Papers by Fernando L Teixeira
We discuss the ab initio rendering of four-dimensional (4-d) spacetime of Maxwell's equations on ... more We discuss the ab initio rendering of four-dimensional (4-d) spacetime of Maxwell's equations on random (irregular) lattices. This rendering is based on casting Maxwell's equations in the framework of the exterior calculus of differential forms, and a translation thereof to a simplicial complex whereby fields and causative sources are represented as differential p-forms and paired with the oriented p-dimensional geometrical objects that comprise the set of spacetime lattice cells (simplices). We pay particular attention to the case of simplicial spacetime lattices because these can serve as building blocks of lattices made of more generic cells (polygons). The generalized Stokes' theorem is used to construct a discrete calculus operations on the lattice based upon combinatorial relations depending solely on the connectivity and relative orientation among simplices. This rendering provides a natural factorization of (lattice) 4-d spacetime Maxwell's equations into a metric-free part and a metric-dependent part. The latter is encoded by discrete Hodge star operators which are built using Whitney forms, i.e. canonical interpolants for discrete differential forms. The derivation of Whitney forms is illustrated here using a geometrical construction based on the concept of barycentric coordinates to represent a point on a simplex, and its generalization to the representation of higher-dimensional objects (lines, areas, volumes, and hypervolumes) in 4-d. We stress the role of the primal lattice, the barycentric dual lattice, and the barycentric decomposition lattice in achieving a complete description of the lattice theory. Lattice Maxwell's equations based on the exterior calculus of differential forms and on the use of Whitney forms inherits the symplectic structure and discrete analogues of conservation laws present in the continuum theory, such as energy and charge conservation. It also provides precise localization rules for the different fields and sources on the lattice, and design principles for constructing consistent numerical solution methods that are free from spurious modes, spectral pollution, and (unconditional) numerical instabilities. We also briefly consider the relationship between lattice 4-d Maxwell's equations and some incarnations of discretization schemes for Maxwell's equations in (3+1)-d, such as finite-differences and finite-elements.
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…, Jan 1, 2010
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JOSA B, Jan 1, 2011
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IEEE Transactions on Antennas and Propagation, 2015
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We derive the key expressions to robustly address the eigenfunction expansion-based analysis of e... more We derive the key expressions to robustly address the eigenfunction expansion-based analysis of electromagnetic (EM) fields produced by current sources within planar non-birefringent anisotropic medium (NBAM) layers. In NBAM, the highly symmetric permeability and permittivity tensors can induce directionally-dependent, but polarization independent, propagation properties supporting "degenerate" characteristic polarizations, i.e. four linearly-independent eigenvectors associated with only two (rather than four) unique, non-defective eigenvalues. We first explain problems that can arise when the source(s) specifically reside within NBAM planar layers when using canonical field expressions. To remedy these problems, we exhibit alternative spectral-domain field expressions, immune to such problems, that form the foundation for a robust eigenfunction expansion-based analysis of time-harmonic EM radiation and scattering within such type of planar-layered media. Numerical results demonstrate the high accuracy and stability achievable using this algorithm.
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Ieee Geosci Remote Sens Lett, 2010
ABSTRACT
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Journal of Lightwave Technology, Sep 1, 2007
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Journal of Electromagnetic Waves and Applications, 1999
Two classes of formulations are prevalent on the perfectly matched layer (PML) concept for the re... more Two classes of formulations are prevalent on the perfectly matched layer (PML) concept for the reflectionless absorption of electromagnetic waves. In the first, additional degrees of freedom modify the curl and div operators of Maxwell's equations. This results in the so-called non-Maxwellian PML. The original Berenger formulation belongs to this class. The non-Maxwellian PML can be systematically derived by an
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We describe a charge-conserving scatter-gather algorithm for particle-in-cell simulations on unst... more We describe a charge-conserving scatter-gather algorithm for particle-in-cell simulations on unstructured grids. Charge conservation is obtained from first principles, i.e., without the need for any post-processing or correction steps. This algorithm recovers, at a fundamental level, the scatter-gather algorithms presented recently by Campos-Pinto et al. [1] (to first-order) and by Squire et al. [2], but it is derived here in a streamlined fashion from a geometric viewpoint. Some ingredients reflecting this viewpoint are (1) the use of (discrete) differential forms of various degrees to represent fields, currents, and charged particles and provide localization rules for the degrees of freedom thereof on the various grid elements (nodes, edges, facets), (2) use of Whitney forms as basic interpolants from discrete differential forms to continuum space, and (3) use of a Galerkin formula for the discrete Hodge star operators (i.e., "mass matrices" incorporating the metric datum of the grid) applicable to generally irregular, unstructured grids. The expressions obtained for the scatter charges and scatter currents are very concise and do not involve numerical quadrature rules. Appropriate fractional areas within each grid element are identified that represent scatter charges and scatter currents within the element, and a simple geometric representation for the (exact) charge conservation mechanism is obtained by such identification. The field update is based on the coupled first-order Maxwell's curl equations to avoid spurious modes with secular growth (otherwise present in formulations that discretize the second-order wave equation). Examples are provided to verify preservation of discrete Gauss' law for all times.
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ABSTRACT
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Journal of Lightwave Technology, Oct 1, 2009
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Proceedings of the 5th European Conference on Antennas and Propagation, 2011
Page 1. Imaging and Tracking of Targets in Clutter Using Differential Time-Reversal Ahmed E. Foud... more Page 1. Imaging and Tracking of Targets in Clutter Using Differential Time-Reversal Ahmed E. Fouda∗, Fernando L .Teixeira∗ and Mehmet E. Yavuz ∗ ElectroScience Laboratory, Dept. of Electrical and Computer Engineering ...
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Papers by Fernando L Teixeira
A finite-difference time-domain (FDTD) algorithm in Cartesian coordinates which combines the PML ABC with piecewise-linear recursive convolution (PLRC) is proposed and implemented, allowing the simulation of electromagnetic fields in inhomogeneous and dispersive media with conductive loss. Two PML-PLRC-FDTD algorithms in cylindrical coordinates are also proposed and implemented. The first is developed through a split-field PML formulation, and the second through a Maxwellian ( unsplit) PML formulation. A comparison is made between numerical properties of these two algorithms.
The PML concept is then studied within the language of differential forms to unify the various PML formulations. Finally, the language of differential forms is also utilized to provide a coordinate-free description and analyze consistency properties of the electromagnetic theory on lattice for PDE solvers such as the finite-difference, finite-volume or finite-element methods.