1D Monoenergetic Neutron Transport Theory

Barry Ganapol, University of Arizona

Did you ever wonder about the variety of solutions to solve the 1D monoenergetic neutron transport equation? Chances are you have not, but if you attend this workshop, you will have the answer — because it’s all about 1D transport solutions. In four hours, we will discuss as many of the following methods as time allows:

  • Converged SN
  • Doubling
  • Matrix Riccati solution
  • Response matrix discrete ordinates Double PN
  • Green’s function method Full-range Caseology
  • FN method.

Our discussion covers the mathematical theory, the numerical formulation and the challenges of each (time permitting). The methods will be theoretically and numerically contrasted to feature their advantages and disadvantages. You might ask, “Why study such a basic transport problem with seemingly little practical value?” One answer is “benchmarking”. Because of simplicity, the 1D monoenergetic transport equation is the most widely solved transport equation in today’s transport community. A second answer is “intellectual enlightenment”. The solutions discussed touch upon a broad range of mathematical and numerical methods taught in the classroom. Specifically, we will discuss what constitutes extreme benchmarks, their application and limitation. Convergence acceleration, central to extreme benchmarks, will be introduced through a specially prepared benchmarking exercise.

If you are a serious student of transport theory and have the burning desire to learn more about analytical solutions from an expert in the field, you certainly do not want to miss this opportunity. The mystery of the 1D transport equation will be unraveled in an understandably consistent way. In addition, each participant, who completes the workshop, will receive a flash drive containing programs and examples of all solutions.

Asymptotic Diffusion Limit of the SN Neutron Transport Equation

Dean Wang, The Ohio State University

Speaker: Dean Wang (The Ohio State University).  It has been well known that the analytic neutron transport solution limits to the analytic solution of a diffusion problem for optically thick systems with small absorption and source. The standard technique for proving the asymptotic diffusion limit is constructing an asymptotic power series of the neutron angular flux in small positive parameter, which is the ratio of a typical mean free path of a particle to a typical dimension of the domain under consideration. In this workshop, we will present a new proof to directly show that the analytical neutron transport solution satisfies the diffusion equation at the asymptotic limit based on a recently obtained closed-form analytical solution of the monoenergetic SN neutron transport equation in slab geometry. In numerical solution of the SN neutron transport equation, a spatial discretization is of practical interest if it possesses the optically thick diffusion limit. Such a numerical scheme will yield accurate solutions for diffusive problems if the spatial mesh size is thin with respect to a diffusion length, whereas the mesh cells are thick in terms of a mean free path. We will present a recently obtained theoretical result on the asymptotic diffusion limit of numerical schemes and what mesh sizes should be used to achieve accurate results. In addition, we will present an interesting implication of the asymptotic diffusion limit on Fourier analysis for CMFD schemes. Audience: anyone.


Alireza Haghihat, Virginia Tech

RAPID (Real-time Analysis for Particle-transport In-situ Detection) is developed based on the MRT (Multi-stage Response-function particle Transport) methodology that enables its real-time simulation capability. The current version of RAPID is capable of simulating nuclear systems such as spent fuel pools, spent fuel casks, and reactor cores. RAPID solves for pin-wise, axially-dependent fission density, critical/subcritical multiplication, and detector response. Recently, new algorithm for 3-D fuel burnup (bRAPID) calculation and reactor kinetics (tRAPID) have been developed and benchmarked for test problems. Experimental benchmarking for these latter algorithms are underway using the Jozef Stefan Institute’s TRIGA research reactor.

Further, a multi-user virtual reality system (VRS) has been developed that provides a web application for input preparation, real-time simulation, and output processing and visualization in a virtual environment. For an introduction, please view the following demo

Topics to be covered:

  • RAPID’s MRT methodology and formulation
  • RAPID code system and benchmarking
  • VRS-RAPID demonstration
  • Hands-on use by participants

Requirements:  There will be access to wireless internet so that the participants can have remote access to VRS- RAPID. The current version of VRS-RAPID is optimized for a Personal Computer using the Google Chrome browser, but it can be accessed through iPad, Tablet, etc. using any other browser.  To facilitate establishing individual accounts, participants are encouraged to contact Prof. Haghighat prior to the workshop.


Wiliam Wieselquist, Oak Ridge National Laboratory

SCALE is a comprehensive modeling and simulation suite for nuclear safety analysis and design developed and maintained by Oak Ridge National Laboratory under contract with the U.S. Nuclear Regulatory Commission, U.S. Department of Energy, and the National Nuclear Security Administration to perform reactor physics, criticality safety, radiation shielding, and spent fuel characterization for nuclear facilities and transportation/storage package designs. This workshop will present recent light water reactor (LWR) and non-LWR applications using SCALE. In LWR space, this includes High Assay Low-enriched Uranium (HALEU) up to 8% enrichment, High Burnup (HBU) up to 80 GWd/MTU, and Accident Tolerant Fuel (ATF) concepts. A hands-on tutorials will be given on estimating isotopic bias and bias uncertainty for 80 GWd/MTU spent fuel. In non-LWR space, this includes neutronics and inventory calculations for the following reactor types: Sodium Fast Reactor (SFR), Fluoride High Temperature Reactor (FHR), High Temperature Gas Reactor (HTGR), and Molten Salt Reactor (MSR). An open-source SCALE model for each reactor type will be provided to participants and a hands-on tutorial will demonstrate a recommended analysis workflow using the SCALE GUI, Fulcrum. The tutorials will be taught with the latest 6.3 beta (or 6.3.0 release if available at time of conference), although the majority of the analyses may be performed with the latest release in the previous 6.2 series.

Scientific Machine Learning for Nuclear Engineering

Xu Wu, North Carolina State University

Machine Learning (ML) is a subset of Artificial Intelligence (AI) that is the study of computer algorithms that improve automatically through experience (data). Deep Learning (DL) is a subset of ML that uses multi-layered neural networks to deliver state-of-the-art accuracy in tasks such as object detection, speech recognition, language translation and others. Scientific Machine Learning (SciML), more specifically, consists of computational technologies that can be trained with scientific data to augment or automate human skills. ML has been very successful in areas such as computer vision, natural language processing, etc. But its application in scientific computing is relatively new, especially in Nuclear Engineering (NE). This workshop aims at augmenting the applications of AI/ML in scientific computing in NE, and promoting ML-based transformative solutions across various DOE missions.

There are many ML/DL libraries/APIs available, such as Keras, TensorFlow, PyTorch, MXNET, etc. Have you had difficulties in understanding their differences and selecting the most appropriate one for your research? Furthermore, ML has a rich taxonomy of different applications, such as Supervised Learning, Unsupervised Learning, Semi-supervised Learning, Reinforcement Learning, Transfer Learning, etc, as well as different algorithms for tasks such as regression, classification, dimensionality reduction and clustering, etc. Did you ever wonder which one of these algorithms is best suitable for your problem at hand? Finally, ML/DL have been providing powerful tools for scientific computing, but in most cases they serve as black-box surrogates. Quantifying the extra uncertainties introduced by ML/DL models is a critical issue. But are there efficient ways to do so?

In this workshop, we will provide (1) introduction, review and comparison of various ML libraries/APIs, (2) instructions on how to select the technique with the best performance for common computational problems in NE, (3) a review of techniques available to quantify the prediction uncertainties in ML/DL models, with a focus on Artificial Neural Networks (ANNs).