3.4.4 Red-Green Refinement

Red-Green refinement is based on the bisection of all edges of a simplex in one step. The resulting mesh topology which defines the connectivity of the split points is chosen such that the refined elements and the original element are self-similar. This is possible for the split element itself (Red refinement) but not for the neighbor elements of which not all edges are split (Green refinement). In such a way the original geometrical element quality can be preserved, but not improved. Hence, such a refinement is only justified when the mesh density needs to be increased according to the control space. The name ``Red-Green refinement'' in literature often refers to the two-dimensional case where the simplex is a triangle. The triangle is split into four triangles of the same shape and the adjacent triangles are each split into two triangles. The same effect could be accomplished with a more universal bisection technique combined with local transformations. Only the desired edges are split. With several adaptation steps including topological modifications the same refinement pattern can be reached as results from the Red-Green technique.

In three dimensions Red-Green refinement with mixed elements has been investigated by [88]. Splitting all edges of the three-dimensional simplex (tetrahedron) introduces six refinement points which define an octahedron (Fig. 3.14).

Figure 3.14: Red-Green refinement using mixed elements in three dimensions.

The neighbor elements (Green region) form the transition from the refined area to the unrefined area. The implementation in AMIGOS combines this refinement pattern with the concept of hierarchical meshes as discussed above. The elements in the Green region are thereby not permanently fixed and are never refined themselves. They will be discarded and replaced by elements forming a Red pattern when further refinement is required.