FMM based solution of electrostatic and based solution of electrostatic and magnetostatic field...

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  • University of Stuttgart Institute for Theory of Electrical Engineering

    FMM based solution of electrostatic and magnetostatic field problems

    Andr BuchauWolfgang Hafla

    Friedemann GrohWolfgang M. Rucker

  • University of Stuttgart Institute for Theory of Electrical Engineering

    Outline

    Introduction

    Octree in practice

    FMM for direct and indirect BEM formulations

    Fast series expansion transformations

    Postprocessing

    Numerical results

    Conclusion

  • University of Stuttgart Institute for Theory of Electrical Engineering

    Introduction

    Fast adaptive multipole boundary element method (FAM-BEM)

    Electrostatic, magnetostatic, and steady current flow field problems

    Direct and indirect BEM formulations

    Dirichlet and Neumann boundary conditions

    8-noded, second order quadrilateral elements

    20-noded, second order hexahedral elements

    GMRES with Jacobi preconditioner

    Fast multipole method

  • University of Stuttgart Institute for Theory of Electrical Engineering

    Octree in practice

    Adaptive meshes

  • University of Stuttgart Institute for Theory of Electrical Engineering

    Octree in practice

    Adaptive meshes

  • University of Stuttgart Institute for Theory of Electrical Engineering

    Octree in practice

    Series expansions

    Multipole expansion

    ( ) ( )100

    1 1 ,4

    L nm mn nn

    n m nu Y M

    r

    += == r

    Local expansion

    ( ) ( )00

    1 ,4

    L nn m mn n

    n m nu r Y L

    = == r

  • University of Stuttgart Institute for Theory of Electrical Engineering

    Octree in practice

    Convergence of the multipole expansion

    0 1 2 3 4 5r (m)

    01020304050u

    (V)

    analyticL = 0L = 1L = 2L = 3

  • University of Stuttgart Institute for Theory of Electrical Engineering

    Octree in practice

    Octree

    xy

    z

    34

    56

    78

    12

  • University of Stuttgart Institute for Theory of Electrical Engineering

    Octree in practice

    Classical near- and far-field definition

    consideredcube

    near-field far-field

  • University of Stuttgart Institute for Theory of Electrical Engineering

    Octree in practice

    Problems caused by higher order elements

    Extremely varying size of the elements

    Inhomogeneous distribution of elements

    Elements can jut out of a cube

    Possible solutions

    Ignore the problems

    Cut the elements at the boundaries of the cubes

    Consider real convergence radii of the cubes

  • University of Stuttgart Institute for Theory of Electrical Engineering

    Octree in practice

    Convergence radius of a cube

    R

    center

    elements

  • University of Stuttgart Institute for Theory of Electrical Engineering

    Octree in practice

    Convergence radius of a cube

    R

  • University of Stuttgart Institute for Theory of Electrical Engineering

    Octree in practice

    Near-field interactions

    1R2R

  • University of Stuttgart Institute for Theory of Electrical Engineering

    Octree in practice

    Far-field interactions

    1R 2R

  • University of Stuttgart Institute for Theory of Electrical Engineering

    FMM for direct and indirect BEM formulations

    Direct BEM formulationElectrostatics

    Steady current flow fields

    Greens theorem

    ( ) ( ) ( ) ( )' 1 1d ' ' d '' ' ' '

    uc u A u A

    n n

    = r

    r r rr r r r

    Dirichlet boundary conditions

    Neumann boundary conditions

  • University of Stuttgart Institute for Theory of Electrical Engineering

    FMM for direct and indirect BEM formulations

    Indirect BEM formulationElectrostatics

    Magnetostatics

    Charge densities

    ( ) ( )0

    '1 d '4 'A

    u A

    =r

    rr r

    Dirichlet boundary conditions

    Neumann boundary conditions

  • University of Stuttgart Institute for Theory of Electrical Engineering

    FMM for direct and indirect BEM formulations

    Classical multipole expansionClassical integral

    ( ) ( )0

    '1 d '4 'A

    u A

    =r

    rr r

    Multipole expansion

    ( ) ( )100

    1 1 ,4

    L nm mn nn

    n m nu Y M

    r

    += == r

    ( ) ( )' ' ', ' d 'm n mn nA

    M r Y A = r

  • University of Stuttgart Institute for Theory of Electrical Engineering

    FMM for direct and indirect BEM formulations

    Fast multipole method for double-layer potentialsIntegral

    ( ) ( ) '0

    1 1' 'd '4 'A

    u A

    = rr r nr r

    Multipole expansion

    ( ) ( )100

    1 1 ,4

    L nm mn nn

    n m nu Y M

    r

    += == r

    ( ) ( ) ( )( )'' ' ' ', ' d 'm n mn nA

    M r Y A = rr n r

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    Fast series expansion transformations

    Series expansion transformations

    Multipole-to-multipole transformation

    Multipole-to-local transformation large CPU-time

    Local-to-local-transformation

  • University of Stuttgart Institute for Theory of Electrical Engineering

    Fast series expansion transformations

    Multipole-to-local transformation

    Classical approach

    ( )( ) 10

    j ,

    1

    m l m ll l m l mL kk k n k nm

    n k k n l mk l k n k

    M A A YL

    A

    +

    + + = = +

    =

    ( )4O L

  • University of Stuttgart Institute for Theory of Electrical Engineering

    Fast series expansion transformations

    Multipole-to-local transformation

    Transformation in z-direction: O(L3)

    Rotation about the z-axisj' em m mn nM M=

    Rotation about the y-axis

    ( )( ) ( ) ( )1 *'

    0

    ' , , ', 1 , , ',n

    mm m mn n n

    m n mM R n m m M R n m m M

    = =

    = + Transformation in z-direction

    ( )( ) ( )( ) ( ) ( ) ( )

    0

    1

    0,0 1 !

    ! ! ! !

    k mLk nm m

    n k k nk m

    Y n kL M

    k m k m n m n m

    ++

    + +=

    +=

    + +

  • University of Stuttgart Institute for Theory of Electrical Engineering

    Fast series expansion transformations

    Multipole-to-local transformation

    Plane waves: O(L2)

    Definition of main-directions: up, down, north, south, east, west

    Rotation of the coordinate system

    Outgoing wave

    ( )( ) ( )

    ,j, j e! !

    l k

    nmL Lmmk n k

    m L n mk

    w MW k ldM dn m n m

    = =

    = +

  • University of Stuttgart Institute for Theory of Electrical Engineering

    Fast series expansion transformations

    Multipole-to-local transformation

    Plane waves: O(L2)

    Incoming wave

    ( ) ( ) ( ) ( )( )0 , 0 ,0 j cos sin, , e e k l k l kk x yzV k l W k l +=Local expansion

    n

    ( ) ( )( )

    ( ),j

    1 1

    j , e! !

    kl k

    m s Mmm k

    nk l

    L V k ldn m n m

    = =

    = +

  • University of Stuttgart Institute for Theory of Electrical Engineering

    Fast series expansion transformations

    In practiceL = 9

    Multipole-to-multipole transformation in z-direction

    Local-to-local transformation in z-direction

    Multipole-to-local transformation in z-direction

  • University of Stuttgart Institute for Theory of Electrical Engineering

    Postprocessing

    ClassicalPotential

    ( ) ( )0

    '1 d '4 'A

    u A

    =r

    rr r

    Field strength

    ( ) ( ) 30

    1 '' d '4 'A

    A =r rE r rr r

  • University of Stuttgart Institute for Theory of Electrical Engineering

    Postprocessing

    FMMOctree for elements and evaluation points

    Same FMM algorithm as for matrix-by-vector-product

    Local expansion

    ( ) ( )00

    1 ,4

    L nn m mn n

    n m nu r Y L

    = == r

    ( )( )00

    1 ,4

    L nn m mn n

    n m nr Y L

    = == E

  • University of Stuttgart Institute for Theory of Electrical Engineering

    Postprocessing

    Meshing strategiesElement size near evaluation points

    Revaluationpoints

  • University of Stuttgart Institute for Theory of Electrical Engineering

    Numerical results

    Experiment in high voltage techniqueGeometrical configuration (adaptive mesh)

  • University of Stuttgart Institute for Theory of Electrical Engineering

    Numerical results

    Experiment in high voltage techniqueGeometrical configuration (fine mesh)

  • University of Stuttgart Institute for Theory of Electrical Engineering

    Numerical results

    Experiment in high voltage techniqueGeometrical configuration (particle on the right spacer)

  • University of Stuttgart Institute for Theory of Electrical Engineering

    Numerical results

    Experiment in high voltage techniquePotential between the electrodes

    -50 -10 30 70x (mm)

    0

    10

    20

    30

    40

    z (m

    m)

  • University of Stuttgart Institute for Theory of Electrical Engineering

    Numerical results

    Experiment in high voltage techniqueElectric field strength above the particle

    -5 -4 -3 -2 -1 0 1y (mm)

    -10000

    -5000

    0

    5000

    Elec

    tric

    field

    stre

    ngth

    (V/m

    )

    ExEyEz

  • University of Stuttgart Institute for Theory of Electrical Engineering

    Numerical results

    Experiment in high voltage techniqueComputer resources

    Coarse mesh Fine meshUnknowns 28857 93409 Memory 932 MByte 1.2 GByte

    CPU-time 41662 s 86385 s Postprocessing 4324 s 1062 s

    Compression rate 85 % 98 %

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