterial wall and a Newtonian approximation of the blood flow. Numerical simulations are performed for physiologic and pathologic axially symmetric geometry models with particular attention to abdominal aortic aneurysms (AAAs). Risk assessment, limitations and perspectives are finally discussed.
Incompressible viscoelastic materials are prevalent in biological applications. In this paper we present a method for incompressible viscoelasticity in which the elasticity of the material is described in Lagrangian form (i.e. in material coordinates), and Eulerian (spatial) coordinates are used for the equations of motion and to enforce the incompressibility condition. The elastic forces are computed directly from an energy functional without the use of stress tensors, and the immersed boundary method is used to communicate between Lagrangian and Eulerian variables. The method is first applied to a warm-up problem, in which a viscoelastic incompressible material fills a two-dimensional periodic domain. For this problem, we study convergence of the velocity field, the deformation map, and the Eulerian force density. The numerical results indicate that the velocity field and deformation map converge strongly at second order and the Eulerian force density converges weakly at second order. Incompressibility is well maintained, as indicated by area conservation in this 2D problem. Finally, the method is applied to a three-dimensional fluid–structure interaction problem with two different materials: an isotropic neo-Hookean model and an anisotropic fiber-reinforced model.
An iterative solution scheme for the incompressible Navier–Stokes equations is presented. It is split into inner and outer iteration cycles, such that the momentum and continuity equations are satisfied within prescribed accuracy. The spatial discretization is based on high-order finite differences which makes it well suited for massively parallel computers. This is demonstrated in a scaling test. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
The immersed boundary (IB) method is both a mathematical formulation and a numerical method for fluid–structure interaction problems, in which immersed incompressible visco-elastic bodies or boundaries interact with an incompressible fluid. Previous formulations of the IB method were not able to treat appropriately immersed materials of finite, nonzero thickness modeled by general hyper-elastic constitutive laws because of the lack of appropriate transmission conditions between the immersed body and the surrounding fluid in the case of a nonzero jump in normal stress at the solid–fluid interface. (Such a jump does not arise when the solid is comprised of fibers that run parallel to the interface, but typically does arise in other cases, e.g., when the solid contains elastic fibers that terminate at the solid–fluid interface). We present a derivation of the IB method that takes into account in an appropriate way the missing term. The derivation presented in this paper starts from a separation of the stress that appears in the principle of virtual work. The stress is divided into its fluid-like and solid-like components, and each of these two terms is treated in its natural framework, i.e., the Eulerian framework for the fluid-like stress and the Lagrangian framework for the solid-like stress. We describe how the IB method can be used in conjunction with standard formulations of continuum mechanics models for immersed incompressible elastic materials and present some illustrative numerical experiments.
The subject of this paper is the flow of a viscous incompressible fluid in a region containing immersed boundaries which move with the fluid and exert forces on the fluid. An example of such a boundary is the flexible leaflet of a human heart valve. It is the main achievement of the present paper that a method for solving the Navier-Stokes equations on a rectangular domain can now be applied to a problem involving this type of immersed boundary. This is accomplished by replacing the boundary by a field of force which is defined on the mesh points of the rectangular domain and which is calculated from the configuration of the boundary. In order to link the representations of the boundary and fluid, since boundary points and mesh points need not coincide, a semi-discrete analog of the δ function is introduced. Because the boundary forces are of order h−1, and because they are sensitive to small changes in boundary configuration, they tend to produce numerical instability. This difficulty is overcome by an implicit method for calculating the boundary forces, a method which takes into account the displacements that will be produced by the boundary forces themselves. The numerical scheme is applied to the two-dimensional simulation of flow around the natural mitral valve.