A perfectly matched layer for the absorption of electromagnetic waves

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

A modified set of Maxwell's equations is presented that includes complex coordinate stretching along the three Cartesian coordinates. The added degrees of freedom in the modified Maxwell's equations allow the specification of absorbing boundaries with zero reflection at all angles of incidence and all frequencies. The modified equations are also related to the perfectly matched layer that was presented recently for 2D wave propagation. Absorbing-material boundary conditions are of particular interest for finite-difference time-domain (FDTD) computations on a single-instruction multiple-data (SIMD) massively parallel supercomputer. A 3D FDTD algorithm has been developed on a connection machine CM-5 based on the modified Maxwell's equations and simulation results are presented to validate the approach. © 1994 John Wiley & Sons, Inc.

A 3D perfectly matched medium from modified maxwell's equations with stretched coordinates

The potential for sea ice-albedo feedback to give rise to nonlinear climate change in the Arctic Ocean – defined as a nonlinear relationship between polar and global temperature change or, equivalently, a time-varying polar amplification – is explored in IPCC AR4 climate models. Five models supplying SRES A1B ensembles for the 21 st century are examined and very linear relationships are found between polar and global temperatures (indicating linear Arctic Ocean climate change), and between polar temperature and albedo (the potential source of nonlinearity). Two of the climate models have Arctic Ocean simulations that become annually sea ice-free under the stronger CO 2 increase to quadrupling forcing. Both of these runs show increases in polar amplification at polar temperatures above-5 o C and one exhibits heat budget changes that are consistent with the small ice cap instability of simple energy balance models. Both models show linear warming up to a polar temperature of-5 o C, well above the disappearance of their September ice covers at about-9 o C. Below-5 o C, surface albedo decreases smoothly as reductions move, progressively, to earlier parts of the sunlit period. Atmospheric heat transport exerts a strong cooling effect during the transition to annually ice-free conditions. Specialized experiments with atmosphere and coupled models show that the main damping mechanism for sea ice region surface temperature is reduced upward heat flux through the adjacent ice-free oceans resulting in reduced atmospheric heat transport into the region.

Geophysical fluid dynamics.

Prior to abour 1990, the modeling of electromagnetic engineering systems was primarily implemented using solution techniques for the sinusoidal steady-state Maxwell's equations. Before about 1960, the principal approaches in this area involved closed-form and infinite-series analytical solutions, with numerical results from these analyses obtained using mechanical calculators. After 1960, the increasing availability of programmable electronic digital computers permitted such frequency-domain approaches to rise markedly in sophistication. Researchers were able to take advantage of the capabilities afforded by powerful new high-level programming languages such as Fortran, rapid random-access storage of large arrags of numbers, and computational speeds that were orders of magnitude faster than possible with mechanical calculators. In this period, the principal computational approaches for Maxwell's equations included the high-frequency asymptotic methods of Keller (1962) as well as Kouyoumjian and Pathak (1974) and the integral equation techniques of Harrington (1968) .

9 – Computational Electromagnetics: The Finite-Difference Time-Domain Method

Construction des conditions aux limites absorbantes pour l'equation des ondes multidimensionnelle a l'aide d'approximations aux differences. Formulation sous forme canonique des conditions aux limites absorbantes analytiques

/pdf/absorbing-boundary-conditions-for-difference-approximations-2jxlhqi64w.pdf

Absorbing boundary conditions for difference approximations to the multi-dimensional wave equation

On developpe une theorie des approximations par differences des conditions aux limites absorbantes pour l'equation d'ondes scalaire a plusieurs dimensions d'espace

/pdf/numerical-absorbing-boundary-conditions-for-the-wave-3i9e7unj58.pdf

Numerical absorbing boundary conditions for the wave equation

Absorbing boundary conditions are needed for computing numerical models of wave motions in unbounded spatial domains. The boundary conditions developed here for elastic waves are generalizations of ones developed earlier for acoustic waves. These conditions are based on compositions of simple first-order differential operators. The formulas can be applied without modification to problems in both two and three dimensions. The boundary conditions are stable for all values of the ratio of P-wave velocity to S-wave velocity, and they are effective near a free surface and in a horizontally stratified medium. The boundary conditions are approximated with simple finite-difference equations that use values of the solution only along grid lines perpendicular to the boundary. This property facilitates implementation, especially near a free surface and at other corners of the computational domain.

Absorbing boundary conditions for elastic waves

In this paper, a class of radiation boundary conditions for two-and three-dimensional elastic wave propagation is developed. The boundary conditions are based on compositions of simple first-order operators.Some recent empirical investigations have suggested that certain earlier radiation boundary conditions may be unstable if the ratio of P-wave velocity to S-wave velocity is sufficiently large. The boundary conditions developed in the present paper are stable for all values of the velocity ratio. The boundary operators used here are defined by relatively simple formulas that apply without modification to problems in both two and three dimensions. Given an arbitrary angle of incidence, some parameters in the boundary conditions can be adjusted to make the conditions perfectly absorbing for P- waves and/or S-waves traveling at that angle of incidence to the boundary.

Radiation boundary conditions for elastic wave propagation

The author develops radiation boundary conditions for the numerical modeling of dispersive waves. During the construction of such boundary conditions, the goal is to simulate the outward radiation of waves near an artificial computational boundary. The velocities of the outgoing waves are typically involved in processes of this nature. A central problem in the dispersive case is that two different types of velocities are present, phase velocity and group velocity, and each can vary with wavenumber and frequency. With the boundary conditions developed in this paper, the user needs to specify some parameters; in the cases that are emphasized here, the parameters can be interpreted in terms of phase velocities of waves that are absorbed exactly at the computational boundary. The amount of reflected error is a continuous function of the parameters, and the performance of the boundary conditions is not sensitive to the choice of parameters. Good performance is obtained in numerical tests involving data that ha...

Robert L. Higdon

Papers

Absorbing boundary conditions for difference approximations to the multi-dimensional wave equation

Numerical absorbing boundary conditions for the wave equation

Absorbing boundary conditions for elastic waves

Radiation boundary conditions for elastic wave propagation

Radiation boundary conditions for dispersive waves