I have been studying the VTK geometric element interpolation algorithm and noticed that the initial values of r, s, and t are set to 0.5 for the program to be solved using Newton’s iteration. The choice of initial values is crucial for the success of the Newtonian iterative method, as inappropriate values can lead to slow divergence or convergence. I am curious as to why the initial values of r, s, and t are specifically set to 0.5 in this algorithm.

Most cells in VTK have their parametric center at (0.5, 0.5, 0.5). Most finite element codes discretize the domain such that curvature within the cell is not large (since large gradients generally correspond to large error bounds on the solution). Since quadratic and other higher-order cells can have gradients that grow quickly with distance from the cell (rather than constant gradients like simple linear cells), it makes sense to start near the center of the cell where the gradient is expected to be smaller.