Shape optimization of an airfoil in the presence of compressible and viscous flows

The objective of this article is to present a step-by-step problem-solving procedure of shape optimization. The procedure is carried out to design an airfoil in the presence of compressible and viscous flows using a control theory approach based on measure theory. An optimal shape design (OSD) problem governed by full Navier-Stokes equations is given. Then, a weak variational form is derived from the linearized governing equations. During the procedure, because the measure theory (MT) approach is implemented using fixed geometry versus moving geometry, a proper bijective transformation is introduced. Finally, an approximating linear programming (LP) problem of the original shape optimization problem is obtained by means of MT approach that is not iterative and does not need any initial guess to proceed. Illustrative examples are provided to demonstrate efficiency of the proposed procedure.