Numerical Optimization of Computing Algorithms of the Variational Nodal Method Based on Transformation of Variables

The properties of the variational nodal method for neutron transport calculations are investigated. The method is generalized for three-dimensional multigroup criticality problems in both hexagonal-z and Cartesian geometries. The method is implemented as part of the Argonne National Laboratory Code DIF3D, and applied to a series of benchmark reactor calculations. Variational nodal methods are compared of nodal transport methods based on both interface-current and discrete ordinate approximations. Model problems are used to examine the effect of running each of the three classes of nodal transport methods on computers with massively parallel architectures.

A perturbation method based on the variational nodal method for solving the neutron transport equation is developed for multidimensional geometries. The method utilizes the solution of the corresponding adjoint transport equation to calculate changes in the critical eigenvalue due to cross-section changes. Both first-order and exact perturbation theory expressions are derived. The adjoint solution algorithm has been formulated and incorporated into the variational nodal option of the Argonne National Laboratory DIF3D production code. To demonstrate the efficacy of the methods, perturbation calculations are performed on the three-dimensional Takeda benchmark problems in both Cartesian and hexagonal geometries. The resulting changes in eigenvalue are also obtained by direct calculation with the variational nodal method and compared with the change approximated by the first-order and exact theory expressions from the perturbation method. Exact perturbation results are in excellent agreement with the actual eigenvalue differences calculated in VARIANT. First-order theory holds well for sufficiently small perturbations. The times required for the perturbation calculations are small compared with those expended for the base-forward and adjoint calculations.

The objective of this project has been to develop computational methods that will enable more effective analysis of Accelerator Driven Systems (ADS). The work is centered at the University of Missouri at Rolla, with a subcontract at Northwestern University, and close cooperation with the Nuclear Engineering Division at Argonne National Laboratory. The work has fallen into three categories. First, the treatment of the source for neutrons originating from the spallation target which drives the neutronics calculations of the ADS. Second, the generalization of the nodal variational method to treat the R-Z geometry configurations frequently needed for scoping calculations in Accelerator Driven Systems. Third, the treatment of void regions within variational nodal methods as needed to treat the accelerator beam tube.

Plane waves and response matrices

The relative ease with which high-throughput data can be collected on the same sampling units has raised interest in relating high-dimensional data sets. This problem can be framed as a multivariate regression model with one data set treated as a response matrix and the other as a covariate matrix. The high-dimensionality of both the response and covariate matrices along with the complex correlation structures within each raises statistical challenges. Here, we review a stochastic partitioning method that addresses this problem by fitting a mixture of regression models with variable selection, thereby uncovering in a unified manner group structures and key relationships between the data sets. The problem of variable selection in this context poses added difficulties compared to standard regression settings, as the membership of objects into different components has to be learned simultaneously with the search of component-specific predictors. We emphasize the practical applicability of the method using real data and illustrate features of the associated R package SPaVS.

Heterogeneous discontinuity factor treatment in Variational Nodal Method

In this novel forecasting algorithm to calculate the unit‐impulse response matrix of an unbounded medium to analyse dynamic medium‐structure interaction in the time domain based on the substructure method, a finite‐element region adjacent to the structure–medium interface consisting of two or three rows only is introduced. Its exterior boundary is chosen to be similar to the interior boundary which coincides with the structure–medium interface. At similar boundaries of the unbounded medium the unit‐impulse response matrices are a function of the dimensionless time. The unit‐impulse response matrix at the exterior boundary can be forecasted from that at the interior boundary via this dimensionless time by taking advantage of the time delay of waves propagating between these two boundaries. The forecasting algorithm based solely on the finite‐element method is exact in the finite‐element sense in modelling the unbounded medium.

Our aim is to characterize those matrices that are the response matrix of a semi-positive definite Schr?dinger operator on a circular planar network. Our findings generalize the known results and allow us to consider both nonsingular and non diagonally dominant matrices as response matrices. To this end, we define the Dirichlet-to-Robin map associated with a Schr?dinger operator on general networks, and we prove that it satisfies the alternating property which is essential to characterize the response matrices.

This paper presents an implementation of the stiffness confinement method in the framework of the variational nodal method (VNM) to solve time-dependent pin resolved neutron transport equations. With frequency-transform, the time-dependent neutron transport equation is decoupled from the precursor equation and transformed into a transient eigenvalue problem (TEVP) which can be solved with the in-house VNM code VITAS-FE. The verification tests were performed using the C5G7-TD benchmark problems. The result agrees well with solutions of the MPACT. The study indicates that the VNM and SCM can be applied to solve the high-fidelity pin-resolved transient problem accurately.

In the zonal method, considerable computational resources are needed to calculate the direct exchange areas (DEA) among the isothermal zones due to integrals with up to six dimensions, while strong singularities occur in the integrands when two zones are adjacent or overlaping (self-irradiation). A special transformation of variables to reduce a double integral into several single integrals is discussed in this paper. This technique was originally presented by Erkku (1959) for calculation of DEA using a uniform zone system in a cylindrical enclosure. However, nonuniform zones are needed for applications with large thermal gradients. Thus we extended this technique to calculate the DEA for non-uniform zones in an axisymmetrical cylinder system. A six-fold reduction in computational time was observed in calculating DEA compared with cases without a variable transformation. It is shown that accuracy and efficiency of estimation of radiation heat flux is improved when using a nonuniform zone system. Reasonable accuracy of all DEA are calculated without resorting to the conservative equations. Results compared well with analytical solutions and numerical results of previous researchers. This technique can be readily extended to rectangular enclosures with similar reduction in computation time expected.

The eigenmodes of the time reversal operator are studied for a single rigid cylinder with elliptical cross-section leading to an asymmetric inter-element response matrix in reflection measurement mode. The elliptical cylinder is of considerable interest in scattering problems and it encompasses within the limit both the circular cross-section case and the strip. Variable cylinder properties will include cross-sectional area, eccentricity and angle of inclination of the semi-major axis with respect to the axis of symmetry of a probing transducer array. Approximations of the exact solutions determining the far-field response using a modal expansion with scattering coefficients computed via Mathieu functions will be discussed. Analysis of simulated results will indicate that for scatterers near the Rayleigh limit it is possible to characterize the aspect ratio of the target using subarray methods.

Image fusion serves as the basis for automatic target recognition; it maps images of teh same scene received from different sensors into a common reference system. A novel fusion method is described that employs image local response and the hybrid evolutionary algorithm (HEA). Given geometric transformation A(V) under parameter vector V (e.g. affine image transformation) of the images subjected to fusion, image local response is defined as image transform components of the parameter vector V are applied to the image, and the corresponding variations of the least squared differences of the gray levels of the two images (i.e. before and after parameter variation) form the image response matrix. The transform R(V) extracts only the dynamic contents of the image, i.e. the salient features that are most sensitive to geometric transformation A(V). Since R(V) maps the image onto itself, the result of the mapping is largely invariant to the type of the sensor that was used to obtain the image. Once the response matrices are built for all images subjected to fusion, HEA is used to map the images into the common reference system.

A Study of Certified Diabetes Educators: A Descriptive Exploration of the Delivery of Diabetes Self-Management Support in Community-based Settings in Ontario

We present a direct slope-based correction algorithm to simultaneously control two deformable mirrors (DMs) in a woofer-tweeter adaptive optics system. A global response matrix was derived from the response matrices of each deformable mirror and the voltages for both deformable mirrors were calculated simultaneously. This control algorithm was tested and compared with a 2-step sequential control method in five normal human eyes using an adaptive optics scanning laser ophthalmoscope. The mean residual total root-mean-square (RMS) wavefront errors across subjects after adaptive optics (AO) correction were 0.128 ± 0.025 μm and 0.107 ± 0.033 μm for simultaneous and 2-step control, respectively (7.75-mm pupil). The mean intensity of reflectance images acquired after AO convergence was slightly higher for 2-step control. Radially-averaged power spectra calculated from registered reflectance images were nearly identical for all subjects using simultaneous or 2-step control. The correction performance of our new simultaneous dual DM control algorithm is comparable to 2-step control, but is more efficient. This method can be applied to any woofer-tweeter AO system.

The authors describe a two-dimensional (vector) regressional model for examining the anisotropic response of ice drift and ocean current velocity (“drift velocity”) to surface wind forcing. Illustration of the method is limited to sea ice response. The principal mathematical and physical properties of the model are outlined, together with estimates of the “response matrices” and the corresponding “response ellipses” (drift velocity response to a unity wind velocity forcing). For each direction, φ, of the wind vector the method describes a corresponding “wind factor” α(φ) (relative drift speed) and “turning angle” θ(φ) (the angle between the drift velocity and wind vector). The major ellipse axis corresponds to the direction of the “effective wind” (φ = φmax) and the minor axis to the direction of the “noneffective” wind. The eigenvectors of the response matrix define wind directions that are the same as the wind-induced drift velocity directions. Depending on the water depth and offshore distance, six analytical cases are possible, ranging from rectilinear response ellipses near the coast to purely circular response ellipses in the open ocean. The model is used to examine ice drift along the western shelf of Sakhalin Island (Sea of Okhotsk). Responses derived from the vector regression (four parameter) method are less constrained and therefore more representative of wind-induced surface motions than those derived using the traditional complex transfer function (two parameter) approach.