Fast 3-D Interconnect Capacitance Extraction and Related Numerical Techniques

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 gt 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
From electro-magnetic analysis to circuit
simulation
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VLSI capacitance extraction

• 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)
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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 BEMs need Krylov subspace iterative
  solverand fast algorithms (multipole
  acceleration, hierarchical, precorrected FFT,
  SVD-based, quasi-multiple medium, )

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Direct BEM for Cap. Extraction

• Physical equations
• Laplace equation within each subregion
• Finite domain model
• Bias voltages set on conductors

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Direct BEM for Cap. Extraction

• Direct boundary element method
• Greens Identity
• Freespace Greens function as weighting function
• The Laplace equation is transformed into the BIE

s is a collocation point
is freespace Greens function, or the
fundamental solution of Laplace equation
More details C. A. Brebbia, The Boundary
Element Method for Engineers, London Pentech
Press, 1978
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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/r3) in discretized
  BIE

• Singular integration
• Non-singular integration
• Dynamic Gauss point selection
• Semi-analytical approach improvescomputational
  speed and accuracy for near singular integration

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Direct BEM for Cap. Extraction

• 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!
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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

• 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.

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Fast algorithms - QMM

• QMM-based capacitance extraction
• Make QMM cutting
• Then, the new structure with manysubregions is
  solved with the BEM

Confirmed in our later experiments
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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

Cutting pair (3, 2)

• Flowchart

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Fast algorithms - QMM

• 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 !

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Fast algorithms - Equ. organ.

• 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
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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 !
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Fast algorithms - Preconditioning

• A brief overview
• Jacobi method (the diagonal preconditioner
  diag(A)-1 )
• Mesh neighbor method (cant 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

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Fast algorithms - Preconditioning

• Principle of the MN method
• The neighbor variables of variable i
• Solve the reduced equation ,
  fill back to ith row of P

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Fast algorithms - Preconditioning

• 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 n1, 2, performs well

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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

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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 (k2
  to 5)
• Finite domain
• C1 is calculated for comparison

4
3
1
2
The 2x2 case
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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 C1 of Raphael with 1M grids
• Error formula

Compar. I
FastCap (1) FastCap (1) FastCap (1) FastCap (1) QMM-BEM QMM-BEM QMM-BEM QMM-BEM QMM-BEM
time mem panel err() time mem panel err() Sp.
2?2 7.9 17.9 1080 1.6 1.0 1.7 1184 2.7 8
3?3 9.2 17.9 1284 2.1 1.3 2.7 1431 2.5 9
4?4 10.0 19.1 1487 3.4 1.6 2.1 1502 1.0 6
5?5 12.5 23.7 1804 2.9 1.5 2.1 1558 1.2 8
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Numerical results (3)
FastCap (2) FastCap (2) FastCap (2) FastCap (2) QMM-BEM QMM-BEM QMM-BEM QMM-BEM QMM-BEM
time mem panel err() time mem panel err() Sp.
2?2 11.5 26.4 1080 2.1 1.0 1.7 1184 2.7 12
3?3 15.1 28.4 1284 2.3 1.3 2.7 1431 2.5 12
4?4 17.5 30.7 1487 2.6 1.6 2.1 1502 1.0 11
5?5 24.3 38.5 1804 3.0 1.5 2.1 1558 1.2 16
Compar. II
Raphael (0.25M) Raphael (0.25M) Raphael (0.25M) Raphael (0.25M) QMM-BEM QMM-BEM QMM-BEM QMM-BEM QMM-BEM
time mem panel err() time mem panel err() Sp.
2?2 78.8 47 - 0.3 1.0 1.7 1184 2.7 79
3?3 67.1 45 - 0.4 1.3 2.7 1431 2.5 52
4?4 88.9 48 - 0.5 1.6 2.1 1502 1.0 56
5?5 81.9 48 - 0.8 1.5 2.1 1558 1.2 55
Compar. III
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Numerical results (4)

• Our QMM-BEM solver
• Panel dont 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
panel Ele_N Var_N Z-val Iter. mem Tgen(s) Tsol(s) Time
QBEM 1502 1896 2435 0.24M 11 2.1 1.02 0.29 1.6
FastCap(1) 1487 1487 1487 - 13 19.1 6.9 2.9 10.0
FastCap(2) 1487 1487 1487 - 9 30.7 13.4 4.0 17.5
Tgen time of generating the linear system Tsol
time of solving the linear system
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Discussion
Contrast
FastCap QBEM
Formulation Single-layer potential formula Direct boundary integral equation
System matrix Dense Dense for single-region, otherwise sparse
Matrix degree N, the number of panels A little larger than N
Acceleration Multipole method less than N2 operations in each matrix-vector product QMM method -- maximize the matrix sparsity much less than N2 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
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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

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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

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Thank you !

• For more information yu-wj_at_tsinghua.edu.cn