Nonparametric Fitting of Aerodynamic Data Using Smoothing Thin Plate Splines

.This paper introduces a nonparametric fitting method for the interpolation of aerodynamic observations over a large range of multiple angles of attack. The method is based on the employment of smoothing thin-plate spline class functions, a well-renewed mathematical tool for multivariate data mining based on the generalization of the univariate natural cubic splines, in which a roughness penalty criterion is used to produce very smooth predictive hypersurfaces. Compared with other methods, such as parametric or even conventional nonparametric methods, the use of a smoothing thin-plate spline is more effective, in that the predictive surface comes directly from the observed points, thus minimizing any intervention of the analyst aimed at introducing model parameters. This forms the basis for a very reliable fitting technique, in which model construction can be relatively easy to implement. An application of the method is carried out on a case study representative of some experimental data coming from a wind-tunnel campaign on a typical three-dimensional fuselage-shaped body, aimed at the acquisition of its aerodynamic coefficients over a rather extensive attitude range. Specifically, the application is focused on the body lift coefficient as a function of both angle of attack and sideslip angle. The data set is also interpolated using concurrent responsesurface methods: namely, a linear model, a bivariate spline, a radial basis function network, a support vector regression technique, a regression kriging, and a moving-least-squares approach, alternatively known as local polynomial regression. Results of data fitting are assessed using a cross-validation approach and reveal a clear superiority of smoothing thin-plate spline over the other methods, leading to a more regular fitted surface and a more reliable prediction tool, even when some observations are omitted. This is important per se, but acquires even more significance when an aerodynamic test campaign is to be planned with the minimum number of experimental observations