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

University of Stuttgart Institute for Theory of Electrical Engineering

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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