A Solenoidal Basis Method For Efficient Inductance Extraction

Hemant MahawarVivek SarinWeiping Shi

Texas A&M UniversityCollege Station, TX

A Solenoidal Basis Method For Efficient Inductance Extraction

Hemant MahawarVivek SarinWeiping Shi

Texas A&M UniversityCollege Station, TX

IntroductionIntroduction

BackgroundBackground

Inductance between current carrying filaments

Kirchoff’s law enforced at each node

Background …Background …

Current density at a point

Linear system for current and potential

Inductance matrix

Kirchoff’s Law

rrr

rJrJΦVd

4π

μjω

V

BVIL jωR

klVr Vrlk

kl dVdVaa

1

4π

μL

kk ll

lk

lk

rr

uu

dT IIB

Linear System of EquationsLinear 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!

dT I

0

V

I

0B

B-L jωR

First Key IdeaFirst Key Idea

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

Solenoidal Basis MethodSolenoidal Basis Method

Linear system

Solenoidal basis Basis for current that satisfies Kirchoff’s law Solenoidal basis matrix P: Current obeying Kirchoff’s law:

Reduced system

Solve via preconditioned Krylov subspace method

0

F

V

I

0B

B-L jωRT

0PBT 0IBPxI T

FPPxL jωRP TT

Local Solenoidal BasisLocal Solenoidal Basis

Cell current k consists of unit current assigned to the four filaments of the kth cell

There are four nonzeros in the kth column of P: 1, 1, -1, -1

Second Key IdeaSecond Key Idea

Observe: where

Approximate reduced system

Approximate by

LPLP~

1PPT

klVr Vrlk

kl dVdVaa

1

4π

ωμL

kk ll

lk rr

1~

2L~

L jωR PPPL jωRP TT ~~

PreconditioningPreconditioning

Preconditioning involves multiplication with

LL jωRL-1~~~~1 M

klVr Vrlk

kl dVdVaa

1

4π

ωμL

kk ll

lk rr

1~

L ω

jLRL 1- ~~~~ 11

highlow MM

1M

Hierarchical ApproximationsHierarchical Approximations

Components of system matrix and preconditioner are dense and large

Hierarchical approximations used to compute matrix-vector products with both 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)

L~

FASTHENRYFASTHENRY

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 of FMM

hierarchy SHELL: filaments within specified radius (expensive)

bWxL jωRW T

WL jωRWZ T ˆˆ

Z

ExperimentsExperiments

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

Ground PlaneGround Plane

Problem SizesProblem Sizes

Mesh Potential Nodes

Current Filaments

Linear System

Solenoidal

functions

33x33 1,089 2,112 3,201 1,024

65x65 4,225 8,320 12,545 4,096

129x129 16,641 33,024 49,665 16,384

257x257 66,049 131,584 197,633 65,536

Comparison with FastHenryComparison with FastHenry

Preconditioned GMRES Iterations (10GHz)

MeshFASTHENRY

DIAGFASTHENRY

CUBESolenoidal

Method

33x33 13 13 5

65x65 16 17 6

129x129 21 19 7

257x257 26 28 9

513x513 - - 14

Comparison …Comparison …

Time and Memory (10GHz)

Mesh FASTHENRY DIAG

FASTHENRY CUBE

Solenoidal Method

Time (sec)

Mem (MB)

Time (sec)

Mem (MB)

Time (sec)

Mem (MB)

33x33 2 10 2 10 2 1

65x65 13 42 17 42 12 5

129x129 95 177 142 177 68 17

257x257 835 734 1364 734 409 69

513x513 - - - - 2925 298

Preconditioner EffectivenessPreconditioner Effectiveness

Preconditioned GMRES iterations

Mesh Filament Length

Frequency (GHz)

1 10 100 1000

33x33 1/32 6 5 5 5

65x65 1/64 6 6 5 5

129x129 1/128 8 7 7 6

256x256 1/256 11 9 8 8

M jω~1

highM

Wire Over Ground PlaneWire Over Ground Plane

Comparison with FastHenryComparison with FastHenry

Preconditioned GMRES Iterations (10GHz)

MeshFASTHENRY

DIAGFASTHENRY

CUBESolenoidal

Method

33x33 13 11 4

65x65 13 14 5

129x129 13 12 6

257x257 3 3 8

513x513 - - 12

Comparison …Comparison …

Time and Memory (10GHz)

Mesh FASTHENRY DIAG

FASTHENRY CUBE

Solenoidal Method

Time (sec)

Mem (MB)

Time (sec)

Mem (MB)

Time (sec)

Mem (MB)

33x33 2 10 2 10 1 1

65x65 12 42 16 42 9 4

129x129 79 178 124 178 55 15

257x257 719 735 2732 735 351 61

513x513 - - - - 2427 260

Preconditioner EffectivenessPreconditioner Effectiveness

Preconditioned GMRES iterations

MeshFilament Length

Frequency (GHz)

1 10 100 1000

33x33 1/32 5 4 4 4

65x65 1/64 6 5 5 5

129x129 1/128 8 6 6 6

257x257 1/256 12 8 8 7

M jω~1

highM

Spiral InductorSpiral Inductor

Preconditioner EffectivenessPreconditioner Effectiveness

Preconditioned GMRES iterations

MeshFilament Length

Frequency (GHz)

1 10 100 1000

33x33 1/32 7 6 6 6

65x65 1/64 8 7 7 7

129x129 1/128 10 9 9 9

257x257 1/256 16 12 11 11

M jω~1

highM

Concluding RemarksConcluding 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: National Science Foundation