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AZTRAN and AZKIND verification with a BWR study case

The AZTLAN Platform project is a national initiative led by the National Institute of Nuclear Research which gather the main public institutions of higher education in Mexico: the National Polytechnic Institute, the National Autonomous University of Mexico and the Autonomous Metropolitan University. It consists in the development of a modeling platform for the analysis and design of nuclear reactors. This project aims to modernize, improve and integrate the neutronic, thermohydraulics and thermomechanical codes, developed in the national institutions, in a platform integrated and developed by Mexican experts. The development of AZTLAN Platform is divided technically into four working groups: neutronics, thermohydraulics, coupling and uncertainty analysis, and users group. The present work is the result of the activities of the Users Group focused on the follow-up, testing and use of the different codes (modules) being developed under the AZTLAN project; with the objective of providing feedback and to suggest development priorities to the team of developers, as well as to promote collaboration and synergy between the participating institutions. In particular, in this work, the AZTRAN and the AZKIND modules were verified and tested against the results of well-known codes of the nuclear industry for boiling water reactor (BWR) models. AZTRAN is a 3D transport code which solves numerically the multi-group time independent neutron transport equation in Cartesian geometry, based on the discrete ordinates (SN) method and the RTN-0 nodal scheme. AZKIND is a 3D diffusion code that solves numerically the time dependent neutron diffusion equations in Cartesian geometry, based on a Galerkin Finite Element Method using the RTN-0 polynomial nodal approximation.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

Collocation methods for nonlinear differential equations on low-rank manifolds

We introduce new methods for integrating nonlinear differential equations on low-rank manifolds. These methods rely on interpolatory projections onto the tangent space, enabling low-rank time integration of vector fields that can be evaluated entry-wise. A key advantage of our approach is that it does not require the vector field to exhibit low-rank structure, thereby overcoming significant limitations of traditional dynamical low-rank methods based on orthogonal projection. To construct the interpolatory projectors, we develop a sparse tensor sampling algorithm based on the discrete empirical interpolation method (DEIM) that parameterizes tensor train manifolds and their tangent spaces with cross interpolation. Using these projectors, we propose two time integration schemes on low-rank tensor train manifolds. The first scheme integrates the solution at selected interpolation indices and constructs the solution with cross interpolation. The second scheme generalizes the well-known orthogonal projector-splitting integrator to interpolatory projectors. We demonstrate the proposed methods with applications to several tensor differential equations arising from the discretization of partial differential equations.

97 MATHEMATICS AND COMPUTING↗

Use of Statistical Analysis of Acoustic Emission Data on Carbon-Epoxy COPV Materials-of-Construction for Enhanced Felicity Ratio Onset Determination

Broadband modal acoustic emission (AE) data were acquired during intermittent load hold tensile test profiles on Toray T1000G carbon fiber-reinforced epoxy (C/Ep) single tow specimens. A novel trend seeking statistical method to determine the onset of significant AE was developed, resulting in more linear decreases in the Felicity ratio (FR) with load, potentially leading to more accurate failure prediction. The method developed uses an exponentially weighted moving average (EWMA) control chart. Comparison of the EWMA with previously used FR onset methods, namely the discrete (n), mean (n (raised bar)), normalized (n%) and normalized mean (n(raised bar)%) methods, revealed the EWMA method yields more consistently linear FR versus load relationships between specimens. Other findings include a correlation between AE data richness and FR linearity based on the FR methods discussed in this paper, and evidence of premature failure at lower than expected loads. Application of the EWMA method should be extended to other composite materials and, eventually, composite components such as composite overwrapped pressure vessels. Furthermore, future experiments should attempt to uncover the factors responsible for infant mortality in C/Ep strands.

Abraham, Arick Reed A.↗

A vector-continuous loading concept for aerodynamic panel methods

An approach to the reduction of discretization errors in aerodynamic panel methods is presented. The approach is based on preventing the occurence of induced velocity singularities at panel slope discontinuities by maintaining continuity of the velocity jump vector across the panels. The approach was implemented in a two-dimensional incompressible panel method formulation and evaluated by application to several external and internal flow problems. The method is shown to exhibit a second order accuracy trend and to produce smaller errors with velocity component boundary conditions imposed on the real flow than with equipotential boundary conditions imposed on the imaginary flow behind the panels. For flows around airfoil sections with either sharp or blunt trailing edges, the method gives excellent agreement with results from a well developed finite difference method. The method is well behaved and is insensitive to irregularities in panel size distribution.

Kemp, W. B., Jr.↗

Comparison of Corcos-based and experimentally-derived coherence factors for BFFs estimation

In this paper, high-spatial-resolution unsteady Pressure Sensitive Paint (uPSP) data are utilized to compare two methods for panel Buffet Forcing Functions (BFF) estimation for the Space Launch System (SLS). Such methods are based on discrete pressure measurements within a panel but employ coherence factors to account for partially-correlated fluctuating pressures across the whole panel. In one method, coherence factors are derived based on the Corcos model, whereas the second method utilizes experimentally-derived coherence factors. To simulate discrete measurements using uPSP data, suitable subsets of the data are extracted. When full uPSP resolution is retained, uPSP data provide a benchmark to assess discrete-measurements-based methods. The analysis focuses on the peak SLS buffet environment located downstream of the Forward Attachment Hardware (FAH) between the core stage and solid rocket boosters. Trends of Corcos-based and experimentally-derived coherence factors are in reasonable agreement with the benchmark. However, at certain frequencies, experimentally-derived coherence factors are sensitive to the separation distance between pressure measurements utilized to compute coherence lengths. Such sensitivity originates from deviation of the experimental-based coherence function from an exponential decay assumption. On the other hand, the present implementation of the Corcos model fails to capture certain non-turbulent boundary layer related environments, such as a subharmonic of FAH vortex-shedding. For all methods presented in this paper, at near transonic conditions, increased pressure coherence and spatial nonuniformity lead to BFF overestimation and sensitivity to the pressure measurement location within the panel.

transonic buffet↗

Comparison of Corcos-Based and Experimentally-Derived Coherence Factors for Buffet Forcing Functions Estimation

In this paper, high-spatial-resolution unsteady Pressure Sensitive Paint (uPSP) data are utilized to compare two methods for panel Buffet Forcing Functions (BFF) estimation for the Space Launch System (SLS). Such methods are based on discrete pressure measurements within a panel but employ coherence factors to account for partially-correlated fluctuating pressures across the whole panel. In one method, coherence factors are derived based on the Corcos model, whereas the second method utilizes experimentally-derived coherence factors. To simulate discrete measurements using uPSP data, suitable subsets of the data are extracted. When full uPSP resolution is retained, uPSP data provide a benchmark to assess discrete-measurements-based methods. The analysis focuses on the peak SLS buffet environment located downstream of the Forward Attachment Hardware (FAH) between the core stage and solid rocket boosters. Trends of Corcos-based and experimentally-derived coherence factors are in reasonable agreement with the benchmark. However, at certain frequencies, experimentally-derived coherence factors are sensitive to the separation distance between pressure measurements utilized to compute coherence lengths. Such sensitivity originates from deviation of the experimental-based coherence function from an exponential decay assumption. On the other hand, the present implementation of the Corcos model fails to capture certain nonturbulent boundary layer related environments, such as a subharmonic of FAH vortex-shedding. For all methods presented in this paper, at near transonic conditions, increased pressure coherence and spatial nonuniformity lead to BFF overestimation and sensitivity to the pressure measurement location within the panel.

buffet↗

Comparison of Corcos-Based and Experimentally-Derived Coherence Factors for Buffet Forcing Function Estimation

In this paper, high-spatial-resolution unsteady Pressure Sensitive Paint (uPSP) data are utilized to compare two methods for panel Buffet Forcing Function (BFF) estimation for the Space Launch System (SLS). Such methods are based on discrete pressure measurements within a panel but employ coherence factors to account for partially-correlated fluctuating pressures across the whole panel. In one method, coherence factors are derived based on the Corcos model, whereas the second method utilizes experimentally-derived coherence factors. To simulate discrete measurements using uPSP data, suitable subsets of the data are extracted. When full uPSP resolution is retained, uPSP data provide a benchmark to assess discrete-measurement-based methods. The analysis focuses on the peak SLS buffet environment located downstream of the Forward Attachment Hardware (FAH) between the core stage and solid rocket boosters. Trends of Corcos-based and experimentally-derived coherence factors are in reasonable agreement with the benchmark. However, at certain frequencies, experimentally-derived coherence factors are sensitive to the separation distance between pressure measurements utilized to compute coherence lengths. Such sensitivity originates from deviation of the experimentally-based coherence function from an exponential decay assumption. On the other hand, the present implementation of the Corcos model fails to capture certain nonturbulent boundary layer related environments, such as a subharmonic of FAH vortex-shedding. For all methods presented in this paper, at near transonic conditions, increased pressure coherence and spatial nonuniformity lead to BFF overestimation and sensitivity to the pressure measurement location within the panel.

transonic buffet↗

Computational methods for optimal linear-quadratic compensators for infinite dimensional discrete-time systems

An abstract approximation theory and computational methods are developed for the determination of optimal linear-quadratic feedback control, observers and compensators for infinite dimensional discrete-time systems. Particular attention is paid to systems whose open-loop dynamics are described by semigroups of operators on Hilbert spaces. The approach taken is based on the finite dimensional approximation of the infinite dimensional operator Riccati equations which characterize the optimal feedback control and observer gains. Theoretical convergence results are presented and discussed. Numerical results for an example involving a heat equation with boundary control are presented and used to demonstrate the feasibility of the method.

Gibson, J. S.↗

A Quantitative Comparison between C0 and C1 elements for Solving the Cahn-Hilliard Equation

The Cahn–Hilliard (CH) equation is a time-dependent fourth-order partial differential equation (PDE). When solving the CH equation via the finite element method (FEM), the domain is discretized by C1-continuous basis functions or the equation is split into a pair of second-order PDEs, and discretized via C0-continuous basis functions. In the current work, a quantitative comparison between C1 Hermite and C0 Lagrange elements is carried out using a continuous Galerkin FEM formulation. The different discretizations are evaluated using the method of manufactured solutions solved with Newton’s method and Jacobian-Free Newton Krylov. It is found that the use of linear Lagrange elements provides the fastest computation time for a given number of elements, while the use of cubic Hermite elements provides the lowest error. The results offer a set of benchmarks to consider when choosing basis functions to solve the CH equation. In addition, an example of microstructure evolution demonstrates the different types of elements for a traditional phase-field model.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

A Comparison of Linear Solvers for Resolving Flow in Three-Dimensional Discrete Fracture Networks

We compare various methods for resolving steady flow within three-dimensional discrete fracture networks, including direct methods, Krylov subspace methods with and without preconditioning, and multi-grid methods. We compared the performance of the methods based on compute times and scaling of the solution as a function of the number of grid nodes and log-variance of the hydraulic aperture. The methods are applied to three test cases: (a) variable density of networks with a truncated power-law distribution of fracture lengths, (b) a fixed network composed of monodisperse fracture sizes but varied permeability/aperture heterogeneity, (c) and a network based on field site in Nevada, US. We chose these cases to allow us to study the impact of the mesh size and flow properties, as well as to demonstrate our conclusions on a large-scale, realistic problem (more than 40 million mesh nodes). A direct solution using Cholesky factorization outperformed other methods for every example but was closely followed in performance by some algebraic multigrid (AMG) preconditioned Krylov subspace methods. Among the Krylov methods, conjugate gradients (CG) with an AMG preconditioner performs the best. Generally, Cholesky factorization is recommended, but CG with an AMG preconditioner may be suitable for very large problems beyond 40 million nodes where the entire linear system cannot reside in memory.

58 GEOSCIENCES↗

Verification of AZTRAN v1.2 code with a PWR-MOX benchmark problem

The AZTLAN platform project is an effort focused on developing a modeling Mexican platform to design and analyze nuclear reactors. It is a national initiative lead by the National Institute of Nuclear Research (ININ), in collaboration with the main Mexico's public universities: the National Polytechnic Institute (IPN), the National Autonomous University of Mexico (UNAM) and the Autonomous Metropolitan University (UAM). The AZTRAN code is a module of AZTLAN that solves the 3D neutron transport equation using the discrete ordinates (SN) method and the nodal scheme RTN-0 on steady state and Cartesian geometry. AZTRAN works with cross sections generated externally by other code. It provides a solution to the eigenvalue problem and determine the neutron distribution for a specific medium at desired conditions. On the other hand, it is important to mention that the Working Group 4 (users group) of AZTLAN platform is responsible of verify and validate the codes development, in order to give feedback to the group of developers: neutronics and thermal-hydraulics. In this work, the AZTRAN v1.2 performance was studied for a PWR-MOX lattice, using as example the Yamamoto's benchmark for LWR fuels, comparing AZTRAN v1.2 results again those obtained with the well-known neutronic code SERPENT-2 (v1.27), and some results from an article presented at PHYSOR-2002. Results from AZTRAN for a BWR fuel lattice is presented in a companion paper in this meeting.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

Extended isogeometric analysis of multi-material and multi-physics problems using hierarchical B-splines

Here, this paper presents an immersed, isogeometric finite element framework to predict the response of multi-material, multi-physics problems with complex geometries using locally refined discretizations. To circumvent the need to generate conformal meshes, this work uses an extended finite element method (XFEM) to discretize the governing equations on non-conforming, embedding meshes. A flexible approach to create truncated hierarchical B-splines discretizations is presented. This approach enables the refinement of each state variable field individually to meet field-specific accuracy requirements. To obtain an immersed geometry representation that is consistent across all hierarchically refined B-spline discretizations, the geometry is immersed into a single mesh, the XFEM background mesh, which is constructed from the union of all hierarchical B-spline meshes. An extraction operator is introduced to represent the truncated hierarchical B-spline bases in terms of Lagrange shape functions on the XFEM background mesh without loss of accuracy. The truncated hierarchical B-spline bases are enriched using a generalized Heaviside enrichment strategy to accommodate small geometric features and multi-material problems. The governing equations are augmented by a formulation of the face-oriented ghost stabilization enhanced for locally refined B-spline bases. We present examples for two- and three-dimensional linear elastic and thermo-elastic problems. The numerical results validate the accuracy of our framework. The results also demonstrate the applicability of the proposed framework to large, geometrically complex problems.

42 ENGINEERING↗

A flexible linear diffusion acceleration to k-eigenvalue neutron transport with SN discontinuous finite element method

In this paper, we derive a flexible linear diffusion acceleration (LDA) for k-eigenvalue neutron transport discretized with discontinuous finite element method (DFEM) and discrete ordinates(SN). This LDA is based on our two pieces of previous works: the flexible non linear diffusion acceleration (NDA) for DFEM-SN and LDA for k-eigenvalue neutron transport using pre-conditioned Jacobian-free Newton-Krylov with self-adjoint angular flux (SAAF), continuous finite element method(CFEM), and SN. We point out the differences between LDA and NDA for DFEM-SN and the difference between DFEM-SN and SAAF-CFEM-SN for LDA. Numerical tests are presented to compare the convergence behaviour of NDA and LDA. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

A New Method for Estimating the Single Scattering Phase Functions of Regolith Grains

Hapke's photometric model has been widely used in solar system remote sensing applications for nearly two decades. Recently, Hapke extended his model to describe the coherent-backscatter opposition effect and multiple-scattering by particles with anisotropic single particle phase functions (SPPF's). A practical difficulty for retrieving Hapke's model parameters from typical planet, satellite, and asteroid photometry data sets is that the model employs a large number of adjustable parameters (at least eight) that can be reliably constrained only for a small number of planetary data sets in which both disk-resolved and whole-disk observations are available from opposition to very large phase angles. The present work aims to reduce the number of adjustable parameters and preserve (or even enhance) the model's accuracy and usefulness by expressing Hapke's parameters in terms of more fundamental physical properties on which they mutually depend. The most difficult part of this task, described here, is to develop a simple method for computing the effective SPPF for structurally complex regolith grains from optical constants, grain-size distribution, and average regolith porosity. The development of light-scattering models for irregularly shaped particles is a large, complex subject and many sophisticated methods, such as Discrete Dipole Approximation (DDA) and Monte-Carlo simulations, have been explored elsewhere. Many of these methods are computationally intensive and probably impractical for routine substitution in Hapke's model. Here, progress is reported in developing a practical, semi-empirical method for estimating the directional scattering behavior (i.e. SPPF) of irregular regolith grains. The method employs Optical Transfer Function (OTF) techniques to model how the structural complexity of regolith particles broaden and attenuate the angular distribution of scattered light relative to that expected from ideal spherical particles of equivalent size and composition.

Helfenstein, P.↗

A Comparison of Groundwater Sampling Technologies, Including Passive Diffusion Sampling, for Radionuclide Contamination - 20205

Using traditional high-flow purge methods for long-term water quality monitoring of deep groundwater wells can be expensive, affect contaminant migration, and produce excessive volumes of discharge water that can be difficult to manage. The use of low-flow pumping methods and depth discrete bailers (DDBs) can reduce the cost of sampling deep groundwater wells. In general, using different pumping methods to obtain reproducible and representative groundwater can be challenging. Passive diffusion samplers (PDSs) have successfully been used in long-term monitoring for volatile organic compounds, major ions, and trace-elements, but application has been limited for stable and radioactive isotopes. Beginning in 2018, the United States Geological Survey (USGS) conducted three sampling events to test the ability of PDSs and DDBs to obtain reproducible and representative groundwater samples. The sampled well is completed in a regional, permeable carbonate aquifer and is one in a set of wells that have historically been used for tritium tracer testing. All samples were obtained at 180 m (590 ft) below land surface (bls) in a 13.97 cm (5.5 inch) uncased well that has a total depth of 202 m (662 ft) bls. The first sampling event deployed a regenerated cellulose dialysis membrane (RCDM) PDS for 14 days with deionized water as the blank. The second sampling event deployed a RCDM for 27 days also with deionized water as the blank. The third sampling event deployed a Dual Membrane (DM) PDS for 65 days using a blank of tritium-dead carbonate water. The DM PDS was used to assess the effect of longer term deployment on tritium concentrations and address whether or not the PDSs reached equilibrium with ambient groundwater. The day after each of the three passive samplers were retrieved a DDB was used to obtain discrete non-integrated groundwater samples. For each DDB sampling day, the bailer was lowered into the well 10 consecutive times to determine if the water chemistry changed from the first to the last bailed sample. Quality assurance samples including blanks and duplicates were obtained during all three sampling events. All blank waters had tritium concentrations less than 21±33 pCi/L. Major ion (e.g. calcium, chloride, sodium, and sulfate) results were compared between all samples obtained with RCDM and DDB. Major ion concentrations showed a coefficient of variation of less than 6% between all RCDM and DDB samples; however, the coefficient of variation between the different deployment times and the two different methods for trace-element concentrations was much larger, particularly for manganese (82%), lead (41%), and zinc (33%). Stable isotope values were compared between the RCDM and DDB samples. The DDB sample results all fell within analytical uncertainty and were considered representative of the formation groundwater. The stable isotope values from the RCDM samples indicated that a longer deployment time was necessary to gain equilibrium and to obtain representative groundwater samples. Tritium results from groundwater samples obtained from the RCDM, DM, and DDB indicate that groundwaters obtained with PDSs produced tritium concentrations 2 to 3 times higher (between 1,426 and 3,060±87 pCi/L) than groundwaters obtained with a DDB (479 to 1,219±51 pCi/L). The longer the passive diffusion samplers were deployed, the higher the tritium concentration, suggesting that equilibrium with tritium was not reached within a 27-day deployment. DDB samples showed tritium results declining from the first to the last bailed sample for all three sampling events. This research suggests that tritium results from groundwater samples obtained from PDSs are more reproducible than samples obtained from DDBs. Also, PDSs likely do not accumulate isotopes of water but rather equilibrate with the ambient groundwater. On the other hand, both PDSs and DDBs were able to provide representative groundwater samples for major ions and have the potential to produce representative groundwater samples for stable isotopes of water (δD and δ{sup 18}O). Utilizing the most effective method for sampling groundwater is important for long-term groundwater monitoring. Quantifying the ability of passive diffusion groundwater samplers to obtain representative radioactive chemistry samples will help determine if these methods can be a useful alternative to pumping and purging wells for long-term monitoring. (authors)

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

Data approximation using a blending type spline construction

Generalized expo-rational B-splines (GERBS) is a blending type spline construction where local functions at each knot are blended together by C{sup k}-smooth basis functions. One way of approximating discrete regular data using GERBS is by partitioning the data set into subsets and fit a local function to each subset. Partitioning and fitting strategies can be devised such that important or interesting data points are interpolated in order to preserve certain features. We present a method for fitting discrete data using a tensor product GERBS construction. The method is based on detection of feature points using differential geometry. Derivatives, which are necessary for feature point detection and used to construct local surface patches, are approximated from the discrete data using finite differences.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

New Methods of Uncertainty Quantification for Mixed Discrete-Continuous Variable Models

The scale and complexity of problems such as designing power grids or planning for climate change is growing rapidly, driving the development of complicated computer models. More complex models have longer run times and incorporate larger numbers of inputs, both continuous and discrete. For example, a detailed physics model may have continuous variables such as temperature, height or pressure along with discrete variables that indicate the choice of a material for a particular piece or the model to be used to calculate air flow. A power grid design model may have continuous variables such as generation capacity, power flow or demand along with discrete variables such as number of generators, number of transmission lines or binary variables to indicate whether or not a node is chosen for generation expansion. A growing awareness of uncertainty and the desire to make risk-informed decisions is causing uncertainty quantification (UQ) to be more routine and often required. UQ provides the underpinnings necessary to establish confidence in models and their use; therefore, much time and effort is being invested in creating efficient approaches for UQ. However, these efforts have been focused on models that take continuous variables as inputs. When discrete inputs are thrown into the mix, the basic approach is to repeat the UQ analysis for each combination of discrete inputs or some subset thereof; this rapidly becomes intractable. Because of the computational complexity inherent in mixed discrete-continuous models, researchers will focus on the uncertainty in their particular problem finding ways to take advantage of symmetries, simplifications or structures. For example, uncertainty propagation in certain dynamical systems can be efficiently carried out after various decomposition steps or uncertainty propagation in stochastic programming is confined to scenario generation. Unfortunately models are not always available for such machinations: models may be embedded in legacy codes, may utilize commercial off the shelf codes or may be created by stringing a series of codes together. It is also time consuming to start each problem from scratch; worse there may not be any simplifications or symmetries to take advantage of. For these situations a UQ method developed for any black box function is necessary. This report documents a new conceptual model for performing UQ for mixed discrete-continuous models which not only applies to any simulator function, but allows the use of the efficient UQ methods that have been developed for continuous inputs only. The conceptual model is presented and an estimation procedure is fleshed out for one class of problems. This is applied to variations of a mixed discrete-continuous optimization test problem. This procedure provides comparable results to a benchmark solution with fewer function evaluations.

Bauman, Lara E.↗

Assessment of the MPACT Resonance Data Generation Procedure

Currently, heterogeneous models are being used to generate resonance self-shielded cross-section tables as a function of background cross sections for important nuclides such as 235 U and 238 U by performing the CENTRM (Continuous Energy Transport Model) slowing down calculation with the MOC (Method of Characteristics) spatial discretization and ESSM (Embedded Self-Shielding Method) calculations to obtain background cross sections. And then the resonance self-shielded cross section tables are converted into subgroup data which are to be used in estimating problem-dependent self-shielded cross sections in MPACT (Michigan Parallel Characteristics Transport Code). Although this procedure has been developed and thus resonance data have been generated and validated by benchmark calculations, assessment has never been performed to review if the resonance data are properly generated by the procedure and utilized in MPACT. This study focuses on assessing the procedure and a proper use in MPACT.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗