Chris L. Pettit

Uncertainty Quantification in Aeroelasticity: Recent Results and Research Challenges

Static and dynamic aeroelasticity considerations are a particularly important component of airframe design because they often control safety and performance. Consequently, the impact of uncertainty on aeroelastic response prediction has begun to receive substantial attention in the research literature. In this paper, general sources of uncertainty that complicate airframe design and testing are briefly described. Recent applications of uncertainty quantification to various aeroelastic problems, for example, flutter flight testing, prediction of limit-cycle oscillations, and design optimization with aeroelastic constraints, are reviewed with an emphasis on new physical insights and promising paths toward improved design methods that have resulted from these studies. Several challenges and needs are explored to suggest future steps that will enable practical application of uncertainty quantification in aeroelasticity design and certification.

Optimization of a Wing Structure for Gust Response and Aileron Effectiveness

Reliability-based weight optimization of a generic, e ghter-like wing structure is conducted for gust response and aileron effectiveness constraints. The formulation accounts for parametric uncertainties in these aeroelastic response quantities. Reliability indices measure the probability of satisfying each constraint, and a preliminary design procedure is developed in which constraints are enforced on these indices. This framework integrates ASTROS for structural and loads analysis, object-oriented MATLAB ® tools for reliability analysis, and DOT for optimization and most probable point estimation. The reliability analysis algorithm takes advantage of adaptive nonlinear approximations to compensate for nonlinearity of the failure surfaces. The wing structure is modeled with e nite elements, each of which is assumed to have random thickness of known standard deviation. Young’ s modulus of the wing skin material is also assumed to be random. Mean thickness values are taken as design variables. Linear unsteady aerodynamics is used to estimate frequency response functions caused by continuous gust loads. Reliability index constraints are enforced for gust-induced bending moment and shear at the wing’ s root, and also for aileron effectiveness. Redistribution of structural mass by the optimizer produces designs with improved aeroelastic performance reliability and relatively small weight penalties.

Polynomial Chaos Expansion Applied to Airfoil Limit Cycle Oscillations

Monte Carlo simulation was used to analyze the aeroelastic response of a typicalsection airfoil subject to uncertainties in key aerodynamic and structural dynamic parameters. When the Monte Carlo results were used to compute the coe‐cients of a Hermite-chaos expansion, di‐culties arose at large times in modeling the probabilistic response of the system in limit-cycle oscillation; in particular, the simulated time histories consistently decayed to zero at large times. A sinusoidal model problem was analyzed to clarify the issues responsible for these di‐culties, which do not appear to have been formally previously in the literature. The projected solution coe‐cients were found to successively gain and lose dominance over other coe‐cients as time increases in a manner that causes any flxed expansion to fail over a simulation time of su‐cient duration. An intuitive explanation of this behavior is ofiered, and a waveletbased stochastic expansion is proposed for future efiorts to improve the convergence of the expansion at large times.

Prediction of Nonlinear Panel Response Using Proper Orthogonal Decomposition

Boundaries of limit-cycle oscillation onset are computed using proper orthogonal decomposition for a nonlinear panel in supersonic flow. The governing structural dynamics equation is the large-deflection, nonlinear, von Karman equation for a pinned panel, and the governing flow equations are the Euler equations. Onset of limit-cycle oscillation is accurately and efficiently predicted with bifurcation analysis, as applied to a coupled set of reduced-order aerodynamic and structural dynamics equations. Nonlinear responses of panels away from critical onset conditions are obtained from reduced-order, time-domain, aeroelastic analyses and shown to compare well with published data. Static bifurcations in the transonic regime are also accurately predicted with reduced-order aeroelastic models. At Mach 0.95, an interesting array of nonlinear behaviors is found to occur over a range of scaled dynamic pressures, including limit-cycle oscillations, pitchfork bifurcation, and limit-point singularity. This paper describes a framework by which reduced-order models can be constructed and then applied to the rapid prediction of static and Hopf bifurcations in aeroelastic systems.

A Direct Method for Quantifying Limit-Cycle Oscillation Response Characteristics in the Presence of Uncertainties

A cyclic method was developed to compute directly the stochastic response of limitcycle oscillations in an aeroelastic system subject to parametric uncertainties. The system is a typical-section airfoil in incompressible flow with nonlinear behavior modeled in the torsional structural coupling. The imposed nonlinearities cause the formation of a subcritical Hopf bifurcation, with the associated presence of a cyclic fold at reduced velocities below the Hopf point, and large-amplitude limit cycles at reduced velocities above the Hopf point. The cubic component of the pitch stiness is treated as a Gaussian random variable, while other parameters are specified deterministically. The method used to evaluate the stochastic response is based on a highly implicit, iterative technique that strongly converges to the complete time-discretized, limit-cycle oscillation without the need for time integration. The method is cast in both intrusive and non-intrusive forms, each relying on polynomial chaos expansions to obtain spectral representations of stochastic system responses. Each form is observed to well characterize stochastic responses at dierent reduced velocities with polynomial chaos expansions containing only two terms, a great improvement over previous time-domain methods. The nonintrusive method is found to be highly ecient, requiring few function evaluations to compute deterministically the limit cycle, and few deterministic samples to compute the expansion coecients.

Integration of Model Reduction and Probabilistic Techniques with Deterministic Multi-physics Models

Over the years, great progress has been made in high fldelity physics models and their solution. Speed, accuracy and veriflcation have all increased substantially, such that models possess a much more complete embodiment of the physics and computational robustness in their resulting equations. The solution times for these equations are continually decreasing. An environment (e.g., a ∞uid and its properties) is never the same every time it’s entered, nor is every system (e.g., a ∞eet of 100 air vehicles) in a manufactured set comprised of identical members, especially as the ∞eet ages. Therefore, the response of a set of manufactured systems in an uncertain environment for a given set of parameters will have a statistical distribution of some sort. In the interest of incorporating these realities into system modeling, many probabilistic techniques have been developed. Monte Carlo Simulation (MCS) is considered the most robust and reliable of these techniques, but also the most computationally expensive. This computational expense occurs because 1) the number of MCS simulations required to describe the system or environment in a probabilistic manner is large, and 2) MCS is typically run on a model the size of the deterministic model. Even today’s high computational speeds do not render the problem of bounding the probabilistic system response computationally tractable. Traditionally, reduced order modeling (ROM) has been used to construct a compact surrogate model of a computationally expensive, full-order model (FOM). Currently, ROM construction from an uncertain FOM has been applied using MCS on the FOM, which does not alleviate the computational expense problem. Once constructed in this manner, the resulting uncer

Wiener-Haar Expansion of Airfoil Limit Cycle Oscillations

Wiener chaos expansions are being evaluated for simulating randomness in the response of nonlinear aeroelastic systems, which often exhibit limit cycles. A two degree-of-freedom airfoil with nonlinear and random pitch stiffness is employed to simulate aeroelastic limit cycles. Polynomial chaos expansions faithfully reproduce the short term dynamics of the airfoil but consistently lose energy at large times relative to the period of oscillation. This is attributed to the continually increasing frequency of the process in the random dimension. The recently developed Wiener-Haar expansion is found to almost entirely eliminate the loss of energy at large times. This is shown initially for a sinusoidal model problem and then for the airfoil at reduced velocities close to and well above the critical value at which a limit cycle is first observed in the mean system. It is also shown that the discrete wavelet transform can be used to efficiently compute the expansion coefficients.

Sensor Placement Optimization for SHM Systems Under Uncertainty

Structural Health Monitoring (SHM) systems that report in real-time a flight vehicles condition are central to meeting the goals of increasing flight vehicle safety and reliability, while reducing operating and maintenance costs. The structural response of flight vehicles is inherently random, requiring deterministic finite element analyses to be augmented with uncertainty quantification methods to compute the response statistics and structural damage probability. To detect damage with maximum probability, sensors must be placed optimally. This requires combining a probabilistic finite element method (FEM) with damage detection algorithms and optimization techniques. This study develops a methodology to integrate the above disciplines into a sensor placement optimization (SPO) methodology for SHM systems under uncertainty. To achieve this, the structural component under consideration is analyzed via FEM and uncertainty of model input quantities is included in the analysis via random processes and fields. In the next two steps probabilistic FEM analyses are performed to determine the model output variability and using these results, damage detection procedures such as feature extraction and state classification are applied to assess the current structural state of the component. Repeating these two steps using both healthy and damaged structural models helps quantify the reliability of a given sensor layout. Finally, SPO is achieved to maximize the reliability of damage detection. The sensor layout design of a thermal protection system component is used as a numerical example.

Uncertainty Quantification for Airframes: Current Status, Needs, and Suggested Directions

Widespread interest in uncertainty quantification methods for airframe design and certification is driven by the desire to realize acquisition and operational cost savings through increased reliance on analysis, the goal being to design and produce more robust airframes. General sources of uncertainty are described that complicate airframe design and testing, thereby contributing to the high cost of airframes. Recent applications of uncertainty quantification are reviewed with the objective of highlighting promising research and transition applications. Finally, several challenges and needs are explored to suggest future steps that must be completed to enable practical application of uncertainty quantification in airframe design and certification.

Detection and simulation of roof-corner pressure transients

Abstract Many practical time series, including pressure signals measured on roof-corners of low-rise buildings in quartering winds, consist of relatively quiescent periods interrupted by intermittent transients. The dyadic wavelet transform is used to detect these transients in pressure time series and a relatively simple pattern classification scheme is used to detect underlying structure in these transients. Statistical analysis of the resulting pattern classes yields a library of signal building blocks, which are useful for detailed characterization of transients inherent to the signals being analyzed. Probability density functions describing the arrival intervals and characteristics of the detected transients are used to synthesize time series that mimic the intermittency of the original signal. In addition, the signal that remains when the detected transients are removed from the original signal is examined to suggest appropriate models for the background noise in the intermittent signal.