1. Technische Universität München Benefits of Structured Cartesian Grids for the Simulation of Fluid- Structure Interactions Miriam Mehl Department of Computer Science TU München

2. Technische Universität München Outline Our Cartesian Grids Requirements Fluid-Structure Interactions Cartesian Grids – CFD Cartesian Grids – Coupling Application Examples Conclusion

3. Technische Universität München Our Cartesian Grids Cartesian grid cells  squares/cubes recursive refinement  tree structure

4. Technische Universität München Our Cartesian Grids Cartesian grid cells  squares/cubes recursive refinement  tree structure

5. Technische Universität München Fluid-Structure Interactions – Requirements complex and changing geometries  flow solver

6. Technische Universität München Fluid-Structure Interactions – Requirements complex and changing geometries  flow solver partitioned approaches  coupling of codes  modularity Structure Solver Flow Solver Coupling

7. Technische Universität München Cartesian Grids – CFD fast and flexible geometry treatment Eulerian Approach + Marker-and-Cell

8. Technische Universität München Cartesian Grids – CFD # grid cellsruntime (sec) 52,662,33748.188 210,666,753168.641 842,687.105662.797 Pentium 4, 2.4 GHz, 512 MB cache fast and flexible geometry treatment

9. Technische Universität München Cartesian Grids – CFD fast and flexible geometry treatment

10. Technische Universität München Cartesian Grids – CFD fast and flexible geometry treatment

11. Technische Universität München Cartesian Grids – CFD recursive cell-tree  local grid changes fast and flexible geometry treatment

12. Technische Universität München Cartesian Grids – CFD hardware + numerical efficiency

13. Technische Universität München Cartesian Grids – CFD cell-oriented operator evaluation  constant difference stencils  no neighbour relations

14. Technische Universität München Cartesian Grids – CFD cell-oriented operator evaluation  constant difference stencils  no neighbour relations i,j i-1,j ½ -1 ½

15. Technische Universität München Cartesian Grids – CFD cell-oriented operator evaluation  constant difference stencils  no neighbour relations i,j i-1,j i,j i-1,j -1 ½ ½

16. Technische Universität München Cartesian Grids – CFD cell-oriented operator evaluation  constant difference stencils  no neighbour relations i-1,j ½ -1 ½

17. Technische Universität München Cartesian Grids – CFD cell-oriented operator evaluation  constant difference stencils  no neighbour relations ½ ½ -1

18. Technische Universität München Cartesian Grids – CFD Peano curve  linearisation of the cell-tree  processing order

19. Technische Universität München Cartesian Grids – CFD Peano curve  linearisation of the cell-tree  processing order

20. Technische Universität München Cartesian Grids – CFD Peano curve + stacks = data access with  locality in space  locality in time

21. Technische Universität München Cartesian Grids – CFD Peano curve + stacks = data access with  locality in space  locality in time

22. Technische Universität München Cartesian Grids – CFD low memory requirements bytes/cellbytes/vertex 2D 62only grid 1420flow solver 3D 102only grid 1828flow solver hardware + numerical efficiency

23. Technische Universität München ==19243== D refs: 7,249,842,728 (4,026,485,237 rd + 3,223,357,491 wr) ==19243== D1 misses: 1,249,032 ( 621,413 rd + 627,619 wr) ==19243== L2d misses: 632,162 ( 301,283 rd + 330,879 wr) ==19243== D1 miss rate: 0.0% ( 0.0% + 0.0% ) ==19243== L2d miss rate: 0.0% ( 0.0% + 0.0% ) ==19243== ==19243== L2 refs: 19,559,185 ( 18,931,566 rd + 627,619 wr) ==19243== L2 misses: 646,343 ( 315,464 rd + 330,879 wr) ==19243== L2 miss rate: 0.0% ( 0.0% + 0.0% ) Cartesian Grids – CFD 2D Poisson equation, 1,000,000 degrees of freedom, Pentium 4, 1MB L2 Cache, Cachegrind simulation hardware + numerical efficiency high cache-efficiency

24. Technische Universität München Cartesian Grids – CFD multigrid dehierarchisation compute residual smooth restrict residual hardware + numerical efficiency

25. Technische Universität München Cartesian Grids – CFD # dyn. refinem.k=0k=1k=2k=3 # iterations91099 accuracy5.972e-24.613e-34.521e-46.771e-5 Poisson equation on a cube, F-cycle hardware + numerical efficiency multigrid

26. Technische Universität München Cartesian Grids – CFD tol. 1.17e-3reg. gridadapt. grid # dofs509.65661.267 hardware + numerical efficiency dynamical adaptivity

27. Technische Universität München Cartesian Grids – CFD dynamically balanced parallelisation 0 1234 567817181920 131415169101112

28. Technische Universität München Cartesian Grids – CFD  connected partitions  quasi-minimal partition surface dynamically balanced parallelisation

29. Technische Universität München Cartesian Grids – CFD dynamically balanced parallelisation

30. Technische Universität München Advantages of Cartesian Grids – CFD dynamically balanced parallelisation

31. Technische Universität München Cartesian Grids – CFD dynamically balanced parallelisation

32. Technische Universität München Cartesian Grids – Coupling efficient data mapping for non-matching grids fluid solver + interpolation struct. solver + interpolation FSI*ce surface coupling Grid administration Data mapping

33. Technische Universität München Cartesian Grids – Coupling

34. Technische Universität München Cartesian Grids – Coupling sphere (8,000 triangles)

35. Technische Universität München Cartesian Grids – Coupling grid resolution# boundary nodesruntime [s] 6418,4820.200 12875,5140.671 256305,3942.594 5121,227,98710.114 sphere (8,000 triangles), Pentium M 1.6 GHz, 2048 kB cache efficient data mapping for non-matching grids

36. Technische Universität München Cartesian Grids – Coupling trianglesruntime [s] 16,00012.7 32,00014.3 64,00016.2 128,00017.4 grid resolution 512, Pentium M 1.6 GHz, 2048 kB cache efficient data mapping for non-matching grids

37. Technische Universität München Application Examples – Cylinder Benchmark

38. Technische Universität München Application Examples – Beam

39. Technische Universität München Application Examples – Drift Ratchet silicon wafer pierced with pores oscillating pressure conditions suspended particles (0.1 – 1.2  m) observation: particle drift

40. Technische Universität München Application Examples – Drift Ratchet

41. Technische Universität München Application Examples – Drift Ratchet

42. Technische Universität München Application Examples – Drift Ratchet frequency=10kHzfrequency=14kHz

43. Technische Universität München Application Examples – Drift Ratchet frequency=7kHz frequency=14kHz

44. Technische Universität München Conclusion applicability of Cartesian Grids fast grid generation / updates memory efficiency numerical efficiency

45. Technische Universität München Persons Hans-Joachim Bungartz Markus Brenk Klaus Daubner Ioan Lucian Muntean Tobias Neckel Tobias Weinzierl