Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Fractional operators”

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 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Analysis of Tempered Fractional Operators

Tempered fractional operators are useful in models for subsurface transport and diffusion due to their ability to capture anomalous diffusion: a behavior which the classical partial differential equation models cannot describe. We analyze tempered fractional operators within the nonlocal vector calculus framework in order to assimilate them to the rigorous mathematical structure developed for nonlocal models. First, we show they are special instances of generalized nonlocal operators in correspondence of a proper choice of nonlocal kernels. Then, we work towards showing tempered fractional operators are equivalent to truncated fractional operators. These truncated operators are useful because they are less computationally intensive than the tempered operators.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Analysis of Tempered Fractional Operators

Tempered fractional operators provide an improved predictive capability for modeling anomalous effects that cannot be captured by standard partial differential equations. These effects include subdiffusion and superdiffusion (i.e. the mean square displacement in a diffusion process is proportional to a fractional power of the time), that often occur in, e.g., geoscience and hydrology. We analyze tempered fractional operators within the nonlocal vector calculus framework in order to assimilate them to the rigorous mathematical structure developed for nonlocal models. First, we show they are special instances of generalized nonlocal operators by means of a proper choice of the nonlocal kernel. Then, we present a plan for showing tempered fractional operators are equivalent to truncated fractional operators. These truncated operators are useful because they are less computationally intensive than the tempered operators.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

A series representation of the discrete fractional Laplace operator of arbitrary order

Although fractional powers of non-negative operators have received much attention in recent years, there is still little known about their behavior if real-valued exponents are greater than one. In this article, we define and study the discrete fractional Laplace operator of arbitrary real-valued positive order. Here a series representation of the discrete fractional Laplace operator for positive non-integer powers is developed. Its convergence to a series representation of a known case of positive integer powers is proven as the power tends to the integer value. Furthermore, we show that the new representation for arbitrary real-valued positive powers of the discrete Laplace operator is consistent with existing theoretical results.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Comparison of Tempered and Truncated Fractional Models

Tempered fractional operators are able to model effects that classical partial differential equations cannot quite capture. For example, this includes the super- and subdiffusion effects that are present in hydrology and geophysics models. However, tempered fractional operators are computationally intensive. We analyze a truncated variation of the fractional operators, which are less computationally intensive, in an effort to use them in place of the more complex tempered variation. In particular, we train parameters of the truncated operator using neural networks in order to optimize the difference of the actions of the two operators.

58 GEOSCIENCES↗
DOE OSTI2022

A fractional model for anomalous diffusion with increased variability: Analysis, algorithms and applications to interface problems

Fractional equations have become the model of choice in several applications where heterogeneities at the microstructure result in anomalous diffusive behavior at the macroscale. Here, we introduce a new fractional operator characterized by a doubly-variable fractional order and possibly truncated interactions. Under certain conditions on the model parameters and on the regularity of the fractional order we show that the corresponding Poisson problem is well-posed. Additionally, we introduce a finite element discretization and describe an efficient implementation of the finite-element matrix assembly in the case of piecewise constant fractional order. Through several numerical tests, we illustrate the improved descriptive power of this new operator across media interfaces. Furthermore, we present one-dimensional and two-dimensional h-convergence results that show that the variable-order model has the same convergence behavior as the constant-order model.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Advances in the long-pulse steady-state high beta H-mode scenario with active controls of divertor heat and particle fluxes in EAST

Since the last IAEA-Fusion Energy Conference, the Experimental Advanced Superconducting Tokamak (EAST) research program has been, in support of ITER and CFETR, focused on development in terms of the long-pulse steady-state (fully noninductive) high beta H-mode scenario with active controls of the stationary and transient divertor heat and particle fluxes. The operational domain of the steady-state H-mode plasma scenario has been significantly extended with ITER-like tungsten mono-block divertor, plasma control and heating schemes. EAST has achieved several important milestones in the development of high β p H-mode scenario and its key physics and technologies. A 60 s-scale long-pulse steady-state high β p H-mode discharge with the major normalized plasma parameters similar to the designed performance of the CFETR 1GW fusion power operation scenario has been successfully established and sustained by pure RF heating and current drive. Several feedback control schemes have been developed for a sustained detachment with good core confinement. This includes control of the total radiation power, target electron temperature, and particle flux measured using divertor Langmuir probes or a combination of the control of target electron temperature and AXUV radiation near the X point. The detachment feedback control schemes have been integrated with small-ELM regimes and high β p scenario via neon seeding, enabling a core and edge compatible integrated high-beta scenario applicable to long-pulse operations. ELM suppression has been achieved using various methods, including resonant magnetic perturbations and impurity seeding. Full suppression of ELMs by using n=4 RMPs has been demonstrated for ITER for the first time in low input torque plasmas in EAST. Finally, EAST has been operated with helium to support the ITER research requirements for the first time. For a long-pulse, high bootstrap current fraction operation, a new lower tungsten divertor with active water-cooling has been installed, along with improvements in the heating and current drive capability.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2024

Variable-spectrum mode control of high poloidal beta discharges

DIII-D experiments demonstrate that high pressure, broad current profile equilibria can be accessed in the high poloidal beta regime by optimizing the MHD mode control poloidal spectrum. A novel, variable spectrum (VS) magnetic feedback scheme implemented using the DIII-D internal non-axisymmetric coils (I-coils) facilitated access to reduced internal inductance $l$ i operation above the no-wall beta limit compared with both no feedback and fixed spectrum feedback. In addition, the VS feedback helped avoid beta collapses caused by marginally unstable resistive wall mode activity. The lower and upper I-coil rows were configured in two independent feedback loops, allowing the feedback field's poloidal spectrum to vary and track changes in the plasma mode structure as the edge safety factor q 95 varied from 11 to 6 during the discharges. The q 95 dependence of the measured phase difference between the lower and upper I-coil rows during VS feedback is qualitatively compatible with ideal MHD simulations of the least-stable plasma kink mode and with plasma response simulations that included kinetic modifications to ideal MHD. The VS feedback approach is a straightforward way to improve resilience to variations in mode structure that occur as plasma parameters change. The demonstrated expansion of the operating space to lower $l$ i is expected to improve the coupling of the plasma kink mode to external fields and beneficial wall eddy currents, and is compatible with high bootstrap fraction operation.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Generalized fractional approach to solving partial differential equations with arbitrary dispersion relations

The dynamics of physical systems are typically modeled by partial differential equations (PDEs). Because of the complexity in physical systems, PDE models rely on approximations or limiting cases. To model the full complexity of physical systems, it is necessary to use multiscale approaches where appropriate models are used at each level. Alternatively, complexity can be directly tackled by fractional integrodifferential equations that must be derived for each system. Here, we argue that complexity can be restored in PDEs by describing them from a fractional calculus perspective. Rather than deriving a fractional integrodifferential equation, we reinterpret the dispersion relation of the system by use of the Riesz definition, which contains the required information relating the energy and momentum space of the system and thus fully describes their dynamics. The approach is demonstrated by two examples: the Landau–Lifshitz equation in a 1D ferromagnetic chain and a modified KdV equation supporting surface gravity waves or Euler dispersion. The presented approach is applicable to fluids, soft matter, and solid-state matter and can be readily generalized to higher dimensions and more complex systems. While numerical calculations are needed to determine the fractional operator, the approach is analytical and can be utilized to determine analytical solutions and investigate nonlinear problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Simultaneous enhancement of tritium burn efficiency and fusion power with low-tritium spin-polarized fuel

This study demonstrates that using spin-polarized deuterium-tritium (D-T) fuel with more deuterium than tritium can increase tritium burn efficiency (TBE) by at least an order of magnitude without compromising fusion power output, compared to unpolarized fuel. Although previous studies show that a low tritium fraction can enhance TBE, this strategy resulted in reduced fusion power density. The surprising improvement in TBE at fixed power reported here is due to the TBE increasing nonlinearly with decreasing tritium fraction but the fusion power density increasing roughly linearly with D-T cross section. A study is performed for an ARC-like tokamak producing 481 MW of fusion power with unpolarized 53:47 D-T fuel, finding the minimum startup tritium inventory (I startup,min ) is 0.69 kg. By spin-polarizing half of the fuel and using a 60:40 D-T mix, I startup,min is reduced to 0.08 kg, and fully spin-polarizing the fuel with a 63:37 D-T mix further reduces I startup,min to 0.03 kg. Some ARC-like scenarios are predicted to achieve plasma ignition with relatively modest spin polarization. These findings indicate that, with advancements in helium divertor pumping efficiency, TBE values of approximately 10%–40% could be achieved using low-tritium-fraction and spin-polarized fuel with minimal power loss. This would dramatically lower tritium startup inventory requirements and reduce the amount of on-site tritium. More generally than just for spin-polarized fuels, increased plasma performance can be used to increase TBE. This strongly motivates the development of spin-polarized fuels and low-tritium-fraction operation for burning plasmas.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2020

Sources of Propane Consumed in California

Project Scope: The objective of this study is to specify the sources of propane consumed in California. It answers the questions, where does the propane used in California come from and how was it produced? The results of this study provide comprehensive, transparent, and verifiable estimates, based on the 2018 market. The information provided in this report is suitable for use to assess the life cycle carbon intensity of propane used as a transportation fuel in California. As the 2009 Low Carbon Fuel Standard (LCFS) aims to reduce California’s greenhouse gas (GHG) emissions and other smog-forming and toxic air pollutants, the appropriate designation of carbon intensity for propane as a transportation fuel is important for evaluating propane’s potential to contribute to GHG goals and understandings in the context of various actions. This study focuses on estimating the shares of total propane consumed in the state of California produced from petroleum refineries, natural gas plants, and bituminous sands sources inside California and elsewhere. Results: An estimated 590 million gallons of propane were consumed in California in 2018, of which, 59.5% originated from refinery production and 40.5% originated from natural gas plants. The majority of this was sourced from refinery production in California, 334 million gallons. Most of the propane imported to California for consumption was sourced from natural gas plants, 113 million gallons, with over half of the imported volume sourced from Canada. The volume sourced from bituminous sand upgrader and fractionator operations was negligible. Details from this analysis are presented in the table below which provides an overview of the propane flows estimated in this study by region and production method. The shares and volumes presented here represent a snapshot for 2018. A significant increase in propane demand, such as could be caused by increased use of propane as a transportation fuel in the state, would affect California’s propane production, imports, and exports. The method and data sources used for the estimates provided in this report also provide the framework which could be used for future updates. Key Method Considerations: The values presented here are based on a two-step approach where the first step was to determine the flows of propane into and out of California from different regions and the second step was to estimate the propane production methods in each region. A volume balance approach is used as the primary method for tracking the volume of propane in and out of California as propane production and import volumes are available by Petroleum Administration of Defense District (PADD) from EIA and neither inter-PADD propane transfers nor state-specific non-prime supplier consumption are available from a public data source. The volume balance performed for this study covered PADD 5 (the West Coast), which includes Arizona, California, Nevada, Oregon, and Washington. The volume balance used all available public datasets to determine propane production, imports, exports, and consumption. Volumes unaccounted for by these datasets were estimated using the resulting volume balance by assuming market equilibrium. Consumption within each state in PADD 5 was estimated based on known import, export, and production volumes and this amount was used to develop the volume balance. EIA only tracks consumption at the state level by prime supplier sales. The volume balance approach provides the basis to correct for additional propane consumed in-states where propane is transferred to California. To determine the California propane sources and trade in 2018. volume of propane consumed in California, the volume balance approach is again used where it was estimated all imported volumes not specifically flagged for re-export were consumed, and the remaining consumption was produced in-state. The California Energy Commission (CEC) provided the total volume of propane imported and exported from California in 2018; this volume data set along with commodity tracking from the Canada Energy Regulator (CER) and the International Trade Commission (ITC) which tracks port of entry and final destination was used to determine where propane originated from and where it was ultimately consumed. For example, the CER tracks propane leaving Canada and entering each state within the U.S. Imported propane from Canada to California – marked for California – is assumed to be consumed in California. When no further data were available, import volumes were assumed to be consumed in California without pass-through (i.e., no propane imported to California was directly sold and exported). In most cases, the production method for each propane source region was applied to the volume of propane transferred to California. In other words, the shares of propane sourced from natural gas and refineries for each production region was assigned to California imports based on their contribution to the total volume flows into California to determine the production method for propane consumed in-state. For volumes imported into California from PADD 4, Washington State, Canada, and the rest of the world (Argentina, Chile, Norway, Peru, South Korea, and Trinidad and Tobago), the volumes sourced from petroleum refineries and natural gas plants reflect the either production ratio for the region or, in cases where the sources specific to the amounts exported to California could be determined, the sources specific to the volumes transferred to California.

03 NATURAL GAS↗
DOE OSTI2024

Full Scale HEPA Filter Encapsulation in Ultra-High-Performance Grout - Proof of Concept

The Hanford Tank Waste Treatment and Immobilization Plant (WTP) currently being constructed to treat radioactive waste, includes vitrification facilities for both the high-level waste (HLW) and low activity waste (LAW) fractions. Operation of the WTP will produce contaminated high-efficiency particulate air filters (HEPA), as part of the solid secondary waste (SSW) stream. The HEPA filters receive off-gas from the vessel vent header and primary off-gas treatment system in LAW Facility and remove particulate contaminants including 99 Tc and 129 I salts. The current disposal method for HEPA filters is encapsulation in metal containers using cementitious material (CM) and disposal in the Integrated Disposal Facility (IDF). Results from the 2017 IDF Performance Assessment (PA) WRPS (2018) demonstrated that while compliance is maintained for the 1000 year compliance period mandated by DOE O 435.1 and its accompanying manual, release of constituents from the HEPA filters result in exceedance of the performance objective imposed as the groundwater regulatory limit at later times. For example, at about 1500 years post-closure, solid secondary waste (SSW), including HEPA filters is predicted to become a dominant contributor to 99 Tc release and over the 10,000-year sensitivity analysis period, SSW is the dominant contributor of 129 I release to the groundwater. The estimated release could potentially be reduced if the HEPA filters are not compacted and waste containers could be distributed throughout a large space, thereby diluting the contaminant release. Additionally, a better cementitious material could be used to encapsulate the HEPA filters. One alternative method for disposal of contaminated HEPA filters is encapsulation of the filters in ultra high-performance grout (UHPG). UHPG is a variation of ultra-high-performance concrete (UHPC) is commonly used in the prestressed concrete industry for large structural members. Recent studies of UHPG show it has excellent properties for containing radionuclides such as 99 Tc and 129 I Nichols and Kaplan (2021). This report presents the results of the first attempt to encapsulate a clean, full-size HEPA filter in UHPG and evaluate the effectiveness of the immobilization process and final waste form. A full-scale proof-of-concept simulated waste form was prepared by encapsulating a HEPA filter in a 110-gallon stainless steel (SS) drum using UHPG. A change from Type I/II PC to Type 1L PLC was made after American Rock Products informed the team that they would no longer be using Type I/II by the end of 2024 and the northwest was phasing out Type I/II PC overall. Type I/II PC used in previous studies of UHPG for encapsulation (Nichols and Kaplan 2021). After the UHPG was cured both the scaled mockup and the full-scale simulated waste forms were sectioned for visual examination. UHPG completely encapsulated the filters and bonded to the external surfaces of materials comprising the filters. No cracks were observed in the sectioned waste forms.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
DOE OSTI2021

An optimization-based approach to parameter learning for fractional type nonlocal models

Nonlocal operators of fractional type are a popular modeling choice for applications that do not adhere to classical diffusive behavior; however, one major challenge in nonlocal simulations is the selection of model parameters. In this study we propose an optimization-based approach to parameter identification for fractional models with an optional truncation radius. We formulate the inference problem as an optimal control problem where the objective is to minimize the discrepancy between observed data and an approximate solution of the model, and the control variables are the fractional order and the truncation length. For the numerical solution of the minimization problem we propose a gradient-based approach, where we enhance the numerical performance by an approximation of the bilinear form of the state equation and its derivative with respect to the fractional order. Several numerical tests in one and two dimensions illustrate the theoretical results and show the robustness and applicability of our method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

An optimization-based approach to parameter learning for fractional type nonlocal models

Nonlocal operators of fractional type are a popular modeling choice for applications that do not adhere to classical diffusive behavior; however, one major challenge in nonlocal simulations is the selection of model parameters. In this work we propose an optimization-based approach to parameter identification for fractional models with an optional truncation radius. We formulate the inference problem as an optimal control problem where the objective is to minimize the discrepancy between observed data and an approximate solution of the model, and the control variables are the fractional order and the truncation length. For the numerical solution of the minimization problem we propose a gradient-based approach, where we enhance the numerical performance by an approximation of the bilinear form of the state equation and its derivative with respect to the fractional order. Several numerical tests in one and two dimensions illustrate the theoretical results and show the robustness and applicability of our method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Moment method and continued fraction expansion in Floquet operator Krylov space

Recursion methods such as Krylov techniques map complex dynamics to an effective noninteracting problem in one dimension. For example, the operator Krylov space for Floquet dynamics can be mapped to the dynamics of an edge operator of the one-dimensional Floquet inhomogeneous transverse field Ising model (ITFIM), where the latter, after a Jordan-Wigner transformation, is a Floquet model of noninteracting Majorana fermions and the couplings correspond to Krylov angles. We present an application of this showing that a moment method exists where given an autocorrelation function, one can construct the corresponding Krylov angles and from that the corresponding Floquet ITFIM. Consequently, when no solutions for the Krylov angles are obtained, it indicates that the autocorrelation is not generated by unitary dynamics. We highlight this by studying certain special cases: stable m-period dynamics derived using the method of continued fractions, exponentially decaying, and power-law decaying stroboscopic dynamics. Remarkably, our examples of stable m-period dynamics correspond to m-period edge modes for the Floquet ITFIM where, deep in the chain, the couplings correspond to a critical phase. Furthermore our results pave the way to engineer Floquet systems with desired properties of edge modes and also provide examples of persistent edge modes in gapless Floquet systems.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2021

Characterization of commercial vehicles’ start-up operations from in-use data

Diesel engines produce disproportionate levels of emissions when the engine and after-treatment systems are operating at low temperatures. This situation arises most commonly when the vehicle is first started after overnight. To quantify emissions attributable to vehicle starts, a sizable collection of on-road commercial vehicle operating data is analyzed to identify start-up events and inspect associated emissions. Data was obtained from the National Renewable Energy Laboratory’s (NREL’s) Fleet DNA and from the Center for Environmental Research & Technology (CE-CERT). Included are 500 + diesel vehicles with more than 42,000 recorded days, drawn from 25 vocational categories across the United States. Analysis shows that vehicle behavior, in terms of engine cold- and warm-operation, starts per day, soak time, and warm-up duration, differs significantly between vehicle vocations. Also, weighting factors for cold- and hot-starts currently used in the U.S. Environmental Protection Agency’s Federal Test Procedure (FTP) for heavy-duty emissions certification accurately represent real-world operations. Although the FTP includes a comparable fraction of cold operation, the hot fraction is much shorter than real-world operation due to limited test duration. The investigation also revealed that real-world engines operate for a significant amount of time when the engine coolant is in the “hot-stabilized” region, but the selective catalytic reduction (SCR) temperature is below its effective operating temperature of 200 °C. Of the vehicles under investigation, almost 20% of their operational time is within this condition. Therefore, novel approaches to raise and maintain SCR temperature are highly required to further reduce engine emissions.

33 ADVANCED PROPULSION SYSTEMS↗
DOE OSTI2021

Identifiability and predictability of integer- and fractional-order epidemiological models using physics-informed neural networks

Here we analyze a plurality of epidemiological models through the lens of physics-informed neural networks (PINNs) that enable us to identify time-dependent parameters and data-driven fractional differential operators. In particular, we consider several variations of the classical susceptible-infectious-removed (SIR) model by introducing more compartments and fractional-order and time-delay models. We report the results for the spread of COVID-19 in New York City, Rhode Island and Michigan states and Italy, by simultaneously inferring the unknown parameters and the unobserved dynamics. For integer-order and time-delay models, we fit the available data by identifying time-dependent parameters, which are represented by neural networks. In contrast, for fractional differential models, we fit the data by determining different time-dependent derivative orders for each compartment, which we represent by neural networks. We investigate the structural and practical identifiability of these unknown functions for different datasets, and quantify the uncertainty associated with neural networks and with control measures in forecasting the pandemic.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2020

Data-driven learning of nonlocal physics from high-fidelity synthetic data

A key challenge to nonlocal models is the analytical complexity of deriving them from first principles, and frequently their use is justified a posteriori. Here, we extract nonlocal models from data, circumventing these challenges and providing data-driven justification for the resulting model form. Extracting data-driven surrogates is a major challenge for machine learning (ML) approaches, due to nonlinearities and lack of convexity — it is particularly challenging to extract surrogates which are provably well-posed and numerically stable. Our scheme not only yields a convex optimization problem, but also allows extraction of nonlocal models whose kernels may be partially negative while maintaining well-posedness even in small-data regimes. To achieve this, based on established nonlocal theory, we embed in our algorithm sufficient conditions on the non-positive part of the kernel that guarantee well-posedness of the learnt operator. These conditions are imposed as inequality constraints to meet the requisite conditions of the nonlocal theory. We demonstrate this workflow for a range of applications, including reproduction of manufactured nonlocal kernels; numerical homogenization of Darcy flow associated with a heterogeneous periodic microstructure; nonlocal approximation to high-order local transport phenomena; and approximation of globally supported fractional diffusion operators by truncated kernels.

42 ENGINEERING↗
DOE OSTI2022

Learning functional priors and posteriors from data and physics

In this work, we develop a new Bayesian framework based on deep neural networks to be able to extrapolate in space-time using historical data and to quantify uncertainties arising from both noisy and gappy data in physical problems. Specifically, the proposed approach has two stages: (1) prior learning and (2) posterior estimation. At the first stage, we employ the physics-informed Generative Adversarial Networks (PI-GAN) to learn a functional prior either from a prescribed function distribution, e.g., Gaussian process, or from historical data and physics. At the second stage, we employ the Hamiltonian Monte Carlo (HMC) method to estimate the posterior in the latent space of PI-GANs. In addition, we use two different approaches to encode the physics: (1) automatic differentiation, used in the physicsinformed neural networks (PINNs) for scenarios with explicitly known partial differential equations (PDEs), and (2) operator regression using the deep operator network (DeepONet) for PDE-agnostic scenarios. We then test the proposed method for (1) meta-learning for one-dimensional regression, and forward/inverse PDE problems (combined with PINNs); (2) PDE-agnostic physical problems (combined with DeepONet), e.g., fractional diffusion as well as saturated stochastic (100-dimensional) flows in heterogeneous porous media; and (3) spatial-temporal regression problems, i.e., inference of a marine riser displacement field using experimental data from the Norwegian Deepwater Programme (NDP). The results demonstrate that the proposed approach can provide accurate predictions as well as uncertainty quantification given very limited scattered and noisy data, since historical data could be available to provide informative priors. In summary, the proposed method is capable of learning flexible functional priors, e.g., both Gaussian and non-Gaussian process, and can be readily extended to big data problems by enabling mini-batch training using stochastic HMC or normalizing flows since the latent space is generally characterized as low dimensional.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗