Laplacian Smoothing

See LocalLaplacianSmoothing() [Cav74]

Laplacian smoothing is a classic smoothing method that works by moving each node to the center of its \(n_i\) neighboring nodes.

\[\bar{x}'_i = \frac{1}{n_i}\sum_{j=1}^{n_i} \bar{x}_j\]

Laplacian smoothing is efficient and effective in many applications, however it has several limitations:

• Repeated iterations can lead to shrinkage if the boundary isn’t fixed

• For volumetric elements (e.g. tetrahedra), smoothing can sometimes lead to element inversions or reduced quality

These limitations have motivated several variants to the classical smoothing algorithm.

Tangential Laplacian Smoothing

See TangentialLaplacianSmoothing() [OBP03]

Tangential Laplacian smoothing mitigates shrinkage by only moving nodes on the plane tangent to the surface at that point, better preserving the original geometry while still smoothing.

The displacement due to local Laplacian smoothing can be calculated as

\[\mathbf{U}_i = \frac{1}{n_i}\sum_{j=1}^{n_i} \left( \bar{x}_j - \bar{x}_i \right)\]

which can then be projected onto the tangent plane by subtracting the vector projection of the displacement \(\mathbf{U}_i\) onto the unit normal vector \(\hat{n}_i\):

\[\mathbf{R}_i = \mathbf{U}_i - \left( \mathbf{U}_i \cdot \hat{n}_i \right)\hat{n}_i\]

The updated node positions are then:

\[\bar{x}'_i = \bar{x} + \mathbf{R}\]

Smart Laplacian Smoothing

See SmartLaplacianSmoothing() [Fre97]

“Smart” Laplacian smoothing follows the same approach as the standard local Laplacian smoothing, but vertex movement is only accepted if the quality of the connected elements is improved. Different strategies can be employed, for example only moving a node if the average element quality improves or if the worst element quality improves.