Μέθοδος Πολυπλέγματoς - .NATIONAL TECHNICAL UNIVERSITY OF ATHENS Parallel CFD

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Transcript of Μέθοδος Πολυπλέγματoς - .NATIONAL TECHNICAL UNIVERSITY OF ATHENS Parallel CFD

  • NATIONAL TECHNICAL UNIVERSITY OF ATHENS

    Parallel CFD & Optimization Unit

    Laboratory of Thermal Turbomachines

    o

    (Multigrid)

    . ,

    , . .

    kgianna@central.ntua.gr

    vasouti@mail.ntua.gr

    mailto:kgianna@central.ntua.grmailto:vasouti@mail.ntua.gr

  • 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

  • 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

  • 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 u

    Update u

    4

  • K.C. Giannakoglou, Parallel CFD & Optimization Unit, NTUA, Greece

    Convergence

    5

  • 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

  • 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

    hihihiHi uuuu 12212 24

    1

    HiHihi

    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

  • 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 hH

    hH rIr

    HHHH ErEA

    HE

    hhh Euu

    hu

    H

    h

    Hh EIE

    8

  • 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)

  • K.C. Giannakoglou, Parallel CFD & Optimization Unit, NTUA, Greece

    Multigrid in 2 dimensions

    Fine grid:

    Coarse grid:

    10

  • 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 jih jih jih 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 jih jih jih jih jiHji uuuuuu 12,212,22,122,122,2,8

    1

    2

    1

    11

  • 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 jiHjiH jiHjih 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

  • 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:

    13

    hhh ubuA

    hhhhhh eAuAbr

    H

    T

    HH rPeA

    Hhh Peuu

    hhh ubuA

    PRAA hH

    Hh

    nnnnRRP hH ,:

    R

  • K.C. Giannakoglou, Parallel CFD & Optimization Unit, NTUA, Greece

    Agglomeration

    14

    Mavriplis D., Mani K., Unstructured Mesh Solution Techniques using the NSU3D Solver, AIAA Paper 2014-0081

  • K.C. Giannakoglou, Parallel CFD & Optimization Unit, NTUA, Greece

    Applications

    Transonic flow around the M6 wing

    M =0.8395

    =3.06

    Re=11.72 106

    Fine grid: ~67k nodes

    Coarse grid: ~11k nodes

    17

    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)

  • 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.