volume | PIER Journals

Abstract

This paper studies a computationally efficient algebraic graph theory engine for simulating time-domain one-dimensional waves in a multi-segment transmission line, such as for reflectometry applications. Efficient simulation of time-domain signals in multi-segment transmission lines is challenging because the number of propagation paths (and therefore the number of operations) increases exponentially with each new interface. We address this challenge through the use of a frequency-domain, algebraic graphical model of wave propagation, which is then converted to the time domain via the Fourier transform. We use this model to achieve an exact, stable, and computationally efficient (O(NQ), where N is the number of segments and Q is the bandwidth) approach for studying one-dimensional wave propagation. Our approach requires the reflection and transmission coefficients for each interface and each segment's complex propagation constant. We compare our simulation results with known analytical solutions.

1. Krautkramer, J. and H. Krautkramer, Ultrasonic Testing of Materials, Springer Science & Business Media, Apr. 2013.

2. Cawley, P., "Practical guided wave inspection and applications to structural health monitoring," Proc. of the Australasian Congress on Applied Mechanics, 10, Brisbane, Dec. 2007. Google Scholar

3. Moilanen, P., "Ultrasonic guided waves in bone," IEEE Trans. Ultrason. Ferroelectr. Freq. Control, Vol. 55, No. 6, 1277-1286, 2008.
doi:10.1109/TUFFC.2008.790 Google Scholar

4. Smith, P., C. Furse, and J. Gunther, "Analysis of spread spectrum time domain reflectometry for wire fault location," IEEE Sens. J., Vol. 5, No. 6, 1469-1478, Dec. 2005.
doi:10.1109/JSEN.2005.858964 Google Scholar

5. Lowe, M. J. S., "Matrix techniques for modeling ultrasonic waves in multilayered media," IEEE Trans. Ultrason. Ferroelectr. Freq. Control, Vol. 42, No. 4, 525-542, Jul. 1995.
doi:10.1109/58.393096 Google Scholar

6. Furse, C., P. Smith, C. Lo, Y. C. Chung, P. Pendayala, and K. Nagoti, "Spread spectrum sensors for critical fault location on live wire networks," Struct. Control Health Monit., Vol. 12, No. 3–4, 257-267, Jul. 2005. Google Scholar

7. Santos, E. J. P. and L. B. M. Silva, "Calculation of scattering parameters in multiple-interface transmission-line transducers," Measurement, Vol. 47, 248-254, Jan. 2014.
doi:10.1016/j.measurement.2013.08.024 Google Scholar

8. Sumithra, P. and D. Thiripurasundari, "Review on computational electromagnetics," Advanced Electromagnetics, Vol. 6, No. 1, 42-55, Mar. 2017.
doi:10.7716/aem.v6i1.407 Google Scholar

9. Chakraborty, A. and S. Gopalakrishnan, "A spectrally formulated finite element for wave propagation analysis in layered composite media," Int. J. Solids Struct., Vol. 41, No. 18, 5155-5183, Sep. 2004.
doi:10.1016/j.ijsolstr.2004.03.011 Google Scholar

10. Alterman, Z. and F. C. Karal, "Propagation of elastic waves in layered media by finite difference methods," Bulletin of the Seismological Society of America, Vol. 58, No. 1, 367-398, Feb. 1968. Google Scholar

11. Hoefer, W. J. R., "The transmission-line matrix method --- Theory and applications," IEEE Trans. Microw. Theory Tech., Vol. 33, No. 10, 882-893, Oct. 1985.
doi:10.1109/TMTT.1985.1133146 Google Scholar

12. Bohlen, T. and E. Saenger, "Accuracy of heterogeneous staggered-grid finite-difference modeling of Rayleigh waves," Geophysics, Vol. 71, No. 4, T109-T115, Jul. 2006.
doi:10.1190/1.2213051 Google Scholar

13. Liu, T., C. Zhao, and Y. Duan, "Generalized transfer matrix method for propagation of surface waves in layered azimuthally anisotropic half-space," Geophysical Journal International, Vol. 190, No. 2, 1204-1212, Aug. 2012.
doi:10.1111/j.1365-246X.2012.05547.x Google Scholar

14. Katsidis, C. C. and D. I. Siapkas, "General transfer-matrix method for optical multilayer systems with coherent, partially coherent, and incoherent interference," Appl. Opt., Vol. 41, No. 19, 3978-3987, Jul. 2002.
doi:10.1364/AO.41.003978 Google Scholar

15. Troparevsky, M. C., A. S. Sabau, A. R. Lupini, and Z. Zhang, "Transfer-matrix formalism for the calculation of optical response in multilayer systems: From coherent to incoherent interference," Opt. Express, Vol. 18, No. 24, 24 715-24 721, Nov. 2010.
doi:10.1364/OE.18.024715 Google Scholar

16. Cormen, T. H., C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms, MIT Press, 2009.

17. Farmaga, I., P. Shmigelskyi, P. Spiewak, and L. Ciupinski, "Evaluation of computational complexity of finite element analysis," Proc. of the International Conference The Experience of Designing and Application of CAD Systems in Microelectronics, 213-214, Feb. 2011. Google Scholar

18. Davidson, D. B. and R. W. Ziolkowski, "Body-of-revolution finite-difference time-domain modeling of space-time focusing by a three-dimensional lens," J. Opt. Soc. Am. A, JOSAA, Vol. 11, No. 4, 1471-1490, Apr. 1994.
doi:10.1364/JOSAA.11.001471 Google Scholar

19. Pozar, D. M., Microwave Engineering, 4th Ed., Wiley, Nov. 2011.

20. Mason, S. J., "Feedback theory --- Some properties of signal flow graphs," Proceedings of the IRE, Vol. 41, No. 9, 1144-1156, Sep. 1953.
doi:10.1109/JRPROC.1953.274449 Google Scholar

21. Schutt-Aine, J. E., "Transient analysis of nonuniform transmission lines," IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, Vol. 39, No. 5, 378-385, May 1992.
doi:10.1109/81.139288 Google Scholar

22. Acar, C., "Nth-order voltage transfer function synthesis using a commercially available active component: Signal-flow graph approach," Electron. Lett., Vol. 32, No. 21, 1933-1934, Oct. 1996.
doi:10.1049/el:19961352 Google Scholar

23. Biggs, N., N. L. Biggs, and E. N. Biggs, Algebraic Graph Theory, Vol. 67, Cambridge University Press, 1993.

24. Li, Z., Z. Duan, G. Chen, and L. Huang, "Consensus of multiagent systems and synchronization of complex networks: A unified viewpoint," IEEE Transactions on Circuits and Systems I: Regular Papers, Vol. 57, No. 1, 213-224, Jan. 2010.
doi:10.1109/TCSI.2009.2023937 Google Scholar

25. Sezer, M. E. and D. D. Iljak, "Nested-decompositions and clustering of complex systems," Automatica, Vol. 22, No. 3, 321-331, 1986.
doi:10.1016/0005-1098(86)90030-0 Google Scholar

26. George, A., J. R. Gilbert, and J. W. Liu, Graph Theory and Sparse Matrix Computation, Vol. 56, Springer Science & Business Media, 2012.

27. Dhillon, I. S. and D. S. Modha, "Concept decompositions for large sparse text data using clustering," Machine Learning, Vol. 42, No. 1–2, 143-175, 2001.
doi:10.1023/A:1007612920971 Google Scholar

28. Shuman, D. I., S. K. Narang, P. Frossard, A. Ortega, and P. Vandergheynst, "The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains," IEEE Signal Processing Magazine, Vol. 30, No. 3, 83-98, May 2013.
doi:10.1109/MSP.2012.2235192 Google Scholar

29. Sandryhaila, A. and J. M. Moura, "Big data analysis with signal processing on graphs: Representation and processing of massive data sets with irregular structure," IEEE Signal Processing Magazine, Vol. 31, No. 5, 80-90, 2014.
doi:10.1109/MSP.2014.2329213 Google Scholar

30. Puschel, M. and J. M. F. Moura, "Algebraic signal processing theory: Foundation and 1-D time," IEEE Trans. Signal Process., Vol. 56, No. 8, 3572-3585, Aug. 2008.
doi:10.1109/TSP.2008.925261 Google Scholar

31. Davis, T. A., S. Rajamanickam, and W. M. Sid-Lakhdar, "A survey of direct methods for sparse linear systems," Acta Numerica, Vol. 25, 383-566, 2016.
doi:10.1017/S0962492916000076 Google Scholar

32. Bunch, J. R. and D. J. Rose, Sparse Matrix Computations, Academic Press, 2014.

33. Luebbers, R., F. P. Hunsberger, K. S. Kunz, R. B. Standler, and M. Schneider, "A frequency-dependent finite-difference time-domain formulation for dispersive materials," IEEE Trans. Electromagn. Compat., Vol. 32, No. 3, 222-227, Aug. 1990.
doi:10.1109/15.57116 Google Scholar

34. Davis, T. A., "Algorithm 832: UMFPACK v4.3 --- An unsymmetric-pattern multifrontal method," ACM Trans. Math. Softw., Vol. 30, No. 2, 196-199, Jun. 2004.
doi:10.1145/992200.992206 Google Scholar

Dr. Joel B. Harley

Dr. Mashad Uddin Saleh

Dr. Samuel Kingston

Dr. Michael A. Scarpulla

Dr. Cynthia Furse