Smooth and Convex Grid Generation over General Plane Regions

A new method to produce convex and smooth grids over general plane regions is introduced. This method belongs to the discrete variational grid generation approach. Theoretical results presented guarantee that a convex grid over a region is obtained when this method is applied; the basic assumption is that at least one convex grid exists. A procedure to control large cells, in addition to smoothness and convexity, is also presented. Experimental results, showing the eeectiveness of these methods, are reported. Grid generation techniques are extensively used in the numerical solution of partial diierential equations. Roughly speaking, a grid (or mesh) over a plane region is some subdivision of it in smaller subregions. We are interested in generating grids usable in nite diierences schemes; therefore the subdivision must be organized in quadrilaterals allowing to use the simplest of these schemes. A useful grid fulllling this objective must have all its quadrilaterals convex. This kind of grid is called convex or unfolded. Among the usual methods for grid generation, we are interested in those called variational. The rst works in this area were developed by Brackbill-Saltzman in 1982 ((3]) and Steinberg-Roache in 1986 ((13]). In those original works the desirable properties of the grid were represented in the form of minimizing some integral functional. Using the Calculus of Variations, the problem reduces to solving some system of partial diierential equations deened over the unit square. This solution is obtained numerically using diierences schemes. The most important contributions in this direction are due to Steinberg and Knupp ((10]). They introduced the area, length, smoothness and area-orthogonality continuous functionals that produce good grids over simple regions; but the main objective is not reached: convexity of the grids is not guaranteed. In another variational approach, the discrete or direct method, the grid properties are directly expressed in terms of the location of grid points, giving the so-called discrete functionals. Then the large scale optimization problem is solved by means of an iterative algorithm. The rst works with this approach are due to Castillo and Steinberg ((5], 7]), who discretized the continuous functionals of area and length producing the corresponding discrete function-als. The resulting grids are very similar to those obtained using the continuous approach, and convexity is still not guaranteed. Using a new form of discretiz-ing the continuous functionals, Barrera, P erez and Castellanos ((2], 4]), made a reformulation of the discrete functionals of area, length, …