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Results for “Finite-Difference Methods”

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 307 records · Page 17
All sourcesNASA NTRSDOE OSTI
NASA NTRS1986

Transonic airfoil calculations including wind tunnel wall-interference effects

The results of Reynolds-averaged time-dependent inviscid and turbulent compressible Navier-Stokes computations using the implicit finite-difference approach of Steger (1978), modified by incorporating a pressure boundary condition, (PBC) to account for wall interference are compared with experimental data on a NACA 64A010 airfoil (Johnson and Bachalo, 1980) in graphs and briefly characterized. The computational approach is the same as that used by King and Johnson (1980), but a 137 x 50 mesh is used instead of a 97 x 35 mesh, and special care is taken in resolving the nose, shock, and trailing-edge regions. Imposition of PBC is shown to improve significantly the accuracy of the computations for the flowfield on the upper surface of the airfoil, shifting the shock forward to its experimentally measured position in the case of turbulent flow. The failure of the method, even with PBC, to match the experimental shock location in the case of a flow with a separation bubble is attributed to inadequacies in the algebraic turbulence model employed (Baldwin and Lomax, 1978).

King, L. S.↗
NASA NTRS1975

Time-dependent transonic flow solutions for axial turbomachinery

Three-dimensional unsteady transonic flow through an axial turbomachine stage is described in terms of a pair of two-dimensional formulations pertaining to orthogonal surfaces, namely, a blade-to-blade surface and a hub-to-casing surface. The resulting systems of nonlinear, inviscid, compressible equations of motion are solved by an explicit finite-difference technique. The blade-to-blade program includes the periodic interaction between rotor and stator blade rows. Treatment of the boundary conditions and of the blade slipstream motion by a characteristic type procedure is discussed in detail. Harmonic analysis of the acoustic far field produced by the blade row interaction, including an arbitrary initial transient, is outlined. Results from the blade-to-blade program are compared with experimental measurements of the rotating pressure field at the tip of a high-speed fan. The hub-to-casing program determines circumferentially averaged flow properties on a meridional plane. Blade row interactions are neglected in this formulation, but the force distributions over the entire blade surface for both the rotor and stator are obtained. Results from the hub-to-casing program are compared with a relaxation method solution for a subsonic rotor. Results are also presented for a quiet fan stage which includes transonic flow in both the rotor and stator and a normal shock in the stator.

Erdos, J.↗
DOE OSTI2026

Benchmarking of massively parallel phase-field codes for directional solidification

We present a detailed benchmark comparing two state-of-the-art phase-field implementations for simulating alloy solidification under experimentally relevant conditions. The study investigates the directional solidification of Al-3wt%Cu under high-velocity solidification conditions and SCN-0.46wt% camphor under microgravity conditions from National Aeronautics and Space Administration (NASA) DECLIC-DSI-R experiments. Both codes, one employing finite-difference discretization with uniform mesh and GPU-acceleration (GPU-PF) and the other one employing finite-element discretization with adaptive-mesh and CPU-parallelization (PRISMS-PF), solve the same quantitative phase-field formulation that incorporates an anti-trapping current for the solidification of dilute alloys. We evaluate the predictions of each code for dendritic morphology, primary spacing, and tip dynamics in both 2D and 3D, as well as their numerical convergence and computational performance. While existing benchmark problems have primarily focused on simplified or small-scale simulations, they do not reflect the computational and modeling challenges posed by employing experimentally relevant time and length scales. Our results provide a practical framework for assessing phase-field code performance as well as validating and facilitating their application in integrated computational materials engineering (ICME) workflows that require integration with realistic experimental data.

36 MATERIALS SCIENCE↗
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

Formation and optical properties of indium nanoparticle arrays for deep-UV plasmonics

We utilize a combined computational-experimental approach to examine the influence of indium nanoparticle (NP) array distributions on deep-ultraviolet (UV) plasmon resonances. For photon energies < 5.7 eV, analysis of ellipsometric spectra reveals an increase in silicon reflectance induced by indium NP arrays on silicon. For various energies in the range 5.7–7.0 eV, a decrease in reflectance is induced by the NP arrays. Similar trends in reflectance are predicted from finite-difference time-domain (FDTD) simulations using NP size distributions extracted from atomic-force micrographs as input. In addition, in the energy range of 7.4–9.2 eV, the FDTD simulations reveal reflectance minima, characteristic of localized surface plasmon resonances. Here, electron energy-loss spectroscopy collected from individual indium NPs reveals the presence of LSPR at ≈ 8 eV, further supporting the promise of indium NP arrays on silicon for deep-UV plasmonics.

36 MATERIALS SCIENCE↗
NASA NTRS1972

Finite element analysis of large transient elastic-plastic deformations of simple structures, with application to the engine rotor fragment containment/deflection problem

Assumed-displacement versions of the finite-element method are developed to predict large-deformation elastic-plastic transient deformations of structures. Both the conventional and a new improved finite-element variational formulation are derived. These formulations are then developed in detail for straight-beam and curved-beam elements undergoing (1) Bernoulli-Euler-Kirchhoff or (2) Timoshenko deformation behavior, in one plane. For each of these categories, several types of assumed-displacement finite elements are developed, and transient response predictions are compared with available exact solutions for small-deflection, linear-elastic transient responses. The present finite-element predictions for large-deflection elastic-plastic transient responses are evaluated via several beam and ring examples for which experimental measurements of transient strains and large transient deformations and independent finite-difference predictions are available.

Wu, R. W.↗
NASA NTRS1974

Liquefaction dynamics of n-octadecane in cylindrical coordinates

The study presents one method of mathematically modeling the effects of gravity-induced free convection upon the melting phenomena of a finite paraffin slab. The theoretical model is based on the numerical computer solution of the solid phase and liquid phase energy equations, coupled with an imposed ideal-viscous flow approximation. An implicit alternating-direction technique is used as the finite-difference approximation for the basic energy equations in the numerical solution. The theoretical predictions are compared to experimental data from an annular test cell, in which n-octadecane, C18H38, was used as the test material. The results show that the model is a good first approximation to the solution of solid-liquid phase change involving gravity-induced free convection.

Bain, R. L.↗
NASA NTRS1975

A finite-difference program for stresses in anisotropic, layered plates in bending

The interlaminar stresses induced in a layered laminate that is bent into a cylindrical surface are studied. The laminate is modeled as a continuum, and the resulting elasticity equations are solved using the finite difference method. The report sets forth the mathematical framework, presents some preliminary results, and provides a listing and explanation of the computer program. Significant among the results are apparent symmetry relationships that will reduce the numerical size of certain problems and an interlaminar stress behavior having a sharp rise at the free edges.

Salamon, N. J.↗
NASA NTRS1975

An investigation of several numerical procedures for time-asymptotic compressible Navier-Stokes solutions

The status of an investigation of four numerical techniques for the time-dependent compressible Navier-Stokes equations is presented. Results for free shear layer calculations in the Reynolds number range from 1000 to 81000 indicate that a sequential alternating-direction implicit (ADI) finite-difference procedure requires longer computing times to reach steady state than a low-storage hopscotch finite-difference procedure. A finite-element method with cubic approximating functions was found to require excessive computer storage and computation times. A fourth method, an alternating-direction cubic spline technique which is still being tested, is also described.

Rudy, D. H.↗
NASA NTRS1978

Numerical solution of the Navier-Stokes equations for super-sonic flows with strong shocks

The numerical solution of the full Navier-Stokes Equations for viscous flows with high Mach numbers and a strong detached bow shock was obtained. Two dimensional flows around a circular cylinder, and a circular cylinder with an aft-body in the form of a fairing, were considered. The solution of the compressible N.S. equations was accomplished by the method of finite differences. An implicit scheme of solution, the S.O.R., was used with the optimum acceleration parameters determined by trial and error. The tensor notation was used in writing the N-S Equations transformed into general curvilinear coordinates. The equations for the generation of the coordinate system were solved, followed by the solution of the N.S. equations, at the end of a set of given number of time steps. "Wiggles", constituted the one major problem that needed to be overcome. These oscillations give rise to quantities such as negative temperatures, which ultimately caused the computational program to break down. Certain dissipative finite-difference schemes damped these oscillations.

Devarayalu, K.↗
NASA NTRS1978

Numerical method for boundary layers with blowing - The exponential box scheme

The paper describes a new numerical scheme based on exponential difference operator concepts combined with Keller's (1968) box scheme approach to produce a stable second-order accurate finite-difference scheme for convection-diffusion problems arising in boundary layer flows in the presence of massive injection through a porous surface. The technique is demonstrated by application to the self-similar boundary layer equations with massive blowing at the surface.

El-Mistikawy, T. M.↗
NASA NTRS1979

Application of numerical methods to heat transfer and thermal stress analysis of aerospace vehicles

The paper describes a thermal-structural design analysis study of a fuel-injection strut for a hydrogen-cooled scramjet engine for a supersonic transport, utilizing finite-element methodology. Applications of finite-element and finite-difference codes to the thermal-structural design-analysis of space transports and structures are discussed. The interaction between the thermal and structural analyses has led to development of finite-element thermal methodology to improve the integration between these two disciplines. The integrated thermal-structural analysis capability developed within the framework of a computer code is outlined.

Wieting, A. R.↗
NASA NTRS1982

Finite-difference solution of the compressible stability eigenvalue problem

A compressible stability analysis computer code is developed. The code uses a matrix finite difference method for local eigenvalue solution when a good guess for the eigenvalue is available and is significantly more computationally efficient than the commonly used initial value approach. The local eigenvalue search procedure also results in eigenfunctions and, at little extra work, group velocities. A globally convergent eigenvalue procedure is also developed which may be used when no guess for the eigenvalue is available. The global problem is formulated in such a way that no unstable spurious modes appear so that the method is suitable for use in a black box stability code. Sample stability calculations are presented for the boundary layer profiles of a Laminar Flow Control (LFC) swept wing.

Malik, M. R.↗
NASA NTRS1982

Incompressible Navier-Stokes and parabolized Navier-Stokes solution procedures and computational techniques

Recent developments with finite-difference techniques are emphasized. The quotation marks reflect the fact that any finite discretization procedure can be included in this category. Many so-called finite element collocation and galerkin methods can be reproduced by appropriate forms of the differential equations and discretization formulas. Many of the difficulties encountered in early Navier-Stokes calculations were inherent not only in the choice of the different equations (accuracy), but also in the method of solution or choice of algorithm (convergence and stability, in the manner in which the dependent variables or discretized equations are related (coupling), in the manner that boundary conditions are applied, in the manner that the coordinate mesh is specified (grid generation), and finally, in recognizing that for many high Reynolds number flows not all contributions to the Navier-Stokes equations are necessarily of equal importance (parabolization, preferred direction, pressure interaction, asymptotic and mathematical character). It is these elements that are reviewed. Several Navier-Stokes and parabolized Navier-Stokes formulations are also presented.

Rubin, S. G.↗
NASA NTRS1981

Mesh generation by conformal and quasiconformal mappings

It is pointed out that many recent advances in the finite-difference solution of elliptic equations have been limited to regions whose boundary contours coincide with coordinate lines of the Cartesian coordinate system. The reason for this is related to the fact that in the case of an arbitrary curvilinear coordinate system the original equation becomes much more complex. However, there is no added complexity if an orthogonal coordinate system is generated from a conformal mapping. In the present investigation, a finite difference method developed for the construction of conformal mappings has been generalized to construct quasi-conformal mappings. It is expected that the use of more sophisticated numerical algorithms could lead to improvements in both speed and accuracy. Quasi-conformal mappings have applications not only in the solution of elliptic equations but also in other areas such as orthogonal mesh generation on surfaces and the solution of certain fluid flow problems.

Mastin, C. W.↗
NASA NTRS1983

An improved finite-difference analysis of uncoupled vibrations of tapered cantilever beams

An improved finite difference procedure for determining the natural frequencies and mode shapes of tapered cantilever beams undergoing uncoupled vibrations is presented. Boundary conditions are derived in the form of simple recursive relations involving the second order central differences. Results obtained by using the conventional first order central differences and the present second order central differences are compared, and it is observed that the present second order scheme is more efficient than the conventional approach. An important advantage offered by the present approach is that the results converge to exact values rapidly, and thus the extrapolation of the results is not necessary. Consequently, the basic handicap with the classical finite difference method of solution that requires the Richardson's extrapolation procedure is eliminated. Furthermore, for the cases considered herein, the present approach produces consistent lower bound solutions.

Subrahmanyam, K. B.↗
NASA NTRS1985

A first-order time-domain Green's function approach to supersonic unsteady flow

A time-domain Green's Function Method for unsteady supersonic potential flow around complex aircraft configurations is presented. The focus is on the supersonic range wherein the linear potential flow assumption is valid. The Green's function method is employed in order to convert the potential-flow differential equation into an integral one. This integral equation is then discretized, in space through standard finite-element technique, and in time through finite-difference, to yield a linear algebraic system of equations relating the unknown potential to its prescribed co-normalwash (boundary condition) on the surface of the aircraft. The arbitrary complex aircraft configuration is discretized into hyperboloidal (twisted quadrilateral) panels. The potential and co-normalwash are assumed to vary linearly within each panel. Consistent with the spatial linear (first-order) finite-element approximations, the potential and co-normalwash are assumed to vary linearly in time. The long range goal of our research is to develop a comprehensive theory for unsteady supersonic potential aerodynamics which is capable of yielding accurate results even in the low supersonic (i.e., high transonic) range.

Freedman, M. I.↗