Local Refinement of Quadrilateral Meshes

Subelement patterns using bisection of edges and schemes for selecting edges to be split are described for the local refinement of elements in all-quadrilateral and quadrilateral-dominant planar meshes. Subelement patterns using trisection of edges are also described for the local refinement of elements in all-quadrilateral planar meshes. These patterns and schemes are easily extendible to quadrilateral surface meshes, provided the surface quadrilaterals are valid (generalization of strictly convex for planar quadrilaterals). Bounds on the quality (shape measure) of the refined elements with respect to the original elements are proven for planar meshes. The theoretical bounds are compared with empirical bounds. 1 Local refinement of 2-D quadrilateral mesh using bisected edges In the local refinement of a 2-D all-quadrilateral or quadrilateral-dominant mesh, certain elements are selected for refinement and other elements are added for refinement due to adjacency to an element to be refined. An all-quadrilateral mesh may be refined to maintain only quadrilaterals or to allow for triangles. The latter case may produce fewer elements in the refined mesh than the former case, due to a smaller extension in the number of elements that must be refined to maintain a conforming mesh of satisfactory quality. A quadrilateral-dominant or mixed mesh contains at least one triangle element and triangles must be allowed in the refined mesh. It is assumed that all quadrilaterals in the mesh are strictly convex, i.e. have positive shape measure. Three quadrilateral and triangle shape measures are defined in [Joe08a]. In this section, we describe subelement patterns for the refinement of elements in a 2-D all-quadrilateral or quadrilateral-dominant mesh using bisection of element edges. The initial refined mesh will have new vertices at edge midpoints and element centroids. If some bisected edges topologically lie on a curve, then a postprocessing step can be used to project some midpoints towards curves while maintaining a valid mesh (in which all elements are counterclockwise-oriented and have positive signed shape measure value). If the refined mesh has any triangles, then flips and merges may be applied to decrease the number of triangles in the mesh [Joe08b]. In order to maintain an all-quadrilateral refined mesh starting from an all-quadrilateral initial mesh, the number of edges bisected for each quadrilateral must be even. If abcd is a quadrilateral to be refined (see Figure 1.1a), then it can be split into 4 subquadrilaterals using bisection of all 4 edges (see Figure 1.1b), 3 subquadrilaterals using bisection of 2 adjacent edges (see Figure 1.1c), or 2 subquadrilaterals using bisection of 2 opposite edges (see Figure 1.1d). If triangles are allowed in the refined mesh, then a quadrilateral abcd can also be split into 2 subquadrilaterals and 1 subtriangle using bisection of 3 edges (see Figures 1.2a and 1.2b) or 1 subquadrilateral and 1 subtriangle using bisection of 1 edge (see Figures 1.2c and 1.2d). The choice of Figure 1.2a versus Figure