To eliminate the triangles remaining in the mesh after the mesh optimization step, a one-level refinement is applied. It consists in introducing a midnode at the center of each edge, triangle, quadrilateral, and pentagon (optionally formed by a remaining triangle and a neighbouring quadrilateral) and consequent splitting each triangle, quadrilateral, and pentagon to three, four, and five quadrilaterals, respectively. This yields the final all-quadrilateral mesh which is optimized by the standard Laplacian smoothing. Note that the newly introduced nodes must satisfy the surface or, if relevant, boundary curve constraint.
Since the refinement procedure actually halves the size of all elements, it is important to generate the initial mixed mesh with the doubled target element size. However, this might not be always possible, especially if the element size is controlled by the surface curvature. In this case, meshes of higher density than desirable are produced. Note also that splitting the quadrilaterals and pentagons with two neighbouring sides at the surface boundary might result in quadrilaterals of poor quality with dihedral angle close to . In such a case it is better to split the quadrilateral to two triangles before the refinement step is performed.