دانلود رایگان مقاله بهینه سازی شکل آیرودینامیکی برای حداقل درگ قوی و محدودیت قابلیت اطمینان آسانسور

Abstract

A methodology for shape optimization of aerodynamic bodies under uncertainties is presented. Flowrelated and geometrical uncertainties are considered and quantified by probability distribution functions. The optimal shape is computed by minimizing a robust estimate of the drag coefficient subject to reliability constraint for the lift coefficient. The robust drag is formulated as a weighted sum of the mean and the standard deviation of the drag coefficient over the space of uncertain parameters. The mean and standard deviation of the drag coefficient are computed using sparse grid techniques. The lift reliability, defined by the probability the lift coefficient is lower than a reference value, is computed using First Order Reliability Method (FORM). A gradient-based optimization algorithm is used to obtain the optimal shape. The sensitivity derivatives of robust drag measure and the lift reliability with respect to the shape controlling and flow related design parameters as well as the uncertain parameters are computed using the adjoint problem for the flow. The methodology is applied to pure aerodynamic shape optimization, comparing optimal designs that arise from the formulation to optimal designs that correspond to special cases, including the case of no uncertainties. A 2D airfoil case is designed based on the Euler equations under uncertain Mach number and angle of attack and geometric variability.

7. Conclusions

A framework for aerodynamic shape optimization under uncertainty was presented. The optimal design minimizes a measure of the drag coefficient that is robust to uncertainties, subject to the lift coefficient reliability constraint. The multi-dimensional inFig. 10. Optimal airfoil shape obtained for the RoRe design cases for either flowrelated uncertainties or both types of uncertainties. Table 3 Comparison of the objective function value, mean value, standard deviation and probability of failure for the RoRe design cases taking into account either flowrelated uncertainties only or both types of uncertainties. CD (μc ) μCD σCD P f flow 0.00369 0.00505 0.00333 0.01 flow & geometry 0.00378 0.00479 0.00256 0.01 tegrals for the mean and standard deviation of the drag coefficient were computed by the sparse grid technique, while the reliability was computed using FORM. The design parameters were the coordinates of the Bézier control points parameterizing the airfoil shape. The uncertain parameters were the flow conditions (Mach number and angle of attack) and the coordinates of the Bézier control points (design variables), accounting through the airfoil contour parameterization for the geometric variabilities of the airfoil. The sensitivities of the robust measure of the drag coefficient and the lift reliability constraint with respect to the design variables and uncertain parameters were computed by the discrete adjoint approach to the Euler equations, substantially reducing the number of flow solutions to the solution on the forward and the adjoint equations, thus making the computational effort independent of the number of design variables and uncertain parameters. In parallel implementation of the proposed optimal shape optimization algorithm, the time-to-solution, which depends on the solution of the forward and adjoint flow equations, scales with the number of iterations required to estimate the FORM “design” point in the uncertain parameter space and it is independent of the number of sparse grid points.