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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 NTRS1974

Solution of an eigenvalue problem for the Laplace operator on a spherical surface

Methods for obtaining approximate solutions for the fundamental eigenvalue of the Laplace-Beltrami operator (also referred to as the membrane eigenvalue problem for the vibration equation) on the unit spherical surface are developed. Two specific types of spherical surface domains are considered: (1) the interior of a spherical triangle, i.e., the region bounded by arcs of three great circles, and (2) the exterior of a great circle arc extending for less than pi radians on the sphere (a spherical surface with a slit). In both cases, zero boundary conditions are imposed. In order to solve the resulting second-order elliptic partial differential equations in two independent variables, a finite difference approximation is derived. The symmetric (generally five-point) finite difference equations that develop are written in matrix form and then solved by the iterative method of point successive overrelaxation. Upon convergence of this iterative method, the fundamental eigenvalue is approximated by iteration utilizing the power method as applied to the finite Rayleigh quotient.

Walden, H.↗
NASA NTRS1977

Numerical determination of the fundamental eigenvalue for the Laplace operator on a spherical domain

Methods for obtaining approximate solutions for the fundamental eigenvalue of the Laplace-Beltrami operator (i.e., the membrane eignevalue problem for the vibration equation) on the unit spherical surface are developed. Two types of spherical surface domains are considered: the interior of a spherical triangle, and the exterior of a great circle arc extending for less than pi radians (a spherical surface with a slit). In both cases, zero boundary conditions are imposed. In order to solve the resulting second-order elliptic partial differential equations in two independent variables, a finite difference approximation is employed. The fundamental eigenvalue is approximated by iteration utilizing the power method and point successive overrelaxation. Some numerical results are given and compared, in certain special cases, with analytical solutions to the eigenvalue problem. The significance of the numerical eigenvalue results is discussed in terms of the singularities in the solution of three-dimensional boundary-value problems near a polyhedral corner of the domain.

Walden, H.↗
NASA NTRS1985

Finite difference operators on unstructured triangular meshes

A new form of the Laplace operator is derived which is shown to be second-order accurate when calculated at mesh triangle nodes if the cells corresponding to the nodes have a high degree of symmetry. It is demonstrated that the Laplace operator directly derived from Stokes' integral theorem is only zeroth-order accurate on these same cells. The construction of gradient and Laplace operators on unstructured triangular grids are discussed from the variational point of view. Two discrete representations of the Laplacian are compared to each other, and conclusions are drawn regarding their accuracy on a pointwise basis. Finally, geometric relations valid on arbitrary polygons are given for completeness in an appendix.

Erlebacher, G.↗
NASA NTRS1967

Polynomial manipulator AP-168

Linear Systems Design Evaluation Program, AP-168 combines the many different analysis techniques used to evaluate and manipulate polynomials. The single program is a pseudo instruction abstraction. It allows the user to enter polynomials of the laplace operators and to manipulate them freely.

Tutt, G. E.↗
NASA NTRS1994

Large-angle cosmic microwave background anisotropies in an open universe

If the universe is open, scales larger than the curvature scale may be probed by observation of large-angle fluctuations in the cosmic microwave background (CMB). We consider primordial adiabatic perturbations and discuss power spectra that are power laws in volume, wavelength, and eigenvalue of the Laplace operator. Such spectra may have arisen if, for example, the universe underwent a period of `frustated' inflation. The resulting large-angle anisotropies of the CMB are computed. The amplitude generally increases as Omega is decreased but decreases as h is increased. Interestingly enough, for all three Ansaetze, anisotropies on angular scales larger than the curvature scale are suppressed relative to the anisotropies on scales smaller than the curvature scale, but cosmic variance makes discrimination between various models difficult. Models with 0.2 approximately less than Omega h approximately less than 0.3 appear compatible with CMB fluctuations detected by Cosmic Background Explorer Satellite (COBE) and the Tenerife experiment and with the amplitude and spectrum of fluctuations of galaxy counts in the APM, CfA, and 1.2 Jy IRAS surveys. COBE normalization for these models yields sigma(sub 8) approximately = 0.5 - 0.7. Models with smaller values of Omega h when normalized to COBE require bias factors in excess of 2 to be compatible with the observed galaxy counts on the 8/h Mpc scale. Requiring that the age of the universe exceed 10 Gyr implies that Omega approximately greater than 0.25, while requiring that from the last-scattering term in the Sachs-Wolfe formula, large-angle anisotropies come primarily from the decay of potential fluctuations at z approximately less than 1/Omega. Thus, if the universe is open, COBE has been detecting temperature fluctuations produced at moderate redshift rather than at z approximately 1300.

Kamionkowski, Marc↗
NASA NTRS1971

Expansion of the planetary disturbing function.

Some methods are described for the expansion of the disturbing function in planetary theory. One method uses the classical binomial expansion theorem or a successive approximation process derived from it. Another method is a direct application of the Laplace series expansions. For both methods it is proposed to first prepare the series to be manipulated by a scaling operation. These methods can be applied either in a literal or in a numerical form, or any combination of both, but they are especially designed for use on a large scale digital computer with standard Poisson series programs. No usage is made of Newcomb operators or derivatives of Laplace coefficients.

Broucke, R.↗
NASA NTRS1996

Dynamics and Control of Articulated Anisotropic Timoshenko Beams

The paper illustrates the use of continuum models in control design for stabilizing flexible structures. A 6-DOF anisotropic Timoshenko beam with discrete nodes where lumped masses or actuators are located provides a sufficiently rich model to be of interest for mathematical theory as well as practical application. We develop concepts and tools to help answer engineering questions without having to resort to ad hoc heuristic ("physical") arguments or faith. In this sense the paper is more mathematically oriented than engineering papers and vice versa at the same time. For instance we make precise time-domain solutions using the theory of semigroups of operators rather than formal "inverse Laplace transforms." We show that the modes arise as eigenvalues of the generator of the semigroup, which are then related to the eigenvalues of the stiffness operator. With the feedback control, the modes are no longer orthogonal and the question naturally arises as to whether there is still a modal expansion. Here we prove that the eigenfunctions yield a biorthogonal Riesz basis and indicate the corresponding expansion. We prove mathematically that the number of eigenvalues is nonfinite, based on the theory of zeros of entire functions. We make precise the notion of asymptotic modes and indicate how to calculate them. Although limited by space, we do consider the root locus problem and show for instance that the damping at first increases as the control gain increases but starts to decrease at a critical value, and goes to zero as the gain increases without bound. The undamped oscillatory modes remain oscillatory and the rigid-body modes go over into deadbeat modes. The Timoshenko model dynamics are translated into a canonical wave equation in a Hilbert space. The solution is shown to require the use of an "energy" norm which is no more than the total energy: potential plus kinetic. We show that, under an appropriate extension of the notion of controllability, rate feedback with a collocated sensor can stabilize the structure in the sense that all modes are damped and the energy decays to zero. An example, non-numeric, is worked out in some detail illustrating the concepts and theory developed.

Balakrishnan, A. V.↗
NASA NTRS1984

Unsteady aerodynamics in time and frequency domains for finite time arbitrary motion of rotary wings in hover and forward flight

Several incompressible finite-time arbitrary-motion airfoil theories suitable for coupled flap-lag-torsional aeroelastic analysis of helicopter rotors in hover and forward flight are derived. These theories include generalized Greenberg's theory, generalized Loewy's theory, and a staggered cascade theory. The generalized Greenberg's and staggered cascade theories were derived directly in Laplace domain considering the finite length of the wake and using operational methods. The load expressions are presented in Laplace, frequency, and time domains. Approximate time domain loads for the various generalized theories, discussed in the paper, are obtained by developing finite state models using the Pade approximant of the appropriate lift deficiency functions. Three different methods for constructing Pade approximants of the lift deficiency functions were considered and the more flexible one was used. Pade approximants of Loewy's lift deficiency function, for various wake spacing and radial location parameters of a helicopter typical rotor blade section, are presented.

Dinyavari, M. A. H.↗
NASA NTRS2002

Variable Order and Distributed Order Fractional Operators

Many physical processes appear to exhibit fractional order behavior that may vary with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. This paper develops the concept of variable and distributed order fractional operators. Definitions based on the Riemann-Liouville definitions are introduced and behavior of the operators is studied. Several time domain definitions that assign different arguments to the order q in the Riemann-Liouville definition are introduced. For each of these definitions various characteristics are determined. These include: time invariance of the operator, operator initialization, physical realization, linearity, operational transforms. and memory characteristics of the defining kernels. A measure (m2) for memory retentiveness of the order history is introduced. A generalized linear argument for the order q allows the concept of "tailored" variable order fractional operators whose a, memory may be chosen for a particular application. Memory retentiveness (m2) and order dynamic behavior are investigated and applications are shown. The concept of distributed order operators where the order of the time based operator depends on an additional independent (spatial) variable is also forwarded. Several definitions and their Laplace transforms are developed, analysis methods with these operators are demonstrated, and examples shown. Finally operators of multivariable and distributed order are defined in their various applications are outlined.

Lorenzo, Carl F.↗
NASA NTRS1974

SUSSA ACTS: A computer program for steady and unsteady, subsonic and supersonic aerodynamics for aerospace complex transportation systems

The computer program SUSSA ACTS (Steady and Unsteady, Subsonic and Supersonic Aerodynamics for Complex Transportation Systems) are presented in the final version. The numerical formulation and the description of the program and numerical results are included. In particular, generalized forces for fully unsteady (complex frequency) aerodynamics for a wing-body configuration, in both subsonic and supersonic flows, are discussed. The mathematical analysis includes completely arbitrary motion. The numerical implementation was limited to steady and oscillatory flows. A more general aerodynamic formulation in the form of a fully transient response for time-domain analysis and the aerodynamic transfer function (Laplace transform of the fully unsteady operator) for frequency-domain analysis is outlined.

Tseng, K.↗
NASA NTRS1975

Fully unsteady subsonic and supersonic potential aerodynamics for complex aircraft configurations with applications to flutter

A general formulation is presented for the analysis of steady and unsteady, subsonic and supersonic aerodynamics for complex aircraft configurations. The theoretical formulation, the numerical procedure, the description of the program SOUSSA (steady, oscillatory and unsteady, subsonic and supersonic aerodynamics) and numerical results are included. In particular, generalized forces for fully unsteady (complex frequency) aerodynamics for a wing-body configuration, AGARD wing-tail interference in both subsonic and supersonic flows as well as flutter analysis results are included. The theoretical formulation is based upon an integral equation, which includes completely arbitrary motion. Steady and oscillatory aerodynamic flows are considered. Here small-amplitude, fully transient response in the time domain is considered. This yields the aerodynamic transfer function (Laplace transform of the fully unsteady operator) for frequency domain analysis. This is particularly convenient for the linear systems analysis of the whole aircraft.

Tseng, K.↗
NASA NTRS1974

An extension of the Laplace transform to Schwartz distributions

A characterization of the Laplace transform is developed which extends the transform to the Schwartz distributions. The class of distributions includes the impulse functions and other singular functions which occur as solutions to ordinary and partial differential equations. The standard theorems on analyticity, uniqueness, and invertibility of the transform are proved by using the characterization as the definition of the Laplace transform. The definition uses sequences of linear transformations on the space of distributions which extends the Laplace transform to another class of generalized functions, the Mikusinski operators. It is shown that the sequential definition of the transform is equivalent to Schwartz' extension of the ordinary Laplace transform to distributions but, in contrast to Schwartz' definition, does not use the distributional Fourier transform. Several theorems concerning the particular linear transformations used to define the Laplace transforms are proved. All the results proved in one dimension are extended to the n-dimensional case, but proofs are presented only for those situations that require methods different from their one-dimensional analogs.

Price, D. R.↗
NASA NTRS1993

Multivariable State-Space Identification in the Delta and Shift Operators

This paper develops algorithms for multivariable state-space identification which can be used to estimate models in any operator of interest i.e., delta-rule, shift, Laplace s, etc. The approach is based on the State-Space from Frequency Data (SSFD) algorithm which was designed specifically to eliminate distortions from windowing effects.

multivariable state-space identification SSFD wind↗
NASA NTRS1982

Assessing the quality of curvilinear coordinate meshes by decomposing the Jacobian matrix

An algebraic decomposition of the Jacobian matrix which relates physical and computational variables is presented. This invertible decomposition parameterizes the mesh by the physically intuitive qualities of cell orientation, cell orthogonality, cell volume, and cell aspect ratio. The decomposition can be used to analyze numerically generated curvilinear coordinate meshes and to assess the contribution of the mesh to the truncation error for any specific differential operator and algorithm. This is worked out in detail for Laplace's equation in nonconservative and conservative forms. The analysis is applied to the solution of the full potential code TAIR, showing grid plots, carpet plots, and truncation error for a NACA 0012 airfoil.

Kerlick, G. D.↗
NASA NTRS1994

Agglomeration multigrid for viscous turbulent flows

Agglomeration multigrid, which has been demonstrated as an efficient and automatic technique for the solution of the Euler equations on unstructured meshes, is extended to viscous turbulent flows. For diffusion terms, coarse grid discretizations are not possible, and more accurate grid transfer operators are required as well. A Galerkin coarse grid operator construction and an implicit prolongation operator are proposed. Their suitability is evaluated by examining their effect on the solution of Laplace's equation. The resulting strategy is employed to solve the Reynolds-averaged Navier-Stokes equations for aerodynamic flows. Convergence rates comparable to those obtained by a previously developed non-nested mesh multigrid approach are demonstrated, and suggestions for further improvements are given.

Mavriplis, D. J.↗