A nonlinear hybrid FDTD/FETD technique based on the parametric quadratic programming method is developed for Maxwell's equations with nonlinear media. The proposed technique allows nonconforming meshes between nonlinear FETD and linear FDTD subdomains. The coarse structured cells of FDTD are used in relatively simple structures with linear media, whereas fine unstructured elements of FETD based on the parametric quadratic programming method are used to simulate complicated structures with nonlinear media. This hybrid technique is particularly suitable for structures with small nonlinear regions in an otherwise linear medium. Numerical results demonstrate the validity of the proposed method.
1. Yee, K. S., "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Transactions on Antennas and Propagation, Vol. 14, 302-307, 1966. doi:10.1109/TAP.1966.1138693
2. Taflove, A., Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House, Norwood, MA, 1995.
3. Kunz, K. S. and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics, CRC, Boca Raton, FL, 1993.
4. Joseph, R. M. and A. Taflove, "Spatial soliton deflection mechanism indicated by FDTD Maxwell's equations modeling," IEEE Photonics Technology Letters, Vol. 6, 1251-1254, 1994. doi:10.1109/68.329654
5. Van, V. and S. K. Chaudhuri, "A hybrid implicit-explicit FDTD scheme for nonlinear optical waveguide modeling," IEEE Transactions on Microwave Theory and Techniques, Vol. 47, 540-545, 1999. doi:10.1109/22.763152
6. Joseph, R. M. and A. Taflove, "FDTD Maxwell's equations models for nonlinear electrodynamics and optics," IEEE Transactions on Antennas and Propagation, Vol. 45, 364-374, 1997. doi:10.1109/8.558652
7. Ziolkowski, R., "Full-wave vector Maxwell equation modeling of the self-focusing of ultrashort optical pulses in a nonlinear kerr medium exhibiting a finite response time," Journal of the Optical Society of America B, Vol. 10, 186-198, 1993. doi:10.1364/JOSAB.10.000186
8. Fisher, A., D. White, and G. Rodrigue, "An efficient vector finite element method for nonlinear electromagnetic modeling," Journal of Computational Physics, Vol. 225, 1331-1346, 2007. doi:10.1016/j.jcp.2007.01.031
9. Chen, J., Q. H. Liu, M. Chai, and J. A. Mix, "A nonspurious 3-D vector discontinuous Galerkin finite-element time-domain method," IEEE Microwave and Wireless Components Letters, Vol. 20, 1-3, 2010. doi:10.1109/LMWC.2009.2035941
10. Zhu, B., J. Chen, W. Zhong, and Q. H. Liu, "A hybrid FETD-FDTD method with nonconforming meshes," Communications in Computational Physics, Vol. 9, 828-842, 2011. doi:10.4208/cicp.230909.140410s
11. Tobon, L. E., Q. Ren, Q. T. Sun, J. Chen, and Q. H. Liu, "New efficient implicit time integration method for DGTD applied to sequential multidomain and multiscale problems," Progress In Electromagnetics Research, Vol. 151, 1-8, 2015. doi:10.2528/PIER14112201
12. Zhu, B., H. Yang, and J. Chen, "A novel finite element time domain method for nonlinear Maxwell's equations based on the parametric quadratic programming method," Microwave and Optical Technology Letters, Vol. 57, 1640-1645, 2015. doi:10.1002/mop.29170
13. Cottle, R. W. and G. B. Dantzig, "Complementary pivot theory of mathematical programming," Linear Algebra Applications, Vol. 1, 103-125, 1982. doi:10.1016/0024-3795(68)90052-9
14. Ferris, M. C. and J. S. Pang, "Engineering and economic applications of complementarity problems," SIAM Review, Vol. 39, 669-713, 1997. doi:10.1137/S0036144595285963
15. Zhang, H. W., S. Y. He, and X. S. Li, "Two aggregate-function-based algorithms for analysis of 3D frictional contact by linear complementarity problem formulation," Computer Methods in Applied Mechanics and Engineering, Vol. 194, 5139-5158, 2005. doi:10.1016/j.cma.2005.01.002
17. Lu, T., P. Zhang, and W. Cai, "Discontinuous Galerkin methods for dispersive and lossy Maxwell equations and PML boundary conditions," Journal of Computational Physics, Vol. 200, 549-580, 2004. doi:10.1016/j.jcp.2004.02.022
18. Lu, T., W. Cai, and P. Zhang, "Discontinuous Galerkin time-domain method for GPR simulation in dispersive media," IEEE Transactions on Geoscience and Remote Sensing, Vol. 43, 72-80, 2005. doi:10.1109/TGRS.2004.838350
19. Mohammadian, A. H., V. Shankar, and W. F. Hall, "Computation of electromagnetic scattering and radiation using a time-domain finite-volume discretization procedure," Computer Physics Communications, Vol. 68, 175-196, 1991. doi:10.1016/0010-4655(91)90199-U
20. Zhu, B., J. Chen, W. Zhong, and Q. H. Liu, "A hybrid finite-element/finite-difference method with an implicit-explicit time-stepping scheme for Maxwell's equations," International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, Vol. 25, 495-506, 2012. doi:10.1002/jnm.1853
21. Luo, M. and Q. H. Liu, "Spectral element method for band structures of three-dimensional anisotropic photonic crystals," Physical Review E, Vol. 80, 1-7, 2009.
22. Fan, G. and Q. H. Liu, "A strongly well-posed PML in lossy media," IEEE Antennas and Wireless Propagation Letters, Vol. 2, 97-100, 2003. doi:10.1109/LAWP.2003.814776