Adjoint-based fluid flow control and optimisation with lattice Boltzmann methods

Abstract

A lattice Boltzmann (LB) framework to solve fluid flow control and optimisation problems numerically is presented. Problems are formulated on a mesoscopic basis. In a side condition, the dynamics of a Newtonian fluid is described by a family of simplified Boltzmann-like equations, namely BGK–Boltzmann equations, which are linked to an incompressible Navier–Stokes equation. It is proposed to solve the non-linear optimisation problem by a line search algorithm. The needed derivatives are obtained by deriving the adjoint equations, referred to as adjoint BGK–Boltzmann equations. The primal equations are discretised by standard lattice Boltzmann methods (LBM) while for the adjoint equations a novel discretisation strategy is introduced. The approach follows the main ideas behind LBM and is therefore referred to as adjoint lattice Boltzmann methods (ALBM). The corresponding algorithm retains most of the basic features of LB algorithms. In particular, it enables a highly-efficient parallel implementation and thus solving large-scale fluid flow control and optimisation problems. The overall solution strategy, the derivation of a prototype adjoint BGK–Boltzmann equation, the novel ALBM and its parallel realisation as well as its validation are discussed in detail in this article. Numerical and performance results are presented for a series of steady-state distributed control problems with up to approximately 1.6 million unknown control parameters obtained on a high performance computer with up to 256 processing units.