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Visualization

Physical Phenomena. Mathematical Model. Visualization. Computational Science. Computing. Numerical Method. Software. Randverdiproblem i 1D u’’(x) = f(x), u(0) = u(1) = 0. D x. x=0. x=1. Deler intervallet [0,1] i N+1 like deler med størrelse D x = 1/(N+1).

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Visualization

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  1. Physical Phenomena Mathematical Model Visualization Computational Science Computing Numerical Method Software

  2. Randverdiproblem i 1D u’’(x) = f(x), u(0) = u(1) = 0. Dx x=0 x=1 Deler intervallet [0,1] i N+1 like deler med størrelse Dx = 1/(N+1). Med xi = i Dx, fås numeriske approksimasjon ui≈u(xi). En Taylor-rekke gir at u”(xi) ≈ (ui+1 – 2ui + ui-1)/Dx2 .

  3. Ved å bruke randbetingelse, u0 = uN+1 = 0, fås likningsystemet Dette kan skrives på formen Au = b Merk båndstrukturen på matrisen!

  4. Ustrukturert grid ”High lift configuration” CRAY T3E – 1450 prosessorer, 25 millioner gridceller University of Wyoming (1998)

  5. Værmelding 4 km oppløsning horisontalt 300 x 500 x 38 gridpunkter tidskritt på 1 min Simulerer 60 timer Bestemmer parametre i 20.5 mrd punkter Roar Skålin, IT-Direktør, met.no

  6. Tsunamien – 26 desember 2004, indiske hav AMRCLAW – adaptiv gridforfining Jan Olav Langseth Dave George Randy LeVeque ”Mesh level 1” 111 km x 111 km ”Mesh level 3” 1.7 km x 1.7 km ”Mesh level 4” 25 m x 25 m

  7. Fakta om simuleringen til venstre... • En million CPU-timer • Et tusen prosessorer • 100.000 GB med data... Joe Werne, Colorado Research Associates DivisionNorthWest Research Associates, Inc.

  8. Numerisk løsning av turbulent miksing forårsaket av en KH-instabilitet. (Rødt/gult – viskøs dissipasjon, blå – termisk dissipasjon) - NWRA/CoRA

  9. Blood Flow Simulations in the Circle of Willis Martin Sandve Alnæs Tor Ingebrigtsen Jørgen Isaksen Kent-Andre Mardal Ola Skavhaug

  10. Navier-Stokes equations are solved with the Finite Element Method, using Featflow Grids are created from a parameterization, using custom written software

  11. Knut Andreas Lie, Sintef anvendt matematikk

  12. Knut Andreas Lie, Sintef anvendt matematikkTo-fase flyt; reservoar

  13. Modelling Geometry Process

  14. Geometry • MR-images • Manual segmentation • Smooth approximation • Computational Mesh

  15. Geometry reconstruction from medical images Goal: - generate grids from MRdata - suitable for FEM - feature/organ sensitive Challenges: - in vivo measurements - the heart beats - image quality - segmentation

  16. Two data sets are generated

  17. Two data sets are generated

  18. Two data sets are generated

  19. A typical data set of torso: 512 x 320 x 40 (x,y,z) images. Body surface, left lung and right lung. A typical data set of heart: 256 x 256 x 10 x 35 (x,y,z,t) images. Heart surface, left ventricle and right ventricle

  20. Raw MR data Manually digitized slices Continuous model 3D grid

  21. A model for human ventriculartissue K. H. W. J. ten Tusscher,1 D. Noble,2 P. J. Noble,2 and A. V. Panfilov1,31Department of Theoretical Biology, Utrecht University, 3584 CH Utrecht, The Netherlands; and 2University Laboratory of Physiology, University of Oxford, Oxford OX1 3PT; and 3Division of Mathematics, University of Dundee, Dundee DD1 4HN, United Kingdom

  22. Figuren kommer fra det å løse store lineære likningssystemer Ax = b som kommer fra PDE-er. ”Improved algorithms and libraries have contributed as much to increases in capability as have improvements in hardware.”

  23. Top 500

  24. LINPACK Benchmarks • Solve a dense NxN system of linear equations, Ax=b • 2/3·N3 + 2·N2 floating point operations • Measure performance in Floating point Operations Per Second (FLOPS) • Maximum performance Rmax for problem size Nmax • Nmax varies between systems.

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