Search NASA⌕ Search

SEARCH · Search NASA

Results for “higher-order derivatives”

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 55 records · Page 3
All sourcesNASA NTRSDOE OSTI
NASA NTRS1987

A generalized procedure for constructing an upwind-based TVD scheme

A generalized formulation for constructing second- and higher-order accurate TVD (total variation diminishing) schemes is presented. A given scheme is made TVD by limiting antidiffusive flux differences with some nonlinear functions, so-called limiters. The general idea of the formulation and its mathematical proof of Harten's TVD conditions is shown by applying the Lax-Wendroff method to a scalar nonlinear equation and constant-coefficient system of conservation laws. For the system of equations, several definitions are derived for the argument used in the limiter function and present their performance to numerical experiments. Then the formulation is formally extended to the nonlinear system of equations. It is demonstrated that use of the present procedure allows easy conversion of existing central or upwind, and second- or higher-order differencing schemes so as to preserve monotonicity and to yield physically admissible solutions. The formulation is simple mathematically as well as numerically; neither matrix-vector multiplication nor Riemann solver is required. Roughly twice as much computational effort is needed as compared to conventional scheme. Although the notion of TVD is based on the initial value problem, application to the steady Euler equations of the formulation is also made. Numerical examples including various ranges of problems show both time- and spatial-accuracy in comparison with exact solutions.

Liou, Meng-Sing↗
NASA NTRS1989

Part A: Nonprincipal-plane scattering from flat plates: Second-order and corner diffractions

Two models of a flat plate for nonprincipal-plane scattering are explored. The first is a revised version of the Physical Optics/Physical Theory of Diffraction (PO/PTD) model with second-order PTD equivalent currents included to account for second-order interactions among the plate edges. The second model uses a heurisitcally derived corner diffraction coefficient to account for the corner scattering mechanism. The patterns obtained using the newer models were compared to the data of previously reported models, the Moment Method (MM), and experimental results. Near normal incidence, all the models agreed; however, near grazing incidence a need for higher-order and corner diffraction mechanisms was noted. In many instances the second-order and corner-scattered fields which were formulated improved the results.

Balanis, Constantine A.↗
NASA NTRS1974

Multiple eigenvalues of sound-absorbing circular and annular ducts

Eigenvalue equations of the form D(lambda) = 0 are studied for acoustic modes in circular and annular ducts without flow. The ducts have locally reacting walls with arbitrary wall admittance. It is shown that circular ducts may have double eigenvalues, or wall admittances where both D(lambda) and D prime (lambda) are zero. These double eigenvalues are formed by the coalescence of the lowest-order eigenfunction with some higher-order eigenfunction. The eigenfunctions, or acoustic modes, associated with the double eigenvalues are found to be orthogonal to themselves so that the standard expansion formula for the acoustic field, which is obtained from a separation-of-variables analysis, is invalid. An alternate expansion formula for the acoustic field is derived. This alternate formula shows that the coalesced acoustic mode is linearly amplified by the transmission distance and exponentially attenuated at the expected rate. The expansion formula also reveals a new eigenfunction, or acoustic mode, which is exponentially attenuated at the same rate as the coalesced eigenfunction.

Zorumski, W. E.↗
NASA NTRS1992

Multiple-scattering effects in quasielastic alpha-He-4 scattering

A multiple-scattering series for describing the quasielastic peak in nucleus-nucleus collisions is derived using the high-energy optical model. The effects of multiple knockout of target nucleons and internal excitation of the projectile are studied and found to be important for large energy loss and momentum transfers in inclusive alpha-He-4 scattering at 7 GeV/c. An approximate evaluation of higher-order inelastic collision terms is considered for forward-peaked wave functions and is demonstrated to be accurate.

Cucinotta, Francis A.↗
NASA NTRS1976

A higher order panel method for general analysis and design applications in subsonic flow

A higher-order panel method is described for numerical solution of boundary-value problems relating to steady inviscid irrotational incompressible subsonic fluid flow in a domain. Both Neumann and Dirichlet boundary conditions are treated; two types of auxiliary conditions are used to remove the degrees of freedom that arise from specifying only the derivative of the perturbation velocity potential. Four general network types and two expansions of the induced potential kernel are employed in the numerical solution. Some results are presented which illustrate the modeling options and numerical characteristics of the method.

Johnson, F. T.↗
NASA NTRS1989

Dynamic stability of unidirectional fiber-reinforced viscoelastic composite plates

This paper deals with a dynamic stability analysis of unidirectional fiber-reinforced composite viscoelastic plates subjected to compressive edge loads. The integrodifferential equations governing the stability problem are obtained by using, in conjunction with a Boltzmann hereditary constitutive law for a three-dimensional viscoelastic medium, a higher-order shear deformation theory of orthotropic plates. Such a theory incorporates transverse shear deformation, transverse normal stress, and rotatory inertia effects. The solution of the stability problem as considered within this paper concerns the determination of the critical in-plane edge loads yielding the asymptotic instability. Numerical applications, based on material properties derived within the framework of Aboudi's micromechanical model, are presented and pertinent conclusions concerning the nature of the loss of stability and the influence of various parameters are outlined.

Chandiramani, N. K.↗
NASA NTRS1987

General surface equations for glancing incidence telescopes

A generalized set of equations are derived for two mirror glancing incidence telescopes using Fermat's principle, a differential form of the law of reflection, the generalized sine condition, and a ray propagation equation described in vector form as a theoretical basis. The resulting formulation groups the possible telescope configurations into three distinct classes which are the Wolter, Wolter-Schwarzschild, and higher-order telescopes in which the Hettrick-Bowyer types are a subset. Eight configurations are possible within each class depending on the sign and magnitude of the parameters.

Saha, Timo T.↗
NASA NTRS1976

Kinetic theory of turbulent compressible flows and comparison with classical theory

A complete set of turbulent correlations is given in terms of expansion coefficients in a double series of Hermite polynomials of a two-particle correlation in phase space. Only two of these coefficients, corresponding to Reynolds stress and turbulent heat flux, are shown to appear in gasdynamic equations of turbulence. This is accomplished by incorporating the phase-space correlation in the collision integral of Boltzmann's equation and by deriving generalized Navier-Stokes and Fourier transport relations. In contrast to the classical formalism, the simple expression remains invariant in form whether the flow is compressible or not. Nevertheless, a bilinear transformation of fluctuating quantities shows that the two formalisms are identical with regard to terms of double correlations. The kinetic theory justifies nonexistence of the higher-order correlations characteristic of compressible-turbulence equations in the classical regime.

Tsuge, S.↗
NASA NTRS1978

A high order accurate finite element algorithm for high Reynolds number flow prediction

A Galerkin-weighted residuals formulation is employed to establish an implicit finite element solution algorithm for generally nonlinear initial-boundary value problems. Solution accuracy, and convergence rate with discretization refinement, are quantized in several error norms, by a systematic study of numerical solutions to several nonlinear parabolic and a hyperbolic partial differential equation characteristic of the equations governing fluid flows. Solutions are generated using selective linear, quadratic and cubic basis functions. Richardson extrapolation is employed to generate a higher-order accurate solution to facilitate isolation of truncation error in all norms. Extension of the mathematical theory underlying accuracy and convergence concepts for linear elliptic equations is predicted for equations characteristic of laminar and turbulent fluid flows at nonmodest Reynolds number. The nondiagonal initial-value matrix structure introduced by the finite element theory is determined intrinsic to improved solution accuracy and convergence. A factored Jacobian iteration algorithm is derived and evaluated to yield a consequential reduction in both computer storage and execution CPU requirements while retaining solution accuracy.

Baker, A. J.↗
NASA NTRS1985

A three-dimensional boundary-layer method for flow over delta wings with leading-edge separation

A three-dimensional, laminar boundary-layer method is applied to the incompressible flow over a slender delta wing at incidence. The predictor-corrector finite-difference scheme of Matsuno is used to difference the governing equations. The method has the advantages that no iterations are required to advance the solution and the cross-flow derivatives are formed independent of the cross-field direction. The difference scheme is demonstrated to yield accurate numerical results when compared to the exact solution of the three-dimensional boundary-layer equations for parabolic flow over a moving flat plate. The method is applied to delta wings of various sweep angles at angles of attack up to 20 deg., with the inviscid solution determined using a higher-order, three-dimensional panel method.

Woodson, S. H.↗
NASA NTRS1992

Linear and nonlinear propagation of water wave groups

Results are presented from a study of the evolution of waveforms with known analytical group shapes, in the form of both transient wave groups and the cloidal (cn) and dnoidal (dn) wave trains as derived from the nonlinear Schroedinger equation. The waveforms were generated in a long wind-wave tank of the Canada Centre for Inland Waters. It was found that the low-amplitude transients behaved as predicted by the linear theory and that the cn and dn wave trains of moderate steepness behaved almost as predicted by the nonlinear Schroedinger equation. Some of the results did not fit into any of the available theories for waves on water, but they provide important insight on how actual groups of waves propagate and on higher-order effects for a transient waveform.

Pierson, W. J., Jr.↗
NASA NTRS1969

Wave propagation in a random medium

A simple technique is used to derive statistical characterizations of the perturbations imposed upon a wave (plane, spherical or beamed) propagating through a random medium. The method is essentially physical rather than mathematical, and is probably equivalent to the Rytov method. The limitations of the method are discussed in some detail; in general they are restrictive only for optical paths longer than a few hundred meters, and for paths at the lower microwave frequencies. Situations treated include arbitrary path geometries, finite transmitting and receiving apertures, and anisotropic media. Results include, in addition to the usual statistical quantities, time-lagged functions, mixed functions involving amplitude and phase fluctuations, angle-of-arrival covariances, frequency covariances, and other higher-order quantities.

Lee, R. W.↗
NASA NTRS1975

High-order numerical solutions using cubic splines

The cubic spline collocation procedure for the numerical solution of partial differential equations was reformulated so that the accuracy of the second-derivative approximation is improved and parallels that previously obtained for lower derivative terms. The final result is a numerical procedure having overall third-order accuracy for a nonuniform mesh and overall fourth-order accuracy for a uniform mesh. Application of the technique was made to the Burger's equation, to the flow around a linear corner, to the potential flow over a circular cylinder, and to boundary layer problems. The results confirmed the higher-order accuracy of the spline method and suggest that accurate solutions for more practical flow problems can be obtained with relatively coarse nonuniform meshes.

Rubin, S. G.↗
NASA NTRS1987

A higher-order theory for geometrically nonlinear analysis of composite laminates

A third-order shear deformation theory of laminated composite plates and shells is developed, the Navier solutions are derived, and its finite element models are developed. The theory allows parabolic description of the transverse shear stresses, and therefore the shear correction factors of the usual shear deformation theory are not required in the present theory. The theory also accounts for the von Karman nonlinear strains. Closed-form solutions of the theory for rectangular cross-ply and angle-ply plates and cross-ply shells are developed. The finite element model is based on independent approximations of the displacements and bending moments (i.e., mixed finite element model), and therefore, only C sup o -approximation is required. The finite element model is used to analyze cross-ply and angle-ply laminated plates and shells for bending and natural vibration. Many of the numerical results presented here should serve as references for future investigations. Three major conclusions resulted from the research: First, for thick laminates, shear deformation theories predict deflections, stresses and vibration frequencies significantly different from those predicted by classical theories. Second, even for thin laminates, shear deformation effects are significant in dynamic and geometrically nonlinear analyses. Third, the present third-order theory is more accurate compared to the classical and firt-order theories in predicting static and dynamic response of laminated plates and shells made of high-modulus composite materials.

Reddy, J. N.↗
NASA NTRS1987

A study of some approximations in the modeling of a coupled rotor/fuselage aeromechanical system

Equations of motion for a three-bladed hingeless rotor coupled to a gimbal-supported fuselage are developed. A brief derivation of the equations is presented, including the assumptions made, the ordering scheme, and a discussion of both the source of the aerodynamic axial load term and the reason for differences due to hinge sequence. The solution of the equations is discussed, including linearization, the use of multiblade coordinates to eliminate periodic coefficients, and specialization to a root-sprung model. The treatment of the blade root offset by using an ordering scheme to neglect higher-order terms is found to affect results in pitch and roll. A small aerodynamic loading term in the blade axial direction has a significant effect at large collective pitch settings, as does the choice of a flap-lag or a lag-flap transformation sequence for blade motion. The results for a hingeless rotor also differ significantly from those for the root-sprung model when the nonrotating natural frequencies are matched.

Blelloch, P. A.↗
NASA NTRS1987

Covariance and the hierarchy of frame bundles

This is an essay on the general concept of covariance, and its connection with the structure of the nested set of higher frame bundles over a differentiable manifold. Examples of covariant geometric objects include not only linear tensor fields, densities and forms, but affinity fields, sectors and sector forms, higher order frame fields, etc., often having nonlinear transformation rules and Lie derivatives. The intrinsic, or invariant, sets of forms that arise on frame bundles satisfy the graded Cartan-Maurer structure equations of an infinite Lie algebra. Reduction of these gives invariant structure equations for Lie pseudogroups, and for G-structures of various orders. Some new results are introduced for prolongation of structure equations, and for treatment of Riemannian geometry with higher-order moving frames. The use of invariant form equations for nonlinear field physics is implicitly advocated.

Estabrook, Frank B.↗
NASA NTRS1989

Derivation of a Tappered p-Version Beam Finite Element

A tapered p-version beam finite element suitable for dynamic applications is derived. The taper in the element is represented by allowing the area moments of inertia to vary as quartic polynomials along the length of the beam, and the cross-sectional area to vary as a quadratic polynomial. The p-version finite-element characteristics are implemented through a set of polynomial shape functions. The lower-order shape functions are identical to the classical cubic and linear shape functions normally associated with a beam element. The higher-order shape functions are a hierarchical set of polynomials that are integrals of orthogonal polynomials. Explicit expressions for the mass and stiffness matrices are presented for an arbitrary value of p. The element has been verified to be numerically stable using shape functions through 22nd order.

Hinnant, Howard E.↗
NASA NTRS1990

TE scattering by a two-dimensional groove in a ground plane using higher order boundary conditions

Higher order boundary conditions involve derivatives of the fields beyond the first and were recently shown to be more effective than traditional first order conditions in modeling dielectric coatings and layers. In this report an application of a third order generalized boundary condition to scattering by a filled rectangular groove is presented. Deficiencies of such higher-order boundary conditions are addressed and a correction is proposed for the present case. As part of the process of examining and correcting the accuracy of the proposed generalized boundary conditions, an exact solution is developed and a comparison is provided with a solution based on the standard impedance boundary condition.

Barkeshli, Kasra↗