Unveiling the gap between continuous and discrete adjoint lattice Boltzmann methods

This work rigorously derives the continuous and discrete adjoint lattice Boltzmann methods (LBMs) for open flow systems, with a special focus on the resulting adjoint boundary conditions. It is found that the adjoint collision and adjoint streaming are exactly the same for the two adjoint methods, while the expressions, the objective derivative term and the execution order of the adjoint boundary conditions are totally different. The numerical performances of the two adjoint methods are comprehensively compared by performing sensitivity analysis on the 2D and 3D pipe bend cases. It is found that inconsistency or singularity exists in the continuous adjoint boundary conditions, leading to much inferior numerical stability of the adjoint solution and obvious numerical error in sensitivity. In contrast, the discrete adjoint LBM with fully consistent adjoint boundary conditions can always achieve exact sensitivity results and the theoretically highest numerical stability, without any increase of computational cost, thus it becomes a perfect solution to the inconsistency and instability issues in the continuous adjoint LBM. 3D microchannel heat sinks under various Reynolds numbers are also designed, and the esthetic and physically reasonable optimized designs are obtained, demonstrating the necessity and versatility of the presented discrete adjoint LBM.