In order to restrict the refinement of elements to certain areas of the finite element mesh, the triangulation has to be closed after the regular refinement of the desired elements. Otherwise there would be so called hanging nodes, which are difficult to handle in calculations. The ``smooth'' transition from regular elements to regularly refined elements is done by the irregular elements. They are generated by a procedure called the green closure.
An edge is called refined if at least one of the neighbouring elements which share this edge is refined regularly. This means that a new midnode is inserted in the middle of the edge. Each tetrahedron has 6 edges which can be either refined or not. Therefore there exist 26=64 edge refinement patterns. One of them leaves all edges unrefined and a second refines all edges, which occurs if the element itself or all its neighbours are refined regularly. Considering symmetry arguments the remaining 62 patterns can be divided into 9 different types. Using these 9 types, any triangulation can be closed. For practical reasons it is sufficient to use four different types and refine those elements regularly which would need one of the missing refinement rules.
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