Optimization Methods for Non-smooth or Noisy Objective Functions in Fluid Design Problems

We present a method for robust optimization of nonsmooth 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 minimizing 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 nonsmooth 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.