A PDE Sensitivity Equation Method for Optimal Aerodynamic Design

Abstract

and burns@icam.vt.edu engineering experience with repeated experimental testing. This is often expensive, motivating computational methodsThe 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 usedWhile there are examples of shape optimization probto provide approximate function evaluations, there are several possible approaches to the problem of computing gradients. One populems 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 equa-tively few design parameters, using design sensitivities, tion method. The sensitivity equation method is presented, conver-quantities which d...

Optimal control of unsteady compressible viscous flows

“…The fourth condition enforces compatibility between the initial velocity ÿeld v 0 and boundary data b − gn. A similar interpretation is valid for the conditions in (12) except that the required spatial regularity of the controls is one order higher. For the purposes of this paper the regularization terms J reg associated with these spaces are more important than the precise meaning of (11) and (12).…”

“…For the purposes of this paper it is su cient to understand that in (11) and (12) ∇g denotes the gradient of g on the boundary, which in our test problem is given by ∇g = g x1 . The ÿrst two conditions in (11) are smoothness conditions on the admissible controls in space and in time, respectively.…”

“…If 1 = 0 we do not enforce smoothness of the control in time. In particular the requirements (@=@t)g ∈ L 2 ((t 0 ; t f ); L 2 ( c )) and g(t 0 ; x) = 0 are dropped from (11), (12). The e ect of the regularization terms on the computed controls are demonstrated in Section 7.…”

“…The papers [1][2][3][4][5][6][7][8][9][10][11] present a sample of recent activities in the former category and References [12][13][14][15][16][17] illustrate activities in the latter category. To the authors' knowledge, the present paper documents the ÿrst approach to the optimal boundary control of unsteady compressible viscous ows.…”

Sensitivity equations for a size-structured population model

“…However, to the best of our knowledge, thus far there is no literature on sensitivity equations and the related analysis for size-structured population models. Sensitivity analysis of dynamical systems has drawn the attention of numerous researchers [1,6,9,10,11,13,14,15,16,17,20,24,25,27,28,35,38,40] for many years because the resulting sensitivity functions can be used in many areas such as optimization and design [16,26,27,34,38], computation of standard errors [9,10,19,21,36], and information theory [12] related quantities (e.g., the Fisher information matrix) as well as control theory, parameter estimation and inverse problems [5,8,9,10,11,40,41]. One of our motivations for investigating sensitivity for size-structured population models derives from our efforts reported in [7], where a shrimp biomass production system and a…”