はじめに・有限要素法入門（Ⅰ） - nkl.cc.u ...

1 I I 2011 I 4820-1027 I 4810-1204 Intro-01 2 ? Intro-01 3 I 4820-1027 I 4810-1204 I 48101204 Intro-01 4 2007 3F 27219 3F 27219 il e-mail 1/2 5 ECCS2008 Intro-01 2/2 6 SMASH Science-Modeling- SMASH ScienceModelingAlgorithm-Software-Hardwa re Intro-01 7

2 Intro-01 1/2 8 Intro-01 2/2 9 Intro-01 10 HPC HPC 2008 2 1990 21 COE 21 Intro-01 HPC High-Performance Computing 11 HPC HighPerformance Computing T2K Intro-01 12 HPC Intro-01 13 HPC 4S HPC 4S ppOpen-HPC ppOpen-HPC One-domain meshUtilitiesPluggable Analysis Modules Static linear DynamicStructureFluidWaveOne-domain meshUtilitiesPluggable Analysis Modules Static linear DynamicStructureFluidWaveiiFDMFEMBEMDEMp pOpen-APPLFVMUser s ProgramPartitioner Dynamic linear Contact Dynamic linear Contact

3 MeshPEEquationsolversVisualizerParallelI /OPartitioned meshiiFTCOMMppOpen-SYSppOpen-HPCV isualization dataGPPViewPEsVisualization dataGPPViewPEsOptimized Application with Optimized ppOpen-APPL, ppOpen-MATHI ntro-014S4S 44 SS SystemSystemStageStageStatus StyleStatus Style14 44 SS SystemSystem StageStage, Status, Style, Status, Style System Stage 4 System SMASH Stage 4 AASSMM dlidliSSciencecience AASSHH AAlgorithmlgorithmMModelingodeling SSMMAASSHHSS oftwareoftwareAAlgorithmlgorithm SSMMAASSHHHH ardwareardwareIntro-014S4S 44 SS SystemSystemStageStageStatus StyleStatus Style15 44 SS SystemSystem Stage, Stage, Status, StyleStatus, Style System Stage 4 System SMASH Stage 4 St t Status HPC Style Style e-Learning Intro-01 16 3 Intro-01 17

4 SMASH 4 Intro-01 2/2 18 Intro-01 19 2008 I I I II FEM FEM 2009 2009 I II II 2010 I II 2011 I I II IIIntro-0120 ? Intro-01 21 Taylor Taylor Intro-01 1/3 22 1/3 =12xxdxdxBFdxdmax22@0,0@0.

5 0=====+ xxBFxBFmax221+ = BF =0 =0 Intro-01 2/3 23 2/3 xxBFxBFmax221+ = X=X XXmax =0X=0X=0 x= =50 Xmax= X 49 0 BF 1 0d0 12255985051200495494912+ + X BF= + = + = Intro-01 24 Finite Difference Method Finite Difference Method i-1 i i+1 x xIntro-01 25 i i+1 i i+1 i-1 i i+1d x x xdxdiii ++ 12/1 x 0 i 11112/12/122xxdxddxddiiiiiiiii+ = = + + + 22xxxdxi = = Intro-01 Taylor 1/3 26 Taylor 1/3 i-1 i i+1 x x()()Kiixxx + + +=+3332221 iiiiixxx +321!3!2 ()()Kiiiiixxxxxx + = 3332221!3!2 iii Intro-01 Taylor 2/3 27 Taylor 2/3 i-1 i i+1 x x()()xx 3322 ()()Kiiiiixxxxxx + + +=+321!

6 3!2 ()() 322 ()()Kiiiiixxxxxx + + = +332221!3!2 x ()()xx 3322 ()()Kiiiiixxxxxx + = 321!3!2 ()()iixx 3221 ()()Kiiiiixxxxxx + + = 321!3!2 x Intro-01 Taylor 3/3 28 Taylor 3/3 i-1 i i+1 x x()() 3322 ()()Kiiiiixxxxxx + + +=+3332221!3!2 ()()Kiixxxxxx + = 3332221!3!2 iiixxx !3!2()x 322 ()Kiiiixxxx + = +311!322 ( x)2 Intro-01 29 i-1 i i+1 x x()()33222/2/ xx ()()K2/132/122/12/11!32/!22/2/+++++ + + +=iiiiixxxxxx ()()K2/13332/12222/12/1!32/!22/2/++++ + =iiiiixxxxxx ()K3321!32/2+ + = iix ( x)2 2/132/1!

7 3++ iixxx() x 3/3 30 1122dddddiiii + 2112/12/1222xxxxxdxdxdxdiiiiii + = = + + )1(0)(2211 NiiBFiii =++ + 022=+BFdxd )1(0)(121)1(0)(122122 NiiBFxxxNiiBFxiii =+ + + + 11)1()()()()(NiiBFiAiAiAiRiDiL = + + + xxx Intro-012221)(,2)(,1)(xiAxiAxiARDL = = =Intro-01 31 Taylor Taylor Fi it Elt M th d FEM Finite Element Method FEM weak form weak solution weak solution Intro-01 32 Handbook of Grid Generation Intro-01 FiniteElement Method (FEM)3 33 Finite-Element Method (FEM) elements (meshes ) & nodes (vertices )()() :22 TT7891314151602222=+ + QyTxT 16 9 ( =1)4567899101112 ( =1) (Q=1) 1 T=01234565678 1 T=0 1123234 Intro-01 Galerkin FEM procedures3 34 Galerkin FEM procedures : :[]022 dVQTTNT []}{ NT= []022= + + dVQyxNV [N].

8 78913141516 [][][][]{} NNNNTT4567899101112[][][][]{}[] + dVyNyNxNxNTV 1234565678[]0=+ dVNQVT1123234 Intro-01 Element Matrix 3 35 Element Matrix eDC[][][][]{} + dVNNNNTT BA{}[]0=+ + dVNQdVyyxxTV =)()()()()()()()()(}{}]{[eeeeeeeeefkkkkf k []0+ dVNQV = )()()()()()()()()()()()(eeBAeeBAeeeeeBDe BCeBBeBAADACABAA fffkkkkkkkkkkkk )()()()()()(eDCeDCeDDeDCeDBeDACDCCCBCA ffkkkkkkkk Intro-01 Global/overall Matrix 3 36 Global/overall Matrix 78913141516 = 2121}{}]{[FFXXXXDXXXXDFK4567899101112 54325432 FFFXXXDXXXXDXXXXXDX1234565678 87658765 FFFXXDXXXXXXXDXXXXXXXXDXXXX1123234 = 111098111098 FFFXXXXDXXXXXXXXDXXXXXXXDXX 1413121114131211 FFFFXDXXXXXDXXXXDXXXXXXXDXXXX 161514161514 FFFDXXXXDXXXXXDXXXXI ntro-01 Global/overall Matrix 3 37 Global/overall Matrix 78913141516 = 2121}{}]

9 {[FFXXXXDXXXXDFK4567899101112 54325432 FFFXXXDXXXXDXXXXXDX1234565678 87658765 FFFXXDXXXXXXXDXXXXXXXXDXXXX1123234 = 111098111098 FFFXXXXDXXXXXXXXDXXXXXXXDXX 1413121114131211 FFFFXDXXXXXDXXXXDXXXXXXXDXXXX 161514161514 FFFDXXXXDXXXXXDXXXXI ntro-01 3 38 ( 1=0) 11 FFXXXXDXXXXD 432432 FFFFXXXDXXXXDXXXXXDXXXXXDX 765765 FFFFXXDXXXXXXXDXXXXXXXXDXXXXXXXDXX = 10981098 FFFFXXXXDXXXXXXXXDXXXXXXXDXXXXDXXX 131211131211 FFFFXDXXXXXDXXXXDXXXXXXXDXXXX 161514161514 FFFDXXXXDXXXXXDXXXXI ntro-01 3 39 22 TT022=+ + QyTxT 78913141516456910111212356781234 Intro-01 40 Intro-01 41 X Intro-01 42 Y Intro-01 43 1950 University of Washington]}

10 University of Washington , NASTRAN NASA MSC PC Intro-01 44 MSC/NASTRAN ABAQUS ANSYS http://www ansys jp/ Intro-01 45 Adaptive Mesh Refinement Intro-01 46 = Re =1062-Lev. Adapted1-Lev. AdaptedInitial Grid2 Lev. Adapted 38342527276925261 Lev. Adapted 793652696650 Initial Grid PE0 137-PE1 137-2769 25262703 252213902524696 650668 652448651PE1 137 PE2 136-PE3 136 -Intro-01 47 movieIntro-0148 ? Intro-01 494 19 CS-01 I 4 26 CS-02 II 5 02 CS03 I 5 02 CS-03 I 5 10 CS-04 II 5 17 CS-05 III 5 24 CS-06 IV 5 31 CS-07 V 1 6 07 6 10 CS-08 I 6 14 CS09 II 6 14 CS-09 II 6 17 CS-10 III 6 21 6 28 CS-11 IV 7 05 CS-12 V 2 7 12 CS-13 1 Intro-01 50 LU GaussSeidel LU Gauss-Seidel C FORTRAN C FORTRAN UNIX ECCS2008 1/2 51 SMASHMM dliodelingSSciencecience SMASH Science AAlgorithmlgorithmMModelingodeling SSoftwareoftwareAAlgorithmlgorithm HHardwareardware