The same concept of the mesh validity as described in Section Mesh Validity (Sequential Mesh Generation) is adopted. The only difference is that the geometrical similarity is verified using exclusively the parametric intersection check (Eq. (2.101)). Note also that when checking the topological compatibility of the mesh with a region, the quadrilaterals bounding hexahedral elements are considered on an abstract level only as they are not physically present in the mesh.
The topological compatibility and geometrical similarity of the mesh with a particular model entity are verified using the parametric tree structure of that model entity. The hierarchical tree structure subjected to the one-level difference rule (or the ``1:2 rule'' if a generalized tree is considered) exhibits similar topological properties as are those required from the mesh. Assuming that individual terminal cells of the tree structure of a model entity of dimension are represented by bounding facets (a segment is bounded by two nodes (), a quadrant by a set of edges (), and an octant by a set of polygons ()) and that the cells and facets are classified to individual entities of the model, then each facet classified to the model entity bounds exactly two terminal cells classified to and each facet classified to the model entity forming times the boundary of the model entity bounds exactly terminal cells classified to . Since this ensures the topological compatibility on the tree level it is only necessary to guarantee the compatibility on the cell level, which is done by a proper construction of templates. Note that the segments used to discretize a curve can be also considered as templates fitted into cells of a binary tree. The verification of the geometrical similarity is also done separately on the tree level and cell level. On the tree level, the geometrical similarity is ensured by the hierarchical structure of the tree, which essentially eliminates any intersection (in the parametric space) between terminal cells of the same tree structure. On the cell level, the geometrical similarity is ensured by an intersection check between the elements forming a template. Assuming that the topological compatibility of the template is satisfied the intersection check is equivalent to a test that the sum of the sizes (lengths, areas, or volumes) of elements forming a template is equal to the size of the cell discretized by that template. This test can be easily accomplished in the parametric space where the templates can be normalized to fit into a unit cell (a unit segment, unit square, or unit cube) in which case the sum is always equal to . Note that this test in necessary only in the phase of the development of a template and once the template has been successfully verified in terms of the geometrical similarity the test can be skipped.