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Fast 3-D Interconnect Capacitance Extraction and Related Numerical Techniques

Fast 3-D Interconnect Capacitance Extraction and Related Numerical Techniques. Wenjian Yu EDA Lab, Dept. Computer Science & Technology, Tsinghua University Nov. 22, 2004. Outline. Background 3-D capacitance extraction with direct BEM

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Fast 3-D Interconnect Capacitance Extraction and Related Numerical Techniques

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

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

  5. VLSI capacitance extraction 1 2 4 3 1V 0V • Capacitance extraction • For m conductors solve mpotential 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 C1i= -Qi(i1)

  6. VLSI capacitance extraction • 3-D methods for capacitance extraction • Finite difference / Finite element • Sparse matrix, but withlarge 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 solverand fast algorithms (multipole acceleration, hierarchical, precorrected FFT, SVD-based, quasi-multiple medium, …)

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

  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 is freespace Green’s function, or the fundamental solution of Laplace equation More details: C. A. Brebbia, The Boundary Element Method for Engineers, London: Pentech Press, 1978

  9. Direct BEM for Cap. Extraction s t j • Discretize domain boundary • Partition quadrilateral elements with constant interpolation • Non-uniform element partition • Integrals (of kernel 1/r and 1/r3) in discretized BIE: • Singular integration • Non-singular integration • Dynamic Gauss point selection • Semi-analytical approach improvescomputational speed and accuracy for near singular integration

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

  11. Fast algorithms - QMM Population of matrix A 3-dielectric structure QMM ! v11 u12 q21 v22 u23 q32 v33 s11 s12 s21 s22 s23 s32 s33 • Quasi-multiple medium method • In each BIE, all variables are within same dielectric region; this leads to sparsity when combining equations for multiple regions • 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.

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

  13. Fast algorithms - QMM with minimal Z-val • 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 Cutting pair: (3, 2) • Flowchart

  14. Fast algorithms - QMM ( Type 1) • The discretized BIE: ai So, bi ( Type 2) • 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 • 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 Need not construct the actual geometry & boundary mesh !

  15. Fast algorithms - Equ. organ. Three stratified medium v11 u12 q21 v22 u23 q32 v33 s11 s12 s21 s22 s23 s32 s33 • 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 12 subregions after applying 22 QMM

  16. Fast algorithms - Preconditioning • 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) • 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 !

  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. Fast algorithms - Preconditioning l1 l2 l 3 Solve, and fillP Var. i l1 l2 l 3 Reduced equation T A P 0 = 1 i 0 • Principle of the MN method • The neighbor variables of variable i: • Solve the reduced equation , fill back to ith row of P

  19. Fast algorithms - Preconditioning 30% or more time reduction, compared with using the Jacobi preconditioner, for more than 100 structures v11 u12 q21 v22 u23 q32 v33 s11 s12 s21 s22 s23 s32 s33 • Extended Jacobi preconditioner • Singular integral is importance • Singular integrals from interface elementsare 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

  20. Fast algorithms -nearly linear m: 2~9, n: 2~6 m: 2~7, n: 2~10 • 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:

  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 • C1 is calculated for comparison 4 3 1 2 The 2x2 case

  22. Numerical results (2) • Computational configuration • FastCap: zeropermittivity is set to the outer-space to represent the Neumman boundary of the finite domain • Criterion: Result C1 of Raphael with 1M grids • Error formula: Compar. I

  23. Numerical results (3) Compar. II Compar. III

  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 Tgen: time of generating the linear system Tsol: time of solving the linear system

  25. Discussion • boundary discretization • stop criterion of 10-2 in GMRES solution • similar preconditioning • almost the same iteration number • Resemblance: Contrast

  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. 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

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