Adjoint Sensitivity Formulation for Discontinuous Galerkin Discretizations in Unsteady Inviscid Flow Problems

Abstract

This paper presents an unsteady discrete adjoint algorithm for high-order implicit discontinuous Galerkin discretizations in time-dependent inviscid flow problems. The major function of the adjoint approach is to obtain the sensitivity information in a time-dependent functional output, which in turn is used to drive an unsteady shape-optimization process to deliver a minimum of the objective functional. A gradient-based optimization strategy is investigated, in which the sensitivity derivatives of the objective functional with respect to input variables are formulated in the context of high-order discontinuous Galerkin discretizations, while special emphasis is given to the variations and linearizations of curvilinear boundary elements. Implicit temporal discretizations consisting of a second-order backward Euler scheme and a fourth-order implicit Runge-Kutta scheme are considered exclusively in this work, where the corresponding adjoint problem is required to be solved in a backward time-integration manner due to the associated transpose operation. Two numerical examples for the unsteady shape design techniques are presented to verify the derived sensitivity formulations and to demonstrate the performance of the adjoint approach; the first involves an inverse shape-optimization case by matching a time-dependent target pressure profile for a two-dimensional periodic vortical gust impinging on a RAE-2822 airfoil, and the second considers minimization of the acoustic noise produced by subsonic flow over a NACA0012 airfoil with a 0.03c thick blunt trailing edge.