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All sourcesNASA NTRSDOE OSTI
NASA NTRS1989

Algebraic multigrid methods applied to problems in computational structural mechanics

The development of algebraic multigrid (AMG) methods and their application to certain problems in structural mechanics are described with emphasis on two- and three-dimensional linear elasticity equations and the 'jacket problems' (three-dimensional beam structures). Various possible extensions of AMG are also described. The basic idea of AMG is to develop the discretization sequence based on the target matrix and not the differential equation. Therefore, the matrix is analyzed for certain dependencies that permit the proper construction of coarser matrices and attendant transfer operators. In this manner, AMG appears to be adaptable to structural analysis applications.

Mccormick, Steve↗
NASA NTRS1990

Physical considerations governing computation of turbulent flows over bodies at large incidence

The physical aspects governing accurate numerical simulation of turbulent flows having large regions of crossflow separation are re-examined. Time-accurate, three-dimensional fine-grid Navier-Stokes solutions were obtained for turbulent subsonic and supersonic flows over a slender ogive-cylinder body of revolution at large angles of attack. These flowfields are complex and contain regions of crossflow separation and an organized leeward-side vortex structure. An algebraic eddy-viscosity turbulence model has been modified to correctly account for the effects of the vortices on the underlying viscous layers. The numerical results show that the vortical flow structure of both subsonic and supersonic high incidence flows are similar, and can be accurately treated by the turbulence model. As the angle of attack is increased, the effectiveness of the model increases since the influence of the vortices on the underlying boundary layer decreases.

Degani, David↗
NASA NTRS1981

Asymptotic form of the longitudinal correlation function for isotropic homogeneous turbulence

An asymptotic form is derived for the longitudinal correlation function for isotropic homogeneous turbulence in an incompressible fluid governed by the Navier-Stokes equation. The result is obtained from an analysis of the algebraic-differential structure of the two-point correlation tensor contained in the complex-valued Fourier transform of the probability measure over the turbulence ensemble, which reveals that the Hopf characteristic functional (the complex-valued Fourier transform of the probability measure) satisfies the Fourier interference inequality. Consequences of the expression obtained, in which the correlation function is a positive definite function of the inverse cube of the spatial coordinate as it approaches infinity, are shown to include the nonexistence of the Loitsianskii invariant, and the solution is shown to be consistent with empirical formulas.

Rosen, G.↗
NASA NTRS2011

Improve Data Mining and Knowledge Discovery Through the Use of MatLab

Data mining is widely used to mine business, engineering, and scientific data. Data mining uses pattern based queries, searches, or other analyses of one or more electronic databases/datasets in order to discover or locate a predictive pattern or anomaly indicative of system failure, criminal or terrorist activity, etc. There are various algorithms, techniques and methods used to mine data; including neural networks, genetic algorithms, decision trees, nearest neighbor method, rule induction association analysis, slice and dice, segmentation, and clustering. These algorithms, techniques and methods used to detect patterns in a dataset, have been used in the development of numerous open source and commercially available products and technology for data mining. Data mining is best realized when latent information in a large quantity of data stored is discovered. No one technique solves all data mining problems; challenges are to select algorithms or methods appropriate to strengthen data/text mining and trending within given datasets. In recent years, throughout industry, academia and government agencies, thousands of data systems have been designed and tailored to serve specific engineering and business needs. Many of these systems use databases with relational algebra and structured query language to categorize and retrieve data. In these systems, data analyses are limited and require prior explicit knowledge of metadata and database relations; lacking exploratory data mining and discoveries of latent information. This presentation introduces MatLab(R) (MATrix LABoratory), an engineering and scientific data analyses tool to perform data mining. MatLab was originally intended to perform purely numerical calculations (a glorified calculator). Now, in addition to having hundreds of mathematical functions, it is a programming language with hundreds built in standard functions and numerous available toolboxes. MatLab's ease of data processing, visualization and its enormous availability of built in functionalities and toolboxes make it suitable to perform numerical computations and simulations as well as a data mining tool. Engineers and scientists can take advantage of the readily available functions/toolboxes to gain wider insight in their perspective data mining experiments.

Shaykhian, Gholam Ali↗
NASA NTRS2011

Improve Data Mining and Knowledge Discovery through the use of MatLab

Data mining is widely used to mine business, engineering, and scientific data. Data mining uses pattern based queries, searches, or other analyses of one or more electronic databases/datasets in order to discover or locate a predictive pattern or anomaly indicative of system failure, criminal or terrorist activity, etc. There are various algorithms, techniques and methods used to mine data; including neural networks, genetic algorithms, decision trees, nearest neighbor method, rule induction association analysis, slice and dice, segmentation, and clustering. These algorithms, techniques and methods used to detect patterns in a dataset, have been used in the development of numerous open source and commercially available products and technology for data mining. Data mining is best realized when latent information in a large quantity of data stored is discovered. No one technique solves all data mining problems; challenges are to select algorithms or methods appropriate to strengthen data/text mining and trending within given datasets. In recent years, throughout industry, academia and government agencies, thousands of data systems have been designed and tailored to serve specific engineering and business needs. Many of these systems use databases with relational algebra and structured query language to categorize and retrieve data. In these systems, data analyses are limited and require prior explicit knowledge of metadata and database relations; lacking exploratory data mining and discoveries of latent information. This presentation introduces MatLab(TradeMark)(MATrix LABoratory), an engineering and scientific data analyses tool to perform data mining. MatLab was originally intended to perform purely numerical calculations (a glorified calculator). Now, in addition to having hundreds of mathematical functions, it is a programming language with hundreds built in standard functions and numerous available toolboxes. MatLab's ease of data processing, visualization and its enormous availability of built in functionalities and toolboxes make it suitable to perform numerical computations and simulations as well as a data mining tool. Engineers and scientists can take advantage of the readily available functions/toolboxes to gain wider insight in their perspective data mining experiments.

Shaykahian, Gholan Ali↗
NASA NTRS1971

On a programming language for graph algorithms

An algorithmic language, GRAAL, is presented for describing and implementing graph algorithms of the type primarily arising in applications. The language is based on a set algebraic model of graph theory which defines the graph structure in terms of morphisms between certain set algebraic structures over the node set and arc set. GRAAL is modular in the sense that the user specifies which of these mappings are available with any graph. This allows flexibility in the selection of the storage representation for different graph structures. In line with its set theoretic foundation, the language introduces sets as a basic data type and provides for the efficient execution of all set and graph operators. At present, GRAAL is defined as an extension of ALGOL 60 (revised) and its formal description is given as a supplement to the syntactic and semantic definition of ALGOL. Several typical graph algorithms are written in GRAAL to illustrate various features of the language and to show its applicability.

Rheinboldt, W. C.↗
NASA NTRS1997

Inherent Conservatism in Deterministic Quasi-Static Structural Analysis

The cause of the long-suspected excessive conservatism in the prevailing structural deterministic safety factor has been identified as an inherent violation of the error propagation laws when reducing statistical data to deterministic values and then combining them algebraically through successive structural computational processes. These errors are restricted to the applied stress computations, and because mean and variations of the tolerance limit format are added, the errors are positive, serially cumulative, and excessively conservative. Reliability methods circumvent these errors and provide more efficient and uniform safe structures. The document is a tutorial on the deficiencies and nature of the current safety factor and of its improvement and transition to absolute reliability.

Verderaime, V.↗
NASA NTRS1993

Robust algebraic image enhancement for intelligent control systems

Robust vision capability for intelligent control systems has been an elusive goal in image processing. The computationally intensive techniques a necessary for conventional image processing make real-time applications, such as object tracking and collision avoidance difficult. In order to endow an intelligent control system with the needed vision robustness, an adequate image enhancement subsystem capable of compensating for the wide variety of real-world degradations, must exist between the image capturing and the object recognition subsystems. This enhancement stage must be adaptive and must operate with consistency in the presence of both statistical and shape-based noise. To deal with this problem, we have developed an innovative algebraic approach which provides a sound mathematical framework for image representation and manipulation. Our image model provides a natural platform from which to pursue dynamic scene analysis, and its incorporation into a vision system would serve as the front-end to an intelligent control system. We have developed a unique polynomial representation of gray level imagery and applied this representation to develop polynomial operators on complex gray level scenes. This approach is highly advantageous since polynomials can be manipulated very easily, and are readily understood, thus providing a very convenient environment for image processing. Our model presents a highly structured and compact algebraic representation of grey-level images which can be viewed as fuzzy sets.

Lerner, Bao-Ting↗
NASA NTRS1968

Embedding of SU/3/ in SU/8/.

SU/3/ embedded in SU/8/, constructing SU/8/ algebra to preserve structure, noting particle assignments and nonleptonic baryon decays

Carlstone, D. S.↗
NASA NTRS1975

Estimation and Analysis of Nonlinear Stochastic Systems

The algebraic and geometric structures of certain classes of nonlinear stochastic systems were exploited in order to obtain useful stability and estimation results. The class of bilinear stochastic systems (or linear systems with multiplicative noise) was discussed. The stochastic stability of bilinear systems driven by colored noise was considered. Approximate methods for obtaining sufficient conditions for the stochastic stability of bilinear systems evolving on general Lie groups were discussed. Two classes of estimation problems involving bilinear systems were considered. It was proved that, for systems described by certain types of Volterra series expansions or by certain bilinear equations evolving on nilpotent or solvable Lie groups, the optimal conditional mean estimator consists of a finite dimensional nonlinear set of equations. The theory of harmonic analysis was used to derive suboptimal estimators for bilinear systems driven by white noise which evolve on compact Lie groups or homogeneous spaces.

Marcus, S. I.↗
NASA NTRS1987

Covariance and the hierarchy of frame bundles

This is an essay on the general concept of covariance, and its connection with the structure of the nested set of higher frame bundles over a differentiable manifold. Examples of covariant geometric objects include not only linear tensor fields, densities and forms, but affinity fields, sectors and sector forms, higher order frame fields, etc., often having nonlinear transformation rules and Lie derivatives. The intrinsic, or invariant, sets of forms that arise on frame bundles satisfy the graded Cartan-Maurer structure equations of an infinite Lie algebra. Reduction of these gives invariant structure equations for Lie pseudogroups, and for G-structures of various orders. Some new results are introduced for prolongation of structure equations, and for treatment of Riemannian geometry with higher-order moving frames. The use of invariant form equations for nonlinear field physics is implicitly advocated.

Estabrook, Frank B.↗
NASA NTRS1989

The weight distribution and randomness of linear codes

Finding the weight distributions of block codes is a problem of theoretical and practical interest. Yet the weight distributions of most block codes are still unknown except for a few classes of block codes. Here, by using the inclusion and exclusion principle, an explicit formula is derived which enumerates the complete weight distribution of an (n,k,d) linear code using a partially known weight distribution. This expression is analogous to the Pless power-moment identities - a system of equations relating the weight distribution of a linear code to the weight distribution of its dual code. Also, an approximate formula for the weight distribution of most linear (n,k,d) codes is derived. It is shown that for a given linear (n,k,d) code over GF(q), the ratio of the number of codewords of weight u to the number of words of weight u approaches the constant Q = q(-)(n-k) as u becomes large. A relationship between the randomness of a linear block code and the minimum distance of its dual code is given, and it is shown that most linear block codes with rigid algebraic and combinatorial structure also display certain random properties which make them similar to random codes with no structure at all.

Cheung, K.-M.↗
NASA NTRS1992

Algebraic surface grid generation in three-dimensional space

An interactive program for algebraic generation of structured surface grids in three dimensional space was developed on the IRIS4D series workstations. Interactive tools are available to ease construction of edge curves and surfaces in 3-D space. Addition, removal, or redistribution of points at arbitrary locations on a general 3-D surface or curve is possible. Also, redistribution of surface grid points may be accomplished through use of conventional surface splines or a method called 'surface constrained transfinite interpolation'. This method allows the user to redistribute the grid points on the edges of a surface patch; the effect of the redistribution is then propagated to the remainder of the surface through a transfinite interpolation procedure where the grid points will be constrained to lie on the surface. The program was written to be highly functional and easy to use. A host of utilities are available to ease the grid generation process. Generality of the program allows the creation of single and multizonal surface grids according to the user requirements. The program communicates with the user through popup menus, windows, and the mouse.

Warsi, Saif↗
NASA NTRS1993

The algebra of supertraces for 2+1 super de Sitter gravity

The algebra of the observables for 2+1 super de Sitter gravity, for one genus of the spatial surface is calculated. The algebra turns out to be an infinite Lie algebra subject to non-linear constraints. The constraints are solved explicitly in terms of five independent complex supertraces. These variables are the true degrees of freedom of the system and their quantized algebra generates a new structure which is referred to as a 'central extension' of the quantum algebra SU(2)q.

Urrutia, L. F.↗
NASA NTRS1993

An automatic multigrid method for the solution of sparse linear systems

An automatic version of the multigrid method for the solution of linear systems arising from the discretization of elliptic PDE's is presented. This version is based on the structure of the algebraic system solely, and does not use the original partial differential operator. Numerical experiments show that for the Poisson equation the rate of convergence of our method is equal to that of classical multigrid methods. Moreover, the method is robust in the sense that its high rate of convergence is conserved for other classes of problems: non-symmetric, hyperbolic (even with closed characteristics) and problems on non-uniform grids. No double discretization or special treatment of sub-domains (e.g. boundaries) is needed. When supplemented with a vector extrapolation method, high rates of convergence are achieved also for anisotropic and discontinuous problems and also for indefinite Helmholtz equations. A new double discretization strategy is proposed for finite and spectral element schemes and is found better than known strategies.

Shapira, Yair↗
NASA NTRS1992

Variable-complexity aerodynamic-structural design of a high-speed civil transport wing

A variable-complexity strategy of combining simple and detailed analysis methods is presented for the design optimization of a high-speed civil transport (HSCT) wing. Two sets of results are shown: the aerodynamic design of the wing using algebraic weight equations for structural considerations, and optimization results of the internal wing structure for a fixed wing configuration. We show example results indicating that using simple analysis methods alone for the calculation of a critical constraint can allow an optimizer to exploit weaknesses in the analysis. The structural optimization results provide a valuable check for the weight equations used in the aerodynamic design. In addition, these results confirm the need for using simple, algebraic models in conjunction with more detailed analysis methods. A strategy of interlaced aerodynanic-structural design is proposed.

Hutchison, M. G.↗