engineering experience with repeated experimental testing. This is often expensive, motivating computational methods
The use of gradient-based optimization algorithms in inverse design is well established as a practical approach to aerodynamic which compute the optimal design directly. These methods design. A typical procedure uses a simulation scheme to evaluate require defining an objective function and an appropriate the objective function (from the approximate states) and its gradient, PDE model of the states of the system. A comparison of then passes this information to an optimization algorithm. Once the several optimal design methods may be found in [13].
simulation scheme (CFD flow solver) has been selected and used While there are examples of shape optimization probto provide approximate function evaluations, there are several possible approaches to the problem of computing gradients. One popu-lems solved using derivative free optimization algorithms lar method is to differentiate the simulation scheme and compute (see, e.g., [15]), many popular approaches couple a gradidesign sensitivities that are then used to obtain gradients. Although ent-based optimization algorithm with function evaluathis black-box approach has many advantages in shape optimization tions provided by a proven simulation scheme. One of problems, one must compute mesh sensitivities in order to compute the disadvantages of these approaches is the expense of the design sensitivity. In this paper, we present an alternative approach using the PDE sensitivity equation to develop algorithms computing the gradient. Using finite differences is often for computing gradients. This approach has the advantage that too costly, even if appropriate step sizes can be found and mesh sensitivities need not be computed. Moreover, when it is the simulation scheme can take advantage of ''nearby'' possible to use the CFD scheme for both the forward problem and solutions (as is the case with iterative solvers for nonlinthe sensitivity equation, then there are computational advantages.
ear equations).
An apparent disadvantage of this approach is that it does not always produce consistent derivatives. However, for a proper combination Two strategies for alleviating the computational expense of discretization schemes, one can show asymptotic consistency of gradient evaluations are adjoint variables [20] and design under mesh refinement, which is often sufficient to guarantee consensitivities [17]. Adjoint methods are advantageous when vergence of the optimal design algorithm. In particular, we show either the problem is self-adjoint or there are a large numthat when asymptotically consistent schemes are combined with a ber of design parameters. However, when there are relatrust-region optimization algorithm, the resulting optimal design method converges. We denote this approach as the sensitivity equatively few design parameters, using design sensitivities, tion method. The sensitivity equation method is presented, converquantities which describe the influence of the design pagence results are given, and the approach is illustrated on two rameters on the states of the system, is an attractive alternaoptimal design problems involving shocks. แฎ 1997 Academic Press tive. In addition to efficient gradient computations, they can be used in some problems to construct an effective update of the approximate Hessian for quasi-Newton opti-