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An algebraic adaptive grid system based on equidistribution law and utilized by the Non-Uniform Rational B-Spline (NURBS) surface for redistribution is presented. A weight function, utilizing a properly weighted boolean sum of various flow field characteristics is developed. Computational examples are presented to demonstrate the success of this technique.
The development of an efficient and robust grid optimization is presented. This algorithm is developed by combining the best characteristics of algebraic, elliptic, and hyperbolic grid generation techniques. This development is based on the following observations and evaluations: (1) algebraic systems are fast and economical; (2) precise spacing control is always achieved; (3) grid generation by elliptic systems is always smooth; and (4) the hyperbolic system preserves the orthogonality at the solid boundary and the point distribution in the field. Computational examples representing practical internal flow configurations are presented to demonstrate the algorithm.
The recovery of a high resolution geopotential from satellite gradiometer observations motivates the examination of high performance computational techniques. The primary subject matter addresses specifically the use of satellite gradiometer and GPS observations to form and invert the normal matrix associated with a large degree and order geopotential solution. Memory resident and out-of-core parallel linear algebra techniques along with data parallel batch algorithms form the foundation of the least squares application structure. A secondary topic includes the adoption of object oriented programming techniques to enhance modularity and reusability of code. Applications implementing the parallel and object oriented methods successfully calculate the degree variance for a degree and order 110 geopotential solution on 32 processors of the Cray T3E. The memory resident gradiometer application exhibits an overall application performance of 5.4 Gflops, and the out-of-core linear solver exhibits an overall performance of 2.4 Gflops. The combination solution derived from a sun synchronous gradiometer orbit produce average geoid height variances of 17 millimeters.
A numerical investigation was conducted to assess the accuracy of two turbulence models when computing non-axisymmetric nozzle-afterbody flows with propulsive jets. Navier-Stokes solutions were obtained for a Convergent-divergent non-axisymmetric nozzle-afterbody and its associated jet exhaust plume at free-stream Mach numbers of 0.600 and 0.938 at an angle of attack of 0 deg. The Reynolds number based on model length was approximately 20 x 10(exp 6). Turbulent dissipation was modeled by the algebraic Baldwin-Lomax turbulence model with the Degani-Schiff modification and by the standard Jones-Launder kappa-epsilon turbulence model. At flow conditions without strong shocks and with little or no separation, both turbulence models predicted the pressures on the surfaces of the nozzle very well. When strong shocks and massive separation existed, both turbulence models were unable to predict the flow accurately. Mixing of the jet exhaust plume and the external flow was underpredicted. The differences in drag coefficients for the two turbulence models illustrate that substantial development is still required for computing very complex flows before nozzle performance can be predicted accurately for all external flow conditions.
FUN3D, a suite of Computational Fluid Dynamics simulation and design tools developed at the NASA Langley Research Center, has undergone continuous development since the late1980s. It contains a large portion of legacy code. Extending it with new capabilities becomes increasingly difficult. To improve the extensibility and reusability, FUN3D is moving toward component-based development. New features, such as Stabilized Finite Elements, Yoga, and Sparse Linear Algebra Toolkit, are integrated into the system as components. Some existing features such as the Node-Centered Finite Volume Solver, are also being refactored to components. The integration of these components poses new requirements on the development workflow. In this paper, we describe the Continuous Integration of FUN3D to support component-based development, and discuss the tools used, the practices followed, and lessons learned during the transition from the traditional approach.
Galerkin finite-element approximations and Newton's method for solving free boundary problems are combined with computer-implemented techniques from nonlinear perturbation analysis to study solidification problems with natural convection in the melt. The Newton method gives rapid convergence to steady state velocity, temperature and pressure fields and melt-solid interface shapes, and forms the basis for algebraic methods for detecting multiple steady flows and assessing their stability. The power of this combination is demonstrated for a two-phase Rayleigh-Benard problem composed of melt and solid in a veritical cylinder with the thermal boundary conditions arranged so that a static melt with a flat melt-solid interface is always a solution. Multiple cellular flows bifurcating from the static state are detected and followed as Rayleigh number is varied. Changing the boundary conditions to approach those appropriate for the vertical Bridgman solidification system causes imperfections that eliminate the static state. The flow structure in the Bridgman system is related to those for the Rayleigh-Benard system by a continuous evolution of the boundary conditions.
An analytical program was conducted to assemble and assess a three-dimensional turbulent viscous flow computer code capable of analyzing the flow field in the transition liners of small gas turbine engines. This code is of the TEACH type with hybrid numerics, and uses the power law and SIMPLER algorithms, an orthogonal curvilinear coordinate system, and an algebraic Reynolds stress turbulence model. The assessments performed in this study, consistent with results in the literature, showed that in its present form this code is capable of predicting trends and qualitative results. The assembled code was used to perform a numerical experiment to investigate the effects of curvature and convergence in the transition liner on the mixing of single and opposed rows of cool dilution jets injected into a hot mainstream flow.
An analytical program was conducted to assemble and assess a three-dimensional turbulent viscous flow computer code capable of analyzing the flow field in the transition liners of small gas turbine engines. This code is of the TEACH type with hybrid numerics, and uses the power law and SIMPLER algorithms, an orthogonal curvilinear coordinate system, and an algebraic Reynolds stress turbulence model. The assessments performed in this study, consistent with results in the literature, showed that in its present form this code is capable of predicting trends and qualitative results. The assembled code was used to perform a numerical experiment to investigate the effects of curvature and convergence in the transition liner on the mixing of single and opposed rows of cool dilution jets injected into a hot mainstream flow.
The present unstructured-grid method follows strictly the basic finite volume forms of the conservation laws of the governing equations for the entire flow domain. High-order spatially accurate formulation has been employed for the numerical solutions of the Navier-Stokes equations. A two-equation k-epsilon turbulence model is also incorporated in the unstructured-grid solver. The convergence of the resulted linear algebraic equation is accelerated with preconditioned Conjugate Gradient method. A statistical spray combustion model has been incorporated into the present unstructured-grid solver. In this model, spray is represented by discrete particles, rather than by continuous distributions. A finite number of computational particles are used to predict a sample of total population of particles. Particle trajectories are integrated using their momentum and motion equations and particles exchange mass, momentum and energy with the gas within the computational cell in which they are located. The interaction calculations are performed simultaneously and eliminate global iteration for the two-phase momentum exchange. A transient spray flame in a high pressure combustion chamber is predicted and then the solution of liquid-fuel combusting flow with a rotating cup atomizer is presented and compared with the experimental data. The major conclusion of this investigation is that the unstructured-grid method can be employed to study very complicated flow fields of turbulent spray combustion. Grid adaptation can be easily achieved in any flow domain such as droplet evaporation and combustion zone. Future applications of the present model can be found in the full three-dimensional study of flow fields of gas turbine and liquid propulsion engine combustion chambers with multi-injectors.
Statistical analysis of test data is currently the preferred method of creating models for uncharacterized side loading in rocket nozzles. However, statistical analysis requires multiple data sets to be reliable; getting these sets requires building and testing multiple rocket motors, which is not feasible given the incredible time and cost required to do so. By using a combination of geometry, simplified finite element analysis (FEA) analogs, and algebra we can create a limited predictive model from only one test data set. Once created, the model requires only the input from one strain gauge located at each yaw and pitch attachment bracket for the actuation system. This process is not computationally intensive, therefore saving an abundance of time and budget over using full 3D FEA models to develop a similar predictive model. To develop the model, conversion factors between the strain at the gauge locations and corresponding actuator load is derived from the data. These conversion factors are used to create a “conversion ellipse” that bounds the constitutive properties of the nozzle as one lumped factor. Using a 2D cross-sectional analog of the nozzle exit plane, we create a function that allows us to determine the azimuthal location of the unexpected load based on the ratio between yaw and pitch strain readings. Combining this with the “conversion ellipse” yields a function that given the strain reading outputs an induced actuator load, regardless of the azimuthal location of the unexpected load. As previously mentioned, this predictive model is limited in its application as there is inaccuracy introduced by the use of data by only one test as well as simplifying assumptions. This approach makes a useful tool to quickly determine whether uncharacterized actuator loading events can damage actuation systems or induce loading that exceeds design limits.
A purely algebraic algorithm is developed for testing positive real character of real rational functions and matrices relative to the unit circle in the complex plane. Since the algorithm is entirely recursive and is performed in finite number of steps, it is suitable for machine computations.
A number of vehicles are currently being proposed for a manned mission to Mars. One of these vehicles has a modified blunt-nosed cone configuration. Experimental results were obtained for this vehicle in 1968. They show lift-over-drag ratios comparable to those needed for Mars entry. Computations are performed to verify the earlier results and to further describe the flight characteristics of this vehicle. An analytical method is used to define the surface of this vehicle. A single-block volume grid is generated around the vehicle using the algebraic Two-Boundary Grid Generation algorithm (TBGG) and transfinite interpolation. Euler solutions are then obtained from a Langley Aerodynamic Upward Relaxation Algorithm (LAURA) at Mach 6.0 and angles of attack of 0, 6, and 12 deg. The lift coefficient determined from the LAURA code agree very well with the experimental results. The drag and pitching moment coefficients, however, are underestimated by the code since viscous effects are not considered. Contour plots of the flowfield show no evidence of separation for angles of attack up to 12 deg.
Some recent work is reviewed which relates families of trees to symbolic algorithms for the exact computation of series which approximate solutions of ordinary differential equations. It turns out that the vector space whose basis is the set of finite, rooted trees carries a natural multiplication related to the composition of differential operators, making the space of trees an algebra. This algebraic structure can be exploited to yield a variety of algorithms for manipulating vector fields and the series and algebras they generate.
Presented here are the main features of a tool package for manipulating and working with sparse matrices. One of the goals of the package is to provide basic tools to facilitate the exchange of software and data between researchers in sparse matrix computations. The starting point is the Harwell/Boeing collection of matrices for which the authors provide a number of tools. Among other things, the package provides programs for converting data structures, printing simple statistics on a matrix, plotting a matrix profile, and performing linear algebra operations with sparse matrices.
A truly two-dimensional algorithm is created for solving the steady-state two-dimensional conservation-law problem. An overdetermined system of algebraic equations is obtained through discretization by finite-volume formulas. These equations are perturbed nonsingularly and are solved by an efficient geometrically oriented l(1) procedure. The basic algorithm and the theory for the linear case f(u) = u are presented, and computational results for the nonlinear case f(u) = sq u are also analyzed. It is noted that the l(1) procedure captures boundary shocks as well as oblige and zigzag interior shocks in bands that are one cell wide, and the solution values are accurate up to the edge of the shock.
An overview of current multigrid techniques for unstructured meshes is given. The basic principles of the multigrid approach are first outlined. Application of these principles to unstructured mesh problems is then described, illustrating various different approaches, and giving examples of practical applications. Advanced multigrid topics, such as the use of algebraic multigrid methods, and the combination of multigrid techniques with adaptive meshing strategies are dealt with in subsequent sections. These represent current areas of research, and the unresolved issues are discussed. The presentation is organized in an educational manner, for readers familiar with computational fluid dynamics, wishing to learn more about current unstructured mesh techniques.
An image-based phase retrieval technique has been developed that can be used on board a space based iterative transformation system. Image-based wavefront sensing is computationally demanding due to the floating-point nature of the process. The discrete Fourier transform (DFT) calculation is presented in "diagonal" form. By diagonal we mean that a transformation of basis is introduced by an application of the similarity transform of linear algebra. The current method exploits the diagonal structure of the DFT in a special way, particularly when parts of the calculation do not have to be repeated at each iteration to converge to an acceptable solution in order to focus an image.
Two types of rolling-element bearings representing radial loaded and thrust loaded bearings were used for this study. Three hundred forty (340) virtual bearing sets totaling 31400 bearings were randomly assembled and tested by Monte Carlo (random) number generation. The Monte Carlo results were compared with endurance data from 51 bearing sets comprising 5321 bearings. A simple algebraic relation was established for the upper and lower L(sub 10) life limits as function of number of bearings failed for any bearing geometry. There is a fifty percent (50 percent) probability that the resultant bearing life will be less than that calculated. The maximum and minimum variation between the bearing resultant life and the calculated life correlate with the 90-percent confidence limits for a Weibull slope of 1.5. The calculated lives for bearings using a load-life exponent p of 4 for ball bearings and 5 for roller bearings correlated with the Monte Carlo generated bearing lives and the bearing data. STLE life factors for bearing steel and processing provide a reasonable accounting for differences between bearing life data and calculated life. Variations in Weibull slope from the Monte Carlo testing and bearing data correlated. There was excellent agreement between percent of individual components failed from Monte Carlo simulation and that predicted.