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DOE OSTI2025

A Particle-in-cell Method for Plasmas with A Generalized Momentum Formulation, Part III: A family of Gauge Conserving Methods

In this paper, we introduce a new family of spatially co-located field solvers for particle-in-cell applications which evolve the potential formulation of Maxwell’s equations under the Lorenz gauge. Our recent work [2] introduced the concept of time-consistency, which connects charge conservation to the preservation of the gauge at the semi-discrete level. It will be shown that there exists a large family of time discretizations which satisfy this property. Additionally, it will be further shown that for large classes of time marching methods, the satisfaction of the gauge condition automatically implies the satisfaction of Gauss’s law for electricity, with the potential formulation ensuring that that Gauss’s law for magnetism is satisfied by definition. We focus on popular time marching methods including centered differences, backward differences, and diagonally-implicit Runge-Kutta methods, which are coupled to a spectral discretization in space. We demonstrate the theory by testing the methods on a relativistic Weibel instability and a drifting cloud of electrons.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2013

GEOS Code Development Road Map

GEOS is a massively parallel computational framework designed to enable HPC-based simulations of subsurface reservoir stimulation activities with the goal of optimizing current operations and evaluating innovative stimulation methods. GEOS will enable coupling of different solvers associated with the various physical processes occurring during reservoir stimulation in unique and sophisticated ways, adapted to various geologic settings, materials and stimulation methods. The overall architecture of the framework includes consistent data structures and will allow incorporation of additional physical and materials models as demanded by future applications. Along with predicting the initiation, propagation and reactivation of fractures, GEOS will also generate a seismic source term that can be linked with seismic wave propagation codes to generate synthetic microseismicity at surface and downhole arrays. Similarly, the output from GEOS can be linked with existing fluid/thermal transport codes. GEOS can also be linked with existing, non-intrusive uncertainty quantification schemes to constrain uncertainty in its predictions and sensitivity to the various parameters describing the reservoir and stimulation operations. We anticipate that an implicit-explicit 3D version of GEOS, including a preliminary seismic source model, will be available for parametric testing and validation against experimental and field data by Oct. 1, 2013.

02 PETROLEUM↗

DOE OSTI2015

Decreasing the temporal complexity for nonlinear, implicit reduced-order models by forecasting

Implicit numerical integration of nonlinear ODEs requires solving a system of nonlinear algebraic equations at each time step. Each of these systems is often solved by a Newton-like method, which incurs a sequence of linear-system solves. Most model-reduction techniques for nonlinear ODEs exploit knowledge of system's spatial behavior to reduce the computational complexity of each linear-system solve. However, the number of linear-system solves for the reduced-order simulation often remains roughly the same as that for the full-order simulation. We propose exploiting knowledge of the model's temporal behavior to (1) forecast the unknown variable of the reduced-order system of nonlinear equations at future time steps, and (2) use this forecast as an initial guess for the Newton-like solver during the reduced-order-model simulation. To compute the forecast, we propose using the Gappy POD technique. As a result, the goal is to generate an accurate initial guess so that the Newton solver requires many fewer iterations to converge, thereby decreasing the number of linear-system solves in the reduced-order-model simulation.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2017

On performance of Krylov smoothing for fully-coupled AMG preconditioners for VMS resistive MHD

Here, this study explores the performance and scaling of a GMRES Krylov method employed as a smoother for an algebraic multigrid (AMG) preconditioned Newton- Krylov solution approach applied to a fully-implicit variational multiscale (VMS) nite element (FE) resistive magnetohydrodynamics (MHD) formulation. In this context a Newton iteration is used for the nonlinear system and a Krylov (GMRES) method is employed for the linear subsystems. The efficiency of this approach is critically dependent on the scalability and performance of the AMG preconditioner for the linear solutions and the performance of the smoothers play a critical role. Krylov smoothers are considered in an attempt to reduce the time and memory requirements of existing robust smoothers based on additive Schwarz domain decomposition (DD) with incomplete LU factorization solves on each subdomain. Three time dependent resistive MHD test cases are considered to evaluate the method. The results demonstrate that the GMRES smoother can be faster due to a decrease in the preconditioner setup time and a reduction in outer GMRESR solver iterations, and requires less memory (typically 35% less memory for global GMRES smoother) than the DD ILU smoother.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2016

Development of the FHR advanced natural circulation analysis code and application to FHR safety analysis

The University of California, Berkeley (UCB) is performing thermal hydraulics safety analysis to develop the technical basis for design and licensing of fluoride-salt-cooled, high-temperature reactors (FHRs). FHR designs investigated by UCB use natural circulation for emergency, passive decay heat removal when normal decay heat removal systems fail. The FHR advanced natural circulation analysis (FANCY) code has been developed for assessment of passive decay heat removal capability and safety analysis of these innovative system designs. The FANCY code uses a one-dimensional, semi-implicit scheme to solve for pressure-linked mass, momentum and energy conservation equations. Graph theory is used to automatically generate a staggered mesh for complicated pipe network systems. Heat structure models have been implemented for three types of boundary conditions (Dirichlet, Neumann and Robin boundary conditions). Heat structures can be composed of several layers of different materials, and are used for simulation of heat structure temperature distribution and heat transfer rate. Control models are used to simulate sequences of events or trips of safety systems. A proportional-integral controller is also used to automatically make thermal hydraulic systems reach desired steady state conditions. A point kinetics model is used to model reactor kinetics behavior with temperature reactivity feedback. The underlying large sparse linear systems in these models are efficiently solved by using direct and iterative solvers provided by the SuperLU code on high performance machines. Input interfaces are designed to increase the flexibility of simulation for complicated thermal hydraulic systems. In conclusion, this paper mainly focuses on the methodology used to develop the FANCY code, and safety analysis of the Mark 1 pebble-bed FHR under development at UCB is performed.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

DOE OSTI2025

Sparse chronology strategy for integrating seasonal energy storage in capacity expansion models

Here, this study develops the sparse chronology method to enhance the representative period framework in capacity expansion models, enabling the effective integration of long-duration energy storage modeling. Traditional representative period methods cannot capture the state of charge of seasonal energy storage systems because they do not establish effective inter-day linkages to connect the state of charge between periods. The sparse chronology approach addresses this limitation by establishing inter-day linkages that allow state of charge to shift inter-seasonally. At the same time, it groups identical representative days into partitions, applying constraints sparsely and implicitly to reduce computational load further. Validation results demonstrate that this method successfully simulates long-duration energy storage patterns, achieving close alignment with a continuous yearly benchmark model, with seasonal trends and state of charge cycles clearly represented. The computational load analysis reveals that the sparse chronology method efficiently applies constraints on maximum and minimum state of charge limits within the representative day framework, eliminating the need for detailed constraints on each individual day. By partitioning representative days and constraining only the start and end of each partition, the method significantly decreases computational requirements. Simulation results show that sparse chronology closely approximates the continuous yearly method's accuracy, even with as few as 20 representative days, achieving correlation values with the benchmark of nearly 0.9 in state of charge plots. Furthermore, it maintains computational efficiency, requiring only 4 % of the solver time compared to the continuous yearly method with 20 representative days. This approach allows capacity expansion models to incorporate long-duration energy storage with high temporal, spatial, and technological resolution, enabling more detailed modeling for large-scale power systems.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2024

Semi-Lagrangian nodal discontinuous Galerkin method for the BGK model

In this work, we propose a semi-Lagrangian (SL) nodal discontinuous Galerkin (DG) solver for the BGK equation. The BGK model was introduced by Bhatnagar, Gross, and Krook [1] as a relaxation model for the fundamental Boltzmann equation [5], which describes the kinetic dynamic of rarefied gases with a probability distribution function. The challenges of designing efficient numerical schemes for the Boltzmann equation mainly come from its high dimensionality and complicated nonlinear collision operator. The BGK model gains interests since it has much lower computational cost, due to the relatively simple structure of the relaxation operator in replacement of the collision operator, while simultaneously preserving several important physical properties, such as macroscopic quantities and dissipation of entropy.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2015

Numerical modeling of the SNS H{sup −} ion source

Ion source rf antennas that produce H- ions can fail when plasma heating causes ablation of the insulating coating due to small structural defects such as cracks. Reducing antenna failures that reduce the operating capabilities of the Spallation Neutron Source (SNS) accelerator is one of the top priorities of the SNS H- Source Program at ORNL. Numerical modeling of ion sources can provide techniques for optimizing design in order to reduce antenna failures. There are a number of difficulties in developing accurate models of rf inductive plasmas. First, a large range of spatial and temporal scales must be resolved in order to accurately capture the physics of plasma motion, including the Debye length, rf frequencies on the order of tens of MHz, simulation time scales of many hundreds of rf periods, large device sizes on tens of cm, and ion motions that are thousands of times slower than electrons. This results in large simulation domains with many computational cells for solving plasma and electromagnetic equations, short time steps, and long-duration simulations. In order to reduce the computational requirements, one can develop implicit models for both fields and particle motions (e.g. divergence-preserving ADI methods), various electrostatic models, or magnetohydrodynamic models. We have performed simulations using all three of these methods and have found that fluid models have the greatest potential for giving accurate solutions while still being fast enough to perform long timescale simulations in a reasonable amount of time. We have implemented a number of fluid models with electromagnetics using the simulation tool USim and applied them to modeling the SNS H- ion source. We found that a reduced, single-fluid MHD model with an imposed magnetic field due to the rf antenna current and the confining multi-cusp field generated increased bulk plasma velocities of > 200 m/s in the region of the antenna where ablation is often observed in the SNS source. We report here on comparisons of simulated plasma parameters and code performance using more accurate physical models, such as two-temperature extended MHD models, for both a related benchmark system describing a inductively coupled plasma reactor, and for the SNS ion source. We also present results from scaling studies for mesh generation and solvers in the USim simulation code.

07 ISOTOPE AND RADIATION SOURCES↗

DOE OSTI2015

brulilo, Version 0.x

brulilo is a Python package for building and evolving thermonuclear reaction networks, such as those used in astrophysical explosion calculations. It is very much a current work in progress, and much of the intended functionality is still under development. The main goal of brulilo is an easy-to-use code for quick calculations or post-processing, as well as a learning tool to test various implementations of solvers on both CPUs and GPUs. Currently, brulilo parses an input text file that specifies the reactions to be included in the network. The notation of the inputs are in the typical astrophysical notation; i.e. hydrogen burning to deuterium in the sun, "p + p + e --> d + neutrino", is given as "p(p,)d." Each reaction is added to a list comprising the reaction network, and once completed, the network contains a list of all isotopes and reactions involved. brulilo then parses publicly available nuclear data (binding energy, spin, partition function data, etc.) for each isotope from the WebNucleo group's data at nucleo.ces.clemson.edu. Fits to reaction rate data for each reaction are obtained from the publicly available data provided by the Joint Institute for Nuclear Astrophysics' Reaclib database. Reverse reactions are calculated from detailed balance. The reaction network then consists of a sparse system of mathematically stiff ordinary differential equations (ODEs). Physically, "stiffness" here means that there are some reaction rates that are occurring much faster than others; the ratio of the fast to slow often being greater than 10^15 in astrophysics. Numerical integration of such systems typically involves implicit integration schemes, which brulilo will leverage from the scientific computing Python package, Numpy, as well as implement its own. There has recently been work from researchers at Oak Ridge National Lab who have found good results from applying quasi-equilibrium and quasi-asymptotic conditions to the stiff system of ODEs, which effectively remove some of the stiffness and allow for efficient explicit integration techniques to be used. The original intent of brulilo was to implement these stiffness-alleviating techniques with explicit integrators and compare the performance to traditional implicit integrations of the full stiff system. This is still underway, as the code is very much in an alpha-release state. Furthermore, explicit integrators are often much easier to parallelize than their implicit counterparts. brulilo will implement parallelization of these techniques, leveraging both the Python implementation of MPI, mpi4py, as well as highly parallelized versions targeted at GPUs with PyOpenCL and/or PyCUDA.

Malone, Chris↗

DOE OSTI2015

brulilo, Version 0.x

Malone, Chris↗

DOE OSTI2023

Simulating Catalysis with Realistic Pellet Geometries Using Mesoflow: A Case Study of Catalytic Propane Dehydrogenation

We present a case study of catalytic propane dehydrogenation with our open-source multiphysics solver, Mesoflow. The solver was developed to simulate reactive flow coupled to heterogeneous catalytic reactions and deactivation in the context of complex, mesoscale geometry. The method leverages cartesian block-structured adaptive mesh refinement to capture realistic catalyst microstructural features acquired directly from X-ray computed tomography data. A kinetic model for propane dehydrogenation and catalyst deactivation was developed based on temporal analysis of products (TAP) reactor experiments. The TAP reactor experiments allow for precise characterization of intrinsic kinetic reaction steps which are implemented into Mesoflow simulations to model the spatial and temporal evolution of reactants, products, and catalyst active sites. The short-term and long-term deactivation behavior is studied by using XCT data collected from fresh and aged catalyst pellets, which exhibit different microstructural features. This study employs time-splitting algorithms to connect disparate reaction and flow timescales, enabling the simulations to achieve realistic deactivation timescales on the order of minutes while the flow time-scales for small particles (100 microns) are several milliseconds. We also introduce a flexible automated python script that writes the necessary files to construct a Mesoflow simulation from user-created chemical mechanisms. We will also introduce a few new features that are added to Mesoflow such as higher order schemes, implicit chemistry integrators and the ability to run on AMD and NVIDIA graphics-processing-units.

AMReX↗

DOE OSTI2016

Center for Extended Magnetohydrodynamics Modeling - Final Technical Report

This project funding supported approximately 74 percent of a Ph.D. graduate student, not including costs of travel and supplies. We had a highly successful research project including the development of a second-order implicit electromagnetic kinetic ion hybrid model [Cheng 2013, Sturdevant 2016], direct comparisons with the extended MHD NIMROD code and kinetic simulation [Schnack 2013], modeling of slab tearing modes using the fully kinetic ion hybrid model and finally, modeling global tearing modes in cylindrical geometry using gyrokinetic simulation [Chen 2015, Chen 2016]. We developed an electromagnetic second-order implicit kinetic ion fluid electron hybrid model [Cheng 2013]. As a first step, we assumed isothermal electrons, but have included drift-kinetic electrons in similar models [Chen 2011]. We used this simulation to study the nonlinear evolution of the tearing mode in slab geometry, including nonlinear evolution and saturation [Cheng 2013]. Later, we compared this model directly to extended MHD calculations using the NIMROD code [Schnack 2013]. In this study, we investigated the ion-temperature-gradient instability with an extended MHD code for the first time and got reasonable agreement with the kinetic calculation in terms of linear frequency, growth rate and mode structure. We then extended this model to include orbit averaging and sub-cycling of the ions and compared directly to gyrokinetic theory [Sturdevant 2016]. This work was highlighted in an Invited Talk at the International Conference on the Numerical Simulation of Plasmas in 2015. The orbit averaging sub-cycling multi-scale algorithm is amenable to hybrid architectures with GPUS or math co-processors. Additionally, our participation in the Center for Extend Magnetohydrodynamics motivated our research on developing the capability for gyrokinetic simulation to model a global tearing mode. We did this in cylindrical geometry where the results could be benchmarked with existing eigenmode calculations. First, we developed a gyrokinetic code capable of simulating long wavelengths using a fluid electron model [Chen 2015]. We benchmarked this code with an eigenmode calculation. Besides having to rewrite the field solver due to the breakdown in the gyrokinetic ordering for long wavelengths, very high radial resolution was required. We developed a technique where we used the solution from the eigenmode solver to specify radial boundary conditions allowing for a very high radial resolution of the inner solution. Using this technique enabled us to use our direct algorithm with gyrokinetic ions and drift kinetic electrons [Chen 2016]. This work was highlighted in an Invited Talk at the American Physical Society - Division of Plasma Physics in 2015.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2019

Cross-scale modeling of storm surge, tide, and inundation in Mid-Atlantic Bight and New York City during Hurricane Sandy, 2012

Driven by high-resolution NAM (North America Mesoscale) and ECMWF (European Centre for Medium-Range Weather Forecasts) atmospheric fields, a 3-D unstructured grid SCHISM (Semi-implicit Cross-scale Hydroscience Integrated System Model) was applied to simulate total water level in the Mid-Atlantic Bight during Hurricane Sandy in 2012. The simulated storm surge, tide, significant wave height, and wave peak period results compared favorably with NOAA observations along US East Coast, Long Island Sound, and New York Harbor. The maximum total water level at The Battery in the New York Harbor was accurately simulated with an absolute error of less than 0.08 m (out of 2.9 m peak surge) and a timing difference less than 10 min, which includes the effects on the order of 0.1–0.3 m (5–10%) from the coastal setup induced by the surface waves. The scenario comparisons of (1) “NAM” versus “ECMWF”, (2) “2-D” versus “3-D”, and (3) “with” versus “without” wind wave model were examined. The 3-D barotropic model forced by ECMWF including the effects of wind waves performs the best, attributed to wave-induced radiation stress and reduction of bottom stress when the 3-D version is used. Simultaneously, the ELCIRC-Sub model, using a 5-m topography/bathymetry sub-grid and a regular 200-m resolution finite volume computational model grid, was developed to simulate the street-level inundation in New York City. The momentum and mass fluxes calculated by the coarser base grid model were effectively coupled with the sub-grid so that running a full-blown high-resolution base model is not required. The ELCIRC-Sub model uses MPI (Message Passing Interface) parallel computing to enlarge the coverage of the land surface and an efficient non-linear solver to improve the accuracy of the wetting-and-drying processes. The temporal comparisons of modeled water level with NOAA's tidal stations and USGS' rapid-deployed gauges showed overall performance with an average error of 0.1 m. Particularly, ELCIRC-Sub captured the profile of the highest peak surge (3.9 m), which rose 2 m within 3 hour at Kings Point, NY. The spatial comparisons of the modeled peak water level at 80 surveyed locations showed an average error less than 0.13 m. The modeled maximum inundation extent also matched well with an 80% hit rate against FEMA's Hurricane Sandy maximum flooding extent.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2018

An assessment of atypical mesh topologies for low-Mach large-eddy simulation

An implicit, low-dissipation, low-Mach, variable density control volume finite element formulation is used to explore foundational understanding of numerical accuracy for large-eddy simulation applications on hybrid meshes. Detailed simulation comparisons are made between low-order hexahedral, tetrahedral, pyramid, and wedge/prism topologies against a third-order, unstructured hexahedral topology. Using smooth analytical and manufactured low-Mach solutions, design-order convergence is established for the hexahedral, tetrahedral, pyramid, and wedge element topologies using a new open boundary condition based on energy-stable methodologies previously deployed within a finite-difference context. A wide range of simulations demonstrate that low-order hexahedral- and wedge-based element topologies behave nearly identically in both computed numerical errors and overall simulation timings. Moreover, low-order tetrahedral and pyramid element topologies also display nearly the same numerical characteristics. Although the superiority of the hexahedral-based topology is clearly demonstrated for trivial laminar, principally-aligned flows, e.g., a 1x2x10 channel flow with specified pressure drop, this advantage is reduced for non-aligned, turbulent flows including the Taylor–Green Vortex, turbulent plane channel flow ( Re τ 395), and buoyant flow past a heated cylinder. With the order of accuracy demonstrated for both homogenous and hybrid meshes, it is shown that solution verification for the selected complex flows can be established for all topology types. Although the number of elements in a mesh of like spacing comprised of tetrahedral, wedge, or pyramid elements increases as compared to the hexahedral counterpart, for wall-resolved large-eddy simulation, the increased assembly and residual evaluation computational time for non-hexahedral is offset by more efficient linear solver times. Lastly, most simulation results indicate that modest polynomial promotion provides a significant increase in solution accuracy.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2019

Time-dependent runaway electron simulations: Ampere–Faraday equations implemented in CQL3D

The runaway electron (RE) distributions driven by a large toroidal electric field induced by the drop in the temperature profile due to disruption or pellets are comprehensively simulated by the 3D Fokker–Planck (FP) solver CQL3D (Harvey and McCoy 1992 Proc. of IAEA TCM ), recently coupled to the Ampere–Faraday (AF) equations. The evolution of the toroidal current in a plasma occurs on a resistive time scale, τ res = 4 πa 2 /( c 2 η ), which is typically of the order of seconds in present tokamaks. Here, a and η are respectively plasma radius or radial extent of a current density perturbation, and Ohmic resistivity. From the Faraday EM equation, the toroidal electric field is proportional to the time derivative of the poloidal magnetic field, which, from the Ampere equation, is proportional to the toroidal current. Thus, the toroidal electric field rapidly increases due to an abrupt temperature drop decrease in conductivity, to prevent change in the toroidal current faster than τ res . This is a example of Lenz's law. For example, in simulations with KPRAD (Whyte et al 2003 J. Nucl. Mater . 313–6 1239) of neon pellet injection into a DIII-D shot, T e drops from 2 keV to 10 eV in 0.1 ms and Z eff increases 1–4, giving that the electric field increases 3500× to 0.8 V cm -1 . As described in Harvey et al (2000 PoP 7 4590), this places much of the tail electron distribution beyond the Dreicer runaway velocity, giving so-called 'hot-tail runaways' which for a time are the dominant source of runaways, more so than the knockon source. In this prior calculation, performed for a single flux surface, the toroidal current density is held constant, on the basis that τ res is large. Most of the initial current can be converted to runaway current, which is then dangerous, particularly for ITER. A more comprehensive A–F model recently implemented in CQL3D, taking into account the time-development of the full-plasma-width toroidal electric field on time-scales of order τ res applies an iterative technique for the toroidal field previously developed for a different application (Kupfer et al 1996 PoP 3 3644), maintaining the implicit-in-time evolution of CQL3D. The degree of runaway current formation is reduced in AF augmented CQL3D, but the basic mechanism of 'hot-tail runaways' remains a dominant contribution to the REs at early times after the T e drop in these simulations. On the other hand, a NIMROD (Sovinec et al 2004 J. Comput. Phys. 195 355 ) simulation of shattered-pellet shutdown of DIII-D plasma (Kim 2018 APS/DPP Meeting), gives a slower thermal quench; when the plasma profiles and electric field are coupled one-way to CQL3D, the 'hot-tail' REs are much less, and growth of RE is dominated by the knockon process.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗