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Journal Club Theme of Oct. 1 2008: Fluid - Structure Interaction, an overview of current trends and challenges

Starting with a solid mechanics background, I have been recently interested in fluid-structure interactions for applications in biomechanics of soft-tissues; and thought that the iMechanica Journal Club forum would be a good opportunity to share my personal experience and start a discussion on this challenging inter-disciplinary subject.

Fluid - structure interaction (FSI) has been a growing field of research and application in the last decades. Initially restricted to simple analytical models, like 2D wing sections with beam-like elastic response, FSI has been developing significantly with the progress of numerical modelling and computing power to finally reach a certain maturity in the last years. The driving applications for the development of FSI models are mostly aerospace, energy production, civil structures (large suspended bridges), naval applications, motor sports, micro-fluidic and last but not least biomedical applications.

As FSI unites two very prolific fields, namely fluid and solid mechanics, a nearly infinite number of problem classes can be identified; basically at least one class for each combination of fluid and structural analysis type. However, the most representative classes of FSI analysis include:

  • the so-called “one way” FSI is typically a “non interaction” problem where the solution of one physics (usually the fluid wall stresses) are projected as a boundary condition to the other physical problem. This type of problem can be useful as a first study for static FSI analysis.
  • static FSI analysis: only the steady state force equilibrium between fluid and solid domains is of interest in this case. Both the fluid and solid models neglect the transient terms of the PDEs. Convergence is checked based on force equilibrium and interface position stationarity.
  • transient FSI analysis: in this case both fluid and structure models include transient terms. This class of problem is probably one of the most complex and for sure the most computationally intensive.
  • FSI instability analysis: typically used in aerospace or large civil structure design, this type of analysis is comparable to buckling analysis in structural mechanics. Instability FSI analysis are mostly based on linear or partially linearized FSI model and treated in frequency or Laplace domains. The typical phenomena of interest (to be avoided in terms of design!) are divergence, represented by a negative eigenvalue in the coupled FSI problem, and flutter, occurring when a coupled eigenmode shows a negative total damping which leads to an unbounded oscillation. In non-linear models, harmonic analysis is used to extract the amplitude of limit cycle oscillations (LCO).

Dowell, E. H. and K. C. Hall (2001). Modeling of fluid-structure interaction. Annual Review of Fluid Mechanics, 33: 445-490.
http://arjournals.annualreviews.org/doi/abs/10.1146%2Fannurev.fluid.33.1.445

Even though both fluid and solid models derive from a common continuum mechanics basis, an intrinsic difficulty in FSI lies in the mismatch of representation of the calculation domains: solid are preferably described in lagrangian form while fluid are mostly represented in eulerian. This difference in formulation represents a typical methodological problem to write a fully coupled set of equations as both domains do not share the same variables (displacement and velocity). Due to this difficulty, several levels of coupling methods can be distinguished.

 Explicitly coupled FSI

In this case, the fluid and solid domains have been formulated in a compatible way and explicit coupling terms are present in the global system of equation to be solved. Despite its evident theoretical advantages, explicitly coupled FSI methods are inherently difficult to develop because of the opposition between Lagrangian and Eulerian formulations which tends to limit the types of constitutive models or non-linear effects that can be implemented. As a result,  explicitly coupled FSI methods still lack in generality but are continuously developing. On the numerical point of view, the explicitly coupled models require significantly more memory and are limited due to their numerical solution complexity and computationnal cost. Among others, interesting explicit coupling approaches have been based on space-time finite element formulations or direct coupling of the fluid and structure formulation at the variationnal level (solid represented in euler frame or fluid solved in terms of displacements).

Torii, R., M. Oshima, et al. (2006). Computer modeling of cardiovascular fluid-structure interactions with the deforming-spatial-domain/stabilized space-time formulation. Computer Methods in Applied Mechanics and Engineering 195(13-16): 1885-1895.
http://dx.doi.org/10.1016/j.cma.2005.05.050

Figueroa, C. A., I. E. Vignon-Clementel, et al. (2006). A coupled momentum method for modeling blood flow in three-dimensional deformable arteries. Computer Methods in Applied Mechanics and Engineering 195(41-43): 5685-5706
http://dx.doi.org/10.1016/j.cma.2005.11.011

Implicitly coupled FSI

In the case of the implicitly coupled approaches, the coupling terms are usually not explicitly formulated and the problem is decomposed in two separate domains with two separate physics, the interaction being only represented by a transfer of boundary conditions between the fluid and solid interfaces. This method is typically used in “code-coupling” algorithms which use existing finite element or finite volume fluid solvers in Arbitrary Lagrangian-Eulerian formulation (ALE) and FE structure solvers (linear or non linear, modal based) to iteratively solve the interaction problem. The actual time integration sequence of fluid – structure solutions can also vary from one method to the other: explicit time stepping with synchronous or asynchronous (half step) fluid-structure solutions (also called weakly coupled FSI) or explicit time stepping with internal fixed point coupling iteration (relaxation).

Typical implicitly coupled FSI algorithm
Time stepping
  Iterative coupling iterations (enabled or disabled)
    Solve fluid problem (update acceleration, velocity & stress fields)
    Project pressure field of solid domain => solid boundary conditions
    Solve solid problem (update displacement, velocity and acceleration fields)
    Project displacement & velocity on fluid domain => fluid boundary conditions
    Update the fluid ALE mesh (laplace diffusion or pseudo-elastic solution)

Depending on the level of sophistication of the time-stepping strategy, this category of FSI method can be rather simple to implement and is now becoming available in modern academic or commercial simulation codes. But unfortunately implicitly coupled and more specifically weakly coupled algorithms (explicit time stepping without coupling iterations) have several significant drawbacks:

  • The convergence is not ensured and the stability of the coupling relies mostly on relaxation techniques; slow convergence rate is usually observed and thus a large number of FSI iterations are required.
  • In dynamics, the computational cost of such methods is usually prohibitive; typically up to 50 fluid & structure solutions are necessary for a single time step of the transient analysis; HPC clusters are typically necessary to run this type of analysis on anything but simple geometries. Reduced models based on simplified flow condition or on modal description of the structure (or fluid flow) have been used until now for that reason.
  • The time-stepping & field interpolation procedures are not always energy preserving, leading to divergence or artificial damping in the solution. The final accuracy is thus questionable and models should be validated with respect to benchmark cases whenever possible. Even if transient response can be studied, stability / artificial damping properties of the coupling algorithm may not allow reaching an accurate solution for steady state limit cycle oscillation problems (LCO, critical for turbine/pump fatigue).
  • Fluid ALE mesh motion propagation from boundary to volume often creates significant and some time critical mesh distortion in presence of large structural deformation. Specific mesh motion algorithms try to circumvent these issues (Radial Basis Functions, pseudo elastic solutions with stiffness gradients) but with limited success. Alternative “fixed grid” methods try to circumvent this issue, but usually at the cost of a lower accuracy on the fluid flow solution: no local mesh refinement around interfaces, no turbulence “wall-functions” and thus limited accuracy on wall shear stress and global flow characteristics (detachment of boundary layer, recirculation zones).

However promising methods are continuously developed to push the limits of this approach of FSI and with the progression of computational power availability (GFlop/$), smart “code-coupling” approach is already and will probably represent the most common trend in the future of FSI.

Among the directions of current research and numerical implementation, the idea of introducing “simplified” tangent operators (coupling term) in implicit coupling appears to be an interesting solution that combines both the advantages of explicit (robustness) and implicit (simpler solution) FSI methods. Additionally, detailed discussions on the properties of time-stepping algorithms are in my opinion a significant step forward to the reliability of FSI simulation.


Matthies, H. G. and J. Steindorf (2003). Partitioned strong coupling algorithms for fluid-structure interaction. Computers & Structures 81(8-11): 805-812.
http://dx.doi.org/10.1016/S0045-7949(02)00409-1

Gerbeau, J. F. and M. Vidrascu (2003). A quasi-Newton algorithm based on a reduced model for fluid-structure interaction problems in blood flows. Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique Et Analyse Numerique 37(4): 631-647.
http://www.esaim-m2an.org/index.php?option=article&access=doi&doi=10.1051/m2an:2003049

Le Tallec, P., J. F. Gerbeau, et al. (2005). Fluid structure interaction problems in large deformation. Comptes Rendus Mecanique 333(12): 910-922.
http://dx.doi.org/10.1016/j.crme.2005.10.009
 

 

Deng, Shouchun

Some of my personal opinions (might not be right):

In my understanding, FSI is a difficult research topic. However, at the same time, it is not getting enough attention because people think it is difficult but not important. Structural analytical methods are mature and accurate, but fluid behaviors are difficult to simulate. If the velocity of fluid is higher, tubulance plays important role. Modeling method of FSI is restricted by fluid models.

Biological simulation might be an important area for this kind of research.

Hope experts in this area can share their knowledge for this interesting topic.

 

====================

Even though I do not have expertise in fluid-solid interaction, I would like to mention the importance of such issue in Bio-MEMS/NEMS. Specifically, micro/nano-mechanical devices for in vitro biomolecular detection are required to be operated in liquid environment, since the interactions between biomolecules occur in water. If the resonance-type mechanical device (e.g. quartz crystal microbalance (QCM), resonant cantilever, etc.) is utilized for in situ biosensor application in liquid, the fluid-solid interaction plays a key role in resonance behavior of such mechanical device. Moreover, the resonant frequency shift for a mechanical device due to biomolecular binding (e.g. protein antigen-antibody binding, DNA hybridization, etc.) is also associated with hydrodynamic loading, since biomolecular interactions is affected by water molecules (e.g. hydrophilicity change of protein due to protein-protein interactions). This phenomenon was found in case of QCM as well as resonant microcantilever. The details of such issue are well described by following references.

[1] T.Y. Kwon, K. Eom, J.H. Park, D.S. Yoon, T.S. Kim, and H.L. Lee (2007) In situ real-time monitoring of biomolecular interactions based on resonating microcantilevers immersed in a viscous fluid. Applied Physics Letters, 90, 223903; Preprint is also available at http://www.imechanica.org/node/1297 

[2] S. Kirstein, M. Mertesdorf, and M. Schonhoff (1998) The influence of a viscous fluid on the vibration dynamics of scanning near-field optical microscopy fiber probes and atomic force microscopy cantilevers. Journal of Applied Physics, 84, 1782.

[3] J. Rickert, A. Brecht, and W. Gopal (1997) QCM operation in liquids: Constant sensitivity during formation of extended protein multilayers by affinity. Analytical Chemistry, 69, 1441 

I believe that theory of solid-fluid interactions may be important for characterization of micro/nano-mechanical devices for their potential application to in situ biosensor operated in liquid environment.

I would like to point out the popular "Immersed Boundary Method " from Charles Peskin's group at the Courant Institute and collaborators.  For really large deformations at high speeds (for both solids and fluids) I'd also like to point out our work at Utah. A preprint (via Scribd) can be viewed below:

In my case (explicit 2D) FSI is not a problem, I came to CSD and FSI

from  CFD (High order Godunov in moving Euler grid)

 I use finite volume (high order Godunov method) for both:

Computational Fluid Dynamics and
Computational Solid Mechanics.

For the border Fluid-Structure  I made an exact Riemann's solver.

For the Fluid  Tate state eq. is used.

For the  Solids (elastic case) -  Hook's law,

there is an exact solution of this problem,

R.P. solver on the border- the same contact velocity of Fluid and Solids,

normal stress is equal to -pressure of fluid,

tangent stress = friction fluid-solids.

 Each step the border is moved in acodernce with the velocity of this R.P. solver,

for 2D explicit case everything is simple:

body fitted mesh is constructed after moving FSI boundaries,

 Fluid and Solids are integrated as in 2D moving Euler.

For for solids predictor step is linear (hook's law), all physics (plasticity, viscoplasticity and etc.

) is introduced in corrector step (splitting technics like for Fluids, it is enough

for 2 order)  

For solids were problems with the big numerical viscosity from the border cells

(not enough accuracy on the border with free or pressure boundaries and so radiation

of the energy).

Now it works well from impacts time (km/sec)  up to milionns steps (vibration

of constructions) . This border viscosity from free or pressure boundary is too big,

but  it is possible to correct this viscosity using modified equation for these border cells,

up to the accuracy of the scheme (2 order)

after that moving Euler for solids became without any viscocity

(similar or even better than Wilkins in lagranzian).

It works well  for the impact and explosion FSI problems

from the FSI up to vibration of construction (milions explicit steps).

So practically the same scheme for Fluids, Solids and FSI and it can be extended

like usual finte volume on different physics.

For 3D explicit case FSI is the same - simple (exact Riemann's solver).

 Problem is how to integrate Solids in 3D in moving Euler for body fitted grids,

the important part is volume calculation- very big influence of the volume errors

on stress tensor.

I think in 3D case the best way is to use nonmoving Euler with

similar to FAVOR technics from FLOW-3D.

abouziar

 

I will try to contact you.

====================

Hi CAEengineer

you coud take some results from our ftp server

server    :  ftp.unn.ru

login       :   mech

password: mech-f.ex. 

file examples.txt is with the discription of ccf.zip with some problems to demonstrate  the method

                
capabilities for CFD, CSD and FSI

if any questions

  e-mail abouziar@dk.mech.unn.ru

or we could disscus in in skype

best wishes

abouziar 

 

 

 

 

                
 

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====================

http://www.comsol.fr/showroom/animations/361/ 

***********************************
CAE Engineer in Automotive Industry

 The interaction between structure and aerodynamics is very difficult. I worked in the area too. Without expressing here all the details (see the links at the end of this message), I can post here the main concepts.

1)  The aerodynamics is considered linear and the effects of the viscosity are not considered.

2) The flow is assumed attached and subsonic. The compressibility can be included. 

3) The structure is a collection of Finite Element plates in the space that simulate a possible equivalent plate model for a planar or non-planar wing system. The structure is geometrically nonlinear.  

4)  The structure takes into account the nonlinearities which can be important even if the deflections are not large. As an example, you can consider a Joined Wing configuration which, in general, is characterized by strong in-plane forces which affect the geometric stiffness matrix. The fact that the deflections are not very large allows one to use linear aerodynamics and assume that there is no separation of the boundary layer

5) Under the above mentioned conditions, the projections of the structural nodes on the reference configuration with no angle of attack can be considered constant. This simplifies the splining a lot because all the matrices used to transfer the aerodynamic loads from the aerodynamic mesh to the structural mesh are constant even if the structure deforms.

6) We are talking about panel codes, so vortex lattice for the steady case (see the book Katz J., and Plotkin A., "Low-Speed Aerodynamics - Second Edition,"
Cambridge University Press, NY, 2001) and Doublet Lattice for the unsteady case (Rodden W. P., Taylor P. F., McIntosh Jr S. C., "Further Refinement of the Subsonic Soublet-Lattice Method", Journal of Aircraft Vol. 35, No. 5, SEptember-October 1998)   

7) For the steady case: the vortex lattice calculates the aerodynamic influence coefficient matrix and after some transformation an aerodynamic tangent matrix (referred to the structural mesh) is calculated. The aerodynamic tangent matrix is then added to the structural tangent matrix to obtain the aeroelastic tangent matrix. As you calculate the buckling load of a "pure" structure you can then calculate the divergence speed by gradually increasing the aerodynamic speed until the aeroelastic tangent matrix becomes singular

8) The unsteady case is more complicated. The main idea is to work with a set of reduced frequencies to get tabulated values for the generalized aerodynamic matrix. Then, Roger procudure can be used. After some other derivations, you move from the frequency domain to the Laplace domain and finally to the time domain. At this point you can numerically integrate the equation using Newmark method. The aerodynamic loads and lag effects affects the response of the system. This procedure is capable of capturing LCO when this is developed from structural nonlinearities (see the work of Attar, Dowell and White "Modeling of a Delta Wing Limit-Cycle Oscillations Using a High-Fidelity Structural Model", Journal of Aircraft Vol. 42, No. 5, September-October 2005)

 You can see some details in the following conference articles:

Demasi L., Livne E., “
Aeroelastic Coupling of Geometrically Nonlinear Structures and Linear Unsteady
Aerodynamics: Two Formulations
”, Presented at the 49th
AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials
Conference, Schaumburg, Illinois, 7-10 April 2008

 

Demasi L., Livne E., “
Dynamic Aeroelasticity of Coupling Full Order Geometrically Nonlinear Structurs
and Full Order Linear Unsteady Aerodynamic - The Joined Wing Case
”,
Presented at the 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics
& Materials Conference, Schaumburg, Illinois, 7-10 April 2008

You can download the PDFs here:

http://attila.sdsu.edu/~demasi/articles/AIAA-2008-1758-644.pdf

http://attila.sdsu.edu/~demasi/articles/AIAA-2008-1818-885.pdf   

 

 

 

http://attila.sdsu.edu/~demasi/default.htm

 

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