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Fast Low-FrequencyImpedance Extraction using a
Volumetric 3D Integral Formulation
A.MAFFUCCI, A. TAMBURRINO, S. VENTRE, F. VILLONEEURATOM/ENEA/CREATE Ass., Università di Cassino, Italy
G. RUBINACCIEURATOM/ENEA/CREATE Association,
Università di Napoli “Federico II”, Italy
Structure of the Talk
•IntroductionAim of the work
“Fast” methods
Aim of the work
Big interconnect delay and coupling increases the importance of interconnect parasitic parameter extraction.
In particular, on-chip inductance effect becomes more and more critical, for the huge element number and high clock speed
Precise simulation of the current distribution is a key issue in the extraction of equivalent frequency dependent R an L for a large scale integration circuit.
Difficulties arise because of the skin-effect and the related proximity effect
Aim of the work
Eddy current volume integral formulations:
Advantages:
– Only the conducting domain meshed
no problems with open boundaries
–“Easy” to treat electrodes and to include electric non linearity.
Disadvantages:
– Dense matrices, with a singular kernel heavy computation
Critical point:
Generation, storage and inversion of large dense matrices
Aim of the work
• Direct methods: O(N3) operations (inversion)• Iterative methods: O(N2) operations per solution
Fast methods: O(N log(N) ) or O(N) scaling
required to solve large-scale problems
“Fast” methods
Two families of approaches:
For regular meshes
FFT based methods (exploiting the translation invariance of the integral
operator, leading to a convolution product on a regular grid) •
For arbitrary shapes
Fast Multipoles Method (FMM)
Block SVD method
Wavelets
…
Basic idea: Separation of long and short range interactions
(Compute large distance field by neglecting source details)
Structure of the talk
•Introduction
•The numerical model
Problem definition
Integral formulation
Problem Definition
S1
S2
AE j
sc AJAA
0dˆ SnJ
'd'
)'(
4ˆ)( 0 r
rr
rJrJA
c
kS on 0n̂J
tt ,, rJrrE
kk SonV
•Set of admissible current densities :
•Integral formulation in terms of the electric vector potential T:
J = T “two components” gauge condition
•Edge element basis functions:
“tree-cotree” decomposition
Integral formulation
N
kkkI
1
)()( xNxJ
kk
div SS on0ˆ,in0),(2 nJJLJ
•Impose Ohm’s law in weak form :
Integral formulation
e
k
N
k S
kks dsVdvj
dvdvj
1
0
ˆ)()(
'')'(
4)()(
nrWArW
rrrJ
rJrW
SS JW ,
Integral formulation
sjω VVFILR
'
'
)'()(
40 dvdvL ji
ij rr
rNrN
dvR jiij )()( rNrN
dvjV siis )()( rArN kS
kiik dSF nrN ˆ)(
dense matrix
sparse matrix
Structure of the talk
•Introduction
•The numerical model
•Solving Large Scale Problems
The Fast Multipoles Method
The block SVD Method
is a real symmetric and sparse NN matrix
is a symmetric and full NN matrix
The solution of by a direct method
requires O(N3) operations
iterative methods
The product needs N2 multiplications
Solving large scale problems
LjωRZ RL
sVVFIZ
IZ
Fast Multipole Method (FMM)
• Goal: computation of the potential due to N charges in the locations of the N charges themselves with O(N) complexity
• Idea: the potential due to a charge far from its source can be accurately approximated by only a few terms of its multipole expansion
p
n
n
nmjj
mnn
j
mn
jini ij
ij Y
r
M
xx
qx
01
,1
),()(
Fast Multipole Method (FMM)
a
rj
Field points
“far” sources
1
01
),()(
p
jj
ip
n
n
nmjj
mnn
j
mn
j r
a
ar
qY
r
Mx
Fast Multipole Method (FMM)
a
rjCoarser level
already computed
1
01
),()(
p
jj
ip
n
n
nmjj
mnn
j
mn
j r
a
ar
qY
r
Mx
Fast Multipole Method (FMM)
1
01
),()(
p
jj
ip
n
n
nmjj
mnn
j
mn
j r
a
ar
qY
r
Mx
a
rjN log(N) algorithm!
Fast Multipole Method (FMM)
• To get a O(N) algorithm: local expansion (potential due to all sources outside a given sphere) inside a target box, rather than evaluation of the far field expansion at target positions
p
n
n
nmjj
mn
nj
mnj YrLx
0
),()(
Fast Multipole Method (FMM)
1. Multipole Expansion (ME) for sources at the finest level
2. ME of coarser levels from ME of finer levels (translation and combination)
3. Local Expansion (LE) at a given level from ME at the same level
4. LE of finer levels from LE of coarser levels
Additional technicalities needed for adaptive algorithm (non-uniform meshes)
Fast Multipole Method (FMM)
• Key point: fast calculation of i-th component of the matrix-vector product
• Compute cartesian components separately:
three scalar computations
dvii NrAIL
farnear AAA
p
n
n
nm
pmn
nmn
far lOYr0
2/)1(10 3),(4
)(
LrA
p
n
n
nm
pmni
mni
neari lO
0
2/)1(1,
0 3),(4
MLILLI
Block SVD Method
r-r’X=source domain
Y=field domain
Xin
YindvdvL
j
i
Y X
jiXYij 0
0,'
'
)'()(
40
N
N
rr
rNrN
Block SVD Method
nr
rm
nm
Y
X
XY
YXXY
)Rdim(
)Qdim(
)Ldim(
RQL
nmr
operationsmnr
operationsnmYYX
YXY
,
)(IRQ
IL
is a low rank matrix
rank r decreases as the separation between X and Y is increased
XYL
Block SVD Method
•The computation of the LI product follows the same lines of the FMM adaptive approach
•Each QR decomposition is obtained by using the modified GRAM-SCHMIDT procedure
•An error threshold is used to stop the procedure for having the smallest rank r for a given approximation
The iterative solver
• The solution of the linear system has been obtained in both cases by using the preconditioned GMRES.
• Preconditioner: sparse matrix Rnear + jLnear,
or with the same sparsity as R, or diagonal
• Incomplete LU factorisation of the preconditioner: dual-dropping strategy (ILUT)
Structure of the talk
•Introduction
•The numerical model
•Solving Large Scale Problems
•Test casesA microstrip line
Critical point: the rather different dimensions of the finite elements in the three dimensions, since the error scales as a/R
A microstrip line
a
R
s=50 elements per box
A microstrip line
s=400 elements per box
A microstrip line
N=11068, S=50, e
The relative error in the LfarI product as a function of the compression rate
N=11068, S=50
Conclusions
• The magnetoquasistationary integral formulation here presented is a flexible tool for the extraction of resistance and inductance of arbitrary 3D conducting structures.
• The related geometrical constraints due to multiply connected domain and to field-circuit coupling are automatically treated.
• FMM and BLOCK SVD are useful methods to reduce the computational cost.
• BLOCK SVD shows superior performances in this case, due to high deviation from regular mesh.