Multi-objective aerodynamic shape optimization using MOGA coupled to advanced adaptive mesh refinement Mohammad Kouhi a,b,⇑ , Dong Seop Lee a,1 , Gabriel Bugeda a,b,1,2 , Eugenio Oñate a,b,1,2 a International Center for Numerical Methods in Engineering (CIMNE), Edificio C1, Gran Capitan, 08034 Barcelona, Spain b Universitat Politècnica de Catalunya (UPC), Gran Capitan, 08034 Barcelona, Spain article info Article history: Received 7 February 2012 Received in revised form 30 July 2013 Accepted 30 August 2013 Available online 15 September 2013 Keywords: Reconstruction/multi-objective optimization Shape optimization Adaptive remeshing Euler equation MOGA abstract This paper demonstrates the big influence of the control of the mesh quality in the final solution of aero- dynamic shape optimization problems. It aims to study the trade-off between the mesh refinement dur- ing the optimization process and the improvement of the optimized solution. This subject is investigated in the transonic airfoil design optimization using an Adaptive Mesh Refinement (AMR) technique coupled to Multi-Objective Genetic Algorithm (MOGA) and an Euler aerodynamic analysis tool. The methodology is implemented to solve three practical design problems; the first test case considers a reconstruction design optimization that minimizes the pressure error between a predefined pressure curve and candi- date pressure distribution. The second test considers the total drag minimization by designing airfoil shape operating at transonic speeds. For the final test case, a multi-objective design optimization is con- ducted to maximize both the lift to drag ratio (L/D) and lift coefficient (Cl). The solutions obtained with and without adaptive mesh refinement are compared in terms of solution improvement and computa- tional cost. Numerical results clearly show that the use of adaptive mesh refinement can improve the solution accuracy while reducing significant computational cost in both single- and multi-objective design optimizations. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction Aerodynamic shape optimization is one of the most important engineering problems which has got a lot of interest due to the large number of requests from the aircraft companies. Many inves- tigations have been carried out over the past two decades in order to speed up the optimization process and improve the optimality of the optimized design. Two main numerical optimization meth- odologies are studied massively by the researchers in the field of aerodynamic shape optimization; gradient-based methods and Genetic Algorithms (GAs). Gradient-based methods rely on sensitivities (gradients) of the objective function respect to the design parameters. Sensitivities are considered as the direction for updating design parameters. Firstly, the traditional finite difference was implemented for eval- uating sensitivities in aerodynamic shape optimization problems [1,2]. This method suffers from the fact that the computational cost is proportional to the number of design variables. The control theory resolved this drawback via presenting a family of sensitivity derivation methodologies called adjoint method. Using the adjoint method, the cost of sensitivity computation is virtually indepen- dent of the number of design variables. As the first attempt, Piron- neau [3] applied the control theory for the shape optimization. Jameson and his group [4–6] derived the adjoint method formula- tion for inviscid/viscous compressible flows with shock waves. Although the continuous adjoint method is studied in [7,8], the dis- crete adjoint method has gained more popularity recently, due to its straightforward formulation and its ability to easily treat the boundary conditions in viscous problems ([9,10]). Even though gradient-based methods require much less objective function eval- uations, they suffer from some unfavorable requirements such as smoothness of the design space, an appropriate initial guess and existence of only one single global optimum [11,12]. Indeed, for handling multi-objective optimization problems, it is necessary to define a global objective function by making a linear combina- tion of the available objective functions through different weights [6,13]. This method is strongly sensitive to the assumed weight coefficients in a way that some optimal solutions may be lost if inappropriate weights are selected. Moreover, multiple optimiza- tion runs are required to be performed in order to compute Pareto fronts. For these reasons, researchers motivated to implement GAs which are based on the process of natural selection instead of 0045-7930/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compfluid.2013.08.015 ⇑ Corresponding author at: International Center for Numerical Methods in Engineering (CIMNE), Edificio C1, Gran Capitan, 08034, Barcelona, Spain. Tel.: +34 932057016/934016200; fax. +34 934016517/934016894. E-mail addresses: kouhi@cimne.upc.edu (M. Kouhi), ds.chris.lee@gmail.com (D.S. Lee), bugeda@cimne.upc.edu (G. Bugeda), onate@cimne.upc.edu (E. Oñate). 1 Tel.: +34 932057016; fax: +34 934016517. 2 Tel.: +34 934016200; fax: +34 934016894. Computers & Fluids 88 (2013) 298–312 Contents lists available at ScienceDirect Computers & Fluids journal homepage: www.elsevier.com/locate/compfluid