Modified Catmull-Clark Methods for Modelling, Reparameterization and Grid Generation

In this paper we demonstrate that subdivision schemes can be used for modelling and grid generation in aeronautical projects such as that in the collaborative research center SFB401 at the RWTH Aachen: F low Modulation and Fluid-Structure Interaction at Airplane Wings. To make the schemes usable for this purposes their rules have to be modified in such a way that the limit surfaces have the desired properties. Furthermore fast algorithms for evaluation and approximation of existing surfaces have to be developed. Introduction We start with a short survey on the history of subdivision. At least fifty years ago the first papers on subdivision were published. In 1947 de Rham’s first articles on corner cutting appeared [1]. He suggested to do corner cutting with ratio 1 : 1 : 1 (compare Figure 1). From the start polygon with vertices P0, P1, . . . ,Pn−1 all the vertices Pi are split the following way (the indices of the old vertices Pi have to be computed modulo n) P0 → { Q2n−1 = (2 ·P0 +Pn−1)/3 Q0 = (2 ·P0 +P1)/3 P1 → { Q1 = (2 ·P1 +P0)/3 Q2 = (2 ·P1 +P2)/3 .. Pi → { Q2i−1 = (2 ·Pi +Pi−1)/3 Q2i = (2 ·Pi +Pi+1)/3 (1)