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At least 1,423 records · Page 79

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DOE OSTI2020

A scalable matrix-free spectral element approach for unsteady PDE constrained optimization using PETSc/TAO

In this work, we provide a new approach for the efficient matrix-free application of the transpose of the Jacobian for the spectral element method for the adjoint-based solution of partial differential equation (PDE) constrained optimization. This results in optimizations of nonlinear PDEs using explicit integrators where the integration of the adjoint problem is not more expensive than the forward simulation. Solving PDE constrained optimization problems entails combining expertise from multiple areas, including simulation, computation of derivatives, and optimization. The Portable, Extensible Toolkit for Scientific computation (PETSc) together with its companion package, the Toolkit for Advanced Optimization (TAO), is an integrated numerical software library that contains an algorithmic/software stack for solving linear systems, nonlinear systems, ordinary differential equations, differential algebraic equations, and large-scale optimization problems and, as such, is an ideal tool for performing PDE-constrained optimization. This paper describes an efficient approach in which the software stack provided by PETSc/TAO can be used for large-scale nonlinear time-dependent problems. Time integration can involve a range of high-order methods, both implicit and explicit. The PDE-constrained optimization algorithm used is gradient-based and seamlessly integrated with the simulation of the physical problem.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1989

The role of shock induced trailing-edge separation in limit cycle oscillations

The potential role of shock induced trailing edge separation (SITES) in limit cycle oscillations (LCO) was established. It was shown that the flip-flop characteristics of transition to and from SITES as well as its hysteresis could couple with wing modes with torsional motion and low damping. This connection led to the formulation of a very simple nonlinear math model using the linear equations of motion with a nonlinear step forcing function with hysteresis. A finite difference solution with time was developed and calculations were made for the F-111 TACT were used to determine the step forcing function due to SITES transition. Since no data were available for the hysteresis, a parameter study was conducted allowing the hysteresis effect to vary. Very small hysteresis effects, which were within expected bounds, were required to obtain reasonable response levels that essentially agreed with flight test results. Also in agreement with wind tunnel tests, LCO calculations for the 1/6 scale F-111 model showed that the model should have not experienced LCO.

Cunningham, Atlee M., Jr.↗

NASA NTRS1985

The effects of molecular weight on the single lap shear creep and constant strain rate behavior of thermoplastic polyimidesulfone adhesive

The bonded shear creep and constant strain rate behaviors of zero, one, and three percent endcapped thermoplastic polyimidesulfone adhesive were examined at room and elevated temperatures. Endcapping was accomplished by the addition of phthalic anhydrides. The primary objective was to determine the effects of molecular weight on the mechanical properties of the adhesive. Viscoelastic and nonlinear elastic constitutive equations were utilized to model the adhesive. Ludwik's and Crochet's relations were used to describe the experimental failure data. The effects of molecular weight changes on the above mentioned mechanical behavior were assessed. The viscoelastic Chase-Goldsmith and elastic nonlinear relations gave a good fit to the experimental stress strain behavior. Crochet's relations based on Maxwell and Chase-Goldsmith models were fit to delayed failure data. Ludwik's equations revealed negligible rate dependence. Ultimate stress levels and the safe levels for creep stresses were found to decrease as molecular weight was reduced.

Dembosky, Stanley K.↗

NASA NTRS1985

Theory of wing rock

Wing rock is one type of lateral-directional instabilities at high angles of attack. To predict wing rock characteristics and to design airplanes to avoid wing rock, parameters affecting wing rock characteristics must be known. A new nonlinear aerodynamic model is developed to investigate the main aerodynamic nonlinearities causing wing rock. In the present theory, the Beecham-Titchener asymptotic method is used to derive expressions for the limit-cycle amplitude and frequency of wing rock from nonlinear flight dynamics equations. The resulting expressions are capable of explaining the existence of wing rock for all types of aircraft. Wing rock is developed by negative or weakly positive roll damping, and sustained by nonlinear aerodynamic roll damping. Good agreement between theoretical and experimental results is obtained.

Hsu, C.-H.↗

NASA NTRS1987

Vectorizable implicit algorithms for the flux-difference split, three-dimensional Navier-Stokes equations

The computational efficiency of four vectorizable implicit algorithms is assessed when applied to calculate steady-state solutions to the three-dimensional, incompressible Navier-Stokes equations in general coordinates. Two of these algorithms are characterized as hybrid schemes; that is, they combine some approximate factorization in two coordinate directions with relaxation in the remaining spatial direction. The other two algorithms utilize an approximate factorization approach which yields two-factor algorithms for three-dimensional systems. All four algorithms are implemented in identical high-resolution upwind schemes for the flux-difference split Navier-Stokes equations. These highly nonlinear schemes are obtained by extending an implicit Total Variation Diminishing (TVD) scheme recently developed for linear one-dimensional systems of hyperbolic conservation laws to the three-dimensional Navier-Stokes equations. The computation of vortical flow over a sharp-edged, thin delta wing has been chosen as a common numerical test case. The convergence of the algorithms is discussed and the accuracy of the computed flow-field results is assessed. The validity of the present results are demonstrated by a comparison with experimental data.

Hartwich, P. M.↗

NASA NTRS2011

Nonlinear Oscillators in Space Physics

We discuss dynamical systems that produce an oscillation without an external time dependent source. Numerical results are presented for nonlinear oscillators in the Em1h's atmosphere, foremost the quasi-biennial oscillation (QBOl. These fluid dynamical oscillators, like the solar dynamo, have in common that one of the variables in a governing equation is strongly nonlinear and that the nonlinearity, to first order, has particular form. of 3rd or odd power. It is shown that this form of nonlinearity can produce the fundamental li'equency of the internal oscillation. which has a period that is favored by the dynamical condition of the fluid. The fundamental frequency maintains the oscillation, with no energy input to the system at that particular frequency. Nonlinearities of 2nd or even power could not maintain the oscillation.

Lester,Daniel↗

NASA NTRS2012

Secondary Instability of Second Modes in Hypersonic Boundary Layers

Second mode disturbances dominate the primary instability stage of transition in a number of hypersonic flow configurations. The highest amplification rates of second mode disturbances are usually associated with 2D (or axisymmetric) perturbations and, therefore, a likely scenario for the onset of the three-dimensionality required for laminar-turbulent transition corresponds to the parametric amplification of 3D secondary instabilities in the presence of 2D, finite amplitude second mode disturbances. The secondary instability of second mode disturbances is studied for selected canonical flow configurations. The basic state for the secondary instability analysis is obtained by tracking the linear and nonlinear evolution of 2D, second mode disturbances using nonlinear parabolized stability equations. Unlike in previous studies, the selection of primary disturbances used for the secondary instability analysis was based on their potential relevance to transition in a low disturbance environment and the effects of nonlinearity on the evolution of primary disturbances was accounted for. Strongly nonlinear effects related to the self-interaction of second mode disturbances lead to an upstream shift in the upper branch neutral location. Secondary instability computations confirm the previously known dominance of subharmonic modes at relatively small primary amplitudes. However, for the Purdue Mach 6 compression cone configuration, it was shown that a strong fundamental secondary instability can exist for a range of initial amplitudes of the most amplified second mode disturbance, indicating that the exclusive focus on subharmonic modes in the previous applications of secondary instability theory to second mode primary instability may not have been fully justified.

Li, Fei↗

NASA NTRS1979

An elliptic representation of coupled boundary layers and inviscid core for computation of separated internal flows

This report presents a viscous-inviscid calculation method for the prediction of turbulent incompressible flows in diffusers with small regions of stall. Integral turbulent boundary layer equations (BLE) are used to provide boundary conditions for a finite difference representation of the inviscid core. The BLE are applied at the floating displacement-thickness line, resulting in a simulataneous set of nonlinear block tridiagonal equations which are solved iteratively using a sucessive line-relaxation technique. Results of the method have been applied to calculate the performance of a variety of asymmetric stalled diffusers.

Ghose, S.↗

NASA NTRS1988

Applications of an exponential finite difference technique

An exponential finite difference scheme first presented by Bhattacharya for one dimensional unsteady heat conduction problems in Cartesian coordinates was extended. The finite difference algorithm developed was used to solve the unsteady diffusion equation in one dimensional cylindrical coordinates and was applied to two and three dimensional conduction problems in Cartesian coordinates. Heat conduction involving variable thermal conductivity was also investigated. The method was used to solve nonlinear partial differential equations in one and two dimensional Cartesian coordinates. Predicted results are compared to exact solutions where available or to results obtained by other numerical methods.

Handschuh, Robert F.↗

NASA NTRS2010

Development and Breakdown of Goertler Vortices in High Speed Boundary Layers

The nonlinear development of G rtler instability over a concave surface gives rise to a highly distorted stationary flow in the boundary layer that has strong velocity gradients in both spanwise and wall-normal directions. This distorted flow is susceptible to strong, high frequency secondary instability that leads to the onset of transition. For high Mach number flows, the boundary layer is also subject to the second mode instability. The nonlinear development of G rtler vortices and the ensuing growth and breakdown of secondary instability, the G rtler vortex interactions with second mode instabilities as well as oblique second mode interactions are examined in the context of both internal and external hypersonic configurations using nonlinear parabolized stability equations, 2-D eigenvalue analysis and direct numerical simulation. For G rtler vortex development inside the Purdue Mach 6 Ludwieg tube wind tunnel, multiple families of unstable secondary eigenmodes are identified and their linear and nonlinear evolution is examined. The computation of secondary instability is continued past the onset of transition to elucidate the physical mechanisms underlying the laminar breakdown process. Nonlinear breakdown scenarios associated with transition over a Mach 6 compression cone configuration are also explored.

Li, Fei↗

NASA NTRS2016

Improved Convergence and Robustness of USM3D Solutions on Mixed-Element Grids

Several improvements to the mixed-elementUSM3Ddiscretization and defect-correction schemes have been made. A new methodology for nonlinear iterations, called the Hierarchical Adaptive Nonlinear Iteration Method, has been developed and implemented. The Hierarchical Adaptive Nonlinear Iteration Method provides two additional hierarchies around a simple and approximate preconditioner of USM3D. The hierarchies are a matrix-free linear solver for the exact linearization of Reynolds-averaged Navier-Stokes equations and a nonlinear control of the solution update. Two variants of the Hierarchical Adaptive Nonlinear Iteration Method are assessed on four benchmark cases, namely, a zero-pressure-gradient flat plate, a bump-in-channel configuration, the NACA 0012 airfoil, and a NASA Common Research Model configuration. The new methodology provides a convergence acceleration factor of 1.4 to 13 over the preconditioner-alone method representing the baseline solver technology.

Pandya, Mohagna J.↗

NASA NTRS2016

Improved Convergence and Robustness of USM3D Solutions on Mixed-Element Grids

Several improvements to the mixed-element USM3D discretization and defect-correction schemes have been made. A new methodology for nonlinear iterations, called the Hierarchical Adaptive Nonlinear Iteration Method, has been developed and implemented. The Hierarchical Adaptive Nonlinear Iteration Method provides two additional hierarchies around a simple and approximate preconditioner of USM3D. The hierarchies are a matrix-free linear solver for the exact linearization of Reynolds-averaged Navier-Stokes equations and a nonlinear control of the solution update. Two variants of the Hierarchical Adaptive Nonlinear Iteration Method are assessed on four benchmark cases, namely, a zero-pressure-gradient flat plate, a bump-in-channel configuration, the NACA 0012 airfoil, and a NASA Common Research Model configuration. The new methodology provides a convergence acceleration factor of 1.4 to 13 over the preconditioner-alone method representing the baseline solver technology.

Pandya, Mohagna J.↗

NASA NTRS1995

On the nonlinear stability of the unsteady, viscous flow of an incompressible fluid in a curved pipe

The stability of the flow of an incompressible, viscous fluid through a pipe of circular cross-section curved about a central axis is investigated in a weakly nonlinear regime. A sinusoidal pressure gradient with zero mean is imposed, acting along the pipe. A WKBJ perturbation solution is constructed, taking into account the need for an inner solution in the vicinity of the outer bend, which is obtained by identifying the saddle point of the Taylor number in the complex plane of the cross-sectional angle co-ordinate. The equation governing the nonlinear evolution of the leading order vortex amplitude is thus determined. The stability analysis of this flow to periodic disturbances leads to a partial differential system dependent on three variables, and since the differential operators in this system are periodic in time, Floquet theory may be applied to reduce this system to a coupled infinite system of ordinary differential equations, together with homogeneous uncoupled boundary conditions. The eigenvalues of this system are calculated numerically to predict a critical Taylor number consistent with the analysis of Papageorgiou. A discussion of how nonlinear effects alter the linear stability analysis is also given, and the nature of the instability determined.

Shortis, Trudi A.↗

NASA NTRS1986

Numerical simulation of a controlled boundary layer

The problem of interest is the boundary layer over a flat plate. The three standard laminar flow control (LFC) techniques are pressure gradient, suction, and heating. The parameters used to describe the amount of control in the context of the boundary layer equations are introduced. The numerical method required to find the mean flow, the linear eigenvalues of the Orr-Sommerfeld equation, and the full, nonlinear, 3-D solution of the Navier-Stokes equations are outlined. A secondary instability exists for the parallel boundary subject to uniform pressure gradient, suction, or heating. Selective control of the spanwise mode reduces the secondary instability in the parallel boundary layer at low Reynolds number.

Zang, Thomas A.↗

NASA NTRS2001

State-Dependent Pseudo-Linear Filter for Spacecraft Attitude and Rate Estimation

This paper presents the development and performance of a special algorithm for estimating the attitude and angular rate of a spacecraft. The algorithm is a pseudo-linear Kalman filter, which is an ordinary linear Kalman filter that operates on a linear model whose matrices are current state estimate dependent. The nonlinear rotational dynamics equation of the spacecraft is presented in the state space as a state-dependent linear system. Two types of measurements are considered. One type is a measurement of the quaternion of rotation, which is obtained from a newly introduced star tracker based apparatus. The other type of measurement is that of vectors, which permits the use of a variety of vector measuring sensors like sun sensors and magnetometers. While quaternion measurements are related linearly to the state vector, vector measurements constitute a nonlinear function of the state vector. Therefore, in this paper, a state-dependent linear measurement equation is developed for the vector measurement case. The state-dependent pseudo linear filter is applied to simulated spacecraft rotations and adequate estimates of the spacecraft attitude and rate are obtained for the case of quaternion measurements as well as of vector measurements.

Bar-Itzhack, Itzhack Y.↗

DOE OSTI2022

Approximation rates of DeepONets for learning operators arising from advection–diffusion equations

Here we present the analysis of approximation rates of operator learning in Chen and Chen (1995) and Lu et al. (2021), where continuous operators are approximated by a sum of products of branch and trunk networks. In this work, we consider the rates of learning solution operators from both linear and nonlinear advection–diffusion equations with or without reaction. We find that the approximation rates depend on the architecture of branch networks as well as the smoothness of inputs and outputs of solution operators.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2019

Laminar-Turbulent Transition Upstream of the Entropy-Layer Swallowing Location in Hypersonic Boundary Layers

Numerical and experimental studies have demonstrated that modal growth of planar Mack modes is responsible for laminar-turbulent transition on sharp cones at hypersonic speeds. However, the physical mechanisms that lead to transition onset upstream of the entropy-layer swallowing location over sufficiently blunt geometries are not well understood as yet. Modal amplification is too weak or nonexistent to initiate transition at moderate-to-large bluntness values. Nonmodal analysis shows that, with increasing nose bluntness, both planar and oblique traveling disturbances that peak within the entropy layer experience appreciable energy amplification. However, because of the relatively weak signature of the nonmodal traveling disturbances within the boundary-layer region, the route to transition onset subsequent to the nonmodal growth remains unclear. Thus, nonlinear parabolized stability equations (NPSE) and direct numerical simulations (DNS) have been used to investigate the potential transition mechanisms over a 7-degree blunt cone that was tested in the AFRL Mach-6 high-Reynoldsnumber facility. Computations are performed to separately follow the nonlinear development of two classes of inflow disturbances, namely, a pair of oblique traveling waves with equal but opposite angles with respect to the mean flow direction and a planar traveling wave. Results in both cases show an excellent agreement between the NPSE and DNS predictions, establishing that the NPSE is an accurate and efficient technique for predicting the nonlinear development for these particular nonmodal traveling disturbances. Computations reveal that the oblique mode interactions lead to the generation of stationary streaks inside the boundary layer that, in turn, facilitate the growth of a subharmonic sinuous disturbance. For relatively modest amplitudes of the inflow disturbance, the oblique-mode breakdown can lead to transition at the measured location of transition onset during the experiment. On the other hand, the nonlinear development of a planar traveling wave leads to the formation of inclined structures just above the boundary-layer edge and these structures are strongly reminiscent of the transitional events observed during blunt cone experiments by using schlieren flow visualizations.

Entropy↗

DOE OSTI2021

Equations of Motion for the Vertical Rigid-Body Rotor: Linear and Nonlinear Cases

Centuries ago, the prolific mathematician Leonhard Euler (1707–1783) wrote down the equations of motion (EOM) for the heavy symmetrical top with one point fixed. The resulting set of equations turned out to be nonlinear and had a limited number of closed-form solutions. Today, tools such as transfer matrix and finite elements enable the calculation of the rotor dynamic properties for rotor-bearing systems. Some of these tools rely on the “linearized” version of the EOM to calculate the eigenvalues, unbalance response, or transients in these systems. In fact, industry standards mandate that rotors be precisely balanced to have safe operational characteristics. However, in some cases, the nonlinear aspect of the EOM should be considered. The purpose of this paper is to show examples of how the linear vs. nonlinear formulations differ. This paper will also show how excessive unbalance is capable of dramatically altering the behavior of the system and can produce chaotic motions associated with the “jump” phenomenon.

42 ENGINEERING↗