fukuoka06 by xiaopangnv

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```									Fast BEM Algorithms for 3D
Interconnect Capacitance and
Resistance Extraction
Wenjian Yu
EDA Lab, Dept. Computer Science
& Technology, Tsinghua University
yu-wj@tsinghua.edu.cn
Direct BEM to solve Laplace Equ.

 Physical     equations                                                A cross-section view        q
   Laplace equation within each subregion u                  
   Same boundary assumption as Raphael RC3                                                2
   Bias voltages set on conductors                                                        1
 2     2u  2u  2u
 u  2  2  2  0 , In i (i  1, , M )              conductor
       x   y      z
u  u0 ,               On u      (u is potential)
q  u n  q0  0,    On q      (q is normal electric field intensity)


 Direct boundary element method
v    u
   Green’s Identity:               (u 2v  v 2u )d   (u
                     n
 v )d
n
   Freespace Green’s function as weighting function
   Laplace equation is transformed into BIE:
cs u s      qs u d      u s q d
*               *
s is a collocation point
2
 i            i
Discretization and integral calculation
A portion of dielectric interface:
   Discretize domain boundary
constant interpolation
•   Non-uniform element partition
•   Integrals (of kernel 1/r and 1/r3) in discretized BIE:
N                   N
cs us   (  q d)u j   (  u d)q j
s
*                   *
s                   s
j                  j
j 1                j 1
P4(x4,y2,z2)       P3(x3,y2,z2)
Y
•   Singular integration                                                             t
j
•   Non-singular integration                                       P1(x1,y1,z1)           P2(x2,y1,z1)

•    Dynamic Gauss point selection
Z       O                             X
•    Semi-analytical approach improves
computational speed and accuracy for near singular integration
3
Locality property of direct BEM
   Write the discretized BIEs as:
H i  ui  G i  q i, (i=1, …, M)
Compatibility equations
along the interface
 a  u a na   b  ub nb
u  u
 a       b
• Non-symmetric large-scale matrix A
Ax  f               • Use GMRES to solve the equation
• Charge on conductor is the sum of q
Medium 1               Med1    Med2 Interface

Interface                      [0]
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 mxn Environment
fictitious subregions, to enlarge the Conductors
matrix sparsity in BEM computation             z

   With iterative equation solver,                                                     x
sparsity brings actual benefit         y
Master Conductor
Master Conductor

A 3-D multi-dielectric case within finite domain,
applied 32 QMM cutting

Strategy of QMM-cutting:
 Uniform spacing
Non-uniform element partition
 Empirical formula to determine (m, n)
on a medium interface
 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
v11 u12 q21 v22 u23 q32 v33
s11
s12                                 Three                   12 subregions
s21                                 stratified              after applying
s22                                 medium                  22 QMM
s23
s32
s33

This ensures a near linear relationship between computing time and non-zero entries
6
Efficient GMRES preconditioning

   Construct MN preconditioner [Vavasis, SIAM J. Matrix,1992]
 PA  I  A P   I  AT pi  ei , i  1, ..., N
T T

   Neighbor set of variable i: L  {l1 , l2 , ... , ln }  {1, 2, ... , N}
   Solve reduced eq. AT pi  ei , fill back to ith row of P
l1 l2 l 3
Var. i                                 Solve, and fill P
l1 l2 l 3

Reduced equation
T
0

A                      A       pi =   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

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