Transcript ICSOC Workshop
Fast BEM Algorithms for 3D Interconnect Capacitance and Resistance Extraction Wenjian Yu
EDA Lab, Dept. Computer Science & Technology, Tsinghua University
Direct BEM to solve Laplace Equ.
Physical equations A cross-section view Laplace equation within each subregion
u
Same boundary assumption as Raphael RC3
2
Bias voltages set on conductors
u q
2
u
u
0
u
2
x
,
n
u
2
q
2
y
0
u
2 0 , Green’s Identity:
z
2
u
2 0 , In On On
u q
(
u
2
v
i
(
i
Direct boundary element method
v
2
u
) 1, ,
(u is potential) (q is normal electric field intensity)
d
M
(
u
)
v
n
v
Freespace Green’s function as weighting function
u
n
)
d
conductor
1
q
Laplace equation is transformed into BIE:
c s u s
i q s
*
u d
i u s
*
q d
s is a collocation point
2
Discretization and integral calculation
A portion of dielectric interface: Discretize domain boundary • Partition quadrilateral elements with constant interpolation • • Non-uniform element partition Integrals (of kernel 1/r and 1/r 3 ) in discretized BIE: • •
c s u s
j N
1
j q s
*
d
)
u j
j N
1
j u s
*
d
)
q j
s P 4 (x 4 ,y 2 ,z 2 ) P 3 (x 3 ,y 2 ,z 2 ) Y
Singular integration Non-singular integration • •
j
P 1 (x 1 ,y 1 ,z 1 ) t
Dynamic Gauss point selection
Z O
Semi-analytical approach improves computational speed and accuracy for near singular integration
P 2 (x 2 ,y 1 ,z 1 ) X
3
Locality property of direct BEM
Write the discretized BIEs as:
H
i
u
i
G
i
q
i
, (
i
=1, …,
M
)
Ax
Compatibility equations along the interface
u a a
u
•
u b a
n
a
b
u b
n
b
Non-symmetric large-scale matrix A
f
•
Use GMRES to solve the equation
•
Charge on conductor is the sum of q Med1 Medium 1 Med2 Interface [0] Interface
A
=
[0] Medium 2 [0] Conductor [0] For problem involving multiple regions, matrix A exhibits sparsity!
4
Quasi-multiple medium method
Quasi-multiple medium (QMM) method Cutting the original dielectric into
m
x
n
fictitious subregions, to enlarge the
Environment Conductors
matrix sparsity in BEM computation
z
With iterative equation solver, sparsity brings actual benefit
y
Master Conductor Master Conductor
x
A 3-D multi-dielectric case within finite domain, applied 3 2 QMM cutting
Non-uniform element partition on a medium interface Strategy of QMM-cutting:
Uniform spacing
Empirical formula to determine (m, n)
Optimal selection of (m, n)
5
Efficient equation organization
Too many subregions produce complexity of equation organizing and storing Bad scheme makes non-zero entries dispersed, and worsens the efficiency of matrix-vector multiplication in iterative solution We order unknowns and collocation points correspondingly; suitable for multi-region problems with arbitrary topology Example of matrix population
s 11 v 11 s 12 s 21 s 22 s 23 s 32 s 33 u 12 q 21 v 22 u 23 q 32 v 33
Three stratified medium 12 subregions after applying 2
2 QMM
Efficient GMRES preconditioning
Construct MN preconditioner [
Vavasis, SIAM J. Matrix,1992
]
PA
T
A P
T
Neighbor set of variable i:
T
A p
l
1
l
2
l
3
A p
i
e
i L i
e
i
,
l l
1 2
i
Var.
i
1, ...,
l n
Solve, and fill
P
N l
1
l
2
l
3
A
Reduced equation
A
T
p
i
= 0 1 0
P i
Our work: for multi-region BEA, propose an approach to get the neighbors, making solution faster for 30% than original Jacobi preconditioner 7
A practical field solver - QBEM
Handling of complex structures Bevel conductor line; conformal dielectric Structure with floating dummy fill Multi-plane dielectric in copper technology Metal with trapezoidal cross section 3-D resistance extraction Complex 3-D structure with multiple vias Improved BEM coupled with analytical formula Extract DC resistance network Hundreds/thousands times fast than Raphael, while maximum error <3% 8