1. Fast 3-D Interconnect Capacitance Extraction and Related Numerical Techniques Wenjian Yu EDA Lab, Dept. Computer Science & Technology, Tsinghua University Nov. 22, 2004

2. 2 Outline Background 3-D capacitance extraction with direct BEM Fast capacitance extraction with QMM acceleration and other numerical techniques Numerical results Conclusion

3. 3 Background Parasitic extraction in SOC Interconnect dominates circuit performance Interconnect delay > device delay Crosstalk, signal integrity, power, reliability Other parasitics Substrate coupling in mixed-signal circuit Thermal parasitics for on-chip thermal analysis Interconnect parasitic extraction Resistance, Capacitance and Inductance Becomes a necessary step for performance verification in the iterative design flow

4. 4 From electro-magnetic analysis to circuit simulation Parasitic extraction / Electromagnetic analysis Thousands of R, L, C Filament with uniform current Panel with uniform charge Model order reduction Reduced circuit

5. 5 VLSI capacitance extraction  Capacitance extraction For m conductors solve m potential problems for the conductor surface charges Electric potential u fulfill: Capacitance is function of wire shape, environment, distance to substrate, distance to surrounding wires Challenges: high accuracy (3-D method), high speed, suitable for complex process 1V 0V 1 2 3 4 C 1i = -Q i (i  1)

6. 6 VLSI capacitance extraction  3-D methods for capacitance extraction Finite difference / Finite element Sparse matrix, but with large number of unknowns Boundary integral formulation (BEM) Fewer unknowns, more accurate, handle complex geometry Two kinds: indirect BEM makes dense matrix direct BEM has localization property Both BEM’s need Krylov subspace iterative solver and fast algorithms (multipole acceleration, hierarchical, precorrected FFT, SVD-based, quasi-multiple medium, …)

7. 7 Direct BEM for Cap. Extraction Physical equations Laplace equation within each subregion Finite domain model Bias voltages set on conductors conductor u is electrical potential q is normal electrical field intensity on boundary

8. 8 Direct BEM for Cap. Extraction Direct boundary element method Green’s Identity Freespace Green’s function as weighting function The Laplace equation is transformed into the BIE: s is a collocation point More details: C. A. Brebbia, The Boundary Element Method for Engineers, London: Pentech Press, 1978 is freespace Green’s function, or the fundamental solution of Laplace equation

9. 9 Direct BEM for Cap. Extraction  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: Singular integration Non-singular integration Dynamic Gauss point selection Semi-analytical approach improves computational speed and accuracy for near singular integrationst jjjj

10. 10 Direct BEM for Cap. Extraction Write the discretized BIEs as:, (i=1, …, M) Non-symmetric large-scale matrix A Use GMRES to solve the equation Charge on conductor is the sum of q Compatibility equations along the interface For problem involving multiple regions, matrix A exhibits sparsity!

11. 11 Fast algorithms - QMM Quasi-multiple medium method In each BIE, all variables are within same dielectric region; this leads to sparsity when combining equations for multiple regions 3-dielectric structure v 11 v 22 v 33 u 12 q 21 u 23 q 32 s 11 s 12 s 21 s 22 s 23 s 32 s 33 Population of matrix A Make fictitious cutting on the normal structure, to enlarge the matrix sparsity in the direct BEM simulation. With iterative equation solver, sparsity brings actual benefit. QMM !

12. 12 Environment Conductors Master Conductor x y z A 3-D multi-dielectric case within finite domain, applied 3  2 QMM cutting Fast algorithms - QMM QMM-based capacitance extraction Make QMM cutting Then, the new structure with many subregions is solved with the BEM Time analysis while the iteration number dose not change a lot Z: number of non-zeros in the final coefficient matrix A Confirmed in our later experiments

13. 13 Fast algorithms - QMM Select optimal cutting pair Empirical formula, or manually specifying Automatic selection, make total computation achieve highest speed; make use of the linear relationship between computational time and the parameter Z Flowchart Cutting pair: (3, 2) with minimal Z-val

14. 14 Fast algorithms - QMM Calculate the Z-value Two types of boundary element Nuemann: one u variable / element Dirichlet: one q variable / element Interface: both u and q variable / element So, The discretized BIE: aiai ( Type 1) ( Type 2) bibi Heuristic rules for set S -- candidates of (m, n) Relatively small size for the sake of saving time Moderate value range of m (along X-axis) and n (along Y-axis) Range is relevant to the dimensions along X/Y-axis Need not construct the actual geometry & boundary mesh !

15. 15  Example of matrix population 12 subregions after applying 2  2 QMM  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 Fast algorithms - Equ. organ. Three stratified medium v 11 v 22 v 33 u 12 q 21 u 23 q 32 s 11 s 12 s 21 s 22 s 23 s 32 s 33

16. 16 Fast algorithms - Preconditioning Basics of the preconditioning technique Aim: improve the condition of the coefficient matrix, so as to obtain faster convergence rate The right-hand preconditioning: Suitable for GMRES a sparer one should be good ! Construct the GMRES preconditioner (matrix P ) should has better spectrum of eigenvalues than should be a brief approximation to To balance the speedup of convergence and the additional consump- tion of the preconditioner (to construct it, multiple it in each iteration)

17. 17 Fast algorithms - Preconditioning A brief overview Jacobi method (the diagonal preconditioner: diag(A) -1 ) Mesh neighbor method: (can’t applied directly) S.A. Vavasis, SIAM J. Matrix Anal. Appl. 1992 K. Chen, SIAM J. Sci. Comput. 1998 K. Chen, SIAM J. Matrix Anal. Appl. 2001 Nearest neighbor method (in FastCap2.0) Coupled with the multipole algorithm Emphasis of our work Suitable for direct boundary element method Simpler and more efficient, since the Jacobi preconditioner has reduced the iterative number down to several tens

18. 18 T = 0 1 0 Reduced equation Fast algorithms - Preconditioning Principle of the MN method The neighbor variables of variable i : Solve the reduced equation, fill back to i th row of P A Var. i l 1 l 2 l 3 P i Solve, and fill P

19. 19 Fast algorithms - Preconditioning Extended Jacobi preconditioner Singular integral is importance Singular integrals from interface elements are not all at the main diagonal Except for row corresponding to interface element, solve a 2  2 reduced equation to involve all singular integrals MN (n) preconditioner n is the number of neighbor elements Scan the ith row, use the absolute value as measure of neighborhood When n=1, 2, performs well v 11 v 22 v 33 u 12 q 21 u 23 q 32 s 11 s 12 s 21 s 22 s 23 s 32 s 33 30% or more time reduction, compared with using the Jacobi preconditioner, for more than 100 structures

20. 20 Fast algorithms - nearly linear Efficient organization and solution technique ensure near linear relationship between the total computing time and non-zero matrix entries (Z-values) For two cases from actual layout: m: 2~9, n: 2~6 m: 2~7, n: 2~10

21. 21 Numerical results (1) Experiment environment SUN UltraSparc II processors (248 MHz) Programs Our QMM-BEM solver: QBEM FastCap 2.0: FastCap(1), FastCap(2) Raphael RC3 (3-D finite difference solver) Test examples k  k crossovers in five layered dielectrics (k=2 to 5) Finite domain C 1 is calculated for comparison The 2x2 case 12 3 4

22. 22 Numerical results (2) Computational configuration FastCap: zero permittivity is set to the outer-space to represent the Neumman boundary of the finite domain Criterion: Result C 1 of Raphael with 1M grids Error formula: FastCap (1)QMM-BEM timemempanelerr(%)timemempanel*err(%)Sp. 2222 7.917.910801.61.01.711842.78 3333 9.217.912842.11.32.714312.59 4444 10.019.114873.41.62.115021.06 555512.523.718042.91.52.115581.28 Compar. I

23. 23 Numerical results (3) Raphael (0.25M)QMM-BEM timemempanelerr(%)timemempanel*err(%)Sp. 2222 78.847-0.31.01.711842.779 3333 67.145-0.41.32.714312.552 4444 88.948-0.51.62.115021.056 555581.948-0.81.52.115581.255 Compar. III FastCap (2)QMM-BEM timemempanelerr(%)timemempanel*err(%)Sp. 2222 11.526.410802.11.01.711842.712 3333 15.128.412842.31.32.714312.512 4444 17.530.714872.61.62.115021.011 555524.338.518043.01.52.115581.216 Compar. II

24. 24 Numerical results (4) Our QMM-BEM solver Panel* don’t count the panels on interfaces between fictitious media The optimal QMM cutting pairs are (4, 4), (5, 5), (3, 3), (3, 3) respectively ; the EJ preconditioner is uesed Comparison IV. Computational details for the 4  4 crossover problem panelEle_NVar_NZ-valIter.memT gen (s)T sol (s)Time QBEM1502189624350.24M112.11.020.291.6 FastCap(1)1487 -1319.16.92.910.0 FastCap(2)1487 -930.713.44.017.5 T gen : time of generating the linear system T sol : time of solving the linear system

25. 25 Discussion FastCapQBEM Formulation Single-layer potential formulaDirect boundary integral equation System matrix DenseDense for single-region, otherwise sparse Matrix degree N, the number of panelsA little larger than N Acceleration Multipole method: less than N 2 operations in each matrix- vector product QMM method -- maximize the matrix sparsity: much less than N 2 operations in each matrix-vector product Other cost Cube partition and multipole expansion are expensive Efficient organizing and storing of sparse matrix make matrix-vector product easy Resemblance: boundary discretization stop criterion of 10 -2 in GMRES solution similar preconditioning almost the same iteration number Contrast

26. 26 Conclusion Numerical techniques in the QMM-BEM solver Analytical / Semi-analytical integration Quasi-multiple medium acceleration (cutting pair selection) Equation organization of discretized direct BEM Preconditioning on the GMRES solver Achieve about 10x speed-up to FastCap Related work Use the blocked Gauss method for capacitance extraction with multiple master conductors Handle problem with floating dummies in area filling Apply the direct BEM to the substrate resistance extraction

27. 27 For more information Wenjian Yu, Zeyi Wang and Jiangchun Gu, “Fast capacitance extraction of actual 3-D VLSI interconnects using quasi-multiple medium accelerated BEM,” IEEE Trans. Microwave Theory Tech., Jan 2003, 51(1): 109-120 Wenjian Yu and Zeyi Wang, “Enhanced QMM-BEM solver for 3-D multiple- dielectric capacitance extraction within the finite domain,” IEEE Trans. Microwave Theory Tech., Feb 2004, 52(2): 560-566 Wenjian Yu, Zeyi Wang and Xianlong Hong, “Preconditioned multi-zone boundary element analysis for fast 3D electric simulation,” Engng. Anal. Bound. Elem., Sep 2004, 28(9): 1035-1044

28. Thank you ! For more information: yu-wj@tsinghua.edu.cn