Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “distribution factor”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 415 records · Page 23

Evaporation from thin porous Coatings: Pore size effects and predictive equation for homogeneous coatings

Evaporation of small water droplets on solids is hindered because surface tension pulls the droplet into a spherical cap that has a small perimeter. Our solution is to coat a solid with a very thin, porous layer into which the droplet flows to create a large-area disk with concomitant high rate of evaporation. We investigate evaporation by varying factors that have not been previously considered: pore size and distribution, contact angle, temperature, and relative humidity (RH). A larger pore size resulted in faster evaporation, which we explain through faster transport within the coating. Even faster evaporation occurred for a bilayer structure with small particles on the air side and larger particles on the solid side. Further, the water advancing contact angle had an insignificant effect in the range from < 10° through to 60°. Our results for different pore sizes, temperature, humidity, and contact angle all collapse onto a single curve when appropriately normalized. This validates an equation that can be used for the evaporation from a homogeneous coating that depends only one empirical factor and the droplet volume. Since the volume is often user-controlled, we envisage that this equation can be used to predict evaporation and guide design of fast-drying coatings.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Estimating the Contributions to Human Error Probability from the Convolution of the Distribution of Time Available and Time Required

As part of their duties, Human Reliability Analysis must often evaluate if crews in nuclear power plants (NPPs) can complete tasks associated with a human-failure event within time limits. For example, the time required in NPP scenarios is determined by systematic and structured walkthroughs, feasibility studies, recorded times from training exercises, and interviews with experienced operators and experts. Typically, a point estimate is derived for the estimate (mean, maximum, or 95th percentile of time required). Using point-estimate values can mask the risk associated with variability among crews, plant conditions and set-up, environmental conditions, and other impact factors under which these actions are executed. While point estimates for time required and time available have served the industry well, without considering the uncertainty they could lead to biased understanding about the risk. The Integrated Human Event Analysis System - General Methodology (IDHEAS-G) model (developed by the US Nuclear Regulatory Commission, NRC) for human error probability calculates human error probability by summing two probabilities: insufficient time and cognitive error. As such, the model takes a more holistic approach by considering the full distributions for time required and time available to calculate the human error probability because the time available to complete the task is insufficient. In this study, we expand on the work of the NRC and discuss methods for estimating these time considerations. For example, for the time required, the impact of Performance Influencing Factors (PIFs) on the distribution was divided into impacts that are aleatory in nature, such as crew-to-crew variability, and those that are epistemic (i.e., the PIFs). Starting with the factors that introduce aleatory uncertainty, a first-order distribution was developed from a large set of time required (i.e., NPP task completion times) data for the range of operator actions that occur in the NPP control room under simulated accident conditions. The first-order distribution can then be adjusted to account for epistemic uncertainty using research associated with the impact of applicable PIFs on the time required. We also develop guidance for analysts to address the probability distributions for the time available. The guidance we developed on how to estimate time required and time available distributions is based on the identification of pertinent research and data, data analyses, and expert knowledge elicitation.

human error probability, human performance, time e↗

Sensitivity of fatigue reliability in wind turbines: effects of design turbulence and the Wöhler exponent

Fatigue assessment of wind turbines involves three main sources of uncertainty: material resistance, load, and the damage accumulation model. Many studies focus on increasing the accuracy of fatigue load assessment to improve the fatigue reliability. Probabilistic modeling of the wind's turbulence standard deviation is an example of an approach used for this purpose. Editions 3 and 4 of the IEC standard for the design of wind energy generation systems (IEC 61400-1) suggest different probability distributions as alternatives for the representative turbulence in the normal turbulence model (NTM) of edition 1. There are debates on whether the suggested distributions provide conservative reliability levels, as the established design safety factors are calibrated based on the representative turbulence approach. The current study addresses the debate by comparing annual reliability based on different scenarios of NTM using a probabilistic approach. More importantly, it elaborates on the relative importance of load assessment accuracy in defining the fatigue reliability. Using the DTU 10 MW reference wind turbine and the first-order reliability method (FORM), we study the changes in the annual reliability level and its sensitivity to the three main random inputs. We perform the study considering the blade root flapwise and the tower base fore–aft moments, assuming different fatigue exponents in each load channel. The results show that integration over distributions of turbulence in each mean wind speed results in less conservative annual reliability levels than representative turbulence. The difference in the reliability levels varies according to turbulence distribution and the fatigue exponent. In the case of the tower base, the difference in the annual reliability index after 20 years can be up to 50 %. However, the model and material uncertainty have much higher effects on the reliability levels compared to load uncertainty. Knowledge about such differences in the reliability levels due to the choice of turbulence distribution is especially important, as it impacts the extent of lifetime extension through reliability reassessments.

17 WIND ENERGY↗

Absolute measurement of the Fano factor using a Skipper-CCD

Skipper-CCD can achieve deep sub-electron readout noise making possible the absolute determination of the exact number of ionized electrons in a large range, from 0 to above 1900 electrons. In this work we present a novel technique that exploits this unique capability to allow self-calibration and the ultimate determination of silicon properties. We performed an absolute measurement of the variance and the mean number of the charge distribution produced by $^{55}$Fe X-rays, getting a Fano factor absolute measurement in Si at 123K and 5.9 keV. A value of 0.119 $\pm$ 0.002 was found and the electron-hole pair creation energy was determined to be (3.749 $\pm$ 0.001) eV. This technology opens the opportunity for direct measurements of the Fano factor at low energies.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

Next-to-next-to-leading power corrections to unpolarized Semi-Inclusive Deep Inelastic Scattering

Semi-Inclusive Deep Inelastic Scattering (SIDIS) is a key tool for exploring the three-dimensional structure of the nucleon through Transverse Momentum Dependent parton distributions and fragmentation functions. While leading-power contributions to the SIDIS cross-section are well established, next-to-leading power (NLP) corrections of order 1/Q and next-to-next-to-leading power (NNLP) corrections of order 1/Q 2 to the hadronic tensor have only recently begun to be systematically investigated. These corrections are essential for reliable phenomenology and interpretation of modern high-precision data. In recent papers by one of the authors, NNLP corrections to the Drell-Yan process were derived using the rapidity factorization formalism. In the present work, we extend this approach to SIDIS and obtain analytic expressions for the unpolarized structure functions. We derive NNLP corrections that include convolutions of unpolarized distributions, f 1 , with unpolarized fragmentation functions, D 1 , and Boer-Mulders functions, ${h}_1^{\perp }$, with Collins fragmentation functions, ${H}_1^{\perp }$. We compare our results with previous formulations, provide numerical studies, confront our predictions with HERMES and COMPASS measurements, and present predictions for future experiments at Jefferson Lab and the Electron-Ion Collider.

deep inelastic scattering↗

Nonlinear simulations of GAEs in NSTX-U

A set of nonlinear simulations has been performed in order to study the nonlinear evolution of unstable global Alfvén eigenmodes in the National Spherical Torus Experiment-Upgrade (NSTX-U). Results of the single toroidal mode number, n, simulations are compared with a full nonlinear simulation (all toroidal harmonics included). In single-n simulations, the conservation of two integrals of motion of a particle in a cyclotron resonance with a monochromatic wave is demonstrated, resulting in a one-dimensional evolution of the particle distribution in (E,μ,pϕ) phase-space. Nonlinear simulations (both single-n and full nonlinear) show a significant redistribution of the resonant fast ions, especially in the pitch parameter. Thus, the changes in the resonant particle's parallel and perpendicular energies can be several times larger than the total particle energy change, with only a small fraction transferring into the excitation of the mode itself. This implies that even a relatively small amplitude mode can significantly modify the beam distribution in the resonant region. For the NSTX-U case considered, the single-n simulation results are close to full nonlinear simulation only for the most unstable mode, in which case the saturation amplitudes and changes in the fast ion distribution are comparable. In contrast, peak amplitudes of subdominant modes in all-n simulations are smaller by a factor of 3–10 compared to single-n runs due to the flattening of the beam ion distribution by the fastest growing mode.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

A Review of Value of Solar Studies In Theory and In Practice

This brief summarizes a collection of state- and utility-commissioned value-of-solar (VoS) studies and related literature, with a focus on who commissioned the study, which value and cost categories were discussed and/or quantified, and the methods used. Our objective is to compile information on prior VoS studies to inform state regulators and other stakeholders that may pursue related studies or integrate findings into rate design. The brief is organized into three parts: 1) an introduction to distributed solar photovoltaic (DPV) compensation; 2) a review of theoretical research on VoS; and 3) a review of VoS studies. The vast majority of VoS studies have served an informational role of quantifying the net benefits of PV. Three studies were commissioned in states or utility service territories that subsequently implemented VoS tariffs in California, New York, and Austin, Texas. When applied as a tariff, VoS aims to compensate PV output as efficiently as possible by doing so at rates that reflect the marginal benefits and costs of PV through value and cost categories that may vary temporally and/or geographically. This could lead to higher compensation in locations and times where more PV output is more valuable and consequently drive adoption in those locations to provide more societal benefits. Value and cost factors can be broadly grouped into five categories: generation, transmission, distribution, other utility, and other social categories. Those conducting VoS studies must weigh various tradeoffs when deciding which categories to include and quantify. Tradeoffs include prioritizing values based on their magnitude of value or cost impact, as well as taking into account the feasibility of data collection and accurate quantification. Values of higher magnitude and estimation feasibility are quantified in the majority of studies, including the earliest of studies conducted in the 2000s and 2010s. Additionally, some values of higher magnitude but low feasibility in the earliest of studies have become quantifiable in recent years. There are some values with low average system-wide levels but very high magnitude in specific locations or hours. The value magnitude in some cases can be tied to DPV penetration with low value in areas with little congestion and/or low penetration and vice versa. In these cases, values that are easier to quantify are often incorporated, while those that are more difficult are often addressed via a placeholder value. The placeholder value is paired with a discussion around data needs and methods to improve future estimates, as well as a conversation about when these value categories may increase in magnitude and necessitate more rigorous quantification. This brief summarizes findings from two meta-analyses of VoS studies that took place between 2005 and 2018, as well as findings from four additional studies published from 2018 to 2023. Table ES-1 summarizes the various value and cost categories included in each respective study and whether they were quantified, discussed, or omitted. Values such as avoided energy, capacity, transmission capacity, line losses, and avoided environmental costs are quantified in every study. Some categories were deemed harder to quantify and less impactful at the time of the study, so they were discussed but not quantified (e.g., ancillary services). Other categories, including many at the distribution level, were very locationally and/or temporally specific and dependent on high DPV penetration. These were sometimes quantified and at other times discussed. Notably, when it came to utility costs, integration costs were discussed in all cases, though they were deemed to have a small impact. Other utility costs were omitted for the most part; however, the utility-commissioned study (by NorthWestern Energy in Montana) included both lost utility revenue and programmatic/administrative cost categories. While there are some similarities across studies, each had fairly unique methods that are detailed in the body of this brief.

14 SOLAR ENERGY↗

Online Model-Free DER Dispatch Via Adaptive Voltage Sensitivity Estimation and Chance Constrained Programming

This paper proposes an online data-driven distributed energy resource management system (DERMS) for distribution system optimal DER dispatch as well as voltage regulation. Here, the key innovation is to leverage the Local Sensitivity Factor (LSF) for transforming the DER control into a computationally efficient linear programming (LP) problem. By taking real-time measurements, the estimation of LSF eliminates the need for an accurate distribution system model as well as full nodal load information, which is difficult to achieve in practice. A robust recursive least squares method is also developed to ensure the robust estimation of LSF, which is initialized using reasonable values from model-derived LSFs. This allows the system to adapt to changing operational conditions effectively. A scenario-based, chance-constrained framework is further employed to ensure voltage remains within acceptable limits in the presence of measurement and estimation uncertainties. Test results on a real-world, 759-node distribution network located in western Colorado, U.S., validate the effectiveness and robustness of the proposed control approach and demonstrate its superior performance as compared to alternative methods.

24 POWER TRANSMISSION AND DISTRIBUTION↗

A new approach to semi-inclusive deep-inelastic scattering with QED and QCD factorization

We present the details of a new factorized approach to semi-inclusive deep-inelastic scattering which treats QED and QCD radiation on equal footing, and provides a systematically improvable approximation to the extraction of transverse momentum dependent parton distributions. We demonstrate how the QED contributions can be well approximated by collinear factorization, and illustrate the application of the factorized approach to QED radiation in inclusive scattering. For semi-inclusive processes, we show how radiation effects prevent a well-defined “photon-nucleon” frame, forcing one to use a two-step process to account for the radiation. We illustrate the utility of the new method by explicit application to the spin-dependent Sivers and Collins asymmetries.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

General mass variable flavor number scheme for 𝑍 boson production in association with a heavy quark at hadron colliders

We present a methodology to streamline implementation of massive-quark radiative contributions in calculations with a variable number of active partons in proton-proton collisions. The methodology introduces subtraction and residual heavy-quark parton distribution functions (PDFs) to implement calculations in the Aivazis–Collins–Olness–Tung (ACOT) factorization scheme and its simplified realization in various processes up to the next-to-the-next-to-leading order in the QCD coupling strength. Interpolation tables for bottom-quark subtraction and residual distributions for CT18 NLO and NNLO PDF ensembles are provided in the common LHAPDF6 format. A numerical calculation of 𝑍-boson production with at least one 𝑏 jet at the Large Hadron Collider beyond the lowest order in QCD is considered for illustration purposes.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

Critical Current Distributions of Recent Bi-2212 Round Wires

Bi 2 Sr 2 CaCu 2 O 8+x (Bi-2212) is the only high-field, high-temperature superconductor (HTS) capable of reaching a critical current density J c (16 T, 4.2 K) of 6500 A·mm –2 in the highly desirable round wire (RW) form. However, state-of-the-art Bi-2212 conductors still have a critical current density (J c ) to depairing current density (J d ) ratio around 20 to 30 times lower than that of state-of-the-art Nb–Ti or REBCO. Previously, we have shown that recent improvements in Bi-2212 RW J c are due to improved connectivity associated with optimization of the heat treatment process, and most recently due to a transition to a finer and more uniform powder manufactured by Engi-Mat. One quantitative measure of connectivity may be the critical current (I c ) distribution, since the local I c in a wire can vary along the length due to variable vortex-microstructure interactions and to factors such as filament shape variations, grain-to-grain connectivity variations and blocking secondary phase distributions. Modeling the experimental V-I transition measured on a low resistance shunt as a complex sum of voltage contributions of individual filament and wire sub-sections allows a numerical extraction of the I c distribution from the d 2 V/dI 2 treatment of the V-I curves. Here we compare ~0.1 m length I c distributions of Bi-2212 RWs with recent state-of-the-art very high-J c Engi-Mat powder and lower J c and older Nexans granulate powder. We do find that the I c spread for Bi-2212 wires is about twice the relative standard of high-J c Nb–Ti well below H irr . We do not yet see any obvious contribution of the Bi-2212 anisotropy to the I c distribution and are rather encouraged that these Bi-2212 round wires show relative I c distributions not too far from high-J c Nb–Ti wires.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

AC Power Flow Based DLMP Calculation and Decomposition Method to Smooth Power Fluctuation of Distributed Renewable Energy Sources

As the penetration of renewable energy sources increases, the growing renewable power variability brings ramping issues to power systems. Meanwhile, the development of distributed energy resources (DERs) makes the distribution systems to provide both energy and ancillary services. To incentivise individual resources and customers to alleviate ramping issues on the demand side, a two-stage distribution locational marginal price (DLMP) calculation and decomposition method is developed to formulate the marginal power ramping price for DERs. In the first stage of the proposed method, a distribution system operator market scheduling model based on AC optimal power flow is designed to estimate the optimal operating point of the distribution system. Subsequently, the voltage and power flow constraints are linearised in stage two to calculate DLMP. Finally, based on the Lagrange function and sensitivity factors, DLMP is decomposed to the marginal costs for active/reactive power, voltage management, power loss and power variability. Case studies demonstrate that the proposed model can effectively smooth the power fluctuation and reduce the ramping flexibility requirements of distribution systems.

AC optimal power flow↗

Finite Element Modeling of the Fiber-Matrix Interface in Polymer Composites

Polymer composites are used in numerous industries due to their high specific strength and high specific stiffness. Composites have markedly different properties than both the reinforcement and the matrix. Of the several factors that govern the final properties of the composite, the interface is an important factor that influences the stress transfer between the fiber and matrix. The present study is an effort to characterize and model the fiber-matrix interface in polymer matrix composites. Finite element models were developed to study the interfacial behavior during pull-out of a single fiber in continuous fiber-reinforced polymer composites. A three-dimensional (3D) unit-cell cohesive damage model (CDM) for the fiber/matrix interface debonding was employed to investigate the effect of interface/sizing coverage on the fiber. Furthermore, a two-dimensional (2D) axisymmetric model was used to (a) analyze the sensitivity of interface stiffness, interface strength, friction coefficient, and fiber length via a parametric study; and (b) study the shear stress distribution across the fiber-interface-matrix zone. It was determined that the force required to debond a single fiber from the matrix is three times higher if there is adequate distribution of the sizing on the fiber. The parametric study indicated that cohesive strength was the most influential factor in debonding. Moreover, the stress distribution model showed the debonding mechanism of the interface. It was observed that the interface debonded first from the matrix and remained in contact with the fiber even when the fiber was completely pulled out.

36 MATERIALS SCIENCE↗

The onset distribution of rotating m , n = 2 , 1 tearing modes and its consequences on the stability of high-confinement-mode plasmas in DIII-D

Abstract Analysis of a multi-scenario database of over 13 000 DIII-D H-mode discharges shows that the m , n = 2 , 1 magnetic islands are dominantly pressure gradient driven, stochastically triggered non-linear instabilities at all edge safety factor ( q 95 ) values. The instability onset time closely follows the exponential distribution in intermediate and high q 95 scenarios and is characterized by near constant onset rate ( λ ), in accordance with Poisson-point processes. This implies that the plasmas are operated in marginally stable conditions, characterized by a small threshold for instability growth and variations in the trigger amplitude and/or the stabilizing mechanisms with temporally uniform random distribution in this database. While the majority of the tearing modes occur in the first current-profile relaxation time of the β N flattop, constant λ throughout the β N flattop shows that the tearing onset is insensitive to the evolution of the equilibrium current profile. In low q 95 scenarios, where a large fraction of the plasmas are operated at low torque, λ increases over the course of the β N flattop, showing that these plasmas evolve toward more unstable conditions. The onset rate rapidly increases with β N , while it does not show a clear dependence on the current gradient at the mode rational surface. Overall, these observations support that the majority of the analyzed 2,1 tearing modes are non-linear, neoclassically driven instabilities and classical stability does not play a dominant role in their onset.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

An Interpretable Index of Social Vulnerability to Environmental Hazards

Index-based measures of social vulnerability to environmental hazards are commonly modeled from composites of population-level risk factors. These models overlook individual context in communities' experiences of environmental hazards, producing metrics that may hinder spatial decision support for mitigating and responding to hazards. This paper introduces an interpretable, high-resolution model for generating an individual-oriented social vulnerability index (IOSVI) for the United States built on synthetic populations that couples individual and social determinants of vulnerability. The IOSVI combines an individual vulnerability index (IVI) that ranks individuals in an area’s synthetic population based on intersecting risk factors, with a social vulnerability index (SVI) based on the population’s cumulative distribution of IVI scores. Interpretability of the IOSVI procedure is demonstrated through examples of national, metropolitan, and neighborhood (census tract) level spatial variation in index scores and IVI themes, as well as an exploratory analysis examining risk factors affecting a specific sub-population (military veterans) in areas of high social and environmental vulnerability.

Tuccillo, Joe↗

Sectoral and growth rate control on elemental uptake by individual calcite crystals

Heterogeneous distribution of trace elements (impurities) within individual calcite crystals is a phenomenon commonly observed in natural and laboratory systems. Changes in thermodynamic intensive parameters (mostly chemical potential and temperature) cannot always explain the inhomogeneous impurity patterns in calcite crystals and it has been suggested that growth rate and crystallographic orientation may exert strong effects on elemental incorporation into calcite. In addition, there are a number of experimental studies on micro-scale element (E) distribution between non-equivalent pairs of calcite vicinal faces (known as sectoral zoning); however, the variability of partition coefficients (e.g., K E = (E/Ca) calcite /(E/Ca) fluid ) within individual crystals remains undetermined. In this study, we have extended the work on elemental distribution between crystal sectors to evaluation of partition coefficients of trace and minor elements (Li, B, Mg, and Sr) in calcite crystal faces (10–14) and (01-12), whose growth rates were assessed. Growth entrapment model (GEM) and lattice strain theory were applied to explain K E heterogeneity by varying near-surface diffusivity of Mg and Sr and by varying surface enrichment factor for Li, B, Mg, and Sr. Decoupling of sectoral and growth rate effects reveals that sectoral zoning plays a key role in elemental distribution. More specifically, K Li and K B vary by more than one order of magnitude and K Mg varies by a factor of two within individual crystal faces. Strontium sectoral distribution is different from those of Li, B, and Mg and K Sr varies by up to a factor of two. Finally, these behaviors likely reflect different mechanisms of incorporation of Li, B, Mg, and Sr into calcite.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Equivalent Fundamental-Mode Source Simulations for Spherical Uranium and Plutonium Systems [Abstract]

The equivalent fundamental-mode source refers to a “source” which is identically distributed in space, energy, and angle as that of a fundamental-mode fission source distribution. It has previously been shown that the system fixed source multiplication ($M_{fs}$) relates to the effective multiplication factor ($\kappa_{eff}$) as $M_{fs} = \frac{g^*}{1-k_{eff}}$ The term $\mathcal{g}$* therefore relates the system source distribution to the system equivalent fundamental-mode source. As previously shown, the $\mathcal{g}$* term is approximately 1 for deeply subcritical systems and then diverges as the system multiplication increases. For a system with a point source located in the center of an assembly (such a sphere), $\mathcal{g}$* will be greater than 1; this happens because there is a higher probability of induced fission when all of the starter neutrons originate from the center of the sphere. For a uniform source, $\mathcal{g}$* is less than 1, but does not diverge from 1 as dramatically as seen by a point source. One reason that $\mathcal{g}$* is an important parameter is that it is used in measurement methods to infer reactor kinetics parameters ($β_{eff}$ being one example). This work will discuss multiple ways to simulate $\mathcal{g}$* using MCNP®6.2, which include use of traditional methods versus a single input file method that was recently published. A discussion related to methods to simulate $M_{fs}$ will be presented. Spherical systems of Highly Enriched Uranium (HEU) will be compared with previous works. In addition, results of plutonium including multiple reflector materials will be investigated. Simulated systems will include both parametric studies as well as configurations used in recent experiments performed at the National Criticality Experiments Research Center (NCERC). These studies will include subtleties associated with system geometry that have not been previously shown. Last, this work will investigate the relationship between $\mathcal{g}$* and reactor kinetics parameters (such as $β_{eff}$).

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

Efficient Fourier transforms for transverse momentum dependent distributions

Hadron production at low transverse momenta in semi-inclusive deep inelastic scattering can be described by transverse momentum dependent (TMD) factorization. This formalism has also been widely used to study the Drell-Yan process and back-to-back hadron pair production in $e^+e^-$ collisions. These processes are the main ones for extractions of TMD parton distribution functions and TMD fragmentation functions, which encode important information about nucleon structure and hadronization. One of the most widely used TMD factorization formalism in phenomenology formulates TMD observables in coordinate $b_\perp$-space, the conjugate space of the transverse momentum. The Fourier transform from $b_\perp$-space back into transverse momentum space is sufficiently complicated due to oscillatory integrands that it requires a careful and computationally intensive numerical treatment in order to avoid potentially large numerical errors. Within the TMD formalism, the azimuthal angular dependence is analytically integrated and the two-dimensional $b_\perp$ integration reduces to a one-dimensional integration over the magnitude $b_\perp$. In this paper we develop a fast numerical Hankel transform algorithm for such a $b_\perp$-integration that improves the numerical accuracy of TMD calculations in all standard processes. Libraries for this algorithm are implemented in Python 2.7 and 3, C++, as well as FORTRAN77. All packages are made available open source.

97 MATHEMATICS AND COMPUTING↗