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NASA NTRS1992

Three-dimensional kinematic reconnection of plasmoids with nulls

The global nonlinear dynamics of magnetic field lines in plasmoids with a pair of nulls, where B = 0, is studied. The aim of this analysis is to describe the separatrix surfaces on which singularities can occur in ideal magnetohydrodynamics because of topological changes in the field. These separatrix surfaces should locate the boundary layers associated with 3D reconnection in the presence of resistivity or inertia. It is found that the field lines exhibit chaotic scattering with several properties in common with plasmoid models without nulls (in which one component of the magnetic field never changes sign). In particular, the singular surfaces can be fractal, implying complex current density structures down to the dissipation scale. These generic features are expected to exist in typical coronal magnetic geometries exhibiting three-dimensional reconnection and the formation of current sheets.

Lau, Yun-Tung↗

NASA NTRS1992

Confining potential in momentum space

A method is presented for the solution in momentum space of the bound state problem with a linear potential in r space. The potential is unbounded at large r leading to a singularity at small q. The singularity is integrable, when regulated by exponentially screening the r-space potential, and is removed by a subtraction technique. The limit of zero screening is taken analytically, and the numerical solution of the subtracted integral equation gives eigenvalues and wave functions in good agreement with position space calculations.

Norbury, John W.↗

NASA NTRS1993

Asteroidal collision probabilities

Several past calculations of collision probabilities between pairs of bodies on independent orbits have yielded inconsistent results. We review the methodologies and identify their various problems. Greenberg's (1982) collision probability formalism (now with a corrected symmetry assumption) is equivalent to Wetherill's (1967) approach, except that it includes a way to avoid singularities near apsides. That method shows that the procedure by Namiki and Binzel (1991) was accurate for those cases where singularities did not arise.

Bottke, William F., Jr.↗

NASA NTRS1993

Matched asymptotic expansion of the Hamilton-Jacobi-Bellman equation for aeroassisted plane-change maneuvers

In this paper we develop a general procedure for constructing a matched asymptotic expansion of the Hamilton-Jacobi-Bellman equation based on the method of characteristics. The development is for a class of perturbation problems whose solution exhibits two-time-scale behavior. A regular expansion for problems of this type is inappropriate since it is not uniformly valid over a narrow range of the independent variable. Of particular interest here is the manner in which matching and boundary conditions are enforced when the expansion is carried out to first order. Two cases are distinguished - one where the left boundary condition coincides with or lies to the right of the singular region and one where the left boundary condition lies to the left of the singular region. A simple example is used to illustrate the procedure, and its potential application to aeroassisted plane change is described.

Calise, Anthony J.↗

NASA NTRS1993

A comparative study of multivariable robustness analysis methods as applied to integrated flight and propulsion control

Three multivariable robustness analysis methods are compared and contrasted. The focus of the analysis is on system stability and performance robustness to uncertainty in the coupling dynamics between two interacting subsystems. Of particular interest is interacting airframe and engine subsystems, and an example airframe/engine vehicle configuration is utilized in the demonstration of these approaches. The singular value (SV) and structured singular value (SSV) analysis methods are compared to a method especially well suited for analysis of robustness to uncertainties in subsystem interactions. This approach is referred to here as the interacting subsystem (IS) analysis method. This method has been used previously to analyze airframe/engine systems, emphasizing the study of stability robustness. However, performance robustness is also investigated here, and a new measure of allowable uncertainty for acceptable performance robustness is introduced. The IS methodology does not require plant uncertainty models to measure the robustness of the system, and is shown to yield valuable information regarding the effects of subsystem interactions. In contrast, the SV and SSV methods allow for the evaluation of the robustness of the system to particular models of uncertainty, and do not directly indicate how the airframe (engine) subsystem interacts with the engine (airframe) subsystem.

Schierman, John D.↗

NASA NTRS1993

Statistics of arcs in clusters of galaxies

Samples of gravitational lens events in clusters show many large arcs compared to arclets, relative to what can be obtained by idealized singular lens models. We describe the probability of image magnification for point sources and for simple but more realistic gravitational lensing models that include a finite core size and an ellipticity. In addition, we explore the changes in the probability distribution of image magnifications, distortions, and angular extents for sources of different sizes as the parameters of the lenses are varied. A finite core in spherically symmetric lens models introduces a discontinuity in the probability distribution at which the relative number of highly magnified images is increased. In elliptical lenses, this discontinuity and its effect are replaced by a continuous increase in the probability of obtaining high-magnification images relative to singular spherically symmetric models. We also find that the finite size of the source causes a further increase in the expected number of images just below the maximum possible magnification.

Bergmann, Anton G.↗

NASA NTRS1993

Symmetric Galerkin boundary formulations employing curved elements

Accounts of the symmetric Galerkin approach to boundary element analysis (BEA) have recently been published. This paper attempts to add to the understanding of this method by addressing a series of fundamental issues associated with its potential computational efficiency. A new symmetric Galerkin theoretical formulation for both the (harmonic) heat conduction and the (biharmonic) elasticity problem that employs regularized singular and hypersingular boundary integral equations (BIEs) is presented. The novel use of regularized BIEs in the Galerkin context is shown to allow straightforward incorporation of curved, isoparametric elements. A symmetric reusable intrinsic sample point (RISP) numerical integration algorithm is shown to produce a Galerkin (i.e., double) integration strategy that is competitive with its counterpart (i.e., singular) integration procedure in the collocation BEA approach when the time saved in the symmetric equation solution phase is also taken into account. This new formulation is shown to be capable of employing hypersingular BIEs while obviating the requirement of C 1 continuity, a fact that allows the employment of the popular continuous element technology. The behavior of the symmetric Galerkin BEA method with regard to both direct and iterative equation solution operations is also addressed. A series of example problems are presented to quantify the performance of this symmetric approach, relative to the more conventional unsymmetric BEA, in terms of both accuracy and efficiency. It is concluded that appropriate implementations of the symmetric Galerkin approach to BEA indeed have the potential to be competitive with, if not superior to, collocation-based BEA, for large-scale problems.

Kane, J. H.↗

NASA NTRS1988

Dynamic structural correlation via nonlinear programming techniques

A solution to the correlation between structural dynamic test results and finite element analyses of the same components is presented in this paper. Basically, the method can be categorized as a Levenberg-Marquardt type Gauss-Newton method which requires only the differences between FE modal analyses and test results and their first derivatives with respect to preassigned design variables. With proper variable normalization and equation scaling, the method has been made numerically better-conditioned and the inclusion of the Levenberg-Marquardt technique overcomes any remaining difficulty encountered in inverting singular or near-singular matrices. An important feature is that each iteration requires only one function evaluation along with the associated design sensitivity analysis and so the procedure is computationally efficient.

Ting, T.↗

NASA NTRS1951

Integrals and integral equations in linearized wing theory

The formulas of subsonic and supersonic wing theory for source, doublet, and vortex distributions are reviewed and a systematic presentation is provided which relates these distributions to the pressure and to the vertical induced velocity in the plane of the wing. It is shown that care must be used in treating the singularities involved in the analysis and that the order of integration is not always reversible. Concepts suggested by the irreversibility of order of integration are shown to be useful in the inversion of singular integral equations when operational techniques are used. A number of examples are given to illustrate the methods presented, attention being directed to supersonic flight speed.

Lomax, Harvard↗

NASA NTRS1956

A special method for finding body distortions that reduce the wave drag of wing and body combinations at supersonic speeds

For a given wing and supersonic Mach number, the problem of shaping an adjoining fuselage so that the combination will have a low wave drag is considered. Only fuselages that can be simulated by singularities (multipoles) distributed along the body axis are studied. However, the optimum variations of such singularities are completely specified in terms of the given wing geometry. An application is made to an elliptic wing having a biconvex section, a thickness-chord ratio equal to 0.05 at the root, and an aspect ratio equal to 3. A comparison of the theoretical results with a wind-tunnel experiment is also presented.

Lomax, Harvard↗

NASA NTRS1993

A wall interference assessment/correction system

A Wall Signature method originally developed by Hackett was selected to be adapted for the Ames 12-ft Wind Tunnel Wall Interference Assessment/Correction (WIAC) System in the project. This method uses limited measurements of the static pressure at the wall, in conjunction with the solid wall boundary condition, to determine the strength and distribution of singularities representing the test article. The singularities are used in turn for estimating wall interference at the model location. The lifting interference will be treated separately by representing in a horseshoe vortex system for the model's lifting effects. The development and implementation of a working prototype will be completed, delivered, and documented with a software manual. The WIAC code will be validated by conducting numerically simulated experiments rather than actual wind tunnel experiments. The simulations will be used to generate both free-air and confined wind-tunnel flow fields for each of the test articles over a range of test configurations. Specifically, the pressure signature at the test section wall will be computed for the tunnel case to provide the simulated 'measured' data. These data will serve as the input for the WIAC method - Wall Signature method. The performance of the WIAC method then may be evaluated by comparing the corrected parameters with those for the free-air simulation. The following two additional tasks are included in the supplement No. 1 to the basic Grant. On-line wall interference calculation: The developed wall signature method (modified Hackett's method) for Ames 12-ft Tunnel will be the pre-computed coefficients which facilitate the on-line calculation of wall interference; and support system effects estimation: The effects on the wall pressure measurements due to the presence of the model support systems will be evaluated.

Lo, Ching F.↗

NASA NTRS1993

Guidance law development for aeroassisted transfer vehicles using matched asymptotic expansions

This report addresses and clarifies a number of issues related to the Matched Asymptotic Expansion (MAE) analysis of skip trajectories, or any class of problems that give rise to inner layers that are not associated directly with satisfying boundary conditions. The procedure for matching inner and outer solutions, and using the composite solution to satisfy boundary conditions is developed and rigorously followed to obtain a set of algebraic equations for the problem of inclination change with minimum energy loss. A detailed evaluation of the zeroth order guidance algorithm for aeroassisted orbit transfer is performed. It is shown that by exploiting the structure of the MAE solution procedure, the original problem, which requires the solution of a set of 20 implicit algebraic equations, can be reduced to a problem of 6 implicit equations in 6 unknowns. A solution that is near optimal, requires a minimum of computation, and thus can be implemented in real time and on-board the vehicle, has been obtained. Guidance law implementation entails treating the current state as a new initial state and repetitively solving the zeroth order MAE problem to obtain the feedback controls. Finally, a general procedure is developed for constructing a MAE solution up to first order, of the Hamilton-Jacobi-Bellman equation based on the method of characteristics. The development is valid for a class of perturbation problems whose solution exhibits two-time-scale behavior. A regular expansion for problems of this type is shown to be inappropriate since it is not valid over a narrow range of the independent variable. That is, it is not uniformly valid. Of particular interest here is the manner in which matching and boundary conditions are enforced when the expansion is carried out to first order. Two cases are distinguished-one where the left boundary condition coincides with, or lies to the right of, the singular region, and another one where the left boundary condition lies to the left of the singular region. A simple example is used to illustrate the procedure where the obtained solution is uniformly valid to O(Epsilon(exp 2)). The potential application of this procedure to aeroassisted plane change is also described and partially evaluated.

Calise, Anthony J.↗

NASA NTRS1993

An object-oriented approach for parallel self adaptive mesh refinement on block structured grids

Self-adaptive mesh refinement dynamically matches the computational demands of a solver for partial differential equations to the activity in the application's domain. In this paper we present two C++ class libraries, P++ and AMR++, which significantly simplify the development of sophisticated adaptive mesh refinement codes on (massively) parallel distributed memory architectures. The development is based on our previous research in this area. The C++ class libraries provide abstractions to separate the issues of developing parallel adaptive mesh refinement applications into those of parallelism, abstracted by P++, and adaptive mesh refinement, abstracted by AMR++. P++ is a parallel array class library to permit efficient development of architecture independent codes for structured grid applications, and AMR++ provides support for self-adaptive mesh refinement on block-structured grids of rectangular non-overlapping blocks. Using these libraries, the application programmers' work is greatly simplified to primarily specifying the serial single grid application and obtaining the parallel and self-adaptive mesh refinement code with minimal effort. Initial results for simple singular perturbation problems solved by self-adaptive multilevel techniques (FAC, AFAC), being implemented on the basis of prototypes of the P++/AMR++ environment, are presented. Singular perturbation problems frequently arise in large applications, e.g. in the area of computational fluid dynamics. They usually have solutions with layers which require adaptive mesh refinement and fast basic solvers in order to be resolved efficiently.

Lemke, Max↗

NASA NTRS1993

Evaluation of advanced materials through experimental mechanics and modelling

Composite materials have been frequently used in aerospace vehicles. Very often defects are inherited during the manufacture and damages are inherited during the construction and services. It becomes critical to understand the mechanical behavior of such composite structure before it can be further used. One good example of these composite structures is the cylindrical bottle of solid rocket motor case with accidental impact damages. Since the replacement of this cylindrical bottle is expensive, it is valuable to know how the damages affects the material, and how it can be repaired. To reach this goal, the damage must be characterized and the stress/strain field must be carefully analyzed. First the damage area, due to impact, is surveyed and identified with a shearography technique which uses the principle of speckle shearing interferometry to measure displacement gradient. Within the damage area of a composite laminate, such as the bottle of solid rocket motor case, all layers are considered to be degraded. Once a lamina being degraded the stiffness as well as strength will be drastically decreased. It becomes a critical area of failure to the whole bottle. And hence the stress/strain field within and around a damage should be accurately evaluated for failure prediction. To investigate the stress/strain field around damages a Hybrid-Numerical method which combines experimental measurement and finite element analysis is used. It is known the stress or strain at the singular point can not be accurately measured by an experimental technique. Nevertheless, if the location is far away from the singular spot, the displacement can be found accurately. Since it reflects the true displacement field locally regardless of the boundary conditions, it is an excellent input data for a finite element analysis to replace the usually assumed boundary conditions. Therefore, the Hybrid-Numerical method is chosen to avoid the difficulty and to take advantage of both experimental technique and finite element analysis. Experimentally, the digital image correlation technique is employed to measure the displacement field. It is done by comparing two digitized images, before and after loading. Numerically, the finite element program, ABAQUS (version 5.2), is used to analyze the stress and strain field. It takes advantage of the high speed and huge memory size of modern supercomputer, CRAY Y-MP, at NASA Marshall Space Flight Center.

Yang, Yii-Ching↗

NASA NTRS1994

A fast numerical solution of scattering by a cylinder: Spectral method for the boundary integral equations

It is known that the exact analytic solutions of wave scattering by a circular cylinder, when they exist, are not in a closed form but in infinite series which converges slowly for high frequency waves. In this paper, we present a fast number solution for the scattering problem in which the boundary integral equations, reformulated from the Helmholtz equation, are solved using a Fourier spectral method. It is shown that the special geometry considered here allows the implementation of the spectral method to be simple and very efficient. The present method differs from previous approaches in that the singularities of the integral kernels are removed and dealt with accurately. The proposed method preserves the spectral accuracy and is shown to have an exponential rate of convergence. Aspects of efficient implementation using FFT are discussed. Moreover, the boundary integral equations of combined single and double-layer representation are used in the present paper. This ensures the uniqueness of the numerical solution for the scattering problem at all frequencies. Although a strongly singular kernel is encountered for the Neumann boundary conditions, we show that the hypersingularity can be handled easily in the spectral method. Numerical examples that demonstrate the validity of the method are also presented.

Hu, Fang Q.↗

NASA NTRS1994

A Galerkin method for linear PDE systems in circular geometries with structural acoustic applications

A Galerkin method for systems of PDE's in circular geometries is presented with motivating problems being drawn from structural, acoustic, and structural acoustic applications. Depending upon the application under consideration, piecewise splines or Legendre polynomials are used when approximating the system dynamics with modifications included to incorporate the analytic solution decay near the coordinate singularity. This provides an efficient method which retains its accuracy throughout the circular domain without degradation at singularity. Because the problems under consideration are linear or weakly nonlinear with constant or piecewise constant coefficients, transform methods for the problems are not investigated. While the specific method is developed for the two dimensional wave equations on a circular domain and the equation of transverse motion for a thin circular plate, examples demonstrating the extension of the techniques to a fully coupled structural acoustic system are used to illustrate the flexibility of the method when approximating the dynamics of more complex systems.

Smith, Ralph C.↗

NASA NTRS1993

Noniterative computation of infimum in H(infinity) optimisation for plants with invariant zeros on the j(omega)-axis

A simple and noniterative procedure for the computation of the exact value of the infimum in the singular H(infinity)-optimization problem is presented, as a continuation of our earlier work. Our problem formulation is general and we do not place any restrictions in the finite and infinite zero structures of the system, and the direct feedthrough terms between the control input and the controlled output variables and between the disturbance input and the measurement output variables. Our method is applicable to a class of singular H(infinity)-optimization problems for which the transfer functions from the control input to the controlled output and from the disturbance input to the measurement output satisfy certain geometric conditions. In particular, the paper extends the result of earlier work by allowing these two transfer functions to have invariant zeros on the j(omega) axis.

Chen, B. M.↗

NASA NTRS1994

A wall interference assessment and correction system

A wall signature method has been selected to be adapted for the Ames 12-ft Wind Tunnel WIAC system. This uses limited measurements of the static pressure at the wall, in conjunction with the solid wall boundary condition, to determine the strength and distribution of singularities representing the test article. The singularities are used to estimate wall interference at the model location. The development and implementation of a working prototype will be completed, delivered and documented with a software manual.

Lo, C. F.↗