Variational grid generation

Recently, variational methods have been used to numerically generate grids on geometometric objects such as plane regions, volumes, and surfaces. This article presents a new method of determining variational problems that can be used to control such properties of the grid as the spacing of the points, area or volume of the cells, and the angles between the grid lines. The methods are applied to curves, surfaces, and volumes in three‐dimensional space; then segments, plane curves, and plane regions appear as special cases of the general discussion. The methods used here are simpler and clearer and provide more direct control over the grid than methods that appear elsewhere. The methods are applicable to any simply connected region or any region that can be made simply connected by inserting artificial boundaries. The methods also generalize easily to solution‐adaptive methods. An important ingredient in our method is the notion of a reference grid. A reference grid is defined on a region that is simpler, but analogous to, the geometric object on which a grid is desired. Variational methods are then used to transfer the reference grid to the geometric object. This gives simple and precise control of the local properties of the grid.