A tensor product B-spline method for numerical grid generation

Abstract

We present a tensor product B-spline method for fast elliptic grid generation. The Cartesian coordinate functions for a block are represented as a sum of tensor product B-spline basis functions defined on the computational domain for the block. The tensor product B-spline basis functions are constructed so that the basis functions and their first partials are continuous on the computational domain for the block. The coordinate functions inherit this smoothness: a grid computed by evaluating the coordinate function along constant parameter lines leads to smooth grid lines with smoothly varying tangents. The expansion coefficients for the coordinates functions are computed by solving the elliptic grid generation equations using collocation. This assures that the computer grid has the smoothness and resolution expected for an elliptic grid with appropriate control. The collocation equations are solved with an ADI solution algorithm analogous to the ADI solution algorithm for the finite difference method. The speed of the method derives from the smoothness of the B-spline basis functions; in effect, a fine grid in the physical domain is obtained by constructing a smooth expansion of the coordinate functions on a coarse grid of knots in the computational domain. Thus, the tensor product B-spline method will be faster than the finite difference method, if a sufficiently smooth fine grid in the physical domain can be obtained for an appropriately coarse grid of knots in the computational domain.