Thin plate spline interpolation on an annulus

Abstract

This work demonstrates a method for surface interpolation on an annulus, with the additional constraint that the surface and its normal derivatives are specified on the annulus boundaries. The method is based on a truncated Fourier series solution to the homogeneous biharmonic equation, combined with Green's functions for the point interpolations. The formulation and solution approach is particularly suitable for the annulus geometry because the boundaries are circles, meaning that the boundary constraints are functions of the angle theta only, and they can be approximated by a truncated Fourier series. The result is that the interpolation method is effectively gridless. The matrix problem resulting from this formulation is highly structured and can be solved in a simple sequential manner. We present results for some test problems to illustrate the effect of the truncation of the Fourier series on the solution and on aspects such as the condition of the matrix problem.