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A Direct Method for Constructing Refined Regions in Unstructured Conforming Triangular–Hexagonal Computational Grids: Application to OLAM

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  • 1 University of Miami, Miami, Florida
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Abstract

A scheme is presented for constructing refined regions of 2D unstructured computational meshes composed of triangular cells. The method preserves the conforming property of the original unrefined mesh and does not produce hanging nodes. The procedure consists of 1) doubling the resolution of triangles inside a specified closed region by adding three edges inside each triangle that connect the midpoints of its three edges; and 2) constructing one or more transition rows immediately outside the refined area by removing and adding edges in order to maintain the conforming property, to regulate the abruptness of the change in resolution, and to keep triangle shapes as close as possible to equilateral. The latter requirement is met partly by restricting the number of edges that meet at any vertex to values of 5, 6, or 7. The method for constructing the transition rows is the main new contribution of this work. Two variants of the construction are described, and for one variant, the number of transition rows is varied from 1 to 5. All construction is noniterative and is therefore extremely rapid, making the method suitable for dynamic mesh refinement. Final adjustment of gridcell shapes is performed iteratively, but this can be limited to only the transition rows and then converges very rapidly. A suitable constraint on triangle shapes that is applied in the adjustment process and satisfies the criterion for a Delaunay mesh naturally extends the mesh refinement algorithm to its dual-Voronoi diagram, which is composed primarily of hexagons plus a few pentagons and heptagons. The refinement method is tested on both Delaunay and Voronoi meshes in the Ocean–Land–Atmosphere Model (OLAM) using shallow-water test case 5. A choice of two or three transition rows is found to be optimal, although using five can increase accuracy slightly.

Corresponding author address: Robert L. Walko, Rosenstiel School of Marine and Atmospheric Science, University of Miami, 4600 Rickenbacker Causeway, Miami, FL 33149-1031. E-mail: rwalko@rsmas.miami.edu

Abstract

A scheme is presented for constructing refined regions of 2D unstructured computational meshes composed of triangular cells. The method preserves the conforming property of the original unrefined mesh and does not produce hanging nodes. The procedure consists of 1) doubling the resolution of triangles inside a specified closed region by adding three edges inside each triangle that connect the midpoints of its three edges; and 2) constructing one or more transition rows immediately outside the refined area by removing and adding edges in order to maintain the conforming property, to regulate the abruptness of the change in resolution, and to keep triangle shapes as close as possible to equilateral. The latter requirement is met partly by restricting the number of edges that meet at any vertex to values of 5, 6, or 7. The method for constructing the transition rows is the main new contribution of this work. Two variants of the construction are described, and for one variant, the number of transition rows is varied from 1 to 5. All construction is noniterative and is therefore extremely rapid, making the method suitable for dynamic mesh refinement. Final adjustment of gridcell shapes is performed iteratively, but this can be limited to only the transition rows and then converges very rapidly. A suitable constraint on triangle shapes that is applied in the adjustment process and satisfies the criterion for a Delaunay mesh naturally extends the mesh refinement algorithm to its dual-Voronoi diagram, which is composed primarily of hexagons plus a few pentagons and heptagons. The refinement method is tested on both Delaunay and Voronoi meshes in the Ocean–Land–Atmosphere Model (OLAM) using shallow-water test case 5. A choice of two or three transition rows is found to be optimal, although using five can increase accuracy slightly.

Corresponding author address: Robert L. Walko, Rosenstiel School of Marine and Atmospheric Science, University of Miami, 4600 Rickenbacker Causeway, Miami, FL 33149-1031. E-mail: rwalko@rsmas.miami.edu
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