INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids (2012) Published online in Wiley Online Library (wileyonlinelibrary.com/journal/nmf). DOI: 10.1002/fld.3722 A method to carry out shape optimization with a large number of design variables Nikhil Kumar, Anant Diwakar, Sandeep Kumar Attree and Sanjay Mittal * ,† Department of Aerospace Engineering, Indian Instituteof Technology Kanpur, Kanpur, U.P. 208016, India SUMMARY A new method for shape optimization with relatively large number of design variables is proposed. It is well known that gradient-based methods converge to a local optimum. As a result, utilization of a richer design space does not necessarily lead to a better design. This is demonstrated via the design of an airfoil for maximum lift for Re D 1000 and ˛ D 4° flow. The airfoil is represented by fourth-order non-uniform rational B-splines, and the control points are used as design variables. Starting with a NACA0012 airfoil, it is found that the optimal airfoil obtained with 13 control points has far superior aerodynamic performance than the ones obtained with 39 and 61 control points. For effective utilization of a richer design space, it is proposed that the number of design variables be increased gradually. The method is demonstrated by designing high lift airfoils for Re D 1000 and 1  10 4 . The objective function is the maximization of the time-averaged lift coefficient for ˛ D 4°. The optimization cycle with 27 control points is initiated with the optimal airfoil obtained with 13 control points. The process is continued with gradual increase in the number of design variables. Beyond a certain number of control points, the optimization leads to a spontaneous appearance of corrugations on the upper surface of the airfoil. The corrugations are responsible for the generation of small vortices that add to the suction on the upper surface of the airfoil and lead to enhanced lift. A stabilized finite element method is used to solve the unsteady flow and adjoint equations. Copyright © 2012 John Wiley & Sons, Ltd. Received 24 September 2011; Revised 4 July 2012; Accepted 23 July 2012 KEY WORDS: shape optimization; adjoint methods; unsteady flows; finite element; airfoil; corrugations; NURBS 1. INTRODUCTION Aerodynamic shape optimization is on its way to becoming an integral part of the design process. Adjoint-based methods [1–5] are especially attractive, as the cost of computing the gradients or sen- sitivities is independent of the number of design variables. They have been utilized in diverse areas such as aerospace [6–9], marine [10] and biomedical engineering [11]. Unlike some other methods, the adjoint-based methods lead to local optima within the design space. The initial guess for the design variables plays a vital role in driving the local optimum toward a global optimum. Srinath and Mittal [12] showed, via the lift maximization for an airfoil for the Re D 250 flow, that the objec- tive function exhibits an oscillatory nature with respect to the design variable(s) and may, therefore, be associated with multiple local optima. The optimizer converges to the closest local minima based on the initial guess. Design of high lift airfoils has received considerable attention in recent years because of its wide range of applications. Increased weight of payloads, short distance take-off and high altitude opera- tions have necessitated the use of high lift airfoil configurations. Unmanned air vehicles and micro air vehicles operate at low Reynolds number. To maintain the required lift, it is desirable to use *Correspondence to: Sanjay Mittal, Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur, U.P. 208016, India. † E-mail: smittal@iitk.ac.in Copyright © 2012 John Wiley & Sons, Ltd.