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At least 577 records · Page 32

Deep Cellular Recurrent Network for Efficient Analysis of Time-Series Data With Spatial Information

Efficient processing of large-scale time series data is an intricate problem in machine learning. Conventional sensor signal processing pipelines with hand engineered feature extraction often involve huge computational cost with high dimensional data. Deep recurrent neural networks have shown promise in automated feature learning for improved time-series processing. However, generic deep recurrent models grow in scale and depth with increased complexity of the data. This is particularly challenging in presence of high dimensional data with temporal and spatial characteristics. Consequently, this work proposes a novel deep cellular recurrent neural network (DCRNN) architecture to efficiently process complex multi-dimensional time series data with spatial information. Here, the cellular recurrent architecture in the proposed model allows for location-aware synchronous processing of time series data from spatially distributed sensor signal sources. Extensive trainable parameter sharing due to cellularity in the proposed architecture ensures efficiency in the use of recurrent processing units with high-dimensional inputs. This study also investigates the versatility of the proposed DCRNN model for classification of multi-class time series data from different application domains. Consequently, the proposed DCRNN architecture is evaluated using two time-series datasets: a multichannel scalp EEG dataset for seizure detection, and a machine fault detection dataset obtained in-house. The results suggest that the proposed architecture achieves state-of-the-art performance while utilizing substantially less trainable parameters when compared to comparable methods in the literature.

60 APPLIED LIFE SCIENCES↗

Microscopy Studies of the Catalyst Inks for PEM Fuel Cells

The microstructure of the catalyst layer in proton exchange membrane fuel cells (PEMFC) is one of the key factors that determine fuel cell performance. The ink preparation, including dispersion solvent, mixing time, and mixing power, are among the less reported parameters that would influence the ionomer distribution on the catalyst surface and the continuity of the carbon network. In this work, novel characterization approaches have been used to investigate the effect of the Pt/C catalyst ink preparation conditions on the morphology and structures of the catalyst layer. Microscopy analysis revealed that longer mixing time led to better performing electrode structures than shorter mixing time. The longer mixing time enabled improved continuity of the ionomer network and high porosity in the cathode layer that contributes to improved proton conductivity and mass transport. This has been reflected in performance and durability tests, where the electrode made from a 5-day catalyst ink displayed improved performance compared to the one made from a 3-day ink. Durability studies showed 26% and 43% loss of the initial mass activity for 5 days and 3 days mixing, respectively. Finally, in addition, the catalyst layer prepared with ethylene glycol as the dispersion solvent showed better durability than water/1-propanol based solution.

25 ENERGY STORAGE↗

Relative-Humidity Dependence of Electrochemically Active Surface Area in Porous Carbon Catalyst Layers

Polymer-electrolyte fuel cells (PEFCs) utilize porous catalyst layers (CLs) formed of carbon supports on which Pt particles are deposited and ionomer films are distributed. Carbon supports themselves have varying degrees of porosity, where high-surface-area carbon (HSC) supports possess nanometer-sized interior pores that are suitable for Pt nanoparticle deposition but prevent deleterious ionomer penetration. However, this requires protons to transport through water pathways inside the pores. To understand the generation of such pathways, we examine the various mechanisms of water uptake by PEFC CLs, and the subsequent impact of water uptake on Pt utilization through developing a multiphysics model of the water wetting phenomena as a function of relative humidity. The model details water uptake via ionomer absorption, capillary condensation in the hydrophilic pores, and surface adsorption using molecular potential that account for various water and surface dipole interactions. The results quantify how mesoporous carbons with highly hydrophilic pores increase Pt utilization through the development of wetted layers, which at the same time enable optimized gas-transport pathways. It also demonstrates the impact of pore-size distribution (PSD) and physical and chemical parameters on the water uptake phenomena, allowing for future CL particle and structure optimization.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Assessing North Atlantic Tropical Cyclone Rainfall Hazard Using Engineered-Synthetic Storms and a Physics-Based Tropical Cyclone Rainfall Model

In this study, we design a statistical method to couple observations with a physics-based tropical cyclone (TC) rainfall model (TCR) and engineered-synthetic storms for assessing TC rainfall hazard. We first propose a bias-correction method to minimize the errors induced by TCR via matching the probability distribution of TCR-simulated historical TC rainfall with gauge observations. Then we assign occurrence probabilities to engineered-synthetic storms to reflect local climatology, through a resampling method that matches the probability distribution of a newly proposed storm parameter named rainfall potential (POT) in the synthetic dataset with that in the observation. POT is constructed to include several important storm parameters for TC rainfall such as TC intensity, duration, and distance and environmental humidity near landfall, and it is shown to be correlated with TCR-simulated rainfall. The proposed method has a satisfactory performance in reproducing the rainfall hazard curve in various locations in the continental United States; it is an improvement over the traditional joint probability method (JPM) for TC rainfall hazard assessment.

54 ENVIRONMENTAL SCIENCES↗

Cloud microphysical structure analysis based on high-resolution insitu measurements

This study investigates the microphysical parameters and shapes of droplet size distributions (DSDs) along three aircraft traverses of developing convective clouds during Cloud Aerosol Interactions and Precipitation Enhancement Experiment (CAIPEEX) Phase III 2015 at a sampling frequency of 25 Hz. The droplet number concentration (Nc, cm -3 ), and liquid water content (LWC, g m -3 ) present steep gradients within zones of a few tens of meters near the cloud edges and relatively gentle gradients in the strong updraft zones. Sometimes, the horizontal LWC distribution resembles a trapezoidal shape with steep LWC and Nc gradients near the cloud edges. The LWC maximums (LWC max ) are lower than the adiabatic LWCs, but the high adiabatic fractions in the cloud core indicate low dilution. High LWC/LWC max , largest droplets, and narrow and similarly shaped DSDs are found in the regions of high updrafts. Zones of low LWC/LWC max are found close to the cloud edges, where DSDs are highly diverse, containing both large and small droplets. Finally, we analyze the mixing diagrams. Significant in-phase turbulent fluctuations in LWC and N c were found. The effective radii change slightly across cloud updraft zones but decrease at the zone of low LWC/LWC max ratio close to cloud edges. The spectra of LWC and N c obey Kolmogorov -5/3 turbulence law. The radii of the correlation of LWC and N c in updraft zones are of several tens of meters. Filaments up to 120–175 m in size are also noticed.

54 ENVIRONMENTAL SCIENCES↗

Triton Field Trials - Changes in Habitats, a Literature Review of Monitoring Technologies

Marine energy devices are installed in highly dynamic environments and have the potential to affect benthic and pelagic habitats around them. Regulatory bodies often require baseline characterization and/or post-installation monitoring to determine whether changes in these habitats are being observed. However, a great diversity of technologies is available for surveying and sampling marine habitats. Selecting the most suitable instrument to identify and measure changes in habitats at marine energy sites can become a daunting task. We conducted a thorough review of journal articles, survey reports, and grey literature to extract information about the technologies used, the data collection and processing methods, and the performance and effectiveness of these instruments. We examined documents related to marine energy development, offshore wind farms, oil and gas offshore sites, and other marine industries around the world over the last 20 years, as well as national and international guidelines for surveying habitats around offshore activities. A total of 120 different technologies were identified across six main habitat categories: seafloor, sediment, infauna, epifauna, pelagic, and biofouling. The technologies were organized into 12 broad technology classes: acoustic, corer, dredge, grab, hook and line, net and trawl, plate, remote sensing, scrape samples, trap, visual, and others. Visual was the most common and the most diverse technology class, with applications across all six habitat categories. Sampling designs varied considerably among the reviewed studies but transect was the predominant design for surveying seafloor, epifauna, and pelagic habitats. The most common data analyses were univariate and multivariate statistical analyses aimed at calculating and comparing biodiversity indices, characterizing faunal assemblages or sediment classes, or modeling the distribution of animals related to abiotic parameters. Technologies and sampling methods adaptable and designed to work efficiently in energetic environments have greater success at marine energy sites. In addition, sampling designs and statistical analyses should be carefully thought through to identify differences in faunal assemblages and spatiotemporal changes in habitats.

16 TIDAL AND WAVE POWER↗

Failure Analysis for Molten Salt Thermal Energy Storage Tanks for In-Service CSP Plants

Thermal Energy Storage (TES) is a fundamental component in concentrating solar power (CSP) plants to increase the plant's dispatchability, capacity factor, while reducing the levelized cost of electricity. In central receivers CSP plants, nitrate molten salts have been used for several years for operation temperatures of up to 565 degrees C. Despite many efforts to advance nitrate salt to higher operation temperatures (even considering a replacement with molten chloride salts) to achieve higher energy conversion efficiencies, the 565 degrees C temperature is currently considered the state-of-the art. Although molten salt tanks have been broadly deployed in commercial CSP plants worldwide, several failures have been reported in these tanks after a few months or years of operation, causing significant economic loss and mistrust in CSP technologies. Most of these failures are associated with the infancy of the technology and multiple issues related to tank design, fabrication, commissioning, and aggressive operation. A technical standard dedicated to the design and fabrication of molten nitrate TES tanks does not exist today. Current in-service molten salt tanks have been generally designed based on the American Petroleum Institute's (API) 650 and ASME Section II standards. The API 650 code provides guidelines for dimensions and fabrication for oil storage tanks up to 260 degrees C. The ASME standard provides allowable stress values for various materials at a range of temperatures and conditions. Both standards seem to be limited for molten salt TES tanks where high temperatures, thermal cycling, and transient conditions are expected. In 2020, NREL released the Concentrating Solar Power Best Practices Study (NREL/TP-5500-75763) that summarized multiple issues in CSP plants, along with potential alternatives and recommendations to address those issues based on information collected from participants representing about 80% of operating CSP plants in the world. One of the recommendations from this study was the development of accurate and validated models to evaluate the plant's transient operation, capable of capturing the effect of short-term clouds and operator response, while being flexible in being adapted to various spatial and temporal resource data. The "Failure Analysis for Molten Salt Thermal Energy Tanks for In-Service CSP Plants" project was inspired on this recommendation and was focused on (1) the development and validation of a physics-based model for a representative, commercial-scale molten salt tank, (2) performing simulations to evaluate the behavior of the tank as a function of typical plant operation conditions, (3) understanding tank failures mechanisms, (4) determining the residual stress and distortion in the tank floor after welding fabrication and evaluating their impact in the stresses developed in the tank during operation, (5) assessing the impact of key operation parameters on the temperature and stress distribution, (6) conduct a preliminary evaluation of design features to reduce stress and improve tank's reliability, and (7) estimate the tank's service life based on the stress developed under diverse operation scenarios. From the analysis conducted in the project and presented in this report, it was found that maximum stresses surpassing the yield strength point of the stainless steel (SS) 347H are developed on the tank floor near the perimeter. These large stresses are strongly influenced by the initial residual stresses and distortion of the tank floor after welding fabrication. During operation, large stresses are developed in the tank floor at high operation temperatures with large salt inventory levels during transient operation. High stresses are also related to elevated temperature gradients in the tank floor that could be attributed to insufficient mixing within the salt inflow and the salt inventory. Based on the analysis, creep is the predominant failure mechanism. However, the large stress levels could favor the plastic deformation into buckles, and crack formation due to stress relaxation cracking during cycle operation. A lifetime below 3 years was estimated for the typical plant operation conditions and a specific initial residual stress and deformation distribution of the tank floor. The estimated life agrees with the service time to failure reported in several commercial molten salt tanks. Desing and operation guidelines can be extracted from the analysis presented in this report, which could be adopted by tank manufacturers and CSP operators to advance toward an ultimate solution for tank failures by reducing residual and operational stresses to achieve a tank service life of more than 30 years. Addressing failures in molten salt TES tanks is fundamental for the CSP industry's survivability, but it is also important for other industrial and power generation applications using this technology, including nuclear and concentrating solar thermal.

14 SOLAR ENERGY↗

Dominant Wave Energy Systems and Conditional Wave Resource Characterization for Coastal Waters of the United States

Opportunities and constraints for wave energy conversion technologies and projects are evaluated by identifying and characterizing the dominant wave energy systems for United States (US) coastal waters using marginal and joint distributions of the wave energy in terms of the peak period, wave direction, and month. These distributions are computed using partitioned wave parameters generated from a 30 year WaveWatch III model hindcast, and regionally averaged to identify the dominant wave systems contributing to the total annual available energy ( A A E ) for eleven distinct US wave energy climate regions. These dominant wave systems are linked to the wind systems driving their generation and propagation. In addition, conditional resource parameters characterizing peak period spread, directional spread, and seasonal variability, which consider dependencies of the peak period, direction, and month, are introduced to augment characterization methods recommended by international standards. These conditional resource parameters reveal information that supports project planning, conceptual design, and operation and maintenance. The present study shows that wave energy resources for the United States are dominated by long-period North Pacific swells (Alaska, West Coast, Hawaii), short-period trade winds and nor’easter swells (East Coast, Puerto Rico), and wind seas (Gulf of Mexico). Seasonality, peak period spread, and directional spread of these dominant wave systems are characterized to assess regional opportunities and constraints for wave energy conversion technologies targeting the dominant wave systems.

16 TIDAL AND WAVE POWER↗

Discovery of a Gamma-Ray Black Widow Pulsar by GPU-accelerated Einstein@Home

We report the discovery of 1.97 ms period gamma-ray pulsations from the 75 minute orbital-period binary pulsar now named PSR J1653-0158. The associated Fermi Large Area Telescope gamma-ray source 4FGL J1653.6-0158 has long been expected to harbor a binary millisecond pulsar. Despite the pulsar-like gamma-ray spectrum and candidate optical/X-ray associations—whose periodic brightness modulations suggested an orbit—no radio pulsations had been found in many searches. The pulsar was discovered by directly searching the gamma-ray data using the GPU-accelerated Einstein@Home distributed volunteer computing system. The multidimensional parameter space was bounded by positional and orbital constraints obtained from the optical counterpart. More sensitive analyses of archival and new radio data using knowledge of the pulsar timing solution yield very stringent upper limits on radio emission. Any radio emission is thus either exceptionally weak, or eclipsed for a large fraction of the time. The pulsar has one of the three lowest inferred surface magnetic-field strengths of any known pulsar with B surf ≈ 4 × 10 7 G. The resulting mass function, combined with models of the companion star's optical light curve and spectra, suggests a pulsar mass gsim 2 M ⊙ . The companion is lightweight with mass ~0.01 M ⊙ , and the orbital period is the shortest known for any rotation-powered binary pulsar. This discovery demonstrates the Fermi Large Area Telescope's potential to discover extreme pulsars that would otherwise remain undetected.

79 ASTRONOMY AND ASTROPHYSICS↗

Bayesian Monte Carlo Evaluation Framework for Imperfect Data and Models [Abstract]

Nuclear data evaluation methods conventionally make the following assumptions: prior and posterior probability distribution functions (PDFs) of all model parameters and data are normal (Gaussian); the linear approximation is sufficiently accurate for minimization of a cost function (even for non-linear models); and that both the model (of, e.g., neutron cross section) and experimental data (including their covariance data) are perfect. These assumptions are inherent to the well-known generalized linear least squares (GLLS) minimization method commonly used for evaluations of resolved resonance region (RRR) neutron cross sections. However, these assumptions are often not justified due to the presence of non-normal PDFs, non-linear models (e.g. R -matrix formalism), and inherent imperfections in data and models (e.g. discrepant data sets, discrepancies between the previous evaluation and newly measured data, or imperfect covariance data). We remove the said assumptions in a mathematical framework of Bayes’ theorem, and implement it using the Metropolis-Hastings Monte Carlo method. Parameters of a new kind are introduced to parameterize inherent imperfections, e.g. , any discrepancies between the theoretical model and measured data. These new parameters enable evaluators to quantify their expert judgement about any discrepancies or imperfections in a reproducible manner. We demonstrate the framework with an ongoing evaluation of 233 U in the eV region using the ENDF-B/VIII library and transmission data measured by Guber, et al. , and compare the posterior parameters to those obtained by conventional evaluation methods.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Reweighting Underlying Event and Colour Reconnection parameter variations in Sherpa

We propose and validate a new method to trace the impact of parameter variations in the simulation of multi-parton interactions and colour reconnections in the Sherpa event generator. They are reflected, at an event-by-event basis, through relative weights with respect to the central production parameters that give rise to the generated events and distributions. Our method facilitates the tuning of the Monte Carlo event generator at a dramatically reduced computational cost, alleviates parameter sensitivity studies, and enables robust quantification of parametric uncertainties on-the-fly, one of the missing ingredients for future simulations of high-energy particle collisions. The method can easily be adapted to and implemented in other event generators. To illustrate its potential, we here consider combined tunes of the multi-parton-interaction and colour-reconnection models in Sherpa using LHC proton-proton collision data at $\sqrt{s}=7\,\text{TeV}$. We furthermore calibrate the energy-scaling behaviour of dimensionful model parameters based on $\sqrt{s}=13\,\text{TeV}$ LHC data and Tevatron data taken at $\sqrt{s}=1.96\,\text{TeV}$.

Pabst, Moritz [Gottingen U.]↗

Remote monitoring system

A remote monitoring system for monitoring parameters of an underground asset, such as an electric cable system, a pipeline serving as a conduit for water, gas, oil, sewage, or the like. Parameter detectors, such as sensors that measure temperature, voltage, current, moisture, etc., are distributed along the asset to provide electrical signals that represent respective monitored parameters of the asset. An underground hub disposed in an underground vault, or manhole, through which the asset passes is coupled to the parameter detectors to acquire the signals provided by the parameter detectors. The hub includes a controller to provide data derived from the acquired signals, the data being transmitted to a remote central location by a transceiver, or modem, via LP-WAN communication. Signals from the transceiver are transmitted directly from underground.

24 POWER TRANSMISSION AND DISTRIBUTION↗

Methods and systems for evaluating and improving distribution-grid observability

This document describes systems and techniques for evaluating and improving distribution-grid observability. These systems and techniques allow engineers to quantify the observability of a distribution grid, which represents an ability to combine actual measurements and various types of computations (e.g., analytics, estimators, forecasters), from a system model. Distribution engineers can also identify islands of observability where operating parameters, including voltages, currents, and power flows, can be determined from available sensor readings. By exclusion, distribution engineers can similarly identify areas of the distribution grid with observability deficiencies that may require additional instrumentation to maintain proper operation. Distribution engineers, using an iterative or automated process, can determine the observability of the system model with new or relocated sensors to generate a sensor allocation plan. The sensor allocation plan can indicate the number and location of sensors to either maximize observability for a fixed sensor cost or minimize sensor cost for predetermined observability.

Reiman, Anthony P.↗

Grassland ecosystem type drives AM fungal diversity and functional guild distribution in North American grasslands

Nutrient exchange forms the basis of the ancient symbiotic relationship that occurs between most land plants and arbuscular mycorrhizal (AM) fungi. Plants provide carbon (C) to AM fungi and fungi provide the plant with nutrients such as nitrogen (N) and phosphorous (P). Nutrient addition can alter this symbiotic coupling in key ways, such as reducing AM fungal root colonization and changing the AM fungal community composition. However, environmental parameters that differentiate ecosystems and drive plant distribution patterns (e.g., pH, moisture), are also known to impact AM fungal communities. Identifying the relative contribution of environmental factors impacting AM fungal distribution patterns is important for predicting biogeochemical cycling patterns and plant-microbe relationships across ecosystems. Here, to evaluate the relative impacts of local environmental conditions and long-term nutrient addition on AM fungal abundance and composition across grasslands, we studied experimental plots amended for 10 years with N, P, or N and P fertilizer in different grassland ecosystem types, including tallgrass prairie, montane, shortgrass prairie, and desert grasslands. Contrary to our hypothesis, we found ecosystem type, not nutrient treatment, was the main driver of AM fungal root colonization, diversity, and community composition, even when accounting for site-specific nutrient limitations. We identified several important environmental drivers of grassland ecosystem AM fungal distribution patterns, including aridity, mean annual temperature, root moisture, and soil pH. This work provides empirical evidence for niche partitioning strategies of AM fungal functional guilds and emphasizes the importance of long-term, large scale research projects to provide ecologically relevant context to nutrient addition studies.

59 BASIC BIOLOGICAL SCIENCES↗

Measurements of $\pi ^\pm $, $K^\pm $, p and $\bar{p}$ spectra in $^7$Be+$^9$Be collisions at beam momenta from 19A to 150A ${\mathrm{Ge} \mathrm{V}}\!/\!c$ with the NA61/SHINE spectrometer at the CERN SPS

The NA61/SHINE experiment at the CERN Super Proton Synchrotron (SPS) studies the onset of deconfinement in hadron matter by a scan of particle production in collisions of nuclei with various sizes at a set of energies covering the SPS energy range. This paper presents results on inclusive double-differential spectra, transverse momentum and rapidity distributions and mean multiplicities of $\pi^, \pm K^\pm$ p and $\bar{p}$ produced in the 20% most central Be+$^9$ Be collisions at beam momenta of 19A, 30A, 40A, 75A and 150A ${\mathrm{Ge} \mathrm{V}}\!/\!c$. The energy dependence of the $K^\pm/ \pi^\pm$ ratios as well as of inverse slope parameters of the $K^\pm$ transverse mass distributions are close to those found in inelastic $p+p$ reactions. The new results are compared to the world data on $p+p and Pb+Pb collisions as well as to predictions of the Epos , U r qmd , Ampt , Phsd and Smash models.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

Magnetic Energy Dissipation and γ-Ray Emission in Energetic Pulsars

Abstract Some of the most energetic pulsars exhibit rotation-modulated γ -ray emission in the 0.1–100 GeV band. The luminosity of this emission is typically 0.1%–10% of the pulsar spin-down power ( γ -ray efficiency), implying that a significant fraction of the available electromagnetic energy is dissipated in the magnetosphere and reradiated as high-energy photons. To investigate this phenomenon we model a pulsar magnetosphere using 3D particle-in-cell simulations with strong synchrotron cooling. We particularly focus on the dynamics of the equatorial current sheet where magnetic reconnection and energy dissipation take place. Our simulations demonstrate that a fraction of the spin-down power dissipated in the magnetospheric current sheet is controlled by the rate of magnetic reconnection at microphysical plasma scales and only depends on the pulsar inclination angle. We demonstrate that the maximum energy and the distribution function of accelerated pairs is controlled by the available magnetic energy per particle near the current sheet, the magnetization parameter. The shape and the extent of the plasma distribution is imprinted in the observed synchrotron emission, in particular, in the peak and the cutoff of the observed spectrum. We study how the strength of synchrotron cooling affects the observed variety of spectral shapes. Our conclusions naturally explain why pulsars with higher spin-down power have wider spectral shapes and, as a result, lower γ -ray efficiency.

79 ASTRONOMY AND ASTROPHYSICS↗

Energy and particle balance during plasma detachment in a long-leg divertor configuration

Abstract Comprehensive studies of energy and particle balances in the transition to plasma detachment in an alternative divertor configuration with long outer legs are shown. Numerical simulations are performed with the 2D code suite SOLPS 4.3, using a disconnected double null grid with narrow, tightly baffled long poloidal leg divertors at the outer lower target and outer upper target. A particle count scan is performed using the ‘closed gas box’ model, where the tunable parameter in the simulations is the total number of deuterium particles in the simulation space and all other parameters are held fixed, including a constant input power and trace neon impurity radiation, to assess the physics of the transition to detachment in the system as the particle count increases. Three main aspects of the physics of divertor detachment are addressed: the criteria for the local onset of divertor detachment in each of the divertors, the distribution of heat flux and other plasma parameters between the four divertors as each divertor transitions to detachment, and the role of perpendicular transport in the transition to the detached regime. A synergistic mechanism by which the cross-field transport is reduced by factors associated with the onset of plasma recombination effects is identified. These results are compared to the existing understanding of the physics of the transition to plasma detachment in standard divertors.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Probability Density Function for the spatial and intensity distribution of neutron-induced defects in Silicon

The ability to model semiconductor device degradation under neutron irradiation depends upon having a robust modeling capability for the neutron-induced collision cascades as well as a means to analytically fit the resulting probability distributions of defect production and ionizing energy deposition for purposes of extrapolation to low-probability, high-consequence scenarios. In this paper, the widely-utilized binary collision approximation codes MARLOWE and SRIM are deployed in conjunction with a critical examination of their parameterizations as benchmarked against higher-fidelity molecular dynamics simulations. A simple 3-parameter form described by the Generalized Logistic Distribution is shown to be a good fit to Frenkel pair and ionization intensity distributions in bulk silicon. The BCA codes are then applied to simulate cascades in 5 nm layers of a representative gate-all-around nanosheet transistor, where joint probability distributions of threshold levels of damage to multiple layers are evaluated.

36 MATERIALS SCIENCE↗