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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 361 records · Page 20
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Control of neutral fueling and helium exhaust to NSTX-U plasmas by means of three-dimensional magnetic control fields

Resonant magnetic perturbations are used to stabilize plasma edge instabilities, so-called edge localized modes (ELMs), in high-performance (H-mode) plasmas explored for fusion energy. These ELMs cause repetitive outburst of confined energy and particles and cause cyclic loading of plasma facing components (PFCs). This endangers the integrity of the PFCs and hence limits the lifetime and hence commercial viability of fusion. It was shown that these ELMs can be stabilized by application of small amplitude resonant magnetic perturbation (RMP) fields. This removes the ELMs as edge instabilities but also induces three-dimensional (3D) effects to the plasma boundary, which was formerly toroidally axisymmetric. The impact of this 3D perturbation on the plasma boundary solution needs to be understood to extrapolate the effects to ITER and towards a fusion reactor, if ELM control by RMP fields would be utilized.

3D Boundary Plasmas↗
DOE OSTI2021

Investigation of species-mass diffusion in binary-species boundary layers at high pressure using direct numerical simulations

Here, direct numerical simulations of single-species and binary-species temporal boundary layers at high pressure are performed with special attention to species-mass diffusion. The working fluids are nitrogen or a mixture of nitrogen and methane. Mean profiles and turbulent fluctuations of mass fraction show that their qualitative characteristics are different from those of streamwise velocity and temperature, due to the different boundary conditions. In a wall-parallel plane near the wall, the streamwise velocity and temperature have streaky patterns and the fields are similar. However, the mass fraction field at the same location is different from the streamwise velocity and temperature fields indicating that species-mass diffusion is not similar to the momentum and thermal diffusion. In contrast, at the centre and near the edge of the boundary layer, the mass fraction and temperature fields have almost the same pattern, indicating that the similarity between thermal and species-mass diffusion holds away from the wall. The lack of similarity near the wall is traced to the Soret effect that induces a temperature-gradient-dependent species-mass flux. As a result, a new phenomenon has been identified for a non-isothermal binary-species system – uphill diffusion, which in its classical isothermal definition can only occur for three or more species. A quadrant analysis for the turbulent mass flux reveals that near the wall the Soret effect enhances the negative contributions of the quadrants. Due to the enhancement of the negative contributions, small species-concentration fluid tends to be trapped near the wall.

42 ENGINEERING↗
DOE OSTI2022

Automatic Calibration of a Geomechanical Model from Sparse Data for Estimating Stress in Deep Geological Formations

Summary In this study, we demonstrate geomechanical modeling with fully automatic parameter calibration to estimate the full geomechanical stress fields of a prospective US carbon dioxide (CO2) storage site, based on sparse measurement data. The goal is to compute full stress tensor field estimates (principal stresses and orientations) that are maximally compatible with observations within the constraints of the model assumptions, thereby extending pointwise, incomplete partial stress measurement to a simulated full formation stress field, as well as a rough assessment of the associated error. We use the Perch site, located in Otsego County, Michigan, USA, as our case study. The input data consist of partial stress tensor information inferred from in-situ borehole tests, geophysical well logs, and processing of seismic data. A static earth model (SEM) of the site was developed, and geomechanical simulation functionality of the open-source MATLAB Reservoir Simulation Toolbox (MRST) was used to model the stress field. Adjoint-based nonlinear optimization was used to adjust boundary conditions and material properties to calibrate simulated results of observations. Results were interpreted through a Bayesian framework. The focus of this paper is to demonstrate how the fully automatic calibration procedure works and discuss the results obtained; it does not attempt a detailed analysis of the stress field in the context of the proposed CO2 storage initiatives. Our work is part of a larger effort to noninvasively determine in-situ stresses in deep formations considered for CO2 storage. Guided by previously published research on geomechanical model calibration, our work presents a novel calibration approach supporting a potentially large number of linear or nonlinear calibration parameters to produce results optimally agreeing with available measurements and thus extend partial pointwise estimates to full tensor fields compatible with the physics of the site.

Engineering↗
DOE OSTI2021

COMBLE ARM Mobile Facility (AMF) Measurements of Ice Nucleating Particles Field Campaign Report

During the Cold-Air Outbreaks in the Marine Boundary Layer Experiment (COMBLE), a project with the overarching goal to shed light on the dynamics and microphysical properties of clouds and precipitation in the high-latitude marine boundary layer during cold-air outbreaks (CAOs), we were tasked with providing and assisting the collection of aerosol filter samples for measuring ice nucleating particle (INP) concentrations. Little is known about the properties of CAO cloud systems, how they vary with surface, environmental, and aerosol conditions, the role of cold-air outbreaks in the global atmospheric and ocean circulation, and the accuracy of the treatment of this atmospheric regime in climate models. Uncertainties in climate projections for the arctic and sub-arctic region are large, in part due to the lack of observational guidance to constrain the treatment of aerosol-cloud-precipitation linkages in climate models.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2021

Physics-informed neural networks for solving forward and inverse flow problems via the Boltzmann-BGK formulation

We report the Boltzmann equation with the Bhatnagar-Gross-Krook collision model (Boltzmann-BGK equation) has been employed to describe multiscale flows, i.e., from the hydrodynamic limit to free molecular flow. In this study we employ physics-informed neural networks (PINNs) to solve forward and inverse problems via the Boltzmann-BGK formulation (PINN-BGK), enabling PINNs to model flows in both the continuum and rarefied regimes. In particular, the PINN-BGK is composed of three sub-networks, i.e., the first for approximating the equilibrium distribution function, the second for approximating the non-equilibrium distribution function, and the third one for encoding the Boltzmann-BGK equation as well as the corresponding boundary/initial conditions. By minimizing the residuals of the governing equations and the mismatch between the predicted and provided boundary/initial conditions, we can approximate the Boltzmann-BGK equation for both continuous and rarefied flows. For forward problems, the PINN-BGK is utilized to solve various benchmark flows given boundary/initial conditions, e.g. Kovasznay flow, Taylor-Green flow, cavity flow, and micro Couette flow for Knudsen number up to 5. For inverse problems, we focus on rarefied flows in which accurate boundary conditions are difficult to obtain. We employ the PINN-BGK to infer the flow field in the entire computational domain given a limited number of interior scattered measurements on the velocity without using the (unknown) boundary conditions. Results for the two-dimensional micro Couette and micro cavity flows with Knudsen numbers ranging from 0.1 to 10 indicate that the PINN-BGK can infer the velocity field in the entire domain with good accuracy. Finally, we also present some results on using transfer learning to accelerate the training process. Specifically, we can obtain a three-fold speedup comparted to the standard training process (e.g., Adam plus L-BFG-B) for the two-dimensional flow problems considered in our work.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Dirichlet absorbing boundary conditions for classical and peridynamic diffusion-type models

Diffusion-type problems in (nearly) unbounded domains play important roles in various fields of fluid dynamics, biology, and materials science. The aim of this paper is to construct accurate absorbing boundary conditions (ABCs) suitable for classical (local) as well as nonlocal peridynamic (PD) diffusion models. The main focus of the present study is on the PD diffusion formulation. The majority of the PD diffusion models proposed so far are applied to bounded domains only. In this study, we propose an effective way to handle unbounded domains both with PD and classical diffusion models. For the former, we employ a meshfree discretization, whereas for the latter the finite element method (FEM) is employed. The proposed ABCs are time-dependent and Dirichlet-type, making the approach easy to implement in the available models. The performance of the approach, in terms of accuracy and stability, is illustrated by numerical examples in 1D, 2D, and 3D.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Inertial dynamics of an interface with interfacial mass flux: Stability and flow fields’ structure, inertial stabilization mechanism, degeneracy of Landau’s solution, effect of energy fluctuations, and chemistry-induced instabilities

This work focuses on the long-standing problem of inertial dynamics of an interface with interfacial mass flux and reports new mechanisms for the interface stabilization and destabilization. The interface is a phase boundary separating fluids of different densities and having interfacial mass flux. To analyze the interface dynamics from a far field, we develop and apply the general matrix method to rigorously solve the boundary value problem involving the governing equations in the fluid bulk and the boundary conditions at the interface and at the outside boundaries of the domain. We find the fundamental solutions for the linearized system of equations and analyze the interplay of interface stability with flow fields’ structure by directly linking rigorous mathematical attributes to physical observables. We find that the interface is stable when the dynamics conserves the fluxes of mass, momentum, and energy; the stabilization is due to an inertial mechanism causing small oscillations of the interface velocity. In the classic Landau’s dynamics, the postulate of perfect constancy of the interface velocity leads to the development of Landau–Darrieus instability. This destabilization is also linked to the imbalance of the perturbed energy at the interface. The classic Landau’s solution is found to have degeneracy; lifting of the degeneracy may lead to singularity and self-similar dynamics. Our results compare well with traditional theories of combustion and propose new experiments to study the dynamics of the interface and the flow fields in combustible systems. We further conduct reactive molecular dynamics simulations to elucidate the complexity of chemical processes, to study the destabilizing effect of energy fluctuations on the interface stability, and to illustrate the chemistry-induced instabilities. In summary, we identify the extreme sensitivity of the interface dynamics to the interfacial boundary conditions, including the formal properties of fundamental solutions and the qualitative and quantitative properties of the flow fields. This provides new opportunities for studies, diagnostics, and control of multiphase flows in a broad range of processes in nature and technology.

42 ENGINEERING↗
DOE OSTI2025

NLR HPC Eagle Jobs Data and Additional Energy Metrics

Overview: Anonymized job-level records from the Eagle high-performance computing (HPC) system at the National Laboratory of the Rockies (NLR). Each record represents a Slurm batch job with scheduling metadata, resource requests, resource utilization, CPU/GPU energy consumption, and efficiency metrics. Sensitive fields (user, account, job name) are replaced with cryptographic hashes. System & Timeframe: Eagle was a 2,000-node, 8-petaflop system operated at NLR from 2019–2024. Data covers the full operational lifetime of the system. Slurm data was processed nightly; timestamps are in Mountain Time. Funding provided by the U.S. Department of Energy, EERE. Files: esif.hpc.eagle.job-anon.zip — Core anonymized job records (Hive-partitioned Parquet) esif.hpc.eagle.job-anon-energy-metrics.zip — Same records with additional iLO and Ganglia energy metrics datacard.md — Full dataset documentation ~13.8 million rows, 62 variables. Readable with PyArrow, pandas, DuckDB, Apache Spark, or any Parquet-compatible tool. Data Collection: Jobs collected via sacct through a pipeline: Eagle Jobs API → Redpanda → StreamSets → HPCMON API → PostgreSQL. Node-level power from iLO (HP Integrated Lights-Out); GPU power from Ganglia monitoring, joined to jobs via node lists and time ranges. Preprocessing: Anonymization of name, user, and account fields via cryptographic hashing Derived columns: queue_wait, cpu_eff, max_mem_eff Simplified job state mapping (e.g., "CANCELLED BY 12345" → "CANCELLED") QoS accounting rules (buy-in, standby, or Slurm QoS value) CPU energy estimated from TDP (200W, Intel Xeon Gold 6154, 18 cores) Timezone-aware columns (_tz) sourced from LEX accounting database to correctly handle DST transitions Key Variables: Scheduling: job_id, partition, state_simple, submit_time_tz, start_time_tz, end_time_tz, queue_waitResources: nodes_req/used, processors_req/used, memory_req, wallclock_req/used, gpus_requested Efficiency: cpu_eff, max_mem_eff Energy: cpu_energy_tdp_estimated_max/used_watt_hours, node_energy_total_watt_hours (iLO), gpu0/1_energy_total_watt_hours (Ganglia) Partitions: bigmem, bigmem-8600, bigscratch, csc, dav, ddn, debug, gpu, haswell, long, mono, short, standard Job States: CANCELLED, COMPLETED, FAILED, NODE_FAIL, OUT_OF_MEMORY, PENDING, RUNNING, TIMEOUT QoS Levels: Unknown, normal, buy-in, debug, penalty, high, standby Important Notes: Non-_tz timestamp columns may be off by one hour across DST boundaries; use _tz columns for time difference calculations Energy fields are null for jobs without monitoring coverage Job step records and raw Slurm JSONB fields are excluded from this extract Do not attempt to re-identify individuals from hashed fields

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Morphologies of dealloying corrosion attack at grain boundaries

Dealloying corrosion at grain boundaries severely compromises the performance of polycrystalline materials across a wide variety of technological applications. The impact of this phenomenon depends upon the morphology and rate of intergranular dealloying, which can range from planar to wormhole-like patterns that rapidly advance into the alloy. Using 2D and 3D multi-phase field simulations, we reveal how diverse microstructures result from a fundamental interplay between alloy composition and a grain boundary migration mechanism that alters diffusional pathways of dealloying. Inside intergranular dealloying channels, corrosion product buildup can spawn new channels that branch into grain interiors, such that alloys can be degraded from the inside out. These processes further lead to an atypical coarsening mode assisted by diffusion in the dealloying agent. We summarize a unifying explanation for dealloying morphology selection in polycrystalline alloys, which provides an important step towards their optimization for dealloying corrosion environments.

Corrosion↗
DOE OSTI2021

Deep Learning Approaches to Surrogates for Solving the Diffusion Equation for Mechanistic Real-World Simulations

In many mechanistic medical, biological, physical, and engineered spatiotemporal dynamic models the numerical solution of partial differential equations (PDEs), especially for diffusion, fluid flow and mechanical relaxation, can make simulations impractically slow. Biological models of tissues and organs often require the simultaneous calculation of the spatial variation of concentration of dozens of diffusing chemical species. One clinical example where rapid calculation of a diffusing field is of use is the estimation of oxygen gradients in the retina, based on imaging of the retinal vasculature, to guide surgical interventions in diabetic retinopathy. Furthermore, the ability to predict blood perfusion and oxygenation may one day guide clinical interventions in diverse settings, i.e., from stent placement in treating heart disease to BOLD fMRI interpretation in evaluating cognitive function (Xie et al., 2019; Lee et al., 2020). Since the quasi-steady-state solutions required for fast-diffusing chemical species like oxygen are particularly computationally costly, we consider the use of a neural network to provide an approximate solution to the steady-state diffusion equation. Machine learning surrogates, neural networks trained to provide approximate solutions to such complicated numerical problems, can often provide speed-ups of several orders of magnitude compared to direct calculation. Surrogates of PDEs could enable use of larger and more detailed models than are possible with direct calculation and can make including such simulations in real-time or near-real time workflows practical. Creating a surrogate requires running the direct calculation tens of thousands of times to generate training data and then training the neural network, both of which are computationally expensive. Often the practical applications of such models require thousands to millions of replica simulations, for example for parameter identification and uncertainty quantification, each of which gains speed from surrogate use and rapidly recovers the up-front costs of surrogate generation. We use a Convolutional Neural Network to approximate the stationary solution to the diffusion equation in the case of two equal-diameter, circular, constant-value sources located at random positions in a two-dimensional square domain with absorbing boundary conditions. Such a configuration caricatures the chemical concentration field of a fast-diffusing species like oxygen in a tissue with two parallel blood vessels in a cross section perpendicular to the two blood vessels. To improve convergence during training, we apply a training approach that uses roll-back to reject stochastic changes to the network that increase the loss function. The trained neural network approximation is about 1000 times faster than the direct calculation for individual replicas. Because different applications will have different criteria for acceptable approximation accuracy, we discuss a variety of loss functions and accuracy estimators that can help select the best network for a particular application. We briefly discuss some of the issues we encountered with overfitting, mismapping of the field values and the geometrical conditions that lead to large absolute and relative errors in the approximate solution.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2023

Advanced Turbulence Models for Large-Scale Atmospheric Boundary Layer Flows

We present high-fidelity large-eddy-simulation (LES) modeling approaches for the turbulent atmospheric boundary layer (ABL) flows. Wind energy is a prime example of an application driven by ABL. Generation of electrical energy from farms of wind turbines at night in the stable ABL is a particularly interesting situation. In this report, we consider the well-known GEWEX (Global Energy and Water Cycle Experiment) Atmospheric Boundary Layer Study (GABLS) stably stratified benchmark LES case. We use a high-order spectral element code Nek5000/RS, which is supported under the DOE's Exascale Computing Project (ECP) Center for Efficient Exascale Discretizations (CEED) project, targeting application simulations on various acceleration-device based exascale computing platforms. In our earlier ANL report, we demonstrated our newly developed subgrid-scale (SGS) models based on high-pass filter (HPF), mean-field eddy viscosity (MFEV), and Smagorinsky (SMG) with no-slip and traction boundary conditions, provided with low-order statistics, convergence and turbulent structure analysis. In this report, we extend the range of our SGS modeling approaches in the context of the mean-field eddy viscosity (MFEV), to include the solution of an SGS turbulent kinetic energy equation (TKE). We demonstrate the model fidelity of Nek5000/RS in comparison to that of AMR-Wind, a block-structured second-order finite-volume code with adaptive-mesh-refinement capabilities, with which we studied scaling performance for both codes in comparison on DOE's leadership computing platforms.

17 WIND ENERGY↗
DOE OSTI2022

Explanation and Application of the Evolving Contact Traction Fields in Shallow Foundation Systems

The present paper provides a qualitative discussion of the evolution of contact traction fields beneath rigid shallow foundations resting on granular materials. A phenomenological similarity is recognized in the measured contact traction fields of rigid footings and at the bases of sandpiles. This observation leads to the hypothesis that the stress distributions are brought about by the same physical phenomena, namely the development of arching effects through force chains and mobilized intergranular friction. A set of semi-empirical equations are suggested for the normal and tangential components of this contact traction based on past experimental measurements and phenomenological assumptions of frictional behaviors at the foundation system scale. These equations are then applied to the prescribed boundary conditions for the analysis of the settlement, resistance, and stress fields in supporting granular materials beneath the footing. A parametric sensitivity study is performed on the proposed modelling method, highlighting solutions to the boundary-value problems in an isotropic, homogeneous elastic half-space.

42 ENGINEERING↗
DOE OSTI2021

nn-PINNs: Non-Newtonian physics-informed neural networks for complex fluid modeling

Time- and rate-dependent material functions in non-Newtonian fluids in response to different deformation fields pose a challenge in integrating different constitutive models into conventional computational fluid dynamic platforms. Considering their relevance in many industrial and natural settings alike, robust data-driven frameworks that enable accurate modeling of these complex fluids are of great interest. The main goal is to solve the coupled Partial Differential Equations (PDEs) consisting of the constitutive equations that relate the shear stress to the deformation and fully capture the behavior of the fluid under various flow protocols with different boundary conditions. Here, in this work, we present non-Newtonian physics-informed neural networks (nn-PINNs) for solving systems of coupled PDEs adopted for complex fluid flow modeling. The proposed nn-PINN method is employed to solve the constitutive models in conjunction with conservation of mass and momentum by benefiting from Automatic Differentiation (AD) in neural networks, hence avoiding the mesh generation step. nn-PINNs are tested for a number of different complex fluids with different constitutive models and for several flow protocols. These include a range of Generalized Newtonian Fluid (GNF) empirical constitutive models, as well as some phenomenological models with memory effects and thixotropic timescales. nn-PINNs are found to obtain the correct solution of complex fluids in spatiotemporal domains with good accuracy compared to the ground truth solution. We also present applications of nn-PINNs for complex fluid modeling problems with unknown boundary conditions on the surface, and show that our approach can successfully recover the velocity and stress fields across the domain, including the boundaries, given some sparse velocity measurements.

42 ENGINEERING↗
DOE OSTI2024

Celestial gluon amplitudes from the outside in

We show that, given a two-dimensional realization of the celestial OPE in self-dual Yang-Mills, we can find a scalar source around which scattering amplitudes replicate correlation functions computed from the 2D ‘gluon’ operators in a limit where a dynamic massless scalar decouples. We derive conditions on the two-dimensional three-point correlation function so that such a source exists and give two particular examples of this construction, one in which gluons are constructed from vertex operators in the semiclassical limit of Liouville theory and another in which the soft gluons arise from generalized free fields. Finally, we identify a bulk dual to the level of the boundary Kac-Moody algebra and discuss moving beyond the decoupling limit.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2025

Tropospheric aerosols over the western North Atlantic Ocean during the winter and summer deployments of ACTIVATE 2020: life cycle, transport, and distribution

The Aerosol Cloud meTeorology Interactions oVer the western ATlantic Experiment (ACTIVATE) is a NASA mission to characterize aerosol–cloud interactions over the western North Atlantic Ocean (WNAO). Such characterization requires understanding of life cycle, composition, transport pathways, and distribution of aerosols over the WNAO. This study uses the GEOS-Chem model to simulate aerosol distributions and properties that are evaluated against aircraft, ground-based, and satellite observations during the winter and summer field deployments in 2020 of ACTIVATE. Transport in the boundary layer (BL) behind cold fronts was a major mechanism for the North American continental outflow of pollution to the WNAO in winter. Turbulent mixing was the main driver for the upward transport of sea salt within and ventilation out of BL in winter. The BL aerosol composition was dominated by sea salt, which increased in the summer, followed by organics and sulfate. Aircraft in situ aerosol measurements provided useful constraints on wet scavenging in GEOS-Chem. The model generally captured observed features such as continental outflow, land–ocean gradient, and mixing of anthropogenic aerosols with sea salt. Model sensitivity experiments with elevated smoke injection heights to the mid-troposphere (versus within BL) better reproduced observations of smoke aerosols from the western US wildfires over the WNAO in the summer. Model analysis suggests strong hygroscopic growth of sea salt particles and their seeding of marine BL clouds over the WNAO (< 35° N). Future modeling efforts should focus on improving parameterizations for aerosol wet scavenging, implementing realistic smoke injection heights, and applying high-resolution models that better resolve vertical transport.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

An finite element analysis surrogate model with boundary oriented graph embedding approach for rapid design

Abstract In this work, we present a boundary oriented graph embedding (BOGE) approach for the graph neural network to assist in rapid design and digital prototyping. The cantilever beam problem has been solved as an example to validate its potential of providing physical field results and optimized designs using only 10 ms. Providing shortcuts for both boundary elements and local neighbor elements, the BOGE approach can embed unstructured mesh elements into the graph and performs an efficient regression on large-scale triangular-mesh-based finite element analysis (FEA) results, which cannot be realized by other machine-learning-based surrogate methods. It has the potential to serve as a surrogate model for other boundary value problems. Focusing on the cantilever beam problem, the BOGE approach with 3-layer DeepGCN model achieves the regression with mean square error (MSE) of 0.011 706 (2.41% mean absolute percentage error) for stress field prediction and 0.002 735 MSE (with 1.58% elements having error larger than 0.01) for topological optimization. The overall concept of the BOGE approach paves the way for a general and efficient deep-learning-based FEA simulator that will benefit both industry and Computer Aided Design (CAD) design-related areas.

42 ENGINEERING↗
DOE OSTI2020

Unraveling magneto-structural coupling of Ni 2 MnGa alloy under the application of stress and magnetic field using in situ polarized neutron diffraction

The magneto-structural evolution of a Ni 2 MnGa single crystal was investigated by in situ polarized neutron diffraction under both magnetic field and mechanical loading. The ability to separate the nuclear and magnetic scatterings by polarized neutrons enables simultaneous observations of the twin reorientation (through twin boundary motion) and magnetic moment configuration. It is found that under a 1.0 T saturating magnetic field, twin reorientation can be activated by compressive stress and is accompanied by a realignment of the magnetic moment, which follows the easy axis of the twin variant. On the other hand, compressive stress as small as 1.0 MPa can freeze the twin reorientation. The evolution of the magnetic scattering amplitudes suggests that the change in magnetization was mainly achieved through the magnetic domain wall motion, while the twin boundary motion was blocked.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Noise and error analysis and optimization in particle-based kinetic plasma simulations

In this paper we analyze the noise in macro-particle methods used in plasma physics and fluid dynamics, leading to approaches for minimizing the total error, focusing on electrostatic models in one dimension. We begin by describing kernel density estimation for continuous values of the spatial variable x, expressing the kernel in a form in which its shape and width are represented separately. The covariance matrix of the noise in the density is computed, first for uniform true density. The bandwidth of the covariance matrix C(x,y) is related to the width of the kernel. A feature that stands out is the presence of constant negative terms in the elements of the covariance matrix both on and off-diagonal. These negative correlations are related to the fact that the total number of particles is fixed at each time step; they also lead to the property ∫C(x,y)dy = 0. We investigate the effect of these negative correlations on the electric field computed by Gauss's law, finding that the noise in the electric field is related to a process called the Ornstein-Uhlenbeck bridge, leading to a covariance matrix of the electric field with variance significantly reduced relative to that of a Brownian process. For non-constant density, p(x), still with continuous x, we analyze the total error in the density estimation and discuss it in terms of bias-variance optimization (BVO). For some characteristic length l, determined by the density and its second derivative, and kernel width h, having too few particles within h leads to too much variance; for h that is large relative to l, there is too much smoothing of the density. The optimum between these two limits is found by BVO. For kernels of the same width, it is shown that this optimum (minimum) is weakly sensitive to the kernel shape. Next, we repeat the analysis for x discretized on a grid. In this case the charge deposition rule is determined by a particle shape. An important property to be respected in the discrete system is the exact preservation of total charge on the grid; this property is necessary to ensure that the electric field is equal at both ends, consistent with periodic boundary conditions. We find that if the particle shapes satisfy a partition of unity property, the particle charge deposited on the grid is conserved exactly. Further, if the particle shape is expressed as the convolution of a kernel with another kernel that satisfies the partition of unity, then the particle shape obeys the partition of unity. This property holds for kernels of arbitrary width, including widths that are not integer multiples of the grid spacing. Furthermore, we show results relaxing the approximations used to do BVO optimization analytically, by doing numerical computations of the total error as a function of the kernel width, on a grid in x. The comparison between numerical and analytical results shows good agreement over a range of particle shapes. We discuss the practical implications of our results, including the criteria for design and implementation of computationally efficient particle shapes that take advantage of the developed theory.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗