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All sourcesNASA NTRSDOE OSTI
NASA NTRS1989

Global properties of pseudospectral methods

The present application of polynomial interpolation methods to function-approximation and numerical solutions for hyperbolic and elliptic PDEs allows the explicit construction of the derivative matrix for a general sequence of collocation points. An evaluation of the effect of several factors on the performance of these methods indicates an inability to interpret global methods in terms of local ones; the accuracy of the approximation will differ when the function's large gradients occur near the center of the region or near the boundary, irrespective of the boundary vicinity's collocation-point density.

Solomonoff, A.↗
NASA NTRS1971

An analysis and comparison of several trajectory optimization methods

The sensitivities of the convergence characteristics of the methods to initially assumed parameters and trial solution, convergence times, computer logic, and storage requirements are discussed. Numerical comparison of the convergence characteristics is made by considering a minimum time, low thrust, Earth-Mars transfer trajectory. A modified quasi-linearization method reduces convergence time by approximately 70% when compared with the generalized Newton-Raphson method and allows the terminal boundary to be specified by a general function of the problem variables. A uniquely specified and easily determined, time dependent weighting matrix for the gradient techniques accelerates the shaping of the optimal control program and improves the convergence characteristics during the terminal iterations. Convergence envelopes, indicating how sensitive the convergence characteristics are to initially assumed parameters, are plotted for the perturbation and quasi-linearization methods. Several iteration schemes are proposed which increase the size of the convergence envelopes and decrease the sensitivity of the method to initially assumed parameters.

Lewallen, J. M.↗
NASA NTRS2008

Simplified Shear Solution for Determination of the Shear Stress Distribution in a Composite Panel from the Applied Shear Resultant

The simplified shear solution method is presented for approximating the through-thickness shear stress distribution within a composite laminate or panel based on laminated beam theory. The method does not consider the solution of a particular boundary value problem; rather it requires only knowledge of the global shear loading, geometry, and material properties of the laminate or panel. It is thus analogous to lamination theory in that ply level stresses can be efficiently determined from global load resultants (as determined, for instance, by finite element analysis) at a given location in a structure and used to evaluate the margin of safety on a ply by ply basis. The simplified shear solution stress distribution is zero at free surfaces, continuous at ply boundaries, and integrates to the applied shear load. Comparisons to existing theories are made for a variety of laminates, and design examples are provided illustrating the use of the method for determining through-thickness shear stress margins in several types of composite panels and in the context of a finite element structural analysis.

Bednarcyk, Brett A.↗
NASA NTRS1993

Development of deterministic transport methods for low energy neutrons for shielding in space

Transport of low energy neutrons associated with the galactic cosmic ray cascade is analyzed in this dissertation. A benchmark quality analytical algorithm is demonstrated for use with BRYNTRN, a computer program written by the High Energy Physics Division of NASA Langley Research Center, which is used to design and analyze shielding against the radiation created by the cascade. BRYNTRN uses numerical methods to solve the integral transport equations for baryons with the straight-ahead approximation, and numerical and empirical methods to generate the interaction probabilities. The straight-ahead approximation is adequate for charged particles, but not for neutrons. As NASA Langley improves BRYNTRN to include low energy neutrons, a benchmark quality solution is needed for comparison. The neutron transport algorithm demonstrated in this dissertation uses the closed-form Green's function solution to the galactic cosmic ray cascade transport equations to generate a source of neutrons. A basis function expansion for finite heterogeneous and semi-infinite homogeneous slabs with multiple energy groups and isotropic scattering is used to generate neutron fluxes resulting from the cascade. This method, called the FN method, is used to solve the neutral particle linear Boltzmann transport equation. As a demonstration of the algorithm coded in the programs MGSLAB and MGSEMI, neutron and ion fluxes are shown for a beam of fluorine ions at 1000 MeV per nucleon incident on semi-infinite and finite aluminum slabs. Also, to demonstrate that the shielding effectiveness against the radiation from the galactic cosmic ray cascade is not directly proportional to shield thickness, a graph of transmitted total neutron scalar flux versus slab thickness is shown. A simple model based on the nuclear liquid drop assumption is used to generate cross sections for the galactic cosmic ray cascade. The ENDF/B V database is used to generate the total and scattering cross sections for neutrons in aluminum. As an external verification, the results from MGSLAB and MGSEMI were compared to ANISN/PC, a routinely used neutron transport code, showing excellent agreement. In an application to an aluminum shield, the FN method seems to generate reasonable results.

Ganapol, Barry↗
NASA NTRS1972

Multipath propagation study for L-band, over-ocean, satellite-aircraft communication link

The results are presented of a study of multipath propagation between an aeronautical relay satellite in synchronous earth orbit and an aircraft flying over the ocean. An analytical model for the reflected power is developed and the results of a computer solution are compared with an approximate solution obtained through integration by the method of steepest descent. The computer solution is shown to differ considerably from the approximate solution. Both methods are compared with the limited experimental results available.

Deal, J. H.↗
NASA NTRS1978

Inviscid, nonadiabatic flow fields over blunt, sonic corner bodies for outer planet entry conditions by a method of integral relations

An investigation has been made into the ability of a method of integral relations to calculate inviscid zero degree angle of attack, radiative heating distributions over blunt, sonic corner bodies for some representative outer planet entry conditions is investigated. Comparisons have been made with a more detailed numerical method, a time asymptotic technique, using the same equilibrium chemistry and radiation transport subroutines. An effort to produce a second order approximation (two-strip) method of integral relations code to aid in this investigation is also described and a modified two-strip routine is presented. Results indicate that the one-strip method of integral relations cannot be used to obtain accurate estimates of the radiative heating distribution because of its inability to resolve thermal gradients near the wall. The two-strip method can sometimes be used to improve these estimates; however, the two-strip method has only a small range of conditions over which it will yield significant improvement over the one-strip method.

Gnoffo, P. A.↗
NASA NTRS2003

High Energy Boundary Conditions for a Cartesian Mesh Euler Solver

Inlets and exhaust nozzles are common place in the world of flight. Yet, many aerodynamic simulation packages do not provide a method of modelling such high energy boundaries in the flow field. For the purposes of aerodynamic simulation, inlets and exhausts are often fared over and it is assumed that the flow differences resulting from this assumption are minimal. While this is an adequate assumption for the prediction of lift, the lack of a plume behind the aircraft creates an evacuated base region thus effecting both drag and pitching moment values. In addition, the flow in the base region is often mis-predicted resulting in incorrect base drag. In order to accurately predict these quantities, a method for specifying inlet and exhaust conditions needs to be available in aerodynamic simulation packages. A method for a first approximation of a plume without accounting for chemical reactions is added to the Cartesian mesh based aerodynamic simulation package CART3D. The method consists of 3 steps. In the first step, a components approach where each triangle is assigned a component number is used. Here, a method for marking the inlet or exhaust plane triangles as separate components is discussed. In step two, the flow solver is modified to accept a reference state for the components marked inlet or exhaust. In the third step, the flow solver uses these separated components and the reference state to compute the correct flow condition at that triangle. The present method is implemented in the CART3D package which consists of a set of tools for generating a Cartesian volume mesh from a set of component triangulations. The Euler equations are solved on the resulting unstructured Cartesian mesh. The present methods is implemented in this package and its usefulness is demonstrated with two validation cases. A generic missile body is also presented to show the usefulness of the method on a real world geometry.

Pandya, Shishir↗
NASA NTRS1993

A new flux conserving Newton's method scheme for the two-dimensional, steady Navier-Stokes equations

A new numerical method is developed for the solution of the two-dimensional, steady Navier-Stokes equations. The method that is presented differs in significant ways from the established numerical methods for solving the Navier-Stokes equations. The major differences are described. First, the focus of the present method is on satisfying flux conservation in an integral formulation, rather than on simulating conservation laws in their differential form. Second, the present approach provides a unified treatment of the dependent variables and their unknown derivatives. All are treated as unknowns together to be solved for through simulating local and global flux conservation. Third, fluxes are balanced at cell interfaces without the use of interpolation or flux limiters. Fourth, flux conservation is achieved through the use of discrete regions known as conservation elements and solution elements. These elements are not the same as the standard control volumes used in the finite volume method. Fifth, the discrete approximation obtained on each solution element is a functional solution of both the integral and differential form of the Navier-Stokes equations. Finally, the method that is presented is a highly localized approach in which the coupling to nearby cells is only in one direction for each spatial coordinate, and involves only the immediately adjacent cells. A general third-order formulation for the steady, compressible Navier-Stokes equations is presented, and then a Newton's method scheme is developed for the solution of incompressible, low Reynolds number channel flow. It is shown that the Jacobian matrix is nearly block diagonal if the nonlinear system of discrete equations is arranged approximately and a proper pivoting strategy is used. Numerical results are presented for Reynolds numbers of 100, 1000, and 2000. Finally, it is shown that the present scheme can resolve the developing channel flow boundary layer using as few as six to ten cells per channel width, depending on the Reynolds number.

Scott, James R.↗
NASA NTRS1982

Analysis and design of steady transonic flow over airfoils by the method of parametric differentiation

The results reported here are based on applying the method of parametric differentiation (MPD) to transform the nonlinear differential equation governing small-disturbance transonic flow to a linear equation. Implicit approximate factorization and monotone methods were used to accelerate convergence of the linear problem by an order of magnitude over successive line over-relaxation. The relative merits of using MPD are discussed in comparison to conventional small-disturbance applications. Several MPD analyses are performed on an array of airfoils. A design procedure utilizing MPD is discussed and demonstrated for two nonlifting cases.

Halt, D. W.↗
NASA NTRS2004

Representation of Ice Geometry by Parametric Functions: Construction of Approximating NURBS Curves and Quantification of Ice Roughness--Year 1: Approximating NURBS Curves

Software was developed to construct approximating NURBS curves for iced airfoil geometries. Users specify a tolerance that determines the extent to which the approximating curve follows the rough ice. The user can therefore smooth the ice geometry in a controlled manner, thereby enabling the generation of grids suitable for numerical aerodynamic simulations. Ultimately, this ability to smooth the ice geometry will permit studies of the effects of smoothing upon the aerodynamics of iced airfoils. The software was applied to several different types of iced airfoil data collected in the Icing Research Tunnel at NASA Glenn Research Center, and in all cases was found to efficiently generate suitable approximating NURBS curves. This method is an improvement over the current "control point formulation" of Smaggice (v.1.2). In this report, we present the relevant theory of approximating NURBS curves and discuss typical results of the software.

Dill, Loren H.↗
NASA NTRS1976

The delta-Eddington approximation for radiative flux transfer

Simple approximations, like the Eddington, are often incapable of coping with the highly asymmetric phase functions typical of particulate scattering. A simple yet accurate method called the delta-Eddington approximation is proposed for determining monochromatic radiative fluxes in an absorbing-scattering atmosphere. In this method, the governing phase function is approximated by a Dirac delta function forward scatter peak and a two-term expansion of the phase function. The fraction of scattering into the truncated forward peak is taken proportional to the square of the phase function asymmetry factor, which distinguishes the delta-Eddington approximation from others of similar nature. The transmission, reflection, and absorption predicted by the delta-Eddington approximation are compared with doubling method calculations for realistic ranges of optical depth, single-scattering albedo, surface albedo, sun angle and asymmetry factor. The approximation is shown to provide an accurate and analytically simple parameterization of radiation to replace the empirism currently encountered in many general circulation and climate models.

Joseph, J. H.↗
NASA NTRS2001

Multigrid approaches to non-linear diffusion problems on unstructured meshes

The efficiency of three multigrid methods for solving highly non-linear diffusion problems on two-dimensional unstructured meshes is examined. The three multigrid methods differ mainly in the manner in which the nonlinearities of the governing equations are handled. These comprise a non-linear full approximation storage (FAS) multigrid method which is used to solve the non-linear equations directly, a linear multigrid method which is used to solve the linear system arising from a Newton linearization of the non-linear system, and a hybrid scheme which is based on a non-linear FAS multigrid scheme, but employs a linear solver on each level as a smoother. Results indicate that all methods are equally effective at converging the non-linear residual in a given number of grid sweeps, but that the linear solver is more efficient in cpu time due to the lower cost of linear versus non-linear grid sweeps.

Mavriplis, Dimitri J.↗
NASA NTRS1980

ACCESS 3. Approximation concepts code for efficient structural synthesis: User's guide

A user's guide is presented for ACCESS-3, a research oriented program which combines dual methods and a collection of approximation concepts to achieve excellent efficiency in structural synthesis. The finite element method is used for structural analysis and dual algorithms of mathematical programming are applied in the design optimization procedure. This program retains all of the ACCESS-2 capabilities and the data preparation formats are fully compatible. Four distinct optimizer options were added: interior point penalty function method (NEWSUMT); second order primal projection method (PRIMAL2); second order Newton-type dual method (DUAL2); and first order gradient projection-type dual method (DUAL1). A pure discrete and mixed continuous-discrete design variable capability, and zero order approximation of the stress constraints are also included.

Fleury, C.↗
NASA NTRS1981

Numerical simulation of steady transonic flow about airfoils

A computer code has been developed that couples a fast transonic full-potential AF2 solver with both an efficient integral boundary-layer method and a viscous wedge approximation of the shock/boundary-layer interaction. The efficiency of the coupled analysis methods and the method of coupling has resulted in a uniquely efficient analysis tool. The airfoil geometry is modified by the displacement thickness before the shock and the displacement thickness plus the viscous wedge thickness after the shock by considering the viscous effects as an equivalent transpiration boundary condition. The flow about conventional and supercritical airfoils under moderately strong shock situations has been calculated. Comparisons with experimental data indicate that this viscous correction method has improved the accuracy of the full-potential analysis. Furthermore, the computer time required to obtain a converged solution has been reduced.

Lee, S. C.↗
NASA NTRS1982

Fitting surfaces to scattered data

A variety of numerical methods for fitting a function to data given at a set of points scattered throughout a domain in the plane are surveyed. Four classes of methods are discussed: (1) global interpolation; (2) local interpolation; (3) global approximation; and (4) local approximation. Also, two-stage methods and contouring are discussed. The surfaces constructed include polynomials, spline functions, and rational functions, among others.

Schumaker, L. L.↗
NASA NTRS1996

Analysis of the Hessian for Aerodynamic Optimization: Inviscid Flow

In this paper we analyze inviscid aerodynamic shape optimization problems governed by the full potential and the Euler equations in two and three dimensions. The analysis indicates that minimization of pressure dependent cost functions results in Hessians whose eigenvalue distributions are identical for the full potential and the Euler equations. However the optimization problems in two and three dimensions are inherently different. While the two dimensional optimization problems are well-posed the three dimensional ones are ill-posed. Oscillations in the shape up to the smallest scale allowed by the design space can develop in the direction perpendicular to the flow, implying that a regularization is required. A natural choice of such a regularization is derived. The analysis also gives an estimate of the Hessian's condition number which implies that the problems at hand are ill-conditioned. Infinite dimensional approximations for the Hessians are constructed and preconditioners for gradient based methods are derived from these approximate Hessians.

Arian, Eyal↗
NASA NTRS1984

Spectral methods for exterior elliptic problems

Spectral approximations for exterior elliptic problems in two dimensions are discussed. As in the conventional finite difference or finite element methods, the accuracy of the numerical solutions is limited by the order of the numerical farfield conditions. A spectral boundary treatment is introduced at infinity which is compatible with the infinite order interior spectral scheme. Computational results are presented to demonstrate the spectral accuracy attainable. Although a simple Laplace problem is examined, the analysis covers more complex and general cases.

Canuto, C.↗