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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 NTRS1994

ASTM E 1559 method for measuring material outgassing/deposition kinetics has applications to aerospace, electronics, and semiconductor industries

The American Society for Testing and Materials has published a new standard test method for characterizing time and temperature-dependence of material outgassing kinetics and the deposition kinetics of outgassed species on surfaces at various temperatures. This new ASTM standard, E 1559(1), uses the quartz crystal microbalance (QCM) collection measurement approach. The test method was originally developed under a program sponsored by the United States Air Force Materials Laboratory (AFML) to create a standard test method for obtaining outgassing and deposition kinetics data for spacecraft materials. Standardization by ASTM recognizes that the method has applications beyond aerospace. In particular, the method will provide data of use to the electronics, semiconductor, and high vacuum industries. In ASTM E 1559 the material sample is held in vacuum in a temperature-controlled effusion cell, while its outgassing flux impinges on several QCM's which view the orifice of the effusion cell. Sample isothermal total mass loss (TML) is measured as a function of time from the mass collected on one of the QCM's which is cooled by liquid nitrogen, and the view factor from this QCM to the cell. The amount of outgassed volatile condensable material (VCM) on surfaces at higher temperatures is measured as a function of time during the isothermal outgassing test by controlling the temperatures of the remaining QCM's to selected values. The VCM on surfaces at temperatures in between those of the collector QCM's is determined at the end of the isothermal test by heating the QCM's at a controlled rate and measuring the mass loss from the end of the QCM's as a function of time and temperature. This reevaporation of the deposit collected on the QCM's is referred to as QCM thermogravimetric analysis. Isothermal outgassing and deposition rates can be determined by differentiating the isothermal TML and VCM data, respectively, while the evaporation rates of the species can be obtained as a function of temperature by differentiating the QCM thermogravimetric analysis data.

Garrett, J. W.↗
NASA NTRS1994

Flexible Method for Inter-object Communication in C++

A method has been developed for organizing and sharing large amounts of information between objects in C++ code. This method uses a set of object classes to define variables and group them into tables. The variable tables presented here provide a convenient way of defining and cataloging data, as well as a user-friendly input/output system, a standardized set of access functions, mechanisms for ensuring data integrity, methods for interprocessor data transfer, and an interpretive language for programming relationships between parameters. The object-oriented nature of these variable tables enables the use of multiple data types, each with unique attributes and behavior. Because each variable provides its own access methods, redundant table lookup functions can be bypassed, thus decreasing access times while maintaining data integrity. In addition, a method for automatic reference counting was developed to manage memory safely.

Curlett, Brian P.↗
NASA NTRS1995

The space-time solution element method: A new numerical approach for the Navier-Stokes equations

This paper is one of a series of papers describing the development of a new numerical method for the Navier-Stokes equations. Unlike conventional numerical methods, the current method concentrates on the discrete simulation of both the integral and differential forms of the Navier-Stokes equations. Conservation of mass, momentum, and energy in space-time is explicitly provided for through a rigorous enforcement of both the integral and differential forms of the governing conservation laws. Using local polynomial expansions to represent the discrete primitive variables on each cell, fluxes at cell interfaces are evaluated and balanced using exact functional expressions. No interpolation or flux limiters are required. Because of the generality of the current method, it applies equally to the steady and unsteady Navier-Stokes equations. In this paper, we generalize and extend the authors' 2-D, steady state implicit scheme. A general closure methodology is presented so that all terms up through a given order in the local expansions may be retained. The scheme is also extended to nonorthogonal Cartesian grids. Numerous flow fields are computed and results are compared with known solutions. The high accuracy of the scheme is demonstrated through its ability to accurately resolve developing boundary layers on coarse grids. Finally, we discuss applications of the current method to the unsteady Navier-Stokes equations.

Scott, James R.↗
NASA NTRS1993

Structured adaptive grid generation using algebraic methods

The accuracy of the numerical algorithm depends not only on the formal order of approximation but also on the distribution of grid points in the computational domain. Grid adaptation is a procedure which allows optimal grid redistribution as the solution progresses. It offers the prospect of accurate flow field simulations without the use of an excessively timely, computationally expensive, grid. Grid adaptive schemes are divided into two basic categories: differential and algebraic. The differential method is based on a variational approach where a function which contains a measure of grid smoothness, orthogonality and volume variation is minimized by using a variational principle. This approach provided a solid mathematical basis for the adaptive method, but the Euler-Lagrange equations must be solved in addition to the original governing equations. On the other hand, the algebraic method requires much less computational effort, but the grid may not be smooth. The algebraic techniques are based on devising an algorithm where the grid movement is governed by estimates of the local error in the numerical solution. This is achieved by requiring the points in the large error regions to attract other points and points in the low error region to repel other points. The development of a fast, efficient, and robust algebraic adaptive algorithm for structured flow simulation applications is presented. This development is accomplished in a three step process. The first step is to define an adaptive weighting mesh (distribution mesh) on the basis of the equidistribution law applied to the flow field solution. The second, and probably the most crucial step, is to redistribute grid points in the computational domain according to the aforementioned weighting mesh. The third and the last step is to reevaluate the flow property by an appropriate search/interpolate scheme at the new grid locations. The adaptive weighting mesh provides the information on the desired concentration of points to the grid redistribution scheme. The evaluation of the weighting mesh is accomplished by utilizing the weight function representing the solution variation and the equidistribution law. The selection of the weight function plays a key role in grid adaptation. A new weight function utilizing a properly weighted boolean sum of various flowfield characteristics is defined. The redistribution scheme is developed utilizing Non-Uniform Rational B-Splines (NURBS) representation. The application of NURBS representation results in a well distributed smooth grid by maintaining the fidelity of the geometry associated with boundary curves. Several algebraic methods are applied to smooth and/or nearly orthogonalize the grid lines. An elliptic solver is utilized to smooth the grid lines if there are grid crossings. Various computational examples of practical interest are presented to demonstrate the success of these methods.

Yang, Jiann-Cherng↗
NASA NTRS1994

Fourth order difference methods for hyperbolic IBVP's

Fourth order difference approximations of initial-boundary value problems for hyperbolic partial differential equations are considered. We use the method of lines approach with both explicit and compact implicit difference operators in space. The explicit operator satisfies an energy estimate leading to strict stability. For the implicit operator we develop boundary conditions and give a complete proof of strong stability using the Laplace transform technique. We also present numerical experiments for the linear advection equation and Burgers' equation with discontinuities in the solution or in its derivative. The first equation is used for modeling contact discontinuities in fluid dynamics, the second one for modeling shocks and rarefaction waves. The time discretization is done with a third order Runge-Kutta TVD method. For solutions with discontinuities in the solution itself we add a filter based on second order viscosity. In case of the non-linear Burger's equation we use a flux splitting technique that results in an energy estimate for certain different approximations, in which case also an entropy condition is fulfilled. In particular we shall demonstrate that the unsplit conservative form produces a non-physical shock instead of the physically correct rarefaction wave. In the numerical experiments we compare our fourth order methods with a standard second order one and with a third order TVD-method. The results show that the fourth order methods are the only ones that give good results for all the considered test problems.

Gustafsson, Bertil↗
NASA NTRS1995

A well-posed numerical method to track isolated conformal map singularities in Hele-Shaw flow

We present a new numerical method for calculating an evolving 2D Hele-Shaw interface when surface tension effects are neglected. In the case where the flow is directed from the less viscous fluid into the more viscous fluid, the motion of the interface is ill-posed; small deviations in the initial condition will produce significant changes in the ensuing motion. This situation is disastrous for numerical computation, as small round-off errors can quickly lead to large inaccuracies in the computed solution. Our method of computation is most easily formulated using a conformal map from the fluid domain into a unit disk. The method relies on analytically continuing the initial data and equations of motion into the region exterior to the disk, where the evolution problem becomes well-posed. The equations are then numerically solved in the extended domain. The presence of singularities in the conformal map outside of the disk introduces specific structures along the fluid interface. Our method can explicitly track the location of isolated pole and branch point singularities, allowing us to draw connections between the development of interfacial patterns and the motion of singularities as they approach the unit disk. In particular, we are able to relate physical features such as finger shape, side-branch formation, and competition between fingers to the nature and location of the singularities. The usefulness of this method in studying the formation of topological singularities (self-intersections of the interface) is also pointed out.

Baker, Gregory↗
NASA NTRS1995

A simple method for verifying the deployment of the TOMS-EP solar arrays

The Total Ozone Mapping Spectrometer-Earth Probe (TOMS-EP) mission relies upon a successful deployment of the spacecraft's solar arrays. Several methods of verification are being employed to ascertain the solar array deployment status, with each requiring differing amounts of data. This paper describes a robust attitude-independent verification method that utilizes telemetry from the coarse Sun sensors (CSS's) and the three-axis magnetometers (TAM's) to determine the solar array deployment status - and it can do so with only a few, not necessarily contiguous, points of data. The method developed assumes that the solar arrays are deployed. Telemetry data from the CSS and TAM are converted to the Sun and magnetic field vectors in spacecraft body coordinates, and the angle between them is calculated. Deployment is indicated if this angle is within a certain error tolerance of the angle between the reference Sun and magnetic field vectors. Although several other methods can indicate a non-deployed state, with this method there is a 70% confidence level in confirming deployment as well as a nearly 100% certainty in confirming a non-deployed state. In addition, the spacecraft attitude (which is not known during the first orbit after launch) is not needed for this algorithm because the angle between the Sun and magnetic field vectors is independent of the spacecraft attitude. This technique can be applied to any spacecraft with a TAM and with CSS's mounted on the solar array(s).

Koppersmith, James R.↗
NASA NTRS1993

A global/local analysis method for treating details in structural design

A method for analyzing global/local behavior of plate and shell structures is described. In this approach, a detailed finite element model of the local region is incorporated within a coarser global finite element model. The local model need not be nodally compatible (i.e., need not have a one-to-one nodal correspondence) with the global model at their common boundary; therefore, the two models may be constructed independently. The nodal incompatibility of the models is accounted for by introducing appropriate constraint conditions into the potential energy in a hybrid variational formulation. The primary advantage of this method is that the need for transition modeling between global and local models is eliminated. Eliminating transition modeling has two benefits. First, modeling efforts are reduced since tedious and complex transitioning need not be performed. Second, errors due to the mesh distortion, often unavoidable in mesh transitioning, are minimized by avoiding distorted elements beyond what is needed to represent the geometry of the component. The method is applied reduced to a plate loaded in tension and transverse bending. The plate has a central hole, and various hole sixes and shapes are studied. The method is also applied to a composite laminated fuselage panel with a crack emanating from a window in the panel. While this method is applied herein to global/local problems, it is also applicable to the coupled analysis of independently modeled components as well as adaptive refinement.

Aminpour, Mohammad A.↗
NASA NTRS1995

Solution of acoustic workshop problems by a spectral multidomain method

We use a new staggered grid Chebyshev spectral multidomain method to solve three of the Workshop benchmark problems. The method defines solution unknowns at the nodes of the Chebyshev Gauss quadrature, and the fluxes at the nodes of the Chebyshev Gauss-Lobatto quadrature. The Chebyshev spectral method gives exponentially convergent phase and dissipation errors. The multidomain approximation gives the method flexibility. Using the method, we solve problems in Categories 1 and 5 of the benchmark problems.

Kopriva, Davis A.↗
NASA NTRS1994

Experience report: Using formal methods for requirements analysis of critical spacecraft software

Formal specification and analysis of requirements continues to gain support as a method for producing more reliable software. However, the introduction of formal methods to a large software project is difficult, due in part to the unfamiliarity of the specification languages and the lack of graphics. This paper reports results of an investigation into the effectiveness of formal methods as an aid to the requirements analysis of critical, system-level fault-protection software on a spacecraft currently under development. Our experience indicates that formal specification and analysis can enhance the accuracy of the requirements and add assurance prior to design development in this domain. The work described here is part of a larger, NASA-funded research project whose purpose is to use formal-methods techniques to improve the quality of software in space applications. The demonstration project described here is part of the effort to evaluate experimentally the effectiveness of supplementing traditional engineering approaches to requirements specification with the more rigorous specification and analysis available with formal methods.

Lutz, Robyn R.↗
NASA NTRS1994

Parallel, adaptive finite element methods for conservation laws

We construct parallel finite element methods for the solution of hyperbolic conservation laws in one and two dimensions. Spatial discretization is performed by a discontinuous Galerkin finite element method using a basis of piecewise Legendre polynomials. Temporal discretization utilizes a Runge-Kutta method. Dissipative fluxes and projection limiting prevent oscillations near solution discontinuities. A posteriori estimates of spatial errors are obtained by a p-refinement technique using superconvergence at Radau points. The resulting method is of high order and may be parallelized efficiently on MIMD computers. We compare results using different limiting schemes and demonstrate parallel efficiency through computations on an NCUBE/2 hypercube. We also present results using adaptive h- and p-refinement to reduce the computational cost of the method.

Biswas, Rupak↗
NASA NTRS1994

The photometric method of extrasolar planet detection revisited

We investigate the geometry concerning the photometric method of extrasolar planet detection, i.e., the detection of dimunition of a parent star's brightness during a planetary transit. Under the assumption that planetary orbital inclinations can be defined by a Gaussian with a sigma of 10 deg centered on the parent star's equatorial plane, Monte Carlo simulations suggest that for a given star observed at an inclination of exactly 90 deg, the probability of at least one Earth-sized or larger planet being suitably placed for transits is approximately 4%. This probability drops to 3% for a star observed at an inclination of 80 deg, and is still approximately 0.5% for a star observed at an inclination of 60 deg. If one can select 100 stars with a pre-determined inclination equal or greater than 80 deg, the probability of at least one planet being suitably configured for transits is 95%. The majority of transit events are due to planets in small-a orbits similar to the Earth and Venus; thus, the photometric method in principle is the method best suited for the detection of Earthlike planets. The photometric method also allows for testing whether or not planets can exist within binary systems. This can ge done by selecting binary systems observed at high orbital inclinations, both eclipsing binaries and wider visual binaries. For a 'real-world' example, we look at the alpha Centauri system (i = 79.2 deg). If we assume that the equatorial planes of both components coincide with the system's orbital plane, Monte Carlo simulations suggest that the probability of at least one planet (of either component) being suitably configured for transits is approximately 8%. In conclusion, we present a non-exhaustive list of solar-type stars, both single and within binary systems, which exhibit a high equatorial inclination. These objects may be considered as preliminary candidates for planetary searches via the photometric method.

Hale, Alan↗
NASA NTRS1995

Third NASA Langley Formal Methods Workshop

This publication constitutes the proceedings of NASA Langley Research Center's third workshop on the application of formal methods to the design and verification of life-critical systems. This workshop brought together formal methods researchers, industry engineers, and academicians to discuss the potential of NASA-sponsored formal methods and to investigate new opportunities for applying these methods to industry problems. contained herein are copies of the material presented at the workshop, summaries of many of the presentations, a complete list of attendees, and a detailed summary of the Langley formal methods program. Much of this material is available electronically through the World-Wide Web via the following URL.

Holloway, C. Michael↗
NASA NTRS1995

Formal methods demonstration project for space applications

The Space Shuttle program is cooperating in a pilot project to apply formal methods to live requirements analysis activities. As one of the larger ongoing shuttle Change Requests (CR's), the Global Positioning System (GPS) CR involves a significant upgrade to the Shuttle's navigation capability. Shuttles are to be outfitted with GPS receivers and the primary avionics software will be enhanced to accept GPS-provided positions and integrate them into navigation calculations. Prior to implementing the CR, requirements analysts at Loral Space Information Systems, the Shuttle software contractor, must scrutinize the CR to identify and resolve any requirements issues. We describe an ongoing task of the Formal Methods Demonstration Project for Space Applications whose goal is to find an effective way to use formal methods in the GPS CR requirements analysis phase. This phase is currently under way and a small team from NASA Langley, ViGYAN Inc. and Loral is now engaged in this task. Background on the GPS CR is provided and an overview of the hardware/software architecture is presented. We outline the approach being taken to formalize the requirements, only a subset of which is being attempted. The approach features the use of the PVS specification language to model 'principal functions', which are major units of Shuttle software. Conventional state machine techniques form the basis of our approach. Given this background, we present interim results based on a snapshot of work in progress. Samples of requirements specifications rendered in PVS are offered to illustration. We walk through a specification sketch for the principal function known as GPS Receiver State processing. Results to date are summarized and feedback from Loral requirements analysts is highlighted. Preliminary data is shown comparing issues detected by the formal methods team versus those detected using existing requirements analysis methods. We conclude by discussing our plan to complete the remaining activities of this task.

Divito, Ben L.↗
NASA NTRS1995

Method for Fabricating Composite Structures Including Continuous Press Forming and Pultrusion Processing

A method for fabricating composite structures at a low-cost, moderate-to-high production rate is disclosed. A first embodiment of the method includes employing a continuous press forming fabrication process. A second embodiment of the method includes employing a pultrusion process for obtaining composite structures. The methods include coating yarns with matrix material, weaving the yarn into fabric to produce a continuous fabric supply, and feeding multiple layers of net-shaped fabrics having optimally oriented fibers into a debulking tool to form an undebulked preform. The continuous press forming fabrication process includes partially debulking the preform, cutting the partially debulked preform, and debulking the partially debulked preform to form a netshape. An electron-beam or similar technique then cures the structure. The pultrusion fabric process includes feeding the undebulked preform into a heated die and gradually debulking the undebulked preform. The undebulked preform in the heated die changes dimension until a desired cross-sectional dimension is achieved. This process further includes obtaining a net-shaped infiltrated uncured preform, cutting the uncured preform to a desired length, and electron-beam curing (or similar technique) the uncured preform. These fabrication methods produce superior structures formed at higher production rates, resulting in lower cost and high structural performance.

Farley, Gary L.↗
NASA NTRS1995

Multigrid Methods for Aerodynamic Problems in Complex Geometries

Work has been directed at the development of efficient multigrid methods for the solution of aerodynamic problems involving complex geometries, including the development of computational methods for the solution of both inviscid and viscous transonic flow problems. The emphasis is on problems of complex, three-dimensional geometry. The methods developed are based upon finite-volume approximations to both the Euler and the Reynolds-Averaged Navier-Stokes equations. The methods are developed for use on multi-block grids using diagonalized implicit multigrid methods to achieve computational efficiency. The work is focused upon aerodynamic problems involving complex geometries, including advanced engine inlets.

Caughey, David A.↗
NASA NTRS1996

An Evaluation of Attitude-Independent Magnetometer-Bias Determination Methods

Although several algorithms now exist for determining three-axis magnetometer (TAM) biases without the use of attitude data, there are few studies on the effectiveness of these methods, especially in comparison with attitude dependent methods. This paper presents the results of a comparison of three attitude independent methods and an attitude dependent method for computing TAM biases. The comparisons are based on in-flight data from the Extreme Ultraviolet Explorer (EUVE), the Upper Atmosphere Research Satellite (UARS), and the Compton Gamma Ray Observatory (GRO). The effectiveness of an algorithm is measured by the accuracy of attitudes computed using biases determined with that algorithm. The attitude accuracies are determined by comparison with known, extremely accurate, star-tracker-based attitudes. In addition, the effect of knowledge of calibration parameters other than the biases on the effectiveness of all bias determination methods is examined.

Hashmall, J. A.↗