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All sources NASA NTRS DOE OSTI

DOE OSTI2021

Internal Conversion between Bright (1 1 B $^+_u$) and Dark (21 A $^-_g$) States in s- trans -Butadiene and s- trans -Hexatriene

Internal conversion (IC) between the two lowest singlet excited states, 1 1 B$_u^+$ and 2 1 A$_g^–$, of s-trans-butadiene and s-trans-hexatriene is investigated using a series of single- and multi- reference wave function and density functional theory (DFT) methodologies. Three independent types of the equation-of-motion coupled-cluster (EOMCC) theory capable of providing an accurate and balanced description of one- as well as two-electron transitions, abbreviated as δ-CR-EOMCC(2,3), DIP-EOMCC(4h2p){N o }, and DEA-EOMCC(4p2h){N u } or DEA-EOMCC(3p1h,4p2h){N u }, consistently predict that the 1 1 B$_u^+$/2 1 A$_g^–$ crossing in both molecules occurs along the bond length alternation coordinate. However, the analogous 1 1 B$_u^+$ and 2 1 A$_g^–$ potentials obtained with some multireference approaches, such as CASSCF and MRCIS(D), as well as with the linear-response formulation of time-dependent DFT (TDDFT), do not cross. Hence, caution needs to be exercised when studying the low-lying singlet excited states of polyenes with conventional multiconfigurational methods and TDDFT. The multistate many-body perturbation theory methods, such as XMCQDPT 2 , do correctly reproduce the curve crossing. Among the simplest and least expensive computational methodologies, the DFT approaches that incorporate the contributions of doubly excited configurations, abbreviated as MRSF (mixed reference spin-flip) TDDFT and SSR( 4,4 ), accurately reproduce our best EOMCC results. This is highly promising for nonadiabatic molecular dynamics simulations in larger systems.

36 MATERIALS SCIENCE↗

DOE OSTI2021

Elaborated thermochemical treatment of HF, CO, N 2 , and H 2 O: Insight into HEAT and its extensions

Here, empirical, highly accurate non-relativistic electronic total atomization energies (eTAEs) are established by combining experimental or computationally converged treatments of the nuclear motion and relativistic contributions with the total atomization energies of HF, CO, N 2 , and H 2 O obtained from the Active Thermochemical Tables. These eTAEs, which have estimated (2σ) uncertainties of less than 10 cm -1 (0.12 kJ mol -1 ), form the basis for an analysis of high-level ab initio quantum chemical calculations that aim at reproducing these eTAEs for the title molecules. The results are then employed to analyze the performance of the high-accuracy extrapolated ab initio thermochemistry, or High-Accuracy Extrapolated Ab Initio Thermochemistry (HEAT), family of theoretical methods. The method known as HEAT-345(Q), in particular, is found to benefit from fortuitous error cancellation between its treatment of the zero-point energy, extrapolation errors in the Hartree-Fock and coupled cluster contributions, neglect of post-(T) core-correlation, and the basis-set error involved in higher-level correlation corrections. In addition to shedding light on a longstanding curiosity of the HEAT protocol—where the cheapest HEAT-345(Q) performs comparably to the theoretically more complete HEAT-456QP procedure—this study lays the foundation for extended HEAT variants that offer substantial improvements in accuracy relative to the established approaches.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2021

A fast multi-fidelity method with uncertainty quantification for complex data correlations: Application to vortex-induced vibrations of marine risers

Here we develop a fast multi-fidelity modeling method for very complex correlations between high- and low-fidelity data by working in modal space to extract the proper correlation function. We apply this method to infer the amplitude of motion of a flexible marine riser in cross-flow, subject to vortex-induced vibrations (VIV). VIV are driven by an absolute instability in the flow, which imposes a frequency (Strouhal) law that requires a matching with the impedance of the structure; this matching is easily achieved because of the rapid parametric variation of the added mass force. As a result, the wavenumber of the riser spatial response is within narrow bands of uncertainty. Hence, an error in wavenumber prediction can cause significant phase-related errors in the shape of the amplitude of response along the riser, rendering correlation between low- and high-fidelity data very complex. Working in modal space as outlined herein, dense data from low-fidelity data, provided by the semi-empirical computer code VIVA, can correlate in modal space with few high-fidelity data, obtained from experiments or fully-resolved CFD simulations, to correct both phase and amplitude and provide predictions that agree very well overall with the correct shape of the amplitude response. We also quantify the uncertainty in the prediction using Bayesian modeling and exploit this uncertainty to formulate an active learning strategy for the best possible location of the sensors providing the high fidelity measurements.

42 ENGINEERING↗

DOE OSTI2023

Pitfalls in the n -mode representation of vibrational potentials

Simulations of anharmonic vibrational motion rely on computationally expedient representations of the governing potential energy surface. The n-mode representation (n-MR)—effectively a many-body expansion in the space of molecular vibrations—is a general and efficient approach that is often used for this purpose in vibrational self-consistent field (VSCF) calculations and correlated analogues thereof. In the present analysis, a lack of convergence in many VSCF calculations is shown to originate from negative and unbound potentials at truncated orders of the n-MR expansion. For cases of strong anharmonic coupling between modes, the n-MR can both dip below the true global minimum of the potential surface and lead to effective single-mode potentials in VSCF that do not correspond to bound vibrational problems, even for bound total potentials. The present analysis serves mainly as a pathology report of this issue. Furthermore, this insight into the origin of VSCF non-convergence provides a simple, albeit ad hoc, route to correct the problem by “painting in” the full representation of groups of modes that exhibit these negative potentials at little additional computational cost. Somewhat surprisingly, this approach also reasonably approximates the results of the next-higher n-MR order and identifies groups of modes with particularly strong coupling. The method is shown to identify and correct problematic triples of modes—and restore SCF convergence—in two-mode representations of challenging test systems, including the water dimer and trimer, as well as protonated tropine.

Chemistry↗

DOE OSTI2020

Gene expression of functionally-related genes coevolves across fungal species: detecting coevolution of gene expression using phylogenetic comparative methods

Researchers often measure changes in gene expression across conditions to better understand the shared functional roles and regulatory mechanisms of different genes. Analogous to this is comparing gene expression across species, which can improve our understanding of the evolutionary processes shaping the evolution of both individual genes and functional pathways. One area of interest is determining genes showing signals of coevolution, which can also indicate potential functional similarity, analogous to co-expression analysis often performed across conditions for a single species. However, as with any trait, comparing gene expression across species can be confounded by the non-independence of species due to shared ancestry, making standard hypothesis testing inappropriate. We compared RNA-Seq data across 18 fungal species using a multivariate Brownian Motion phylogenetic comparative method (PCM), which allowed us to quantify coevolution between protein pairs while directly accounting for the shared ancestry of the species. Our work indicates proteins which physically-interact show stronger signals of coevolution than randomly-generated pairs. Interactions with stronger empirical and computational evidence also showing stronger signals of coevolution. We examined the effects of number of protein interactions and gene expression levels on coevolution, finding both factors are overall poor predictors of the strength of coevolution between a protein pair. Simulations further demonstrate the potential issues of analyzing gene expression coevolution without accounting for shared ancestry in a standard hypothesis testing framework. Furthermore, our simulations indicate the use of a randomly-generated null distribution as a means of determining statistical significance for detecting coevolving genes with phylogenetically-uncorrected correlations, as has previously been done, is less accurate than PCMs, although is a significant improvement over standard hypothesis testing. These methods are further improved by using a phylogenetically-corrected correlation metric. Our work highlights potential benefits of using PCMs to detect gene expression coevolution from high-throughput omics scale data. This framework can be built upon to investigate other evolutionary hypotheses, such as changes in transcription regulatory mechanisms across species.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2020

Radioactive Source Localization via Bayesian Particle Filter

In the event of a misplaced radioactive source or other emergency situation, measuring a radiation field, mapping its distribution, and determining a source location are essential tasks to ameliorating the situation. However, radiation fields may be extremely hazardous to human surveyors and minimizing received radiation doses is just as essential. Robots appear to be a potential solution to these problems. Beyond simply measuring radiation, the robot's computer processing capabilities offer a way to apply complex data analysis methods to radiation measurements in real-time. Methods which predict likely source locations can then feed this information into other processes, potentially improving path planning and enabling more efficient measurements. Given a robot mounted with a gamma-ray detector, can we: develop a methodology to account for detector performance across a wide range of source angles, distances, and photon energies? operate an autonomously navigating robot to effectively survey and characterize an area of interest? implement a data analysis method, conventionally used in measurements of motion, for source localization purposes? An open-source TurtleBot 3 robot, running Robot Operating System (ROS) on Ubuntu 16.04 LTS, was fitted with a Kromek GR1{sup R} Cadmium Zinc Telluride (CZT) solid-state gamma-ray detector. As a part of ROS, the packages OpenSlam, gmapping, and amcl were used to perform Simultaneous Localization and Mapping (SLAM), determining the robot's position and mapping the surrounding area. Data was acquired via Lidar mounted on top the TurtleBot 3. Detector Calibration Fit: The equation was fit to 365 counts of various energies, distances, and angles. A MATLAB{sup R} program was written to simulate measurements taken a robot on a random walk, with count data and positions discretized into finite element pixels. Using this program, a sample of 100 runs was performed on a map with a simulated source at the center, with a total of 200 of 2 pixels each. Similarly, multiple runs of the filter were performed on recorded robot measurement data. In both simulation and real tests, when corrected for errors (particles placed outside of bounds or on the robot, and simulation-specific errors), corresponding t-tests of predicted x and y-coordinates were within a 95% confidence interval of the actual position. For the real trial, these positions are slightly skewed right in the x-axis as the robot remained mainly to the left side of the source within the sample area. These simulations demonstrate potential validity for the usage of a particle filter as method of radioactive source localization. In the future, true real-time implementation and data fusion may further augment the performance of the robot to localize lost sources. Additionally, identification of multiple sources, determination of source types, and usage of a collimator are areas to potentially be explored.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

DOE OSTI2021

A stochastic solver based on the residence time algorithm for crystal plasticity models

Abstract The deformation of crystalline materials by dislocation motion takes place in discrete amounts determined by the Burgers vector. Dislocations may move individually or in bundles, potentially giving rise to intermittent slip. This confers plastic deformation with a certain degree of variability that can be interpreted as being caused by stochastic fluctuations in dislocation behavior. However, crystal plasticity (CP) models are almost always formulated in a continuum sense, assuming that fluctuations average out over large material volumes and/or cancel out due to multi-slip contributions. Nevertheless, plastic fluctuations are known to be important in confined volumes at or below the micron scale, at high temperatures, and under low strain rate/stress deformation conditions. Here, we develop a stochastic solver for CP models based on the residence-time algorithm that naturally captures plastic fluctuations by sampling among the set of active slip systems in the crystal. The method solves the evolution equations of explicit CP formulations, which are recast as stochastic ordinary differential equations and integrated discretely in time. The stochastic CP model is numerically stable by design and naturally breaks the symmetry of plastic slip by sampling among the active plastic shear rates with the correct probability. This can lead to phenomena such as intermittent slip or plastic localization without adding external symmetry-breaking operations to the model. The method is applied to body-centered cubic tungsten single crystals under a variety of temperatures, loading orientations, and imposed strain rates.

42 ENGINEERING↗

DOE OSTI2020

Stability Analysis of Inline ZFP Compression for Floating-Point Data in Iterative Methods

Currently, the dominating constraint in many high performance computing applications is data capacity and bandwidth, in both internode communications and even moreso in intranode data motion. A new approach to address this limitation is to make use of data compression in the form of a compressed data array. Storing data in a compressed data array and converting to standard IEEE-754 types as needed during a computation can reduce the pressure on bandwidth and storage. However, repeated conversions (lossy compression and decompression) introduce additional approximation errors, which need to be shown to not significantly affect the simulation results. Here, we extend recent work that analyzed the error of a single use of compression and decompression of the ZFP compressed data array representation to the case of time-stepping and iterative schemes, where an advancement operator is repeatedly applied in addition to the conversions. We show that the accumulated error for iterative methods involving fixed-point and time evolving iterations is bounded under standard constraints. An upper bound is established on the number of additional iterations required for the convergence of stationary fixed-point iterations. An additional analysis of traditional forward and backward error of stationary iterative methods using ZFP compressed arrays is also presented. The results of several 1D, 2D, and 3D test problems are provided to demonstrate the correctness of the theoretical bounds.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Computer-controlled finishing via dynamically constraint position-velocity-time scheduler

In a Computer Numerical Controlled (CNC) finishing process, the target material removal from an optical surface is guided by the convolution between the influence function of a machine tool and its dwell time at certain points over the surface. To reduce dynamics stressing and increase machining efficiency, the dwell time must be converted to varying velocities, which are the actual inputs to the machine tool controller. Conventionally, the conversion assumed constant acceleration and relied on linear motion interpolation, which caused discontinuities in velocities. This unsmooth motion affects the material removal distribution, and, thus, the accuracy of the finished surface shape. Many modern CNC machines support the smoother, cubic-polynomial interpolated Position-Velocity-Time (PVT) motion mode; however, the conventional scheduler may fail to provide suitable velocities for the PVT. Here in this study answers this challenge by proposing a novel PVT-based velocity scheduler that achieves smooth motion while considering CNC dynamic limits. Firstly, the principle of the PVT is explained, and the PVT-based velocity scheduler is formulated. Secondly, a quadratic programming is used to optimize the velocities by imposing the CNC dynamic constraints and the C 1 continuities (zeroth and first derivatives are continuous) simultaneously. Thirdly, the smoothness and accuracy of the scheduled velocities are studied on different kinds of tool paths via simulation. Finally, a sub-0.3 nm level surface finishing experiment using ion beam figuring is demonstrated to verify the feasibility of the proposed method. The PVT-based scheduler and simulator code is open-sourced.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Bulk Spray and Individual Plume Characterization of LPG and Iso-Octane Sprays at Engine-Like Conditions

This study presents experimental and numerical examination of directly injected (DI) propane and iso-octane, surrogates for liquified petroleum gas (LPG) and gasoline, respectively, at various engine like conditions with the overall objective to establish the baseline with regards to fuel delivery required for future high efficiency DI-LPG fueled heavy-duty engines. Sprays for both iso-octane and propane were characterized and the results from the optical diagnostic techniques including high-speed Schlieren and planar Mie scattering imaging were applied to differentiate the liquid-phase regions and the bulk spray phenomenon from single plume behaviors. The experimental results, coupled with high-fidelity internal nozzle-flow simulations were then used to define best practices in CFD Lagrangian spray models. Optical imaging revealed that unlike iso-octane, propane’s spray propagation was fed by its flash boiling, spray collapse, and high degree of vaporization, resulting in a direct proportionality of propane’s penetration length to temperature. These unique features of propane and its variation from iso-octane’s spray pattern, contributed to its classification as an unconventional spray. Appropriate corrections to the injection and breakup models were developed to reproduce the under-expanded jet dynamics and to mimic the flash boiling-driven spray development observed with propane sprays. The simulation results were found to be sensitive to cone and inclusion angles of the blob injector. The current work represents a first assessment of the capability of the commonly available models for engine-spray simulations and highlights the fact that, despite the reasonable agreement obtained in the fuel vapor morphology, the representation of the liquid phase lacks accuracy and requires further model development.

02 PETROLEUM↗

DOE OSTI2021

Toward a First-Principles Framework for Predicting Collective Properties of Electrolytes

Conspectus Given the universal importance of electrolyte solutions, it is natural to expect that we have a nearly complete understanding of the fundamental properties of these solutions (e.g., the chemical potential) and that we can therefore explain, predict, and control the phenomena occurring in them. In fact, reality falls short of these expectations. But, recent advances in the simulation and modeling of electrolyte solutions indicate that it should soon be possible to make progress toward these goals. In this Account, we will discuss the use of first-principles interaction potentials based in quantum mechanics (QM) to enhance our understanding of electrolyte solutions. Specifically, we will focus on the use of quantum density functional theory (DFT) combined with molecular dynamics simulation (DFT-MD) as the foundation for our approach. The overarching concept is to understand and accurately reproduce the balance between local or short-ranged (SR) structural details and long-range (LR) correlations, allowing the prediction of the thermodynamics of both single ions in solution as well as the collective interactions characterized by activity/osmotic coefficients. In doing so, relevant collective motions and driving forces characterized by chemical potentials can be determined. Here, in this Account, we will make the case that understanding electrolyte solutions requires a faithful QM representation of the SR nature of the ion–ion, ion–water, and water–water interactions. However, the number of molecules that is required for collective behavior makes the direct application of high-level QM methods that contain the best SR physics untenable, making methods that balance accuracy and efficiency a practical goal. Alternatives such as continuum solvent models (CSMs) and empirically based classical molecular dynamics have been extensively employed to resolve this problem but without yet overcoming the fundamental issue of SR accuracy. We will demonstrate that accurately describing the SR interaction is imperative for predicting both intrinsic properties, namely, at infinite dilution, and collective properties of electrolyte solutions. DFT has played an important role in our understanding of condensed phase systems, e.g., bulk liquid water, the air–water interface, ions in bulk, and at the air–water interface. This approach holds huge promise to provide benchmark calculations of electrolyte solution properties that will allow for the development and improvement of more efficient methods, as well as an enhanced understanding of fundamental phenomena. However, the standard protocol using the generalized gradient approximation with van der Waals (vdW) correction requires improvement in order to achieve a high level of quantitative accuracy. Simply simulating with higher level DFT functionals may not be the best route considering the significant computational cost. Alternative methods of incorporating information from higher levels of QM should be explored; e.g., using force matching techniques on small clusters, where high level benchmark calculations are possible, to develop ideal correction terms to the DFT functional is a promising possibility. We argue that DFT with statistical mechanics is becoming an increasingly useful framework enabling the prediction of collective electrolyte properties.

Duignan, Timothy T.↗

DOE OSTI2023

Experiment and simulation of high-speed gas jet penetration into a semicircular fluidized bed

This work marks the third in a series of experiments that were in a semi-circular, gas-fluidized bed with side jets. In this work, the particles are (nominally) 1 mm ceramic beads. The bed is operated just at and slightly above and below the minimum fluidization velocity and additional fluidization is provided by two high-speed gas located on the sides of the bed near the flat, front face of the unit. Two primary measurements are taken: high-speed video recording of the front of the bed and bed pressure drop from a tap in the back of the bed. Particle Image Velocimetry (PIV) is used to determine particle motion, characterized as a mean Froude number, from the high-speed video. A CFD-DEM model of the bed is presented using the recently released MFIX-Exa code. Four model subvariants are considered using two methods of representing the jets and two drag models, both of which are calibrated to exactly match the experimentally measured minimum fluidization velocity. Although it is more difficult to determine the jet penetration depths in a straightforward manner as in the previous works using Froude number contours, the CFD-DEM results compare quite well to the PIV measurements, particularly for submodel flow Syam. Unfortunately, the good agreement of the solids-phase is overshadowed by significant disagreement in the gas-phase data. Specifically, the predicted time averaged standard deviation of the pressure drop is found to be over an order of magnitude larger than measured. Due to the low value of the measurements, just 1% of the mean bed pressure drop, it seems possible that the data is in error. On the other hand, the model may not be accurately capturing pressure attenuation through an under-fluidized region in the back of the bed. Without the possibility additional experiments to test the validity of the data, this work is simply being reported “as is” without being able to indicate which, either the simulation or the experiment, is more correct.<br>

Fullmer, William D.↗

DOE OSTI2023

Diffusion–convection model of runaway electrons due to large magnetohydrodynamic perturbations in post-thermal quench plasmas

Systematic test particle tracing simulations for runaway electrons (REs) are performed for six post-thermal quench equilibria from DIII-D and ITER, where large scale, kink-like n = 1 (n is the toroidal mode number) magnetohydrodynamic (MHD) instabilities are found. The modeled particle guiding center orbits allow extraction of the effective diffusion–convection coefficients of REs in the presence of large three-dimensional (3D) perturbations up to 10% of the equilibrium toroidal field. With a fixed spatial distribution of the field perturbation, the RE transport coefficients along the plasma radial coordinate track reasonably well with the surface-averaged perturbation level. A substantial variation in the value of the transport coefficients—by three orders of magnitude in most cases, however, occurs with varying launching location of REs along the plasma radius. Large 3D perturbations almost always lead to comparable diffusion and convection processes, meaning that diffusion alone is insufficient to describe the particle motion. At lower (but still high) level of perturbation, the RE convection is found to be dominant over diffusion. A similar observation is made when the perturbation is too strong. In the presence of large perturbation, the dependence of the RE transport on the particle energy is sensitive to the spatial distribution of the perturbation. Based on numerically obtained RE transport coefficients, an analytic fitting model is proposed to quantify the particle diffusion and convection processes due to large MHD events in post-thermal quench plasmas. The model is shown to reasonably well reproduce the direct test particle tracing results for the RE loss fraction and can, thus, be useful for incorporating into other kinetic RE codes in order to simulate the RE beam evolution in the presence of large 3D perturbations.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2024

Failure investigation and mitigation after experimental research reactor fuel plate deformation in an irradiation device

Experimental research reactor fuel testing is conducted in the Belgian Reactor 2 (BR2) of the Belgian Nuclear Research Centre (SCK CEN) in dedicated irradiation vehicles or rigs. One such vehicle allows flat full-size fuel plates to be irradiated by inserting them into slotted baskets that captures a narrow portion of the longitudinal edges of the plates. The motion of the fuel plates within the baskets is possible within the narrow slots and thus, the plate is considered to be unattached. The design intentionally omits fixing mechanisms of the fuel plates to the baskets to facilitate the inspection and repositioning of the plates between the irradiation cycles and the accommodation of thermal expansion of the plates in the lateral direction. However, loosely inserted fuel plates have weak structural boundary conditions allowing for larger out-of-plane deflections caused by hydrodynamic loads exerted by the flowing coolant, as compared to those of fixed plates. Unexpected large deformations of plates occurred in several irradiation cycles that further resulted in a loss of cladding integrity. These deformations could not be attributed to a single source. This triggered a series of thermal hydraulic, structural, and fluid-structure interaction analyses aiming at understanding the observed phenomenon. The analyses revealed that, for a certain combination of unfavorable manufacturing and assembly tolerances, fuel plate edges could escape out of the slots in the irradiation basket due to the hydrodynamic load. Subsequently, the plate could become wedged inside the basket coolant channel opening. This resulted in reduced coolant flow and accelerated temperature increase and thermal expansion of the plate while under irradiation. This unfavorable feedback loop could then lead to excessive plate surface temperatures, deformed plates and cladding failure, as was observed in the experiments. These analyses not only provided a probable cause of the fuel plate failures, but also resulted in a new and improved design of the irradiation basket to avoid these issues in the future. In conclusion, a series of recent successful irradiations confirm that the sources of failures were identified correctly, and the implemented mitigations were adequate.

BR2↗

DOE OSTI2025

Review of High‐Speed Digital Image Correlation: Advancements and Good Practices

This paper reviews the current state of the art in high‐speed (HS) and ultrahigh‐speed (UHS) digital image correlation (DIC) techniques, emphasizing their critical role in experimental research across various scientific domains. HS and UHS DIC have evolved significantly, driven by advancements in camera systems, image processing algorithms and experimental methodologies. These developments have opened new avenues for capturing and analysing dynamic events with unprecedented temporal and spatial detail, but not without introducing challenges such as optical distortions, motion blur and lighting issues that can affect measurement quality. This review advocates for standardized reporting practices in HS/UHS DIC methodologies to improve reproducibility and reliability across studies, drawing on guidelines from the International Digital Image Correlation Society (iDICs). Through a comprehensive analysis of over 150 articles, this review identifies key advancements in imaging technology and their application in six research domains: material characterization, test development, fracture mechanics, model validation, ballistic and explosive phenomena assessment and measurement uncertainties. Distinctions between two‐dimensional (2D) and stereo‐DIC applications are explored, offering insights into their practical implementation and trade‐offs. Good practices for HS/UHS DIC applications are proposed along with suggestions for future directions for this evolving field, highlighting the indispensable role of technological innovation in expanding the capabilities of optical metrology.

experimental mechanics↗

DOE OSTI2024

A Particle-in-Cell Method for Plasmas with a Generalized Momentum Formulation, Part II: Enforcing the Lorenz Gauge Condition

In a previous paper Christlieb et al. (A particle-in-cell method for plasmas with a generalized momentum formulation, part I: Model formulation, 2024), we developed a new particle-in-cell (PIC) method for the relativistic Vlasov–Maxwell system in which the electromagnetic fields and the equations of motion for the particles were cast in terms of scalar and vector potentials through a Hamiltonian formulation. This new method evolved the potentials under the Lorenz gauge using integral equation methods. New methods to construct spatial derivatives of the potentials that converge at the same rates as the fields were also presented. The new particle method was compared against standard explicit discretizations, including the well-known FDTD-PIC method, for a range of applications involving sheaths and particle beams. Here, this paper extends this new class of methods by focusing on the enforcement the Lorenz gauge condition in both exact and approximate forms using co-located meshes. A time-consistency property of the proposed field solver for the vector potential form of Maxwell’s equations is established, which is shown to preserve the equivalence between the semi-discrete Lorenz gauge condition and the analogous semi-discrete continuity equation. Using this property, we present three methods to enforce a semi-discrete gauge condition. The first method introduces an update for the continuity equation that is consistent with the discretization of the Lorenz gauge condition. Both the finite difference and spectral implementations satisfy this discrete gauge condition to machine precision. The second approach we propose enforces a semi-discrete continuity equation using the boundary integral solution to the field equations. The potential benefit of this approach is that it eliminates spatial derivatives that appear on the particle data, namely the current density, which is often calculated by linear combinations of low-order spline basis functions. This method is ideally suited to boundary integral equation methods that invert multi-dimensional operators without dimensional splitting techniques and will be the subject of future work. The third approach introduces a gauge correcting method that makes direct use of the gauge condition to modify the scalar potential and uses local maps for both the charge and current densities. This results in a gauge error, as the maps do not enforce the continuity equation. The vector potential coming from the current density is taken to be exact, and using the Lorenz gauge, we compute a correction to the scalar potential that makes the two potentials satisfy the gauge condition. This method also enforces the gauge condition to machine precision. We demonstrate two of the proposed methods in the context of periodic domains. Problems defined on bounded domains, including those with complex geometric features remain an ongoing effort. However, this work shows that it is possible to design computationally efficient methods that can effectively enforce the Lorenz gauge condition in a non-staggered PIC formulation.

97 MATHEMATICS AND COMPUTING↗