Surface Blending Using a Power Series Solution to Fourth Order Partial Differential Equations

In our previous work, we proposed a general fourth order partial differential equation (PDE) for the generation of blending surfaces together with an accurate closed form solution for special boundary conditions where closed form solutions exist. In general, however, a closed form solution of a PDE is often either not existent or not obtainable depending on the boundary conditions and the coefficients of the PDE. In order to solve a wider range of surface blending problems using the new form of PDE, we propose in this paper a much more general resolution method using power series expansion. The main idea is to decompose the boundary conditions into a number of basic functions and separate them into two groups, one with and the other without closed form solutions. For the first group, the closed form solutions are produced with our previously proposed method. For the second group, a composite function is constructed which combines these basic functions with the power series of one parametric variable. The unknown constants of the composite function are determined by making use of both the boundary conditions and the weighted residual method, which minimizes the surface errors. Compared with existing resolution methods, the proposed method exhibits a number of advantages: 1) it is a general resolution method and hence able to solve a much wider class of surface blending problems; 2) the boundary conditions are always satisfied exactly that is crucial to the quality of blending surfaces; and 3) only a small number of collocation points and unknown constants are needed, making our method computationally efficient. Several examples given in the paper have demonstrated that this method is not only accurate, but fast enough for interactive applications.