Ferienakademie 2007 Partitioned approach for Fluid-Structure-Interaction (FSI) Atanas Gegov TU München

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München2 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 Sept.-5.Okt. 2007Atanas Gegov, TU München3 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 – FSI*ce

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München4 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 Sept.-5.Okt. 2007Atanas Gegov, TU München5 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 Sept.-5.Okt. 2007Atanas Gegov, TU München6 What is FSI | Examples of different FSI occurrences Tacoma Narrows Bridge collapse in 1940 source:

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München7 What is FSI | Examples of different FSI occurrences Hydraulic ram pump source:

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München8 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 –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 Sept.-5.Okt. 2007Atanas Gegov, TU München9 What is FSI | Examples of different FSI occurrences Flow around elastic structures Eulerian source: Dunne, Heidelberg

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München10 What is FSI | Examples of different FSI occurrences Flow around elastic structures ALE source: Dunne, Heidelberg

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München11 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 – FSI*ce

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München12 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 Sept.-5.Okt. 2007Atanas Gegov, TU München13 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

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München14 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

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München15 Different approaches for solving FSI problems Partitioned approach | Idea Interface Building Surface (structure), Wind Last (fluid) Interface Dam Surface (structure), Water (fluid) 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 Sept.-5.Okt. 2007Atanas Gegov, TU München16 Different approaches for solving FSI problems Partitioned approach | Idea m1 m2 Interface System 1 System 2 m2 Partitioning m1 Whole system (Two single mass swings) Partitioned system source: Group Prof.Rank, TUM

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München17 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

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München18 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

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München19 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

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München20 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) => partitioned approach –commercial environment, rigid deliverable timetable, massive software development resources, global interaction effects => monolithic approach

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München21 Different approaches for solving FSI problems Partitioned approach | Example of the basic idea Backward Euler integration: Monolithic approach: Simple partitioned solution: source: paper C. A. Felippa

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München22 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) 1 x 0 x 0 y Step 2 Step 3 Step 4 Step 1 2 x 2 y - With two predictors (both x and y) both programs advance concurrently - better for parallel computer 0 x 0 y 2 x 2 y Step 1 1 x Step 2

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München23 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

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München24 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 => 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 Sept.-5.Okt. 2007Atanas Gegov, TU München25 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 – FSI*ce

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München26 Algorithmical improvements of the partitioned approach Subiteration in detail –Initial approximation z 0 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 Sept.-5.Okt. 2007Atanas Gegov, TU München27 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: z j → z j+1 = C(z j ), 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 Sept.-5.Okt. 2007Atanas Gegov, TU München28 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

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München29 Algorithmical improvements of the partitioned approach Multi-Grid FSI converged end h h 2h 4h Initialization t=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,Tuj

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München30 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

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München31 Algorithmical improvements of the partitioned approach 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’ (z i )*(z i -z i+1 ) =R (z i ) A * x = b

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München32 Algorithmical improvements of the partitioned approach Interface-GMRES/Newton-Krylov –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 K m = R m (exact solution found) –however, after a small number of iterations (relative to m), the vector x n already a good approximation –GMRES method developed by Yousef Saad and Martin H. Schultz in 1986

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München33 Algorithmical improvements of the partitioned approach Interface-GMRES/Newton-Krylov –Further improvement reuse of Krylov vectors in subsequent Newton steps => Interface-GMRESR => can result in considerable computational savings (example for z with two points)

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München34 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 => 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)

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München35 Outline What is FSI Different approaches for solving FSI problems Algorithmical improvements of the partitioned approach How partitioned FSI can be realized – FSI*ce –Requirements –Design –FSI*ce in use

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München36 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 Sept.-5.Okt. 2007Atanas Gegov, TU München37 How partitioned FSI can be realized – FSI*ce 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 Sept.-5.Okt. 2007Atanas Gegov, TU München38 How partitioned FSI can be realized – FSI*ce 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 Sept.-5.Okt. 2007Atanas Gegov, TU München39 How partitioned FSI can be realized – FSI*ce Design –The communiction Sockets transport a message from one process to another MPI

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München40 How partitioned FSI can be realized – FSI*ce Design | The communication Server programs are serialServer programs are parallel Communication with Sockets / distibuted application Communication with MPI

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München41 How partitioned FSI can be realized – FSI*ce 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 Sept.-5.Okt. 2007Atanas Gegov, TU München42 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 Sept.-5.Okt. 2007Atanas Gegov, TU München43 Bibliography (II) Internet: –FSI in general: –Eulerian and Lagrangian fluid description: heidelberg.de/Research/dunne.html –Tacoma Narrows Bridge: –Hydraulic ram pump: –Newton’s method: –Partition solution of coupled systems: muenchen.de/~kollmannsberger/SoftLab2005CoupledSystems/Files/third_presentation.ppt –GMRES approach: –GMRES approach: –Krylov subspace: –Linear span: –Forschergruppe 493: –MPI exercises:

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München44 Thank you for your attention!

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München45

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München46 Backup slides

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München47 Algorithmical improvements of the partitioned approach Interface-GMRES/Newton-Krylov –Further improvement 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 (example for z with two points)

Ferienakademie Sept.-5.Okt. 2007Atanas Gegov, TU München48 How partitioned FSI can be realized – FSI*ce 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 #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.cmpicc % mpirun -np 4 helloworldmpirun 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 %