Improved adjoint lattice Boltzmann method for topology optimization of laminar convective heat transfer

Solving flow-related inverse problems such as topology optimization problems is intricate but significant in various engineering fields. The lattice Boltzmann method (LBM) and the related adjoint method are highly suitable to perform sensitivity analysis in flow-related inverse problems thanks to their strong capability to handle complex structures and excellent parallel scalability. However, the current continuous adjoint LBM shows theoretical inconsistency and poor numerical stability for open flow systems. To solve these issues, the present work develops the fully consistent adjoint boundary conditions from the discrete adjoint LBM. For the first time, the gap between the two adjoint LBMs is unveiled by rigorously deriving both the continuous and discrete adjoint LBMs and comprehensively evaluating their numerical performances in the 2D and 3D pipe bend optimization cases. It is revealed that theoretical inconsistency or singularity exists in the continuous adjoint boundary conditions for open flow systems, corresponding to a much inferior numerical stability of the adjoint solution and an obvious numerical error in sensitivity. Fully consistent adjoint boundary conditions in elegant local form are derived from the discrete adjoint LBM in this work, which can always acquire exact sensitivity results and the theoretically highest numerical stability, with a 10 times higher Reynolds number (Re) achieved while without any increase of computational cost. 3D microchannel heat sinks under various Re are designed, and the esthetic and physically reasonable optimized designs are obtained under various parameter settings, demonstrating the necessity and versatility of the presented discrete adjoint LBM.