The root locus method - A survey
Asymptotic behavior of polynomial root locus in presence/absence of time lag, outlining procedures for constructing root locus diagrams
Krall, A. M.↗
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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
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A pseudospectral Legendre method for hyperbolic equations with an improved stability condition
A new pseudospectral method is introduced for solving hyperbolic partial differential equations. This method uses different grid points than previously used pseudospectral methods: in fact the grid points are related to the zeroes of the Legendre polynomials. The main advantage of this method is that the allowable time step is proportional to the inverse of the number of grid points 1/N rather than to 1/n(2) (as in the case of other pseudospectral methods applied to mixed initial boundary value problems). A highly accurate time discretization suitable for these spectral methods is discussed.
Tal-Ezer, Hillel↗
Direct Numerical Simulation of Incompressible Pipe Flow Using a B-Spline Spectral Method
A numerical method based on b-spline polynomials was developed to study incompressible flows in cylindrical geometries. A b-spline method has the advantages of possessing spectral accuracy and the flexibility of standard finite element methods. Using this method it was possible to ensure regularity of the solution near the origin, i.e. smoothness and boundedness. Because b-splines have compact support, it is also possible to remove b-splines near the center to alleviate the constraint placed on the time step by an overly fine grid. Using the natural periodicity in the azimuthal direction and approximating the streamwise direction as periodic, so-called time evolving flow, greatly reduced the cost and complexity of the computations. A direct numerical simulation of pipe flow was carried out using the method described above at a Reynolds number of 5600 based on diameter and bulk velocity. General knowledge of pipe flow and the availability of experimental measurements make pipe flow the ideal test case with which to validate the numerical method. Results indicated that high flatness levels of the radial component of velocity in the near wall region are physical; regions of high radial velocity were detected and appear to be related to high speed streaks in the boundary layer. Budgets of Reynolds stress transport equations showed close similarity with those of channel flow. However contrary to channel flow, the log layer of pipe flow is not homogeneous for the present Reynolds number. A topological method based on a classification of the invariants of the velocity gradient tensor was used. Plotting iso-surfaces of the discriminant of the invariants proved to be a good method for identifying vortical eddies in the flow field.
Loulou, Patrick↗
Measurement of very large flow angles with non-nulling seven-hole probes
This paper describes a method for measuring local direction and total and static pressures of a flow by means of a fixed probe, provided that the local air flow does not make an angle of more than 80 degrees with the axis of the probe. The probe is easily manufactured from standard-sized tubing materials. The power series calibration method used with the probe results in explicit polynomial expressions for the desired aerodynamic properties. The calibration method is easily programmed on a data acquisition system. This paper includes an example of a complete incompressible calibration.
Gallington, R. W.↗
Simplified computational approach for dual-probe heat-pulse method
Two equations are currently available for estimating soil volumetric heat capacity (pc) with the dual-probe heat-pulse (DPHP) method. One is simple but gives only approximate results because it assumes that the DPHP sensor releases an impulse of heat instantaneously. The other explicitly accounts for the finite duration of heating and gives exact results. Unfortunately, the equation that gives exact results involves the exponential integral function, which is not available in most computer spreadsheet software packages or data logger function libraries. In this note we introduce an approximation of the exact equation that contains only simple algebraic functions. The approximation consists of the first five terms of a Taylor series, which are written as a telescoped polynomial for computational purposes. For most applications of the DPHP method, the polynomial approximation gives estimates of pc that are at least an order of magnitude more accurate than estimates obtained from the simple equation based on instantaneous heating.
NASA Center JSC↗
Convergence acceleration for vector sequences and applications to computational fluid dynamics
Some recent developments in acceleration of convergence methods for vector sequences are reviewed. The methods considered are the minimal polynomial extrapolation, the reduced rank extrapolation, and the modified minimal polynomial extrapolation. The vector sequences to be accelerated are those that are obtained from the iterative solution of linear or nonlinear systems of equations. The convergence and stability properties of these methods as well as different ways of numerical implementation are discussed in detail. Based on the convergence and stability results, strategies that are useful in practical applications are suggested. Two applications to computational fluid mechanics involving the three dimensional Euler equations for ducted and external flows are considered. The numerical results demonstrate the usefulness of the methods in accelerating the convergence of the time marching techniques in the solution of steady state problems.
Sidi, Avram↗
Convergence acceleration for vector sequences and applications to computational fluid dynamics
Some recent developments in acceleration of convergence methods for vector sequences are reviewed. The methods considered are the minimal polynomial extrapolation, the reduced rank extrapolation, and the modified minimal polynomial extrapolation. The vector sequences to be accelerated are those that are obtained from the iterative solution of linear or nonlinear systems of equations. The convergence and stability properties of these methods as well as different ways of numerical implementation are discussed in detail. Based on the convergence and stability results, strategies that are useful in practical applications are suggested. Two applications to computational fluid mechanics involving the three dimensional Euler equations for ducted and external flows are considered. The numerical results demonstrate the usefulness of the methods in accelerating the convergence of the time marching techniques in the solution of steady state problems.
Sidi, Avram↗
High-Order Space-Time Methods for Conservation Laws
Current high-order methods such as discontinuous Galerkin and/or flux reconstruction can provide effective discretization for the spatial derivatives. Together with a time discretization, such methods result in either too small a time step size in the case of an explicit scheme or a very large system in the case of an implicit one. To tackle these problems, two new high-order space-time schemes for conservation laws are introduced: the first is explicit and the second, implicit. The explicit method here, also called the moment scheme, achieves a Courant-Friedrichs-Lewy (CFL) condition of 1 for the case of one-spatial dimension regardless of the degree of the polynomial approximation. (For standard explicit methods, if the spatial approximation is of degree p, then the time step sizes are typically proportional to 1/p(exp 2)). Fourier analyses for the one and two-dimensional cases are carried out. The property of super accuracy (or super convergence) is discussed. The implicit method is a simplified but optimal version of the discontinuous Galerkin scheme applied to time. It reduces to a collocation implicit Runge-Kutta (RK) method for ordinary differential equations (ODE) called Radau IIA. The explicit and implicit schemes are closely related since they employ the same intermediate time levels, and the former can serve as a key building block in an iterative procedure for the latter. A limiting technique for the piecewise linear scheme is also discussed. The technique can suppress oscillations near a discontinuity while preserving accuracy near extrema. Preliminary numerical results are shown
Huynh, H. T.↗
Uniform high order spectral methods for one and two dimensional Euler equations
Uniform high order spectral methods to solve multi-dimensional Euler equations for gas dynamics are discussed. Uniform high order spectral approximations with spectral accuracy in smooth regions of solutions are constructed by introducing the idea of the Essentially Non-Oscillatory (ENO) polynomial interpolations into the spectral methods. The authors present numerical results for the inviscid Burgers' equation, and for the one dimensional Euler equations including the interactions between a shock wave and density disturbance, Sod's and Lax's shock tube problems, and the blast wave problem. The interaction between a Mach 3 two dimensional shock wave and a rotating vortex is simulated.
Cai, Wei↗
Uniform high-order spectral methods for one- and two-dimensional Euler equations
Uniform high order spectral methods to solve multi-dimensional Euler equations for gas dynamics are discussed. Uniform high order spectral approximations with spectral accuracy in smooth regions of solutions are constructed by introducing the idea of the Essentially Non-Oscillatory (ENO) polynomial interpolations into the spectral methods. The authors present numerical results for the inviscid Burgers' equation, and for the one-dimensional Euler equations including the interactions between a shock wave and density disturbance, Sod's and Lax's shock tube problems, and the blast wave problem. The interaction between a Mach 3 two-dimensional shock wave and a rotating vortex is simulated.
Cai, Wei↗
Discontinuous Galerkin and Related Methods for ODE
A defining feature of the discontinuous Galerkin (DG) method for ODE is that the piecewise polynomial solution can have a jump discontinuity at the beginning of each step. Starting from the standard integral formulation, the DG method is derived here in differential form. The key ingredient is a polynomial called the correction function, which helps ‘correct’ the discontinuous solution by approximating the jump and yields a continuous one. Under the right Radau quadrature, this continuous solution is identical to the solutions by the right Radau collocation and the continuous Galerkin (CG) methods. Next, the correction function facilitates the construction of the associated implicit Runge-Kutta schemes (IRK-DG). Different quadratures for DG result in different IRK-DG methods: left Radau quadrature in Radau IA, right Radau quadrature in Radau IIA or right Radau collocation, and Gauss quadrature in a method called DG-Gauss. The construction of IRK-DG clarifies the meaning and facilitates the proofs of various 𝐵(𝑝), 𝐶(𝜂), and 𝐷(𝜁) conditions for accuracy. The two consequences of these conditions are that all 𝑠-stage IRK-DG methods are accurate to order 2𝑠 − 1, and the IRK-DG methods of Radau type are unique. Numerical examples showing the behavior of the DG solutions are provided. In all, the correction function plays a key role and helps establish the relations among the DG, IRK-DG, collocation, and CG schemes.
numerical methods↗
Acceleration of convergence of vector sequences
A general approach to the construction of accelerated convergence methods for vector sequences is proposed. A simplified version of minimal polynomial extrapolation is emphasized. The convergence of this method is analyzed and it is shown that it is especially suitable for accelerating the convergence of vector sequences that are obtained when one solves linear systems of equations iteratively.
Sidi, A.↗
Design Under Uncertainty for Conceptual Aircraft Design Leveraging Analytical Gradients
The purpose of this paper is to extend previously demonstrated methodologies for design under uncertainty, leveraging analytical gradients to higher fidelity analysis for use in conceptual aircraft design. Previous work developed methods to generate analytical derivatives through polynomial chaos expansion, eliminating the need to estimate derivatives via complex step or finite difference. In this research, the authors build upon the methods to include physics-based aircraft design codes for aircraft design under uncertainty. This extends the previous work’s case study, which employed analytical aerodynamics and Breguet range estimations for wing design, to a higher fidelity level. In addition, this work extends previous work on interface development between the Uncertainty Quantification with Polynomial Chaos Expansion (UQPCE) software and Model-Based Systems Analysis and Engineering (MBSA&E) frameworks. This paper will discuss the development work necessary to perform multidisciplinary design under uncertainty as well as demonstrate the mechanics of interfacing UQPCE and conceptual aircraft design tools such as NASA’s Aviary code. In a case study, a conceptual aircraft design under uncertainty was conducted and compared against a traditional deterministic design. When given information about the uncertainty space from UQPCE, the optimizer was able to shape the output distribution and produce a more robust design
UQ↗
Design Under Uncertainty for Conceptual Aircraft Design Leveraging Analytical Gradients
The purpose of this paper is to extend previously demonstrated methodologies for design under uncertainty, leveraging analytical gradients to higher fidelity analysis for use in conceptual aircraft design. Previous work developed methods to generate analytical derivatives through polynomial chaos expansion, eliminating the need to estimate derivatives via complex step or finite difference. In this research, the authors build upon the methods to include physics-based aircraft design codes for aircraft design under uncertainty. This extends the previous work’s case study, which employed analytical aerodynamics and Breguet range estimations for wing design, to a higher fidelity level. In addition, this work extends previous work on interface development between the Uncertainty Quantification with Polynomial Chaos Expansion (UQPCE) software and Model-Based Systems Analysis and Engineering (MBSA&E) frameworks. This paper will discuss the development work necessary to perform multidisciplinary design under uncertainty as well as demonstrate the mechanics of interfacing UQPCE and conceptual aircraft design tools such as NASA’s Aviary code. In a case study, a conceptual aircraft design under uncertainty was conducted and compared against a traditional deterministic design. When given information about the uncertainty space from UQPCE, the optimizer was able to shape the output distribution and produce a more robust design.
UQ↗
Radar image registration and rectification
Two techniques for radar image registration and rectification are presented. In the registration method, a general 2-D polynomial transform is defined to accomplish the geometric mapping from one image into the other. The degree and coefficients of the polynomial are obtained using an a priori found tiepoint data set. In the second part of the paper, a rectification procedure is developed that models the distortion present in the radar image in terms of the radar sensor's platform parameters and the topographic variations of the imaged scene. This model, the ephemeris data and the digital topographic data are then used in rectifying the radar image. The two techniques are then used in registering and rectifying two examples of radar imagery. Each method is discussed as to its benefits, shortcomings and registration accuracy.
Naraghi, M.↗
On adaptive weighted polynomial preconditioning for Hermitian positive definite matrices
The conjugate gradient algorithm for solving Hermitian positive definite linear systems is usually combined with preconditioning in order to speed up convergence. In recent years, there has been a revival of polynomial preconditioning, motivated by the attractive features of the method on modern architectures. Standard techniques for choosing the preconditioning polynomial are based only on bounds for the extreme eigenvalues. Here a different approach is proposed, which aims at adapting the preconditioner to the eigenvalue distribution of the coefficient matrix. The technique is based on the observation that good estimates for the eigenvalue distribution can be derived after only a few steps of the Lanczos process. This information is then used to construct a weight function for a suitable Chebyshev approximation problem. The solution of this problem yields the polynomial preconditioner. In particular, we investigate the use of Bernstein-Szego weights.
Fischer, Bernd↗
Method reduces computer time for smoothing functions and derivatives through ninth order polynomials
Analysis presented is an efficient technique to adjust previously calculated orthogonal polynomial coefficients for an odd number of equally spaced data points. The adjusting technique derivation is for a ninth order polynomial. It reduces computer time for smoothing functions.
Glauz, R. D.↗
Computation of aerodynamic interference effects on oscillating airfoils with controls in ventilated subsonic wind tunnels
Lift interference effects are discussed based on Bland's (1968) integral equation. A mathematical existence theory is utilized for which convergence of the numerical method has been proved for general (square-integrable) downwashes. Airloads are computed using orthogonal airfoil polynomial pairs in conjunction with a collocation method which is numerically equivalent to Galerkin's method and complex least squares. Convergence exhibits exponentially decreasing error with the number n of collocation points for smooth downwashes, whereas errors are proportional to 1/n for discontinuous downwashes. The latter can be reduced to 1/n to the m+1 power with mth-order Richardson extrapolation (by using m = 2, hundredfold error reductions were obtained with only a 13% increase of computer time). Numerical results are presented showing acoustic resonance, as well as the effect of Mach number, ventilation, height-to-chord ratio, and mode shape on wind-tunnel interference. Excellent agreement with experiment is obtained in steady flow, and good agreement is obtained for unsteady flow.
Fromme, J. A.↗