Singularitytreated quadratureevaluated method of moments solver for 3D capacitance extraction

1
Singularity-treatedquadrature-evaluated method
ofmoments solver for3-D capacitance extraction

• Jinsong Zhao
• June 08, 2000

Cadence Design Systems, Inc.
2
Outlines

• Introduction
• QMMS
• Implications
• Numerical examples
• Conclusions and future work

3
Introduction

• Extraction
• critical piece in physical design and
  verification
• on-the-fly extraction
• database building
• Requirement
• accuracy
• speed
• capacity

4
Research progress

• Classic partial differential equations
• Finite-element
• Finite-difference (absorbing boundary condition,
  MEI)
• Solve in the 3-D space, hit the curse of
  dimension
• Integral equation method
• Integral equation boundary condition
• Smaller (compared to domain method) but full
  matrix
• Fast integral equation solver

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Fast integral equation solvers

• Multipole-accelerated FastCap (K. Nabors, 1991)
• Pre-corrected FFT technique (J. Phillips, 1995)
• SVD-compressing IES3 (S. Kapur, 1996)
• Monopole-based (W. Shi, 1998)
• Wavelet-type multiscale (J. Tausch, 1999)

means acceleration method is Kernel-independent
Now we are able to solve 100000 unknowns!! ...But
WHY??
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Reason for the difficulty

• Why the numerical extraction is so difficult?
• Hand-calculation examples
• plate capacitor
• a metal ball
• A cube becomes a hard problem
• Convergence is slow
• Because corners and edges create SINGULARITIES!

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Singularity structure

• Luckily, we know the singularity structure!

q
Which reminds us to use Gaussian-Jacobi
quadrature to evaluate the integral
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Gaussian Quadrature

• Gaussian quadrature is the best way to evaluate
  the integral of a smooth function.

W(x) can be considered as a Precondition
function that best handles singularities
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QMMS

• Quadrature evaluated method of moments solver
• Sample points, rather than coefficients, are
  unknowns. This is known as pseudo-spectral
  method, as powerful as high-order, yet much
  simpler to implement.
• Testing functions handle the singularity in the
  Greens function quadrature handles the
  singularity in the solution space.

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Comparison to Nystrom method

• Nystrom method (Kapur Long, 1998)
• For each r, special quadrature rules are created
• collocation on both r and r
• singularity solution space is not handled
• QMMS
• For each testing function, a general
  Gaussian-Jacobi rule is used. Testing function
  weakens the singularity in Greens function
• collocation on r only
• Implications
• Gaussian nodes are the best ways to allocate
  nodes and discretization.

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A square plate example

• Charge distribution

Smooth-part ofcharge distribution
True charge distribution
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Convergence comparison

• Gaussian node allocation high-order

0.1
uniform
Uniform(2)
Uniform(3)
0.01
0.001
Gaussian(1)
0.0001
Gaussian(2)
Gaussian(3)
350
400
50
100
150
200
250
300
To get 1 accuracy, we only need 16 points for a
plate and 24 points for a cube. It takes no time
to compute.
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Interdigitated capacitor
3000
1000
500
With 500 points, results match 21908 adaptive
panels!
Each panel is discretized based on Gaussian nodes
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Some numerical observations

• SVD-compression technique is used on top of QMMS
• So far, NO preconditioner is found to be better
  than without preconditioner
• high-order testing degrades convergence and a
  better preconditioner is needed for really large
  problems.

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Conclusion

• QMMS is presented
• Charge singularity structure at corners and edges
• Method of Moments testing handles Greens
  function singularity, Guassian-Jacobi quadrature
  handles solution singularity
• High-order testing and collocation in solution
• Numerical implication
• Within 1 accuracy, regular LU is good and fast
• More accuracy is achieved through SVD-accelerated
  method
• Future work
• Inductance
• full-wave