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At least 865 records · Page 48
All sourcesNASA NTRSDOE OSTI
NASA NTRS1993

Sensitivity analysis of Lyapunov and Riccati equations with application to controls-structures integrated design

Controls-structure integrated design is a complicated multidisciplinary design optimization problem which involves the state equations pertaining to open-loop eigenvalues and control laws. In order to alleviate the intensity of the computation, this study uses the adjoint variable method to derive sensitivity equations for the eigenvalue, Liapunov, and Riccati equations. These individual sensitivity equations are then combined together to form the multidisciplinary sensitivity equations for the control structure integrated design problems. A set of linear sensitivity equations, proportional in number to the number of performance functions involved in the optimization process, are solved. This proposed approach may provide a great saving in computer resources. The validity of the newly developed sensitivity equations is verified by numerical examples.

Hou, Gene↗
NASA NTRS1994

Variations in the modal characteristics of a telescopically deploying beam

The equations of motion for a two-segment deploying telescopic beam are derived through application of Lagrange's equation. The outer tube of the beam is fixed at one end and the inner tube slides freely relative to the fixed segment. The resulting nonlinear, non-autonomous set of equations is linearized and simplified to the standard Euler-Bernoulli partial differential equations for an elastic beam by freezing the deployment process at various stages of deployment, and examining the small amplitude and natural modes of vibration of the resulting configuration. Application of the natural boundary conditions and compatibility of motion relations for the two segments in their common region of overlap leads to a transcendental characteristic equation in the frequency parameter Beta(L). Numerical solution of the equation for the characteristic roots determines the modal frequencies, and the corresponding mode shapes are obtained from the general solution of the Euler-Bernoulli equation tailored to the natural boundary conditions. Sample results of modal frequencies and shapes are presented for various stages of deployment and discussed. It is shown that for all intermediate stages of deployment (between 0 and 100 percent) the spectral distribution is drastically altered by the appearance of regions of very closely spaced modal frequencies. The sources of this modal agglomeration are explored.

Amos, Anthony K.↗
NASA NTRS1988

On a family of nonoscillatory equations y double prime = phi(x)y

The oscillation or nonoscillation of a class of second-order linear differential equations is investigated analytically, with a focus on cases in which the functions phi(x) and y are complex-valued. Two linear transformations are introduced, and an asymptotic-decomposition procedure involving Shur triangularization is applied. The relationship of the present analysis to the nonoscillation criterion of Kneser (1896) and other more recent results is explored in two examples.

Gingold, H.↗
NASA NTRS1983

Inlet flow distortion in turbomachinery - Comparison of theory and experiment in a transonic fan stage

Consideration is given to both velocity and temperature circumferential inlet distortions at upstream infinity (Seidel et al., 1980). The blade rows here are modeled as semiactuator disks, and losses and quasi-steady deviation angle correlations are included in the analysis. The governing equations are linearized, and the perturbations in stagnation pressure and stagnation temperature at upstream infinity are represented as Fourier series. The flow in the rotor is modeled as inviscid, one-dimensional, unsteady, and compressible. the flow is steady elsewhere. The deviation angles for the rotor and stator are taken to be functions of the relative inlet angle and Mach number, and use is made of the correlations contained in Johnson and Bullock (1965). It is assumed that the losses in relative stagnation pressure in the rotor and stator occur across the trailing edge. Boundary conditions applied at the various stations furnish the equations that make it possible to solve for the several quantities introduced in the linearization of the governing equations.

Seidel, B. S.↗
NASA NTRS1993

An algorithm for solving the perturbed gas dynamic equations

The present application of a compact, higher-order central-difference approximation to the linearized Euler equations illustrates the multimodal character of these equations by means of computations for acoustic, vortical, and entropy waves. Such dissipationless central-difference methods are shown to propagate waves exhibiting excellent phase and amplitude resolution on the basis of relatively large time-steps; they can be applied to wave problems governed by systems of first-order partial differential equations.

Davis, Sanford↗
DOE OSTI2025

Greedy emulators for nuclear two-body scattering

Applications of reduced basis method emulators are increasing in low-energy nuclear physics because they enable fast and accurate sampling of high-fidelity calculations, enabling robust uncertainty quantification. Here, in this paper, we develop, implement, and test two model-driven emulators based on the (Petrov-)Galerkin projection using the prototypical test case of two-body scattering with the Minnesota potential and a more realistic local chiral potential. The high-fidelity scattering equations are solved with the matrix Numerov method, a reformulation of the popular Numerov recurrence relation for solving special second-order differential equations as a linear system of coupled equations. A novel error estimator based on reduced-space residuals is applied to an active learning approach (a greedy algorithm) to choosing training samples (“snapshots”) for the emulator and contrasted with a proper orthogonal decomposition (POD) approach. Both approaches allow for computationally efficient offline-online decompositions, but the greedy approach requires many fewer snapshot calculations. These developments set the groundwork for emulating scattering observables based on chiral nucleon-nucleon and three-nucleon interactions and optical models, where computational speed-ups are necessary for Bayesian uncertainty quantification. Our emulators and error estimators are widely applicable to linear systems.

Bayesian methods↗
DOE OSTI2021

Shear reflectivity compensation in full-waveform inversion using least-squares reverse-time migration

SUMMARY The computational cost of elastic-waveform inversion is too high for inverting PP reflections, while using acoustic full-waveform inversion (FWI) is inaccurate because it does not depend on the shear modulus/velocity/impedance that affects elastic PP wavefield amplitudes. To solve this problem, we develop a waveform inversion method that uses acoustic least-squares reverse-time migration (LSRTM) to compensate the shear reflectivity for acoustic FWI. Our method is based on the quasi-elastic-wave equation developed by Chapman et al. (2014). The quasi-elastic-wave equation uses a linearized acoustic-wave equation with shear modulus μ as a virtual source to correct the acoustic PP wavefield amplitudes toward elastic ones. Our waveform inversion method inverts for elastic parameters by minimizing the L2 norm of the difference between recorded and predicted PP reflections modelled using the quasi-elastic-wave equation. Numerical tests on synthetic and field data show that our method can properly handle the amplitudes of elastic PP reflections and provides an accurate estimate of the P- and S-wave velocities/impedances and, in some cases, the density. The method does not need the computationally expensive numerical solution to the elastic-wave equation. It also gives a better estimate of elastic parameters than a pure LSRTM method for elastic PP reflections.

Feng, Zongcai↗
NASA NTRS1971

A dynamic model of the human postural control system.

Description of a digital simulation of the pitch axis dynamics of a stick man. The difficulties encountered in linearizing the equations of motion are discussed; the conclusion reached is that a completely linear simulation is of such restricted validity that only a nonlinear simulation is of any practical use. Typical simulation results obtained from the full nonlinear model are illustrated.

Hill, J. C.↗
NASA NTRS1972

A dynamic model of the human postural control system

A digital simulation of the pitch axis dynamics of a stick man of figures is described. Difficulties encountered in linearizing the equations of motion are discussed; the conclusion reached is that a completely linear simulation is of such restricted validity that only a nonlinear simulation is of any practical use. Typical simulation results obtained from the full nonlinear model are presented.

Hill, J. C.↗
NASA NTRS1974

Aeroelastic stability of coupled flap-lag motion of hingeless helicopter blades at arbitrary advance ratios

Equations for large amplitude coupled flap-lag motion of a hingeless elastic helicopter blade in forward flight are derived. Only a torsionally rigid blade excited by quasi-steady aerodynamic loads is considered. The effects of reversed flow together with some new terms due to radial flow are included. Using Galerkin's method the spatial dependence is eliminated and the equations are linearized about a suitable equilibrium position. The resulting system of homogeneous periodic equations is solved using multivariable Floquet-Liapunov theory, and the transition matrix at the end of the period is evaluated by two separate methods. Computational efficiency of the two numerical methods is compared. Results illustrating the effects of forward flight and various important blade parameters on the stability boundaries are presented.

Friedmann, P.↗
NASA NTRS1984

The influence of dynamic inflow and torsional flexibility on rotor damping in forward flight from symbolically generated equations

The combined effects of blade torsion and dynamic inflow on the aeroelastic stability of an elastic rotor blade in forward flight are studied. The Helicopter Equations for Stability and Loads (HESL) program is extended to derive the governing equations of motion for the blade, and a Lagrangian formulation is used to obtain the equations in generalized coordinates. The program generates the steady-state and linearized perturbation equations in symbolic form and then codes them into FORTRAN subroutines. The coefficients for each equation and for each mode are identified through a numerical program; the latter can also be used to obtain the harmonic balance equations. The governing multiblade equations are derived explicitly using HESL. These equations can accommodate any number of elastic blade modes. Stability results are presented for several hingeless rotor blade structural models, and the influence of dynamic inflow in forward flight with an elastic hingeless rotor is investigated.

Reddy, T. S. R.↗
NASA NTRS1985

Linear analysis of an axially grooved rectangular gyrotron for harmonic operation

In an axially grooved rectangular waveguide the linearized Vlasov equation is solved to find the perturbed distribution function resulting from the electromagnetic forces on the electrons. The resulting beam current and the propagating electromagnetic waves of the cold tube are used in the inhomogeneous Maxwell's equation to derive the general dispersion equation. This equation is then transformed into the electron beam frame and the resulting linear growth rate of amplification is calculated. By maximizing the linear growth rate, the operational parameters of the gyrotron are then optimized.

Ferendeci, A. M.↗
NASA NTRS1988

Problems in fluid dynamics

A scheme was developed for the parametric differentiation and integration of gas dynamics equations. A numerical integration of the gas dynamics equations is necessarily performed for a specific set of parameter values. The linear variational equations are obtained by differentiating the exact equations with respect to each of the relevant parameters. The resulting matrix of flow quantities is referred to as the Jacobi matrix. The subsequent procedure is then straightforward. The method was tested for two dimensional supersonic flow past an airfoil, with airfoil thickness, camber, and angle of attack varied. This approach has great potential value for rapidly assessing the effect of design changes. The other focus of the work was on problems in fluid stability, bifurcations, and turbulence.

Sirovich, Lawrence↗
NASA NTRS1994

Transfer function modeling of damping mechanisms in distributed parameter models

This work formulates a method for the modeling of material damping characteristics in distributed parameter models which may be easily applied to models such as rod, plate, and beam equations. The general linear boundary value vibration equation is modified to incorporate hysteresis effects represented by complex stiffness using the transfer function approach proposed by Golla and Hughes. The governing characteristic equations are decoupled through separation of variables yielding solutions similar to those of undamped classical theory, allowing solution of the steady state as well as transient response. Example problems and solutions are provided demonstrating the similarity of the solutions to those of the classical theories and transient responses of nonviscous systems.

Slater, J. C.↗
NASA NTRS2016

On the Assessment of Acoustic Scattering and Shielding by Time Domain Boundary Integral Equation Solutions

Based on the time domain boundary integral equation formulation of the linear convective wave equation, a computational tool dubbed Time Domain Fast Acoustic Scattering Toolkit (TD-FAST) has recently been under development. The time domain approach has a distinct advantage that the solutions at all frequencies are obtained in a single computation. In this paper, the formulation of the integral equation, as well as its stabilization by the Burton-Miller type reformulation, is extended to cases of a constant mean flow in an arbitrary direction. In addition, a "Source Surface" is also introduced in the formulation that can be employed to encapsulate regions of noise sources and to facilitate coupling with CFD simulations. This is particularly useful for applications where the noise sources are not easily described by analytical source terms. Numerical examples are presented to assess the accuracy of the formulation, including a computation of noise shielding by a thin barrier motivated by recent Historical Baseline F31A31 open rotor noise shielding experiments. Furthermore, spatial resolution requirements of the time domain boundary element method are also assessed using point per wavelength metrics. It is found that, using only constant basis functions and high-order quadrature for surface integration, relative errors of less than 2% may be obtained when the surface spatial resolution is 5 points-per-wavelength (PPW) or 25 points-per-wavelength squared (PPW2).

Hu, Fang Q.↗
NASA NTRSDate not supplied

Sparse Linear Algebra Toolkit for Computational Aerodynamics

Finding solutions to sparse linear systems of equations is an essential step in Computational Engineering applications of interest to NASA. Linear systems of equations are composed and solved in almost every computational engineering application. The characteristics of linear systems vary greatly from one application to another. Accordingly, there are a wide variety of methods for the solution of linear systems of equations. The operations and methods prepared by the authors are focused on linear systems of interest to NASA, primarily those associated with Computational Fluid Dynamics (CFD), Aeroelasticity, and Aeroacoustics. The Sparse Linear Algebra Toolkit (SLAT) is a coordinated collection of software featuring operations, methods, and data structures that are useful when solving sparse linear systems of equations on modern computer architectures. The implemented operations and methods are designed and tuned for parallelism in shared memory, in distributed memory, and across the hybrid combination of distributed-shared memory. The toolkit includes novel methods and implementations for modern architectures and facilitates development of new approaches for meeting NASA’s evolving computational engineering challenges using evolving computer architectures that are not available in vendor libraries. In this paper, significant features and interfaces within SLAT are presented and verified for simulations performed with NASA’s CFD solver, FUN3D. The runtime and scaling performance of the Generalized Minimum Residual (GMRES) method implemented in SLAT is analyzed for the linear subproblems within the solution of turbulent Navier-Stokes equations employed in the simulation of high-lift configurations. Prior to this work, the SPARSKIT GMRES implementation was the only Krylov subspace method available within FUN3D. A strong scaling study shows the SLAT GMRES implementation facilitates accurate Reynolds-averaged Navier-Stokes CFD solutions between 15% and 56% faster than the SPARSKIT GMRES implementation.

Stephen L Wood↗
NASA NTRS1978

Forced oscillations in quadratically damped systems

Bayliss (1975) has studied the question whether in the case of linear differential equations the relationship between the stability of the homogeneous equations and the existence of almost periodic solutions to the inhomogeneous equation is preserved by finite difference approximations. In the current investigation analogous properties are considered for the case in which the damping is quadratic rather than linear. The properties of the considered equation for arbitrary forcing terms are examined and the validity is proved of a theorem concerning the characteristics of the unique solution. By using the Lipschitz continuity of the mapping and the contracting mapping principle, almost periodic solutions can be found for perturbations of the considered equation. Attention is also given to the Lipschitz continuity of the solution operator and the results of numerical tests which have been conducted to test the discussed theory.

Bayliss, A.↗