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Results for “Finite-difference approximation”

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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NASA NTRS1956

Extrapolation techniques applied to matrix methods in neutron diffusion problems

A general matrix method is developed for the solution of characteristic-value problems of the type arising in many physical applications. The scheme employed is essentially that of Gauss and Seidel with appropriate modifications needed to make it applicable to characteristic-value problems. An iterative procedure produces a sequence of estimates to the answer; and extrapolation techniques, based upon previous behavior of iterants, are utilized in speeding convergence. Theoretically sound limits are placed on the magnitude of the extrapolation that may be tolerated. This matrix method is applied to the problem of finding criticality and neutron fluxes in a nuclear reactor with control rods. The two-dimensional finite-difference approximation to the two-group neutron fluxes in a nuclear reactor with control rods. The two-dimensional finite-difference approximation to the two-group neutron-diffusion equations is treated. Results for this example are indicated.

Mccready, Robert R↗

DOE OSTI2025

HFBTHO-AD: Differentiation of a nuclear energy density functional code

The HFBTHO code implements a nuclear energy density functional solver to model the structure of atomic nuclei. HFBTHO has previously been used to calibrate energy functionals and perform sensitivity analysis by using derivative-free methods. To enable derivative-based optimization and uncertainty quantification approaches, we must compute the derivatives of HFBTHO outputs with respect to the parameters of the energy functional, which are a subset of all input parameters of the code. Here, we use the algorithmic/automatic differentiation (AD) tool Tapenade to differentiate HFBTHO. We compare the derivatives obtained using AD against finite-difference approximation and examine the performance of the derivative computation.

Algorithmic differentiation↗

NASA NTRS1995

Analysis of developing laminar flows in circular pipes using a higher-order finite-difference technique

A higher-order finite-difference technique is developed to calculate the developing-flow field of steady incompressible laminar flows in the entrance regions of circular pipes. Navier-Stokes equations governing the motion of such a flow field are solved by using this new finite-difference scheme. This new technique can increase the accuracy of the finite-difference approximation, while also providing the option of using unevenly spaced clustered nodes for computation such that relatively fine grids can be adopted for regions with large velocity gradients. The velocity profile at the entrance of the pipe is assumed to be uniform for the computation. The velocity distribution and the surface pressure drop of the developing flow then are calculated and compared to existing experimental measurements reported in the literature. Computational results obtained are found to be in good agreement with existing experimental correlations and therefore, the reliability of the new technique has been successfully tested.

Gladden, Herbert J.↗

NASA NTRS1972

An implicit finite-difference solution to the viscous shock layer, including the effects of radiation and strong blowing

An implicit finite-difference scheme is developed for the fully coupled solution of the viscous, radiating stagnation-streamline equations, including strong blowing. Solutions are presented for both air injection and injection of carbon-phenolic ablation products into air at conditions near the peak radiative heating point in an earth entry trajectory from interplanetary return missions. A detailed radiative-transport code that accounts for the important radiative exchange processes for gaseous mixtures in local thermodynamic and chemical equilibrium is utilized in the study. With minimum number of assumptions for the initially unknown parameters and profile distributions, convergent solutions to the full stagnation-line equations are rapidly obtained by a method of successive approximations. Damping of selected profiles is required to aid convergence of the solutions for massive blowing. It is shown that certain finite-difference approximations to the governing differential equations stabilize and improve the solutions. Detailed comparisons are made with the numerical results of previous investigations. Results of the present study indicate lower radiative heat fluxes at the wall for carbonphenolic ablation than previously predicted.

Garrett, L. B.↗

NASA NTRS1973

An economical method of analyzing transient motion of gas-lubricated rotor-bearing systems.

A method of economically evaluating the hydrodynamic forces generated in a gas-lubricated tilting-pad bearing is presented. The numerical method consists of solving the case of the infinite width bearing and then converting this solution to the case of the finite bearing by accounting for end leakage. The approximate method is compared to the finite-difference solution of Reynolds equation and yields acceptable accuracy while running about one-hundred times faster. A mathematical model of a gas-lubricated tilting-pad vertical rotor systems is developed. The model is capable of analyzing a two-bearing-rotor system in which the rotor center of mass is not at midspan by accounting for gyroscopic moments. The numerical results from the model are compared to actual test data as well as analytical results of other investigators.

Falkenhagen, G. L.↗

DOE OSTI2025

Derivative-free stochastic optimization via adaptive sampling strategies

In this paper, we present a novel derivative-free framework for solving unconstrained stochastic optimization problems. Many problems in fields ranging from simulation optimization to reinforcement learning to quantum computing involve settings where only stochastic function values are obtained via a zeroth-order oracle, which has no available gradient information and necessitates the usage of derivative-free optimization methodologies. Our approach includes estimating gradients using stochastic function evaluations and integrating adaptive sampling techniques to control the accuracy in these stochastic approximations. Our framework encapsulates several gradient estimation techniques, including standard finite-difference, Gaussian smoothing, sphere smoothing, randomized coordinate finite-difference, and randomized subspace finite-difference methods. We provide theoretical convergence guarantees for our framework and analyze the worst-case iteration and sample complexities associated with each gradient estimation method. Finally, we demonstrate the empirical performance of the methods on logistic regression and nonlinear least squares problems.

Adaptive sampling↗

DOE OSTI2025

Scalable freeform optimization of wide-aperture 3D metalenses by zoned discrete axisymmetry

We introduce a novel framework for design and optimization of 3D freeform metalenses that attains nearly linear scaling of computational cost with diameter, by breaking the lens into a sequence of radial “zones” with 𝑛-fold discrete axisymmetry, where 𝑛 increases with radius. This allows vastly more design freedom than imposing continuous axisymmetry, while avoiding the compromises of the locally periodic approximation (LPA) or scalar diffraction theory. Using a GPU-accelerated finite-difference time-domain (FDTD) solver in cylindrical coordinates, we perform full-wave simulation and topology optimization within each supra-wavelength zone. We validate our approach by designing millimeter and centimeter-scale, poly-achromatic, 3D freeform metalenses which outperform the state of the art. By demonstrating the scalability and resulting optical performance enabled by our “zoned discrete axisymmetry” (ZDA) and supra-wavelength domain decomposition, we highlight the potential of our framework to advance large-scale meta-optics and next-generation photonic technologies.

Sun, Mengdi [Wesleyan University]↗

NASA NTRS1972

Prediction of far flow field in trailing vortices.

A finite-difference machine code is brought to bear on the wake-vortex problem in the quasi-cylindrical boundary-layer approximation. A turbulent-energy model containing new features is developed. Parameters of the model are evaluated by comparison of calculated velocities and turbulent intensities with measurements in an axisymmetric wake. Comparisons are made with a previous calculation of the decay of an isolated vortex and with wind tunnel and flight measurements in trailing vortices. A self-similar solution develops at large axial distance that decays with the square root of distance. A slower decay occurs in the preceding transition region.

Baldwin, B. S.↗

DOE OSTI2025

Effect of activation temperature on quantum efficiency and lifetime of NEA truncated nanocone array GaAs photocathode

This study investigates the quantum efficiency (QE) and operational lifetime of a negative electron affinity GaAs truncated nanocone array (TNCA) photocathode benchmarked against a conventional flat GaAs photocathode under varying activation temperatures. The TNCA structure demonstrated a QE of up to 13.6% at 590 nm with room temperature (RT) activation—approximately 1.5 times higher than its flat counterpart. This enhancement is due to Mie resonance effects within the nanostructure, as confirmed by finite-difference time-domain simulations. Moreover, the TNCA photocathode exhibits significantly extended charge lifetime, with enhancement factors of ∼6.1 and ∼19.8 under RT and 50 °C activations, respectively. These gains are primarily attributed to increased effective surface area and optimized dipole layer formation at elevated temperatures. In addition, shorter excitation wavelengths further contribute to lifetime improvements. These findings underscore the TNCA GaAs photocathode’s potential as a high QE, long lifetime electron source for many large-scale electron accelerators.

Cs-NF3 activation↗

DOE OSTI2025

Decoupling Carrier Dynamics and Energy Transport in Ultrafast Near-Field Nanoscopy

Ultrafast near-field optical nanoscopy has emerged as a powerful platform to characterize low-dimensional materials. While analytical and numerical models have been established to account for photoexcited carrier dynamics, quantitative evaluation of the associated pulsed laser heating remains elusive. Here, we decouple the photocarrier density and temperature increase in near-field nanoscopy by integrating the two-temperature model (TTM) with finite-difference time-domain (FDTD) simulations. These results reveal that the electron–phonon coupling in a silicon film after femtosecond laser excitation is most pronounced within approximately 3 ps–substantially shorter than the photocarrier decay time scale at tens of picoseconds. Moreover, the coupled TTM-FDTD method indicates that ultrafast laser heating can cause up to a 14% variation in the near-field signal at a 220 μJ/cm 2 pump pulse fluence. Our numerical results are further validated by transient experiments, highlighting the potential of this method for investigations of carrier and thermal phenomena in emerging nanomaterials and nanodevices.

77 NANOSCIENCE AND NANOTECHNOLOGY↗

NASA NTRS1973

Apollo 15 measurement of lunar surface brightness temperatures - Thermal conductivity of the upper 1.5 meters of regolith.

In situ measurements of lunar surface brightness temperatures made as a part of the Apollo Lunar Surface Experiments Package at the Apollo 15 Hadley Rille landing site are reported. Data derived from five thermocouples of the Heat Flow Experiment, which are lying on or just above the surface, are used to examine the thermal properties of the upper 15 cm of the lunar regolith using eclipse and nighttime cool-down temperatures. Application of finite-difference techniques in modeling the lunar soil shows that the thermocouple data are best fit by a model consisting of a low-density and low-thermal conductivity surface layer approximately 2 cm thick overlying a region increasing in conductivity and density with depth. Conductivities on the order of 0.00001 W per cm per deg K are postulated for the upper layer, with conductivity increasing to the order of 0.0001 W per cm per deg K at depths exceeding 20 cm. An increase in mean temperature with depth indicates that the ratio of radiative to conductive transfer at 350 K is 2.7 for at least the upper few centimeters of lunar soil; this value is nearly twice that measured for returned lunar fines.

Keihm, S. J.↗

NASA NTRS1973

Decay of Far-Flowfield in Trailing Vortices

Methods for reduction of velocities in trailing vortices of large aircraft are of current interest for the purpose of shortening the waiting time between landings at central airports. We have made finite-difference calculations of the flow in turbulent wake vortices as an aid to interpretation of wind-tunnel and flight experiments directed toward that end. Finite-difference solutions are capable of adding flexibility to such investigations if they are based on an adequate model of turbulence. Interesting developments have been taking place in the knowledge of turbulence that may lead to a complete theory in the future. In the meantime, approximate methods that yield reasonable agreement with experiment are appropriate. The simplified turbulence model we have selected contains features that account for the major effects disclosed by more sophisticated models in which the parameters are not yet established. Several puzzles are thereby resolved that arose in previous theoretical investigations of wake vortices.

Baldwin, B. S.↗

DOE OSTI2025

Systematic study of the validity of the eikonal model including uncertainties

Nuclear reactions at intermediate beam energies are often interpreted using the eikonal model. In the analysis of complex reaction probes, where few-body reaction methods are needed, the eikonal method may be used as an efficient way for describing the fragment-target reaction process. In this work, we perform a systematic study to test the validity of the eikonal approximation for nucleon-nucleus reactions. We also quantify uncertainties due to the nucleon optical potential on reaction observables. We inspect the validity of the eikonal model and its semiclassical correction by comparing it to exact solutions (obtained from solving the optical-model equation with a finite-differences method) for a wide range of reactions. We also study the effect of relativistic corrections, both kinematic and dynamic, by effectively incorporating the relativistic effects at intermediate energies. The uncertainties from a Bayesian global optical potential (KDUQ) are propagated to the observables of interest. Our study includes neutron and proton reactions on 27 Al , 40 Ca , 90 Zr , and 208 Pb , for a wide range of energies 𝐸 lab = 0–400 MeV. We calculate neutron-total cross sections (elastic and reactions) as well as proton-absorption cross sections as a function of beam energy, using the eikonal model, the eikonal model with a semiclassical correction, and the exact solution. Here, we also compute angular distributions for the methods above. Our results show that for the proton-absorption cross section, the eikonal model can be used down to around 60 MeV and the semiclassical correction extends its use to 30 MeV. However, the validity of the eikonal model for the neutron-total cross section only goes down to ≈120 MeV, a range extended to ≈ 50 MeV when using the semiclassical correction. We find the semiclassical correction to the eikonal model to be less effective in describing the angular distributions. The 1⁢𝜎 uncertainty intervals on the observables we studied is less than 5% for most of the energies considered, but increases rapidly for higher energies, namely energies outside the range of KDUQ (𝐸 lab > 200MeV).

Cluster models↗

DOE OSTI2025

Laser damage of crazed electron-beam high-reflectors following infrared and ultraviolet irradiation in the nanosecond pulse regime

Laser damage of optical components can be a limiting factor in scaling the energetics of high-peak and average power laser systems. Specifically for optical coatings, damage under nanosecond pulsed irradiation is initiated by pre-existing defects in the coating layers, including those that cause discontinuities in the structure, like craze lines. Crazing or cracking in a multilayer dielectric optical coating is induced when the overall coating stress is sufficiently tensile, and is an occasionally observed issue when employing more porous deposition techniques like electron-beam evaporation. Here, in this study, electron-beam high-reflectors were fabricated utilizing process parameters that are known to induce crazing based on prior processing history to systematically evaluate the impact of crazing on reflector damage performance for 1064 and 355 nm lasers. The crazing that was observed was apparently nucleated at nodular defects. When the cross-section of these nodules was investigated, it was observed that there were cracks into the fused silica substrate of approximately 5 µm in depth. The craze lines were irradiated with 1064 and 355 nm light at fluences slightly above the onset of damage initiation fluence of the coating. The 1064 nm irradiated sub-apertures exhibit laser damage but with no spatial correlation with the craze line, whereas the 355 nm irradiated area exhibited many damage sites along the craze line. Finite-difference time-domain electric-field simulations were conducted, and ∼2× field amplification in hafnia was observed for the 355 nm wavelength case. The laser damage can be attributed to a slight electric-field intensification coincidental with an area where UV damage-prone precursors are known to occur. The 355 nm laser damage in uncoated fused silica substrates has been previously correlated to initiate through localized UV absorption at the broken silica bonds in the tips of fractures.

Harthcock, Colin [Lawrence Livermore National Labo↗

NASA NTRS1973

On the numerical simulation of three-dimensional transonic flow with application to the C-141 wing

Results computed by a finite-difference, relaxation algorithm are presented for the supercritical flow (M = 0.825) about the C-141 airplane wing, which has sweep, taper, and twist. Comparisons with both wind-tunnel and flight data indicate that computed solutions of the classical transonic small disturbance equation can accurately simulate high Reynolds number flows when the shock sweep angle is small. It is also shown that this equation poorly approximates the complete potential equation when embedded shock waves are swept at angles greater than about 15 deg. Hence, a more consistent small disturbance equation is derived for use in more general cases.

Lomax, H.↗

DOE OSTI2025

A robust fourth-order finite-difference discretization for the strongly anisotropic transport equation in magnetized plasmas

We propose a second-order temporally implicit, fourth-order-accurate spatial discretization scheme for the strongly anisotropic heat transport equation characteristic of hot, fusion-grade plasmas. Following Du Toit et al. (2018), the scheme transforms mixed-derivative diffusion fluxes (which are responsible for the lack of a discrete maximum principle) into nonlinear advective fluxes, amenable to nonlinear-solver-friendly monotonicity-preserving limiters. The scheme enables accurate multi-dimensional heat transport simulations with up to seven orders of magnitude of heat-transport-coefficient anisotropies with low cross-field numerical error pollution and excellent algorithmic performance, with the number of linear iterations scaling very weakly with grid resolution and grid anisotropy, and scaling with the square-root of the implicit timestep. We propose a multigrid preconditioning strategy based on a lower-order approximation that renders the scheme efficient and scalable under grid refinement. Several numerical tests are presented that display the expected spatial convergence rates and strong algorithmic performance, including fully nonlinear magnetohydrodynamics simulations of kink instabilities in a Bennett pinch in 2D helical geometry and of ITER in 3D toroidal geometry.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2025

A Particle-in-Cell Method for Plasmas with a Generalized Momentum Formulation, Part I: Model Formulation

Here, this paper formulates a new particle-in-cell method for the Vlasov–Maxwell system. Under the Lorenz gauge condition, Maxwell’s equations for the electromagnetic fields can be written as a collection of scalar and vector wave equations. The use of potentials for the fields motivates the adoption of a Hamiltonian formulation for particles that employs the generalized (conjugate) momentum. A notable advantage offered by the Hamiltonian formulation is the elimination of time derivatives in the Lorenz gauge formulation that are required by the standard Newton–Lorentz treatment of the particles. This allows the fields to retain the full time-accuracy guaranteed by the field solver. The resulting updates for particles require only knowledge of the fields and their spatial derivatives. An analytical method for constructing these spatial derivatives is presented that exploits the underlying integral solution used in the field solver for the wave equations. Moreover, these derivatives are demonstrated to converge at the same rate as the fields in both time and space. The Method of Lines Transpose field solver we consider in this work is globally first-order accurate in time and high-order accurate in space (e.g., fourth- and fifth-order) and belongs to a larger class of methods which are unconditionally stable, can address geometry, and leverage $\mathcal {O}(N)$ fast summation methods for efficiency. We demonstrate the method on several well-established benchmark problems on bounded domains, including a plasma sheath as well as a relativistic particle beam. The efficacy of the proposed formulation is established by comparing with a second-order accurate finite-difference time-domain method that employs a leapfrog time advance for particles and a charge conserving map suitable for bounded domains. The new method shows mesh-independent numerical heating properties even in cases where the plasma Debye length is smaller than the grid spacing. This is an important feature of the new method for problems defined on bounded domains, because it permits the use of coarser grids in space in the representation of the fields. Such a capability has significant implications for the simulation of plasmas in bounded domains with complex geometry, where the ratio between the largest and smallest cells can vary significantly. The use of high-order spatial approximations in the new method also means that fewer grid points are required in order to achieve a fixed accuracy. Our results also suggest that the new method can be used with fewer simulation particles per cell compared to the benchmark explicit method, which permits further computational savings.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1973

Nonlinear vibrations of rectangular plates.

A finite-difference method is developed to determine the large amplitude dynamic responses of thin elastic plates subjected to uniform pressure pulse-type loads. Four different sets of boundary conditions are considered. Some specific problems are solved. The results are compared with approximate solutions obtained by Yamaki (1961). The numerical method presented provides an accurate and efficient approximate solution to the problem, and should be useful as a check on other approximate methods. The grid-size and the time-step necessary for obtaining numerical stability depend on the particular problem. For many cases the method converges rapidly and a rather large grid-size and time-step is adequate.

Bayles, D. J.↗