fukuoka06

							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
    •   Partition quadrilateral elements with
        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