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Stable 3D FDTD method for arbitrary fully electric and magnetic anisotropic Maxwell's equations
Author(s) -
Nehls John,
Dineen Colm,
Liu Jinjie,
Poole Cody,
Brio Moysey,
Moloney Jerome V.
Publication year - 2018
Publication title -
international journal of numerical modelling: electronic networks, devices and fields
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.249
H-Index - 30
eISSN - 1099-1204
pISSN - 0894-3370
DOI - 10.1002/jnm.2521
Subject(s) - maxwell's equations , finite difference time domain method , piecewise , solver , electromagnetic field solver , physics , boundary value problem , mathematical analysis , anisotropy , eigenvalues and eigenvectors , electric field , classical mechanics , mathematics , optics , optical field , mathematical optimization , inhomogeneous electromagnetic wave equation , quantum mechanics
We have developed a new fully anisotropic 3D FDTD Maxwell solver for arbitrary electrically and magnetically anisotropic media for piecewise constant electric and magnetic materials that are co‐located over the primary computational cells. Two numerical methods were developed that are called nonaveraged and averaged methods, respectively. The nonaveraged method is first‐order accurate, while the averaged method is second‐order accurate for smoothly‐varying materials and reduces to first order for discontinuous material distributions. For the standard FDTD field locations with the co‐location of the electric and magnetic materials at the primary computational cells, the averaged method required development of the different inversion algorithms of the constitutive relations for the electric and magnetic fields. We provide a mathematically rigorous stability proof followed by extensive numerical testing that includes long‐time integration, eigenvalue analysis, tests with extreme, randomly placed material parameters, and various boundary conditions. For accuracy evaluation, we have constructed a test case with an explicit analytic solution. Using transformation optics, we have constructed complex, spatially inhomogeneous geometrical object with fully‐anisotropic materials and a large dynamic range of ε _ and μ _ , such that a plane wave incident on the object is perfectly reconstructed downstream.