3.4.6 Mesh Smoothing

Another way of adapting a mesh is the relaxation or smoothing of the mesh points. The position of the mesh points is modified, but the mesh topology remains unchanged. Shifting points can have a drastic effect on the quality of a mesh and it is more efficient than refinement and collapsing points especially when the translation amplitudes are small. A global optimum is reached by sequentially updating the mesh points in a Gauss-Seidel-type iteration [49,9]. The location of each mesh point is derived from a local optimization of a certain criterion while holding all other mesh points fixed. Usually, only a few sweeps through the mesh points are required to achieve convergence, and the local optimization problem is solved with a damped Newton method.

More advanced approaches define the local optimum as a maximum of a combination of a geometrical quality and a solution dependent quality [9]. The local displacement of a mesh point is then naturally constrained by the geometrical quality of the incident elements. Other approaches use translation forces to define attraction and repulsion between mesh points depending on the distance and direction. The force function is crucial for convergence and is often defined similarly to the physical binding forces between atoms [190]. Consistency checks are necessary to ensure that the displacement of a mesh point does not result in overlapping or zero volume elements.

A simple and straight-forward method can be derived from a finite difference approximation of the Laplace operator and is called Laplacian smoothing [62]. A mesh point is moved to the centroid of the surrounding mesh points which are topologically connected.

$$P = \frac{1}{n} \sum_{i=1}^{n} \alpha_{i} P_{i}$$ (3.27)

Several sweeps through all mesh points are required. If a set of surrounding mesh points forms an extremely non-convex polyhedron/polygon its center of mass may lie outside and overlapping elements would be generated. In such cases the mesh point is skipped. The weighting factors $\alpha_{i}$ could be set to one, or for example be derived from the volume/area of the incident elements where $\sum \alpha_{i}$ is equal to $n$. Laplacian smoothing results in more homogeneous meshes and is not suitable to achieve or maintain an anisotropic mesh density. Only for the trivial case of an homogeneous anisotropic mesh, e.g. an ortho-product grid with constant spacing $h_{x}, h_{y}$ over the domain where $h_{x} \ll h_{y}$, would the anisotropy remain unaffected after smoothing.

Another simple method aims to equally distribute the angle of the elements incident at each mesh point [129]. A general limitation of smoothing techniques is a bad mesh topology. If for example in two dimensions the number of edges incident at a given mesh point is too large or too small, it will become impossible to improve the angles. Most powerful will be a combination of refinement, smoothing, and local transformations.