Surface Splines Generalization and Large Deflection Interpolation

Abstract

T HE surface spline presented by Harder and Desmarais [1] ingeniously has been generally applied in aeroelastic analysis since the comment by Rodden et al. [2], and has been a standard method of function interpolation [3]. It is based on the smalldeflection equation of an infinite plate, and so is also called infiniteplate spline (IPS). However, as a mathematical approach of data fitting, it is not limited to the hypothesis of small deflection. In this paper, we give a generalization of IPS to fit a vector value function with arbitrary number of variables, and apply it to the deformation interpolation of large deflection structures. In [2], Rodden et al. presented the importance of interpolation methods concerning the coupling of aerodynamic/structure by quoting the words of Hitch, and reviewed the development of 2-D interpolation methods before 1970s. After that, there are two major progresses in the surface spline method. The first one is called the finite-surface spline by Appa [4], which is based upon the finite element method of a finite plate. It improves the extrapolation of the infinite-surface spline, but it is less applicable in aeroelastic analysis. The other is a kind of generalized IPS, named thin-plate spline (TPS), which extends the number of variables from two to three [5], and is applied in ZAERO [6]. The deformation of a very flexible structure has more than one component, and so amethod to interpolate a vector function has to be established. In this note, the data fitting method between arbitrary dimension spaces is carried out by the further generalization of TPS. The transformation matrix defines a mapping from the original structure configuration to the set of displacement, or to the final configuration. Through the interpolation, the tangentmapping can be obtained as well to calculate the tangent or normal vectors of the configuration. This method provides a general interpolation scheme not limited to the structural and aerodynamic interface, but also applicable to interpolate any smooth data and curve/surface reconstruct. II. Mathematical Analysis