Analytic Hessian derivation for the quasi-one-dimensional Euler equations

Authors:
Eyal Arian;Angelo Iollo
Affiliations:
Computational Mathematics, The Boeing Company, MC 7L-21 P.O. Box 3707, Seattle, WA 98124-2207, USA;Institut de Mathématiques de Bordeaux, UMR 5251 CNRS, Université Bordeaux 1, INRIA Bordeaux Sud-Ouest, Equipe-Projet MC2, 33405 Talence cedex, France
Venue:
Journal of Computational Physics
Year:
2009

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ANALYSIS OF THE HESSIAN FOR AERODYNAMIC OPTIMIZATION: INVISCID FLOW

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The effect of shocks on second order sensitivities for the quasi-one-dimensional Euler equations

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Abstract

The Hessian for the quasi-one-dimensional Euler equations is derived. A pressure minimization problem and a pressure matching inverse problem are considered. The flow sensitivity, adjoint sensitivity, gradient and Hessian are calculated analytically using a direct approach that is specific to the model problems. For the pressure minimization problem we find that the Hessian exists and it contains elements with significantly larger values around the shock location. For the pressure matching inverse problem we find at least one case for which the gradient as well as the Hessian do not exist. In addition, two formulations for calculating the Hessian are proposed and implemented for the given problems. Both methods can be implemented in industrial applications such as large scale aerodynamic optimization.