Dramatic Improvements in the Matrix Solution Time for Method of ...

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Dramatic Improvements in the Matrix Solution Time for Method of Moment Problems Involving Stripline Interconnects Xing Wang, Zhaohui Zhu, Member, IEEE, Yi Cao, Steven L. Dvorak, Member, IEEE, and John L. Prince, Fellow, IEEE

Abstract—Method of moments (MoM) reaction matrices are typically thought of as being full matrices in most applications. However, in this paper, we demonstrate for the first time that sparse reaction matrices are produced when modeling stripline interconnects provided that a parallel-plate Green’s function is employed in the analysis. This is demonstrated by investigating the sparse nature of the MoM reaction matrices that are produced when using the full-wave layered interconnect solver (UA-FWLIS) to model stripline interconnects. In order to explain the sparse nature of the reaction matrices, the electric fields that are excited by horizontal and vertical electric dipole sources are briefly overviewed, and the cutoff mode behavior of these electric fields is studied. Then the variations of the reaction elements with distance are studied, and this information is used to provide a cutoff criterion for the reaction element calculations. Once the reasons for the matrix sparsity have been explained, then we test various matrix solution algorithms in order to determine their efficiencies. We found that by applying sparse matrix storage techniques and a sparse matrix solver, it is possible to dramatically improve the matrix solution time when compared with a commercial MoM-based simulator. Index Terms—Conjugate gradient, integral equation, interconnects, method of moments (MOM), sparse matrix, stripline.

mode [i.e., the transverse electromagnetic (TEM) mode], they provide more accurate results than conventional 2-D analysis tools [1], which are based on the assumption that only a TEM mode exists. The method of moments (MoM) is a popular full-wave technique that has been applied to the solution of many electromagnetic problems over the years [2]. In this method, Maxwell’s equations are used together with boundary conditions to form an integral equation for the problems. The unknown currents are then expanded as a series of expansion functions with unknown current amplitudes, i.e.,

(1) After substituting (1) into the integral equation, inner products are formed with a set of weighting functions, thereby reducing the integral equation to a matrix equation

I. INTRODUCTION (2)

T

HE requirement to simulate larger and more complex interconnect circuits is being driven by the rapid developments that are taking place in the integrated circuit industry, where more complex circuits are continually being designed. A rigorous full-wave approach is required to fully account for all the interconnect phenomena, e.g., issues like crosstalk, signal delay, and distortion. Since full-wave analyses account for all the higher order modes, in addition to the transmission line

Manuscript received March 20, 2006; revised January 10, 2007. This work was supported by the Semiconductor Research Corporation (SRC) under contract 2005-KC-1292.024 and was carried out at the University of Arizona. This paper was presented in part at the 56th Electronic Components and Technology ConferenceSan Diego, CA,May2006. X. Wang is with NVIDIA Corporation, 2701 San Tomas Expressway, Santa Clara, CA 95050 USA. Z. Zhu is with Intel Corporation, Chandler, AZ 85226 USA. Y. Cao and S. L. Dvorak are with the Department of Electrical and Computer Engineering, The University of Arizona, Tucson, AZ 85721 USA (e-mail: [email protected]). J. L. Prince, deceased, was with the Department of Electrical and Computer Engineering, The University of Arizona, Tucson, AZ 85721 USA. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TADVP.2007.898638

reaction matrix, is a where [ ] represents an source voltage vector, and is a vector that contains the unknown current amplitudes. In applications where either free-space or layered microstrip Green’s functions are employed, the MoM will yield dense reacfor convention matrices [3]. The computational cost of tional Gaussian elimination and LU factorization methods [4] becomes prohibitively high when the matrix dimension goes beyond a few thousand unknowns. Therefore, various methods have been proposed to reduce the MoM matrix solution time. For example, the fast multipole method (FMM) [5], which is typically used with the free-space Green’s function, groups the reaction of all cells into nearby elements and employs multipole approximations for the distant elements. The FMM reduces to the matrix–vector multiplications from conventional per iteration, and further reductions to per iteration are achieved with the multilevel fast multipole algorithm (MLFMA) [6]. Another way to generate sparse matrices in MoM applications is by defining a special set of basis functions, called wavelet functions [7], to represent the unknown electric current. However, the wavelet basis functions have been applied with only one variable to solve 2-D problems so far [8].

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WANG et al.: DRAMATIC IMPROVEMENTS IN THE MATRIX SOLUTION TIME FOR METHOD OF MOMENT PROBLEMS

In contrast, it was shown in [9] that the closely spaced ground planes cutoff the propagation of all the higher order modes in the parallel-plate waveguide in stripline packaging problems. Furthermore, in previous field studies in stripline environments, it has been shown [10], [11] that the fields associated with horizontal current elements decay very rapidly as the observation point moves away from the source point. Since a reaction element results from a linear operation on the electric field, only closely-spaced cells will yield large reaction elements. Therefore, a sparse reaction matrix will result when the MoM is utilized with a parallel-plate Green’s function in packaging problems involving stripline interconnects. In this paper, we provide the theory and numerical results that show that sparse reaction matrices do occur when simulating stripline interconnect problems. An integral-equation-based, full-wave layered interconnect simulator (UA-FWLIS) [9], [12], which employs a parallel-plate Green’s function, has recently been developed for the fast calculation of the reaction elements for stripline interconnects. This technique first obtains the Green’s function in the spectral domain by applying boundary conditions. Then the MoM [2] is applied to the integral equation to obtain an approximate solution for the unknown currents on the traces. The reaction elements, which have the form of 2-D inverse Fourier transform integrals, are simplified by changing the rectangular coordinate system to a polar system, thereby yielding Sommerfeld-type integrals [13], which are highly oscillatory and slowly convergent. In order to avoid numerical integration, Cauchy’s Residue theory [14] was employed to express these integrals in terms of residue series, where the residues are calculated at the pole locations associated with the parallel-plate Green’s function. The resulting residue series [9], [12], which is free from numerical integration, involves a summation over special functions, i.e., Bessel functions and incomplete Lipschitz–Hankel integrals. We then employed rapidly computable series expansions [15]–[18] for the special functions, thereby allowing for the efficient computation of the residue series and the reaction elements. However, when simulating large circuits that involve a large number of unknowns, the matrix solution time becomes the dominant factor in the total CPU time, thus overwhelming the progress that has been gained in the matrix filling time. Fortunately, by recognizing that sparse matrices are obtained when analyzing stripline interconnects, we can continue making progress towards dramatic reductions in MoM solution times by implementing a sparse matrix solver in UA-FWLIS. In this paper, we demonstrate for the first time that the sparse nature of the reaction matrices in stripline packaging problems can lead to dramatic improvements in matrix solution times. In Section II, we first provide a summary of the closed-form modal solutions that are used to efficiently fill the matrix in UA-FWLIS. Then in Sections III and IV, the behaviors of the electric fields excited by electric dipoles are briefly overviewed and the reasons for the sparse reaction matrix are discussed. The variation of the reaction elements with respect to the distance between the cells is also investigated. Based on these results, we provide a cutoff criterion, wherein reaction elements are set to zero for separation distances that are larger than this criterion.

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Since numerical noise appears when computing the highly attenuated reaction elements, this cutoff criterion is beneficial for both the sake of efficiency and accuracy. In order to take advantage of the sparse matrix behavior in stripline problems, we use linked-list data structures [19] to store the sparse reaction matrix in order to reduce the memory usage. We then implemented the conjugate gradient (CG) method [20] to solve the sparse matrix. Numerical results that demonstrate the improvements in the efficiency and accuracy are included in Sections V and VI. Finally, the conclusions follow in Section VII. II. FILLING THE REACTION MATRIX As previously discussed, UA-FWLIS employs rapidly computable residue series expansions to compute the required reaction elements. For example, the reaction between horizontal rooftop test and expansion functions is expressed as [9], [12]:

(3) where

(4)

(5)

(6)

(7) (8) (9) symbol in the superscripts denotes that the reaction is The between a horizontal test function and a horizontal expansion function, i.e., the current is parallel to the ground planes.

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Fig. 1. Problem geometry showing x-directed and z -directed dipole sources in a homogeneously-filled stripline structure.

Note that the angular integration in (6) can be calculated by expressing it in terms of incomplete Lipschitz–Hankel integrals and then employing rapidly computable series expansions for these special functions [15]–[18]. This analytical representation then allows the semi-infinite integrals to be evaluated by using Cauchy’s residue theorem [9], [12]. III. FIELD PHENOMENOLOGY AND COUPLING BETWEEN TRACES In this section, we first review the behavior of the fields excited by horizontal electric dipole (HED) and vertical electric dipole (VED) sources in a parallel-plate waveguide (see Fig. 1 and [10], [11]). Because the reaction elements are directly related to the electric fields by

Fig. 2. Integration contour for the application of residue theory and the pole locations.

integrals and finally express the reaction element as the summation of the residues evaluated at all the pole locations. and functions, which are Recall that the expressed in (8) and (9), have poles that are located at (11)

(10) the investigation of the field behavior will show why the reaction matrix is sparse. A. Field Variation With Respect to the Separation Distance in the – Plane mode As discussed in the previous field study [11], the is not excited by a horizontal source in the homogeneously-filled waveguide case. Therefore, the total field is predominately made mode. Since all the and modes decay up of the exponentially with increasing distance from the source for small , the total field also decays very rapidly plate spacings with increasing distance in electronic packaging applications. For the vertical dipole source [10], the - and -components of the electric field once again decay exponentially since there is mode contribution for these field components. Howno mode, which is excited by a VED, conever, since the tributes to the -component, the total electric field decays fairly slowly for separation distances beyond a turning point, i.e., the distance beyond which all higher order modes are negligible mode. when compared with the B. Reaction Variation With Respect to the Separation Distance in the x–y Plane The reaction element in (3) is analytically calculated by extending the semi-infinite integrals in (4) and (5) to infinite integrals [12]. Then we apply Cauchy’s residue theory to the infinite

in the complex

plane. Since (12)

plane as shown in these poles are located in the complex and Fig. 2. If we look at the first pole locations for , i.e., the case, we find that their limiting cases approaches zero, which corresponds to the when mode, are finite values, i.e.,

(13)

(14) is a removable pole for both and Therefore, , thus indicating that there is no mode contribution to the horizontal-horizontal reactions in homogeneouslyfilled stripline structures. , The above analysis was also applied to the derivation of and , where the superscripts and denote horizontal and vertical current cells, respectively. Out of all the types case only yields a of reactions, it was found that the non-removable pole for the function in . Therefore, the reaction between a vertical expansion and a vertical test

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after the turning point. Therefore, the decays slowly as coupling between two vertical reacting cells is stronger at larger distances and may need to be accounted for. IV. SPARSE NATURE OF MoM REACTION MATRICES IN PACKAGING PROBLEMS INVOLVING STRIPLINE INTERCONNECTS If the parallel-plate spacing is much less than half of a mode will propagate inside wavelength, then only the the stripline structure and all the higher order modes are cutoff and attenuate rapidly with distance. As shown in [10] and [11], the electric field expressions for HED and VED excitations in. When the volve summations over Hankel functions, becomes large, the dominant exponential imaginary part of . behavior associated with the cutoff modes is given by As shown in Fig. 2, only the pole corresponding the mode has a small imaginary part of , which is caused by a nonzero loss tangent. All the other poles that are associated with the higher order modes have very large negative values for the imaginary part of , thereby causing a very rapid cutoff befor these modes. The requirement that havior for large is much smaller than half the wavelength is almost always true for packaging problem. For instance, for applications up to 10 , a half wavelength is GHz in a FR4-filled stripline around 14.3 mm, which greatly exceeds the spacing between the power and ground planes in modern printed circuit boards (e.g., is typically around 0.6–1.2 mm). Since the electric fields become insignificant when the distance between the expansion and the test function cells becomes , beyond which all large, we define a maximum distance, higher order modes can be neglected. The threshold value for can be set as [9]:

Fig. 3. Reaction elements plotted as a function of separation distance  be(y = 0). tween two rooftop cells. (a) jZ j. plotted as a function of x (b) jZ j plotted as a function of y (x = 0).

(15)

function contains a contribution from the mode together with the higher order modes, while all other types of reactions are made up solely of higher order modes contributions. Thus ) between the with large increases in the distance (i.e., , expansion and test functions, all of the reactions and become negligible. However, since the -component of the electric field excited by a VED has a mode con, which may tribution, there will be increased coupling for have to be taken into account even when the cells are far apart. In Fig. 3, we plot the four types of reactions as a function of the distance between the expansion and test functions in the and -directions. As expected, the results in Fig. 3 indicate that , and decay exponentially the magnitudes of with increasing distance between the reacting cells. Therefore, ), then the when the reacting cells are far apart (e.g., coupling is negligible. However, for two vertical reacting cells, there will still be some coupling even when they are far apart, mode that contributes to the reaction because the

where SD is the desired number of significant digits of accuracy. Since far away reactions are highly attenuated, numerical noise might result if they are computed. Therefore, to improve both the computational efficiency and avoid numerical noise, it and is advantageous to skip computing reactions for just set them to zero. While it is possible to ignore the far away , , and , since the mode reactions for does not play a role in these cases, this criterion is not suitable reactions since the mode contributes to for the such reactions. MoM reaction matrices are full matrices in most common applications, thereby limiting the number of unknowns that can be employed in a problem. As previously discussed, in order to overcome this limitation, the Fast multiple-pole method (FMM) is often employed. However, in stripline packaging problems, , the evanescent mode behavior directly leads to where sparse reaction matrices. Therefore, the sparse matrices that result in packaging applications provide the opportunity to employ sparse matrix techniques. In this paper, we study two examples:

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Fig. 5. Reaction matrix fill-time comparison at f = 1 GHz. (a) Multiple parallel lines. (b) Multiple crossover lines.

Fig. 4. Sparse reaction matrix in packaging problems obtained using UA-FWLIS. (a) Contour plot for jZ j that shows the sparse reaction matrix for a stripline structure that consists of three parallel lines. (b) Contour plot for jZ j that shows the sparse reaction matrix for a stripline structure that consists of threex-directed and three y -directed crossing lines. (c) Variation of the matrix density with the number of unknowns.

parallel signal lines and crossover signal lines. The reason that we chose parallel and crossover lines is because it is relatively easy to create the large geometrical input files for these structures that are necessary inputs for both UA-FWLIS and Agilent Momentum. We also believe that parallel lines yield the sparsest

matrices and crossover lines yield the densest matrices. Therefore, we have used these two extreme cases for our studies. Almodes, the reactions between the though vias will launch connected traces in different layers in multilayer stripline structures is zero, thereby leading to even greater sparsity in the reaction matrices. More complicated structures, such as multilayered stripline structures, will be analyzed in the future. As an example, Fig. 4(a) shows a contour plot for the magnitudes of the reaction matrix elements associated with three and lengths of parallel lines with widths of . The spacing between the ground planes is and the lines are placed 800 apart from each other in the middle between the two planes. The dielectric filling between and the ground planes has the parameters , and we assumed an operating frequency of 1 GHz. We employed 126 unknowns for each line, thereby giving a total reactions. Out of these of 378 unknowns and reactions, there are only 13,884 nonzero elements that satisfy , which occupy only 9.72% of the matrix. Therefore, the reaction matrix shows a modest sparsity. Note that the

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Fig. 6. Reaction matrix solution time comparison at f = 1 GHz. (a) Multiple parallel lines. (b) Multiple crossover lines.

banded nature shown in Fig. 4(a) results from the coupling between the various lines. The diagonal in the matrix, which is associated with the self-coupling terms, exhibits the largest amplitudes. The two adjacent sidebands result from the coupling between nearby traces, and are reduced in amplitude as expected. Finally, the outer sidebands, which have even smaller amplitudes, are associated with the coupling between the outer traces. As additional horizontal lines are added, additional bands with decreasing amplitudes are also added to the reaction matrix. Above phenomenon and analysis also apply to Fig. 4(b), where we show a contour plot for the reaction matrix associated with a set of three -directed and three -directed crossing lines. For the multiple crossover line example, all the parameters remain the same as those for parallel lines except that we place and . lines in two layers, i.e., at In Fig. 4(c), we plot the matrix density as a percentage versus the number of unknowns. We observe that with increases in the size of the simulated structure (i.e., the number of unknowns), the matrices become more and more sparse.

Fig. 7. Matrix solution efficiency improvement factor provided by the conjugate gradient method. (a) Compared with momentum. (b) Compared with the Fortran IMSL full matrix solver. (c) Compared with the Fortran IMSL sparse matrix solver.

V. IMPROVEMENT IN THE MATRIX SOLUTION TIME In order to exploit the matrix sparsity, we implemented the conjugate gradient (CG) algorithm [20] with the point Jacobi preconditioner [21] in order to solve the system of (2) that are used to calculate the currents (1) on the traces. We carried out all the numerical tests on a personal computer with an Intel Pentium

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Fig. 8. Number of iterations required by the CG method versus the number of unknowns.

Fig. 10. Comparisons for the two crossover line layouts. (a) Two different crossover line layouts. (b) Matrix solution time. Fig. 9. Relative error versus the number of iterations with the CG method.

IV 3.0 GHz HT technology processor and 512 MB of memory. In Fig. 5, we plot the matrix filling times required when using UA-FWLIS and Agilent Momentum [22] for multiple parallel lines and multiple crossover lines. In Fig. 6, we plot the solution times required when using different matrix solving routines for both multiple parallel lines and multiple crossover lines. Both axes are plotted using logarithmic scales in order to display the computational cost more clearly. We see from Fig. 6 that the matrix solving efficiency has been improved greatly by employing the CG method in UA-FWLIS. Since the CG curves have the for parallel lines and for smallest slopes (i.e., crossover lines), the advantage of the CG method grows as the number of unknowns increases. Note that if one of the lines in Fig. 6 ends, then it indicates that the solver could not handle larger problems. Also note that Agilent Momentum automatically adds 1500 unknowns when simulating multiple crossover

lines. That is why the Momentum’s curve begins at 2000 in Fig. 6(b). The advantages provided by using the CG method can be seen more clearly in Fig. 7, where we plot the efficiency improvement factor for the matrix solution time obtained with the CG method versus the other solvers. The efficiency improvement goes as high as 300 times versus the Fortran full matrix solver, and 100 times versus the Agilent Momentum solver for the cases we tested. Since we are the first to show that stripline structures yield sparse matrices, we compare with full-matrix solvers because we believe that commercial simulators utilize full-matrix solvers. In fact, the slope of the curve for the Agilent Momentum results verifies this hypothesis. In Fig. 7(c), we also compare the CG method with a Fortran sparse matrix solver that was taken from the IMSL library, which employs direct LU factorization on sparse matrices by using the symmetric Markowitz strategy [23] to choose pivots that are most likely to reduce fill-ins while maintaining numerical stability. It is clear

WANG et al.: DRAMATIC IMPROVEMENTS IN THE MATRIX SOLUTION TIME FOR METHOD OF MOMENT PROBLEMS

Fig. 11. Accuracy comparison: S-parameters for multiple parallel lines. (a) S

from Fig. 7(c) that the nonstationary iterative CG method is also more efficient than the direct sparse solver for problems that require more than 2000 unknowns, since the efficiency improvement gets as high as 30 times. Even larger efficiency improvements would have resulted if the other simulators could handle larger problems like UA-FWLIS. As can been seen in Figs. 8 and 9, the CG method that we applied for the stripline reaction matrix application has some nice features. The first thing to note is that it employs an almost fixed number of iterations irregardless of the number of unknowns ). For each iteration in the CG for a set relative error (i.e., method, the computational cost is proportional to the number of nonzero elements in the matrices that employ linked list storage formats [19]. For the case of multiple parallel lines, adding an additional line would not affect the faraway lines. Therefore, each line has a fixed number of nonzero elements, thus causing , which is the optimal the required CPU time to scale as case. However, for the multiple crossover line case, each additional line that is added will interact with all the crossing lines, thereby causing the matrix to be denser than for the parallel line , which is the worst case, and the CPU time to scale as case scenario. Other layouts will yield results that lie in between these two extreme cases.

magnitude. (b) S

phase. (c) S

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magnitude. (d) S

phase.

In Fig. 9, we plot the relationship between the number of iterations and the relative error. We observe that the magnitude of the relative error decreases linearly with the number of iterations. In Fig. 10(b), we compare the matrix solution times for the two different layouts of crossover lines shown in Fig. 10(a). and add In layout 1, we use fixed line lengths of more lines to increase the number of unknowns. In layout 2, we maintain a square shape in the - and -directions and increase the line lengths as additional lines are added. We used layout 1 for all of the previous tests. When increasing the number of unknowns, the crossover lines in layout 2 maintain a square shape, which is more like what occurs in practical applications. However, Fig. 10(b) shows that it really does not matter how the crossover lines are placed since the matrix solution times remain about the same. VI. ACCURACY COMPARISONS Thus far we have shown that the CG technique provides a dramatic improvement in the computational efficiency when the MoM is applied to the modeling of stripline interconnects. However, these improvements in the efficiency does not mean much if the results are not also accurate. In order to validate the accuracy of UA-FWLIS, we ran the case with 16 parallel lines (i.e., a

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total of 2034 cells) and found the far-end crosstalk between two adjacent lines. Because our calculations of the S-parameters are based on the extraction of the dominant forward and backward traveling-wave modes at a port, the locations of the ports have to be far enough away from discontinuities to avoid the interference caused by the higher order modes that are created by the open-end discontinuities on other lines. Therefore, we have added extension lines with lengths of 12 mm to the two ports. We then de-embedded the S-parameters at locations that were chosen 5.04 mm from the ends of these extended lines. The simulation results that are produced by UA-FWLIS are plotted in Fig. 11, along with the results that are obtained by Agilent Momentum. We observe an excellent agreement for both the magnitude and phase of the S-parameters, indicating that UA-FWLIS, with the built in CG sparse solver, does properly capture the crosstalk. Excellent agreement was also obtained for the case of the crossover lines. However, these results are not shown because of space limitations.

VII. CONCLUSION In this paper, we first discussed the reason why stripline interconnects produce sparse reaction matrices. Since all the higher order modes are cutoff in stripline packaging structures, the electric fields decay exponentially with increasing distance from the source. Therefore, reactions between widely spaced test and expansion functions are negligible, thereby resulting in sparse reaction matrices in stripline packaging problems. On the other hand, since microstrip has an open region that allows for radiation and surface-wave modes, microstrip interconnects would not exhibit this same cutoff behavior and larger reactions will occur for widely-space expansion and test functions. A nonstationary, iterative CG method was then implemented to take advantage of this sparse matrix phenomenology. Comparisons with the Agilent Momentum solver, the IMSL full matrix solver, and a direct IMSL sparse solver demonstrated the advantages of using the CG method. For example, the matrix solution time was improved by a factor of 20-250 when compared with Agilent Momentum for multiple parallel lines, and a factor of 10–100 for multiple crossover lines. The computa, while Agtional cost for the CG method is of . Thereilent Momentum has a computational cost of fore, the larger the structure, the larger the improvement factor gained by employing CG method will be. The improvements in the matrix solving time that are offered by the CG method means that full-wave simulations of complex, realistic stripline interconnect structures are feasible.

REFERENCES [1] W. Cao, R. Harrington, J. Mautz, and T. Sarkar, “Multiconductor transmission lines in multilayered dielectric media,” IEEE Trans. Microwave Theory Tech., vol. MTT-32, pp. 439–450, Apr. 1984. [2] R. F. Harrington, Field Computation by Moment Methods. Piscataway, NJ: IEEE Press, 1993. [3] L. Gürel and M. I. Aksun, “Electromagnetic scattering solution of conducting strips in layered media using the fast multipole method,” IEEE Microw. Guided Wave Lett., vol. 6, no. 8, pp. 277–279, Aug. 1996.

[4] M. Crow, Computational Methods for Electric Power Systems. Boca Raton, FL: CRC Press, 2003. [5] J. M. Song and W. C. Chew, “Fast multipole method solution using parametric geometry,” Microw. Optical Technol. Lett., vol. 7, no. 16, pp. 760–765, Nov. 1994. [6] J. M. Song and W. C. Chew, “Multilevel fast-multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microw.Optical Technol. Lett., vol. 10, no. 1, pp. 14–19, Sep. 1995. [7] B. Z. Steinberg and Y. Leviatan, “On the use of wavelet expansions in the method of moments,” IEEE Trans. Antennas Propag., vol. 41, no. 5, pp. 610–619, May 1993. [8] S. M. Rao and N. Balakrishnan, “Computational electromagnetics—A review,” Current Sci., vol. 77, no. 10, pp. 1343–1347, 1999. [9] S. Kabir, S. L. Dvorak, and J. L. Prince, “Reaction analysis in stripline circuits,” IEEE Trans. Adv. Packag., vol. 24, no. 3, pp. 347–356, Aug. 2001. [10] X. Wang, S. Kabir, J. Weber, S. L. Dvorak, and J. L. Prince, “A study of the fields associated with vertical dipole sources in stripline circuits,” IEEE Trans. Adv. Packag., vol. 25, no. 2, pp. 272–279, May 2002. [11] X. Wang, S. Kabir, J. Weber, S. L. Dvorak, and J. L. Prince, “A study of the fields associated with horizontal dipole sources in stripline circuits,” IEEE Trans. Adv. Packag., vol. 25, no. 2, pp. 280–287, May 2002. [12] Z. Zhu, Q. Li, X. Wang, S. L. Dvorak, and J. L. Prince, “Extension of an efficient moment method based, full-wave layered-interconnect simulator to finite-width expansion functions,” IEEE Trans. Adv. Packag., submitted for publication. [13] A. Sommerfeld, Partial Differential Equations in Physics. New York: Academic, 1949. [14] E. B. Saff and A. D. Snider, Fundamentals of Complex Analysis for Mathematics, Science, and Engineering. Englewood Cliffs, NJ: Prentice-Hall, 1976. [15] M. M. Mechaik and S. L. Dvorak, “Series expansions for the incomplete Lipschitz-Hankel integral J e (a; z ),” Radio Sci., vol. 30, no. 5, pp. 1393–1404, 1995. [16] M. M. Mechaik and S. L. Dvorak, “Series expansions for the incomplete Lipschitz-Hankel integral Y e (a; z ),” Radio Sci., vol. 31, no. 2, pp. 409–422, 1996. [17] D. Heckmann and S. L. Dvorak, “Numerical computation of Hankel functions of integer order and complex-valued arguments,” Radio Sci., vol. 36, no. 6, pp. 1265–1270, 2001. [18] Z. Zhu, S. L. Dvorak, D. L. Heckmann, and J. L. Prince, “Numerical computation of incomplete Lipschitz-Hankel integrals of the Hankel type for complex-valued arguments,” Radio Sci., vol. 40, pp. 1–17, 2005, 10.1029/2004RS003231, RS6009. [19] S. Pissanetzky, Sparse Matrix Technology. London: Academic, 1984. [20] M. F. Cátedra, R. P. Torres, J. Basterrechea, and E. Gago, The CG-FFT Method, Application of Signal Processing Techniques to Electromagnetics. Boston, MA: Artech, 1995. [21] G. Forsythe and E. Strauss, “On best conditioned matrices,” Pro. Am. Math. Soc., vol. 6, pp. 340–345, 1955. [22] “Agilent Momentum User Manual,” ver. 2003A, Agilent, Santa Clara, CA. [23] I. S. Duff, A. M. Erisman, and J. K. Reid, Direct Methods for Sparse Matrices. New York: Oxford Univ.Press, 1986.

Xing Wang received the B.S. degree from the Department of Electronics, Tsinghua University, Beijing, China, in 1999, and the M.S. and Ph.D. degrees from the Department of Electrical and Computer Engineering, the University of Arizona, Tucson, in 2002 and 2006, respectively. His major was in the interconnect packaging simulation/design areas, with special emphasis on employing full-wave simulation methods. He is now working in the Signal Integrity Research Laboratory at nVIDIA Corporation, Santa Clara, CA. His research interests are in the areas of packaging design and simulation, highspeed/frequency IC interconnect modeling, simulation, and methodologies.

WANG et al.: DRAMATIC IMPROVEMENTS IN THE MATRIX SOLUTION TIME FOR METHOD OF MOMENT PROBLEMS

Zhaohui Zhu (M’06) received the B.S. degree in electrical engineering from the University of Science and Technology of China, in 1991, the M.S. degree in signal processing from the Institute of Electronics, Academia Sinica, in 1994, and the Ph.D. degree from the Department of Electrical and Computer Engineering, the University of Arizona, Tucson, in 2005. From 1994 to 2000, she was with Telecommunication Department of the China National Clearing Center, where she worked as a Telecommunication System Engineer. She is currently working as a Packaging Engineer for ATTD, Intel Corporation. Her research interests include signal integrity, electromagnetic modeling of high-speed circuits, electromagnetic transients, wave propagation, and theoretical and computational electromagnetics.

Yi Cao received the B.S.E.E., M.S., and the Ph.D. degrees from Shanghai Jiao Tong University, Shanghai, China, in 1994, 1996, and 1999, respectively. Since 2004, he has been a postdoctoral researcher at the University of Arizona, Tucson. His main research area is the modeling of high-speed interconnects and packaging, including numerical techniques for solving field problems and circuit simulations.

Steven L. Dvorak (M’86) received the B.S. and Ph.D. degrees in electrical engineering from the University of Colorado, Boulder, in 1984 and 1989. He is currently a Professor in the Department of Electrical and Computer Engineering at the University of Arizona, Tucson. He served as an Assistant Professor in this Department from 1989 to 1996 and an Associate Professor from 1996 to 2004. He previously held a position with TRW Space and Technology Group from 1984 to 1989. His principal interests include electromagnetic modeling

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of high-speed interconnects, electromagnetic transients, wave propagation, theoretical and computational electromagnetics, optics, geophysical applications of electromagnetics, applied mathematics, and microwave measurements. Dr. Dvorak is an elected member of the International Union of Radio Science Commissions B and F, and a member of Tau Beta Pi. He received the Antennas and Propagation Society S. A. Schelkunoff Prize Paper Award in 1997 and the URSI Young Scientist Award in 1996. He was also awarded the Department of Electrical and Computer Engineering IEEE and HKN Outstanding Teaching Award and the Andersen Consulting Teaching Award in 1994.

John L. Prince (S’65–M’68–SM’78–F’90) received the B.S.E.E. degree from Southern Methodist University, and as an NSF Graduate Fellow received the M.S.E.E. and Ph.D. degrees in electrical engineering from North Carolina State University. He was a Professor of Electrical and Computer Engineering and Director of the Center for Electronic Packaging Research at the University of Arizona. He came to the University of Arizona in 1983. He was the Principal Investigator of the Semiconductor Research Corporation (SRC) Program in VLSI Packaging and Interconnection Research at the university from 1984 until his death in December of 2005. In 1991–1992, he was Acting Director, Packaging Sciences at SRC. He had extensive industrial experience. He was active in consulting work in both the reliability and packaging areas. He was coauthor on two books in the field of electronic packaging, Simultaneous Switching Noise of CMOS Devices and Systems, by Senthinathan and Prince, and Electronic Packaging: Design, Materials, Processing and Reliability, by Lau, Wong, Prince, and Nakayama. He taught courses in electronic packaging and the University of Arizona. His research interests centered on developing modeling and simulation techniques for switching noise in packages and MCMs, on modeling and simulation techniques for mixed-signal system packaging, and on the development of high-frequency measurements on packaging structures. He authored or co-authored over 210 papers in the field of electronic packaging and 35 papers in the fields of semiconductor device physics, process development, and reliability.