A Solenoidal Basis Method For Efficient Inductance Extraction Hemant Mahawar Vivek Sarin Weiping Shi

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A Solenoidal Basis Method For Efficient
Inductance ExtractionHemant MahawarVivek
SarinWeiping ShiTexas AM UniversityCollege
Station, TX
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Introduction

• Fast and accurate inductance extraction is
  critical for design and verification of high
  speed circuits
• Inductance extraction algorithms
• Accurate formulation - Loop inductance
  (FastHenry)
• Approximations - Partial inductance (PEEC)
• Solenoidal basis method
• Originally developed for fluid flows
• Applicable to loop inductance formulation
• Near-optimal preconditioning schemes
• Significantly faster

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Background

• Loop inductance formulation
• Inductance between current carrying filaments
• Kirchoffs law enforced at each node

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Background

• Current density at a point
• Linear system for current and potential
• Inductance matrix
• Kirchoffs Law

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Linear System of Equations

• Characteristics
• Extremely large R, B sparse L dense
• Matrix-vector products with L use hierarchical
  approximations
• Solution methodology
• Solved by preconditioned Krylov subspace methods
• Robust and effective preconditioners are critical
  
• Developing good preconditioners is a challenge
  because system is never computed explicitly!

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Current Components

• Fixed current satisfying external condition Id
  (left)
• Linear combination of cell currents (right)

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Solenoidal Basis Method

• Linear system with modified RHS
• Solenoidal basis
• Basis for current that satisfies Kirchoffs law
• Solenoidal basis matrix P
• Current obeying Kirchoffs law
• Reduced system
• Solve via preconditioned Krylov subspace method

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Local Solenoidal Basis

• Current in the kth cell consists of unit current
  assigned to the four filaments of the cell
• There are four nonzeros in the kth column ofP
  1, 1, 1, 1

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Preconditioning

• Observe where
• Approximate reduced system
• where is the average filament
  resistance in each cell
• Approximate by

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Preconditioning

• Preconditioning step involves multiplicationwith

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Hierarchical Approximations

• System matrix and preconditioner are large,dense
  matrices
• Hierarchical approximations used to compute
  matrix-vector products with L and
• Used for fast decaying Greens functions, such as
  1/r (r distance from origin)
• Reduced accuracy at lower cost
• Examples
• Fast Multipole Method O(n)
• Barnes-Hut O(nlogn)

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FASTHENRY

• Uses mesh currents to generate a reduced system
• Approximation to reduced system computed by
  sparsification of inductance matrix
• Preconditioner derived from
• Sparsification strategies
• DIAG self inductance of filaments only
• CUBE filaments in the same oct-tree cube ofFMM
  hierarchy
• SHELL filaments within specified radius
  (expensive)

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Experiments

• Benchmark problems
• Ground plane
• Wire over plane
• Spiral inductor
• Operating frequencies 1GHz-1THz
• Strategy
• Uniform two-dimensional mesh
• Solenoidal function method
• Preconditioned GMRES for reduced system
• Comparison
• FASTHENRY with CUBE DIAG preconditioners

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Ground Plane

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Problem Sizes

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Comparison with FastHenry

• Preconditioned GMRES Iterations (10GHz)

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Comparison

• Time and Memory (10GHz)

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Preconditioner Effectiveness

• Preconditioned GMRES iterations

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Wire Over Ground Plane
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Comparison with FastHenry

• Preconditioned GMRES Iterations (10GHz)

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Comparison

• Time and Memory (10GHz)

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Preconditioner Effectiveness

• Preconditioned GMRES iterations

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Spiral Inductor
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Preconditioner Effectiveness

• Preconditioned GMRES iterations

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Concluding Remarks

• Preconditioned solenoidal method is very
  effective for linear systems in inductance
  extraction
• Near-optimal preconditioning assures fast
  convergence rates that are nearly independent of
  frequency and mesh width
• Significant improvement over FastHenry w.r.t.
  time and memory
• Acknowledgements NSF