Discrete adjoint for coupled conjugate heat transfer

The typical method to solve multi-physics problems such as Conjugate Heat Transfer (CHT) is the partitioned approach which couples separate solvers through boundary conditions. Effective gradient-based optimisation of partitioned CHT problems requires the adjoint of the coupling to maintain the efficiency of the original multi-physics coupling, which is a significant challenge. The use of automatic differentiation (AD) has the potential to ease this burden and leads to generic gradient computation methods. In this paper, we present how to automate the generation of adjoint fluid and solid solvers for a CHT adjoint using Automatic Differentiation (AD). The derivation of the adjoint of the loose coupling algorithms is shown for three fixed-point coupling algorithms. Application of the coupled adjoint algorithm is shown to two CHT optimisation benchmark cases for inverse design and shape optimisation. The results demonstrate that Robin-based coupling algorithms have faster runtimes and are an attractive option for CHT optimisation problems.