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
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
rθ ϕ
πε += =−
= ∑ ∑r
Local expansion
( ) ( )00
1 ,4
L nn m mn n
n m nu r Y Lθ ϕ
πε = =−
= ∑ ∑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
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
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 '4πε 'A
u Aσ
=−∫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 '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
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 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
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
( )( ) 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
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 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
++
+ +=
− +=
− + − +∑
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
α λ=− =
= − +
∑ ∑
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 ,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 −
= =
= − − +
∑ ∑
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 '4πε 'A
u Aσ
=−∫r
rr r
Field strength
( ) ( ) 30
1 '' d '4πε 'A
Aσ −=
−∫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 Lθ ϕ
πε = =−
= ∑ ∑r
( )( )00
1 ,4
L nn m mn n
n m nr Y Lθ ϕ
πε = =−
= − ∇∑ ∑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 %
University of Stuttgart Institute for Theory of Electrical Engineering
Numerical results
Chip on a printed circuit boardGeometrical configuration
University of Stuttgart Institute for Theory of Electrical Engineering
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 %
University of Stuttgart Institute for Theory of Electrical Engineering
Numerical results
ContactorGeometrical configuration
University of Stuttgart Institute for Theory of Electrical Engineering
Numerical results
Contactor
Number of unknowns: 43949
CPU time: 1 day
Non-linear iterations steps: 9
Memory requirements: 990 MByte
Compression rate: 93 %
University of Stuttgart Institute for Theory of Electrical Engineering
Numerical results
Steady current flow field problemGeometrical configuration
University of Stuttgart Institute for Theory of Electrical Engineering
Numerical results
Steady current flow field problemPotential inside the conductor
University of Stuttgart Institute for Theory of Electrical Engineering
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
University of Stuttgart Institute for Theory of Electrical Engineering
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
Top Related