The Laplacian smoothing technique changes the position of nodes without modifying the topology of the mesh. Each internal surface node is moved to a new position given by the average of nodes connected to it by an edge. This works fine for all-triangular or all-quadrilateral meshes. For mixed meshes, however, this simple scheme behaves not fully satisfactorily. Let's assume, for example, the mesh depicted in Fig. 7 on the left consisting of ideal elements only (squares and equilateral triangles). Employing the standard Laplacian smoothing will move the nodes on the interface between the quadrilaterals and triangles towards the triangles and consequently deteriorate the quality of the elements. This effect can be eliminated by introducing weighted Laplacian smoothing with different weights for nodes connected to the smoothed node by an edge shared
These weights can be determined using a ``do not harm'' concept , the idea of which is to not move the node shared by elements of ideal shape (Fig. 7).
The Laplacian smoothing can be further extended by weights accounting for the valence (connectivity number) of nodes connected to the smoothed node. It is well known that nodes with a valence larger (smaller) than the ideal one tend to attract (repulse) the smoothed node. Again, the setup of the weights for all-triangular and all-quadrilateral meshes is quite easy, because the ideal valence is evident. In a mixed mesh, however, the determination of the ideal valence is not straightforward. The following formulas for the ideal valence of an internal node and of a boundary node in a mixed mesh proved to be beneficial
where and stand for number of quadrilaterals and triangles sharing the node and is the angle (in degrees) in the surface tangent plane at the boundary node filled in by the surface.
Note that the repositioning of a node is likely to shift the node out of the surface (unless the surface is planar). Therefore the projection must be employed to satisfy the surface constraint.