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

University of Stuttgart Institute for Theory of Electrical Engineering

FMM based solution of electrostatic and magnetostatic field problems

André BuchauWolfgang Hafla

Friedemann GrohWolfgang M. Rucker


Outline

Introduction

Octree in practice

FMM for direct and indirect BEM formulations

Fast series expansion transformations

Postprocessing

Numerical results

Conclusion


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


Octree in practice

Adaptive meshes


Octree in practice

Series expansions

Multipole expansion

( ) ( )100

1 1 ,4

L nm mn nn

n m nu Y M

rθ ϕ

πε += =−

= ∑ ∑r

Local expansion

( ) ( )00

1 ,4

L nn m mn n

n m nu r Y Lθ ϕ

πε = =−

= ∑ ∑r


Octree in practice

Convergence of the multipole expansion

0 1 2 3 4 5r (m)

01020304050u

(V)

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


Octree in practice

Octree

xy

z

34

56

78

12


Octree in practice

Classical near- and far-field definition

consideredcube

near-field far-field


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


Octree in practice

Convergence radius of a cube

R

center

elements


Octree in practice

Convergence radius of a cube

R


Octree in practice

Near-field interactions

1R2R


Octree in practice

Far-field interactions

1R 2R



Direct BEM formulationElectrostatics

Steady current flow fields

Green’s theorem

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

uc u A u A

n n∂ ∂

= −∂ − ∂ −∫ ∫r

r r rr r r r

Dirichlet boundary conditions

Neumann boundary conditions



Indirect BEM formulationElectrostatics

Magnetostatics

Charge densities

( ) ( )0

'1 d '4πε 'A

u Aσ

=−∫r

rr r

Dirichlet boundary conditions

Neumann boundary conditions



Classical multipole expansionClassical integral

( ) ( )0

'1 d '4πε 'A

u Aσ

=−∫r

rr r

Multipole expansion

( ) ( )100

1 1 ,4

L nm mn nn

n m nu Y M

rθ ϕ

πε += =−

= ∑ ∑r

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

A

M r Y Aσ θ ϕ−= ∫ r



Fast multipole method for double-layer potentialsIntegral

( ) ( ) '0

1 1' 'd '4 'A

u Aτπε

= ∇ ⋅−∫ rr r nr r

Multipole expansion

( ) ( )100

1 1 ,4

L nm mn nn

n m nu Y M

rθ ϕ

πε += =−

= ∑ ∑r

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

A

M r Y Aτ θ ϕ−= ⋅∇∫ rr n r



Series expansion transformations

Multipole-to-multipole transformation

Multipole-to-local transformation ← large CPU-time

Local-to-local-transformation



Multipole-to-local transformation

Classical approach

( )( ) 1

0

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

Aµ ν

ρ

− − − −+

+ + −= =− +

=−

∑∑

( )4O L




Transformation in z-direction: O(L3)

Rotation about the z-axisj' em m m

n nM M β=

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 Mα α

−

=− =

= − +∑ ∑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 mρ

++

+ +=

− +=

− + − +∑




“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

α λ=− =

= − +

∑ ∑




“Plane waves”: O(L2)

Incoming wave

( ) ( ) ( ) ( )( )0 , 0 ,0j cos sin, , e e k l k l kk

x yzV k l W k l λ α αλ +−=

Local expansionn

( ) ( )( )

( ),j

1 1

j , e! !

kl k

m s Mmm k

nk l

L V k ldn m n m

εαλ −

= =

= − − +

∑ ∑



In practiceL = 9

Multipole-to-multipole transformation in z-direction

Local-to-local transformation in z-direction

Multipole-to-local transformation in z-direction


Postprocessing

ClassicalPotential

( ) ( )0

'1 d '4πε 'A

u Aσ

=−∫r

rr r

Field strength

( ) ( ) 30

1 '' d '4πε 'A

Aσ −=

−∫r rE r rr r


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 Lθ ϕ

πε = =−

= ∑ ∑r

( )( )00

1 ,4

L nn m mn n

n m nr Y Lθ ϕ

πε = =−

= − ∇∑ ∑E


Postprocessing

Meshing strategiesElement size near evaluation points

Revaluationpoints


Numerical results

Experiment in high voltage techniqueGeometrical configuration (adaptive mesh)


Numerical results

Experiment in high voltage techniqueGeometrical configuration (fine mesh)


Numerical results

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


Numerical results

Experiment in high voltage techniquePotential between the electrodes

-50 -10 30 70x (mm)

0

10

20

30

40

z (m

m)


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


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 %


Numerical results

Chip on a printed circuit boardGeometrical configuration


Numerical results

Chip on a printed circuit boardComputer resources

Coarse mesh Fine meshUnknowns 20964 56980 Memory 195 MByte 832 MByte

CPU-time 25657 s 344877 s Compression rate 94 % 99.6 %


Numerical results

ContactorGeometrical configuration


Numerical results

Contactor

Number of unknowns: 43949

CPU time: 1 day

Non-linear iterations steps: 9

Memory requirements: 990 MByte

Compression rate: 93 %


Numerical results

Steady current flow field problemGeometrical configuration


Numerical results

Steady current flow field problemPotential inside the conductor


Numerical results

Steady current flow field problem3720 second order boundary elements

11244 unknowns

160 linear iteration steps

Compression rate: 88.3 %

CPU-time: 1 hour and 8 minutes (Pentium III 1 GHz)

113 MByte (instead of 965 MByte)

Computation of the potential in 17220 evaluation points in 145 s


Conclusion

Static electric and magnetic field problems

Direct and indirect boundary element method

Volume integral equations for non-linear problems

Fast adaptive multilevel multipole method

Adaptive meshes

High compression rates and accuracy

Fast postprocessing

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

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