A spatial and angular moments analysis of the monoenergetic pencil beam problem

The analysis of characteristic methods in Cartesian geometry performed by M. L. Adams, T. A. Wareing and W. F. Walters showed that characteristic methods on triangular cells meet the asymptotic diffusion limit. We currently working on the development of a linear characteristic method in RZ geometry on triangular grids. The grid is formed on the basis of a rectangular grid in (r, z) space. Each rectangular cell of this grid is divided into four triangular cells. The discretization method approximates the angular flux in each triangular cell with linear functions along the cell edge and in the interior of the cell. The unknowns are edge-average and cell-average angular fluxes, first spatial moments of the angular flux over the cell and edges. The scattering term is interpolated with a linear function in each cell by means of the cell-average scalar flux and its linear spatial moments. The main research issue is the calculation of linear spatial moments of the scalar flux based on the linear spatial moments of the angular flux in a cell in (ξ, z) coordinates that will enable one to construct a linear approximation of the cell-wise scalar flux with desired properties.

The linear Boltzmann equation for the transport of neutral particles is investigated with the objective of generating a benchmark-quality calculation for the three-dimensional searchlight problem in a semi-infinite medium. The derivation assumes stationarity, one energy group, and isotropic scattering. The scalar flux (both surface and interior) and the current at the surface are the quantities of interest. The source considered is a pencil-beam incident at a point on the surface of a semi-infinite medium. The scalar flux will have two-dimensional variation only if the beam is normal; otherwise, it is three-dimensional. The solutions are obtained by using Fourier and Laplace transform methods. The transformed transport equation is formulated so that it can be related to a one-dimensional pseudo problem, thus providing some analytical leverage for the inversions. The numerical inversions use standard numerical techniques such as Gauss-Legendre quadrature, summation of infinite series, H-function iteration and evaluation, and Euler-Knopp acceleration. The numerical evaluations of the scalar flux and current at the surface are relatively simple, and the interior scalar flux is relatively difficult to calculate because of the embedded two-dimensional Fourier transform inversion, Laplace transform inversion, and H-function evaluation. Comparisons of these numerical solutions to results from the MCNP probabilistic code and the THREEDANT discrete ordinates code are provided and help confirm proper operation of the analytical code.

In order to improve the efficiency and accuracy of numerical simulation of gravity anomalies in large-scale complex models, this paper proposes a method for three-dimensional numerical simulation of gravity anomalies under arbitrary terrain, and implements its CPU-GPU parallel acceleration scheme. By performing a two-dimensional Fourier transform along the horizontal direction, three-dimensional partial differential satisfied by gravitational potential are transformed into one-dimensional ordinary differential equations in different wavenumbers, reducing computational and storage requirements. Moreover, the ordinary differential equations in different wavenumbers are independent of each other and have good parallelism. By retaining the vertical direction as the spatial domain and introducing traveling wave decomposition, the effects caused by upper and lower boundaries are eliminated. The vertical grid is arbitrary. A two-dimensional quasi-complete information Fourier transform is utilized to enhance both the accuracy and efficiency of the Fourier transform process, allowing both uniform and non-uniform flexible sampling. Based on the high parallelism, one-dimensional ordinary differential equations are solved in parallel using CPU, and quasi-complete information Fourier transform are computed in parallel using GPU. An abnormal sphere is designed to analyze the wavenumber spectral distribution characteristics of the anomalous potential and summarize the wavenumber sampling rules. The logarithmic uniform sampling in the wavenumber domain has high accuracy. Compared with Gauss-FFT, the quasi-complete information Fourier transform selects fewer wavenumbers and exhibits higher accuracy. The computational efficiency is greatly enhanced through the use of single-precision CPU-GPU parallel scheme, and a model with tens of millions of nodes only takes a few seconds. Two terrain models are designed to verify the algorithm’s adaptability to arbitrary complex terrains. This is significant for achieving fine inversion imaging and interpretation of gravity anomalies under large-scale complex conditions.

An exponentially-fitted method for singularly-perturbed ordinary differential equations with turning points and parabolic problems

Equations governing flow and transport in randomly heterogeneous porous media are stochastic and scale dependent. In the moment equation (ME) method, exact deterministic equations for the leading moments of state variables are obtained at the same support scale as the governing equations. Computable approximations of the MEs can be derived via perturbation expansion in orders of the standard deviation of the random model parameters. As such, their convergence is guaranteed only for standard deviation smaller than one. Here, we consider steady‐state saturated flow in a porous medium with random second‐order stationary conductivity field. We show it is possible to identify a support scale , where the typically employed approximate formulations of MEs yield accurate (statistical) moments of a target state variable. Therefore, at support scale and larger, MEs present an attractive alternative to slowly convergent Monte Carlo (MC) methods whenever lead‐order statistical moments of a target state variable are needed. We also demonstrate that a surrogate model for statistical moments can be constructed from MC simulations at larger support scales and be used to accurately estimate moments at smaller scales, where MC simulations are expensive and the ME method is not applicable.

Measurement of the neutron current and its use for the localisation of a neutron source

59 - Exact Second Order Equations

A pencil beam incident on a translucent material causes a luminous circular spot of volume-reflected light. Its spectral radiance L (r) decreases with radius r. Lc, the value close to the centre is mainly determined by the scattering parameter s(1-g) of material. Le, the value at the edge of the spot, is determined by s(1-g) and the absorption coefficient a. The instrument (the CTM) employs three fibre bundles to measure Lc and Le, both as a function of λ (400-700 nm). A central bundle (2 mm p) of - 1300 fibres each of 50 um g is randomly divided in a bundle for illumination and a bundle for measurement of Lc. A concentric ring (i.d. 4 mm, o.d. 5 mm) of - 2500 fibres is used to measure Le. This instrument was tested with aqueous suspensions of latex and a dissolved, non-latex adsorbing, red dye. Thus s(1-g) (scattering) and a (absorption) could be independently varied. For a=0, both Lc and Le increased with s(1-g), reached a maximum and decreased. The maxima for Lc and Le were at s(1 g) = 1 and 0.06 mm -1, respectively. At a constant scattering, increase of absorption decreased Le much stronger than it did Lc. This absorption-caused decrease depended only weakly on the scattering coefficient: a variation of scattering of a factor 20 caused only a few percent change in absorption-caused decrease of Lc and only a factor 2 in absorption-caused decrease of Le. At s(1-g) = 0.3 mm-1, Le depended much more strongly on a than did the overall regular reflection spectrum of the suspension as measured with a Hunter spectrophotometer under 0°/45° geometry. The readings of Lc and Le with this instrument can be used to determine s(1-g) and a and the reflection spectrum. Only small samples are needed in comparison to the regular reflection spectrometry. To obtain absolute values the instrument has to be calibrated on the specific type of material under investigation.© (1989) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Structural analysis for designing rotor blades with integral actuators

In this paper, an accurate yet computationally efficient beam model based on hierarchical Legendre expansion functions is developed for the analysis of non-uniform or restrained torsion problems of Functionally Graded (FG) beams with solid or thin-walled section. The mechanical properties of the FG beams studied in this paper, such as Young’s modulus and shear modulus, are assumed to continuously vary along either the length or thickness direction following a power-law distribution. The proposed beam model is based on the assumption that the beam’s cross-section is infinitely rigid in its own plane. However, the longitudinal displacement field over the beam’s cross-section is enriched in an element-wise manner by the unknown longitudinal displacement parameters multiplying with hierarchical Legendre expansion functions. The proposed modeling methodology has two novel aspects: First, it allows the torsional or twisting angle, which is a priori defined as an unknown kinematic variable, to be directly captured without a post-processing recovery step, even if there exists a strong flexual-torsional coupling within the beam; second, the longitudinal warping response of the beam, triggered by stretching, bending, twisting, or the coupling between them, can be captured without the pre-determination of warping modes and at a lower level of DOFs. The strong-form governing equations of non-uniform or restrained torsion problems of the FG beam are derived based on the principle of minimum potential energy and is directly solved by the high-quality general-purpose ordinary differential equation (ODE) solver, i.e. COLSYS ODE solver. Also, a more efficient Rayleigh-Ritz energy method is applied to provide the weak-form solutions. The resulting beam model is suitable to a more general cross-section, such as solid section, branched open section, or half open-half closed section. The accuracy and efficiency of the proposed beam model are validated extensively by comparing with the previously published results in literature. Effects of the power-law index, taper ratio, and section-type on the torsional response of FG beams with material gradation along length or thickness direction are studied with various numerical examples.

Chapter 7 - Ordinary Differential Equations

The plastic collapse bending moment in a pipe cross-section with a circumferential crack is defined in ASME B&PV Code Section XI, Appendix C using simplified equilibrium equations by approximating the pipe mean radius Rm and the neutral axis angle β. In previous papers it was demonstrated by the authors that, for externally cracked pipes, those simplified equilibrium equations are not conservative and hence improved equations were developed and proposed which account for the cracked pipe ligament mean radius Rmc. In this paper, it is demonstrated that the accuracy of the collapse bending moment equation can be refined by taking into account the neutral axis position Yna of the cracked pipe section. This leads to exact collapse bending moment equations without any approximation on the pipe mean radius Rm nor on the neutral axis angle β. In this framework, it is shown that, for externally cracked pipes, the Appendix C equations could lead to more than 20% less conservative collapse bending moment than with the exact equations. An extended finite element method analysis completes this study to assess the relevance of the model used to determine the plastic collapse bending moment.

: The motion of elastic finned projectiles has been analyzed by various approximate theories. In this report the exact equations of small amplitude motion are derived for a symmetric missile. The aerodynamic and elastic symmetries are used to allow the use of complex variables to describe the lateral motion in a non-rotating coordinate system. Although the resulting equations are both ordinary and partial differential equations, frequencies and damping rates of free oscillations are obtained from an ordinary differential equation with boundary conditions. Equations for a permanently deformed bent missile are derived, and an ordinary differential equation for the forced motion of a bent missile is obtained. Sample calculations for a finned projectile with a fineness ratio of 20 show resonant motion at the aerodynamic frequency as well as at each elastic frequency. The nonlinear roll moment associated with a bent missile is computed and the location of possible spin-yaw lock-in is determined. The flight motion of an elastic missile is shown to be the sum of two elliptical motions: a low frequency pitching motion and a higher frequency flexing motion. The induced drag coefficients for both motions are computed as functions of the missile's elasticity.

Here we studied the light transport initiated by oblique illumination of a turbid cylinder on its top or barrel with a pencil beam. The pencil beam was approximated by an isotropic point source according to the incident point and angle. Solutions were derived for the steady-state, frequency, and time domains. Time- and spatially resolved reflectance results with different incident angles agreed well with those of Monte Carlo simulations. A method to determine optical properties using the time- and spatially resolved reflectance results along the z direction was proposed for the light incident on the cylinder barrel. Analytical solutions can improve analysis of turbid cylindrical media with incident illumination and effective optical properties reconstruction.

The efficiency of a SAGD operation depends on developing a uniform steam chamber and maintaining an optimal subcool along the length of the well pair. Heterogeneity in reservoir properties may lead to suboptimal subcool levels. Recently, Model Predictive Control (MPC) based on the real-time production, temperature, and pressure data along with other well and surface constraint information has been proposed for a real-time feedback control of SAGD well pairs. Reservoir dynamics in MPC is represented using either linear step response model or one-dimensional ordinary differential equation. However, such simplified models are insufficient in MPC since SAGD is more complex and highly nonlinear process. Therefore, MPC framework that represents nonlinear behaviour of SAGD over an extended control period is required to achieve optimized subcool and steam conformance. In this research, two novel workflows are proposed to handle nonlinear reservoir dynamics in MPC. First approach known as Adaptive MPC includes recursive estimations at each control interval based on system identification theory. This allows evolution of the coefficients of a fixed model structure such that the updated system identification model in MPC controller reflects current reservoir dynamics adequately. Another approach, Gain-Scheduled MPC, decomposes the subcool control problem in a parallel manner and uses a bank of multiple controllers rather than only one controller. This ensures effective control of the nonlinear reservoir system even in adverse control situations by employing aggressive variations in input parameters. Suggested workflows are implemented using history-matched numerical model of a reservoir located in northern Alberta. Steam injection rates and liquid production rate are considered as input variables in MPC, constrained to available surface facilities. Well-pair is divided into multiple sections and subcool of each section is considered as an output variable. Optimum set-point for subcool is considered as 20°C. Results are compared with actual field data (in which no control algorithm is used) and analyzed based on two criteria: 1) Do all subcools track optimum set-point while maintaining stability in input variables and 2) Does net present value (NPV) of oil improve in case of Adaptive and Gain-Scheduled MPC? In general, we conclude that both Adaptive and Gain-Scheduled MPC provide superior tracking of subcool set-point and hence better steam conformance due to adequate representation of reservoir dynamics by recursive estimation of coefficients and multiple controllers. In addition, results indicate stability in input parameters and improvement in economic performance. NPV is improved by 23.69% and 10.36% in case of Adaptive and Gain-Scheduled MPC, respectively. Under current economic scenario, proposed workflows can improve the NPV of a SAGD reservoir by optimizing the well operational parameters while considering constraints of surface facilities and minimizing environmental footprints.