The discontinuous Galerkin's (DG) method is an efficient technique for packaging problems. It divides an original computational region into several subdomains, i.e., splits a large linear system into several smaller and balanced matrices. Once the spatial discretization is solved, an optimal time integration method is necessary. For explicit time stepping schemes, the smallest edge length in the entire discretized domain determines the maximal time step interval allowed by the stability criterion, thus they require a large number of time steps for packaging problems. Implicit time stepping schemes are unconditionally stable, thus domains with small structures can use a large time step interval. However, this approach requires inversion of matrices which are generally not positive definite as in explicit shemes for the first-order Maxwell's equations and thus becomes costly to solve for large problems. This work presents an algorithm that exploits the sequential way in which the subdomains are usually placed for layered structures in packaging problems. Specifically, a reordering of interface and volume unknowns combined with a block LDU (Lower-Diagonal-Upper) decomposition allows improvements in terms of memory cost and time of execution, with respect to previous DGTD implementations.
2. Xiao, T. and Q. H. Liu, "Three-dimensional unstructured-grid discontinuous Galerkin method for Maxwell’s equations with well-posed perfectly matched layer," Microw. Opt. Technol. Lett., Vol. 46, No. 5, 459-463, 2005.
3. Hesthaven, J. S. and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Vol. 54, Vol. 54, Springer, 2007.
4. Lee, J.-H. and Q. H. Liu, "A 3-D spectral-element time-domain method for electromagnetic simulation," IEEE Trans. Microw. Theory Techn., Vol. 55, No. 5, 983-991, 2007.
5. Lee, J.-H., J. Chen, and Q. H. Liu, "A 3-D discontinuous spectral element time-domain method for Maxwell’s equations," IEEE Trans. Antennas Propag., Vol. 57, No. 9, 2666-2674, 2009.
6. Chen, J. and Q. H. Liu, "A hybrid spectral-element/finite-element method with the implicit-explicit Runge-Kutta time stepping scheme for multiscale computation," IEEE Intl. Symposium on Antennas and Propagation (APSURSI), 1-4, 2010.
7. Chen, J., Q. H. Liu, M. Chai, and J. A. Mix, "A nonspurious 3-D vector discontinuous Galerkin finite-element time-domain method," IEEE Micro. Wireless Comp. Lett., Vol. 20, No. 1, 1-3, 2010.
8. Chen, J., L. Tobon, M. Chai, J. Mix, and Q. H. Liu, "Efficient implicit-explicit time stepping scheme with domain decomposition for multiscale modeling of layered structures," IEEE Trans. Compon. Packag. Manuf. Technol., Vol. 1, No. 9, 1438-1446, 2011.
9. Tobon, L., J. Chen, and Q. H. Liu, "Multilayer microwave filter design using a locally implicit discontinuous Galerkin finite-element time-domain (DG-FETD) method," 2011 IEEE Intl. Symposium on Antennas and Propagation (APSURSI), 2972-2975, 2011.
10. Courant, R., K. Friedrichs, and H. Lewy, "On the partial difference equations of mathematical physics," IBM. J. Res. Dev., Vol. 11, No. 2, 215-234, 1967.
11. Sun, G. and C. W. Trueman, "Unconditionally stable Crank-Nicolson scheme for solving two-dimensional Maxwell’s equations," Electron. Lett., Vol. 39, No. 7, 595-597, 2003.
12. Sun, G. and C. W. Trueman, "Unconditionally-stable FDTD method based on Crank-Nicolson scheme for solving three-dimensional Maxwell equations," Electron. Lett., Vol. 40, No. 10, 2004.
13. Sun, G. and C. W. Trueman, "Efficient implementations of the Crank-Nicolson scheme for the finite-difference time-domain method," IEEE Trans. Microw. Theory Techn., Vol. 54, No. 5, 2275-2284, 2006.
14. Yang, Y., R. S. Chen, and E. K. N. Yung, "The unconditionally stable Crank-Nicolson FDTD method for three-dimensional Maxwell’s equations," Micro. Opt. Techn. Lett., Vol. 48, No. 8, 1619-1622, 2006.
15. Chen, R. S., L. Du, Z. Ye, and Y. Yang, "An efficient algorithm for implementing the Crank-Nicolson scheme in the mixed finite-element time-domain method," IEEE Trans. Antennas Propag., Vol. 57, No. 10, 3216-3222, 2009.
16. Nedelec, J. C., "Mixed finite elements in R3," Numer. Math., Vol. 35, No. 3, 315-341, 1980.
17. Bossavit, A., "Whitney forms: A class of finite elements for three-dimensional computations in electromagnetism," Proc. Inst. Elect. Eng., Vol. 135, No. 8, 493-500, 1988.
18. Lee, J.-H., T. Xiao, and Q. H. Liu, "A 3-D spectral-element method using mixed-order curl conforming vector basis functions for electromagnetic fields," IEEE Trans. Microw. Theory Tech., Vol. 54, No. 1, 437-444, 2006.
19. Chen, J. and Q. H. Liu, "A non-spurious vector spectral element method for Maxwell’s equations," Progress In Electromagnetics Research, Vol. 96, 205-215, 2009.
20. Shankar, V., A. H. Mohammadian, and W. F. Hall, "A time-domain, finite-volume treatment for the Maxwell equations," Electromagnetics, Vol. 10, 127-145, 1990.
21. Mohammadian, A. H., V. Shankar, and W. F. Hall, "Computation of electromagnetic scattering and radiation using a time-domain finite-volume discretization procedure," Comput. Phys. Commun., Vol. 68, 175-196, 1991.
22. Liu, Q. H., "The PSTD algorithm: A time-domain method requiring only two cells per wavelength," Microw. Opt. Techn. Lett., Vol. 15, No. 3, 158-165, 1997.