Fast Low-Frequency
Impedance Extraction using a
Volumetric 3D Integral Formulation
A.MAFFUCCI, A. TAMBURRINO, S. VENTRE, F. VILLONE
EURATOM/ENEA/CREATE Ass., Università di Cassino, Italy
G. RUBINACCI
EURATOM/ENEA/CREATE Association,
Università di Napoli “Federico II”, Italy
Structure of the Talk
•Introduction
Aim 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
E   jA  
A  Ac J   A s
Ac J (r) ˆ 0  J(r' ) dr'

S1
4  r  r'
S2
 J  nˆ dS  0


J  nˆ  0 on   Sk
  Vk on Sk
Er, t    r Jr, t 
Integral formulation
•Set of admissible current densities :

S  J  L2div (),   J  0 in , J  nˆ  0 on    S k
k

•Integral formulation in terms of the electric vector potential T:
J = T
“two components” gauge condition
•Edge element basis functions:
N
J ( x)   I k   N k ( x)
k 1
“tree-cotree” decomposition
Integral formulation
•Impose Ohm’s law in weak form :

0 J (r ' ) 
 W(r)  J (r)  j 4  r  r' dv'dv
Ne
  j  W (r )  A s dv  Vk  W (r )  nˆ k ds

k 1
Sk
W  S , J  S
Integral formulation
R  jωLI  FV  V s
dense matrix
sparse matrix
0   N i (r )    N j (r ' )
Lij 
dvdv'


4  
r  r'
Rij     N i (r )   N j (r )dv

Fik     N i (r )  nˆ k dS
Sk
Vs i   j    N i (r )  A s (r )dv

Structure of the talk
•Introduction
•The numerical model
•Solving Large Scale Problems
The Fast Multipoles Method
The block SVD Method
Solving large scale problems
Z  R  jωL
R
L
is a real symmetric and sparse NN matrix
is a symmetric and full NN matrix
The solution of ZI  FV  V s
requires O(N3) operations

The product
by a direct method
iterative methods
ZI needs N2 multiplications
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
( x j )  
i 1,n
i j
p
n
qi
M nm m
   n1 Yn ( j ,  j )
x j  xi n0 m n rj
Fast Multipole Method (FMM)
p
( x j ) 
n
M nm
n 1
r
n 0 m   n j


qi  a 
m
 
Yn ( j ,  j ) 
r j  a  r j 
p 1
“far” sources
rj
Field points
a
Fast Multipole Method (FMM)
p
( x j ) 
n
M nm
n 1
r
n 0 m   n j


qi  a 
m
 
Yn ( j ,  j ) 
r j  a  r j 
rj
a
p 1
Coarser level
already computed
Fast Multipole Method (FMM)
p
( x j ) 
n
M nm
n 1
r
n 0 m   n j


qi  a 
m
 
Yn ( j ,  j ) 
r j  a  r j 
rj
a
p 1
N 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
 ( x j )    Lmn rj Ynm ( j , j )
n 0 m   n
n
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
•
LI i   Ar     N i dv
Compute cartesian components separately:
 three scalar computations
A  A near  A far
0 p n m n m
1 ( p 1) / 2


A (r ) 
L
r
Y
(

,

)

O
l
3


n
n
4 n0 m n
far
p
n

near
LI i  L I i  0   LmnM i,mn ( , )  Ol 1 3( p1) / 2 
4 n0 m n
Block SVD Method
0   Ni (r )    N j (r ' )
L 
dvdv',


4 Y X
r  r'
XY
ij
N i  0 in Y
N j  0 in X
Y=field domain
r-r’
X=source domain
Block SVD Method
XY
L is a low rank matrix
rank r decreases as the separation between X and Y is increased
L
XY
Q R
X
Y
dim( L )  m  n
XY
dim( Q )  m  r
X
dim( R )  r  n
Y
L I  m  n operations
XY Y
Q R I  r  (n  m) operations
X
Y Y
r  m, n
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 + jLnear, 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 cases
A microstrip line
A microstrip line
Critical point: the rather different dimensions of
the finite elements in the three dimensions,
since the error scales as a/R
a
R
A microstrip line
s=50 elements per box
A microstrip line
s=400 elements per box
N=11068, S=50, e1.e4
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.