Robust Computation of Dipole Electromagnetic Fields in Arbitrarily-Anisotropic, Planar-Stratified Environments

We develop a general-purpose formulation, based on two-dimensional spectral integrals, for computing electromagnetic fields produced by arbitrarily-oriented dipoles in planar-stratified environments, where each layer may exhibit arbitrary and independent anisotropy in both the (complex) permittivity and permeability. Among the salient features of our formulation are (1) computation of eigenmodes (characteristic plane waves) supported in arbitrarily anisotropic media in a numerically robust fashion, (2) implementation of an hp-adaptive refinement for the numerical integration to evaluate the radiation and weakly-evanescent spectra contributions, and (3) development of an adaptive extension of an integral convergence acceleration technique to compute the strongly-evanescent spectrum contribution. While other semianalytic techniques exist to solve this problem, none have full applicability to media exhibiting arbitrary double anisotropies in each layer, where one must account for the whole range of possible phenomena such as mode coupling at interfaces and non-reciprocal mode propagation. Brute-force numerical methods can tackle this problem but only at a much higher computational cost. The present formulation provides an efficient and robust technique for field computation in arbitrary planar-stratified environments. We demonstrate the formulation for a number of problems related to geophysical exploration.Comment: 39 pages, 17 figure

Polygons with Parallel Opposite Sides

In this paper we consider planar polygons with parallel opposite sides. This type of polygons can be regarded as discretizations of closed convex planar curves by taking tangent lines at samples with pairwise parallel tangents. For this class of polygons, we define discrete versions of the area evolute, central symmetry set, equidistants and area parallels and show that they behave quite similarly to their smooth counterparts.Comment: 17 pages, 11 figure

Stable pseudoanalytical computation of electromagnetic fields from arbitrarily-oriented dipoles in cylindrically stratified media

Computation of electromagnetic fields due to point sources (Hertzian dipoles) in cylindrically stratified media is a classical problem for which analytical expressions of the associated tensor Green's function have been long known. However, under finite-precision arithmetic, direct numerical computations based on the application of such analytical (canonical) expressions invariably lead to underflow and overflow problems related to the poor scaling of the eigenfunctions (cylindrical Bessel and Hankel functions) for extreme arguments and/or high-order, as well as convergence problems related to the numerical integration over the spectral wavenumber and to the truncation of the infinite series over the azimuth mode number. These problems are exacerbated when a disparate range of values is to be considered for the layers' thicknesses and material properties (resistivities, permittivities, and permeabilities), the transverse and longitudinal distances between source and observation points, as well as the source frequency. To overcome these challenges in a systematic fashion, we introduce herein different sets of range-conditioned, modified cylindrical functions (in lieu of standard cylindrical eigenfunctions), each associated with non-overlapped subdomains of (numerical) evaluation to allow for stable computations under any range of physical parameters. In addition adaptively-chosen integration contours are employed in the complex spectral wavenumber plane to ensure convergent numerical integration in all cases. We illustrate the application of the algorithm to problems of geophysical interest involving layer resistivities ranging from 1000 $\Omega \cdot$m to 10$^{-8} \Omega \cdot$m, frequencies of operation ranging from 10 MHz down to the low magnetotelluric range of 0.01 Hz, and for various combinations of layer thicknesses.Comment: 33 pages, 23 figures. This v2 is slightly condensed and has some material moved to the Appendice

Affine Properties of Convex Equal-Area Polygons

In this paper we discuss some affine properties of convex equal-area polygons, which are convex polygons such that all triangles formed by three consecutive vertices have the same area. Besides being able to approximate closed convex smooth curves almost uniformly with respect to affine length, convex equal-area polygons admit natural definitions of the usual affine differential geometry concepts, like affine normal and affine curvature. These definitions lead to discrete analogous of the six vertices theorem and an affine isoperimetric inequality. One can also define discrete counterparts of the affine evolute, parallels and the affine distance symmetry set preserving many of the properties valid for smooth curves.Comment: 16 pages, 10 figure

Schr\"odinger formalism for a particle constrained to a surface in R13\mathbb{R}_1^3

In this work it is studied the Schr\"odinger equation for a non-relativistic particle restricted to move on a surface $S$ in a three-dimensional Minkowskian medium $\mathbb{R}_1^3$, i.e., the space $\mathbb{R}^3$ equipped with the metric $\text{diag}(-1,1,1)$. After establishing the consistency of the interpretative postulates for the new Schr\"odinger equation, namely the conservation of probability and the hermiticity of the new Hamiltonian built out of the Laplacian in $\mathbb{R}_1^3$, we investigate the confining potential formalism in the new effective geometry. Like in the well-known Euclidean case, it is found a geometry-induced potential acting on the dynamics $V_S = - \frac{\hbar^{2}}{2m} \left(\varepsilon H^2-K\right)$ which, besides the usual dependence on the mean ($H$) and Gaussian ($K$) curvatures of the surface, has the remarkable feature of a dependence on the signature of the induced metric of the surface: $\varepsilon= +1$ if the signature is $(-,+)$, and $\varepsilon=1$ if the signature is $(+,+)$. Applications to surfaces of revolution in $\mathbb{R}^3_1$ are examined, and we provide examples where the Schr\"odinger equation is exactly solvable. It is hoped that our formalism will prove useful in the modeling of novel materials such as hyperbolic metamaterials, which are characterized by a hyperbolic dispersion relation, in contrast to the usual spherical (elliptic) dispersion typically found in conventional materials.Comment: 26 pages, 1 figure; comments are welcom

Parâmetros físicos e químicos da laranja pera na região de Manaus, AM.

bitstream/item/32012/1/CPATU-BP109.pd