Adjoint parameter sensitivity analysis for the hydrodynamic lattice Boltzmann method with applications to design optimization

We present an adjoint parameter sensitivity analysis formulation and solution strategy for the lattice Boltzmann method (LBM). The focus is on design optimization applications, in particular topology optimization. The lattice Boltzmann method is briefly described with an in-depth discussion of solid boundary conditions. We show that a porosity model is ideally suited for topology optimization purposes and models no-slip boundary conditions with sufficient accuracy when compared to interpolation bounce-back conditions. Augmenting the porous boundary condition with a shaping factor, we define a generalized geometry optimization formulation and derive the corresponding sensitivity analysis for the single relaxation LBM for both topology and shape optimization applications. Using numerical examples, we verify the accuracy of the analytical sensitivity analysis through a comparison with finite differences. In addition, we show that for fluidic topology optimization a scaled volume constraint should be used to obtain the desired “0-1” optimal solutions.