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NATIONAL TECHNICAL UNIVERSITY OF ATHENS
Parallel CFD & Optimization Unit
Laboratory of Thermal Turbomachines
Μέθοδος Πολυπλέγματoς
(Multigrid)
Κυριάκος Χ. Γιαννάκογλου, Καθηγητής ΕΜΠ
Βαρβάρα Ασούτη, Δρ. Μηχ. Μηχανικός ΕΜΠ
[email protected]
[email protected]
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K.C. Giannakoglou, Parallel CFD & Optimization Unit, NTUA, Greece
Example
Problem:
Boundary Conditions:
Initialization:
02 11 iii uuu
00 Muu
0 1 M M-1
M
ikui
sin
Mi 0
2
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K.C. Giannakoglou, Parallel CFD & Optimization Unit, NTUA, Greece
Discretization
)(22
1
)()(2)(
02
0
1111
2
2
1111
2
11
2
2
iiiiii
iiiiii
ii
uuuuuxrhsu
xrhsuuuuuu
x
uuu
x
u
3
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K.C. Giannakoglou, Parallel CFD & Optimization Unit, NTUA, Greece
dx = 1.d0/dfloat(nx)
c1p = -1.d0
c1m = -1.d0
cc = 2.d0
do ic=0,ncyc
do i=0,nx
du(i) = 0.d0
enddo
restot = 0.d0
do i=1,nx-1
au = -u(i-1) + 2.d0*u(i) - u(i+1)
resid(i) = rhs(i) - au
du(i) = (dx*dx*rhs(i)-au - c1m*du(i-1) - c1p*du(i+1))/cc
restot = restot + resid(i)*resid(i)
enddo
restot = dsqrt(restot)/dfloat(nx+1)
error = 0.d0
do i=1,nx-1
u(i) = u(i) + du(i)
error = u(i)*u(i)
enddo
enddo
Coding
Compute Δu
Update u
4
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K.C. Giannakoglou, Parallel CFD & Optimization Unit, NTUA, Greece
Convergence
5
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K.C. Giannakoglou, Parallel CFD & Optimization Unit, NTUA, Greece
Geometric Multigrid: The Idea
Different grid sizes:
Ability to solve all wave lengths
Rapid convergence
6
x
2x
4x
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K.C. Giannakoglou, Parallel CFD & Optimization Unit, NTUA, Greece
Operators
Restriction
Mapping from the fine grid to the coarse grid
Injection
Full weighting
Prolongation
Mapping from the coarse grid to the fine grid
h
i
H
i uu 2
h
i
h
i
h
i
H
i uuuu 12212 24
1
H
i
H
i
h
i
H
i
h
i
uuu
uu
112
2
2
1
7
hh
H
H
hHh
H
I
I
:
HH
h
h
HhH
h
I
I
:
h: Fine grid
H: Coarse grid
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K.C. Giannakoglou, Parallel CFD & Optimization Unit, NTUA, Greece
V-cycle using two grids (fine/h, coarse/H) :
1. Solve (2-3 Jacobi or Gauss-Seidel iterations)
2. Restrict the residual to the coarse grid
3. Solve (2-3 Jacobi or Gauss-Seidel iterations)
4. Prolongate (interpolate) to fine grid
and correct
5. Go to step 1, using the updated
Multigrid Algorithm
hhh ubuA
hhhh uAbr h
H
hH rIr
HHHH ErEA
HE
hhh Euu
hu
H
h
Hh EIE
8
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K.C. Giannakoglou, Parallel CFD & Optimization Unit, NTUA, Greece
Multigrid Schemes
9
4 (h)
3 (2h)
2 (4h)
1 (8h)
V-cycles
and
W-cycles
Full Multigrid
(FMG-cycles) 4 (h)
3 (2h)
2 (4h)
1 (8h)
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K.C. Giannakoglou, Parallel CFD & Optimization Unit, NTUA, Greece
Multigrid in 2 dimensions
Fine grid:
Coarse grid:
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K.C. Giannakoglou, Parallel CFD & Optimization Unit, NTUA, Greece
2D Restriction
Injection:
Weighting:
Half
Full
0 0 0
0 0 1
0 0 0
0 0 1/8
1/8 1/8 1/2
0 0 1/8
1/16 1/16 1/8
1/8 1/8 1/4
1/16 1/16 1/8
h
ji
H
ji uu 2,2,
h
ji
h
ji
h
ji
h
ji
h
ji
h
ji
h
ji
h
ji
h
ji
H
ji
uuuu
uuuuuu
12,1212,1212,1212,12
12,212,22,122,122,2,
16
1
8
1
4
1
h
ji
h
ji
h
ji
h
ji
h
ji
H
ji uuuuuu 12,212,22,122,122,2,8
1
2
1
11
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K.C. Giannakoglou, Parallel CFD & Optimization Unit, NTUA, Greece
2D Prolongation
1/4 1/4 1/2
1/2 1/2 1
1/4 1/4 1/2
H
ji
H
ji
H
ji
H
ji
h
ji
H
ji
H
ji
h
ji
H
ji
H
ji
h
ji
H
ji
h
ji
uuuuu
uuu
uuu
uu
1,11,,1,12,12
1,,12,2
,1,2,12
,2,2
4
1
2
1
2
1
12
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K.C. Giannakoglou, Parallel CFD & Optimization Unit, NTUA, Greece
Algebraic Multigrid (AMG)
Method to solve linear systems based on multigrid principles, without
explicit knowledge of the problem geometry (only matrix coefficients)
Determines coarse “grids”, inter-grid transfer operators and the coarse grid
equations are exclusively based on the matrix entries
Two-grid algorithm:
1. Solve
2. Compute residual
3. Solve with
4. Correct
5. Solve
Prolongation: matrix
Restriction:
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hhh ubuA
hhhhhh eAuAbr
H
T
HH rPeA
Hhh Peuu
hhh ubuA
PRAA hH
Hh
nnnnRRP hH ,:
R
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K.C. Giannakoglou, Parallel CFD & Optimization Unit, NTUA, Greece
Agglomeration
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Mavriplis D., Mani K., “Unstructured Mesh Solution Techniques using the NSU3D Solver”, AIAA Paper 2014-0081
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K.C. Giannakoglou, Parallel CFD & Optimization Unit, NTUA, Greece
Applications
Transonic flow around the M6 wing
M ∞ =0.8395
α ∞ =3.06ο
Re=11.72 106
Fine grid: ~67k nodes
Coarse grid: ~11k nodes
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Lambropoulos N., “Multigrid techniques and parallel
processing for the numerical prediction of flowfields thtough
thermal turbomachines, using unstructured grids”, Phd
Thesis, NTUA, 2005 (in greek)
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K.C. Giannakoglou, Parallel CFD & Optimization Unit, NTUA, Greece
References
18
Brandt A. “Multi-level Adaptive Solutions to Boundary Value Problems”
Mathematics of Computation, 31:333-390, 1977.
Brandt A. “Guide to Multigrid Development” 1984.
Hackbusch W., “Multigrid Methods and Applications” Springer – Verlag 1985.
Briggs W, Henson, V., McCormick S., “A Multigrid Tutorial”, 2nd edition, SIAM
publications 2000.