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For conservative remapping, the areas of these polygons must match the Gauss-Lobatto weight of each Gauss-Lobatto node. If we construct this dual grid in the usual way, by connecting the centers of all the subcells shown in the subcell grid, the areas of the cells will in general not match the Gauss-Lobatto weights. We thus have to perform an iteration, similar to spring dynamics, tweaking the polygons until the areas are correct. We have a couple of algorithms to do this, but the resulting polygons can be a little odd. The result of one of these algorithms is shown below. For this figure, we have allowed pentagons and hexagons to so that the algorithm will converge faster and to more uniform cells, but now this means that some of them are will be slightly non-convex. ESMF can handle non-convex cells, but other utilities may require they be convex.
If we only allow triangles and quads we get convex polygons, but they are less regular:
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