Optimization methods for non-smooth or noisy objective functions in fluid design problems

Abstract

Convergence difficulties may arise during derivative-based optimization involving nonsmooth or noisy objective functions. Such may be the case whcSh shape design problems are attempted using analysis codes for fluid flows. For example, the interaction between the discretization of the design problem and a shock wave in the flow solution may cause the objective function to be non-smooth. In this work, we present a method for robust optimization of non-smoot h objective functions. The optimization begins with the construction of a response surface that smoothly approximates the objective function. Here the response surface is a least-squares polynomial fit to carefully selected design points. By minimiging the response surface we can obtain a first guess for a reasonable design. Optimization may continue in one of two ways. In the first method, we probe a small region of the design space around the minimum and perform another response surface minimization. In the second method, we use a derivative-based optimization where estimates of the sensitivity derivatives are obtained by means of the discrete direct or adjoint formulations. To overcome difficulties associated with the non-smooth objective function, the sensitivity equations are regularized by adding artificial dissipative terms, whereas the flow solution and the objective function are unmodified. Two design problems involving inviscid flow with shock waves are formulated to demonstrate the efficacy and robustness of the two methods.