Adjoint-based optimization methods for flow problems

Abstract

Over the last decade, adjoint sensitivity analysis has become an established technique

for the task of shape optimisation when many degrees of freedom are present. The success stems from the fact that the adjoint approach only needs one flow simulation for both the primal and the adjoint system, no matter how many design parameters are present. The derivation of the continuous adjoint approach is based on an augmented cost function which inheres the primal governing equations (here the RANS-equations) as constraints which have to be satisfied in the computational domain. Accordingly, the primal RANS equations are augmented with Lagrange multipliers and added to the thermal-fluid dynamic cost function. For shape optimisation, the variational formulation of the augmented cost function indicates the behaviour of the cost function with the variation of the shape, i.e. the variation of the surface mesh in normal direction. We present the derivation and application of the continuous adjoint approach for the incom- pressible Reynolds-averaged Navier-Stokes (RANS) equations augmented with heat transfer. The derived approach is implemented into the framework of the C++ CFD toolbox OpenFOAM in order to derive a complete design cycle for shape optimisation. The derived optimisation process is applied to dimpled surface geometries in order to optimise cooling devices.