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Partitioned approach for Fluid-Structure-Interaction (FSI)

Partitioned approach for Fluid-Structure-Interaction (FSI). Atanas Gegov TU M ünchen. Outline. What is FSI Different approaches for solving FSI problems Algorithmical improvements of the partitioned approach How partitioned FSI can be realized – FSI*ce. Outline. What is FSI

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Partitioned approach for Fluid-Structure-Interaction (FSI)

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  1. Partitioned approach for Fluid-Structure-Interaction (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 – FSI*ce Atanas Gegov, TU München

  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 – FSI*ce Atanas Gegov, TU München

  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 Atanas Gegov, TU München

  5. 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 Atanas Gegov, TU München

  6. What is FSI | Examples of different FSI occurrences • Tacoma Narrows Bridge collapse in 1940 source: http://en.wikipedia.org Atanas Gegov, TU München

  7. What is FSI | Examples of different FSI occurrences • Hydraulic ram pump source: http://schou.dk/animation/ Atanas Gegov, TU München

  8. 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 What is FSI | Examples of different FSI occurrences • ALE (Arbitrary Lagrangian-Eulerian) description Atanas Gegov, TU München

  9. What is FSI | Examples of different FSI occurrences • Flow around elastic structures • Eulerian source: Dunne, Heidelberg Atanas Gegov, TU München

  10. What is FSI | Examples of different FSI occurrences • Flow around elastic structures • ALE source: Dunne, Heidelberg Atanas Gegov, TU München

  11. 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 Atanas Gegov, TU München

  12. 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 Atanas Gegov, TU München

  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 Atanas Gegov, TU München

  14. 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 Atanas Gegov, TU München

  15. 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 Atanas Gegov, TU München

  16. System 1 Partitioning m1 m1 Interface m2 m2 System 2 Different approaches for solving FSI problems • Partitioned approach | Idea Whole system (Two single mass swings) Partitioned system source: Group Prof.Rank, TUM Atanas Gegov, TU München

  17. 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 Atanas Gegov, TU München

  18. 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 Atanas Gegov, TU München

  19. 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 Different approaches for solving FSI problems source: paper C. A. Felippa Atanas Gegov, TU München

  20. 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 Atanas Gegov, TU München

  21. 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: Atanas Gegov, TU München

  22. Step 2 Step 2 x x x x x x 1 1 0 0 2 2 Step 1 Step 3 Step 1 Step 4 Step 2 y y y y 0 0 2 2 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 Atanas Gegov, TU München

  23. 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 Atanas Gegov, TU München

  24. Different approaches for solving FSI problems • Partitioned approach | Loosely-coupled and strongly-coupled partitioned approaches • 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) • Strongly-coupled methods • alternate fluid and structure solutions within a time stepuntil convergence • treat the interaction between the fluid and the structure synchronously • maintain conservation • disadvantage: greater computational cost per time step • => algorithmical improvements possible Atanas Gegov, TU München

  25. 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 Atanas Gegov, TU München

  26. Algorithmical improvements of the partitioned approach • Subiteration in detail • Initial approximation z0Є 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 Atanas Gegov, TU München

  27. 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 → 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) Atanas Gegov, TU München

  28. 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 Atanas Gegov, TU München

  29. Algorithmical improvements of the partitioned approach t=t+Δt • Multi-Grid N N FSIconverged t=t_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 Atanas Gegov, TU München

  30. 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 Atanas Gegov, TU München

  31. 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’ (zi)*(zi-zi+1) =R (zi) A * x = b Atanas Gegov, TU München

  32. 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 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 Atanas Gegov, TU München

  33. 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) Atanas Gegov, TU München

  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 => 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) Atanas Gegov, TU München

  35. 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 Atanas Gegov, TU München

  36. 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 Atanas Gegov, TU München

  37. 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 Atanas Gegov, TU München

  38. 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 Atanas Gegov, TU München

  39. How partitioned FSI can be realized – FSI*ce • Design • The communiction • Sockets transport a message from one process to another • MPI Atanas Gegov, TU München

  40. How partitioned FSI can be realized – FSI*ce • Design | The communication Communication with MPI Server programs are serial Server programs are parallel Communication with Sockets / distibuted application Atanas Gegov, TU München

  41. 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 Atanas Gegov, TU München

  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 Atanas Gegov, TU München

  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/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 Atanas Gegov, TU München

  44. Thank you for your attention! Atanas Gegov, TU München

  45. Atanas Gegov, TU München

  46. Backup slides Atanas Gegov, TU München

  47. 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) Atanas Gegov, TU München

  48. 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 <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 % Atanas Gegov, TU München

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