Μέθοδος Πολυπλέγματoς (Multigrid)

Transcription

1 NATIONAL TECNICAL UNIVERSITY OF ATENS Parallel CFD & Optmzaton Unt Laboratory of Termal Trbomacnes Μέθοδος Πολυπλέγματoς (Mltgrd) Κυριάκος Χ. Γιαννάκογλου, Καθηγητής ΕΜΠ Βαρβάρα Ασούτη, Δρ. Μηχ. Μηχανικός ΕΜΠ

2 K.C. Gannakoglo, Parallel CFD & Optmzaton Unt, NTUA, Greece Example 0 M- M Problem: 0 0 M Bondary Condtons: 0 M 0 Intalzaton: k sn M

3 K.C. Gannakoglo, Parallel CFD & Optmzaton Unt, NTUA, Greece Dscretzaton ) ( ) ( ) ( ) ( 0 0 x rs x rs x x 3

4 K.C. Gannakoglo, Parallel CFD & Optmzaton Unt, NTUA, Greece 4 Codng dx =.d0/dfloat(nx) cp = -.d0 cm = -.d0 cc =.d0 do c=0,ncyc do =0,nx d() = 0.d0 enddo restot = 0.d0 do =,nx- a = -(-) +.d0*() - (+) resd() = rs() - a d() = (dx*dx*rs()-a - cm*d(-) - cp*d(+))/cc restot = restot + resd()*resd() enddo restot = dsqrt(restot)/dfloat(nx+) error = 0.d0 do =,nx- () = () + d() error = ()*() enddo enddo Update Compte Δ

5 K.C. Gannakoglo, Parallel CFD & Optmzaton Unt, NTUA, Greece 5 Convergence

6 K.C. Gannakoglo, Parallel CFD & Optmzaton Unt, NTUA, Greece 6 Geometrc Mltgrd: Te Idea Dfferent grd szes: Ablty to solve all wave lengts Rapd convergence x x 4x

7 K.C. Gannakoglo, Parallel CFD & Optmzaton Unt, NTUA, Greece Operators Restrcton Mappng from te fne grd to te coarse grd Inecton Fll wegtng Prolongaton Mappng from te coarse grd to te fne grd 4 7 I I : I I : : Fne grd : Coarse grd

8 K.C. Gannakoglo, Parallel CFD & Optmzaton Unt, NTUA, Greece 8 Mltgrd Algortm V-cycle sng two grds (fne/, coarse/) :. Solve (-3 Jacob or Gass-Sedel teratons). Restrct te resdal to te coarse grd 3. Solve (-3 Jacob or Gass-Sedel teratons) 4. Prolongate (nterpolate) to fne grd and correct 5. Go to step, sng te pdated r b E E A A A E E I b E r I r r E

9 K.C. Gannakoglo, Parallel CFD & Optmzaton Unt, NTUA, Greece 9 Mltgrd Scemes 4 () 3 () (4) (8) V-cycles and W-cycles 4 () 3 () Fll Mltgrd (FMG-cycles) (4) (8)

10 K.C. Gannakoglo, Parallel CFD & Optmzaton Unt, NTUA, Greece 0 Mltgrd n dmensons Fne grd: Coarse grd:

11 D Restrcton /8 0 /6 /8 /6 0 0 /8 / /8 /8 /4 / /8 0 /6 /8 /6 Inecton:,, Wegtng: alf Fll, 4 6, 8 8,,,,,,,,,,,,,, K.C. Gannakoglo, Parallel CFD & Optmzaton Unt, NTUA, Greece

12 K.C. Gannakoglo, Parallel CFD & Optmzaton Unt, NTUA, Greece D Prolongaton /4 /4 / / / /4 /4 /,,,,,,,,,,,,, 4

13 K.C. Gannakoglo, Parallel CFD & Optmzaton Unt, NTUA, Greece 3 Algebrac Mltgrd (AMG) Metod to solve lnear systems based on mltgrd prncples, wtot explct knowledge of te problem geometry (only matrx coeffcents) Determnes coarse grds, nter-grd transfer operators and te coarse grd eqatons are exclsvely based on te matrx entres Two-grd algortm:. Solve. Compte resdal 3. Solve wt 4. Correct 5. Solve A b r b T A e P r Pe A b A A A e RA P Prolongaton: Restrcton: P : R R n n R, n n matrx

14 K.C. Gannakoglo, Parallel CFD & Optmzaton Unt, NTUA, Greece 4 Agglomeraton Mavrpls D., Man K., Unstrctred Mes Solton Tecnqes sng te NSU3D Solver, AIAA Paper

15 K.C. Gannakoglo, Parallel CFD & Optmzaton Unt, NTUA, Greece 7 Applcatons Transonc flow arond te M6 wng M = α =3.06 ο Re= Fne grd: ~67k nodes Coarse grd: ~k nodes Lambropolos N., Mltgrd tecnqes and parallel processng for te nmercal predcton of flowfelds ttog termal trbomacnes, sng nstrctred grds, Pd Tess, NTUA, 005 (n greek)

16 K.C. Gannakoglo, Parallel CFD & Optmzaton Unt, NTUA, Greece 8 References Brandt A. Mlt-level Adaptve Soltons to Bondary Vale Problems Matematcs of Comptaton, 3: , 977. Brandt A. Gde to Mltgrd Development 984. ackbsc W., Mltgrd Metods and Applcatons Sprnger Verlag 985. Brggs W, enson, V., McCormck S., A Mltgrd Ttoral, nd edton, SIAM pblcatons 000.