Mathematical derivation of a unified equations for adjoint lattice Boltzmann method in airfoil and wing aerodynamic shape optimization

Abstract

Unified equations of the adjoint lattice Boltzmann method (ALBM) are derived for five applicable objective functions in 2D/3D aerodynamic shape optimization problems. The derived equations include the adjoint equation, boundary condition, terminal condition and gradient of the cost function. In this research, firstly, these relations are extracted for each objective in details and then the general form of ALBM equations are presented for all defined practical aerodynamic objective function. Five applicable cost functions which are the most important objectives in optimization of aerodynamic geometries include desired pressure and viscous shear stress (VSS) inverse design, drag and moment at fixed lift and finally lift to drag ratio at fixed angle of attack. The new extracted relations are based on the circular and spherical function scheme, and are valid for viscous/inviscid, compressible/incompressible and 2D/3D flows in all continuous flow regimes. Proof of new extracted general relations have been performed by authors.