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All sourcesNASA NTRSDOE OSTI
NASA NTRS1986

Desingularization of periodic vortex sheet roll-up

An analytical approach is used in an attempt to model the evolution of a vortex sheet past the critical time by means of a desingularization method. Evolution of the sheet, which is embedded in a two-dimensional flow, is described by approximating the total circulation between a fixed material point and an imaginary point on the curve of the flow. A linear dispersion relation is defined which shows that the short wavelength modes of the sheet are not unstable and therefore do adversely effect the computations as artifacts of round-off errors. The desingularization approach is demonstrated to converge beyond the critical time for the vortex sheet. Application of the technique for the study of the vortex sheet shed from an elliptically loaded wing is indicated.

Krasny, R.↗
NASA NTRS1987

A numerical simulation to verify the stress-free growth of silicon crystal ribbon

Thermal stresses developed during the growth of silicon crystal ribbon have been shown to be negligible, thus eliminating residual stresses and dislocations, if the temperature profile satisfies a second-order partial differential equation inside the ribbon. This has been numerically verified through a finite element model, an outline of which is presented here. This model shows that, for homogeneous isotropic material with temperature independent thermal expansion coefficients, thermal stresses will vanish if the temperature profile satisfies the Laplacian. A comparison of stresses due to uniform and nonuniform temperature gradients in the plane of the ribbon is also presented. The strategies employed to control the round-off error and to validate the computer model are discussed.

Ray, Sujit K.↗
NASA NTRS1987

Three dimensional numerical scattering from ocean-like surfaces

A new method of calculating the electric field scattered from two-dimensional conducting rough surfaces is presented. This method uses the operator conjugate gradients technique and the extended boundary condition equation. This avoids the build-up of round-off errors encountered in the inversion of large matrices. It also avoids the singularities of the integral kernels usually encountered in the method of moments. Techniques for improving the rate of convergence by applying physical constraints are discussed. This method is presently being used to calculate the polarization signatures from simulated ocean-like surfaces.

Rodriguez, Ernesto↗
NASA NTRS1987

Computation of vortex sheet roll-up in the Trefftz plane

Two vortex-sheet evolution problems arising in aerodynamics are studied numerically. The approach is based on desingularizing the Cauchy principal value integral which defines the sheet's velocity. Numerical evidence is presented which indicates that the approach converges with respect to refinement in the mesh-size and the smoothing parameter. For elliptic loading, the computed roll-up is in good agreement with Kaden's asymptoic spiral at early times. Some aspects of the solution's instability to short-wavelength perturbations, for a small value of the smoothing parameter, are inferred by comparing calculations performed with different levels of computer round-off error. The tip vortices' deformation, due to their mutual interaction, is shown in a long-time calculation. Computations for a simulated fuselage-flap configuration show a complicated process of roll-up, deformation and interaction involving the tip vortex and the inboard neighboring vortices.

Krasny, Robert↗
NASA NTRS1989

On the equivalence of Gaussian elimination and Gauss-Jordan reduction in solving linear equations

A novel general approach to round-off error analysis using the error complexity concepts is described. This is applied to the analysis of the Gaussian Elimination and Gauss-Jordan scheme for solving linear equations. The results show that the two algorithms are equivalent in terms of our error complexity measures. Thus the inherently parallel Gauss-Jordan scheme can be implemented with confidence if parallel computers are available.

Tsao, Nai-Kuan↗
NASA NTRS1990

Simulation Of Unsteady, Viscous, Incompressible Flow

Method for numerical solution of Navier-Stokes equations of viscous, incompressible flow developed based on use of fractional-step procedure. Accurate to second order in both space and time. Attempt made to minimize Poisson-equation difficulties by choosing pressures at centers and volume fluxes across faces of computational cells as dependent variables instead of familiar Cartesian components of velocity. Choice ensures satisfaction of discrete equation of conservation of mass to within round-off errors in any coordinate system and has favorable effects on convergence properties.

Rosenfeld, Moshe↗
NASA NTRS1992

Schroedinger's radial equation - Solution by extrapolation

A high-accuracy numerical method for the solution of a 1D Schroedinger equation that is suitable for a diatomic molecule, obtained by combining a finite-difference method with iterative extrapolation to the limit, is presently shown to have several advantages over more conventional methods. Initial guesses for the term values are obviated, and implementation of the algorithm is straightforward. The method is both less sensitive to round-off error, and faster than conventional methods for equivalent accuracy. These advantages are illustrated through the solution of Schroedinger's equation for a Morse potential function suited for HCl and a numerically derived Rydberg-Klein-Rees potential function for the X 1Sigma(+) state of CO.

Goorvitch, D.↗
NASA NTRS1993

On the 'delta-equations' for vortex sheet evolution

We use a set of equations, sometimes referred to as the 'delta-equations', to approximate the two-dimensional inviscid motion of an initially circular vortex sheet released from rest in a cross-flow. We present numerical solutions of these equations for the case with delta-square = 0 (for which the equations are exact) and for delta-square greater than 0. For small values of the smoothing parameter delta, a spectral filter must be used to eliminate spurious instabilities due to round-off error. Two singularities appear simultaneously in the vortex sheet when delta-square = 0 at a critical time t(c). After t(c), the solutions do not converge as the computational mesh is refined. With delta-square greater than 0, converged solutions were found for all values of delta-square when t is less than t(c), and for all but the two smallest values of delta-square used when t is greater than t(c). Our results show that, when delta-square is greater than 0, the vortex sheet deforms into two doubly branched spirals some time after t(c). The limiting solution as delta approaching 0 clearly exists and equals the delta = 0 solution when t is less than t(c).

Rottman, James W.↗
NASA NTRS1988

An alternative to Guyan reduction of finite-element models

Structural modeling is a key part of structural system identification for large space structures. Finite-element structural models are commonly used in practice because of their general applicability and availability. The initial models generated by using a standard computer program such as NASTRAN, ANSYS, SUPERB, STARDYNE, STRUDL, etc., generally contain tens of thousands of degrees of freedom. The models must be reduced for purposes of identification. Not only does the magnitude of the identification effort grow exponentially as a function of the number of degrees of freedom, but numerical procedures may also break down because of accumulated round-off errors. Guyan reduction is usually applied after a static condensation. Misapplication of Guyan reduction can lead to serious modeling errors. It is quite unfortunate and disappointing, since the accuracy of the original detailed finite-element model one tries very hard to achieve is lost by the reduction. First, why and how Guyan reduction always causes loss of accuracy is examined. An alternative approach is then introduced. The alternative can be thought of as an improvement of Guyan reduction, the Rayleigh-Ritz method, and in particular the recent algorithm of Wilson, Yuan, and Dickens. Unlike Guyan reduction, the use of the alternative does not need any special insight, experience, or skill for partitioning the structural degrees of freedom. In addition to model condensation, this alternative approach can also be used for predicting analytically, quickly, and economically, what are those structural modes that are excitable by a force actuator at a given trial location. That is, in the excitation of the structural modes for identification, it can be used for guiding the placement of the force actuators.

Lin, Jiguan Gene↗
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 NTRS2001

Numerical Study of Buoyancy and Different Diffusion Effects on the Structure and Dynamics of Triple Flames

Numerical simulations of 2-D triple flames under gravity force have been implemented to identify the effects of gravity on triple flame structure and propagation properties and to understand the mechanisms of instabilities resulting from both heat release and buoyancy effects. A wide range of gravity conditions, heat release, and mixing widths for a scalar mixing layer are computed for downward-propagating (in the same direction with the gravity vector) and upward-propagating (in the opposite direction of the gravity vector) triple flames. Results of numerical simulations show that gravity strongly affects the triple flame speed through its contribution to the overall flow field. A simple analytical model for the triple flame speed, which accounts for both buoyancy and heat release, is developed. Comparisons of the proposed model with the numerical results for a wide range of gravity, heat release and mixing width conditions, yield very good agreement. The analysis shows that under neutral diffusion, downward propagation reduces the triple flame speed, while upward propagation enhances it. For the former condition, a critical Froude number may be evaluated, which corresponds to a vanishing triple flame speed. Downward-propagating triple flames at relatively strong gravity effects have exhibited instabilities. These instabilities are generated without any artificial forcing of the flow. Instead disturbances are initiated by minute round-off errors in the numerical simulations, and subsequently amplified by instabilities. A linear stability analysis on mean profiles of stable triple flame configurations have been performed to identify the most amplified frequency in spatially developed flows. The eigenfunction equations obtained from the linearized disturbance equations are solved using the shooting method. The linear stability analysis yields reasonably good agreements with the observed frequencies of the unstable triple flames. The frequencies and amplitudes of disturbances increase with the magnitude of the gravity vector. Moreover, disturbances appear to be most amplified just downstream of the premixed branches. The effects of mixing width and differential diffusion are investigated and their roles on the flame stability are studied.

Chen, Jyh-Yuan↗
NASA NTRS1994

Numerical Generation of Asymmetric Flows about Slender Bodies of Revolution at Extreme Incidence

It is well known that slender bodies of revolution will develop an asymmetric, unsteady flow pattern in experimental tests, if the angle of incidence to the oncoming stream is above a critical value. It has been suggested that the origin of these asymmetric flows may stem from geometric imperfections of the model being tested, or from disturbances in the oncoming stream. In numerical simulations, it is possible to generate bodies of revolution which are perfectly symmetric about their longitudinal axis, and to impose uniform flow conditions which are free from disturbances. The current work presents numerical simulations of the flow about an ogive-cylinder configuration at 40 and 60 degree angle of incidence. These simulations. were performed using numerical algorithms which are also symmetric about the lateral plane of the cylinder body. The flowfields at 40 degree angle of attack were seen to remain symmetric to the round-off accuracy of the computer. At 60 degree angle of attack, a lateral force coefficient developed of O(1) which progressed to an alternate vortex shedding in time. The nature of this lateral force generation and vortex shedding was dependent on the choice of numerical algorithm. The origin of the asymmetries observed in the 60 degree angle of attack computations were traced to round-off errors in the implicit block-matrix inverter. A means of inverting the implicit operator matrices, which maintains the symmetry of the overall numerical algorithm was implemented.

Murman, Scott M.↗
NASA NTRS2007

The alpha(3) Scheme - A Fourth-Order Neutrally Stable CESE Solver

The conservation element and solution element (CESE) development is driven by a belief that a solver should (i) enforce conservation laws in both space and time, and (ii) be built from a non-dissipative (i.e., neutrally stable) core scheme so that the numerical dissipation can be controlled effectively. To provide a solid foundation for a systematic CESE development of high order schemes, in this paper we describe a new 4th-order neutrally stable CESE solver of the advection equation Theta u/Theta + alpha Theta u/Theta x = 0. The space-time stencil of this two-level explicit scheme is formed by one point at the upper time level and three points at the lower time level. Because it is associated with three independent mesh variables u(sup n) (sub j), (u(sub x))(sup n) (sub j) , and (uxz)(sup n) (sub j) (the numerical analogues of u, Theta u/Theta x, and Theta(exp 2)u/Theta x(exp 2), respectively) and four equations per mesh point, the new scheme is referred to as the alpha(3) scheme. As in the case of other similar CESE neutrally stable solvers, the alpha(3) scheme enforces conservation laws in space-time locally and globally, and it has the basic, forward marching, and backward marching forms. These forms are equivalent and satisfy a space-time inversion (STI) invariant property which is shared by the advection equation. Based on the concept of STI invariance, a set of algebraic relations is developed and used to prove that the alpha(3) scheme must be neutrally stable when it is stable. Moreover it is proved rigorously that all three amplification factors of the alpha(3) scheme are of unit magnitude for all phase angles if |v| <= 1/2 (v = alpha delta t/delta x). This theoretical result is consistent with the numerical stability condition |v| <= 1/2. Through numerical experiments, it is established that the alpha(3) scheme generally is (i) 4th-order accurate for the mesh variables u(sup n) (sub j) and (ux)(sup n) (sub j); and 2nd-order accurate for (uxx)(sup n) (sub j). However, in some exceptional cases, the scheme can achieve perfect accuracy aside from round-off errors.

Chang, Sin-Chung↗
NASA NTRS2019

A Mixed Real and Floating-Point Solver

Reasoning about mixed real and floating-point constraints is essential for developing accurate analysis tools for floating-point pro- grams. This paper presents FPRoCK, a prototype tool for solving mixed real and floating-point formulas. FPRoCK transforms a mixed formula into an equisatisfiable one over the reals. This formula is then solved using an off-the-shelf SMT solver. FPRoCK is also integrated with the PRECiSA static analyzer, which computes a sound estimation of the round-off error of a floating-point program. It is used to detect infeasible computational paths, thereby improving the accuracy of PRECiSA.

Salvia, Rocco↗
NASA NTRS2019

Provably Correct Floating-Point Implementation of a Point-In-Polygon Algorithm

The problem of determining whether or not a point lies inside a given polygon occurs in many applications. In air traffic management concepts, a correct solution to the point-in-polygon problem is critical to geofencing systems for Unmanned Aerial Vehicles and in weather avoidance applications. Many mathematical methods can be used to solve the point-in-polygon problem. Unfortunately, a straightforward floating- point implementation of these methods can lead to incorrect results due to round-off errors. In particular, these errors may cause the control flow of the program to diverge with respect to the ideal real-number algorithm. This divergence potentially results in an incorrect point-in- polygon determination even when the point is far from the edges of the polygon. This paper presents a provably correct implementation of a point-in-polygon method that is based on the computation of the winding number. This implementation is mechanically generated from a source- to-source transformation of the ideal real-number specification of the algorithm. The correctness of this implementation is formally verified within the Frama-C analyzer, where the proof obligations are discharged using the Prototype Verification System (PVS).

Moscato, Mariano M.↗
NASA NTRS1986

A theoretical analysis of the round-off propagation in different Kalman filter implementations

A theoretical analysis is made of the error propagation due to numerical round-off for four different Kalman filter implementations: the conventional Kalman filter, the square root covariance filter, the square root information filter and the Chandrasekhar square root filter. From these error models, new insights about the applicability of the different filters and their sensitivity to round-off, is obtained. It is shown that the CKF may become completely unreliable when the original plant is unstable, and that this is easily circumvented by a number of techniques. The square root filters, often quoted to possess a conditioning or sensitivity that is the square root of that of the CKF, are shown to possess this property only for the computation of the covariance of the filtered signal and not for the computation of the Kalman gain or the filtered estimate. Finally, the Chandrasekhar filter is shown to be numerically unstable.

Verhaegen, M.↗
NASA NTRS1998

Cascade Error Projection with Low Bit Weight Quantization for High Order Correlation Data

In this paper, we reinvestigate the solution for chaotic time series prediction problem using neural network approach. The nature of this problem is such that the data sequences are never repeated, but they are rather in chaotic region. However, these data sequences are correlated between past, present, and future data in high order. We use Cascade Error Projection (CEP) learning algorithm to capture the high order correlation between past and present data to predict a future data using limited weight quantization constraints. This will help to predict a future information that will provide us better estimation in time for intelligent control system. In our earlier work, it has been shown that CEP can sufficiently learn 5-8 bit parity problem with 4- or more bits, and color segmentation problem with 7- or more bits of weight quantization. In this paper, we demonstrate that chaotic time series can be learned and generalized well with as low as 4-bit weight quantization using round-off and truncation techniques. The results show that generalization feature will suffer less as more bit weight quantization is available and error surfaces with the round-off technique are more symmetric around zero than error surfaces with the truncation technique. This study suggests that CEP is an implementable learning technique for hardware consideration.

Duong, Tuan A.↗