Tensor Product B-Spline Method for 3D Multi-Block Elliptic Grid Generation.

摘要

We formulate a tensor product b-spline method for multi-block numerical grid generation. The Cartesian coordinate functions for a block are represented as a sum of tensor product b-spline basis functions defined on a parameter space 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 parameter space. The coordinate functions inherit this smoothness: a grid computed by evaluating the coordinate functions along constant parameter lines leads to smooth grid lines with smoothly varying tangents. The expansion coefficients for the coordinate functions are computed by solving the usual elliptic grid generation equations using simple collocation. This assures that the computed grid has the smoothness and resolution expected for an elliptic grid with appropriate control. An important result of the formulation is that the dimension of the collocation equations is the number of distinct knots for the tensor product b-spline basis functions. Combining this results with the smoothness of the b-spline representation makes it possible to reduce the dimension of the tensor product method with respect to the finite difference method, simply by using fewer knots than grid points. We formulate the expansion of the Cartesian coordinate functions as a sum of tensor product b-spline basis functions, and then we derive the collocation and boundary condition equations for the usual elliptic grid generation equations. We investigate the structure of the system of equations for the expansion coefficients and then formulate a solution algorithm to compute the coefficients. Finally, we describe the implementation of the method in a 2D multi-block grid code and discuss the performance of the method for several grids. 3 refs., 11 figs.