Smoothing methods based on coordinate transformation in a linear space and application in airfoil aerodynamic design optimization

In detailed aerodynamic design optimization, a large number of design variables in geometry parameterization are required to provide sufficient flexibility and obtain the potential optimum shape. However, with the increasing number of design variables, it becomes difficult to maintain the smoothness on the surface which consequently makes the optimization process progressively complex. In this paper, smoothing methods based on B-spline functions are studied to improve the smoothness and design efficiency. The wavelet smoothing method and the least square smoothing method are developed through coordinate transformation in a linear space constructed by B-spline basis functions. In these two methods, smoothing is achieved by a mapping from the linear space to itself such that the design space remains unchanged. A design example is presented where aerodynamic optimization of a supercritical airfoil is conducted with smoothing methods included in the optimization loop. Affirmative results from the design example confirm that these two smoothing methods can greatly improve quality and efficiency compared with the existing conventional non-smoothing method.