The standard Laplacian smoothing technique, that changes the position of a surface node to a new position given by the average of nodes connected to it by an edge, works fine for all-triangular and even for all-quadrilateral meshes. For mixed meshes, however, this simple scheme behaves not fully satisfactorily. This can be demonstrated, for example, on the mesh depicted in Figure 63 on the left consisting of ideal elements only (squares and equilateral triangles). The application of the standard Laplacian smoothing moves the nodes on the interface between the quadrilaterals and triangles towards the triangles and consequently deteriorates the quality of the elements. This effect is eliminated by adopting the weighted Laplacian smoothing given by Eq. 83 where weight is different for nodes connected to the smoothed node by an edge shared by two quadrilaterals ( ) or by one quadrilateral and one triangle ( ) or by two triangles (). These weights can be determined using a ``do not harm'' concept , the idea of which is to not move nodes shared by elements of the ideal shape. The mesh schemes for the determination of the smoothing weights are depicted in Figure 63.
The Laplacian smoothing can be further extended by weights accounting for the mesh density and for the nodal connectivity as was described in Subsection Smoothing of Section Direct Triangulation of 3D Surfaces. The setup of the weights is given by Eqs 84 and 85. However, contrary to all-triangular and all-quadrilateral meshes, the determination of the ``optimal'' valence for mixed meshes is not straightforward. The following formula for the ``optimal'' valence in a mixed mesh proved to be beneficial