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

摘要

This paper presents an unsteady discrete adjoint algorithm for high-order implicit discontinuous Galerkinndiscretizations in time-dependent inviscid flow problems. Themajor function of the adjoint approach is to obtain thensensitivity information in a time-dependent functional output, which in turn is used to drive an unsteady shape-noptimization process to deliver a minimum of the objective functional. A gradient-based optimization strategy isninvestigated, in which the sensitivity derivatives of the objective functional with respect to input variables arenformulated in the context of high-order discontinuousGalerkin discretizations, while special emphasis is given to thenvariations and linearizations of curvilinear boundary elements. Implicit temporal discretizations consisting of ansecond-order backward Euler scheme and a fourth-order implicit Runge–Kutta scheme are considered exclusivelynin this work, where the corresponding adjoint problem is required to be solved in a backward time-integrationnmanner due to the associated transpose operation.Two numerical examples for the unsteady shape design techniquesnare presented to verify the derived sensitivity formulations and to demonstrate the performance of the adjointnapproach; the first involves an inverse shape-optimization case bymatching a time-dependent target pressure profilenfor a two-dimensional periodic vortical gust impinging on aRAE-2822 airfoil, and the second considersminimizationnof the acoustic noise produced by subsonic flow over a NACA0012 airfoil with a 0:03c thick blunt trailing edge.