Partitioned approach for Fluid-Structure-Interaction (FSI)

Ferienakademie 2007

Partitioned approach for Fluid-Structure-Interaction (FSI)

Atanas GegovTU München

23.Sept.-5.Okt. 2007 Atanas Gegov, TU München 2

Ferienakademie 2007

Outline

• What is FSI

• Different approaches for solving FSI problems

• Algorithmical improvements of the partitioned approach

• How partitioned FSI can be realized – FSI*ce


Ferienakademie 2007

Outline

• What is FSI

– Why is FSI simulation interesting

– Examples of different FSI occurrences





Ferienakademie 2007

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


Ferienakademie 2007

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


Ferienakademie 2007

What is FSI | Examples of different FSI occurrences

• Tacoma Narrows Bridge collapse in 1940

source: http://en.wikipedia.org


Ferienakademie 2007


• Hydraulic ram pump

source: http://schou.dk/animation/


Ferienakademie 2007


• Flow around elastic structures

• Lagrangian description– each fluid particle carries its own

properties such as density, momentum, etc

– ρ(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

– ρ(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


Ferienakademie 2007



• Eulerian

source: Dunne, Heidelberg


Ferienakademie 2007



• ALE

source: Dunne, Heidelberg


Ferienakademie 2007

Outline

• What is FSI


– Monolithic approach

– Partitioned approach• Idea• Terminology• Pros and contras• Example of the basic idea• Loosely-coupled and strongly-coupled partitioned approach




Ferienakademie 2007

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


Ferienakademie 2007


• Partitioned approach

– Very popular for solving FSI

– The idea is universal for coupled systems

• Applications in– thermomechanics– FSI– control-structure-Interaction


Ferienakademie 2007


• 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


Ferienakademie 2007



Interface

Building Surface (structure), Wind Last (fluid)

S Fl

Interface

Dam Surface (structure), Water (fluid)

S

Fl

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

source: Group Prof.Rank, TUM


Ferienakademie 2007



m1

m2

Interface

System 1

System 2

m2

Partitioning

m1

Whole system (Two single mass swings)

Partitioned system

source: Group Prof.Rank, TUM


Ferienakademie 2007



– 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


Ferienakademie 2007


• 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


Ferienakademie 2007


• 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



Ferienakademie 2007


• 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) => partitioned approach – commercial environment, rigid deliverable timetable, massive software

development resources, global interaction effects => monolithic approach


Ferienakademie 2007


• Partitioned approach | Example of the basic idea

Backward Euler integration:

•Monolithic approach:

•Simple partitioned solution:



Ferienakademie 2007



…Simple partitioned solution:

– Suppose two communicating programs(staggered solution procedure)

– One predictor (y)

1x0x

0y

Step 2

Step 3

Step 41y

Step

1

2x

2y

- With two predictors (both x and y) both programs advance concurrently

- better for parallel computer

0x

0y

2x

2y

Step 1

1x

1y

Step 2

Step 2


Ferienakademie 2007



– partitioned analysis gives alternative algorithm and implementation possibilities

- subcycling



Ferienakademie 2007


• 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

=> 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)


Ferienakademie 2007

Outline

• What is FSI



– Multi-Grid

– Interface-GMRES(R)/ Newton-Krylov



Ferienakademie 2007

Algorithmical improvements of the partitioned approach

• Subiteration in detail

– Initial approximation z0 Є 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


Ferienakademie 2007


• 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 → zj+1 = C(zj), C nonlinear operator induced from (1) to (4) (not explicitly available)

– The fixed point is where ż: Cż = ż

– 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)


Ferienakademie 2007


• 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


Ferienakademie 2007


• Multi-Grid

FSIconverged

end

h h

2h

4h

Initializationt=t_end

N

Y

N

Y

t=t+Δt

Computation of flow field

(finite volumes)

Computation of modified

mesh

Computation of wall forces

Computation of deformations(finite elements)

grid

Fw

p,vj,T uj


Ferienakademie 2007


• 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


Ferienakademie 2007


• Interface-GMRES/Newton-Krylov

– Generalized Minimal RESidual

– The nonlinear problem Cż = ż • Cż –ż = 0 • Rż=0 with R=C-I

– After some transformations: R’ (zi)*(zi-zi+1) =R (zi)

A * x = b


Ferienakademie 2007



– A*x=b 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


Ferienakademie 2007



– Further improvement

• reuse of Krylov vectors in subsequent Newton steps => Interface-GMRESR

=> can result in considerable computational savings



Ferienakademie 2007




• disadvantage: need of storing the search-direction vectors used by now (N, if problem N-dimensional)

• advantage: less Newton- subiterations (evaluations of R) needed => 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



Ferienakademie 2007

Outline

• What is FSI




– Requirements

– Design

– FSI*ce in use


Ferienakademie 2007

How partitioned FSI can be realized – FSI*ce

• 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


Ferienakademie 2007


• 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


Ferienakademie 2007


• 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


Ferienakademie 2007


• Design

– The communiction

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


Ferienakademie 2007


• Design | The communication

Server programs are serial Server programs are parallel

Communication with Sockets / distibuted application

Communication with MPI


Ferienakademie 2007


• FSI*ce 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


Ferienakademie 2007

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


Ferienakademie 2007

Bibliography (II)

• 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– GMRES approach: 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


Ferienakademie 2007

Thank you for your attention!


Ferienakademie 2007


Ferienakademie 2007

Backup slides


Ferienakademie 2007




• reuse of Krylov vectors in subsequent Newton steps => Interface-GMRESR

• once vector reused, search space formally no longer a Krylov space => 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

=> can result in considerable computational savings



Ferienakademie 2007


• 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 <stdio.h> #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

%

Documents

Partitioned approach for Fluid-Structure-Interaction (FSI)