Grid Generation Techniques

To date two techniques have been used to create the grids used to develop the surface plots: simple transfinite interpolation and a tensor product spline algorithm developed by one of the authors [2]. The tensor product algorithm uses the mapping

$${\bf T}(\xi,\eta)= \left(\begin{array}{c} x(\xi,\eta) \\ ... ...j=1}^{n} \beta_{ij} B_{ij}(\xi,\eta) \\ \end{array} \right),$$ (1)

where $0 \leq \xi,\eta \leq 1$ and each $B_{ij}$ is the tensor product of cubic B-splines. Hence, $B_{ij}(\xi,\eta) = B_i(\xi)B_j(\eta)$ where $B_i$ and $B_j$ are elements of cubic B-spline sequences associated with finite nondecreasing knot sequences, say, $\{s_i\}_1^{m+4}$ and $\{t_j\}_1^{n+4}$, respectively. The initial coefficients are chosen so that the mapping approximates transfinite blending function interpolation. More specifically, the coefficients are selected to produce a variation diminishing spline approximation to the transfinite blending function interpolant. In short, this means the coefficients are obtained by evaluating the interpolant at average knot values as discussed in [5]. If more orthogonality and smoothness are needed, the coefficients can be adjusted to minimize the discrete functional

$$G = \sum_{i,j} w_1 \left[\left(\frac{J_{i+1,j}-J_{ij}} {\triangle\xi}... ...J_{i,j+1}-J_{ij}} { \triangle\eta} \right)^2 \right] \triangle\xi \triangle\eta$$ (2)

$$+ \sum_{i,j} w_2 Dot_{ij}^2 \triangle\xi \triangle\eta$$

where $J_{ij}$ is the Jacobian value and $Dot_{ij}$ is the dot product of $\partial \bf T/\partial \xi$ and $\partial \bf T/\partial \eta$ at mesh point $(\xi_{i},\eta_{j})$ on the unit square. The code has been updated to handle larger problems. Both the transfinite code and the tensor product spline code allow the user to easily change the size of the grid while guaranteeing that certain boundary grid points are fixed to maintain the accuracy of the boundary approximation.