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With increasing miniaturization of diagnostic devices for more effective detection of blood-borne pathogens for example, Poiseuille molecular flow in micro channels has become increasingly relevant. Since continuum mechanics no longer applies for Poiseuille flow when the Knudson number is near or larger than unity, kinetic theory is required to capture the microscopic molecular scattering responsible for channel molecular flow and the velocity profile across a channel. Here, we apply a response matrix solution to the 1D Poiseuille flow with a BGK approximation featuring simplicity with precision by following a consistent numerical formulation leading to high precision, 8-place benchmarks., Comment: 28 pgs 1 fig 8 tables

As more and more numerical and analytical solutions to the linear neutron transport equation become available, verification of numerical results is increasingly important. This presentation concerns the development of another benchmark for the linear neutron transport equation. There are numerous ways of solving the transport equation, such as the Wiener-Hopf method based on analyticity, method of singular eigenfunctions, Laplace and Fourier transforms and ana-lytical discrete ordinates, which is arguably one of the most straightforward, to name a few. Another potential method is the PN method, where the solution is expanded in terms of full range orthogonal Legendre polynomials and with orthogonality and truncation, the moments form a set of second order ODEs. Because of the half-range boundary conditions for incoming particles however, full range Legendre expansions are inaccurate near material discontinuities. For this reason, a double PN (DPN) expansion is more appropriate, where the incoming and exiting flux distributions are expanded separately to preserve the discontinuity at material interfaces. Here, a new method of solution for the DPN equations is proposed and demonstrated for an isotropically scattering medium., Comment: 19 pgs 2 tablws 1 figure

Extreme benchmarks of ten or more places for the point kinetics equations for time dependent nuclear reactor power transients are rare. Therefore, to establish an extreme benchmark, we will employ a Taylor series with continuous analytical continuation (CAC) to solve the ordinary differential equations of point kinetics including feedback. Non-linear Wynn-epsilon convergence acceleration confirms the highly precise solutions for neutron and precursor densities. Through adaptive partitioning of time intervals, the proposed Converged Accelerated Taylor Series, or CATS algorithm [1] in double precision, automatically performs successive mesh refinement to obtain high precision initial conditions for each sub-interval, with the intent to reduce propagation error. Confirmation of ten to twelve places comes from comparison to the BEFD (Backward Euler Finite Difference) algorithm [2] in quadruple precision also developed by the author. We report benchmark results for common cases found in the literature including step, ramp, zigzag and sinusoidal prescribed reactivity insertions and insertions with non-linear adiabatic Doppler feedback. We also establish a suite of new prescribed reactivity insertions and insertions with feedback, based on reactivities with Taylor series as suggested by the CATS algorithm., Comment: 60 pgs

The convergence of Boltzmann Fokker Planck solution can become arbitrarily slow with iterative procedures like source iteration. This paper derives and investigates a nonlinear diffusion acceleration scheme for the solution of the Boltzmann Fokker Planck equation in slab geometry. This method is a conventional high order low order scheme with a traditional diffusion-plus-drift low-order system. The method, however, differs from the earlier variants as the definition of the low order equation, which is adjusted according to the zeroth and first moments of the Boltzmann Fokker Planck equation. For the problems considered, we observe that the NDA-accelerated solution follows the unaccelerated well and provides roughly an order of magnitude savings in iteration count and runtime compared to source iteration., Comment: This draft has been submitted to Physor 2024 for review

Here, we are concerned with a new, highly precise, numerical solution to the 1D neutron transport equation based on Cases analytical singular eigenfunction expansion (SEE). While a considerable number numerical solutions currently exist, understandably, because of its complexity, even in 1D, there are only a few truly analytical solutions to the neutron transport equation. In 1960, Case introduced a consistent theory of the SEE for a variety of idealized transport problems and forever changed the landscape of analytical transport theory. Several numerical methods including the FN method were based on the theory. What is presented is yet another, called the Lagrange order N method (LNM), featuring the simplicity and precision of the FN method, but for a more convenient and natural Lagrangian polynomial basis., Comment: 22 pgs 4 tables

In recent years, the first author has developed three successful numerical methods to solve the 1D radiative transport equation yielding highly precise benchmarks. The second author has shown a keen interest in novel solution methodologies and an ability for their implementation. Here, we combine talents to generate yet another high precision solution, the Matrix Riccati Equation Method (MREM). MREM features the solution to two of the four matrix Riccati ODEs that arise from the interaction principle of particle transport. Through interaction coefficients, the interaction principle describes how particles reflect from- and transmit through- a single slab. On combination with Taylor series and doubling, a high quality numerical benchmark, to nearly seven places, is established., Comment: 31 pages

The analytical solution of neutron transport equation has fascinated mathematicians and physicists alike since the Milne half-space problem was introduce in 1921 [1]. Numerous numerical solutions exist, but understandably, there are only a few analytical solutions, with the prominent one being the singular eigenfunction expansion (SEE) introduced by Case [2] in 1960. For the half-space, the method, though yielding, an elegant analytical form resulting from half-range completeness, requires numerical evaluation of complicated integrals. In addition, one finds closed form analytical expressions only for the infinite medium and half-space cases. One can find the flux in a slab only iteratively. That is to say, in general one must expend a considerable numerical effort to get highly precise benchmarks from SEE. As a result, investigators have devised alternative methods, such as the CN [3], FN [4] and Greens Function Method (GFM) [5] based on the SEE have been devised. These methods take the SEE at their core and construct a numerical method around the analytical form. The FN method in particular has been most successful in generating highly precise benchmarks. No method yielding a precise numerical solution has yet been based solely on a fundamental discretization until now. Here, we show for the albedo problem with a source on the vacuum boundary of a homogeneous medium, a precise numerical solution is possible via Lagrange interpolation over a discrete set of directions., Comment: None

We develop a multiphysics-based model to predict the response of localized tumors to combined-hyperthermia-radiotherapy (CHR) treatment. This procedure combines hyperthermia (tumor heating) with standard radiotherapy to improve efficacy of the overall treatment. In addition to directly killing tumor cells, tumor heating amends several parameters within the tumor microenvironment. This leads to radiosensitization, which improves the performance of radiotherapy while reducing the side-effects of excess radiation in the surrounding normal tissue. Existing tools to model this kind of treatment consider each of the physics separately. The model presented in this paper accounts for the synergy between hyperthermia and radiotherapy providing a more realistic and holistic approach to simulate CHR treatment. Our model couples radiation transport and heat-transfer with cell population dynamics., Comment: 10 pages, 3 figures, submitted to ANS Topical Meeting

Particle transport through a duct by Lambertian reflection from duct walls is again considered. This popular transport example has been solved by most numerical transport methods except notably one- the method of doubling. We shall show that the method of doubling provides every bit as, or more, accurate reflectances and transmittances as the numerical discrete ordinates (NDO) and analytical discrete ordinates (ADO) methods with less mathematical and numerical effort., Comment: 39 pages

Nearly 45 years ago, Ken Case published his seminal paper on the singular eigenfunction solution for the Green's function of the monoenergetic neutron transport equation with isotropic scattering. Previously, the solution had been obtained by Fourier transform. While it is apparent the two had to be equivalent, a convincing equivalence proof for general anisotropic scattering remained a challenge until now., Comment: 34 pages,2 figs, to be submitted to either Nucl. sci & Eng or J. Math Phys

Proteinaceous aggregation occurs through self-assembly-- a process not entirely understood. In a recent article [1], an analytical theory for amyloid fibril growth via secondary rather than primary nucleation was presented. Remarkably, with only a single kinetic parameter, the authors were able to unify growth characteristics for a variety of experimental data. In essence, they seem to have uncovered the underlying allometric laws governing the evolution of filament elongation simply from two coupled non-linear ordinary differential equations (ODEs) stemming from a master equation. While this work adds significantly to our understanding of filament self-assembly, it required an approximate analytical solution representation. Here, we show that the same results are found by purely numerical means once a straightforward and reliable numerical solution to the set of ODEs has been established., Comment: 28 pages,8 figures

To explicate the sources of uncertainty in the prediction of biophysical variables over space, consider the general equation: where z is a variable with values on some nominal, ordinal, interval or ratio scale; y is a vector of input variables; u is the spatial support of y and z ; x and u are the spatial locations of y and z , respectively; f is a model and B is the vector of the parameters of this model. Any y or z has a value and a spatial extent which is called its support. Viewed in this way, categories of uncertainty are from variable (e.g. measurement), parameter, positional. support and model (e.g. structural) sources. The prediction of Leaf Area Index (LAI) and the fraction of absorbed photosynthetically active radiation (fPAR) are examples of z variables predicted using model(s) as a function of y variables and spatially constant parameters. The MOD15 algorithm is an example of f, called f(sub 1), with parameters including those defined by one of six biome types and solar and view angles. The Leaf Canopy Model (LCM)2, a nested model that combines leaf radiative transfer with a full canopy reflectance model through the phase function, is a simpler though similar radiative transfer approach to f(sub 1). In a previous study, MOD15 and LCM2 gave similar results for the broadleaf forest biome. Differences between these two models can be used to consider the structural uncertainty in prediction results. In an effort to quantify each of the five sources of uncertainty and rank their relative importance for the LAI/fPAR prediction problem, we used recent data for an EOS Core Validation Site in the broadleaf biome with coincident surface reflectance, vegetation index, fPAR and LAI products from the Moderate Resolution Imaging Spectrometer (MODIS). Uncertainty due to support on the input reflectance variable was characterized using Landsat ETM+ data. Input uncertainties were propagated through the LCM2 model and compared with published uncertainties from the MOD15 algorithm.

An important contribution to coral reef research is to improve spectral distinction between various health states of coral species in areas subject to harmful anthropogenic activity and climate change. New insights into radiative transfer properties of corals under healthy and stressed conditions can advance understandings of ecological processes on reefs and allow better assessments of the impacts of large-scale bleaching and disease events, Our objective was to examine the spectral and spatial properties of hyperspectral sensors that may be used to remotely sense changes in reef community health. We compare in situ reef environment spectra (healthy coral, stressed coral, dead coral, algae, and sand) with airborne hyperspectral data to identify important spectral characteristics and indices. Additionally, spectral measurements over a range of water depths, relief, and bottom types are compared to help quantify bottom-water column influences. In situ spectra were collected in July and August 2002 at the Long Rock site in the Andros Island, Bahamas coastal zone coral reef. Our primary emphasis was on Acropora palmata (or elkhorn coral), a major reef building coral, which is prevalent in the study area, but is suffering from white band disease. A. palmata is currently being, proposed as an endangered species because its populations have severely declined in many areas of the Caribbean. In addition to the A. palmata biotope, we have collected spectra of at least seven other coral biotopes that exist within the study area, each with different coral community composition, density of corals, relief, and size of corals. Coral spectral reflectance was then input into a radiative transfer model, CORALMOD (CM1), which is based on a leaf radiative transfer model. In CM1, input coral reflectance measurements produce modeled reflectance through an inversion at each visible wavelength to provide the absorption spectrum. Initially, we imposed a scattering baseline that is the same regardless of the coral condition and that coral is optically thick and no light is transmitted through coral. Here we will focus on methodology, experimental design, and initial findings of the in situ spectral measurements and preliminary output from the radiative transfer model.

Transport of low energy neutrons associated with the galactic cosmic ray cascade is analyzed in this dissertation. A benchmark quality analytical algorithm is demonstrated for use with BRYNTRN, a computer program written by the High Energy Physics Division of NASA Langley Research Center, which is used to design and analyze shielding against the radiation created by the cascade. BRYNTRN uses numerical methods to solve the integral transport equations for baryons with the straight-ahead approximation, and numerical and empirical methods to generate the interaction probabilities. The straight-ahead approximation is adequate for charged particles, but not for neutrons. As NASA Langley improves BRYNTRN to include low energy neutrons, a benchmark quality solution is needed for comparison. The neutron transport algorithm demonstrated in this dissertation uses the closed-form Green's function solution to the galactic cosmic ray cascade transport equations to generate a source of neutrons. A basis function expansion for finite heterogeneous and semi-infinite homogeneous slabs with multiple energy groups and isotropic scattering is used to generate neutron fluxes resulting from the cascade. This method, called the FN method, is used to solve the neutral particle linear Boltzmann transport equation. As a demonstration of the algorithm coded in the programs MGSLAB and MGSEMI, neutron and ion fluxes are shown for a beam of fluorine ions at 1000 MeV per nucleon incident on semi-infinite and finite aluminum slabs. Also, to demonstrate that the shielding effectiveness against the radiation from the galactic cosmic ray cascade is not directly proportional to shield thickness, a graph of transmitted total neutron scalar flux versus slab thickness is shown. A simple model based on the nuclear liquid drop assumption is used to generate cross sections for the galactic cosmic ray cascade. The ENDF/B V database is used to generate the total and scattering cross sections for neutrons in aluminum. As an external verification, the results from MGSLAB and MGSEMI were compared to ANISN/PC, a routinely used neutron transport code, showing excellent agreement. In an application to an aluminum shield, the FN method seems to generate reasonable results.

Nontrivial benchmark solutions are developed for the galactic heavy ion transport equations in the straightahead approximation with energy and spatial coupling. Analytical representations of the ion fluxes are obtained for a variety of sources with the assumption that the nuclear interaction parameters are energy independent. The method utilizes an analytical LaPlace transform inversion to yield a closed form representation that is computationally efficient. The flux profiles are then used to predict ion dose profiles, which are important for shield design studies.

A formalism for photon transport in leaf canopies with finite-dimensional scattering centers that mutually cross-shade was recently proposed resulting in an equation set which is rather complicated. In this paper, we propose a simplification of the correction factors that describe the correlation in photon fates. These simplified correction factors permit the derivation of a modified integro-differential transfer equation with two additional terms that arise from considerations of finite size scatterers. A numerical procedure for the solution of this transfer equation in the framework of a successive collisions approach is also presented.