Grid orthogonalization for curvilinear alternating-direction techniques

A method is developed for an a posteriori iterative improvement to an arbitrary computational grid. Local corrections to the coordinates of the grid points are used to form a global cost function which is minimized with respect to a single parameter. The local corrections and cost function can be constructed to maximize the local smoothness and/or the local orthogonality of the grid. The advantage of this method is that it allows the user to generate an initial grid using any inexpensive method, and then the grid can be improved with respect to both orthogonality and smoothness.This technique was used to generate grids for a finite element alternating-direction method which uses curved elements. A sample transient diffusion problem was solved on a series of grids to investigate the sensitivity of the curvilinear alternating-direction method to grid orthogonalization. The initial grid was highly nonorthogonal and each grid produced by the automatic grid generation program was smoother and more orthogonal.This work shows that the adaptive grid program can be easily used to generate nearly orthogonal grids and it shows that the curvilinear alternating-direction technique is not highly sensitive to nonorthogonality of the grid. It is shown that as long as a grid is somewhat reasonable, the alternating-direction method will perform quite well.