A Floating Random-Walk Algorithm Based on Iterative Perturbation Theory: Solution of the 2D, Vector-Potential Maxwell-Helmholtz Equation
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A Floating Random-Walk Algorithm Based on Iterative Perturbation Theory: Solution of the 2D, Vector-Potential Maxwell-Helmholtz EquationAbstract
At present multi-GHz operating frequencies, the elec-
trical properties of high-end, multilevel IC intercon-
nects must be described with Maxwell’s equations.
We have developed an entirely new floating random-
walk (RW) algorithm to solve the 2D time-harmonic
Maxwell-Helmholtz equation. The algorithm requires
no numerical mesh, thus consuming a minimum of
computational memory—even in complicated prob-
lem domains, such as those encountered in IC inter-
connects. The major theoretical challenge of deriving
an analytical Green’s functions in arbitrary heteroge-
neous problem domains has been successfully re-
solved by means of an accurate approximation: itera-
tive perturbation theory. Initial numerical verification
of the algorithm has been achieved for the case of a
“skin-effect” problem within a uniform circular con-
ductor cross section, and also for a heterogeneous
“split-conductor” problem, where one segment of a
square domain is conducting material, while the other
segment is insulating. As an example of electrical
parameter extraction using this algorithm, we have
extracted the frequency-dependent impedance of the
uniform circular cross-section previously mentioned.
Excellent agreement has been obtained between the
analytical and RW solutions, supporting the theoreti-
cal formulation presented here
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References
Y.L. Le Coz and R.B. Iverson, “A Stochastic
Algorithm for High Speed Capacitance Extrac-
tion in Integrated Circuits”, Solid-State Electron-
ics, vol. 35, pp. 1005–12, 1992.
Ilya M. Sobol, A Primer for the Monte Carlo
Method, Boca Raton, FL: CRC Press, 1994.
K.K. Sabelfeld, Monte Carlo Methods in Bound-
ary Value Problems, New York, NY: Springer-
Verlag, 1991.
K. Chatterjee, Development of a Floating Ran-
dom-Walk Algorithm for Solving Maxwell’s
Equations in Complex IC-Interconnect Struc-
tures, pp. 28–31, Doctoral Thesis, Rensselaer
Polytechnic Institute, Troy, NY, April 2002.
D.M. Pozar, Microwave Engineering, 2nd
Edition, pp. 16-7, New York, NY: John Wiley &
Sons, 1998.
J.D. Jackson, Classical Electrodynamics, 3rd
Edition, pp. 241-2, New York, NY: John Wiley
& Sons, 1999.
Van Dyke, Perturbation Methods in Fluid Me-
chanics. Stanford, CA: Parabolic Press, 1975.
R. Haberman, Elementary Applied Partial Dif-
ferential Equations with Fourier Series and
CHATTERJEE, LE COZ: A FLOATING RANDOM-WALK ALGORITHM
Boundary Value Problems, 3rd Edition, pp. 155-
, Upper Saddle River, NJ: Prentice Hall, 1998.
R. Haberman, Elementary Applied Partial Dif-
ferential Equations with Fourier Series and
Boundary Value Problems, 3rd Edition, pp. pp.
-1, Upper Saddle River, NJ: Prentice Hall,
R. Haberman, Elementary Applied Partial Dif-
ferential Equations with Fourier Series and
Boundary Value Problems, 3rd Edition, pp. 406-
, Upper Saddle River, NJ: Prentice Hall, 1998.
R. Haberman, Elementary Applied Partial Dif-
ferential Equations with Fourier Series and
Boundary Value Problems, 3rd Edition, pp. 422-
, Upper Saddle River, NJ: Prentice Hall, 1998.
S. Ramo, J. R. Whinnery, and T.V. Duzer, Fields
and Waves in Communication Electronics, 3rd
Edition, pp. 180-4, New York, NY: John Wiley
& Sons, 1993.
K. Chatterjee, Development of a Floating Ran-
dom-Walk Algorithm for Solving Maxwell’s
Equations in Complex IC-Interconnect Struc-
tures, pp. 153-4, Doctoral Thesis, Rensselaer
Polytechnic Institute, Troy, NY, April 2002.
A. Papoulis and S.U. Pillai, Probability, Random
Variable and Stochastic Processes, 4th Edition,
McGraw Hill, 2001.
