First- and Second-Order Aerodynamic Sensitivity Derivatives via Automatic Differentiation with Incremental Iterative Methods

✍ Scribed by Laura L. Sherman; Arthur C. Taylor III; Larry L. Green; Perry A. Newman; Gene W. Hou; Vamshi Mohan Korivi

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 370 KB
• Volume: 129
• Category: Article
• ISSN: 0021-9991
• DOI: 10.1006/jcph.1996.0252

✦ Synopsis

discipline sensitivity derivatives (SDs). The incremental iterative method (IIM) was proposed and demonstrated

The straightforward automatic-differentiation and the hand-differentiated incremental iterative methods are interwoven to produce a to provide such first-order SDs from a two-dimensional hybrid scheme that captures some of the strengths of each strategy. (2D) thin-layer Navier-Stokes (TLNS) flow code for both With this compromise, discrete aerodynamic sensitivity derivatives geometric (shape) and nongeometric (flow) design variare calculated with the efficient incremental iterative solution algoables in Refs. [1, 2]. The IIM allows accurate, consistent rithm of the original flow code. Moreover, the principal advantage discrete SDs to be obtained with computational efficiency of automatic differentiation is retained (i.e., all complicated source code for the derivative calculations is constructed quickly with accu-(with respect to both computational time and memory racy). The basic equations for second-order sensitivity derivatives requirements). Furthermore, the IIM also allows the use are presented, which results in a comparison of four different methof approximate matrix operators for further efficiency, ods. Each of these four schemes for second-order derivatives reparallelization, or robustness, etc. Results for first-order quires that large systems are solved first for the first-order deriva-SDs from an IIM for three-dimensional (3D) Euler codes tives and, in all but one method, for the first-order adjoint variables.

(Refs. [3][4][5]) have also been presented. In all of the Of these latter three schemes, two require no solutions of large systems thereafter. For the other two for which additional systems above cited works, the discretized flow residuals were are solved, the equations and solution procedures are analogous differentiated by hand (also called the quasi-analytical to those for the first-order derivatives. From a practical viewpoint, (QA) method) and assembled to obtain the first-order implementation of the second-order methods is feasible only with SDs by an IIM.

software tools such as automatic differentiation, because of the

In the present study, numerical results are given for extreme complexity and large number of terms. First-and secondorder sensitivities are calculated accurately for two airfoil problems, the application of automatic differentiation (AD) [6][7][8] to including a turbulent-flow example. In each of these two sample obtain first-order aerodynamic SDs from an IIM for the problems, three dependent variables (coefficients of lift, drag, and same 2D TLNS code and sample problems studied in [1].

pitching-moment) and six independent variables (three geometric-

The numerical results are compared on the basis of accushape and three flow-condition design variables) are considered.

racy and computational time and memory. Previous first-Several different procedures are tested, and results are compared on the basis of accuracy, computational time, and computer mem-order SDs from the hand-differentiated IIM and the central ory. For first-order derivatives, the hybrid incremental iterative finite-difference (CD) method [1] are compared with newly scheme obtained with automatic differentiation is competitive with obtained AD results for both the straightforward and the the best hand-differentiated method. Furthermore, it is at least two incremental-iterative-form applications. This latter apto four times faster than central finite differences, without an overproach is new. That is, previous straightforward applicawhelming penalty in computer memory.