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

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At least 1,279 records · Page 71
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Pressure falloff testing to characterize CO 2 plume and dry-out zone during CO 2 injection in saline aquifers

The fluid mobilities and the lateral extent of the CO 2 plume and the dry-out zone are among important unknowns required to effectively manage CO 2 storage in saline aquifers projects. These unknowns can be derived from pressure falloff testing which has been widely used to characterize the subsurface. This paper provides a three-region analytical solution for falloff pressure during the infinite-acting period of CO 2 injection in saline aquifers. The solution is derived using Laplace transformation technique of the governing diffusivity equations and the accompanying initial and boundary conditions. The exact solution is validated against numerical results obtained from a black-oil reservoir simulator for two cases. The solution is approximated into simple forms in real-time domain considering different time periods. Then, a graphical semi-log analysis technique based on the approximate solutions is provided as an interpretation tool of falloff tests where the fluid mobilities and the extents of dry-out zone and CO 2 plume can be inferred. The interpretation technique is applied to synthetic falloff test data to investigate its potential and reliability.

02 PETROLEUM↗
DOE OSTI2022

CO 2 Pipeline risk assessment and comparison for the midcontinent United States

For this work, a comprehensive quantitative risk assessment for the construction and operation of CO 2 transportation networks considered for the Midcontinent United States was conducted. The results showed risks associated with CO 2 pipelines were significantly less than those of other pipeline types. The assessment used four conceptual pipelines of different lengths to discuss risks operators may see. The assessment evaluated the risk associated with construction and operation using data from the US Occupational Safety Health Administration to determine the risk of injury or death for pipeline workers and data from the US Pipeline and Hazardous Materials Safety Administration for CO 2 , natural gas distribution, natural gas transmission/gathering, and non-CO 2 hazardous liquid pipelines to develop quantitative likelihood and severity values leading to risk values. The data for the assessment covered incidents from 2010 to 2017 for CO 2 pipelines. The average risk for construction and 30 years of operation for four CO 2 pipeline configurations ranging between 79 and 1,546 miles in length was found. The construction and operational risk averaged between $\$1,400,521$ (approximately $\$0.02$/tonne of CO 2 ) for the shorter pipeline (79 miles) and $\$27,481,939$ (approximately $\$0.10$/tonne of CO 2 ) for a longer pipeline (1,546 miles). The largest risks of fatality for CO 2 pipelines comes from vehicle transport. The largest operational risk to the pipeline was due to leakage. Public pipeline opposition is also a significant risk; it was not quantified but is addressed.

03 NATURAL GAS↗
DOE OSTI2026

Basin-scale analysis of Mokelumne River Formation for multi-well CO 2 injection

Large-scale carbon sequestration will likely require multiple projects injecting CO 2 into the same subsurface formation, raising concerns about safe operation and efficient use of storage capacity. This study evaluates the long-term response of the Mokelumne River Formation in California’s Sacramento Basin to multi-megaton CO 2 injection using three geologic models of the formation and the open-source simulator GEOS. The analysis focuses on three aspects of reservoir performance: (1) average pressure increase and dissolved CO 2 mass after 30 years for varying well counts and injection rates, (2) pressure interference in a multi-well configuration, and (3) dynamic storage capacity with identification of overpressure-prone regions. The results show that average formation pressure increases linearly with injected mass, while CO 2 dissolution exhibits mixed scaling: approximately linear with the number of wells but sublinear with injection rate, indicating that distributing injection across more wells enhances dissolution more effectively than increasing per-well rates. Pressure-interference effects are significant, with lower-permeability conditions delaying their onset but amplifying their magnitude at later times. Dynamic capacity, defined by the first occurrence of pressure exceeding the local overburden-based limit anywhere in the formation, varies across geologic models and assumed overburden pressure gradients. A lower fidelity geologic model predicts nearly twice the storage capacity of the two higher fidelity models, which consistently estimate approximately 1 Gt under the upper-bound overburden pressure gradient considered for the Sacramento Basin. In all model scenarios, overpressure develops away from injection wells, particularly in higher-elevation regions, highlighting the importance of basin-scale modelling for identifying risks beyond the immediate well vicinity.

Basin-scale↗
DOE OSTI2022

Reticulated porous lanthanum strontium manganite structures for solar thermochemical hydrogen production

Porous ceramic foams are utilized in various fields and applications due to their advantageous properties such as high porosity, high specific surface area, and controlled permeability. Non-stoichiometric oxide foams are of particular interest for the solar thermochemical production of hydrogen in directly irradiated solar receivers due to their enhanced heat transfer capabilities and structural integrity. Here we report the fabrication of lanthanum strontium manganite replica foams and their solar thermochemical water splitting performance. Highly porous (>80%) foams were produced with controlled structure and tested in a high temperature water splitting furnace to demonstrate stability and overall performance. The temperature profile of the burnout and sintering of the foam structure were altered based on simultaneous differential scanning calorimetry and thermal gravimetry analysis. Two-step sintering was used both to retain micron-sized pores (D50 = 0.4 μm) on the foam's struts and to limit grain growth to a grain size of ~1 μm. The structural characteristics were confirmed by porosimetry and scanning electron microscopy. The foams were repeatedly cycled, producing approximately 200 μmol/g of H 2 per cycle. Water splitting performance was tested for 50 cycles and approximately 35 h at 1400 °C. No statistical difference in the effective hydrogen production per cycle was observed, which is promising for long-term redox cycle stability.

08 HYDROGEN↗
DOE OSTI2024

Investigation of precooling unit design options in hydrogen refueling station for heavy-duty fuel-cell electric vehicles

Precooling gaseous hydrogen fuel to a cold temperature before refueling a heavy-duty (HD) hydrogen fuel cell electric vehicle (HFCEV) is essential to avoid overheating the vehicle tank, as well as achieving a high state of charge (SOC). Because a large volume of hydrogen is dispensed during each fill, the need for a shorter fill time amplifies the need to precool each load for refueling a HFCEV. Thus, the design and operation of a precooling unit (PCU), as well as the associated capital and operating costs, plays a pivotal role in any plans for heavy-duty hydrogen refueling stations. Here, in this paper, we present a thermodynamic and technoeconomic analysis of a PCU in a gaseous hydrogen refueling station (HRS) for HD HFCEVs. By employing Argonne National Laboratory's hydrogen station cost optimization and performance evaluation (H2SCOPE) model, the refueling of 50 kg of hydrogen on-board type IV tank at ambient temperatures of 15–45 °C and varying fill rates is simulated. The required degree of precooling temperature to obtain either 100% or the maximum possible SOC is obtained from the simulation. Additionally, the simulation results demonstrate that the average flow rate of hydrogen is approximately 40% lower than the maximum flow rate during a typical fill; which motivates further evaluation of the instantaneous hydrogen mass flow rate profile and suggests the scope of improving precooling unit design. Accordingly, a hybrid strategy of precooling hydrogen has been proposed to address the cooling load by sizing the refrigeration unit for the average flow rate of hydrogen, while supplementing the above average peak hydrogen flow cooling load through thermal buffering. The combined technique enables the downsizing of the original PCU capacity by 25–40% and demonstrates a potential cost reduction of the PCU by approximately 30%, which translates to an installed cost reduction of ∼$125,000 per dispenser.

25 ENERGY STORAGE↗
DOE OSTI2022

Analytic model of dislocation density evolution in fcc polycrystals accounting for dislocation generation, storage, and dynamic recovery mechanisms

Here, an analytic model of the evolution of dislocation density in fcc polycrystals is described. The evolution equations approximately account for most known dislocation storage, dynamic recovery, and dislocation generation mechanisms in fcc polycrystals. Specifically, the model incorporates network (forest) and grain boundary storage, mobile-network and mobile–mobile annihilation, screw–screw annihilation via athermal and thermal single cross-slip, generation by double cross-slip (Koehler mechanism, including dipole formation), Frank-Read sources, grain boundary nucleation, and mobile–immobile dislocation nucleation due to shock loading. Single cross-slip is assumed to proceed through the Friedel–Escaig (FE) mechanism; the corresponding activation energy is calculated using a modified FE model. The activation energy for double cross-slip is calculated for the first time by extending the FE model. The exact evolution equations are integro-differential equations, and as such are difficult to implement in a code; hence, the evolution equations are simplified by making several approximations. Preliminary results on copper are presented, including comparisons to experimental data.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Unsupervised learning from three-component accelerometer data to monitor the spatiotemporal evolution of meso-scale hydraulic fractures

Enhanced geothermal systems can provide a substantial share of the global energy demand. There exist several hurdles in the engineering implementations of such geothermal systems. One such hurdle is the accurate monitoring of the fracture networks created in subsurface through hydraulic stimulation of these systems. Micro seismicity associated with the stimulation is the primary means to locate the event hypocenters for estimating the stimulated rock volume. Existing methods for location the hypocenters are restricted to only the highest amplitude impulsive signals that are simultaneously detected on several sensors. Consequently, a large portion (usually ~99%) of the measurements are left unused. In this paper, an unsupervised manifold-approximation followed by clustering of 3-component accelerometer data is used to analyze the seismicity recorded on a monitoring well. With this method, a larger portion of the measured signal is used for the monitoring of the hydraulic fracture network. We analyze the EGS Collab experiment 1 microseismic data, recorded at the Sanford Underground Research Facility, South Dakota. Using the data from a single three-component accelerometer, the polarization features viz. Azimuth, incidence, rectilinearity, and planarity are used as inputs for the unsupervised manifold approximation followed by clustering. Our study shows that density-based clusters in the projected 3D space correspond to distinct types of hydraulically fractured zones around the injection point. Finally, we show that the temporal evolution of these clusters can be used to track fracture creation and propagation.

58 GEOSCIENCES↗
DOE OSTI2022

Screening and evaluation of biomass upgrading strategies for sustainable transportation fuel production with biomass-derived volatile fatty acids

Biomass conversion to fuels and chemicals is crucial to decarbonization, but choosing an advantageous upgrading pathway out of many options is challenging. Rigorously evaluating all candidate pathways (process simulation, product property testing) requires a prohibitive amount of research effort; even simple upgrading schemes have hundreds of possible permutations. We present a method enabling high-throughput screening by approximating upgrading unit operations and drop-in compatibility of products (e.g., fuel properties) and apply it to volatile fatty acid (VFA) conversion to liquid transportation fuels via a MATLAB script, VFA Upgrading to Liquid Transportation fUels Refinery Estimation (VULTURE). VULTURE selects upgrading configurations that maximize fuel blend bio-derived content. We validate VULTURE's approximations through surrogate fuel property testing and process simulation. Techno-economic and life cycle analyses suggest that VFA upgrading processes down-selected by VULTURE are profitable and have low carbon intensities, demonstrating the potential for the strategy to accelerate process development timelines at decreased costs.

09 BIOMASS FUELS↗
DOE OSTI2024

Superconductivity in WBe 2

WBe 2 , which occurs in space group 194, with hexagonal symmetry P6 3 /mmc, is prepared by arc-melting at temperatures above 2200 C, where Be vapor loss is significant. This study is motivated by recent work on MoB 2 and WB 2 , both superconductors (T c =32 and 17 K respectively) under high (∼70 GPa) pressure. In order to avoid the known Be-rich superconducting phases (WBe 13 and WBe 22 ) in the complex phase diagram, both known to be superconducting at 4.1 K, the sample was prepared with a slight (∼5 %) excess of W. The resultant sample, prepared using high purity (99.999 %) Be, is essentially single phase WBe 2 , with some spread in its superconducting properties due to the known homogeneity range. (WBe 2 forms in space group 194 between approximately W 1.02 Be 1.98 and W 0.88 Be 2.12 .) Characterization was carried out with x-ray diffraction, electrical resistivity, ρ, in zero and applied magnetic fields, and specific heat. The resistivity in zero and applied fields and specific heat data indicate that our sample of WBe 2 is a bulk superconductor at ambient pressure with a T c onset in ρ at 1.05 K, and T c (ρ→0) at ∼0.86 K. Furthermore, there is no signature of superconductivity in ρ at 4.1 K, indicating successful avoidance of WBe 13 and WBe 22 . The ρ data in field indicate an upper critical field of approximately 400 gauss.

Electrical transport↗
DOE OSTI2020

Creating Brønsted acidity at the SiO 2 -Nb 2 O 5 interface

Catalytically active acid sites associated with the silica-niobia interface were probed with a series of overcoated SiO 2 on Nb 2 O 5 (SiO 2 /Nb 2 O 5 ) mixed oxide materials prepared by deposition of tetraethyl orthosilicate onto niobic acid (Nb 2 O 5 nH 2 O) or calcined niobia (Nb 2 O 5 ). NH 3 TPD and pyridine DRIFTS studies indicated that the speciation of acid sites in the materials evolved as a function of SiO 2 loading, impacting the quantity and stability of Brønsted sites. Catalyst activity was highly dependent on SiO 2 loading in the liquid phase hydroalkoxylation of dihydropyran with n-octanol. At SiO 2 surface densities corresponding to approximately 1 Si per 2 surface Nb, the activity of these catalysts passed through a maximum approximately 20 times higher than the activity of calcined Nb 2 O 5 . We found that apparent reaction barriers measured over the most active SiO 2 /Nb 2 O 5 catalysts were 10 kJ/mol lower than those measured over niobic acid, suggesting that the OH features unique to the SiO 2 -Nb 2 O 5 interface were slightly more reactive than those on niobic acid.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

A panel analysis of groundwater use in California

We report groundwater is a relevant source of drinking and agricultural water in many regions of the world, but many aquifers have been unsustainably over-drafted and polluted. This has significant environmental, health, and economic implications. We rely on panel analysis, with small-sample corrections for cluster-robust variance estimation and hypothesis testing, to investigate the dynamics of groundwater extraction. We focus on California, yet our approach could be helpful to analyze the dynamics of groundwater extraction in other groundwater-reliant regions of the world. In California, over-reliance on groundwater has led to significant overdraft, affecting long-term water supply reliability and groundwater pumping costs. It further caused subsidence and infrastructure damage, harmed groundwater-dependent ecosystems, and threatened the sustainability of groundwater resources in the state. We use panel data of the 56 California Water Plan planning areas over the 1998–2015 period. We concentrate on agricultural and urban water use and the major water projects in the state, to provide a better understanding of the relationships between groundwater extraction and water use and supply. Results suggest that reducing agricultural water in Central California and urban water in Southern California could reduce groundwater extraction in these regions by approximately the same amount of the reduced water. Other opportunities to reduce the stress on the groundwater resources in the state are available for other regions, yet with lower benefits. Results also suggest that a decrease in deliveries from the Central Valley Project to the southern part of the Central Valley would increase groundwater extraction by approximately the same proportion. Changes in deliveries from the major water projects in the state, as well as from other sources of surface water, would also have some, yet lower impacts on groundwater extraction in the Central and Southern California.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2020

A quadratic programming flux correction method for high-order DG discretizations of S transport

In this work, we present a new flux-fixup approach for arbitrarily high-order discontinuous Galerkin (DG) discretizations of the S N transport equation, and we demonstrate the compatibility of this approach with the Variable Eddington Factor (VEF) method. The new fixup approach is sweep-compatible: during a transport sweep (block Gauss-Seidel iteration in which the scattering source is lagged), a local quadratic programming (QP) problem is solved in each spatial element to ensure that the solution satisfies certain physical constraints, including local particle balance. In this paper, we describe two choices of physical constraints, resulting in two variants of the method: QP Zero (QPZ) and QP Maximum Principle (QPMP). In QPZ, the finite element coefficients of the solution are constrained to be nonnegative. In QPMP, they are constrained to adhere to an approximate discrete maximum principle. There are two primary takeaways in this paper. First, when the positive Bernstein basis is used for DG discretization, the QPMP method eliminates negativities, preserves high-order accuracy for smooth problems, and significantly dampens unphysical oscillations in the solution. The latter feature – the dampening of unphysical oscillations – is an improvement upon standard, simpler fixup approaches such as the approach described in (denoted as the “zero and rescale” (ZR) method in this paper). This improvement comes at a moderate computational cost, but it is not prohibitive. Our results show that, even in an unrealistic worst-case scenario where 83% of the spatial elements require a fixup, the computational cost of performing a transport sweep with fixup is only ~31% greater than performing one without fixup. The second takeaway is that the VEF method can be used to accelerate the convergence of transport sweeps even when a fixup is applied. When optically thick regions are present, transport sweeps converge slowly, regardless of whether a fixup is applied, and acceleration is needed. However, attempting to apply standard diffusion synthetic acceleration (DSA) to fixed-up transport sweeps results in divergence for optically thick problems. Our results show that the same is not true for VEF. When VEF is combined with fixed-up transport sweeps, the result is a scheme that produces a nonnegative solution, converges independently of the mean free path, and, in the case of the QPMP fixup, adheres to an approximate discrete maximum principle.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Improvements to a class of hybrid methods for radiation transport: Nyström reconstruction and defect correction methods

In this study, two modifications are introduced for improving the accuracy, versatility, and robustness of a class of hybrid methods for radiation transport. In general, such methods are constructed by splitting the radiative flux into collided and uncollided components to which low- and high-resolution angular approximations are applied, respectively. In this work we focus on discrete ordinates discretizations of high and low order. The first modification we introduce changes the way in which the collided component is mapped into the uncollided component at the end of each time step in a simulation. The new mapping is a Nyström-type reconstruction that is applicable to arbitrary discrete ordinates quadratures, is guaranteed to preserve positivity of the solution provided that all ordinate weights are positive, is significantly more accurate than previous methods, and can be readily extended to other discretizations such as moment methods, finite element methods, and diffusion approximations. The second modification leverages integral deferred correction (IDC) to iteratively correct for the splitting error introduced by the inconsistency in angular discretization between the collided and uncollided components, in addition to improving the accuracy of the low-order temporal error that is treated by traditional IDC methods. Numerical tests in one- and two-dimensional geometries are used to demonstrate the increased accuracy and efficiency of the proposed modifications. It is found that the two techniques combined yield methods with solution accuracy and memory requirements comparable to that of monolithic discrete ordinates methods while reducing runtime by as much as a factor of between two and ten, depending on the problem.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2021

B-PINNs: Bayesian Physics-informal Neural Networks for Forward and Inverse PDE Problems with Noisy Data

We propose a Bayesian physics-informed neural network (B-PINN) to solve both forward and inverse nonlinear problems described by partial differential equations (PDEs) and noisy data. In this Bayesian framework, the Bayesian neural network (BNN) combined with a PINN for PDEs serves as the prior while the Hamiltonian Monte Carlo (HMC) or the variational inference (VI) could serve as an estimator of the posterior. B-PINNs make use of both physical laws and scattered noisy measurements to provide predictions and quantify the aleatoric uncertainty arising from the noisy data in the Bayesian framework. Compared with PINNs, in addition to uncertainty quantification, B-PINNs obtain more accurate predictions in scenarios with large noise due to their capability of avoiding overfitting. We conduct a systematic comparison between the two different approaches for the B-PINNs posterior estimation (i.e., HMC or VI), along with dropout used for quantifying uncertainty in deep neural networks. Our experiments show that HMC is more suitable than VI with mean field Gaussian approximation for the B-PINNs posterior estimation, while dropout employed in PINNs can hardly provide accurate predictions with reasonable uncertainty. Finally, we replace the BNN in the prior with a truncated Karhunen-Loève (KL) expansion combined with HMC or a deep normalizing flow (DNF) model as posterior estimators. The KL is as accurate as BNN and much faster but this framework cannot be easily extended to high-dimensional problems unlike the BNN based framework.

Non linear PDEs, Noisy data, Bysian physics inform↗
DOE OSTI2020

Deep learning of free boundary and Stefan problems

Free boundary problems appear naturally in numerous areas of mathematics, science and engineering. These problems present a great computational challenge because they necessitate numerical methods that can yield an accurate approximation of free boundaries and complex dynamic interfaces. In this work, we propose a multi-network model based on physics-informed neural networks to tackle a general class of forward and inverse free boundary problems called Stefan problems. Specifically, we approximate the unknown solution as well as any moving boundaries by two deep neural networks. Besides, we formulate a new type of inverse Stefan problems that aim to reconstruct the solution and free boundaries directly from sparse and noisy measurements. We demonstrate the effectiveness of our approach in a series of benchmarks spanning different types of Stefan problems, and illustrate how the proposed framework can accurately recover solutions of partial differential equations with moving boundaries and dynamic interfaces. All code and data accompanying this manuscript are publicly available at https://github.com/PredictiveIntelligenceLab/DeepStefan.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Multiscale formulation of frictional contact mechanics at the pore scale

Direct numerical simulation (DNS) yields the highest fidelity predictions of mechanical deformation at the pore scale, but is prohibitively expensive for analyzing large or many samples. Discrete element methods (DEM) are an efficient alternative, but are limited to granular media and incapable of estimating or controlling prediction errors. We present a pore-level multiscale method (PLMM) that approximates DNS efficiently and with controllable accuracy. We focus on the linear elastic response of a consolidated geologic porous medium with arbitrary microstructure, heterogeneous mineralogy, containing cracks or defects. PLMM decomposes the solid phase into non-overlapping subdomains, on which local basis functions are constructed. The bases are then coupled with a global interface problem that accounts for slip or stick contact conditions between the subdomains. PLMM produces an initial, but accurate, approximation to DNS that can be iteratively improved. It is amenable to parallelism and allows for different mesh, models, and physics in each subdomain. An algebraic interpretation of PLMM as a preconditioner is also presented to allow non-intrusive implementation into existing solvers. Lastly, this work extends previous developments of PLMM in fluid dynamics to solid mechanics and enables future extensions towards modeling coupled flow and mechanics problems.

42 ENGINEERING↗
DOE OSTI2021

A Feynman-Kac based numerical method for the exit time probability of a class of transport problems

The exit time probability, which gives the likelihood that an initial condition leaves a prescribed region of the phase space of a dynamical system at, or before, a given time, is arguably one of the most natural and important transport problems. In this work, we present an accurate and efficient numerical method for computing this probability for systems described by non-autonomous (time-dependent) stochastic differential equations (SDEs) or their equivalent Fokker-Planck partial differential equations. The method is based on the direct approximation of the Feynman-Kac formula that establishes a link between the adjoint Fokker-Planck equation and the forward SDE. The Feynman-Kac formula is approximated using the Gauss-Hermite quadrature rules and piecewise cubic Hermite interpolating polynomials, and a GPU accelerated matrix representation is used to compute the entire time evolution of the exit time probability using a single pass of the algorithm. The method is unconditionally stable, exhibits second order convergence in space, first order convergence in time, and it is straightforward to parallelize. Applications are presented to the advection diffusion of a passive tracer in a fluid flow exhibiting chaotic advection, and to the runaway acceleration of electrons in a plasma in the presence of an electric field, collisions, and radiation damping. Benchmarks against analytical solutions as well as comparisons with explicit and implicit finite difference standard methods for the adjoint Fokker-Planck equation are presented.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder

Traditional linear subspace reduced order models (LS-ROMs) are able to accelerate physical simulations in which the intrinsic solution space falls into a subspace with a small dimension, i.e., the solution space has a small Kolmogorov n-width. However, for physical phenomena not of this type, e.g., any advection-dominated flow phenomena such as in traffic flow, atmospheric flows, and air flow over vehicles, a low-dimensional linear subspace poorly approximates the solution. To address cases such as these, we have developed a fast and accurate physics-informed neural network ROM, namely nonlinear manifold ROM (NM-ROM), which can better approximate high-fidelity model solutions with a smaller latent space dimension than the LS-ROMs. Our method takes advantage of the existing numerical methods that are used to solve the corresponding full order models. The efficiency is achieved by developing a hyper-reduction technique in the context of the NM-ROM. Numerical results show that neural networks can learn a more efficient latent space representation on advection-dominated data from 1D and 2D Burgers' equations. A speedup of up to 2.6 for 1D Burgers' and a speedup of 11.7 for 2D Burgers' equations are achieved with an appropriate treatment of the nonlinear terms through a hyper-reduction technique. Lastly, a posteriori error bounds for the NM-ROMs are derived that take account of the hyper-reduced operators.

97 MATHEMATICS AND COMPUTING↗