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NASA NTRS1991

Periodic trim solutions with HP-version finite elements in time

Finite Element in Time has been proven to be a powerful alternative solving strategy for the rotor craft trim problem. Additionally, Finite Element Method in Time has been developed in various versions like time-marching framework, Galerkin framework, Rayleigh-Ritz framework, and mixed formulation. Recently, this method was applied to the rotorcraft trim problem to obtain linearized solutions. The rotorcraft trim problem consists of trying to find a period solution for period-coefficient, differential equations subject to side constraints where certain force and momentum balance equations are forced to be equal to zero. There are free (or trim) parameters that are chosen to meet these side constraints. This project aims at expanding the application, in terms of the rotorcraft trim problem, from a linearized solution to nonlinear solution.

Source record↗

NASA NTRS2017

Stabilization of Hypersonic Boundary Layers by Linear and Nonlinear Optimal Perturbations

The effect of stationary, finite-amplitude, linear and nonlinear optimal perturbations on the modal disturbance growth in a Mach 6 axisymmetric flow over a 7 deg. half-angle cone with 0:126 mm nose radius and 0:305 m length is investigated. The freestream parameters (M = 6, Re(exp 1) = 18 x 10(exp. 6) /m) are selected to match the flow conditions of a previous experiment in the VKI H3 hypersonic tunnel. Plane-marching parabolized stability equations are used in conjunction with a partial-differential equation based planar eigenvalue analysis to characterize the boundary layer instability in the presence of azimuthally periodic streaks. The streaks are observed to stabilize nominally planar Mack mode instabilities, although oblique Mack mode and first-mode disturbances are destabilized. Experimentally measured transition onset in the absence of any streaks correlates with an amplification factor of N = 6 for the planar Mack modes. For high enough streak amplitudes, the transition threshold of N = 6 is not reached by the Mack mode instabilities within the length of the cone; however, subharmonic first-mode instabilities, which are destabilized by the presence of the streaks, do reach N = 6 near the end of the cone. The highest stabilization is observed at streak amplitudes of approximately 20 percent of the freestream velocity. Because the use of initial disturbance profiles based on linear optimal growth theory may yield suboptimal control in the context of nonlinear streaks, the computational predictions are extended to nonlinear optimal growth theory. Results show that by using nonlinearly optimal perturbation leads to slightly enhanced stabilization of plane Mack mode disturbances as well as reduced destabilization of subharmonic first-mode disturbances.

Incompressible boundary layer↗

NASA NTRS1976

A theoretical study of the acoustic impedance of orifices in the presence of a steady grazing flow

An analysis of the oscillatory fluid flow in the vicinity of a circular orifice with a steady grazing flow is presented. The study is similar to that of Hersh and Rogers but with the addition of the grazing flow. Starting from the momentum and continuity equations, a considerably simplified system of partial differential equations is developed with the assumption that the flow can be described by an oscillatory motion superimposed upon the known steady flow. The equations are seen to be linear in the region where the grazing flow effects are dominant, and a solution and the resulting orifice impedance are presented for this region. The nonlinearity appears to be unimportant for the usual conditions found in aircraft noise suppressors. Some preliminary conclusions of the study are that orifice resistance is directly proportional to grazing flow velocity (known previously from experimental data) and that the orifice inductive (mass reactance) end correction is not a function of grazing flow. This latter conclusion is contrary to the widely held notion that grazing flow removes the effect of the orifice inductive end correction. This conclusion also implies that the experimentally observed total inductance reduction with grazing flow might be in the flow within the orifice rather than in the end correction.

Rice, E. J.↗

NASA NTRS1983

Spectral methods for time dependent partial differential equations

The theory of spectral methods for time dependent partial differential equations is reviewed. When the domain is periodic Fourier methods are presented while for nonperiodic problems both Chebyshev and Legendre methods are discussed. The theory is presented for both hyperbolic and parabolic systems using both Galerkin and collocation procedures. While most of the review considers problems with constant coefficients the extension to nonlinear problems is also discussed. Some results for problems with shocks are presented.

Gottlieb, D.↗

NASA NTRS2015

Projection-Based Reduced Order Modeling for Spacecraft Thermal Analysis

This paper presents a mathematically rigorous, subspace projection-based reduced order modeling (ROM) methodology and an integrated framework to automatically generate reduced order models for spacecraft thermal analysis. Two key steps in the reduced order modeling procedure are described: (1) the acquisition of a full-scale spacecraft model in the ordinary differential equation (ODE) and differential algebraic equation (DAE) form to resolve its dynamic thermal behavior; and (2) the ROM to markedly reduce the dimension of the full-scale model. Specifically, proper orthogonal decomposition (POD) in conjunction with discrete empirical interpolation method (DEIM) and trajectory piece-wise linear (TPWL) methods are developed to address the strong nonlinear thermal effects due to coupled conductive and radiative heat transfer in the spacecraft environment. Case studies using NASA-relevant satellite models are undertaken to verify the capability and to assess the computational performance of the ROM technique in terms of speed-up and error relative to the full-scale model. ROM exhibits excellent agreement in spatiotemporal thermal profiles (<0.5% relative error in pertinent time scales) along with salient computational acceleration (up to two orders of magnitude speed-up) over the full-scale analysis. These findings establish the feasibility of ROM to perform rational and computationally affordable thermal analysis, develop reliable thermal control strategies for spacecraft, and greatly reduce the development cycle times and costs.

0000↗

NASA NTRS1975

On the convergence of certain finite-difference schemes by an inverse-matrix method

The inverse-matrix method of analyzing the convergence of the solution of a given system of finite-difference equations to the solution of the corresponding system of partial-differential equations is discussed and generalized. The convergence properties of a time- and space-centered differencing of the diffusion equation are analyzed as well as a staggered grid differencing of the Cauchy-Riemann equations. These two schemes are significant since they serve as simplified model algorithms for two recently developed methods used to calculate nonlinear aerodynamic flows.

Steger, J. L.↗

NASA NTRS1987

Lyapunov exponents for infinite dimensional dynamical systems

Classically it was held that solutions to deterministic partial differential equations (i.e., ones with smooth coefficients and boundary data) could become random only through one mechanism, namely by the activation of more and more of the infinite number of degrees of freedom that are available to such a system. It is only recently that researchers have come to suspect that many infinite dimensional nonlinear systems may in fact possess finite dimensional chaotic attractors. Lyapunov exponents provide a tool for probing the nature of these attractors. This paper examines how these exponents might be measured for infinite dimensional systems.

Mhuiris, Nessan Mac Giolla↗

NASA NTRS2015

Kodiak: An Implementation Framework for Branch and Bound Algorithms

Recursive branch and bound algorithms are often used to refine and isolate solutions to several classes of global optimization problems. A rigorous computation framework for the solution of systems of equations and inequalities involving nonlinear real arithmetic over hyper-rectangular variable and parameter domains is presented. It is derived from a generic branch and bound algorithm that has been formally verified, and utilizes self-validating enclosure methods, namely interval arithmetic and, for polynomials and rational functions, Bernstein expansion. Since bounds computed by these enclosure methods are sound, this approach may be used reliably in software verification tools. Advantage is taken of the partial derivatives of the constraint functions involved in the system, firstly to reduce the branching factor by the use of bisection heuristics and secondly to permit the computation of bifurcation sets for systems of ordinary differential equations. The associated software development, Kodiak, is presented, along with examples of three different branch and bound problem types it implements.

Smith, Andrew P.↗

NASA NTRS1997

Nonlinear Waves and Particle Heating in the Auroral Ionosphere

In the two year period, the researchers have developed innovative 2-D simulation codes for modeling the interaction between electron beams and plasma waves and particles in the auroral ionosphere. These new simulations include kinetic wave particles as well as wave-wave interactions near and above approximately 500 km. The principal numerical model consists of partial-differential equations which evolve large amplitude magnetized 2-D Langmuir waves self-consistently with the electron distribution function. Progress was also made in developing 2-D Particle in Cell (PIC) codes.

Goldman, Martin↗

NASA NTRS1984

The effect of surface contamination on thermocapillary flow in a two-dimensional slot

In an analysis of the effect of insoluble surfactants on the steady capillary flow in a shallow slot, the problem is formulated as a two-dimensional channel heated differentially from the side, with a free surface on which an insoluble surface species is adsorbed. The vertical boundaries are considered to be isothermal and impenetrable and the lower horizontal boundary to be insulated and impenetrable. Lubrication theory and matched asymptotic expansions are used to reduce the problem to a single nonlinear integral-algebraic equation. For small elasticity E, the surface velocity is nearly constant, with the result that, for small Pe, the surface concentration is nearly uniform while, for large Pe, it develops a boundary layer near the stagnation point at the cold wall, resulting in high local surface concentrations. Condensed surfactant layers will form only with E much less than one and Pe much greater than one.

Homsy, G. M.↗

NASA NTRS1985

Reducing the number of variational equations in the implementation of multiple shooting

The standard method of multiple shooting for a system of n first-order differential equations with k unknown initial conditions requires the integration of k sets of variational equations on the first shot and n sets of variational equations on every shot thereafter. This paper describes a variant of multiple shooting that requires the solution of k sets of variational equations on every shot. The technique applies to both linear and nonlinear boundary-value problems. Techniques to deal with difficulties unique to the solution of nonlinear problems are suggested.

Krogh, F. T.↗

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 NTRS1993

Control of nonlinear systems represented in quasilinear form

Methods to synthesize controllers for nonlinear systems are developed by exploiting the fact that under mild differentiability conditions, systems of the form: x-dot = f(x) + G(x)u can be represented in quasilinear form, viz: x-dot = A(x)x + B(x)u. Two classes of control methods are investigated. The first is zero-look-ahead control, where the control input depends only on the current values of A(x) and B(x). For this case the control input is computed by continuously solving a matrix Riccati equation as the system progresses along a trajectory. The second is controllers with look-ahead, where the control input depends on the future behavior of A(x) and B(x). These controllers use the similarity between quasilinear systems and linear time varying systems to find approximate solutions to optimal control type problems. The methods that are developed are not guaranteed to be globally stable. However in simulation studies they were found to be useful alternatives for synthesizing control laws for a general class of nonlinear systems.

Coetsee, Josef A.↗

NASA NTRS1987

A numerical solution of Duffing's equations including the prediction of jump phenomena

Numerical methodology for the solution of Duffing's differential equation is presented. Algorithms for the prediction of multiple equilibrium solutions and jump phenomena are developed. In addition, a filtering algorithm for producing steady state solutions is presented. The problem of a rigidly clamped circular plate subjected to cosinusoidal pressure loading is solved using the developed algorithms (the plate is assumed to be in the geometrically nonlinear range). The results accurately predict regions of solution multiplicity and jump phenomena.

Moyer, E. T., Jr.↗

NASA NTRS1995

Analysis of discretization errors in LES

All numerical simulations of turbulence (DNS or LES) involve some discretization errors. The integrity of such simulations therefore depend on our ability to quantify and control such errors. In the classical literature on analysis of errors in partial differential equations, one typically studies simple linear equations (such as the wave equation or Laplace's equation). The qualitative insight gained from studying such simple situations is then used to design numerical methods for more complex problems such as the Navier-Stokes equations. Though such an approach may seem reasonable as a first approximation, it should be recognized that strongly nonlinear problems, such as turbulence, have a feature that is absent in linear problems. This feature is the simultaneous presence of a continuum of space and time scales. Thus, in an analysis of errors in the one dimensional wave equation, one may, without loss of generality, rescale the equations so that the dependent variable is always of order unity. This is not possible in the turbulence problem since the amplitudes of the Fourier modes of the velocity field have a continuous distribution. The objective of the present research is to provide some quantitative measures of numerical errors in such situations. Though the focus of this work is LES, the methods introduced here can be just as easily applied to DNS. Errors due to discretization of the time-variable are neglected for the purpose of this analysis.

Ghosal, Sandip↗

NASA NTRS1999

Low-Storage, Explicit Runge-Kutta Schemes for the Compressible Navier-Stokes Equations

The derivation of storage explicit Runge-Kutta (ERK) schemes has been performed in the context of integrating the compressible Navier-Stokes equations via direct numerical simulation. Optimization of ERK methods is done across the broad range of properties, such as stability and accuracy efficiency, linear and nonlinear stability, error control reliability, step change stability, and dissipation/dispersion accuracy, subject to varying degrees of memory economization. Following van der Houwen and Wray, 16 ERK pairs are presented using from two to five registers of memory per equation, per grid point and having accuracies from third- to fifth-order. Methods have been assessed using the differential equation testing code DETEST, and with the 1D wave equation. Two of the methods have been applied to the DNS of a compressible jet as well as methane-air and hydrogen-air flames. Derived 3(2) and 4(3) pairs are competitive with existing full-storage methods. Although a substantial efficiency penalty accompanies use of two- and three-register, fifth-order methods, the best contemporary full-storage methods can be pearl), matched while still saving two to three registers of memory.

Kennedy, Chistopher A.↗

NASA NTRS2015

In-Flight Aeroelastic Stability of the Thermal Protection System on the NASA HIAD, Part II: Nonlinear Theory and Extended Aerodynamics

Conical shell theory and a supersonic potential flow aerodynamic theory are used to study the nonlinear pressure buckling and aeroelastic limit cycle behavior of the thermal protection system for NASA's Hypersonic Inflatable Aerodynamic Decelerator. The structural model of the thermal protection system consists of an orthotropic conical shell of the Donnell type, resting on several circumferential elastic supports. Classical Piston Theory is used initially for the aerodynamic pressure, but was found to be insufficient at low supersonic Mach numbers. Transform methods are applied to the convected wave equation for potential flow, and a time-dependent aerodynamic pressure correction factor is obtained. The Lagrangian of the shell system is formulated in terms of the generalized coordinates for all displacements and the Rayleigh-Ritz method is used to derive the governing differential-algebraic equations of motion. Aeroelastic limit cycle oscillations and buckling deformations are calculated in the time domain using a Runge-Kutta method in MATLAB. Three conical shell geometries were considered in the present analysis: a 3-meter diameter 70 deg. cone, a 3.7-meter 70 deg. cone, and a 6-meter diameter 70 deg. cone. The 6-meter configuration was loaded statically and the results were compared with an experimental load test of a 6-meter HIAD. Though agreement between theoretical and experimental strains was poor, the circumferential wrinkling phenomena observed during the experiments was captured by the theory and axial deformations were qualitatively similar in shape. With Piston Theory aerodynamics, the nonlinear flutter dynamic pressures of the 3-meter configuration were in agreement with the values calculated using linear theory, and the limit cycle amplitudes were generally on the order of the shell thickness. The effect of axial tension was studied for this configuration, and increasing tension was found to decrease the limit cycle amplitudes when the circumferential elastic supports were neglected, but resulted in more complex behavior when the supports were included. The nominal flutter dynamic pressure of the 3.7-meter configuration was significantly lower than that of the 3-meter, and it was found that two sets of natural modes coalesce to flutter modes near the same dynamic pressure. This resulted in a significant drop in the limit cycle frequencies at higher dynamic pressures, where the flutter mode with the lower frequency becomes more critical. Pre-buckling pressure loads and the aerodynamic pressure correction factor were studied for all geometries, and these effects resulted in significantly lower flutter boundaries compared with Piston Theory alone. The maximum dynamic pressure predicted by aerodynamic simulations of a proposed 3.7-meter HIAD vehicle was still lower than any of the calculated flutter dynamic pressures, suggesting that aeroelastic effects for this vehicle are of little concern.

Goldman, Benjamin D.↗

NASA NTRS1974

Estimation and detection of signals in multiplicative noise

We define a class of detection-estimation problems on matrix Lie groups in which the observation noise is multiplicative in nature. By examining the differential versions of the hypotheses, which are bilinear, we are able to derive the relevant likelihood ratio formula and the associated optimal estimation equations for the signal given the observations and the assumption that the signal is present. These estimation equations are of interest in their own right, in that they represent a finite-dimensional optimal solution to a nonlinear estimation problem and consist of a Kalman-Bucy filter along with the on-line computation of the solution of the associated Riccati equation, which is driven by the observations. The usefulness of these results is illustrated via an example concerning the detection of an actuator failure in a rigid-body rotational control system.

Willsky, A. S.↗