# Adjoint Lattice Bolztmann for Optimal Control

Grzegorz Gruszczyński

^{1}, Łukasz Łaniewki-Wołłk^{1}, Jacek Szumbarski^{1}^{1}

*Warsaw University of Technology, The Faculty of Power and Aeronautical Engineering*

In many industrial processes mixing of fluids play a crucial role. For instance, it is often desired to enforce a uniform distribution of the temperature or a chemical reactant in the domain using minimal amount of work during a prescribed time.

To solve the optimal control problem the Lattice Bolztmann method is used. The method is is characterized by an inherited time dependence, easy implementation and almost ideal scalability on parallel supercomputers.

The author presents a discrete adjoint formulation to calculate sensitivity of the LBM solution. The derivatives are calculated from the code of the standard LBM solver using the AutomaticDifferentiation. The adjoint problem has similar properties to the primal one. These includes locality and explicit time-stepping, thus it can be straightforwardly incorporated to the original solver. This approach allows to solve the optimal control problem keeping the key scheme of LBM solver intact.

To solve the optimal control problem the Lattice Bolztmann method is used. The method is is characterized by an inherited time dependence, easy implementation and almost ideal scalability on parallel supercomputers.

The author presents a discrete adjoint formulation to calculate sensitivity of the LBM solution. The derivatives are calculated from the code of the standard LBM solver using the AutomaticDifferentiation. The adjoint problem has similar properties to the primal one. These includes locality and explicit time-stepping, thus it can be straightforwardly incorporated to the original solver. This approach allows to solve the optimal control problem keeping the key scheme of LBM solver intact.

**Keywords:**Computational methods, Convection

**Figure 1:**The figure illustrates the process of mixing of a passive scalar droplet in a lid driven cavity flow.