Transcript Document
A Solenoidal Basis Method For Efficient Inductance Extraction
H
emant Mahawar Vivek Sarin Weiping Shi Texas A&M University College Station, TX
Introduction
Background
Inductance between current carrying filaments
Kirchoff’s law enforced at each node
Background …
Current density at a point J
j ω V μ 4 π
r J
r
d V Φ
Linear system for current and potential
R j ω L I BV
Inductance matrix
L kl μ 1 4 π a k a l r k V k r l V l
u k r k
u l r l
dV l dV k
Kirchoff’s Law
B T I I d
Linear System of Equations
R j B T ω L B 0 I V 0 I d
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!
First Key Idea
Current Components
Fixed current satisfying external condition I d (left)
Linear combination of cell currents (right)
Solenoidal Basis Method
Linear system
R j B T ω L B 0 I V F 0
Solenoidal basis
Basis for current that satisfies Kirchoff’s law
Solenoidal basis matrix P:
Current obeying Kirchoff’s law:
I B T P Px 0 B T I 0
Reduced system
P T R j ω L Px P T F
Solve via preconditioned Krylov subspace method
Local 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 Idea
~ L ~ L kl ωμ 4 π 1 a k a l r k V k r l V l
r k
1
r l
dV l dV k
Approximate reduced system
P T R j ω L P P T P ~ R j ω ~ L 1 ~ L
Preconditioning
M
1 ~ L ~ R j ω ~ L -1 ~ L ~ L kl ωμ 4 π 1 a k a l r k V k r l V l
r k
1
r l
dV l dV k
M low
1 ~ L ~ R 1 ~ L
M high
1 j ω ~ L
Preconditioning involves multiplication with
M
1
Hierarchical Approximations
Components of system matrix and preconditioner are dense and large
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)
FASTHENRY
Uses mesh currents to generate a reduced system
W T R j ω L Wx b
Approximation to reduced system computed by sparsification of inductance matrix
W T R j ω ˆ W
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)
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
Ground Plane
Problem Sizes Mesh 33x33 65x65 129x129 257x257 Potential Nodes Current Filaments 1,089 4,225 16,641 66,049 2,112 8,320 33,024 131,584 Linear System 3,201 12,545 49,665 197,633 Solenoidal functions 1,024 4,096 16,384 65,536
Comparison with FastHenry Preconditioned GMRES Iterations (10GHz) Mesh 33x33
FASTHENRY
DIAG 13
FASTHENRY
CUBE 13 Solenoidal Method 5 65x65 129x129 16 21 17 19 6 7 257x257 513x513 26 28 9 14
Comparison … Mesh 33x33 65x65 129x129 257x257 513x513 Time and Memory (10GHz)
FASTHENRY
DIAG Time (sec) Mem (MB) 2 10 13 42 95 835 177 734 -
FASTHENRY
CUBE Time (sec) Mem (MB) 2 10 17 42 142 1364 177 734 Solenoidal Method Time (sec) Mem (MB) 2 1 12 5 68 409 2925 17 69 298
Preconditioner Effectiveness Mesh Preconditioned GMRES iterations Frequency (GHz) Filament Length 1 10 100 33x33 65x65 129x129 256x256 1/32 1/64 1/128 1/256 6 6 8 11 5 6 7 9
M high
1 j ω ~ M
5 5 7 8 1000 5 5 6 8
Wire Over Ground Plane
Comparison with FastHenry Preconditioned GMRES Iterations (10GHz) Mesh 33x33
FASTHENRY
DIAG 13
FASTHENRY
CUBE 11 Solenoidal Method 4 65x65 129x129 13 13 14 12 5 6 257x257 513x513 3 3 8 12
Comparison … Mesh 33x33 65x65 129x129 257x257 513x513 Time and Memory (10GHz)
FASTHENRY
DIAG Time (sec) Mem (MB) 2 10 12 42 79 719 178 735 -
FASTHENRY
CUBE Time (sec) Mem (MB) 2 10 16 42 124 2732 178 735 Solenoidal Method Time (sec) Mem (MB) 1 1 9 4 55 351 2427 15 61 260
Preconditioner Effectiveness Mesh Preconditioned GMRES iterations Frequency (GHz) Filament Length 1 10 100 1000 33x33 65x65 129x129 257x257 1/32 1/64 1/128 1/256 5 6 8 4 5 6 12 8
M high
1 j ω ~ M
4 5 6 8 4 5 6 7
Spiral Inductor
Preconditioner Effectiveness Mesh Preconditioned GMRES iterations Frequency (GHz) Filament Length 1 10 100 1000 33x33 65x65 129x129 257x257 1/32 1/64 1/128 1/256 7 8 10 6 7 9 16 12
M high
1 j ω ~ M
6 7 9 11 6 7 9 11
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: National Science Foundation