Discrete adjoint sensitivity analysis for fluid flow topology optimization based on the generalized lattice Boltzmann method

A discrete adjoint sensitivity analysis for fluid flow topology optimization based on the lattice Boltzmann method (LBM) with multiple-relaxation-times (MRT) is developed. The lattice Boltzmann fluid solver is supplemented by a porosity model using a Darcy force. The continuous transition from fluid to solid facilitates a gradient based optimization process of the design topology of fluidic channels. The adjoint LBM equation, which is used to compute the gradient of the optimization objective with respect to the design variables, is derived in moment space and found to be as simple as the original LBM. The moment based spatial momentum derivatives used to express the discrete objective functional (cost function) have the advantage that the local stress tensor is a local quantity avoiding the numerical computation of gradients of the discrete velocity field. This is particularly useful if dissipation is a design criterion as demonstrated in this paper. The method is validated by a detailed comparison with results obtained by Borrvall et al. for Stokes flow. While their approach is only valid for Stokes flow (i.e. very low Reynolds numbers) our approach in its present form can in principle be applied for flows of different Reynolds numbers just like the Navier–Stokes equation based approaches. This point is demonstrated with a bending pipe example for various Reynolds numbers.