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Combinatorial Auction-Based Strategic Deconfliction of Federated UTM Airspace

Unmanned Aerial Vehicles (UAVs) have become commonly used to perform a wide range of commercial activities such as cinematography and medical supply delivery. Consequently, regulators have become interested in designing UAV Traffic Management systems (UTMs) to coordinate UAV traffic among a collection of UAV operators. One framework which has been recently proposed for a UTM system is a combinatorial auction. In this framework, airspace is modelled as a 4D grid of space-time cells. UAV operators bid on cells which collectively form paths for their UAVs. Ideally, an airspace auction should reveal information about the current price of flight paths to bidders, allowing bidders to identify and bid on a select number of paths instead of placing as many bids as possible in the hopes of stumbling on a cheap path. Revealing too much information, however, can allow bad actors to place bids which are intended not to win but to raise the price that a rival bidder must pay. We address these twin challenges with a new information revelation framework which provides bidders with wide-ranging pricing information while suppressing bad actors. We evaluate our framework on scenarios based on a Japan Aerospace Exploration Agency (JAXA) case study and find that it can scale to thousands of bids.

Christopher J C Leet↗

NASA NTRS2001

Numerical Modeling of Liquid-Vapor Phase Change

We implemented a two- and three-dimensional finite difference/front tracking technique to solve liquid-vapor phase change problems. The mathematical and the numerical features of the method were explained in great detail in our previous reports, Briefly, we used a single formula representation which incorporated jump conditions into the governing equations. The interfacial terms were distributed as singular terms using delta functions so that the governing equations would be the same as conventional conservation equations away from the interface and in the vicinity of the interface they would provide correct jump conditions. We used a fixed staggered grid to discretize these equations and an unstructured grid to explicitly track the front. While in two dimensions the front was simply a connection of small line segments, in three dimensions it was represented by a connection of small triangular elements. The equations were written in conservative forms and during the course of computations we used regriding to control the size of the elements of the unstructured grid. Moreover, we implemented a coalescence in two dimensions which allowed the merging of different fronts or two segments of the same front when they were sufficiently close. We used our code to study thermocapillary migration of bubbles, burst of bubbles at a free surface, buoyancy-driven interactions of bubbles, evaporation of drops, rapid evaporation of an interface, planar solidification of an undercooled melt, dendritic solidification, and a host of other problems cited in the reference.

Esmaeeli, Asghar↗

NASA NTRS2003

Toward Immersed Boundary Simulation of High Reynolds Number Flows

In the immersed boundary (IB) method, the surface of an object is reconstructed with forcing terms in the underlying flow field equations. The surface may split a computational cell removing the constraint of the near wall gridlines to be aligned with the surface. This feature greatly simplifies the grid generation process which is cumbersome and expensive in particular for structured grids and complex geometries. The IB method is ideally suited for Cartesian flow solvers. The flow equations written in Cartesian coordinates appear in a very simple form and several numerical algorithms can be used for an efficient solution of the equations. In addition, the accuracy of numerical algorithms is dependent on the underlying grid and it usually deteriorates when the grid deviates from a Cartesian mesh. The challenge for the IB method lies in the representation of the wall boundaries and in providing an adequate near wall flow field resolution. The issue of enforcing no-slip boundary conditions at the immersed surface has been addressed by several authors by imposing a local reconstruction of the solution. Initial work by Verzicco et al. was based on a simple linear, one-dimensional operator and this approach proved to be accurate for boundaries largely aligned with the grid lines. Majumdar et al. used various multidimensional and high order polynomial interpolations schemes. These high order schemes, however, are keen to introduce wiggles and spurious extrema. Iaccarino & Verzicco and Kalitzin & Iaccarino proposed a tri-linear reconstruction for the velocity components and the turbulent scalars. A modified implementation that has proven to be more robust is reported in this paper. The issue of adequate near wall resolution in a Cartesian framework can initially be addressed by using a non-uniform mesh which is stretched near the surface. In this paper, we investigate an unstructured approach for local grid refinement that utilizes Cartesian mesh features. The computation of high Reynolds number wall bounded flows is particularly challenging as it requires the consideration of thin turbulent boundary layers, i.e. near wall regions with large gradients of the flow field variables. For such flows, the representation of the wall boundary has a large impact on the accuracy of the computation. It is also critical for the robustness and convergence of the flow solver.

Kalitzin, Georgi↗

NASA NTRS1998

Developing Information Power Grid Based Algorithms and Software

This was an exploratory study to enhance our understanding of problems involved in developing large scale applications in a heterogeneous distributed environment. It is likely that the large scale applications of the future will be built by coupling specialized computational modules together. For example, efforts now exist to couple ocean and atmospheric prediction codes to simulate a more complete climate system. These two applications differ in many respects. They have different grids, the data is in different unit systems and the algorithms for inte,-rating in time are different. In addition the code for each application is likely to have been developed on different architectures and tend to have poor performance when run on an architecture for which the code was not designed, if it runs at all. Architectural differences may also induce differences in data representation which effect precision and convergence criteria as well as data transfer issues. In order to couple such dissimilar codes some form of translation must be present. This translation should be able to handle interpolation from one grid to another as well as construction of the correct data field in the correct units from available data. Even if a code is to be developed from scratch, a modular approach will likely be followed in that standard scientific packages will be used to do the more mundane tasks such as linear algebra or Fourier transform operations. This approach allows the developers to concentrate on their science rather than becoming experts in linear algebra or signal processing. Problems associated with this development approach include difficulties associated with data extraction and translation from one module to another, module performance on different nodal architectures, and others. In addition to these data and software issues there exists operational issues such as platform stability and resource management.

Dongarra, Jack↗

NASA NTRS1991

Cost-effective accurate coarse-grid method for highly convective multidimensional unsteady flows

A fundamentally multidimensional convection scheme is described based on vector transient interpolation modeling rewritten in conservative control-volume form. Vector third-order upwinding is used as the basis of the algorithm; this automatically introduces important cross-difference terms that are absent from schemes using component-wise one-dimensional formulas. Third-order phase accuracy is good; this is important for coarse-grid large-eddy or full simulation. Potential overshoots or undershoots are avoided by using a recently developed universal limiter. Higher order accuracy is obtained locally, where needed, by the cost-effective strategy of adaptive stencil expansion in a direction normal to each control-volume face; this is controlled by monitoring the absolute normal gradient and curvature across the face. Higher (than third) order cross-terms do not appear to be needed. Since the wider stencil is used only in isolated narrow regions (near discontinuities), extremely high (in this case, seventh) order accuracy can be achieved for little more than the cost of a globally third-order scheme.

Leonard, B. P.↗

NASA NTRS2004

Numerical Simulation of Two-grid Ion Optics Using a 3D Code

A three-dimensional ion optics code has been developed under NASA's Project Prometheus to model two grid ion optics systems. The code computes the flow of positive ions from the discharge chamber through the ion optics and into the beam downstream of the thruster. The rate at which beam ions interact with background neutral gas to form charge exchange ions is also computed. Charge exchange ion trajectories are computed to determine where they strike the ion optics grid surfaces and to determine the extent of sputter erosion they cause. The code has been used to compute predictions of the erosion pattern and wear rate on the NSTAR ion optics system; the code predicts the shape of the eroded pattern but overestimates the initial wear rate by about 50%. An example of use of the code to estimate the NEXIS thruster accelerator grid life is also presented.

ion optics↗

NASA NTRS1999

A Pseubo-Temporal Multi-Grid Relaxation Scheme for Solving the Parabolized Navier-Stokes Equations

A multi-grid, flux-difference-split, finite-volume code, VULCAN, is presented for solving the elliptic and parabolized form of the equations governing three-dimensional, turbulent, calorically perfect and non-equilibrium chemically reacting flows. The space marching algorithms developed to improve convergence rate and or reduce computational cost are emphasized. The algorithms presented are extensions to the class of implicit pseudo-time iterative, upwind space-marching schemes. A full approximate storage, full multi-grid scheme is also described which is used to accelerate the convergence of a Gauss-Seidel relaxation method. The multi-grid algorithm is shown to significantly improve convergence on high aspect ratio grids.

Morrison, J. H.↗

NASA NTRS1999

A Pseudo-Temporal Multi-Grid Relaxation Scheme for Solving the Parabolized Navier-Stokes Equations

White, J. A.↗

NASA NTRS1986

Finite volume solultion of the compressible boundary layer equations

A box-type finite volume discretization is applied to the integral form of the compressible boundary layer equations. Boundary layer scaling is introduced through the grid construction: streamwise grid lines follow eta = y/h = const., where y is the normal coordinate and h(x) is a scale factor proportional to the boundary layer thickness. With this grid, similarity can be applied explicitly to calculate initial conditions. The finite volume method preserves the physical transparency of the integral equations in the discrete approximation. The resulting scheme is accurate, efficient, and very simple. Computations for similar and non-similar flows show excellent agreement with tabulated results, solutions computed with Keller's Box scheme, and experimental data.

Loyd, B.↗

NASA NTRS1986

Finite volume solution of the compressible boundary-layer equations

A box-type finite volume discretization is applied to the integral form of the compressible boundary layer equations. Boundary layer scaling is introduced through the grid construction: streamwise grid lines follow eta = y/h = const., where y is the normal coordinate and h(x) is a scale factor proportional to the boundary layer thickness. With this grid, similarity can be applied explicity to calculate initial conditions. The finite volume method preserves the physical transparency of the integral equations in the discrete approximation. The resulting scheme is accurate, efficient, and conceptually simple. Computations for similar and non-similar flows show excellent agreement with tabulated results, solutions computed with Keller's Box scheme, and experimental data.

Loyd, B.↗

NASA NTRS1985

Intercomparison of global SST fields derived from satellite sensors and ship observations

Near global comparisons were made between binned forms of SST obtained from different instruments, ship meteorological observations, and the output of a meteorological model. Binning hereinafter means gridding by simple summation of data values in a bin, which in this study was a 2 deg. x 2 deg. box or a 10 deg. x 10 box surrounding a grid node. In the binned comparisons, the question is asked: are the binned data from two instruments drawn from the same population? The first two measures of statistical moments of the populations are used to test the hypothesis that they are. Comparisons were also attempted between raw forms of the various SST retrievals, using a structure function analysis. The results of this analysis were relatively uninteresting and are therefore omitted.

Pazan, S. E.↗

NASA NTRS1997

Implicit/Multigrid Algorithms for Incompressible Turbulent Flows on Unstructured Grids

An implicit code for computing inviscid and viscous incompressible flows on unstructured grids is described. The foundation of the code is a backward Euler time discretization for which the linear system is approximately solved at each time step with either a point implicit method or a preconditioned Generalized Minimal Residual (GMRES) technique. For the GMRES calculations, several techniques are investigated for forming the matrix-vector product. Convergence acceleration is achieved through a multigrid scheme that uses non-nested coarse grids that are generated using a technique described in the present paper. Convergence characteristics are investigated and results are compared with an exact solution for the inviscid flow over a four-element airfoil. Viscous results, which are compared with experimental data, include the turbulent flow over a NACA 4412 airfoil, a three-element airfoil for which Mach number effects are investigated, and three-dimensional flow over a wing with a partial-span flap.

Anderson, W. Kyle↗

NASA NTRS1991

Development of a grid-independent approximate Riemannsolver

A grid-independent approximate Riemann solver for use with the Euler and Navier-Stokes equations was introduced and explored. The two-dimensional Euler and Navier-Stokes equations are described in Cartesian and generalized coordinates, as well as the traveling wave form of the Euler equations. The spatial and temporal discretization are described for both explicit and implicit time-marching schemes. The grid-aligned flux function of Roe is outlined, while the 5-wave grid-independent flux function is derived. The stability and monotonicity analysis of the 5-wave model are presented. Two-dimensional results are provided and extended to three dimensions. The corresponding results are presented.

Rumsey, Christopher Lockwood↗

NASA NTRS1982

Recent progress in terrestrial photovoltaic collector technology

The U.S. Photovoltaic Research and Development Program has the objective to develop the technology necessary to foster widespread grid-competitive electric power generation by the late 1980s. The flat-plate and the concentrator collector activities form the nucleus of the program. The project is concerned with the refining of silicon, silicon sheet production, solar cell processing and fabrication, encapsulation materials development, and collector design and production. The Large-Area Silicon Sheet Task has the objective to develop and demonstrate the feasibility of several methods for producing large area silicon sheet material suitable for fabricating low-cost, high-efficiency solar cells. It is expected that a variety of economic flat-plate and concentrator collectors will become commercially available for grid-connected applications.

Ferber, R. R.↗

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 NTRS1985

Navier-Stokes simulations of rotor-stator interaction using patched and overlaid grids

An accurate numerical analysis of the flows associated with rotor-stator configurations in turbomachinery can be very helpful in optimizing the performance of turbomachinery. In this study the unsteady, thin-layer, Navier-Stokes equations are solved using a system of patched and overlaid grids for a rotor-stator configuration of an axial turbine. The equations necessary for an accurate transfer of information between the several grids are briefly described within the framework of an iterative, implicit algorithm. Results in the form of Mach number contours, time-averaged pressures, unsteady pressures, amplitudes, and phase are presented. The numerical results are also compared with experimental data and the agreement is found to be good.

Rai, M. M.↗

NASA NTRS1993

An advancing-front Delaunay-triangulation algorithm designed for robustness

The following topics, which are associated with computational fluid dynamics, are discussed: unstructured mesh generation; the advancing front methodology; failures of the advancing front methodology; Delaunay triangulation; the Tanamua-Merriam algorithm; Yet Another Grid Generator (YAGG); and advancing front-Delaunay triangulation. The discussion is presented in viewgraph form.

D J Mavriplis↗

NASA NTRS1993

Three-dimensional unsteady flow calculations in an advanced gas generator turbine

This paper deals with the application of a three-dimensional, unsteady Navier-Stokes code for predicting the unsteady flow in a single stage of an advanced gas generator turbine. The numerical method solves the three-dimensional thin-layer Navier-Stokes equations, using a system of overlaid grids, which allow for relative motion between the rotor and stator airfoils. Results in the form of time averaged pressures and pressure amplitudes on the airfoil surfaces will be shown. In addition, instantaneous contours of pressure, Mach number, etc. will be presented in order to provide a greater understanding of the inviscid as well as the viscous aspects of the flowfield. Also, relevant secondary flow features such as cross-plane velocity vectors and total pressure contours will be presented. Prior work in two-dimensions has indicated that for the advanced designs, the unsteady interactions can play a significant role in turbine performance. These interactions affect not only the stage efficiency but can substantially alter the time-averaged features of the flow. This work is a natural extension of the work done in two-dimensions and hopes to address some of the issues raised by the two-dimensional calculations. These calculations are being performed as an integral part of an actual design process and demonstrate the value of unsteady rotor-stator interaction calculations in the design of turbomachines.

Rangwalla, Akil A.↗