13 14 Randverdiproblem i 1D u’’(x) = f(x), u(0) = u(1) = 0. x=0 x=1 xx Deler intervallet [0,1] i N+1 like deler med størrelse.

Presentasjon om: "13 14 Randverdiproblem i 1D u’’(x) = f(x), u(0) = u(1) = 0. x=0 x=1 xx Deler intervallet [0,1] i N+1 like deler med størrelse."— Utskrift av presentasjonen:

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14 14 Randverdiproblem i 1D u’’(x) = f(x), u(0) = u(1) = 0. x=0 x=1 xx Deler intervallet [0,1] i N+1 like deler med størrelse  x = 1/(N+1). Med x i = i  x, fås numeriske approksimasjon u i ≈u(x i ). En Taylor-rekke gir at u”(x i ) ≈ (u i+1 – 2u i + u i-1 )/  x 2.

15 15 Ved å bruke randbetingelse, u 0 = u N+1 = 0, fås likningsystemet Dette kan skrives på formen Au = b Merk båndstrukturen på matrisen!

16 16 Ustrukturert grid ”High lift configuration” CRAY T3E – 1450 prosessorer, 25 millioner gridceller University of Wyoming (1998)

17 17 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

18 18 Tsunamien – 26 desember 2004, indiske hav AMRCLAW – adaptiv gridforfining ”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 Jan Olav Langseth Dave George Randy LeVeque

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20 20 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.

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

22 22 Blood Flow Simulations in the Circle of Willis Martin Sandve Alnæs Tor Ingebrigtsen Jørgen Isaksen Kent-Andre Mardal Ola Skavhaug

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25 25 Navier-Stokes equations are solved with the Finite Element Method, using Featflow Grids are created from a parameterization, using custom written software

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27 Knut Andreas Lie, Sintef anvendt matematikk

28 Knut Andreas Lie, Sintef anvendt matematikk To-fase flyt; reservoar

29 29 Modelling Geometry Process

30 30 Geometry MR-images Manual segmentation Smooth approximation Computational Mesh

31 31 Geometry reconstruction from medical images Goal: - generate grids from MR data - suitable for FEM - feature/organ sensitive Challenges: - in vivo measurements - the heart beats - image quality - segmentation

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36 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

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46 Raw MR dataManually digitized slices 3D gridContinuous model

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

48 48 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.”

49 Top 500 49

50 LINPACK Benchmarks Solve a dense NxN system of linear equations, Ax=b 2/3·N 3 + 2·N 2 floating point operations Measure performance in Floating point Operations Per Second (FLOPS) Maximum performance R max for problem size N max N max varies between systems. 50

51 ENIAC 330 FLOPS One thousand times faster than electro- mechanical machines Greatest leap ever in computing power 51

52 Thinking Machines CM-5/1024 1. top500 list, June 1993 Los Alamos National Laboratory R max = 59.7 GigaFLOPS 1024 cores 52

53 JAGUAR Top 500 no 1, nov 2009 National Centre for Computer Science, Oak Ridge R max = 1.75 PetaFLOPS 224 162 Cores 53

54 MODULO Simula Research Laboratory 64 cores

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58 R max is less than theoretical peak performance R peak R peak for JAGUAR is 2.3 PetaFLOPS The Linpack benchmark uses dense matrix operations, this puts R max closer to R peak than is typically observed in practice. 58

59 Boussinesq Equations CPUTimeSpeedup 1166.66N/A 283.611.99 444.453.75 820.168.27 1611.4314.58