Mohammad Kurdi

Uncertainty Quantification of the Goland+ Wing's Flutter Boundary

Accurate numerical prediction of flutter boundary for fighter aircraft is of great importance. Existing models are deterministic, and do not allow for inherent variations in the system parameters. These variations (e.g. structural dimensions, aerodynamic flow field, stores properties) propagate to uncertainty in the model predictions. In this paper we examine variations in structural dimensions of a “heavy” version of the Goland wing on the flutter boundaries. Initially, the large number of random quantities (component thicknesses and areas) are efficiently reduced by conducting a sensitivity analysis of the baseline wing. Next, an optimization study is carried out to provide a design of the wing that maximizes its first natural frequency while constraining the frequency of the remaining nine modes to no less than their baseline wing counterpart values. The sensitivity study enables selection of a random variable set of the wing components having significant impact on the wing natural frequencies. Monte Carlo simulation is used to propagate the variation in the dimensional properties of the selected set of random quantities of the designed wing. The effect of correlation between random variables is considered. A modal analysis of each realization is evaluated using MSC.Nastran. Flutter boundaries of the propagated sample are predicted based on linear aerodynamic theory (ZAERO R ©), resulting in a “banded” stability boundary. Results indicate the high sensitivity of the flutter speed to small changes in the structure with an apparent switching in the failure modes.

Spectral element method in time for rapidly actuated systems

In this paper, the spectral element (SE) method is applied in time to find the entire time-periodic or transient solution of time-dependent differential equations. The time-periodic solution is computed by enforcing periodicity of the element set. Of particular interest are periodic forcing functions possessing high frequency content. To maintain the spectral accuracy for such forcing functions, an h-refinement scheme is employed near the semi-discontinuity without increasing the number of degrees of freedom.Time discretization by spectral elements is applied initially to a standard form of a set of linear, first-order differential equations subject to harmonic excitation and an excitation admitting rapid variation. Other case studies include the application of the SE approach to parabolic and hyperbolic partial differential equations. The first-order form of these equations is obtained through semi-discretization using conventional finite-element, spectral element and finite-difference schemes. Element clustering (h-refinement) is applied to maintain the high accuracy and efficiency in the region of the forcing function admitting rapid variation. The convergence in time of the method is demonstrated. In some cases, machine precision is obtained with 25 degrees of freedom per cycle. Finally the method is applied to a weakly nonlinear problem with time-periodic solution to demonstrate its future applicability to the analysis of limit-cycle oscillations in aeroelastic systems.

Optimization of Dynamic Response Using Temporal Spectral Element Method

The design of systems for dynamic response may involve constraints that need to be satisfied over an entire time interval or objective functions evaluated over the interval. Efficiently performing the constrained optimization is challenging, since the typical response is implicitly linked to the design variables through a numerical integration of the governing differential equations. Evaluating constraints is costly, as is the determination of sensitivities to variations in the design variables. In this paper, we investigate the application of a temporal spectral element method to the optimization of transient and time-periodic responses of fundamental engineering systems. Through the spectral discretization, the response is computed globally, thereby enabling a more explicit connection between the response and design variables and facilitating the efficient computation of response sensitivities. Furthermore, the response is captured in a higher order manner to increase analysis accuracy. Two applications of the coupling of dynamic response optimization with the temporal spectral element method are demonstrated. The first application, a one-degree-of-freedom, linear, impact absorber, is selected from the auto industry, and tests the ability of the method to treat transient constraints over a large-time interval. The second application, a related mass-spring-damper system, shows how the method can be used to obtain work and amplitude optimal time-periodic control force subject to constraints over a periodic time interval.

Work-Amplitude Optimal Actuation of Nonlinear Resonant System

In this study, we examine a new approach for actuation of dynamical systems with minimum work while maintaining constraints on system response and actuation force. Recently, we applied the approach to a lightly damped, linear oscillator. Here, we extend the method to nonlinear, resonant systems. Two methodology issues are addressed in the paper: sensitivity analysis about the nonlinear transient response and exploration of the strongly nonlinear relationship between the objective function and the actuation design variables. The optimization analysis is carried out on a lightly damped Duffing system of hardening and softening nonlinearities. With a hardening nonlinear spring, the optimization formulation is assessed for dependence of nonlinear response on initial conditions. The formulation of the two-objective problem is found ideally suited to resolve the difficulty of dependence of response on initial conditions. The tradeoff curve of minimum work and maximum amplitude is computed. In comparison to harmonic actuation, the optimal actuation is found to yield the target amplitude with 2.2% savings in expended work and 17% reduction in the force amplitude. In some other designs, work savings amount to 135%. The optimal actuation strove to compensate for the limited force amplitude by an abrupt change in the force in time. With a spring with both softening and hardening, similar work savings and robustness of the optimization were demonstrated. In addition, the optimization search taps into regions of the response deemed discontinuous using harmonic actuation.