Search NASA⌕ Search

SEARCH · Search NASA

Results for “Finite difference method”

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 253 records · Page 14
All sourcesNASA NTRSDOE OSTI
NASA NTRS1971

Similarity and generalized finite-difference solutions of parabolic partial differential equations.

Techniques are presented for obtaining generalized finite-difference solutions to partial differential equations of the parabolic type. It is shown that the advantages of similarity in the solution of similar problems are generally not lost if the solution to the original partial differential equations is effected in the physical plane by finite-difference methods. The analysis results in a considerable saving in computational effort in the solution of both similar and nonsimilar problems. Several examples, including both the heat-conduction equation and the boundary-layer equations, are given. The analysis also provides a practical means of estimating the accuracy of finite-difference solutions to parabolic equations.

Clausing, A. M.↗
NASA NTRS1988

Results of an integrated structure-control law design sensitivity analysis

Next generation air and space vehicle designs are driven by increased performance requirements, demanding a high level of design integration between traditionally separate design disciplines. Interdisciplinary analysis capabilities have been developed, for aeroservoelastic aircraft and large flexible spacecraft control for instance, but the requisite integrated design methods are only beginning to be developed. One integrated design method which has received attention is based on hierarchal problem decompositions, optimization, and design sensitivity analyses. This paper highlights a design sensitivity analysis method for Linear Quadratic Cost, Gaussian (LQG) optimal control laws, which predicts change in the optimal control law due to changes in fixed problem parameters using analytical sensitivity equations. Numerical results of a design sensitivity analysis for a realistic aeroservoelastic aircraft example are presented. In this example, the sensitivity of the optimally controlled aircraft's response to various problem formulation and physical aircraft parameters is determined. These results are used to predict the aircraft's new optimally controlled response if the parameter was to have some other nominal value during the control law design process. The sensitivity results are validated by recomputing the optimal control law for discrete variations in parameters, computing the new actual aircraft response, and comparing with the predicted response. These results show an improvement in sensitivity accuracy for integrated design purposes over methods which do not include changess in the optimal control law. Use of the analytical LQG sensitivity expressions is also shown to be more efficient that finite difference methods for the computation of the equivalent sensitivity information.

Gilbert, Michael G.↗
NASA NTRS1989

Results of an integrated structure/control law design sensitivity analysis

A design sensitivity analysis method for Linear Quadratic Cost, Gaussian (LQG) optimal control laws, which predicts change in the optimal control law due to changes in fixed problem parameters using analytical sensitivity equations is discussed. Numerical results of a design sensitivity analysis for a realistic aeroservoelastic aircraft example are presented. In this example, the sensitivity of the optimally controlled aircraft's response to various problem formulation and physical aircraft parameters is determined. These results are used to predict the aircraft's new optimally controlled response if the parameter was to have some other nominal value during the control law design process. The sensitivity results are validated by recomputing the optimal control law for discrete variations in parameters, computing the new actual aircraft response, and comparing with the predicted response. These results show an improvement in sensitivity accuracy for integrated design purposes over methods which do not include changes in the optimal control law. Use of the analytical LQG sensitivity expressions is also shown to be more efficient than finite difference methods for the computation of the equivalent sensitivity information.

Gilbert, Michael G.↗
NASA NTRS1985

A hybrid numerical technique for predicting the aerodynamic and acoustic fields of advanced turboprops

A hybrid numerical procedure is presented for the prediction of the aerodynamic and acoustic performance of advanced turboprops. A hybrid scheme is proposed which in principle leads to a consistent simultaneous prediction of both fields. In the inner flow a finite difference method, the Approximate-Factorization Alternating-Direction-Implicit (ADI) scheme, is used to solve the nonlinear Euler equations. In the outer flow the linearized acoustic equations are solved via a Boundary-Integral Equation (BIE) method. The two solutions are iteratively matched across a fictitious interface in the flow so as to maintain continuity. At convergence the resulting aerodynamic load prediction will automatically satisfy the appropriate free-field boundary conditions at the edge of the finite difference grid, while the acoustic predictions will reflect the back-reaction of the radiated field on the magnitude of the loading source terms, as well as refractive effects in the inner flow. The equations and logic needed to match the two solutions are developed and the computer program implementing the procedure is described. Unfortunately, no converged solutions were obtained, due to unexpectedly large running times. The reasons for this are discussed and several means to alleviate the situation are suggested.

Homicz, G. F.↗
NASA NTRS1976

Numerical and approximate solution of the high Reynolds number small separation problem

Several possible methods of solving the small separation problem at high Reynolds number are investigated. In addition to using analytical methods, there are several numerical approaches which are used. High Reynolds number laminar two dimensional problems are used for simplicity. A brief discussion is given of the finite difference methods since these methods are discussed in detail. Most of the emphasis is placed on developing an approximate integral method. As a model problem the supersonic compression ramp problem is chosen since several numerical solutions along with experimental data are available. The techniques discussed are modified and applied to other similar type wall geometries.

Davis, R. T.↗
NASA NTRS1983

Numerical approximations to nonlinear conservation laws with locally varying time and space grids

Numerical approximations to the initial value problem for nonlinear systems of conservation laws are considered. The considered system is said to be hyperbolic when all eigenvalues of every real linear combination of the Jacobian matrices are real. Solutions may develop discontinuities in finite time, even when the initial data are smooth. In the investigation, explicit finite difference methods which use locally varying time grids are considered. The global CFL restriction is replaced by a local restriction. The numerical flux function is studied from a finite volume viewpoint, and a differencing technique is developed at interface points between regions of distinct time increments.

Osher, S.↗
NASA NTRS1982

Finite difference modeling of rotor flows including wake effects

Rotary wing finite difference methods are investigated. The main concern is the specification of boundary conditions to properly account for the effect of the wake on the blade. Examples are given of an approach where wake effects are introduced by specifying an equivalent angle of attack. An alternate approach is also given where discrete vortices are introduced into the finite difference grid. The resulting computations of hovering and high advance ratio cases compare well with experiment. Some consideration is also given to the modeling of low to moderate advance ratio flows.

Caradonna, F. X.↗
NASA NTRS1978

Trends and techniques for space base electronics

A system was developed for depositing aluminum and aluminum alloys by the D.C. sputtering technique. This system which was designed for a high level of cleanliness and ion monitoring the deposition parameters during film preparation is ready for studying the deposition and annealing parameters upon double level metal preparation. The finite element method was studied for use in the computer modeling of two dimensional MOS transistor structures. An algorithm was developed for implementing a computer study which is based upon the finite difference method. The program was modified and used to calculate redistribution data for boron and phosphorous which had been predeposited by ion implantation with range and straggle conditions typical of those used at MSFC. Data were generated for 111 oriented SOS films with redistribution in N2, dry O2 and steam ambients. Data are given showing both two dimensional effects and the evolution of the junction depth, sheet resistance and integrated dose with redistribution time.

Gassaway, J. D.↗
NASA NTRS1975

Numerical computation of viscous flows on the lee side of blunt shapes flying at supersonic speeds

A numerical method for solving the parabolic approximation to the steady-state compressible Navier-Stokes equations is examined. The approximation neglects only the streamwise gradients of shear stress. An implicit finite difference method is used which advances the solution downstream from an initial data surface and determines the complete viscous-inviscid flow between the body and bow shock wave. It is necessary that the inviscid portion of the flow field be supersonic. Crossflow separation is determined as part of the solution. The method is applied to a 15 deg sphere-cone at 15 deg angle of attack, and the results are compared with an inviscid method-of-characteristics calculation.

Rakich, J. V.↗
NASA NTRS1978

A comparison of the Method of Lines to finite difference techniques in solving time-dependent partial differential equations

Steady state solutions to two time dependent partial differential systems have been obtained by the Method of Lines (MOL) and compared to those obtained by efficient standard finite difference methods: (1) Burger's equation over a finite space domain by a forward time central space explicit method, and (2) the stream function - vorticity form of viscous incompressible fluid flow in a square cavity by an alternating direction implicit (ADI) method. The standard techniques were far more computationally efficient when applicable. In the second example, converged solutions at very high Reynolds numbers were obtained by MOL, whereas solution by ADI was either unattainable or impractical. With regard to 'set up' time, solution by MOL is an attractive alternative to techniques with complicated algorithms, as much of the programming difficulty is eliminated.

Kurtz, L. A.↗
NASA NTRS1993

High order finite difference and multigrid methods for spatially evolving instability in a planar channel

The fourth-order finite-difference scheme with fully implicit time-marching presently used to computationally study the spatial instability of planar Poiseuille flow incorporates a novel treatment for outflow boundary conditions that renders the buffer area as short as one wavelength. A semicoarsening multigrid method accelerates convergence for the implicit scheme at each time step; a line-distributive relaxation is developed as a robust fast solver that is efficient for anisotropic grids. Computational cost is no greater than that of explicit schemes, and excellent agreement with linear theory is obtained.

Liu, C.↗
NASA NTRS1959

Some Finite Difference Solutions of the Laminar Compressible Boundary Layer Showing the Effects of Upstream Transpiration Cooling

Three numerical solutions of the partial differential equations describing the compressible laminar boundary layer are obtained by the finite difference method described in reports by I. Flugge-Lotz, D.C. Baxter, and this author. The solutions apply to steady-state supersonic flow without pressure gradient, over a cold wall and over an adiabatic wall, both having transpiration cooling upstream, and over an adiabatic wall with upstream cooling but without upstream transpiration. It is shown that for a given upstream wall temperature, upstream transpiration cooling affords much better protection to the adiabatic solid wall than does upstream cooling without transpiration. The results of the numerical solutions are compared with those of approximate solutions. The thermal results of the finite difference solution lie between the results of Rubesin and Inouye, and those of Libby and Pallone. When the skin-friction results of one finite difference solution are used in the thermal analysis of Rubesin and Inouye, improved agreement between the thermal results of the two methods of solution is obtained.

Howe, John T.↗
NASA NTRS1979

The accurate solution of Poisson's equation by expansion in Chebyshev polynomials

A Chebyshev expansion technique is applied to Poisson's equation on a square with homogeneous Dirichlet boundary conditions. The spectral equations are solved in two ways - by alternating direction and by matrix diagonalization methods. Solutions are sought to both oscillatory and mildly singular problems. The accuracy and efficiency of the Chebyshev approach compare favorably with those of standard second- and fourth-order finite-difference methods.

Haidvogel, D. B.↗
NASA NTRS1988

Accurate solutions, parameter studies and comparisons for the Euler and potential flow equations

Parameter studies are conducted using the Euler and potential flow equation models for steady and unsteady flows in both two and three dimensions. The Euler code is an implicit, upwind, finite volume code which uses the Van Leer method of flux vector splitting which has been recently extended for use on dynamic meshes and maintain all the properties of the original splitting. The potential flow code is an implicit, finite difference method for solving the transonic small disturbance equations and incorporates both entropy and vorticity corrections into the solution procedures thereby extending its applicability into regimes where shock strength normally precludes its use. Parameter studies resulting in benchmark type calculations include the effects of spatial and temporal refinement, spatial order of accuracy, far field boundary conditions for steady flow, frequency of oscillation, and the use of subiterations at each time step to reduce linearization and factorization errors. Comparisons between Euler and potential flow results are made, as well as with experimental data where available.

Anderson, W. Kyle↗
NASA NTRS1993

Preconditioning and the limit to the incompressible flow equations

The use of preconditioning methods to accelerate the convergence to a steady state for both the incompressible and compressible fluid dynamic equations are considered. The relation between them for both the continuous problem and the finite difference approximation is also considered. The analysis relies on the inviscid equations. The preconditioning consists of a matrix multiplying the time derivatives. Hence, the steady state of the preconditioned system is the same as the steady state of the original system. For finite difference methods the preconditioning can change and improve the steady state solutions. An application to flow around an airfoil is presented.

Turkel, E.↗
NASA NTRS1995

Buckling Analysis of Debonded Sandwich Panel Under Compression

A sandwich panel with initial through-the-width debonds is analyzed to study the buckling of its faceskin when subject to an in-plane compressive load. The debonded faceskin is modeled as a beam on a Winkler elastic foundation in which the springs of the elastic foundation represent the sandwich foam. The Rayleigh-Ritz and finite-difference methods are used to predict the critical buckling load for various debond lengths and stiffnesses of the sandwich foam. The accuracy of the methods is assessed with a plane-strain finite-element analysis. Results indicate that the elastic foundation approach underpredicts buckling loads for sandwich panels with isotropic foam cores.

Sleight, David W.↗
NASA NTRS1991

Time-marching methods for three-dimensional steady and unsteady viscous imcompressible flows

An implicit algorithm for the solution of three-dimensional, steady and unsteady, viscous, incompressible flows is presented. The algorithm is based on an upwind-relaxation finite-difference method. Steady-state solutions are carried out using a time-marching solution technique in combination with a local time-stepping strategy. To obtain time-accurate solutions, a subiterative procedure is employed at each physical time step using a global time step to ensure the divergence-free condition. Steady-state flows in several straight ducts and in a square duct with a 90-degree bend are computed and compared with analytical and experimental results. The classical problem of starting flow in a circular pipe is chosen to verify the time accuracy of the present scheme. Finally, the three-dimensional bubble-type vortex breakdown of a slender cylindrical vortex in an unbounded flow is investigated.

Hsu, C.-H.↗
NASA NTRS1976

Calculation of transonic flows using an extended integral equation method

An extended integral equation method for transonic flows is developed. In the extended integral equation method velocities in the flow field are calculated in addition to values on the aerofoil surface, in contrast with the less accurate 'standard' integral equation method in which only surface velocities are calculated. The results obtained for aerofoils in subcritical flow and in supercritical flow when shock waves are present compare satisfactorily with the results of recent finite difference methods.

Nixon, D.↗