DN1215 vt 12 för ME. Numeriska metoder DN1240 numi12 1

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1 DN1215 vt 12 för ME Numeriska metoder DN1240 numi12 1

2 F1 Översikt: Lektionsundervisning och datorlaborationer Lärare Jesper Oppelstrup Eamination 2 Matlab-lab, muntlig redovisning, 1 projekt, skriftlig Tenta som alla Numme GK (1212,1214, ) 1 kryssfrågedel för betyg E 1 med räkneuppgifter DN1240 numi12 2

3 A(n in)convenient truth It is hard to understand an ocean because it is too big. It is hard to understand a molecule because it is too small. It is hard to understand nuclear physics because it is too fast. It is hard to understand the greenhouse effect because it is too slow. [Super]Computers break these barriers to understanding. They, in effect, shrink oceans, zoom in on molecules, slow down physics, and fast-forward climates. Clearly a scientist who can see natural phenomena at the right size and the right speed learns more than one who is faced with a blur. Al Gore, 1990, Scientific Computing DN1240 numi12 3

4 Wikipedia: Scientific computing (or computational science) is concerned with constructing mathematical models and numerical solution techniques and [ ] using computers to analyze and solve scientific, social scientific, and engineering problems. Scientific Computing Numerical analysis is the study of algorithms for the problems of continuous mathematics [ ] to compute quantities that are typically uncomputable, [ ] with lightning speed (L.N.Trefethen 1992) The purpose of computing is insight, not numbers. (R.W.Hamming) "The Unreasonable Effectiveness of Mathematics" (1980) DN1212 F

5 Computer simulation is the third paradigm of science Simulation has become recognized as the third paradigm of science, the first two being eperimentation and theory. High Performance Computing and Communications: Foundation for America's Information Future (Supplement to the President s FY 1996 Budget) DN1212 F

6 Antenna in rear-view mirror: Directivity, EMC, Dendritic solidification Mawell simulator (T.Rylander & al, CTU) Phase-field simulator (G.Amberg & al, KTH) There are three kinds of lies: Lies, damn lies, and colorful computer pictures (P.Colella) DN1212 F

7 Differential Equation Models Shortest path Micro-wave tomography Tsunami DN1212 F

8 Computational Materials Science From quantum mechanics to structures: Goals: σ ij i = µ(u, j Predict macroscopic properties from first principles Design new materials (e. nano-technology) Scale: space 10 8, time QM: Many-D 1 A HΨ=EΨ DFT MD ij σ, j = Fi Dislocation field + u,i j ) λu,k k δ ij Navier 1 cm DN1212 F

9 Success Story I: MATLAB Major tool for engineering computing, Numerical analysis, visualization 1974 Matri Laboratory C.Moler Now: MathWorks > 1000 pers. 20th anniversary 2005: Success Story II: comsol Multiphysics FE software S.Littmarck 1995: MATLAB PDEToolBo Now: Comsol > 150 pers. DN1212 F

10 Numerical Methods 101" Why numerical methods? What can be computed? 2. Approimation, iteration, linearization, algorithm 3. Ordinary differential equations: simulation, eamples 1. Partial differential equations: simulation, eempel DN1212 F

11 Most solutions are approimations! Compute but how? = ± 1. Calculator... but how does it do it? 2. Mathematics: 2 = 2 2 ±1.414 (1+ ) 1/ 2 = 1/ 2 k =1+ 1 k k=0 DN1212 F

12 (2) 1/ 2 = ( )1/ 2 = 5 4 ( )1/ 2 = = ( ) = 1.25 = (Puh despite the convergence acceleration) DN1212 F

13 DN1212 F Linearisation and iteration: Newton's method ( ) ( ) + = = = = = + + n n n n n n n f f f f : ) (..., 0 : ) ( Mmm!

14 Linjära ekvationssystem Finn = ( 1, 2,, n ) så att A = b, A = (a ij ) NAM: 1.4, 1.5, 1.6 (inte 1.6.2), 1.7, 1.8 eller minstakvadrat-lösning min A b 2 NAM 2 Olinjär ekvation Finn så att f()=0 NAM 6 Olinjära ekvationssystem Finn så att f() = 0 NAM 6.8, 6.9 eller minstakvadrat-lösning min f() 2 NAM DN1240 numi12 14

15 Initialvärdesproblem för ordinära differentialekvationer Finn y(t) för t > 0 då dy/dt = f(t,y), y(0) = c NAM plus etra Randvärdesproblem för ordinära differentialekvationer Finn y() för a < < b då dy/d = f(,y), G(y(a),y(b)) = 0 NAM 8.7 plus etra DN1240 numi12 15

16 Kvadratur Beräkna b I = f ( ) d a I = f, y) ddy Beräkna f ( ) då f ( i ) = yi, i = 1,2,..., m ( Interpolation / Approimation D - Tabeller NAM 5 Inte 5.2.4,5.2.5; - Polynom P k () Optimering - Spline-funktioner Finn ma. för f() i - Bézier-kurvor a < < b NAM 7 (inte 7.1.2) plus etra DN1240 numi12 16

17 Linjära ekvationssystem Finn = ( 1, 2,, n ) så att A = b, A = (a ij ) Formulering Gauss-elimination Pivotering Arbetsvolym A NAM p 7,8 Blandningsproblem: Råvara R i (i=1 n) har halterna a ij av grundpigment j (j=1 n) Hur ska man blanda, halt j av R j för att få en produkt med halt w k av pigment k? n = w : aij j = wi, i = 1,2,..., n j= j DN1240 numi12 17

18 MATLAB % NAMp7 om färger lind = [ ]'; gras = [ ]'; moss = [ ]'; hav = [ ]'; A = [lind gras moss hav] wanted = [ ]'; = A\wanted pause [ ] gör radvektor - transponerar till kolumn A = [lind gras moss hav] - gör 44 matris genom att ställa kolumnerna bredvid varann wanted = [ ] - gör högerled = A \ wanted - lös med Gauss-elimination pause - stanna DN1240 numi12 18

19 MATLAB Vilka färger kan blandas? Simulera! Gör N blandningar där man väljer i som positiva slumptal som summerar till 1 ( i = halt) Lagra i en 4 N tabell % generera slumptal som summerar till 1; N = 1000; = rand(4,n); % rand ger slumptal mellan 0 och 1, % 4 rader, N kolonner s = sum(,1); % Summera kolonner = *diag(1./s); % Dividera kolonn med dess summa % genom multiplikation med diagonalmatris % från höger y = A*; % Blanda! DN1240 numi12 19

20 Plotta! Plotta alla punkterna som ligger i en 4D simple i R 4 som fyra projektioner på de tre-dimensionella rummen med alar (2,3,4),(1,3,4),(1,2,4),(1,2,3): Alla punkter i projektionerna i 3D simplear = tetraedrar. Tetraederkanterna plottas DN1240 numi12 20

21 % now plot anum = 1:4; for k = 1:4 ypl = y; ypl(k,:)= []; anumpl = anum; anumpl(k)=[]; Apl = A; Apl(k,:)=[]; subplot(1,4,k); end MATLAB %clf hold on plot3(ypl(1,:),ypl(2,:),ypl(3,:),'.k'); label(['x',num2str(anumpl(1))]) ylabel(['x',num2str(anumpl(2))]) zlabel(['x',num2str(anumpl(3))]) for j = 1:4 a = Apl(:,j); for l = j+1:4 b = Apl(:,l); plot3([a(1),b(1)],[a(2),b(2)],[a(3),b(3)],'k'); end end ais([ ]) Plotta i delfönster, 1 rad 4 kolonner, i k:te delfönstret Plotta i 3D som svarta punkter Välj ael-storlek Resten av programmet är speciellt ; titta på det senare DN1240 numi12 21