Parallel calculation of sensitivity derivatives for aircraft design using automatic differentiation

Abstract

Sensitivity derivative (SD) calculation via automatic differentiation (AD) typical of that required for the aerodynamic design of a transport-type aircraft is considered. Two ways of computing SD via code generated by the ADIFOR automatic differentiation tool are compared for efficiency and applicability to problems involving large numbers of design variables.A vector implementation on a Cray Y-MP computer is compared with a coarse-grained parallel implementation on an IBM SP1 computer, employing a Fortran M wrapper.The SD are computed for a swept transport wing in turbulent, transonic flow; the number of geometric design variables varies from 1 to 60 with coupling between a wing grid generation program and a state-of-the-art, 3-D computational fluid dynamics program, both augmented for derivative computation via AD. For a small number of design variables, the Cray Y-MP implementation is much faster.As the number of design variables grows, however, the IBM SP1 becomes an attractive alternative in terms of compute speed, job turnaround time, and total memory available for solutions with large numbers of design variables. The coarse-grained parallel implementa-

Sensitivity analysis of a mixed‐phase chemical mechanism using automatic differentiation

“…Using ADIFOR "out of the box," it was feasible to compute this large number of sensitivities with little human effort, and the time required to compute the derivatives mattered little in comparison to the effort required to interpret the 27.7 Mbytes of data produced by one run.We note, however, that AD and the associativity of the chain rule allow for various ways of decreasing the computational cost, both with respect to memory and run time. Derivatives can be computed in parallel[Bischof et at. 1994;Bischof and Wu, 1997], cost can be reduced by techniques such as interface contraction[Hovtand et at., 1997; Bischof and Haghighat, 1996], the number of derivatives can be reduced via "shaped" perturbations [Bischof et at., 1996c], or the understanding of the mathematical underpinnings of a particular algorithm may make it possible to avoid differentiation of parts of the code[Griewank et at., 1993; Carte and Fagan, 1996].Given the sensitivities of individual species, the sensitivity of lumped concentrations of two (or more) species (e.g., RxO v, defined inTable 1) can also be calculated as the sum of products of the concentration fraction and the sensitivity coefficient of each species:•Sij, = ql t•(Ci + Cj•) _. C•i •il + Cj •jl (the average concentrations and percent changes in concentrations of 12 major gas phase species and total sulfate and nitrate formed in all phases during the last hour simulation under various conditions.…”

MDO for Hypersonic Scramjet Vehicle Development (Invited)

“…The Complex Derivative (CD) method [39][40][41][42] finds a first derivative with one function evaluation and is orders of magnitude more accurate than FD (albeit at twice the cost), but can only compute one derivative at a time and cannot be applied to code that utilizes imaginary numbers. Automatic Differentiation (AD) uses source code transformation [38] is more efficient than CD, but requires preprocessing of the code [43][44][45][46][47][48]. This work highlightes the Quasi-Complex Gradient (QCG) method, which computes all derivatives (the entire Jacobian, Hessian, or higher-order) using a single function evaluation and maintains the accuracy of CD, while allowing preexisting complex operations.…”

Efficient and accurate derivatives for a software process chain in airfoil shape optimization

“…An airfoil optimization is performed in [8] where the drag coefficient is minimized for constant lift using an Euler solver. Further references where automatic differentiation has been applied to problems from computational fluid dynamics include [9][10][11].…”