Partitioned approach for FluidStructureInteraction FSI

1
Partitioned approach for Fluid-Structure-Interact
ion (FSI)

• Atanas GegovTU München

2
Outline

• What is FSI
• Different approaches for solving FSI problems
• Algorithmical improvements of the partitioned
  approach
• How partitioned FSI can be realized FSIce

3
Outline

• What is FSI
• Why is FSI simulation interesting
• Examples of different FSI occurrences
• Different approaches for solving FSI problems
• Algorithmical improvements of the partitioned
  approach
• How partitioned FSI can be realized FSIce

4
What is FSI

• Fluid-Structure-Interaction (in German Fluid-
  Struktur- Wechselwirkung )
• Describes interaction between fluid (liquid or
  gas) and solid body (structure) in a system
• fluid interacts with a solid structure, exerting
  pressure that may cause deformation or
  displacement in the structure and, thus, alter
  the flow of the fluid itself
• Typically connected with bad things
• fluttering of airplanes
• deformations
• vibrations
• even collapse of buildings
• Interesting for many researchers in physics,
  mathematics and computer science

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What is FSI Why is FSI simulation interesting

• Possibilities due to high-performance computing
• Simulation describing or predicting the state of
  the system under specified conditions. A set of
  states ordered according to time is a response.
• Extensive experimental testing
• costly
• time-consuming
• Growing demand for the accurate and efficient
  numerical solution of FSI problems in various
  engineering disciplines

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What is FSI Examples of different FSI
occurrences

• Tacoma Narrows Bridge collapse in 1940

source http//en.wikipedia.org
7
What is FSI Examples of different FSI
occurrences

• Hydraulic ram pump

source http//schou.dk/animation/
8
What is FSI Examples of different FSI
occurrences

• Flow around elastic structures
• Lagrangian description
• each fluid particle carries its own properties
  such as density, momentum, etc
• ?(p,t) , V(p,t), P(p,t),...
• computationally expensive
• neutrally swimming probe is an example of a
  Lagrangian measuring device

• Eulerian description
• record the evolution of the flow properties at
  every point in space as time varies
• ?(x,t) , V(x,t), P(x,t),...
• good for FSI
• probe fixed in space is an example of an Eulerian
  measuring device

• ALE (Arbitrary Lagrangian-Eulerian) description

9
What is FSI Examples of different FSI
occurrences

• Flow around elastic structures
• Eulerian

source Dunne, Heidelberg
10
What is FSI Examples of different FSI
occurrences

• Flow around elastic structures
• ALE

source Dunne, Heidelberg
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Outline

• What is FSI
• Different approaches for solving FSI problems
• Monolithic approach
• Partitioned approach
• Idea
• Terminology
• Pros and contras
• Example of the basic idea
• Loosely-coupled and strongly-coupled partitioned
  approach
• Algorithmical improvements of the partitioned
  approach
• How partitioned FSI can be realized FSIce

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Different approaches for solving FSI problems

• Monolithic approach
• Treats coupled fluid and structure equations
  simultaneously
• System is in general nonlinear,
• solution involves a Newton method
• Advantages
• high accuracy
• Disadvantages
• expensive computation of derivatives (Jacobian
  matrix)
• loss of software modularity due to the
  simultaneous solution of fluid and structure

13
Different approaches for solving FSI problems

• Partitioned approach
• Very popular for solving FSI
• The idea is universal for coupled systems
• Applications in
• thermomechanics
• FSI
• control-structure-Interaction

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Different approaches for solving FSI problems

• Partitioned approach Idea
• Systems spatially decomposed into partitions
• Solution is separately advanced in time over each
  partition
• Partitions interact on their interface (mesh
  structure that is closed, e.g. airplane)
• Interaction by transmission and synchronization
  of coupled state variables

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Different approaches for solving FSI problems

• Partitioned approach Idea

• The behaviour of each region (structure and
  fluid) can be described by differential equations
• The interaction is happening on the interface by
  information exchange

Interface
Interface
Building Surface (structure), Wind Last (fluid)
Dam Surface (structure), Water (fluid)
source Group Prof.Rank, TUM
16
Different approaches for solving FSI problems

• Partitioned approach Idea

Whole system (Two single mass swings)
Partitioned system
source Group Prof.Rank, TUM
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Different approaches for solving FSI problems

• Partitioned approach Idea
• Systems analyzed by decomposition
• Decompositions called partitions are suitable for
  computer simulation
• Partitioning process of spatial separation of a
  discrete model into interacting components
  generically called partitions
• Decomposition driven by
• physical
• functional
• computational considerations
• Example flight simulation
• multilevel partition hierarchy coupled system,
  structure, substructure, subdomain and element
  typical of present practice in modeling and
  computational technology

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Different approaches for solving FSI problems

• Partitioned approach Terminology
• coupled system one in which physically or
  computationally heterogeneous mechanical
  components interact dynamically
• Decomposition of a complex coupled system for
  simulation is hierarchical with two to four
  levels. At the first level two types of
  subsystems with the generic term field
• physical subsystems (fields) mathematical model
  described by field equations Examples solids,
  fluids, heat, electromagnetics
• artificial subsystems
• incorporated for computational convenience
• For computational treatment,
• fields are discretized in space
  (partitioning)
• and time (splitting)

source paper C. A. Felippa
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Different approaches for solving FSI problems

• Partitioned approach Terminology
• Algebraic partitioning
• the complete coupled system is spatially
  discretized, then decomposed
• originally developed for matched meshes, typical
  for Structure-Str.-Inter.

• Differential partitioning
• the decomposition is done first and each field
  then discretized separately
• leads to nonmatched meshes, typical for FSI

source paper C. A. Felippa
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Different approaches for solving FSI problems

• Partitioned approach Pros and contras
• Advantages
• customization
• independent modeling
• software reuse
• modularity
• Disadvantages
• partitioned approach requires careful formulation
  and implementation to avoid serious degradation
  in stability and accuracy
• parallel implementations are error-prone
• Summary
• research environment, access to existing
  software, localized interaction effects (e.g.
  surface vs volume) gt partitioned approach
• commercial environment, rigid deliverable
  timetable, massive software development
  resources, global interaction effects gt
  monolithic approach

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Different approaches for solving FSI problems

• Partitioned approach Example of the basic idea

Backward Euler integration

• Monolithic approach

source paper C. A. Felippa

• Simple partitioned solution

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Different approaches for solving FSI problems

• Partitioned approach Example of the basic idea
• Simple partitioned solution
• Suppose two communicating programs(staggered
  solution procedure)
• One predictor (y)

• With two predictors (both x and y) both programs
  advance concurrently
• better for parallel computer

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Different approaches for solving FSI problems

• Partitioned approach Example of the basic idea
• partitioned analysis gives alternative algorithm
  and implementation possibilities

- subcycling
source paper C. A. Felippa
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Different approaches for solving FSI problems

• Partitioned approach Loosely-coupled and
  strongly-coupled partitioned approaches

• Strongly-coupled methods
• alternate fluid and structure solutions within a
  time step until convergence
• treat the interaction between the fluid and the
  structure synchronously
• maintain conservation
• disadvantage greater computational cost per
  time step
• gt algorithmical improvements possible

• Loosely-coupled methods
• single (one time for the fluid program and one
  for the structure) solution per time step
• disadvantage loss of conservation properties of
  the continuum fluid-structure system (energy
  increasing, unstable)
• time step is usually smaller
• improvements by predictors (accuarcy and
  stability)

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Outline

• What is FSI
• Different approaches for solving FSI problems
• Algorithmical improvements of the partitioned
  approach
• Multi-Grid
• Interface-GMRES(R)/ Newton-Krylov
• How partitioned FSI can be realized FSIce

26
Algorithmical improvements of the partitioned
approach

• Subiteration in detail
• Initial approximation z0 ? Z of the structure
  solution (the structure displacement at the
  interface) for j 1, 2 . . .
• (1) Solve the kinematic condition
• fluid velocity at the interface velocity
  of the interface
• Constitutes a boundary condition for the
  initial-boundary-value problem of the fluid
• (2) Solve the fluid
• the result is the flow velocity and
  pressure fields
• (3) Solve the dynamic condition
• the result is the fluid pressure (the
  forces) acting on the structure surface
• (4) Solve the structure
• the result is the displacement of every
  point on the structure surface

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Algorithmical improvements of the partitioned
approach

• Subiteration in detail
• no simultaneous treatment of the fluid and the
  structure
• reduces the complexity of solving the aggregated
  fluid-structure equations to a sequence of
  standard problems
• Subiteration process as mapping from one
  structural interface displacement to the next,
  i.e.
• C zj ? zj1 C(zj), C nonlinear
  operator induced from (1) to (4) (not explicitly
  available)
• The fixed point is where z Cz z
• Drawbacks
• subiteration converges slowly or even diverges
  for problems with large computational time steps
• subiteration generally solves a sequence of
  similar problems (but without reuse)

(example for z with two points)
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Algorithmical improvements of the partitioned
approach

• Multi-Grid
• makes subiterations, but the they are done one
  more than one grids
• from the top-level (the main grid where the FSI
  has to be solved) down to levels with lower
  resolution
• iteration less expensive due to the reduced
  dimension
• gathered information is propagated again to the
  top levels
• makes therefore their iterations more efficient

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Algorithmical improvements of the partitioned
approach
tt?t

• Multi-Grid

N
N
FSIconverged
tt_end
h
h
Y
Y
Initialization
2h
end
Computation of flow field (finite volumes)
Computation of modified mesh
grid
4h
p,vj,T
uj
Computation of wall forces
Computation of deformations (finite elements)
Fw
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Algorithmical improvements of the partitioned
approach

• Multi-Grid
• multiple grids have to be created
• very complex, if generated manually (with
  generator tool)
• involving hierarchical approach (e.g octree) is
  better
• therefore, although the idea of Multi-Grid is
  good, it is not so easy to be realized in
  practical applications

31
Algorithmical improvements of the partitioned
approach

• Interface-GMRES/Newton-Krylov
• Generalized Minimal RESidual
• The nonlinear problem Cz z
• Cz z 0
• Rz0 with RC-I
• After some transformations R (zi)(zi-zi1) R
  (zi)
• A x b

32
Algorithmical improvements of the partitioned
approach

• Interface-GMRES/Newton-Krylov
• Axb solved by the GMRES method
• iterative method for the numerical solution of a
  system of linear equations
• approximates the solution by the vector in a
  Krylov subspace with minimal residual
• every subspace contained in the next subspace,
  the residual decreases monotonically in every
  iteration
• after m iterations (m - size of A) the Krylov
  space Km Rm (exact solution found)
• however, after a small number of iterations
  (relative to m), the vector xn already a good
  approximation
• GMRES method developed by Yousef Saad and Martin
  H. Schultz in 1986

33
Algorithmical improvements of the partitioned
approach

• Interface-GMRES/Newton-Krylov
• Further improvement
• reuse of Krylov vectors in subsequent Newton
  steps gt Interface-GMRESR
• gt can result in considerable computational
  savings

(example for z with two points)
34
Algorithmical improvements of the partitioned
approach

• Interface-GMRES/Newton-Krylov
• Further improvement
• disadvantage need of storing the
  search-direction vectors used by now
• (N, if problem N-dimensional)
• advantage less Newton- subiterations
  (evaluations of R) needed
• gt significant increase in efficiency
• computational expense of Interface-GMRESR method
  may be comparable to loosely-coupled partitioned
  methods (single fluid and structure solution per
  time step) by more stability and accuracy

(example for z with two points)
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Outline

• What is FSI
• Different approaches for solving FSI problems
• Algorithmical improvements of the partitioned
  approach
• How partitioned FSI can be realized FSIce
• Requirements
• Design
• FSIce in use

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How partitioned FSI can be realized FSIce

• Requirements
• Exisiting
• CFD ( computational fluid dynamics, viz. fluid
  solver program )
• CSD ( computational structure dynamics, viz.
  structure solver program)
• plug-in mechanism for the CFD/CSD programs,
  simple replacement ability for the components
• implementation of the coupling schema outside
  from the CFD/CSD simulation programs

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How partitioned FSI can be realized FSIce

• Design
• Direct communication vs. Client-Server scheme

• coupling scheme inside the programs
• application calls the other for new boundary
  conditions
• synchronization of the time steps required

• applications as servers
• requests from client
• concept fulfills the two requirements

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How partitioned FSI can be realized FSIce

• Design
• independent representation of the coupling
  geometry
• Vertex-edge-face Graph (vef-Graph)
• Closed body (airplane, u-boat)
• Data structure FSI_mesh stores
• coordinates
• data associated with the vertices or the faces

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How partitioned FSI can be realized FSIce

• Design
• The communiction

• Sockets transport a message from one process to
  another
• MPI

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How partitioned FSI can be realized FSIce

• Design The communication

Communication with MPI
Server programs are serial
Server programs are parallel
Communication with Sockets / distibuted
application
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How partitioned FSI can be realized FSIce

• FSIce in use
• already successfully tested with programs
  developed in scientific environment that allow
  access to the source code

• a first significant step in the partitioned
  solution of FSI problems
• will be further develpoed

42
Bibliography (I)

• 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

43
Bibliography (II)

• Internet
• FSI in general http//www.win.tue.nl/fsi/
• Eulerian and Lagrangian fluid description
  http//numerik.iwr.uni-heidelberg.de/Research/dunn
  e.html
• Tacoma Narrows Bridge http//en.wikipedia.org/wik
  i/Tacoma_Narrows_Bridge
• Hydraulic ram pump http//schou.dk/animation/
• Newtons method http//en.wikipedia.org/wiki/Newt
  on's_method
• Partition solution of coupled systems
  http//www.inf.bauwesen.tu-muenchen.de/kollmanns
  berger/SoftLab2005CoupledSystems/Files/third_prese
  ntation.ppt
• GMRES approach http//de.wikipedia.org/wiki/GMRES
  -Verfahren
• GMRES approach http//en.wikipedia.org/wiki/GMRES
  
• Krylov subspace http//de.wikipedia.org/wiki/Kryl
  ow-Unterraum
• Linear span http//de.wikipedia.org/wiki/Lineare_
  HC3BClle
• Forschergruppe 493 http//fsw.informatik.tu-muenc
  hen.de/index.php
• MPI exercises http//www-unix.mcs.anl.gov/mpi/tut
  orial/mpiexmpl/contents.html

44

• Thank you for your attention!

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(No Transcript)
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Backup slides

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Algorithmical improvements of the partitioned
approach

• Interface-GMRES/Newton-Krylov
• Further improvement
• reuse of Krylov vectors in subsequent Newton
  steps gt Interface-GMRESR
• once vector reused, search space formally no
  longer a Krylov space gt search directions do not
  necessarily constitute preferential search
  directions
• typically fewer Krylov vectors added to the
  reused space than generated for a reconstructed
  Krylov space
• gt can result in considerable computational
  savings

(example for z with two points)
48
How partitioned FSI can be realized FSIce

• Excursus MPI
• quasi- standard for message passing between
  parallel programs
• programs built as SPMD (Single Program Multiple
  Data)
• execution starts many instances of the program
  (processes)
• include ltstdio.hgt
• include "mpi.h
• int main( int argc, char argv )
• int rank, size
• MPI_Init( argc, argv )
• MPI_Comm_size( MPI_COMM_WORLD, size )
• MPI_Comm_rank( MPI_COMM_WORLD, rank )
• printf( "Hello world from process d of d\n",
  rank, size )
• MPI_Finalize()
• return 0

mpicc -o helloworld helloworld.c mpirun -np
4 helloworld Hello world from process 0 of 4
Hello world from process 3 of 4 Hello world from
process 1 of 4 Hello world from process 2 of 4