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NASA NTRS1992

High-performance equation solvers and their impact on finite element analysis

The role of equation solvers in modern structural analysis software is described. Direct and iterative equation solvers which exploit vectorization on modern high-performance computer systems are described and compared. The direct solvers are two Cholesky factorization methods. The first method utilizes a novel variable-band data storage format to achieve very high computation rates and the second method uses a sparse data storage format designed to reduce the number od operations. The iterative solvers are preconditioned conjugate gradient methods. Two different preconditioners are included; the first uses a diagonal matrix storage scheme to achieve high computation rates and the second requires a sparse data storage scheme and converges to the solution in fewer iterations that the first. The impact of using all of the equation solvers in a common structural analysis software system is demonstrated by solving several representative structural analysis problems.

Poole, Eugene L.↗

NASA NTRS1972

Applications of remote sensing to stream discharge predictions

A feasibility study has been initiated on the use of remote earth observations for augmenting stream discharge prediction for the design and/or operation of major reservoir systems, pumping systems and irrigation systems. The near-term objectives are the interpolation of sparsely instrumented precipitation surveillance networks and the direct measurement of water loss by evaporation. The first steps of the study covered a survey of existing reservoir systems, stream discharge prediction methods, gage networks and the development of a self-adaptive variation of the Kentucky Watershed model, SNOPSET, that includes snowmelt. As a result of these studies, a special three channel scanner is being built for a small aircraft, which should provide snow, temperature and water vapor maps for the spatial and temporal interpolation of stream gages.

Krause, F. R.↗

NASA NTRS1996

Methodology for Sensitivity Analysis, Approximate Analysis, and Design Optimization in CFD for Multidisciplinary Applications

An incremental iterative formulation together with the well-known spatially split approximate-factorization algorithm, is presented for solving the large, sparse systems of linear equations that are associated with aerodynamic sensitivity analysis. This formulation is also known as the 'delta' or 'correction' form. For the smaller two dimensional problems, a direct method can be applied to solve these linear equations in either the standard or the incremental form, in which case the two are equivalent. However, iterative methods are needed for larger two-dimensional and three dimensional applications because direct methods require more computer memory than is currently available. Iterative methods for solving these equations in the standard form are generally unsatisfactory due to an ill-conditioned coefficient matrix; this problem is overcome when these equations are cast in the incremental form. The methodology is successfully implemented and tested using an upwind cell-centered finite-volume formulation applied in two dimensions to the thin-layer Navier-Stokes equations for external flow over an airfoil. In three dimensions this methodology is demonstrated with a marching-solution algorithm for the Euler equations to calculate supersonic flow over the High-Speed Civil Transport configuration (HSCT 24E). The sensitivity derivatives obtained with the incremental iterative method from a marching Euler code are used in a design-improvement study of the HSCT configuration that involves thickness. camber, and planform design variables.

Taylor, Arthur C., III↗

NASA NTRS2006

Improved Oxygen-Beam Texturing of Glucose-Monitoring Optics

An improved method has been devised for using directed, hyperthermal beams of oxygen atoms and ions to impart desired textures to the tips of polymethylmethacrylate [PMMA] optical fibers to be used in monitoring the glucose content of blood. The improved method incorporates, but goes beyond, the method described in Texturing Blood-Glucose- Monitoring Optics Using Oxygen Beams (LEW-17642-1), NASA Tech Briefs, Vol. 29, No. 4 (April 2005), page 11a. The basic principle of operation of such a glucose-monitoring sensor is as follows: The textured surface of the optical fiber is coated with chemicals that interact with glucose in such a manner as to change the reflectance of the surface. Light is sent down the optical fiber and is reflected from, the textured surface. The resulting change in reflectance of the light is measured as an indication of the concentration of glucose. The required texture on the ends of the optical fibers is a landscape of microscopic cones or pillars having high aspect ratios (microscopic structures being taller than they are wide). The average distance between hills must be no more than about 5 mso that blood cells (which are wider) cannot enter the valleys between the hills, where they would interfere with optical sensing of glucose in the blood plasma. On the other hand, the plasma is required to enter the valleys, and high aspect ratio structures are needed to maximize the surface area in contact with the plasma, thereby making it possible to obtain a given level of optical glucose-measurement sensitivity with a relatively small volume of blood. There is an additional requirement that the hills be wide enough that a sufficient amount of light can propagate into them and, after reflection, can propagate out of them. The method described in the cited prior article produces a texture comprising cones and pillars that conform to the average-distance and aspect-ratio requirements. However, a significant fraction of the cones and pillars are so narrow that not enough light can propagate along them. The improved method makes it possible to form wider cones and pillars while still satisfying the average-distance and aspect-ratio requirements. In the improved method, as in the previously reported method, multiple optical fibers are first bundled together for simultaneous texturing of their distal tips. However, prior to texturing by exposure to an oxygen beam, the tips are first coated by vapor deposition of a thin, sparse layer of aluminum: The exposure to the aluminum vapor must be short enough (typically of the order of seconds) so that the aluminum nucleates into islands separated by uncoated areas. The coated tips are textured by exposure to a directed beam of hyperthermal (kinetic energy >1 eV) oxygen atoms and/or ions in a vacuum chamber, as in the previously reported method. The aluminum islands partially shield the underlying PMMA from oxidation and erosion by the beam, so that the cones or pillars remaining after texturing are wider than they would otherwise be. To some extent, the dimensions of the hills and the distances between them can be tailored through choice of the thickness of the aluminum coat and/or the oxygen-beam fluence. The figure illustrates an example of texturing of the tip of a PMMA optical fiber without and with prior aluminum coating.

Banks, Bruce A.↗

NASA NTRS1993

Discrete range clustering using Monte Carlo methods

For automatic obstacle avoidance guidance during rotorcraft low altitude flight, a reliable model of the nearby environment is needed. Such a model may be constructed by applying surface fitting techniques to the dense range map obtained by active sensing using radars. However, for covertness, passive sensing techniques using electro-optic sensors are desirable. As opposed to the dense range map obtained via active sensing, passive sensing algorithms produce reliable range at sparse locations, and therefore, surface fitting techniques to fill the gaps in the range measurement are not directly applicable. Both for automatic guidance and as a display for aiding the pilot, these discrete ranges need to be grouped into sets which correspond to objects in the nearby environment. The focus of this paper is on using Monte Carlo methods for clustering range points into meaningful groups. One of the aims of the paper is to explore whether simulated annealing methods offer significant advantage over the basic Monte Carlo method for this class of problems. We compare three different approaches and present application results of these algorithms to a laboratory image sequence and a helicopter flight sequence.

Chatterji, G. B.↗

NASA NTRS1994

Preconditioned domain decomposition scheme for three-dimensional aerodynamic sensitivity analysis

A discrete sensitivity analysis algorithm had previously been developed and applied to two-dimensional aerodynamic optimization problems, where the computational domains were discretized by using single grids. The sparse, unsymmetric systems of linear equations resulting from this algorithm were solved by a direct matrix inversion matrix. However, for large two-dimensional problems and, practically, all three-dimensional problems, direct inversion methods become inapplicable, primarily due to the prohibitive computer storage needed. In an attempt to alleviate such hindrances, the sensitivity analysis with domain decomposition (SADD) scheme was developed. This scheme divides the computational domain into smaller and nonoverlapping subdomains (multiblock grids) that are solved separately. Then, the final solution is constructed from the subdomain solutions. As the number of grid points in the interface boundaries of the subdomains becomes large, the computer memory required to store the effective coefficient matrix of these interface points starts to increase. Presented in this Technical Note is a preconditioned iterative procedure to overcome this particular problem.

Eleshaky, Mohamed E.↗

NASA NTRS1991

A pressure flux-split technique for computation of inlet flow behavior

A method for calculating the flow field in aircraft engine inlets is presented. The phenomena of inlet unstart and restart are investigated. Solutions of the reduced Navier-Stokes (RNS) equations are obtained with a time consistent direct sparse matrix solver that computes the transient flow field both internal and external to the inlet. Time varying shocks and time varying recirculation regions can be efficiently analyzed. The code is quite general and is suitable for the computation of flow for a wide variety of geometries and over a wide range of Mach and Reynolds numbers.

Pordal, H. S.↗

NASA NTRS1983

Preconditioners for the spectral multigrid method

The systems of algebraic equations which arise from spectral discretizations of elliptic equations are full and direct solutions of them are rarely feasible. Iterative methods are an attractive alternative because Fourier transform techniques enable the discrete matrix-vector products to be computed with nearly the same efficiency as is possible for corresponding but sparse finite difference discretizations. For realistic Dirichlet problems preconditioning is essential for acceptable convergence rates. A brief description of Chebyshev spectral approximations and spectral multigrid methods for elliptic problems is given. A survey of preconditioners for Dirichlet problems based on second-order finite difference methods is made. New preconditioning techniques based on higher order finite differences and on the spectral matrix itself are presented. The preconditioners are analyzed in terms of their spectra and numerical examples are presented.

Phillips, T. N.↗

NASA NTRS1986

Preconditioners for the spectral multigrid method

The systems of algebraic equations which arise from spectral discretizations of elliptic equations are full and direct solutions of them are rarely feasible. Iterative methods are an attractive alternative because Fourier transform techniques enable the discrete matrix-vector products to be computed with nearly the same efficiency as is possible for corresponding but sparse finite difference discretizations. For realistic Dirichlet problem preconditioning is essential for acceptable convergence rates. A brief description of Chebyshev spectral approximations and spectral multigrid methods for elliptic problems is given. A survey of preconditioners for Dirichlet problems based on second-order finite difference methods is made. New preconditioning techniques based on higher order finite differences and on the spectral matrix itself are presented. The preconditioners are analyzed in terms of their spectra and numerical examples are presented.

Phillips, T. N.↗

NASA NTRS2008

Application of Fast Multipole Methods to the NASA Fast Scattering Code

The NASA Fast Scattering Code (FSC) is a versatile noise prediction program designed to conduct aeroacoustic noise reduction studies. The equivalent source method is used to solve an exterior Helmholtz boundary value problem with an impedance type boundary condition. The solution process in FSC v2.0 requires direct manipulation of a large, dense system of linear equations, limiting the applicability of the code to small scales and/or moderate excitation frequencies. Recent advances in the use of Fast Multipole Methods (FMM) for solving scattering problems, coupled with sparse linear algebra techniques, suggest that a substantial reduction in computer resource utilization over conventional solution approaches can be obtained. Implementation of the single level FMM (SLFMM) and a variant of the Conjugate Gradient Method (CGM) into the FSC is discussed in this paper. The culmination of this effort, FSC v3.0, was used to generate solutions for three configurations of interest. Benchmarking against previously obtained simulations indicate that a twenty-fold reduction in computational memory and up to a four-fold reduction in computer time have been achieved on a single processor.

Dunn, Mark H.↗

NASA NTRS2023

Efficient Two-Way Coupled Analysis of Steady-State Particle-Laden Hypersonic Flows

A direct solution approach for surface erosion in particle-laden hypersonic flows is extended for use in low-cost two-way coupled solutions of dilute gas-particle flows. The Trajectory Control Volume method, which uses a sparse set of probe particles to predict surface erosion distributions on general vehicles, is reformulated for the solution of source terms by mean trajectory subdivision and computing a flux differencing. The approach is verified successfully against a boundary layer solution and shown to agree well with experimental measurements. A representative Mars entry case, with conditions and geometry based on the ExoMars Schiaparelli capsule, is solved with the approach to study the impact of two-way coupling on surface heating and erosion. Results indicate that for realistic loading conditions, heating is largely unmodified compared to one-way coupled results at peak heating trajectory conditions, and no measureable difference is observed in the surface erosion rate. At exaggerated loading conditions high enough to observe coupling effects, the worst case collisional heating can increase heating by up to 60%.

Particle Laden Flow↗

NASA NTRS2016

Investigation of Unsteady Pressure-Sensitive Paint (uPSP) and a Dynamic Loads Balance to Predict Launch Vehicle Buffet Environments

This NESC assessment examined the accuracy of estimating buffet loads on in-line launch vehicles without booster attachments using sparse unsteady pressure measurements. The buffet loads computed using sparse sensor data were compared with estimates derived using measurements with much higher spatial resolution. The current method for estimating launch vehicle buffet loads is through wind tunnel testing of models with approximately 400 unsteady pressure transducers. Even with this relatively large number of sensors, the coverage can be insufficient to provide reliable integrated unsteady loads on vehicles. In general, sparse sensor spacing requires the use of coherence-length-based corrections in the azimuthal and axial directions to integrate the unsteady pressures and obtain reasonable estimates of the buffet loads. Coherence corrections have been used to estimate buffet loads for a variety of launch vehicles with the assumption methodology results in reasonably conservative loads. For the Space Launch System (SLS), the first estimates of buffet loads exceeded the limits of the vehicle structure, so additional tests with higher sensor density were conducted to better define the buffet loads and possibly avoid expensive modifications to the vehicle design. Without the additional tests and improvements to the coherence-length analysis methods, there would have been significant impacts to the vehicle weight, cost, and schedule. If the load estimates turn out to be too low, there is significant risk of structural failure of the vehicle. This assessment used a combination of unsteady pressure-sensitive paint (uPSP), unsteady pressure transducers, and a dynamic force and moment balance to investigate the integration schemes used with limited unsteady pressure data by comparing them with direct integration of extremely dense fluctuating pressure measurements. An outfall of the assessment was to evaluate the potential of using the emerging uPSP technique in a production test environment for future launch vehicles. The results show that modifications to the current technique can improve the accuracy of buffet estimates. More importantly, the uPSP worked remarkably well and, with improvements to the frequency response, sensitivity, and productivity, will provide an enhanced method for measuring wind tunnel buffet forcing functions (BFFs).

Schuster, David M.↗

NASA NTRS1979

FORTRAN subroutines for out-of-core solutions of large complex linear systems

The design and usage of two main subprograms using direct methods to solve large linear complex systems, of the form Ax = b, whose coeffficient matrices are too large to be stored in core are described. The first main subprogram is for systems whose coefficient matrices are of a particular sparse structure, namely, the matrix A can be written in the form B + D, where B is a block-banded system, and D has only a few columns of nonzeros. Key elements of the algorithms used in the subprograms include: the data structure, the strategy for preserving numerical stability, the adaptability of the algorithms for dense systems as well as for block-profile systems.

Yip, E. L.↗

NASA NTRS2014

Pressure Sensitive Paint Applied to Flexible Models Project

One gap in current pressure-measurement technology is a high-spatial-resolution method for accurately measuring pressures on spatially and temporally varying wind-tunnel models such as Inflatable Aerodynamic Decelerators (IADs), parachutes, and sails. Conventional pressure taps only provide sparse measurements at discrete points and are difficult to integrate with the model structure without altering structural properties. Pressure Sensitive Paint (PSP) provides pressure measurements with high spatial resolution, but its use has been limited to rigid or semi-rigid models. Extending the use of PSP from rigid surfaces to flexible surfaces would allow direct, high-spatial-resolution measurements of the unsteady surface pressure distribution. Once developed, this new capability will be combined with existing stereo photogrammetry methods to simultaneously measure the shape of a dynamically deforming model in a wind tunnel. Presented here are the results and methodology for using PSP on flexible surfaces.

model deformation↗

NASA NTRS1985

Variation in efficiency of parallel algorithms

The present study has the objective to investigate some iterative parallel-processor linear equation solving algorithms with respect to efficiency for analyses of typical linear engineering systems. Attention is given to a set of n linear equations, Ku = p, where K = an n x n positive definite, sparsely populated, symmetric matrix, u = an n x 1 vector of unknown responses, and p = an n x 1 vector of prescribed constants. This study is concerned with a hybrid method in which iteration is used to solve the problem, while a direct method is used on the local processor level. Variations in the efficiency of parallel algorithms are explored. Measures of the efficiency are based on computer experiments regarding the algorithms. For all the algorithms, the wall clock time is found to decrease as the number of processors increases.

Hayashi, A.↗

NASA NTRS2004

Finite Element Analysis in Concurrent Processing: Computational Issues

The purpose of this research is to investigate the potential application of new methods for solving large-scale static structural problems on concurrent computers. It is well known that traditional single-processor computational speed will be limited by inherent physical limits. The only path to achieve higher computational speeds lies through concurrent processing. Traditional factorization solution methods for sparse matrices are ill suited for concurrent processing because the null entries get filled, leading to high communication and memory requirements. The research reported herein investigates alternatives to factorization that promise a greater potential to achieve high concurrent computing efficiency. Two methods, and their variants, based on direct energy minimization are studied: a) minimization of the strain energy using the displacement method formulation; b) constrained minimization of the complementary strain energy using the force method formulation. Initial results indicated that in the context of the direct energy minimization the displacement formulation experienced convergence and accuracy difficulties while the force formulation showed promising potential.

Sobieszczanski-Sobieski, Jaroslaw↗

NASA NTRS2021

Locating the Isolator Shock-Train Leading Edge with Limited Pressure Information

Real-time detection and control of the isolator shock-train leading edge (STLE) is important to the performance of high-speed air-breathing engines, such as dual-mode scramjets. Typically, the STLE location is determined using wall static-pressure measurements, but there are often restrictions on the placement and overall number of the pressure transducers, reducing the viability and accuracy of such approaches. To address these issues, we introduce the adaptive pressure profile (APP) method for estimating the STLE location. This method does not require extensive prior characterization of the isolator or engine model. Instead, it uses real-time pressure measurements from a small number of transducers to adaptively learn the isolator pressure profile and subsequently uses this deduced profile to estimate the STLE location in a data-driven manner. The APP method works well in situations with sparse transducer placement. It produces accurate estimates when the STLE location is 1) not bounded by two or more transducers or 2) between two transducers that are several isolator duct heights apart. We demonstrate the efficacy of the APP method using simulations and experimental data from direct-connect isolator models. This validation shows that the APP method is accurate and robust for different flow regimes, transducer configurations, and model geometries.

Gregory J. Hunt↗

NASA NTRS2016

The Global Drivers of Photosynthesis and Light Use Efficiency Seasonality: A Granger Frequency Causality Analysis

Photosynthesis and light use efficiency (LUE) are major factors in the evolution of the continental carbon cycle due to their contribution to gross primary production (GPP). However, while the drivers of photosynthesis and LUE on a plant or canopy scale can often be identified, significant uncertainties exist when modeling these on a global scale. This is due to sparse observations in regions such as the tropics and the lack of a direct global observation dataset. Although others have attempted to address this issue using correlations (Beer, 2010) or calculating GPP from vegetation indices (Running, 2004), in this study we take a new approach. We combine the statistical method of Granger frequency causality and partial Granger frequency causality with remote sensing data products (including sun-induced fluorescence used as a proxy for GPP) to determine the main environmental drivers of GPP across the globe.

photosynthesis↗