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DOE OSTI2015

Temperature Profile in Fuel and Tie-Tubes for Nuclear Thermal Propulsion Systems

A finite element method to calculate temperature profiles in heterogeneous geometries of tie-tube moderated LEU nuclear thermal propulsion systems and HEU designs with tie-tubes is developed and implemented in MATLAB. This new method is compared to previous methods to demonstrate shortcomings in those methods. Typical methods to analyze peak fuel centerline temperature in hexagonal geometries rely on spatial homogenization to derive an analytical expression. These methods are not applicable to cores with tie-tube elements because conduction to tie-tubes cannot be accurately modeled with the homogenized models. The fuel centerline temperature directly impacts safety and performance so it must be predicted carefully. The temperature profile in tie-tubes is also important when high temperatures are expected in the fuel because conduction to the tie-tubes may cause melting in tie-tubes, which may set maximum allowable performance. Estimations of maximum tie-tube temperature can be found from equivalent tube methods, however this method tends to be approximate and overly conservative. A finite element model of heat conduction on a unit cell can model spatial dependence and non-linear conductivity for fuel and tie-tube systems allowing for higher design fidelity of Nuclear Thermal Propulsion.

33 ADVANCED PROPULSION SYSTEMS↗
DOE OSTI2023

Machine learning simulation of finite element analysis in augmented reality

Media, method and system for approximating a finite element analysis texture map for an object. To accomplish this, the object is converted to a computer generated model and finite element analysis is performed for a plurality of different simulated inputs to generate a plurality of simulated mappings. Each simulated mapping is converted into a simulated texture map. A machine learning model is trained on the simulated inputs and simulated texture maps to generate a texture map which approximates a finite element analysis. The machine learning model receives a user input and generates the texture map therefrom. The texture map is then wrapped to the object and displayed.

Scherer, Derek Carl↗
DOE OSTI2023

DeepSurveySim: Simulation Software and Benchmark Challenges for Astronomical Observation Scheduling

Modern astronomical surveys have multiple competing scientific goals. Optimizing the observation schedule for these goals presents significant computational and theoretical challenges, and state-of-the-art methods rely on expensive human inspection of simulated telescope schedules. Automated methods, such as reinforcement learning, have recently been explored to accelerate scheduling. However, there do not yet exist benchmark data sets or user-friendly software frameworks for testing and comparing these methods. We present DeepSurveySim -- a high-fidelity and flexible simulation tool for use in telescope scheduling. DeepSurveySim provides methods for tracking and approximating sky conditions for a set of observations from a user-supplied telescope configuration. We envision this tool being used to produce benchmark data sets and for evaluating the efficacy of ground-based telescope scheduling algorithms, particularly for machine learning algorithms that would suffer in efficacy if limited to real data for training.We introduce three example survey configurations and related code implementations as benchmark problems that can be simulated with DeepSurveySim.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2018

Shrinking Self-Interaction Errors with the Fermi–Löwdin Orbital Self-Interaction-Corrected Density Functional Approximation

The self-interaction error (SIE) is one of the major drawbacks of practical exchange-correlation functionals for Kohn-Sham density functional theory. Despite this, the use of methods that explicitly remove SIE from approximate density functionals is scarce in the literature due to their relatively high computational cost and lack of consistent improvement over standard modern functionals. Here, we assess the performance of a novel approach recently proposed by Pederson, Ruzsinszky, and Perdew for performing self-interaction free calculations in density functional theory based on Fermi-orbitals. To this end, we employ test sets consisting of reaction energies that are considered particularly sensitive to SIE. We found that the parameter-free Fermi-Löwdin orbital self-interaction correction method combined with standard local spin density approximation (LSDA) and Perdew-Burke-Ernzerhof (PBE) functionals gives a much better estimate of reaction energies compared to their parent LSDA and PBE functionals for most of the reactions in these two sets. They also perform on par with the global PBE0 and range-separated LC-ωPBE hybrids, which partially eliminate the SIE by including Hartree-Fock exchange. This shows the potential of the Fermi-Löwdin orbital self-interaction corrected (FLOSIC) method for practical density functional calculations without SIE.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2019

Eigenvalue Problems for Exponential-Type Kernels

Here, We study approximations of eigenvalue problems for integral operators associated with kernel functions of exponential type. We show convergence rate | λ k - λ k , h | ≤ C k h 2 in the case of lowest order approximation for both Galerkin and Nystrom methods, where h is the mesh size, λ k and λ k , h are the exact and approximate k th largest eigenvalues, respectively. We prove that the two methods are numerically equivalent in the sense that | λ k , h ( G ) - λ k , h ( N ) | ≤ C h 2 , where λ k , h ( G ) and λ k , h ( N ) denote the k th largest eigenvalues computed by Galerkin and Nystrom methods, respectively, and C is a eigenvalue independent constant. The theoretical results are accompanied by a series of numerical experiments.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Variational multiscale element-free Galerkin method combined with the moving Kriging interpolation for solving some partial differential equations with discontinuous solutions

Solving partial differential equations with discontinuous solutions is an important challenging problem in numerical analysis. To this end, there are some methods such as finite volume method, discontinuous Galerkin approach and particle technique that are able to solve these problems. In the current paper, the moving Kriging element-free Galerkin method has been combined with the variational multiscale algorithm to obtain acceptable and high-resolution solutions. For testing this technique, we select some PDEs with discontinuous solution such as Burgers’, Sod’s shock tube, advection–reaction–diffusion, Kuramoto–Sivashinsky, Boussinesq and shallow water equations. First, we obtain a time-discrete scheme by approximating time derivative via finite difference technique. Then we introduce the moving Kriging interpolation and also obtain their shape functions. We use the element-free Galerkin method for approximating the spatial derivatives. This method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the classical element-free Galerkin method test and trial functions are moving least squares approximation (MLS) shape functions. Since the shape functions of moving least squares (MLS) approximation do not have Kronecker delta property, we cannot implement the essential boundary condition, directly. Thus, we employ the shape functions of moving Kriging interpolation and radial point interpolation technique which have the mentioned property. Also, in the element-free Galekin method, we do not use any triangular, quadrangular or other type of meshes. The element-free Galerkin method is a global method while finite element method is a local one. This technique employs a background mesh for integration which makes it different from the truly mesh procedures. Several test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed schemes.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

A meshless Galerkin method for non-local diffusion using localized kernel bases

Here, we introduce a meshless method for solving both continuous and discrete variational formulations of a volume constrained, nonlocal diffusion problem. We use the discrete solution to approximate the continuous solution. Our method is nonconforming and uses a localized Lagrange basis that is constructed out of radial basis functions. By verifying that certain inf-sup conditions hold, we demonstrate that both the continuous and discrete problems are well-posed, and also present numerical and theoretical results for the convergence behavior of the method. The stiffness matrix is assembled by a special quadrature routine unique to the localized basis. Combining the quadrature method with the localized basis produces a well-conditioned, symmetric matrix. This then is used to find the discretized solution.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Embedded, Real-Time, and Distributed Traveling Wave Fault Location Method Using Graph Convolutional Neural Networks

This work proposes and develops an implementation of a fault location method to provide a fast and resilient protection scheme for power distribution systems. The method analyzes the transient dynamics of traveling waves (TWs) to generate features using the discrete wavelet transform (DWT), which are then used to train several graph convolutional network (GCN) models. Faults are simulated in the IEEE 34-node system, which is divided into three protection zones (PZs). The goal is to identify the PZ in which the fault occurs. The GCN models create a distributed protection scheme, as all nodes are able to retrieve a prediction. Given that message-passing between nodes occurs both during training and in the execution of the model, the resiliency of such schemes to communication losses was analyzed and demonstrated. One of the models, which only uses voltage measurements, was implemented on a Texas Instruments F28379D development board. The execution times were monitored to assess the speed of the protection scheme. It is shown that the proposed method can be executed in approximately a millisecond, which is comparable to existing TW protection in the transmission system. For experimental purposes, a DWT-based detection method is employed. A design of a setup to playback TWs using two development boards is also addressed.

Jiménez-Aparicio, Miguel (ORCID:000000016864461X)↗
DOE OSTI2019

Interplay between Approximation Theory and Renormalization Group

The review presents general methods for treating complicated problems that cannot be solved exactly and whose solution encounters two major difficulties. First, there are no small parameters allowing for the safe use of perturbation theory in powers of these parameters, and even when small parameters exist, the related perturbative series are strongly divergent. Second, such perturbative series in powers of these parameters are rather short, so that the standard resummation techniques either yield bad approximations or are not applicable at all. The emphasis in the review is on the methods advanced and developed by the author. One of the general methods is Optimized Perturbation Theory now widely employed in various branches of physics, chemistry, and applied mathematics. The other powerful method is Self-Similar Approximation Theory allowing for quite simple and accurate summation of divergent series. These theories share many common features with the method of renormalization group, which is briefly sketched in order to stress the similarities in their ideas and their mutual interconnection.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2024

Monte Carlo Explicitly Correlated Second-Order Many-Body Green’s Function Calculations of Semiconductor Band Gaps

A systematically converging series of ab initio, post-density-functional, size-consistent, electron-correlated approximations is desired for predictive computing of felectronic band structures of insulating, semiconducting, and metallic solids. A series that meets all of these desiderata (except the applicability to metals) is ab initio many-body Green's function theory based on Gaussian-type-orbital (GTO) basis sets. Here, its leading-order approximation, the second-order Green's function (GF2) method in the diagonal and frequency-independent approximations with the aug-cc-pVDZ basis set, is applied to the fundamental band gaps of three semiconductors (diamond, silicon, and silicon carbide in the zincblende structure) using cluster models. Corrections are made to the basis-set-incompleteness errors by the explicit-correlation (F12) ansatz (GF2-F12) for the valence band edges. The crystals are modeled as surface-passivated clusters of increasing sizes, whose wave functions are expanded by up to 2709 GTO basis functions. Immense computational costs of these calculations are overcome by the highly scalable stochastic algorithm of the Monte Carlo GF2-F12 method, whose operation cost per state increases only as a cubic power of system size, which has a tiny memory footprint and easily achieves near-perfect parallel efficiency on thousands of CPUs or on hundreds of GPUs. The correlated, F12-corrected highest-occupied and lowest-unoccupied molecular-orbital energy (HOMO-LUMO) gap is 5.78 ± 0.07 eV for C 87 H 76 as compared with the experimental value of the fundamental (indirect) band gap of bulk diamond at 5.48 eV. The correlated, F12-corrected HOMO-LUMO gaps for Si 75 H 76 and Si 32 C 43 H 76 are 2.56 ± 0.15 eV and 3.50 ± 0.12 eV, respectively, which are expected to decrease further with increasing cluster sizes. As a result, the experimental fundamental (indirect) band gaps of bulk silicon and silicon carbide are 1.17 eV and 2.42 eV, respectively.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2020

Two-particle correlation functions in cluster perturbation theory: Hubbard spin susceptibilities

Cluster perturbation theory (CPT) is a computationally economic method commonly used to estimate the momentum- and energy-resolved single-particle Green's function. It has been used extensively in direct comparisons with experiments that effectively measure the single-particle Green's function, e.g., angle-resolved photoemission spectroscopy. However, many experimental observables are given by two-particle correlation functions. CPT can be extended to compute two-particle correlation functions by approximately solving the Bethe-Salpeter equation. We implement this method and focus on the transverse spin susceptibility, measurable via inelastic neutron scattering or with optical probes of atomic gases in optical lattices. Finally, we benchmark the method with the one-dimensional Fermi-Hubbard model by comparing with known results.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2017

Surrogate Models for TREAT Transient Calculations

The Transient Reactor Test Facility (TREAT) at Idaho National Laboratory (INL) is currently preparing for resumption of operation after more than 20 years in a long-term standby state. In parallel with preparation for operations and a new transient testing program, the DOENE Advanced Modeling and Simulation program (NEAMS) has been supporting the development of three-dimensional transient multi-physics modeling capabilities within the reactor physics application MAMMOTH. Recently, the MAMMOTH reactor physics team has begun development of uncertainty and sensitivity analysis for the TREAT core. This research will be used to support the analysis of reactor operation and experiment planning. Traditional approaches, i.e., Monte Carlo simulations, for such tasks require a large number of model evaluations under both steady state and transient conditions. However, exhaustive sets of sampling runs are simply not practical for TREAT since evaluations of each model can take a substantial amount of time. A solution to this problem is to build a surrogate model employing a smaller number of model evaluations. The surrogate model is designed to provide an accurate and fast approximation to the detailed TREAT model. In this manner, the surrogate model can be used in place of the TREAT transient model for uncertainty and sensitivity analysis at a fraction of the computational cost. The high dimensional model representation (HDMR) method is employed here for approximating the responses of interest in terms of functions of lower dimensions. This method has been implemented in RAVEN and all TREAT transient calculations are performed using the multiphysics code MAMMOTH. In addition, a code interface has also been developed in order to couple RAVEN with MAMMOTH. The surrogate model is then tested and compared for TREAT startup testing program Transient Test no. 15 core, and its accuracy is measured by its ability to estimate the mean, coefficient of variance, skewness, excess kurtosis and probability density function of the outputs of Transient Test no. 15 model, as well as root mean square error metric.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2024

Laplace's method applied to functional integrals

In statistical thermodynamics, partition functions are required to evaluate the current state. It is typically impossible to analytically calculate the resulting configuration integrals, but it is sometimes possible to obtain approximations in the limit of steep potential energies. This approach directly leverages Laplace’s method for approximating integrals and has been quite successful in recent applications. Motivated by models with an infinite number of degrees of freedom, this approach should be extended to the statistical thermodynamics of fields. To that end, Laplace’s method is developed for functional integrals.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Surrogate Models for TREAT Transient Calculations

The Transient Reactor Test Facility (TREAT) at Idaho National Laboratory (INL) is currently preparing for resumption of operation after more than 20 years in a long-term standby state. In parallel with preparation for operations and a new transient testing program, the DOE-NE Advanced Modeling and Simulation program (NEAMS) has been supporting the development of three-dimensional transient multi-physics modeling capabilities within the reactor physics application MAMMOTH [1]. Recently, the MAMMOTH reactor physics team has begun development of uncertainty and sensitivity analysis for the TREAT core. This research will be used to support the analysis of reactor operation and experiment planning. Traditional approaches, i.e., Monte Carlo simulations, for such tasks require a large number of model evaluations under both steady state and transient conditions. However, exhaustive sets of sampling runs are simply not practical here since evaluations of each model can take a substantial amount of time. A solution to this problem is to build a surrogate model employing a smaller number of model evaluations. The surrogate model is designed to provide an accurate and fast approximation to the detailed TREAT model. In this manner, the surrogate model can be used in place of the TREAT transient model for uncertainty and sensitivity analysis at a fraction of the computational cost. The high dimensional model representation (HDMR) [2] method is employed here for approximating the responses of interest in terms of functions of lower dimensions. This method has been implemented in RAVEN [3] and all TREAT transient calculations are performed using the multiphysics code MAMMOTH. In addition, a code interface has also been developed in order to couple RAVEN with MAMMOTH. The surrogate model is then tested and compared for the MAMMOTH model of the TREAT startup testing program Transient Test #15 core [4], and its accuracy is measured by its ability to estimate the mean, coefficient of variance, skewness, excess kurtosis and probability density function of the outputs of Transient Test #15 model, as well as root mean square error metric.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2017

Final Report of the Project "From the finite element method to the virtual element method"

The Finite Element Method (FEM) is a powerful numerical tool that is being used in a large number of engineering applications. The FEM is constructed on triangular/tetrahedral and quadrilateral/hexahedral meshes. Extending the FEM to general polygonal/polyhedral meshes in straightforward way turns out to be extremely difficult and leads to very complex and computationally expensive schemes. The reason for this failure is that the construction of the basis functions on elements with a very general shape is a non-trivial and complex task. In this project we developed a new family of numerical methods, dubbed the Virtual Element Method (VEM) for the numerical approximation of partial differential equations (PDE) of elliptic type suitable to polygonal and polyhedral unstructured meshes. We successfully formulated, implemented and tested these methods and studied both theoretically and numerically their stability, robustness and accuracy for diffusion problems, convection-reaction-diffusion problems, the Stokes equations and the biharmonic equations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2015

Electron impact excitation of helium in Debye plasma

The probability, differential, and integral scattering cross sections of the 1{sup 1}S→2{sup 1}S and 1{sup 1}S→2{sup 1}P transitions of helium have been calculated in the first Born approximation. The projectile-target interactions depending on the temperature and the density of plasma are described by the Debye-Hückel model. Wave functions of the target before and after collision were modeled by non orthogonal Hartree-Fock orbitals. The wave functions parameters are calculated with the Ritz variational method. We improve our unscreened first Born approximation integral cross sections by using the BE-scaled (B stands for binding energy and E excitation energy) method. The second Born approximation has also been used to calculate the excitation cross sections in Debye plasma. Our calculations are compared to other theoretical and experimental results where applicable.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2018

Sufficient Conditions for the Equivalent Convergence of Sequences of Different Approximants for Two-Dimensional Continued Fractions

By the method of fundamental inequalities, we study the equivalent convergence of two figured approximants for two-dimensional continued fractions, namely, the approximants obtained from the problem of correspondence of two-dimensional continued fractions to some formal double power series and the problem of equivalence of two-dimensional continued fractions. We also establish sufficient conditions under which the indicated two approximants of two-dimensional continued fractions converge to the same limit.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Exploiting Kronecker structure in exponential integrators: Fast approximation of the action of φ $-$functions of matrices via quadrature

Here, in this article, we propose an algorithm for approximating the action of φ $-$ functions of matrices against vectors, which is a key operation in exponential time integrators. In particular, we consider matrices with Kronecker sum structure, which arise from problems admitting a tensor product representation. The method is based on quadrature approximations of the integral form of the φ $-$ functions combined with a scaling and modified squaring method. Owing to the Kronecker sum representation, only actions of 1D matrix exponentials are needed at each quadrature node and assembly of the full matrix can be avoided. Additionally, we derive a priori bounds for the quadrature error, which show that, as expected by classical theory, the rate of convergence of our method is supergeometric. Guided by our analysis, we construct a fast and robust method for estimating the optimal scaling factor and number of quadrature nodes that minimizes the total cost for a prescribed error tolerance. We investigate the performance of our algorithm by solving several linear and semilinear time-dependent problems in 2D and 3D. The results show that our method is accurate and orders of magnitude faster than the current state-of-the-art.

97 MATHEMATICS AND COMPUTING↗