Paul E. Crittenden

Fast-converging steady-state heat conduction in a rectangular parallelepiped

Abstract A Greens function approach for precisely computing the temperature and the three components of the heat flux in a rectangular parallelepiped is presented. Each face of the parallelepiped may have a different, but spatially uniform, boundary condition. Uniform volume energy generation is also treated. Three types of boundary conditions are included: type 1, a specified temperature; type 2, a specified flux; or type 3, a specified convection boundary condition. A general form of the Greens function covering all three types of boundary conditions is given. An algorithm is presented to obtain the temperature and flux at high accuracy with a minimal number of calculations for points in the interior as well as on any of the faces. Heat flux on type 1 boundaries, impossible to evaluate with traditional Fourier series, is found by factoring out lower-dimensional solutions. A numerical example is given. This research and resulting computer program was part of a code verification project for Sandia National Laboratories.

Steady-periodic heating of a cylinder

Steady periodic heating is an important experimental technique for measurement of thermal properties. In these methods the thermal properties are deduced from a systematic comparison between the data (such as temperature) and a detailed thermal model. This paper addresses steady-periodic heat transfer on cylindrical geometries with application to thermal-property measurements. The method of Greens functions is used to provide a comprehensive collection of exact analytical expressions for temperature in cylinders. Five kinds of boundary conditions are treated for one-, two-, and three-dimensional geometries. For some geometries an alternate form of the Greens function is given, which can be used for improvement of series convergence and for checking purposes to produce highly accurate numerical values. Numerical examples are given.

Design of Experiments for Thermal Characterization of Metallic Foam

Metallic foams are being investigated for possible use in the thermal protection systems of reusable launch vehicles. As a result, the performance of these materials needs to be characterized over a wide range of temperatures and pressures. In this paper a radiation/conduction model is presented for heat transfer in metallic foams. Candidates for the optimal transient experiment to determine the intrinsic properties of the model are found by two methods. First, an optimality criterion is used to nd an experiment to nd all of the parameters using one heating event. Second, a pair of heating events is used to determine the parameters in which one heating event is optimal for nding the parameters related to conduction, while the other heating event is optimal for nding the parameters associated with radiation. Simulated data containing random noise was analyzed to determine the parameters using both methods. In all cases the parameter estimates could be improved by analyzing a larger data record than suggested by the optimality criterion.

Scattering cross sections for multiple scale rough surfaces based on a unified full wave approach

The radar cross sections for random rough surfaces are evaluated. The rough surface height autocorrelation functions and the corresponding rough surface height spectral density functions are assumed in these examples to be Gaussian. The rough surface is regarded as an ensemble of patches of random rough surfaces with arbitrary orientations and the total radar cross sections are obtained by averaging over the random slopes of the ensemble of patches. The size of the patches determine the scales of roughness within each patch (or pixel) as well as the larger scale surfaces upon which the arbitrarily oriented patches lie. It is shown that the values of the full wave radar cross sections are stationary over a wide range of pixel sizes.

Backscatter cross sections of composite random rough surfaces based on the selection of variational parameters to decompose the surface height spectral density function

Scattering from composite two scale models of rough sea surfaces is analyzed. Unlike the conventional hybrid perturbation-physical optics solutions, the unified full wave approach used here is not restricted by the small Rayleigh roughness parameter nor are the two cross sections added in an ad hoc manner. The total surface height spectral density function is decomposed in a continuous smooth manner, into of a larger and a smaller scale surface height spectral density function. The variational parameter, proportional to the ratio of the mean square height (or slopes) of the larger scale surface and the mean square height (or slope) of the total surface, is increased from zero to unity. Thus, in the limit, the unified full wave solutions for the scatter cross sections reduce to the physical optics solution and the small slope, original full wave solution. It is shown that the unified full wave solutions for the scatter cross sections are stationary over a very wide range of values of the variational parameter. The slope modulation tends to significantly increase the backscatter cross sections for the horizontally polarized waves

Near-field and far-field characterization of stratified chiral structures

Full wave expression for the electromagnetic fields scattered by a rough interface between two chiral materials with laterally varying electromagnetic properties are obtained from generalized telegraphists equation for irregular media. The telegraphists equation are a set of coupled differential equations for the forward and backward wave amplitudes of the transverse components of the magnetic field and the electric field. They can be used to determine the electromagnetic near and far fields scattered above and below the interface. This has direct applications to the detection of chiral materials, the discrimination between different chiral media and the optimization of desired electromagnetic characteristics of artificial chiral materials. To derive the generalized telegraphists equations, no simplfying assumptions are made about the characteristics of the rough interface, the frequency of the source, or the locations of the source and observation points. Therefore, they provide advantageous starting points for deriving solutions to a broad variety of physical problems. In electrical engineering possible applications include integrated optic devices, polarization transformers, modulators and directional couplers. In all these applications, sub-wavelength fluctuations at the interfaces between the media can significantly affect the physical characteristics of the chiral structure. The analysis can be used in the detection, characterization and design of chiral structures consisting of complex media with engineering, biomedical, agricultural and biosecurity applications.

A modal solution for reflection and transmission at a chiral-chiral interface

In this paper a modal solution for the reflection and transmission of electromagnetic waves excited by magnetic or electric line sources, above or below an interface between two chiral materials is derived. The modal solution is found by first finding a harmonic solution using the standard Fourier transform in the lateral variable. The harmonic solution is converted into a modal solution by deforming the contour of integration for the inverse transform, in the complex plane. The complete expansion for the fields is expressed as the sum of four integrals along branch cuts and two residue contributions The wave species associated with the residue contributions and each branch cut are identified For the branch cut integrals, the species are identified through asymptotic analysis of each term of the integrand. Direct, reflected lateral and surface waves of various combinations of polarizations along each segment of their path are identified. Since there are twice as many branch cut integrals and residue contributions as there are in the achiral case, many more wave species are encountered in the chiral case.

Electromagnetic characterization of stratified chiral structures in nanotechnology

The circularly polarized wave decomposition of Maxwells equations for electromagnetic wave propagation in chiral materials is the starting point for this analysis. The Fourier transforms of the Greens functions for the electromagnetic waves on both sides of a flat interface between two semi-infinite chiral materials are derived. These harmonic solutions are expressed in terms of the characteristic right and left circularly polarized waves. Through a path deformation in the complex plane, the Greens functions are converted into alternate, modal, representations that are suitable for the complete expansion of the electromagnetic fields above and below a rough interface between two chiral materials with laterally varying material properties. From these representations, generalized Fourier teransform pairs are derived. The generalized Fourier transforms can be used to obtain two sets of coupled ordinary differential equations for the field transforms in terms of the forward and backward wave amplitudes of the transverse fields. Iterative solutions of these generalized telegraphists equations are found. From these solutions the fields can be found under appropriate assumptions. Since no a priori assumptions are made about the surface height, the frequency of the source, or the material parameter this work could be applied to nanotechnology involving stratified chiral structures.

Stationary solutions for the rough surface radar backscatter cross sections based on a two scale full wave approach

Using a full wave approach, the rough surface radar backscatter cross sections are expressed as weighted sums of two cross sections. The radar cross sections associated with the larger scale surface are reduced by a factor equal to the square of the characteristic function of the smaller scale surface (that rides upon the larger scale surface). The Bragg scatter contributions from the small scale surface are modulated by the slopes of the large scale surface. The backscatter cross sections are obtained by regarding the rough surface as an ensemble of patches of random rough surfaces with arbitrary orientations. It is shown that the full wave solutions are stationary over a wide range of patch sizes. The patches are characterized by the spectral wavenumber that separates the large scale surface from the small scale surface. The mean square heights and slopes of the large (and small) scale surfaces depend on the choice of patch sizes.

One- and two-dimensionally rough-surface radar backscatter cross section based on a stationary two-scale full-wave approach

The full wave approach is applied to one and two dimensionally rough surfaces that are characterized by Gaussian surface height probability density functions. The full wave solutions are compared with published analytical and numerical solutions for one dimensional rough surfaces. The decomposition of the rough surface into smaller and larger rough scale surfaces is not restricted by the small perturbation limitations when the two-scale full wave approach is used. Thus the mean square height of the smaller scale surface is not restricted to small values. In the small slope limit, the total rough surface is regarded as a small scale surface and the corresponding solution is given by the single scatter original full wave solution. In the high frequency limit, the total rough surface is regarded as a large scale surface and the full wave solution reduces to the physical optics solution. For the intermediate two-scale case, the radar cross sections are obtained by regarding the rough surface as an ensemble of arbitrarily oriented patches of small scale surfaces that ride upon the large scale surface. The rough surface radar cross sections are expressed as weighted sums of two cross sections. It is shown that the full wave solutions are stationary over a wide range of patch sizes.