Partitioned approach for Fluid-Structure

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Partitioned approach for Fluid-Structure-Interaction (FSI)
Atanas Gegov, TUM
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< [email protected] >
Ferienakademie- Autumn 2007
What is FSI
Different approaches for solving FSI problems
Algorithmical improvements of the partitioned approach
How partitioned FSI can be realized – FSI*ce
Bibliography
What is FSI
FSI (German: “Fluid- Struktur- Wechselwirkung”) describes the phenomenon of interaction between
fluid (liquid or gas) and solid body (structure) in a system.
a) Why is FSI simulation interesting
To determine the effects of fluid-structure interaction for a given system, engineering design often
involves extensive experimental testing. However, experiments may be costly, time consuming and in
some cases even infeasible. As numerical models and techniques have matured over the last decades to
deliver more accurate predictions, and with the advent of increasing computing power for affordable
prices, numerical simulation has become more established in the design process to support or even
replace experimental testing.
b) Examples of different FSI problems
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One of the most prominent when talking about FSI. The collapse
of the Tacoma Narrows Bridge in 1940.
The hydraulic ram pump. Pumping is possible without
maintenance and power supply!
Flow around elastic structures (airplanes, submarines etc.).
Examples for three methods for describing fluids: Lagrangian,
Eulerian, ALE (Arbitrary Lagrangian- Eulerian) description:
Different approaches for solving FSI problems
a) Monolithic approach
Monolithic solution methods treat the coupled fluid and structure equations simultaneously, i.e., they
directly operate on the aggregated fluid and structure equations. As this system is in general nonlinear,
the solution procedure typically involves a Newton method.
b) Partitioned approach
Very popular in solving FSI. It is not only used in FSI but the idea can be applied to different problems
and situations. Important applications are provided by thermomechanics,
fluid-structure interaction and control-structure interaction.
Idea: In the partitioned solution approach, systems are spatially decomposed into partitions. This
decomposition is driven by physical or computational considerations. The solution is separately
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advanced in time over each partition. Interaction
effects are accounted for by transmission and
synchronization of coupled state variables. The
partitions interact with each other on their
interface (typically a mesh structure that is closed,
for example airplane).
The term partitioning identifies the process of
spatial separation of a discrete model into
interacting components generically called
partitions. The decomposition may be driven by
physical, functional, or computational
considerations. For example, the structure of a complete airplane can be decomposed into
substructures such as wings and fuselage according to function. Substructures can be further
decomposed into submeshes or subdomains to accomodate parallel computing requirements.
Subdomains are composed of individual elements. Going the other way, if that flexible airplane is part
of a flight simulation, a top-level partition driven by physics is into fluid and structure models. This
kind of multilevel partition hierarchy: coupled system, structure, substructure, subdomain and element,
is typical of present practice in modeling and computational technology.
Terminology: A coupled system is one in which physically or computationally heterogeneous
mechanical components interact dynamically. Physical Subsystems. Subsystems are called physical
fields when their mathematical model is described by field equations. Examples are mechanical and
non-mechanical objects treated by continuum theories: solids, fluids, heat, electromagnetics. Artificial
Subsystems. Sometimes artificial subsystems are incorporated for computational convenience. A
coupled system is characterized as two-field, three-field, etc., according to the number of different
fields that appear in the first-level decomposition.
For computational treatment of a dynamical coupled system, fields are discretized in space and time. A
field partition is a field-by-field decomposition of the space discretization. A splitting is a
decomposition of the time discretization of a field within its time step interval.
Pros and contras: Customization. This means that each field can be treated by discretization
techniques and solution algorithms that are known to perform well for the isolated system.
Independent Modeling. The partitioned approach facilitates the use of non-matching models.
Software Reuse. The partitioned approach facilitates taking advantage of existing code.
Modularity. New methods and models may be used in a modular fashion according to project needs.
Disadvantages: The partitioned approach requires careful formulation and implementation to avoid
serious degradation in stability and accuracy. Parallel implementations are particularly delicate. In
summary, circumstances that favor the partitioned approach for tackling a new coupled problem are: a
research environment with few delivery constraints, access to existing software, localized interaction
effects (e.g. surface versus volume), and widespread spatial/temporal component characteristics. The
opposite circumstances favor a monolithic approach.
Loosely- coupled and strongly- coupled partitioned approaches: If only a single (one time
for the fluid program and one for the structure) solution per time step are carried out, such partitioned
methods are commonly referred to as loosely-coupled partitioned methods. Their essential
disadvantage pertains to the loss of the conservation properties of the continuum fluid-structure
system. Although the order of the incurred error can be improved by predictors, loosely-coupled
methods can never be exactly conservative.
Partitioned methods which solve the fluid-structure system by repeating within a time step alternate
fluid and structure solutions until convergence are called strongly-coupled partitioned methods.
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To improve the order of the numerical evaluation error incurred by loosely-coupled
partitioned methods, prediction techniques are used. For example, instead of integrating
the fluid equations based on the position of the structure boundary in the previous time
interval, a prediction can be used for the position of the structure boundary in the
current time interval.
Strongly-coupled methods have a greater computational cost per time step than loosely-coupled
methods. However, strongly-coupled methods can maintain conservation at the fluid-structure
interface, which renders them unconditionally stable. In contrast, loosely-coupled methods are
typically energy increasing and, hence, numerically unstable.
Algorithmical improvements of the partitioned approach
As mentioned above, the numerical solution of fluid-structure interaction problems
commonly employs subiteration, i.e., fluid and structure equations are solved
alternately according to the interface conditions. This process is repeated until
convergence. Subiteration sidesteps a simultaneous treatment of the coupled fluidstructure equations and the resulting difficulties.
The following steps define the subiteration process: for j = 1, 2 . . .
(1) Solve the kinematic condition: Constitutes a boundary condition for the initialboundary-value problem of the fluid
(2) Solve the fluid: Result is the flow velocity and pressure fields
(3) Solve the dynamic condition: Result is the fluid pressure (the forces) acting on the structure surface
(4) Solve the structure: Result is the displacement of every point on the structure surface.
The subiteration process can then be seen as a mapping from one structural interface displacement to
the next, i.e. C: zj → zj+1 = Czj. The fixed point is where .z: C.z = .z.
Although subiteration is a good solver for many problems, it suffers from two essential drawbacks:
Firstly, subiteration converges only slowly or even diverges for problems with large computational
time steps. Subiteration is only conditionally stable. Secondly, subiteration is generally employed in a
sequential time-integration process and, hence, solves a sequence of similar problems. However, the
method cannot exploit this property and reuse generated information, for instance, for preconditioning
purposes. Therefore, subiteration is to be considered inefficient.
It is possible to overcome these drawbacks by combining subiteration with GMRES acceleration. A
Multi-Grid solution method is another approach for achieving better efficiency in the subiterations.
a) Multi-Grid
It also makes subiterations, but the point here is that the iterations are done one more than one grids.
One begins from the top-level (the main grid where the FSI has to be solved) and goes down to levels
with lower resolution, where the iteration is accordingly less expensive due to the reduced dimension.
Than the gathered information is propagated again to the top levels and makes therefore their iterations
also more efficient. Although the idea of Multi-Grid is good, it is not so easy to be realized in practical
applications.
b) Interface-GMRES(R)/ Newton-Krylov
The nonlinear problem Cz =z can be formulated as Cz-z=0 and Rz=0 with R=C-I.
After some transformations we get R’ (z (i))*(z (i)-z (i+1)) =R (z (i)). One can see the shape A*x=b
which is an LSE. This system can be solved be the GMRES method. GMRES approximates the exact
solution of Ax = b by the vector xn e Kn that minimizes the norm of the residual Axn − b. The nth iterate
minimizes the residual in the Krylov subspace Kn. The idea is that after a small number of iterations,
the vector xn is already a good approximation to the exact solution.
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With Interface- GMRES we are performing better than with the fix-point iteration, but there is still one
more optimization possible. As each Newton step invokes the solution of a linear system by a Krylov
method (GMRES), the Newton-Krylov method lends itself naturally to reuse of Krylov vectors in
subsequent Newton steps. In our context we refer to such reuse as Interface-GMRESR. Typically,
much fewer Krylov vectors need to be added to the reused space than are generated for a reconstructed
Krylov space, which can result in considerable computational savings. The search in direction of the
zero-point is not more “random” but more specific and directed. Therefore there are less subiterations
needed and this means a significant increase in efficiency. It appears that the reuse of the Krylov
vectors can make the computational expense of the Interface-GMRESR method comparable to
loosely-coupled partitioned methods which perform only a single fluid and structure solution per time
step. However, in contrast to such loosely-coupled methods, the Interface-GMRESR method enables
conservation and improves stability and accuracy.
How partitioned FSI can be realized – FSI*ce
A real environment for making the partitioned FSI
simulation on the computer. It is called FSI*ce and is
developed by the teams of Prof. Rank and Prof. Bungartz
(TUM).
Requirements: This is firstly a “Plug-In” mechanism for the CFD/CSD programs (this means simple
replacement ability for the components) and secondly- implementation of the coupling schema (for
example Interface-GMRES or simply single iteration per time step) outside from the CFD/CSD
simulation programs.
Design: The full flexibility of the partitioned approach is achieved in FSIce with the Client-ServerModel. In FSIce a vertex-edge-face Graph (vef-Graph) is used for the coupling geometry. FSIce
allows communication via MPI or Sockets. FSIce is already successfully tested with programs
developed in scientific environment that allow access to the source code. However, FSIce is not
completely developed and is going to be further improved.
Bibliography
Books:
 “Efficient Numerical Methods for Fluid-Structure Interaction” by Christian Michler, Netherlands 2005
Papers:
 “Partitioned analysis of coupled mechanical systems” by Carlos A. Felippa, K.C. Park, Charbel Farhat, USA 1999
 Paper about FSIce (title to be defined) by TUM Lehrstuhl V (Dipl.-Geophys. Markus Brenk), Germany, to appear
Internet:
 FSI in general: http://www.win.tue.nl/fsi/
 Eulerian and Lagrangian fluid description:
http://numerik.iwr.uni-heidelberg.de/Research/dunne.html
 Tacoma Narrows Bridge: http://en.wikipedia.org/wiki/Tacoma_Narrows_Bridge
 Hydraulic ram pump: http://schou.dk/animation/
 Newton’s method: http://en.wikipedia.org/wiki/Newton's_method
 Partition solution of coupled systems: http://www.inf.bauwesen.tu-muenchen.de/~kollmannsberger/
SoftLab2005CoupledSystems/Files/third_presentation.ppt
 GMRES approach: http://de.wikipedia.org/wiki/GMRES-Verfahren or http://en.wikipedia.org/wiki/GMRES
 Krylov subspace: http://de.wikipedia.org/wiki/Krylow-Unterraum
 Linear span: http://de.wikipedia.org/wiki/Lineare_H%C3%BClle
 Forschergruppe 493: http://fsw.informatik.tu-muenchen.de/index.php
 MPI exercises: http://www-unix.mcs.anl.gov/mpi/tutorial/mpiexmpl/contents.html
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