The purpose of inverse design is to achieve a particular shape so that the resultant shape satisfies the target pressure distribution on its boundaries. In this thesis, a novel inverse design algorithm called Elastic Surface Algorithm (ESA) is presented for airfoil shape design in subsonic and transonic flow regimes. Also, flexible membrane method (MGM) is presented in this thesis and this method is compared to elastic surface algorithm. ESA is a physically based iterative inverse design method including two parts: an efficient inverse design algorithm, called elastic surface algorithm, and a 2D flow analysis code. In this thesis, two flow solvers are used: 1- an inviscid flow analysis code, in which the Euler equations are solved using the advection upstream splitting method (AUSM) for flow field analysis. 2- Fluent software. In the proposed method, the airfoil wall shape is changed under a physical algorithm based on the deformation of a flexible beam. The difference between the target and current pressure distributions causes to deform the flexible beam at each shape modification step. Having achieved to the target shape, the difference between the target and current pressure distribution vanishes and finally the beam deformation is stopped and then, the internal stresses are set to zero. This method is based on the finite element formulation and can be used to design and optimize the airfoils shape. In this investigation, various airfoils in subsonic and transonic flow regimes are validated by this method, which show the robustness of the method in flow regime with separation and normal shock. Also, some design examples in subsonic and transonic regimes are presented here, which show the flexibility of the proposed method. Moreover, the convergence rate of the method is compared to MGM and BSA methods, which shows ESA considerably increases the convergence rate in transonic flow regimes. Hence, ESA is a Powerful technique for airfoil design in transonic regime. One of the advantages of the presented method is a physical and quick converging approach and can efficiently utilize flow analysis codes as a black box. Despite mathematical based methods such as MGM that require arbitrary choice of parameters, the beam characteristics such as thickness, elastic modulus, shear modulus, etc. control the solution stability in ESA. Therefore, the proper choice of them increases the convergence rate of the design process. Keywords: Inverse Design, Airfoil, Elastic Surface Algorithm, Viscous and Invicid Flow Regime.
