On the Optimality System for a 1-D Euler Flow Problems

Eugene M. Cliff
Aerospace and Ocean Engineering Deptartment
Virginia Polytechnic Institute and State University

Matthias Heinkenschloss
Department of Computational and Applied Mathematics
Rice University

Ajit R. Shenoy
Aerospace and Ocean Engineering Deptartment
Virginia Polytechnic Institute and State University


AIAA Paper 96-3993, September 1996

Abstract

In this paper, a problem of shape design for a duct with the flow governed by the one-dimensional Euler equations is analyzed. The flow is assumed to be transonic, in the sense that we have a shock embedded in the flow. The presence of the shock introduces analytical and numerical difficulties that are overcome by introducing the shock location as an explicit state variable. We justify the use of adjoint variables by establishing conditions to ensure that the linearized constraint is surjective. These adjoint variables are used to state necessary optimality conditions and to compute gradients. In addition, our characterization of the adjoint variables is of interest for comparison to adjoints associated with shock-capturing formulations.

Keywords

Euler flow equations, adjoint equations, optimal control.

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