Simulated annealing (SA) algorithms and genetic algorithms (GA) have the advantages of yielding a global minimum of the objective function and therefore overcome the limitations of deterministic gradient-based search methods. However, they require a large number of function evaluations and relative long computation times. An attractive alternative to SA and GA is the simultaneous perturbation stochastic approximation (SPSA) method. The SPSA algorithm starts by iterating from an initial guess of the optimal (p-dimensional) vector of design variables (design vector). Next, a p-dimensional random perturbation vector is constructed and two measurements (or calculations) of the objective function are obtained based on simultaneous perturbations (with opposite signs) around the given design vector. From the results, the gradient of the objective function can be approximately calculated. From the gradient, a new design vector is calculated using a general recursive SA form. The essential feature of SPSA is the underlying gradient approximation, which requires only two measurements of the objective function to approximate the gradient regardless of the dimension of the optimization problem. It only uses objective function calculations and does not require direct calculations of the gradient of the objective function. This feature results in a significant decrease in the cost of optimization. In this study, the SPSA method has been successfully implemented for inverse and constrained airfoil design (accomplishment of a given pressure distribution and transonic drag minimization without lowering lift and cross section area) and axisymmetric nozzle design (thrust maximization). The objective functions are calculated during each iteration using a finite volume computational fluid dynamics technique for solving the compressible Navier-Stokes equations. Results show that SPSA is able to find the optima of the design optimization problems considered in this paper and that the optimization algorithm can be easily implemented and integrated with the CFD code. The SPSA method generally shows higher convergence speed compared with the SA method. However, for the inverse airfoil design problem, the satisfaction of the termination criteria appears to show that the SPSA method is slightly less accurate than the SA method. SPSA results in an objective function value reaching a converged value by oscillating slightly about some mean value. This can be improved by replacing the SPSA method with other methods, such as local search method, when the objective function starts oscillation.
Application of simultaneous perturbation stochastic approximation method for aerodynamic shape design optimization
Anwendung eines stochastischen Näherungsverfahrens zur Optimierung aerodynamischer Profilie
AIAA Journal ; 43 , 2 ; 284-294
2005
11 Seiten, 13 Bilder, 42 Quellen
Article (Journal)
English
Druckverteilung , Gradient , inverses Problem , iterativer Optimierungsalgorithmus , Konvergenz , Methodenvergleich , numerische Strömungssimulation , Profil (Kontur) , Rechenzeit , rechnerunterstützter Entwurf , Schubdüse , stochastisches Verfahren , Tragflügel , Zeiteinsparung , Optimierungssimulation
British Library Conference Proceedings | 2002
|